1 00:00:00,070 --> 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 --> 00:00:06,880 Your support will help MIT OpenCourseWare continue 4 00:00:06,880 --> 00:00:10,740 to offer high quality educational resources for free. 5 00:00:10,740 --> 00:00:13,360 To make a donation or view additional materials 6 00:00:13,360 --> 00:00:17,237 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,237 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:21,276 --> 00:00:21,900 PROFESSOR: OK. 9 00:00:21,900 --> 00:00:25,400 First, welcome everybody and good afternoon. 10 00:00:25,400 --> 00:00:29,120 So the question is about the master equation. 11 00:00:29,120 --> 00:00:31,440 We know is that the most general master 12 00:00:31,440 --> 00:00:34,040 equation has to have the Lindblad form. 13 00:00:34,040 --> 00:00:37,710 And the Lindblad form has a jump operator 14 00:00:37,710 --> 00:00:39,930 which is providing dissipation. 15 00:00:39,930 --> 00:00:42,890 In its simplest case, a jump operator 16 00:00:42,890 --> 00:00:46,430 is just the sigma minus operator for spontaneous emission, which 17 00:00:46,430 --> 00:00:49,400 takes us from the excited state to the current state. 18 00:00:49,400 --> 00:00:54,740 And when we have the Linblad form, the jump operator 19 00:00:54,740 --> 00:00:58,210 and it's commission conjugate, like sigma plus, sigma minus, 20 00:00:58,210 --> 00:01:02,220 appear on the right-hand side of the master equation 21 00:01:02,220 --> 00:01:06,200 with the statistical operator here, here, and in the middle. 22 00:01:06,200 --> 00:01:09,110 There are plus-minus signs in factors of two. 23 00:01:09,110 --> 00:01:13,240 So I don't have a intuitive explanation 24 00:01:13,240 --> 00:01:14,420 where the factors come from. 25 00:01:14,420 --> 00:01:15,670 They come from the derivation. 26 00:01:15,670 --> 00:01:18,210 But you have a question about it, Nancy? 27 00:01:18,210 --> 00:01:23,110 AUDIENCE: So after writing the Linblad form, 28 00:01:23,110 --> 00:01:27,356 we get gamma [INAUDIBLE], which we are saying for the r dot 29 00:01:27,356 --> 00:01:31,510 vector, we were writing [INAUDIBLE]. 30 00:01:31,510 --> 00:01:33,350 And we said that they are directly 31 00:01:33,350 --> 00:01:36,796 coming from the first one? 32 00:01:36,796 --> 00:01:37,420 PROFESSOR: Yes. 33 00:01:41,130 --> 00:01:43,914 So let's go right here. 34 00:01:43,914 --> 00:01:46,080 That's actually what we were going to discuss today. 35 00:01:49,466 --> 00:01:51,840 We want to discuss, today, solutions of the optical Bloch 36 00:01:51,840 --> 00:01:53,140 equations. 37 00:01:53,140 --> 00:01:59,440 And just to remind you, we derived a master equation 38 00:01:59,440 --> 00:02:00,920 for the density matrix. 39 00:02:00,920 --> 00:02:04,100 And this master equation, through the interaction 40 00:02:04,100 --> 00:02:07,560 with the reservoir, is acting a term 41 00:02:07,560 --> 00:02:09,360 to the Hamiltonian evolution. 42 00:02:09,360 --> 00:02:11,060 And this is showing right here. 43 00:02:11,060 --> 00:02:13,495 These are the interactions with the reservoir. 44 00:02:13,495 --> 00:02:17,890 And in the case of the optical Bloch equation 45 00:02:17,890 --> 00:02:23,850 and spontaneous emission, we have 46 00:02:23,850 --> 00:02:26,110 only one Lindblad operator, which 47 00:02:26,110 --> 00:02:28,330 is Lindblad operator, sigma minus, 48 00:02:28,330 --> 00:02:30,460 for spontaneous emission. 49 00:02:30,460 --> 00:02:35,360 So this here is the master equation 50 00:02:35,360 --> 00:02:39,070 for the two-level system interacting with a vacuum 51 00:02:39,070 --> 00:02:41,010 through spontaneous emission. 52 00:02:41,010 --> 00:02:50,180 And by substituting the two-level density matrix, 53 00:02:50,180 --> 00:02:54,250 has three non-trivial components-- sure, it's a 2 54 00:02:54,250 --> 00:02:57,140 by 2 matrix, but the craziest one-- and so 55 00:02:57,140 --> 00:03:00,650 we can transform from the matrix elements of the density matrix 56 00:03:00,650 --> 00:03:04,000 to three other coefficient, which come in very handy, 57 00:03:04,000 --> 00:03:07,020 because they allow a simple geometric interpretation. 58 00:03:07,020 --> 00:03:08,540 And this is the Bloch vector. 59 00:03:08,540 --> 00:03:11,200 And so the master equation, which is shown here, 60 00:03:11,200 --> 00:03:14,580 turns into a differential equation for the Bloch vector. 61 00:03:14,580 --> 00:03:16,520 And those two equations are identical, 62 00:03:16,520 --> 00:03:19,030 we've just done those substitution. 63 00:03:19,030 --> 00:03:21,228 But your question is now about-- 64 00:03:21,228 --> 00:03:26,120 AUDIENCE: Plus gamma, because we have negative gamma over 2, 65 00:03:26,120 --> 00:03:29,024 which is from the first and last term. 66 00:03:29,024 --> 00:03:30,960 But the gamma term should be positive, right? 67 00:03:30,960 --> 00:03:32,180 PROFESSOR: This one here? 68 00:03:32,180 --> 00:03:33,112 AUDIENCE: No. 69 00:03:33,112 --> 00:03:34,026 PROFESSOR: This one? 70 00:03:34,026 --> 00:03:34,609 AUDIENCE: Yes. 71 00:03:34,609 --> 00:03:39,768 Because that's negative gamma over 2 times negative 2. 72 00:03:39,768 --> 00:03:40,610 PROFESSOR: No. 73 00:03:40,610 --> 00:03:43,690 I mean, first of all, this is a differential equation 74 00:03:43,690 --> 00:03:44,680 for r dot. 75 00:03:47,310 --> 00:03:48,810 OK. 76 00:03:48,810 --> 00:03:51,420 This is a differential equation for the Bloch vector. 77 00:03:51,420 --> 00:03:58,060 And we have here three different relaxation rates, 78 00:03:58,060 --> 00:04:00,700 which are gamma, gamma over 2, and gamma over 2. 79 00:04:00,700 --> 00:04:02,325 And they are all negative, because they 80 00:04:02,325 --> 00:04:03,840 are relaxation rates. 81 00:04:03,840 --> 00:04:07,380 This here is not a perfecter of r. 82 00:04:07,380 --> 00:04:09,030 It is sort of a constant. 83 00:04:09,030 --> 00:04:11,700 And it means, in the long time limit-- I mean, 84 00:04:11,700 --> 00:04:13,950 what happens if you have a system which spontaneously 85 00:04:13,950 --> 00:04:15,660 decay after a while? 86 00:04:15,660 --> 00:04:18,190 The r vector is not 0. 87 00:04:18,190 --> 00:04:20,430 The r vector is hanging down. 88 00:04:20,430 --> 00:04:22,640 The system is in its ground state. 89 00:04:22,640 --> 00:04:23,850 And this is exactly that. 90 00:04:28,510 --> 00:04:29,010 Wait. 91 00:04:29,010 --> 00:04:30,710 Am I r dot? 92 00:04:33,650 --> 00:04:34,280 Yes. 93 00:04:34,280 --> 00:04:37,370 The effect is the system, the differential equation, 94 00:04:37,370 --> 00:04:40,980 has to relax towards the equilibrium. 95 00:04:40,980 --> 00:04:44,110 You can actually re-write this equation, 96 00:04:44,110 --> 00:04:51,300 r dot equals a matrix times r minus r equilibrium. 97 00:04:51,300 --> 00:04:55,880 So here is the matrix of the optical Bloch equation. 98 00:04:55,880 --> 00:05:00,060 So if you re-write that that r dot is the matrix times r 99 00:05:00,060 --> 00:05:03,910 minus r equilibrium, then r equilibrium times 100 00:05:03,910 --> 00:05:05,970 the matrix gives something constant. 101 00:05:05,970 --> 00:05:08,850 And this is exactly this here. 102 00:05:08,850 --> 00:05:10,330 But it's a mathematical identity. 103 00:05:10,330 --> 00:05:12,270 Just look at it. 104 00:05:12,270 --> 00:05:13,380 There is no assumption. 105 00:05:13,380 --> 00:05:15,215 It's an exact re-writing of the equation. 106 00:05:22,090 --> 00:05:28,940 So it is this equation which will 107 00:05:28,940 --> 00:05:32,490 be in the focus of our attention not only today, 108 00:05:32,490 --> 00:05:38,730 but also when we talk about light forces. 109 00:05:38,730 --> 00:05:41,630 The first thing we did is we wanted 110 00:05:41,630 --> 00:05:44,480 to discuss the Mollow triplet. 111 00:05:44,480 --> 00:05:50,030 And just as a reminder, we have talked about, 112 00:05:50,030 --> 00:05:51,990 at the beginning of last class, we 113 00:05:51,990 --> 00:05:54,080 talked about the fluorescence. 114 00:05:54,080 --> 00:05:57,390 An atom is excited by mono-chromatic laser light. 115 00:05:57,390 --> 00:06:02,040 And at low intensity, a delta function 116 00:06:02,040 --> 00:06:05,790 emits exactly the same frequency of light 117 00:06:05,790 --> 00:06:08,820 which is provided by the laser light. 118 00:06:08,820 --> 00:06:10,760 So the incoming and outgoing photons 119 00:06:10,760 --> 00:06:14,120 have exactly the same frequency. 120 00:06:14,120 --> 00:06:16,740 And this is simply a consequence of energy conservation. 121 00:06:16,740 --> 00:06:20,070 But at higher intensity, we have sidebands. 122 00:06:20,070 --> 00:06:22,170 And we understand intuitively that we 123 00:06:22,170 --> 00:06:25,400 have sidebands because the system is Rabi oscillation. 124 00:06:25,400 --> 00:06:27,880 So same, classically, we have an emitter 125 00:06:27,880 --> 00:06:31,760 which has some modulation that creates sidebands. 126 00:06:31,760 --> 00:06:42,990 But is rather easy to understand is, 127 00:06:42,990 --> 00:06:45,570 by diagonalizing the 2 by 2 matrix, 128 00:06:45,570 --> 00:06:52,990 we find that we have a splitting of excited state with n photons 129 00:06:52,990 --> 00:06:55,560 and grounds it with n plus 1 photons. 130 00:06:55,560 --> 00:06:58,120 So we have sort of a plus and minus state. 131 00:06:58,120 --> 00:07:00,590 And if we look at the possible transitions, 132 00:07:00,590 --> 00:07:02,850 we immediately find the explanation 133 00:07:02,850 --> 00:07:04,910 for the Mollow triplet. 134 00:07:04,910 --> 00:07:06,660 There are three different frequencies 135 00:07:06,660 --> 00:07:10,960 which can be emitted in transitions 136 00:07:10,960 --> 00:07:13,460 between rest states. 137 00:07:13,460 --> 00:07:16,800 What is much more subtle, what cannot be obtained from 138 00:07:16,800 --> 00:07:21,590 a preterbative treatment, is the width, 139 00:07:21,590 --> 00:07:25,590 how wide are those peaks in the Mollow triplet. 140 00:07:25,590 --> 00:07:34,390 And there is tens and tens of pages 141 00:07:34,390 --> 00:07:36,470 in atom photo interaction. 142 00:07:36,470 --> 00:07:39,020 But at least, for two limiting cases, 143 00:07:39,020 --> 00:07:43,670 I could show you what the width of the peak is by first saying, 144 00:07:43,670 --> 00:07:46,120 if you have delta equals 0 and g equals 145 00:07:46,120 --> 00:07:49,380 0, the matrix, which determines the dynamics, 146 00:07:49,380 --> 00:07:52,115 has three eigenvalues, gamma over 2, gamma over 2, 147 00:07:52,115 --> 00:07:53,280 and gamma. 148 00:07:53,280 --> 00:07:58,430 And now, if we detune, we add a rotation along z. 149 00:07:58,430 --> 00:08:03,290 If you drive the system strong, it corresponds to rotation 150 00:08:03,290 --> 00:08:05,370 around x. 151 00:08:05,370 --> 00:08:07,980 You probably remember, from 8.421 152 00:08:07,980 --> 00:08:09,940 the spin in a magnetic field. 153 00:08:09,940 --> 00:08:10,440 It rotates. 154 00:08:16,040 --> 00:08:18,190 In a frame, rotating with omega, it 155 00:08:18,190 --> 00:08:20,180 rotates at the detuning delta. 156 00:08:20,180 --> 00:08:23,750 But if you drive it, we make it flip along the x-axis. 157 00:08:23,750 --> 00:08:25,680 So we find exactly that here. 158 00:08:25,680 --> 00:08:31,010 And then, it's an intuitive way to sort of matrix, if you want, 159 00:08:31,010 --> 00:08:32,870 solve the matrix mathematically. 160 00:08:32,870 --> 00:08:36,750 But intuitively, what happens is, if you have a z-rotation, 161 00:08:36,750 --> 00:08:38,950 we don't change in this eigenvalue. 