1 00:00:00,070 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,207 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,207 --> 00:00:17,832 at ocw.mit.edu. 8 00:00:20,490 --> 00:00:22,890 PROFESSOR: OK. 9 00:00:22,890 --> 00:00:28,440 Well, the result of which we obtained on Wednesday 10 00:00:28,440 --> 00:00:32,299 for spontaneous emission for the Einstein A coefficient 11 00:00:32,299 --> 00:00:36,250 regarded as an accomplishment as a highlight of the course. 12 00:00:36,250 --> 00:00:39,700 We've worked hard to talk about atoms 13 00:00:39,700 --> 00:00:41,800 and electromagnetic fields. 14 00:00:41,800 --> 00:00:45,680 And ultimately, to deal with spontaneous emission, 15 00:00:45,680 --> 00:00:50,140 it was not enough to put a semi-classical light atom 16 00:00:50,140 --> 00:00:55,620 interaction, dipole Hamiltonian, Rabi oscillation and such 17 00:00:55,620 --> 00:00:57,570 to put that into the picture. 18 00:00:57,570 --> 00:01:00,240 We really needed a quantized version 19 00:01:00,240 --> 00:01:01,900 of the electromagnetic field. 20 00:01:01,900 --> 00:01:07,050 And this is a result when an atom is excited and interact 21 00:01:07,050 --> 00:01:10,430 with all of the empty mods of the vacuum. 22 00:01:10,430 --> 00:01:13,970 And be summed up the probability that photon 23 00:01:13,970 --> 00:01:16,610 is immediate in any of those modes. 24 00:01:16,610 --> 00:01:18,550 And by doing, kind of, all of the ever 25 00:01:18,550 --> 00:01:20,340 reaching with intensity of state, 26 00:01:20,340 --> 00:01:24,430 and for all the possibility of actions we obtained. 27 00:01:24,430 --> 00:01:28,870 The famous result for the Einstein A coefficient, 28 00:01:28,870 --> 00:01:30,895 which is also the natural aligned width 29 00:01:30,895 --> 00:01:36,895 of the atomic excited state. 30 00:01:36,895 --> 00:01:41,250 Do you have any questions about the derivation 31 00:01:41,250 --> 00:01:42,760 or what we did last week? 32 00:01:49,870 --> 00:01:53,435 Then I think I will just continue and interpret 33 00:01:53,435 --> 00:01:55,100 the result. 34 00:01:55,100 --> 00:01:57,520 So we go to result for an Einstein A coefficient. 35 00:01:57,520 --> 00:02:00,690 And well, the question is, how big is it? 36 00:02:00,690 --> 00:02:08,830 Well it has a number of constants. 37 00:02:08,830 --> 00:02:16,930 And if it is-- let's discuss it now in atomic units. 38 00:02:16,930 --> 00:02:20,990 Well, if we assume the frequency or the energy 39 00:02:20,990 --> 00:02:23,260 is on the order of Rydberg-- that's 40 00:02:23,260 --> 00:02:26,330 sort of the measure for an electronic excitation 41 00:02:26,330 --> 00:02:32,070 in the atom-- we assume the dipole matrix element is one. 42 00:02:32,070 --> 00:02:35,530 That means one per radius. 43 00:02:35,530 --> 00:02:38,330 Since we have pretty much set everything one and expressed 44 00:02:38,330 --> 00:02:42,580 everything in atomic units, it means that the speed of light 45 00:02:42,580 --> 00:02:45,600 is-- remember? 46 00:02:45,600 --> 00:02:50,040 The velocity of the atom in any correspondent 47 00:02:50,040 --> 00:02:53,110 was alpha times smaller than the speed of light. 48 00:02:53,110 --> 00:02:56,190 But the velocity of the atom is one atomic unit. 49 00:02:56,190 --> 00:02:59,370 So therefore, the speed of light in atomic units 50 00:02:59,370 --> 00:03:02,070 is one over alpha. 51 00:03:02,070 --> 00:03:05,290 And that means that if you look at the formula, 52 00:03:05,290 --> 00:03:08,110 there is the speed of light to the power of 3 53 00:03:08,110 --> 00:03:09,740 in the denominator. 54 00:03:09,740 --> 00:03:13,120 And that means that in atomic units 55 00:03:13,120 --> 00:03:15,920 the Einstein A coefficient is alpha 56 00:03:15,920 --> 00:03:22,370 to the 3, which is 3 times 10 to the minus 7. 57 00:03:26,260 --> 00:03:31,600 So that means that the ratio of this spontaneous emission 58 00:03:31,600 --> 00:03:34,610 rate, which is also the inverse lifetime 59 00:03:34,610 --> 00:03:36,867 and, therefore, the natural alignments of the excited 60 00:03:36,867 --> 00:03:37,366 state. 61 00:03:42,930 --> 00:03:47,880 Relative to the transition frequency, so the damping 62 00:03:47,880 --> 00:03:49,970 of the harmonic oscillator or the two level 63 00:03:49,970 --> 00:03:55,340 system relative to the NFC spacing of the oscillator. 64 00:03:55,340 --> 00:03:56,260 It's small. 65 00:03:56,260 --> 00:03:59,770 It's actually alpha cube. 66 00:03:59,770 --> 00:04:04,850 So if you take this 3 times 10 to the minus 7 67 00:04:04,850 --> 00:04:07,790 and multiply it with the atomic unit of frequency, 68 00:04:07,790 --> 00:04:08,690 which is 2 Rydbergs. 69 00:04:08,690 --> 00:04:12,180 We obtain on the order of 10 to 9. 70 00:04:16,680 --> 00:04:23,390 And it's a rate of 10 to the 9 per second. 71 00:04:23,390 --> 00:04:28,290 And that means that the lifetime of a tubercle atomic level 72 00:04:28,290 --> 00:04:29,770 is on the order of 1 nanosecond. 73 00:04:32,350 --> 00:04:41,880 Well, often it's 10 200 nanosecond 74 00:04:41,880 --> 00:04:49,150 because many transition frequencies are smaller 75 00:04:49,150 --> 00:04:52,870 by quite a factor than the atomic unit of the transition 76 00:04:52,870 --> 00:04:54,000 frequency. 77 00:04:54,000 --> 00:04:57,950 Remember, the Rydberg frequency would be deep in the UV. 78 00:04:57,950 --> 00:05:00,245 But a lot of atoms have transitions in the visible. 79 00:05:13,030 --> 00:05:17,380 I highlighted already when I derived it 80 00:05:17,380 --> 00:05:23,540 that the spontaneous emission has this famous omega cube 81 00:05:23,540 --> 00:05:25,420 dependents. 82 00:05:25,420 --> 00:05:29,560 And that this actually important to understand 83 00:05:29,560 --> 00:05:37,130 why lower lying levels-- excited hyperfine levels-- 84 00:05:37,130 --> 00:05:39,470 do not radiate. 85 00:05:39,470 --> 00:05:43,250 So let me just, kind of, formalize it. 86 00:05:43,250 --> 00:05:48,210 If I would now estimate what is the radiative lifetime 87 00:05:48,210 --> 00:05:52,782 for a transition, which is not as I just assumed in the UV 88 00:05:52,782 --> 00:05:53,680 or in the visible. 89 00:05:53,680 --> 00:05:57,370 Let me estimate what is the radiative lifetime to emit 90 00:05:57,370 --> 00:05:59,300 a microwave photon at a few gigahertz? 91 00:06:05,310 --> 00:06:14,410 Well, the microwave frequency of the gigahertz 10 to the nine 92 00:06:14,410 --> 00:06:17,990 is five orders of magnitude smaller than the frequency 10 93 00:06:17,990 --> 00:06:19,930 to the 14 of an optical transition. 94 00:06:23,860 --> 00:06:29,640 So therefore, this is 10 to the 15 times longer. 95 00:06:29,640 --> 00:06:31,880 And if you have, typically, one or ten 96 00:06:31,880 --> 00:06:34,400 nanosecond for an electronic transition. 97 00:06:34,400 --> 00:06:38,360 That means that this spontaneous lifetime for a microwave 98 00:06:38,360 --> 00:06:42,180 transition is seven months. 99 00:06:42,180 --> 00:06:50,250 If in addition we factor in that hyperfine transitions have 100 00:06:50,250 --> 00:06:53,950 an operator which is a Bohr magneton and magnetic type 101 00:06:53,950 --> 00:07:04,230 of operator, not an electric dipole and we 102 00:07:04,230 --> 00:07:08,210 discussed that the Bohr magneton is actually 103 00:07:08,210 --> 00:07:11,020 when we discuss multiple transitions we discuss 104 00:07:11,020 --> 00:07:13,820 that the Bohr magneton is alpha times smaller 105 00:07:13,820 --> 00:07:16,360 than a typical electric dipole moment. 106 00:07:16,360 --> 00:07:19,840 So therefore, a magnetic dipole transition 107 00:07:19,840 --> 00:07:27,160 is alpha times weaker than an electronic dipole transition. 108 00:07:27,160 --> 00:07:30,880 And that means now, if you multiply months, which 109 00:07:30,880 --> 00:07:33,100 we obtain by the frequency scaling, 110 00:07:33,100 --> 00:07:35,400 again, by alpha square for the weakness 111 00:07:35,400 --> 00:07:41,340 of the magnetic dipole, we find that atomic hyperfine levels 112 00:07:41,340 --> 00:07:48,630 have a lifetime, which is on the order of 1,000 years. 113 00:07:48,630 --> 00:07:51,034 And this is why it's very safe to neglect 114 00:07:51,034 --> 00:07:52,450 those transition in the laboratory 115 00:07:52,450 --> 00:07:56,970 and assume that all hyperfine states in the ground state 116 00:07:56,970 --> 00:08:01,750 manifold pretty much don't decay and are long lived. 117 00:08:07,476 --> 00:08:07,976 Questions? 118 00:08:14,940 --> 00:08:20,330 OK, so with that we have discussed spontaneous emission. 119 00:08:24,444 --> 00:08:26,110 Let's go through a few clicker questions 120 00:08:26,110 --> 00:08:39,630 to discuss the subject and verify your understanding. 121 00:08:39,630 --> 00:08:56,540 So the first question is can an E2 transition, 122 00:08:56,540 --> 00:09:07,920 which is a quadruple transition, can you 123 00:09:07,920 --> 00:09:09,885 drive it by a plane wave? 124 00:09:12,930 --> 00:09:14,700 Or does it need a laser beam which 125 00:09:14,700 --> 00:09:18,150 has an intensity gradient such as a focus laser beam. 126 00:09:28,310 --> 00:09:29,240 Yes or no? 127 00:09:59,836 --> 00:10:00,335 OK. 128 00:10:03,260 --> 00:10:05,790 Well the answer is yes. 129 00:10:05,790 --> 00:10:08,330 You can just use your laser beam. 130 00:10:08,330 --> 00:10:12,460 If a quadrupole transition would require a gradient, 131 00:10:12,460 --> 00:10:14,550 it would really require a gradient 132 00:10:14,550 --> 00:10:16,010 over the size of the atom. 133 00:10:16,010 --> 00:10:19,400 And that would be extremely hard to achieve. 134 00:10:19,400 --> 00:10:22,860 Fortunately, this is not the case because what happens 135 00:10:22,860 --> 00:10:25,980 is we actually assumed in the derivation 136 00:10:25,980 --> 00:10:29,370 that we had a plane wave into the IKR. 137 00:10:29,370 --> 00:10:32,680 And then do the tailor expansion. 138 00:10:32,680 --> 00:10:35,920 And it was these part of the tailor expansion 139 00:10:35,920 --> 00:10:39,950 of a plain wave, which gave rise to the matrix element 140 00:10:39,950 --> 00:10:42,000 for the quadrupole transition. 