1 00:00:00,060 --> 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 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:24,979 --> 00:00:26,020 PROFESSOR: Ready to roll? 9 00:00:28,620 --> 00:00:30,350 OK, good afternoon. 10 00:00:32,990 --> 00:00:37,510 Count down four lectures and you know everything 11 00:00:37,510 --> 00:00:40,670 you have to know about atomic and optical physics, 12 00:00:40,670 --> 00:00:44,910 at least for those of you who take part two-- for those 13 00:00:44,910 --> 00:00:47,760 of you who've taken part one and part two-- for the other ones, 14 00:00:47,760 --> 00:00:52,850 well, there is one more semester which is the following spring. 15 00:00:52,850 --> 00:00:57,340 So we have discussed light forces, first 16 00:00:57,340 --> 00:00:59,040 with the optical Bloch equations, 17 00:00:59,040 --> 00:01:02,120 and then we discussed the stimulated light force 18 00:01:02,120 --> 00:01:04,410 using the dressed atom picture. 19 00:01:04,410 --> 00:01:09,850 So this is sort of finishing the presentation and derivation 20 00:01:09,850 --> 00:01:14,420 of they light forces. 21 00:01:14,420 --> 00:01:20,700 Today, I want to continue to discuss for the aspects. 22 00:01:20,700 --> 00:01:23,440 We've discussed dipole traps, different ways how 23 00:01:23,440 --> 00:01:26,210 we can understand why there is a dipole potential 24 00:01:26,210 --> 00:01:28,840 from the harmonic oscillator to the dressed atom 25 00:01:28,840 --> 00:01:33,500 to the refraction of light by small spheres. 26 00:01:33,500 --> 00:01:38,830 So in all cases, we realize red detuned light traps. 27 00:01:38,830 --> 00:01:41,930 Today I want to discuss what is at work here, 28 00:01:41,930 --> 00:01:44,380 electrical magnetic forces, and then I 29 00:01:44,380 --> 00:01:45,870 want to address what several of you 30 00:01:45,870 --> 00:01:49,320 have asked, where is energy conservation? 31 00:01:49,320 --> 00:01:53,170 Where does the energy go when we cool with a stimulated force? 32 00:01:53,170 --> 00:01:55,130 But this will only take 10 minutes. 33 00:01:55,130 --> 00:01:57,800 The main part of today's lecture will 34 00:01:57,800 --> 00:02:01,750 be techniques to go to very low temperature 35 00:02:01,750 --> 00:02:03,190 to ultra-low temperature. 36 00:02:03,190 --> 00:02:06,880 This is sub-Doppler cooling, sub-recoil cooling, 37 00:02:06,880 --> 00:02:09,130 evaporative cooling, and magnetic trapping 38 00:02:09,130 --> 00:02:10,039 as a technique. 39 00:02:10,039 --> 00:02:12,890 So this is the menu today. 40 00:02:12,890 --> 00:02:16,870 Again, a feature-picked menu. 41 00:02:16,870 --> 00:02:20,300 Now talking about electric and magnetic forces, 42 00:02:20,300 --> 00:02:22,110 you may think it's a trick question, 43 00:02:22,110 --> 00:02:24,920 but it's rather subtle. 44 00:02:24,920 --> 00:02:31,520 All optical forces were derived from the electric dipole 45 00:02:31,520 --> 00:02:35,650 Hamiltonian D dot E. This is what 46 00:02:35,650 --> 00:02:38,790 we used as a starting point for light forces. 47 00:02:38,790 --> 00:02:42,890 So my question is, what about the Lorentz force? 48 00:02:42,890 --> 00:02:45,840 The Lorentz force comes from the magnetic part 49 00:02:45,840 --> 00:02:48,100 of the electromagnetic field. 50 00:02:48,100 --> 00:02:51,690 Now, is there Lorentz force on the atoms? 51 00:02:51,690 --> 00:02:55,380 And, if yes, does it contribute to the light forces or is 52 00:02:55,380 --> 00:02:56,610 it negligibly small? 53 00:03:00,190 --> 00:03:02,030 Actually, the question I'm asking you, 54 00:03:02,030 --> 00:03:04,630 I don't think it's addressed in any textbook 55 00:03:04,630 --> 00:03:06,390 and I think you can go to conferences 56 00:03:06,390 --> 00:03:08,450 and ask some of your colleagues. 57 00:03:08,450 --> 00:03:11,920 You may get a wide range of answers. 58 00:03:11,920 --> 00:03:14,790 So but anyway, has anybody of you thought about it? 59 00:03:14,790 --> 00:03:18,240 What about, is there Lorentz force on neutral atoms? 60 00:03:18,240 --> 00:03:19,670 If no, you're done. 61 00:03:19,670 --> 00:03:23,100 If yes, does it contribute, is it negligible, 62 00:03:23,100 --> 00:03:24,270 or is it even dominant? 63 00:03:30,186 --> 00:03:32,651 AUDIENCE: [INAUDIBLE] temperature 64 00:03:32,651 --> 00:03:35,609 or is equivalent to the T minus A picture, 65 00:03:35,609 --> 00:03:39,553 and the T minus A picture has the Lorentz force, so 66 00:03:39,553 --> 00:03:41,525 [INAUDIBLE]. 67 00:03:41,525 --> 00:03:42,540 So. 68 00:03:42,540 --> 00:03:45,320 PROFESSOR: So it should be included 69 00:03:45,320 --> 00:03:48,520 because we are talking about two different representations. 70 00:03:48,520 --> 00:03:50,585 Collin? 71 00:03:50,585 --> 00:03:51,210 AUDIENCE: Yeah. 72 00:03:54,050 --> 00:03:58,070 PROFESSOR: OK, so yes. 73 00:03:58,070 --> 00:04:00,080 The Lorentz force is included. 74 00:04:00,080 --> 00:04:02,400 Also, when you ride on D dot E, you 75 00:04:02,400 --> 00:04:05,880 think it's the electric force of an electric dipole. 76 00:04:05,880 --> 00:04:07,300 We come to that in a moment. 77 00:04:07,300 --> 00:04:09,900 The next question is, do you have any idea 78 00:04:09,900 --> 00:04:13,195 if the Lorentz force is important 79 00:04:13,195 --> 00:04:14,320 or whether it's negligible? 80 00:04:17,442 --> 00:04:19,089 AUDIENCE: For neutral, or? 81 00:04:19,089 --> 00:04:20,362 PROFESSOR: For neutral atoms. 82 00:04:25,322 --> 00:04:27,802 AUDIENCE: [INAUDIBLE] 83 00:04:27,802 --> 00:04:32,930 PROFESSOR: OK, who of you is working on optical lattices 84 00:04:32,930 --> 00:04:36,160 or wants to work on optical lattices? 85 00:04:36,160 --> 00:04:38,390 OK, just to catch your attention, 86 00:04:38,390 --> 00:04:41,390 when the atom goes up and down the optical lattice, if feels 87 00:04:41,390 --> 00:04:43,080 the lattice potential. 88 00:04:43,080 --> 00:04:46,350 The force is 100% the Lorentz force. 89 00:04:46,350 --> 00:04:50,400 There is no contribution from the electric force. 90 00:04:50,400 --> 00:04:52,200 So that's what I want to show you now. 91 00:04:55,200 --> 00:05:02,850 So we have two fundamental forces, the electric force 92 00:05:02,850 --> 00:05:07,700 and-- if you have two charges separated in the field, 93 00:05:07,700 --> 00:05:10,960 two different electric fields-- we derive from the Coulomb 94 00:05:10,960 --> 00:05:12,780 force, the dipole force. 95 00:05:12,780 --> 00:05:15,580 And the force on the electric dipole 96 00:05:15,580 --> 00:05:19,255 is-- that potential is-- D dot E. The gradient of it 97 00:05:19,255 --> 00:05:25,080 is D times gradient E-- with correct Victorian notation-- 98 00:05:25,080 --> 00:05:29,170 whereas the Lorentz force is V cross B. 99 00:05:29,170 --> 00:05:31,458 So these are the two forces we have to consider. 100 00:05:36,970 --> 00:05:42,350 Let me show two configurations where we use dipole forces, 101 00:05:42,350 --> 00:05:44,230 the stimulated light force. 102 00:05:44,230 --> 00:05:47,820 One is here where we focus a laser beam, 103 00:05:47,820 --> 00:05:52,690 and let's just assume for the sake of the discussion 104 00:05:52,690 --> 00:05:57,030 that the laser beam-- it's propagating here 105 00:05:57,030 --> 00:06:00,390 and the linear polarization goes up and down. 106 00:06:00,390 --> 00:06:03,320 So now we have a dipole moment of the atom, which 107 00:06:03,320 --> 00:06:09,210 oscillates up and down, which is parallel to the electric field. 108 00:06:09,210 --> 00:06:15,330 So now we have D dot E, and indeed if you take the atom 109 00:06:15,330 --> 00:06:19,640 and move it into the laser beam, it 110 00:06:19,640 --> 00:06:22,020 will experience an attractive force 111 00:06:22,020 --> 00:06:24,680 and this attractive force is purely electric. 112 00:06:27,300 --> 00:06:30,790 But let's now come to the situation which many of you 113 00:06:30,790 --> 00:06:34,820 work on in the laboratory, would that we have two plane waves-- 114 00:06:34,820 --> 00:06:37,070 so the laser beams are infinity extended-- 115 00:06:37,070 --> 00:06:41,940 we have two plane waves and they form an optical lattice. 116 00:06:41,940 --> 00:06:48,450 Now, the dipole, the oscillating dipole, the electric dipole, 117 00:06:48,450 --> 00:06:51,090 is again driven by linearly polarized light. 118 00:06:51,090 --> 00:06:56,940 It points up, but the gradient of the electric field 119 00:06:56,940 --> 00:07:00,680 is in the direction of the interference of the lattice, 120 00:07:00,680 --> 00:07:03,010 so the gradient of the electric field 121 00:07:03,010 --> 00:07:06,650 is perpendicular to the dipole moment. 122 00:07:06,650 --> 00:07:11,520 So therefore, D dot E is exactly 0, 123 00:07:11,520 --> 00:07:15,310 and therefore, the electric part of the potential-- the Coulomb 124 00:07:15,310 --> 00:07:18,470 force which is microscopically behind it-- 125 00:07:18,470 --> 00:07:21,220 will not contribute anything. 126 00:07:21,220 --> 00:07:24,720 So I could stop here and say, well, OK, 127 00:07:24,720 --> 00:07:27,200 since we have only the Coulomb force and the Lorentz force, 128 00:07:27,200 --> 00:07:29,240 I've proven to you it's not the Coulomb force, 129 00:07:29,240 --> 00:07:31,820 so therefore it's a Lorentz force. 130 00:07:31,820 --> 00:07:33,220 But let me just tell you why. 131 00:07:36,560 --> 00:07:42,060 So what I'm really telling you is it is actually remarkable 132 00:07:42,060 --> 00:07:45,980 that if you go along the Z direction 133 00:07:45,980 --> 00:07:50,560 and we take the Z derivative of the AC Stock's shift potential, 134 00:07:50,560 --> 00:07:53,370 and the AC Stock's shift potential is one half, 135 00:07:53,370 --> 00:07:55,290 alpha is now the polarizability-- 136 00:07:55,290 --> 00:07:58,370 we use alpha in many places-- times E square. 137 00:07:58,370 --> 00:08:00,590 So this is the AC Stock's shift potential, 138 00:08:00,590 --> 00:08:03,490 and the spatial derivative of the AC Stock's shift potential 139 00:08:03,490 --> 00:08:06,510 is a force, but it's the Lorentz force. 140 00:08:06,510 --> 00:08:07,550 Actually, I was amazed. 141 00:08:07,550 --> 00:08:09,425 I had to work it out myself because I haven't 142 00:08:09,425 --> 00:08:11,900 seen it discussed anywhere, but you can simply 143 00:08:11,900 --> 00:08:14,520 take the electric field of a standing wave, 144 00:08:14,520 --> 00:08:16,820 perform this derivative, and what you find 145 00:08:16,820 --> 00:08:19,010 is you get the Lorentz force. 146 00:08:19,010 --> 00:08:24,390 If you wonder how you get it without going through all 147 00:08:24,390 --> 00:08:29,370 the Victorian notation, the Lorentz force 148 00:08:29,370 --> 00:08:34,110 is V cross B. The velocity of the charge 149 00:08:34,110 --> 00:08:36,520 is related to the derivative of the dipole moment. 150 00:08:36,520 --> 00:08:39,720 The dipole moment is charge times position, 151 00:08:39,720 --> 00:08:41,380 and the derivative of the dipole moment 152 00:08:41,380 --> 00:08:43,570 is charge times velocity. 153 00:08:43,570 --> 00:08:45,750 So don't let yourself fool yourself. 154 00:08:45,750 --> 00:08:49,640 When you have an oscillating field, 155 00:08:49,640 --> 00:08:53,650 there are charges which move their currents inside the atom. 156 00:08:53,650 --> 00:08:57,290 So our V cross B, the V is pretty much D dot. 157 00:08:57,290 --> 00:09:00,520 The D dot is polarizability times E dot, 158 00:09:00,520 --> 00:09:03,220 this is alpha polarizability-- frequency 159 00:09:03,220 --> 00:09:05,590 times the electric field. 160 00:09:05,590 --> 00:09:11,170 But, as Cody said, we can go form P dot A to D dot E-- 161 00:09:11,170 --> 00:09:15,670 or, to be precise-- the magnetic field is the curl of the vector 162 00:09:15,670 --> 00:09:20,590 potential, and the curl gives us the [INAUDIBLE]-- 163 00:09:20,590 --> 00:09:23,520 I'm suppressing Victorian notation, here-- 164 00:09:23,520 --> 00:09:27,570 and you know that E, the electric field, 165 00:09:27,570 --> 00:09:32,710 is the temporal derivative of A, so therefore we get that. 166 00:09:32,710 --> 00:09:39,260 So if you now multiply D dot with the curl of A, 167 00:09:39,260 --> 00:09:42,850 you get something which is K alpha E square. 168 00:09:42,850 --> 00:09:47,220 K is for a standing wave, the spatial derivative. 169 00:09:47,220 --> 00:09:48,720 So I've shown you-- at least you can 170 00:09:48,720 --> 00:09:51,160 see by dimensional analysis, or by just pointing 171 00:09:51,160 --> 00:09:54,820 to the different terms, that if you take the AC Stock's shift 172 00:09:54,820 --> 00:09:56,420 potential through the derivative, 173 00:09:56,420 --> 00:09:58,711 you get everything which you have in the Lorentz force. 174 00:10:02,480 --> 00:10:05,142 It would be a 10 minute, easy homework assignment, 175 00:10:05,142 --> 00:10:07,350 but we're running out of homework assignments-- we're 176 00:10:07,350 --> 00:10:11,730 at the end of the semester-- to show that, explicitly. 177 00:10:11,730 --> 00:10:13,600 So it is really the Lorentz force 178 00:10:13,600 --> 00:10:15,424 which provides the trapping potential 179 00:10:15,424 --> 00:10:16,340 in an optical lattice. 180 00:10:19,780 --> 00:10:21,210 Questions? 181 00:10:21,210 --> 00:10:21,710 [INAUDIBLE] 182 00:10:21,710 --> 00:10:23,336 AUDIENCE: So in the simple picture 183 00:10:23,336 --> 00:10:26,108 you have [INAUDIBLE] the lattice, 184 00:10:26,108 --> 00:10:29,826 you're saying that the transverse confinement is 185 00:10:29,826 --> 00:10:32,526 provided by the electric force, but what 186 00:10:32,526 --> 00:10:34,245 about the perpendicular direction? 