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 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,240 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,240 --> 00:00:17,865 at ocw.mit.edu. 8 00:00:27,400 --> 00:00:30,770 PROFESSOR: So good afternoon. 9 00:00:30,770 --> 00:00:33,790 Our main topic today is to understand 10 00:00:33,790 --> 00:00:38,040 light forces, in particular the stimulated force using 11 00:00:38,040 --> 00:00:40,030 the dressed atom picture. 12 00:00:40,030 --> 00:00:44,690 And I hope once you understand how light forces can 13 00:00:44,690 --> 00:00:46,680 be described in the dressed atom picture 14 00:00:46,680 --> 00:00:49,010 you will love the dressed atom picture, 15 00:00:49,010 --> 00:00:50,920 as most atomic physicists do. 16 00:00:50,920 --> 00:00:53,400 It is really the language we use, 17 00:00:53,400 --> 00:00:57,110 the intuition we apply to atomic systems. 18 00:00:57,110 --> 00:01:00,930 Whenever we have a laser beam which drives the atom 19 00:01:00,930 --> 00:01:04,790 it is, in most situations, more appropriate 20 00:01:04,790 --> 00:01:09,420 and better to use dressed states as your basis states 21 00:01:09,420 --> 00:01:12,400 and not the naked, the bare states, which 22 00:01:12,400 --> 00:01:15,010 are ground and excited state. 23 00:01:15,010 --> 00:01:17,620 Now I know last week I quickly went 24 00:01:17,620 --> 00:01:19,370 though the dressed atom picture. 25 00:01:19,370 --> 00:01:24,520 Let me sort of go through it in a fly-over 26 00:01:24,520 --> 00:01:26,890 by not doing any of the derivation, 27 00:01:26,890 --> 00:01:29,440 but making a few comments here and there. 28 00:01:29,440 --> 00:01:32,650 Also, because I want that some of the results 29 00:01:32,650 --> 00:01:35,320 are fresh in your head when we apply it 30 00:01:35,320 --> 00:01:38,220 to mechanical forces of light. 31 00:01:38,220 --> 00:01:43,460 So I told you that in the dressed atom picture 32 00:01:43,460 --> 00:01:46,730 we have two kinds of interactions, the atom 33 00:01:46,730 --> 00:01:49,460 interact with a vacuum, with a reservoir 34 00:01:49,460 --> 00:01:53,120 of empty modes with spontaneous emission. 35 00:01:53,120 --> 00:01:56,810 And we also interact with a laser field. 36 00:01:56,810 --> 00:01:59,260 And the new feature about the dressed atom picture 37 00:01:59,260 --> 00:02:02,495 is that we first solve this exactly, 38 00:02:02,495 --> 00:02:06,360 we rediagonalize the Hamiltonian for that Hilbert space, 39 00:02:06,360 --> 00:02:10,710 and then we allow spontaneous emission to happen. 40 00:02:10,710 --> 00:02:15,940 Now the derivation is pretty straightforward. 41 00:02:15,940 --> 00:02:21,690 After a number of assumptions, we simply 42 00:02:21,690 --> 00:02:25,740 came to the two-by-two matrix, where 43 00:02:25,740 --> 00:02:29,300 we have two levels of the atoms coupled with the Rabi 44 00:02:29,300 --> 00:02:30,940 frequency. 45 00:02:30,940 --> 00:02:34,910 I derived it here for you using the quantized electromagnetic 46 00:02:34,910 --> 00:02:35,900 field. 47 00:02:35,900 --> 00:02:40,380 But in your homework you derive the same two-by-two matrix 48 00:02:40,380 --> 00:02:43,880 using a classical electromagnetic field, 49 00:02:43,880 --> 00:02:45,610 which drives the atoms. 50 00:02:45,610 --> 00:02:49,430 And we've talked about the similarities and differences 51 00:02:49,430 --> 00:02:50,070 last week. 52 00:02:52,499 --> 00:02:54,540 If you have any questions, please raise your hand 53 00:02:54,540 --> 00:02:57,030 or interrupt me. 54 00:02:57,030 --> 00:02:59,940 Now two-by-two matrix is fairly trivial. 55 00:02:59,940 --> 00:03:02,170 We can immediately diagonalize it. 56 00:03:02,170 --> 00:03:04,630 And we find Eigensolutions, which 57 00:03:04,630 --> 00:03:07,750 are just sine theta, cosine theta, amplitude, 58 00:03:07,750 --> 00:03:10,530 linear superpositions of the naked states. 59 00:03:10,530 --> 00:03:14,225 And the naked states are ground and excited states, state A 60 00:03:14,225 --> 00:03:18,866 and state B, with N or N plus 1 photons in the laser field. 61 00:03:21,710 --> 00:03:23,100 So it's simple. 62 00:03:23,100 --> 00:03:25,215 It's just a two-by-two matrix. 63 00:03:25,215 --> 00:03:28,100 But the insight we can get out of it 64 00:03:28,100 --> 00:03:30,960 is pretty profound and pretty subtle. 65 00:03:30,960 --> 00:03:34,120 So what we first discussed is-- and we 66 00:03:34,120 --> 00:03:37,140 will need that to understand forces and energy 67 00:03:37,140 --> 00:03:41,970 conservation-- we can now look at spontaneous emission. 68 00:03:41,970 --> 00:03:46,820 Spontaneous emission coupling to the empty modes of the vacuum 69 00:03:46,820 --> 00:03:51,120 can only connect an excited state B with a ground state A. 70 00:03:51,120 --> 00:03:53,940 And therefore, we have diachromatically 71 00:03:53,940 --> 00:03:56,780 those four combinations, represented here 72 00:03:56,780 --> 00:03:58,600 by those four arrows. 73 00:03:58,600 --> 00:04:01,510 And if you look for the matrix element, 74 00:04:01,510 --> 00:04:05,430 you collect cosine theta and sine theta factors. 75 00:04:05,430 --> 00:04:07,680 And that's what we did here. 76 00:04:07,680 --> 00:04:11,940 So we immediately get in the dressed atom picture, 77 00:04:11,940 --> 00:04:14,860 an expression which tells us what 78 00:04:14,860 --> 00:04:18,850 are the matrix elements for the carrier, for the blue 79 00:04:18,850 --> 00:04:23,150 and for the red detuned sideband of the Mollow triplet. 80 00:04:23,150 --> 00:04:25,870 But then there's something else, and I 81 00:04:25,870 --> 00:04:28,630 edit this with the lecture notes in green. 82 00:04:28,630 --> 00:04:32,190 We can also, now, if you know what the transition rates are, 83 00:04:32,190 --> 00:04:35,550 we can set up rate equations and can solve them. 84 00:04:35,550 --> 00:04:39,190 And once you solve a set of differential equations for rate 85 00:04:39,190 --> 00:04:42,080 equations, you know what the populations are. 86 00:04:42,080 --> 00:04:46,390 But now, today, I want to tell you one more subtlety. 87 00:04:46,390 --> 00:04:50,800 You know already that you cannot, 88 00:04:50,800 --> 00:04:54,270 in the most general situation, formulate an equation with just 89 00:04:54,270 --> 00:04:55,660 a population. 90 00:04:55,660 --> 00:04:58,350 You know that we have the master equation, 91 00:04:58,350 --> 00:05:00,240 we have the Optical Bloch Equation. 92 00:05:00,240 --> 00:05:03,400 This is an equation for the density matrix. 93 00:05:03,400 --> 00:05:07,070 And usually we get equations where the populations 94 00:05:07,070 --> 00:05:09,020 are mixed with coherences. 95 00:05:09,020 --> 00:05:13,200 In other words, you cannot just say the diagonal matrix element 96 00:05:13,200 --> 00:05:15,290 of the density matrix is a population, 97 00:05:15,290 --> 00:05:19,460 and the change of population is what goes out of this state 98 00:05:19,460 --> 00:05:21,260 and what arrives in this state. 99 00:05:21,260 --> 00:05:23,400 Coherences play a role. 100 00:05:23,400 --> 00:05:26,390 But, and this is what the dressed atom picture does 101 00:05:26,390 --> 00:05:31,020 for you, in the limit that we have a strong Rabi frequency-- 102 00:05:31,020 --> 00:05:33,460 and this is the limit you want to consider here-- 103 00:05:33,460 --> 00:05:41,120 the matrix equation decouples into an equation 104 00:05:41,120 --> 00:05:44,330 for populations and an equation for coherences. 105 00:05:44,330 --> 00:05:47,650 And if something decouples that should tell you we 106 00:05:47,650 --> 00:05:49,920 all on the right track, we have the right description. 107 00:05:49,920 --> 00:05:52,970 In this description things decouple, things are simpler. 108 00:05:52,970 --> 00:05:54,740 And that means this description is 109 00:05:54,740 --> 00:05:59,510 most appropriate for the physics we want to understand. 110 00:05:59,510 --> 00:06:03,060 Or, to say it more specifically-- and all 111 00:06:03,060 --> 00:06:04,645 the details of course are in the book 112 00:06:04,645 --> 00:06:08,110 Atom Photon Interaction-- you can rewrite the Optical Bloch 113 00:06:08,110 --> 00:06:13,710 Equations not in the bare bases but in the dressed atom basis. 114 00:06:13,710 --> 00:06:15,210 And in the dressed atom basis you 115 00:06:15,210 --> 00:06:19,116 find this wonderful separation between populations 116 00:06:19,116 --> 00:06:19,740 and coherences. 117 00:06:24,200 --> 00:06:27,820 So we have the simple rate equations. 118 00:06:27,820 --> 00:06:29,740 By setting the left hand side to 0 119 00:06:29,740 --> 00:06:32,810 we find the steady state populations. 120 00:06:32,810 --> 00:06:36,160 This will be very important to understand the light forces. 121 00:06:36,160 --> 00:06:40,680 But you should just remember the steady state solution such 122 00:06:40,680 --> 00:06:41,620 as we know them. 123 00:06:41,620 --> 00:06:45,850 They're just exactly given by this angle theta, which 124 00:06:45,850 --> 00:06:48,360 tells us what kind of superposition 125 00:06:48,360 --> 00:06:51,800 of the bare states from the dressed states. 126 00:06:51,800 --> 00:06:55,960 Remember in the case of theta equals 45 degrees, 127 00:06:55,960 --> 00:07:01,640 the dressed states are just symmetric and antisymmetric 128 00:07:01,640 --> 00:07:04,960 superposition of the naked states. 129 00:07:04,960 --> 00:07:09,320 Now what is also important is those rate equations, 130 00:07:09,320 --> 00:07:13,500 or those rate equations have a time constant. 131 00:07:13,500 --> 00:07:18,280 The time constant is here, the relaxation time 132 00:07:18,280 --> 00:07:19,740 for populations. 133 00:07:19,740 --> 00:07:22,020 And we know exactly what it is. 134 00:07:22,020 --> 00:07:24,260 It's a little bit more complicated to find 135 00:07:24,260 --> 00:07:27,540 the equivalent equation for the coherences. 136 00:07:27,540 --> 00:07:30,200 But just take my word, the density matrix 137 00:07:30,200 --> 00:07:33,670 in the dressed atom basis decouples into equation 138 00:07:33,670 --> 00:07:37,420 for population, which is simple, equation for coherences, 139 00:07:37,420 --> 00:07:40,370 little bit harder to derive but looks the same. 140 00:07:40,370 --> 00:07:44,950 And from this equation we have now an expression for what 141 00:07:44,950 --> 00:07:47,810 is the relaxation time for coherences. 142 00:07:47,810 --> 00:07:50,690 And just to show you one non-trivial example, 143 00:07:50,690 --> 00:07:54,220 if you add up cosine square plus sine square you get 1. 144 00:07:54,220 --> 00:07:56,130 But based on what we derived we have 145 00:07:56,130 --> 00:07:59,440 to take cosine to the power of 4, sine to the power of 4. 146 00:07:59,440 --> 00:08:03,650 And this gives us effect of 1/2, which will appear somewhere 147 00:08:03,650 --> 00:08:05,290 in an important place in a few minutes. 148 00:08:10,190 --> 00:08:14,180 So with that we can now address, we know now something 149 00:08:14,180 --> 00:08:17,300 about the intensities of this Mollow triplet, 150 00:08:17,300 --> 00:08:21,872 because the intensity is nothing else than the population, 151 00:08:21,872 --> 00:08:25,270 what is the population in the initial state times 152 00:08:25,270 --> 00:08:28,070 the matrix element squared. 153 00:08:28,070 --> 00:08:31,190 But what is also-- what comes out very naturally-- 154 00:08:31,190 --> 00:08:34,340 is that we have different widths. 155 00:08:34,340 --> 00:08:36,390 The features have different widths. 156 00:08:36,390 --> 00:08:38,429 The sidebands have a different width 157 00:08:38,429 --> 00:08:41,539 from the central part of the Mollow triplet. 158 00:08:41,539 --> 00:08:44,732 And well, this is now very naturally interpreted 159 00:08:44,732 --> 00:08:47,740 at one has a width, which is a relaxation 160 00:08:47,740 --> 00:08:49,250 rate for the population. 161 00:08:49,250 --> 00:08:51,144 The other one, is the sidebands have a width 162 00:08:51,144 --> 00:08:53,060 which is a relaxation rate for the coherences. 163 00:08:58,370 --> 00:09:02,620 I showed you, and this was just a reminder of a few weeks ago, 164 00:09:02,620 --> 00:09:06,350 that when we looked at the linear matrix equation, which 165 00:09:06,350 --> 00:09:08,470 are the Optical Bloch Equation, we 166 00:09:08,470 --> 00:09:11,820 discussed complex eigenvalues mathematically. 167 00:09:11,820 --> 00:09:16,040 And those complex eigenvalues, the imaginary part 168 00:09:16,040 --> 00:09:18,310 is the sideband frequency. 169 00:09:18,310 --> 00:09:21,730 And the real part was the widths. 170 00:09:21,730 --> 00:09:24,750 What I have now, sort of-- I can get everything 171 00:09:24,750 --> 00:09:26,100 from the dressed atom picture. 172 00:09:26,100 --> 00:09:29,430 But what I've gotten in a very natural and simple way 173 00:09:29,430 --> 00:09:33,310 is this case, which is the case where the dressed atom 174 00:09:33,310 --> 00:09:35,460 picture is particularly simple. 175 00:09:35,460 --> 00:09:37,550 It's called the secular case, where 176 00:09:37,550 --> 00:09:40,700 the population and coherences decouple. 177 00:09:40,700 --> 00:09:45,680 And so, what I just showed you, that the population relaxation 178 00:09:45,680 --> 00:09:47,820 time is gamma over 2. 179 00:09:47,820 --> 00:09:55,550 This explains now why, in this case, 180 00:09:55,550 --> 00:09:58,300 the widths of the central feature 181 00:09:58,300 --> 00:10:03,940 is 1/2 of what it is when we're on resonance. 182 00:10:03,940 --> 00:10:06,425 So everything is sort of nicely playing out here. 183 00:10:10,390 --> 00:10:16,740 OK, any questions about dressed level populations, 184 00:10:16,740 --> 00:10:22,670 rate equations, the Mollow triplet? 185 00:10:22,670 --> 00:10:25,610 Because we want to apply it now, in a few minutes, 186 00:10:25,610 --> 00:10:28,040 to the stimulated light force. 187 00:10:32,870 --> 00:10:35,810 But before I do that, let me show you 188 00:10:35,810 --> 00:10:44,260 one nice piece of physics, which can be formulated 189 00:10:44,260 --> 00:10:47,750 both in the bare, in the uncoupled, 190 00:10:47,750 --> 00:10:50,280 and in the dressed state basis. 