1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,880 --> 00:00:23,410 PROFESSOR: --equal forces of light. 9 00:00:23,410 --> 00:00:25,060 But before we get started with it, 10 00:00:25,060 --> 00:00:28,860 do you have any question about the last lecture, which 11 00:00:28,860 --> 00:00:34,360 was on open quantum system-- Quantum Monte Carlo methods-- 12 00:00:34,360 --> 00:00:38,530 and in particular, the conceptual things we discussed 13 00:00:38,530 --> 00:00:40,760 is what happens when you do a measurement, 14 00:00:40,760 --> 00:00:43,330 or when you don't detect a photon, 15 00:00:43,330 --> 00:00:46,010 how does it change the wave function? 16 00:00:46,010 --> 00:00:48,055 And this was actually at the heart 17 00:00:48,055 --> 00:00:51,020 of realizing similar quantum systems, 18 00:00:51,020 --> 00:00:55,870 and simulating them using quantum Monte Carlo methods. 19 00:00:55,870 --> 00:00:56,600 Any question? 20 00:00:56,600 --> 00:01:01,441 Any points for discussion? 21 00:01:01,441 --> 00:01:01,940 OK. 22 00:01:04,750 --> 00:01:11,630 Today is, actually conceptually, a simple lecture, 23 00:01:11,630 --> 00:01:13,790 because I'm not really introducing 24 00:01:13,790 --> 00:01:15,870 some subtleties of quantum physics. 25 00:01:15,870 --> 00:01:20,500 We know our Hamiltonian-- the dipole Hamiltonian-- 26 00:01:20,500 --> 00:01:23,900 this is the fully quantized Hamiltonian, 27 00:01:23,900 --> 00:01:28,940 how atoms interact with the electromagnetic field. 28 00:01:28,940 --> 00:01:30,500 And, of course, it's fully quantized 29 00:01:30,500 --> 00:01:33,360 because E perpendicular is the operator 30 00:01:33,360 --> 00:01:35,610 of the quantized electromagnetic field, 31 00:01:35,610 --> 00:01:38,790 and it's the sum of a plus a [? dega. ?] 32 00:01:38,790 --> 00:01:43,040 But when it comes to forces, it's actually fairly simple. 33 00:01:43,040 --> 00:01:44,560 We don't introduce anything else. 34 00:01:44,560 --> 00:01:46,080 This is our Hamiltonian. 35 00:01:46,080 --> 00:01:47,410 This is the energy. 36 00:01:47,410 --> 00:01:49,790 And the force is nothing else than the gradient 37 00:01:49,790 --> 00:01:52,320 of this energy. 38 00:01:52,320 --> 00:01:56,460 So I decided it today-- it's rather unusual-- 39 00:01:56,460 --> 00:02:01,170 but I will have everything pre-written, because I think 40 00:02:01,170 --> 00:02:04,185 the concentration of new ideas is not so high today, 41 00:02:04,185 --> 00:02:06,160 so I want to go a little bit faster. 42 00:02:06,160 --> 00:02:08,545 But I invite everybody if you slow me down 43 00:02:08,545 --> 00:02:10,295 if you think I'm going too fast and if you 44 00:02:10,295 --> 00:02:12,710 think I'm skipping something. 45 00:02:12,710 --> 00:02:15,950 Also another justification is at least 46 00:02:15,950 --> 00:02:19,150 the first 30%, 40% of this lecture, 47 00:02:19,150 --> 00:02:21,970 you've already done as a homework assignment, when 48 00:02:21,970 --> 00:02:27,290 you looked at the classical limit of mechanical forces. 49 00:02:27,290 --> 00:02:29,580 You actually had already, as a preparation 50 00:02:29,580 --> 00:02:31,960 for this class, a homework assignment 51 00:02:31,960 --> 00:02:36,440 on the spontaneous scattering force and the stimulated dipole 52 00:02:36,440 --> 00:02:37,450 force. 53 00:02:37,450 --> 00:02:39,940 PROFESSOR: Well, the only subtlety today 54 00:02:39,940 --> 00:02:43,130 is how do we deal with a fully quantized 55 00:02:43,130 --> 00:02:45,040 electromagnetic field? 56 00:02:45,040 --> 00:02:47,660 But what I want to show you in the next few minutes is 57 00:02:47,660 --> 00:02:50,810 in the end, we actually do approximations 58 00:02:50,810 --> 00:02:53,430 that the quantized electromagnetic field pretty 59 00:02:53,430 --> 00:02:55,560 much disappears from the equation. 60 00:02:55,560 --> 00:02:56,900 We have a classical field. 61 00:02:56,900 --> 00:03:01,260 And then, almost everything you have done in your homework 62 00:03:01,260 --> 00:03:03,240 directly applies. 63 00:03:03,240 --> 00:03:06,860 The only other thing we do today is the dipole moment. 64 00:03:06,860 --> 00:03:10,500 Well, we use, for the dipole moment of the atom, 65 00:03:10,500 --> 00:03:13,770 the solution of the Optical Bloch Equation. 66 00:03:13,770 --> 00:03:15,530 So I think now you know already everything 67 00:03:15,530 --> 00:03:17,655 we do in the first 30, 40 minutes. 68 00:03:22,810 --> 00:03:26,650 PROFESSOR: So we get rid of all of the quantum character 69 00:03:26,650 --> 00:03:29,870 of the electromagnetic field in two stages. 70 00:03:32,950 --> 00:03:36,900 One is the following, well, we want 71 00:03:36,900 --> 00:03:40,760 to describe the interactions of atoms with laser fields. 72 00:03:40,760 --> 00:03:43,080 And that means the electromagnetic field 73 00:03:43,080 --> 00:03:45,110 is in a coherent state. 74 00:03:45,110 --> 00:03:46,850 And you know from several weeks ago, 75 00:03:46,850 --> 00:03:48,840 the coherent state is a superposition 76 00:03:48,840 --> 00:03:51,050 of a [INAUDIBLE] states, coherent superposition 77 00:03:51,050 --> 00:03:53,520 and such, so they're really photons inside. 78 00:03:53,520 --> 00:03:58,710 It's a fully quantized description of the laser beam. 79 00:03:58,710 --> 00:04:01,450 But now-- and this is something if you're not familiar, 80 00:04:01,450 --> 00:04:03,170 you should really look up-- there 81 00:04:03,170 --> 00:04:09,360 is an exact canonical transformation 82 00:04:09,360 --> 00:04:13,060 where you transform your Hamiltonian-- 83 00:04:13,060 --> 00:04:15,670 you do, so to speak, a basis transformation-- 84 00:04:15,670 --> 00:04:17,430 to another Hamiltonian. 85 00:04:17,430 --> 00:04:23,100 And the result is, after this unitary transformation, 86 00:04:23,100 --> 00:04:26,510 the coherent state is transformed to the vacuum 87 00:04:26,510 --> 00:04:28,000 state. 88 00:04:28,000 --> 00:04:31,510 So in other words, the quantized electromagnetic field 89 00:04:31,510 --> 00:04:33,140 is no longer in the coherent state, 90 00:04:33,140 --> 00:04:36,330 it's in the vacuum state. 91 00:04:36,330 --> 00:04:41,000 But what appears instead is a purely 92 00:04:41,000 --> 00:04:43,670 classical electromagnetic field. 93 00:04:43,670 --> 00:04:46,270 A c number. 94 00:04:46,270 --> 00:04:49,250 So you can sort of say-- I'm waving my hands now-- 95 00:04:49,250 --> 00:04:50,970 the coherent state is the vacuum, 96 00:04:50,970 --> 00:04:53,760 with a displacement operator. 97 00:04:53,760 --> 00:04:56,420 All the quantumness of the electromagnetic field 98 00:04:56,420 --> 00:04:58,130 is in the vacuum state, and that's 99 00:04:58,130 --> 00:04:59,940 something we have to keep because this 100 00:04:59,940 --> 00:05:02,300 is responsible for spontaneous emission quantum 101 00:05:02,300 --> 00:05:04,100 fluctuations and such. 102 00:05:04,100 --> 00:05:06,080 But the displacement operator, which 103 00:05:06,080 --> 00:05:09,130 gives us an arbitrary coherent state, that 104 00:05:09,130 --> 00:05:11,807 can be absorbed by completely treating 105 00:05:11,807 --> 00:05:13,390 the electromagnetic field classically. 106 00:05:18,290 --> 00:05:22,520 So therefore, we take the fully quantized electromagnetic field 107 00:05:22,520 --> 00:05:28,000 and we replace it now by a classical field, and simply 108 00:05:28,000 --> 00:05:30,120 the vacuum fluctuation. 109 00:05:30,120 --> 00:05:32,690 So there are no photons around anymore. 110 00:05:32,690 --> 00:05:36,770 There is the vacuum ready to absorb the photons, 111 00:05:36,770 --> 00:05:39,470 and then there's a classical electromagnetic field, 112 00:05:39,470 --> 00:05:40,980 which drives the atoms. 113 00:05:40,980 --> 00:05:44,980 And, of course, the classical electromagnetic field 114 00:05:44,980 --> 00:05:47,740 is not changing because of absorption emission, 115 00:05:47,740 --> 00:05:51,860 because it's a c number in our Hamiltonian. 116 00:05:51,860 --> 00:05:54,400 So this is an important conceptual step, 117 00:05:54,400 --> 00:05:56,030 and it's the first step in how we 118 00:05:56,030 --> 00:05:58,965 get rid of the quantum nature of the electromagnetic field 119 00:05:58,965 --> 00:06:01,060 when it interacts with atoms mechanically. 120 00:06:05,061 --> 00:06:05,560 OK. 121 00:06:10,240 --> 00:06:13,365 So, by the way, everything I'm telling you today-- 122 00:06:13,365 --> 00:06:15,270 actually, in the first half of today-- 123 00:06:15,270 --> 00:06:17,840 can be found in Atom-Photon Interaction. 124 00:06:17,840 --> 00:06:19,770 Now, so this takes care actually, 125 00:06:19,770 --> 00:06:22,300 of most of the electromagnetic field. 126 00:06:22,300 --> 00:06:24,210 We come back to that in a moment. 127 00:06:24,210 --> 00:06:27,440 The next thing is we want to use the full quantum 128 00:06:27,440 --> 00:06:30,060 solution for the atomic dipole operator, 129 00:06:30,060 --> 00:06:33,660 and that means we want to remind ourselves 130 00:06:33,660 --> 00:06:37,780 of the solution of the Optical Bloch Equations. 131 00:06:37,780 --> 00:06:45,741 PROFESSOR: Now, talking about the Optical Bloch Equations, 132 00:06:45,741 --> 00:06:49,020 I just have to remind you of what we did in class, 133 00:06:49,020 --> 00:06:51,520 but I have to apologize it's a little bit 134 00:06:51,520 --> 00:06:53,320 exercise in notation. 135 00:06:53,320 --> 00:06:55,690 Because in Atom-Photon Interaction 136 00:06:55,690 --> 00:06:59,350 they use a slightly different notation then 137 00:06:59,350 --> 00:07:01,250 you find in other sources. 138 00:07:01,250 --> 00:07:03,990 On the other hand, I think I've nicely prepared for you, 139 00:07:03,990 --> 00:07:06,080 and I give you the translation table. 140 00:07:06,080 --> 00:07:11,180 In Atom-Photon Interaction, they regard the Bloch vector, 141 00:07:11,180 --> 00:07:14,070 and we have introduced the components, u, v, 142 00:07:14,070 --> 00:07:17,830 and w, as a fictitious spin, 1/2. 143 00:07:17,830 --> 00:07:22,130 Therefore, you find with this kind of old-fashioned letters, 144 00:07:22,130 --> 00:07:26,880 Sx, Sy, Sc, in Atom-Photon Interaction. 145 00:07:26,880 --> 00:07:28,710 The other thing, if you open the book, 146 00:07:28,710 --> 00:07:33,100 Atom-Photon Interaction is that their operators, 147 00:07:33,100 --> 00:07:36,620 density matrix-- the atomic density matrix-- sigma, 148 00:07:36,620 --> 00:07:37,990 and sigma head. 149 00:07:37,990 --> 00:07:43,830 One is in the rotating frame, and one is in the lab frame. 150 00:07:43,830 --> 00:07:46,500 So we have to go back and forth to do that, 151 00:07:46,500 --> 00:07:52,790 but the result is fairly trivial. 152 00:07:52,790 --> 00:07:55,100 So this is simply the definition. 153 00:07:55,100 --> 00:07:57,700 First of all, the density matrix is the full description 154 00:07:57,700 --> 00:07:59,660 of the atomic system, but instead 155 00:07:59,660 --> 00:08:03,020 of dealing with matrix elements of the density matrix, 156 00:08:03,020 --> 00:08:05,240 it's more convenient for two level system 157 00:08:05,240 --> 00:08:09,160 to use a fictitious spin or the Bloch vector. 158 00:08:09,160 --> 00:08:15,160 So the important equation-- the are the Optical Block 159 00:08:15,160 --> 00:08:20,990 Equations-- is the equation A20 in API. 160 00:08:20,990 --> 00:08:24,020 And it's simply the first order differential equation 161 00:08:24,020 --> 00:08:26,370 for the components of the Optical Bloch 162 00:08:26,370 --> 00:08:30,070 vector, u v, and w. 163 00:08:30,070 --> 00:08:32,549 Notation is as usual, the laser detuning, 164 00:08:32,549 --> 00:08:36,600 delta L. Omega 1 is the [INAUDIBLE] frequency. 165 00:08:36,600 --> 00:08:38,669 And this is the parametrization of 166 00:08:38,669 --> 00:08:44,520 the classical electromagnetic field which drives the system. 167 00:08:44,520 --> 00:08:48,360 Now, this equation is identical to the equation 168 00:08:48,360 --> 00:08:51,790 we discussed in the unit on solutions of Optical Bloch 169 00:08:51,790 --> 00:08:53,720 Equations. 170 00:08:53,720 --> 00:08:56,790 We just called the [INAUDIBLE] frequency, g, 171 00:08:56,790 --> 00:08:59,240 delta L was called delta. 172 00:08:59,240 --> 00:09:03,620 And then, vectors of 2r was confusing. 173 00:09:03,620 --> 00:09:09,290 The components, [? r,x,y,z, ?] are two times u, v, w. 174 00:09:09,290 --> 00:09:12,540 And the reason for that is some people 175 00:09:12,540 --> 00:09:15,490 prefer that it represents spin 1/2, 176 00:09:15,490 --> 00:09:18,240 so they want to normalize things to 1/2. 177 00:09:18,240 --> 00:09:20,330 Or if you want to have unit vectors, 178 00:09:20,330 --> 00:09:22,770 your throw in a vector of two. 179 00:09:22,770 --> 00:09:26,450 But that is all, so if you're confused about a vector of two, 180 00:09:26,450 --> 00:09:30,390 it's exactly this substitution which has been made. 181 00:09:30,390 --> 00:09:34,950 So these are the Optical Bloch Equations, and what we need now 182 00:09:34,950 --> 00:09:36,195 is the solution. 183 00:09:42,920 --> 00:09:47,380 What we need now is the solution for the dipole moment, 184 00:09:47,380 --> 00:09:51,390 because it is the dipole moment which 185 00:09:51,390 --> 00:09:55,420 is responsible for the mechanical forces. 186 00:09:55,420 --> 00:10:00,460 The dipole operator is, of course, the non-diagonal matrix 187 00:10:00,460 --> 00:10:04,560 elements, the coherencies of the density matrix. 188 00:10:04,560 --> 00:10:09,640 So all we want to take now from the previous unit, 189 00:10:09,640 --> 00:10:12,750 the expectation value of the dipole moment, which 190 00:10:12,750 --> 00:10:17,071 is nothing else than the trace of the statistical operator 191 00:10:17,071 --> 00:10:18,695 with the operator of the dipole moment. 192 00:10:24,020 --> 00:10:27,250 At this point, just to guide you through, 193 00:10:27,250 --> 00:10:31,540 we go from the lab frame-- from the density matrix in the lab 194 00:10:31,540 --> 00:10:34,510 frame-- to the density matrix in the rotating frame, 195 00:10:34,510 --> 00:10:37,960 and this is when the laser frequency appears, 196 00:10:37,960 --> 00:10:40,530 is into the i omega l t. 197 00:10:40,530 --> 00:10:43,240 And therefore, we want to describe things 198 00:10:43,240 --> 00:10:44,500 in the lab frame. 199 00:10:44,500 --> 00:10:50,380 If you use the u and v part of the Optical Bloch 200 00:10:50,380 --> 00:10:55,140 vector obtained in the rotating frame, in the lab frame 201 00:10:55,140 --> 00:10:58,580 everything rotates at the laser frequency. 