1 00:00:00,080 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,238 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,238 --> 00:00:17,863 at ocw.mit.edu. 8 00:00:20,877 --> 00:00:21,960 PROFESSOR: Good afternoon. 9 00:00:24,800 --> 00:00:28,530 We have talked in the last class about magnetic trapping. 10 00:00:28,530 --> 00:00:31,910 Today, I want to finish our discussion on magnetic traps 11 00:00:31,910 --> 00:00:36,300 with two demonstrations of classical forms 12 00:00:36,300 --> 00:00:37,880 of magnetic trapping. 13 00:00:37,880 --> 00:00:39,590 But I told you that magnetic trapping 14 00:00:39,590 --> 00:00:42,010 is a purely classical phenomenon. 15 00:00:42,010 --> 00:00:44,490 The only quantum mechanical aspect 16 00:00:44,490 --> 00:00:51,030 is that this angle at which the magnetic dipole is oriented 17 00:00:51,030 --> 00:00:55,370 with the magnetic field is quantized. 18 00:00:55,370 --> 00:00:57,785 I also mentioned to you, and this 19 00:00:57,785 --> 00:01:01,620 was Wilks' Theorem, that the total magnetic field can 20 00:01:01,620 --> 00:01:05,150 have only local minima, not maxima. 21 00:01:05,150 --> 00:01:10,540 And therefore, we you can do magnetic trapping only 22 00:01:10,540 --> 00:01:12,570 for weak field seeking states. 23 00:01:12,570 --> 00:01:17,170 These are states which lower their energy 24 00:01:17,170 --> 00:01:19,370 with photomagnetic fields. 25 00:01:19,370 --> 00:01:22,020 And as a result, since the spin is up, 26 00:01:22,020 --> 00:01:25,090 the magnetic moment is anti-parallel 27 00:01:25,090 --> 00:01:27,440 to the magnetic field, those states 28 00:01:27,440 --> 00:01:32,240 can always lower their energy by flipping the spin. 29 00:01:32,240 --> 00:01:34,650 I know it's a simple demonstration, 30 00:01:34,650 --> 00:01:37,980 but, well, I can't bring a real magnetic trap 31 00:01:37,980 --> 00:01:39,450 into the classroom. 32 00:01:39,450 --> 00:01:42,320 So what I have here, this is just two little ring magnets, 33 00:01:42,320 --> 00:01:44,570 refrigerator magnets. 34 00:01:44,570 --> 00:01:47,430 And there is a tube. 35 00:01:47,430 --> 00:01:50,520 So I just want to demonstrate the one dimensional 36 00:01:50,520 --> 00:01:52,980 form of magnetic trapping to you. 37 00:01:52,980 --> 00:01:54,160 And this is our atom. 38 00:01:54,160 --> 00:01:58,550 It's in strong magnet, elongated, 39 00:01:58,550 --> 00:02:04,510 and it can be spin up or it can be spin down. 40 00:02:04,510 --> 00:02:09,370 So what you're seeing is now you're seeing the two magnets. 41 00:02:09,370 --> 00:02:15,780 And you're seeing our dipole, our atom, 42 00:02:15,780 --> 00:02:18,070 which is in a stable trapping configuration. 43 00:02:22,360 --> 00:02:24,910 And you can-- am I blocking it? 44 00:02:24,910 --> 00:02:31,210 You can see that it's stable by-- I just move it. 45 00:02:31,210 --> 00:02:34,320 And you see how it always comes back. 46 00:02:34,320 --> 00:02:35,780 So that's one position. 47 00:02:35,780 --> 00:02:37,630 I pull it over. 48 00:02:37,630 --> 00:02:43,350 Here, you have a second position where our magnet is trapped. 49 00:02:43,350 --> 00:02:46,260 Here is a third one. 50 00:02:46,260 --> 00:02:48,110 And here's a fourth one. 51 00:02:48,110 --> 00:02:49,880 So there are four positions where 52 00:02:49,880 --> 00:02:52,570 we have one dimensional trapping. 53 00:02:52,570 --> 00:02:55,410 Well, this is one for one orientation. 54 00:02:55,410 --> 00:02:59,330 Now I take the magnet and flip it over. 55 00:02:59,330 --> 00:03:02,680 So we are changing now in the laboratory frame. 56 00:03:02,680 --> 00:03:05,830 The sign of the magnetic moment. 57 00:03:05,830 --> 00:03:08,270 And there is actually a very nice minimum. 58 00:03:08,270 --> 00:03:12,580 You see how clearly it is trapped. 59 00:03:12,580 --> 00:03:16,110 There is one here in the middle. 60 00:03:16,110 --> 00:03:18,920 I can also just show you, it really stays put. 61 00:03:18,920 --> 00:03:20,870 There's a weak restoring force. 62 00:03:20,870 --> 00:03:23,720 And then here's number three. 63 00:03:23,720 --> 00:03:25,400 So it seems very, very rich. 64 00:03:25,400 --> 00:03:29,000 I've shown you seven different minima 65 00:03:29,000 --> 00:03:31,400 where trapping has happened. 66 00:03:31,400 --> 00:03:36,020 And well, just to fill you in, if this is our magnet, 67 00:03:36,020 --> 00:03:38,350 these are ring magnets, the magnetic field 68 00:03:38,350 --> 00:03:41,560 is about something like that. 69 00:03:41,560 --> 00:03:47,950 And if I plot the magnetic field for this configuration, 70 00:03:47,950 --> 00:03:56,270 I find indeed that for one orientation, 71 00:03:56,270 --> 00:04:00,520 there are four maxima. 72 00:04:00,520 --> 00:04:06,040 So for one orientation-- just give me one second. 73 00:04:19,550 --> 00:04:22,870 Yeah, there are four maxima of the magnetic field 74 00:04:22,870 --> 00:04:24,710 for this orientation. 75 00:04:24,710 --> 00:04:26,360 And therefore, there are four minima 76 00:04:26,360 --> 00:04:29,820 of the trapping potential. 77 00:04:29,820 --> 00:04:32,110 So maybe I should have shown it like this. 78 00:04:32,110 --> 00:04:35,110 There are one, two, three, four minima. 79 00:04:35,110 --> 00:04:40,530 And for the opposite configuration, 80 00:04:40,530 --> 00:04:44,300 there are one, two, three trapping minima. 81 00:04:44,300 --> 00:04:48,090 But now if you look, there are one, two, three, four, five, 82 00:04:48,090 --> 00:04:52,870 six ones marked in red, which are for strong field seekers, 83 00:04:52,870 --> 00:04:56,240 because these are maxima at the magnetic field. 84 00:04:56,240 --> 00:04:58,170 And there is only one where the magnetic field 85 00:04:58,170 --> 00:05:00,260 has a local minimum. 86 00:05:00,260 --> 00:05:03,140 So in the last part, I have shown 87 00:05:03,140 --> 00:05:08,250 you the absolute value of the magnetic field. 88 00:05:08,250 --> 00:05:11,070 And you realize there are one, two, three, four, five, 89 00:05:11,070 --> 00:05:12,340 six maxima. 90 00:05:12,340 --> 00:05:15,440 And this is responsible for six trapping configurations. 91 00:05:15,440 --> 00:05:19,230 And the seventh one was a minimum of the magnetic field. 92 00:05:19,230 --> 00:05:21,450 So in one dimension, there is no problem. 93 00:05:21,450 --> 00:05:24,070 In one dimension, I can get a restoring force 94 00:05:24,070 --> 00:05:26,880 for the strong and for the weak field seeker 95 00:05:26,880 --> 00:05:29,540 for any orientation of the magnetic dipole. 96 00:05:29,540 --> 00:05:32,820 But if we want to have a three dimensional trap, 97 00:05:32,820 --> 00:05:37,860 we have to look at the three dimensional stability. 98 00:05:37,860 --> 00:05:42,240 And it would be only this one configuration, 99 00:05:42,240 --> 00:05:45,700 which can be stabilised in x and y. 100 00:05:45,700 --> 00:05:47,950 What I've shown you are symmetric fields. 101 00:05:47,950 --> 00:05:51,190 So all these minima and maxima only settle points. 102 00:05:51,190 --> 00:05:52,950 They trap in one dimension. 103 00:05:52,950 --> 00:05:56,190 But with suitable radial magnetic field, 104 00:05:56,190 --> 00:06:00,040 you could turn this one here into a real three 105 00:06:00,040 --> 00:06:01,210 dimensional trap. 106 00:06:01,210 --> 00:06:04,840 But the other ones you cannot, because this would violate 107 00:06:04,840 --> 00:06:05,610 Wilks' Theorem. 108 00:06:11,690 --> 00:06:14,900 I find it amazing that this demonstration 109 00:06:14,900 --> 00:06:19,550 has seven different magnetic trapping configurations. 110 00:06:19,550 --> 00:06:21,460 That's just how it turns out to be. 111 00:06:21,460 --> 00:06:24,130 Questions? 112 00:06:24,130 --> 00:06:25,046 Collin. 113 00:06:25,046 --> 00:06:26,837 AUDIENCE: So if you look at the clover leaf 114 00:06:26,837 --> 00:06:35,846 coil, the parts of the field that sort of axial [INAUDIBLE] 115 00:06:35,846 --> 00:06:39,270 provided by [INAUDIBLE], those are the inner coils? 116 00:06:39,270 --> 00:06:42,110 PROFESSOR: Yeah, so if you look at a clover leaf trap, 117 00:06:42,110 --> 00:06:44,730 the pinch coils create such a minimum, 118 00:06:44,730 --> 00:06:48,020 which provides confinement in the z direction. 119 00:06:48,020 --> 00:06:52,750 But then you add a field, a radially outward quadrupole 120 00:06:52,750 --> 00:06:59,745 field, and that overcomes-- in this demonstration, 121 00:06:59,745 --> 00:07:01,120 everything is actually symmetric. 122 00:07:01,120 --> 00:07:04,040 So this is a minimum for this direction. 123 00:07:04,040 --> 00:07:08,140 But if you would now plot it as a function of x and y and z, 124 00:07:08,140 --> 00:07:09,760 it would be a saddle point. 125 00:07:09,760 --> 00:07:12,800 So this configuration is trapping along z, 126 00:07:12,800 --> 00:07:15,600 but anti-trapping along x and y. 127 00:07:15,600 --> 00:07:17,450 But if you add a quadrupole field, 128 00:07:17,450 --> 00:07:20,470 a linear field which points outward with these clover 129 00:07:20,470 --> 00:07:24,720 leaves or the [INAUDIBLE] bars, then they 130 00:07:24,720 --> 00:07:27,960 would provide confinement, which is stronger 131 00:07:27,960 --> 00:07:30,840 than the anti-confinement of the saddle point. 132 00:07:30,840 --> 00:07:33,580 If you look at some equations I showed you in the last class, 133 00:07:33,580 --> 00:07:36,940 you will actually see that in one of the slides, 134 00:07:36,940 --> 00:07:39,860 I had the expression for the magnetic field, 135 00:07:39,860 --> 00:07:43,050 where you see how the [INAUDIBLE] bars, 136 00:07:43,050 --> 00:07:46,930 the radially linear confinement overcomes 137 00:07:46,930 --> 00:07:53,230 the anti-trapping feature of the settle point. 138 00:07:53,230 --> 00:07:53,920 Other question? 139 00:07:56,860 --> 00:08:03,360 Then finally, let me give you a demonstration for the following 140 00:08:03,360 --> 00:08:05,940 effect. 141 00:08:05,940 --> 00:08:10,600 I mentioned to you that magnetic traps become unstable 142 00:08:10,600 --> 00:08:13,830 when the magnetic field is very low. 143 00:08:13,830 --> 00:08:16,360 And I told you, well, at very low magnetic field, 144 00:08:16,360 --> 00:08:18,810 the energy difference between spin up and spin down 145 00:08:18,810 --> 00:08:20,350 is very small. 146 00:08:20,350 --> 00:08:22,900 At zero magnetic field, it becomes zero. 147 00:08:22,900 --> 00:08:28,320 And then, the atom cannot adiabatically stay in its MF 148 00:08:28,320 --> 00:08:33,970 state, because we can violate the adiabaticity condition. 149 00:08:33,970 --> 00:08:36,850 So let me now show you an example of purely 150 00:08:36,850 --> 00:08:38,659 classical trapping. 151 00:08:38,659 --> 00:08:42,690 And I think a number of you now have seen the levitron toy. 152 00:08:42,690 --> 00:08:48,340 Actually, a few years ago when my group was one of the first 153 00:08:48,340 --> 00:08:50,900 to use magnetic trapping, in every single talk 154 00:08:50,900 --> 00:08:53,150 I explained how magnetic trap worked. 155 00:08:53,150 --> 00:08:56,050 And I even showed this demonstration live. 156 00:08:56,050 --> 00:08:58,470 But in order to demonstrate it really well, 157 00:08:58,470 --> 00:09:00,460 I was building my own levitron. 158 00:09:00,460 --> 00:09:01,950 I used machine parts. 159 00:09:01,950 --> 00:09:03,490 I used the motor drive. 160 00:09:03,490 --> 00:09:05,970 So I hope I can show you one aspect of the levitron 161 00:09:05,970 --> 00:09:12,010 demonstration, which is usually not known. 162 00:09:12,010 --> 00:09:15,880 So let me first give you the punchline. 163 00:09:15,880 --> 00:09:20,020 Magnetic trapping happens with an orientation 164 00:09:20,020 --> 00:09:23,810 of the magnetic dipole which could always 165 00:09:23,810 --> 00:09:26,230 lower its energy by flipping over. 166 00:09:26,230 --> 00:09:29,140 The reason why it doesn't flip over-- well, 167 00:09:29,140 --> 00:09:30,770 quantum mechanically, it adiabatically 168 00:09:30,770 --> 00:09:33,920 stays in a quantum state-- but classically, 169 00:09:33,920 --> 00:09:37,920 if we have angular momentum, it is gyroscopically stabilized. 170 00:09:37,920 --> 00:09:42,220 The dipole cannot flip over, because it's a gyroscope. 171 00:09:42,220 --> 00:09:43,950 It has angular momentum. 172 00:09:43,950 --> 00:09:45,750 And actually, it also has angular momentum 173 00:09:45,750 --> 00:09:46,610 in quantum physics. 174 00:09:46,610 --> 00:09:49,650 So the two explanations sounded different 175 00:09:49,650 --> 00:09:52,290 when I said stays in a quantum state, 176 00:09:52,290 --> 00:09:53,800 is gyroscopically stabilized. 177 00:09:53,800 --> 00:09:56,220 But if you think about it more deeply, 178 00:09:56,220 --> 00:09:59,440 they have much more in common than my language suggests. 179 00:09:59,440 --> 00:10:03,190 So I think you know how the levitron works. 180 00:10:03,190 --> 00:10:05,250 You spin up the magnet. 181 00:10:05,250 --> 00:10:08,890 And I had this nice motor tool to spin it up. 182 00:10:08,890 --> 00:10:11,220 And when you have prepared the system. 183 00:10:11,220 --> 00:10:13,940 Your atom has now-- atom in quotes-- angular momentum 184 00:10:13,940 --> 00:10:15,170 in the magnetic moment. 185 00:10:15,170 --> 00:10:18,870 And you bring it to the position where the three dimensional 186 00:10:18,870 --> 00:10:22,630 magnetic field fulfills the stability condition. 187 00:10:22,630 --> 00:10:25,380 And you can now enjoy magnetic trapping. 188 00:10:25,380 --> 00:10:29,830 This is exactly what your atoms do in your magnetic trap. 189 00:10:29,830 --> 00:10:33,415 It is gyroscopically stabilized magnetic levitation. 190 00:10:36,640 --> 00:10:40,020 The only difference is that gravity has to be compensated. 191 00:10:40,020 --> 00:10:41,770 Gravity is a major player. 192 00:10:41,770 --> 00:10:45,040 So the stability point in the three dimensional 193 00:10:45,040 --> 00:10:49,440 magnetic field configuration includes the compensation 194 00:10:49,440 --> 00:10:51,040 for gravity. 195 00:10:51,040 --> 00:10:53,250 But I have to say, my group was also at some point 196 00:10:53,250 --> 00:10:56,430 trapping a Bose-Einstein condensate in a few hertz 197 00:10:56,430 --> 00:11:00,450 weak trap, where gravity was the strongest force of all. 198 00:11:00,450 --> 00:11:04,990 So what you have seen is an exact demonstration 199 00:11:04,990 --> 00:11:08,800 of the principle for magnetic trapping. 200 00:11:08,800 --> 00:11:10,490 But now comes my question. 201 00:11:10,490 --> 00:11:12,440 What would you expect, what would 202 00:11:12,440 --> 00:11:19,070 happen to our magnetic trap when we spin the levitron faster? 203 00:11:19,070 --> 00:11:23,860 Does it help or does it hurt to spin up the levitron, to spin 204 00:11:23,860 --> 00:11:26,800 up the gyroscope, to higher angular velocities? 205 00:11:30,490 --> 00:11:31,790 So three possibility. 206 00:11:31,790 --> 00:11:32,480 Nothing happens. 