191 00:10:50,280 --> 00:10:53,350 And this is the process of spontaneous emission. 192 00:10:53,350 --> 00:10:55,410 If an atom is in the laser field, 193 00:10:55,410 --> 00:10:57,970 it just emits photon, rapid fire photons. 194 00:10:57,970 --> 00:11:01,080 And I want to show you now, how we would describe it 195 00:11:01,080 --> 00:11:03,370 in the bare basis, how we describe it 196 00:11:03,370 --> 00:11:05,760 in the dressed basis. 197 00:11:05,760 --> 00:11:09,780 And we get two very different physical pictures out of it. 198 00:11:09,780 --> 00:11:12,270 Well, in the bare basis, maybe just 199 00:11:12,270 --> 00:11:14,150 think of quantum Monte Carlo simulation, 200 00:11:14,150 --> 00:11:16,100 a photon has been emitted, and that 201 00:11:16,100 --> 00:11:18,440 means we are in the ground state. 202 00:11:18,440 --> 00:11:22,330 Now it takes a Rabi cycle before the ground state 203 00:11:22,330 --> 00:11:24,700 atom is in the excited state. 204 00:11:24,700 --> 00:11:28,030 And when it is in the excited state, it can emit again 205 00:11:28,030 --> 00:11:29,820 and goes back to the ground state. 206 00:11:29,820 --> 00:11:32,220 And then things start over. 207 00:11:32,220 --> 00:11:37,300 So the picture you obtain here is the picture of antibunching, 208 00:11:37,300 --> 00:11:40,560 that the atom emits a photon, but then it 209 00:11:40,560 --> 00:11:44,840 takes the inverse Rabi frequency to load the gun again. 210 00:11:44,840 --> 00:11:47,910 And then it can fire again and emit the next photon. 211 00:11:47,910 --> 00:11:52,610 So therefore, the probability of getting another photon right 212 00:11:52,610 --> 00:11:57,280 after you've observed the first photon is reduced, 213 00:11:57,280 --> 00:12:00,040 or T equals 0, it is exactly 0. 214 00:12:00,040 --> 00:12:02,490 And this is antibunching, the photons are antibunched. 215 00:12:05,140 --> 00:12:09,450 So this is how we would describe the radiative cascade, how 216 00:12:09,450 --> 00:12:13,200 we go from N to N minus 1, to N minus 2, how we lose photons 217 00:12:13,200 --> 00:12:15,430 from the laser field and dump them 218 00:12:15,430 --> 00:12:18,200 into the empty modes of the vacuum. 219 00:12:18,200 --> 00:12:22,560 That's how we describe it in the uncoupled basis. 220 00:12:22,560 --> 00:12:25,143 Well in the dressed basis we have sort of 221 00:12:25,143 --> 00:12:29,800 diagonalized the atoms interacting 222 00:12:29,800 --> 00:12:31,460 with the strong laser field. 223 00:12:31,460 --> 00:12:34,350 And now we would have the following picture. 224 00:12:34,350 --> 00:12:36,630 We have this manifold here. 225 00:12:36,630 --> 00:12:38,700 We can emit from this manifold. 226 00:12:38,700 --> 00:12:41,800 And we can go to the upper or lower dressed state. 227 00:12:41,800 --> 00:12:45,010 And then, the next thing will happen again. 228 00:12:45,010 --> 00:12:48,770 But what we realize here, if we are in the upper dressed state, 229 00:12:48,770 --> 00:12:51,420 we can either emit on the carrier 230 00:12:51,420 --> 00:12:53,970 or we can emit a blue detune sideband. 231 00:12:53,970 --> 00:12:57,890 But once we have emitted a blue detune sideband, 232 00:12:57,890 --> 00:13:01,550 the next sideband has to be red detuned because we are now 233 00:13:01,550 --> 00:13:06,240 in the lower dressed state, and we need a lower sideband 234 00:13:06,240 --> 00:13:08,710 to go back to the upper dressed state. 235 00:13:08,710 --> 00:13:13,630 So what this picture gives us, it gives us the sidebands 236 00:13:13,630 --> 00:13:16,550 because we are resolving the splitting in this picture. 237 00:13:16,550 --> 00:13:19,360 And secondly, it tells us that there 238 00:13:19,360 --> 00:13:22,980 is an alternation between the blue detuned photons 239 00:13:22,980 --> 00:13:24,306 and the red detuned photon. 240 00:13:29,270 --> 00:13:32,880 Now you would say, well aren't those two pictures 241 00:13:32,880 --> 00:13:33,940 really different? 242 00:13:33,940 --> 00:13:35,940 In one case we have antibunching, 243 00:13:35,940 --> 00:13:37,920 and all of the photons are the same. 244 00:13:37,920 --> 00:13:40,720 And over here I talk about an alternation 245 00:13:40,720 --> 00:13:43,640 between the blue and the red sideband. 246 00:13:43,640 --> 00:13:48,060 Well again, the two pictures are complementary. 247 00:13:48,060 --> 00:13:51,870 In this case, if I want to resolve the antibunching 248 00:13:51,870 --> 00:13:53,940 I need a temporal resolution which 249 00:13:53,940 --> 00:13:57,130 is better than the inverse Rabi frequency. 250 00:13:57,130 --> 00:14:01,430 But if my temporal resolution is better than the inverse Rabi 251 00:14:01,430 --> 00:14:05,740 frequency then I don't have the spectral resolution 252 00:14:05,740 --> 00:14:09,180 to observe the sidebands, and vice versa. 253 00:14:09,180 --> 00:14:11,070 If I want to resolve the sidebands, 254 00:14:11,070 --> 00:14:15,990 if I want to distinguish the red from the blue detuned sideband 255 00:14:15,990 --> 00:14:19,470 of the Mollow triplet I cannot run the experiment with 256 00:14:19,470 --> 00:14:22,880 the spectral resolution, which would be required to observe 257 00:14:22,880 --> 00:14:25,120 the antibunching. 258 00:14:25,120 --> 00:14:27,370 So these are sort of two different pictures. 259 00:14:27,370 --> 00:14:29,890 One is sort of in terms of energy eigenstates 260 00:14:29,890 --> 00:14:31,670 when we have spectral resolution. 261 00:14:31,670 --> 00:14:34,210 And this other picture is more appropriate 262 00:14:34,210 --> 00:14:37,810 when we're interested in the short time domain. 263 00:14:37,810 --> 00:14:39,310 So in that sense, you need both. 264 00:14:39,310 --> 00:14:41,040 You need the bare basis. 265 00:14:41,040 --> 00:14:42,760 You need to dressed basis. 266 00:14:42,760 --> 00:14:47,530 And you have to just know for which physical process, which 267 00:14:47,530 --> 00:14:52,013 of them allows the more direct description. 268 00:14:52,013 --> 00:14:52,513 Questions? 269 00:15:00,640 --> 00:15:07,730 OK, so this was partly review, partly preparation 270 00:15:07,730 --> 00:15:10,390 for the next big chapter, namely we 271 00:15:10,390 --> 00:15:13,520 want to understand the dipole forces within the dressed atom 272 00:15:13,520 --> 00:15:14,990 picture. 273 00:15:14,990 --> 00:15:19,670 The book Atom Photon Interaction has a nice summary of it. 274 00:15:19,670 --> 00:15:23,640 But the more detailed picture, and I heavily draw from this, 275 00:15:23,640 --> 00:15:27,450 is in the truly seminal article by Jean Dalibard and Claude 276 00:15:27,450 --> 00:15:28,890 Cohen-Tannoudji. 277 00:15:28,890 --> 00:15:33,280 I've linked this article to our website. 278 00:15:33,280 --> 00:15:45,000 So I've summarized here pretty much what I just told you. 279 00:15:45,000 --> 00:15:48,960 This is sort of the summary from the previous section. 280 00:15:48,960 --> 00:15:52,040 What we have to know is to understand light forces 281 00:15:52,040 --> 00:15:54,650 in the dressed atom picture is number one, 282 00:15:54,650 --> 00:15:57,191 the energy speaking between the two. 283 00:15:57,191 --> 00:16:03,170 The splitting between the two is the generalized Rabi frequency. 284 00:16:03,170 --> 00:16:05,950 If there's anything which is not-- 285 00:16:05,950 --> 00:16:09,130 well I shouldn't say 100%-- but at least 90% clear to you, 286 00:16:09,130 --> 00:16:10,310 ask me question. 287 00:16:10,310 --> 00:16:12,130 I mean these are the results we really 288 00:16:12,130 --> 00:16:14,180 need for the following discussion. 289 00:16:14,180 --> 00:16:16,200 So number one is the dressed energy 290 00:16:16,200 --> 00:16:20,680 levels are split by the generalized Rabi frequency. 291 00:16:20,680 --> 00:16:26,306 Number two is with this angle theta, which 292 00:16:26,306 --> 00:16:27,930 came from the solution of diagonalizing 293 00:16:27,930 --> 00:16:31,400 the two-by-two matrix, we have now the steady state 294 00:16:31,400 --> 00:16:35,930 populations in the upper and in the lower dressed state. 295 00:16:35,930 --> 00:16:40,970 And the third thing is when we have a population when 296 00:16:40,970 --> 00:16:46,060 we are not in steady state, then the steady state populations 297 00:16:46,060 --> 00:16:50,680 are obtained with the time constant, which is given here. 298 00:16:50,680 --> 00:16:54,710 The time constant is not just gamma, 299 00:16:54,710 --> 00:16:59,630 it's gamma times these trigonometric functions. 300 00:16:59,630 --> 00:17:01,570 So these are simple ingredients. 301 00:17:01,570 --> 00:17:03,910 But by using those ingredients we 302 00:17:03,910 --> 00:17:11,450 can get a very nice formulation for the stimulated force. 303 00:17:19,819 --> 00:17:22,140 So what I've shown here is, and this 304 00:17:22,140 --> 00:17:30,380 is the situation we want to focus on for a while. 305 00:17:30,380 --> 00:17:34,070 Assume you focus a laser beam. 306 00:17:34,070 --> 00:17:38,290 Then plot it as a function of position. 307 00:17:38,290 --> 00:17:40,590 The atom is outside the laser beam, 308 00:17:40,590 --> 00:17:42,495 it goes into the laser beam where 309 00:17:42,495 --> 00:17:45,560 it experiences the electromagnetic field. 310 00:17:45,560 --> 00:17:48,320 And if it goes further it has crossed the laser beam 311 00:17:48,320 --> 00:17:52,080 and is out of the laser beam again. 312 00:17:52,080 --> 00:17:56,800 So what happens is the splitting between the two dressed 313 00:17:56,800 --> 00:18:00,380 energy levels is the generalized Rabi frequency. 314 00:18:00,380 --> 00:18:04,150 And this is outside the laser beam, simply the detuning. 315 00:18:04,150 --> 00:18:06,520 But then when the laser beam is on, 316 00:18:06,520 --> 00:18:08,746 the generalized Rabi frequency gets larger. 317 00:18:08,746 --> 00:18:11,120 We have a larger lar splitting between the dressed energy 318 00:18:11,120 --> 00:18:14,160 levels, and so on. 319 00:18:14,160 --> 00:18:18,100 What is also important is that if the laser is 320 00:18:18,100 --> 00:18:24,510 red detuned, which is here, the lower dressed level correlates, 321 00:18:24,510 --> 00:18:26,750 corresponds to the ground state. 322 00:18:26,750 --> 00:18:38,180 Whereas for blue detuning the ground state in this limit, 323 00:18:38,180 --> 00:18:41,380 the ground state correlates with the upper dressed level. 324 00:18:41,380 --> 00:18:42,830 So it's important. 325 00:18:42,830 --> 00:18:46,920 So therefore an atom in this situation-- because atoms 326 00:18:46,920 --> 00:18:48,430 are always more in the ground state 327 00:18:48,430 --> 00:18:50,280 than in the excited state-- an atom 328 00:18:50,280 --> 00:18:52,849 will have more population in the upper dressed level. 329 00:18:52,849 --> 00:18:55,140 And here it will have more population the lower dressed 330 00:18:55,140 --> 00:18:55,640 level. 331 00:18:58,720 --> 00:19:01,050 So what we want to do now is we want 332 00:19:01,050 --> 00:19:06,830 to rewrite the stimulated light force, or the dipole force, 333 00:19:06,830 --> 00:19:11,770 as we want to express it through the forces acting 334 00:19:11,770 --> 00:19:13,800 on the dressed atom level. 335 00:19:13,800 --> 00:19:19,080 Well, I told you that we have a splitting of the dressed energy 336 00:19:19,080 --> 00:19:21,640 levels by the generalized Rabi frequency. 337 00:19:21,640 --> 00:19:24,735 So therefore, if I just cover that, 338 00:19:24,735 --> 00:19:28,172 the energy level of the two dressed 339 00:19:28,172 --> 00:19:32,630 levels is h bar over 2 plus minus the generalized Rabi 340 00:19:32,630 --> 00:19:33,760 frequency. 341 00:19:33,760 --> 00:19:37,590 And the force if an atom is in one of the dressed levels 342 00:19:37,590 --> 00:19:39,720 is simply the gradient of that. 343 00:19:39,720 --> 00:19:43,000 So therefore we have the picture that the two dressed energy 344 00:19:43,000 --> 00:19:45,540 levels, atoms in those two dressed levels, 345 00:19:45,540 --> 00:19:48,360 always experience opposite forces. 346 00:19:51,080 --> 00:19:54,453 Then an expression for the net force 347 00:19:54,453 --> 00:19:58,130 is that the average dipole force is 348 00:19:58,130 --> 00:20:00,830 the force in dressed level one times 349 00:20:00,830 --> 00:20:03,660 the probability that the atom is in dressed level one. 350 00:20:03,660 --> 00:20:05,670 And the same for two. 351 00:20:05,670 --> 00:20:09,270 And if you sum it up we get exactly the expression 352 00:20:09,270 --> 00:20:13,910 we have derived earlier for the stimulated light force. 353 00:20:13,910 --> 00:20:18,080 But let's just sort of look at it in the following picture. 354 00:20:18,080 --> 00:20:22,320 If we are blue detuned, and let's assume 355 00:20:22,320 --> 00:20:24,230 we are not saturating the transition, 356 00:20:24,230 --> 00:20:27,670 so therefore the atom will be mainly in the ground state. 357 00:20:27,670 --> 00:20:31,790 So you have more population in that state. 358 00:20:31,790 --> 00:20:35,910 But, well, if the laser power is 0, 359 00:20:35,910 --> 00:20:37,890 we have 100% population here. 360 00:20:37,890 --> 00:20:40,400 But when the atom moves into the laser beam 361 00:20:40,400 --> 00:20:44,130 there is a superposition of ground and excited state. 362 00:20:44,130 --> 00:20:46,440 But that also means there's a superposition 363 00:20:46,440 --> 00:20:49,420 of upper and lower dressed level. 364 00:20:49,420 --> 00:20:52,320 But we have always, of course, more population 365 00:20:52,320 --> 00:20:55,300 in the dressed level which connects to the ground state 366 00:20:55,300 --> 00:21:02,140 because an atom will-- well, it also comes out of the solution. 367 00:21:02,140 --> 00:21:07,350 But it's natural, just assume you make the laser beam weak, 368 00:21:07,350 --> 00:21:09,440 you start in the ground state and everything 369 00:21:09,440 --> 00:21:10,929 is sort of connected. 