202 00:10:58,580 --> 00:11:02,330 If you drive an atom, if you drive a harmonic oscillator 203 00:11:02,330 --> 00:11:05,390 with a laser frequency, the harmonic oscillator 204 00:11:05,390 --> 00:11:09,720 responds at the laser frequency, not at the atomic frequency. 205 00:11:09,720 --> 00:11:11,940 We discussed that a while ago. 206 00:11:11,940 --> 00:11:14,640 So therefore, the dipole moment oscillates 207 00:11:14,640 --> 00:11:17,690 at the laser frequency. 208 00:11:17,690 --> 00:11:23,510 And their two components, u and v-- since our laser field 209 00:11:23,510 --> 00:11:30,330 was parametrized by definition-- epsilon 0 times cosine omega t. 210 00:11:30,330 --> 00:11:33,595 We have now nicely separated, through the Optical Bloch 211 00:11:33,595 --> 00:11:37,750 vector the in phase part of the dipole moment, 212 00:11:37,750 --> 00:11:42,470 and the in quadrature phase of dipole moment. 213 00:11:42,470 --> 00:11:45,840 If you didn't pay attention what I said the last five 214 00:11:45,840 --> 00:11:48,430 minutes, that's OK, you can just start here, 215 00:11:48,430 --> 00:11:52,230 saying the expectation value of the dipole moment oscillates 216 00:11:52,230 --> 00:11:54,010 with a laser frequency. 217 00:11:54,010 --> 00:11:56,930 And as for any harmonic oscillator, or harmonic 218 00:11:56,930 --> 00:11:59,770 oscillator type system, there is one component 219 00:11:59,770 --> 00:12:01,432 which is in phase, and one component 220 00:12:01,432 --> 00:12:02,390 which is in quadrature. 221 00:12:06,710 --> 00:12:08,760 Fast forward. 222 00:12:08,760 --> 00:12:09,910 OK. 223 00:12:09,910 --> 00:12:13,020 And you know from any harmonic oscillator, 224 00:12:13,020 --> 00:12:20,970 when it comes to the absorbed power, 225 00:12:20,970 --> 00:12:23,990 that it's only the quadrature component which 226 00:12:23,990 --> 00:12:25,420 absorbs the power. 227 00:12:25,420 --> 00:12:27,410 And you can immediately see that when 228 00:12:27,410 --> 00:12:31,140 we say-- just use what is written in black 229 00:12:31,140 --> 00:12:35,500 now-- the energy is nothing else than the charge 230 00:12:35,500 --> 00:12:37,960 times the displacement. 231 00:12:37,960 --> 00:12:40,930 When we divide by delta t, you'll 232 00:12:40,930 --> 00:12:49,070 find that the absorbed power is the electric field 233 00:12:49,070 --> 00:12:53,410 times the derivative of the dipole moment. 234 00:12:53,410 --> 00:12:56,220 Well, if you ever reach over one cycle, 235 00:12:56,220 --> 00:12:59,710 we are only interested in d dot, which 236 00:12:59,710 --> 00:13:02,630 oscillates with cosine omega t. 237 00:13:02,630 --> 00:13:08,360 But that means since it's d dot, d oscillates with sine omega t. 238 00:13:08,360 --> 00:13:12,200 And this was the in quadrature component, v. 239 00:13:12,200 --> 00:13:16,500 So it is only the part, v, of the Optical Bloch vector which 240 00:13:16,500 --> 00:13:18,990 is responsible for exchanging energy 241 00:13:18,990 --> 00:13:21,694 with the electromagnetic field. 242 00:13:21,694 --> 00:13:31,070 And we can also, by dividing by the energy of a photon, 243 00:13:31,070 --> 00:13:33,490 find out what is the number of absorbed photons. 244 00:13:33,490 --> 00:13:35,810 But we did that already when we discussed Optical Bloch 245 00:13:35,810 --> 00:13:36,309 Equations. 246 00:13:38,910 --> 00:13:43,160 OK, so that was our look back on the Optical Bloch Equation, 247 00:13:43,160 --> 00:13:45,940 the solution for the oscillating dipole moment, 248 00:13:45,940 --> 00:13:50,080 and now we are ready to put everything together. 249 00:13:50,080 --> 00:13:52,800 So we want to know the force. 250 00:13:52,800 --> 00:13:57,140 The force, using Heisenberg's Equation of Motion 251 00:13:57,140 --> 00:14:01,340 for the momentum derivative is the expectation value 252 00:14:01,340 --> 00:14:08,180 of the operator, which is the gradient of the Hamiltonian. 253 00:14:08,180 --> 00:14:08,820 OK. 254 00:14:08,820 --> 00:14:11,460 So now we have to take the gradient of the Hamiltonian. 255 00:14:14,450 --> 00:14:16,250 This is the only expression, actually, 256 00:14:16,250 --> 00:14:19,405 today, I would've liked to just write it down, and build it 257 00:14:19,405 --> 00:14:20,470 up piece by piece. 258 00:14:20,470 --> 00:14:22,520 But let me step you through. 259 00:14:22,520 --> 00:14:24,085 We want to take the gradient. 260 00:14:29,680 --> 00:14:32,920 The dipole operator of the atom does not have an r dependence, 261 00:14:32,920 --> 00:14:35,830 it's just x on the atom, wherever it is. 262 00:14:35,830 --> 00:14:40,130 But the gradient with respect to the center of mass position 263 00:14:40,130 --> 00:14:42,870 from the atoms comes from the operator 264 00:14:42,870 --> 00:14:45,450 of the electromagnetic field. 265 00:14:45,450 --> 00:14:47,520 So in other words, what this involves 266 00:14:47,520 --> 00:14:52,430 is the gradient of the operator of the fully quantized 267 00:14:52,430 --> 00:14:54,300 electromagnetic field. 268 00:14:54,300 --> 00:14:58,770 But we have already written at the electromagnetic field 269 00:14:58,770 --> 00:15:05,080 as an external electromagnetic field-- 270 00:15:05,080 --> 00:15:07,640 classical electromagnetic field-- plus a vacuum 271 00:15:07,640 --> 00:15:09,050 fluctuations. 272 00:15:09,050 --> 00:15:12,980 But the vacuum fluctuations are symmetric. 273 00:15:12,980 --> 00:15:14,980 There is nothing which tells vacuum fluctuations 274 00:15:14,980 --> 00:15:16,600 what is left and what is right. 275 00:15:16,600 --> 00:15:18,580 So therefore, the derivative only 276 00:15:18,580 --> 00:15:22,840 acts on the classical electromagnetic field. 277 00:15:22,840 --> 00:15:24,880 So this is how we have now, you know, 278 00:15:24,880 --> 00:15:27,260 we first throw out the coherent state, 279 00:15:27,260 --> 00:15:29,980 and now we throw out the vacuum state. 280 00:15:29,980 --> 00:15:33,465 And therefore, it is only the classical part 281 00:15:33,465 --> 00:15:35,790 of the electromagnetic field which 282 00:15:35,790 --> 00:15:37,790 is responsible for the forces. 283 00:15:44,140 --> 00:15:51,380 Now we make one way assumption which greatly simplifies it. 284 00:15:51,380 --> 00:15:55,570 R is the center of mass position of the atom, 285 00:15:55,570 --> 00:15:57,790 and we have to take the electromagnetic field 286 00:15:57,790 --> 00:15:59,500 as the center of mass position. 287 00:15:59,500 --> 00:16:03,200 And if the atom is localized-- wave packet 288 00:16:03,200 --> 00:16:05,480 with a long [? dipole ?] wave-- we sort of 289 00:16:05,480 --> 00:16:08,350 have to involve the atomic wave function 290 00:16:08,350 --> 00:16:12,800 when we evaluate this operator of the electromagnetic field. 291 00:16:12,800 --> 00:16:15,970 However, if the atom is very well 292 00:16:15,970 --> 00:16:20,262 localized-- you would actually want the atom 293 00:16:20,262 --> 00:16:26,170 to be localized to within an optical wave length-- 294 00:16:26,170 --> 00:16:29,455 then, under those circumstances, you 295 00:16:29,455 --> 00:16:33,610 can replace the parameter, r, which 296 00:16:33,610 --> 00:16:36,990 is a position of the atom, by the center of the atomic wave 297 00:16:36,990 --> 00:16:38,430 packet. 298 00:16:38,430 --> 00:16:40,900 So therefore, the kind of wave nature 299 00:16:40,900 --> 00:16:42,690 of the atoms-- which means the atom 300 00:16:42,690 --> 00:16:47,500 is smeared out-- as long as this wave packet is small, 301 00:16:47,500 --> 00:16:49,520 compared to the wave lengths, you 302 00:16:49,520 --> 00:16:53,030 can just evaluate the electromagnetic field 303 00:16:53,030 --> 00:16:57,380 at the center of the wave packet. 304 00:16:57,380 --> 00:17:04,290 This assumption requires-- since the scale 305 00:17:04,290 --> 00:17:07,040 of the electromagnetic field is set by the wave lengths-- 306 00:17:07,040 --> 00:17:11,760 this requires that the atom is localized better 307 00:17:11,760 --> 00:17:13,680 than an optical wave length. 308 00:17:13,680 --> 00:17:16,310 And that would mean that the energy 309 00:17:16,310 --> 00:17:19,130 of the atom, or the temperature, has 310 00:17:19,130 --> 00:17:22,230 to be larger than the recoil in it. 311 00:17:22,230 --> 00:17:24,980 And if you're localized the, to within a wave length, 312 00:17:24,980 --> 00:17:28,760 your momentum spread is larger, by Heisenberg's Uncertainty 313 00:17:28,760 --> 00:17:30,970 Relation than h bar k, and that means 314 00:17:30,970 --> 00:17:34,270 your energy is larger than the recoil energy. 315 00:17:34,270 --> 00:17:39,690 So if you want to have a description or light 316 00:17:39,690 --> 00:17:46,660 forces in that limit, you may have to revise this point. 317 00:17:46,660 --> 00:17:50,210 But for most of laser cooling, when you start with a hot cloud 318 00:17:50,210 --> 00:17:52,850 and cool it down to micro-Kelvin temperatures, 319 00:17:52,850 --> 00:17:55,360 this is very [INAUDIBLE]. 320 00:17:55,360 --> 00:17:55,870 Colin? 321 00:17:55,870 --> 00:17:57,578 AUDIENCE: Vacuum field doesn't contribute 322 00:17:57,578 --> 00:18:01,434 in sort of in the sense that the expectation value is zero. 323 00:18:01,434 --> 00:18:05,772 But wouldn't the rms value be non-zero? 324 00:18:05,772 --> 00:18:08,510 So what's the--? 325 00:18:08,510 --> 00:18:09,550 PROFESSOR: Exactly. 326 00:18:09,550 --> 00:18:11,840 I mean, in the end, you will see that in a moment, 327 00:18:11,840 --> 00:18:15,230 when we talk about cooling limits, 328 00:18:15,230 --> 00:18:18,680 fluctuations, spontaneous emission, provides heating. 329 00:18:18,680 --> 00:18:20,980 It doesn't contribute if you want 330 00:18:20,980 --> 00:18:24,210 to find the expectation value for the force. 331 00:18:24,210 --> 00:18:26,220 The vacuum and the fluctuations-- I mean, 332 00:18:26,220 --> 00:18:29,380 that's also logical-- the vacuum has only fluctuations, 333 00:18:29,380 --> 00:18:30,910 it has no net force. 334 00:18:30,910 --> 00:18:32,940 But the fluctuation's heat. 335 00:18:32,940 --> 00:18:35,200 And for heating, it will be very important. 336 00:18:38,450 --> 00:18:42,156 But we'll come to that in a little bit later. 337 00:18:42,156 --> 00:18:43,530 AUDIENCE: So what is [INAUDIBLE]? 338 00:18:52,229 --> 00:18:55,130 PROFESSOR: If the light is squeezed, yes. 339 00:18:55,130 --> 00:18:57,490 That's a whole different thing. 340 00:18:57,490 --> 00:19:02,920 We really assumed here that the light is in a coherent state. 341 00:19:02,920 --> 00:19:04,976 Now-- 342 00:19:04,976 --> 00:19:08,420 AUDIENCE: The transformation that we had [INAUDIBLE]. 343 00:19:16,790 --> 00:19:24,820 PROFESSOR: Well, you know, you can squeeze the vacuum. 344 00:19:24,820 --> 00:19:26,020 And then it's squeezed. 345 00:19:26,020 --> 00:19:28,375 But there's so few photons, you can't really 346 00:19:28,375 --> 00:19:29,860 laser cool with that. 347 00:19:29,860 --> 00:19:32,530 So the only way how you could possibly laser cool 348 00:19:32,530 --> 00:19:37,720 is if you take the squeezed vacuum, and displace it. 349 00:19:37,720 --> 00:19:40,230 So now you have, instead of a little circle, 350 00:19:40,230 --> 00:19:42,190 which is a coherent state, you have 351 00:19:42,190 --> 00:19:45,540 an ellipse, which is displaced. 352 00:19:45,540 --> 00:19:47,500 I haven't done the math-- maybe that would 353 00:19:47,500 --> 00:19:49,820 be an excellent homework assignment for the next time I 354 00:19:49,820 --> 00:19:52,570 teach the course-- you could probably 355 00:19:52,570 --> 00:19:56,340 show that the displacement operator, which is now 356 00:19:56,340 --> 00:20:00,350 displacing the squeezed vacuum, again can be transformed 357 00:20:00,350 --> 00:20:04,940 into a c number, and the ellipse this 358 00:20:04,940 --> 00:20:09,610 squeezed vacuum fluctuation-- the ellipse has 359 00:20:09,610 --> 00:20:13,540 an even symmetry, has an 180 degree rotation symmetry. 360 00:20:13,540 --> 00:20:20,400 And I could imagine that it will not contribute to the force. 361 00:20:20,400 --> 00:20:24,020 So in other words, if you have a displaced, squeezed vacuum, 362 00:20:24,020 --> 00:20:27,510 my gut feeling is nothing would change here. 363 00:20:27,510 --> 00:20:29,570 But what would change, aren't the fluctuations, 364 00:20:29,570 --> 00:20:31,730 but which would possibly change is the heating. 365 00:20:39,600 --> 00:20:45,250 OK, so we within those approximations, 366 00:20:45,250 --> 00:20:49,050 Heisenberg's Equation of Motion for the force 367 00:20:49,050 --> 00:20:52,290 becomes an equation for the acceleration 368 00:20:52,290 --> 00:20:55,140 for the atomic wave packet. 369 00:20:55,140 --> 00:20:59,080 And it involves now, two terms. 370 00:20:59,080 --> 00:21:01,910 One is the atomic dipole moment as 371 00:21:01,910 --> 00:21:03,890 obtained from the Optical Bloch Equations, 372 00:21:03,890 --> 00:21:06,750 and the other one is nothing else 373 00:21:06,750 --> 00:21:10,310 than the gradient of the classical part 374 00:21:10,310 --> 00:21:12,880 of the electromagnetic field. 375 00:21:12,880 --> 00:21:18,270 And with that we are pretty much at the equations which 376 00:21:18,270 --> 00:21:20,270 you looked at in your homework assignment, which 377 00:21:20,270 --> 00:21:23,220 was a classical model for the light force. 378 00:21:23,220 --> 00:21:25,980 Because this part is completely classical, 379 00:21:25,980 --> 00:21:30,340 and the only quantum aspect which we still have 380 00:21:30,340 --> 00:21:33,830 is that we evaluate the dipole moment, 381 00:21:33,830 --> 00:21:38,070 not necessarily, it's just a driven harmonic oscillator. 382 00:21:38,070 --> 00:21:40,520 When we take the Optical Bloch Equations, 383 00:21:40,520 --> 00:21:45,530 we will also find saturation a two-level system can 384 00:21:45,530 --> 00:21:48,290 be saturated, whereas a harmonic oscillator can never 385 00:21:48,290 --> 00:21:50,880 be saturated. 386 00:21:50,880 --> 00:21:53,730 So let me just repeat, so this is completely 387 00:21:53,730 --> 00:21:55,730 classical except for the expectation 388 00:21:55,730 --> 00:21:57,970 value for the dipole moment. 389 00:21:57,970 --> 00:22:01,910 But for no excitation, of course-- and we've 390 00:22:01,910 --> 00:22:04,310 discussed it many, many times-- a two-level system 391 00:22:04,310 --> 00:22:06,065 is nothing else for weak excitation, 392 00:22:06,065 --> 00:22:08,870 than a weakly excited harmonic oscillator. 393 00:22:08,870 --> 00:22:11,240 So then we are completely classical. 