207 00:11:32,480 --> 00:11:34,670 It doesn't matter as long as it spins. 208 00:11:34,670 --> 00:11:38,070 The second one, the trap becomes more stable. 209 00:11:38,070 --> 00:11:42,590 The third possibility, the trap becomes unstable. 210 00:11:42,590 --> 00:11:44,560 Do you want to offer any opinion? 211 00:12:00,110 --> 00:12:00,636 Collin. 212 00:12:00,636 --> 00:12:02,760 AUDIENCE: Well, we have an angular momentum, right, 213 00:12:02,760 --> 00:12:05,636 of the really large omega. 214 00:12:05,636 --> 00:12:07,548 When you apply torque, it's going 215 00:12:07,548 --> 00:12:09,938 to get torque [INAUDIBLE], right. 216 00:12:09,938 --> 00:12:11,850 It's going to be the prefactor of omega. 217 00:12:11,850 --> 00:12:17,389 So we have a giant omega, we imagine that [INAUDIBLE] small, 218 00:12:17,389 --> 00:12:18,383 [INAUDIBLE] torques. 219 00:12:21,318 --> 00:12:22,359 No, no, no wait a second. 220 00:12:22,359 --> 00:12:23,400 I did this the other way. 221 00:12:23,400 --> 00:12:26,350 It was [? healing. ?] Never mind. 222 00:12:26,350 --> 00:12:28,750 PROFESSOR: So do we have to do the experiment? 223 00:12:28,750 --> 00:12:30,680 Maybe. 224 00:12:30,680 --> 00:12:32,900 So now I put the motor controller to full speed. 225 00:12:32,900 --> 00:12:34,830 And I speed it up much faster. 226 00:12:34,830 --> 00:12:36,672 You can hear the sound. 227 00:12:36,672 --> 00:12:38,380 I really want to do a careful experiment. 228 00:12:38,380 --> 00:12:41,830 So we wait until everything is quiet and has died out. 229 00:12:41,830 --> 00:12:45,100 And now, we want to try if we can can do magnetic trapping. 230 00:12:45,100 --> 00:12:46,920 And you see it's impossible. 231 00:12:46,920 --> 00:12:49,060 The system, when it reaches the point where 232 00:12:49,060 --> 00:12:52,840 magnetic trapping would occur, it's unstable. 233 00:12:52,840 --> 00:12:55,230 But then, well, just to prove that it is only 234 00:12:55,230 --> 00:12:58,420 the speed of rotation which has caused the instability, 235 00:12:58,420 --> 00:13:01,370 I just wait until friction has slowed down 236 00:13:01,370 --> 00:13:02,800 the angular velocity. 237 00:13:02,800 --> 00:13:05,040 And now again, it works perfectly. 238 00:13:07,860 --> 00:13:14,860 So you see, if you rotate the gyroscope too fast, it's bad. 239 00:13:14,860 --> 00:13:16,850 It makes the magnetic trap unstable. 240 00:13:19,720 --> 00:13:21,340 Convinced? 241 00:13:21,340 --> 00:13:22,560 How would you explain that? 242 00:13:41,660 --> 00:13:44,020 If we have a gyro-- Collin? 243 00:13:44,020 --> 00:13:46,520 AUDIENCE: Why aren't we working in the limit where we assume 244 00:13:46,520 --> 00:13:50,885 that the magnetic field generated by our magnet 245 00:13:50,885 --> 00:13:53,795 is sort of weak compared to the trap. 246 00:13:53,795 --> 00:13:57,700 So the magnet's really modifying [INAUDIBLE]. 247 00:13:57,700 --> 00:13:59,700 PROFESSOR: No, we assume here-- and I 248 00:13:59,700 --> 00:14:01,660 can give you the analysis-- but yes. 249 00:14:01,660 --> 00:14:03,320 No, these are permanent magnets. 250 00:14:03,320 --> 00:14:08,400 So the floating magnet is just, you can say, a probe, a test 251 00:14:08,400 --> 00:14:12,760 object which is put into the permanent magnetic field 252 00:14:12,760 --> 00:14:16,458 of the stronger magnets of the stationary magnet. 253 00:14:16,458 --> 00:14:19,452 AUDIENCE: Get an additional torque 254 00:14:19,452 --> 00:14:22,446 to-- because it gets an additional force 255 00:14:22,446 --> 00:14:23,943 into the upper state. 256 00:14:23,943 --> 00:14:27,935 So it gets an additional torque in towards the center. 257 00:14:30,940 --> 00:14:35,069 PROFESSOR: It's not necessarily the additional torque. 258 00:14:35,069 --> 00:14:37,360 Let's put two things together, it's really fascinating, 259 00:14:37,360 --> 00:14:38,780 from different principles. 260 00:14:38,780 --> 00:14:45,070 The first thing is magnetic trapping 261 00:14:45,070 --> 00:14:47,586 requires-- if you have a magnetic trap 262 00:14:47,586 --> 00:14:49,460 and you have an inhomogeneous magnetic field. 263 00:14:49,460 --> 00:14:52,100 And of course, you need an inhomogeneous magnetic field, 264 00:14:52,100 --> 00:14:55,810 the angle cosine theta between the spinning dipole 265 00:14:55,810 --> 00:14:59,240 and the magnetic field should stay the same. 266 00:14:59,240 --> 00:15:01,190 And this means quantum mechanically, 267 00:15:01,190 --> 00:15:02,990 we stay in the same quantum state. 268 00:15:02,990 --> 00:15:04,900 So therefore, a magnetic trap only 269 00:15:04,900 --> 00:15:09,560 works because the rapidly precessing spin, 270 00:15:09,560 --> 00:15:12,430 when the magnetic field always precesses 271 00:15:12,430 --> 00:15:13,770 around the magnetic field. 272 00:15:13,770 --> 00:15:15,930 And when the magnetic field tips, 273 00:15:15,930 --> 00:15:21,170 the precession keeps the dipole, the magnetic moment, 274 00:15:21,170 --> 00:15:23,520 aligned with the magnetic field. 275 00:15:23,520 --> 00:15:27,380 Now, what happens in a gyroscope with a precession frequency 276 00:15:27,380 --> 00:15:29,160 when you spin the gyroscope faster? 277 00:15:34,490 --> 00:15:36,700 We've seen that in your classical mechanical 278 00:15:36,700 --> 00:15:39,530 demonstration, if you had the spinning gyroscope, which 279 00:15:39,530 --> 00:15:42,490 was only [INAUDIBLE] suspended with one rope. 280 00:15:42,490 --> 00:15:46,260 And then it was precessing in the Earth magnetic field. 281 00:15:46,260 --> 00:15:50,010 Does this precession frequency get faster or slower 282 00:15:50,010 --> 00:15:51,570 when you spin the gyroscope faster? 283 00:15:57,410 --> 00:15:58,670 Pardon? 284 00:15:58,670 --> 00:16:01,050 When the gyroscope spins faster, what 285 00:16:01,050 --> 00:16:04,312 happens to the precession frequency? 286 00:16:04,312 --> 00:16:05,720 AUDIENCE: Slower. 287 00:16:05,720 --> 00:16:06,870 PROFESSOR: Slower. 288 00:16:06,870 --> 00:16:10,340 Because the torque per unit time adds some angular momentum. 289 00:16:10,340 --> 00:16:13,430 This angular momentum adds to the existing angular momentum. 290 00:16:13,430 --> 00:16:15,510 But the more angular momentum exists, 291 00:16:15,510 --> 00:16:16,760 the smaller is the change. 292 00:16:16,760 --> 00:16:19,620 And this lowers the precession. 293 00:16:19,620 --> 00:16:20,920 I can give you another example. 294 00:16:20,920 --> 00:16:23,430 If you rotate a coin. 295 00:16:23,430 --> 00:16:27,170 When does the coin really wobble very, very quickly? 296 00:16:27,170 --> 00:16:30,000 Just when it has slowed down and is about to fall. 297 00:16:30,000 --> 00:16:32,990 And this rapid wobbling is the precession frequency. 298 00:16:32,990 --> 00:16:36,660 So the lower the gyroscopic angular momentum is, 299 00:16:36,660 --> 00:16:40,640 angular velocity is, the faster is the precession frequency. 300 00:16:40,640 --> 00:16:45,070 And fast precession is important for adiabatic following. 301 00:16:45,070 --> 00:16:48,700 So in other words, what you saw here in this demonstration 302 00:16:48,700 --> 00:16:50,640 was a classical analogy for Majorana Flops. 303 00:16:53,480 --> 00:17:00,780 Now, if we would translate from our classical demonstration 304 00:17:00,780 --> 00:17:06,420 to a real atom, what feature, what 305 00:17:06,420 --> 00:17:10,470 parameter characterizing the atom, are we changing? 306 00:17:10,470 --> 00:17:15,569 So when I spin the magnet faster, 307 00:17:15,569 --> 00:17:19,300 what would that correspond to in atomic properties? 308 00:17:19,300 --> 00:17:22,036 AUDIENCE: Higher mu. 309 00:17:22,036 --> 00:17:22,910 PROFESSOR: Higher mu? 310 00:17:22,910 --> 00:17:25,030 I'm not changing the permanent magnetic moment 311 00:17:25,030 --> 00:17:27,339 of the magnet by spinning it faster-- higher 312 00:17:27,339 --> 00:17:29,640 angular momentum. 313 00:17:29,640 --> 00:17:32,145 But in this the equation, what corresponds to higher angular 314 00:17:32,145 --> 00:17:32,645 momentum? 315 00:17:36,278 --> 00:17:38,174 AUDIENCE: [INAUDIBLE] omega alpha, 316 00:17:38,174 --> 00:17:41,030 the precession frequency around the static beam field? 317 00:17:41,030 --> 00:17:41,780 PROFESSOR: Yes. 318 00:17:41,780 --> 00:17:45,860 But what I mean-- so the Larmor frequency, 319 00:17:45,860 --> 00:17:48,110 this is a precision frequency, becomes lower. 320 00:17:48,110 --> 00:17:51,750 So what becomes lower in the atomic property? 321 00:17:51,750 --> 00:17:53,910 AUDIENCE: Sort of the external magnetic-- 322 00:17:53,910 --> 00:17:54,660 PROFESSOR: Pardon? 323 00:17:54,660 --> 00:17:58,060 AUDIENCE: h bar? 324 00:17:58,060 --> 00:18:00,950 PROFESSOR: No, let's not mess around with h bar here. 325 00:18:00,950 --> 00:18:03,210 h bar is given by nature. 326 00:18:03,210 --> 00:18:04,680 We can't change that. 327 00:18:04,680 --> 00:18:07,040 But, I mean, OK. 328 00:18:07,040 --> 00:18:07,990 Multiple choice. 329 00:18:07,990 --> 00:18:09,370 B, no, no, no, no. 330 00:18:09,370 --> 00:18:09,870 It's g. 331 00:18:09,870 --> 00:18:10,730 Yeah, it's g. 332 00:18:10,730 --> 00:18:13,750 So what you have seen is, you've seen a demonstration where 333 00:18:13,750 --> 00:18:19,050 in front of your eyes, I've changed the atomic g factor. 334 00:18:19,050 --> 00:18:21,270 And now you sort of see, let me put it together. 335 00:18:21,270 --> 00:18:24,530 What happens is, the mechanical magnet 336 00:18:24,530 --> 00:18:26,720 has a given magnetic moment. 337 00:18:26,720 --> 00:18:31,030 And if I put much more angular momentum into it, 338 00:18:31,030 --> 00:18:33,790 it can sort of-- it has, quantum mechanically, 339 00:18:33,790 --> 00:18:35,870 speaking more intermediate states. 340 00:18:35,870 --> 00:18:38,080 Because it can change its angular momentum 341 00:18:38,080 --> 00:18:39,280 in steps of one. 342 00:18:39,280 --> 00:18:43,630 So if I spin it faster, it has many more intermediate states. 343 00:18:43,630 --> 00:18:48,470 And each energy separation has become smaller. 344 00:18:48,470 --> 00:18:51,110 And smaller energy separation means 345 00:18:51,110 --> 00:18:56,760 I'm getting closer to degeneracy where adiabaticity breaks down. 346 00:18:56,760 --> 00:18:58,160 Anyway, think about it. 347 00:18:58,160 --> 00:19:01,281 The analogy is really deep. 348 00:19:01,281 --> 00:19:01,780 Questions? 349 00:19:05,270 --> 00:19:05,770 OK. 350 00:19:09,720 --> 00:19:20,050 So that's all I wanted to tell you about magnetic trapping. 351 00:19:20,050 --> 00:19:20,550 Collin. 352 00:19:20,550 --> 00:19:21,920 AUDIENCE: When you increase the angular momentum, 353 00:19:21,920 --> 00:19:25,030 you don't necessarily change the spacing between the levels, 354 00:19:25,030 --> 00:19:25,900 though. 355 00:19:25,900 --> 00:19:26,483 PROFESSOR: No. 356 00:19:26,483 --> 00:19:29,410 The energy levels is when the magnet is aligned, 357 00:19:29,410 --> 00:19:33,750 it has an energy absolute value of mu times b. 358 00:19:33,750 --> 00:19:37,550 Here, it has minus absolute value of mu times b. 359 00:19:37,550 --> 00:19:41,940 And mu is simply the magnetic moment of the permanent magnet. 360 00:19:41,940 --> 00:19:44,230 So I go from here to there. 361 00:19:44,230 --> 00:19:47,290 And the number of energy levels in between 362 00:19:47,290 --> 00:19:51,140 is the total angular momentum divided by h bar. 363 00:19:51,140 --> 00:19:57,320 So when I give it more angular momentum, in one energy level, 364 00:19:57,320 --> 00:20:02,320 in one transition, there is less energy which will be released. 365 00:20:02,320 --> 00:20:05,060 And it is actually, you can say, the big note 366 00:20:05,060 --> 00:20:07,090 between energy levels, or the difference 367 00:20:07,090 --> 00:20:10,660 between energy levels, which is a Larmor frequency. 368 00:20:10,660 --> 00:20:13,310 So therefore, the precession, quantum mechanically, 369 00:20:13,310 --> 00:20:17,130 is the energy difference between adjacent energy levels. 370 00:20:17,130 --> 00:20:20,610 And if you give it more angular momentum, 371 00:20:20,610 --> 00:20:23,550 this energy difference becomes smaller. 372 00:20:26,750 --> 00:20:30,410 And this low precession frequency, this small energy 373 00:20:30,410 --> 00:20:32,465 difference, is bad for adiabaticity. 374 00:20:36,670 --> 00:20:38,639 Yes. 375 00:20:38,639 --> 00:20:41,597 AUDIENCE: Does that have much of an effect on f equals 1 376 00:20:41,597 --> 00:20:42,583 and f equals 2? 377 00:20:42,583 --> 00:20:45,050 Do you get more [INAUDIBLE] if f equals 2? 378 00:20:45,050 --> 00:20:47,340 PROFESSOR: Yes. 379 00:20:47,340 --> 00:20:50,470 In f equals 2, we have twice the magnetic movement 380 00:20:50,470 --> 00:20:54,560 than in f equals 1. 381 00:20:54,560 --> 00:20:58,310 But in f equals 2, we have five levels. 382 00:20:58,310 --> 00:21:01,050 In f equals 1, we have three levels. 383 00:21:01,050 --> 00:21:04,515 So the-- 384 00:21:04,515 --> 00:21:06,950 AUDIENCE: Is it that significant of a difference? 385 00:21:06,950 --> 00:21:09,100 PROFESSOR: So I think it just cancels out. 386 00:21:09,100 --> 00:21:11,870 The g factor-- I showed you the formula. 387 00:21:11,870 --> 00:21:14,450 What matters really is the g factor. 388 00:21:14,450 --> 00:21:19,310 And the g factor on f equals 2 is 1/2. 389 00:21:19,310 --> 00:21:21,430 In f equals minus 1, it's minus 1/2. 390 00:21:24,020 --> 00:21:25,555 What happens is the magnetic moment 391 00:21:25,555 --> 00:21:29,570 in f equals 2 is larger, because everything is stretched. 392 00:21:29,570 --> 00:21:31,660 All angular momenta are aligned. 393 00:21:31,660 --> 00:21:34,600 But the multiplicity-- you have five levels 394 00:21:34,600 --> 00:21:37,090 versus three levels-- and the two effects just cancel. 395 00:21:42,810 --> 00:21:45,180 Other questions? 396 00:21:45,180 --> 00:21:45,680 Yes. 397 00:21:45,680 --> 00:21:48,940 AUDIENCE: What breaks the system when the magnet spinning gets 398 00:21:48,940 --> 00:21:50,790 slower and slower. 399 00:21:50,790 --> 00:21:53,690 Now we know why it destabilizes when you spin it too fast. 400 00:21:53,690 --> 00:21:58,650 But if you don't spin it at all, it also floats, right? 401 00:21:58,650 --> 00:22:00,820 PROFESSOR: OK, what happens is yes. 402 00:22:00,820 --> 00:22:04,320 If I stop spinning it, it will no longer work. 403 00:22:04,320 --> 00:22:08,840 So what we have is we have a hierarchy of frequencies. 404 00:22:08,840 --> 00:22:12,670 The fastest frequency has to be the spinning frequency. 405 00:22:12,670 --> 00:22:14,800 Then we have the trapping frequency. 406 00:22:14,800 --> 00:22:16,760 And the precession frequency is one 407 00:22:16,760 --> 00:22:18,667 over the spinning frequency. 408 00:22:18,667 --> 00:22:20,375 So you want that the precession frequency 409 00:22:20,375 --> 00:22:24,140 is between spinning and one over spinning, 410 00:22:24,140 --> 00:22:26,200 because this is the precession. 