141 00:10:42,000 --> 00:10:46,679 So a plane wave laser beam is sufficient to drive 142 00:10:46,679 --> 00:10:47,845 higher multiple transitions. 143 00:10:50,590 --> 00:10:51,180 Next question. 144 00:10:55,880 --> 00:11:06,240 Can spontaneous emission be described 145 00:11:06,240 --> 00:11:18,185 as a stimulated emission process by the zero point field. 146 00:11:23,130 --> 00:11:26,539 So by the zero point field, we know the electromagnetic wave 147 00:11:26,539 --> 00:11:27,580 is a harmonic oscillator. 148 00:11:27,580 --> 00:11:28,565 And a harmonic oscillator has a ground state. 149 00:11:28,565 --> 00:11:31,270 And in the ground state you have zero point motion. 150 00:11:31,270 --> 00:11:33,250 So there is an electric field, even when 151 00:11:33,250 --> 00:11:34,830 we have the vacuum state. 152 00:11:34,830 --> 00:11:42,010 And the question is, can spontaneous emission 153 00:11:42,010 --> 00:11:49,850 be described as simply being stimulated emission but now do 154 00:11:49,850 --> 00:11:53,170 to the silver point fluctuations of the electromagnetic field. 155 00:12:17,331 --> 00:12:17,830 OK. 156 00:12:20,670 --> 00:12:23,690 The answer is it depends. 157 00:12:23,690 --> 00:12:26,260 It depends if you just want to make a qualitative hand waving 158 00:12:26,260 --> 00:12:26,760 argument. 159 00:12:29,260 --> 00:12:31,500 Then I would say you are correct. 160 00:12:31,500 --> 00:12:35,090 You can say that the electromagnetic field 161 00:12:35,090 --> 00:12:37,630 of the vacuum stimulates a transition. 162 00:12:37,630 --> 00:12:42,340 But when I said described, I meant 163 00:12:42,340 --> 00:12:45,290 if you can get it quantitatively correct. 164 00:12:45,290 --> 00:12:48,090 And there the answer is actually no 165 00:12:48,090 --> 00:12:52,030 because the energy of the electromagnetic field 166 00:12:52,030 --> 00:12:55,660 is n plus 1/2 h bar omega. 167 00:12:55,660 --> 00:13:02,910 Whereas, this spontaneous emission of eight is n plus 1. 168 00:13:02,910 --> 00:13:06,580 So you have half a photon verse of extra energy. 169 00:13:06,580 --> 00:13:08,690 But this spontaneous emission is sort 170 00:13:08,690 --> 00:13:13,240 of like the spontaneous emission is the rate, which 171 00:13:13,240 --> 00:13:17,320 would be stimulated by an extra energy of h bar omega. 172 00:13:17,320 --> 00:13:20,240 So in other words, you would get the answer 173 00:13:20,240 --> 00:13:21,760 wrong by a factor of two. 174 00:13:24,650 --> 00:13:27,400 I think decoding deeper in the electrodynamics description 175 00:13:27,400 --> 00:13:29,180 of spontaneous emission you would 176 00:13:29,180 --> 00:13:32,560 identify two terms for spontaneous emission. 177 00:13:32,560 --> 00:13:36,030 One is actually the stimulation by the vacuum field. 178 00:13:36,030 --> 00:13:39,610 But there is another term called radiation reaction. 179 00:13:39,610 --> 00:13:41,830 So there's, sort of, two terms. 180 00:13:41,830 --> 00:13:42,370 Trust me. 181 00:13:42,370 --> 00:13:44,590 If not, there are hundreds of pages 182 00:13:44,590 --> 00:13:46,780 in [INAUDIBLE], which is books written about it. 183 00:13:46,780 --> 00:13:50,910 And in the ground state, the two terms destructively interfere. 184 00:13:50,910 --> 00:13:52,740 Therefore, you have no spontaneous emission 185 00:13:52,740 --> 00:13:55,690 in the current state, which is reassuring. 186 00:13:55,690 --> 00:13:58,730 But then in the excited state the two terms 187 00:13:58,730 --> 00:14:00,580 constructively interfere. 188 00:14:00,580 --> 00:14:02,530 And therefore, you get spontaneous emission, 189 00:14:02,530 --> 00:14:06,100 which is twice as much as you would get if you just 190 00:14:06,100 --> 00:14:10,380 look at the stimulation by the vacuum field. 191 00:14:10,380 --> 00:14:14,150 So the answer is not quantitative but half of it, 192 00:14:14,150 --> 00:14:18,770 yes, can be regarded as stimulated emission 193 00:14:18,770 --> 00:14:21,410 by the vacuum fluctuations of the electromagnetic field. 194 00:14:26,861 --> 00:14:27,360 OK. 195 00:14:32,060 --> 00:14:35,920 We emphasized that spontaneous emission 196 00:14:35,920 --> 00:14:37,930 is proportional to omega cube. 197 00:14:46,110 --> 00:14:53,720 The question is now what is the dependence in one dimension? 198 00:14:53,720 --> 00:14:56,860 If everything the atom can only emit in one dimension, 199 00:14:56,860 --> 00:15:01,090 everything is one dimensional, put the atom into a waveguide. 200 00:15:01,090 --> 00:15:17,330 So your choices are omega cube, omega square, or omega-- well, 201 00:15:17,330 --> 00:15:21,082 if you press D, none of the above. 202 00:15:21,082 --> 00:15:23,750 But I can already tell you it's one of those three. 203 00:15:34,950 --> 00:15:37,590 So everything the same. 204 00:15:37,590 --> 00:15:40,510 But we are in one dimension. 205 00:15:40,510 --> 00:15:43,804 The world seen by the atom and by the electromagnetic waves 206 00:15:43,804 --> 00:15:44,595 is one dimensional. 207 00:15:52,910 --> 00:15:55,100 Yes, it's correct. 208 00:15:55,100 --> 00:15:59,730 As you remember, out of the omega cube dependence. 209 00:15:59,730 --> 00:16:03,200 Omega square came from the density of states. 210 00:16:03,200 --> 00:16:05,580 And what is omega square in three dimension 211 00:16:05,580 --> 00:16:09,300 becomes omega in two dimensions and constant density 212 00:16:09,300 --> 00:16:11,250 of state in one dimension. 213 00:16:11,250 --> 00:16:14,800 So therefore, in one dimension, we 214 00:16:14,800 --> 00:16:18,875 are only left with the omega dependence. 215 00:16:25,090 --> 00:16:28,530 OK, so there is one factor of omega, 216 00:16:28,530 --> 00:16:32,460 which does not come from the density of state. 217 00:16:32,460 --> 00:16:45,560 And the next question is where does the other power of omega 218 00:16:45,560 --> 00:16:46,210 come from? 219 00:16:50,230 --> 00:16:52,500 As we discussed, it's not the density of states. 220 00:16:55,440 --> 00:16:59,530 So we have three choices. 221 00:16:59,530 --> 00:17:04,430 One is it comes from the atomic matrix element, 222 00:17:04,430 --> 00:17:09,550 it comes from the dipole approximation, 223 00:17:09,550 --> 00:17:12,640 or it comes from the quantization 224 00:17:12,640 --> 00:17:14,130 of the electromagnetic field. 225 00:17:47,790 --> 00:17:50,560 OK, the majority got it right. 226 00:17:50,560 --> 00:17:51,766 It's a field quantization. 227 00:17:55,830 --> 00:17:59,490 Sort of remember when you write down the electric dipole 228 00:17:59,490 --> 00:18:02,830 Hamiltonian, in the quantized version, 229 00:18:02,830 --> 00:18:05,870 there is a perfecter, which is electric field 230 00:18:05,870 --> 00:18:08,430 of a single photon. 231 00:18:08,430 --> 00:18:13,860 So if you have a single photon, it 232 00:18:13,860 --> 00:18:18,450 gives rise to an electric field squared, 233 00:18:18,450 --> 00:18:20,262 which is proportionate to h bar omega. 234 00:18:20,262 --> 00:18:22,220 And this is, sort of, the normalization factor. 235 00:18:34,070 --> 00:18:34,890 Two more questions. 236 00:18:44,400 --> 00:18:47,910 We talked a lot about the rotating wave approximation. 237 00:18:47,910 --> 00:18:52,710 And we also talked about it for a spinning system 238 00:18:52,710 --> 00:18:54,560 driven by magnetic field. 239 00:18:54,560 --> 00:18:56,460 If you have a rotating magnetic field, 240 00:18:56,460 --> 00:18:59,180 we do not need the rotating wave approximation 241 00:18:59,180 --> 00:19:01,490 because if you drive a spin system with a rotating 242 00:19:01,490 --> 00:19:06,820 magnetic field, we have only the co-rotating term. 243 00:19:06,820 --> 00:19:11,680 The question I have now for you is whether the same is correct 244 00:19:11,680 --> 00:19:14,520 or not for an electronic transition. 245 00:19:14,520 --> 00:19:23,640 So therefore, the question is for electronic transitions 246 00:19:23,640 --> 00:19:35,025 do we always get the counter rotating term. 247 00:19:37,610 --> 00:19:40,890 And if you want to have a simple Hamiltonian, 248 00:19:40,890 --> 00:19:43,640 then we do the rotating wave approximation. 249 00:19:43,640 --> 00:19:47,680 So the question is is the rotating wave approximation 250 00:19:47,680 --> 00:19:51,560 necessary because we always get the counter rotating 251 00:19:51,560 --> 00:19:53,610 term for the electronic transition, 252 00:19:53,610 --> 00:19:55,530 then the answer is yes. 253 00:19:55,530 --> 00:20:02,390 Or are there examples where the system is exactly 254 00:20:02,390 --> 00:20:04,160 described by only one term? 255 00:20:04,160 --> 00:20:05,444 The core rotating term. 256 00:20:26,230 --> 00:20:28,987 I will come back to that later in the class. 257 00:20:28,987 --> 00:20:30,445 But I thought it's a good question. 258 00:20:35,410 --> 00:20:39,290 OK, let me give you the answer. 259 00:20:39,290 --> 00:20:41,680 I actually coincide with everybody in the class 260 00:20:41,680 --> 00:20:49,420 here because I would tend to say no because there are situations 261 00:20:49,420 --> 00:20:53,910 where the counter rotating term can be zero 262 00:20:53,910 --> 00:21:00,620 due to angular momentum selection rules. 263 00:21:04,050 --> 00:21:08,380 However, if you have an electronic transition 264 00:21:08,380 --> 00:21:11,050 and you have a sigma plus transition to one state, 265 00:21:11,050 --> 00:21:14,090 there's always a possibility for sigma minus transition. 266 00:21:14,090 --> 00:21:16,370 So you usually get both. 267 00:21:16,370 --> 00:21:20,550 But if you apply an infinitely strong magnetic field, 268 00:21:20,550 --> 00:21:23,060 then the m equals minus 1 state can 269 00:21:23,060 --> 00:21:24,690 be moved out of the picture. 270 00:21:24,690 --> 00:21:27,530 You have only, let's say, the m equals plus 1 state. 271 00:21:27,530 --> 00:21:31,500 And then selection holds mean that the counter rotating 272 00:21:31,500 --> 00:21:36,690 term is vanishingly small. 