394 00:22:11,240 --> 00:22:14,175 But we want to discuss also saturation and that's 395 00:22:14,175 --> 00:22:16,880 been the Optical Bloch Equations, of course, come 396 00:22:16,880 --> 00:22:17,580 in very handy. 397 00:22:23,390 --> 00:22:25,220 OK. 398 00:22:25,220 --> 00:22:29,130 Next approximation, we want to know-- 399 00:22:29,130 --> 00:22:32,630 approximate-- the dipole moment-- the expectation 400 00:22:32,630 --> 00:22:38,680 value of the dipole moment-- with the steady state solution. 401 00:22:38,680 --> 00:22:40,882 The steady state solution-- analytic expression-- 402 00:22:40,882 --> 00:22:43,340 we just plug it in, and you can have a wonderful discussion 403 00:22:43,340 --> 00:22:44,900 about light forces. 404 00:22:44,900 --> 00:22:48,940 The question is, are we allowed to do that? 405 00:22:48,940 --> 00:22:51,520 Now I hope you'll remember that's actually one reason why 406 00:22:51,520 --> 00:23:01,300 I did that when we discussed the master equation. 407 00:23:01,300 --> 00:23:05,960 And we discussed an [? atomic ?] [? cavity. ?] I introduced 408 00:23:05,960 --> 00:23:09,600 to you the adiabatic elimination of variables. 409 00:23:09,600 --> 00:23:12,240 We adiabatically eliminated [? of dipole ?] 410 00:23:12,240 --> 00:23:13,700 matrix elements. 411 00:23:13,700 --> 00:23:17,410 Whenever you have something which relaxes sufficiently 412 00:23:17,410 --> 00:23:21,370 fast, and you're not interested in this short time scale, 413 00:23:21,370 --> 00:23:24,750 you can always say, I replace this quantity 414 00:23:24,750 --> 00:23:27,430 by its steady state value. 415 00:23:27,430 --> 00:23:33,040 Steady state with regard to the slowly changing parameter. 416 00:23:33,040 --> 00:23:35,820 And we want to do exactly the same here. 417 00:23:35,820 --> 00:23:37,170 So let me just translate. 418 00:23:37,170 --> 00:23:39,090 It's the same idea, exactly the same idea, 419 00:23:39,090 --> 00:23:40,460 but let me translate. 420 00:23:40,460 --> 00:23:47,000 If an atom moves, we have two aspects, two times [? case. ?] 421 00:23:47,000 --> 00:23:49,330 One is the motion of the atom. 422 00:23:49,330 --> 00:23:51,690 The atom may change its velocity. 423 00:23:51,690 --> 00:23:57,160 The atom may move from an area of strong electric field 424 00:23:57,160 --> 00:24:00,700 to an area of weak electric field. 425 00:24:00,700 --> 00:24:04,280 But this is really all that is related to the motion, 426 00:24:04,280 --> 00:24:07,840 to the change of momentum of a heavy object. 427 00:24:07,840 --> 00:24:11,620 But now inside the atoms, you play with the, you know, 428 00:24:11,620 --> 00:24:14,340 Bloch vector, excited state. 429 00:24:14,340 --> 00:24:15,930 If the electric field is higher, you 430 00:24:15,930 --> 00:24:18,050 have a larger population in the excited 431 00:24:18,050 --> 00:24:19,760 state than in the ground state. 432 00:24:19,760 --> 00:24:25,520 But this usually adjusts with the damping time of gamma, 433 00:24:25,520 --> 00:24:27,070 with a spontaneous emission rate. 434 00:24:27,070 --> 00:24:29,030 And that's usually very fast. 435 00:24:29,030 --> 00:24:30,840 So you can make the assumption when 436 00:24:30,840 --> 00:24:34,650 an atom moves to a changing electric field, at every point 437 00:24:34,650 --> 00:24:37,910 in the electric field, it very, very quickly-- 438 00:24:37,910 --> 00:24:40,910 the dipole moment-- assumes the steady state 439 00:24:40,910 --> 00:24:45,780 solution for the local electric field. 440 00:24:45,780 --> 00:24:48,870 And this would now be called an adiabatic elimination 441 00:24:48,870 --> 00:24:52,320 of the dynamics of the atom. 442 00:24:52,320 --> 00:24:57,630 And we simply replace the expectation value 443 00:24:57,630 --> 00:25:02,219 for the dipole moment by the steady state solution 444 00:25:02,219 --> 00:25:03,510 of the Optical Bloch Equations. 445 00:25:07,700 --> 00:25:10,350 Well, this is, of course, an approximation. 446 00:25:10,350 --> 00:25:14,175 And I want to take this approximation 447 00:25:14,175 --> 00:25:18,190 and who you what simple conclusions we 448 00:25:18,190 --> 00:25:19,150 can draw from that. 449 00:25:19,150 --> 00:25:21,690 But I also want you to know, I mean 450 00:25:21,690 --> 00:25:23,560 this is what I focus off in this course, 451 00:25:23,560 --> 00:25:26,380 to sort of think about them and get a feel where 452 00:25:26,380 --> 00:25:28,520 those approximations break down. 453 00:25:28,520 --> 00:25:34,650 So this approximation requires that the internal motion, that 454 00:25:34,650 --> 00:25:36,780 this, you know, two-level physics, 455 00:25:36,780 --> 00:25:39,120 that the internal density matrix, 456 00:25:39,120 --> 00:25:41,940 it has a relaxation rate of gamma. 457 00:25:41,940 --> 00:25:45,520 Whereas the external motion, we will actually 458 00:25:45,520 --> 00:25:48,820 see that very soon when we talk about molasses and damping, 459 00:25:48,820 --> 00:25:54,640 has a characteristic damping time which 460 00:25:54,640 --> 00:25:57,740 is one over the recoil energy. 461 00:25:57,740 --> 00:25:59,890 The recoiled energy involves the mass. 462 00:25:59,890 --> 00:26:02,480 The heavier the object, the slower is the damping time. 463 00:26:02,480 --> 00:26:03,860 It has all the right scaling. 464 00:26:03,860 --> 00:26:07,360 And you will actually see in the data 465 00:26:07,360 --> 00:26:11,680 today, that we will naturally obtain this as the time scale 466 00:26:11,680 --> 00:26:14,140 over which atomic motions is changing. 467 00:26:17,260 --> 00:26:22,690 So, is that approximation, this hierarchy of time scales, 468 00:26:22,690 --> 00:26:25,470 is that usually fulfilled? 469 00:26:25,470 --> 00:26:30,380 It is fulfilled in almost all of the atoms you are working on. 470 00:26:30,380 --> 00:26:34,410 For instance, in sodium, the ratio of those two time scales 471 00:26:34,410 --> 00:26:37,110 is 400. 472 00:26:37,110 --> 00:26:41,670 But if you push it, you will find examples 473 00:26:41,670 --> 00:26:44,390 like, helium in the triplets statement-- 474 00:26:44,390 --> 00:26:47,140 metastable helium in the triplet state-- where the two time 475 00:26:47,140 --> 00:26:48,590 scales are comparable. 476 00:26:48,590 --> 00:26:56,110 Helium is a very light, so therefore, it's 477 00:26:56,110 --> 00:27:00,450 faster for a light atom to change its motion by exchanging 478 00:27:00,450 --> 00:27:02,670 momentum with electromagnetic field. 479 00:27:02,670 --> 00:27:06,290 And at the same time, the transition in helium 480 00:27:06,290 --> 00:27:09,750 is narrower, and therefore slower. 481 00:27:09,750 --> 00:27:14,980 So for metastable helium, the two time scales are comparable. 482 00:27:14,980 --> 00:27:18,260 | if you really want to do quantitative experiments 483 00:27:18,260 --> 00:27:24,110 with metastable helium, you may need a more complicated theory. 484 00:27:26,810 --> 00:27:29,990 But, of course, if something is sort of simple in one case, 485 00:27:29,990 --> 00:27:32,760 the richer situation provides another opportunity 486 00:27:32,760 --> 00:27:33,810 for research. 487 00:27:33,810 --> 00:27:36,580 And I know, for instance, Hal Metcalf, a good colleague 488 00:27:36,580 --> 00:27:40,540 and friend of mine, he focused for a while 489 00:27:40,540 --> 00:27:43,690 on studying laser cooling of metastable helium, 490 00:27:43,690 --> 00:27:45,800 because he was interested what happens 491 00:27:45,800 --> 00:27:50,030 if those simple approximations don't work anymore? 492 00:27:50,030 --> 00:27:53,059 Then the internal motion and the external motion 493 00:27:53,059 --> 00:27:54,350 becomes sort of more entangled. 494 00:27:54,350 --> 00:27:57,710 You have to treat them together, you cannot separate them 495 00:27:57,710 --> 00:27:59,840 by a hierarchy of time scales. 496 00:28:03,630 --> 00:28:05,130 Any questions about that? 497 00:28:14,680 --> 00:28:19,760 OK, now we have done, you know, everything which is complicated 498 00:28:19,760 --> 00:28:21,870 has been approximated away now. 499 00:28:21,870 --> 00:28:23,718 And now we have a very simple result. 500 00:28:27,390 --> 00:28:29,780 So I'm pretty much repeating what 501 00:28:29,780 --> 00:28:34,510 you have done your homework, but now with expressions which 502 00:28:34,510 --> 00:28:40,480 have still the quantum character of the Optical Bloch Equations. 503 00:28:40,480 --> 00:28:43,580 So let's assume that we have an external electric field. 504 00:28:51,340 --> 00:29:03,250 it's parametrized like that, and at this point, by providing it 505 00:29:03,250 --> 00:29:07,310 like this, I leave it open whether we have a standing 506 00:29:07,310 --> 00:29:10,630 wave-- if you have a traveling wave, of course, 507 00:29:10,630 --> 00:29:12,860 we have a phase factor, kr, here-- 508 00:29:12,860 --> 00:29:15,900 if you have a standing wave, we don't have this phase factor. 509 00:29:15,900 --> 00:29:18,970 So I have parametrized the electromagnetic field 510 00:29:18,970 --> 00:29:24,780 in such a way that we can derive a general expression. 511 00:29:24,780 --> 00:29:28,490 And then we will discuss what happens in a traveling wave, 512 00:29:28,490 --> 00:29:29,990 and what happens in a standing wave. 513 00:29:32,900 --> 00:29:38,090 I make one more assumption here, namely that the polarization 514 00:29:38,090 --> 00:29:41,032 is independent of position. 515 00:29:41,032 --> 00:29:43,480 I'm just lazy if I take derivatives 516 00:29:43,480 --> 00:29:44,760 of the electric field. 517 00:29:44,760 --> 00:29:46,920 I want to take a derivative with the amplitude 518 00:29:46,920 --> 00:29:48,450 of the electric field, and I want 519 00:29:48,450 --> 00:29:50,040 to take a derivative of the phase 520 00:29:50,040 --> 00:29:51,740 of the electromagnetic field. 521 00:29:51,740 --> 00:29:55,070 I do not want to take a derivative of the polarization. 522 00:29:55,070 --> 00:29:56,800 Of course, what I'm throwing out here, 523 00:29:56,800 --> 00:29:59,190 is all of the interesting cooling mechanisms 524 00:29:59,190 --> 00:30:01,460 of polarization gradient cooling, when 525 00:30:01,460 --> 00:30:03,790 the polarization of the light field 526 00:30:03,790 --> 00:30:07,330 spirals around in three dimensions. 527 00:30:07,330 --> 00:30:09,550 We'll talk a little bit about that later on, 528 00:30:09,550 --> 00:30:11,614 I just want to keep things simple here. 529 00:30:16,700 --> 00:30:19,320 OK, so now if you take the gradient 530 00:30:19,320 --> 00:30:23,100 of the electromagnetic field, finally, we 531 00:30:23,100 --> 00:30:25,200 have approximated everything which 532 00:30:25,200 --> 00:30:26,870 complicates our life away. 533 00:30:26,870 --> 00:30:28,320 And we have two terms. 534 00:30:28,320 --> 00:30:30,920 One is the gradient x on the amplitude 535 00:30:30,920 --> 00:30:34,080 of the electromagnetic field, or the gradient x on the phase. 536 00:30:36,950 --> 00:30:38,960 And since we have taken a derivative, when 537 00:30:38,960 --> 00:30:40,890 we take the derivative of the phase, 538 00:30:40,890 --> 00:30:47,150 because of the chain rule, the cosine changes into sine. 539 00:30:47,150 --> 00:30:51,230 So therefore, the gradient of the electromagnetic field 540 00:30:51,230 --> 00:30:55,920 has a cosine in phase and an in quadrature component. 541 00:30:55,920 --> 00:31:00,150 And now, for the force, we have to multiply 542 00:31:00,150 --> 00:31:02,510 this with the dipole moment. 543 00:31:02,510 --> 00:31:04,240 And, of course, we will average over 544 00:31:04,240 --> 00:31:07,120 one cycle of the electromagnetic wave. 545 00:31:07,120 --> 00:31:09,260 And, of course, that's what we discussed. 546 00:31:09,260 --> 00:31:12,970 There's is a u and v component of the Bloch vector, which 547 00:31:12,970 --> 00:31:15,980 gives rise to an in phase and in quadrature component 548 00:31:15,980 --> 00:31:18,720 of the dipole moment. 549 00:31:18,720 --> 00:31:22,060 And now, if you combine-- if you multiply-- 550 00:31:22,060 --> 00:31:29,860 the gradient of the electric field with the dipole moment, 551 00:31:29,860 --> 00:31:30,940 and average. 552 00:31:30,940 --> 00:31:33,250 The cosine part goes with the cosine part, 553 00:31:33,250 --> 00:31:35,220 the sine part goes with the sine part, 554 00:31:35,220 --> 00:31:38,910 and the cos terms average out. 555 00:31:38,910 --> 00:31:42,130 So therefore, we have the simple result 556 00:31:42,130 --> 00:31:46,710 that we have two contributions to the [? old ?] 557 00:31:46,710 --> 00:31:49,530 mechanical force of light. 558 00:31:49,530 --> 00:31:55,280 And the u component-- the in phase component-- 559 00:31:55,280 --> 00:32:00,110 of the dipole moment goes with the amplitude of the gradient. 560 00:32:00,110 --> 00:32:02,910 And the in quadrature component goes 561 00:32:02,910 --> 00:32:05,900 with the gradient of the phase. 562 00:32:05,900 --> 00:32:09,100 So in other words, when it comes to light forces, 563 00:32:09,100 --> 00:32:12,640 the cosine component and the sine component 564 00:32:12,640 --> 00:32:15,160 of the dipole moment talk, actually, 565 00:32:15,160 --> 00:32:18,590 to two different quantities associated 566 00:32:18,590 --> 00:32:21,280 with the electromagnetic field. 567 00:32:21,280 --> 00:32:25,860 It's just in phase and in quadrature. 568 00:32:25,860 --> 00:32:26,430 OK. 569 00:32:26,430 --> 00:32:36,920 Now I just have to tell you a few definitions and names. 570 00:32:36,920 --> 00:32:40,640 This component, which goes with the in phase component, 571 00:32:40,640 --> 00:32:43,160 is called the reactive force. 572 00:32:43,160 --> 00:32:49,060 Whereas the other one is called the dissipative force. 573 00:32:49,060 --> 00:32:52,010 The name, of course, comes that if you have a harmonic 574 00:32:52,010 --> 00:32:56,860 oscillator, the component which oscillates in quadrature 575 00:32:56,860 --> 00:33:01,050 absorbs directly energy, whereas the in phase component 576 00:33:01,050 --> 00:33:04,480 does not absorb energy. 577 00:33:04,480 --> 00:33:06,500 I'll leave it for a little bit later 578 00:33:06,500 --> 00:33:10,770 in our discussion how you can have a force and not 579 00:33:10,770 --> 00:33:12,370 exchange energy. 580 00:33:12,370 --> 00:33:14,140 That's a little bit a mystery when 581 00:33:14,140 --> 00:33:17,130 we talk about cooling with a reactive force. 582 00:33:17,130 --> 00:33:21,520 We will have to scrutinize what happens with energy, 583 00:33:21,520 --> 00:33:24,350 because at least in the most basic, 584 00:33:24,350 --> 00:33:28,580 the in phase component of the oscillating system 585 00:33:28,580 --> 00:33:32,750 does not extract energy from the electromagnetic field. 