187 00:10:34,245 --> 00:10:37,620 [INAUDIBLE] electric field like this, but the other direction? 188 00:10:37,620 --> 00:10:40,120 PROFESSOR: OK, I can discuss with you two simple geometries. 189 00:10:40,120 --> 00:10:42,310 In one case, in one direction it's 190 00:10:42,310 --> 00:10:44,500 the pure Coulomb force at work. 191 00:10:44,500 --> 00:10:47,010 In the other direction, it's the pure Lorentz force. 192 00:10:47,010 --> 00:10:50,670 But in general, when you go through an AC Stock's shift 193 00:10:50,670 --> 00:10:52,880 potential, you have both. 194 00:10:52,880 --> 00:10:55,720 So you may wonder that the Coulomb force, when 195 00:10:55,720 --> 00:10:58,240 you integrate it, would not be integrable. 196 00:10:58,240 --> 00:11:00,620 It does not give rise to potential. 197 00:11:00,620 --> 00:11:03,740 Also, the Lorentz force does not give rise to potential. 198 00:11:03,740 --> 00:11:06,800 But if you add the Coulomb force and the Lorentz force, 199 00:11:06,800 --> 00:11:11,750 you can integrate it up to the optical dipole potential. 200 00:11:11,750 --> 00:11:15,060 I find it remarkable, but it is what it is. 201 00:11:23,105 --> 00:11:23,605 Questions? 202 00:11:26,230 --> 00:11:30,490 So energy conservation, yes. 203 00:11:30,490 --> 00:11:32,490 That's a good thing to discuss now. 204 00:11:38,170 --> 00:11:42,450 Let me first discuss I want to have-- energy conservation 205 00:11:42,450 --> 00:11:47,270 has to be discussed with two different perspectives. 206 00:11:47,270 --> 00:11:49,797 One is a transient phenomenon-- which 207 00:11:49,797 --> 00:11:52,380 I'll do in a few minutes-- but let me first discuss something, 208 00:11:52,380 --> 00:11:56,270 which is CW, and this is the following. 209 00:11:56,270 --> 00:12:00,690 You have atoms which move over a standing wave, 210 00:12:00,690 --> 00:12:04,220 and I use the dressed atom picture to show you 211 00:12:04,220 --> 00:12:07,880 that-- for a strong, blue detuned standing wave-- 212 00:12:07,880 --> 00:12:09,830 we have a friction force. 213 00:12:09,830 --> 00:12:12,790 So the atom-- this was the Sisyphus cooling-- 214 00:12:12,790 --> 00:12:17,360 that the atom is climbing up hills, there's some transition, 215 00:12:17,360 --> 00:12:19,650 climbs up the next hill, but on average the atom 216 00:12:19,650 --> 00:12:23,390 climbs up more hills than it scoots down, 217 00:12:23,390 --> 00:12:25,485 and therefore there is work done. 218 00:12:25,485 --> 00:12:30,150 There is a net force, and this net force provides friction. 219 00:12:30,150 --> 00:12:33,090 It slows down the velocity of the atom. 220 00:12:33,090 --> 00:12:37,280 My question to you is, where has this energy gone? 221 00:12:37,280 --> 00:12:41,200 The kinetic energy lost by atom, where does this energy appear? 222 00:12:48,570 --> 00:12:49,471 Which fields? 223 00:12:49,471 --> 00:12:52,700 AUDIENCE: [INAUDIBLE] the laser beam? 224 00:12:52,700 --> 00:12:56,580 PROFESSOR: Well, we have two fields, the laser fields, 225 00:12:56,580 --> 00:13:01,440 and we have maybe a spontaneous emission? 226 00:13:01,440 --> 00:13:05,940 But the force is only a stimulated force due to-- yeah? 227 00:13:09,970 --> 00:13:11,618 Which fields? 228 00:13:11,618 --> 00:13:13,100 AUDIENCE: The laser field? 229 00:13:13,100 --> 00:13:16,064 Because when you stimulate into a laser field, 230 00:13:16,064 --> 00:13:20,030 your [INAUDIBLE] modulator [INAUDIBLE]. 231 00:13:20,030 --> 00:13:24,470 PROFESSOR: Well, I have a little bit problem with that 232 00:13:24,470 --> 00:13:27,530 because when the atoms-- the force 233 00:13:27,530 --> 00:13:30,190 comes from distributing photons from one standing 234 00:13:30,190 --> 00:13:33,900 wave to the other one, and these are photons of equal frequency. 235 00:13:36,510 --> 00:13:44,230 So in that sense, the net effect is that those two laser beams-- 236 00:13:44,230 --> 00:13:45,880 actually, the atom moves, here. 237 00:13:45,880 --> 00:13:48,840 So what should happen to the laser beams? 238 00:13:48,840 --> 00:13:52,930 Photons are not disappearing. 239 00:13:59,872 --> 00:14:02,330 I will later tell you, in the transient picture-- and this, 240 00:14:02,330 --> 00:14:04,913 maybe you've already done your homework assignment number 10-- 241 00:14:04,913 --> 00:14:06,970 there is something happening to the laser beam. 242 00:14:06,970 --> 00:14:09,905 But this would only happen when atoms fly in 243 00:14:09,905 --> 00:14:12,030 and fly out of a laser beam. 244 00:14:12,030 --> 00:14:14,060 Then we have a different situation. 245 00:14:14,060 --> 00:14:18,120 But here, if the atoms stay in the laser beam the laser beam-- 246 00:14:18,120 --> 00:14:22,050 so to speak-- sees a constant index of refraction medium. 247 00:14:22,050 --> 00:14:24,500 It just goes through and nothing has 248 00:14:24,500 --> 00:14:28,630 happened to the photons except that-- occasionally-- photons 249 00:14:28,630 --> 00:14:30,040 are spontaneously emitted. 250 00:14:33,760 --> 00:14:34,470 Well, OK. 251 00:14:34,470 --> 00:14:40,320 If it's not the laser beams, what remains now? 252 00:14:43,754 --> 00:14:45,170 AUDIENCE: Spontaneous [INAUDIBLE]. 253 00:14:45,170 --> 00:14:47,100 PROFESSOR: Spontaneous emission. 254 00:14:47,100 --> 00:14:55,170 Now how can spontaneous emission carry the energy? 255 00:14:57,720 --> 00:14:59,940 What is the spectrum of spontaneous emission? 256 00:15:04,740 --> 00:15:05,676 AUDIENCE: Triplets? 257 00:15:05,676 --> 00:15:07,050 PROFESSOR: Yes, the mono-triplet. 258 00:15:07,050 --> 00:15:11,000 We have the central carrier, the central carrier 259 00:15:11,000 --> 00:15:13,750 is not carrying away any energy because it 260 00:15:13,750 --> 00:15:16,320 is at the laser frequency. 261 00:15:16,320 --> 00:15:18,420 Now we have two side bands. 262 00:15:18,420 --> 00:15:21,810 Can the side bands carry away energy? 263 00:15:21,810 --> 00:15:23,190 AUDIENCE: Interlacing them. 264 00:15:23,190 --> 00:15:24,110 PROFESSOR: Huh? 265 00:15:24,110 --> 00:15:26,410 AUDIENCE: Interlacing with the [INAUDIBLE]. 266 00:15:26,410 --> 00:15:29,178 PROFESSOR: Asymmetry in what sense? 267 00:15:29,178 --> 00:15:32,440 AUDIENCE: If the side bands are [INAUDIBLE] escape. 268 00:15:32,440 --> 00:15:35,020 PROFESSOR: Well, but didn't we discuss, 269 00:15:35,020 --> 00:15:38,580 with a radiative cascade, that the blue detuned side 270 00:15:38,580 --> 00:15:41,060 bend comes when you go from the upper dressed level 271 00:15:41,060 --> 00:15:42,870 to the dressed level? 272 00:15:42,870 --> 00:15:44,710 And in the lower dressed level, you 273 00:15:44,710 --> 00:15:48,554 can only emit a red detuned side band, not a blue detuned side 274 00:15:48,554 --> 00:15:49,090 bands. 275 00:15:49,090 --> 00:15:50,960 What we said in the radiative cascade 276 00:15:50,960 --> 00:15:55,800 after one blue detuned photon, there is a red detuned photon. 277 00:15:55,800 --> 00:15:59,290 So the number of photons emitted on the red side band 278 00:15:59,290 --> 00:16:01,530 and on the blue side band are exactly equal. 279 00:16:04,470 --> 00:16:04,970 [INAUDIBLE]? 280 00:16:04,970 --> 00:16:07,630 AUDIENCE: But then you have the [INAUDIBLE] cooling picture, 281 00:16:07,630 --> 00:16:11,796 where the red detuned side then was emitted in the lower field 282 00:16:11,796 --> 00:16:12,490 region? 283 00:16:12,490 --> 00:16:13,470 PROFESSOR: Yes. 284 00:16:13,470 --> 00:16:15,530 So the situation is the following. 285 00:16:15,530 --> 00:16:17,900 The atom emits an equal number red, blue. 286 00:16:17,900 --> 00:16:21,110 Red detuned and blue detuned, red detuned and blue detuned. 287 00:16:21,110 --> 00:16:23,125 But when it emits the blue detuned, 288 00:16:23,125 --> 00:16:26,210 the side bend spacing is larger because it 289 00:16:26,210 --> 00:16:29,490 emits the blue detuned photon when it's on top of the hill. 290 00:16:29,490 --> 00:16:32,260 And the red detuned photon is emitted 291 00:16:32,260 --> 00:16:34,860 when the atom is more in the dark, 292 00:16:34,860 --> 00:16:37,670 and therefore, the side bend spacing is lower. 293 00:16:37,670 --> 00:16:40,400 So it alternates blue, red, blue, red. 294 00:16:40,400 --> 00:16:44,280 But the blue detuned photon is emitted when the generalized RB 295 00:16:44,280 --> 00:16:47,810 frequency is larger, when the atoms is at an anti-node 296 00:16:47,810 --> 00:16:51,300 And the red detuned is more preferably emitted 297 00:16:51,300 --> 00:16:58,050 when the photon is at a node of the standing wave. 298 00:16:58,050 --> 00:17:01,170 So it is the difference of these modulations, 299 00:17:01,170 --> 00:17:04,420 the side bands are modulated, and there is a preference 300 00:17:04,420 --> 00:17:06,920 to emit the blue detuned side band when the side band 301 00:17:06,920 --> 00:17:09,829 spacing is larger than in the red detuned case. 302 00:17:09,829 --> 00:17:13,670 And this is where the energy goes. 303 00:17:13,670 --> 00:17:15,690 AUDIENCE: You could make the argument and that 304 00:17:15,690 --> 00:17:19,396 would give a simple picture for the maximum cooling rate? 305 00:17:19,396 --> 00:17:20,020 PROFESSOR: Yes. 306 00:17:20,020 --> 00:17:22,810 AUDIENCE: Where you have omega times your [? scattering ?] 307 00:17:22,810 --> 00:17:23,750 rate? 308 00:17:23,750 --> 00:17:24,520 PROFESSOR: Yes. 309 00:17:24,520 --> 00:17:29,180 That's exactly how, last class, I calculated the cooling rate. 310 00:17:29,180 --> 00:17:32,180 The cooling rate, the energy removed from the system, 311 00:17:32,180 --> 00:17:46,960 is, well, yeah, I said it is the height of the lattice, 312 00:17:46,960 --> 00:17:48,650 but the height of the lattice is also 313 00:17:48,650 --> 00:17:53,710 the amount by which the-- the lattice is 314 00:17:53,710 --> 00:17:56,230 nothing else than the generalized RB frequency. 315 00:17:56,230 --> 00:17:58,860 This is the energy level of one crest level, 316 00:17:58,860 --> 00:18:01,760 and therefore the lattice is the increment 317 00:18:01,760 --> 00:18:04,090 by which we modulate blue and red side bands. 318 00:18:04,090 --> 00:18:06,620 Exactly. 319 00:18:06,620 --> 00:18:08,870 This can be quantitatively worked out. 320 00:18:08,870 --> 00:18:11,620 If I had another half hour, I could do it for you. 321 00:18:11,620 --> 00:18:13,480 It's in the references, but this is 322 00:18:13,480 --> 00:18:17,990 the physical picture for cooling. 323 00:18:17,990 --> 00:18:26,870 OK, but so you realize, this is one of the subtleties. 324 00:18:26,870 --> 00:18:29,450 We can understand the force simply 325 00:18:29,450 --> 00:18:33,090 from a simple picture, energy levels, Sisyphus cooling 326 00:18:33,090 --> 00:18:35,120 and such, but in order to understand 327 00:18:35,120 --> 00:18:38,330 where the energy comes from, we need the mono-triplet. 328 00:18:38,330 --> 00:18:40,580 We need spontaneous emission. 329 00:18:40,580 --> 00:18:44,690 So the force comes from the redistribution of photons. 330 00:18:44,690 --> 00:18:46,135 It's the stimulated redistribution 331 00:18:46,135 --> 00:18:48,600 of photons which is responsible for the force, 332 00:18:48,600 --> 00:18:51,940 but the energy balance comes from, 333 00:18:51,940 --> 00:18:56,330 what can be neglected for the momentum balance, namely 334 00:18:56,330 --> 00:19:00,270 the few spontaneous emission events, and especially 335 00:19:00,270 --> 00:19:03,870 those which involve the blue and red side bands. 336 00:19:03,870 --> 00:19:05,880 And it beautifully works together 337 00:19:05,880 --> 00:19:09,840 and all the equation of physics are obeyed. 338 00:19:09,840 --> 00:19:14,610 OK, but that tells you now that the cooling effect 339 00:19:14,610 --> 00:19:19,440 hinges on spontaneous emission. 340 00:19:19,440 --> 00:19:21,541 And yes, spontaneous emission is dissipation. 341 00:19:21,541 --> 00:19:23,540 And whenever you want the friction coefficient-- 342 00:19:23,540 --> 00:19:26,300 you want dissipation-- you need an open system 343 00:19:26,300 --> 00:19:28,550 and the open system is spontaneous emission 344 00:19:28,550 --> 00:19:29,870 into those nodes. 345 00:19:29,870 --> 00:19:32,840 So let's now talk about an opposite situation 346 00:19:32,840 --> 00:19:36,290 where we do not have any spontaneous emission. 347 00:19:36,290 --> 00:19:38,010 And I think almost all the people 348 00:19:38,010 --> 00:19:41,330 who work with optical lattices use infrared lasers for them, 349 00:19:41,330 --> 00:19:44,300 or [INAUDIBLE] detuned laser where spontaneous emission 350 00:19:44,300 --> 00:19:47,395 happens only once every 20, 30 seconds 351 00:19:47,395 --> 00:19:51,410 or 100 seconds per atom, so we can really neglect it. 352 00:19:51,410 --> 00:19:53,350 So now the question is the following. 353 00:19:53,350 --> 00:19:55,830 Assume you have a focused laser beam 354 00:19:55,830 --> 00:19:58,290 and you load a Bose-Einstein condensate 355 00:19:58,290 --> 00:20:00,830 at the edge of the cloud, and now the condensate 356 00:20:00,830 --> 00:20:05,150 is accelerated, sucked into the laser beam. 357 00:20:05,150 --> 00:20:08,550 So now the atom has kinetic energy. 358 00:20:08,550 --> 00:20:13,510 The question is, where does this kinetic energy come from? 359 00:20:13,510 --> 00:20:16,210 And this time, we do not have spontaneous emission 360 00:20:16,210 --> 00:20:19,630 as we have when we do frictional cooling. 