273 00:21:36,690 --> 00:21:40,130 But it's an artificial situation. 274 00:21:40,130 --> 00:21:44,630 So you can all claim credit for your answer. 275 00:21:44,630 --> 00:21:52,140 Finally, the last question is about the Lamb shift. 276 00:22:00,062 --> 00:22:04,720 We are now talking about electronic transitions. 277 00:22:04,720 --> 00:22:14,960 And the question is Lamb shift-- if it's 278 00:22:14,960 --> 00:22:16,660 due to the counter rotating term. 279 00:22:23,050 --> 00:22:26,450 In other words, if you have a situation where the counter 280 00:22:26,450 --> 00:22:29,700 rotating term is zero, as we just 281 00:22:29,700 --> 00:22:31,340 discussed in the previous example 282 00:22:31,340 --> 00:22:32,960 that there may be situations. 283 00:22:32,960 --> 00:22:35,210 Somewhat artificially but you could arrange for it. 284 00:22:35,210 --> 00:22:40,640 The set then implies that there is no lamb shift. 285 00:22:40,640 --> 00:22:44,060 So yes or no. 286 00:22:44,060 --> 00:22:52,610 Is the lamb shift caused by the counter rotating term 287 00:22:52,610 --> 00:22:54,100 involved in electronic transitions? 288 00:23:06,370 --> 00:23:06,870 OK. 289 00:23:15,120 --> 00:23:19,440 OK, well what else is the lamb shift? 290 00:23:19,440 --> 00:23:24,290 It is the AC stock effect of the counter rotating term. 291 00:23:24,290 --> 00:23:32,300 So is it due to the counter rotating term? 292 00:23:32,300 --> 00:23:33,125 Yes, of course. 293 00:23:35,690 --> 00:23:44,950 The lamb shift is the AC stock effect 294 00:23:44,950 --> 00:23:46,530 caused by the vacuum fluctuations. 295 00:23:55,070 --> 00:23:56,720 That's what it is. 296 00:23:56,720 --> 00:23:58,880 But we come to that because I want 297 00:23:58,880 --> 00:24:04,270 to discuss later today some aspects of the fully 298 00:24:04,270 --> 00:24:05,560 quantized Hamiltonian. 299 00:24:05,560 --> 00:24:09,390 And we will, again, in the fully quantized picture 300 00:24:09,390 --> 00:24:11,930 see the operators, which are responsible 301 00:24:11,930 --> 00:24:14,217 for the core rotating for the counter rotating turn. 302 00:24:14,217 --> 00:24:15,800 And then I will point to the operator, 303 00:24:15,800 --> 00:24:18,460 which causes a lamb shift. 304 00:24:18,460 --> 00:24:23,920 But before I continue, any questions about the questions? 305 00:24:23,920 --> 00:24:24,420 Collin. 306 00:24:24,420 --> 00:24:28,980 AUDIENCE: When you derive the amplitude in the electric field 307 00:24:28,980 --> 00:24:30,390 due to the single photon-- 308 00:24:30,390 --> 00:24:31,082 PROFESSOR: Yep. 309 00:24:31,082 --> 00:24:33,040 AUDIENCE: I always get the factor of two wrong. 310 00:24:33,040 --> 00:24:38,326 So you wrote h bar omega is 2 epsilon 0 [INAUDIBLE] squared. 311 00:24:38,326 --> 00:24:42,041 Now there's a contribution that comes from the electric field 312 00:24:42,041 --> 00:24:44,290 and magnetic field because you have one factor of two. 313 00:24:44,290 --> 00:24:47,272 Then there's always that other factor of two. 314 00:24:47,272 --> 00:24:50,162 Are you getting that from using one half h 315 00:24:50,162 --> 00:24:51,745 bar because of the vacuum fluctuation. 316 00:24:55,224 --> 00:24:57,160 PROFESSOR: I'm not going back to the formula 317 00:24:57,160 --> 00:24:59,670 because I run the risk that it was wrong. 318 00:24:59,670 --> 00:25:04,620 But all I want to say is what I really mean is use Jackson. 319 00:25:04,620 --> 00:25:08,540 Put in a volume V-- an electromagnetic field-- 320 00:25:08,540 --> 00:25:10,530 with h bar omega energy. 321 00:25:10,530 --> 00:25:15,840 And the electric field squared of this photon, 322 00:25:15,840 --> 00:25:18,220 that's what I mean. 323 00:25:18,220 --> 00:25:22,450 And if you find a factor of two mistakes in my E square, 324 00:25:22,450 --> 00:25:26,110 I can still, you know, get out of theory exit 325 00:25:26,110 --> 00:25:30,480 by the rear-entrance door by saying that there is also 326 00:25:30,480 --> 00:25:33,420 a difference whether E square is E square RNS 327 00:25:33,420 --> 00:25:35,356 or whether E square is the amplitude. 328 00:25:35,356 --> 00:25:37,730 You know I mean there are risk factors of two everywhere. 329 00:25:37,730 --> 00:25:41,150 But what I mean is really the electric field 330 00:25:41,150 --> 00:25:43,090 caused by one photon. 331 00:25:43,090 --> 00:25:45,340 And of course, the argument stands. 332 00:25:45,340 --> 00:25:49,070 I don't need any factors of two or any subtleties 333 00:25:49,070 --> 00:25:51,080 of the electromagnetic field energy. 334 00:25:51,080 --> 00:25:55,040 We know that the energy is n plus 1/2 but emission is 335 00:25:55,040 --> 00:25:56,110 n plus 1. 336 00:25:56,110 --> 00:25:59,670 And these shows that the stimulation by the vacuum field 337 00:25:59,670 --> 00:26:02,340 cannot quantitatively account for spontaneous emission. 338 00:26:02,340 --> 00:26:06,268 AUDIENCE: So the quantity that you set equal to 339 00:26:06,268 --> 00:26:11,669 is h bar omega 1/2, not the fluctuation but the real-- 340 00:26:11,669 --> 00:26:14,627 PROFESSOR: OK, if you want to know, 341 00:26:14,627 --> 00:26:16,210 let's not compare apples with oranges. 342 00:26:16,210 --> 00:26:17,334 You want an electric field. 343 00:26:17,334 --> 00:26:19,540 And you can pick whether it's the RMS field 344 00:26:19,540 --> 00:26:22,150 or whether it is the maximum amplitude. 345 00:26:22,150 --> 00:26:23,430 You can pick what you want. 346 00:26:23,430 --> 00:26:27,470 But now we are comparing what is the e-square 347 00:26:27,470 --> 00:26:29,940 for the vacuum-- for single-mode-- vacuum. 348 00:26:29,940 --> 00:26:32,860 And what is the e-square for single photon? 349 00:26:32,860 --> 00:26:35,250 The two answers differ by a factor of 2. 350 00:26:35,250 --> 00:26:38,520 A single photon is twice as strong 351 00:26:38,520 --> 00:26:43,130 in e-square as the vacuum fluctuations in the same mode. 352 00:26:43,130 --> 00:26:45,750 That's what it means. 353 00:26:45,750 --> 00:26:46,250 Yes? 354 00:26:46,250 --> 00:26:50,218 AUDIENCE: I have a question about the quantum emission 355 00:26:50,218 --> 00:26:51,210 rate. 356 00:26:51,210 --> 00:26:54,186 The explanation that it had-- quantum mechanic derivation 357 00:26:54,186 --> 00:26:59,146 that we have, do people not know the formula, 358 00:26:59,146 --> 00:27:02,122 how to describe spontaneous emission [INAUDIBLE]? 359 00:27:08,074 --> 00:27:09,880 PROFESSOR: I think so. 360 00:27:09,880 --> 00:27:12,690 I have not gone deeply back into the story. 361 00:27:12,690 --> 00:27:15,510 But a lot of credit is given to Einstein. 362 00:27:15,510 --> 00:27:19,770 And as I mentioned last week that Einstein actually 363 00:27:19,770 --> 00:27:23,443 had spontaneous emission in his derivation for the Einstein A 364 00:27:23,443 --> 00:27:26,680 and B coefficient in this famous paper. 365 00:27:26,680 --> 00:27:32,270 And so he found that there must be spontaneous emission based 366 00:27:32,270 --> 00:27:34,160 on a thermodynamic argument. 367 00:27:34,160 --> 00:27:36,520 It's only spontaneous emission, which 368 00:27:36,520 --> 00:27:42,820 brings the internal population of an atom into equilibrium. 369 00:27:42,820 --> 00:27:45,758 So I think it is correct to say. 370 00:27:45,758 --> 00:27:48,091 AUDIENCE: Can you derive it from that stagnant condition 371 00:27:48,091 --> 00:27:49,959 of getting [INAUDIBLE]? 372 00:27:49,959 --> 00:27:51,590 PROFESSOR: That's what Einstein did. 373 00:27:51,590 --> 00:27:54,980 And the answer is, by comparison with the Planck law, 374 00:27:54,980 --> 00:28:00,330 you get an expression for the Einstein A and B coefficient. 375 00:28:00,330 --> 00:28:05,390 Now of course, you can go the other way around. 376 00:28:05,390 --> 00:28:11,140 You can see if you just use classical physics you would 377 00:28:11,140 --> 00:28:13,300 actually expect-- now it depends. 378 00:28:13,300 --> 00:28:14,880 If you use the Bohr model, you would 379 00:28:14,880 --> 00:28:18,920 expect that the electron is radiating and it was a mystery. 380 00:28:18,920 --> 00:28:21,360 How can you have an atom in the ground state, which 381 00:28:21,360 --> 00:28:28,090 is circling around a nucleolus, and not radiating at all? 382 00:28:28,090 --> 00:28:30,280 On the other hand, in quantum mechanics, 383 00:28:30,280 --> 00:28:33,030 we are not assuming that the atom is circulating. 384 00:28:33,030 --> 00:28:36,320 And we have an accelerated charge 385 00:28:36,320 --> 00:28:38,820 and then we have a time dependent charge distribution. 386 00:28:38,820 --> 00:28:42,090 We use the steady state wave function. 387 00:28:42,090 --> 00:28:43,860 So I'm not sure if there is maybe 388 00:28:43,860 --> 00:28:47,770 an argument, which would say there should 389 00:28:47,770 --> 00:28:50,090 be some spontaneous emission based 390 00:28:50,090 --> 00:28:52,220 on a purely classic argument. 391 00:28:52,220 --> 00:28:54,470 But this would not be the whole story 392 00:28:54,470 --> 00:28:57,910 because a classic argument would then deal with the difficulty. 393 00:28:57,910 --> 00:28:59,630 Why is there difference between n 394 00:28:59,630 --> 00:29:03,830 equals 1, which does not radiate in n equals 2, which radiates. 395 00:29:03,830 --> 00:29:07,130 So my understanding is that it is only the physics 396 00:29:07,130 --> 00:29:10,640 either through the perspective of Einstein 397 00:29:10,640 --> 00:29:15,430 by just using equilibration or our microscopic derivation 398 00:29:15,430 --> 00:29:20,320 using filed quantization, which allows us to understand 399 00:29:20,320 --> 00:29:22,490 the phenomenon of a spontaneous emission. 400 00:29:28,254 --> 00:29:28,920 Other questions? 401 00:29:34,540 --> 00:29:43,830 OK, then before we talk about some really cute and nice 402 00:29:43,830 --> 00:29:47,750 aspects of the fully quantised Hamiltonian, 403 00:29:47,750 --> 00:29:54,090 I want to spend a few minutes talking 404 00:29:54,090 --> 00:29:55,760 about degeneracy factors. 405 00:30:03,940 --> 00:30:05,920 I've already given you my opinion. 