162 00:08:38,950 --> 00:08:43,090 And the rotation is just aiding i omega-- omega 163 00:08:43,090 --> 00:08:46,990 is the rotation-- to x and y, because x and y are now 164 00:08:46,990 --> 00:08:48,560 rotating. 165 00:08:48,560 --> 00:08:58,470 So therefore, we obtained that, in the case of detuning 166 00:08:58,470 --> 00:09:01,570 the z-rotation, we have minus gamma, minus gamma over 2, 167 00:09:01,570 --> 00:09:02,830 minus gamma over 2. 168 00:09:02,830 --> 00:09:07,160 And here, what appears here is the rotation frequency. 169 00:09:07,160 --> 00:09:12,980 And then, if you rotate around the x-axis, then, of course, 170 00:09:12,980 --> 00:09:15,840 nothing happens, but what used to be the x-axis. 171 00:09:15,840 --> 00:09:19,220 So we get one eigenvalue with minus gamma over 2. 172 00:09:19,220 --> 00:09:25,210 But the y and z eigenvector are rotated. 173 00:09:25,210 --> 00:09:28,990 And hand-wavingly, I can tell you. 174 00:09:28,990 --> 00:09:30,820 But you can show it mathematically 175 00:09:30,820 --> 00:09:34,760 that this means the two new eigenvectors have 176 00:09:34,760 --> 00:09:37,540 the average of those two damping terms. 177 00:09:37,540 --> 00:09:46,230 And this is 3/4 gamma. 178 00:09:46,230 --> 00:09:48,580 So at least, for the two cases for large detuning 179 00:09:48,580 --> 00:09:53,970 and strong drive, I was, with a little bit of intuition, 180 00:09:53,970 --> 00:09:57,540 showing you and deriving for you where 181 00:09:57,540 --> 00:10:01,179 the non-trivial bits of the Mollow triplet comes from. 182 00:10:01,179 --> 00:10:02,720 So I hope you're, at least, impressed 183 00:10:02,720 --> 00:10:05,210 that these optical Bloch equations are powerful 184 00:10:05,210 --> 00:10:07,230 and allow us to make predictions which 185 00:10:07,230 --> 00:10:10,240 would be difficult to obtain otherwise. 186 00:10:10,240 --> 00:10:12,970 Any questions about that? 187 00:10:12,970 --> 00:10:16,690 About the Bloch vector, the differential equation, 188 00:10:16,690 --> 00:10:19,970 and the spectrum of the immediate light? 189 00:10:25,160 --> 00:10:28,890 OK, we want to now discuss one other aspect, which is actually 190 00:10:28,890 --> 00:10:30,010 fairly simple. 191 00:10:30,010 --> 00:10:33,300 We want to discuss steady state solution. 192 00:10:33,300 --> 00:10:36,540 For the steady state solution, we just go to the differential 193 00:10:36,540 --> 00:10:40,620 equation and say, the left-hand side is 0. 194 00:10:40,620 --> 00:10:43,760 That immediately gives us a solution for the Bloch vector. 195 00:10:48,790 --> 00:10:53,090 It gives us a Lorentzian. 196 00:10:53,090 --> 00:10:56,030 But for strong drive, it has a term here, 197 00:10:56,030 --> 00:11:00,570 which we will immediately realize 198 00:11:00,570 --> 00:11:02,960 that this gives us power broadening. 199 00:11:02,960 --> 00:11:07,200 And then the steady state Bloch vector is g delta, 200 00:11:07,200 --> 00:11:13,730 g gamma over 2, delta square plus gamma square over 4. 201 00:11:13,730 --> 00:11:17,670 So we can now use this solution for steady state 202 00:11:17,670 --> 00:11:22,970 and discuss everything we want what happens in steady state. 203 00:11:22,970 --> 00:11:25,280 So one question is, if you have an atom laser 204 00:11:25,280 --> 00:11:27,560 beam in steady state, how much light is absorbed? 205 00:11:33,220 --> 00:11:42,610 Well, the basic equation is, if you have a charged q which 206 00:11:42,610 --> 00:11:51,710 moves in an electric field, that means work is done. 207 00:11:55,330 --> 00:12:01,400 So therefore, the absorbed power is related to that. 208 00:12:01,400 --> 00:12:08,260 And we can now combine q dr/dt. 209 00:12:08,260 --> 00:12:10,860 And this is nothing else than the derivative 210 00:12:10,860 --> 00:12:11,870 of the dipole moment. 211 00:12:16,470 --> 00:12:19,320 And then we calculate the average. 212 00:12:19,320 --> 00:12:25,190 So with that, we find the result they the absorbed 213 00:12:25,190 --> 00:12:44,460 power is given by-- just one second-- e0? 214 00:12:44,460 --> 00:12:45,420 Yeah. 215 00:12:45,420 --> 00:12:50,650 We express now e0 by the Rabi frequency. 216 00:12:50,650 --> 00:12:52,850 And h bar appears in omega. 217 00:12:52,850 --> 00:12:59,270 But the important thing is that we need now the dipole moment 218 00:12:59,270 --> 00:13:01,070 or the derivative of the dipole moment. 219 00:13:01,070 --> 00:13:07,180 And the dipole moment is given by the coherences 220 00:13:07,180 --> 00:13:08,840 of the density matrix. 221 00:13:08,840 --> 00:13:11,840 And that is, if you look at the substitution, 222 00:13:11,840 --> 00:13:18,220 is the x and y component of the optical Bloch vector. 223 00:13:18,220 --> 00:13:20,340 But now what we need, of course, we 224 00:13:20,340 --> 00:13:26,540 need only the component of the dipole moment, which 225 00:13:26,540 --> 00:13:29,590 gives a non-vanishing term with cosine omega t. 226 00:13:29,590 --> 00:13:31,510 That means, what we need is-- I mean, 227 00:13:31,510 --> 00:13:33,060 that's like an harmonic oscillator. 228 00:13:33,060 --> 00:13:34,570 The component of the motion which 229 00:13:34,570 --> 00:13:36,630 is responsible for absorption is the one 230 00:13:36,630 --> 00:13:41,540 which is in quadrature, which is assigned only the d-component. 231 00:13:41,540 --> 00:13:45,040 And that, you have to go back and look by inspection. 232 00:13:45,040 --> 00:13:52,210 But this is exactly what the y-component of the Bloch vector 233 00:13:52,210 --> 00:13:54,320 does for us. 234 00:13:54,320 --> 00:13:58,270 It will become important later on for optical forces. 235 00:13:58,270 --> 00:14:03,280 So we have the optical Bloch vector. 236 00:14:03,280 --> 00:14:05,070 We are in the rotating frame. 237 00:14:05,070 --> 00:14:10,370 And the one component, which is the x-component, 238 00:14:10,370 --> 00:14:12,110 is in phase with the light. 239 00:14:12,110 --> 00:14:16,030 And the y-phase is 90 degree out of phases in quadrature. 240 00:14:16,030 --> 00:14:18,980 And for absorption it's, of course, the in quadrature 241 00:14:18,980 --> 00:14:21,890 component which is relevant and which 242 00:14:21,890 --> 00:14:23,145 determines light scattering. 243 00:14:26,090 --> 00:14:28,780 OK, this is the absorbed light. 244 00:14:28,780 --> 00:14:34,620 And we don't measure power in watt, 245 00:14:34,620 --> 00:14:38,390 we measure power in the number of absorbed photons. 246 00:14:38,390 --> 00:14:40,190 This is the natural unit here. 247 00:14:40,190 --> 00:14:45,000 And for that, we have to divide the expression by h bar omega. 248 00:14:45,000 --> 00:14:51,310 And that means now that the number of photons absorbed 249 00:14:51,310 --> 00:14:53,790 per unit time is given by the Rabi 250 00:14:53,790 --> 00:14:59,000 frequency and the y-component of the Bloch vector. 251 00:14:59,000 --> 00:15:02,030 Well, we can ask another question 252 00:15:02,030 --> 00:15:04,485 and ask, what is the number of emitted photons? 253 00:15:11,260 --> 00:15:15,580 Well, the number of emitted photons 254 00:15:15,580 --> 00:15:19,830 is related to the population in the excited state. 255 00:15:19,830 --> 00:15:21,960 We take the population in the excited state 256 00:15:21,960 --> 00:15:23,610 and multiply with gamma. 257 00:15:23,610 --> 00:15:28,020 This is spontaneous emission out of the excited state. 258 00:15:28,020 --> 00:15:33,060 Now, the excited state can be-- the excited state population, 259 00:15:33,060 --> 00:15:35,110 you have to look up at the substitution, 260 00:15:35,110 --> 00:15:40,744 but is related to the z-component 261 00:15:40,744 --> 00:15:41,910 of the optical Bloch vector. 262 00:15:41,910 --> 00:15:44,300 To remind you, the z-component is the difference 263 00:15:44,300 --> 00:15:47,650 of population between excited and ground state. 264 00:15:47,650 --> 00:15:49,680 But the sum of the two is one. 265 00:15:49,680 --> 00:15:54,900 So therefore, this is now the excited state population. 266 00:15:54,900 --> 00:16:00,740 And here is our steady state solution. 267 00:16:00,740 --> 00:16:04,470 Here is our steady state solution for the z-component. 268 00:16:07,220 --> 00:16:15,510 And by using that now, we find that we 269 00:16:15,510 --> 00:16:24,080 have the result with the Lorentzian denominator. 270 00:16:24,080 --> 00:16:30,310 And what we have here is gamma g square over 4. 271 00:16:30,310 --> 00:16:33,830 And now, of course, if you compare 272 00:16:33,830 --> 00:16:40,430 that with the solution for our y, 273 00:16:40,430 --> 00:16:44,080 for the number of absorbed photons, you find-- well, 274 00:16:44,080 --> 00:16:45,870 it shouldn't come as a big surprise-- 275 00:16:45,870 --> 00:16:48,520 that the number of absorbed photon in steady state 276 00:16:48,520 --> 00:16:52,960 is equal to the number of emitted photon. 277 00:16:52,960 --> 00:16:57,490 So this is one case, one important part of the solution. 278 00:17:08,230 --> 00:17:13,170 And I should have shown that to you earlier. 279 00:17:13,170 --> 00:17:16,770 When we look at the two components, the x 280 00:17:16,770 --> 00:17:28,520 and y-component, the y-component is 281 00:17:28,520 --> 00:17:30,135 absorbed, if it is Lorentzian. 282 00:17:32,810 --> 00:17:36,145 Whereas, the x-component is dispersive. 283 00:17:50,270 --> 00:17:54,890 Just by inspection if these two expressions, here is a delta. 284 00:17:54,890 --> 00:18:00,770 And the delta makes the x-component anti-symmetric. 285 00:18:00,770 --> 00:18:04,050 And the y-component is symmetric with respect to the origin. 286 00:18:06,620 --> 00:18:09,470 So what we see here is we have a Lorentzian. 287 00:18:09,470 --> 00:18:15,190 The Lorentzian has a natural line broadening by gamma. 288 00:18:15,190 --> 00:18:16,600 So the full width at half maximum 289 00:18:16,600 --> 00:18:21,960 is gamma, but only in the limit of 0 drive. 290 00:18:21,960 --> 00:18:26,000 When we drive it, we have an additional broadening, 291 00:18:26,000 --> 00:18:27,520 which is called power broadening. 292 00:18:30,340 --> 00:18:35,060 So for absorption, the full width at half maximum 293 00:18:35,060 --> 00:18:46,090 is 2 times gamma square over 4 plus g square over 2. 294 00:18:46,090 --> 00:18:52,560 And this last term goes by the name power broadening 295 00:18:52,560 --> 00:18:53,655 or saturation broadening. 296 00:18:58,610 --> 00:19:05,300 Just to avoid common confusion, if you go to very high power, 297 00:19:05,300 --> 00:19:08,390 that least term doesn't play a role. 298 00:19:08,390 --> 00:19:12,645 How does the line width scale with power in power broadening? 299 00:19:20,800 --> 00:19:22,409 Linear in power? 300 00:19:22,409 --> 00:19:23,200 Quadratic in power? 301 00:19:23,200 --> 00:19:24,200 Square root of power? 302 00:19:24,200 --> 00:19:25,565 AUDIENCE: Square root? 303 00:19:25,565 --> 00:19:26,930 PROFESSOR: Square root of power. 304 00:19:26,930 --> 00:19:27,430 Yeah. 305 00:19:27,430 --> 00:19:30,617 So in that sense, yes, power broadening 306 00:19:30,617 --> 00:19:32,200 only goes to the square root of power. 307 00:19:32,200 --> 00:19:34,590 It's, in essence, the Rabi frequency. 308 00:19:34,590 --> 00:19:37,390 If you drive the system at the Rabi frequency, 309 00:19:37,390 --> 00:19:40,580 it gets power broadened by the Rabi frequency. 310 00:19:40,580 --> 00:19:43,380 And I think that's a pre-factor which is 2 or square root 2. 311 00:19:43,380 --> 00:19:46,610 But in essence, power broadening scales with the Rabi frequency. 312 00:19:51,220 --> 00:19:56,770 So let us quickly discuss the case 313 00:19:56,770 --> 00:20:00,770 when we go to very high power, very high Rabi frequency. 