361 00:20:19,630 --> 00:20:23,140 So the question is now, where does the energy of the atom 362 00:20:23,140 --> 00:20:23,640 come from? 363 00:20:28,520 --> 00:20:31,936 AUDIENCE: But if we were talking about a dielectric medium, 364 00:20:31,936 --> 00:20:37,304 it's coming from those light waves that are reflected or not 365 00:20:37,304 --> 00:20:39,760 reflected. 366 00:20:39,760 --> 00:20:42,220 PROFESSOR: If comes from those light rays, 367 00:20:42,220 --> 00:20:45,590 but-- the atom is a dielectric medium, that's good-- 368 00:20:45,590 --> 00:20:52,560 but the reflection and deflection of light rays-- 369 00:20:52,560 --> 00:20:55,670 at least in leading order-- you would say photons of the same 370 00:20:55,670 --> 00:20:57,702 energy are deflected-- 371 00:20:57,702 --> 00:21:01,638 AUDIENCE: Wouldn't the [INAUDIBLE] of the atom 372 00:21:01,638 --> 00:21:07,060 just get absorbed [INAUDIBLE] and the [INAUDIBLE]? 373 00:21:07,060 --> 00:21:09,540 PROFESSOR: I would hate to use the picture of absorption 374 00:21:09,540 --> 00:21:11,090 and emission. 375 00:21:11,090 --> 00:21:13,610 What really happens is it's just scattering. 376 00:21:13,610 --> 00:21:14,740 It's scattering. 377 00:21:14,740 --> 00:21:17,670 you scatter a photon which has one frequency 378 00:21:17,670 --> 00:21:19,017 into a different direction. 379 00:21:19,017 --> 00:21:20,225 It's really light scattering. 380 00:21:23,540 --> 00:21:30,056 Well, you have a homework assignment on that, 381 00:21:30,056 --> 00:21:32,430 and I don't need to go into details because it can easily 382 00:21:32,430 --> 00:21:36,790 be worked out, but what happens is the following. 383 00:21:36,790 --> 00:21:39,290 The atom is a dielectric medium. 384 00:21:39,290 --> 00:21:41,210 It has an index of refraction. 385 00:21:41,210 --> 00:21:45,050 And if you put an index-- if you suddenly 386 00:21:45,050 --> 00:21:48,990 put an index of refraction into a laser beam-- 387 00:21:48,990 --> 00:21:52,770 then you change the phase of the laser beam. 388 00:21:52,770 --> 00:21:56,760 The atoms act as an electro-optical modulator, 389 00:21:56,760 --> 00:22:01,600 and they change the frequency of the transmitted photons. 390 00:22:01,600 --> 00:22:03,780 You have a homework assignment on that. 391 00:22:03,780 --> 00:22:06,580 So if you increase the number of atoms in your laser beam, 392 00:22:06,580 --> 00:22:10,540 because they move in, this is-- the phase 393 00:22:10,540 --> 00:22:13,960 of the transmitted photons is shifted. 394 00:22:13,960 --> 00:22:17,740 If the phase of the photons is shifted during a certain time, 395 00:22:17,740 --> 00:22:20,390 phase over time is frequency. 396 00:22:20,390 --> 00:22:23,630 So therefore you will see, when the atoms move in and gain 397 00:22:23,630 --> 00:22:28,880 kinetic energy, the transmitted photons of the laser beam 398 00:22:28,880 --> 00:22:31,250 have lost some energy. 399 00:22:31,250 --> 00:22:34,100 They are slightly shifted to the red. 400 00:22:34,100 --> 00:22:36,800 So in other words, it's funny but if you have a dipole trap 401 00:22:36,800 --> 00:22:39,020 and the laser beam is focused and atoms 402 00:22:39,020 --> 00:22:41,750 slosh through the trap, if you would carefully 403 00:22:41,750 --> 00:22:44,590 analyze the frequency of the transmitted beam, 404 00:22:44,590 --> 00:22:46,890 you will find that it's a little bit blue, a little bit 405 00:22:46,890 --> 00:22:48,390 red, a little bit blue, a little bit 406 00:22:48,390 --> 00:22:52,130 red, and this compensates for the kinetic energy. 407 00:22:52,130 --> 00:22:54,800 Again, I don't think you'll find it in any textbook. 408 00:22:54,800 --> 00:22:57,900 It confused me for a while until I could work it out. 409 00:22:57,900 --> 00:23:01,280 But the homework assignment gives you an exact dissoluble 410 00:23:01,280 --> 00:23:04,060 model where you can exactly show what I told you. 411 00:23:07,580 --> 00:23:10,680 So these are the two situations, here. 412 00:23:10,680 --> 00:23:12,230 It's either spontaneous emission, 413 00:23:12,230 --> 00:23:14,160 which is responsible for cooling, 414 00:23:14,160 --> 00:23:17,320 but here we have a conservative potential. 415 00:23:17,320 --> 00:23:19,510 And in this conservative potential, 416 00:23:19,510 --> 00:23:22,660 the only player is the laser beam which is transmitted. 417 00:23:22,660 --> 00:23:27,640 And, indeed, it is the photons which have been transmitted, 418 00:23:27,640 --> 00:23:29,850 and usually you wouldn't put that into your picture. 419 00:23:29,850 --> 00:23:32,640 Remember, the dipole force can be explained 420 00:23:32,640 --> 00:23:35,710 by eliminating the coherent field of the laser 421 00:23:35,710 --> 00:23:39,280 field-- for canonical transformation-- 422 00:23:39,280 --> 00:23:41,000 and then we have a C number. 423 00:23:41,000 --> 00:23:42,910 We have a classical electric field. 424 00:23:42,910 --> 00:23:46,090 This is your Hamiltonian, and from this Hamiltonian, 425 00:23:46,090 --> 00:23:49,610 you derive that you have a conservative potential. 426 00:23:49,610 --> 00:23:52,750 But if you now ask where the energy goes, 427 00:23:52,750 --> 00:23:54,840 the energy-- really, in this situation-- 428 00:23:54,840 --> 00:23:58,250 goes into a frequency shift of the photons. 429 00:23:58,250 --> 00:24:00,470 Also, in your Hamiltonian, you have 430 00:24:00,470 --> 00:24:04,440 assumed that the electric field is an external classical field, 431 00:24:04,440 --> 00:24:10,110 E naught times cosine omega T. 432 00:24:10,110 --> 00:24:13,070 Well, if you want my advice, whenever 433 00:24:13,070 --> 00:24:17,130 you use the AC Stock's shift for trapping, 434 00:24:17,130 --> 00:24:19,420 just use the AC Stock's shift and regard it 435 00:24:19,420 --> 00:24:22,370 as a classic potential, and just completely forget 436 00:24:22,370 --> 00:24:25,470 that there are photons behind because if you want to account 437 00:24:25,470 --> 00:24:28,310 what really happens to the photons-- what really happens 438 00:24:28,310 --> 00:24:31,522 to the energy in this system-- it really gets ugly 439 00:24:31,522 --> 00:24:32,855 and it can get really confusing. 440 00:24:37,586 --> 00:24:38,086 Questions? 441 00:24:44,528 --> 00:24:48,625 OK, we are done with the stimulated force. 442 00:24:48,625 --> 00:24:51,660 We are now in preparation for next week 443 00:24:51,660 --> 00:24:54,340 when I want to tell you about degenerate Fermi gases 444 00:24:54,340 --> 00:24:57,440 and degenerate Bose gases. 445 00:24:57,440 --> 00:25:00,920 Those gases require nanokelvin temperatures, 446 00:25:00,920 --> 00:25:03,840 so today, in the next hour, I want 447 00:25:03,840 --> 00:25:05,830 to tell you what are the techniques 448 00:25:05,830 --> 00:25:10,000 to create such ultra-low temperatures, 449 00:25:10,000 --> 00:25:11,860 and the techniques I want to discuss 450 00:25:11,860 --> 00:25:14,770 is one addition to laser cooling. 451 00:25:14,770 --> 00:25:17,950 We have so far discussed laser cooling to the Doppler limit, 452 00:25:17,950 --> 00:25:20,250 but now I have to tell you that that's not 453 00:25:20,250 --> 00:25:21,900 where laser cooling stops. 454 00:25:21,900 --> 00:25:26,310 You can go sub-Doppler and sub-recoil. 455 00:25:26,310 --> 00:25:29,990 And, at least when the Nobel Prize 456 00:25:29,990 --> 00:25:32,610 was given to Bill Phillips, Steve Chu, and Claude 457 00:25:32,610 --> 00:25:36,750 Cohen-Tannoudji, well, if you go to the Nobel website 458 00:25:36,750 --> 00:25:40,964 and read the report of the Nobel Committee-- well, 459 00:25:40,964 --> 00:25:42,630 they should have given it to those three 460 00:25:42,630 --> 00:25:44,560 people for the many, many wonderful things 461 00:25:44,560 --> 00:25:47,130 they have done, and I've made frequent references 462 00:25:47,130 --> 00:25:47,910 to their work. 463 00:25:47,910 --> 00:25:50,020 Bill Phillips invented Zeeman slower, 464 00:25:50,020 --> 00:25:52,760 Claude Cohen-Tannoudji invented the dressed atom picture, 465 00:25:52,760 --> 00:25:54,790 they have many, many contributions, 466 00:25:54,790 --> 00:25:58,265 but the Nobel Committee tried to justify their choice 467 00:25:58,265 --> 00:26:01,240 with a coherent story, and the coherent story 468 00:26:01,240 --> 00:26:04,750 was Steve Chu cooled to the Doppler limit in molasses, 469 00:26:04,750 --> 00:26:07,700 Bill Phillips discovered sub-Doppler cooling, 470 00:26:07,700 --> 00:26:11,600 and Claude Cohen-Tannoudji realized sub-recoil cooling. 471 00:26:11,600 --> 00:26:15,005 So the storyline was cold, colder, the coldest. 472 00:26:15,005 --> 00:26:16,660 And that's what we want to talk today. 473 00:26:19,630 --> 00:26:24,620 So sub-Doppler cooling. 474 00:26:24,620 --> 00:26:27,770 Well, I could have spent-- or, 10 years ago, I 475 00:26:27,770 --> 00:26:29,710 spent a whole lecture on sub-Doppler cooling, 476 00:26:29,710 --> 00:26:32,210 polarization gradient cooling. 477 00:26:32,210 --> 00:26:33,150 Wonderful! 478 00:26:33,150 --> 00:26:38,200 The kind of epiphany of elegance in [INAUDIBLE] 479 00:26:38,200 --> 00:26:40,250 a mechanical description of an atom. 480 00:26:40,250 --> 00:26:43,130 But I have to say-- and then I explained, 481 00:26:43,130 --> 00:26:45,380 in another lecture, sub-recoil cooling, 482 00:26:45,380 --> 00:26:48,080 how you can even cool below the recoil limit. 483 00:26:48,080 --> 00:26:53,200 Well, this year, 2013, I spent 15 minute on it. 484 00:26:53,200 --> 00:26:55,920 The reason is the following. 485 00:26:55,920 --> 00:27:00,420 When I joined atomic physics and was a post-doc 486 00:27:00,420 --> 00:27:02,460 in the '90s and assistant professor, 487 00:27:02,460 --> 00:27:05,180 the conference's [INAUDIBLE] of [INAUDIBLE], 488 00:27:05,180 --> 00:27:09,690 [INAUDIBLE] was full on intents to find new ways to laser cool. 489 00:27:09,690 --> 00:27:12,770 Ideas of how to get to lower and lower temperatures. 490 00:27:12,770 --> 00:27:16,120 There were many, many different-- sub-Doppler, 491 00:27:16,120 --> 00:27:18,200 sub-recoil techniques were discussed. 492 00:27:18,200 --> 00:27:19,990 This was the main topic. 493 00:27:19,990 --> 00:27:23,160 But then suddenly, in 1995, evaporative 494 00:27:23,160 --> 00:27:26,770 cooling-- an intellectually boring cooling technique, 495 00:27:26,770 --> 00:27:29,160 just have atoms collide and evaporate-- 496 00:27:29,160 --> 00:27:31,770 this led to the lowest temperature ever, 497 00:27:31,770 --> 00:27:34,110 and it was almost a sudden transition. 498 00:27:34,110 --> 00:27:37,930 Within a few months there was no research, no papers anymore, 499 00:27:37,930 --> 00:27:39,940 on advanced methods of laser cooling. 500 00:27:39,940 --> 00:27:42,860 Evaporative cooling it just completely wiped out 501 00:27:42,860 --> 00:27:45,200 this area over atomic physics. 502 00:27:45,200 --> 00:27:47,940 The reason being because all cooling schemes which 503 00:27:47,940 --> 00:27:51,590 have been discussed had, in the end, some problems. 504 00:27:51,590 --> 00:27:53,990 At some point, photons heat. 505 00:27:53,990 --> 00:27:58,770 Heat because they excite an atom and if an excited atom collides 506 00:27:58,770 --> 00:28:01,980 with another atom, there is a heating mechanism and such. 507 00:28:01,980 --> 00:28:07,100 So even all the optimistic proposals for laser cooling 508 00:28:07,100 --> 00:28:10,930 reached very low temperature, but never at high density. 509 00:28:10,930 --> 00:28:14,270 And evaporative cooling just did everything for you. 510 00:28:14,270 --> 00:28:18,700 But anyway, I think it-- not just because a Nobel Prize was 511 00:28:18,700 --> 00:28:21,760 given for sub-Doppler and sub-recoil cooling, 512 00:28:21,760 --> 00:28:24,900 this is really an accomplishment to understand how can you 513 00:28:24,900 --> 00:28:27,710 go through conventional cooling limits, 514 00:28:27,710 --> 00:28:31,800 and at least every graduate student 515 00:28:31,800 --> 00:28:34,280 who happens to graduate in AMO physics at MIT 516 00:28:34,280 --> 00:28:36,090 should know what are the concepts 517 00:28:36,090 --> 00:28:40,030 behind sub-Doppler and sub-recoil cooling. 518 00:28:40,030 --> 00:28:43,110 So what I need in order to introduce those cooling methods 519 00:28:43,110 --> 00:28:47,450 for you is I have to remind you that optical pumping is 520 00:28:47,450 --> 00:28:52,170 a cooling scheme, and one could actually say, to some extent, 521 00:28:52,170 --> 00:28:57,010 every cooling you do with lasers is based on optical pumping. 522 00:28:57,010 --> 00:28:58,790 And so let me explain that. 523 00:28:58,790 --> 00:29:02,200 Optical pumping happens that-- let's 524 00:29:02,200 --> 00:29:05,130 say you have a level structure with a circularly 525 00:29:05,130 --> 00:29:09,340 polarized light where you can go maybe from M equals 1 to M 526 00:29:09,340 --> 00:29:10,540 equals zero. 527 00:29:10,540 --> 00:29:13,550 And then there can be spontaneous emission back to M 528 00:29:13,550 --> 00:29:17,020 equals 1 and back to M equals minus 1, but the laser-- 529 00:29:17,020 --> 00:29:19,820 because it has angular momentum-- 530 00:29:19,820 --> 00:29:24,000 cannot excite this state to any excited state. 531 00:29:24,000 --> 00:29:26,220 So what happens is, of course, pretty clear. 