370 00:21:10,929 --> 00:21:12,345 The ground state is simply dressed 371 00:21:12,345 --> 00:21:16,630 by a small excited state contribution. 372 00:21:16,630 --> 00:21:20,850 So that explains it, why for blue detuned light, 373 00:21:20,850 --> 00:21:23,380 the physics is mainly in the upper dressed level. 374 00:21:23,380 --> 00:21:27,590 And we see the repulsive dipole force of a blue detuned laser. 375 00:21:27,590 --> 00:21:30,850 Whereas, for red detuned light, the dominant population 376 00:21:30,850 --> 00:21:33,180 is in the lower dressed level, and we 377 00:21:33,180 --> 00:21:36,560 have an attractive force. 378 00:21:36,560 --> 00:21:38,080 Well that's simple. 379 00:21:41,638 --> 00:21:46,420 We haven't gained so much, but you'll see, 380 00:21:46,420 --> 00:21:50,560 we can now rapidly gain insight by take it to the next level. 381 00:21:50,560 --> 00:21:53,690 Let me first mention a trivial example. 382 00:21:53,690 --> 00:21:56,390 And this is when we are on resonance. 383 00:21:56,390 --> 00:22:00,410 When we are on resonance we have equal population in the two 384 00:22:00,410 --> 00:22:04,190 dressed states, they are resonantly mixed. 385 00:22:04,190 --> 00:22:06,846 And therefore if the population in the two 386 00:22:06,846 --> 00:22:10,370 dressed states is the same, but the atoms experience 387 00:22:10,370 --> 00:22:14,220 opposite forces, that means the dipole force average out to 0. 388 00:22:17,342 --> 00:22:21,970 In a few minutes I want to calculate for you 389 00:22:21,970 --> 00:22:24,630 what the heating is. 390 00:22:24,630 --> 00:22:26,810 The dipole force is 0. 391 00:22:26,810 --> 00:22:31,180 But you know the atom in this radiative cascade 392 00:22:31,180 --> 00:22:35,500 is making transitions between upper and lower dressed level. 393 00:22:35,500 --> 00:22:37,670 So what does it mean for the atom? 394 00:22:37,670 --> 00:22:41,100 When it is in one dressed level it experience the force 395 00:22:41,100 --> 00:22:42,160 from this side. 396 00:22:42,160 --> 00:22:44,300 If the atom is in the other dressed level, 397 00:22:44,300 --> 00:22:47,290 it's experience a force from that side. 398 00:22:47,290 --> 00:22:53,350 So the atom is actually subject to an average force which is 0. 399 00:22:53,350 --> 00:22:56,570 But if you take these picture literally, and you can, 400 00:22:56,570 --> 00:22:59,600 the atom is really whacked left, right, left, right, left, 401 00:22:59,600 --> 00:23:00,100 right. 402 00:23:00,100 --> 00:23:01,950 And there is fluctuations. 403 00:23:01,950 --> 00:23:05,930 And we can use those fluctuating forces to immediately calculate 404 00:23:05,930 --> 00:23:07,250 what is the heating. 405 00:23:07,250 --> 00:23:11,410 The force average is 0, but the fluctuations of the force 406 00:23:11,410 --> 00:23:12,990 cause heating. 407 00:23:12,990 --> 00:23:16,430 So that's immediately the next step of physical insight 408 00:23:16,430 --> 00:23:19,120 we will get out of this picture. 409 00:23:19,120 --> 00:23:22,490 But before I do that I want to do something else. 410 00:23:22,490 --> 00:23:29,730 I want to discuss what happens now 411 00:23:29,730 --> 00:23:34,010 to the force when the atom is slowly moving. 412 00:23:34,010 --> 00:23:41,440 Remember, there is still one phenomenon 413 00:23:41,440 --> 00:23:43,630 which I owe you an explanation for. 414 00:23:43,630 --> 00:23:46,480 And this is when we discussed molasses. 415 00:23:46,480 --> 00:23:48,960 We always had the effect in optical molasses 416 00:23:48,960 --> 00:23:51,000 with a spontaneous light force that we 417 00:23:51,000 --> 00:23:55,330 need red detuning of the laser beam to get cooling. 418 00:23:55,330 --> 00:23:58,280 The simple picture is the atom absorbs a red detuned photon 419 00:23:58,280 --> 00:24:00,870 and emits away a resonant photon. 420 00:24:00,870 --> 00:24:03,460 And then that energy difference is taken away 421 00:24:03,460 --> 00:24:06,260 from the kinetic energy, and this is cooling. 422 00:24:06,260 --> 00:24:08,680 But then I told you that when you 423 00:24:08,680 --> 00:24:12,690 go to higher power of the standing wave, 424 00:24:12,690 --> 00:24:17,330 the sine of alpha of the friction coefficient reverses. 425 00:24:17,330 --> 00:24:20,950 And that means cooling is now done by blue detuned light, 426 00:24:20,950 --> 00:24:23,287 heating is done by red detuned light. 427 00:24:23,287 --> 00:24:24,870 And that's what we want to understand. 428 00:24:29,530 --> 00:24:35,090 So in order to get any cooling effect, or heating effect, 429 00:24:35,090 --> 00:24:39,450 we have to now change the equation 430 00:24:39,450 --> 00:24:41,890 we had for the average dipole force. 431 00:24:41,890 --> 00:24:44,690 Remember also, the average dipole force, 432 00:24:44,690 --> 00:24:47,575 we did that earlier, can be derived from a potential. 433 00:24:47,575 --> 00:24:49,620 It's a gradient over potential. 434 00:24:49,620 --> 00:24:52,180 So therefore, when you fly through a laser beam 435 00:24:52,180 --> 00:24:55,230 you go down and up or up and down the potential, 436 00:24:55,230 --> 00:24:57,560 but nothing has happened to you. 437 00:24:57,560 --> 00:25:00,310 But the assumption we had for that 438 00:25:00,310 --> 00:25:05,810 is that the dipole force was calculated with a steady state 439 00:25:05,810 --> 00:25:07,500 population. 440 00:25:07,500 --> 00:25:10,860 So now we want to say if the atom is slowly 441 00:25:10,860 --> 00:25:15,410 moving it may not be fully adiabatic, 442 00:25:15,410 --> 00:25:18,120 and we want to calculate the first correction 443 00:25:18,120 --> 00:25:21,150 to the non-adiabaticity. 444 00:25:21,150 --> 00:25:24,310 So in other words, we have the following picture. 445 00:25:24,310 --> 00:25:26,950 When an atom flies through the laser beam 446 00:25:26,950 --> 00:25:32,210 it needs a certain time to adjust its population. 447 00:25:32,210 --> 00:25:35,230 And the question, what is this time? 448 00:25:35,230 --> 00:25:38,870 Well, it is the relaxation time for the populations 449 00:25:38,870 --> 00:25:40,990 among the dressed states to adjust. 450 00:25:40,990 --> 00:25:44,400 And this is exactly the inverse of the rate gamma 451 00:25:44,400 --> 00:25:46,575 pop, which we have calculated earlier. 452 00:25:51,590 --> 00:25:56,190 So the physical picture now is that in the dressed level 453 00:25:56,190 --> 00:26:00,250 one and two, the population at any given moment 454 00:26:00,250 --> 00:26:03,970 is almost the steady state. 455 00:26:03,970 --> 00:26:06,990 But there is a time lag, a time lag 456 00:26:06,990 --> 00:26:09,960 by the time it takes to relax. 457 00:26:09,960 --> 00:26:12,160 So in other words, the atom is always a little bit 458 00:26:12,160 --> 00:26:15,140 behind in adjusting to the laser field 459 00:26:15,140 --> 00:26:19,190 because it takes a finite amount of time. 460 00:26:19,190 --> 00:26:25,190 So let's now try to understand, in this picture which 461 00:26:25,190 --> 00:26:27,980 we have obtained with a laser beam 462 00:26:27,980 --> 00:26:33,450 the two dressed levels and the atom flying through, 463 00:26:33,450 --> 00:26:41,900 let's try to understand what this extra lag time means. 464 00:26:41,900 --> 00:26:46,170 So the atom is flying into the laser beam. 465 00:26:46,170 --> 00:26:48,620 It starts out in the ground state, 466 00:26:48,620 --> 00:26:53,330 which means 100% in the upper dressed energy level. 467 00:26:53,330 --> 00:26:57,100 Now, in the laser beam the higher the laser power is, 468 00:26:57,100 --> 00:27:01,980 the more there is a mixture of the other dressed energy level. 469 00:27:01,980 --> 00:27:04,870 But what happens is, when the atom is here, 470 00:27:04,870 --> 00:27:09,530 when it just assume it flies in, and there was not enough time 471 00:27:09,530 --> 00:27:12,330 to adjust the population, that would 472 00:27:12,330 --> 00:27:16,190 mean that the atom-- and this is indicated with the red dots-- 473 00:27:16,190 --> 00:27:18,450 that the atom has a little bit of more 474 00:27:18,450 --> 00:27:20,680 population in the upper dressed level 475 00:27:20,680 --> 00:27:23,450 than it would have in equilibrium 476 00:27:23,450 --> 00:27:25,290 But what does it mean for the force 477 00:27:25,290 --> 00:27:29,570 now, if there is more population here than there? 478 00:27:29,570 --> 00:27:33,120 Well, the more population you have in the upper state, 479 00:27:33,120 --> 00:27:35,030 the more you're climbing up a hill, 480 00:27:35,030 --> 00:27:37,320 the more you get a force to the left. 481 00:27:37,320 --> 00:27:41,560 So this red arrow shows due to the non-adiabaticity, 482 00:27:41,560 --> 00:27:44,410 due to the time lag, there is a little extra force 483 00:27:44,410 --> 00:27:47,390 to the left hand side. 484 00:27:47,390 --> 00:27:49,710 So this is on the up hill. 485 00:27:49,710 --> 00:27:52,810 Let's now understand what happens on the down hill. 486 00:27:52,810 --> 00:27:54,980 Well, we just have to keep track of signs. 487 00:27:54,980 --> 00:27:59,630 Here, the atom, in a fairly high laser field, 488 00:27:59,630 --> 00:28:02,620 is in the upper dressed level with a mixture in the lower 489 00:28:02,620 --> 00:28:03,580 dressed level. 490 00:28:03,580 --> 00:28:07,040 But now it flies out of the laser beam, 491 00:28:07,040 --> 00:28:10,490 but it cannot immediately adjust its population. 492 00:28:10,490 --> 00:28:14,160 And that means it wants to have now 493 00:28:14,160 --> 00:28:17,280 more population in the upper dressed level 494 00:28:17,280 --> 00:28:19,250 but it hasn't fully adjusted. 495 00:28:19,250 --> 00:28:23,700 So therefore, instead of having the blue bullet, which 496 00:28:23,700 --> 00:28:27,270 represents equilibrium, the atom has the red bullet. 497 00:28:27,270 --> 00:28:30,440 It has a little bit lower population in that, 498 00:28:30,440 --> 00:28:32,940 and higher population there. 499 00:28:32,940 --> 00:28:36,430 But lower population here and hire population here 500 00:28:36,430 --> 00:28:39,660 means-- if you have a little bit higher population than you 501 00:28:39,660 --> 00:28:43,500 should have here it means you are climbing up the hill, 502 00:28:43,500 --> 00:28:49,040 and there is an extra force, which is again, to the left. 503 00:28:49,040 --> 00:28:53,690 So we realize the n the time lag, 504 00:28:53,690 --> 00:28:57,860 means when an atom goes up the hill there's a little bit 505 00:28:57,860 --> 00:28:59,180 extra force to the left. 506 00:28:59,180 --> 00:29:00,720 And when it goes down the hill there 507 00:29:00,720 --> 00:29:03,080 is a little bit extra force to the left. 508 00:29:03,080 --> 00:29:07,300 And after it has flown through the steady state 509 00:29:07,300 --> 00:29:10,410 part of the force roller coaster up down, which 510 00:29:10,410 --> 00:29:14,110 can be derived from a potential, has integrated to 0. 511 00:29:14,110 --> 00:29:17,530 But what is left is this non-adiabatic component 512 00:29:17,530 --> 00:29:22,340 of the force which is slowing down the atom. 513 00:29:22,340 --> 00:29:28,750 In other words, I have shown you in this simple picture 514 00:29:28,750 --> 00:29:34,370 that the time lag in adjusting the dressed energy population 515 00:29:34,370 --> 00:29:39,370 means that there is a small differential force which 516 00:29:39,370 --> 00:29:41,860 is slowing down the atom. 517 00:29:41,860 --> 00:29:45,460 And what I assumed here was blue detuning. 518 00:29:45,460 --> 00:29:50,820 So we have cooling, we have friction for blue detuned light 519 00:29:50,820 --> 00:29:52,515 when it comes to the stimulated force. 520 00:29:56,190 --> 00:30:02,710 Now this picture, this equation that the population is 521 00:30:02,710 --> 00:30:07,080 a steady state at the given position minus a time 522 00:30:07,080 --> 00:30:11,880 lag, we can solve it. 523 00:30:11,880 --> 00:30:13,820 And I'm not doing it here because it 524 00:30:13,820 --> 00:30:15,920 involves a few mathematical equations. 525 00:30:15,920 --> 00:30:19,160 But by just using these answers, and the rest is straight 526 00:30:19,160 --> 00:30:23,030 forward, you can then arrive at an expression 527 00:30:23,030 --> 00:30:26,330 for the stimulated for the dipole force. 528 00:30:26,330 --> 00:30:27,074 Colin? 529 00:30:27,074 --> 00:30:28,782 AUDIENCE: So, in the spirit of last time, 530 00:30:28,782 --> 00:30:31,945 keeping track of the energy, the energy here 531 00:30:31,945 --> 00:30:34,510 would be going into the fact that now I 532 00:30:34,510 --> 00:30:37,265 have-- if the atom's a harmonic oscillator 533 00:30:37,265 --> 00:30:39,535 it's a little behind the drive field? 534 00:30:39,535 --> 00:30:40,160 PROFESSOR: Yes. 535 00:30:40,160 --> 00:30:43,073 AUDIENCE: So you imprint some fluctuation on the laser beam, 536 00:30:43,073 --> 00:30:45,073 and that's where the energy's being carried off, 537 00:30:45,073 --> 00:30:46,067 in the laser beam? 538 00:30:50,242 --> 00:30:52,700 PROFESSOR: Well the energy's carried off by the laser beam. 539 00:30:52,700 --> 00:30:55,570 I have actually prepared something 540 00:30:55,570 --> 00:30:57,070 which I want to show you in 10 or 15 541 00:30:57,070 --> 00:30:58,700 minutes, where the energy goes. 542 00:30:58,700 --> 00:31:04,820 So let's maybe do that in a moment. 543 00:31:04,820 --> 00:31:06,190 So this is a physical picture. 544 00:31:06,190 --> 00:31:09,490 And I think what is also important I want to later tell 545 00:31:09,490 --> 00:31:12,530 you, for other laser cooling mechanisms, which 546 00:31:12,530 --> 00:31:16,970 are too complicated to describe, but they also have a time lag. 547 00:31:16,970 --> 00:31:20,870 And I think, I try to teach you basic concepts 548 00:31:20,870 --> 00:31:25,360 by using this simplest physical picture I can have. 549 00:31:25,360 --> 00:31:29,310 And what can be simpler than two levels of an atom and one laser 550 00:31:29,310 --> 00:31:30,070 beam? 551 00:31:30,070 --> 00:31:33,730 But what you understand here with this stimulated force, 552 00:31:33,730 --> 00:31:36,180 which is also called blue molasses, 553 00:31:36,180 --> 00:31:39,390 is for the first time how a time lag 554 00:31:39,390 --> 00:31:42,730 in adjusting to the local laser field, 555 00:31:42,730 --> 00:31:44,516 how that leads to cooling. 