314 00:20:00,770 --> 00:20:04,770 At that moment, you can just see it in the solution on the page 315 00:20:04,770 --> 00:20:05,850 above. 316 00:20:05,850 --> 00:20:11,660 The z-component of the optical Bloch vector becomes 0. 317 00:20:11,660 --> 00:20:14,200 That means there is no population difference 318 00:20:14,200 --> 00:20:20,460 between excited and ground state anymore. 319 00:20:24,130 --> 00:20:31,840 We often describe the limit of high power 320 00:20:31,840 --> 00:20:36,520 by introducing a saturation parameter, which 321 00:20:36,520 --> 00:20:37,610 is defined here. 322 00:20:45,830 --> 00:20:49,850 Saturation is discussed in more details in Part 1 323 00:20:49,850 --> 00:20:51,490 of the course. 324 00:20:51,490 --> 00:20:54,240 But I want to relate it here to the solution 325 00:20:54,240 --> 00:20:55,920 with the optical Bloch vector. 326 00:20:55,920 --> 00:21:00,300 The z-component, which is the difference 327 00:21:00,300 --> 00:21:05,060 between ground and excited state population, 328 00:21:05,060 --> 00:21:11,290 now has a very simple form, 1 over 1 plus s. 329 00:21:11,290 --> 00:21:15,780 The population to be in the excited state is 1/2. 330 00:21:15,780 --> 00:21:18,350 That's what we get at infinite power. 331 00:21:18,350 --> 00:21:20,580 And then, if the power is finite, 332 00:21:20,580 --> 00:21:25,030 it is multiplied by s times s plus 1. 333 00:21:25,030 --> 00:21:30,230 In the power broadened line width is the natural line 334 00:21:30,230 --> 00:21:34,040 width, which gives the line width 335 00:21:34,040 --> 00:21:35,900 in the limit of low power, and then 336 00:21:35,900 --> 00:21:38,660 you multiply with the square root 1 plus s. 337 00:21:47,890 --> 00:21:55,430 So what does it mean to have a saturation parameter of 1? 338 00:21:55,430 --> 00:22:00,260 Well, if you just look how it was defined, 339 00:22:00,260 --> 00:22:03,730 if you have the ground and the excited state at a saturation 340 00:22:03,730 --> 00:22:08,050 parameter of-- first, at a saturation of infinite, 341 00:22:08,050 --> 00:22:11,010 you have 50-50 population. 342 00:22:11,010 --> 00:22:16,720 And at a saturation parameter of 1, you are 1/2 way to 50-50. 343 00:22:16,720 --> 00:22:21,080 That means you have 3/4 population here 344 00:22:21,080 --> 00:22:25,660 and 1/4 population there. 345 00:22:25,660 --> 00:22:36,260 So at this case, the number of photons emitted per unit time, 346 00:22:36,260 --> 00:22:39,210 which is always given by this formula, 347 00:22:39,210 --> 00:22:43,950 becomes now gamma over 4. 348 00:22:43,950 --> 00:22:45,950 So at a saturation parameter of 1, 349 00:22:45,950 --> 00:22:48,470 the emission rate is gamma over 4. 350 00:22:48,470 --> 00:22:51,600 At a saturation parameter of infinity, 351 00:22:51,600 --> 00:22:53,895 the spontaneous emission rate is gamma over 2. 352 00:22:59,950 --> 00:23:05,860 Any questions about steady state solution? 353 00:23:05,860 --> 00:23:06,412 [INAUDIBLE]? 354 00:23:06,412 --> 00:23:07,995 AUDIENCE: So what we derived right now 355 00:23:07,995 --> 00:23:09,547 is a spontaneous emission rate. 356 00:23:09,547 --> 00:23:11,380 Does the optical Bloch equation say anything 357 00:23:11,380 --> 00:23:16,007 about the stimulated emission? 358 00:23:16,007 --> 00:23:18,442 Or is that the fact that it just doesn't have [INAUDIBLE]? 359 00:23:21,364 --> 00:23:25,750 PROFESSOR: Stimulated emission is built in the optical Bloch 360 00:23:25,750 --> 00:23:30,460 equations for negligible damping contained Rabi oscillation. 361 00:23:30,460 --> 00:23:34,340 And Rabi oscillation is the stimulated emission. 362 00:23:34,340 --> 00:23:36,910 So this is being built in. 363 00:23:36,910 --> 00:23:41,950 However, it seems to be a describing the drive 364 00:23:41,950 --> 00:23:47,490 field by a classical field by a Rabi frequency. 365 00:23:47,490 --> 00:23:52,550 We are not really accounting for the number 366 00:23:52,550 --> 00:23:54,920 of photons exchanged with that. 367 00:23:54,920 --> 00:23:59,280 So in a way, the driving field is always 368 00:23:59,280 --> 00:24:02,280 treated in the undepleted approximation. 369 00:24:02,280 --> 00:24:03,350 It's a C number. 370 00:24:03,350 --> 00:24:05,980 But in the drive through system, but of course, 371 00:24:05,980 --> 00:24:08,660 you can immediately lead from the Rabi oscillation 372 00:24:08,660 --> 00:24:12,657 the exchange of excitations between the atomic system 373 00:24:12,657 --> 00:24:13,490 and the drive field. 374 00:24:16,420 --> 00:24:19,590 So yes, the optical Bloch equations contains that, 375 00:24:19,590 --> 00:24:20,840 but not in the photon picture. 376 00:24:23,560 --> 00:24:25,550 AUDIENCE: If you have Rabi oscillations, 377 00:24:25,550 --> 00:24:27,299 you can kind of guess that, every time you 378 00:24:27,299 --> 00:24:29,330 go down you simulated the events. 379 00:24:29,330 --> 00:24:31,392 In space, say, you don't have Rabi oscillations. 380 00:24:31,392 --> 00:24:33,430 Can you still somehow extract the rate? 381 00:24:36,430 --> 00:24:39,530 PROFESSOR: I hope the question we answer will become crystal 382 00:24:39,530 --> 00:24:41,610 when we discuss quantum Monte Carlo simulations. 383 00:24:41,610 --> 00:24:42,151 AUDIENCE: OK. 384 00:24:42,151 --> 00:24:44,200 PROFESSOR: In quantum Monte Carlo simulation, 385 00:24:44,200 --> 00:24:46,000 we have an ensemble of atom. 386 00:24:46,000 --> 00:24:48,410 And the atoms all do Rabi oscillations. 387 00:24:48,410 --> 00:24:51,470 But in steady state, they do it out of phase. 388 00:24:51,470 --> 00:24:53,400 So you have atoms in your ensemble 389 00:24:53,400 --> 00:24:56,930 which, at a given time, absorb. 390 00:24:56,930 --> 00:24:59,410 And others emit in a stimulated way. 391 00:24:59,410 --> 00:25:01,430 And the effect cancels out in steady state. 392 00:25:04,300 --> 00:25:05,770 Other questions? 393 00:25:05,770 --> 00:25:06,530 Yes? 394 00:25:06,530 --> 00:25:08,647 AUDIENCE: So you're saying, in steady state, 395 00:25:08,647 --> 00:25:09,605 they kind of de-cohere. 396 00:25:09,605 --> 00:25:12,970 And you don't have any net change in those populations. 397 00:25:12,970 --> 00:25:15,002 Do you still get the triplet lines 398 00:25:15,002 --> 00:25:21,830 if you don't have a modulation of your intensity beam? 399 00:25:21,830 --> 00:25:22,600 PROFESSOR: Yes. 400 00:25:22,600 --> 00:25:23,100 Absolutely. 401 00:25:23,100 --> 00:25:26,269 The Mollow triplet is a feature in steady state. 402 00:25:26,269 --> 00:25:28,060 And maybe let me explain that, because it's 403 00:25:28,060 --> 00:25:30,110 an interesting discussion. 404 00:25:30,110 --> 00:25:34,160 What will happen is, in steady state in your ensemble, 405 00:25:34,160 --> 00:25:36,280 you don't have Rabi oscillations. 406 00:25:36,280 --> 00:25:39,280 But if you would look at one atom, 407 00:25:39,280 --> 00:25:41,130 you see Rabi oscillations. 408 00:25:41,130 --> 00:25:45,231 It's just when you average over all atoms in your ensemble, 409 00:25:45,231 --> 00:25:49,280 the phase of the Rabi oscillation has averaged out. 410 00:25:49,280 --> 00:25:52,500 So in other words, for instance, what would happen 411 00:25:52,500 --> 00:25:55,000 is the following. 412 00:25:55,000 --> 00:25:57,710 If you take a steady state solution 413 00:25:57,710 --> 00:26:05,180 and you ask, what is the dipole moment as a function of time 414 00:26:05,180 --> 00:26:09,580 in steady state, you will find that this is 0. 415 00:26:09,580 --> 00:26:11,270 So you would then ask, hey, where 416 00:26:11,270 --> 00:26:13,820 is the light emitted, because isn't the light emitted 417 00:26:13,820 --> 00:26:17,720 by a time-dependent dipole moment? 418 00:26:17,720 --> 00:26:21,040 What you really have to do is, when 419 00:26:21,040 --> 00:26:26,050 you want to study light emission in steady state, 420 00:26:26,050 --> 00:26:28,060 you can't look at expectation values 421 00:26:28,060 --> 00:26:31,100 because they're not changing as a function of time. 422 00:26:31,100 --> 00:26:33,810 What you have to do is you have to look at the correlation 423 00:26:33,810 --> 00:26:36,200 function. 424 00:26:36,200 --> 00:26:37,780 And the correlation function sort of 425 00:26:37,780 --> 00:26:41,330 tells you-- let me just use it for Rabi oscillation. 426 00:26:41,330 --> 00:26:45,890 If one atom is in the excited state, a Rabi period later, 427 00:26:45,890 --> 00:26:47,860 it would be again in the excited state. 428 00:26:47,860 --> 00:26:50,110 So although you are in steady state, 429 00:26:50,110 --> 00:26:55,280 you have correlations because, if you look at your ensemble 430 00:26:55,280 --> 00:26:59,220 and see there is a fluctuation where the atom is excited, 431 00:26:59,220 --> 00:27:01,652 you will actually find a co-related fluctuation 432 00:27:01,652 --> 00:27:03,620 a Rabi period later. 433 00:27:03,620 --> 00:27:06,920 So in steady state, when you want to study dynamics 434 00:27:06,920 --> 00:27:10,090 like the Mollow triplet, you should not 435 00:27:10,090 --> 00:27:12,350 look at the solution for the dipole. 436 00:27:12,350 --> 00:27:15,730 You should look for the solution of the correlation 437 00:27:15,730 --> 00:27:19,180 of the correlation function. 438 00:27:19,180 --> 00:27:22,670 And I made this cryptic remark on Wednesday 439 00:27:22,670 --> 00:27:26,540 that this correlation function follows the same differential 440 00:27:26,540 --> 00:27:28,830 equation as the expectation value. 441 00:27:28,830 --> 00:27:32,400 Therefore, when I try to discuss for you the line 442 00:27:32,400 --> 00:27:34,820 width of the Mollow triplet, I simply 443 00:27:34,820 --> 00:27:38,130 use the matrix for the Bloch vector 444 00:27:38,130 --> 00:27:41,390 knowing that it is the same matrix, the same differential 445 00:27:41,390 --> 00:27:45,580 equation which will describe this correlation function. 446 00:27:48,980 --> 00:27:52,600 But it is important here that, in steady state, 447 00:27:52,600 --> 00:27:55,890 the average values are 0, but we have fluctuations. 448 00:27:55,890 --> 00:27:57,720 And it is the fluctuations which emit 449 00:27:57,720 --> 00:28:01,110 the light, the fluctuations which emit the Mollow triplet. 450 00:28:01,110 --> 00:28:05,410 And fluctuations are described by quantities like this. 451 00:28:05,410 --> 00:28:08,500 So it's a little bit beyond what I want to discuss in the book, 452 00:28:08,500 --> 00:28:13,154 but atom photon indirection has a wonderful chapter on that. 453 00:28:13,154 --> 00:28:13,820 Other questions? 454 00:28:19,870 --> 00:28:23,570 There is another important case of the optical Bloch equation, 455 00:28:23,570 --> 00:28:34,300 which is the weak excitation limit, which is simple. 456 00:28:34,300 --> 00:28:39,460 But I've already asked you to look at it in your homework 457 00:28:39,460 --> 00:28:39,960 assignment. 458 00:28:44,620 --> 00:28:51,590 OK, let's now come to the third aspect of master equation 459 00:28:51,590 --> 00:28:54,170 solutions of optical Bloch or master equation. 460 00:28:54,170 --> 00:29:04,770 And these are damped vacuum Rabi oscillations. 461 00:29:23,240 --> 00:29:25,460 Yes. 462 00:29:25,460 --> 00:29:27,630 I really like this example in this chapter, 463 00:29:27,630 --> 00:29:31,240 so I hope-- well, I think we have many highlights, 464 00:29:31,240 --> 00:29:33,054 but this is a really nice example. 465 00:29:33,054 --> 00:29:34,720 I learned it from Professor [? Schwan ?] 466 00:29:34,720 --> 00:29:36,050 when he introduced it. 467 00:29:36,050 --> 00:29:38,070 And what I like about it is I can 468 00:29:38,070 --> 00:29:42,880 use it to show you what other Lindblad operators may 469 00:29:42,880 --> 00:29:43,960 be important. 