532 00:29:26,220 --> 00:29:29,530 You have whatever distribution you have in state one and two. 533 00:29:29,530 --> 00:29:32,930 You switch on your laser, and after a few cycles 534 00:29:32,930 --> 00:29:35,680 everything has fallen into the dark state. 535 00:29:35,680 --> 00:29:39,210 You have 100% population in stage two. 536 00:29:39,210 --> 00:29:41,910 So this is the simplest example of optical 537 00:29:41,910 --> 00:29:45,650 pumping using the three-level scheme. 538 00:29:45,650 --> 00:29:49,890 Well, you may ask now, what has that to do with cooling? 539 00:29:49,890 --> 00:29:54,400 Well, temperature is a Boltzmann factor, 540 00:29:54,400 --> 00:29:58,880 and if you-- we can define a temperature by saying 541 00:29:58,880 --> 00:30:04,770 the population between two levels 542 00:30:04,770 --> 00:30:07,160 is given by a Boltzmann factor. 543 00:30:07,160 --> 00:30:10,100 Let's just introduce an energy-splitting delta 544 00:30:10,100 --> 00:30:14,340 E-- which is somewhat arbitrary here-- but in any event, 545 00:30:14,340 --> 00:30:18,000 you see if you completely pump out a level, 546 00:30:18,000 --> 00:30:20,810 if you have all the population in one state, 547 00:30:20,810 --> 00:30:24,010 this corresponds to zero temperature. 548 00:30:24,010 --> 00:30:27,540 So if you optically pump the atoms, preferentially, 549 00:30:27,540 --> 00:30:30,710 into certain states, you have lowered 550 00:30:30,710 --> 00:30:33,320 the entropy of the system, the atoms are no longer distributed 551 00:30:33,320 --> 00:30:35,400 over as many states as before, and this 552 00:30:35,400 --> 00:30:38,960 corresponds to lower temperature. 553 00:30:38,960 --> 00:30:40,700 And the message I want to give you 554 00:30:40,700 --> 00:30:46,360 is that you can understand laser cooling as optical pumping 555 00:30:46,360 --> 00:30:49,760 in translation space in velocity space. 556 00:30:49,760 --> 00:30:53,920 When you laser cool, you excite the atoms at high velocity 557 00:30:53,920 --> 00:30:56,590 and then because of the mechanisms we discussed, 558 00:30:56,590 --> 00:30:59,860 spontaneous emission leads to lower velocities, 559 00:30:59,860 --> 00:31:03,750 and this is one form of optical pumping. 560 00:31:03,750 --> 00:31:07,340 I also like the word-- optical pumping is, in some sense, 561 00:31:07,340 --> 00:31:09,060 a spontaneous Raman process. 562 00:31:09,060 --> 00:31:11,360 You go up with one photon, you go down 563 00:31:11,360 --> 00:31:13,260 with-- you go up with the laser photon, 564 00:31:13,260 --> 00:31:16,390 you go down with the spontaneously emitted photon. 565 00:31:16,390 --> 00:31:18,660 So this is a spontaneous Raman process, 566 00:31:18,660 --> 00:31:22,790 and laser cooling-- even if you have just a single ground 567 00:31:22,790 --> 00:31:26,150 state-- can be regarded as a Raman process 568 00:31:26,150 --> 00:31:29,340 where the initial and final state differ 569 00:31:29,340 --> 00:31:31,120 in velocity or momentum. 570 00:31:31,120 --> 00:31:32,900 You have the same internal state, 571 00:31:32,900 --> 00:31:34,820 but you have a different external state. 572 00:31:34,820 --> 00:31:38,274 So you can see laser cooling is spontaneous Raman scattering 573 00:31:38,274 --> 00:31:39,690 between different momentum states. 574 00:31:43,110 --> 00:31:54,300 OK, now how can we achieve sub-Doppler cooling? 575 00:31:54,300 --> 00:32:02,270 Sub-Doppler cooling was discovered in 1988, 576 00:32:02,270 --> 00:32:05,320 and I remember I was at the [? IKA ?] Conference in Paris 577 00:32:05,320 --> 00:32:07,800 when Bill Phillips' group said, we've 578 00:32:07,800 --> 00:32:09,620 carefully measured the temperature 579 00:32:09,620 --> 00:32:11,780 and in sodium we measure a temperature which 580 00:32:11,780 --> 00:32:14,930 is lower than the Doppler limit, which was rigorously derived. 581 00:32:14,930 --> 00:32:18,510 I derived for you the Doppler limit of molasses. 582 00:32:18,510 --> 00:32:20,710 So what went wrong here? 583 00:32:20,710 --> 00:32:24,160 Well, it became clear the only loophole-- 584 00:32:24,160 --> 00:32:26,440 if you have a theory which predicts something 585 00:32:26,440 --> 00:32:28,770 and then you find a violation, you carefully 586 00:32:28,770 --> 00:32:30,510 have to check the assumption. 587 00:32:30,510 --> 00:32:32,420 And the assumption which was made-- 588 00:32:32,420 --> 00:32:34,200 and which we have made in this course-- 589 00:32:34,200 --> 00:32:37,390 is that you had a two-level system. 590 00:32:37,390 --> 00:32:41,520 Now, atoms have hyperfine structure-- 591 00:32:41,520 --> 00:32:43,550 and for pedagogical reasons, because I 592 00:32:43,550 --> 00:32:46,070 want to explain to you why in sub-Doppler cooling scheme 593 00:32:46,070 --> 00:32:48,590 I assume this hyperfine structure with F 594 00:32:48,590 --> 00:32:51,260 equals one half, F equals three half-- 595 00:32:51,260 --> 00:32:54,340 but let me first tell you what the novel feature is 596 00:32:54,340 --> 00:33:02,370 about a multi-level atom which has hyperfine structure. 597 00:33:02,370 --> 00:33:06,430 Now, the one thing which happens is 598 00:33:06,430 --> 00:33:10,820 that-- instead of just going between our single ground state 599 00:33:10,820 --> 00:33:14,950 and our single excited state-- we now have transitions. 600 00:33:14,950 --> 00:33:17,225 You may want to call them Raman transitions, 601 00:33:17,225 --> 00:33:21,070 or optical pumping, between the hyperfine atoms. 602 00:33:21,070 --> 00:33:24,900 So there are now transitions not only up to an excited 603 00:33:24,900 --> 00:33:27,860 state which rapidly decays, there 604 00:33:27,860 --> 00:33:31,270 are now transitions-- Raman transitions-- 605 00:33:31,270 --> 00:33:39,310 between ground states which have a very low widths. 606 00:33:39,310 --> 00:33:42,290 The width of the excited state is gamma. 607 00:33:42,290 --> 00:33:46,890 Well, what is the width of the ground state? 608 00:33:46,890 --> 00:33:49,580 Well, if you don't have a laser beam, 609 00:33:49,580 --> 00:33:52,770 the width is zero, but if you have 610 00:33:52,770 --> 00:33:56,670 laser beams the atom doesn't stay there forever. 611 00:33:56,670 --> 00:34:01,550 There will be a time, which is estimated here-- depending 612 00:34:01,550 --> 00:34:04,700 on the RB frequency and the detuning-- this is the time, 613 00:34:04,700 --> 00:34:07,150 this is simply nothing else than the scattering 614 00:34:07,150 --> 00:34:09,510 rate which we derived before. 615 00:34:09,510 --> 00:34:13,170 So that's the rate at which you scatter photons. 616 00:34:13,170 --> 00:34:17,820 And then, depending what the bunching ratio is, 617 00:34:17,820 --> 00:34:21,090 this is also the time in which you may optically 618 00:34:21,090 --> 00:34:23,909 pump to another ground state. 619 00:34:23,909 --> 00:34:26,719 So therefore, what we have to take into account 620 00:34:26,719 --> 00:34:32,590 now is that we have narrower widths in our system, 621 00:34:32,590 --> 00:34:38,250 and those widths correspond to another two-level system which 622 00:34:38,250 --> 00:34:41,710 is driven by two photons in which connects the two ground 623 00:34:41,710 --> 00:34:43,280 states. 624 00:34:43,280 --> 00:34:47,590 I will not give you any derivation how this exactly 625 00:34:47,590 --> 00:34:50,540 leads to a lower temperature, but I 626 00:34:50,540 --> 00:34:55,739 want to give you two pictures which I have already 627 00:34:55,739 --> 00:34:57,400 introduced to you. 628 00:34:57,400 --> 00:35:03,860 One is, remember, when we discussed Doppler cooling, 629 00:35:03,860 --> 00:35:07,350 the final temperature was proportionate to gamma, 630 00:35:07,350 --> 00:35:09,770 the width of the transition. 631 00:35:09,770 --> 00:35:12,620 So now I wave my hands and say, if you have transitions 632 00:35:12,620 --> 00:35:15,990 between ground states which have a much, much smaller width-- 633 00:35:15,990 --> 00:35:17,970 which is the rate of optical pumping-- 634 00:35:17,970 --> 00:35:19,820 and for low laser power this width 635 00:35:19,820 --> 00:35:22,200 can become very, very narrow. 636 00:35:22,200 --> 00:35:24,342 But you have at least one ingredient 637 00:35:24,342 --> 00:35:25,925 which can lead to a lower temperature. 638 00:35:28,520 --> 00:35:34,370 OK, in all truth in advertising, for the Sisyphus-- 639 00:35:34,370 --> 00:35:36,780 for the polarization gradient cooling scheme-- 640 00:35:36,780 --> 00:35:38,530 the final temperature is not given 641 00:35:38,530 --> 00:35:42,090 by gamma prime, the rate of optical pumping. 642 00:35:42,090 --> 00:35:46,470 It is a proportionality factor, but there is another factor. 643 00:35:46,470 --> 00:35:47,975 If you want to learn about it, you 644 00:35:47,975 --> 00:35:51,410 have to read some of the classic papers. 645 00:35:51,410 --> 00:35:54,240 But there is another picture which I can also 646 00:35:54,240 --> 00:35:56,830 use to tell you-- and that's actually 647 00:35:56,830 --> 00:36:01,940 related-- why optical pumping at least gives 648 00:36:01,940 --> 00:36:04,270 the possibility for lower temperature. 649 00:36:04,270 --> 00:36:07,365 I explained to you with the stimulated force in [? blue ?] 650 00:36:07,365 --> 00:36:11,370 molasses-- that the only reason why we have cooling-- 651 00:36:11,370 --> 00:36:15,010 is that the atom has a lag time. 652 00:36:15,010 --> 00:36:19,060 It cannot instantaneously adjust itself to the light field. 653 00:36:19,060 --> 00:36:21,220 It lags behind. 654 00:36:21,220 --> 00:36:23,410 And you can say, in Doppler cooling, 655 00:36:23,410 --> 00:36:27,260 the lag time is the spontaneous emission time, 656 00:36:27,260 --> 00:36:30,870 and the inverse of it is gamma. 657 00:36:30,870 --> 00:36:34,110 Here, in this case, the lag time can 658 00:36:34,110 --> 00:36:37,140 be the very long time to optically pump. 659 00:36:37,140 --> 00:36:40,920 So therefore, if the atom is in one hyperfine state and now 660 00:36:40,920 --> 00:36:45,640 moves into an area where the polarization of the laser beams 661 00:36:45,640 --> 00:36:49,940 is different, it may take a long time to adjust. 662 00:36:49,940 --> 00:36:54,700 And it is this lag time to which the friction coefficient 663 00:36:54,700 --> 00:36:56,360 was proportional. 664 00:36:56,360 --> 00:37:00,350 So this is a new feature, long delay times, narrow widths, 665 00:37:00,350 --> 00:37:02,780 and resonances between ground state levels. 666 00:37:06,050 --> 00:37:07,940 OK, a little bit show and tell now. 667 00:37:14,540 --> 00:37:15,915 Nobody thought about polarization 668 00:37:15,915 --> 00:37:18,610 creating cooling, nobody thought about Doppler cooling, 669 00:37:18,610 --> 00:37:21,780 but it was discovered when people simply 670 00:37:21,780 --> 00:37:24,610 did Doppler molasses and it's one 671 00:37:24,610 --> 00:37:27,100 of those big violations of Murphy's law 672 00:37:27,100 --> 00:37:30,810 where cooling worked much better than everybody had thought, 673 00:37:30,810 --> 00:37:32,920 and the result is the following. 674 00:37:32,920 --> 00:37:36,450 That-- when I drew for you the blue curve, which 675 00:37:36,450 --> 00:37:39,220 is force versus velocity-- it's pretty much 676 00:37:39,220 --> 00:37:41,780 the subtraction of two [? Lorentzian ?]. 677 00:37:41,780 --> 00:37:45,780 What happened is-- for sodium atoms, or for any alkali atom-- 678 00:37:45,780 --> 00:37:48,350 it's the red curve. 679 00:37:48,350 --> 00:37:51,430 For large velocity you have the Doppler cooling mechanism, 680 00:37:51,430 --> 00:37:57,020 but for smaller velocities you have a steepening of the slope, 681 00:37:57,020 --> 00:37:59,970 and the steeper the slope, the larger is your friction 682 00:37:59,970 --> 00:38:01,570 coefficient, alpha. 683 00:38:01,570 --> 00:38:09,550 And this is the mechanism of polarization gradient cooling. 684 00:38:09,550 --> 00:38:13,220 Let me just take this famous paper 685 00:38:13,220 --> 00:38:17,150 and show you one way how polarization gradient cooling 686 00:38:17,150 --> 00:38:20,860 works, which is the most famous form of sub-Doppler cooling, 687 00:38:20,860 --> 00:38:22,780 and this goes as follows. 688 00:38:22,780 --> 00:38:25,690 If you have molasses with two laser beams 689 00:38:25,690 --> 00:38:28,760 and the two laser beams are lin-perp-lin, 690 00:38:28,760 --> 00:38:30,900 linear polarization perpendicular 691 00:38:30,900 --> 00:38:32,670 to the other linear polarization. 692 00:38:32,670 --> 00:38:35,722 So you have two laser beams with these polarization. 693 00:38:35,722 --> 00:38:38,870 When these two polarizations overlap and they 694 00:38:38,870 --> 00:38:42,450 have the same phase, you get light at 45 degree, 695 00:38:42,450 --> 00:38:45,930 but if these two polarization have 90 degree out of phase, 696 00:38:45,930 --> 00:38:48,320 you get circularly polarized light. 697 00:38:48,320 --> 00:38:53,040 So as these two laser beams counter-propagate, 698 00:38:53,040 --> 00:38:56,710 you periodically go from linear polarized sigma minus, 699 00:38:56,710 --> 00:38:58,740 linear polarized sigma plus. 700 00:38:58,740 --> 00:39:03,280 So you have spatial-- at any given point 701 00:39:03,280 --> 00:39:06,910 you have a polarization, but the polarization changes. 702 00:39:06,910 --> 00:39:08,880 You have a-- you would say, naively, 703 00:39:08,880 --> 00:39:13,080 lin-perp-lin don't interfere, don't form a standing wave. 704 00:39:13,080 --> 00:39:16,540 Well, they do not form a standing wave in intensity. 