406 00:30:05,920 --> 00:30:09,180 You should not think in almost all situations 407 00:30:09,180 --> 00:30:11,470 about levels, which have a degeneracy. 408 00:30:11,470 --> 00:30:12,680 Just think about states. 409 00:30:12,680 --> 00:30:16,880 A state is a state, and it counts as one. 410 00:30:16,880 --> 00:30:19,500 And if you have a level which has triple degeneracy, well, 411 00:30:19,500 --> 00:30:20,510 it has three states. 412 00:30:20,510 --> 00:30:23,600 Just kind of count the states and look at the states. 413 00:30:23,600 --> 00:30:30,880 However, there are formula for which 414 00:30:30,880 --> 00:30:33,600 involves degeneracy factors. 415 00:30:33,600 --> 00:30:36,860 And just to remind you, when we had the discussion 416 00:30:36,860 --> 00:30:39,500 of Einstein's A and B coefficient, 417 00:30:39,500 --> 00:30:44,650 the Einstein A coefficient was proportionate to the B 418 00:30:44,650 --> 00:30:48,460 coefficient responsible for stimulated emission 419 00:30:48,460 --> 00:30:51,560 from the excited to the ground state. 420 00:30:51,560 --> 00:30:56,930 But the Einstein B coefficient for absorption 421 00:30:56,930 --> 00:30:59,210 was related to the Einstein B coefficient 422 00:30:59,210 --> 00:31:04,710 for stimulated emission by involving 423 00:31:04,710 --> 00:31:06,390 these degeneracy factors. 424 00:31:10,290 --> 00:31:19,580 So degeneracies appear and in some formal layer 425 00:31:19,580 --> 00:31:22,320 that it makes a lot of sense to use them. 426 00:31:22,320 --> 00:31:28,010 So I've always said for a fundamental understanding, 427 00:31:28,010 --> 00:31:30,910 you should just assume all degeneracies are one. 428 00:31:30,910 --> 00:31:32,410 This is how you can avoid, sort of, 429 00:31:32,410 --> 00:31:35,310 some baggage in deriving equations. 430 00:31:35,310 --> 00:31:37,680 And I'm still standing to my statement. 431 00:31:37,680 --> 00:31:39,320 I want to show you now a situation 432 00:31:39,320 --> 00:31:43,140 where it becomes useful to consider degeneracy factors. 433 00:31:43,140 --> 00:31:46,490 So let me give you an example. 434 00:31:46,490 --> 00:31:52,150 We can now look at the situation where we have an excited P 435 00:31:52,150 --> 00:31:56,660 state and a ground state, which is S. 436 00:31:56,660 --> 00:32:01,520 Or I can look at the opposite situation 437 00:32:01,520 --> 00:32:07,630 where we have an S state, which can radiate to a P state. 438 00:32:10,190 --> 00:32:14,890 Well by symmetry, the different p states 439 00:32:14,890 --> 00:32:18,020 and plus 1 and minus 1 m equals 0 are just 440 00:32:18,020 --> 00:32:20,340 connected by spatial rotations. 441 00:32:20,340 --> 00:32:26,920 So therefore, their lifetime of the 3 P states 442 00:32:26,920 --> 00:32:30,430 and the rate of spontaneous emission are the same. 443 00:32:35,120 --> 00:32:41,050 But if you now assume that you have absorption, 444 00:32:41,050 --> 00:32:45,030 you go from the S state to the P state. 445 00:32:45,030 --> 00:32:51,126 Then you find that the Einstein B coefficient there 446 00:32:51,126 --> 00:32:53,090 are now three possible ways. 447 00:32:53,090 --> 00:32:56,270 Not just one polarization or 3 polarization. 448 00:32:56,270 --> 00:33:03,370 And you will find that this is proportional to three times r. 449 00:33:03,370 --> 00:33:15,964 However, in this situation, it's a reverse 450 00:33:15,964 --> 00:33:17,130 but let me just finish here. 451 00:33:21,750 --> 00:33:26,140 So here the natural align rates and the rate 452 00:33:26,140 --> 00:33:30,150 of stimulated emission described by the coefficient 453 00:33:30,150 --> 00:33:32,460 from the excited state to the ground state 454 00:33:32,460 --> 00:33:35,330 is proportionate to R. 455 00:33:35,330 --> 00:33:39,665 Whereas, in the other situation, if you have absorption now, 456 00:33:39,665 --> 00:33:43,590 well, each of those levels, there's only one transition, 457 00:33:43,590 --> 00:33:45,830 one pass way. 458 00:33:45,830 --> 00:33:51,540 Therefore, you will find that the coefficient for absorption 459 00:33:51,540 --> 00:33:57,680 is proportionate to R. Whereas, gamma 460 00:33:57,680 --> 00:34:05,410 and the stimulated emission, which is now BSP, 461 00:34:05,410 --> 00:34:08,138 is proportionate to three R because there 462 00:34:08,138 --> 00:34:08,929 are three pathways. 463 00:34:14,639 --> 00:34:22,239 So depending what the situation is, you have to be careful. 464 00:34:22,239 --> 00:34:27,330 And you would say-- but if it's an S to P transition, 465 00:34:27,330 --> 00:34:31,550 it maybe connected by the same matrix element. 466 00:34:31,550 --> 00:34:33,880 And therefore, you would say shouldn't there 467 00:34:33,880 --> 00:34:36,679 be align strings, which is independent 468 00:34:36,679 --> 00:34:40,429 whether you go from S to P or P to S, which just describes 469 00:34:40,429 --> 00:34:43,909 in a natural way what is really the coupling between S and P 470 00:34:43,909 --> 00:34:45,050 state? 471 00:34:45,050 --> 00:34:49,639 And yes indeed, there is in the literature 472 00:34:49,639 --> 00:34:57,650 some definition of line strings where the lines strings 473 00:34:57,650 --> 00:35:05,760 S would be proportionate to the sum of all 474 00:35:05,760 --> 00:35:12,310 of the eights between an initial and the final state. 475 00:35:12,310 --> 00:35:15,200 And do sum over all. 476 00:35:15,200 --> 00:35:20,320 So therefore, when you use this formula for the line strings, 477 00:35:20,320 --> 00:35:26,910 whether you have the situation on the left side 478 00:35:26,910 --> 00:35:35,610 or on the right side, you will do always 479 00:35:35,610 --> 00:35:38,880 the sum over the 3 possible transitions. 480 00:35:38,880 --> 00:35:42,010 So the lines things is the same for both situations. 481 00:35:42,010 --> 00:35:43,825 It's just generic for an S to P transition. 482 00:35:48,290 --> 00:35:52,470 So if you use this definition but then you 483 00:35:52,470 --> 00:35:56,990 have the situation that spontaneous emission is always 484 00:35:56,990 --> 00:36:00,060 given by the line strings but you 485 00:36:00,060 --> 00:36:04,340 have to multiply now by the multiplicity of the excited 486 00:36:04,340 --> 00:36:05,420 state. 487 00:36:05,420 --> 00:36:08,440 If you have a P state, the whole line strings 488 00:36:08,440 --> 00:36:10,650 is distributed over three states. 489 00:36:10,650 --> 00:36:15,030 And each state has only a spontaneous emission rate, 490 00:36:15,030 --> 00:36:19,130 which is a third of what the line strings gives you. 491 00:36:19,130 --> 00:36:20,550 I don't want to beat it to death, 492 00:36:20,550 --> 00:36:22,590 because I hate degeneracy factors. 493 00:36:22,590 --> 00:36:25,210 But I just thought this example with the P to S 494 00:36:25,210 --> 00:36:29,470 and S to P transition tells you why they necessarily 495 00:36:29,470 --> 00:36:33,070 have to appear in derivations like Einstein's A and B 496 00:36:33,070 --> 00:36:33,570 coefficient. 497 00:36:40,386 --> 00:36:42,760 I hope there are no further questions about degeneracies. 498 00:36:45,890 --> 00:36:48,780 But you know, making this comment also 499 00:36:48,780 --> 00:37:04,285 allows me to say, well, when I derived the Einstein A 500 00:37:04,285 --> 00:37:08,240 coefficient-- what we did last class-- I did not 501 00:37:08,240 --> 00:37:09,600 use any degeneracy factors. 502 00:37:12,720 --> 00:37:15,520 Well, this is correct. 503 00:37:15,520 --> 00:37:19,740 Our derivation assumed that there 504 00:37:19,740 --> 00:37:26,890 was-- we assumed that there is only one final state. 505 00:37:26,890 --> 00:37:29,730 We did not include degeneracy factors. 506 00:37:29,730 --> 00:37:33,290 We also assumed that we had a dipole matrix 507 00:37:33,290 --> 00:37:41,160 element, which was along the z-axis. 508 00:37:41,160 --> 00:37:46,120 And so by those definitions, I have implicitly 509 00:37:46,120 --> 00:37:49,710 picked a geometry, which can be represented 510 00:37:49,710 --> 00:37:52,270 by that we have an exciting piece 511 00:37:52,270 --> 00:37:57,580 state in the m equals 0 state. 512 00:37:57,580 --> 00:38:00,960 And we have a pie transition with linear polarization 513 00:38:00,960 --> 00:38:03,690 to this s state. 514 00:38:03,690 --> 00:38:05,657 And by doing that, I did not have 515 00:38:05,657 --> 00:38:06,990 to account for any degeneracies. 516 00:38:18,570 --> 00:38:26,200 But in general, if you derive microscopically 517 00:38:26,200 --> 00:38:28,570 an equation for spontaneous emission, 518 00:38:28,570 --> 00:38:31,520 you may have to take into account that your excited 519 00:38:31,520 --> 00:38:34,680 state has different transitions-- sigma plus 520 00:38:34,680 --> 00:38:37,510 and sigma minus transitions-- to different states. 521 00:38:37,510 --> 00:38:39,380 And you have to be careful how you 522 00:38:39,380 --> 00:38:43,370 do the sum over all possible finer states. 523 00:38:43,370 --> 00:38:48,957 And this is where degeneracies would eventually matter? 524 00:38:48,957 --> 00:38:49,456 Questions? 525 00:38:54,790 --> 00:38:58,970 OK, so then lets go from P counting or accounting 526 00:38:58,970 --> 00:39:01,940 for the number of states to something, 527 00:39:01,940 --> 00:39:05,920 which is hopefully more exciting. 528 00:39:05,920 --> 00:39:11,880 We want to talk about the fully quantized Hamiltonian. 529 00:39:30,690 --> 00:39:33,710 So what we are working towards now 530 00:39:33,710 --> 00:39:36,660 and it may spill over into the Wednesday class 531 00:39:36,660 --> 00:39:42,750 is I want to give you the sort of paradigmatic example 532 00:39:42,750 --> 00:39:47,690 of cavity QED where an atom within an excited state 533 00:39:47,690 --> 00:39:50,020 is in an empty cavity. 534 00:39:50,020 --> 00:39:52,920 And now it can emit a photon into the mortification 535 00:39:52,920 --> 00:39:54,140 mode of the cavity. 536 00:39:54,140 --> 00:39:56,890 But these photon can be reabsorbed. 537 00:39:56,890 --> 00:40:01,490 So this is a phenomenon of vacuum Rabi oscillations. 