411 00:22:26,200 --> 00:22:29,080 And if you would take the spinning 412 00:22:29,080 --> 00:22:34,391 to lower and lower values, you would violate that hierarchy. 413 00:22:34,391 --> 00:22:34,890 Yes. 414 00:22:38,620 --> 00:22:41,670 OK, evaporative cooling. 415 00:22:41,670 --> 00:22:46,590 Evaporative cooling is a powerful cooling scheme 416 00:22:46,590 --> 00:22:48,550 to reach nanokelvin. 417 00:22:48,550 --> 00:22:52,200 Actually, I forgot to update this slide. 418 00:22:52,200 --> 00:22:54,200 I wanted to say, this is the only technique 419 00:22:54,200 --> 00:22:56,010 so far to reach quantum degeneracy 420 00:22:56,010 --> 00:22:57,700 for bosons and fermions. 421 00:22:57,700 --> 00:23:01,400 Very recently, people have demonstrated laser cooling 422 00:23:01,400 --> 00:23:03,770 of atomic strontium to quantum degeneracy. 423 00:23:03,770 --> 00:23:06,560 But if you read the paper, it was laser cooling 424 00:23:06,560 --> 00:23:09,790 aided not by evaporative cooling, 425 00:23:09,790 --> 00:23:12,410 but by collisional distribution of atoms. 426 00:23:12,410 --> 00:23:15,610 It's likely evaporation where you evaporate into-- 427 00:23:15,610 --> 00:23:18,250 and you keep-- read the paper. 428 00:23:18,250 --> 00:23:18,750 It's-- 429 00:23:18,750 --> 00:23:22,020 [LAUGHTER] 430 00:23:22,020 --> 00:23:30,710 The scheme only worked because collisions-- 431 00:23:30,710 --> 00:23:34,470 how to say-- you cooled one region and another region 432 00:23:34,470 --> 00:23:35,800 was cooled by collisions. 433 00:23:35,800 --> 00:23:37,682 And this brings you pretty much back 434 00:23:37,682 --> 00:23:39,140 to further [INAUDIBLE] evaporation. 435 00:23:39,140 --> 00:23:42,590 Anyway, what I want to say is there is a small footnote. 436 00:23:42,590 --> 00:23:44,470 The field is evolving. 437 00:23:44,470 --> 00:23:47,390 You can find now paper's laser cooling to be easy. 438 00:23:47,390 --> 00:23:48,960 But if you read the paper carefully, 439 00:23:48,960 --> 00:23:50,810 or if you talk to me, I will tell you 440 00:23:50,810 --> 00:23:53,780 that there were still collisions necessary, 441 00:23:53,780 --> 00:23:55,180 the same kind of collisions which 442 00:23:55,180 --> 00:23:57,070 drive the reparative cooling. 443 00:23:57,070 --> 00:24:00,620 But before we go into an expert discussion about variants 444 00:24:00,620 --> 00:24:02,139 of evaporate cooling, I should first 445 00:24:02,139 --> 00:24:03,680 tell you what evaporative cooling is. 446 00:24:03,680 --> 00:24:06,270 But somebody raised his hand. 447 00:24:06,270 --> 00:24:08,238 AUDIENCE: Oh, no I just [INAUDIBLE]. 448 00:24:08,238 --> 00:24:11,682 AUDIENCE: You wrote the paper on [INAUDIBLE], for example, 449 00:24:11,682 --> 00:24:16,110 that if your quantum degeneracy [INAUDIBLE]. 450 00:24:16,110 --> 00:24:21,550 And that would be just purely laser cooling, right? 451 00:24:21,550 --> 00:24:24,030 PROFESSOR: Sub-recoil cooling has not-- 452 00:24:24,030 --> 00:24:27,570 any form of laser cooling to sub-recoil temperatures was not 453 00:24:27,570 --> 00:24:30,190 compatible with high atomic densities. 454 00:24:30,190 --> 00:24:32,420 It only worked at such low densities 455 00:24:32,420 --> 00:24:35,460 that they stayed far away from quantum degeneracy. 456 00:24:35,460 --> 00:24:37,250 So nanokelvin, yes. 457 00:24:37,250 --> 00:24:39,970 Temperature in the nanokelvin range 458 00:24:39,970 --> 00:24:42,110 his been reached by laser cooling, 459 00:24:42,110 --> 00:24:45,730 but not at sufficiently high densities. 460 00:24:45,730 --> 00:24:48,760 The densities, high density, causes collisions. 461 00:24:48,760 --> 00:24:52,670 Those collisions are screwing up laser 462 00:24:52,670 --> 00:24:54,870 cooling that it doesn't work anymore. 463 00:24:54,870 --> 00:24:57,920 And the only technique which can reach nanokelvin temperatures 464 00:24:57,920 --> 00:25:01,610 at sufficiently high density is evaporative cooling. 465 00:25:01,610 --> 00:25:03,270 And that applies to strontium. 466 00:25:03,270 --> 00:25:05,350 Strontium was laser cooled at low density, 467 00:25:05,350 --> 00:25:07,280 and then the low temperature was collisionally 468 00:25:07,280 --> 00:25:10,270 transferred to high density region. 469 00:25:10,270 --> 00:25:15,270 So you can say, in a way, it's an oddity of nature 470 00:25:15,270 --> 00:25:18,640 that we are now using quantum gases to reveal 471 00:25:18,640 --> 00:25:21,910 new features of quantum physics using ultra-cold atoms. 472 00:25:21,910 --> 00:25:25,980 But the cooling techniques which gets us 473 00:25:25,980 --> 00:25:28,140 into the quantum degenerate regime 474 00:25:28,140 --> 00:25:30,090 is pretty much classical. 475 00:25:35,070 --> 00:25:37,130 So this is a cartoon picture how you 476 00:25:37,130 --> 00:25:39,690 could think evaporative cooling works. 477 00:25:39,690 --> 00:25:42,160 You have a thermal distribution. 478 00:25:42,160 --> 00:25:46,940 You remove the high energy tail of the thermal distribution. 479 00:25:46,940 --> 00:25:50,850 And then, you allow the distribution to relax. 480 00:25:50,850 --> 00:25:54,360 And it will relax to a Boltzmann distribution, which 481 00:25:54,360 --> 00:25:57,390 is a little bit shifted towards the cold, 482 00:25:57,390 --> 00:26:00,270 or the low energy, side compared to the original dash 483 00:26:00,270 --> 00:26:01,890 distribution. 484 00:26:01,890 --> 00:26:04,480 And if you do it again and again and again, you 485 00:26:04,480 --> 00:26:07,120 wind up with a distribution of atoms which 486 00:26:07,120 --> 00:26:11,390 is colder and colder, because every time you axe away 487 00:26:11,390 --> 00:26:15,730 the high energy tail, you remove atoms 488 00:26:15,730 --> 00:26:19,580 which have, on average, more than the average energy. 489 00:26:19,580 --> 00:26:22,350 And therefore, the average energy per atom 490 00:26:22,350 --> 00:26:24,140 drops and drops and drops. 491 00:26:24,140 --> 00:26:27,320 Of course, the atom number also drops rapidly. 492 00:26:27,320 --> 00:26:29,800 And I was only able to draw it in this way 493 00:26:29,800 --> 00:26:33,600 because I've been on-- I think I-- actually, did 494 00:26:33,600 --> 00:26:37,280 I. I forget what the normalization here 495 00:26:37,280 --> 00:26:38,360 is in the plot. 496 00:26:38,360 --> 00:26:39,770 AUDIENCE: What is n? 497 00:26:39,770 --> 00:26:41,460 PROFESSOR: n is a number of steps. 498 00:26:41,460 --> 00:26:45,470 So after 25 removals of the high energy tail, I'm here. 499 00:26:45,470 --> 00:26:49,250 And after 50, I have a very, very narrow distribution. 500 00:26:49,250 --> 00:26:51,470 So it tells you already something very powerful, 501 00:26:51,470 --> 00:26:53,590 which people were not fully aware 502 00:26:53,590 --> 00:26:56,600 before evaporative cooling of atoms was invented, 503 00:26:56,600 --> 00:26:58,410 that it doesn't take so long. 504 00:26:58,410 --> 00:27:00,840 It doesn't take so many steps. 505 00:27:00,840 --> 00:27:04,090 It takes 50 rethermalization steps. 506 00:27:04,090 --> 00:27:06,700 And each rethermalization takes two or three 507 00:27:06,700 --> 00:27:08,540 elastic collisions. 508 00:27:08,540 --> 00:27:10,610 So what this already demonstrates, 509 00:27:10,610 --> 00:27:14,220 if you can keep your atoms and evaporatively 510 00:27:14,220 --> 00:27:17,210 cool by removing the high energy atom, 511 00:27:17,210 --> 00:27:21,256 after a time which corresponds to a few hundred 512 00:27:21,256 --> 00:27:24,780 elastic collision times, just a few hundred collisions, 513 00:27:24,780 --> 00:27:26,605 you can go way down in temperature. 514 00:27:29,290 --> 00:27:32,660 So if you have this, I actually used this cartoon picture 515 00:27:32,660 --> 00:27:36,630 to write one of the-- to develop a mathematical model 516 00:27:36,630 --> 00:27:39,060 for evaporative cooling, which is still the simplest 517 00:27:39,060 --> 00:27:40,570 model for evaporative cooling which 518 00:27:40,570 --> 00:27:42,560 you can find in the literature. 519 00:27:42,560 --> 00:27:44,130 So in any event, but if you think 520 00:27:44,130 --> 00:27:47,160 in that way, every time you remove 521 00:27:47,160 --> 00:27:50,710 a few percent of the atoms, you realize 522 00:27:50,710 --> 00:27:53,820 that you should think in a logarithmic way. 523 00:27:53,820 --> 00:27:57,110 Every time you lose a certain percentage of your atoms, 524 00:27:57,110 --> 00:27:59,690 you decrease the temperature by a certain percentage. 525 00:28:02,370 --> 00:28:05,800 So in the end, if you think either in discrete steps 526 00:28:05,800 --> 00:28:08,430 or continuously as a function of time, 527 00:28:08,430 --> 00:28:11,010 things should happen exponentially. 528 00:28:11,010 --> 00:28:15,270 So if you want to characterize what happens into this system, 529 00:28:15,270 --> 00:28:17,340 we should correlate the percentage 530 00:28:17,340 --> 00:28:19,450 of temperature change to the percentage 531 00:28:19,450 --> 00:28:20,930 of the change in number. 532 00:28:20,930 --> 00:28:22,840 And here we have a coefficient. 533 00:28:22,840 --> 00:28:25,080 And this coefficient would give us 534 00:28:25,080 --> 00:28:29,130 an exponential-- would give us the power law, how 535 00:28:29,130 --> 00:28:31,140 temperature and number are related. 536 00:28:31,140 --> 00:28:35,550 Or mathematically alpha, which characterizes how much cooling 537 00:28:35,550 --> 00:28:38,500 do you get for which loss in the atom number 538 00:28:38,500 --> 00:28:43,610 is the logarithmic derivative of number with temperature. 539 00:28:43,610 --> 00:28:49,630 All other quantities also scale as power loss of the number. 540 00:28:49,630 --> 00:28:52,680 Let me just assume we are in a potential. 541 00:28:52,680 --> 00:28:56,160 We have T dimensions. 542 00:28:56,160 --> 00:28:59,720 So if you have an harmonic oscillator, it's r square. 543 00:28:59,720 --> 00:29:02,500 If you have a linear trap, it's r to the one. 544 00:29:02,500 --> 00:29:04,630 And for reasons for simplicity, I 545 00:29:04,630 --> 00:29:06,930 took d, the number of dimensions, out. 546 00:29:06,930 --> 00:29:10,100 So you choose delta to get the 1d, 2d, 547 00:29:10,100 --> 00:29:14,546 or whatever to get a harmonic oscillator or linear potential. 548 00:29:14,546 --> 00:29:19,620 OK, Because of equal partition, the temperature, 549 00:29:19,620 --> 00:29:21,740 which is a measure for kinetic energy, 550 00:29:21,740 --> 00:29:23,910 is equal to the potential energy. 551 00:29:23,910 --> 00:29:28,860 And the potential energy is r to the power d over delta. 552 00:29:28,860 --> 00:29:32,380 So therefore, the size of the atoms in the trap 553 00:29:32,380 --> 00:29:35,340 scales with temperature with the power law. 554 00:29:35,340 --> 00:29:37,780 But if the temperature scales with the power law 555 00:29:37,780 --> 00:29:43,770 over the number, then the volume of the atoms, the temperature, 556 00:29:43,770 --> 00:29:46,540 everything scales with the number of atoms 557 00:29:46,540 --> 00:29:48,450 to some exponent. 558 00:29:48,450 --> 00:29:50,660 So everything is sort of exponential 559 00:29:50,660 --> 00:29:53,610 according to power laws. 560 00:29:53,610 --> 00:29:58,160 So I mentioned already, the volume and what I've shown here 561 00:29:58,160 --> 00:30:02,700 is just the three parameters which determine-- 562 00:30:02,700 --> 00:30:05,720 or the two parameters-- which determine all exponential. 563 00:30:05,720 --> 00:30:08,880 The volume goes with the product of delta alpha. 564 00:30:08,880 --> 00:30:10,870 The density goes with something. 565 00:30:10,870 --> 00:30:13,790 The phase space density is density over temperature 566 00:30:13,790 --> 00:30:15,100 to the 3/2. 567 00:30:15,100 --> 00:30:18,230 The collision rate is n sigma v. So 568 00:30:18,230 --> 00:30:24,490 everything scales with n to the alpha. 569 00:30:24,490 --> 00:30:27,080 And all the coefficients are given here. 570 00:30:36,180 --> 00:30:38,210 So the question is, what is alpha? 571 00:30:38,210 --> 00:30:40,860 So it seems delta is our trapping potential. 572 00:30:40,860 --> 00:30:43,210 Once we know alpha, we have a clear prediction 573 00:30:43,210 --> 00:30:46,560 what evaporative cooling can do for us. 574 00:30:46,560 --> 00:30:50,730 Well, alpha was, remember, it told us 575 00:30:50,730 --> 00:30:53,170 how much we lower temperature or energy 576 00:30:53,170 --> 00:30:55,720 when we lose a number of atoms. 577 00:30:55,720 --> 00:30:58,510 So therefore, all we have to figure out 578 00:30:58,510 --> 00:31:01,780 is, when we evaporate an atom, how much energy 579 00:31:01,780 --> 00:31:03,610 does it take with it? 580 00:31:03,610 --> 00:31:08,600 And therefore, alpha will be determined by some people 581 00:31:08,600 --> 00:31:10,110 call it the knife edge. 582 00:31:10,110 --> 00:31:13,910 If you truncate the trap at eta kt-- 583 00:31:13,910 --> 00:31:16,650 I showed you a cartoon where I used the x 584 00:31:16,650 --> 00:31:19,570 to chop off the tail of the Maxwell-Boltzmann distribution 585 00:31:19,570 --> 00:31:20,740 at 4kt. 586 00:31:20,740 --> 00:31:22,700 So eta would have been 4. 587 00:31:22,700 --> 00:31:25,380 So this is what we control experimentally. 588 00:31:25,380 --> 00:31:30,380 At what energy do we allow atoms to leak out of the trap. 589 00:31:30,380 --> 00:31:31,030 All right. 590 00:31:31,030 --> 00:31:34,690 So if you set a threshold of, let's say, 4kt, 591 00:31:34,690 --> 00:31:38,690 4kt is the minimum energy for atoms to leak out. 592 00:31:38,690 --> 00:31:40,730 But some will be a little bit faster. 593 00:31:40,730 --> 00:31:43,980 In other words, they are not creeping over the edge. 594 00:31:43,980 --> 00:31:47,370 They are jumping, they are zipping over the edge. 595 00:31:47,370 --> 00:31:50,380 But because everything is thermal, 596 00:31:50,380 --> 00:31:54,600 this extra energy is on the order of kt, or 1/2 kt. 597 00:31:54,600 --> 00:31:57,960 And in the first analysis, we can neglect it. 598 00:31:57,960 --> 00:32:02,840 So in other words, we can say that each atom, when 599 00:32:02,840 --> 00:32:07,590 it escapes, takes away the energy eta kt. 600 00:32:07,590 --> 00:32:10,180 And eta is the famous eta parameter, 601 00:32:10,180 --> 00:32:11,920 which we determine experimentally 602 00:32:11,920 --> 00:32:14,780 when we evaporate. 603 00:32:14,780 --> 00:32:16,160 Any questions so far? 604 00:32:19,090 --> 00:32:22,110 Right now, I've pretty much gone through definitions. 605 00:32:22,110 --> 00:32:24,190 And now, I simply look at energy conservation. 606 00:32:26,950 --> 00:32:31,170 So now let's look at what is the change of energy 607 00:32:31,170 --> 00:32:33,030 during evaporation. 608 00:32:33,030 --> 00:32:36,990 Well, for n atoms, this is the kinetic energy. 