586 00:33:32,750 --> 00:33:35,060 So maybe just believe that for now, 587 00:33:35,060 --> 00:33:40,190 and we will scrutinize it later. 588 00:33:40,190 --> 00:33:42,980 Any questions? 589 00:33:42,980 --> 00:33:43,850 Am I going too fast? 590 00:33:47,950 --> 00:33:48,960 Am I going too slow? 591 00:33:53,100 --> 00:33:55,230 Does it mean about right? 592 00:33:55,230 --> 00:33:55,730 OK. 593 00:33:59,730 --> 00:34:05,180 So we have now obtained the reactive force, which is also 594 00:34:05,180 --> 00:34:07,550 called the dipole force or stimulated force, 595 00:34:07,550 --> 00:34:10,000 and the dissipated force, which is also 596 00:34:10,000 --> 00:34:11,840 called the spontaneous force. 597 00:34:14,820 --> 00:34:19,190 The next thing is purely nomenclature. 598 00:34:19,190 --> 00:34:23,510 We want to introduce appropriate vectors 599 00:34:23,510 --> 00:34:26,559 which point along the gradient of the phase, 600 00:34:26,559 --> 00:34:31,909 and which point along the gradient of the amplitude. 601 00:34:31,909 --> 00:34:34,570 And this is done here. 602 00:34:34,570 --> 00:34:36,659 So we want to say that the reactive 603 00:34:36,659 --> 00:34:39,800 and the dissipative force are written 604 00:34:39,800 --> 00:34:44,852 in a very sort of similar way, but the one off them 605 00:34:44,852 --> 00:34:47,409 points in the direction of a vector, alpha, 606 00:34:47,409 --> 00:34:51,110 the other one in the direction of a vector, beta. 607 00:34:51,110 --> 00:34:55,730 The beta vector is the gradient of the phase. 608 00:34:55,730 --> 00:34:59,660 If the phase is kr, the gradient of the phase 609 00:34:59,660 --> 00:35:03,200 is simply k, the k vector of the electromagnetic wave. 610 00:35:03,200 --> 00:35:06,130 So that will come in handy in just a few seconds. 611 00:35:06,130 --> 00:35:10,510 Whereas the reactive force points 612 00:35:10,510 --> 00:35:15,160 into the direction of the gradient of the amplitude, 613 00:35:15,160 --> 00:35:18,220 but we never want to talk about electromagnetic field 614 00:35:18,220 --> 00:35:18,720 amplitude. 615 00:35:18,720 --> 00:35:23,760 For us, we always use the electric field in terms of you 616 00:35:23,760 --> 00:35:25,590 parametrize the Rabi frequency. 617 00:35:25,590 --> 00:35:28,190 The Rabi frequency is really the atomic unit 618 00:35:28,190 --> 00:35:29,710 of the electric field. 619 00:35:29,710 --> 00:35:32,470 So therefore, this vector which involves now 620 00:35:32,470 --> 00:35:36,810 the gradient of the amplitude, becomes a normalized gradient 621 00:35:36,810 --> 00:35:40,100 of the Rabi frequency. 622 00:35:40,100 --> 00:35:44,557 So these are the two fundamental light forces, the reactive 623 00:35:44,557 --> 00:35:45,640 and the dissipative force. 624 00:35:53,190 --> 00:35:55,320 OK. 625 00:35:55,320 --> 00:35:59,710 Now we are going to have a quick look at those two forces. 626 00:35:59,710 --> 00:36:03,370 So therefore, I'm now discussing the two 627 00:36:03,370 --> 00:36:06,740 simple, but already very characteristic 628 00:36:06,740 --> 00:36:08,650 cases for those forces. 629 00:36:08,650 --> 00:36:14,090 One is we want to look at a plane travelling wave, 630 00:36:14,090 --> 00:36:18,190 and then we want to look at a pure standing wave. 631 00:36:18,190 --> 00:36:22,090 And the beauty of it is in a plane travelling wave, 632 00:36:22,090 --> 00:36:25,690 you've a plain wave, the amplitude of the electric field 633 00:36:25,690 --> 00:36:29,120 is constant everywhere, it just oscillates. 634 00:36:29,120 --> 00:36:33,380 And therefore, this alpha vector is zero. 635 00:36:33,380 --> 00:36:36,710 And in a few minutes, we look at this standing wave, 636 00:36:36,710 --> 00:36:39,830 and in the standing wave, the beta vector is zero. 637 00:36:39,830 --> 00:36:44,660 So the travelling wave and the standing wave allows us now 638 00:36:44,660 --> 00:36:49,080 to look at the two forces separately in two physical 639 00:36:49,080 --> 00:36:53,110 situations which are-- as you will see later-- 640 00:36:53,110 --> 00:36:56,276 highly relevant for experiments. 641 00:36:56,276 --> 00:37:00,376 Well, I don't think I have to tell anybody in this room-- 642 00:37:00,376 --> 00:37:01,750 maybe except some people who take 643 00:37:01,750 --> 00:37:03,640 the course for [? Brett's ?] requirement-- 644 00:37:03,640 --> 00:37:06,620 that standing waves or optical lattices 645 00:37:06,620 --> 00:37:08,260 are common in all labs. 646 00:37:08,260 --> 00:37:10,960 And similar, traveling wave beams 647 00:37:10,960 --> 00:37:15,100 are used, for instance, for decelerating atomic beams. 648 00:37:15,100 --> 00:37:17,270 So we illustrate the two forces now, 649 00:37:17,270 --> 00:37:21,615 but we already discussing two experimentally very important 650 00:37:21,615 --> 00:37:22,115 geometries. 651 00:37:27,010 --> 00:37:29,850 OK, plain travelling waves. 652 00:37:29,850 --> 00:37:34,860 I mean, I just really plug now the electromagnetic field 653 00:37:34,860 --> 00:37:37,930 for plain travelling wave, I plug it into the equations 654 00:37:37,930 --> 00:37:39,700 above. 655 00:37:39,700 --> 00:37:42,090 The important thing is-- and which simplifies things 656 00:37:42,090 --> 00:37:45,390 a lot-- is that the alpha vector is zero. 657 00:37:45,390 --> 00:37:47,420 The amplitude of the plain wave is-- 658 00:37:47,420 --> 00:37:49,810 that's what a plain wave is-- constant. 659 00:37:49,810 --> 00:37:54,040 And the beta vector is the gradient of the Rabi frequency 660 00:37:54,040 --> 00:37:58,630 is just the k vector of the light. 661 00:37:58,630 --> 00:38:04,030 So therefore, the dissipative or spontaneous light force 662 00:38:04,030 --> 00:38:10,680 was the Rabi frequency times the v component of the Optical 663 00:38:10,680 --> 00:38:15,340 Bloch vector times h bar k. 664 00:38:15,340 --> 00:38:21,800 And this has a very simple interpretation 665 00:38:21,800 --> 00:38:24,590 because the steady state solution of the Optical Bloch 666 00:38:24,590 --> 00:38:28,100 Equation is nothing else than gamma times the excited state 667 00:38:28,100 --> 00:38:29,770 population. 668 00:38:29,770 --> 00:38:35,420 And since the excited state scatters, or emits photons 669 00:38:35,420 --> 00:38:38,880 at a rate, gamma, this is nothing else 670 00:38:38,880 --> 00:38:43,020 than the number of absorbed emitted 671 00:38:43,020 --> 00:38:47,680 scattered photons times h bar k. 672 00:38:47,680 --> 00:38:50,470 We are in steady state here, so the number 673 00:38:50,470 --> 00:38:54,840 of photons which are emitted into all space, into the vacuum 674 00:38:54,840 --> 00:38:57,450 has to be the number of photons which 675 00:38:57,450 --> 00:39:00,870 has been absorbed into the laser beam. 676 00:39:00,870 --> 00:39:04,010 So therefore, the interpretation here is very simple, 677 00:39:04,010 --> 00:39:05,410 you have a laser beam. 678 00:39:05,410 --> 00:39:09,150 Every time the atom absorbs a photon, it 679 00:39:09,150 --> 00:39:13,360 receives a recoil transfer, h bar k. 680 00:39:13,360 --> 00:39:16,910 Now, afterwards, the photon is scattered, 681 00:39:16,910 --> 00:39:20,710 but the scattering is symmetric, and does not 682 00:39:20,710 --> 00:39:23,350 impart the force onto the atom. 683 00:39:23,350 --> 00:39:25,320 And this is what we discussed earlier. 684 00:39:25,320 --> 00:39:27,820 The quantum part of the electromagnetic field 685 00:39:27,820 --> 00:39:29,090 is symmetric. 686 00:39:29,090 --> 00:39:32,450 And here, we see sort of what it means visually. 687 00:39:32,450 --> 00:39:35,140 Spontaneous emission goes equally 688 00:39:35,140 --> 00:39:39,770 probable in opposite direction, and there is no net force, 689 00:39:39,770 --> 00:39:41,720 but there is heating as we discussed later. 690 00:39:45,080 --> 00:39:54,120 So we can plug in the solution-- the steady state solution-- 691 00:39:54,120 --> 00:39:56,200 from the Optical Bloch Equation. 692 00:39:56,200 --> 00:39:58,540 That's our Lorentzian. 693 00:39:58,540 --> 00:40:00,710 We have discussed power broadening already 694 00:40:00,710 --> 00:40:03,710 when we discussed the Optical Bloch Equation. 695 00:40:03,710 --> 00:40:07,260 And, of course, I just remind you 696 00:40:07,260 --> 00:40:11,830 if you saturate the system with a strong laser power, what you 697 00:40:11,830 --> 00:40:14,630 can obtain is that half of the population 698 00:40:14,630 --> 00:40:17,115 is in the excited state, and half of the population 699 00:40:17,115 --> 00:40:18,880 is in the ground state. 700 00:40:18,880 --> 00:40:24,290 Under those circumstances, do you get the maximum force. 701 00:40:24,290 --> 00:40:26,440 And the maximum spontaneous force 702 00:40:26,440 --> 00:40:31,080 is the momentum per photon times the maximum rate at which 703 00:40:31,080 --> 00:40:33,070 a two-level atom can scatter photons. 704 00:40:36,820 --> 00:40:39,100 So that's, in a nutshell, I think 705 00:40:39,100 --> 00:40:42,960 all you have to know about the spontaneous light force 706 00:40:42,960 --> 00:40:46,600 as far as the force is concerned. 707 00:40:46,600 --> 00:40:50,640 But we still have to discuss the heating associated 708 00:40:50,640 --> 00:40:53,225 with a spontaneous force, which are the force fluctuations. 709 00:40:55,617 --> 00:40:56,116 Questions? 710 00:41:01,780 --> 00:41:07,280 Well, then let's do a similar discussion 711 00:41:07,280 --> 00:41:09,015 with reactive forces. 712 00:41:12,390 --> 00:41:20,260 Which is a little bit richer because, well, you will see. 713 00:41:20,260 --> 00:41:23,380 So a standing wave is parametrized here. 714 00:41:30,260 --> 00:41:32,880 We only have an alpha vector, not a beta vector 715 00:41:32,880 --> 00:41:36,220 now, because everything is in phase, 716 00:41:36,220 --> 00:41:39,990 there is no phase phi of r. 717 00:41:39,990 --> 00:41:45,630 So therefore, there is no dissipative force, 718 00:41:45,630 --> 00:41:47,730 the only force is a reactive force. 719 00:41:50,260 --> 00:41:54,160 Since the reactive force depends only on u, 720 00:41:54,160 --> 00:41:58,200 there is no exchange of energy. 721 00:41:58,200 --> 00:42:04,870 So that raises the question, what is really going on it is 722 00:42:04,870 --> 00:42:08,970 only the reactive part with u, how can the atom really-- 723 00:42:08,970 --> 00:42:13,480 How can the motion change if it cannot exchange energy? 724 00:42:13,480 --> 00:42:17,070 Well, we'll come to that in different pieces. 725 00:42:17,070 --> 00:42:20,080 The one part I want to sort of discuss here 726 00:42:20,080 --> 00:42:25,772 is that you should-- I want to show you 727 00:42:25,772 --> 00:42:27,455 that at the level of our discussion 728 00:42:27,455 --> 00:42:31,000 we can already understand that the atom is actually 729 00:42:31,000 --> 00:42:35,700 doing a redistribution of energy and momentum. 730 00:42:35,700 --> 00:42:39,030 Well, if I only hold onto my standing wave, 731 00:42:39,030 --> 00:42:42,520 there is no energy exchange because the atom is always 732 00:42:42,520 --> 00:42:44,580 in phase with the standing wave-- 733 00:42:44,580 --> 00:42:48,880 the atomic dipole-- which is this one [INAUDIBLE] force. 734 00:42:48,880 --> 00:42:54,580 But I can now do what everybody does in the lab. 735 00:42:54,580 --> 00:42:58,022 We generate the standing wave as a superposition of two 736 00:42:58,022 --> 00:42:58,730 travelling waves. 737 00:43:03,210 --> 00:43:08,600 And if I have a superposition of two traveling waves, 738 00:43:08,600 --> 00:43:12,560 there is a point in space where the phases of the traveling 739 00:43:12,560 --> 00:43:14,150 waves are the same. 740 00:43:14,150 --> 00:43:16,450 Then one phase is advanced in phase, 741 00:43:16,450 --> 00:43:20,390 and the other side one of the waves is lagging in phase. 742 00:43:20,390 --> 00:43:25,300 And for pedagogical reasons, I pick the point where the two 743 00:43:25,300 --> 00:43:29,110 travelling waves are 90 degree phase shifted. 744 00:43:34,860 --> 00:43:36,480 OK. 745 00:43:36,480 --> 00:43:45,550 Now the situation is that we have our electric field 746 00:43:45,550 --> 00:43:50,330 and what is responsible for the reactive light force is the u 747 00:43:50,330 --> 00:43:53,780 part of the dipole moment of the Optical Bloch vector, which 748 00:43:53,780 --> 00:43:56,370 oscillates in phase. 749 00:43:56,370 --> 00:44:00,900 But nobody prevents me from analyzing it with respect 750 00:44:00,900 --> 00:44:03,310 to E1 and E2. 751 00:44:03,310 --> 00:44:06,780 And so if I ask now, I say, I have one laser beam. 752 00:44:06,780 --> 00:44:08,000 I have another laser beam. 753 00:44:08,000 --> 00:44:11,140 And if you want, after the laser beams have crossed, 754 00:44:11,140 --> 00:44:14,260 you can put in two photo-diodes, and you can not only 755 00:44:14,260 --> 00:44:15,855 ask, what happens to the standing 756 00:44:15,855 --> 00:44:20,165 wave, you can also ask, what happens to each traveling wave? 757 00:44:20,165 --> 00:44:21,040 Are photons absorbed? 758 00:44:21,040 --> 00:44:23,110 Or what happens? 759 00:44:23,110 --> 00:44:28,040 And now, what is obvious is if you have an harmonic 760 00:44:28,040 --> 00:44:31,360 oscillator, and you have an oscillating dipole moment, 761 00:44:31,360 --> 00:44:35,180 if the dipole moment is withing 0 and 180 762 00:44:35,180 --> 00:44:39,650 degrees with a dry field, you absorb power. 763 00:44:39,650 --> 00:44:44,840 If it is in the two other quadrants, 764 00:44:44,840 --> 00:44:47,390 it emits, or delivers power. 765 00:44:47,390 --> 00:44:50,610 And now you see, that we are in a situation-- 766 00:44:50,610 --> 00:44:56,100 which I've peak here-- where one travelling wave loses energy, 767 00:44:56,100 --> 00:44:59,900 and the other traveling wave gains energy. 768 00:44:59,900 --> 00:45:01,590 And there is a classic experiment, 769 00:45:01,590 --> 00:45:04,880 which was done by Bill Phillips a while ago-- he had atoms 770 00:45:04,880 --> 00:45:09,190 in an optical lattice, and they were sloshing. 771 00:45:09,190 --> 00:45:13,410 And she could really measure that when the atoms were 772 00:45:13,410 --> 00:45:18,500 accelerated this way, one of the laser beams gained power, 773 00:45:18,500 --> 00:45:20,260 and the other one lost power. 774 00:45:20,260 --> 00:45:22,310 So while the atoms where sloshing 775 00:45:22,310 --> 00:45:26,850 in momentum space, the power between the two laser beams 776 00:45:26,850 --> 00:45:29,660 was distributed back and forth. 