538 00:40:01,490 --> 00:40:04,180 And so I want to set up the Hamiltonian and then 539 00:40:04,180 --> 00:40:06,380 the equation to demonstrate to you 540 00:40:06,380 --> 00:40:08,890 the vacuum Rabi oscillations. 541 00:40:08,890 --> 00:40:11,250 And for me, the vacuum Rabi oscillations 542 00:40:11,250 --> 00:40:16,050 are the demonstration, that spontaneous emission, 543 00:40:16,050 --> 00:40:19,550 has no randomness, no spontaneity, 544 00:40:19,550 --> 00:40:25,260 so to speak because you can observe coherent oscillation. 545 00:40:25,260 --> 00:40:30,070 A coherent time evolution of the whole system 546 00:40:30,070 --> 00:40:36,380 and which is possible only due to spontaneous emission. 547 00:40:36,380 --> 00:40:37,570 So let's go there. 548 00:40:41,620 --> 00:40:51,600 So just to make the connection, a few lectures ago, 549 00:40:51,600 --> 00:40:54,380 we had a semi classical Hamiltonian. 550 00:40:59,230 --> 00:41:01,170 This is when I wanted to show you 551 00:41:01,170 --> 00:41:06,170 that the two level electronic system can be mapped 552 00:41:06,170 --> 00:41:12,360 onto a spin one half system driven by magnetic field. 553 00:41:12,360 --> 00:41:15,565 So this was when we only looked at the stimulated term 554 00:41:15,565 --> 00:41:18,990 when we only did perturbation theory. 555 00:41:18,990 --> 00:41:26,040 And in that situation, we had the electronic excitation. 556 00:41:26,040 --> 00:41:31,540 And then we had the drive field, which 557 00:41:31,540 --> 00:41:35,940 was assumed to be purely classical like a rotating 558 00:41:35,940 --> 00:41:40,390 magnetic field which drives spin up spin down transitions 559 00:41:40,390 --> 00:41:42,250 magnetically. 560 00:41:42,250 --> 00:41:47,980 And we concluded that, yes, if you use a laser field, 561 00:41:47,980 --> 00:41:51,520 it does exactly the same to a two level atom what 562 00:41:51,520 --> 00:41:54,250 a magnetic field does to spin up spin down. 563 00:41:54,250 --> 00:41:56,140 But now we are one step further. 564 00:41:56,140 --> 00:41:58,310 We've quantized the electromagnetic field. 565 00:41:58,310 --> 00:42:00,100 And we have spontaneous emission. 566 00:42:00,100 --> 00:42:02,960 And this is something, for reasons I just mentioned, 567 00:42:02,960 --> 00:42:05,260 you will never find in spin up spin down 568 00:42:05,260 --> 00:42:07,790 because it will take 1,000 years for spontaneous emission 569 00:42:07,790 --> 00:42:08,750 to happen. 570 00:42:08,750 --> 00:42:13,080 So now we want to actually go beyond this semi classical 571 00:42:13,080 --> 00:42:15,500 picture, which is fully analogous to the precession 572 00:42:15,500 --> 00:42:18,064 and rotation of the spin in a magnetic field. 573 00:42:18,064 --> 00:42:19,730 And we want to add spontaneous emission. 574 00:42:30,830 --> 00:42:35,360 So what we had here is the Rabi frequency 575 00:42:35,360 --> 00:42:38,000 was a matrix element-- the dipole matrix 576 00:42:38,000 --> 00:42:42,790 element-- times a classic electric field. 577 00:42:42,790 --> 00:42:47,760 And we want to replace that now by the electric field 578 00:42:47,760 --> 00:42:49,550 at the position of the atom. 579 00:42:49,550 --> 00:42:52,500 But we want to use the fully quantized version 580 00:42:52,500 --> 00:42:53,520 of the electric field. 581 00:42:56,929 --> 00:43:04,930 And it also becomes useful to look at the sigma x operator, 582 00:43:04,930 --> 00:43:11,480 which actually has two matrix elements of [INAUDIBLE], which 583 00:43:11,480 --> 00:43:15,830 connects ground excited and excited ground state. 584 00:43:15,830 --> 00:43:21,980 And one of them is going from the excited to the ground 585 00:43:21,980 --> 00:43:22,910 state. 586 00:43:22,910 --> 00:43:25,460 So this is, sort of, lowering the energy 587 00:43:25,460 --> 00:43:27,700 to sigma minus operator. 588 00:43:27,700 --> 00:43:30,780 And the other one will be a raising operator. 589 00:43:30,780 --> 00:43:33,510 It raises the excitation of the atom. 590 00:43:33,510 --> 00:43:35,430 And we will refer to it as sigma plus. 591 00:43:38,330 --> 00:43:45,370 So the electric field is replaced 592 00:43:45,370 --> 00:43:49,567 by the operator obtained from the fully quantized picture. 593 00:43:52,700 --> 00:43:59,240 Here we have the prefactor, which 594 00:43:59,240 --> 00:44:03,400 is the electric field of a single photon or half a photon, 595 00:44:03,400 --> 00:44:04,160 whatever. 596 00:44:04,160 --> 00:44:07,789 But it's factors of 2 r square over 2. 597 00:44:07,789 --> 00:44:08,830 We have the polarization. 598 00:44:08,830 --> 00:44:17,280 And now if you would take the previous result 599 00:44:17,280 --> 00:44:20,210 and would look at it. 600 00:44:20,210 --> 00:44:22,430 Well we want to go to the Schrodinger picture. 601 00:44:22,430 --> 00:44:24,760 And I mentioned that in the Schrodinger picture 602 00:44:24,760 --> 00:44:27,010 the operators are time independent. 603 00:44:27,010 --> 00:44:29,970 So we cancelled the e to the i omega t term. 604 00:44:29,970 --> 00:44:33,490 If you would go to the result we had last week 605 00:44:33,490 --> 00:44:37,320 and would simply get rid of the e to the i omega t term, 606 00:44:37,320 --> 00:44:43,220 you would now find operators a and dagger. 607 00:44:43,220 --> 00:44:47,220 But they would have factors of i in front of it. 608 00:44:47,220 --> 00:44:50,810 That's a equation we had when we derived it. 609 00:44:50,810 --> 00:44:56,220 Well I prefer note to use something which looks nicer. 610 00:44:56,220 --> 00:44:59,970 Just use a and a dagger. 611 00:44:59,970 --> 00:45:09,750 And you can obtain it by shifting the origin of time. 612 00:45:14,380 --> 00:45:17,790 So we're not looking e to the i omega t or t equals 0. 613 00:45:17,790 --> 00:45:22,430 We wait a quarter period into e to the i omega t just gives us 614 00:45:22,430 --> 00:45:26,800 factors of i, which conveniently cancel the other factors of i. 615 00:45:26,800 --> 00:45:31,300 So what I'm doing is just for convenience. 616 00:45:31,300 --> 00:45:40,296 And let me write down that this is in the Schrodinger picture. 617 00:45:48,336 --> 00:45:48,835 OK. 618 00:45:53,620 --> 00:46:02,980 So we want to absorb all constant by in one constant 619 00:46:02,980 --> 00:46:07,600 now, which is the single photon Rabi frequency. 620 00:46:07,600 --> 00:46:12,730 We have the type or matrix element of the atom. 621 00:46:12,730 --> 00:46:16,920 There's a dot product with the polarization of the light. 622 00:46:16,920 --> 00:46:19,570 And then we have the electric field amplitude 623 00:46:19,570 --> 00:46:20,770 of a single photon. 624 00:46:20,770 --> 00:46:25,170 h bar omega over 2 epsilon 0 v. 625 00:46:25,170 --> 00:46:29,210 So this is what appears in the coupling. 626 00:46:29,210 --> 00:46:35,200 And we want to write it s h bar omega 1 over 2. 627 00:46:35,200 --> 00:46:40,930 And this omega 1 is the single photon Rabi frequency. 628 00:46:40,930 --> 00:46:44,730 And with that, we have now a Hamiltonian, 629 00:46:44,730 --> 00:46:47,700 which is really a classic Hamiltonian, written down 630 00:46:47,700 --> 00:46:49,430 in the standard form. 631 00:46:51,940 --> 00:46:59,820 It has the excitation energy times the sigma z matrix. 632 00:46:59,820 --> 00:47:04,712 It has the single photon Rabi frequency. 633 00:47:04,712 --> 00:47:13,150 The single photon Rabi frequency appears. 634 00:47:13,150 --> 00:47:16,690 You know, this is the single photon Rabi frequency. 635 00:47:16,690 --> 00:47:19,650 But then the operator for the electric field, 636 00:47:19,650 --> 00:47:23,520 after getting rid off the i's, is simply h plus h dagger. 637 00:47:29,460 --> 00:47:33,510 h plus h dagger. 638 00:47:33,510 --> 00:47:37,130 So this takes care of the photon field. 639 00:47:37,130 --> 00:47:39,680 And the operator which acts on the atoms 640 00:47:39,680 --> 00:47:43,920 are the raising and lowering operator sigma plus and sigma 641 00:47:43,920 --> 00:47:45,390 minus. 642 00:47:45,390 --> 00:47:50,420 And finally, we have the Hamiltonian, 643 00:47:50,420 --> 00:47:52,870 which describes the photon field which 644 00:47:52,870 --> 00:48:01,170 is h bar omega times a dagger a the photon number operator. 645 00:48:05,170 --> 00:48:06,330 Any questions? 646 00:48:06,330 --> 00:48:07,770 Yes? 647 00:48:07,770 --> 00:48:08,730 AUDIENCE: [INAUDIBLE]? 648 00:48:20,076 --> 00:48:21,450 PROFESSOR: I mean, we are looking 649 00:48:21,450 --> 00:48:24,870 at the interaction with an atom, which is at rest at the origin. 650 00:48:24,870 --> 00:48:27,040 Therefore, e to the ikr is 0. 651 00:48:27,040 --> 00:48:31,900 We will only consider the spatial dependence e 652 00:48:31,900 --> 00:48:36,230 to the ikr when we allow the atom to move. 653 00:48:36,230 --> 00:48:38,830 As long as the atom is stationary for convenience, 654 00:48:38,830 --> 00:48:41,090 we put the atom at i equals 0. 655 00:48:41,090 --> 00:48:46,370 But in 8.422 when we talk about light forces and laser cooling, 656 00:48:46,370 --> 00:48:51,130 then it becomes essential to allow the photon to move. 657 00:48:51,130 --> 00:48:55,040 And this is actually where the recoil and the light forces 658 00:48:55,040 --> 00:48:55,704 come into play. 659 00:48:55,704 --> 00:48:57,870 But as long as we're not interested in light forces, 660 00:48:57,870 --> 00:49:01,570 only in the internal dynamics-- calm and excited state-- 661 00:49:01,570 --> 00:49:06,550 we can conveniently neglect our spatial dependencies. 662 00:49:06,550 --> 00:49:07,216 Other questions? 663 00:49:18,710 --> 00:49:20,430 So this is really a famous Hamiltonian. 664 00:49:23,190 --> 00:49:25,490 And you also see how natural the definition 665 00:49:25,490 --> 00:49:28,180 of the single-photon Rabi frequency. 666 00:49:28,180 --> 00:49:33,010 So we have one half h bar omega for the diagonal sigma z 667 00:49:33,010 --> 00:49:33,680 matrix. 668 00:49:33,680 --> 00:49:35,280 This is the atomic excitation. 