609 00:32:36,990 --> 00:32:40,650 And the extra potential energy for harmonic oscillator, 610 00:32:40,650 --> 00:32:41,680 it would be the same. 611 00:32:41,680 --> 00:32:46,200 Equal partition for another power law potential 612 00:32:46,200 --> 00:32:48,800 that we have to introduce as delta parameter, which 613 00:32:48,800 --> 00:32:52,520 defines the power law of the trapping potential. 614 00:32:52,520 --> 00:32:59,500 So that means the following, that originally, this 615 00:32:59,500 --> 00:33:03,200 describes the number of the total energy. 616 00:33:03,200 --> 00:33:06,310 After an evaporation step, delta N is negative. 617 00:33:06,310 --> 00:33:09,300 I've lost some atoms and delta T is negative. 618 00:33:09,300 --> 00:33:12,560 I am now at a lower temperature and a lower atom number. 619 00:33:12,560 --> 00:33:17,530 And the difference is simply the energy taken away 620 00:33:17,530 --> 00:33:21,045 by the number of atoms TN which have evaporated. 621 00:33:23,670 --> 00:33:27,860 So with that, by just rewriting that, 622 00:33:27,860 --> 00:33:30,410 I get a result for the alpha coefficient. 623 00:33:30,410 --> 00:33:32,320 The alpha coefficient, which tells 624 00:33:32,320 --> 00:33:37,630 us what is the percentage in temperature which-- 625 00:33:37,630 --> 00:33:40,200 how many percent is the temperature 626 00:33:40,200 --> 00:33:42,640 lowered when I lose a certain percentage 627 00:33:42,640 --> 00:33:44,160 of the number of atoms. 628 00:33:44,160 --> 00:33:46,870 Actually, 1% drop of the number of atoms 629 00:33:46,870 --> 00:33:49,554 gives alpha percent in change in temperature. 630 00:33:49,554 --> 00:33:50,970 And this is the alpha coefficient. 631 00:33:54,070 --> 00:33:56,380 So I have an analytic expression for that. 632 00:33:56,380 --> 00:34:01,050 And sure, you realize if you put your cut eta 633 00:34:01,050 --> 00:34:04,180 not at high energy, if you cut at lower energy, 634 00:34:04,180 --> 00:34:06,830 than your alpha coefficient can even turn negative. 635 00:34:06,830 --> 00:34:09,760 Because then, the atoms which evaporate 636 00:34:09,760 --> 00:34:12,100 do not have more than the average energy. 637 00:34:12,100 --> 00:34:15,210 But actually, then the model breaks down. 638 00:34:15,210 --> 00:34:17,330 OK. 639 00:34:17,330 --> 00:34:22,560 So alpha characterizes how much more than the average energy 640 00:34:22,560 --> 00:34:25,590 is removed by escaping atoms. 641 00:34:25,590 --> 00:34:28,083 So [INAUDIBLE] are very simple once we 642 00:34:28,083 --> 00:34:34,150 know what is the threshold of the trap, eta kt. 643 00:34:34,150 --> 00:34:36,699 At what energy do we leak out atoms? 644 00:34:36,699 --> 00:34:40,120 We have a complete description what is the energy, the phase 645 00:34:40,120 --> 00:34:42,449 space density, and such and such after we've 646 00:34:42,449 --> 00:34:44,750 lost a number of atoms. 647 00:34:44,750 --> 00:34:46,889 But you realize, at least for all of you 648 00:34:46,889 --> 00:34:49,060 who do the experiment, something is missing here 649 00:34:49,060 --> 00:34:50,770 that sounds almost too ideal. 650 00:34:50,770 --> 00:34:51,810 We just evaporate. 651 00:34:51,810 --> 00:34:57,685 And we can freely peek what is the energy, 652 00:34:57,685 --> 00:34:59,570 or what is the energy threshold and such. 653 00:34:59,570 --> 00:35:01,870 The experiment is more constrained. 654 00:35:01,870 --> 00:35:06,230 And we have to work harder to get 655 00:35:06,230 --> 00:35:08,900 into this good regime of evaporation. 656 00:35:08,900 --> 00:35:12,050 But let me introduce the experimental constraint which 657 00:35:12,050 --> 00:35:15,540 is very important by asking the question how efficient can 658 00:35:15,540 --> 00:35:17,150 evaporative cooling be? 659 00:35:17,150 --> 00:35:19,490 So based on the idealistic model, 660 00:35:19,490 --> 00:35:23,800 which I've presented to you so far, what is the highest 661 00:35:23,800 --> 00:35:25,835 efficiency of evaporation you can imagine? 662 00:35:31,050 --> 00:35:31,680 Collin. 663 00:35:31,680 --> 00:35:34,584 AUDIENCE: I guess in principle, you could remove just one atom. 664 00:35:34,584 --> 00:35:36,985 Then you'd save [INAUDIBLE]. 665 00:35:36,985 --> 00:35:37,610 PROFESSOR: Yes. 666 00:35:37,610 --> 00:35:38,940 You remove one atom. 667 00:35:38,940 --> 00:35:42,980 You wait until you have one atom, which 668 00:35:42,980 --> 00:35:45,350 has pretty much all the energy of the system. 669 00:35:45,350 --> 00:35:47,790 One atom evaporates and your whole system 670 00:35:47,790 --> 00:35:50,510 is as cold as you want to have it. 671 00:35:50,510 --> 00:35:52,450 Of course, you are all laughing because this 672 00:35:52,450 --> 00:35:54,865 will take much longer than the dwell time 673 00:35:54,865 --> 00:35:56,920 of a graduate student at MIT. 674 00:35:56,920 --> 00:36:00,240 In other words, what you realize, time is a premium. 675 00:36:00,240 --> 00:36:03,510 And it's not just the dwell time of a graduate student. 676 00:36:03,510 --> 00:36:05,080 It's not your patience. 677 00:36:05,080 --> 00:36:10,430 What happens is in a real experiment, there are losses. 678 00:36:10,430 --> 00:36:14,390 There is some form of technical heating. 679 00:36:14,390 --> 00:36:16,640 Since you don't have a perfect vacuum, 680 00:36:16,640 --> 00:36:19,050 residual gas coalitions cause losses. 681 00:36:19,050 --> 00:36:23,390 And its clear you have a time budget which is set by losses. 682 00:36:23,390 --> 00:36:26,290 And either you evaporate in this time budget, 683 00:36:26,290 --> 00:36:29,890 or you've lost your atoms for other reasons. 684 00:36:29,890 --> 00:36:33,810 So that's now what we want to bring in. 685 00:36:33,810 --> 00:36:36,900 We can't make a realistic model of evaporative cooling 686 00:36:36,900 --> 00:36:40,040 without putting in the constraint of time. 687 00:36:40,040 --> 00:36:45,860 And the time constraint is usually determined by losses, 688 00:36:45,860 --> 00:36:48,300 by unavoidable atom losses. 689 00:36:48,300 --> 00:36:50,874 So now we want to understand what 690 00:36:50,874 --> 00:36:52,040 is the speed of evaporation. 691 00:36:55,900 --> 00:36:56,960 So we assume. 692 00:36:56,960 --> 00:36:58,595 We truncate. 693 00:36:58,595 --> 00:37:05,370 We remove an amount dT dN of atoms above this threshold. 694 00:37:05,370 --> 00:37:09,030 And then the question is, how fast can we do it again? 695 00:37:09,030 --> 00:37:12,520 How long does it take for collisions 696 00:37:12,520 --> 00:37:15,420 to replace the tail of the Maxwell-Boltzmann distribution? 697 00:37:20,850 --> 00:37:26,080 But now, you can make an analytic model. 698 00:37:26,080 --> 00:37:29,340 I was very pleased when I saw that it is so easy to actually 699 00:37:29,340 --> 00:37:33,650 get a precise analytic model of that. 700 00:37:33,650 --> 00:37:37,860 If either in the asymptotic limit that eta is very high, 701 00:37:37,860 --> 00:37:40,670 the number of collisions, there is a certain number 702 00:37:40,670 --> 00:37:43,950 of collisions which replenish the tail. 703 00:37:43,950 --> 00:37:46,940 And you want to know how fast does it happen. 704 00:37:46,940 --> 00:37:49,570 But now you can use detail balance. 705 00:37:49,570 --> 00:37:53,090 In an equilibrium situation, those atoms 706 00:37:53,090 --> 00:37:56,240 will collide live with the bulk of the distribution. 707 00:37:56,240 --> 00:38:00,450 And because they are in a highly improbable state, 708 00:38:00,450 --> 00:38:02,670 most of the outcomes of the collision 709 00:38:02,670 --> 00:38:04,870 will put those energetic atoms back 710 00:38:04,870 --> 00:38:08,160 into the bulk of the Maxwell-Boltzmann distribution. 711 00:38:08,160 --> 00:38:12,090 So therefore, the number of particles 712 00:38:12,090 --> 00:38:17,140 which arrive in this tail in equilibrium 713 00:38:17,140 --> 00:38:19,770 is identical to the number of particles 714 00:38:19,770 --> 00:38:21,620 which will leave this tail. 715 00:38:24,380 --> 00:38:27,240 And so all we have to do is we have 716 00:38:27,240 --> 00:38:30,690 to calculate how many such collisions happen. 717 00:38:30,690 --> 00:38:34,190 This is an expression for the fraction of atoms, 718 00:38:34,190 --> 00:38:36,220 with the exponential Boltzmann factor, 719 00:38:36,220 --> 00:38:38,610 which you can find in this tail. 720 00:38:38,610 --> 00:38:42,310 And those atoms collide with a velocity, 721 00:38:42,310 --> 00:38:44,590 which is not the thermal velocity. 722 00:38:44,590 --> 00:38:46,670 The velocity is larger by square root eta, 723 00:38:46,670 --> 00:38:48,430 because those are fast. 724 00:38:48,430 --> 00:38:51,600 So by simply multiplying the fraction of the atoms 725 00:38:51,600 --> 00:39:00,780 with the collision rate, we find how many atoms per unit time 726 00:39:00,780 --> 00:39:02,840 are removed from this tail. 727 00:39:02,840 --> 00:39:06,245 And in detail balance, it means the same number of atoms 728 00:39:06,245 --> 00:39:08,650 is replenished into the tail. 729 00:39:08,650 --> 00:39:11,290 So if you now switch to a continuous model 730 00:39:11,290 --> 00:39:13,730 of evaporation, where we constantly 731 00:39:13,730 --> 00:39:16,120 evaporate the atoms which are produced 732 00:39:16,120 --> 00:39:20,630 through elastic collisions with an energy larger than eta kt, 733 00:39:20,630 --> 00:39:25,830 then this here is our rate of evaporation. 734 00:39:25,830 --> 00:39:28,830 Since we want to think in terms of time constant, 735 00:39:28,830 --> 00:39:32,330 this rate of evaporation is described 736 00:39:32,330 --> 00:39:35,010 by a time constant for evaporation. 737 00:39:35,010 --> 00:39:38,290 And this time constant is now expressed here 738 00:39:38,290 --> 00:39:42,960 by our experimental control parameter eta. 739 00:39:42,960 --> 00:39:44,328 Nancy. 740 00:39:44,328 --> 00:39:47,821 AUDIENCE: So when we are saying that the collisions are putting 741 00:39:47,821 --> 00:39:50,680 atoms back into the lower velocity states, 742 00:39:50,680 --> 00:39:53,323 are we saying that the collisions are more 743 00:39:53,323 --> 00:39:56,790 defined than Maxwell-Boltzmann distribution? 744 00:39:56,790 --> 00:39:58,958 So when you let the system [INAUDIBLE], 745 00:39:58,958 --> 00:40:01,990 it automatically goes into a new Maxwell-Boltzmann distribution, 746 00:40:01,990 --> 00:40:03,797 and that's what determines the tail. 747 00:40:03,797 --> 00:40:06,323 But then we are saying that the collisions are putting 748 00:40:06,323 --> 00:40:08,687 the atoms on the table back into the lower velocities. 749 00:40:08,687 --> 00:40:12,110 So the collisions are not [INAUDIBLE] 750 00:40:12,110 --> 00:40:14,467 Maxwell-Boltzmann distribution? 751 00:40:14,467 --> 00:40:15,050 PROFESSOR: No. 752 00:40:15,050 --> 00:40:19,620 We assume here that the truncation is only 753 00:40:19,620 --> 00:40:22,680 weakly perturbing the Maxwell-Boltzmann distribution. 754 00:40:22,680 --> 00:40:25,940 And at least the easiest way to figure out 755 00:40:25,940 --> 00:40:31,350 how many atoms are produced per unit time, 756 00:40:31,350 --> 00:40:34,260 if atoms in this truncated Maxwell-Boltzmann distribution 757 00:40:34,260 --> 00:40:37,850 collide, they produce, with a certain time constant, 758 00:40:37,850 --> 00:40:40,090 atoms which will populate the tail. 759 00:40:40,090 --> 00:40:45,290 And I can estimate what is this number of atoms which 760 00:40:45,290 --> 00:40:48,070 are per unit time fed into the tail 761 00:40:48,070 --> 00:40:51,190 by assuming I do not have a truncated Boltzmann 762 00:40:51,190 --> 00:40:51,860 distribution. 763 00:40:51,860 --> 00:40:54,440 I have a full Boltzmann distribution. 764 00:40:54,440 --> 00:40:57,520 And I simply calculate what is the total elastic collision 765 00:40:57,520 --> 00:41:00,070 rate of the atoms in this tail. 766 00:41:00,070 --> 00:41:02,310 So in other words, I want to know 767 00:41:02,310 --> 00:41:04,870 how many atoms are fed into the tail. 768 00:41:07,890 --> 00:41:12,620 I get this number of collisions by saying in detail balance, 769 00:41:12,620 --> 00:41:15,360 this number of collisions is the same 770 00:41:15,360 --> 00:41:17,770 as the number of collisions in the full Boltzmann 771 00:41:17,770 --> 00:41:20,180 distribution which goes backward. 772 00:41:20,180 --> 00:41:22,150 And with that argument, I can immediately 773 00:41:22,150 --> 00:41:25,660 write down an expression for what 774 00:41:25,660 --> 00:41:28,760 is the collision rate which produces high energy atoms. 775 00:41:33,080 --> 00:41:33,860 Think about it. 776 00:41:33,860 --> 00:41:36,490 It's subtle, but it's fairly straightforward. 777 00:41:36,490 --> 00:41:39,570 I make the assumption here that eta is sufficiently large, 778 00:41:39,570 --> 00:41:45,090 that I can use properties of the equilibrium Maxwell-Boltzmann 779 00:41:45,090 --> 00:41:48,670 distribution to estimate those eight constants. 780 00:41:48,670 --> 00:41:52,260 And actually, when I found the analytic expression, 781 00:41:52,260 --> 00:41:56,220 I could compare to theory, which was much more complicated 782 00:41:56,220 --> 00:41:58,530 and used truncated Boltzmann distribution. 783 00:41:58,530 --> 00:42:00,770 And in the asymptotic limit of large eta, 784 00:42:00,770 --> 00:42:02,950 I was in full agreement with the other results. 785 00:42:09,740 --> 00:42:12,400 So yes, we have an expression now 786 00:42:12,400 --> 00:42:15,550 for the time constant of evaporation, 787 00:42:15,550 --> 00:42:19,430 how fast evaporation happens because of elastic collisions 788 00:42:19,430 --> 00:42:21,740 which populate the high energy tail. 789 00:42:21,740 --> 00:42:24,030 But usually, when you have a time constant, 790 00:42:24,030 --> 00:42:27,100 you want to express it by another physical time. 791 00:42:27,100 --> 00:42:30,860 And the physical time which characterizes a gas 792 00:42:30,860 --> 00:42:34,550 is the rate of elastic collisions per atom in a gas. 793 00:42:34,550 --> 00:42:40,490 So therefore, I want to express the rate for evaporation. 794 00:42:40,490 --> 00:42:44,920 The rate at which atoms are produced in the high energy 795 00:42:44,920 --> 00:42:48,620 tail is a ratio lambda with the time 796 00:42:48,620 --> 00:42:51,450 between elastic collisions. 797 00:42:51,450 --> 00:42:56,150 And so we realize, of course, that the atoms which 798 00:42:56,150 --> 00:42:58,580 have enough energy to evaporate are not 799 00:42:58,580 --> 00:43:00,970 produced in every elastic collisions. 800 00:43:00,970 --> 00:43:03,190 Actually, there is an exponential factor e 801 00:43:03,190 --> 00:43:06,770 to the eta, because it's only a small part 802 00:43:06,770 --> 00:43:08,400 of those elastic collisions which 803 00:43:08,400 --> 00:43:12,090 happen which produces an high energy atom which can escape. 804 00:43:15,880 --> 00:43:18,670 OK, so with that, we know how many atoms 805 00:43:18,670 --> 00:43:21,810 we can lose by evaporation. 806 00:43:21,810 --> 00:43:24,710 And this is our expression. 