777 00:45:29,660 --> 00:45:32,650 So this is the character of the dissipative force, 778 00:45:32,650 --> 00:45:38,080 that it redistributes momentum and energy between the two 779 00:45:38,080 --> 00:45:38,890 traveling waves. 780 00:45:48,422 --> 00:45:49,005 Any questions? 781 00:45:54,590 --> 00:45:55,090 Boris? 782 00:45:55,090 --> 00:45:56,631 AUDIENCE: But if there's a net change 783 00:45:56,631 --> 00:45:58,823 in the patterns of motion, the energy still 784 00:45:58,823 --> 00:46:00,066 has to come from somewhere. 785 00:46:00,066 --> 00:46:01,557 Right? 786 00:46:01,557 --> 00:46:03,048 Can you make that [INAUDIBLE] 787 00:46:16,979 --> 00:46:18,520 PROFESSOR: This question has bothered 788 00:46:18,520 --> 00:46:21,010 in many different iterations. 789 00:46:21,010 --> 00:46:25,310 And at some point, I found a very, very easy answer to it. 790 00:46:25,310 --> 00:46:27,830 And this is one of your homework assignments 791 00:46:27,830 --> 00:46:29,410 for homework number 10. 792 00:46:29,410 --> 00:46:31,360 There's a very simple example where 793 00:46:31,360 --> 00:46:33,380 you will realize where the energy comes. 794 00:46:36,840 --> 00:46:40,270 What happens is, in steady state, 795 00:46:40,270 --> 00:46:42,270 if everything is stationary, of course, 796 00:46:42,270 --> 00:46:44,070 there can't be an exchange. 797 00:46:44,070 --> 00:46:46,125 But, if you're for instance, saying-- 798 00:46:46,125 --> 00:46:50,300 and I give you no part of the solution-- if you have atoms, 799 00:46:50,300 --> 00:46:54,240 and they are attracted by the dipole force, 800 00:46:54,240 --> 00:46:58,260 there's a strong laser beam and the atoms are accelerated in, 801 00:46:58,260 --> 00:47:02,450 and you're now asking, where does the energy come from? 802 00:47:02,450 --> 00:47:06,110 Because all of what the atoms do, at least according 803 00:47:06,110 --> 00:47:09,010 to what I'm telling you, they redistribute 804 00:47:09,010 --> 00:47:12,810 photons of energy, h bar omega, into photons 805 00:47:12,810 --> 00:47:14,260 of energy, h bar omega. 806 00:47:14,260 --> 00:47:17,080 So where does the energy come from? 807 00:47:17,080 --> 00:47:19,590 You have to go higher in the approximation here 808 00:47:19,590 --> 00:47:22,471 to understand where the energy comes from. 809 00:47:22,471 --> 00:47:23,970 Let me just tell you one thing, this 810 00:47:23,970 --> 00:47:27,690 is actually something which I've encountered several times. 811 00:47:27,690 --> 00:47:31,850 And it can be really confusing, I've 812 00:47:31,850 --> 00:47:35,130 really proven to mathematically the this is false. 813 00:47:35,130 --> 00:47:38,000 So in this level of treatment, with these very simple 814 00:47:38,000 --> 00:47:41,960 concepts, we correctly obtained the optical force. 815 00:47:41,960 --> 00:47:47,880 But if you're now asking, where does the energy go? 816 00:47:47,880 --> 00:47:49,920 You have to go deeper. 817 00:47:49,920 --> 00:47:51,900 And, for instance, what happens is 818 00:47:51,900 --> 00:47:54,760 if you take a cloud of atoms, and the cloud of atoms 819 00:47:54,760 --> 00:47:57,310 moves in an out of a laser beam-- 820 00:47:57,310 --> 00:48:01,100 just think of it as center of mass dipole 821 00:48:01,100 --> 00:48:06,150 oscillations of a Boson-Einstein condensate in an optical dipole 822 00:48:06,150 --> 00:48:10,540 trap-- what happens is when the atoms move in and out, 823 00:48:10,540 --> 00:48:14,830 they actually act as a phase modulator for the laser beam. 824 00:48:14,830 --> 00:48:18,270 The index of the refraction of your laser beam is changing. 825 00:48:18,270 --> 00:48:21,530 So therefore-- and we don't deal with that 826 00:48:21,530 --> 00:48:24,170 when he replace the electromagnetic field by a c 827 00:48:24,170 --> 00:48:29,940 number-- your laser beam, when the atoms move in will actually 828 00:48:29,940 --> 00:48:33,380 show a frequency modulation. 829 00:48:33,380 --> 00:48:36,440 And you will find-- and it's a wonderful homework assignment, 830 00:48:36,440 --> 00:48:38,270 I'm pretty proud that we created it, 831 00:48:38,270 --> 00:48:40,020 and that it works out so easily-- you 832 00:48:40,020 --> 00:48:42,760 will find that the kinetic energy of the atoms 833 00:48:42,760 --> 00:48:45,740 is exactly compensated by the energy shift 834 00:48:45,740 --> 00:48:49,550 the photons which have passed through the atoms. 835 00:48:49,550 --> 00:48:51,940 But I'm giving you a very advanced answer. 836 00:48:51,940 --> 00:48:55,130 It's often very difficult if you have a simple picture 837 00:48:55,130 --> 00:48:58,260 for the force to figure out what happens to the energy. 838 00:49:01,470 --> 00:49:05,030 Actually, let me give you the other example. 839 00:49:05,030 --> 00:49:07,190 When we have the previous example, 840 00:49:07,190 --> 00:49:09,990 where we have atoms and we have a laser beam 841 00:49:09,990 --> 00:49:13,550 which cools with a radiation pressure. 842 00:49:13,550 --> 00:49:26,790 This was our previous example. 843 00:49:26,790 --> 00:49:28,640 Where does the energy go? 844 00:49:28,640 --> 00:49:32,150 I have a wonderful correct-- within the assumptions-- 845 00:49:32,150 --> 00:49:34,610 an exact expression for the force. 846 00:49:34,610 --> 00:49:35,700 Where does the energy go? 847 00:49:39,660 --> 00:49:44,530 The atoms have a counter-propagating laser beam. 848 00:49:44,530 --> 00:49:49,080 They scatter photons with this rate. 849 00:49:49,080 --> 00:49:52,590 Every time they scatter a photon, they slow down 850 00:49:52,590 --> 00:49:53,750 by h bar k. 851 00:49:53,750 --> 00:49:56,430 They lose energy. 852 00:49:56,430 --> 00:49:58,310 We have a complete description what 853 00:49:58,310 --> 00:50:03,189 happens in the momentum picture and in forces. 854 00:50:03,189 --> 00:50:04,480 But where does the energy come? 855 00:50:04,480 --> 00:50:05,646 Or where does the energy go? 856 00:50:05,646 --> 00:50:07,964 The energy where the atoms have lost? 857 00:50:07,964 --> 00:50:10,374 AUDIENCE: They absorb a slightly high frequency 858 00:50:10,374 --> 00:50:12,302 with a Doppler shift. 859 00:50:12,302 --> 00:50:16,158 And then slow down and then re-release 860 00:50:16,158 --> 00:50:18,429 a photon with slightly lower energy. 861 00:50:18,429 --> 00:50:19,220 PROFESSOR: Exactly. 862 00:50:19,220 --> 00:50:21,850 So in that case, it actually depends which way. 863 00:50:21,850 --> 00:50:25,160 When the atom loses energy, the energy 864 00:50:25,160 --> 00:50:27,760 goes into blue-shifted photons. 865 00:50:27,760 --> 00:50:30,730 I'm just saying, in order to address the question, what 866 00:50:30,730 --> 00:50:34,290 happens to the energy, we have to bring in our physics, which 867 00:50:34,290 --> 00:50:37,830 we didn't even consider to address here. 868 00:50:37,830 --> 00:50:42,120 It is now the physics of the spontaneously emitted photons. 869 00:50:42,120 --> 00:50:45,380 We said for the force, spontaneous emission vacuum 870 00:50:45,380 --> 00:50:47,710 fluctuations can be eliminated. 871 00:50:47,710 --> 00:50:52,530 But if you try to understand where does the energy go, 872 00:50:52,530 --> 00:50:55,690 we have to go to those terms, and we even 873 00:50:55,690 --> 00:50:59,990 find the physics of Doppler shifts, which we didn't put in, 874 00:50:59,990 --> 00:51:01,510 which we didn't even assume here. 875 00:51:01,510 --> 00:51:05,630 But, of course, it has to be self consistent. 876 00:51:05,630 --> 00:51:10,210 So therefore, you sometimes have to go to very different physics 877 00:51:10,210 --> 00:51:12,350 which isn't even covered by your equation 878 00:51:12,350 --> 00:51:14,890 to see the flip side of the coin. 879 00:51:14,890 --> 00:51:17,710 One side is forces, which are very simple, 880 00:51:17,710 --> 00:51:21,740 but the other side of the coin can be more difficult. 881 00:51:21,740 --> 00:51:26,790 We'll actually come back to that when we talk about cooling 882 00:51:26,790 --> 00:51:28,120 [INAUDIBLE]. 883 00:51:28,120 --> 00:51:31,540 Cooling with a stimulated force. 884 00:51:31,540 --> 00:51:36,890 How can a stimulated force cool, because it's all u. 885 00:51:36,890 --> 00:51:38,360 Well, we'll get there. 886 00:51:38,360 --> 00:51:43,060 And it will again be-- the energy-- 887 00:51:43,060 --> 00:51:46,720 will be extracted through spontaneous emission 888 00:51:46,720 --> 00:51:48,860 from all those side [INAUDIBLE] and all that. 889 00:51:48,860 --> 00:51:50,845 So we have to trace down the energy, 890 00:51:50,845 --> 00:51:52,220 and we successfully will do that. 891 00:51:52,220 --> 00:51:55,180 But energy can be subtle. 892 00:51:55,180 --> 00:51:55,930 Forces are simple. 893 00:52:02,804 --> 00:52:03,470 Other questions? 894 00:52:17,600 --> 00:52:21,200 Actually, since I'm in chatty mood, 895 00:52:21,200 --> 00:52:25,150 when I explained to you where the energy goes 896 00:52:25,150 --> 00:52:28,740 in a dipole trap-- that it's a phase modulation for the laser 897 00:52:28,740 --> 00:52:31,220 beam-- I have been using optical dipole traps 898 00:52:31,220 --> 00:52:33,940 in my laboratory for many, many years, 899 00:52:33,940 --> 00:52:36,770 before I really figured out where the energy goes. 900 00:52:36,770 --> 00:52:39,680 And I'm almost 99% certain if you 901 00:52:39,680 --> 00:52:42,740 go to DAMOP, and ask some of the experts 902 00:52:42,740 --> 00:52:45,420 on cooling and trapping, and ask them to figure out 903 00:52:45,420 --> 00:52:48,030 this problem, most of the people will not 904 00:52:48,030 --> 00:52:49,900 be able to give you this answer. 905 00:52:49,900 --> 00:52:53,740 I've actually not found the answer in any standard textbook 906 00:52:53,740 --> 00:52:56,940 of laser cooling, until I eventually found it myself 907 00:52:56,940 --> 00:53:01,620 and posted and prepared it as a homework assignment. 908 00:53:01,620 --> 00:53:04,290 So many, many people may not be able to give you 909 00:53:04,290 --> 00:53:07,340 the correct answer, where does the energy go 910 00:53:07,340 --> 00:53:12,040 when atoms slosh around in a dipole trap. 911 00:53:12,040 --> 00:53:15,510 Try it out at DAMOP, it may be fun. 912 00:53:15,510 --> 00:53:16,010 OK. 913 00:53:16,010 --> 00:53:22,690 So back to the simple physics of forces. 914 00:53:22,690 --> 00:53:24,520 We don't need to understand the energy, 915 00:53:24,520 --> 00:53:27,220 because we know the forces, but we'll come back to that. 916 00:53:27,220 --> 00:53:30,200 So we have the reactive force. 917 00:53:30,200 --> 00:53:33,840 The reactive force is written here. 918 00:53:33,840 --> 00:53:35,330 What have we done? 919 00:53:35,330 --> 00:53:38,240 Well, you remember the alpha vector 920 00:53:38,240 --> 00:53:41,210 was the gradient of the Rabi frequency, 921 00:53:41,210 --> 00:53:45,520 and we have to multiply with the u component of the steady state 922 00:53:45,520 --> 00:53:47,840 solution of the Optical Bloch Equation. 923 00:53:47,840 --> 00:53:49,540 Here it is, just put together. 924 00:53:52,600 --> 00:53:54,486 It's a nice expression. 925 00:53:54,486 --> 00:53:58,340 There are two things you should know about it. 926 00:53:58,340 --> 00:54:02,850 The first one is that you can actually 927 00:54:02,850 --> 00:54:05,955 write this force exactly as the gradient of a potential. 928 00:54:10,670 --> 00:54:13,620 So there is a dipole potential, and is 929 00:54:13,620 --> 00:54:15,860 exactly this dipole potential which 930 00:54:15,860 --> 00:54:18,920 is used as a trapping potential when you have dipole traps. 931 00:54:22,900 --> 00:54:27,700 Probably 99% of you use dipole traps in the limit 932 00:54:27,700 --> 00:54:30,050 where the detuning is large. 933 00:54:30,050 --> 00:54:33,780 And then the logarithm of 1 plus x simply becomes x, 934 00:54:33,780 --> 00:54:37,050 and you have the simple expression you are using. 935 00:54:37,050 --> 00:54:39,410 But in a way, this is remarkable. 936 00:54:39,410 --> 00:54:45,220 The dipole force can be derived from a potential, 937 00:54:45,220 --> 00:54:48,560 even if the detuning is small, and you 938 00:54:48,560 --> 00:54:52,090 have a hell of a lot of spontaneous scattering, which 939 00:54:52,090 --> 00:54:57,010 is usually not regarded as being due to conservative potential. 940 00:54:57,010 --> 00:54:59,010 So this expression, that the reactive force 941 00:54:59,010 --> 00:55:02,760 is a gradient of a potential, even applies to a situation 942 00:55:02,760 --> 00:55:07,490 where gamma spontaneous scattering [INAUDIBLE] 943 00:55:07,490 --> 00:55:09,610 And you're not simply in-- and this 944 00:55:09,610 --> 00:55:12,300 is what this last term is-- in the perturbative limit 945 00:55:12,300 --> 00:55:15,270 of the AC Stark Effect. 946 00:55:15,270 --> 00:55:15,770 OK. 947 00:55:15,770 --> 00:55:18,140 That's number one you should know about this expression. 948 00:55:18,140 --> 00:55:25,020 The second thing is if you look at this expression, 949 00:55:25,020 --> 00:55:28,640 the question is, what is the maximum reactive force? 950 00:55:28,640 --> 00:55:33,770 We talked about the maximum spontaneous, or dissipative 951 00:55:33,770 --> 00:55:37,110 force, which is h bar k, the momentum 952 00:55:37,110 --> 00:55:40,190 of a photon times gamms over 2. 953 00:55:40,190 --> 00:55:43,090 What is the maximum force you can get out 954 00:55:43,090 --> 00:55:45,750 of the reactive force? 955 00:55:45,750 --> 00:55:48,600 Well, of course, if you want a lot of force, 956 00:55:48,600 --> 00:55:50,660 you want to use a lot of laser power. 957 00:55:50,660 --> 00:55:54,750 Therefore, you want to use a high Rabi frequency. 958 00:55:54,750 --> 00:55:58,010 And now, if you want to use your laser power wisely, 959 00:55:58,010 --> 00:56:01,500 the question is, how should you pick the detuning? 960 00:56:01,500 --> 00:56:04,425 And you realize if you pick the detuning small, 961 00:56:04,425 --> 00:56:07,370 there is a prefector which goes to zero, 962 00:56:07,370 --> 00:56:09,815 but if you pick your detuning extremely large, 963 00:56:09,815 --> 00:56:12,410 the denominator kills you. 964 00:56:12,410 --> 00:56:17,200 And well, if you analyze it and or stare at this expression 965 00:56:17,200 --> 00:56:19,390 for more than a second, you realize 966 00:56:19,390 --> 00:56:23,710 that the optimum detuning is when the detuning is 967 00:56:23,710 --> 00:56:26,810 on the order of the Rabi frequency. 