669 00:49:35,280 --> 00:49:38,740 This is the unperturbed Hamiltonian of the atom. 670 00:49:38,740 --> 00:49:41,290 This is the unperturbed Hamiltonian of the photon. 671 00:49:41,290 --> 00:49:43,340 And now the two are coupled. 672 00:49:43,340 --> 00:49:46,640 And the coupling is a product of an operator acting 673 00:49:46,640 --> 00:49:50,250 on the photon field plus minus one photon. 674 00:49:50,250 --> 00:49:54,730 And the other one is an operator acting on the atoms. 675 00:49:54,730 --> 00:49:58,910 And it is plus, minus, and atomic excitation. 676 00:49:58,910 --> 00:50:01,570 So let me just remind you of that. 677 00:50:01,570 --> 00:50:04,130 The sigma plus and sigma minus operator. 678 00:50:07,320 --> 00:50:09,620 The sigma plus is the atomic raising operator, 679 00:50:09,620 --> 00:50:12,440 which takes a ground to the excited state. 680 00:50:12,440 --> 00:50:17,360 And the sigma minus operator is the atomic lowering operator, 681 00:50:17,360 --> 00:50:19,910 which takes the atom from the excited to the ground state. 682 00:50:23,450 --> 00:50:24,860 So this is our Hamiltonian. 683 00:50:28,696 --> 00:50:34,500 And to hear about space on which this Hamiltonian acts 684 00:50:34,500 --> 00:50:38,520 is the product space of the atom. 685 00:50:38,520 --> 00:50:43,050 Direct product of the states of the light. 686 00:50:46,080 --> 00:50:49,310 Or in other words, the basis state 687 00:50:49,310 --> 00:50:53,360 would be that we use for the atoms. 688 00:50:53,360 --> 00:50:57,700 The states which have zero or one quantum of excitation. 689 00:50:57,700 --> 00:51:02,110 So we use excited state or ground state. 690 00:51:02,110 --> 00:51:08,500 And for the photon, we can just use the Fock states 691 00:51:08,500 --> 00:51:13,140 where the occupation number is n. 692 00:51:13,140 --> 00:51:14,510 Questions about that? 693 00:51:19,980 --> 00:51:25,860 So it's a very-- just look at it with some enjoyment 694 00:51:25,860 --> 00:51:27,100 for a few seconds. 695 00:51:27,100 --> 00:51:31,970 I mean, this is a Hamiltonian, which has just a few terms. 696 00:51:31,970 --> 00:51:33,640 But what is behind it is, of course, 697 00:51:33,640 --> 00:51:35,560 a power of all the definitions. 698 00:51:35,560 --> 00:51:37,980 I mean, each symbol has so much meaning. 699 00:51:37,980 --> 00:51:44,360 But in the end, by having this formalism of operators 700 00:51:44,360 --> 00:51:46,570 quantized electromagnetic field. 701 00:51:46,570 --> 00:51:52,170 We can write down-- we can catch many, many aspects 702 00:51:52,170 --> 00:51:55,640 or we can, pretty much, fully describe how a two level 703 00:51:55,640 --> 00:51:58,510 system interacts a quantized electromagnetic field 704 00:51:58,510 --> 00:52:00,646 with that set of equations. 705 00:52:03,420 --> 00:52:07,340 Of course, the fact is not that everything is so simple. 706 00:52:07,340 --> 00:52:11,200 The fact is that we have, by understanding the physics, 707 00:52:11,200 --> 00:52:13,410 we have skillfully made definitions, 708 00:52:13,410 --> 00:52:17,160 which allow us to write everything down 709 00:52:17,160 --> 00:52:18,205 in this compact form. 710 00:52:20,730 --> 00:52:23,380 So often something is simple to write down. 711 00:52:23,380 --> 00:52:26,050 But if there's a lot of physics insight, 712 00:52:26,050 --> 00:52:28,470 we spend some time in discussing it. 713 00:52:28,470 --> 00:52:31,380 And the first thing I want to just point out and discuss 714 00:52:31,380 --> 00:52:33,730 is this interaction term. 715 00:52:33,730 --> 00:52:36,240 We have the product of sigma plus and sigma 716 00:52:36,240 --> 00:52:39,560 minus with a and a dagger. 717 00:52:39,560 --> 00:52:43,724 So what we have here is we have an interaction term. 718 00:52:48,230 --> 00:53:00,320 And this interaction part has actually four terms 719 00:53:00,320 --> 00:53:03,494 in a very natural way. 720 00:53:03,494 --> 00:53:06,960 Well, let me just write them down. 721 00:53:06,960 --> 00:53:12,080 It's sigma plus with a. 722 00:53:12,080 --> 00:53:14,260 Sigma minus with a dagger. 723 00:53:17,330 --> 00:53:20,490 Sigma plus with a dagger. 724 00:53:20,490 --> 00:53:25,590 And sigma minus with a. 725 00:53:25,590 --> 00:53:29,880 OK, so let's discuss those. 726 00:53:29,880 --> 00:53:33,050 Sigma plus with a dagger. 727 00:53:33,050 --> 00:53:38,980 Sigma plus is actually an absorption process. 728 00:53:38,980 --> 00:53:44,850 a reduces the photo number by one, 729 00:53:44,850 --> 00:53:48,860 and increases the atomic excitation 730 00:53:48,860 --> 00:53:50,560 from the column to the excited state. 731 00:53:56,120 --> 00:54:01,790 The other term looks naturally, intuitively like emission. 732 00:54:01,790 --> 00:54:06,690 The a dagger operator takes us from n to n plus 1. 733 00:54:06,690 --> 00:54:10,800 And sigma minus takes us from the excited state 734 00:54:10,800 --> 00:54:13,820 to the current state. 735 00:54:13,820 --> 00:54:16,580 So these are the two terms, which 736 00:54:16,580 --> 00:54:25,590 we would call intuitive terms because they make sense. 737 00:54:25,590 --> 00:54:29,050 The other terms are somewhat more tricky. 738 00:54:29,050 --> 00:54:34,450 Sigma plus and a dagger means we create a photon 739 00:54:34,450 --> 00:54:37,240 and we create an excitation. 740 00:54:37,240 --> 00:54:40,530 So in other words, it's not that, like the other term, 741 00:54:40,530 --> 00:54:44,110 quantum of excitation disappears from the field, appears 742 00:54:44,110 --> 00:54:46,080 in the atom, and vice versa. 743 00:54:46,080 --> 00:54:52,150 Sigma plus a dagger means we have an atom excitation 744 00:54:52,150 --> 00:54:55,980 takes us from the ground to the excited state. 745 00:54:55,980 --> 00:54:59,600 Plus, we emit a photon at the same time. 746 00:54:59,600 --> 00:55:02,520 And sigma minus a dagger means that we 747 00:55:02,520 --> 00:55:05,470 go from the excited to the ground state. 748 00:55:05,470 --> 00:55:09,065 So we have an atom d excitation. 749 00:55:09,065 --> 00:55:10,530 And I would say, well, if the atom 750 00:55:10,530 --> 00:55:13,990 is d excited it should emit a photon. 751 00:55:13,990 --> 00:55:20,765 But instead, the photon disappears. 752 00:55:27,140 --> 00:55:31,170 So we have those processes. 753 00:55:31,170 --> 00:55:37,520 The last two are sometimes referred to 754 00:55:37,520 --> 00:55:39,490 in the theoretical literature. 755 00:55:39,490 --> 00:55:42,450 They are off shell. 756 00:55:42,450 --> 00:55:44,750 Under shell is energy conservation. 757 00:55:44,750 --> 00:55:48,860 Off shell means they cannot conserve energy. 758 00:55:48,860 --> 00:55:51,510 But nevertheless, these are terms 759 00:55:51,510 --> 00:55:52,855 which appear in the operator. 760 00:55:57,828 --> 00:56:00,385 But you should be used to if you have often terms 761 00:56:00,385 --> 00:56:03,900 in the operator which cannot drive a resonant transition. 762 00:56:03,900 --> 00:56:07,740 When you looked at the DC stock effect or when we looked 763 00:56:07,740 --> 00:56:10,970 at the AC stock effect for low frequency photons, 764 00:56:10,970 --> 00:56:13,945 those low frequency photons cannot excite an atom 765 00:56:13,945 --> 00:56:15,960 to the excited state. 766 00:56:15,960 --> 00:56:18,960 So they are not causing a transition, 767 00:56:18,960 --> 00:56:22,500 but they led to energy shifts in second order perturbation 768 00:56:22,500 --> 00:56:25,260 theory. 769 00:56:25,260 --> 00:56:32,280 So therefore, those terms this language now cannot drive 770 00:56:32,280 --> 00:56:33,340 transitions. 771 00:56:33,340 --> 00:56:37,111 They can only drive transitions to virtual states, which 772 00:56:37,111 --> 00:56:39,610 would mean they can only appear in second order perturbation 773 00:56:39,610 --> 00:56:42,840 theory that you go up to a so-called virtual state 774 00:56:42,840 --> 00:56:45,460 but you immediately go down. 775 00:56:45,460 --> 00:56:49,590 And those terms give only rise to shifts. 776 00:56:53,740 --> 00:56:56,230 No transitions because you couldn't conserve energy 777 00:56:56,230 --> 00:56:57,750 in the transition. 778 00:56:57,750 --> 00:57:00,540 But you can do shifts in second order. 779 00:57:05,010 --> 00:57:07,850 And one example, which we discussed in the clicker 780 00:57:07,850 --> 00:57:16,720 question is that those shifts are actually lamb shifts. 781 00:57:16,720 --> 00:57:19,770 And in other places, especially in the context 782 00:57:19,770 --> 00:57:29,770 of microwave fields, they are called Bloch-Siegert shifts 783 00:57:29,770 --> 00:57:35,380 And let's just look at one specific state. 784 00:57:35,380 --> 00:57:39,030 And this is the simplest of all. 785 00:57:39,030 --> 00:57:42,670 We have the vacuum no photons. 786 00:57:42,670 --> 00:57:45,980 And the atom is in the ground state. 787 00:57:45,980 --> 00:57:49,380 If you look at the four possibilities 788 00:57:49,380 --> 00:57:54,830 of the interaction term, there is only one non vanishing term. 789 00:57:54,830 --> 00:57:58,220 The photon is at the bottom off all possible states. 790 00:57:58,220 --> 00:58:01,890 The atom is at the bottom of the possible states. 791 00:58:01,890 --> 00:58:04,250 So when we act with the four terms on it, 792 00:58:04,250 --> 00:58:05,820 the only term which contributes is 793 00:58:05,820 --> 00:58:08,360 where those is where those are raised 794 00:58:08,360 --> 00:58:09,840 because all the others are 0. 795 00:58:13,490 --> 00:58:23,120 The only non vanishing term is where 796 00:58:23,120 --> 00:58:30,030 we create a virtual atomic excitation and also 797 00:58:30,030 --> 00:58:34,090 a virtual excitation of the photon field. 798 00:58:34,090 --> 00:58:37,870 And we know that when we have an atom in the ground 799 00:58:37,870 --> 00:58:42,650 state in the vacuum that the only manifestation 800 00:58:42,650 --> 00:58:45,600 of the electromagnetic field is, of course, not 801 00:58:45,600 --> 00:58:47,720 spontaneous emission but the lamb shift. 