705 00:39:16,540 --> 00:39:20,850 They form a standing wave in polarization. 706 00:39:20,850 --> 00:39:22,850 So now what happens is the following. 707 00:39:22,850 --> 00:39:27,940 If you have a multi-level atom and I use this simple scheme 708 00:39:27,940 --> 00:39:31,450 here, the different polarizations, 709 00:39:31,450 --> 00:39:35,370 linear polarization drives, the pi transition, 710 00:39:35,370 --> 00:39:37,170 sigma plus and sigma minus [INAUDIBLE] 711 00:39:37,170 --> 00:39:39,270 different transitions, and those transitions 712 00:39:39,270 --> 00:39:41,310 have different strengths. 713 00:39:41,310 --> 00:39:44,570 So an atom here-- when it experience 714 00:39:44,570 --> 00:39:46,800 this polarization-- reaches cycle, 715 00:39:46,800 --> 00:39:50,446 but when it experiences the other circular polarization, 716 00:39:50,446 --> 00:39:54,500 it will be optically pumped over here and then it cycles. 717 00:39:54,500 --> 00:39:59,250 So therefore, an atom which from sigma plus polarization and was 718 00:39:59,250 --> 00:40:02,160 here, it flies over to an area where you've 719 00:40:02,160 --> 00:40:06,450 sigma minus polarization, it will actually be pumped over. 720 00:40:06,450 --> 00:40:09,810 So the atom will constantly be pumped back and forth 721 00:40:09,810 --> 00:40:12,440 between those hyperfine states. 722 00:40:12,440 --> 00:40:18,770 And this actually gives rise to a beautiful form of Sisyphus 723 00:40:18,770 --> 00:40:22,190 cooling that the atom experiences sigma plus 724 00:40:22,190 --> 00:40:24,750 light in one ground state. 725 00:40:24,750 --> 00:40:29,340 It climbs up the hill, it sees the AC Stock's shift 726 00:40:29,340 --> 00:40:31,740 potential-- and the AC Stock's shift potential, 727 00:40:31,740 --> 00:40:36,440 because of the, for instance, for one hyperfine state, sigma 728 00:40:36,440 --> 00:40:38,550 plus drives the strongest transition-- 729 00:40:38,550 --> 00:40:42,550 and therefore we have sigma plus light, 730 00:40:42,550 --> 00:40:44,840 one ground state has the deepest potential. 731 00:40:44,840 --> 00:40:47,200 Where we have sigma minus light, it 732 00:40:47,200 --> 00:40:50,370 is the other ground state which has the deepest potential. 733 00:40:50,370 --> 00:40:54,260 And so what happens is that the atom is in one hyperfine state. 734 00:40:54,260 --> 00:40:56,550 It climbs up the hill, and then it's 735 00:40:56,550 --> 00:40:58,880 optically pumped to the other hyperfine state, 736 00:40:58,880 --> 00:41:05,850 and we have exactly the same kind of Sisyphus cooling. 737 00:41:05,850 --> 00:41:07,980 I just show you pictures and you sort of match it 738 00:41:07,980 --> 00:41:09,290 with what you know. 739 00:41:09,290 --> 00:41:11,520 This is more complicated because this 740 00:41:11,520 --> 00:41:15,130 involves optical pumping between hyperfine states. 741 00:41:15,130 --> 00:41:17,660 The different hyperfine states have different [INAUDIBLE] 742 00:41:17,660 --> 00:41:19,200 coefficients and such. 743 00:41:19,200 --> 00:41:22,900 It's not really complicated but more complex. 744 00:41:22,900 --> 00:41:25,370 What I explained to you is how Sisyphus cooling 745 00:41:25,370 --> 00:41:27,950 works in the dressed atom picture 746 00:41:27,950 --> 00:41:30,030 just for a two-level system. 747 00:41:30,030 --> 00:41:34,350 And here you find a more subtle form of Sisyphus cooling, 748 00:41:34,350 --> 00:41:37,290 but this form is more important because whenever 749 00:41:37,290 --> 00:41:40,500 you operate a magneto-optic trap, 750 00:41:40,500 --> 00:41:42,610 you get this cooling mechanism for free. 751 00:41:49,360 --> 00:41:58,030 OK, so that's all I want to tell you about sub-Doppler cooling. 752 00:41:58,030 --> 00:42:02,210 OK, sub-recoil cooling, we can quickly deal with it 753 00:42:02,210 --> 00:42:06,300 because I want to prove to you here that sub-recoil cooling is 754 00:42:06,300 --> 00:42:08,100 impossible. 755 00:42:08,100 --> 00:42:10,730 And unless you tell me what this wrong in my derivation, 756 00:42:10,730 --> 00:42:12,480 I don't need to discuss sub-recoil cooling 757 00:42:12,480 --> 00:42:14,760 because I've convinced you that it's impossible . 758 00:42:14,760 --> 00:42:19,010 OK, so let's assume our atom has an initial kinetic energy 759 00:42:19,010 --> 00:42:21,910 and then we absorb one photon. 760 00:42:21,910 --> 00:42:25,560 That would mean the momentum has now changed by the laser photon 761 00:42:25,560 --> 00:42:29,710 by the recoil-- and then it emits a photon, 762 00:42:29,710 --> 00:42:33,420 so the momentum gets reduced by the immediate photon-- 763 00:42:33,420 --> 00:42:36,740 and all I do is I ask, what is the difference 764 00:42:36,740 --> 00:42:39,560 between the initial kinetic energy 765 00:42:39,560 --> 00:42:44,530 and the final kinetic energy after two photons-- a laser 766 00:42:44,530 --> 00:42:49,100 photon and the spontaneously photon-- have been exchanged? 767 00:42:49,100 --> 00:42:51,590 And what you find is-- and this is just exact-- 768 00:42:51,590 --> 00:42:55,625 that the change in energy is two times the recoil energy. 769 00:42:55,625 --> 00:43:01,860 The recoil energy is h-bar square K square over 2m, plus-- 770 00:43:01,860 --> 00:43:07,300 and this is now the cross-term-- plus KL-- 771 00:43:07,300 --> 00:43:11,510 the K vector of the laser-- minus K spontaneous emission 772 00:43:11,510 --> 00:43:13,590 times the velocity. 773 00:43:13,590 --> 00:43:16,690 And now, of course, we assume-- which is correct-- 774 00:43:16,690 --> 00:43:20,220 that spontaneous emission goes randomly in all directions. 775 00:43:20,220 --> 00:43:22,670 Therefore, if we average over many cycles, 776 00:43:22,670 --> 00:43:25,040 this does not contribute. 777 00:43:25,040 --> 00:43:29,660 So therefore we find now that the average energy exchanged 778 00:43:29,660 --> 00:43:33,750 by an absorption and emission event is two times 779 00:43:33,750 --> 00:43:41,840 the recoil energy plus h-bar K laser times [? width ?], 780 00:43:41,840 --> 00:43:45,770 and we want to make it negative because we want to cool. 781 00:43:45,770 --> 00:43:49,300 Well, we make it negative by arranging the laser beam-- 782 00:43:49,300 --> 00:43:51,700 surprise, surprise-- counter-propagating 783 00:43:51,700 --> 00:43:52,800 to the velocity. 784 00:43:52,800 --> 00:43:55,640 This is how we can get the best cooling. 785 00:43:55,640 --> 00:43:58,290 Well, surprise, that's what we expected. 786 00:43:58,290 --> 00:44:08,080 But you find, of course, that when this velocity is a very 787 00:44:08,080 --> 00:44:15,480 small then K V is smaller than K squared. 788 00:44:15,480 --> 00:44:20,200 So therefore, once the velocity is smaller than the recoil 789 00:44:20,200 --> 00:44:26,410 velocity of a single photon, you cannot choose-- 790 00:44:26,410 --> 00:44:32,830 you cannot make this equation or this expression go negative. 791 00:44:32,830 --> 00:44:34,600 In other words, what I've shown to you 792 00:44:34,600 --> 00:44:38,920 is if there is an atom which has a velocity which is smaller 793 00:44:38,920 --> 00:44:44,315 than the recoil velocity of a photon, whenever 794 00:44:44,315 --> 00:44:48,550 this atom scatters a single photon, 795 00:44:48,550 --> 00:44:52,300 it will be hotter than it was before. 796 00:44:55,902 --> 00:44:57,860 So is it clear, what I've shown you, in energy? 797 00:44:57,860 --> 00:45:00,080 When an atom has a velocity which 798 00:45:00,080 --> 00:45:03,040 is smaller than the recoil velocity, 799 00:45:03,040 --> 00:45:07,340 any further photon scattering will not cool. 800 00:45:07,340 --> 00:45:09,460 It will lead to an energy transfer delta 801 00:45:09,460 --> 00:45:12,440 E, which is large, which is possible. 802 00:45:16,530 --> 00:45:21,907 So that shows you that sub-recoil cooling 803 00:45:21,907 --> 00:45:22,490 is impossible. 804 00:45:30,220 --> 00:45:32,160 Any idea how we can sub-recoil cool? 805 00:45:36,190 --> 00:45:40,530 Any idea why Claude Cohen-Tannoudji 806 00:45:40,530 --> 00:45:41,610 got the Nobel Prize? 807 00:45:44,436 --> 00:45:46,030 AUDIENCE: Further down in the lattice, 808 00:45:46,030 --> 00:45:47,777 you can have an effective mass that 809 00:45:47,777 --> 00:45:51,910 is much heavier than [? bare ?] mass? 810 00:45:51,910 --> 00:45:54,190 PROFESSOR: OK, great idea. 811 00:45:54,190 --> 00:45:57,230 We put the atom in a lattice, a lattice has band structure, 812 00:45:57,230 --> 00:45:59,610 in a band structure we have an effective mass, 813 00:45:59,610 --> 00:46:04,200 and heavier atoms can be cooled to lower and lower temperature. 814 00:46:04,200 --> 00:46:07,520 Actually, it's well known that, in Doppler cooling 815 00:46:07,520 --> 00:46:10,295 and sub-Doppler cooling-- especially 816 00:46:10,295 --> 00:46:13,470 in sub-Doppler cooling-- cesium an rubidium reach 817 00:46:13,470 --> 00:46:17,510 a microkelvin, sodium reaches only 25 microkelvin. 818 00:46:17,510 --> 00:46:20,530 So making the atom heavier is OK, 819 00:46:20,530 --> 00:46:24,540 but I have to say you're only rescaling your recoil limit, 820 00:46:24,540 --> 00:46:27,370 you're not breaking through the recoil limit which is now 821 00:46:27,370 --> 00:46:31,140 defined with a heavier effective mass. 822 00:46:31,140 --> 00:46:34,880 AUDIENCE: [INAUDIBLE] emission direction-- 823 00:46:34,880 --> 00:46:35,505 PROFESSOR: Yes? 824 00:46:35,505 --> 00:46:37,445 AUDIENCE: --means that the atom must 825 00:46:37,445 --> 00:46:40,585 have [INAUDIBLE] responding is [INAUDIBLE]? 826 00:46:40,585 --> 00:46:41,210 PROFESSOR: Yes. 827 00:46:41,210 --> 00:46:43,160 AUDIENCE: And [INAUDIBLE]? 828 00:46:43,160 --> 00:46:46,510 PROFESSOR: That was the assumption. 829 00:46:46,510 --> 00:46:49,982 AUDIENCE: [INAUDIBLE] at least for this cooling, 830 00:46:49,982 --> 00:46:56,430 it needs the time much larger than the Doppler cooling 831 00:46:56,430 --> 00:47:00,164 [INAUDIBLE] spontaneous emission time? 832 00:47:00,164 --> 00:47:02,080 PROFESSOR: Well, that was the assumption here. 833 00:47:02,080 --> 00:47:07,520 We scatter several times, a spontaneously emitted photon 834 00:47:07,520 --> 00:47:13,170 do not contribute the recoil of the spontaneously emitted 835 00:47:13,170 --> 00:47:17,970 photons cancels out because spontaneous emission-- and this 836 00:47:17,970 --> 00:47:21,020 is correct, this is not wrong-- spontaneous emission goes 837 00:47:21,020 --> 00:47:26,080 equally probable in the plus X and the minus X direction. 838 00:47:26,080 --> 00:47:30,630 Well, I thought I want to show you a demonstration how-- 839 00:47:30,630 --> 00:47:33,810 mechanical demonstration-- how you can sub-recoil cool. 840 00:47:33,810 --> 00:47:36,250 It's not a demonstration in velocity space, 841 00:47:36,250 --> 00:47:38,430 it's a demonstration in position space. 842 00:47:38,430 --> 00:47:40,710 What I have here is a Plexiglas tray 843 00:47:40,710 --> 00:47:43,380 and it has a little hole in the middle. 844 00:47:43,380 --> 00:47:45,700 And I have a bunch of ball bearings 845 00:47:45,700 --> 00:47:48,332 and I put those ball bearings in. 846 00:47:48,332 --> 00:47:49,040 Let me translate. 847 00:47:53,020 --> 00:47:58,570 I can blindfold myself and all I do is I shake the tray. 848 00:47:58,570 --> 00:48:01,860 So whenever I shake it, I kick the atoms 849 00:48:01,860 --> 00:48:06,570 randomly in position space from here to here. 850 00:48:06,570 --> 00:48:10,420 So the amount of position control I have over the atoms 851 00:48:10,420 --> 00:48:13,230 is on the order of this size. 852 00:48:13,230 --> 00:48:16,110 But the question is, can I-- without having 853 00:48:16,110 --> 00:48:21,410 any control of the transfer of position I give to them-- 854 00:48:21,410 --> 00:48:24,130 can I steer all the atoms into an area 855 00:48:24,130 --> 00:48:28,110 here into the hole in the middle, which is very, very 856 00:48:28,110 --> 00:48:31,100 narrow in position space? 857 00:48:31,100 --> 00:48:34,350 So the translation is, if I randomly scatter photons 858 00:48:34,350 --> 00:48:36,720 and they kick the atoms around with h-bar K, 859 00:48:36,720 --> 00:48:38,710 I don't have any control about momentum 860 00:48:38,710 --> 00:48:40,820 transfer smaller than h-bar K, is it 861 00:48:40,820 --> 00:48:44,530 possible to localize the atoms in momentum space 862 00:48:44,530 --> 00:48:46,810 to a momentum around zero, which is 863 00:48:46,810 --> 00:48:49,186 much, much smaller than h-bar K? 864 00:48:49,186 --> 00:48:51,810 Well, let's do the experiment. 865 00:48:51,810 --> 00:48:56,080 I just close my eyes and I shake it, and I shake it, 866 00:48:56,080 --> 00:49:01,020 and I just shake it for a while and I continue shaking it. 867 00:49:01,020 --> 00:49:05,740 And, well, zero temperature! 868 00:49:05,740 --> 00:49:09,410 So without controlling the motion on the scale 869 00:49:09,410 --> 00:49:15,950 I'm interested in, I manage to cool into a target region which 870 00:49:15,950 --> 00:49:18,470 is much, much narrower. 871 00:49:18,470 --> 00:49:22,210 So in other words, what you need is-- all you 872 00:49:22,210 --> 00:49:28,460 need is-- you need some dark state which 873 00:49:28,460 --> 00:49:30,770 is velocity selective. 874 00:49:30,770 --> 00:49:35,240 If you scatter light but you create a situation where 875 00:49:35,240 --> 00:49:41,020 once the atoms are close to zero velocity, they're not excited. 