470 00:29:43,960 --> 00:29:46,410 So you suddenly understand, in a bigger context, 471 00:29:46,410 --> 00:29:48,350 what the master equation is. 472 00:29:48,350 --> 00:29:52,460 And then we continue with an atomic cavity. 473 00:29:52,460 --> 00:29:55,490 And the damping is no longer by spontaneous emission. 474 00:29:55,490 --> 00:29:58,690 The damping is by photons sneaking out of the cavity. 475 00:29:58,690 --> 00:30:00,770 So we really learn something which 476 00:30:00,770 --> 00:30:02,840 is similar to optical Bloch equation, 477 00:30:02,840 --> 00:30:04,350 but in another context. 478 00:30:04,350 --> 00:30:07,780 And often, if you see two different realizations 479 00:30:07,780 --> 00:30:12,430 of similar physics, you maybe realize more 480 00:30:12,430 --> 00:30:14,810 what is more generic and what is special for instance 481 00:30:14,810 --> 00:30:17,480 of spontaneous emission, so I really like that. 482 00:30:17,480 --> 00:30:20,310 But also, I found that this example 483 00:30:20,310 --> 00:30:24,010 allows me to introduce to you the concept of the quantum Zeno 484 00:30:24,010 --> 00:30:26,730 effect and the concept of atypical elimination 485 00:30:26,730 --> 00:30:27,480 of coherences. 486 00:30:27,480 --> 00:30:30,290 So it's a wonderful example which connects us 487 00:30:30,290 --> 00:30:34,270 with a number of really neat concepts. 488 00:30:34,270 --> 00:30:41,720 So with that promise, let me remind you 489 00:30:41,720 --> 00:30:47,100 that, in the master equation, when we derived it, 490 00:30:47,100 --> 00:31:01,140 we got an exact expression when we did second order 491 00:31:01,140 --> 00:31:02,620 perturbation theory. 492 00:31:02,620 --> 00:31:08,760 And the structure which came was this double commutator 493 00:31:08,760 --> 00:31:12,460 between the interaction operator and the steady state operator. 494 00:31:12,460 --> 00:31:18,590 Just to remind you, in the interaction picture-- first 495 00:31:18,590 --> 00:31:20,700 in the non-interacting, in the normal picture, 496 00:31:20,700 --> 00:31:24,270 the derivative of the density matrix is a commutator with h. 497 00:31:24,270 --> 00:31:26,430 In the interaction picture, it's a commutator 498 00:31:26,430 --> 00:31:28,380 with v with the interaction. 499 00:31:28,380 --> 00:31:31,560 But when we iterate an equation to second order, 500 00:31:31,560 --> 00:31:34,330 we plot the first result into the equation. 501 00:31:34,330 --> 00:31:38,310 Then we get, in second order, this structure. 502 00:31:38,310 --> 00:31:48,850 And if I, by expanding the commutator, 503 00:31:48,850 --> 00:31:52,160 we obtain the following structure. 504 00:31:52,160 --> 00:31:58,390 And that gave rise to the Lindblad form 505 00:31:58,390 --> 00:32:05,900 of the master equation, which is rho dot equals a Hamiltonian 506 00:32:05,900 --> 00:32:07,750 part. 507 00:32:07,750 --> 00:32:13,000 But then we have now a sum, v, the interaction 508 00:32:13,000 --> 00:32:16,380 with the environment, can be a sum of mu 509 00:32:16,380 --> 00:32:19,740 dot b and b-- and so an external b field 510 00:32:19,740 --> 00:32:22,880 times magnetization and dipole moment times an electric field. 511 00:32:22,880 --> 00:32:25,920 It can have a number of terms. 512 00:32:25,920 --> 00:32:31,270 And therefore, in general, we have more than one Lindblad 513 00:32:31,270 --> 00:32:32,150 operator. 514 00:32:32,150 --> 00:32:39,190 But the structure is always given 515 00:32:39,190 --> 00:32:47,030 by this commutator in the following form. 516 00:32:47,030 --> 00:32:47,890 OK. 517 00:32:47,890 --> 00:32:52,180 So we have this general derivation. 518 00:32:52,180 --> 00:32:57,060 But until now, we have only looked 519 00:32:57,060 --> 00:33:04,040 at a system which is a simple atom. 520 00:33:04,040 --> 00:33:08,390 And the environment was the vacuum. 521 00:33:08,390 --> 00:33:10,720 So this is our environment. 522 00:33:10,720 --> 00:33:15,510 And also, the environment was always in the vacuum state. 523 00:33:15,510 --> 00:33:16,950 So these were our assumptions. 524 00:33:19,730 --> 00:33:24,470 And that means that the only Lindblad operator, 525 00:33:24,470 --> 00:33:27,020 the only jump operator in this sum 526 00:33:27,020 --> 00:33:29,725 is sigma minus for spontaneous emission. 527 00:33:32,480 --> 00:33:36,860 And by inserting this jump operator 528 00:33:36,860 --> 00:33:40,280 into the Lindblad form of the master equation, 529 00:33:40,280 --> 00:33:42,270 we find the optical Bloch equation. 530 00:33:45,020 --> 00:33:49,420 But now let's do cavity QED. 531 00:33:54,010 --> 00:33:56,800 And there are lots of experiments 532 00:33:56,800 --> 00:34:04,580 going on in the research group of Professor Vuletic. 533 00:34:04,580 --> 00:34:08,100 And in this situation, our system is actually not 534 00:34:08,100 --> 00:34:12,290 just the atoms, it's the atoms and a single mode 535 00:34:12,290 --> 00:34:14,230 of the cavity. 536 00:34:14,230 --> 00:34:16,400 This is our system. 537 00:34:16,400 --> 00:34:20,710 And the environment are all other modes. 538 00:34:27,370 --> 00:34:37,880 So in other words, the atoms interacting 539 00:34:37,880 --> 00:34:43,920 with a single mode of the cavity, this is our system. 540 00:34:43,920 --> 00:34:54,400 And the environment is now accessed 541 00:34:54,400 --> 00:34:59,220 by emitting-- the atom is emitting. 542 00:34:59,220 --> 00:35:03,010 Or there are photons leaking out of the cavity, 543 00:35:03,010 --> 00:35:12,970 because the cavity mirrors do not have 100.0% reflectivity. 544 00:35:12,970 --> 00:35:16,820 I will immediately reduce it to one Lindblad operator. 545 00:35:16,820 --> 00:35:24,700 But we could now say that this system, in its full glory, 546 00:35:24,700 --> 00:35:27,950 has four different Lindblad operators which 547 00:35:27,950 --> 00:35:31,050 provide dissipation, which provide coupling 548 00:35:31,050 --> 00:35:32,690 to the environment. 549 00:35:32,690 --> 00:35:38,820 The first one is simply what we had before, 550 00:35:38,820 --> 00:35:42,050 spontaneous emission. 551 00:35:42,050 --> 00:35:52,850 However, if we allow that the environment is thermal, 552 00:35:52,850 --> 00:35:53,732 we have to multiply. 553 00:35:53,732 --> 00:35:55,940 Well, whenever you have emission and you have already 554 00:35:55,940 --> 00:35:58,700 population in a mode, and thermal 555 00:35:58,700 --> 00:36:03,090 is a number of thermal photons in every mode, on average, 556 00:36:03,090 --> 00:36:07,200 then you have an extra bosonic or photonic stimulation term. 557 00:36:07,200 --> 00:36:10,470 That's stimulated emission, but not by the laser beam. 558 00:36:10,470 --> 00:36:15,050 It's now stimulated emission by a thermal occupation 559 00:36:15,050 --> 00:36:16,380 of all the other modes. 560 00:36:16,380 --> 00:36:18,510 So just assume, for that purpose, 561 00:36:18,510 --> 00:36:19,690 that you have a cavity here. 562 00:36:19,690 --> 00:36:26,000 But the cavity is now in a black body cavity. 563 00:36:26,000 --> 00:36:28,690 And everything, all modes in the vacuum, 564 00:36:28,690 --> 00:36:29,932 are no longer vacuum modes. 565 00:36:29,932 --> 00:36:31,640 They are occupied within thermal photons. 566 00:36:34,400 --> 00:36:38,230 Of course, if this is the case, we 567 00:36:38,230 --> 00:36:40,700 have also the opposite effect. 568 00:36:40,700 --> 00:36:44,090 You can see by unitality, namely, 569 00:36:44,090 --> 00:36:54,080 that we can have-- oops, n should be-- 570 00:36:54,080 --> 00:37:01,080 but that we have an absorption process. 571 00:37:01,080 --> 00:37:04,640 We absorb a photon from the thermal background. 572 00:37:04,640 --> 00:37:06,780 But now, we have two more processes. 573 00:37:06,780 --> 00:37:10,290 And this is photons leaking out of the cavity. 574 00:37:10,290 --> 00:37:13,810 When a photon leaks out of the cavity, this leakage 575 00:37:13,810 --> 00:37:17,720 is not coupled to the atom, it's coupled to the mode. 576 00:37:17,720 --> 00:37:21,880 We describe the mode by a mode operator, a. 577 00:37:21,880 --> 00:37:25,550 So now, the leakage out with the rate, 578 00:37:25,550 --> 00:37:28,640 kappa, is presented by that. 579 00:37:28,640 --> 00:37:33,070 But in the case of thermal photons, 580 00:37:33,070 --> 00:37:36,650 we have stimulation by the thermal photons. 581 00:37:36,650 --> 00:37:41,100 And also, if we have thermal photons, 582 00:37:41,100 --> 00:37:43,830 photons cannot only leak out of the cavity, 583 00:37:43,830 --> 00:37:47,400 they can also leak into the cavity. 584 00:37:47,400 --> 00:37:54,196 So we have assumed here that we have n thermal photons 585 00:37:54,196 --> 00:38:01,855 in each mode of the environment. 586 00:38:04,670 --> 00:38:08,800 So in other words, if you want to look at, for instance, 587 00:38:08,800 --> 00:38:12,510 Serge Haroche does experiment in the wonderful experiments 588 00:38:12,510 --> 00:38:14,130 in the microwave domain. 589 00:38:14,130 --> 00:38:19,520 He has cavities at, I think, gigahertz frequencies. 590 00:38:19,520 --> 00:38:21,760 And even at cryogenic temperature, 591 00:38:21,760 --> 00:38:25,310 there are occasional thermal photons in this mode. 592 00:38:25,310 --> 00:38:27,630 But you are now expert enough that you 593 00:38:27,630 --> 00:38:29,430 would know how to set up a master 594 00:38:29,430 --> 00:38:32,030 equation for that situation. 595 00:38:32,030 --> 00:38:35,230 But I want to discuss simple limiting cases. 596 00:38:35,230 --> 00:38:38,620 So I think you are glad to hear that we 597 00:38:38,620 --> 00:38:43,230 are choosing 0 thermal photons. 598 00:38:43,230 --> 00:38:48,190 So therefore, we assume we have a cavity in vacuum. 599 00:38:48,190 --> 00:38:54,370 And therefore, we only need the Lindblad operators, L1 and L2. 600 00:38:57,480 --> 00:39:08,430 And we want to describe the system which has only one 601 00:39:08,430 --> 00:39:10,565 x, which has, maximally, one excitation. 602 00:39:17,270 --> 00:39:19,930 We try to keep the density matrix small, 603 00:39:19,930 --> 00:39:24,380 so we only want to consider three states. 604 00:39:24,380 --> 00:39:27,070 So when I said there is one excitation, 605 00:39:27,070 --> 00:39:29,240 it is, of course, the state which 606 00:39:29,240 --> 00:39:32,950 has no photon in the excited state. 607 00:39:32,950 --> 00:39:38,070 The state two is we have the ground state and one photon. 608 00:39:38,070 --> 00:39:42,560 And those two states are coupled by the cavity. 609 00:39:42,560 --> 00:39:46,330 And since I really want to discuss the simplest case which 610 00:39:46,330 --> 00:39:48,700 already illustrates all the concepts I want to do 611 00:39:48,700 --> 00:39:54,300 is they are coupled by the vacuum Rabi oscillation. 612 00:39:54,300 --> 00:39:56,590 And we have discussed vacuum Rabi oscillation 613 00:39:56,590 --> 00:39:58,200 in part one of the course. 614 00:39:58,200 --> 00:39:59,340 But it's very simple. 615 00:39:59,340 --> 00:40:00,830 It's just two levels coupled. 616 00:40:00,830 --> 00:40:03,470 It's a 2 by 2 matrix, so this is something 617 00:40:03,470 --> 00:40:05,140 you are familiar with. 618 00:40:07,980 --> 00:40:15,300 But now, the master equation, the Lindblad operator 619 00:40:15,300 --> 00:40:18,970 bring in the third level, which is 620 00:40:18,970 --> 00:40:20,550 the ground state without photons. 621 00:40:23,180 --> 00:40:25,710 And this can happen in two ways. 622 00:40:25,710 --> 00:40:29,360 One is spontaneous emission. 