807 00:43:24,710 --> 00:43:30,180 But now we have a complete pretty realistic but wonderful 808 00:43:30,180 --> 00:43:34,470 toy model to discuss all aspects of evaporative cooling. 809 00:43:34,470 --> 00:43:37,310 We have our control parameter eta, 810 00:43:37,310 --> 00:43:41,700 which sets the threshold at which atoms can evaporate. 811 00:43:41,700 --> 00:43:46,670 And this factor eta determines the two relevant parameters-- 812 00:43:46,670 --> 00:43:50,190 alpha, which is the efficiency of evaporation, 813 00:43:50,190 --> 00:43:54,070 and lambda, which is the speed of evaporation. 814 00:43:54,070 --> 00:43:58,200 If we set eta very high, following Collin's suggestion, 815 00:43:58,200 --> 00:44:07,040 we can put it so high that one lost atom, one evaporated atom, 816 00:44:07,040 --> 00:44:09,870 can cool all the other atoms to very low temperature. 817 00:44:09,870 --> 00:44:13,890 But we know that this would take too much time. 818 00:44:13,890 --> 00:44:15,850 So in other words, we have a compromise. 819 00:44:15,850 --> 00:44:19,930 If you set eta very high, each atom which evaporates 820 00:44:19,930 --> 00:44:22,010 provides a lot of cooling power. 821 00:44:22,010 --> 00:44:25,520 But high eta means we have exponential slow down 822 00:44:25,520 --> 00:44:26,950 in the evaporation rate. 823 00:44:26,950 --> 00:44:29,080 And we have to wait longer and longer, 824 00:44:29,080 --> 00:44:34,150 or we never get into evaporation because inelastic collisions 825 00:44:34,150 --> 00:44:37,960 and losses has taken its effect. 826 00:44:37,960 --> 00:44:41,850 So therefore, it seems clear that this interplay 827 00:44:41,850 --> 00:44:47,250 between efficiency and speed is asking for compromise. 828 00:44:47,250 --> 00:44:48,840 And this is what we have to realize 829 00:44:48,840 --> 00:44:50,173 in the experimental realization. 830 00:44:53,530 --> 00:44:56,730 OK, there is one addition we have to make to the model. 831 00:44:59,270 --> 00:45:01,730 And this is the following. 832 00:45:01,730 --> 00:45:04,000 We have to introduce losses, losses 833 00:45:04,000 --> 00:45:06,090 which do not come from evaporation. 834 00:45:06,090 --> 00:45:08,620 It can be losses due to background gas collisions, 835 00:45:08,620 --> 00:45:12,450 or losses due to inelastic collisions. 836 00:45:12,450 --> 00:45:16,955 So let me just show you how I introduce that. 837 00:45:16,955 --> 00:45:19,410 I mean, I told you that everything 838 00:45:19,410 --> 00:45:24,750 is the logarithmic derivative. 839 00:45:24,750 --> 00:45:27,310 The logarithmic change of any quantity 840 00:45:27,310 --> 00:45:29,310 goes with the logarithmic change in the atom 841 00:45:29,310 --> 00:45:31,750 number times the coefficient. 842 00:45:31,750 --> 00:45:34,360 And for reasons which become clear in a moment, 843 00:45:34,360 --> 00:45:37,330 I'm now not looking at the temperature, or the density, 844 00:45:37,330 --> 00:45:39,410 or the phase space density. 845 00:45:39,410 --> 00:45:42,050 I'm really interested in the collision rate, 846 00:45:42,050 --> 00:45:43,920 because the collisions rate is what 847 00:45:43,920 --> 00:45:45,550 drives evaporative cooling. 848 00:45:45,550 --> 00:45:48,390 As long as we have collisions, the cooling can go on. 849 00:45:48,390 --> 00:45:51,860 So I want to focus now on how does the collision rate 850 00:45:51,860 --> 00:45:54,520 change during evaporation. 851 00:45:54,520 --> 00:45:57,110 And during evaporation, what we are changing 852 00:45:57,110 --> 00:46:01,070 is the number of atoms, because we evaporated. 853 00:46:01,070 --> 00:46:07,170 So by just putting everything we have said together, 854 00:46:07,170 --> 00:46:08,350 I have this expression. 855 00:46:11,160 --> 00:46:13,540 I assume that the number of atoms 856 00:46:13,540 --> 00:46:18,670 changes as a function of time with the rate at which 857 00:46:18,670 --> 00:46:21,890 high energy atoms are produced. 858 00:46:21,890 --> 00:46:24,940 The time constant for this was lambda, the ratio 859 00:46:24,940 --> 00:46:27,640 between the evaporation time and the elastic collision time 860 00:46:27,640 --> 00:46:29,730 times the elastic collision time. 861 00:46:29,730 --> 00:46:34,340 So this is just re-writing what we have discussed so far. 862 00:46:34,340 --> 00:46:37,830 But now I introduce that there is another loss 863 00:46:37,830 --> 00:46:41,260 rate due to inelastic collision, technical problems and such, 864 00:46:41,260 --> 00:46:44,860 which has a time constant of tau loss. 865 00:46:44,860 --> 00:46:49,090 And if I now do introduce the famous ratio 866 00:46:49,090 --> 00:46:51,880 of good to bad collisions, good collisions 867 00:46:51,880 --> 00:46:54,330 are elastic collisions which drive evaporation. 868 00:46:54,330 --> 00:46:56,760 Bad collisions are collisions where atoms are just 869 00:46:56,760 --> 00:47:01,880 lost due to technical reasons and inelastic collisions. 870 00:47:01,880 --> 00:47:05,430 So if I define this ratio of good to bad collisions, 871 00:47:05,430 --> 00:47:25,400 I have now this equation here which tells me 872 00:47:25,400 --> 00:47:32,030 how the collision rate changes with time. 873 00:47:32,030 --> 00:47:36,200 Just wondering, is a dT missing here or not? 874 00:47:36,200 --> 00:47:36,990 Yes. 875 00:47:36,990 --> 00:47:39,100 So there should be a dT. 876 00:47:39,100 --> 00:47:42,000 So this tells me how the relative 877 00:47:42,000 --> 00:47:45,160 or how the collision rate changes 878 00:47:45,160 --> 00:47:47,250 with the function of time. 879 00:47:47,250 --> 00:47:51,519 But now remember, since this is not 880 00:47:51,519 --> 00:47:53,060 the derivative of the collision rate, 881 00:47:53,060 --> 00:47:54,560 it's the derivative of the collision 882 00:47:54,560 --> 00:47:55,768 rate over the collision rate. 883 00:47:55,768 --> 00:47:57,380 It's a logarithmic derivative. 884 00:47:57,380 --> 00:48:01,430 What we are talking here about it, please add dT to it, 885 00:48:01,430 --> 00:48:05,140 we're talking about is the collision rate exponentially 886 00:48:05,140 --> 00:48:08,520 growing when this coefficient is positive. 887 00:48:08,520 --> 00:48:11,530 Or is it exponentially decaying when this coefficient 888 00:48:11,530 --> 00:48:13,170 is negative? 889 00:48:13,170 --> 00:48:17,880 So you realize that we obtain a threshold condition 890 00:48:17,880 --> 00:48:22,400 when this coefficient is larger or smaller than zero. 891 00:48:22,400 --> 00:48:26,610 And this is called the threshold for runaway evaporation. 892 00:48:26,610 --> 00:48:30,060 So let me just summarize the physics of power loss. 893 00:48:30,060 --> 00:48:33,760 The physics of exponential increase and decrease 894 00:48:33,760 --> 00:48:39,910 actually means that the experimental situation is often 895 00:48:39,910 --> 00:48:41,600 talking about a threshold. 896 00:48:41,600 --> 00:48:44,700 If you're above threshold and you get evaporative cooling 897 00:48:44,700 --> 00:48:48,020 going, you have a positive exponent. 898 00:48:48,020 --> 00:48:50,910 And it will go faster, and faster, and faster. 899 00:48:50,910 --> 00:48:53,240 If this exponent is negative, you 900 00:48:53,240 --> 00:48:56,640 have slowed down evaporation, you can evaporate a little bit. 901 00:48:56,640 --> 00:48:59,950 But it will pretty much come to a stand still. 902 00:48:59,950 --> 00:49:03,860 So this is why quite often, the experimental realization 903 00:49:03,860 --> 00:49:08,150 of evaporative cooling requires to put enough atoms 904 00:49:08,150 --> 00:49:11,570 from the laser cooling stage at sufficiently low temperature 905 00:49:11,570 --> 00:49:15,060 into a magnetic trap that you fulfill the threshold 906 00:49:15,060 --> 00:49:16,890 condition for runaway evaporation. 907 00:49:23,220 --> 00:49:24,410 Any questions about that? 908 00:49:26,990 --> 00:49:29,080 So in other words, what we have found out, 909 00:49:29,080 --> 00:49:32,100 we have found here an expression for the threshold 910 00:49:32,100 --> 00:49:33,630 of runaway evaporation. 911 00:49:33,630 --> 00:49:37,150 And it tells us that we need a minimum ratio 912 00:49:37,150 --> 00:49:38,890 of good to bad collisions. 913 00:49:38,890 --> 00:49:42,590 We may need 100 elastic collision 914 00:49:42,590 --> 00:49:45,100 until we have one inelastic collision. 915 00:49:45,100 --> 00:49:47,570 And then our ratio is 100. 916 00:49:47,570 --> 00:49:50,530 And we will see in a minute if this ratio is 100, 917 00:49:50,530 --> 00:49:54,352 if then we can make right hand side in such a way 918 00:49:54,352 --> 00:49:55,560 that we run away evaporation. 919 00:49:58,600 --> 00:50:05,480 OK, so the left hand side is the ratio of good to bad collisions 920 00:50:05,480 --> 00:50:08,280 is maybe determined how good our vacuum is. 921 00:50:08,280 --> 00:50:11,280 What determines the right hand side? 922 00:50:15,169 --> 00:50:16,210 Well, we talked about it. 923 00:50:16,210 --> 00:50:20,350 Delta is our trapping potential, linear or quadratic. 924 00:50:20,350 --> 00:50:24,390 Alpha and lambda depend on the threshold eta. 925 00:50:24,390 --> 00:50:28,380 How aggressive are we in setting a threshold in energy 926 00:50:28,380 --> 00:50:30,710 for the evaporating atoms? 927 00:50:30,710 --> 00:50:34,080 And what I'm showing you here is the condition. 928 00:50:34,080 --> 00:50:37,460 I'm varying the threshold eta. 929 00:50:37,460 --> 00:50:46,471 And now I'm figuring out what is the-- if I vary the ratio eta, 930 00:50:46,471 --> 00:50:49,830 I call this expression R min. 931 00:50:49,830 --> 00:50:52,000 And my ratio of good to bad collision 932 00:50:52,000 --> 00:50:54,150 has to be better than that. 933 00:50:54,150 --> 00:51:04,270 So if you just look at the solid curve for parabolic trap, 934 00:51:04,270 --> 00:51:07,400 this shows you that you have to pick your parameter 935 00:51:07,400 --> 00:51:10,610 eta between five and seven. 936 00:51:10,610 --> 00:51:14,590 If you pick eta too fast, evaporation is too slow. 937 00:51:14,590 --> 00:51:18,040 And you need a much better ratio of good to bad collisions. 938 00:51:18,040 --> 00:51:21,010 In other words, you need a better vacuum, for instance. 939 00:51:21,010 --> 00:51:24,370 But if you put eta too low, you're 940 00:51:24,370 --> 00:51:25,930 not cooling enough, because you're 941 00:51:25,930 --> 00:51:27,555 cutting too deep into the distribution. 942 00:51:30,140 --> 00:51:33,960 You also realize that if you take a linear power 943 00:51:33,960 --> 00:51:38,740 law, like a quadrupole trap, you get the dashed line. 944 00:51:38,740 --> 00:51:43,120 And the dashed line has a much lower ratio, 945 00:51:43,120 --> 00:51:45,490 has a much lower value of R min. 946 00:51:45,490 --> 00:51:48,470 In other words, if your vacuum is not good enough 947 00:51:48,470 --> 00:51:51,110 and you have losses, in a linear trap, 948 00:51:51,110 --> 00:51:54,190 you can still overcome it by picking your eta 949 00:51:54,190 --> 00:51:57,080 in this regime, whereas for a parabolic trap, 950 00:51:57,080 --> 00:52:00,090 you need at least two or three times 951 00:52:00,090 --> 00:52:02,880 better vacuum to get into the runaway regime. 952 00:52:05,680 --> 00:52:09,030 Anyway, this is how you can look at those equations 953 00:52:09,030 --> 00:52:16,290 and figure out what is needed to get into evaporative cooling. 954 00:52:16,290 --> 00:52:16,790 Collin. 955 00:52:16,790 --> 00:52:19,730 AUDIENCE: Do you need to be in the runway regime to get PEC? 956 00:52:19,730 --> 00:52:21,380 PROFESSOR: Not necessarily. 957 00:52:21,380 --> 00:52:23,670 Some early experiments have done, 958 00:52:23,670 --> 00:52:27,970 I think, the first experiment of Eric Cornell, 959 00:52:27,970 --> 00:52:30,050 I think they never saw this speed up. 960 00:52:30,050 --> 00:52:32,760 They had sort of constant evaporation. 961 00:52:32,760 --> 00:52:37,800 Maybe the cooling rate was even slightly going down. 962 00:52:37,800 --> 00:52:41,940 So you don't necessarily need the exponential speedup. 963 00:52:41,940 --> 00:52:46,860 But you have to be in a regime where at least, 964 00:52:46,860 --> 00:52:48,550 if you're not gaining speed, you're 965 00:52:48,550 --> 00:52:49,940 not losing too much speed. 966 00:52:56,260 --> 00:52:56,760 But, yeah. 967 00:52:56,760 --> 00:52:58,384 I mean, you can just take the equations 968 00:52:58,384 --> 00:53:04,870 and analyze them and figure out if you're 969 00:53:04,870 --> 00:53:05,990 in a favorable regime. 970 00:53:05,990 --> 00:53:09,220 And ultimately, it's fairly easy to integrate those equations 971 00:53:09,220 --> 00:53:12,920 as a function of time and have completely realistic models. 972 00:53:12,920 --> 00:53:17,030 But what I presented to you here is a simple analytic model. 973 00:53:17,030 --> 00:53:20,720 And I used the criterion for runaway evaporation 974 00:53:20,720 --> 00:53:24,371 to discuss how do you have to pick your truncation parameter, 975 00:53:24,371 --> 00:53:26,870 and what happens if you have a different trapping potential. 976 00:53:30,900 --> 00:53:37,330 So based on those models, you will find out 977 00:53:37,330 --> 00:53:44,530 that if you truncated an eta parameter of six, 978 00:53:44,530 --> 00:53:47,100 every truncation of the Maxwell-Boltzmann distribution 979 00:53:47,100 --> 00:53:51,660 means about 1% loss in the atom number. 980 00:53:51,660 --> 00:53:58,570 And after 600 collisions, after 600 elastic collision times, 981 00:53:58,570 --> 00:54:01,620 you have lowered the atom number by 100. 982 00:54:01,620 --> 00:54:03,970 But your phase space density has increased 983 00:54:03,970 --> 00:54:06,800 by six orders of magnitude. 984 00:54:06,800 --> 00:54:09,190 And that means if you have two orders of magnitude 985 00:54:09,190 --> 00:54:13,830 in the number and get six orders of magnitude in the phase space 986 00:54:13,830 --> 00:54:17,620 density, then your gamma factor, which I haven't really 987 00:54:17,620 --> 00:54:20,990 defined it here, but it's the factor 988 00:54:20,990 --> 00:54:23,580 which tells you how the phase space density increase. 989 00:54:23,580 --> 00:54:25,720 Every order of magnitude in the number 990 00:54:25,720 --> 00:54:28,330 boosts the phase space density by three orders of magnitude. 991 00:54:28,330 --> 00:54:31,280 And that's regarded as very favorable. 992 00:54:31,280 --> 00:54:34,150 So in other words, we've talked a lot about laser cooling. 993 00:54:34,150 --> 00:54:38,730 In laser cooling, the standard laser cooling schemes, 994 00:54:38,730 --> 00:54:41,150 you're typically six orders of magnitude away 995 00:54:41,150 --> 00:54:43,540 form quantum degeneracy. 996 00:54:43,540 --> 00:54:46,200 And this tells you how evaporative cooling 997 00:54:46,200 --> 00:54:47,470 can get you there. 998 00:54:47,470 --> 00:54:52,050 You should expect to lose approximately a number of 100 999 00:54:52,050 --> 00:54:55,210 in the number of atoms. 