876 00:49:41,020 --> 00:49:45,460 You're not steering the atoms in a deterministic way, 877 00:49:45,460 --> 00:49:50,490 you just wait until-- by random chance-- one photon is emitted, 878 00:49:50,490 --> 00:49:53,910 and by the random chance the atom hits the hole. 879 00:49:53,910 --> 00:49:57,120 And then, it will never be re-excited again. 880 00:49:57,120 --> 00:50:00,330 This was the idea of this demonstration. 881 00:50:00,330 --> 00:50:04,010 And I could, yes, in the old days 882 00:50:04,010 --> 00:50:06,650 I may have spent two hours on teaching it, 883 00:50:06,650 --> 00:50:10,250 but by using Raman resonances between ground states-- 884 00:50:10,250 --> 00:50:14,740 which are terribly narrow-- or by using VSCPT-- velocity 885 00:50:14,740 --> 00:50:18,600 selective coherent population trapping-- you can create 886 00:50:18,600 --> 00:50:21,100 such narrow dark resonances which 887 00:50:21,100 --> 00:50:24,290 have the effect that the atoms scatter, scatter, scatter, 888 00:50:24,290 --> 00:50:27,450 but the moment they reach a very narrow region around zero 889 00:50:27,450 --> 00:50:29,840 velocity, they stop scattering. 890 00:50:29,840 --> 00:50:32,750 So what was wrong in my proof is that here we 891 00:50:32,750 --> 00:50:38,430 have a situation where we stop the laser cooling exactly 892 00:50:38,430 --> 00:50:41,220 at the time when the atom, by random chance, 893 00:50:41,220 --> 00:50:45,250 happens to be at low velocity, at very low velocity. 894 00:50:45,250 --> 00:50:47,250 And then you don't need any control, 895 00:50:47,250 --> 00:50:49,630 you just accumulate in the same way 896 00:50:49,630 --> 00:50:51,670 as I accumulated the ball bearings. 897 00:51:00,130 --> 00:51:07,110 OK, this was sub-recoil cooling in five minutes. [? Timo ?]? 898 00:51:07,110 --> 00:51:09,400 AUDIENCE: So I wasn't here last class. 899 00:51:09,400 --> 00:51:11,367 But just to summarize, the Doppler cooling rate 900 00:51:11,367 --> 00:51:12,908 gets us to a temperature on the order 901 00:51:12,908 --> 00:51:14,330 of gamma, which is natural alignment. 902 00:51:14,330 --> 00:51:14,955 PROFESSOR: Yes. 903 00:51:14,955 --> 00:51:18,936 AUDIENCE: And the recoil limit is the recoil energy 904 00:51:18,936 --> 00:51:21,870 which is usually tens of kilograms. 905 00:51:21,870 --> 00:51:23,920 But if you have a really, really narrow line, 906 00:51:23,920 --> 00:51:26,624 would you use the Doppler cooling limit? 907 00:51:26,624 --> 00:51:29,970 You could, in principle, beat the recoil limit 908 00:51:29,970 --> 00:51:31,404 with Doppler cooling, no? 909 00:51:31,404 --> 00:51:32,850 If you have a-- 910 00:51:32,850 --> 00:51:35,230 PROFESSOR: No, you can't, because if you do Doppler 911 00:51:35,230 --> 00:51:37,430 cooling-- [? Timo's ?] question is, 912 00:51:37,430 --> 00:51:42,390 if you simply do molasses with a very, very narrow line, 913 00:51:42,390 --> 00:51:53,410 we derived that the limit of Doppler cooling is Kt equals 914 00:51:53,410 --> 00:51:55,390 gamma, and what you are saying now, 915 00:51:55,390 --> 00:51:57,740 if you use narrower and narrower lines, 916 00:51:57,740 --> 00:52:00,990 can we reach [? arbitrary ?] low temperature? 917 00:52:00,990 --> 00:52:02,140 This is not the case. 918 00:52:06,524 --> 00:52:09,340 In our derivation of Doppler cooling, 919 00:52:09,340 --> 00:52:11,740 we made a continuum assumptions when 920 00:52:11,740 --> 00:52:14,290 we plotted the force versus velocity 921 00:52:14,290 --> 00:52:16,230 and we had the friction force, we 922 00:52:16,230 --> 00:52:19,580 assumed that an atom, when it scatters a photon, 923 00:52:19,580 --> 00:52:22,230 stays, let's say, within the linear part 924 00:52:22,230 --> 00:52:24,500 of the force versus velocity. 925 00:52:24,500 --> 00:52:28,580 So we had a hidden assumption which 926 00:52:28,580 --> 00:52:36,240 required that the recoil energy is smaller than h-bar gamma. 927 00:52:36,240 --> 00:52:39,110 But there is a lot of literature where people looked into it, 928 00:52:39,110 --> 00:52:43,040 and it turns out that when you have a very narrow line, 929 00:52:43,040 --> 00:52:46,860 you can go to the recoil limit, but you can't go beyond. 930 00:52:46,860 --> 00:52:49,610 If you want to go-- and I think this is what I showed you 931 00:52:49,610 --> 00:52:51,570 in the last five minutes-- if you want 932 00:52:51,570 --> 00:52:53,990 to go below the recoil limit, you 933 00:52:53,990 --> 00:52:56,410 need some velocity-selective dark state, 934 00:52:56,410 --> 00:53:00,620 and you would not have that in this situation. 935 00:53:00,620 --> 00:53:03,490 On the other hand, if you have a very narrow resonance, 936 00:53:03,490 --> 00:53:05,993 you can probably engineer a dark state 937 00:53:05,993 --> 00:53:09,830 that there is destructive interference in some excitation 938 00:53:09,830 --> 00:53:12,140 in a very, very narrow velocity class, 939 00:53:12,140 --> 00:53:14,050 and this narrow velocity class is simply 940 00:53:14,050 --> 00:53:17,160 selected by the Doppler effect. 941 00:53:17,160 --> 00:53:25,420 So you can use the narrow line to engineer sub-recoil cooling, 942 00:53:25,420 --> 00:53:27,690 but the simple arrangement of just having molasses 943 00:53:27,690 --> 00:53:30,290 with narrower and narrower line will not work. 944 00:53:30,290 --> 00:53:31,180 [? Collin ?]? 945 00:53:31,180 --> 00:53:32,763 AUDIENCE: I think the group that laser 946 00:53:32,763 --> 00:53:36,806 cooled to BEC just had-- their dark state was just a Stock's 947 00:53:36,806 --> 00:53:40,587 shift, [INAUDIBLE] and that was their-- and it shook. 948 00:53:40,587 --> 00:53:43,450 There were resonances in kilohertz or something. 949 00:53:43,450 --> 00:53:46,860 PROFESSOR: Thy used a very-- did they 950 00:53:46,860 --> 00:53:48,800 go below the recoil limit in laser cooling? 951 00:53:48,800 --> 00:53:50,630 I'm not sure. 952 00:53:50,630 --> 00:53:52,630 They may have had a situation where 953 00:53:52,630 --> 00:53:55,290 it was enough to go to the recoil limit. 954 00:53:55,290 --> 00:53:57,090 I have to check out the paper, but-- 955 00:53:57,090 --> 00:53:59,922 AUDIENCE: I remember they did have some sort of dark state. 956 00:53:59,922 --> 00:54:03,698 Maybe it was to increase density or-- they had the high-- 957 00:54:03,698 --> 00:54:07,180 PROFESSOR: Anyway, let me just be clear. 958 00:54:07,180 --> 00:54:09,710 Sub-recoil cooling requires that you 959 00:54:09,710 --> 00:54:12,340 have a dark region in velocity space. 960 00:54:12,340 --> 00:54:15,080 For that, you need some narrow line widths. 961 00:54:15,080 --> 00:54:16,480 That's necessary. 962 00:54:16,480 --> 00:54:18,400 But I think you just don't get it 963 00:54:18,400 --> 00:54:20,390 by having counter-propagating laser beams, 964 00:54:20,390 --> 00:54:21,640 you have to do something more. 965 00:54:23,980 --> 00:54:24,605 Other question? 966 00:54:31,040 --> 00:54:36,320 OK, so I've told you about sub-Doppler cooling, 967 00:54:36,320 --> 00:54:40,525 sub-recoil cooling, but what really got atomic physicists 968 00:54:40,525 --> 00:54:43,900 to nanokelvin temperature were two other techniques. 969 00:54:43,900 --> 00:54:47,620 Laser cooling was used as pre-cooling, 970 00:54:47,620 --> 00:54:50,070 but then the final trapping and cooling 971 00:54:50,070 --> 00:54:53,880 was done by magnetic trapping and evaporative cooling. 972 00:54:53,880 --> 00:54:56,840 So I want to give you now, in the last half 973 00:54:56,840 --> 00:55:01,860 hour, a quick overview of magnetic trapping 974 00:55:01,860 --> 00:55:04,590 and evaporative cooling. 975 00:55:04,590 --> 00:55:10,240 So I think any discussions of magnetic trapping starts 976 00:55:10,240 --> 00:55:14,400 with a theorem, a theorem which tells you 977 00:55:14,400 --> 00:55:16,980 that not everything is possible you would like to do. 978 00:55:16,980 --> 00:55:22,420 And this is the following, then if you have a region without 979 00:55:22,420 --> 00:55:26,260 charges and currents without-- through an empty space, 980 00:55:26,260 --> 00:55:30,930 and what this theorem says-- in empty space, 981 00:55:30,930 --> 00:55:35,780 you cannot have a local minimum-- 982 00:55:35,780 --> 00:55:38,530 you cannot have a local maximum of your electric and magnetic 983 00:55:38,530 --> 00:55:39,330 fields. 984 00:55:39,330 --> 00:55:42,010 So if you take the strengths of the electric field 985 00:55:42,010 --> 00:55:43,950 or the strengths of the magnetic field, 986 00:55:43,950 --> 00:55:45,404 you cannot have a maximum. 987 00:55:45,404 --> 00:55:46,570 You can only have a minimum. 988 00:55:49,080 --> 00:55:52,520 Well, and this is important when it comes to magnetic traps. 989 00:55:52,520 --> 00:55:55,830 Depending on the magnetic moment-- spin up, 990 00:55:55,830 --> 00:55:59,620 spin down-- we could create a trapping potential 991 00:55:59,620 --> 00:56:02,480 around a maximum of the magnetic field 992 00:56:02,480 --> 00:56:04,560 or the trapping potential around the minimum 993 00:56:04,560 --> 00:56:05,890 of the magnetic field. 994 00:56:05,890 --> 00:56:08,865 You want to be at the minimum of the potential, 995 00:56:08,865 --> 00:56:10,240 but if you have a minus sign, you 996 00:56:10,240 --> 00:56:13,020 may want to be at the maximum of a magnetic field. 997 00:56:13,020 --> 00:56:15,790 But [? Bing's ?] theorem says that only one of them 998 00:56:15,790 --> 00:56:18,000 is possible. 999 00:56:18,000 --> 00:56:21,180 The proof really goes in two lines. 1000 00:56:21,180 --> 00:56:25,610 You really show, if you assume that there is a maximum, that 1001 00:56:25,610 --> 00:56:31,590 would mean that if you add a field to it, you add a field-- 1002 00:56:31,590 --> 00:56:34,620 and no matter in which direction you add a field-- 1003 00:56:34,620 --> 00:56:36,425 the total field strengths become smaller. 1004 00:56:40,600 --> 00:56:44,515 But that requires a violation of the Laplace equation. 1005 00:56:47,660 --> 00:56:50,310 It's easier for you-- you can probably read it at home-- 1006 00:56:50,310 --> 00:56:53,370 but you simply assume you have this situation, 1007 00:56:53,370 --> 00:56:56,280 so you add-- and you show you have an incremental field, 1008 00:56:56,280 --> 00:56:59,360 delta E. This field, delta E, you 1009 00:56:59,360 --> 00:57:01,710 are at the maximum of an electric field, 1010 00:57:01,710 --> 00:57:04,440 and then delta E over R is the small difference 1011 00:57:04,440 --> 00:57:05,930 of the electric field. 1012 00:57:05,930 --> 00:57:12,140 And you can show that this expression is only negative. 1013 00:57:12,140 --> 00:57:18,750 That means you have a maximum of the electric field when 1014 00:57:18,750 --> 00:57:24,060 the electric field would not fulfill the Laplace equation. 1015 00:57:24,060 --> 00:57:26,920 So the gist of the argument is the following. 1016 00:57:26,920 --> 00:57:29,290 Each component of the electric and magnetic field 1017 00:57:29,290 --> 00:57:31,140 fulfills the Laplace equation. 1018 00:57:31,140 --> 00:57:34,060 And the Laplace equation-- I don't know if you've probably 1019 00:57:34,060 --> 00:57:37,020 heard about it-- if something fulfills the Laplace equation, 1020 00:57:37,020 --> 00:57:40,630 it says at any given point the function which 1021 00:57:40,630 --> 00:57:44,190 fulfills the Laplace equation is equal to the average 1022 00:57:44,190 --> 00:57:49,190 of function, averaged over a small sphere around it. 1023 00:57:49,190 --> 00:57:54,860 And that means-- if something fulfills the Laplace equation-- 1024 00:57:54,860 --> 00:57:57,440 you cannot have a local maximum or a local minimum 1025 00:57:57,440 --> 00:58:00,810 because if the value here equals the average, 1026 00:58:00,810 --> 00:58:06,080 that would mean in one direction the value gets higher, 1027 00:58:06,080 --> 00:58:08,800 in the other direction the value gets lower. 1028 00:58:08,800 --> 00:58:11,870 So if something fulfills the Laplace equation, 1029 00:58:11,870 --> 00:58:15,150 you cannot have a local maximum or local minimum. 1030 00:58:15,150 --> 00:58:19,060 But now, we're not asking for maximum or minimum 1031 00:58:19,060 --> 00:58:22,380 in one component of the electric or magnetic field. 1032 00:58:22,380 --> 00:58:26,860 We ask for a maximum or minimum in the total value, 1033 00:58:26,860 --> 00:58:29,540 or the square of the electric field. 1034 00:58:29,540 --> 00:58:32,900 But then you show, when you assume that you have a maximum, 1035 00:58:32,900 --> 00:58:35,290 the field points along the Z direction, 1036 00:58:35,290 --> 00:58:39,640 that then the Z component of the field cannot fulfill 1037 00:58:39,640 --> 00:58:42,480 the Laplace equation. 1038 00:58:42,480 --> 00:58:45,090 So maxima are not possible. 1039 00:58:45,090 --> 00:58:48,640 Minima are possible. 1040 00:58:48,640 --> 00:58:49,350 Good! 1041 00:58:49,350 --> 00:58:51,320 So the magnetic trapping potential 1042 00:58:51,320 --> 00:58:54,530 comes because we have a magnetic dipole moment which 1043 00:58:54,530 --> 00:58:57,620 interacts with a magnetic field. 1044 00:58:57,620 --> 00:59:00,220 Magnetic traps are actually classical. 1045 00:59:00,220 --> 00:59:03,200 You can have a classical model for a magnetic trap, 1046 00:59:03,200 --> 00:59:06,220 and in that case, you say this potentially 1047 00:59:06,220 --> 00:59:08,760 is mu times B times cosine theta. 