968 00:56:26,810 --> 00:56:30,980 And if you plug that in, you find 969 00:56:30,980 --> 00:56:33,420 that under those optimum conditions, 970 00:56:33,420 --> 00:56:37,720 the reactive force can be written again 971 00:56:37,720 --> 00:56:41,150 as the momentum of a photon-- it must 972 00:56:41,150 --> 00:56:45,480 be, the momentum of the photon is the unit of force, 973 00:56:45,480 --> 00:56:47,650 the quantum of force, which has to appear-- 974 00:56:47,650 --> 00:56:50,500 but not the frequency is not gamma, 975 00:56:50,500 --> 00:56:55,940 but it is the Rabi frequency. 976 00:56:55,940 --> 00:56:58,300 So the picture you should sort of have 977 00:56:58,300 --> 00:57:00,880 is that under those situations, just 978 00:57:00,880 --> 00:57:03,530 assume the standing wave consists of two traveling 979 00:57:03,530 --> 00:57:06,160 waves, and the atom is Rabi flopping. 980 00:57:06,160 --> 00:57:09,690 But it is Rabi flopping in a way that it goes up 981 00:57:09,690 --> 00:57:12,680 by taking a photon from one laser beam, 982 00:57:12,680 --> 00:57:15,520 and then it goes down by emitting the photon 983 00:57:15,520 --> 00:57:17,080 into the other laser beam. 984 00:57:17,080 --> 00:57:21,210 So during each Rabi flux cycle, it 985 00:57:21,210 --> 00:57:24,650 exchanges a momentum of 2 h bar k 986 00:57:24,650 --> 00:57:28,030 with the electromagnetic field. 987 00:57:28,030 --> 00:57:32,270 So therefore, the reactive force-- the stimulated force-- 988 00:57:32,270 --> 00:57:38,690 never saturates if you increase the Rabi frequency-- depending, 989 00:57:38,690 --> 00:57:41,220 of course, on your detuning, but if you choose the detuning 990 00:57:41,220 --> 00:57:43,630 correctly-- you can get a force which 991 00:57:43,630 --> 00:57:45,530 is just going further and further. 992 00:58:01,900 --> 00:58:02,770 Questions? 993 00:58:02,770 --> 00:58:03,617 Yes, Jen? 994 00:58:03,617 --> 00:58:06,102 AUDIENCE: Is this the same concept as the Raman constants, 995 00:58:06,102 --> 00:58:08,587 except that the frequencies are just the same? 996 00:58:08,587 --> 00:58:09,581 Or is it just a-- 997 00:58:13,570 --> 00:58:15,590 PROFESSOR: It is a Raman process. 998 00:58:15,590 --> 00:58:19,330 Actually, everything is a Raman process here. 999 00:58:19,330 --> 00:58:22,160 Because in laser cooling, in everything 1000 00:58:22,160 --> 00:58:26,125 which involves the mechanical-- the motion, the external degree 1001 00:58:26,125 --> 00:58:30,560 freedom of the atoms-- we have an atom in one momentum state. 1002 00:58:30,560 --> 00:58:33,760 We go-- light scattering always has to involve the excited 1003 00:58:33,760 --> 00:58:39,940 state-- and then we go down to a different momentum state. 1004 00:58:39,940 --> 00:58:42,760 So therefore, laser cooling is nothing else 1005 00:58:42,760 --> 00:58:47,329 than Raman processes in the external degree of freedom 1006 00:58:47,329 --> 00:58:48,960 of the atom. 1007 00:58:48,960 --> 00:58:51,780 You may be used to [INAUDIBLE] Raman process 1008 00:58:51,780 --> 00:58:54,105 more when you start in one hyperfine state, 1009 00:58:54,105 --> 00:58:57,140 or for molecules in one vibration rotation state, 1010 00:58:57,140 --> 00:59:00,950 and you go to another one, but, well, 1011 00:59:00,950 --> 00:59:03,180 for vibration, and rotational states, 1012 00:59:03,180 --> 00:59:06,640 and hyperfine states of it, Raman is used all the time. 1013 00:59:06,640 --> 00:59:09,970 I usually used it also for the external motion, 1014 00:59:09,970 --> 00:59:13,630 because I think it clearly brings out 1015 00:59:13,630 --> 00:59:15,790 common features between all those different Raman 1016 00:59:15,790 --> 00:59:17,850 processes. 1017 00:59:17,850 --> 00:59:25,280 So you can actually say that the reactive force is a stimulated 1018 00:59:25,280 --> 00:59:27,990 Raman process between two momentum states, where 1019 00:59:27,990 --> 00:59:31,350 both legs are stimulated by laser beams. 1020 00:59:31,350 --> 00:59:34,100 In a standing wave, it would be-- the two legs-- 1021 00:59:34,100 --> 00:59:36,990 would be stimulated by the two travelling wave components 1022 00:59:36,990 --> 00:59:38,950 of the standing wave. 1023 00:59:38,950 --> 00:59:44,300 Whereas the dissipative force is spontaneous Raman process, 1024 00:59:44,300 --> 00:59:47,327 where one leg is driven by the laser, and the other one 1025 00:59:47,327 --> 00:59:48,660 comes with spontaneous emission. 1026 00:59:52,870 --> 00:59:53,370 Nicky? 1027 00:59:53,370 --> 00:59:55,950 AUDIENCE: [INAUDIBLE] when the atoms 1028 00:59:55,950 --> 00:59:57,410 slow down, the [INAUDIBLE] 1029 01:00:07,078 --> 01:00:09,036 PROFESSOR: So the spontaneously emitted photon? 1030 01:00:09,036 --> 01:00:11,516 AUDIENCE: Yeah, I'm just thinking 1031 01:00:11,516 --> 01:00:15,484 of actually the stimulated force. 1032 01:00:15,484 --> 01:00:17,468 Because the atom must be stimulated. 1033 01:00:17,468 --> 01:00:20,940 [INAUDIBLE] I'm confused how when 1034 01:00:20,940 --> 01:00:24,908 the energy of the emitted photon must be slightly different 1035 01:00:24,908 --> 01:00:27,884 due to the phase modulation? 1036 01:00:27,884 --> 01:00:36,075 [INAUDIBLE] stimulated emission can be added to the frequency? 1037 01:00:36,075 --> 01:00:37,060 [INAUDIBLE] 1038 01:00:37,060 --> 01:00:40,820 PROFESSOR: I mean we had the discussion after [INAUDIBLE] 1039 01:00:40,820 --> 01:00:47,780 question, if you go up and down, and you are stimulated, 1040 01:00:47,780 --> 01:00:50,250 you go up in a stimulated way, you go down 1041 01:00:50,250 --> 01:00:57,560 in a stimulated way, and both laser fields 1042 01:00:57,560 --> 01:01:02,010 which stimulate the transition have the same frequency, omega, 1043 01:01:02,010 --> 01:01:09,460 you would actually see that there is no energy exchange. 1044 01:01:09,460 --> 01:01:13,090 So at our current level of description 1045 01:01:13,090 --> 01:01:16,530 we have described where the force comes from, 1046 01:01:16,530 --> 01:01:19,440 but we don't understand yet where the energy goes, 1047 01:01:19,440 --> 01:01:22,800 or where the energy comes from. 1048 01:01:22,800 --> 01:01:25,550 And this is really more sophisticated, 1049 01:01:25,550 --> 01:01:28,820 and I don't want to sort of continue the discussion. 1050 01:01:28,820 --> 01:01:32,460 We will encounter one situation, and this 1051 01:01:32,460 --> 01:01:36,050 is in the [INAUDIBLE] atom picture, where 1052 01:01:36,050 --> 01:01:40,160 we will discuss in a week or so, sisyphus cooling. 1053 01:01:40,160 --> 01:01:42,220 And we will find out that there is 1054 01:01:42,220 --> 01:01:44,290 symmetries in the mono-triplet. 1055 01:01:44,290 --> 01:01:46,610 So again, we will find the missing energy 1056 01:01:46,610 --> 01:01:49,000 in spontaneous emission. 1057 01:01:49,000 --> 01:01:52,890 But if you don't have spontaneous emission, 1058 01:01:52,890 --> 01:01:55,550 if you just have atoms moving in an optical lattice 1059 01:01:55,550 --> 01:01:58,840 without spontaneous emission, it is really what I said, 1060 01:01:58,840 --> 01:02:00,290 the phase modulation. 1061 01:02:00,290 --> 01:02:02,050 And I would probably ask you at this point 1062 01:02:02,050 --> 01:02:04,514 to do the homework assignment, and maybe 1063 01:02:04,514 --> 01:02:06,430 have a discussion afterwards, because then you 1064 01:02:06,430 --> 01:02:08,040 know exactly what I'm talking about. 1065 01:02:11,170 --> 01:02:15,000 Does it at least-- You can take that as a preliminary answer. 1066 01:02:17,960 --> 01:02:19,570 So these are the two limiting cases. 1067 01:02:19,570 --> 01:02:21,630 One is spontaneous emission, and the other one 1068 01:02:21,630 --> 01:02:25,100 is phase modulation at a certain frequency. 1069 01:02:25,100 --> 01:02:27,980 One can be sustained in steady state, 1070 01:02:27,980 --> 01:02:30,920 you can spontaneously emit in sort of an atom 1071 01:02:30,920 --> 01:02:33,750 goes to a standing wave, and this is sustainable. 1072 01:02:33,750 --> 01:02:38,190 Whereas the phase modulation thing is actually a transient. 1073 01:02:38,190 --> 01:02:43,580 It's an oscillation which is added in. 1074 01:02:43,580 --> 01:02:52,509 AUDIENCE: [INAUDIBLE] You're adding the force 1075 01:02:52,509 --> 01:02:55,216 to the other one [INAUDIBLE] 1076 01:02:55,216 --> 01:02:56,341 PROFESSOR: Yeah, actually-- 1077 01:02:56,341 --> 01:02:57,299 AUDIENCE: [INAUDIBLE] 1078 01:02:57,299 --> 01:03:00,640 PROFESSOR: If that helps you, but it's the following. 1079 01:03:00,640 --> 01:03:04,850 I would sometimes say you know, I 1080 01:03:04,850 --> 01:03:07,860 like sort of intuitive explanations of quantum 1081 01:03:07,860 --> 01:03:08,600 physics. 1082 01:03:08,600 --> 01:03:09,970 Let's assume I'm an atom. 1083 01:03:09,970 --> 01:03:12,090 I'm in [INAUDIBLE] standing wave. 1084 01:03:12,090 --> 01:03:15,060 And you don't know it yet, but I do sisyphus cooling. 1085 01:03:15,060 --> 01:03:19,780 By just exchanging photons, a stimulated force, 1086 01:03:19,780 --> 01:03:21,407 I'm actually cooling. 1087 01:03:21,407 --> 01:03:22,990 And you would say, how is it possible? 1088 01:03:22,990 --> 01:03:27,600 Because all the photon exchanges involve the same photon, 1089 01:03:27,600 --> 01:03:28,460 the same energy. 1090 01:03:28,460 --> 01:03:31,050 How can I get rid of energy? 1091 01:03:31,050 --> 01:03:33,560 And I think what really happens is the following, 1092 01:03:33,560 --> 01:03:38,280 the atoms can lose momentum without energy, 1093 01:03:38,280 --> 01:03:40,600 but due to Heisenberg's Uncertainty Relationship 1094 01:03:40,600 --> 01:03:41,660 this is possible. 1095 01:03:41,660 --> 01:03:44,830 So the atom is first taking care of its momentum-- 1096 01:03:44,830 --> 01:03:47,680 is losing momentum-- by stimulated force, 1097 01:03:47,680 --> 01:03:50,250 and eventually, before it's too late, 1098 01:03:50,250 --> 01:03:52,590 it has to do some spontaneous emission where 1099 01:03:52,590 --> 01:03:57,250 it is paying back it debts in energy to Heisenberg. 1100 01:03:57,250 --> 01:04:02,210 So and of course, after a certain time, everything is OK. 1101 01:04:02,210 --> 01:04:04,900 The force was provided by exchanging photons 1102 01:04:04,900 --> 01:04:06,400 of identical frequency. 1103 01:04:06,400 --> 01:04:10,410 And the energy is provided by an occasional spontaneous emission 1104 01:04:10,410 --> 01:04:13,870 event, using one of the [INAUDIBLE]. 1105 01:04:13,870 --> 01:04:16,010 So everything is there. 1106 01:04:16,010 --> 01:04:18,970 Everything is a perfect picture, but you 1107 01:04:18,970 --> 01:04:21,610 have to sort of, in this description, 1108 01:04:21,610 --> 01:04:24,770 allow a certain uncertainty that the force is 1109 01:04:24,770 --> 01:04:27,420 the exchange of identical photons, 1110 01:04:27,420 --> 01:04:30,330 and the energy balance is reconciled 1111 01:04:30,330 --> 01:04:32,550 in spontaneous emission. 1112 01:04:32,550 --> 01:04:39,110 And on any longer time scale there are enough events 1113 01:04:39,110 --> 01:04:42,040 that the balance is perfectly matched. 1114 01:04:42,040 --> 01:04:43,950 But if you would take the position, 1115 01:04:43,950 --> 01:04:45,440 no, this is not possible. 1116 01:04:45,440 --> 01:04:50,830 You know, the atom can only-- I mean, it's sort of in diagrams. 1117 01:04:50,830 --> 01:04:54,290 It's not that one diagram which scatters a photon 1118 01:04:54,290 --> 01:04:55,910 has to conserve energy. 1119 01:04:55,910 --> 01:04:57,160 You have a little bit of time. 1120 01:04:57,160 --> 01:04:59,630 You have Heisenberg's uncertainty time 1121 01:04:59,630 --> 01:05:02,510 to make sure that another diagram jumps 1122 01:05:02,510 --> 01:05:06,560 in, and reconciles energy conservation. 1123 01:05:06,560 --> 01:05:08,740 That's at least my way of looking into it, 1124 01:05:08,740 --> 01:05:11,750 but it's a very maybe, my personal interpretation 1125 01:05:11,750 --> 01:05:15,120 of how all these diagrams and photon scattering 1126 01:05:15,120 --> 01:05:16,670 events work together. 1127 01:05:21,100 --> 01:05:21,600 OK. 1128 01:05:21,600 --> 01:05:24,300 Let's do something simpler now. 1129 01:05:24,300 --> 01:05:29,220 So I've explained to you the dissipative force, 1130 01:05:29,220 --> 01:05:36,290 the reactive force, and in the next unit 1131 01:05:36,290 --> 01:05:40,100 I want to show you simply applications 1132 01:05:40,100 --> 01:05:41,890 of the spontaneous force. 1133 01:05:41,890 --> 01:05:44,500 In every experiment on cold atoms, 1134 01:05:44,500 --> 01:05:47,530 the spontaneous force is center stage. 1135 01:05:47,530 --> 01:05:50,990 It is necessary to slow down atomic beams. 1136 01:05:50,990 --> 01:05:55,610 It provides molasses, which is the colling of atoms 1137 01:05:55,610 --> 01:06:00,230 to micro-Kelvin temperature, and the spontaneous force 1138 01:06:00,230 --> 01:06:03,460 is also responsible for the Magneto-optical trap. 1139 01:06:03,460 --> 01:06:06,830 So what I want to do here is, again, 1140 01:06:06,830 --> 01:06:13,210 by showing you the relevant equations, how 1141 01:06:13,210 --> 01:06:16,040 the spontaneous force, which we have just discussed, 1142 01:06:16,040 --> 01:06:18,830 how these spontaneous force leads to those three 1143 01:06:18,830 --> 01:06:21,340 applications. 1144 01:06:21,340 --> 01:06:24,200 So this is one of the most experimental sections 1145 01:06:24,200 --> 01:06:27,740 of this course, because this equation has it all in it. 1146 01:06:27,740 --> 01:06:31,700 And I just want to show you how this equation can be applied 1147 01:06:31,700 --> 01:06:35,180 to three different important experimental geometries. 1148 01:06:45,450 --> 01:06:47,770 OK. 1149 01:06:47,770 --> 01:06:51,500 So this is our equation. 1150 01:06:51,500 --> 01:06:55,210 It has the momentum transfer per photon. 1151 01:06:55,210 --> 01:06:58,430 It has the maximum scatter rate, gamma over two. 1152 01:06:58,430 --> 01:07:05,930 And then, here, it has the Lorentzian line shape. 1153 01:07:05,930 --> 01:07:11,050 And the important thing is when we talk about molasses 1154 01:07:11,050 --> 01:07:14,910 and beam slowing is that the detuning is the laser 1155 01:07:14,910 --> 01:07:17,270 detuning and the Doppler detuning. 1156 01:07:17,270 --> 01:07:20,850 So the velocity dependence enters now 1157 01:07:20,850 --> 01:07:24,780 the spontaneous force through the detuning in the Lorentzian 1158 01:07:24,780 --> 01:07:28,910 denominator, and it enters through the Doppler effect. 1159 01:07:28,910 --> 01:07:30,750 So you can pretty much say it like this, 1160 01:07:30,750 --> 01:07:32,360 if you have a bunch of laser beam, 1161 01:07:32,360 --> 01:07:36,760 and slow and cool your atoms, how can the lasers do the job? 1162 01:07:36,760 --> 01:07:39,640 Well, they measure the velocity of the atoms 1163 01:07:39,640 --> 01:07:41,100 through the Doppler shift. 