556 00:31:47,770 --> 00:31:49,900 I'm summarizing here, simply taking 557 00:31:49,900 --> 00:31:51,745 this picture, this equation, and you 558 00:31:51,745 --> 00:31:53,610 know, turning the crank on it. 559 00:31:53,610 --> 00:31:56,670 The paper by Jean Dalibard and Claude Cohen-Tannoudji 560 00:31:56,670 --> 00:32:00,650 calculates what is the dipole force in steady state. 561 00:32:00,650 --> 00:32:04,840 We get almost the same result with the Optical Bloch 562 00:32:04,840 --> 00:32:06,910 Equations, but we are discussing here 563 00:32:06,910 --> 00:32:10,850 the limit that the Rabi frequency is larger than gamma. 564 00:32:10,850 --> 00:32:14,810 So therefore, if you're missing occurrences of gamma, 565 00:32:14,810 --> 00:32:16,920 they don't appear here because we've always 566 00:32:16,920 --> 00:32:19,310 assumed omega is larger than gamma in this picture. 567 00:32:22,740 --> 00:32:26,710 Now I want to apply this picture now, 568 00:32:26,710 --> 00:32:30,120 not just going through one hump of the laser field, 569 00:32:30,120 --> 00:32:32,070 but going through a standing wave. 570 00:32:32,070 --> 00:32:34,870 So we have a standing wave of light, which is blue detuned. 571 00:32:42,100 --> 00:32:47,530 Of course, in a standing wave of light, 572 00:32:47,530 --> 00:32:51,940 if you ever reach over one optical wavelengths, 573 00:32:51,940 --> 00:32:54,350 you go up the hill, you go down the hill, 574 00:32:54,350 --> 00:32:56,550 the lambda average dipole force is 0. 575 00:32:59,690 --> 00:33:05,300 But now you find that if you put in this time lag, 576 00:33:05,300 --> 00:33:10,240 you find a friction force, with a friction coefficient alpha. 577 00:33:10,240 --> 00:33:17,240 And this friction coefficient is cooling for blue detuned light. 578 00:33:17,240 --> 00:33:21,790 Now I don't want to elaborate on this expression. 579 00:33:21,790 --> 00:33:26,250 But if you take the treatment of Jean Dalibard and Claude 580 00:33:26,250 --> 00:33:28,110 Cohen-Tonnoudji, which is very transparent. 581 00:33:28,110 --> 00:33:31,405 It reads wonderfully easily, as the book Atom Photon 582 00:33:31,405 --> 00:33:33,230 Interaction. 583 00:33:33,230 --> 00:33:35,850 In the limit of large detuning, which is the simplest 584 00:33:35,850 --> 00:33:39,390 possibly limit, you find the friction coefficient 585 00:33:39,390 --> 00:33:42,260 which looks highly nontrivial. 586 00:33:42,260 --> 00:33:46,580 It depends to the sixth power of the Rabi frequency and such. 587 00:33:46,580 --> 00:33:50,240 The only reason why I put it here, in 20 minutes 588 00:33:50,240 --> 00:33:53,470 I want to give you a very, very simple picture where 589 00:33:53,470 --> 00:33:56,090 I derive that for you. 590 00:33:56,090 --> 00:34:00,100 So that's just a check mark. 591 00:34:00,100 --> 00:34:02,570 Any questions right now about-- because I 592 00:34:02,570 --> 00:34:05,910 want to go on-- about the stimulated force 593 00:34:05,910 --> 00:34:07,790 in the dressed atom picture? 594 00:34:07,790 --> 00:34:10,870 And if we allowed deviations from the steady state 595 00:34:10,870 --> 00:34:13,250 population we get a cooling effect, 596 00:34:13,250 --> 00:34:16,650 and this explains the effect of blue molasses. 597 00:34:16,650 --> 00:34:18,909 And we've applied it to a standing wave. 598 00:34:18,909 --> 00:34:19,409 Cody? 599 00:34:19,409 --> 00:34:24,110 AUDIENCE: Just [INAUDIBLE] the upper alpha 600 00:34:24,110 --> 00:34:26,920 are a friction constant and the lower one 601 00:34:26,920 --> 00:34:28,376 is the polarizability? 602 00:34:28,376 --> 00:34:28,959 Or, something. 603 00:34:28,959 --> 00:34:31,540 PROFESSOR: No, oh, thanks for asking. 604 00:34:31,540 --> 00:34:34,940 Alpha, yeah, we [INAUDIBLE]. 605 00:34:34,940 --> 00:34:37,639 Here it's the friction coefficient. 606 00:34:37,639 --> 00:34:40,940 Here, remember when we had the light force-- 607 00:34:40,940 --> 00:34:44,290 the spontaneous light force and the stimulated light force-- 608 00:34:44,290 --> 00:34:47,760 one was directed along a vector alpha, 609 00:34:47,760 --> 00:34:50,049 and the other one along a vector beta. 610 00:34:50,049 --> 00:34:53,580 Alpha was pointing into the direction 611 00:34:53,580 --> 00:34:56,130 of the gradient of the laser intensity, 612 00:34:56,130 --> 00:35:00,400 and beta was the phase gradient of the laser beam. 613 00:35:00,400 --> 00:35:02,080 So those two have nothing to do-- 614 00:35:02,080 --> 00:35:03,630 this is friction coefficient. 615 00:35:03,630 --> 00:35:06,980 This is the direction for the stimulated light force. 616 00:35:06,980 --> 00:35:09,580 And yes, a little bit later today, 617 00:35:09,580 --> 00:35:12,160 I will use alpha again, but for the polarizability. 618 00:35:15,426 --> 00:35:16,358 Yes. 619 00:35:16,358 --> 00:35:18,222 AUDIENCE: The [INAUDIBLE] track the atoms 620 00:35:18,222 --> 00:35:21,138 in the [INAUDIBLE] standing wave, [INAUDIBLE]. 621 00:35:27,480 --> 00:35:30,100 PROFESSOR: You're talking about trap in here, 622 00:35:30,100 --> 00:35:39,530 that if you have a standing wave of blue detuned light-- 623 00:35:39,530 --> 00:35:42,550 well if the standing wave is infinitely extended then 624 00:35:42,550 --> 00:35:44,400 the atom is always in the standing wave. 625 00:35:44,400 --> 00:35:47,390 But if you use two Gaussian laser beams to form a standing 626 00:35:47,390 --> 00:35:50,200 wave, there is always a net force 627 00:35:50,200 --> 00:35:53,160 to expel the atoms away from blue detuned light, 628 00:35:53,160 --> 00:35:56,510 because blue detuned light has an average dipole 629 00:35:56,510 --> 00:35:58,260 force which is repulsive. 630 00:35:58,260 --> 00:36:00,230 So whenever you want to manipulate atoms 631 00:36:00,230 --> 00:36:05,300 with blue detuned light you need an additional magnetic trap, 632 00:36:05,300 --> 00:36:07,799 or an additional red detuned optical trap. 633 00:36:07,799 --> 00:36:10,090 Because whatever you want to do with atoms, if you want 634 00:36:10,090 --> 00:36:12,230 to do it with atoms in steady state 635 00:36:12,230 --> 00:36:14,350 you have to keep the atoms together. 636 00:36:14,350 --> 00:36:15,980 And what keeps the atoms together 637 00:36:15,980 --> 00:36:20,620 is red detuned optical traps, or magnetic traps. 638 00:36:20,620 --> 00:36:23,560 The blue detuned light can be used 639 00:36:23,560 --> 00:36:25,900 to create a blue detuned lattice. 640 00:36:25,900 --> 00:36:29,599 Or it can be used at least, if you 641 00:36:29,599 --> 00:36:31,890 want to do laser cooling with blue detuned light it can 642 00:36:31,890 --> 00:36:34,600 be used for laser cooling, as I've 643 00:36:34,600 --> 00:36:36,080 started to describe for you now. 644 00:36:38,750 --> 00:36:47,250 But yes, I mean, in that sense as an experimentalist, 645 00:36:47,250 --> 00:36:49,865 you can run experiments where you have only red detuned 646 00:36:49,865 --> 00:36:51,490 light-- red detuned light for trapping, 647 00:36:51,490 --> 00:36:53,260 red detuned light for cooling. 648 00:36:53,260 --> 00:36:57,870 But if you use blue detuned light for a standing wave 649 00:36:57,870 --> 00:37:00,815 or for blue molasses, in addition 650 00:37:00,815 --> 00:37:03,250 you will always use red detuned light 651 00:37:03,250 --> 00:37:04,840 to keep the clouds together. 652 00:37:04,840 --> 00:37:06,920 So pretty much every experiment usually 653 00:37:06,920 --> 00:37:10,090 has some red detuned light for confinement. 654 00:37:10,090 --> 00:37:13,100 And you have to then match blue detuned light 655 00:37:13,100 --> 00:37:15,350 with red detuned light to get the two things you want. 656 00:37:22,270 --> 00:37:25,690 We can now use the dressed atom picture 657 00:37:25,690 --> 00:37:28,450 to calculate the heating. 658 00:37:28,450 --> 00:37:32,770 Remember the final temperature you can reach 659 00:37:32,770 --> 00:37:37,240 was always the ratio between a heating coefficient, which 660 00:37:37,240 --> 00:37:39,430 is actually the momentum diffusion coefficient, 661 00:37:39,430 --> 00:37:40,930 over alpha. 662 00:37:40,930 --> 00:37:44,250 So right now we have understood the alpha part. 663 00:37:44,250 --> 00:37:48,360 We know the physical picture, why alpha 664 00:37:48,360 --> 00:37:50,670 provides cooling, why we have a friction 665 00:37:50,670 --> 00:37:53,450 force for blue detuned light. 666 00:37:53,450 --> 00:37:56,090 But let's now take it to the next step, 667 00:37:56,090 --> 00:38:00,070 and calculate what is the heating. 668 00:38:00,070 --> 00:38:04,020 And I mentioned it already in the context of simple molasses, 669 00:38:04,020 --> 00:38:06,850 which we discussed, I think, two weeks ago, 670 00:38:06,850 --> 00:38:12,560 that we describe heating by the momentum diffusion coefficient. 671 00:38:12,560 --> 00:38:15,230 Well the momentum diffusion coefficient 672 00:38:15,230 --> 00:38:18,180 tells you that P square is simply 673 00:38:18,180 --> 00:38:19,830 growing as a function of time. 674 00:38:19,830 --> 00:38:22,170 If you have a cold cloud and you heat it up, 675 00:38:22,170 --> 00:38:25,420 you can describe the heating by the temporal growth 676 00:38:25,420 --> 00:38:27,440 of P square. 677 00:38:27,440 --> 00:38:29,480 If you don't like momentum, divide P 678 00:38:29,480 --> 00:38:31,940 squared by 2m you have the kinetic energy, which 679 00:38:31,940 --> 00:38:33,140 increases. 680 00:38:33,140 --> 00:38:36,390 So it is this momentum diffusion coefficient. 681 00:38:36,390 --> 00:38:40,075 And if you have a cloud which moves with an average momentum, 682 00:38:40,075 --> 00:38:43,410 you only want to have the P square minus P average square. 683 00:38:43,410 --> 00:38:46,560 But if you think about a cloud which is being cooled, 684 00:38:46,560 --> 00:38:49,750 the second term is 0. 685 00:38:49,750 --> 00:38:51,690 So the momentum diffusion coefficient 686 00:38:51,690 --> 00:38:54,280 is the derivative of P square. 687 00:38:54,280 --> 00:38:58,030 But the derivative of P is simply the force. 688 00:38:58,030 --> 00:39:02,160 So therefore we can absorb the derivative by replacing P-- 689 00:39:02,160 --> 00:39:04,920 at least one of the P's-- by the force. 690 00:39:04,920 --> 00:39:07,900 And we can now get rid of to second P 691 00:39:07,900 --> 00:39:09,610 of the second occurrence of momentum 692 00:39:09,610 --> 00:39:12,510 by saying well momentum is derivative of force. 693 00:39:16,120 --> 00:39:18,580 Sorry, derivative of momentum is force, 694 00:39:18,580 --> 00:39:21,890 or momentum is the integral of force. 695 00:39:21,890 --> 00:39:26,550 So we can exactly rewrite that momentum diffusion coefficient 696 00:39:26,550 --> 00:39:31,480 by an integral over correlation function of forces. 697 00:39:31,480 --> 00:39:34,430 And then we have the nice physical picture 698 00:39:34,430 --> 00:39:37,410 that with increase of kinetic energy, 699 00:39:37,410 --> 00:39:39,420 the momentum diffusion coefficient 700 00:39:39,420 --> 00:39:42,871 comes because we have fluctuations of the force. 701 00:39:42,871 --> 00:39:43,370 Yes? 702 00:39:43,370 --> 00:39:45,328 AUDIENCE: With the last time, the previous page 703 00:39:45,328 --> 00:39:48,240 the factor [INAUDIBLE]. 704 00:39:48,240 --> 00:39:49,701 PROFESSOR: The factor of 2 here? 705 00:39:49,701 --> 00:39:51,170 AUDIENCE: No, the next slide. 706 00:39:51,170 --> 00:39:52,089 PROFESSOR: Here is 2. 707 00:39:52,089 --> 00:39:53,130 AUDIENCE: For both terms. 708 00:39:53,130 --> 00:39:54,295 PROFESSOR: Here is 2. 709 00:39:54,295 --> 00:39:56,780 And here it disappears, because we have a factor of 2 710 00:39:56,780 --> 00:39:58,980 and a factor of 2, and the two cancel each other. 711 00:39:58,980 --> 00:40:02,340 AUDIENCE: No, I meant, there's a 2 multiplied the first average. 712 00:40:02,340 --> 00:40:04,740 And there's a 2 for the second one. 713 00:40:04,740 --> 00:40:07,150 Are those for the first term? 714 00:40:07,150 --> 00:40:08,670 PROFESSOR: I think this is correct. 715 00:40:08,670 --> 00:40:10,470 AUDIENCE: [INAUDIBLE]. 716 00:40:10,470 --> 00:40:17,970 PROFESSOR: What happens is P-- you use-- 717 00:40:17,970 --> 00:40:18,940 AUDIENCE: [INAUDIBLE]. 718 00:40:18,940 --> 00:40:21,217 AUDIENCE: No, it's supposed to be-- 719 00:40:21,217 --> 00:40:22,550 PROFESSOR: Wait, is one missing? 720 00:40:25,900 --> 00:40:26,840 Maybe [INAUDIBLE]. 721 00:40:26,840 --> 00:40:28,254 I think you're right. 722 00:40:28,254 --> 00:40:29,170 AUDIENCE: [INAUDIBLE]. 723 00:40:33,167 --> 00:40:35,250 PROFESSOR: Let's assume this is what it should be. 724 00:40:40,640 --> 00:40:45,190 So here is the reminder, the kinetic energy. 725 00:40:45,190 --> 00:40:49,390 If you have a process which causes momentum diffusion, 726 00:40:49,390 --> 00:40:51,390 the momentum diffusion coefficient 727 00:40:51,390 --> 00:40:53,480 gives you the heating rate. 728 00:40:53,480 --> 00:40:58,280 And the cooling rate is given by our friction coefficient alpha. 729 00:40:58,280 --> 00:41:01,960 And as we discussed in the context of molasses-- 730 00:41:01,960 --> 00:41:05,400 of red detuned molasses, but it applies to any cooling scheme-- 731 00:41:05,400 --> 00:41:07,440 the finite temperature is when you 732 00:41:07,440 --> 00:41:10,460 have detail balance between heating and cooling 733 00:41:10,460 --> 00:41:13,530 And therefore its ratio of the momentum diffusion 734 00:41:13,530 --> 00:41:17,450 coefficient over the friction coefficient. 735 00:41:17,450 --> 00:41:17,979 Kensie? 736 00:41:17,979 --> 00:41:19,479 AUDIENCE: In the previous step, what 737 00:41:19,479 --> 00:41:25,692 was the significance of choosing f of 0 basically? 