1048 00:59:08,760 --> 00:59:11,840 [? Crendo ?] mechanically, of course, the angle cosine theta 1049 00:59:11,840 --> 00:59:15,580 is quantized and we have the different M F levels, 1050 00:59:15,580 --> 00:59:17,950 the different orientations of the spin, 1051 00:59:17,950 --> 00:59:18,910 relative to the Z-axis. 1052 00:59:21,740 --> 00:59:26,790 OK, the fact that the magnetic field can only 1053 00:59:26,790 --> 00:59:32,070 have minima and not maxima means that we 1054 00:59:32,070 --> 00:59:36,120 want to make sure that the magnetic moment times the G 1055 00:59:36,120 --> 00:59:39,750 factor times M F is negative, or cosine theta 1056 00:59:39,750 --> 00:59:42,960 is negative in the classical picture. 1057 00:59:42,960 --> 00:59:46,930 And if we absorb whatever we choose for cosine theta-- 1058 00:59:46,930 --> 00:59:51,320 or what we call mu Bohr G M F, the magnetic moment-- 1059 00:59:51,320 --> 00:59:54,360 we want that the magnetic moment of the particle 1060 00:59:54,360 --> 01:00:00,490 is anti-parallel to the magnetic field. 1061 01:00:00,490 --> 01:00:02,750 Then we have a magnetic trapping potential, 1062 01:00:02,750 --> 01:00:06,750 which is-- I've taken care of the sin now-- mu times B, 1063 01:00:06,750 --> 01:00:10,250 and if B has a minimum this potential has a minimum, 1064 01:00:10,250 --> 01:00:13,190 and that is a magnetic trap. 1065 01:00:13,190 --> 01:00:15,990 But so the consequence of [? Bing's ?] theorem 1066 01:00:15,990 --> 01:00:20,430 is that we can only trap particles 1067 01:00:20,430 --> 01:00:24,960 which are anti-parallel with a magnetic field. 1068 01:00:24,960 --> 01:00:27,840 And they can always lower their energy 1069 01:00:27,840 --> 01:00:31,030 by flipping to the other state. 1070 01:00:31,030 --> 01:00:34,150 So therefore, a magnetic trap does not 1071 01:00:34,150 --> 01:00:37,966 allow us to trap particles in the absolute ground state. 1072 01:00:37,966 --> 01:00:41,810 There is always the possibility of spin-flip collisions 1073 01:00:41,810 --> 01:00:44,730 which lead to an anti-trap state which 1074 01:00:44,730 --> 01:00:46,470 is expelled from the trapping region. 1075 01:00:50,370 --> 01:00:52,990 Now, spin-flip collisions can happen 1076 01:00:52,990 --> 01:00:55,610 when you have particles at sufficiently high density 1077 01:00:55,610 --> 01:00:58,560 and they collide. 1078 01:00:58,560 --> 01:01:02,860 For the experts, there can be spin relaxation, 1079 01:01:02,860 --> 01:01:04,620 there can be dipolar relaxation, they 1080 01:01:04,620 --> 01:01:07,050 are two different kinds of spin-flip collisions. 1081 01:01:07,050 --> 01:01:11,180 Fortunately, in magnetic traps, they only 1082 01:01:11,180 --> 01:01:14,980 become relevant when you pick the wrong spin state. 1083 01:01:14,980 --> 01:01:18,650 But for suitable choices, magnetic traps 1084 01:01:18,650 --> 01:01:21,160 are very long-lived. 1085 01:01:21,160 --> 01:01:26,120 Dipolar relaxation is often-- can be a limiting process 1086 01:01:26,120 --> 01:01:28,130 for Bose-Einstein condensates, for instance, 1087 01:01:28,130 --> 01:01:30,200 for high [? dose ?] Bose-Einstein condensation, 1088 01:01:30,200 --> 01:01:35,530 dipolar relaxation was-- limited the number and density of atoms 1089 01:01:35,530 --> 01:01:37,410 in the Bose-Einstein condensate. 1090 01:01:37,410 --> 01:01:40,080 But I don't want to talk about cold collisions, here. 1091 01:01:40,080 --> 01:01:43,660 You should just know, based on the fundamental theorem, 1092 01:01:43,660 --> 01:01:45,440 you cannot trap in the ground state. 1093 01:01:45,440 --> 01:01:48,410 You have to trap into a state which 1094 01:01:48,410 --> 01:01:51,590 has more energy than other states, and, in principle, 1095 01:01:51,590 --> 01:02:00,860 it's possible to flip the spin and go to a lower state. 1096 01:02:00,860 --> 01:02:04,400 Now, what we have assumed here is-- 1097 01:02:04,400 --> 01:02:06,600 when I wrote the trapping potential like this-- 1098 01:02:06,600 --> 01:02:10,320 I assumed that the atom stays in a given hyperfine state. 1099 01:02:10,320 --> 01:02:15,050 So we put the atom in one state and it experiences potential. 1100 01:02:15,050 --> 01:02:18,590 Classically it means that the angle of the dipole, 1101 01:02:18,590 --> 01:02:22,160 with respect to the magnetic field, stays constant. 1102 01:02:22,160 --> 01:02:25,070 And this is the case due to rapid precession. 1103 01:02:25,070 --> 01:02:27,870 Classically, the magnetic dipole precesses 1104 01:02:27,870 --> 01:02:29,870 around the magnetic field, and when 1105 01:02:29,870 --> 01:02:32,530 the direction of the magnetic field changes, 1106 01:02:32,530 --> 01:02:36,390 the rapid precession keeps the dipole aligned. 1107 01:02:36,390 --> 01:02:40,320 Or, in other words, for slow changes of the magnetic field 1108 01:02:40,320 --> 01:02:46,300 direction, cosine theta is an adiabatic invariant. 1109 01:02:46,300 --> 01:02:49,660 So the word adiabatic is important, either classically 1110 01:02:49,660 --> 01:02:52,780 or you stay in a quantum state as long as you're adiabatic. 1111 01:02:55,370 --> 01:03:00,150 Well if an atom would rapidly go through a region 1112 01:03:00,150 --> 01:03:03,870 of very weak magnetic field, then the precession-- 1113 01:03:03,870 --> 01:03:06,190 the [INAUDIBLE] precession-- of the atom is very slow, 1114 01:03:06,190 --> 01:03:10,210 and this adiabatic condition can be violated. 1115 01:03:10,210 --> 01:03:13,610 And this violation of an adiabaticity condition 1116 01:03:13,610 --> 01:03:16,840 is called Majorana flops. 1117 01:03:16,840 --> 01:03:19,100 So if you operate a magnetic trap 1118 01:03:19,100 --> 01:03:23,145 at very low magnetic field, you may destabilized the trap 1119 01:03:23,145 --> 01:03:24,520 because you violate adiabaticity. 1120 01:03:27,780 --> 01:03:30,850 OK, so these are the consequences of the fact that 1121 01:03:30,850 --> 01:03:34,260 we cannot create maxima, we can only create minima 1122 01:03:34,260 --> 01:03:36,410 of the magnetic field in free space, 1123 01:03:36,410 --> 01:03:40,690 and therefore we have to deal with those two possible loss 1124 01:03:40,690 --> 01:03:46,380 processes, but we have learned what kind of magnetic trap, 1125 01:03:46,380 --> 01:03:49,870 what kind of atom to pick, and in general these are not big 1126 01:03:49,870 --> 01:03:52,370 problems. 1127 01:03:52,370 --> 01:03:56,810 OK, so we have to-- so this shows here 1128 01:03:56,810 --> 01:03:59,520 the typical hyperfine structure of an alkali atom. 1129 01:03:59,520 --> 01:04:04,450 It could be rubidium 87 or sodium 23. 1130 01:04:04,450 --> 01:04:07,740 The fact that we have only a local minimum 1131 01:04:07,740 --> 01:04:15,090 of the magnetic field means we can only trap hyperfine 1132 01:04:15,090 --> 01:04:19,010 states-- which I've marked here in green-- where the slope is 1133 01:04:19,010 --> 01:04:21,760 positive, and these are the atoms where 1134 01:04:21,760 --> 01:04:23,630 the magnetic moment is inter-parallel 1135 01:04:23,630 --> 01:04:26,800 with the magnetic field. 1136 01:04:26,800 --> 01:04:29,920 And, well, for stability reasons we 1137 01:04:29,920 --> 01:04:33,440 usually pick the highest state of the lower manifold, 1138 01:04:33,440 --> 01:04:35,900 or the highest state of the upper manifold. 1139 01:04:35,900 --> 01:04:38,370 In principle, it would be also possible to trap 1140 01:04:38,370 --> 01:04:41,670 here-- you have the correct slope-- 1141 01:04:41,670 --> 01:04:44,960 but often those states suffer form 1142 01:04:44,960 --> 01:04:47,320 spin relaxation and collision loss. 1143 01:04:49,629 --> 01:04:50,170 [? Mickey ?]? 1144 01:04:50,170 --> 01:04:53,520 AUDIENCE: [INAUDIBLE] the picture you showed before, 1145 01:04:53,520 --> 01:04:57,790 is it actually possible to make a magnetic trap with state 1146 01:04:57,790 --> 01:05:03,225 number four from the top so that they expel from the center? 1147 01:05:03,225 --> 01:05:05,660 From [INAUDIBLE] a-- 1148 01:05:05,660 --> 01:05:08,000 PROFESSOR: Yeah, there is a peculiarity here, 1149 01:05:08,000 --> 01:05:12,220 you have the transition from the weak field to the strong field 1150 01:05:12,220 --> 01:05:13,430 region. 1151 01:05:13,430 --> 01:05:16,630 Whenever the magnetic moment is constant, 1152 01:05:16,630 --> 01:05:19,960 you need a local minimum of the magnetic field. 1153 01:05:19,960 --> 01:05:23,750 But if the magnetic moment would change 1154 01:05:23,750 --> 01:05:26,680 and you have a gradient of the magnetic field, 1155 01:05:26,680 --> 01:05:29,630 you could actually trap here. 1156 01:05:29,630 --> 01:05:35,060 So in this case, you have a magnetic field gradient 1157 01:05:35,060 --> 01:05:39,240 but you have a spatial variation of the magnetic moment. 1158 01:05:39,240 --> 01:05:41,730 This has been discussed in one special paper 1159 01:05:41,730 --> 01:05:43,690 in the literature, but it has not really 1160 01:05:43,690 --> 01:05:45,862 found any [? lucent ?] realization. 1161 01:05:45,862 --> 01:05:48,222 AUDIENCE: Wouldn't it help to put the Majorana 1162 01:05:48,222 --> 01:05:49,638 flops in the center? 1163 01:05:49,638 --> 01:05:51,580 Because they're repelled from the center. 1164 01:05:51,580 --> 01:05:54,290 PROFESSOR: Well, yeah, but these atoms 1165 01:05:54,290 --> 01:05:55,910 would undergo spin-flip collisions 1166 01:05:55,910 --> 01:05:59,090 so it's not a good choice. 1167 01:05:59,090 --> 01:06:00,920 It would help against Majorana flops, 1168 01:06:00,920 --> 01:06:04,550 but we have many solutions against Majorana flops. 1169 01:06:04,550 --> 01:06:07,360 And here, you would solve the Majorana flops problem 1170 01:06:07,360 --> 01:06:10,930 but you would solve it with another problem. 1171 01:06:10,930 --> 01:06:14,660 OK, so we need minima of the magnetic field. 1172 01:06:14,660 --> 01:06:19,230 This provides magnetic trapping potential, 1173 01:06:19,230 --> 01:06:24,680 and there are two kinds of possible trap configuration. 1174 01:06:24,680 --> 01:06:29,470 One is where the minima is at zero magnetic field, 1175 01:06:29,470 --> 01:06:33,480 and one is where the minima is at finite magnetic field. 1176 01:06:33,480 --> 01:06:36,400 Now, the zero magnetic field minimum one 1177 01:06:36,400 --> 01:06:40,850 can be simply created with anti-Helmholtz coil. 1178 01:06:40,850 --> 01:06:44,590 What happens is the magnetic field, 1179 01:06:44,590 --> 01:06:49,210 as a function of position, would actually cross through zero. 1180 01:06:49,210 --> 01:06:52,620 It's just a field gradient which crosses through zero, 1181 01:06:52,620 --> 01:06:55,227 but, of course, since we are interested 1182 01:06:55,227 --> 01:06:57,060 in the absolute value of the magnetic field, 1183 01:06:57,060 --> 01:07:00,420 we get the V-shape potential. 1184 01:07:00,420 --> 01:07:03,020 This V-shape potential was what was 1185 01:07:03,020 --> 01:07:06,620 used for the first demonstration for Bose-Einstein condensation 1186 01:07:06,620 --> 01:07:10,650 because a V-shape potential is much, much more 1187 01:07:10,650 --> 01:07:15,050 confining in this cusp than in harmonic oscillator potential. 1188 01:07:15,050 --> 01:07:18,240 And this had advantages for tight confinement 1189 01:07:18,240 --> 01:07:20,520 and rapid evaporative cooling. 1190 01:07:20,520 --> 01:07:23,875 So those quadrupoled traps with the V-shape potential 1191 01:07:23,875 --> 01:07:27,380 are the best confinement you can get, 1192 01:07:27,380 --> 01:07:29,190 the best confinement for the buck, 1193 01:07:29,190 --> 01:07:33,720 and the buck here is your power supply and your coils. 1194 01:07:33,720 --> 01:07:36,020 However, they have a problem when 1195 01:07:36,020 --> 01:07:37,990 the magnetic field is zero. 1196 01:07:37,990 --> 01:07:42,050 We violate adiabaticity because the different spin 1197 01:07:42,050 --> 01:07:44,210 configurations become degenerate, 1198 01:07:44,210 --> 01:07:47,610 and in degenerates you can't have adiabaticity. 1199 01:07:47,610 --> 01:07:49,800 So therefore, the two first demonstrations 1200 01:07:49,800 --> 01:07:53,620 of Bose-Einstein condensation avoided this cusp 1201 01:07:53,620 --> 01:07:55,510 in two different ways. 1202 01:07:55,510 --> 01:08:00,560 One way was to use a rotating magnetic field, 1203 01:08:00,560 --> 01:08:04,410 and to use a perpendicular field but which was time-dependent, 1204 01:08:04,410 --> 01:08:08,330 and the MIT solution was to use a blue detuned laser beam, 1205 01:08:08,330 --> 01:08:11,870 use the optical dipole trap-- optical dipole potential-- 1206 01:08:11,870 --> 01:08:16,050 to push the atoms away from the dangerous [INAUDIBLE] 1207 01:08:16,050 --> 01:08:18,100 field region. 1208 01:08:18,100 --> 01:08:25,540 I should say rotating-- none of those-- most traps which 1209 01:08:25,540 --> 01:08:29,080 are now used are the other kind of trap, the trap which 1210 01:08:29,080 --> 01:08:34,649 has an harmonic oscillator potential, which doesn't have 1211 01:08:34,649 --> 01:08:40,569 the cusp and where the minimum is at a finite magnetic field. 1212 01:08:40,569 --> 01:08:42,569 Let me just make one comment. 1213 01:08:42,569 --> 01:08:44,500 The rotating trap was very popular 1214 01:08:44,500 --> 01:08:47,859 because it led to the first BEC and a lot of people built that, 1215 01:08:47,859 --> 01:08:49,770 but to the best of my knowledge, it's 1216 01:08:49,770 --> 01:08:53,060 only used in different places. 