623 00:40:29,360 --> 00:40:32,110 The excited state emits, but not into the cavity. 624 00:40:32,110 --> 00:40:33,990 It emits to the side. 625 00:40:33,990 --> 00:40:36,040 So spontaneous emission can take us 626 00:40:36,040 --> 00:40:41,630 down here, or when the excitation is in the cavity. 627 00:40:41,630 --> 00:40:44,760 We have one photon in the cavity, 628 00:40:44,760 --> 00:40:46,490 this photon can leak out. 629 00:40:46,490 --> 00:40:49,390 And the rate that is kappa. 630 00:40:54,740 --> 00:40:59,650 And since we have talked about spontaneous emission 631 00:40:59,650 --> 00:41:04,990 so much with the optical Bloch equation, 632 00:41:04,990 --> 00:41:09,530 we want to discuss now the case where gamma can be neglected. 633 00:41:09,530 --> 00:41:11,530 And we want to understand what happens 634 00:41:11,530 --> 00:41:14,060 when the only dissipation of the system 635 00:41:14,060 --> 00:41:16,595 comes by photons leaking out of the cavity. 636 00:41:22,800 --> 00:41:24,900 Any questions about the system and the motivation? 637 00:41:29,130 --> 00:41:32,670 So our system is now an atom with a cavity. 638 00:41:32,670 --> 00:41:34,920 The environment is a vacuum. 639 00:41:34,920 --> 00:41:39,860 But the process of dissipation relaxation 640 00:41:39,860 --> 00:41:41,980 is no longer spontaneous emission, 641 00:41:41,980 --> 00:41:44,910 it is the photon leaking out of the cavity. 642 00:41:44,910 --> 00:41:47,770 And this realizes different physics. 643 00:41:47,770 --> 00:41:51,510 And I hope, by experiencing these different physics, 644 00:41:51,510 --> 00:41:54,450 you have sort of a nicer picture what is dissipation 645 00:41:54,450 --> 00:41:57,070 in an open system and what is similar, 646 00:41:57,070 --> 00:41:59,363 but what is also different or spontaneous emission. 647 00:41:59,363 --> 00:41:59,862 [? Koren? ?] 648 00:41:59,862 --> 00:42:04,500 AUDIENCE: If you're considering only kappa, isn't there 649 00:42:04,500 --> 00:42:07,060 an equivalent process of order kappa that takes you 650 00:42:07,060 --> 00:42:09,732 to ground of two photons in the cavity? 651 00:42:09,732 --> 00:42:11,790 Don't you have to consider that state as well? 652 00:42:22,122 --> 00:42:25,074 That term would be of order kappa as well. 653 00:42:25,074 --> 00:42:26,140 PROFESSOR: So great. 654 00:42:26,140 --> 00:42:32,100 The question is I restrict the Hilbert 655 00:42:32,100 --> 00:42:35,370 space to maximally one excitation. 656 00:42:35,370 --> 00:42:37,910 So this is how we start out with our differential equation. 657 00:42:37,910 --> 00:42:40,610 AUDIENCE: Oh, this isn't in a thermal bath? 658 00:42:40,610 --> 00:42:43,230 I thought we were assuming we had a-- 659 00:42:43,230 --> 00:42:45,290 PROFESSOR: Oh, sorry. 660 00:42:45,290 --> 00:42:47,910 I first made everything complicated. 661 00:42:47,910 --> 00:42:49,830 But then I said, now, let's make it simple. 662 00:42:49,830 --> 00:42:51,930 We assumed the thermal bath is 0. 663 00:42:51,930 --> 00:42:53,080 So we reduced-- 664 00:42:53,080 --> 00:42:57,500 In a way, yes, I told you here are four Lindblad operators, 665 00:42:57,500 --> 00:42:59,690 and you can describe everything you want. 666 00:42:59,690 --> 00:43:02,800 But then I said, hey, two of them become 0, 667 00:43:02,800 --> 00:43:04,780 because we eliminate the thermal bath. 668 00:43:04,780 --> 00:43:07,810 And now I make the approximation that spontaneous emission 669 00:43:07,810 --> 00:43:08,870 is negligible. 670 00:43:08,870 --> 00:43:10,540 And now we are back to the situation 671 00:43:10,540 --> 00:43:12,910 which is nice for classroom discussion. 672 00:43:12,910 --> 00:43:15,140 We've only one term left, and this 673 00:43:15,140 --> 00:43:17,930 is the leaking of the photon out of the cavity. 674 00:43:17,930 --> 00:43:19,780 So I will show to you what happens. 675 00:43:19,780 --> 00:43:23,305 But ultimately, when we make the cavity better and better 676 00:43:23,305 --> 00:43:27,120 and kappa smaller and smaller, I will say, wait a moment. 677 00:43:27,120 --> 00:43:29,281 We now have to check that our solution is still 678 00:43:29,281 --> 00:43:31,780 consistent with our assumption that spontaneous emission can 679 00:43:31,780 --> 00:43:33,197 be neglected. 680 00:43:33,197 --> 00:43:34,780 So that's sort of what you want to do. 681 00:43:43,195 --> 00:43:43,695 Yes. 682 00:43:52,540 --> 00:43:54,870 Let's just get two more pages. 683 00:44:02,870 --> 00:44:03,640 OK. 684 00:44:03,640 --> 00:44:07,430 So what we want to do is we want to look 685 00:44:07,430 --> 00:44:10,030 for the dynamics of the system. 686 00:44:10,030 --> 00:44:12,580 We start out in state one. 687 00:44:12,580 --> 00:44:17,780 So we inject one excited atom into the cavity. 688 00:44:17,780 --> 00:44:21,500 And what we want to learn now is-- 689 00:44:21,500 --> 00:44:25,750 and it has a lot of interesting physics in it-- 690 00:44:25,750 --> 00:44:37,065 what will happen as a function of this new dissipation, kappa. 691 00:44:41,960 --> 00:44:45,980 Well, qualitatively, it should be pretty clear. 692 00:44:45,980 --> 00:44:51,040 If kappa is 0, we have vacuum Rabi oscillation 693 00:44:51,040 --> 00:44:53,740 between state one and state two. 694 00:44:53,740 --> 00:44:56,380 If you then put a little bit kappa into the system, 695 00:44:56,380 --> 00:45:00,380 you have a Rabi oscillation, but they are getting damped. 696 00:45:00,380 --> 00:45:04,270 And like in any oscillator, if you crank up the damping, 697 00:45:04,270 --> 00:45:05,660 you go into an overdamped regime. 698 00:45:09,550 --> 00:45:12,410 And this is exactly what the equations give us, 699 00:45:12,410 --> 00:45:22,480 Rabi oscillation, damped Rabi oscillation, and then 700 00:45:22,480 --> 00:45:23,280 overdamped regime. 701 00:45:27,110 --> 00:45:29,120 And actually, the most interesting regime for us 702 00:45:29,120 --> 00:45:30,370 will be the overdamped regime. 703 00:45:30,370 --> 00:45:32,950 That's something we haven't encountered. 704 00:45:32,950 --> 00:45:43,240 But let me just, before we go there, write down 705 00:45:43,240 --> 00:45:47,090 the master equation for you. 706 00:45:47,090 --> 00:45:58,100 So without the photon leaking out of the cavity, 707 00:45:58,100 --> 00:46:02,850 we have the following terms. 708 00:46:02,850 --> 00:46:05,730 Without the photon leaking out of the cavity, of course, 709 00:46:05,730 --> 00:46:08,070 state three is not involved. 710 00:46:08,070 --> 00:46:11,890 We simply have Rabi oscillation between state one and state 711 00:46:11,890 --> 00:46:13,990 two. 712 00:46:13,990 --> 00:46:18,880 And this is actually also like the optical Bloch equation. 713 00:46:18,880 --> 00:46:22,440 Without damping, it's simply the Jaynes-Cummings model. 714 00:46:22,440 --> 00:46:37,750 Minus, that's this. 715 00:46:37,750 --> 00:46:44,120 And for the coherences, we get omega 0 716 00:46:44,120 --> 00:46:49,590 rho 1, 1 minus rho 2, 2. 717 00:46:49,590 --> 00:46:52,300 But now, we put in the terms with kappa. 718 00:46:56,120 --> 00:46:58,260 The state, which is the state two, which 719 00:46:58,260 --> 00:47:00,850 is the ground state with the photon, 720 00:47:00,850 --> 00:47:04,790 has now a damping term, because the photon can leak out. 721 00:47:08,510 --> 00:47:12,150 We get, of course, an equation for the ground 722 00:47:12,150 --> 00:47:15,150 state, the population, the photon leaks out. 723 00:47:15,150 --> 00:47:21,630 And we populate state three. 724 00:47:21,630 --> 00:47:27,650 And the damping also affects the coherences. 725 00:47:27,650 --> 00:47:33,640 And the form of the master equation, the Lindblad form, 726 00:47:33,640 --> 00:47:37,660 gives us the following damping term. 727 00:47:46,010 --> 00:47:48,610 So there are two new concepts which 728 00:47:48,610 --> 00:47:50,960 I promised you to introduce. 729 00:47:50,960 --> 00:48:05,315 One is the idiopathic elimination of coherences. 730 00:48:16,350 --> 00:48:21,745 And the second one is the quantum Zeno effect. 731 00:48:26,540 --> 00:48:31,430 So first of all, if you look at the differential equation, 732 00:48:31,430 --> 00:48:34,710 and when kappa is small, in particular, kappa 733 00:48:34,710 --> 00:48:39,270 is smaller than 2 times the vacuum Rabi frequency, 734 00:48:39,270 --> 00:48:48,280 then the population to be in the excited state, which is rho 2, 735 00:48:48,280 --> 00:48:53,420 2, undergoes Rabi oscillation. 736 00:48:53,420 --> 00:48:56,810 And those Rabi oscillations, this population 737 00:48:56,810 --> 00:48:59,710 is damped by e to the minus kappa t. 738 00:49:03,730 --> 00:49:07,845 So what I want to discuss now is the overdamped case. 739 00:49:13,350 --> 00:49:18,350 And actually, before I derive it for you, 740 00:49:18,350 --> 00:49:20,940 I wanted to ask you some clicker question. 741 00:49:20,940 --> 00:49:24,290 I'm afraid I put the books out, but somebody put it away. 742 00:49:24,290 --> 00:49:27,370 So nobody took clickers? 743 00:49:27,370 --> 00:49:32,170 So why don't we just pass them around? 744 00:49:35,520 --> 00:49:37,925 And I can already formulate the question. 745 00:49:46,070 --> 00:49:59,080 I think you know by now that I have a preference 746 00:49:59,080 --> 00:50:02,260 to maybe ask you questions about really simple quantum physics. 747 00:50:02,260 --> 00:50:05,060 Quantum physics is sadly enough that nobody fully 748 00:50:05,060 --> 00:50:07,030 understands it. 749 00:50:07,030 --> 00:50:09,630 And we want to improve on that. 750 00:50:09,630 --> 00:50:36,110 So I want to test your intuition by looking at the situation. 751 00:50:36,110 --> 00:50:38,320 We have an excited state without photon. 752 00:50:41,980 --> 00:50:43,020 We have a ground state. 753 00:50:43,020 --> 00:50:47,410 So I want to test your intuition in the following way. 754 00:50:47,410 --> 00:50:51,525 We have Rabi oscillations between two levels. 755 00:50:55,240 --> 00:50:58,420 The case in the cavity is, it is Rabi oscillation between state 756 00:50:58,420 --> 00:51:00,320 two and state one, but it doesn't matter. 757 00:51:00,320 --> 00:51:07,630 You can also assume that it is Rabi oscillation 758 00:51:07,630 --> 00:51:09,910 between hyperfine levels. 759 00:51:09,910 --> 00:51:13,740 And the Rabi oscillation is driven by an RF field. 760 00:51:13,740 --> 00:51:15,690 And the one thing we introduce now 761 00:51:15,690 --> 00:51:21,820 is we introduce damping kappa to this level. 762 00:51:21,820 --> 00:51:25,420 But we start out the system with all 763 00:51:25,420 --> 00:51:27,030 of the population in this state. 764 00:51:29,640 --> 00:51:36,436 And in the overdamped case, when kappa is sufficiently strong, 765 00:51:36,436 --> 00:51:38,310 then we know an oscillator will be overdamped 766 00:51:38,310 --> 00:51:41,406 and will no longer oscillate. 767 00:51:41,406 --> 00:51:47,510 And in this situation, the probability 768 00:51:47,510 --> 00:51:51,435 to be in the initial state will decay. 769 00:51:54,530 --> 00:52:03,020 e to the minus kap-- e to the minus-- how should we call it? 770 00:52:03,020 --> 00:52:04,430 gamma t. 771 00:52:04,430 --> 00:52:09,630 So it decays with the decay rate kappa. 772 00:52:12,470 --> 00:52:18,490 And my question for you is, A, B and C, whether the decay 773 00:52:18,490 --> 00:52:26,770 rate, kappa, is proportional to the damping rate, gamma? 774 00:52:26,770 --> 00:52:30,310 Proportional to the inverse of gamma? 775 00:52:30,310 --> 00:52:32,440 Or independent of gamma? 776 00:52:37,720 --> 00:52:41,030 So we've exacted this system. 777 00:52:41,030 --> 00:52:44,390 The system has possibility of Rabi oscillation. 