738 00:41:25,692 --> 00:41:31,159 Like, the time arguments were 0 and D. So we said that f of 0 739 00:41:31,159 --> 00:41:33,650 was DPDT? 740 00:41:33,650 --> 00:41:39,810 PROFESSOR: Oh, we assumed that the laser beam's Hamiltonian is 741 00:41:39,810 --> 00:41:42,140 invariant against time translation, 742 00:41:42,140 --> 00:41:45,800 and we simply evaluate that at T equals 0. 743 00:41:45,800 --> 00:41:50,520 I mean, that's of the argument you have fluctuating forces. 744 00:41:50,520 --> 00:41:52,680 And you could have fluctuating forces 745 00:41:52,680 --> 00:41:55,720 at time T and time T prime. 746 00:41:55,720 --> 00:42:00,600 But then you use the augment that in a steady state solution 747 00:42:00,600 --> 00:42:02,590 the correlation function does not 748 00:42:02,590 --> 00:42:05,910 depend on T and T prime separately, but only 749 00:42:05,910 --> 00:42:07,180 on the difference. 750 00:42:07,180 --> 00:42:09,644 And then you simply set one argument to 0, 751 00:42:09,644 --> 00:42:11,685 and the other argument becomes a time difference. 752 00:42:15,154 --> 00:42:15,820 Other questions? 753 00:42:26,310 --> 00:42:29,780 So now, by understanding that momentum diffusion is 754 00:42:29,780 --> 00:42:33,720 nothing else than the integral over the force fluctuation, 755 00:42:33,720 --> 00:42:37,300 we can now use this physical picture which I gave you 756 00:42:37,300 --> 00:42:45,886 earlier that an atom which cascades 757 00:42:45,886 --> 00:42:49,610 through upper, lower, upper, lower dressed states 758 00:42:49,610 --> 00:42:54,000 will experience opposite but equal forces, form the right 759 00:42:54,000 --> 00:42:56,220 and from the left. 760 00:42:56,220 --> 00:43:02,310 So I can simply evaluate that by assuming we are on resonance. 761 00:43:02,310 --> 00:43:09,020 On resonance the force is h bar over 2 times the derivative 762 00:43:09,020 --> 00:43:11,520 of the generalized Rabi frequency. 763 00:43:11,520 --> 00:43:14,980 And the force correlation, what is it? 764 00:43:14,980 --> 00:43:19,480 Well, it's the force f of 0 times f of T. 765 00:43:19,480 --> 00:43:22,290 But as long as the atom is in one dressed state 766 00:43:22,290 --> 00:43:23,940 the force doesn't change. 767 00:43:23,940 --> 00:43:37,812 So I simply take here the force, I take the force squared, 768 00:43:37,812 --> 00:43:44,610 and then when I integrate over that, well the moment the atom 769 00:43:44,610 --> 00:43:47,040 makes a transition there is no longer 770 00:43:47,040 --> 00:43:48,670 any correlation between the force. 771 00:43:48,670 --> 00:43:53,630 Everything becomes random and the integral averages out to 0. 772 00:43:53,630 --> 00:43:55,180 So in other words, that's the way 773 00:43:55,180 --> 00:43:56,770 how you should look at this integral. 774 00:43:56,770 --> 00:43:59,590 You start with the force f0, as long 775 00:43:59,590 --> 00:44:01,520 as the force is correlated. 776 00:44:01,520 --> 00:44:04,670 Because the atom hasn't done spontaneous emission, 777 00:44:04,670 --> 00:44:06,920 you have sort of f of 0, f of 0. 778 00:44:06,920 --> 00:44:10,200 And you integrate it, nominally to infinity. 779 00:44:10,200 --> 00:44:13,980 But you simply integrate it until some spontaneous emission 780 00:44:13,980 --> 00:44:16,000 randomizes the system. 781 00:44:16,000 --> 00:44:22,330 So therefore, this integral over the correlation function 782 00:44:22,330 --> 00:44:26,740 is nothing else then the force squared times the typical time 783 00:44:26,740 --> 00:44:29,850 scale for a transition. 784 00:44:29,850 --> 00:44:33,630 So if you put that in on resonance, 785 00:44:33,630 --> 00:44:36,470 we have the force here, we square the force. 786 00:44:36,470 --> 00:44:40,860 And then we multiply with the spontaneous emission rate, 787 00:44:40,860 --> 00:44:45,620 which for resonance system is gamma over 2. 788 00:44:45,620 --> 00:44:49,830 And with that we get now the heating coefficient, 789 00:44:49,830 --> 00:44:54,910 the momentum diffusion coefficient 790 00:44:54,910 --> 00:44:58,280 for an atom which is exposed to resonant light 791 00:44:58,280 --> 00:44:59,800 due to the stimulated light force. 792 00:45:03,040 --> 00:45:05,430 The paper, the references I've given you, 793 00:45:05,430 --> 00:45:09,320 is simply taking this augment and extending that 794 00:45:09,320 --> 00:45:10,890 to arbitrary detuning. 795 00:45:10,890 --> 00:45:14,480 For arbitrary detuning, I told you what happens on resonance. 796 00:45:14,480 --> 00:45:20,400 The atom spends equal amount in each dressed state. 797 00:45:20,400 --> 00:45:24,154 If you detune the atom spends more time in one dressed state, 798 00:45:24,154 --> 00:45:25,820 then quickly in the other dressed state, 799 00:45:25,820 --> 00:45:28,810 and then more time in the other dressed state again. 800 00:45:28,810 --> 00:45:32,730 So you can, by simply understanding 801 00:45:32,730 --> 00:45:36,550 what are the rates for cascading between the dressed levels, 802 00:45:36,550 --> 00:45:40,320 you could pretty much write down the expression, find it, 803 00:45:40,320 --> 00:45:43,570 and here is the solution taken out of the paper. 804 00:45:43,570 --> 00:45:45,670 So we have to correct prefactor, but then 805 00:45:45,670 --> 00:45:48,750 if the detuning is non-vanishing we 806 00:45:48,750 --> 00:45:50,660 get a contribution from the second factor. 807 00:45:50,660 --> 00:45:51,160 Colin? 808 00:45:51,160 --> 00:45:55,624 AUDIENCE: Are you assuming the exponential correlation-- 809 00:45:55,624 --> 00:45:58,104 exponential behavior to the correlation. 810 00:45:58,104 --> 00:46:01,080 Because then you might have different prefactors, 811 00:46:01,080 --> 00:46:03,808 if it were something like if you were in a [INAUDIBLE]. 812 00:46:08,024 --> 00:46:16,450 PROFESSOR: Well what happens is the following. 813 00:46:16,450 --> 00:46:20,680 I think if you're in one dressed state 814 00:46:20,680 --> 00:46:22,910 and the force pushes you in one direction 815 00:46:22,910 --> 00:46:28,710 and you integrate over it you have an exponential decay. 816 00:46:28,710 --> 00:46:33,740 But instead of integrating over the exponential decay 817 00:46:33,740 --> 00:46:36,000 you can just take the force squared 818 00:46:36,000 --> 00:46:39,665 and multiply it with the average time the atom stays. 819 00:46:44,530 --> 00:46:47,700 In other words, if I have an exponentially decaying function 820 00:46:47,700 --> 00:46:50,900 I can always approximate it by the function of T equals 821 00:46:50,900 --> 00:46:53,340 0 times the time. 822 00:46:53,340 --> 00:46:54,920 And what you are asking me now is 823 00:46:54,920 --> 00:46:58,100 is the time the one over E time, or is it an average time? 824 00:46:58,100 --> 00:46:59,962 What is the correct time to use here? 825 00:46:59,962 --> 00:47:01,837 AUDIENCE: I guess when the time scale depends 826 00:47:01,837 --> 00:47:06,228 on how the correlations decay for [INAUDIBLE] different. 827 00:47:09,610 --> 00:47:13,610 PROFESSOR: Yes, OK, I agree. 828 00:47:13,610 --> 00:47:16,540 The shape of the decay of the correlation function 829 00:47:16,540 --> 00:47:17,460 would matter. 830 00:47:17,460 --> 00:47:19,240 I'm not going into that. 831 00:47:19,240 --> 00:47:21,590 But what I'm sort of assuming here 832 00:47:21,590 --> 00:47:25,470 is that the correlation function indicated with time 833 00:47:25,470 --> 00:47:29,130 is the correlation function of T equals 0 times the correlation 834 00:47:29,130 --> 00:47:30,290 time. 835 00:47:30,290 --> 00:47:32,940 And how the correlation time is related 836 00:47:32,940 --> 00:47:34,890 to the exponential time, or if there's 837 00:47:34,890 --> 00:47:39,090 a small prefactor and such, that may depend on details. 838 00:47:39,090 --> 00:47:40,305 Yes, you're right. 839 00:47:40,305 --> 00:47:41,646 I've swept this under the rug. 840 00:47:53,502 --> 00:47:54,210 How are we doing? 841 00:47:54,210 --> 00:47:54,710 Good. 842 00:47:57,970 --> 00:47:59,640 Let me just summarize. 843 00:47:59,640 --> 00:48:02,630 The dressed atom picture has given us 844 00:48:02,630 --> 00:48:06,200 two pieces of major insight into the stimulated force. 845 00:48:06,200 --> 00:48:12,040 The first one is that the atom experience is always 846 00:48:12,040 --> 00:48:15,880 opposite forces in the two dressed levels. 847 00:48:15,880 --> 00:48:17,940 And it is the population imbalance 848 00:48:17,940 --> 00:48:22,100 between the two dressed levels which results in a net force. 849 00:48:22,100 --> 00:48:24,210 And this requires either red detuning, 850 00:48:24,210 --> 00:48:25,890 then the force is repulsive, or blue 851 00:48:25,890 --> 00:48:27,710 detuning-- sorry, red detuning then 852 00:48:27,710 --> 00:48:29,834 the force is attractive, or blue detuning then it's 853 00:48:29,834 --> 00:48:31,570 a force repulsive. 854 00:48:31,570 --> 00:48:34,650 Second, with this time lag thing, 855 00:48:34,650 --> 00:48:38,210 we understand that time lag, non-adiabaticity, 856 00:48:38,210 --> 00:48:40,380 is the way to get cooling. 857 00:48:40,380 --> 00:48:43,335 And the third thing we have learned, when the atom does 858 00:48:43,335 --> 00:48:47,790 a radiative cascade the force-- when it goes from the upper 859 00:48:47,790 --> 00:48:49,930 to the lower dressed level-- the force 860 00:48:49,930 --> 00:48:52,260 is suddenly reversing it's sign. 861 00:48:52,260 --> 00:48:56,070 And this causes fluctuations, this shakes up atom, 862 00:48:56,070 --> 00:48:57,990 and this causes heating. 863 00:48:57,990 --> 00:49:02,730 And this has nothing to do with the other heating mechanism we 864 00:49:02,730 --> 00:49:05,300 discussed earlier, namely every time there 865 00:49:05,300 --> 00:49:07,590 is spontaneous emission there's a random recoil. 866 00:49:11,080 --> 00:49:14,740 In strong laser fields the fluctuations 867 00:49:14,740 --> 00:49:16,209 of the stimulated force, which has 868 00:49:16,209 --> 00:49:17,875 nothing to do with spontaneous emission, 869 00:49:17,875 --> 00:49:20,590 are much, much larger than the photon recoil. 870 00:49:20,590 --> 00:49:22,630 So therefore for strong laser fields, 871 00:49:22,630 --> 00:49:26,450 it is this alternation of the force when 872 00:49:26,450 --> 00:49:28,730 atoms switch dressed energy levels, which 873 00:49:28,730 --> 00:49:30,460 is responsible for most of the heating. 874 00:49:46,460 --> 00:49:49,270 I want to come now to another physical situation, which 875 00:49:49,270 --> 00:49:50,580 is actually wonderful. 876 00:49:50,580 --> 00:49:52,860 You learn about Sisyphus cooling. 877 00:49:52,860 --> 00:49:55,090 Everybody in cold atoms speaks about Sisyphus 878 00:49:55,090 --> 00:50:00,070 cooling because it's a very, very elegant cooling scheme. 879 00:50:00,070 --> 00:50:04,040 We usually apply in the laboratory Sisyphus cooling 880 00:50:04,040 --> 00:50:06,090 when we do polarization gradient cooling. 881 00:50:06,090 --> 00:50:08,050 I will say you a few things later. 882 00:50:08,050 --> 00:50:11,250 But this is much, much harder to understand than the Sisyphus 883 00:50:11,250 --> 00:50:16,250 cooling, which takes place for a two level atom moving 884 00:50:16,250 --> 00:50:19,790 through a blue detuned standing wave. 885 00:50:19,790 --> 00:50:21,850 So in other words, I try to explain now 886 00:50:21,850 --> 00:50:24,420 to you what Sisyphus cooling is. 887 00:50:24,420 --> 00:50:29,260 The relevant application is multi-level polarization 888 00:50:29,260 --> 00:50:31,600 gradient cooling, which I give you an idea, 889 00:50:31,600 --> 00:50:33,480 but it's hard to fully describe. 890 00:50:33,480 --> 00:50:37,790 But here, in our current discussion on the two level 891 00:50:37,790 --> 00:50:42,590 atom interacting with one laser beam, or one standing wave, 892 00:50:42,590 --> 00:50:45,255 we find the simplest physical realization 893 00:50:45,255 --> 00:50:47,320 of Sisyphus cooling. 894 00:50:47,320 --> 00:50:50,020 So just to make sure that you are following the argument, 895 00:50:50,020 --> 00:50:54,200 I have so far explained to you why blue detuned light cools 896 00:50:54,200 --> 00:50:57,580 in the limit of very small velocities. 897 00:50:57,580 --> 00:51:01,200 Very small velocity means we did a first order Taylor 898 00:51:01,200 --> 00:51:04,600 expansion in this lag time. 899 00:51:04,600 --> 00:51:08,820 But now I want to discuss with you a different regime, namely 900 00:51:08,820 --> 00:51:11,730 that the atom moves with a velocity such 901 00:51:11,730 --> 00:51:18,040 that it can go up and down several periods of the standing 902 00:51:18,040 --> 00:51:22,060 wave before it does spontaneous emission. 903 00:51:22,060 --> 00:51:25,130 So before when we discussed the friction coefficient, 904 00:51:25,130 --> 00:51:26,950 the velocity was infinitesimal. 905 00:51:26,950 --> 00:51:29,620 But now I assume that the velocity is such 906 00:51:29,620 --> 00:51:31,980 that the atom can surf a few waves 907 00:51:31,980 --> 00:51:34,590 and then it will spontaneously emit. 908 00:51:34,590 --> 00:51:37,820 And let me assume that we are blue detuned. 909 00:51:37,820 --> 00:51:43,970 So the atom, let's say here, where the splitting is minimum, 910 00:51:43,970 --> 00:51:45,940 well this is where we don't have light. 911 00:51:45,940 --> 00:51:48,960 The generalized Rabi frequency is just the detuning. 912 00:51:48,960 --> 00:51:51,560 So we are in a node of the standing wave. 913 00:51:51,560 --> 00:51:54,220 And then the atom starts in the ground state, 914 00:51:54,220 --> 00:51:57,660 and it experiences the periodic potential 915 00:51:57,660 --> 00:52:00,700 due to the standing wave. 916 00:52:00,700 --> 00:52:05,720 But we want to now ask, OK, when will spontaneous emission 917 00:52:05,720 --> 00:52:07,800 happen in this picture? 918 00:52:07,800 --> 00:52:11,040 Well, it cannot happen here, when the atom is in the ground 919 00:52:11,040 --> 00:52:12,670 state, when there is not light. 920 00:52:12,670 --> 00:52:16,910 But where there is light, when the atoms move to an anti-node, 921 00:52:16,910 --> 00:52:21,560 there is a mixture, indicated in red, of the excited state. 