1217 01:08:53,060 --> 01:08:57,880 And the only real application it has 1218 01:08:57,880 --> 01:09:02,340 is-- since you have some rotating magnetic field-- 1219 01:09:02,340 --> 01:09:04,790 by some modification you can actually 1220 01:09:04,790 --> 01:09:06,970 make of rotating potential. 1221 01:09:06,970 --> 01:09:08,859 And this is nice if you want to study 1222 01:09:08,859 --> 01:09:11,410 Bose-Einstein condensation on a rotating frame, 1223 01:09:11,410 --> 01:09:14,800 create vortices and things like that. 1224 01:09:14,800 --> 01:09:17,790 But for simply creating a Bose-Einstein condensate, 1225 01:09:17,790 --> 01:09:21,740 this trap is used, by far, most frequently. 1226 01:09:21,740 --> 01:09:25,540 But there is actually a renaissance of this trap. 1227 01:09:25,540 --> 01:09:27,149 One reason why people use this trap 1228 01:09:27,149 --> 01:09:29,859 and why my own research group immediately 1229 01:09:29,859 --> 01:09:31,330 switched to this trap-- and we had 1230 01:09:31,330 --> 01:09:34,260 the idea how to built it-- is, well, harmonic potential 1231 01:09:34,260 --> 01:09:34,760 is nice. 1232 01:09:34,760 --> 01:09:37,029 Every physicist loves an harmonic potential. 1233 01:09:37,029 --> 01:09:39,490 You can solve, immediately, thermodynamics 1234 01:09:39,490 --> 01:09:40,880 in a harmonic potential. 1235 01:09:40,880 --> 01:09:43,630 Who wants to deal with that potential? 1236 01:09:43,630 --> 01:09:46,420 Also, if the laser beam just drifts by a micrometer, 1237 01:09:46,420 --> 01:09:49,750 this symmetric potential becomes asymmetric. 1238 01:09:49,750 --> 01:09:51,970 So you characterize your potential today 1239 01:09:51,970 --> 01:09:54,450 and a few hours later you have a different potential. 1240 01:09:54,450 --> 01:09:57,140 Whereas a magnetic potential, harmonic potential, 1241 01:09:57,140 --> 01:09:59,880 once it's characterized-- as long as you don't change 1242 01:09:59,880 --> 01:10:01,480 the current in your power supplies-- 1243 01:10:01,480 --> 01:10:04,550 it's the same week after week, months after months. 1244 01:10:04,550 --> 01:10:08,430 Well, there is a renaissance now because a lot of groups 1245 01:10:08,430 --> 01:10:10,520 now doing experiments in optical lattices, 1246 01:10:10,520 --> 01:10:14,430 they don't care what shape the potential is because they don't 1247 01:10:14,430 --> 01:10:16,210 do physics in this potential. 1248 01:10:16,210 --> 01:10:19,010 They just take the Bose-Einstein condensate or the cooled Fermi 1249 01:10:19,010 --> 01:10:22,220 gases and transfer them to an optical lattice. 1250 01:10:22,220 --> 01:10:24,960 So in that case, it doesn't play any role, 1251 01:10:24,960 --> 01:10:26,940 and then the advantage of this potential 1252 01:10:26,940 --> 01:10:30,147 is you get more bang for the buck. 1253 01:10:30,147 --> 01:10:35,636 AUDIENCE: [INAUDIBLE] you were saying before [INAUDIBLE] more 1254 01:10:35,636 --> 01:10:38,640 rapid evaporation in V-shape versus the [INAUDIBLE]. 1255 01:10:38,640 --> 01:10:43,840 Where is the differences in evap time? 1256 01:10:43,840 --> 01:10:46,390 PROFESSOR: It depends. 1257 01:10:46,390 --> 01:10:47,930 I may run out of time today. 1258 01:10:47,930 --> 01:10:50,700 I needed-- I can probably-- I wanted 1259 01:10:50,700 --> 01:10:52,300 to go through evaporative cooling, 1260 01:10:52,300 --> 01:10:55,230 and I will talk to you that, for evaporative cooling, 1261 01:10:55,230 --> 01:10:57,520 there could be a threshold in confinement 1262 01:10:57,520 --> 01:10:59,680 where you go into a runaway regime 1263 01:10:59,680 --> 01:11:02,250 that evaporative cooling is speeding up. 1264 01:11:02,250 --> 01:11:08,650 So a little bit of confinement can make the difference 1265 01:11:08,650 --> 01:11:11,240 without never getting into a runaway regime 1266 01:11:11,240 --> 01:11:12,940 or being in that runaway regime. 1267 01:11:12,940 --> 01:11:14,900 So confinement, extra confinement, 1268 01:11:14,900 --> 01:11:17,340 can make the difference between getting BEC and not 1269 01:11:17,340 --> 01:11:20,230 getting a BEC. 1270 01:11:20,230 --> 01:11:21,220 So it really depends. 1271 01:11:21,220 --> 01:11:23,630 It's highly nonlinear, but I will show you 1272 01:11:23,630 --> 01:11:28,200 later on that when you put a system together, 1273 01:11:28,200 --> 01:11:31,700 the kind of threshold density at which the cloud can evaporate 1274 01:11:31,700 --> 01:11:34,745 to BEC is much lower here than it is there. 1275 01:11:38,140 --> 01:11:39,780 But-- and this is the next thing I 1276 01:11:39,780 --> 01:11:41,540 want to explain-- it's not really 1277 01:11:41,540 --> 01:11:43,960 that this is a linear potential and this is a [INAUDIBLE] 1278 01:11:43,960 --> 01:11:49,380 potential because those finite magnetic field-- 1279 01:11:49,380 --> 01:11:52,540 these traps which have a minimum at a finite magnetic field-- 1280 01:11:52,540 --> 01:11:56,510 are usually done in the following way. 1281 01:11:56,510 --> 01:11:59,940 Those traps are called Ioffe-Pritchard trap. 1282 01:11:59,940 --> 01:12:04,330 Ioffe actually suggested such a magnetic field configuration 1283 01:12:04,330 --> 01:12:07,530 for confinement of plasma, and Dave Pritchard 1284 01:12:07,530 --> 01:12:10,670 was the first to point out that such a configuration would 1285 01:12:10,670 --> 01:12:13,460 be a good choice for neutral atoms. 1286 01:12:13,460 --> 01:12:16,730 So I sometimes joke and I say, well, plasma physics 1287 01:12:16,730 --> 01:12:18,800 is the study of very hot matter. 1288 01:12:18,800 --> 01:12:21,540 Cold atoms is the study of very cold matter. 1289 01:12:21,540 --> 01:12:25,520 But when matter is either too hot or too cold that you cannot 1290 01:12:25,520 --> 01:12:28,440 put it in ordinary container, you want it confined with 1291 01:12:28,440 --> 01:12:29,750 magnetic fields. 1292 01:12:29,750 --> 01:12:32,030 And it happens that, for plasma, you 1293 01:12:32,030 --> 01:12:35,690 need a minimum of the magnetic field, and for neutral atoms 1294 01:12:35,690 --> 01:12:37,710 you need a minimum of the magnetic field. 1295 01:12:37,710 --> 01:12:40,960 So therefore here is something which ultra-cold atoms have 1296 01:12:40,960 --> 01:12:43,432 in common with plasma physics. 1297 01:12:43,432 --> 01:12:45,890 I don't think there is a lot the two fields have in common, 1298 01:12:45,890 --> 01:12:49,450 but when it comes to magnetic field configurations, yes. 1299 01:12:49,450 --> 01:12:53,760 A similar magnetic field configuration 1300 01:12:53,760 --> 01:12:59,230 can confine a plasma and can confine neutral atoms. 1301 01:12:59,230 --> 01:13:05,480 OK, so the generic way of how these magnetic fields are 1302 01:13:05,480 --> 01:13:14,720 generated is you want to have two coils, called pinch coils. 1303 01:13:14,720 --> 01:13:18,230 You can say each of them creates a magnetic field which decays, 1304 01:13:18,230 --> 01:13:20,570 a magnetic field which decays, and now in the middle 1305 01:13:20,570 --> 01:13:23,370 you have the parabolic minimum. 1306 01:13:23,370 --> 01:13:27,800 But then you have to add the so-called four Ioffe bars which 1307 01:13:27,800 --> 01:13:30,520 create a linear potential, and this is shown here. 1308 01:13:30,520 --> 01:13:33,640 So the pinch coil is simply creating a local minimum 1309 01:13:33,640 --> 01:13:36,270 along the Z-axis, and if you only 1310 01:13:36,270 --> 01:13:38,780 want to trap in one dimension, you would be done, 1311 01:13:38,780 --> 01:13:41,060 But you want to trap in 3 dimensions. 1312 01:13:41,060 --> 01:13:43,532 And so what happens is you add now 1313 01:13:43,532 --> 01:13:47,940 an anti-Helmholtz-- or quadrupold-- field in X and Y, 1314 01:13:47,940 --> 01:13:52,260 and this blue field is done via the green bars, 1315 01:13:52,260 --> 01:13:53,650 by the Ioffe bars. 1316 01:13:53,650 --> 01:13:58,150 So you add together a harmonic quadratic field in the Z 1317 01:13:58,150 --> 01:14:03,000 direction with the linear quadrupoled field in X and Y. 1318 01:14:03,000 --> 01:14:05,530 And what matters for the atoms is-- 1319 01:14:05,530 --> 01:14:07,580 and what matters for the Zeeman energy 1320 01:14:07,580 --> 01:14:11,060 is-- the absolute value of the magnetic field. 1321 01:14:11,060 --> 01:14:13,380 So you add those things in quadrature, 1322 01:14:13,380 --> 01:14:15,430 and because you add it in quadrature, 1323 01:14:15,430 --> 01:14:18,380 you have now an harmonic oscillator potential 1324 01:14:18,380 --> 01:14:21,100 in X, Y, and Z, so you have an harmonic 1325 01:14:21,100 --> 01:14:24,830 trap in three directions. 1326 01:14:24,830 --> 01:14:30,510 However, if this field is not larger than this-- 1327 01:14:30,510 --> 01:14:33,350 if these field becomes larger than this field, 1328 01:14:33,350 --> 01:14:35,250 so then when you add it in quadrature, 1329 01:14:35,250 --> 01:14:38,150 you actually get a linear potential. 1330 01:14:38,150 --> 01:14:40,950 So in other words, the Ioffe-Pritchard trap 1331 01:14:40,950 --> 01:14:44,580 is quadratic for smaller values of X and Y, 1332 01:14:44,580 --> 01:14:47,170 but if you go out to large X and Y, 1333 01:14:47,170 --> 01:14:49,330 you get the linear potential. 1334 01:14:49,330 --> 01:14:51,980 So in the end-- and this is maybe in response 1335 01:14:51,980 --> 01:14:56,080 to Matt's question-- for a high temperature, 1336 01:14:56,080 --> 01:14:58,160 the Ioffe-Pritchard trap is actually not 1337 01:14:58,160 --> 01:14:59,880 harmonic in three dimensions. 1338 01:14:59,880 --> 01:15:01,640 It's linear in two dimensions and 1339 01:15:01,640 --> 01:15:03,960 harmonic in the third dimension. 1340 01:15:03,960 --> 01:15:07,480 So, therefore the hit you take in evaporative cooling 1341 01:15:07,480 --> 01:15:11,430 is not as large as we'd initially assumed. 1342 01:15:11,430 --> 01:15:14,500 And the moment we realized that, we 1343 01:15:14,500 --> 01:15:17,750 were building a Ioffe-Pritchard trap. 1344 01:15:17,750 --> 01:15:19,620 And this has been the traps we' have 1345 01:15:19,620 --> 01:15:23,340 been using at MIT ever since. 1346 01:15:23,340 --> 01:15:27,300 Well, just as a warning, this is sort of the simplified 1347 01:15:27,300 --> 01:15:30,110 description, but if you have curvature, 1348 01:15:30,110 --> 01:15:33,100 the curvature has to fulfill Maxwell's equation. 1349 01:15:33,100 --> 01:15:36,290 And you cannot have a curvature only along Z. 1350 01:15:36,290 --> 01:15:38,710 Maxwell's equations are three-dimensional, 1351 01:15:38,710 --> 01:15:41,500 so you get sort of all sorts of curvature terms. 1352 01:15:41,500 --> 01:15:43,640 If you really build a magnetic trap, 1353 01:15:43,640 --> 01:15:45,160 you should understand those. 1354 01:15:45,160 --> 01:15:47,895 If you just want to understand why magnetic traps work, 1355 01:15:47,895 --> 01:15:49,270 the previous slide is sufficient. 1356 01:15:51,930 --> 01:15:54,860 The different ways to build those traps, 1357 01:15:54,860 --> 01:15:57,370 this is the design we invented at MIT. 1358 01:15:57,370 --> 01:15:59,410 It's called a clover-leaf trap. 1359 01:15:59,410 --> 01:16:05,800 So you have one coil package here, 1360 01:16:05,800 --> 01:16:08,110 to another coil package here, and what 1361 01:16:08,110 --> 01:16:11,360 you see is the pinch coils, and when you have two of them 1362 01:16:11,360 --> 01:16:15,350 you create this parabolic minimum. 1363 01:16:15,350 --> 01:16:19,170 We don't like Ioffe bars because if you have a vacuum chamber 1364 01:16:19,170 --> 01:16:21,670 you either have to put the Ioffe bars into the vector vacuum 1365 01:16:21,670 --> 01:16:24,030 chamber or you drill holes through your vacuum chamber 1366 01:16:24,030 --> 01:16:26,127 to put the Ioffe bars back and forth. 1367 01:16:26,127 --> 01:16:27,710 Don't laugh, some people have done it. 1368 01:16:27,710 --> 01:16:29,459 Of course, they put little tubes around it 1369 01:16:29,459 --> 01:16:31,520 so it was a very highly-engineered vacuum 1370 01:16:31,520 --> 01:16:34,740 chamber where they could string Ioffe bars through the chamber. 1371 01:16:34,740 --> 01:16:37,880 But we realized that the same field as Ioffe bars 1372 01:16:37,880 --> 01:16:41,600 can be generated by taking the Ioffe bars 1373 01:16:41,600 --> 01:16:45,500 and flipping them out, and after flipping them out, 1374 01:16:45,500 --> 01:16:47,590 they had the shape of clover leaves. 1375 01:16:47,590 --> 01:16:53,010 It's just a variant of creating the same field geometry. 1376 01:16:53,010 --> 01:16:57,670 Yes, that's all I want to tell you about magnetic trapping. 1377 01:16:57,670 --> 01:17:00,180 I think I should not start with evaporative cooling. 1378 01:17:00,180 --> 01:17:02,100 This will just take 10 or 15 minutes, 1379 01:17:02,100 --> 01:17:06,060 but I'll do that on Monday. 1380 01:17:06,060 --> 01:17:07,720 So since we're on time, do you have 1381 01:17:07,720 --> 01:17:12,150 any questions about magnetic trapping? 1382 01:17:12,150 --> 01:17:14,020 Different forms of magnetic trapping? 1383 01:17:20,520 --> 01:17:22,200 OK. 1384 01:17:22,200 --> 01:17:25,390 Final announcement, next week is the due date 1385 01:17:25,390 --> 01:17:26,195 for the term paper. 1386 01:17:31,360 --> 01:17:34,320 Yes, and the term paper is due on the date 1387 01:17:34,320 --> 01:17:37,160 of the last class, which is Friday. 1388 01:17:37,160 --> 01:17:38,820 Any questions about that? 1389 01:17:41,580 --> 01:17:42,310 OK. 1390 01:17:42,310 --> 01:17:43,860 Good.