1164 01:07:41,100 --> 01:07:43,240 And if is the Doppler shift which tells 1165 01:07:43,240 --> 01:07:45,181 the laser beams what to do, so to speak. 1166 01:07:45,181 --> 01:07:46,680 And this is how laser cooling works. 1167 01:07:50,050 --> 01:07:52,590 So just as an experimentalist, you 1168 01:07:52,590 --> 01:07:55,790 should actually know what the scale is. 1169 01:08:00,930 --> 01:08:02,740 If somebody asked you, how strong 1170 01:08:02,740 --> 01:08:05,620 is the spontaneous scattering force? 1171 01:08:05,620 --> 01:08:08,170 Well, a way to connect it with real time 1172 01:08:08,170 --> 01:08:10,880 units-- with real life units-- is, 1173 01:08:10,880 --> 01:08:13,400 what is the maximum deceleration? 1174 01:08:13,400 --> 01:08:17,740 Well, this is 10 to the 5 G. You can ask, 1175 01:08:17,740 --> 01:08:19,470 is 10 to the 5 G a lot? 1176 01:08:19,470 --> 01:08:20,660 Or not? 1177 01:08:20,660 --> 01:08:23,520 Well, for an astronaut, it would be a lot. 1178 01:08:23,520 --> 01:08:24,620 It would. 1179 01:08:24,620 --> 01:08:28,870 No living organism can sustain 10 to the 5 G. 1180 01:08:28,870 --> 01:08:33,109 But I'm really surprised when I did this calculation. 1181 01:08:33,109 --> 01:08:36,510 When you compare it to the electric force 1182 01:08:36,510 --> 01:08:40,540 on an ionized atom, the same force, 1183 01:08:40,540 --> 01:08:43,540 which is provided by the spontaneous light force, 1184 01:08:43,540 --> 01:08:46,040 would be provided by an electric field 1185 01:08:46,040 --> 01:08:49,050 of one millivolt per centimeter. 1186 01:08:49,050 --> 01:08:51,235 So it's not easy if you have a stainless steel 1187 01:08:51,235 --> 01:08:53,359 chamber to avoid petch effects, which 1188 01:08:53,359 --> 01:08:55,359 create those electric fields. 1189 01:08:55,359 --> 01:08:57,109 And if you've a battery with 9 volt, 1190 01:08:57,109 --> 01:09:00,120 a centimeter apart-- just a 9 volt battery-- 1191 01:09:00,120 --> 01:09:03,950 would accelerate an ion four orders of magnitude 1192 01:09:03,950 --> 01:09:10,819 faster than your beloved strong spontaneous light force. 1193 01:09:10,819 --> 01:09:13,720 So in that sense, 10 to the 5 G is 1194 01:09:13,720 --> 01:09:16,210 a lot in the macroscopic world, but if you 1195 01:09:16,210 --> 01:09:18,950 would look at microscopic forces-- which are often 1196 01:09:18,950 --> 01:09:21,750 electric forces-- it's absolutely tiny. 1197 01:09:21,750 --> 01:09:24,140 And sometimes I say, the fact that you 1198 01:09:24,140 --> 01:09:31,580 can do 100 kilovolt per centimeter, 1199 01:09:31,580 --> 01:09:33,840 that you can make electric forces which 1200 01:09:33,840 --> 01:09:36,439 are seven, eight, or nine, orders of magnitude 1201 01:09:36,439 --> 01:09:41,050 stronger, that's actually the reason why ion traps were 1202 01:09:41,050 --> 01:09:44,950 invented before trapping of neutral particles. 1203 01:09:44,950 --> 01:09:47,809 So some developments in the field 1204 01:09:47,809 --> 01:09:49,970 of trapping particles-- and eventually 1205 01:09:49,970 --> 01:09:53,569 laser cooling them-- it first started with ion traps, 1206 01:09:53,569 --> 01:09:56,290 and then it proceeded to neutral atoms. 1207 01:09:56,290 --> 01:09:58,255 And the reason is the ion trappers 1208 01:09:58,255 --> 01:10:00,892 have forces at their disposal which 1209 01:10:00,892 --> 01:10:02,850 are eight or nine orders of magnitude stronger. 1210 01:10:06,920 --> 01:10:09,070 OK. 1211 01:10:09,070 --> 01:10:09,835 Optical molasses. 1212 01:10:13,790 --> 01:10:17,340 I know some of you will hate it, because I've just 1213 01:10:17,340 --> 01:10:23,080 explained to you that in a standing wave, 1214 01:10:23,080 --> 01:10:25,250 we don't have any spontaneous light force, 1215 01:10:25,250 --> 01:10:29,120 all we have is the reactive force, the stimulated force, 1216 01:10:29,120 --> 01:10:31,250 because it is the u, the in phase 1217 01:10:31,250 --> 01:10:35,120 oscillation of the atomic dipole operator which 1218 01:10:35,120 --> 01:10:38,170 is responsible for everything. 1219 01:10:38,170 --> 01:10:40,940 But now I'm just, you know, wearing another hat, 1220 01:10:40,940 --> 01:10:47,420 and I tell you that there is a limit where the total force can 1221 01:10:47,420 --> 01:10:50,810 be regarded with some of the two forces. 1222 01:10:50,810 --> 01:10:54,940 So therefore, I'm pretending now that if you 1223 01:10:54,940 --> 01:10:59,040 have near resonant light in a standing wave, 1224 01:10:59,040 --> 01:11:02,670 that the force in the standing wave-- which you know 1225 01:11:02,670 --> 01:11:06,080 is a purely reactive force-- can now 1226 01:11:06,080 --> 01:11:10,400 be written as the sum of the two propagating waves. 1227 01:11:10,400 --> 01:11:12,580 And, of course, each propagating wave 1228 01:11:12,580 --> 01:11:17,330 does a purely dissipative force. 1229 01:11:17,330 --> 01:11:19,900 That is, what I'm telling you can be mathematically proven. 1230 01:11:22,670 --> 01:11:28,410 That the stimulated force in a standing wave is equal 1231 01:11:28,410 --> 01:11:32,420 to the sum of the two dissipative forces or each 1232 01:11:32,420 --> 01:11:34,870 travelling wave. 1233 01:11:34,870 --> 01:11:37,470 I just keep that in mind whenever 1234 01:11:37,470 --> 01:11:40,690 you think you can rigorously distinguish 1235 01:11:40,690 --> 01:11:44,340 between the dissipative force and the stimulated force. 1236 01:11:44,340 --> 01:11:47,090 Keep in mind that a standing wave-- 1237 01:11:47,090 --> 01:11:49,490 which has only a stimulated force, 1238 01:11:49,490 --> 01:11:55,740 as I rigorously proved to you-- can alternatively be described 1239 01:11:55,740 --> 01:11:58,730 as the sum of two spontaneous light forces, 1240 01:11:58,730 --> 01:12:00,980 each of which provided by one of the travelling waves. 1241 01:12:04,250 --> 01:12:07,550 Well, it sort of makes sense in a perturbative limit. 1242 01:12:07,550 --> 01:12:10,069 You can just take one beam, you can take the other beam, 1243 01:12:10,069 --> 01:12:11,985 and the combined effect of the two laser beams 1244 01:12:11,985 --> 01:12:12,990 is higher order. 1245 01:12:12,990 --> 01:12:16,817 So the fact that in some low intensity limit 1246 01:12:16,817 --> 01:12:18,400 this has to be valid, is pretty clear. 1247 01:12:23,450 --> 01:12:29,220 Anyway, but just a warning, if you really 1248 01:12:29,220 --> 01:12:32,100 want to use a stimulated force here, you're in big trouble. 1249 01:12:32,100 --> 01:12:34,470 It's much, much harder to get this result out 1250 01:12:34,470 --> 01:12:35,860 of the stimulated force. 1251 01:12:35,860 --> 01:12:39,720 Because to get a stimulated force with velocity dependence 1252 01:12:39,720 --> 01:12:43,270 requires you to take solutions of the Optical Bloch 1253 01:12:43,270 --> 01:12:46,980 Equations, which are not steady state, but non-adiabatic. 1254 01:12:46,980 --> 01:12:47,990 So just a warning. 1255 01:12:47,990 --> 01:12:50,900 You can do it, if you want with a stimulated force, 1256 01:12:50,900 --> 01:12:53,900 but I would strongly advise you to first use 1257 01:12:53,900 --> 01:12:56,760 the simpler formalism. 1258 01:12:56,760 --> 01:12:59,750 You have to go to very different approximation schemes and much 1259 01:12:59,750 --> 01:13:01,700 more technical complexity if you want 1260 01:13:01,700 --> 01:13:04,630 to get it out of the stimulated force. 1261 01:13:04,630 --> 01:13:05,250 OK. 1262 01:13:05,250 --> 01:13:09,310 So with that assumption, you'll remember 1263 01:13:09,310 --> 01:13:13,310 we had the dissipative force of each laser beam. 1264 01:13:13,310 --> 01:13:15,270 Now we have two laser beams. 1265 01:13:15,270 --> 01:13:16,460 We summed them up. 1266 01:13:16,460 --> 01:13:19,400 Because the two laser beams come from different directions, 1267 01:13:19,400 --> 01:13:20,940 we have a minus sign here. 1268 01:13:20,940 --> 01:13:24,040 And for the same reason, we have plus and minus sign 1269 01:13:24,040 --> 01:13:26,350 in the Doppler effect. 1270 01:13:26,350 --> 01:13:29,240 So what we have is the following, 1271 01:13:29,240 --> 01:13:34,205 each force of each laser beam has this Lorentzian envelope, 1272 01:13:34,205 --> 01:13:39,010 shown in black, but the other force of the other laser beam 1273 01:13:39,010 --> 01:13:40,400 has the opposite sign. 1274 01:13:40,400 --> 01:13:43,980 And the two have opposite Doppler shifts. 1275 01:13:43,980 --> 01:13:49,410 So if I add up the two black forces, I get the red force. 1276 01:13:49,410 --> 01:13:53,180 And the red force is anti-symmetric with respect 1277 01:13:53,180 --> 01:13:54,540 to velocity. 1278 01:13:54,540 --> 01:13:58,350 So therefore, in the limit of small velocities 1279 01:13:58,350 --> 01:14:02,100 I get a force which is minus alpha times v. 1280 01:14:02,100 --> 01:14:06,120 And this is the force of friction. 1281 01:14:06,120 --> 01:14:09,580 And in a whimsical way, people called this arrangement 1282 01:14:09,580 --> 01:14:13,690 of two traveling waves Optical Molasses 1283 01:14:13,690 --> 01:14:18,820 because the atom literally gets stuck 1284 01:14:18,820 --> 01:14:21,550 in this configuration like a tiny ball 1285 01:14:21,550 --> 01:14:24,060 bearing you throw into honey. 1286 01:14:24,060 --> 01:14:27,340 So this is optical molasses. 1287 01:14:27,340 --> 01:14:29,980 Actually, I think some of the Europeans 1288 01:14:29,980 --> 01:14:32,790 had to learn the word molasses for it. 1289 01:14:32,790 --> 01:14:35,110 In the US, everybody knows what molasses is. 1290 01:14:35,110 --> 01:14:37,200 Well, in Europe, we know what honey is, 1291 01:14:37,200 --> 01:14:39,490 but molasses is not a standard staple. 1292 01:14:39,490 --> 01:14:43,210 But anyway, it's optical molasses. 1293 01:14:43,210 --> 01:14:44,840 OK. 1294 01:14:44,840 --> 01:14:46,220 Very simple result. 1295 01:14:46,220 --> 01:14:50,390 And, again, it's not easy to get it out of the stimulated force. 1296 01:14:50,390 --> 01:14:52,990 But it's possible. 1297 01:14:52,990 --> 01:14:55,390 I will actually tell you how we get cooling out 1298 01:14:55,390 --> 01:14:57,659 of the stimulated force later in some limit. 1299 01:14:57,659 --> 01:15:00,154 AUDIENCE: Is there [INAUDIBLE] why 1300 01:15:00,154 --> 01:15:05,643 this assumption of dissipative forces is easier [INAUDIBLE] 1301 01:15:05,643 --> 01:15:08,637 and the stimulated force will be made difficult [INAUDIBLE]? 1302 01:15:20,792 --> 01:15:22,250 PROFESSOR: I'm not sure if there is 1303 01:15:22,250 --> 01:15:25,010 a simple answer why it's easier. 1304 01:15:25,010 --> 01:15:26,880 Well, it can't be easier than that. 1305 01:15:32,875 --> 01:15:35,125 AUDIENCE: I mean, there has to be something wrong with 1306 01:15:35,125 --> 01:15:38,990 our assumptions of stimulated forces so that we cannot get 1307 01:15:38,990 --> 01:15:39,490 [INAUDIBLE] 1308 01:15:39,490 --> 01:15:40,156 PROFESSOR: Yeah. 1309 01:15:40,156 --> 01:15:42,880 I think what happens is what I briefly said. 1310 01:15:42,880 --> 01:15:45,700 To the best of my knowledge, but I'm not 100% certain. 1311 01:15:45,700 --> 01:15:48,060 We have the simply expression because we 1312 01:15:48,060 --> 01:15:52,000 have used the steady state solution of the Optical Bloch 1313 01:15:52,000 --> 01:15:52,640 Equation. 1314 01:15:52,640 --> 01:15:54,460 In other words, we have a laser beam, 1315 01:15:54,460 --> 01:15:57,650 and it's just the power of the laser which tells us 1316 01:15:57,650 --> 01:15:59,080 how much light scattering happens. 1317 01:15:59,080 --> 01:16:01,020 And we use the steady state solution. 1318 01:16:01,020 --> 01:16:04,200 And we sort of have folded that into the force for one laser 1319 01:16:04,200 --> 01:16:04,720 beam. 1320 01:16:04,720 --> 01:16:06,094 And then we have the same package 1321 01:16:06,094 --> 01:16:07,500 for the other laser beam. 1322 01:16:07,500 --> 01:16:10,100 And we have never considered that the two laser 1323 01:16:10,100 --> 01:16:11,870 beam may have some cross talk. 1324 01:16:11,870 --> 01:16:15,810 And this is indeed valid for all low laser power. 1325 01:16:15,810 --> 01:16:18,080 However, if you want to get cooling out 1326 01:16:18,080 --> 01:16:23,270 off the stimulated force, I will show some of you later, 1327 01:16:23,270 --> 01:16:26,100 you can no longer use the steady state solution 1328 01:16:26,100 --> 01:16:29,790 because it would miss the effect here. 1329 01:16:29,790 --> 01:16:32,360 You wouldn't get any cooling out of it. 1330 01:16:32,360 --> 01:16:34,990 In order to get something which is dissipative, 1331 01:16:34,990 --> 01:16:36,670 out of a reactive force-- reactive 1332 01:16:36,670 --> 01:16:38,830 force is by definition not dissipative-- 1333 01:16:38,830 --> 01:16:40,820 you need a dissipation mechanism. 1334 01:16:40,820 --> 01:16:43,950 And the dissipation mechanism is that you're not 1335 01:16:43,950 --> 01:16:48,150 quite steady state, there is a relaxation time, a time lag. 1336 01:16:48,150 --> 01:16:51,870 The atom is not instantaneously in its steady state solution. 1337 01:16:51,870 --> 01:16:54,200 It needs a little bit of time to adjust. 1338 01:16:54,200 --> 01:17:00,440 And it is these time lag of the atom which eventually gives 1339 01:17:00,440 --> 01:17:02,920 rise to an alpha coefficient in the stimulated force. 1340 01:17:02,920 --> 01:17:04,808 AUDIENCE: So then physically, you are, 1341 01:17:04,808 --> 01:17:07,168 during cooling experiments, you can take care 1342 01:17:07,168 --> 01:17:09,370 that you are actually not in the steady state limit? 1343 01:17:15,010 --> 01:17:18,500 PROFESSOR: What I'm telling you is, for the spontaneous force, 1344 01:17:18,500 --> 01:17:22,180 we get in leading order the effect by assuming 1345 01:17:22,180 --> 01:17:25,210 the steady state limit, but if you want to get it out 1346 01:17:25,210 --> 01:17:27,735 of the stimulated force, as far as I know, 1347 01:17:27,735 --> 01:17:29,875 you don't get it with the steady state limit. 1348 01:17:29,875 --> 01:17:30,416 AUDIENCE: OK. 1349 01:17:34,520 --> 01:17:36,830 PROFESSOR: If you use different approaches-- 1350 01:17:36,830 --> 01:17:39,500 I mean this is why you want to pick your approach. 1351 01:17:39,500 --> 01:17:42,570 Sometimes you get something already in lower order. 1352 01:17:42,570 --> 01:17:44,760 Sometimes you get it in higher order. 