922 00:52:21,560 --> 00:52:24,950 So in other words, when an atom does this roller coaster, 923 00:52:24,950 --> 00:52:29,800 it will most likely emit a photon-- either the carrier 924 00:52:29,800 --> 00:52:32,190 or one of the sidebands-- when it 925 00:52:32,190 --> 00:52:37,290 is at the maximum of the potential corresponding 926 00:52:37,290 --> 00:52:42,740 to an anti-node where it experiences the light. 927 00:52:42,740 --> 00:52:46,180 OK, when the atom emits on the carrier 928 00:52:46,180 --> 00:52:48,790 it goes from the upper dressed level 929 00:52:48,790 --> 00:52:50,480 to the upper dressed level. 930 00:52:50,480 --> 00:52:51,920 Then sort of nothing has happened 931 00:52:51,920 --> 00:52:54,290 because the atom is in the same dressed level. 932 00:52:54,290 --> 00:52:57,650 Let me now discuss the case when something happens. 933 00:52:57,650 --> 00:53:02,130 And this is when the atom decays from the upper dressed level 934 00:53:02,130 --> 00:53:04,830 to the lower dressed level. 935 00:53:04,830 --> 00:53:08,580 And at least for weak excitation the lower dressed level 936 00:53:08,580 --> 00:53:11,410 corresponds to the excited state. 937 00:53:11,410 --> 00:53:14,770 In other words, when there is no light, 938 00:53:14,770 --> 00:53:17,190 the upper dressed level was ground state, 939 00:53:17,190 --> 00:53:21,160 the lower dressed level is excited state. 940 00:53:21,160 --> 00:53:23,870 But what happens now is actually interesting, 941 00:53:23,870 --> 00:53:26,710 the atom had to climb up the hill. 942 00:53:26,710 --> 00:53:29,720 But the spontaneous emission to the lower dressed level 943 00:53:29,720 --> 00:53:32,110 takes the atom to the bottom. 944 00:53:32,110 --> 00:53:35,600 So you can see, on average, it has climbed a hill here, 945 00:53:35,600 --> 00:53:37,690 but now it is at the bottom. 946 00:53:37,690 --> 00:53:40,810 And now we're going to ask how does the atom get out 947 00:53:40,810 --> 00:53:42,850 of the lower dressed level? 948 00:53:42,850 --> 00:53:45,490 Well, the lower dressed level is mainly 949 00:53:45,490 --> 00:53:50,830 excited-- is 100% excited if there is no light. 950 00:53:50,830 --> 00:53:54,460 But if there is light, and light means the generalized Rabi 951 00:53:54,460 --> 00:53:57,920 frequency is larger, the excited state has now in it 952 00:53:57,920 --> 00:54:00,340 mixture of the ground state because the light 953 00:54:00,340 --> 00:54:03,580 mixes ground and excited state. 954 00:54:03,580 --> 00:54:06,475 So now, of course, whenever there is red 955 00:54:06,475 --> 00:54:08,460 you can spontaneously emit. 956 00:54:08,460 --> 00:54:11,960 But there is a probability that the atom 957 00:54:11,960 --> 00:54:15,770 is more likely to emit when you are in a purely excited state, 958 00:54:15,770 --> 00:54:17,910 and not in a mixed state. 959 00:54:17,910 --> 00:54:20,930 So now, talking about the lower manifold, 960 00:54:20,930 --> 00:54:22,860 the atom does it's roller coaster. 961 00:54:22,860 --> 00:54:24,960 It has a velocity that it can cover 962 00:54:24,960 --> 00:54:29,280 several periods of the standing wave in one lifetime. 963 00:54:29,280 --> 00:54:32,150 But there is a higher probability for the atom 964 00:54:32,150 --> 00:54:36,060 now to emit when it is on top of the hill. 965 00:54:36,060 --> 00:54:37,610 So that's remarkable. 966 00:54:37,610 --> 00:54:39,710 We have the upper dressed level. 967 00:54:39,710 --> 00:54:41,450 We have the lower dressed level. 968 00:54:41,450 --> 00:54:44,100 The atom is doing the roller coaster. 969 00:54:44,100 --> 00:54:48,950 But whenever it emits a sideband, whenever it switches 970 00:54:48,950 --> 00:54:52,300 state, it mainly does it by emitting 971 00:54:52,300 --> 00:54:54,450 at the top of the hill. 972 00:54:54,450 --> 00:54:56,300 And after the spontaneous emission 973 00:54:56,300 --> 00:54:59,000 it finds itself at the bottom of the hill. 974 00:54:59,000 --> 00:55:02,950 So on average the atom is climbing more up hills 975 00:55:02,950 --> 00:55:05,510 than down hills. 976 00:55:05,510 --> 00:55:07,240 And this is called Sisyphus cooling. 977 00:55:07,240 --> 00:55:10,070 I think you've all heard about the Greek myth of King 978 00:55:10,070 --> 00:55:13,670 Sisyphus, who was condemned. 979 00:55:13,670 --> 00:55:15,770 He challenged the gods and he got his punishment. 980 00:55:15,770 --> 00:55:17,920 And his punishment was that he always 981 00:55:17,920 --> 00:55:20,210 has to roll a stone up the hill. 982 00:55:20,210 --> 00:55:22,600 And when he's done the stone falls down 983 00:55:22,600 --> 00:55:24,740 and he has to roll the stone up. 984 00:55:24,740 --> 00:55:27,710 So in other words, we have condemned the atoms 985 00:55:27,710 --> 00:55:28,880 to the same verdict. 986 00:55:28,880 --> 00:55:31,480 They always have to go up hill to work. 987 00:55:31,480 --> 00:55:33,810 Then they fall down hill with the help 988 00:55:33,810 --> 00:55:35,480 of a spontaneous photon. 989 00:55:35,480 --> 00:55:38,210 But then they have to climb up hill. 990 00:55:38,210 --> 00:55:43,090 And it is the uphill climb where the experience and net friction 991 00:55:43,090 --> 00:55:47,720 force, which cools the atoms down and reduces 992 00:55:47,720 --> 00:55:48,650 their kinetic energy. 993 00:55:56,090 --> 00:55:58,980 So this is Sisyphus cooling. 994 00:55:58,980 --> 00:56:03,640 And since I need it in a few minutes, 995 00:56:03,640 --> 00:56:07,950 I just wrote down for you, based on this simple and wonderfully 996 00:56:07,950 --> 00:56:10,515 elegant picture, how much energy does 997 00:56:10,515 --> 00:56:12,230 the atom lose per unit time? 998 00:56:12,230 --> 00:56:14,160 What is the cooling rate? 999 00:56:14,160 --> 00:56:17,200 Well it's clear that the cooling rate 1000 00:56:17,200 --> 00:56:20,120 comes when the atom switches from one dressed energy 1001 00:56:20,120 --> 00:56:23,650 level to the next, because nothing happens on the carrier. 1002 00:56:23,650 --> 00:56:26,900 So it is the transition state between dressed level one 1003 00:56:26,900 --> 00:56:28,920 and dressed level two. 1004 00:56:28,920 --> 00:56:31,620 And then we multiply with mu naught 1005 00:56:31,620 --> 00:56:34,710 the depths of the optical lattice, because it 1006 00:56:34,710 --> 00:56:39,440 is this energy difference which is extracted from the atom. 1007 00:56:42,000 --> 00:56:46,440 You may ask, OK what is time limiting? 1008 00:56:46,440 --> 00:56:50,550 It's really the transition rate form the first dressed level, 1009 00:56:50,550 --> 00:56:55,510 which is mainly ground state, to the other dressed level, which 1010 00:56:55,510 --> 00:56:57,550 is mainly exciting state. 1011 00:56:57,550 --> 00:57:00,570 Because once the atom is in this level, which is mainly 1012 00:57:00,570 --> 00:57:03,800 the excited state, it will pretty much 1013 00:57:03,800 --> 00:57:06,900 do the next transition at a rate which 1014 00:57:06,900 --> 00:57:09,500 corresponds to the spontaneous emission rate. 1015 00:57:09,500 --> 00:57:14,810 So in other words, if you're not fully on resonance, 1016 00:57:14,810 --> 00:57:17,070 or if you're not situated in laser power, 1017 00:57:17,070 --> 00:57:20,860 the rate, one, two, is slower than the rate two, one. 1018 00:57:20,860 --> 00:57:23,311 And therefore this rate is the rate limiting step. 1019 00:57:26,610 --> 00:57:32,690 Let me show you just a few pictures. 1020 00:57:32,690 --> 00:57:37,230 What I'm just explaining to you was explored very early on, 1021 00:57:37,230 --> 00:57:40,450 just have to sort of put you back that it was in the mid 1022 00:57:40,450 --> 00:57:44,640 '80s, '82, '83, '84, that people developed [INAUDIBLE] slowing. 1023 00:57:44,640 --> 00:57:48,880 It was in '85 that optical molasses with red detuned light 1024 00:57:48,880 --> 00:57:51,230 was demonstrated for the first time. 1025 00:57:51,230 --> 00:57:54,500 It was in '86 that the dipole trap 1026 00:57:54,500 --> 00:57:56,290 was demonstrated by Steve Chu. 1027 00:57:56,290 --> 00:57:58,840 And in '87 that Steve Chu and Dave Pritchard 1028 00:57:58,840 --> 00:58:00,790 introduced the magneto optic trap. 1029 00:58:00,790 --> 00:58:04,580 So the mid to late '80s were fantastic times, 1030 00:58:04,580 --> 00:58:09,070 where one important technique was realized and demonstrated. 1031 00:58:09,070 --> 00:58:13,000 And this was actually the first major experimental paper 1032 00:58:13,000 --> 00:58:15,920 of a new laser cooling group, which 1033 00:58:15,920 --> 00:58:18,460 was founded by Claude Cohen-Tonnoudji at the Ecole 1034 00:58:18,460 --> 00:58:20,170 Normale Superieure in Paris. 1035 00:58:20,170 --> 00:58:25,440 And it was exactly about cooling atoms with stimulated emission, 1036 00:58:25,440 --> 00:58:27,570 and this is blue molasses. 1037 00:58:27,570 --> 00:58:31,210 And, I mean, you exactly see here out of this paper, 1038 00:58:31,210 --> 00:58:32,685 the Sisyphus cooling mechanism. 1039 00:58:35,430 --> 00:58:40,860 And addressing the question about red detuned light here, 1040 00:58:40,860 --> 00:58:42,950 this was actually beam experiment. 1041 00:58:42,950 --> 00:58:45,430 You head a blue detuned standing wave, 1042 00:58:45,430 --> 00:58:48,260 and you were sending an atomic beam through. 1043 00:58:48,260 --> 00:58:51,350 And if this atomic beam was cooled, 1044 00:58:51,350 --> 00:58:54,550 the divergence of the atomic beam was reduced. 1045 00:58:54,550 --> 00:58:59,030 So this experiment had only one blue detuned standing wave 1046 00:58:59,030 --> 00:59:02,630 and nothing else, because it was not a trapping experiment. 1047 00:59:02,630 --> 00:59:05,770 It was a beam collimation experiment. 1048 00:59:05,770 --> 00:59:08,670 And what is shown here is that you 1049 00:59:08,670 --> 00:59:12,230 have the original atomic beam profile. 1050 00:59:12,230 --> 00:59:17,920 And for blue detuned light you make it narrower, 1051 00:59:17,920 --> 00:59:19,390 so this is cooling. 1052 00:59:19,390 --> 00:59:22,210 Whereas for red detuned light you have a defocusing, 1053 00:59:22,210 --> 00:59:23,500 which corresponds to heating. 1054 00:59:28,770 --> 00:59:29,375 Any question? 1055 00:59:45,340 --> 00:59:50,150 I want to wrap up the discussion of the stimulated light force 1056 00:59:50,150 --> 00:59:58,300 by discussing pretty much everything again, 1057 00:59:58,300 --> 01:00:01,430 but in the simplest possibly limit. 1058 01:00:01,430 --> 01:00:03,160 That's actually something I haven't 1059 01:00:03,160 --> 01:00:04,770 found in any text books. 1060 01:00:04,770 --> 01:00:08,450 But when I derived it for myself it 1061 01:00:08,450 --> 01:00:10,450 helped me to really understand it. 1062 01:00:10,450 --> 01:00:13,960 It's sort of the perturbative result of the dressed atom 1063 01:00:13,960 --> 01:00:14,890 picture. 1064 01:00:14,890 --> 01:00:16,640 So everything is simple. 1065 01:00:16,640 --> 01:00:18,556 And by telling you, look at this. 1066 01:00:18,556 --> 01:00:20,601 You know this is just EC stock shift. 1067 01:00:20,601 --> 01:00:22,100 But in the dressed atom picture it's 1068 01:00:22,100 --> 01:00:24,480 a generalized Rabi frequency, you suddenly 1069 01:00:24,480 --> 01:00:27,600 understand the trivial perturbative result 1070 01:00:27,600 --> 01:00:30,770 and how it translates into the dressed atom picture. 1071 01:00:30,770 --> 01:00:34,230 So what I'm presenting you now in the next 10 minutes, 1072 01:00:34,230 --> 01:00:38,180 it provides a lot of insight because it 1073 01:00:38,180 --> 01:00:41,490 connects simple pictures. 1074 01:00:41,490 --> 01:00:44,280 So I'm discussing things here in the limit 1075 01:00:44,280 --> 01:00:45,670 that the detuning is large. 1076 01:00:45,670 --> 01:00:48,870 The detuning is larger than the Rabi frequency, 1077 01:00:48,870 --> 01:00:51,020 and spontaneous emission rate is the smallest 1078 01:00:51,020 --> 01:00:52,300 of the three rates. 1079 01:00:52,300 --> 01:00:56,670 In that case, I can simply do perturbation theory, 1080 01:00:56,670 --> 01:00:58,680 and I want to show you. 1081 01:00:58,680 --> 01:01:00,940 Also, because this is pedagogical, 1082 01:01:00,940 --> 01:01:03,950 I just want to show you how the effects come together. 1083 01:01:03,950 --> 01:01:07,230 I neglect all factors on the order of unity. 1084 01:01:07,230 --> 01:01:10,920 And I set h bar to 1. 1085 01:01:10,920 --> 01:01:13,580 So let's assume we have a standing wave. 1086 01:01:13,580 --> 01:01:15,950 A standing wave is cosine KX. 1087 01:01:21,090 --> 01:01:23,000 So we have an electric field. 1088 01:01:23,000 --> 01:01:25,660 But we describe the electric field with a Rabi frequency. 1089 01:01:25,660 --> 01:01:29,200 So the Rabi frequency forms a standing wave. 1090 01:01:29,200 --> 01:01:31,430 Now what are the two dressed states? 1091 01:01:31,430 --> 01:01:35,900 The dressed states 1 and 2 are the bare states, 1092 01:01:35,900 --> 01:01:37,670 ground and excited state. 1093 01:01:37,670 --> 01:01:40,090 And then there a perturbative mixture. 1094 01:01:40,090 --> 01:01:43,440 And in first order perturbation theory 1095 01:01:43,440 --> 01:01:46,720 we take the matrix element over the detuning. 1096 01:01:46,720 --> 01:01:48,520 So one dressed state is the ground 1097 01:01:48,520 --> 01:01:50,980 state with a little bit of excited stated mixed. 1098 01:01:50,980 --> 01:01:54,300 The other dressed state is the excited state, minus sign, 1099 01:01:54,300 --> 01:01:56,270 little bit of ground stated mix. 1100 01:01:56,270 --> 01:02:00,717 These are our two dressed states in the trivial limit 1101 01:02:00,717 --> 01:02:02,300 that we can apply perturbation theory. 1102 01:02:08,120 --> 01:02:10,840 What are now the transition rates 1103 01:02:10,840 --> 01:02:13,850 between the two dressed states? 1104 01:02:13,850 --> 01:02:16,490 Well, we want to make spontaneous emission 1105 01:02:16,490 --> 01:02:19,480 from dressed state 1 to dressed state 2. 