802 00:58:51,760 --> 00:58:56,050 So therefore, if you would apply this operator 803 00:58:56,050 --> 00:58:59,910 to the bound state of an electron in an atom, 804 00:58:59,910 --> 00:59:04,550 the complicated 1s wave function of hydrogen and sum 805 00:59:04,550 --> 00:59:08,500 this operator over all modes of the electromagnetic field. 806 00:59:08,500 --> 00:59:12,070 Then you would have done a first principle QED 807 00:59:12,070 --> 00:59:14,500 calculation of the lamb shift. 808 00:59:14,500 --> 00:59:16,480 I'm not doing it but you should understand 809 00:59:16,480 --> 00:59:20,490 that this operator-- sigma plus a dagger-- 810 00:59:20,490 --> 00:59:23,052 is you operator for the Lamb shift. 811 00:59:28,110 --> 00:59:28,610 Questions? 812 00:59:31,471 --> 00:59:31,970 Yes? 813 00:59:31,970 --> 00:59:32,962 AUDIENCE: [INAUDIBLE]? 814 00:59:39,410 --> 00:59:41,500 PROFESSOR: Oh, no, everything is. 815 00:59:41,500 --> 00:59:45,050 If you have a two level system, this Hamiltonian 816 00:59:45,050 --> 00:59:50,180 captures everything which appears in nature 817 00:59:50,180 --> 00:59:52,710 if you have a two level system interacting 818 00:59:52,710 --> 00:59:55,870 with the electromagnetic field. 819 00:59:55,870 --> 00:59:58,220 That's it. 820 00:59:58,220 --> 01:00:00,410 A radiation reaction is just something 821 01:00:00,410 --> 01:00:02,040 we can pull out of here. 822 01:00:02,040 --> 01:00:04,520 Stimulated emission we can pull out of here. 823 01:00:04,520 --> 01:00:08,190 The way how vacuum fluctuations create a lamb shift or the way 824 01:00:08,190 --> 01:00:12,150 how vacuum fluctuations affect an atom in the excited state, 825 01:00:12,150 --> 01:00:14,460 everything is included in here. 826 01:00:14,460 --> 01:00:16,820 The question is just can we solve it. 827 01:00:16,820 --> 01:00:19,260 And the calculations can get involved. 828 01:00:19,260 --> 01:00:23,280 But this is the full QED Hamiltonian 829 01:00:23,280 --> 01:00:26,110 for a two level system. 830 01:00:26,110 --> 01:00:27,150 That's a full picture. 831 01:00:27,150 --> 01:00:30,070 I mean, that's why I sort of said before be proud of it. 832 01:00:30,070 --> 01:00:34,220 You understand the full picture of how two level systems 833 01:00:34,220 --> 01:00:36,660 interact with electromagnetic radiation. 834 01:00:36,660 --> 01:00:40,280 The only complication is, yes, if you put more levels into it 835 01:00:40,280 --> 01:00:43,530 and such and things can get richer and richer. 836 01:00:43,530 --> 01:00:57,190 And-- yes, we have also made the dipole approximation, 837 01:00:57,190 --> 01:01:02,674 which we're just wondering how critical it is. 838 01:01:02,674 --> 01:01:04,632 Well, we use the electric field a and a dagger, 839 01:01:04,632 --> 01:01:12,360 but my gut feeling is it doesn't really matter what we have. 840 01:01:12,360 --> 01:01:16,015 Here is the most generic term, which can create and annihilate 841 01:01:16,015 --> 01:01:17,890 photons, and we have the a and a dagger term. 842 01:01:26,100 --> 01:01:28,600 Actually, I don't know what would 843 01:01:28,600 --> 01:01:32,294 happen if you don't make the dipole approximation. 844 01:01:32,294 --> 01:01:33,710 Well, if you have two levels which 845 01:01:33,710 --> 01:01:37,340 are coupled by magnetic dipole, then you 846 01:01:37,340 --> 01:01:39,140 have the same situation. 847 01:01:39,140 --> 01:01:43,790 It is just your prefactor, the semi photon Rabi frequency, 848 01:01:43,790 --> 01:01:47,420 is now alpha times smaller because of the smaller dipole 849 01:01:47,420 --> 01:01:48,770 matrix element. 850 01:01:48,770 --> 01:01:57,670 So I think you can pick, pretty much, any level you want. 851 01:01:57,670 --> 01:02:01,710 And this is why I actually discussed matrix elements 852 01:02:01,710 --> 01:02:03,970 at the beginning of the unit. 853 01:02:03,970 --> 01:02:05,745 For, pretty much, all of the discussion 854 01:02:05,745 --> 01:02:07,510 you're going to have, it doesn't really 855 01:02:07,510 --> 01:02:09,980 matter what kind of transition you have as long 856 01:02:09,980 --> 01:02:13,590 as the transition creates or annihilates a photon. 857 01:02:13,590 --> 01:02:19,000 And all the physics of the multiplicity of the transition, 858 01:02:19,000 --> 01:02:22,380 magnetic, dipole, electric, quadrupole, or whatever 859 01:02:22,380 --> 01:02:25,530 just defines what this the semi photon Rabi frequency is. 860 01:02:27,875 --> 01:02:29,750 You've put me on the spot, but the only thing 861 01:02:29,750 --> 01:02:35,010 which comes to my mind now is if you would formulate QED 862 01:02:35,010 --> 01:02:38,070 not in the dipole approximation but through with the p 863 01:02:38,070 --> 01:02:41,700 minus a formulation. 864 01:02:41,700 --> 01:02:43,780 Then we have an a-square term. 865 01:02:43,780 --> 01:02:46,620 And then we have the possibility that one transition 866 01:02:46,620 --> 01:02:48,440 can emit two photons. 867 01:02:48,440 --> 01:02:50,158 So that's not included here. 868 01:02:50,158 --> 01:02:53,651 AUDIENCE: So that's higher-- 869 01:02:53,651 --> 01:02:56,670 PROFESSOR: This would be something higher order. 870 01:02:56,670 --> 01:02:58,300 On the other hand, we can shoulder 871 01:02:58,300 --> 01:03:00,910 the canonical transformation that the p 872 01:03:00,910 --> 01:03:04,120 minus a formalization with the a-square term 873 01:03:04,120 --> 01:03:07,370 is equivalent to dipole approximation. 874 01:03:07,370 --> 01:03:10,010 So the question whether you have a transition which 875 01:03:10,010 --> 01:03:14,460 emits two photons simultaneously or two photons sequentially 876 01:03:14,460 --> 01:03:20,810 eventually by going through an immediate state, 877 01:03:20,810 --> 01:03:23,610 this is not a fundamental distinction. 878 01:03:23,610 --> 01:03:26,130 You can have one description of your quantum system 879 01:03:26,130 --> 01:03:28,830 via two photons automated in one transition. 880 01:03:28,830 --> 01:03:31,860 You have another description of your quantum system where 881 01:03:31,860 --> 01:03:34,980 photons cannot-- only one photon can be emitted. 882 01:03:34,980 --> 01:03:37,800 And then you have to lend an intermediate state. 883 01:03:37,800 --> 01:03:40,710 And you would say, well, either two photons at once 884 01:03:40,710 --> 01:03:42,370 or one photon at a time. 885 01:03:42,370 --> 01:03:44,420 This is two different kinds of physics. 886 01:03:44,420 --> 01:03:46,770 But we can show that the two pictures are connected 887 01:03:46,770 --> 01:03:48,790 with economical transformation. 888 01:03:48,790 --> 01:03:52,290 So therefore, you have two descriptions here. 889 01:03:52,290 --> 01:03:55,090 But anyway, I'm going a little bit beyond my knowledge. 890 01:03:55,090 --> 01:03:57,380 I'm just telling you bits and pieces I know. 891 01:03:57,380 --> 01:04:07,700 But this Hamiltonian is either generally exact. 892 01:04:07,700 --> 01:04:10,740 I just don't know how to prove it. 893 01:04:10,740 --> 01:04:14,510 But it really captures in all of the QED aspects 894 01:04:14,510 --> 01:04:16,570 of the system we want to get into. 895 01:04:22,110 --> 01:04:29,490 So OK. 896 01:04:29,490 --> 01:04:34,940 So in many situations we may decide 897 01:04:34,940 --> 01:04:39,270 that the off shell terms of the interaction 898 01:04:39,270 --> 01:04:43,480 just create level shifts, Lamb shifts, Bloch-Siegert shifts. 899 01:04:43,480 --> 01:04:46,540 And we may simply absorb those lamb shifts 900 01:04:46,540 --> 01:04:51,410 in our atomic energy levels, omega e and omega g. 901 01:04:51,410 --> 01:04:54,546 So therefore, for the dynamic of the system, 902 01:04:54,546 --> 01:04:56,170 if you include all of those lamb shifts 903 01:04:56,170 --> 01:05:00,210 in the atomic description, you do not need those off shell 904 01:05:00,210 --> 01:05:02,120 counter intuitive terms. 905 01:05:02,120 --> 01:05:04,860 These are actually also the counter-rotating terms 906 01:05:04,860 --> 01:05:07,140 in the semi classical approximation. 907 01:05:07,140 --> 01:05:11,220 We only keep the intuitive terms. 908 01:05:11,220 --> 01:05:17,560 And that's called, again, the rotating wave approximation. 909 01:05:17,560 --> 01:05:20,600 Just to remind you, we do not have rotating waves here. 910 01:05:20,600 --> 01:05:21,970 Everything is operators. 911 01:05:21,970 --> 01:05:26,830 But the same kind of physics-- co- and counter-rotating-- 912 01:05:26,830 --> 01:05:30,220 appears here that we have four terms. 913 01:05:30,220 --> 01:05:33,490 Two are the fully quantized version 914 01:05:33,490 --> 01:05:35,230 of the co-rotating terms. 915 01:05:35,230 --> 01:05:37,620 And the other two-- the off shell terms-- 916 01:05:37,620 --> 01:05:41,600 are the quantized version of the counter rotating term. 917 01:05:41,600 --> 01:05:49,600 So therefore, if you neglect those two off shell terms, 918 01:05:49,600 --> 01:05:58,165 we have now the fully quantized Hamiltonian 919 01:05:58,165 --> 01:05:59,623 in the rotating wave approximation. 920 01:06:03,930 --> 01:06:07,150 So let me just write it down because it's also 921 01:06:07,150 --> 01:06:09,100 a beautiful line. 922 01:06:09,100 --> 01:06:13,450 We have the electronic system. 923 01:06:13,450 --> 01:06:19,400 We have the interaction Hamiltonian, 924 01:06:19,400 --> 01:06:22,220 which has now owned the two terms. 925 01:06:22,220 --> 01:06:24,490 When we raise the atomic excitation, 926 01:06:24,490 --> 01:06:28,640 we lower the photon excitation and vice versa. 927 01:06:31,230 --> 01:06:41,550 And we have the Hamiltonian for the photon field a dagger a. 928 01:06:46,610 --> 01:06:52,240 And this is apart from those lamb shift terms. 929 01:06:52,240 --> 01:06:54,555 The full QED description of the system. 930 01:06:57,290 --> 01:07:05,470 And if we only consider one mode-- here, of course, 931 01:07:05,470 --> 01:07:06,980 in general, the general Hamiltonian 932 01:07:06,980 --> 01:07:08,549 has to be sent over a modes. 