1353 01:17:44,760 --> 01:17:47,060 And I think one favorite example is 1354 01:17:47,060 --> 01:17:50,300 you solve this problem about [INAUDIBLE] scattering 1355 01:17:50,300 --> 01:17:53,380 and Thompson scattering, pick your Hamiltonian, d dot e, 1356 01:17:53,380 --> 01:17:55,650 or p minus a. 1357 01:17:55,650 --> 01:17:58,221 In one case, you have to work harder than in the other case. 1358 01:17:58,221 --> 01:17:59,470 And I think here it's similar. 1359 01:18:02,790 --> 01:18:09,680 Anyway, let's just put in-- I think that's, yeah, 1360 01:18:09,680 --> 01:18:12,130 we can do in the next few minutes. 1361 01:18:12,130 --> 01:18:18,190 So we now want to put in the effect 1362 01:18:18,190 --> 01:18:20,070 that the force fluctuates. 1363 01:18:20,070 --> 01:18:23,170 And that means we have heating. 1364 01:18:23,170 --> 01:18:27,840 Before I do that, let me just tell you what the solution is 1365 01:18:27,840 --> 01:18:32,320 for force equals minus alpha times v. Well, 1366 01:18:32,320 --> 01:18:35,890 it means we extract energy out of the system 1367 01:18:35,890 --> 01:18:39,150 at a rate-- Well, energy per unit time 1368 01:18:39,150 --> 01:18:42,490 is force times velocity. 1369 01:18:42,490 --> 01:18:46,355 Force times velocity is now minus alpha times v square. 1370 01:18:46,355 --> 01:18:49,760 But v square is the energy, the kinetic energy. 1371 01:18:49,760 --> 01:18:53,110 In other words, this equation tells us 1372 01:18:53,110 --> 01:18:59,030 that the atomic motion, the kinetic energy, 1373 01:18:59,030 --> 01:19:04,310 is exponentially [? damned. ?] And if there were nothing else, 1374 01:19:04,310 --> 01:19:07,510 you just need two laser beams-- optical molasses-- 1375 01:19:07,510 --> 01:19:10,980 and you would go not just to micro-Kelvin, but to nano- 1376 01:19:10,980 --> 01:19:12,750 and pico- Kelvin temperatures. 1377 01:19:12,750 --> 01:19:16,910 It's an exponential decay to absolute zero. 1378 01:19:16,910 --> 01:19:19,440 However, we don't each reach nano- 1379 01:19:19,440 --> 01:19:22,389 and pico- Kelvin temperatures in laser cooling 1380 01:19:22,389 --> 01:19:23,805 because there are other processes. 1381 01:19:27,450 --> 01:19:30,900 And what is important here is spontaneous emission. 1382 01:19:35,560 --> 01:19:36,520 OK. 1383 01:19:36,520 --> 01:19:39,460 So the way you treat spontaneous emission is 1384 01:19:39,460 --> 01:19:44,770 the following, every time an atom emits spontaneously, there 1385 01:19:44,770 --> 01:19:49,740 is a random momentum kick of h bar k. 1386 01:19:49,740 --> 01:19:54,900 If you have n photons scattered because the momentum kicks 1387 01:19:54,900 --> 01:19:58,140 going random direction, they only add up in a random box. 1388 01:19:58,140 --> 01:20:01,280 You get square root N. Or if you ask, 1389 01:20:01,280 --> 01:20:06,700 what is the average of p square due to spontaneously emission? 1390 01:20:06,700 --> 01:20:09,500 It is the momentum of the photon squared 1391 01:20:09,500 --> 01:20:12,310 times the number of scattering events. 1392 01:20:12,310 --> 01:20:15,340 So therefore, in the form of a differential equation, 1393 01:20:15,340 --> 01:20:18,200 the heating rate, or the derivative of-- the TEMPO 1394 01:20:18,200 --> 01:20:20,770 derivative-- of p square goes now 1395 01:20:20,770 --> 01:20:22,530 with the number of photons per unit 1396 01:20:22,530 --> 01:20:23,990 time, which is the scattering rate. 1397 01:20:27,590 --> 01:20:29,286 OK. 1398 01:20:29,286 --> 01:20:34,870 If you would stop here, and that's 1399 01:20:34,870 --> 01:20:43,190 what many people do who explain heating in this situation, 1400 01:20:43,190 --> 01:20:45,510 you would miss half of the heating, 1401 01:20:45,510 --> 01:20:48,850 because what you have treated here 1402 01:20:48,850 --> 01:20:52,950 is only the photon transfer and spontaneous emission. 1403 01:20:52,950 --> 01:20:57,150 However, there is also fluctuation and absorption. 1404 01:20:57,150 --> 01:21:00,860 I mean just look at two atoms in the [INAUDIBLE]. 1405 01:21:00,860 --> 01:21:03,070 Kind of me and another atom. 1406 01:21:03,070 --> 01:21:05,200 And we both scatter photons. 1407 01:21:05,200 --> 01:21:09,620 On average, in the same laser beam, we absorb in photons 1408 01:21:09,620 --> 01:21:11,670 and get in momentum kicks. 1409 01:21:11,670 --> 01:21:13,732 But there is Poissonian statistics 1410 01:21:13,732 --> 01:21:16,620 how many photons I absorb, and how many 1411 01:21:16,620 --> 01:21:19,100 photons my twin brother absorbs. 1412 01:21:19,100 --> 01:21:22,500 So therefore, due to the randomness in absorption, 1413 01:21:22,500 --> 01:21:25,010 or the fluctuations in absorption. 1414 01:21:25,010 --> 01:21:28,550 There is another square root n variance 1415 01:21:28,550 --> 01:21:34,120 in the recoil kicks which comes from the absorption process. 1416 01:21:34,120 --> 01:21:37,600 And it so happens, for exactly that reason, 1417 01:21:37,600 --> 01:21:41,630 that the heating-- the derivative of kinetic energy, 1418 01:21:41,630 --> 01:21:45,010 or the difference of momentum squared due to absorption 1419 01:21:45,010 --> 01:21:46,515 is exactly the same as in emission. 1420 01:21:49,880 --> 01:21:52,100 Well, if you now make different assumptions, 1421 01:21:52,100 --> 01:21:54,830 spontaneous emission has a dipole pattern. 1422 01:21:54,830 --> 01:22:01,790 And you can sort of factor on the order of unity, depending 1423 01:22:01,790 --> 01:22:03,870 what the pattern of the spontaneous emission 1424 01:22:03,870 --> 01:22:08,540 is, whether you make a 1D or 2D model of spontaneous emission 1425 01:22:08,540 --> 01:22:11,180 that the photons can only go in one dimension, in two 1426 01:22:11,180 --> 01:22:12,480 dimension, or three dimensions. 1427 01:22:12,480 --> 01:22:15,170 So you have to get other prefectors, 1428 01:22:15,170 --> 01:22:19,680 but the picture is that without going into numerical factors 1429 01:22:19,680 --> 01:22:24,820 on the order of unity, you have fluctuations in the absorption, 1430 01:22:24,820 --> 01:22:27,090 you have a randomness in spontaneous emission, 1431 01:22:27,090 --> 01:22:30,760 and they both equally contribute. 1432 01:22:30,760 --> 01:22:42,300 So that means now the following, that we have heating rate. 1433 01:22:42,300 --> 01:22:46,730 The heating rate we just talked about. 1434 01:22:46,730 --> 01:22:48,580 The increase in p square. 1435 01:22:48,580 --> 01:22:51,060 Well, if you divide by 2 times the mass, 1436 01:22:51,060 --> 01:22:53,790 it's increasing kinetic energy. 1437 01:22:53,790 --> 01:22:56,060 So the increase in kinetic energy 1438 01:22:56,060 --> 01:22:57,325 is given by this expression. 1439 01:23:00,140 --> 01:23:04,130 And it is common if you have a heating process 1440 01:23:04,130 --> 01:23:06,290 to introduce a momentum diffusion 1441 01:23:06,290 --> 01:23:07,680 coefficient in that way. 1442 01:23:07,680 --> 01:23:09,310 It's just the definition. 1443 01:23:09,310 --> 01:23:14,150 It shows you how p square increases per unit time. 1444 01:23:14,150 --> 01:23:18,320 And now we can get the cooling limit for spontaneous emission, 1445 01:23:18,320 --> 01:23:21,890 namely by saying, in steady state when 1446 01:23:21,890 --> 01:23:24,470 we have-- due to photon scattering-- 1447 01:23:24,470 --> 01:23:27,400 the same amount of heating and the same amount of cooling, 1448 01:23:27,400 --> 01:23:29,530 the temperature will have asymptotically reached 1449 01:23:29,530 --> 01:23:30,890 a steady state value. 1450 01:23:33,760 --> 01:23:36,890 The heating rate was parametrized 1451 01:23:36,890 --> 01:23:40,970 by momentum diffusion coefficient, d over m. 1452 01:23:40,970 --> 01:23:45,090 So this is sort of independent of the velocity of the atom. 1453 01:23:45,090 --> 01:23:47,670 Whereas you'll remember the cooling rate 1454 01:23:47,670 --> 01:23:51,000 was proportional to the energy because it 1455 01:23:51,000 --> 01:23:56,810 was an exponential approach to zero energy. 1456 01:23:56,810 --> 01:24:00,770 So therefore, when we have the heating equals the cooling, 1457 01:24:00,770 --> 01:24:03,340 we have the energy of the atoms there, 1458 01:24:03,340 --> 01:24:06,280 and therefore, we find that the energy 1459 01:24:06,280 --> 01:24:11,110 of the atoms in steady state is given by, actually, 1460 01:24:11,110 --> 01:24:15,260 the ratio of heating versus cooling. 1461 01:24:15,260 --> 01:24:17,120 It's a simple expression. 1462 01:24:17,120 --> 01:24:22,045 The energy or the limiting temperature for molasses 1463 01:24:22,045 --> 01:24:24,850 is the ratio of heating versus cooling. 1464 01:24:24,850 --> 01:24:28,800 Heating is described by momentum diffusion coefficient and alpha 1465 01:24:28,800 --> 01:24:32,330 is described by the damping force. 1466 01:24:32,330 --> 01:24:35,080 I want you, actually, to keep this expression in mind. 1467 01:24:35,080 --> 01:24:37,520 A limit in temperature in laser cooling 1468 01:24:37,520 --> 01:24:40,690 is always-- actually, in other processes in laser cooling-- 1469 01:24:40,690 --> 01:24:43,720 is the ratio of heating over cooling. 1470 01:24:43,720 --> 01:24:47,550 And momentum diffusion coefficient due to heating, 1471 01:24:47,550 --> 01:24:50,430 over a friction force due to cooling. 1472 01:24:50,430 --> 01:24:55,010 Because we will later find polarization gradient cooling, 1473 01:24:55,010 --> 01:24:56,460 cooling in blue molasses, we will 1474 01:24:56,460 --> 01:24:58,490 find other cooling schemes, where 1475 01:24:58,490 --> 01:25:02,440 we will calculate-- with the appropriate model-- 1476 01:25:02,440 --> 01:25:04,390 the heating and the cooling. 1477 01:25:04,390 --> 01:25:06,340 And I don't have to repeat this part. 1478 01:25:06,340 --> 01:25:08,510 The moment I calculate the heating, the momentum 1479 01:25:08,510 --> 01:25:10,980 diffusion, and I calculate the friction, 1480 01:25:10,980 --> 01:25:14,540 I know what the limiting temperatures is. 1481 01:25:14,540 --> 01:25:20,775 Again, when I said, the energy is kt over 2, you know, 1482 01:25:20,775 --> 01:25:24,640 of course, kinetic energy is kt over 2 times the number 1483 01:25:24,640 --> 01:25:25,820 of degrees of freedom. 1484 01:25:25,820 --> 01:25:27,740 I assumed 1D, here. 1485 01:25:27,740 --> 01:25:29,755 So in everything I've said on this page, 1486 01:25:29,755 --> 01:25:31,990 there are numerical factors which 1487 01:25:31,990 --> 01:25:35,245 may change whether you assume one or two or three dimension. 1488 01:25:38,960 --> 01:25:39,460 OK. 1489 01:25:39,460 --> 01:25:43,930 I think that's the last thing I want to tell you. 1490 01:25:43,930 --> 01:25:47,100 And then on Wednesday we do Zeeman's slowing 1491 01:25:47,100 --> 01:25:48,410 and Magneto-optical trapping. 1492 01:25:53,000 --> 01:25:59,565 We had an expression for alpha. 1493 01:26:03,630 --> 01:26:07,490 Remember we had this Lorentz profile, for the other Lorentz 1494 01:26:07,490 --> 01:26:09,434 profile, and alpha was just the slope. 1495 01:26:09,434 --> 01:26:10,600 I mean, everything is known. 1496 01:26:10,600 --> 01:26:12,370 I just didn't bother to calculate it. 1497 01:26:12,370 --> 01:26:15,460 But it just involves derivative of Lorentzian. 1498 01:26:15,460 --> 01:26:19,070 And now we can ask, what is the lowest temperature 1499 01:26:19,070 --> 01:26:20,720 we can reach? 1500 01:26:20,720 --> 01:26:25,140 Well, you can now analyze your expression for alpha 1501 01:26:25,140 --> 01:26:30,360 for a two-level system solved by the Optical Bloch Equation. 1502 01:26:30,360 --> 01:26:35,630 And you find that you will have the most favorable conditions 1503 01:26:35,630 --> 01:26:37,920 for low temperature-- you get the minimum possible 1504 01:26:37,920 --> 01:26:41,980 temperature-- in the limit of low laser power. 1505 01:26:41,980 --> 01:26:44,370 And your detuning-- your optimum detuning-- 1506 01:26:44,370 --> 01:26:46,990 is half a line width away. 1507 01:26:46,990 --> 01:26:50,850 And then you'll find this famous result, 1508 01:26:50,850 --> 01:26:54,900 that if you cool a two-level atom, 1509 01:26:54,900 --> 01:26:57,850 the limiting temperature is simply 1510 01:26:57,850 --> 01:27:01,030 given by the spontaneous emission rate, 1511 01:27:01,030 --> 01:27:05,030 or the line width of your transition, gamma. 1512 01:27:05,030 --> 01:27:09,120 And this is the famous result for the Doppler limit. 1513 01:27:09,120 --> 01:27:12,600 For sodium atoms, it's 250 micro-Kelvin, 1514 01:27:12,600 --> 01:27:17,920 for rubidium and cesium, because of the heavier mass, 1515 01:27:17,920 --> 01:27:21,265 it's lower, is 10's of micro-Kelvin, 50 1516 01:27:21,265 --> 01:27:23,190 or 100 micro-Kelvin. 1517 01:27:23,190 --> 01:27:25,440 So this is a famous limit. 1518 01:27:25,440 --> 01:27:27,710 And the physics of it is the following, 1519 01:27:27,710 --> 01:27:32,070 the narrower the line width is. 1520 01:27:32,070 --> 01:27:38,400 Then you can say the better you can cool particle down 1521 01:27:38,400 --> 01:27:40,960 to zero temperature-- and I think, 1522 01:27:40,960 --> 01:27:43,490 Jenny, your idea of the Raman process 1523 01:27:43,490 --> 01:27:46,820 comes now in very handy-- I can have an atom, 1524 01:27:46,820 --> 01:27:50,360 and I can have a Raman process where I scatter photons 1525 01:27:50,360 --> 01:27:52,780 and go to higher energy or I scatter photons 1526 01:27:52,780 --> 01:27:54,300 and got to lower energy. 1527 01:27:54,300 --> 01:27:56,750 And the difference between the two processes 1528 01:27:56,750 --> 01:27:59,890 is actually, the one that whether I cool 1529 01:27:59,890 --> 01:28:03,080 or whether I heat comes from the Doppler effect. 1530 01:28:03,080 --> 01:28:06,810 So the more I can discriminate between through the Doppler 1531 01:28:06,810 --> 01:28:10,260 effect, the better I can cool. 1532 01:28:10,260 --> 01:28:12,640 But the Doppler effect-- the Doppler shift, 1533 01:28:12,640 --> 01:28:16,320 kv-- how well I can resolve it, depends 1534 01:28:16,320 --> 01:28:18,520 on the width of the atomic transition. 1535 01:28:18,520 --> 01:28:20,840 The narrower the atomic transition, 1536 01:28:20,840 --> 01:28:25,910 the more the Doppler effect can steer the laser cooling-- 1537 01:28:25,910 --> 01:28:28,190 the Raman process-- towards lower velocity, 1538 01:28:28,190 --> 01:28:30,470 not towards higher velocity. 1539 01:28:30,470 --> 01:28:33,510 So that's the natural result that the-- or an expected 1540 01:28:33,510 --> 01:28:37,711 result-- that the natural line width appears here. 1541 01:28:37,711 --> 01:28:38,210 OK. 1542 01:28:38,210 --> 01:28:39,543 I've talked a few extra minutes. 1543 01:28:39,543 --> 01:28:43,470 Is there any question? 1544 01:28:43,470 --> 01:28:46,190 Then I see you on Wednesday.