778 00:52:44,390 --> 00:52:46,380 I showed you the damped Rabi oscillation 779 00:52:46,380 --> 00:52:48,940 and in the limit of small kappa. 780 00:52:48,940 --> 00:52:51,380 But now you go to the overdamped regime. 781 00:52:51,380 --> 00:52:55,040 And my question is, if the system is overdamped, 782 00:52:55,040 --> 00:53:01,580 will this decay rate increase with kappa? 783 00:53:01,580 --> 00:53:02,710 Decrease with kappa? 784 00:53:02,710 --> 00:53:04,615 Or will it be independent of a damped regime? 785 00:53:35,480 --> 00:53:36,305 All right. 786 00:53:42,750 --> 00:53:43,250 OK. 787 00:53:43,250 --> 00:53:45,870 There's a clear neutrality. 788 00:53:45,870 --> 00:53:50,770 Let me not yet discuss the answer. 789 00:53:50,770 --> 00:53:54,700 Let me maybe try to give you another problem. 790 00:53:54,700 --> 00:53:58,460 And you should maybe figure out if those two problems 791 00:53:58,460 --> 00:54:01,290 are related. 792 00:54:01,290 --> 00:54:03,680 I want to ask you now something simpler. 793 00:54:03,680 --> 00:54:06,800 There are not three levels involved, 794 00:54:06,800 --> 00:54:09,750 there are two levels involved. 795 00:54:09,750 --> 00:54:14,020 But one of the level is broadened by kappa. 796 00:54:16,700 --> 00:54:18,770 Call it an unstable state. 797 00:54:18,770 --> 00:54:21,510 Because it can decay with the rate, kappa, and we 798 00:54:21,510 --> 00:54:24,600 often show something which is broadened. 799 00:54:24,600 --> 00:54:31,750 And now we have a matrix element between the discrete level 800 00:54:31,750 --> 00:54:35,430 and the broadened state, which I call omega. 801 00:54:35,430 --> 00:54:38,940 It's a matrix element or Rabi frequency. 802 00:54:38,940 --> 00:54:40,620 And now you should figure out what 803 00:54:40,620 --> 00:54:43,630 is the physical picture which describes 804 00:54:43,630 --> 00:54:49,580 the coupling of a discrete initial state to something 805 00:54:49,580 --> 00:54:51,980 which is broadened. 806 00:54:51,980 --> 00:55:03,060 It's now the transition rate from the original state. 807 00:55:03,060 --> 00:55:08,280 Does the original state decay with a collectivistic rate, 808 00:55:08,280 --> 00:55:16,235 which is omega? 809 00:55:19,890 --> 00:55:30,720 Or is it omega squared over kappa? 810 00:55:33,650 --> 00:55:37,010 Or is it none of the above? 811 00:55:44,210 --> 00:55:49,370 So coupling between two levels, very, very simple. 812 00:55:49,370 --> 00:55:50,565 But one level is broadened. 813 00:56:10,791 --> 00:56:11,290 OK. 814 00:56:16,110 --> 00:56:20,450 Yes, it's Fermi's Golden Rule, what I'm asking you. 815 00:56:20,450 --> 00:56:23,440 This is nothing else than Fermi's Golden Rule. 816 00:56:23,440 --> 00:56:28,890 We couple from one level into some continuum of states. 817 00:56:28,890 --> 00:56:33,200 If I take one state and smear it out over a width, 818 00:56:33,200 --> 00:56:37,880 kappa, the density of states, the density of modes which 819 00:56:37,880 --> 00:56:41,340 appears in Fermi's Golden Rule, is 1 over kappa. 820 00:56:41,340 --> 00:56:45,010 And Fermi's Golden Rule tells us that the coupling 821 00:56:45,010 --> 00:56:50,380 is the matrix element squared times the density of states, 822 00:56:50,380 --> 00:56:51,470 which is 1 over kappa. 823 00:56:56,930 --> 00:57:02,970 So this is nothing else than Fermi's Golden Rule. 824 00:57:10,200 --> 00:57:12,240 OK. 825 00:57:12,240 --> 00:57:14,160 Why don't we come back to the first question 826 00:57:14,160 --> 00:57:18,240 now where we have two discrete levels, maybe 827 00:57:18,240 --> 00:57:22,450 two hyperfine states, which are coupled by a Rabi frequency? 828 00:57:22,450 --> 00:57:25,390 But then one of the hyperfine states 829 00:57:25,390 --> 00:57:28,740 decays because there is some leakage, or we-- 830 00:57:28,740 --> 00:57:31,890 I don't know-- we interrogate this state with laser 831 00:57:31,890 --> 00:57:32,780 beams or such. 832 00:57:32,780 --> 00:57:34,380 We make it unstable. 833 00:57:34,380 --> 00:57:38,700 We allow this state to decay to another state. 834 00:57:38,700 --> 00:57:44,120 And the question is, if we make the decay of that level 835 00:57:44,120 --> 00:57:50,110 stronger and stronger, does the decay of the initial state 836 00:57:50,110 --> 00:57:50,960 increase? 837 00:57:50,960 --> 00:57:51,510 Decrease? 838 00:57:51,510 --> 00:57:52,512 Or is of that? 839 00:57:58,010 --> 00:57:59,990 Good. 840 00:57:59,990 --> 00:58:03,820 So what you have learned here is that, actually, the more we 841 00:58:03,820 --> 00:58:08,070 damp the system on this side, the longer-lived 842 00:58:08,070 --> 00:58:11,260 is the initial state. 843 00:58:11,260 --> 00:58:12,280 But it's trivial. 844 00:58:12,280 --> 00:58:13,610 It's just Fermi's Golden Rule. 845 00:58:13,610 --> 00:58:16,760 When you couple into a level which is broadened, 846 00:58:16,760 --> 00:58:19,784 the coupling becomes weaker and weaker because, well, that's 847 00:58:19,784 --> 00:58:21,200 what Fermi's Golden Rule tells us. 848 00:58:24,030 --> 00:58:28,010 I want to now derive this situation for you for our atom 849 00:58:28,010 --> 00:58:30,240 in the cavity. 850 00:58:30,240 --> 00:58:33,950 And I will say that this is one example for the quantum Zeno 851 00:58:33,950 --> 00:58:35,180 effect. 852 00:58:35,180 --> 00:58:36,860 But let me already explain to you 853 00:58:36,860 --> 00:58:39,260 what the quantum Zeno effect is. 854 00:58:39,260 --> 00:58:41,490 Zeno is a Greek philosopher. 855 00:58:41,490 --> 00:58:46,620 And the Greek philosophers had some deep thoughts 856 00:58:46,620 --> 00:58:49,910 about the nature of time, the nature of motion. 857 00:58:49,910 --> 00:58:52,220 And I think, in the Greek philosophy-- sorry. 858 00:58:54,820 --> 00:58:59,635 I didn't go-- I mean, I don't have a strong background 859 00:58:59,635 --> 00:59:01,780 in philosophy, but I think the idea 860 00:59:01,780 --> 00:59:05,640 is that motion is when something moves around. 861 00:59:05,640 --> 00:59:07,550 But it takes some time to move. 862 00:59:07,550 --> 00:59:10,180 So if you observe it, it can't move, 863 00:59:10,180 --> 00:59:13,514 because every time you observe it, you localize something. 864 00:59:13,514 --> 00:59:14,180 That's the idea. 865 00:59:14,180 --> 00:59:17,610 If you see an arrow, the moment you observe it, it's localized. 866 00:59:17,610 --> 00:59:21,170 And if you localize something to often, it cannot move. 867 00:59:21,170 --> 00:59:23,590 And this is sort of what you can say here. 868 00:59:23,590 --> 00:59:26,700 The system wants to evolve to a second state. 869 00:59:26,700 --> 00:59:30,010 But kappa can actually be a measurement process. 870 00:59:30,010 --> 00:59:32,640 We can just shine a strong laser on this state 871 00:59:32,640 --> 00:59:35,360 and figure out has something arrived here. 872 00:59:35,360 --> 00:59:41,340 And the more often we look, the stronger our measurement is, 873 00:59:41,340 --> 00:59:44,340 the less the system can evolve. 874 00:59:44,340 --> 00:59:47,970 So a measurement of what arrives in this state 875 00:59:47,970 --> 00:59:51,570 slows down the dynamics to that state. 876 00:59:51,570 --> 00:59:53,720 And this is called the quantum Zeno effect. 877 00:59:53,720 --> 00:59:56,450 It also goes by the name of quantum Zeno paradox. 878 00:59:56,450 --> 01:00:00,680 But it's just quantum mechanics, it's not a paradox. 879 01:00:00,680 --> 01:00:03,930 In the popular world, it is sometimes paraphrased 880 01:00:03,930 --> 01:00:09,470 as if you observe the tea kettle, the water never boils. 881 01:00:09,470 --> 01:00:13,220 But you see now where it comes from. 882 01:00:13,220 --> 01:00:15,200 OK, so this is the quantum Zeno effect. 883 01:00:15,200 --> 01:00:18,470 And we want to now, by looking at the master 884 01:00:18,470 --> 01:00:20,990 equation for an atom in a cavity, 885 01:00:20,990 --> 01:00:22,910 I want to show you the limit that it really 886 01:00:22,910 --> 01:00:25,640 comes out of the master equation. 887 01:00:25,640 --> 01:00:26,850 Questions about that? 888 01:00:43,260 --> 01:00:49,720 OK, so this is the quantum Zeno effect. 889 01:00:49,720 --> 01:00:56,680 And well, one of the nicest papers on the quantum Zeno 890 01:00:56,680 --> 01:01:00,820 effect-- well, I'm biased here-- but it was written by my group, 891 01:01:00,820 --> 01:01:04,240 because we used the Bose-Einstein condensate. 892 01:01:04,240 --> 01:01:08,450 We had an RF coupling to another hyperfine state. 893 01:01:08,450 --> 01:01:15,840 And then we used a laser beam and observed the population 894 01:01:15,840 --> 01:01:17,050 in the final state. 895 01:01:17,050 --> 01:01:21,070 And we saw that the effect of the RF drive 896 01:01:21,070 --> 01:01:23,820 became weaker and weaker and weaker, 897 01:01:23,820 --> 01:01:28,278 the stronger we made the measurement in the final state. 898 01:01:28,278 --> 01:01:32,860 And this is the reference. 899 01:01:32,860 --> 01:01:37,160 And our work was the first quantitative comparison. 900 01:01:37,160 --> 01:01:38,780 You can now do a measurement by just 901 01:01:38,780 --> 01:01:42,530 using a strong laser in, maybe, every millisecond or so. 902 01:01:42,530 --> 01:01:45,570 Or you can use a weaker laser beam continuously. 903 01:01:45,570 --> 01:01:50,010 So we showed in this work that the quantum Zeno effect 904 01:01:50,010 --> 01:01:53,910 is the same, whether you do a pulsed observation, which 905 01:01:53,910 --> 01:01:56,600 is a strong measurement, or continuous weak measurement. 906 01:02:01,180 --> 01:02:01,680 OK. 907 01:02:01,680 --> 01:02:05,660 But let's now, after this short interlude, 908 01:02:05,660 --> 01:02:07,610 go back to the master equation. 909 01:02:07,610 --> 01:02:19,550 So our job is now to solve this master equation 910 01:02:19,550 --> 01:02:22,240 for the limit of strong damping. 911 01:02:22,240 --> 01:02:23,860 And what we will find, actually, is 912 01:02:23,860 --> 01:02:27,180 we will find both the quantum Zeno effect and another tidbit. 913 01:02:27,180 --> 01:02:29,700 We will also find the Purcell effect, 914 01:02:29,700 --> 01:02:33,570 which is enhanced spontaneous emission to a cavity. 915 01:02:33,570 --> 01:02:41,570 Now the way how we solve this master equation 916 01:02:41,570 --> 01:02:48,450 in this limiting case is by the method 917 01:02:48,450 --> 01:02:52,430 of idiopathic elimination of coherences. 918 01:02:52,430 --> 01:02:56,110 And let me explain it to you with that equation. 919 01:02:56,110 --> 01:03:01,120 What we have here is the derivative of the coherences. 920 01:03:01,120 --> 01:03:03,240 And when you look at the right-hand side, 921 01:03:03,240 --> 01:03:07,550 the derivative of the coherences is kappa, a damping terms, 922 01:03:07,550 --> 01:03:09,550 times the coherence. 923 01:03:09,550 --> 01:03:11,820 Well, if you have an equation which 924 01:03:11,820 --> 01:03:16,890 says a dot equals minus strong damping times a, 925 01:03:16,890 --> 01:03:19,575 you have a very rapid exponential damping. 926 01:03:22,110 --> 01:03:27,680 The system immediately gets into an equilibrium. 927 01:03:27,680 --> 01:03:31,360 And the equilibrium is given by the first term. 928 01:03:31,360 --> 01:03:36,990 So if you have a situation that the population do not rapidly 929 01:03:36,990 --> 01:03:42,710 change, then this equation can be 930 01:03:42,710 --> 01:03:47,820 written that we have a very rapid damping. 