1000 00:54:55,210 --> 00:54:56,760 And that's what it takes you to go 1001 00:54:56,760 --> 00:54:58,910 from laser cooling to quantum degeneracy. 1002 00:55:04,200 --> 00:55:06,180 And you can estimate what your time 1003 00:55:06,180 --> 00:55:09,840 is by asking what is the elastic collision rate 1004 00:55:09,840 --> 00:55:11,750 right after laser cooling. 1005 00:55:11,750 --> 00:55:14,920 If your elastic collision rate is two seconds, 1006 00:55:14,920 --> 00:55:18,250 and you take 600 collisions to get to PEC, 1007 00:55:18,250 --> 00:55:21,400 you know that it would take you 20 minutes. 1008 00:55:21,400 --> 00:55:24,430 And you better work on a vacuum which has 20 minutes lifetime. 1009 00:55:24,430 --> 00:55:27,920 Or alternatively, you improve your laser cooling. 1010 00:55:27,920 --> 00:55:30,220 Or you do some adiabatic compression 1011 00:55:30,220 --> 00:55:32,430 in your magnetic trap to make sure 1012 00:55:32,430 --> 00:55:35,340 that your elastic collision time is faster, 1013 00:55:35,340 --> 00:55:39,830 that you can afford 600 collisions within the lifetime 1014 00:55:39,830 --> 00:55:41,160 given by other parameters. 1015 00:55:47,482 --> 00:55:48,065 Any questions? 1016 00:55:54,610 --> 00:56:01,570 OK, so evaporative cooling happens in the everyday world. 1017 00:56:01,570 --> 00:56:05,770 If you have water and you blow at the water, 1018 00:56:05,770 --> 00:56:07,310 the water evaporates. 1019 00:56:07,310 --> 00:56:11,110 And by evaporation, the water gets colder. 1020 00:56:11,110 --> 00:56:12,890 So the water gets colder. 1021 00:56:12,890 --> 00:56:15,340 And it has, of course, this process 1022 00:56:15,340 --> 00:56:18,390 has a lot of common to the evaporative cooling of atoms 1023 00:56:18,390 --> 00:56:19,830 in an atom trap. 1024 00:56:19,830 --> 00:56:24,600 But I want to ask you now why is evaporative 1025 00:56:24,600 --> 00:56:26,620 cooling more dramatic with atoms? 1026 00:56:26,620 --> 00:56:29,010 We can get the atoms really cold. 1027 00:56:29,010 --> 00:56:32,370 But I don't think you've ever seen that you can blow at water 1028 00:56:32,370 --> 00:56:34,070 and the water freezes. 1029 00:56:34,070 --> 00:56:36,200 So what is the difference? 1030 00:56:36,200 --> 00:56:41,360 What is different in evaporation how you encounter it 1031 00:56:41,360 --> 00:56:43,850 in every day life, and evaporation 1032 00:56:43,850 --> 00:56:50,500 in the way how I just described it, how we apply it to atoms? 1033 00:56:50,500 --> 00:56:52,460 AUDIENCE: Your control of eta. 1034 00:56:52,460 --> 00:56:55,400 Well, in atoms we can really control eta. 1035 00:56:55,400 --> 00:57:01,290 But by blowing on it, it's just one level we evaporate. 1036 00:57:01,290 --> 00:57:02,780 PROFESSOR: That's very close. 1037 00:57:02,780 --> 00:57:06,060 We can pick our eta. 1038 00:57:06,060 --> 00:57:08,905 But even more so, what is constant? 1039 00:57:11,760 --> 00:57:15,300 Or what is the parameter which describes evaporation in water? 1040 00:57:18,483 --> 00:57:19,566 AUDIENCE: Surface tension. 1041 00:57:19,566 --> 00:57:20,040 PROFESSOR: Pardon? 1042 00:57:20,040 --> 00:57:20,990 AUDIENCE: The surface tension? 1043 00:57:20,990 --> 00:57:22,323 PROFESSOR: Some surface tension. 1044 00:57:22,323 --> 00:57:25,790 But the surface tension always turns into a work function. 1045 00:57:25,790 --> 00:57:30,520 It tells us if we have water, what is the energy, the work 1046 00:57:30,520 --> 00:57:34,400 function, for a water molecule to escape? 1047 00:57:34,400 --> 00:57:37,160 And this work function, it's an energy, 1048 00:57:37,160 --> 00:57:40,320 would correspond to eta kt. 1049 00:57:40,320 --> 00:57:43,390 If the water evaporates, well, the work function 1050 00:57:43,390 --> 00:57:44,480 stays the same sort. 1051 00:57:44,480 --> 00:57:47,450 It's sort of electron, whatever fraction of an electron volt 1052 00:57:47,450 --> 00:57:48,480 or whatever it is. 1053 00:57:48,480 --> 00:57:50,510 But the temperature gets lower. 1054 00:57:50,510 --> 00:57:53,440 And therefore, the number of molecules which, 1055 00:57:53,440 --> 00:57:55,220 water molecules, which can evaporate, 1056 00:57:55,220 --> 00:57:58,170 becomes exponentially smaller. 1057 00:57:58,170 --> 00:58:01,390 I mean, I've sort of lured you into thinking that keeping eta 1058 00:58:01,390 --> 00:58:03,980 constant is the most natural thing in the world. 1059 00:58:03,980 --> 00:58:06,030 Yes, it's the most natural world for us 1060 00:58:06,030 --> 00:58:09,160 who want to-- efficient evaporation of atoms. 1061 00:58:09,160 --> 00:58:12,050 But for normal substances, it's the work function 1062 00:58:12,050 --> 00:58:13,410 which is constant. 1063 00:58:13,410 --> 00:58:15,900 So therefore, as the system cools, 1064 00:58:15,900 --> 00:58:17,830 your eta becomes larger and larger. 1065 00:58:17,830 --> 00:58:21,710 And everything turns into a standstill. 1066 00:58:21,710 --> 00:58:26,100 So we are actively tuning the work function of our system 1067 00:58:26,100 --> 00:58:28,295 to sustain a high rate of evaporation. 1068 00:58:38,650 --> 00:58:40,070 How do we define eta? 1069 00:58:40,070 --> 00:58:45,920 How do we select the energy threshold for evaporation? 1070 00:58:45,920 --> 00:58:50,750 Well, for many, many years, Bose-Einstein condensation 1071 00:58:50,750 --> 00:58:52,660 was mainly done in magnetic traps. 1072 00:58:52,660 --> 00:58:54,070 And there were two methods. 1073 00:58:54,070 --> 00:58:56,460 One is just lower the magnetic fields. 1074 00:58:56,460 --> 00:58:59,000 But lowering the magnetic fields is pretty bad, 1075 00:58:59,000 --> 00:59:02,540 because it weakens the magnetic trap. 1076 00:59:02,540 --> 00:59:05,280 And if you weaken the trap, you lower the density. 1077 00:59:05,280 --> 00:59:10,000 And therefore, just because of that, the elastic collision 1078 00:59:10,000 --> 00:59:11,780 rate slows down. 1079 00:59:11,780 --> 00:59:15,380 So what turned out to be by far the superior choice 1080 00:59:15,380 --> 00:59:20,020 is to remove atoms with our F induced evaporation. 1081 00:59:20,020 --> 00:59:23,620 So if you have a magnetic trap, you can tune in our F 1082 00:59:23,620 --> 00:59:28,330 spin flip transition to a certain frequency. 1083 00:59:28,330 --> 00:59:30,570 But the frequency depends on magnetic field. 1084 00:59:30,570 --> 00:59:33,010 The magnetic field in a trap depends on position. 1085 00:59:33,010 --> 00:59:35,850 So what you're doing is, you're selecting with your RF 1086 00:59:35,850 --> 00:59:38,900 frequency a certain point in space 1087 00:59:38,900 --> 00:59:41,700 where the atoms can leave the trap 1088 00:59:41,700 --> 00:59:45,600 and are transferred by a spin flip RF transition 1089 00:59:45,600 --> 00:59:52,580 to an anti-trap state, and are then rejected from the trap. 1090 00:59:52,580 --> 00:59:54,150 I mean, this is very flexible. 1091 00:59:54,150 --> 00:59:57,380 You can change the depths of your magnetic trap 1092 00:59:57,380 --> 00:59:59,880 by just using an RF synthesizer. 1093 00:59:59,880 --> 01:00:01,400 And you can change the trap depths. 1094 01:00:01,400 --> 01:00:04,722 You can lower the trap depths without weakening 1095 01:00:04,722 --> 01:00:05,805 the confinement potential. 1096 01:00:08,360 --> 01:00:10,330 I don't have time to go into details, 1097 01:00:10,330 --> 01:00:14,060 but what I'm showing here is that there 1098 01:00:14,060 --> 01:00:18,784 is two regimes when you have this magnetic trap. 1099 01:00:18,784 --> 01:00:20,200 Red is spin up, blue is spin down. 1100 01:00:20,200 --> 01:00:22,180 And this is your RF transition. 1101 01:00:22,180 --> 01:00:27,380 There is a regime where you have strong RF, 1102 01:00:27,380 --> 01:00:30,010 you should now use dressed energy levels. 1103 01:00:30,010 --> 01:00:33,360 We've discussed dressed energy levels in the optical domain. 1104 01:00:33,360 --> 01:00:36,620 These are now dressed energy levels in the RF domain. 1105 01:00:36,620 --> 01:00:38,300 And there's a wonderful chapter about it 1106 01:00:38,300 --> 01:00:40,200 in atom photon interaction. 1107 01:00:40,200 --> 01:00:42,600 So it really means in the dressed energy 1108 01:00:42,600 --> 01:00:45,020 levels become something like this. 1109 01:00:45,020 --> 01:00:50,810 So you have a potential which looks like an inverted W. 1110 01:00:50,810 --> 01:00:53,350 And you really realize this potential. 1111 01:00:53,350 --> 01:00:59,420 But when the RF is weak, you have a certain probability 1112 01:00:59,420 --> 01:01:03,030 when the atoms go back and forth through the transition region 1113 01:01:03,030 --> 01:01:05,800 that sometimes they will fall down to the lower state. 1114 01:01:05,800 --> 01:01:07,380 And this looks more like this. 1115 01:01:07,380 --> 01:01:09,120 The diabatic picture looks more like 1116 01:01:09,120 --> 01:01:12,190 that you have a little leak and atoms are trickling down 1117 01:01:12,190 --> 01:01:15,720 to the non-confining state. 1118 01:01:15,720 --> 01:01:17,070 Anyway, there are two regimes. 1119 01:01:17,070 --> 01:01:21,020 And the experiment is usually somewhere in between. 1120 01:01:21,020 --> 01:01:24,380 It is not necessary to go to the fully adiabatic. 1121 01:01:24,380 --> 01:01:28,210 It doesn't pay off to go to the fully adiabatic limit. 1122 01:01:28,210 --> 01:01:31,200 So this is how RF evaporation is implemented. 1123 01:01:31,200 --> 01:01:35,630 But I should say in these days, a lot of evaporation 1124 01:01:35,630 --> 01:01:37,380 is now done in optical traps. 1125 01:01:37,380 --> 01:01:39,910 And in optical traps, the method of choice 1126 01:01:39,910 --> 01:01:45,140 is you just ramp down the optical trapping potential. 1127 01:01:45,140 --> 01:01:47,660 Now, when you ramp down the optical trapping potential, 1128 01:01:47,660 --> 01:01:50,220 addressing Collin's question, you usually 1129 01:01:50,220 --> 01:01:52,770 do not go into the runaway regime. 1130 01:01:52,770 --> 01:01:55,900 Because I haven't included that my model assumed 1131 01:01:55,900 --> 01:01:57,810 we have a constant trap. 1132 01:01:57,810 --> 01:02:01,410 But if you now add to the model that you are continuously 1133 01:02:01,410 --> 01:02:04,960 opening up, weakening the trap, you 1134 01:02:04,960 --> 01:02:07,870 have another effect which makes the exponent 1135 01:02:07,870 --> 01:02:10,836 for runaway evaporation more and more negative. 1136 01:02:10,836 --> 01:02:12,210 And I think ultimately, you don't 1137 01:02:12,210 --> 01:02:14,530 get any runaway evaporation anymore. 1138 01:02:14,530 --> 01:02:17,960 On the other hand, optical traps are often more tightly 1139 01:02:17,960 --> 01:02:19,840 confining than magnetic traps. 1140 01:02:19,840 --> 01:02:23,430 And you have sufficiently high density to begin with. 1141 01:02:23,430 --> 01:02:27,240 So therefore, you can tolerate a slow down of evaporation 1142 01:02:27,240 --> 01:02:30,692 and still reach the destination. 1143 01:02:30,692 --> 01:02:32,400 But anyway, I'm now getting more and more 1144 01:02:32,400 --> 01:02:34,210 into technical aspects. 1145 01:02:34,210 --> 01:02:36,750 I think I've given you the concepts. 1146 01:02:36,750 --> 01:02:39,160 But let me just flash you one picture. 1147 01:02:39,160 --> 01:02:42,060 This of course, assumes an idealized model 1148 01:02:42,060 --> 01:02:45,130 where we have only two levels, spin up and spin down. 1149 01:02:45,130 --> 01:02:48,590 You all know that atoms have hyperfine structure. 1150 01:02:48,590 --> 01:02:51,690 Sodium or barium has n equals 2. 1151 01:02:51,690 --> 01:02:54,250 And if you draw now the stress levels 1152 01:02:54,250 --> 01:02:57,350 for five hyperfine levels in RF transition, 1153 01:02:57,350 --> 01:02:59,470 it looks fairly complicated. 1154 01:02:59,470 --> 01:03:01,970 But the result is fairly beautiful. 1155 01:03:01,970 --> 01:03:06,140 When an atom in F equals 2 reaches this point, 1156 01:03:06,140 --> 01:03:10,290 and in the dressed atom picture, the point of evaporation, 1157 01:03:10,290 --> 01:03:14,190 the point where the atom is in resonance with the RF 1158 01:03:14,190 --> 01:03:18,370 is the point where this potential bends down. 1159 01:03:18,370 --> 01:03:22,330 The atom is adiabatically transferred in a dressed state 1160 01:03:22,330 --> 01:03:27,460 from MF plus 2 to MF minus 2. 1161 01:03:27,460 --> 01:03:30,420 So when you evaporate at this point, 1162 01:03:30,420 --> 01:03:32,500 without maybe you noticing it, you actually 1163 01:03:32,500 --> 01:03:36,280 do a four photon transition in the dressed atom picture. 1164 01:03:36,280 --> 01:03:38,330 Again, it's an example where when 1165 01:03:38,330 --> 01:03:40,740 we teach about the schemes, we can completely 1166 01:03:40,740 --> 01:03:42,670 neglect about hyperfine structure. 1167 01:03:42,670 --> 01:03:46,340 And it's just wonderful to see that the actual implementation 1168 01:03:46,340 --> 01:03:49,470 works as well for complicated atoms 1169 01:03:49,470 --> 01:03:53,590 then it does for our idealized two level atom. 1170 01:03:57,310 --> 01:03:59,092 Final remark. 1171 01:03:59,092 --> 01:04:01,175 What is the cooling limit for evaporative cooling? 1172 01:04:05,760 --> 01:04:08,140 When we talked about laser cooling, 1173 01:04:08,140 --> 01:04:10,440 I derived for you the Doppler limit. 1174 01:04:10,440 --> 01:04:14,400 And we talked about even improved cooling limits 1175 01:04:14,400 --> 01:04:17,930 when we discussed sub-Doppler and sub-recoil cooling. 1176 01:04:17,930 --> 01:04:21,240 So what is the cooling limit for evaporative cooling? 1177 01:04:21,240 --> 01:04:25,400 Well, the answer is there is no fundamental limit. 1178 01:04:25,400 --> 01:04:33,290 There is no h bar or quantized limit there. 1179 01:04:33,290 --> 01:04:37,950 There is a practical limit, which usually depends 1180 01:04:37,950 --> 01:04:40,380 on residual heating, on inelastic collisions, 1181 01:04:40,380 --> 01:04:41,420 and all that. 1182 01:04:41,420 --> 01:04:44,690 But we have reached an evaporative cooling 1183 01:04:44,690 --> 01:04:47,930 temperatures as low as 400 picokelvin. 1184 01:04:47,930 --> 01:04:51,090 And the limit was not set by anything fundamental. 1185 01:04:51,090 --> 01:04:53,940 It was more set by our patience. 1186 01:04:53,940 --> 01:04:58,000 The lower the temperature, the slower the process becomes. 1187 01:04:58,000 --> 01:05:01,244 And also, by the sensitivity to technical noise 1188 01:05:01,244 --> 01:05:02,410 and technical perturbations. 1189 01:05:08,870 --> 01:05:10,750 Of course, just a final comment which 1190 01:05:10,750 --> 01:05:13,810 is a segue to what we hopefully do on Friday. 1191 01:05:13,810 --> 01:05:16,390 For evaporative cooling, I've always 1192 01:05:16,390 --> 01:05:22,350 assumed that the energy in the trap is continuous. 