1106 01:02:19,480 --> 01:02:23,880 We have to go from this small mixture, which is the excited 1107 01:02:23,880 --> 01:02:27,480 stated mixture, to the ground state here. 1108 01:02:27,480 --> 01:02:30,600 So therefore we get the product of the two amplitudes, 1109 01:02:30,600 --> 01:02:32,990 and then we multiply with gamma. 1110 01:02:32,990 --> 01:02:36,220 Of course, most of-- yeah. 1111 01:02:36,220 --> 01:02:39,020 So this is the rate to go from dressed state 1112 01:02:39,020 --> 01:02:41,210 1 to dressed state 2. 1113 01:02:41,210 --> 01:02:43,720 What is the inverse rate to go back from dressed state 1114 01:02:43,720 --> 01:02:45,640 2 to dressed state 1? 1115 01:02:45,640 --> 01:02:49,200 Well, you go from the excited state to the ground state, 1116 01:02:49,200 --> 01:02:53,130 multiply with gamma, and that's what you get. 1117 01:02:53,130 --> 01:02:58,400 Sure we have, in perturbation theory, the coefficient is 1. 1118 01:02:58,400 --> 01:03:00,620 It's only in higher order perturbation theory 1119 01:03:00,620 --> 01:03:05,490 that the coefficient of the bare state becomes less than 1. 1120 01:03:05,490 --> 01:03:06,650 So isn't that simple? 1121 01:03:06,650 --> 01:03:09,200 We just discussed it with all the cosine to the 4, 1122 01:03:09,200 --> 01:03:10,050 sine to the 4. 1123 01:03:10,050 --> 01:03:12,040 But now in perturbation theory this 1124 01:03:12,040 --> 01:03:16,380 is the rate for going from dressed state 1 to 2. 1125 01:03:16,380 --> 01:03:19,910 And this is the rate to go the other way around. 1126 01:03:19,910 --> 01:03:24,650 So what is our dressed state potential? 1127 01:03:24,650 --> 01:03:26,900 Well, the dressed state potential 1128 01:03:26,900 --> 01:03:30,300 is simply the EC stock shift, which 1129 01:03:30,300 --> 01:03:33,770 is opposite for the ground and for the excited state. 1130 01:03:33,770 --> 01:03:36,920 And the EC stock shift in perturbation theory 1131 01:03:36,920 --> 01:03:40,830 is nothing else than matrix elements squared over detuning. 1132 01:03:40,830 --> 01:03:42,840 So this is mu naught. 1133 01:03:42,840 --> 01:03:46,740 This is the potential of the standing wave experienced 1134 01:03:46,740 --> 01:03:47,240 by the atom. 1135 01:03:52,730 --> 01:03:54,780 Now we want to do Sisyphus cooling. 1136 01:03:54,780 --> 01:03:58,330 What is the cooling rate in Sisyphus cooling? 1137 01:03:58,330 --> 01:04:00,790 Well, remember, Sisyphus cooling we 1138 01:04:00,790 --> 01:04:02,840 assume that the atom is fast enough 1139 01:04:02,840 --> 01:04:07,250 to go over several hills and valleys. 1140 01:04:07,250 --> 01:04:13,610 The cooling rate is determined by the rate at which the atoms 1141 01:04:13,610 --> 01:04:16,650 switch dressed levels times mu naught. 1142 01:04:16,650 --> 01:04:20,400 And I can scroll back, but mu naught was the EC stock effect, 1143 01:04:20,400 --> 01:04:24,350 and gamma one, two was this expression we had before. 1144 01:04:24,350 --> 01:04:26,136 So therefore, in the simple picture, 1145 01:04:26,136 --> 01:04:27,510 while everything is perturbative, 1146 01:04:27,510 --> 01:04:32,220 we have a nice expression for the cooling rate. 1147 01:04:32,220 --> 01:04:36,770 Well, now let's get something non-trivial out of it. 1148 01:04:36,770 --> 01:04:40,070 What is the friction coefficient? 1149 01:04:40,070 --> 01:04:42,690 Well, in many situations I've plotted 1150 01:04:42,690 --> 01:04:46,710 for you, the force versus velocity. 1151 01:04:46,710 --> 01:04:51,200 Now I want to plot for you the cooling rate versus velocity. 1152 01:04:51,200 --> 01:04:54,030 That means its force times velocity. 1153 01:04:54,030 --> 01:04:57,670 And as we just discussed, when the velocity is sufficiently 1154 01:04:57,670 --> 01:05:01,390 fast that the atom can go over several hills and valleys 1155 01:05:01,390 --> 01:05:04,030 in a spontaneous lifetime, the cooling rate 1156 01:05:04,030 --> 01:05:07,590 will saturate because it is limited 1157 01:05:07,590 --> 01:05:10,500 by the rate at which the atom switches 1158 01:05:10,500 --> 01:05:11,530 between dressed levels. 1159 01:05:15,810 --> 01:05:17,630 But now the question is, if we want 1160 01:05:17,630 --> 01:05:20,180 to understand what happens at low velocities, 1161 01:05:20,180 --> 01:05:23,560 how should we connect the two? 1162 01:05:23,560 --> 01:05:27,010 Well, you can say we know, for analytical reasons, 1163 01:05:27,010 --> 01:05:30,850 that the force versus velocity has to be linear. 1164 01:05:30,850 --> 01:05:34,030 So the cooling rate, which is another power of velocity, 1165 01:05:34,030 --> 01:05:35,610 should be quadratic. 1166 01:05:35,610 --> 01:05:39,160 And since I neglect all factors on the order of unity, 1167 01:05:39,160 --> 01:05:43,330 this is now the force is friction coefficient times V. 1168 01:05:43,330 --> 01:05:46,060 So therefore, what I do in red here 1169 01:05:46,060 --> 01:05:48,080 is a parabolic approximation, which 1170 01:05:48,080 --> 01:05:51,880 is alpha the cooling rate times V squared. 1171 01:05:51,880 --> 01:05:57,350 And now, if I want to know what is alpha, 1172 01:05:57,350 --> 01:06:00,310 I just sort of connect this parabola 1173 01:06:00,310 --> 01:06:03,740 with the saturated value, which I know. 1174 01:06:03,740 --> 01:06:07,300 And I know the transition happens at the velocity when 1175 01:06:07,300 --> 01:06:09,900 the atom moves further than one wavelength in one 1176 01:06:09,900 --> 01:06:12,020 spontaneous emission time. 1177 01:06:12,020 --> 01:06:14,620 So therefore, based on this very physical picture, 1178 01:06:14,620 --> 01:06:18,970 and on the inside how low velocities and high velocities 1179 01:06:18,970 --> 01:06:23,820 connect together, I can actually obtain the friction coefficient 1180 01:06:23,820 --> 01:06:28,210 in this blue detuned standing wave by taking the cooling rate 1181 01:06:28,210 --> 01:06:33,680 and dividing it by the velocity at which the atom moves 1182 01:06:33,680 --> 01:06:36,590 one wavelength per spontaneous emission time. 1183 01:06:40,860 --> 01:06:44,190 And if I do that, I obtain this result, 1184 01:06:44,190 --> 01:06:46,650 which I have already quoted earlier to you 1185 01:06:46,650 --> 01:06:48,950 from the paper of Cohen-Tonnoudji and Jean 1186 01:06:48,950 --> 01:06:49,980 Dalibard. 1187 01:06:49,980 --> 01:06:54,040 It looks fairly non-trivial with power to the 6, power to the 5. 1188 01:06:54,040 --> 01:06:57,362 But this is now the simple perturbative results 1189 01:06:57,362 --> 01:06:58,570 for the friction coefficient. 1190 01:07:02,800 --> 01:07:05,360 Now we have the friction coefficient. 1191 01:07:05,360 --> 01:07:06,580 Let's take it further. 1192 01:07:06,580 --> 01:07:09,420 I want to give you, in this perturbative limit, 1193 01:07:09,420 --> 01:07:12,510 I want to give you the momentum diffusion coefficient. 1194 01:07:12,510 --> 01:07:15,570 And then we will find in the perturbative limit what 1195 01:07:15,570 --> 01:07:18,480 is the lowest temperature to which we can cool. 1196 01:07:18,480 --> 01:07:20,050 So pretty much I'll do everything 1197 01:07:20,050 --> 01:07:23,550 as we've done before, but I use the simple perturbative limit. 1198 01:07:29,790 --> 01:07:34,450 So again, in order to calculate the diffusion coefficient-- 1199 01:07:34,450 --> 01:07:36,740 the momentum diffusion coefficient-- and the heating 1200 01:07:36,740 --> 01:07:40,490 we need the fluctuations of the force. 1201 01:07:40,490 --> 01:07:43,600 And now, just remember, we are in the perturbative limit. 1202 01:07:43,600 --> 01:07:47,510 The atom is mainly in the ground state, 1203 01:07:47,510 --> 01:07:51,630 which is the upper dressed level. 1204 01:07:51,630 --> 01:07:54,620 It has only a little excited stated mixture, 1205 01:07:54,620 --> 01:07:56,760 so it will mainly stay in this state. 1206 01:07:56,760 --> 01:08:00,350 But then with this excited stated mixture and this, 1207 01:08:00,350 --> 01:08:03,610 power of the Rabi frequency, it makes the transition 1208 01:08:03,610 --> 01:08:05,210 to the other dressed state. 1209 01:08:05,210 --> 01:08:09,030 But the other dressed state is almost 100% excited state. 1210 01:08:09,030 --> 01:08:11,040 So it will leave the other dressed state 1211 01:08:11,040 --> 01:08:12,990 almost immediately. 1212 01:08:12,990 --> 01:08:16,540 So this could be a Quantum Monte Carlo wave function result 1213 01:08:16,540 --> 01:08:19,290 that the atom experiences the force 1214 01:08:19,290 --> 01:08:21,460 in the upper dressed level. 1215 01:08:21,460 --> 01:08:24,689 Then it goes to the lower dressed level, which is mainly 1216 01:08:24,689 --> 01:08:28,060 the excited state, just 4 times 1 over gamma. 1217 01:08:28,060 --> 01:08:30,939 And then it returns. 1218 01:08:30,939 --> 01:08:35,340 So the picture now is that the atom for almost all the time 1219 01:08:35,340 --> 01:08:38,410 experiences the steady state force, which 1220 01:08:38,410 --> 01:08:40,130 is the blue dashed line. 1221 01:08:40,130 --> 01:08:43,029 And there are only, when it goes to the other dressed state, 1222 01:08:43,029 --> 01:08:48,090 these small spikes where the force has changed sign. 1223 01:08:48,090 --> 01:08:52,250 And if you now calculate the fluctuations of the force-- 1224 01:08:52,250 --> 01:08:58,750 the integral of f square-- once we calculate the square 1225 01:08:58,750 --> 01:09:02,170 the small deviations from the average force 1226 01:09:02,170 --> 01:09:03,439 does not contribute. 1227 01:09:03,439 --> 01:09:06,689 What contributes are the spikes. 1228 01:09:06,689 --> 01:09:15,399 So all I do is now I take the 4 square in one of the spikes. 1229 01:09:23,000 --> 01:09:26,220 And the force is nothing else than the derivat-- remember 1230 01:09:26,220 --> 01:09:28,580 we have a standing wave a mu naught. 1231 01:09:28,580 --> 01:09:31,390 The derivative of the standing wave is the force. 1232 01:09:31,390 --> 01:09:33,790 But if you have a sinusoid lattice 1233 01:09:33,790 --> 01:09:36,810 the force is K times the amplitude of the lattice. 1234 01:09:36,810 --> 01:09:41,779 I square it, so this is the spike squared. 1235 01:09:41,779 --> 01:09:43,960 Then I have to multiply with the time 1 1236 01:09:43,960 --> 01:09:49,740 over gamma, over which the force stays the same. 1237 01:09:49,740 --> 01:09:53,029 So this is, you can see, the correlation time of the force. 1238 01:09:53,029 --> 01:09:55,030 But then I have to multiply, also, 1239 01:09:55,030 --> 01:09:57,810 when I do an average over many, sort of, you can see, 1240 01:09:57,810 --> 01:10:02,440 trajectories, I have to average over the probability 1241 01:10:02,440 --> 01:10:08,760 that-- I have to multiply with the probability 1242 01:10:08,760 --> 01:10:10,490 that the atom is in the dressed state 1243 01:10:10,490 --> 01:10:13,390 1 and not in the dressed state 2. 1244 01:10:13,390 --> 01:10:15,190 It may take you a few minutes just 1245 01:10:15,190 --> 01:10:17,310 to think about the combinatorics. 1246 01:10:17,310 --> 01:10:20,790 But in the end all I do is I take the 4 square, 1247 01:10:20,790 --> 01:10:24,650 I multiply with correlation time when the atom is in state 2, 1248 01:10:24,650 --> 01:10:27,810 and get the probability that the atom is in state 2. 1249 01:10:27,810 --> 01:10:31,560 And when the atom is in state 1, in my ensemble, 1250 01:10:31,560 --> 01:10:33,970 it doesn't contribute at all, because in state 1 1251 01:10:33,970 --> 01:10:35,860 you're so close to the average force 1252 01:10:35,860 --> 01:10:39,280 that it doesn't contribute with the fluctuation. 1253 01:10:39,280 --> 01:10:41,780 Anyway, it's simple but subtle. 1254 01:10:41,780 --> 01:10:44,860 But it's a one-liner. 1255 01:10:44,860 --> 01:10:47,170 And now based on this perturbative 1256 01:10:47,170 --> 01:10:52,010 picture we have perturbatively exact an expression 1257 01:10:52,010 --> 01:10:55,040 for the heating rate. 1258 01:10:55,040 --> 01:10:58,000 But now it's interesting, the rate 1259 01:10:58,000 --> 01:11:04,580 gamma one, two, the rate to switch between dressed energy 1260 01:11:04,580 --> 01:11:11,230 levels always also appears in the expression for the friction 1261 01:11:11,230 --> 01:11:12,660 coefficient. 1262 01:11:12,660 --> 01:11:17,620 So therefore, if you're now asking 1263 01:11:17,620 --> 01:11:22,430 what is the ultimate temperature to which we can cool the atom? 1264 01:11:22,430 --> 01:11:26,030 It is the ratio of this analytic result for the heating 1265 01:11:26,030 --> 01:11:28,340 coefficient divided by the analytic result 1266 01:11:28,340 --> 01:11:29,760 for the friction coefficient. 1267 01:11:29,760 --> 01:11:33,450 Pretty much everything cancels out, and what remains 1268 01:11:33,450 --> 01:11:35,940 is mu naught. 1269 01:11:35,940 --> 01:11:38,300 And this is a remarkable result. 1270 01:11:38,300 --> 01:11:41,880 It's highly non-trivial, and I really enjoyed showing to you 1271 01:11:41,880 --> 01:11:45,320 how simply it can be derived by-- it's really 1272 01:11:45,320 --> 01:11:46,810 just perturbation theory. 1273 01:11:46,810 --> 01:11:50,730 But you have to put in the right concepts from the dressed atom 1274 01:11:50,730 --> 01:11:52,100 picture. 1275 01:11:52,100 --> 01:11:54,940 So what we learn is the following. 1276 01:11:54,940 --> 01:11:57,780 If you have a standing wave, and we 1277 01:11:57,780 --> 01:12:01,770 cool with a stimulated force, the lowest temperature 1278 01:12:01,770 --> 01:12:05,652 is-- with a prefactor, which I haven't calculated-- 1279 01:12:05,652 --> 01:12:08,330 the amplitude of the standing wave. 1280 01:12:08,330 --> 01:12:13,120 And that means that it is impossible to ever localize 1281 01:12:13,120 --> 01:12:15,820 atoms within a standing wave. 1282 01:12:15,820 --> 01:12:19,900 Laser cooling cannot be used to create atoms in an optical 1283 01:12:19,900 --> 01:12:20,790 lattice. 1284 01:12:20,790 --> 01:12:23,270 The temperature of the atoms is always 1285 01:12:23,270 --> 01:12:26,330 comparable to the depths of the lattice. 