933 01:07:08,549 --> 01:07:10,840 And then you'll get spontaneous emission and everything 934 01:07:10,840 --> 01:07:11,760 we want. 935 01:07:11,760 --> 01:07:15,160 But if you have a situation where you only 936 01:07:15,160 --> 01:07:21,170 look at one single mode, then you have what 937 01:07:21,170 --> 01:07:32,960 is called the famous chains Cummings model 938 01:07:32,960 --> 01:07:38,405 And very important result of this James Cummings model 939 01:07:38,405 --> 01:07:40,310 are the vacuum Rabi oscillations, 940 01:07:40,310 --> 01:07:42,478 which I want to discuss now. 941 01:07:51,720 --> 01:07:52,220 OK. 942 01:07:58,280 --> 01:08:03,280 So let me just-- it's called James Cummings Model. 943 01:08:03,280 --> 01:08:06,070 So let me describe to you why it is a model. 944 01:08:06,070 --> 01:08:09,030 Well it assumes a two level system, 945 01:08:09,030 --> 01:08:13,790 which we find a lot of candidates among the atoms 946 01:08:13,790 --> 01:08:14,710 we want. 947 01:08:14,710 --> 01:08:17,910 Sure our atoms have hyperfine states. 948 01:08:17,910 --> 01:08:21,130 But we can always select a situation 949 01:08:21,130 --> 01:08:23,779 where, essentially, we only couple two states. 950 01:08:23,779 --> 01:08:26,370 We can prepare initial state by optical pumping, 951 01:08:26,370 --> 01:08:28,740 and then use circularly polarized slide 952 01:08:28,740 --> 01:08:30,390 on a cycling transition. 953 01:08:30,390 --> 01:08:33,439 And this is how we prepare in the laboratory a two level 954 01:08:33,439 --> 01:08:34,540 system. 955 01:08:34,540 --> 01:08:38,890 So that's one assumption of this model with a two level system. 956 01:08:38,890 --> 01:08:46,500 But the second assumption is that the atom only 957 01:08:46,500 --> 01:08:48,250 interacts with a similar mode. 958 01:08:48,250 --> 01:08:50,580 And that requires a little bit of engineering 959 01:08:50,580 --> 01:08:52,420 because it means we need a cavity. 960 01:08:59,850 --> 01:09:01,414 So let me just set up the system. 961 01:09:04,710 --> 01:09:08,240 So our laboratory is a big box of volume 962 01:09:08,240 --> 01:09:14,120 v. And this is where we maybe quantize electromagnetic field 963 01:09:14,120 --> 01:09:16,899 to calculate spontaneous emission. 964 01:09:16,899 --> 01:09:21,020 And our atom here may actually decay 965 01:09:21,020 --> 01:09:23,990 with the rate gamma, which is given by the Einstein A 966 01:09:23,990 --> 01:09:25,229 coefficient. 967 01:09:25,229 --> 01:09:28,859 And in order to describe this spontaneous emission, 968 01:09:28,859 --> 01:09:35,319 be quantized electromagnetic field in the large volume v. 969 01:09:35,319 --> 01:09:39,249 But now we have a cavity with two mirrors. 970 01:09:44,460 --> 01:09:52,979 And those two mirrors define one mode 971 01:09:52,979 --> 01:09:55,300 of the electromagnetic field, which 972 01:09:55,300 --> 01:10:02,460 will be in resonance on your resonance with the atom. 973 01:10:02,460 --> 01:10:11,200 Well there will be some losses out of the cavity, which 974 01:10:11,200 --> 01:10:14,980 eventually coupe the electromagnetic mode 975 01:10:14,980 --> 01:10:18,200 inside the cavity to the other awards modes in the speaker 976 01:10:18,200 --> 01:10:22,500 volume v. And this is described by a cavity damping 977 01:10:22,500 --> 01:10:23,520 constant kappa. 978 01:10:29,000 --> 01:10:35,770 What is also important is when we use cavity to single out one 979 01:10:35,770 --> 01:10:37,790 mode of the electromagnetic field, 980 01:10:37,790 --> 01:10:40,850 the cavity volume is v prime. 981 01:10:40,850 --> 01:10:44,560 And we often make it very small by putting the atoms 982 01:10:44,560 --> 01:10:48,360 in the cavity where the mirror spacing is extremely small. 983 01:10:50,995 --> 01:10:51,495 OK. 984 01:10:54,620 --> 01:10:57,060 We know, and I'm not writing it down again, 985 01:10:57,060 --> 01:11:01,140 what the Einstein A coefficient is. 986 01:11:01,140 --> 01:11:05,540 The Rabi frequency-- the single photon Rabi frequency-- 987 01:11:05,540 --> 01:11:14,120 which couples the atom to the one mode of the cavity 988 01:11:14,120 --> 01:11:20,710 has this important perfecter, which 989 01:11:20,710 --> 01:11:26,210 was or is the electric field of one photon in the cavity. 990 01:11:26,210 --> 01:11:29,950 And importantly, it involves the electric field 991 01:11:29,950 --> 01:11:34,920 of the photon in the cavity value, which is B prime. 992 01:11:34,920 --> 01:11:38,480 So now in addition to using, you know-- 993 01:11:38,480 --> 01:11:41,510 now you see what our experimental handle is. 994 01:11:41,510 --> 01:11:45,640 If you make this volume very small, 995 01:11:45,640 --> 01:11:54,690 then we can enter this strong coupling regime 996 01:11:54,690 --> 01:11:59,540 where the single photon Rabi frequency for this one mode 997 01:11:59,540 --> 01:12:05,090 selected by the cavity becomes much larger then 998 01:12:05,090 --> 01:12:09,410 the spontaneous emission into all the many other modes. 999 01:12:09,410 --> 01:12:13,440 So the interaction with one mode due to the cavity 1000 01:12:13,440 --> 01:12:15,820 and the smallness of the volume is, sort of, 1001 01:12:15,820 --> 01:12:20,380 outperforming all these many, many modes of the surroundings. 1002 01:12:20,380 --> 01:12:24,520 And that would mean that an atom in an excited state 1003 01:12:24,520 --> 01:12:28,580 is more likely to emit into the mode between the two 1004 01:12:28,580 --> 01:12:34,340 cavity mirrors than to any other modes to the side. 1005 01:12:34,340 --> 01:12:37,820 Secondly, of course, when the photon 1006 01:12:37,820 --> 01:12:39,380 has been emitted into the cavity, 1007 01:12:39,380 --> 01:12:43,270 the photon can still couple to the other modes 1008 01:12:43,270 --> 01:12:46,090 by cavity losses kappa. 1009 01:12:46,090 --> 01:12:51,660 And now we assume that we have such high reflectivity mirrors 1010 01:12:51,660 --> 01:12:55,308 that kappa is smaller that the single photon Rabi frequency. 1011 01:13:02,780 --> 01:13:07,990 And this is called the strong coupling regime of cavity QED. 1012 01:13:12,580 --> 01:13:19,830 So then we can at least observe for a limited time 1013 01:13:19,830 --> 01:13:25,610 the interplay between a single mode of the cavity and a two 1014 01:13:25,610 --> 01:13:26,500 level system. 1015 01:13:26,500 --> 01:13:29,250 And this is a James Cummings model. 1016 01:13:29,250 --> 01:13:30,531 The James Cummings model. 1017 01:13:33,300 --> 01:13:40,510 So in that situation, the Hamiltonian, 1018 01:13:40,510 --> 01:13:49,200 the fully quantized Hamiltonian, and the QED Hamiltonian couples 1019 01:13:49,200 --> 01:14:00,670 only pairs of states which we label 1020 01:14:00,670 --> 01:14:03,390 those states the manifold n. 1021 01:14:03,390 --> 01:14:08,440 So we have an excited state with n photons. 1022 01:14:08,440 --> 01:14:17,060 And it is coupled to the ground state with one more photon. 1023 01:14:17,060 --> 01:14:22,320 Our Hamiltonian has two coupling terms. 1024 01:14:22,320 --> 01:14:25,600 Remember the other tool where you clicked it in the rotating 1025 01:14:25,600 --> 01:14:30,060 wave approximation and we can go from left to right 1026 01:14:30,060 --> 01:14:33,620 with sigma minus a data plus. 1027 01:14:33,620 --> 01:14:39,590 And we can go from right to left with the operator sigma plus 1028 01:14:39,590 --> 01:14:41,907 and the annihilation of the [INAUDIBLE] a. 1029 01:14:47,690 --> 01:14:55,560 So as long as we have a detuning delta, 1030 01:14:55,560 --> 01:14:56,875 which is relatively small. 1031 01:15:03,930 --> 01:15:08,250 As long as detuning is small, the rotating wave approximation 1032 01:15:08,250 --> 01:15:09,960 is excellent. 1033 01:15:09,960 --> 01:15:19,960 So let me just conclude by writing down 1034 01:15:19,960 --> 01:15:22,770 the Hamiltonian for the situation I just discussed. 1035 01:15:22,770 --> 01:15:25,445 And then we'll discuss the Hamiltonian on Wednesday. 1036 01:15:29,440 --> 01:15:35,680 So if this is energy, we have two levels. 1037 01:15:35,680 --> 01:15:39,130 The excited state with n photons, 1038 01:15:39,130 --> 01:15:42,780 the ground state with n plus 1 photons. 1039 01:15:42,780 --> 01:15:47,270 If the photons are on resonance, the two levels are degenerate. 1040 01:15:47,270 --> 01:15:50,881 But if you have a detuning delta, 1041 01:15:50,881 --> 01:15:52,975 the two levels are split by delta. 1042 01:15:56,650 --> 01:16:00,350 And what we are doing right now is for the [INAUDIBLE] 1043 01:16:00,350 --> 01:16:02,840 the Hamiltonian, we shift the origin 1044 01:16:02,840 --> 01:16:05,290 so the zero of the energy is just halfway 1045 01:16:05,290 --> 01:16:06,770 between those two states. 1046 01:16:06,770 --> 01:16:09,240 That's natural. 1047 01:16:09,240 --> 01:16:19,470 So this avoids just off sets in our equations. 1048 01:16:19,470 --> 01:16:22,730 So our Hamiltonian has now the splitting 1049 01:16:22,730 --> 01:16:28,930 of plus minus delta over two. 1050 01:16:28,930 --> 01:16:32,770 The coupling has the perfecter, which is the single photon Rabi 1051 01:16:32,770 --> 01:16:34,430 frequency. 1052 01:16:34,430 --> 01:16:51,470 And then the a and a dagger terms 1053 01:16:51,470 --> 01:16:54,030 depends on n square root n plus 1. 1054 01:16:54,030 --> 01:16:56,980 So what I wrote down now is the Hamiltonian rotating wave 1055 01:16:56,980 --> 01:17:00,450 approximation, which interacts, which 1056 01:17:00,450 --> 01:17:03,490 describes only one pair of states. 1057 01:17:03,490 --> 01:17:05,970 But we have sort of a cause in our Hilbert space. 1058 01:17:05,970 --> 01:17:09,610 One pair of states for each label n. 1059 01:17:09,610 --> 01:17:11,600 But each of them is, sort of, described 1060 01:17:11,600 --> 01:17:15,510 by the decoupled Hamiltonian. 1061 01:17:15,510 --> 01:17:19,520 So that's what I wanted to present you today. 1062 01:17:19,520 --> 01:17:23,210 And I will show you on Wednesday how this Hamiltonian needs 1063 01:17:23,210 --> 01:17:28,310 to Rabi oscillations not induced by an external field 1064 01:17:28,310 --> 01:17:29,440 but induced by the vacuum. 1065 01:17:32,740 --> 01:17:34,590 Any questions?