1193 01:05:22,350 --> 01:05:26,020 In other words, if you have an harmonic oscillator trapping 1194 01:05:26,020 --> 01:05:30,120 potential, I've neglected the discrete level structure. 1195 01:05:30,120 --> 01:05:32,730 And this is an excellent approximation, 1196 01:05:32,730 --> 01:05:36,340 because many, many atom traps have dressed frequencies 1197 01:05:36,340 --> 01:05:38,390 of a few hertz or kilohertz. 1198 01:05:38,390 --> 01:05:42,595 And at very low temperature, even 1199 01:05:42,595 --> 01:05:46,480 at nanokelvin temperatures, you populate many levels. 1200 01:05:46,480 --> 01:05:50,720 When it comes to the discrete nature of trapping levels, 1201 01:05:50,720 --> 01:05:52,530 we should use a quantum description 1202 01:05:52,530 --> 01:05:54,780 of the motion of atoms in the trap. 1203 01:05:54,780 --> 01:05:57,550 And this is the regime of sideband cooling. 1204 01:05:57,550 --> 01:06:00,470 Sideband cooling is much more important 1205 01:06:00,470 --> 01:06:05,430 for ions, charged particles, and for neutral atoms. 1206 01:06:05,430 --> 01:06:10,560 So therefore, we will discuss sideband cooling on Friday 1207 01:06:10,560 --> 01:06:12,230 when we discuss ion traps. 1208 01:06:17,970 --> 01:06:20,150 Any questions about evaporative cooling? 1209 01:06:27,416 --> 01:06:29,638 AUDIENCE: So may be technical, but once you 1210 01:06:29,638 --> 01:06:31,368 get to the temperatures of picokelvins, 1211 01:06:31,368 --> 01:06:36,308 how do you maintain-- do you keep 1212 01:06:36,308 --> 01:06:39,290 cooling to maintain that temperature? 1213 01:06:39,290 --> 01:06:42,170 PROFESSOR: Yeah, usually when we reach very low temperature, 1214 01:06:42,170 --> 01:06:44,440 there is some form of technical heating. 1215 01:06:44,440 --> 01:06:47,490 And we've often seen when we prepare cloud 1216 01:06:47,490 --> 01:06:49,570 at nanokelvin temperature, we can only 1217 01:06:49,570 --> 01:06:54,320 keep it when we allow a little bit of building evaporation. 1218 01:06:54,320 --> 01:06:58,170 We've sometimes seen that when we just 1219 01:06:58,170 --> 01:07:01,480 keep the atoms in an atom trap, they just slowly heat up. 1220 01:07:01,480 --> 01:07:05,140 But if you keep on evaporating them at a very slow rate, 1221 01:07:05,140 --> 01:07:07,370 we can maintain low temperatures for much longer. 1222 01:07:16,844 --> 01:07:17,510 Other questions? 1223 01:07:21,130 --> 01:07:21,630 OK. 1224 01:07:25,710 --> 01:07:27,290 So already pretty late. 1225 01:07:27,290 --> 01:07:30,870 We just have 10 or 15 minutes left. 1226 01:07:30,870 --> 01:07:32,390 I was hoping that I could already 1227 01:07:32,390 --> 01:07:35,780 start earlier today to talk about Bose gasses. 1228 01:07:35,780 --> 01:07:40,720 Today, I was hoping to talk mainly about Bose gasses, 1229 01:07:40,720 --> 01:07:44,250 on Wednesday on Fermi gasses, on Friday on ion traps. 1230 01:07:44,250 --> 01:07:46,340 We are sort of half an hour behind schedule, 1231 01:07:46,340 --> 01:07:48,530 and I have to figure out how I make up for it. 1232 01:07:48,530 --> 01:07:53,170 But let me just start now with Bose gasses. 1233 01:07:53,170 --> 01:07:55,880 I've taught about this subject several times 1234 01:07:55,880 --> 01:07:57,100 at summer schools. 1235 01:07:57,100 --> 01:08:00,860 And what I give you here is a compressed version on it. 1236 01:08:00,860 --> 01:08:03,190 I will not talk too much about the ideal Bose 1237 01:08:03,190 --> 01:08:04,690 gas, because most of you have seen 1238 01:08:04,690 --> 01:08:06,550 that in statistical mechanics. 1239 01:08:06,550 --> 01:08:09,980 And I will also omit superfluid hydrodynamics, 1240 01:08:09,980 --> 01:08:13,660 because well, it's interesting, but it's 1241 01:08:13,660 --> 01:08:15,690 more special than the other topics. 1242 01:08:15,690 --> 01:08:18,670 So what I want to cover here is to give you 1243 01:08:18,670 --> 01:08:21,510 the main ingredients for the description 1244 01:08:21,510 --> 01:08:24,700 of weekly interacting homogeneous and inhomogeneous 1245 01:08:24,700 --> 01:08:26,210 Bose gases. 1246 01:08:26,210 --> 01:08:28,960 And then as a second part, talk about 1247 01:08:28,960 --> 01:08:31,750 the superfluid Mott-insulator transition. 1248 01:08:31,750 --> 01:08:33,740 But well, for those of you who are not 1249 01:08:33,740 --> 01:08:37,560 working with ultra-cold Bose gasses, maybe some of it 1250 01:08:37,560 --> 01:08:39,270 sounds like jargon to you. 1251 01:08:39,270 --> 01:08:43,899 But there is one sort of overarching concept, which 1252 01:08:43,899 --> 01:08:47,359 I want to emphasize in the theoretical description. 1253 01:08:47,359 --> 01:08:51,609 And this is some form of mean field approximation. 1254 01:08:51,609 --> 01:08:55,040 When we go through the weakly interacting Bose gas, 1255 01:08:55,040 --> 01:08:57,560 then we go through superfluid Fermi gasses, 1256 01:08:57,560 --> 01:09:01,439 and we discuss the superfluid Mott-insulator transition. 1257 01:09:01,439 --> 01:09:06,816 One theme will be if you have a product of four operators, 1258 01:09:06,816 --> 01:09:09,149 and that's what you get when you have interacting atoms, 1259 01:09:09,149 --> 01:09:10,710 you cannot solve anything. 1260 01:09:10,710 --> 01:09:14,450 And you need a method to go from the product of four operators 1261 01:09:14,450 --> 01:09:16,279 to the product of two operators. 1262 01:09:16,279 --> 01:09:18,319 And then you solve a quadratic equation. 1263 01:09:18,319 --> 01:09:20,920 And the step to go from four to two 1264 01:09:20,920 --> 01:09:22,819 is called a mean field approximation. 1265 01:09:22,819 --> 01:09:25,750 And I want to show you three kinds of mean field 1266 01:09:25,750 --> 01:09:28,990 approximation, for the one you have often 1267 01:09:28,990 --> 01:09:32,100 seen for the weakly interacting Bose-Einstein condensate. 1268 01:09:32,100 --> 01:09:34,660 But then I want to show you mean field approximation 1269 01:09:34,660 --> 01:09:38,240 for fermions, where it is a pairing field which 1270 01:09:38,240 --> 01:09:41,490 is a mean field, not your usual mean field energy. 1271 01:09:41,490 --> 01:09:46,029 And then I want to talk about before that even, I 1272 01:09:46,029 --> 01:09:48,160 want to talk about the superfluid Mott-insulator 1273 01:09:48,160 --> 01:09:50,510 transition, where we do a very different mean field 1274 01:09:50,510 --> 01:09:51,819 approximation. 1275 01:09:51,819 --> 01:09:53,390 So maybe you'll realize a little bit 1276 01:09:53,390 --> 01:09:56,670 by repeating that scheme how theory is done 1277 01:09:56,670 --> 01:10:00,910 and how you can deal with simple Hamiltonians, 1278 01:10:00,910 --> 01:10:02,960 but your can't solve them because they 1279 01:10:02,960 --> 01:10:05,930 contain products of four operators. 1280 01:10:05,930 --> 01:10:07,460 So there is sort of a [INAUDIBLE] 1281 01:10:07,460 --> 01:10:09,300 through those three chapters and sort 1282 01:10:09,300 --> 01:10:12,060 of showing you how you can do interesting many body 1283 01:10:12,060 --> 01:10:15,370 physics by doing the right approximation. 1284 01:10:15,370 --> 01:10:19,320 It also teaches you how many body physics is done. 1285 01:10:19,320 --> 01:10:21,660 You have an Hamiltonian which you can't solve. 1286 01:10:21,660 --> 01:10:25,540 And you have to guess the solution and put half of it 1287 01:10:25,540 --> 01:10:26,730 in an approximation. 1288 01:10:26,730 --> 01:10:28,800 And once you've done the right approximation, 1289 01:10:28,800 --> 01:10:31,010 the rest becomes [INAUDIBLE]. 1290 01:10:31,010 --> 01:10:33,390 So with that spirit, I want to go with you 1291 01:10:33,390 --> 01:10:40,260 through the Bogoliubov approximation 1292 01:10:40,260 --> 01:10:44,141 for weakly interacting Bose gas. 1293 01:10:44,141 --> 01:10:45,640 I don't think I have to say too much 1294 01:10:45,640 --> 01:10:48,120 about the ideal Bose-Einstein condensate, 1295 01:10:48,120 --> 01:10:52,040 because it's dealt in pretty much all undergraduate 1296 01:10:52,040 --> 01:10:55,220 or graduate text books. 1297 01:10:55,220 --> 01:10:57,370 There are just two things to remember 1298 01:10:57,370 --> 01:10:59,150 in terms of a system description. 1299 01:11:01,770 --> 01:11:04,590 First, whether Bose-Einstein condensation 1300 01:11:04,590 --> 01:11:08,610 occurs or not depends on the density of states. 1301 01:11:08,610 --> 01:11:11,710 And that depends on dimension and confinement. 1302 01:11:11,710 --> 01:11:13,940 So the fact that you are in a trap, 1303 01:11:13,940 --> 01:11:15,666 it changes the density of states. 1304 01:11:15,666 --> 01:11:18,165 And it changes the criterion for Bose-Einstein condensation. 1305 01:11:20,740 --> 01:11:23,050 But then in terms of a system description, 1306 01:11:23,050 --> 01:11:27,460 if you want to describe your Bose gas and its properties, 1307 01:11:27,460 --> 01:11:32,420 there are aspects of Bose-Einstein condensation 1308 01:11:32,420 --> 01:11:34,540 which are pretty close to the ideal gas, 1309 01:11:34,540 --> 01:11:37,780 and others which require many body physics. 1310 01:11:37,780 --> 01:11:41,240 What is always close to the ideal gas is the transition 1311 01:11:41,240 --> 01:11:45,220 temperature and the condensate fraction, because what happens 1312 01:11:45,220 --> 01:11:47,480 is in almost all experiments, when 1313 01:11:47,480 --> 01:11:49,950 you reach Bose-Einstein condensation, 1314 01:11:49,950 --> 01:11:55,310 your gas is to a good approximation non-interacting. 1315 01:11:55,310 --> 01:11:58,490 Because kt, the transition temperature, 1316 01:11:58,490 --> 01:12:01,940 is much larger than the interaction energy in the gas. 1317 01:12:01,940 --> 01:12:06,350 So before condensation, or at the onset of condensation, 1318 01:12:06,350 --> 01:12:09,750 your gas is like an ideal gas. 1319 01:12:09,750 --> 01:12:11,940 And you will only find a few percent corrections 1320 01:12:11,940 --> 01:12:14,240 to the formula for the ideal gas. 1321 01:12:14,240 --> 01:12:17,644 So therefore, if you want to know at what temperature 1322 01:12:17,644 --> 01:12:19,060 do you reach the transition point, 1323 01:12:19,060 --> 01:12:21,080 or if you're below the transition point, 1324 01:12:21,080 --> 01:12:23,420 if you're 50% below the transition temperature, what 1325 01:12:23,420 --> 01:12:25,850 is your condensate fraction, you can simply 1326 01:12:25,850 --> 01:12:28,900 look up the original formula by Einstein. 1327 01:12:28,900 --> 01:12:31,420 And it gives you a reasonably accurate answer. 1328 01:12:33,960 --> 01:12:37,690 However, for the condensed gas, for the fraction 1329 01:12:37,690 --> 01:12:40,990 of atoms which are both condensed, 1330 01:12:40,990 --> 01:12:43,970 those are atoms in one quantum state. 1331 01:12:43,970 --> 01:12:48,660 For them, there is no other scale 1332 01:12:48,660 --> 01:12:51,310 than the energy between the atoms. 1333 01:12:51,310 --> 01:12:55,220 So therefore, for the uncondensed gas, 1334 01:12:55,220 --> 01:12:57,780 you can get away with an ideal gas approximation. 1335 01:12:57,780 --> 01:12:59,570 For the condensate itself, we have 1336 01:12:59,570 --> 01:13:01,900 to put in the many body physics of the interaction. 1337 01:13:09,500 --> 01:13:13,790 Well, this slide shows here shadow images 1338 01:13:13,790 --> 01:13:16,200 of expanding Bose-Einstein condensates. 1339 01:13:16,200 --> 01:13:18,820 We do evaporative cooling in a magnetic trap. 1340 01:13:18,820 --> 01:13:22,100 You see the shadow picture of the thermal cloud. 1341 01:13:22,100 --> 01:13:25,690 And you see the onset of Bose-Einstein condensation 1342 01:13:25,690 --> 01:13:30,190 as the sudden appearance of, it looks like, a pit in a cherry. 1343 01:13:30,190 --> 01:13:39,580 There is a cool down more confined distribution 1344 01:13:39,580 --> 01:13:41,570 of atoms in the Bose condensed state. 1345 01:13:46,370 --> 01:13:47,860 I think it should play again. 1346 01:13:53,650 --> 01:13:55,610 So this gas is pretty much described 1347 01:13:55,610 --> 01:13:58,730 by an ideal gas where you can put 1348 01:13:58,730 --> 01:14:02,240 in the g2 function, the onset of quantum degeneracy. 1349 01:14:02,240 --> 01:14:04,710 But it looks almost like an ideal gas, 1350 01:14:04,710 --> 01:14:09,250 whereas in an ideal gas, the Bose-Einstein condensate 1351 01:14:09,250 --> 01:14:12,700 should be in the lowest energy state of the trap. 1352 01:14:12,700 --> 01:14:17,290 And I will show you in the next few minutes on Wednesday 1353 01:14:17,290 --> 01:14:19,340 that we are far away from that. 1354 01:14:19,340 --> 01:14:23,500 So interactions are negligible for the normal component, 1355 01:14:23,500 --> 01:14:28,410 for the thermal cloud, but are very important for the Bose 1356 01:14:28,410 --> 01:14:31,640 condensed gas. 1357 01:14:31,640 --> 01:14:35,600 So when you take cross sections through this cloud, or 2-D 1358 01:14:35,600 --> 01:14:40,270 pictures, you see how the broad thermal distribution 1359 01:14:40,270 --> 01:14:43,880 turns into a Bose-Einstein condensate. 1360 01:14:43,880 --> 01:14:46,380 And if you look at such a profile, 1361 01:14:46,380 --> 01:14:50,260 you can clearly see the normal component. 1362 01:14:50,260 --> 01:14:53,460 The normal component can be, with good accuracy, 1363 01:14:53,460 --> 01:14:56,500 be fitted by a non-interacting model, 1364 01:14:56,500 --> 01:15:00,820 whereas central peak is a Bose-Einstein condensate, which 1365 01:15:00,820 --> 01:15:05,412 requires the description I want to show you now. 1366 01:15:05,412 --> 01:15:08,960 The condensate fraction is shown here. 1367 01:15:08,960 --> 01:15:10,870 And again, with fairly good accuracy, 1368 01:15:10,870 --> 01:15:14,950 it follows the description of Einstein, 1369 01:15:14,950 --> 01:15:17,320 because the condensate fraction is not 1370 01:15:17,320 --> 01:15:19,120 a property of the condensate. 1371 01:15:19,120 --> 01:15:21,670 1 minus the condensate fraction is 1372 01:15:21,670 --> 01:15:23,880 a property of the thermal gas. 1373 01:15:23,880 --> 01:15:26,080 In the spirit of Einstein, Einstein 1374 01:15:26,080 --> 01:15:32,650 calculated how many atoms can be in the thermal component 1375 01:15:32,650 --> 01:15:35,380 at a given temperature. 1376 01:15:35,380 --> 01:15:37,790 So Einstein called it the saturated gas. 1377 01:15:37,790 --> 01:15:45,510 At a given temperature, you can only 1378 01:15:45,510 --> 01:15:48,750 keep in thermal equilibrium a certain number 1379 01:15:48,750 --> 01:15:50,440 of atoms in your gas. 1380 01:15:50,440 --> 01:15:55,140 If you have more atoms, they condense into the ground state. 1381 01:15:55,140 --> 01:15:57,400 This is sort of the statistical description. 