931 01:03:47,820 --> 01:03:51,640 And ultimately, the coherences settle 932 01:03:51,640 --> 01:03:55,070 to an quasi-equilibrium value in a time 1 933 01:03:55,070 --> 01:03:59,300 over kappa, which is given by the first term. 934 01:03:59,300 --> 01:04:02,010 In other words, you set the time derivative 935 01:04:02,010 --> 01:04:04,110 of the left-hand side, 0. 936 01:04:04,110 --> 01:04:07,680 And then the coherences are expressed 937 01:04:07,680 --> 01:04:10,800 in terms of the slowly changing quantity, which 938 01:04:10,800 --> 01:04:14,010 are the populations. 939 01:04:14,010 --> 01:04:15,260 So let me repeat. 940 01:04:15,260 --> 01:04:19,760 If coherences are rapidly damped, 941 01:04:19,760 --> 01:04:23,280 we can neglect the time derivative 942 01:04:23,280 --> 01:04:25,070 of the left-hand side. 943 01:04:25,070 --> 01:04:28,550 And at any given moment, the coherences 944 01:04:28,550 --> 01:04:32,910 will be given by an expression which involves the population. 945 01:04:32,910 --> 01:04:36,700 And if the populations slowly change, 946 01:04:36,700 --> 01:04:40,110 the coherences will slowly change. 947 01:04:40,110 --> 01:04:43,160 Or in other words, the coherences 948 01:04:43,160 --> 01:04:44,860 follow the population. 949 01:04:44,860 --> 01:04:48,550 And the lag time is, at most, 1 over kappa. 950 01:04:48,550 --> 01:04:51,530 And we are in the limit of strong kappa. 951 01:04:51,530 --> 01:04:54,950 This principle of eliminating coherences and we 952 01:04:54,950 --> 01:04:57,910 simply get a master equation for population 953 01:04:57,910 --> 01:05:01,700 is called the idiopathic elimination of coherences. 954 01:05:01,700 --> 01:05:04,760 It's also called by Eric Hargan in his famous work 955 01:05:04,760 --> 01:05:09,010 about synergy by the principle that, if you have rapidly 956 01:05:09,010 --> 01:05:13,140 damped modes, they are slaved by the slow modes. 957 01:05:13,140 --> 01:05:16,280 In other words, the rapidly damped degrees of freedom 958 01:05:16,280 --> 01:05:18,890 follow the slow degrees of freedom. 959 01:05:18,890 --> 01:05:22,090 And this is what I just said, that the instantaneous value 960 01:05:22,090 --> 01:05:24,815 of the coherences is always given by the population. 961 01:05:27,510 --> 01:05:30,620 So let me just write that down and see where it takes us. 962 01:05:34,390 --> 01:05:41,640 So we have the situation that we want 963 01:05:41,640 --> 01:05:43,940 to now look at the case of strong damping. 964 01:05:46,890 --> 01:05:48,510 This is the overdamped case. 965 01:05:51,260 --> 01:06:05,680 And we use now the method of idiopathic elimination 966 01:06:05,680 --> 01:06:13,820 of coherences, because they are rapidly damped. 967 01:06:20,610 --> 01:06:22,390 And there is a wonderful discussion 968 01:06:22,390 --> 01:06:27,180 about this method in Atom-Photon interaction. 969 01:06:27,180 --> 01:06:35,800 It's the Exercise 18 on page 601. 970 01:06:35,800 --> 01:06:38,910 So mathematically, what we assume 971 01:06:38,910 --> 01:06:48,030 is that the population vary slowly. 972 01:06:48,030 --> 01:06:50,305 For instance, they vary at the Rabi frequency. 973 01:06:54,590 --> 01:07:04,410 So therefore, we can say that over the rapid time evolution 974 01:07:04,410 --> 01:07:15,160 of the coherences-- rho 1, 2 plus rho 2, 1-- 975 01:07:15,160 --> 01:07:18,860 so this is the rapid time evolution of the coherences. 976 01:07:18,860 --> 01:07:27,740 But for times which are on the order of 1 over kappa, 977 01:07:27,740 --> 01:07:29,300 the population are not changing. 978 01:07:29,300 --> 01:07:32,660 The population change over the much longer time scale, 979 01:07:32,660 --> 01:07:35,610 which is the period of the Rabi oscillation. 980 01:07:35,610 --> 01:07:40,140 So therefore, for the short time scale, 981 01:07:40,140 --> 01:07:44,570 I can sort of pretend that this term is constant. 982 01:07:44,570 --> 01:07:50,200 And for obvious reasons, let me now call this constant term, 983 01:07:50,200 --> 01:07:58,220 or slowly varying term, it is, at least for short time scales, 984 01:07:58,220 --> 01:08:00,280 the equilibrium value. 985 01:08:00,280 --> 01:08:02,180 So the way how we should read it is 986 01:08:02,180 --> 01:08:04,740 that the coherences will very, very rapidly 987 01:08:04,740 --> 01:08:08,140 damp to this equilibrium value. 988 01:08:08,140 --> 01:08:15,240 And this equilibrium value was given 989 01:08:15,240 --> 01:08:20,769 by the population, which may slowly change. 990 01:08:29,020 --> 01:08:35,680 But that means, now, by eliminating coherences, 991 01:08:35,680 --> 01:08:40,930 we obtain a rate equation for the populations only. 992 01:08:51,640 --> 01:08:54,100 So therefore, if you neglect short transients 993 01:08:54,100 --> 01:08:58,760 over the short time 1 over kappa, 994 01:08:58,760 --> 01:09:01,649 the coherences are always given by 995 01:09:01,649 --> 01:09:04,939 this quasi-equilibrium value, which 996 01:09:04,939 --> 01:09:07,319 is expressed by the population. 997 01:09:07,319 --> 01:09:10,850 And therefore, we can go to our master equation, which 998 01:09:10,850 --> 01:09:14,120 was a master equation for the of diagram matrix 999 01:09:14,120 --> 01:09:16,990 elements of the density matrix, and we can now 1000 01:09:16,990 --> 01:09:20,729 replace the coherences by an expression which 1001 01:09:20,729 --> 01:09:22,390 involves only population. 1002 01:09:22,390 --> 01:09:27,149 So we get now closed equations for only the population. 1003 01:09:27,149 --> 01:09:29,729 I'm sort of emphasizing that because you may have asked 1004 01:09:29,729 --> 01:09:33,219 yourself, when you have the rate equation a la Einstein, 1005 01:09:33,219 --> 01:09:36,009 with the a and b coefficient, these are just 1006 01:09:36,009 --> 01:09:38,590 rate equation for populations. 1007 01:09:38,590 --> 01:09:41,609 But here, in our class, we've always talked about the density 1008 01:09:41,609 --> 01:09:44,270 matrix, how important the coherences are. 1009 01:09:44,270 --> 01:09:45,979 The coherences are important. 1010 01:09:45,979 --> 01:09:50,689 Without coherences, you would never absorb or emit light. 1011 01:09:50,689 --> 01:09:54,340 But if the coherences can be expressed by populations, 1012 01:09:54,340 --> 01:09:56,340 you don't need a differential equation for them. 1013 01:10:01,140 --> 01:10:02,020 OK. 1014 01:10:02,020 --> 01:10:09,090 So with that now, we have our master equation 1015 01:10:09,090 --> 01:10:11,680 for the population. 1016 01:10:11,680 --> 01:10:22,230 And by just inserting that, we find this. 1017 01:10:22,230 --> 01:10:30,520 And we were interested in the decay of rho 1, 1. 1018 01:10:30,520 --> 01:10:33,400 Or this is also the population of the atom in the excited 1019 01:10:33,400 --> 01:10:36,110 state. 1020 01:10:36,110 --> 01:10:39,990 If we initially start where rho 1, 1 is large, 1021 01:10:39,990 --> 01:10:42,260 we're interested in the initial decay, 1022 01:10:42,260 --> 01:10:47,560 then the population in the ground state is small. 1023 01:10:47,560 --> 01:10:52,090 And we find, indeed, that there is exponential decay 1024 01:10:52,090 --> 01:10:54,970 of the initial distribution. 1025 01:10:54,970 --> 01:11:00,940 And the decay rate, gamma z, gamma cavity, 1026 01:11:00,940 --> 01:11:02,435 is given by this expression. 1027 01:11:05,120 --> 01:11:08,796 So therefore, what we find is-- but you know it 1028 01:11:08,796 --> 01:11:13,910 by now-- the effect that the stronger we damp the cavity, 1029 01:11:13,910 --> 01:11:17,850 the smaller, the weaker, is the decay of the initial state. 1030 01:11:26,360 --> 01:11:28,940 And I've already explained to you 1031 01:11:28,940 --> 01:11:33,690 that one picture to explain that is that we have the excited 1032 01:11:33,690 --> 01:11:37,780 state, we have the ground state. 1033 01:11:37,780 --> 01:11:42,640 The ground state gets damped by kappa. 1034 01:11:42,640 --> 01:11:48,350 And therefore, the time evolution, 1035 01:11:48,350 --> 01:11:51,360 which is given by the vacuum Rabi oscillation, slows down. 1036 01:11:56,780 --> 01:11:57,280 OK. 1037 01:12:04,560 --> 01:12:05,100 Yes. 1038 01:12:05,100 --> 01:12:05,683 Any questions? 1039 01:12:12,260 --> 01:12:18,010 So we have now resolved for gamma cavity. 1040 01:12:18,010 --> 01:12:21,980 We have a result how the atom decays 1041 01:12:21,980 --> 01:12:23,480 in the presence of the cavity. 1042 01:12:26,990 --> 01:12:30,840 And I want to rewrite this for you in a very nice way. 1043 01:12:30,840 --> 01:12:34,230 Namely, I want to show you the famous result 1044 01:12:34,230 --> 01:12:38,070 that, when an atom emits in a cavity, 1045 01:12:38,070 --> 01:12:43,140 that the atom can decay faster than the spontaneous decay. 1046 01:12:43,140 --> 01:12:45,440 Because if the cavity has a resonance, 1047 01:12:45,440 --> 01:12:49,230 the cavity changes the vacuum around the atom. 1048 01:12:49,230 --> 01:12:51,450 It increases the density of modes. 1049 01:12:51,450 --> 01:12:53,160 And therefore, there is a stimulation 1050 01:12:53,160 --> 01:12:56,120 of the atom into the cavity. 1051 01:12:56,120 --> 01:12:58,450 So I don't have to do any other math 1052 01:12:58,450 --> 01:13:03,550 than just rewriting our result of gamma cavity in other units, 1053 01:13:03,550 --> 01:13:04,470 in other ways. 1054 01:13:04,470 --> 01:13:08,419 And then we can compare it to the natural decay of an atom 1055 01:13:08,419 --> 01:13:09,210 without the cavity. 1056 01:13:11,770 --> 01:13:14,600 So what we want to do now is we want 1057 01:13:14,600 --> 01:13:25,300 to take the cavity-induced decay of the excited state, 1058 01:13:25,300 --> 01:13:33,170 and we want to compare with the natural decay rate, gamma. 1059 01:13:33,170 --> 01:13:38,350 So I just have to rewrite things in a few units. 1060 01:13:38,350 --> 01:13:43,250 The cavity-induced decay is the vacuum Rabi oscillation 1061 01:13:43,250 --> 01:13:45,025 squared over kappa. 1062 01:13:49,450 --> 01:13:57,350 So let me remind you that the Rabi frequency is always 1063 01:13:57,350 --> 01:14:01,320 given by an dipole moment times electric field. 1064 01:14:01,320 --> 01:14:03,710 It's a matrix element. 1065 01:14:03,710 --> 01:14:05,770 Well, there's one of those factors of 2. 1066 01:14:05,770 --> 01:14:08,680 And H bar keeps the units correct. 1067 01:14:08,680 --> 01:14:13,030 And in the case of the vacuum Rabi oscillation, 1068 01:14:13,030 --> 01:14:18,660 the electric field is the electric field, so to speak, 1069 01:14:18,660 --> 01:14:24,230 of one photon in the cavity volume. 1070 01:14:24,230 --> 01:14:27,460 And there is a factor of 2, so this is now 1071 01:14:27,460 --> 01:14:30,600 the volume of the cavity. 1072 01:14:34,320 --> 01:14:44,555 So therefore, the vacuum Rabi frequency 1073 01:14:44,555 --> 01:14:54,330 is a dipole moment times 2 atomic resonance 1074 01:14:54,330 --> 01:14:58,570 frequency epsilon 0 h bar v. 1075 01:14:58,570 --> 01:15:02,250 And that means now-- and this is a nice result-- 1076 01:15:02,250 --> 01:15:09,540 that the cavity-induced decay is omega 0 square over kappa. 1077 01:15:09,540 --> 01:15:13,870 And I just use the above result for omega 0. 1078 01:15:13,870 --> 01:15:19,260 And for kappa, I introduce the Q factor 1079 01:15:19,260 --> 01:15:22,900 of the cavity, which is the number of oscillations 1080 01:15:22,900 --> 01:15:24,130 before it is damped. 1081 01:15:24,130 --> 01:15:26,310 It's omega over kappa. 