1286 01:12:26,330 --> 01:12:27,950 So therefore the atoms will never 1287 01:12:27,950 --> 01:12:30,691 be localized in one well of the lattice. 1288 01:12:33,460 --> 01:12:35,550 And this is generally valid. 1289 01:12:35,550 --> 01:12:37,650 The assumption we have made here is 1290 01:12:37,650 --> 01:12:40,900 that we have a two level atom. 1291 01:12:40,900 --> 01:12:43,360 A lot of you work with atoms to optical lattices. 1292 01:12:43,360 --> 01:12:46,040 And you often use evaporative cooling, take a Bose Einstein 1293 01:12:46,040 --> 01:12:49,110 condensate, which is pretty much a [INAUDIBLE] temperature put 1294 01:12:49,110 --> 01:12:50,490 it in optical lattice. 1295 01:12:50,490 --> 01:12:55,160 But before '95, people studied laser cooling and atoms 1296 01:12:55,160 --> 01:12:56,700 in optical lattices. 1297 01:12:56,700 --> 01:12:59,760 And this was only possible because atoms 1298 01:12:59,760 --> 01:13:02,220 can be cooled to lower temperatures 1299 01:13:02,220 --> 01:13:06,370 than we have derived here because they are not two level 1300 01:13:06,370 --> 01:13:07,660 atoms. 1301 01:13:07,660 --> 01:13:10,760 So laser cooling in a standing wave, 1302 01:13:10,760 --> 01:13:14,350 localization of atoms in a standing wave by laser cooling, 1303 01:13:14,350 --> 01:13:18,811 is only possibly by physics beyond what we have discussed. 1304 01:13:18,811 --> 01:13:19,310 Colin? 1305 01:13:19,310 --> 01:13:23,675 AUDIENCE: If you use something like Raman's sideband cooling, 1306 01:13:23,675 --> 01:13:26,342 you could-- like, yeah, you'd kind of need 1307 01:13:26,342 --> 01:13:29,980 three levels for that, but you eliminated the third. 1308 01:13:29,980 --> 01:13:32,890 Really, it's just that it's two levels, right? 1309 01:13:32,890 --> 01:13:36,430 PROFESSOR: If you do Raman's sideband cooling, 1310 01:13:36,430 --> 01:13:39,580 he posted-- I think, in the next 10 minutes, 1311 01:13:39,580 --> 01:13:41,440 I cover what you want to see. 1312 01:13:41,440 --> 01:13:44,360 Because now it involves two ground states. 1313 01:13:44,360 --> 01:13:46,630 And I want to show you, if you have two ground states, 1314 01:13:46,630 --> 01:13:51,120 to Hyper fine-- Raman's sideband cooling-- if you have 1315 01:13:51,120 --> 01:13:54,180 two different ground states there is no physics. 1316 01:13:54,180 --> 01:13:57,230 And I briefly want to touch upon that. 1317 01:13:57,230 --> 01:13:59,790 We'll just kind of fill in one thing here, 1318 01:13:59,790 --> 01:14:02,430 and it is, well, if you take everything 1319 01:14:02,430 --> 01:14:05,590 I've told you seriously, you would say OK, too bad. 1320 01:14:05,590 --> 01:14:08,110 We cannot localize atoms in an optical lattice 1321 01:14:08,110 --> 01:14:10,390 because the temperature is always comparable 1322 01:14:10,390 --> 01:14:12,140 to the amplitude of the lattice. 1323 01:14:12,140 --> 01:14:17,880 But by making the lattice smaller and smaller and smaller 1324 01:14:17,880 --> 01:14:21,050 I can reach lower and lower temperatures. 1325 01:14:21,050 --> 01:14:24,920 Well, yes, the some extent that's possible. 1326 01:14:24,920 --> 01:14:28,240 But what we have done here is we've done approximations. 1327 01:14:28,240 --> 01:14:31,160 We've only focused on the stimulated force. 1328 01:14:31,160 --> 01:14:33,440 We have not looked at the recoil which 1329 01:14:33,440 --> 01:14:35,620 comes in spontaneous emission. 1330 01:14:35,620 --> 01:14:39,780 And therefore, the moment we would make the optical lattice 1331 01:14:39,780 --> 01:14:43,720 smaller than gamma then we have to include heating 1332 01:14:43,720 --> 01:14:45,560 by the spontaneous light force. 1333 01:14:45,560 --> 01:14:50,800 And the result will be that blue molasses cannot reach lower 1334 01:14:50,800 --> 01:14:53,850 temperatures than red detuned molasses. 1335 01:14:53,850 --> 01:14:56,530 And for red detuned molasses we have already 1336 01:14:56,530 --> 01:15:00,040 discussed the famous Doppler limit, which is gamma over 2. 1337 01:15:09,840 --> 01:15:11,360 So that's what I wanted to present 1338 01:15:11,360 --> 01:15:13,560 to you with the stimulated force. 1339 01:15:13,560 --> 01:15:16,820 First, a more general discussion using the real 1340 01:15:16,820 --> 01:15:18,510 dressed atom picture. 1341 01:15:18,510 --> 01:15:22,194 But now in the end, to fly over using sort of the baby 1342 01:15:22,194 --> 01:15:24,610 dressed atom picture, which is simple perturbation theory. 1343 01:15:31,050 --> 01:15:39,010 OK, so we've done the hard work. 1344 01:15:39,010 --> 01:15:42,260 Now we want to have some fun discussing the physics. 1345 01:15:42,260 --> 01:15:45,970 And I'm realizing I've only five, six, seven minutes left. 1346 01:15:45,970 --> 01:15:48,630 I want to discuss three points. 1347 01:15:48,630 --> 01:15:54,100 One is I want to give you all the different pictures to find 1348 01:15:54,100 --> 01:15:56,220 the potential of a dipole trap, because we've 1349 01:15:56,220 --> 01:15:58,110 encountered several. 1350 01:15:58,110 --> 01:16:00,930 The second thing is I want to have a discussion with you 1351 01:16:00,930 --> 01:16:05,260 if the stimulated force is due to electric or magnetic forces. 1352 01:16:05,260 --> 01:16:08,420 And eventually, and we'll be running out of time, 1353 01:16:08,420 --> 01:16:11,120 I have to do that on Wednesday to discuss 1354 01:16:11,120 --> 01:16:13,440 the question of energy conservation. 1355 01:16:13,440 --> 01:16:16,630 If you cool the atoms where does the energy go? 1356 01:16:19,550 --> 01:16:25,620 So if you use the stimulated force, 1357 01:16:25,620 --> 01:16:27,970 it gives rise to dipole potential. 1358 01:16:27,970 --> 01:16:30,870 And using the Optical Bloch Equations or the dressed atom 1359 01:16:30,870 --> 01:16:34,340 picture, we found this expression 1360 01:16:34,340 --> 01:16:37,790 for the dipole potential. 1361 01:16:37,790 --> 01:16:47,590 And there are at least four ways how we could have derived it. 1362 01:16:47,590 --> 01:16:51,050 One is the Optical Bloch Equations, we did that. 1363 01:16:51,050 --> 01:16:53,020 The dressed atom picture, we did that today. 1364 01:16:55,830 --> 01:16:59,100 Another picture is very straightforward and simple. 1365 01:16:59,100 --> 01:17:02,370 It just says that the potential is 1366 01:17:02,370 --> 01:17:05,560 the electric potential of a dipole. 1367 01:17:05,560 --> 01:17:11,570 And then all you have to know is from the EC polarizability, 1368 01:17:11,570 --> 01:17:15,740 if you drive an atom with a laser field 1369 01:17:15,740 --> 01:17:18,510 below resonance-- I mean that's just 1370 01:17:18,510 --> 01:17:21,045 harmonic oscillator-- below resonance 1371 01:17:21,045 --> 01:17:25,170 the harmonic oscillator is in phase with the force. 1372 01:17:25,170 --> 01:17:28,670 Above resonance, if you drive a harmonic oscillator fast, 1373 01:17:28,670 --> 01:17:32,350 the response of the harmonic oscillator is in opposite, 1374 01:17:32,350 --> 01:17:33,980 it's out of phase. 1375 01:17:33,980 --> 01:17:37,310 So therefore, if your potential is minus the E, 1376 01:17:37,310 --> 01:17:41,310 you are in phase-- it is a negative attractive potential. 1377 01:17:41,310 --> 01:17:44,160 If the harmonic oscillator is driven above resonance 1378 01:17:44,160 --> 01:17:45,760 there is another minus sign, you get 1379 01:17:45,760 --> 01:17:49,500 plus the E, which becomes a repulsive potential. 1380 01:17:49,500 --> 01:17:51,574 So it's, look and say, just the physics 1381 01:17:51,574 --> 01:17:52,615 of a harmonic oscillator. 1382 01:17:56,420 --> 01:17:58,810 There's one thing I like, which is sometimes not 1383 01:17:58,810 --> 01:18:01,210 taught in an atomic physics course 1384 01:18:01,210 --> 01:18:05,945 because it is the force on a macroscopic dielectric object. 1385 01:18:08,630 --> 01:18:12,640 So let's assume we have an atom-- no, we 1386 01:18:12,640 --> 01:18:13,660 don't have an atom. 1387 01:18:13,660 --> 01:18:15,750 We have something which is bigger. 1388 01:18:15,750 --> 01:18:18,340 Let's assume a tiny polystyrenes sphere. 1389 01:18:18,340 --> 01:18:21,430 You can assume a small sphere of glass. 1390 01:18:21,430 --> 01:18:24,420 What happens if I shine a laser beam 1391 01:18:24,420 --> 01:18:27,200 on a small sphere of glass? 1392 01:18:27,200 --> 01:18:33,420 Well, let me assume that the laser beam is-- well, 1393 01:18:33,420 --> 01:18:38,670 if the laser beam hits this sphere right on, 1394 01:18:38,670 --> 01:18:40,670 it's symmetric, there is no force. 1395 01:18:40,670 --> 01:18:43,540 So what we want to do is we want to have a laser beam, shown 1396 01:18:43,540 --> 01:18:46,560 here by this arrow, which has a maximum here 1397 01:18:46,560 --> 01:18:48,580 so the profile is like this. 1398 01:18:48,580 --> 01:18:52,310 And we want to discuss whether this sphere is sucked 1399 01:18:52,310 --> 01:18:54,870 into the laser beam, this would be an attractive force, 1400 01:18:54,870 --> 01:18:57,170 or whether this sphere is repelled. 1401 01:18:57,170 --> 01:18:59,459 And all we want to use is optics, 1402 01:18:59,459 --> 01:19:00,500 the optics of refraction. 1403 01:19:03,240 --> 01:19:06,450 When we have a laser beam which is weak, medium, strong, 1404 01:19:06,450 --> 01:19:09,460 strong here, and for red detuning 1405 01:19:09,460 --> 01:19:12,640 we have an index of refraction which is larger than 1. 1406 01:19:12,640 --> 01:19:15,470 So the sphere acts like a lens. 1407 01:19:15,470 --> 01:19:17,980 So that leaves those photons up and in, 1408 01:19:17,980 --> 01:19:20,670 and those photons up and in. 1409 01:19:20,670 --> 01:19:25,330 And when a photon is bent in, the momentum of the photon 1410 01:19:25,330 --> 01:19:26,300 has changed. 1411 01:19:26,300 --> 01:19:29,150 And the momentum, of course, the opposite momentum transfer, 1412 01:19:29,150 --> 01:19:31,870 is imparted to the sphere. 1413 01:19:31,870 --> 01:19:36,320 So in this case, we have sort of-- we 1414 01:19:36,320 --> 01:19:39,470 have more photons in this part of the laser beam, which 1415 01:19:39,470 --> 01:19:42,500 are bent up, than photons are bent down. 1416 01:19:42,500 --> 01:19:45,670 And this means the net force is that the sphere is 1417 01:19:45,670 --> 01:19:48,110 sucked into the laser beam. 1418 01:19:48,110 --> 01:19:50,790 And if you go for blue detuned light, 1419 01:19:50,790 --> 01:19:53,250 where the index of refraction is smaller than 1, 1420 01:19:53,250 --> 01:19:55,690 it is the opposite. 1421 01:19:55,690 --> 01:19:58,030 So the picture which we have discussed here 1422 01:19:58,030 --> 01:20:01,010 was for atoms form microscopic objects. 1423 01:20:01,010 --> 01:20:05,700 But you have, actually, similar physics 1424 01:20:05,700 --> 01:20:09,810 of a stimulated force for mesoscopic and macroscopic 1425 01:20:09,810 --> 01:20:10,740 objects. 1426 01:20:10,740 --> 01:20:15,170 And then what matters whether you trap or anti-trap, 1427 01:20:15,170 --> 01:20:17,870 is whether the index of refraction is larger than 1 1428 01:20:17,870 --> 01:20:20,090 or smaller than one. 1429 01:20:20,090 --> 01:20:23,350 And you know, two level system for blue detuned light 1430 01:20:23,350 --> 01:20:26,000 has an index of refraction which is smaller than 1. 1431 01:20:26,000 --> 01:20:28,880 And for red detuned light our atomic clouds 1432 01:20:28,880 --> 01:20:32,520 have an index of refraction which is larger than 1. 1433 01:20:32,520 --> 01:20:36,210 But this picture that you can use the same set up, 1434 01:20:36,210 --> 01:20:38,650 the same focus beam and a similar dipole 1435 01:20:38,650 --> 01:20:43,250 force to trap larger objects is actually 1436 01:20:43,250 --> 01:20:47,940 exploited in a different field of physics, 1437 01:20:47,940 --> 01:20:52,160 in biology, in the form of optical tweezers. 1438 01:20:52,160 --> 01:20:54,210 When actually in the early '80s, when 1439 01:20:54,210 --> 01:20:57,850 Steve Chu came to Bell Labs and had discussions with Art 1440 01:20:57,850 --> 01:21:03,380 Ashkin, they actually discussed optical trapping, both of atoms 1441 01:21:03,380 --> 01:21:09,590 and both of biological objects, little microspheres, or cells. 1442 01:21:09,590 --> 01:21:13,780 And it's sort of remarkable that both approaches 1443 01:21:13,780 --> 01:21:16,230 were realized within the same year. 1444 01:21:16,230 --> 01:21:18,440 So atoms were trapped in '86. 1445 01:21:18,440 --> 01:21:21,740 And macroscopic objects were manipulated, 1446 01:21:21,740 --> 01:21:23,670 I think it's exactly the same year. 1447 01:21:23,670 --> 01:21:26,630 And as you know, the optical tweezers 1448 01:21:26,630 --> 01:21:32,030 is now an important technology in biology and biophysics. 1449 01:21:32,030 --> 01:21:34,970 For instance, you can focus a laser beam 1450 01:21:34,970 --> 01:21:37,660 and manipulate objects within a living cell. 1451 01:21:40,780 --> 01:21:44,650 So I think I should stop here, but maybe take your questions. 1452 01:21:49,360 --> 01:21:52,030 Then let me conclude with an outlook. 1453 01:21:52,030 --> 01:21:57,490 We have this week and next week, and then the semester is over. 1454 01:21:57,490 --> 01:22:01,750 So this week we have one class on Wednesday. 1455 01:22:01,750 --> 01:22:04,190 I want to finish this discussion and teach 1456 01:22:04,190 --> 01:22:07,560 about magnetic trapping and evaporative cooling. 1457 01:22:07,560 --> 01:22:10,840 And the last week, because of a makeup class on Friday, 1458 01:22:10,840 --> 01:22:13,970 we have three more classes, Monday, Wednesday, Friday. 1459 01:22:13,970 --> 01:22:15,970 By the way, I just learned that the Friday class 1460 01:22:15,970 --> 01:22:18,170 is in these room. 1461 01:22:18,170 --> 01:22:19,940 And on Monday, Wednesday, Friday we 1462 01:22:19,940 --> 01:22:22,070 talk about, well, it's really featured 1463 01:22:22,070 --> 01:22:26,070 pick about Bose gasses, lecture one, Fermi gasses, lecture two. 1464 01:22:26,070 --> 01:22:29,350 And the third lecture is about quantum logic with ions. 1465 01:22:31,920 --> 01:22:34,050 OK, see you Wednesday.