1382 01:15:57,400 --> 01:15:58,980 So therefore, what I'm plotting here 1383 01:15:58,980 --> 01:16:02,810 is the condensate fraction is actually 1384 01:16:02,810 --> 01:16:04,580 a property of the normal gas. 1385 01:16:04,580 --> 01:16:07,850 It shows that the normal gas is saturated, 1386 01:16:07,850 --> 01:16:10,430 can only hold a certain number of atoms, 1387 01:16:10,430 --> 01:16:12,660 and the remainder of the number of atoms 1388 01:16:12,660 --> 01:16:14,863 has to be in the condensate. 1389 01:16:14,863 --> 01:16:16,904 AUDIENCE: I thought in the three dimensional gas, 1390 01:16:16,904 --> 01:16:19,716 it was three halves. 1391 01:16:19,716 --> 01:16:21,590 PROFESSOR: Yes, but this is an harmonic trap. 1392 01:16:21,590 --> 01:16:23,756 And the harmonic trap changes the density of states. 1393 01:16:30,040 --> 01:16:33,040 Let's talk about the homogeneous Bose-Einstein condensate 1394 01:16:33,040 --> 01:16:33,985 and weak interactions. 1395 01:16:41,510 --> 01:16:45,440 If you write down the Hamiltonian 1396 01:16:45,440 --> 01:16:51,030 for the interactions, it will appear many, many times. 1397 01:16:51,030 --> 01:16:55,190 The general way to write down interactions 1398 01:16:55,190 --> 01:16:58,930 between two particles is you annihilate particles 1399 01:16:58,930 --> 01:17:01,620 in momentum k and p. 1400 01:17:01,620 --> 01:17:06,000 And then, they reappear at different momenta. 1401 01:17:06,000 --> 01:17:08,790 So one momentum gets upshifted by q. 1402 01:17:08,790 --> 01:17:11,350 And one momentum gets downshifted by q. 1403 01:17:11,350 --> 01:17:13,830 This guarantees momentum conservation. 1404 01:17:13,830 --> 01:17:16,330 So what I'm showing you here is the elementary process 1405 01:17:16,330 --> 01:17:20,270 of scattering two particles with momentum k and p scatter, 1406 01:17:20,270 --> 01:17:21,360 disappear. 1407 01:17:21,360 --> 01:17:23,510 That's why we have the annihilation operator. 1408 01:17:23,510 --> 01:17:25,930 And then they reappear to a new momenta. 1409 01:17:25,930 --> 01:17:29,850 That's the most general form of a binary interaction. 1410 01:17:29,850 --> 01:17:32,170 And now we have to make approximations. 1411 01:17:32,170 --> 01:17:34,030 Nobody can solve this Hamiltonian 1412 01:17:34,030 --> 01:17:35,320 in the most general way. 1413 01:17:35,320 --> 01:17:39,980 So one approximation we make is that since the range 1414 01:17:39,980 --> 01:17:43,390 of the atomic interactions is much smaller than the distance 1415 01:17:43,390 --> 01:17:46,260 between atoms or the thermal [INAUDIBLE] wavelengths, 1416 01:17:46,260 --> 01:17:50,390 we approximate the potential by a short range potential, 1417 01:17:50,390 --> 01:17:52,180 or delta function. 1418 01:17:52,180 --> 01:17:57,010 Well, the Fourier of a delta function is constant. 1419 01:17:57,010 --> 01:18:00,460 And that would mean in momentum space, this momentum 1420 01:18:00,460 --> 01:18:04,750 dependent matrix element squared-- yeah, 1421 01:18:04,750 --> 01:18:08,010 matrix element-- is just constant. 1422 01:18:08,010 --> 01:18:13,120 So therefore, we can approximate the Hamiltonian 1423 01:18:13,120 --> 01:18:15,920 by a constant interaction parameter. 1424 01:18:15,920 --> 01:18:19,150 And then we have the sum over all these creation annihilation 1425 01:18:19,150 --> 01:18:20,650 operators. 1426 01:18:20,650 --> 01:18:24,020 I don't want to go into details of low energy scattering 1427 01:18:24,020 --> 01:18:27,600 physics, but it is most convenient to describe 1428 01:18:27,600 --> 01:18:31,110 this parameter by u knot, which is the Fourier 1429 01:18:31,110 --> 01:18:34,650 transform of the interaction potential. 1430 01:18:34,650 --> 01:18:38,770 Or very often we parametrize it with the s wave scattering 1431 01:18:38,770 --> 01:18:41,410 lengths, which is the only relevant parameter 1432 01:18:41,410 --> 01:18:44,890 for elastic collisions at low temperature. 1433 01:18:44,890 --> 01:18:46,950 So with that, we have now a Hamiltonian 1434 01:18:46,950 --> 01:18:49,150 which has kinetic energy. 1435 01:18:49,150 --> 01:18:54,700 And here is the potential energy due to the interaction 1436 01:18:54,700 --> 01:18:56,400 between the atoms. 1437 01:18:56,400 --> 01:19:00,330 And we have taken this constant Fourier transform u knot out 1438 01:19:00,330 --> 01:19:02,230 of the summation. 1439 01:19:02,230 --> 01:19:05,700 Now, I mentioned already to you that a product of four 1440 01:19:05,700 --> 01:19:06,970 operators cannot be solved. 1441 01:19:06,970 --> 01:19:09,010 You need an approximation where you 1442 01:19:09,010 --> 01:19:13,540 reduce the number of operators from four to two. 1443 01:19:13,540 --> 01:19:15,510 And then you solve a quadratic equation. 1444 01:19:15,510 --> 01:19:17,830 And the solution is Bogoliubov solution. 1445 01:19:17,830 --> 01:19:20,720 So how do we reduce now this product 1446 01:19:20,720 --> 01:19:23,060 of four operators to two? 1447 01:19:23,060 --> 01:19:27,310 Well, when we have a condensate where many, many, atoms are 1448 01:19:27,310 --> 01:19:30,040 in one quantum state, we can make the Bogoliubov 1449 01:19:30,040 --> 01:19:31,080 approximation. 1450 01:19:31,080 --> 01:19:33,750 The Bogoliubov approximation is, well, 1451 01:19:33,750 --> 01:19:37,380 if the creation annihilation operator in the zero momentum 1452 01:19:37,380 --> 01:19:43,090 state for the condensate has the following matrix element. 1453 01:19:43,090 --> 01:19:45,870 And now you can see, if N knot, the number 1454 01:19:45,870 --> 01:19:48,460 of atoms in the condensate is large, 1455 01:19:48,460 --> 01:19:54,980 well, we can neglect the difference between N knot, 1456 01:19:54,980 --> 01:19:57,620 N knot plus 1, and N knot minus 1. 1457 01:19:57,620 --> 01:20:00,270 And we simply make the approximation 1458 01:20:00,270 --> 01:20:03,020 that the operator a knot and a knot dagger 1459 01:20:03,020 --> 01:20:06,150 is replaced by the square root of N knot. 1460 01:20:06,150 --> 01:20:09,890 And then, in this sum of [? our quartic ?] terms, 1461 01:20:09,890 --> 01:20:21,550 we only keep those terms which have at least two occurrences 1462 01:20:21,550 --> 01:20:25,460 of the index zero, because the terms which 1463 01:20:25,460 --> 01:20:29,400 are for instance new occurrence of the index zero 1464 01:20:29,400 --> 01:20:31,450 don't get this multiplier N knot. 1465 01:20:31,450 --> 01:20:35,130 So we sort of do an expansion in powers of N knot, 1466 01:20:35,130 --> 01:20:38,580 and we stop here. 1467 01:20:38,580 --> 01:20:43,380 So therefore, now if we make sure that any combination 1468 01:20:43,380 --> 01:20:48,540 of those two involves the index zero, we factor out N knot. 1469 01:20:48,540 --> 01:20:51,590 And then we have products of k minus k with dagger, 1470 01:20:51,590 --> 01:20:54,860 dagger k minus k annihilation operator, or we 1471 01:20:54,860 --> 01:20:59,340 have mixed term a dagger k, a of k, a minus k dagger, a minus k. 1472 01:20:59,340 --> 01:21:02,300 But this is what we get. 1473 01:21:02,300 --> 01:21:04,290 The next step is purely technical. 1474 01:21:04,290 --> 01:21:08,420 We want to get rid of N knot and replace it by N. 1475 01:21:08,420 --> 01:21:12,200 So N is N knot plus the sum of the population 1476 01:21:12,200 --> 01:21:13,365 or other momentum states. 1477 01:21:16,070 --> 01:21:19,260 And that's now our Hamiltonian, which still looks complicated. 1478 01:21:19,260 --> 01:21:22,890 But it can immediately be solved, because all it involves 1479 01:21:22,890 --> 01:21:25,880 is a quadratic product of operators. 1480 01:21:29,390 --> 01:21:30,760 Let me finish this derivation. 1481 01:21:30,760 --> 01:21:32,210 It takes four to five minutes. 1482 01:21:32,210 --> 01:21:34,470 I don't think people come in on Mondays. 1483 01:21:34,470 --> 01:21:36,700 They always come in after us on Wednesday. 1484 01:21:36,700 --> 01:21:39,510 Is that correct? 1485 01:21:39,510 --> 01:21:40,700 So let me just continue. 1486 01:21:40,700 --> 01:21:43,600 I would like to reach the final result with the Bogoliubov 1487 01:21:43,600 --> 01:21:45,770 transformation. 1488 01:21:45,770 --> 01:21:48,030 So what I want to show you is that the moment you have 1489 01:21:48,030 --> 01:21:50,760 bilinear operators, all you have to do is in essence 1490 01:21:50,760 --> 01:21:52,910 you have to solve a quadratic equation. 1491 01:21:52,910 --> 01:21:58,050 And because with all the indices and constant, 1492 01:21:58,050 --> 01:21:59,970 it looks a little bit complicated. 1493 01:21:59,970 --> 01:22:04,320 But so let me just say that the structure of this Hamiltonian 1494 01:22:04,320 --> 01:22:10,850 involves sums which are a of k and a of minus k. k minus k is 1495 01:22:10,850 --> 01:22:12,750 of course important, because that's 1496 01:22:12,750 --> 01:22:14,800 important for momentum conservation. 1497 01:22:14,800 --> 01:22:20,420 So let me now call a of k a, and a of minus k b. 1498 01:22:20,420 --> 01:22:27,654 Then this Hamiltonian has the following structure. 1499 01:22:27,654 --> 01:22:31,470 It has terms a dagger a, b dagger b, b dagger b. 1500 01:22:31,470 --> 01:22:35,100 But then it has other terms a dagger b dagger plus ba. 1501 01:22:35,100 --> 01:22:36,590 Now, let me put it this way. 1502 01:22:36,590 --> 01:22:39,790 If that wouldn't exist, we would be done. 1503 01:22:39,790 --> 01:22:43,530 Because an Hamiltonian which has e knot a dagger 1504 01:22:43,530 --> 01:22:46,280 a is an harmonic oscillator Hamiltonian. 1505 01:22:46,280 --> 01:22:48,060 It is diagonalized. 1506 01:22:48,060 --> 01:22:51,640 a and a dagger are just eigenoperators, 1507 01:22:51,640 --> 01:22:55,900 which create quasi-particles with energy e knot. 1508 01:22:55,900 --> 01:22:59,420 So if you could eliminate this term, we would be done. 1509 01:22:59,420 --> 01:23:03,240 So therefore, let's follow Bogoliubov 1510 01:23:03,240 --> 01:23:07,440 and say that we introduce new operators, alpha beta. 1511 01:23:07,440 --> 01:23:12,270 And the alpha beta operators are linear combinations of a and b. 1512 01:23:12,270 --> 01:23:15,360 Or vice versa, a and b are linear combinations 1513 01:23:15,360 --> 01:23:18,320 of the new operators, alpha beta. 1514 01:23:18,320 --> 01:23:20,350 And since we have a bosonic system, 1515 01:23:20,350 --> 01:23:24,570 and we think it's a good idea to keep the system bosonic, 1516 01:23:24,570 --> 01:23:27,790 we require that those new operators 1517 01:23:27,790 --> 01:23:31,060 fulfill bosonic commutation relations. 1518 01:23:31,060 --> 01:23:33,360 Those bosonic commutation relations 1519 01:23:33,360 --> 01:23:38,880 are fulfilled if u square minus v square, u and v 1520 01:23:38,880 --> 01:23:41,770 are the linear coefficient which express ab 1521 01:23:41,770 --> 01:23:44,850 in terms of alpha and beta if that is 1. 1522 01:23:44,850 --> 01:23:49,050 So in other words, u and v are now two new parameters. 1523 01:23:49,050 --> 01:23:53,310 One condition for u and v is used up 1524 01:23:53,310 --> 01:23:57,190 to ensure the bosonic character of alpha and beta. 1525 01:23:57,190 --> 01:23:59,640 But now, we have a second condition. 1526 01:23:59,640 --> 01:24:02,470 We can have two conditions for two parameters. 1527 01:24:02,470 --> 01:24:06,740 So what we do is we just rewrite this Hamiltonian 1528 01:24:06,740 --> 01:24:08,410 in terms of alpha beta. 1529 01:24:08,410 --> 01:24:09,730 And that's what we get. 1530 01:24:09,730 --> 01:24:12,770 It's a bilinear Hamiltonian where the linear transformation 1531 01:24:12,770 --> 01:24:14,610 stays bilinear. 1532 01:24:14,610 --> 01:24:19,310 But now, our second condition which we can impose on u and v 1533 01:24:19,310 --> 01:24:23,940 is that the prefactor of this cross product is zero. 1534 01:24:23,940 --> 01:24:26,400 So then, by using those two conditions 1535 01:24:26,400 --> 01:24:30,280 for u and v-- and you find the equations in many textbooks. 1536 01:24:30,280 --> 01:24:32,580 I'm not discussing them in great detail here. 1537 01:24:32,580 --> 01:24:35,970 We have obtained an Hamiltonian of this kind. 1538 01:24:35,970 --> 01:24:38,140 And this is diagonalized. 1539 01:24:38,140 --> 01:24:41,930 We know now that alpha beta create quasi-particles 1540 01:24:41,930 --> 01:24:42,920 at a certain energy. 1541 01:24:46,390 --> 01:24:50,370 So now, by going back, I'm in alpha beta 1542 01:24:50,370 --> 01:24:52,570 where expressed by a and b. 1543 01:24:52,570 --> 01:24:56,360 A was a of k. b was a of minus k. 1544 01:24:56,360 --> 01:24:59,220 I just go back to the original nomenclature. 1545 01:24:59,220 --> 01:25:01,450 But we have diagonalized the Hamiltonian. 1546 01:25:01,450 --> 01:25:04,160 So what we have achieved now is our Hamiltonian 1547 01:25:04,160 --> 01:25:08,380 is written in this following way. 1548 01:25:08,380 --> 01:25:12,250 Actually, I've recycled a now. a is no longer 1549 01:25:12,250 --> 01:25:13,910 a particle in a given momentum state. 1550 01:25:13,910 --> 01:25:15,850 It's now a quasi-particle. 1551 01:25:15,850 --> 01:25:18,200 So we have now diagonalized the Hamiltonian 1552 01:25:18,200 --> 01:25:20,440 for the weakly interacting Bose gas. 1553 01:25:20,440 --> 01:25:22,780 And what we have obtained is, well, we 1554 01:25:22,780 --> 01:25:23,840 have the full solution. 1555 01:25:23,840 --> 01:25:26,585 Everything you want to know about this system we know. 1556 01:25:26,585 --> 01:25:28,850 And in particular, we know what are 1557 01:25:28,850 --> 01:25:31,580 the characteristic excitation energies for quasi-particle. 1558 01:25:36,260 --> 01:25:40,360 And this quasi-particle energy gives us 1559 01:25:40,360 --> 01:25:45,200 the energy as a function of momentum 1560 01:25:45,200 --> 01:25:49,150 by replacing the parameter u knot by c. 1561 01:25:49,150 --> 01:25:51,320 c is the speed of sound. 1562 01:25:51,320 --> 01:25:53,390 I can be rewrite it like this. 1563 01:25:53,390 --> 01:25:56,430 And you see that it makes sense immediately, 1564 01:25:56,430 --> 01:26:00,060 because we have now a dispersion relation, which 1565 01:26:00,060 --> 01:26:05,240 in the limit of high momenta is just the normal kinetic energy. 1566 01:26:05,240 --> 01:26:07,950 So the quasi-particle are free particles, 1567 01:26:07,950 --> 01:26:11,990 whereas at low momentum, when this term dominates, 1568 01:26:11,990 --> 01:26:14,820 the dispersion relation is linear. 1569 01:26:14,820 --> 01:26:18,750 And linear means sound and phonons. 1570 01:26:18,750 --> 01:26:22,590 Bose-Einstein condensation is a low energy phenomena. 1571 01:26:22,590 --> 01:26:27,220 So you should not expect that you change any characteristics 1572 01:26:27,220 --> 01:26:29,210 of high energy quasi-particles. 1573 01:26:29,210 --> 01:26:31,870 If you wreck a particle in the Bose-Einstein condensate 1574 01:26:31,870 --> 01:26:36,220 with high energy, it flies out at a high energy particle. 1575 01:26:36,220 --> 01:26:39,970 But a low energy excitation creates sound waves. 1576 01:26:39,970 --> 01:26:43,182 So this is what we have found now. 1577 01:26:43,182 --> 01:26:47,730 And I will show you on Wednesday how it is observed.