1082 01:15:26,310 --> 01:15:31,200 And with that, I can rewrite the cavity-induced decay 1083 01:15:31,200 --> 01:15:38,300 as 2d squared epsilon null H bar v times Q. 1084 01:15:38,300 --> 01:15:41,620 And if you remember, from part 1 of the course, 1085 01:15:41,620 --> 01:15:45,420 or standard textbook result, that spontaneous decay is 1086 01:15:45,420 --> 01:15:59,923 a dipole moment squared times frequency to the power 3, 1087 01:15:59,923 --> 01:16:06,680 then we obtain the famous result, 1088 01:16:06,680 --> 01:16:13,220 which is called the Purcell factor, namely 1089 01:16:13,220 --> 01:16:17,410 that the decay in the cavity, compared 1090 01:16:17,410 --> 01:16:22,570 to the decay in free space, is proportional to the Q 1091 01:16:22,570 --> 01:16:23,295 of the cavity. 1092 01:16:27,400 --> 01:16:30,550 There is a numerical prefector. 1093 01:16:30,550 --> 01:16:33,080 But then, what enters is the wavelength 1094 01:16:33,080 --> 01:16:35,240 of the photon cubed over the volume. 1095 01:16:42,670 --> 01:16:47,290 So the case of eta larger than 1-- 1096 01:16:47,290 --> 01:16:51,010 and it's actually interesting that this result is completely 1097 01:16:51,010 --> 01:16:52,645 independent of atomic properties. 1098 01:16:58,280 --> 01:17:00,810 So in other words, every atom you put in a cavity, 1099 01:17:00,810 --> 01:17:03,340 no matter what its dipole moment is, 1100 01:17:03,340 --> 01:17:11,360 it will decay with a rate which is Q times larger. 1101 01:17:14,387 --> 01:17:15,970 There are many ways how to look at it. 1102 01:17:15,970 --> 01:17:19,720 You can say, if the cavity consists of two mirrors, 1103 01:17:19,720 --> 01:17:24,960 you create Q mirror images of your atom. 1104 01:17:24,960 --> 01:17:26,390 And you have sort of an atom which 1105 01:17:26,390 --> 01:17:28,280 has a Q times stronger dipole moment. 1106 01:17:28,280 --> 01:17:31,570 And therefore, because the mirror images also radiate, 1107 01:17:31,570 --> 01:17:33,710 and therefore you get a Q times faster decay. 1108 01:17:33,710 --> 01:17:36,990 So this limit makes sense in many, many different pictures. 1109 01:17:36,990 --> 01:17:39,720 And it's a general result that, if an atom emits 1110 01:17:39,720 --> 01:17:43,050 in a cavity with finesse Q, you get a Q times 1111 01:17:43,050 --> 01:17:46,780 enhancement of spontaneous decay. 1112 01:17:46,780 --> 01:17:55,680 So this is called cavity-enhanced spontaneous 1113 01:17:55,680 --> 01:17:57,680 emission. 1114 01:17:57,680 --> 01:18:04,420 It goes back to Purcell in 1946. 1115 01:18:04,420 --> 01:18:11,850 And you can see the whole field of cavity QED 1116 01:18:11,850 --> 01:18:16,730 depends on this effect, that you can enhance spontaneous decay. 1117 01:18:16,730 --> 01:18:19,580 Or it's the opposite. 1118 01:18:19,580 --> 01:18:21,430 If an atom has a resonance frequency 1119 01:18:21,430 --> 01:18:24,450 between two modes of the cavity, the spontaneous decay 1120 01:18:24,450 --> 01:18:26,930 is inhibited. 1121 01:18:26,930 --> 01:18:29,710 And this is sort of what is exploited in cavity QED. 1122 01:18:32,770 --> 01:18:37,800 Let me just conclude here by saying 1123 01:18:37,800 --> 01:18:40,656 that our assumptions have to be consistent. 1124 01:18:43,860 --> 01:18:49,160 In our derivation, we neglected gamma. 1125 01:18:49,160 --> 01:18:51,780 When I said the two Lindblad operators, 1126 01:18:51,780 --> 01:18:53,910 we only look at the leakage out of the photon. 1127 01:18:53,910 --> 01:18:57,750 We neglect spontaneous decay. 1128 01:18:57,750 --> 01:19:02,120 So of course, this requires that eta is larger 1. 1129 01:19:02,120 --> 01:19:09,390 So this is the consistent range of validity of our result. 1130 01:19:09,390 --> 01:19:16,410 Secondly, we discussed the limit of overdamping, 1131 01:19:16,410 --> 01:19:20,420 which requires that kappa has to be larger than the vacuum Rabi 1132 01:19:20,420 --> 01:19:21,900 frequency. 1133 01:19:21,900 --> 01:19:32,090 And therefore, the above result is valued for large Q. 1134 01:19:32,090 --> 01:19:34,770 Q has to be sufficiently large, but cannot be larger than this 1135 01:19:34,770 --> 01:19:35,270 value. 1136 01:19:39,970 --> 01:19:41,330 Any questions? 1137 01:19:43,910 --> 01:19:44,805 Nancy? 1138 01:19:44,805 --> 01:19:49,170 AUDIENCE: For kappa, is that the loss in the cavity-- 1139 01:19:49,170 --> 01:19:51,595 is there any difference between the loss and the cavity 1140 01:19:51,595 --> 01:19:53,050 because it's a cavity? 1141 01:19:53,050 --> 01:19:57,415 Or the loss is into other modes of the cavity? 1142 01:19:57,415 --> 01:19:59,810 PROFESSOR: Kappa is the loss. 1143 01:19:59,810 --> 01:20:03,745 You can say kappa is the rate at which a photon in the cavity 1144 01:20:03,745 --> 01:20:04,430 is lost. 1145 01:20:04,430 --> 01:20:06,269 AUDIENCE: Is lost from the cavity totally? 1146 01:20:06,269 --> 01:20:09,213 Or is lost into another mode inside the cavity? 1147 01:20:09,213 --> 01:20:10,213 Is there any difference? 1148 01:20:10,213 --> 01:20:15,143 Is v here just one mode inside the cavity. 1149 01:20:15,143 --> 01:20:18,290 PROFESSOR: Well, we have not an element-- 1150 01:20:18,290 --> 01:20:22,390 OK, if the mirror has imperfections, 1151 01:20:22,390 --> 01:20:25,850 we usually think of light leaking out of the mirror. 1152 01:20:25,850 --> 01:20:28,780 If the mirror is absorbing because the coating has 1153 01:20:28,780 --> 01:20:31,440 a tiny little bit of absorption, you 1154 01:20:31,440 --> 01:20:33,530 know, from the beam splitter analogy, 1155 01:20:33,530 --> 01:20:36,830 that a small absorption is equivalent here 1156 01:20:36,830 --> 01:20:39,740 to a small transmission factor. 1157 01:20:39,740 --> 01:20:43,200 Usually, we don't consider that the light transforms 1158 01:20:43,200 --> 01:20:45,030 from one mode of the cavity to the other. 1159 01:20:45,030 --> 01:20:47,350 It's either absorbed or leaking out. 1160 01:20:47,350 --> 01:20:50,460 But if you assume that the mirrors are not super polished 1161 01:20:50,460 --> 01:20:52,440 and they scatter light a little bit, 1162 01:20:52,440 --> 01:20:54,940 then you would have a mechanism that the light in the cavity 1163 01:20:54,940 --> 01:20:56,330 scatters into other modes. 1164 01:20:56,330 --> 01:20:58,780 And all those losses are summarized 1165 01:20:58,780 --> 01:21:00,730 in one coupling constant, kappa. 1166 01:21:04,690 --> 01:21:08,860 Let me finish the class today and this week 1167 01:21:08,860 --> 01:21:14,050 by showing to you the recent nature paper by the Innsbrook 1168 01:21:14,050 --> 01:21:14,960 group. 1169 01:21:14,960 --> 01:21:17,860 We've talked so much about Lindblad operators. 1170 01:21:17,860 --> 01:21:22,430 And the Lindblad operators, the dissipation, 1171 01:21:22,430 --> 01:21:25,230 tells us what is the effect of the environment. 1172 01:21:25,230 --> 01:21:28,210 And usually, you would say, well, if you have an atom, 1173 01:21:28,210 --> 01:21:30,270 and it interacts with the environment, 1174 01:21:30,270 --> 01:21:33,110 what is the equilibrium state? 1175 01:21:33,110 --> 01:21:35,060 Well, if you just have the vacuum, 1176 01:21:35,060 --> 01:21:36,710 the atom goes to the ground state. 1177 01:21:36,710 --> 01:21:39,090 And the ground state is the attractor, 1178 01:21:39,090 --> 01:21:41,050 the stable state of the system. 1179 01:21:41,050 --> 01:21:43,960 If you drive the system with one laser beam, 1180 01:21:43,960 --> 01:21:46,970 the attractor state where the system goes into 1181 01:21:46,970 --> 01:21:50,560 is the steady state solution of the optical Bloch equation. 1182 01:21:50,560 --> 01:21:52,700 That all sounds a little bit boring. 1183 01:21:52,700 --> 01:21:56,190 But in this paper, based on this theoretical suggestion 1184 01:21:56,190 --> 01:21:59,020 of Peter Zola and an experimental realization 1185 01:21:59,020 --> 01:22:02,930 with trapped ions at Innsbrook, they actually 1186 01:22:02,930 --> 01:22:08,120 engineered an environment, engineered Lindblad operators 1187 01:22:08,120 --> 01:22:11,770 in such a way that the system of two ions 1188 01:22:11,770 --> 01:22:15,930 was relaxing into a Bell state. 1189 01:22:15,930 --> 01:22:18,290 So you engineer now the environment 1190 01:22:18,290 --> 01:22:23,020 that therm-- I wanted to say thermal equilibrium, no-- 1191 01:22:23,020 --> 01:22:26,740 the equilibrium state with the environment, the dissipation, 1192 01:22:26,740 --> 01:22:29,640 leads to a Bell state. 1193 01:22:29,640 --> 01:22:32,740 Well, it's a little bit complicated how it is done, 1194 01:22:32,740 --> 01:22:39,340 but I just want to just look at a few highlighted key messages. 1195 01:22:39,340 --> 01:22:44,120 So the idea is, here, engineering of dissipation, 1196 01:22:44,120 --> 01:22:47,170 creating experimentally dissipative operators. 1197 01:22:47,170 --> 01:22:50,530 We usually manipulate quantum system with Hamilton operators. 1198 01:22:50,530 --> 01:22:55,850 But here, it is manipulated not by the H operator, 1199 01:22:55,850 --> 01:22:58,670 but by the L operator. 1200 01:22:58,670 --> 01:23:04,810 And so the idea here is to have an evolution of the density 1201 01:23:04,810 --> 01:23:07,900 matrix, a linear mapping that the density matrix evolves 1202 01:23:07,900 --> 01:23:10,820 in a certain way. 1203 01:23:10,820 --> 01:23:12,850 I think you'll recognize partial trace. 1204 01:23:12,850 --> 01:23:16,400 You'll recognize a few equations, 1205 01:23:16,400 --> 01:23:19,840 a few results which we have discussed. 1206 01:23:19,840 --> 01:23:23,650 And what you see here is exactly the Lindblad form. 1207 01:23:23,650 --> 01:23:26,540 And what they did is, by using a number of laser 1208 01:23:26,540 --> 01:23:33,090 beams in quantum operation, they designed operators, z, 1209 01:23:33,090 --> 01:23:36,300 jump operators, Lindblad operators, in such a way 1210 01:23:36,300 --> 01:23:42,080 that the system was damping out into a Bell state. 1211 01:23:42,080 --> 01:23:46,730 So that seems a new frontier for research. 1212 01:23:46,730 --> 01:23:50,160 It goes by the title that we can do quantum simulations, 1213 01:23:50,160 --> 01:23:53,880 using trapped ions and cold atoms of Hamiltonians. 1214 01:23:53,880 --> 01:23:58,200 But we can also quantum simulate very special environments. 1215 01:23:58,200 --> 01:24:00,560 Maybe environments which it would 1216 01:24:00,560 --> 01:24:03,190 be hard to find them in nature, but they 1217 01:24:03,190 --> 01:24:07,260 are quantum-mechanically allowed possible environments. 1218 01:24:07,260 --> 01:24:09,040 And they may have new features which 1219 01:24:09,040 --> 01:24:10,290 have not been explored so far. 1220 01:24:13,960 --> 01:24:15,780 I will add this paper to the website, 1221 01:24:15,780 --> 01:24:17,930 if you want to read more about it. 1222 01:24:17,930 --> 01:24:20,330 Any question? 1223 01:24:20,330 --> 01:24:21,150 Then a reminder. 1224 01:24:21,150 --> 01:24:23,260 No class next week. 1225 01:24:23,260 --> 01:24:25,210 Monday is Patriot's Day. 1226 01:24:25,210 --> 01:24:27,850 I'm out of town on Wednesday, but you 1227 01:24:27,850 --> 01:24:30,330 have a homework to solve. 1228 01:24:30,330 --> 01:24:33,510 I rather feel you should continue to have a homework 1229 01:24:33,510 --> 01:24:35,400 assignment every week, because then you 1230 01:24:35,400 --> 01:24:38,350 have more time at the end of the term for the term paper. 1231 01:24:38,350 --> 01:24:40,100 I always have 10 homework assignments. 1232 01:24:40,100 --> 01:24:43,045 So when we spread them out more, there 1233 01:24:43,045 --> 01:24:45,420 is less time left at the end of the semester for the term 1234 01:24:45,420 --> 01:24:46,190 paper. 1235 01:24:46,190 --> 01:24:49,290 So I decided, since we have covered enough material, 1236 01:24:49,290 --> 01:24:53,750 we have a wonderful homework assignment for next week.