1 00:00:00,060 --> 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 --> 00:00:17,200 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,200 --> 00:00:17,825 at ocw.mit.edu. 8 00:00:27,060 --> 00:00:29,490 PROFESSOR: OK. 9 00:00:29,490 --> 00:00:35,460 Back to cavity QED, back to the fully quantized radiation 10 00:00:35,460 --> 00:00:38,125 field, back to vacuum Rabi oscillation. 11 00:00:42,700 --> 00:00:45,730 Let me just recapitulate and sort of make the transition 12 00:00:45,730 --> 00:00:48,650 from this intense discussion about homework 13 00:00:48,650 --> 00:00:52,560 to the intellectually stimulating discussion 14 00:00:52,560 --> 00:00:55,060 about atoms and photons. 15 00:00:55,060 --> 00:00:57,490 So in the semiclassical description 16 00:00:57,490 --> 00:01:00,230 of the electromagnetic field, photons 17 00:01:00,230 --> 00:01:02,450 can only be emitted because we have 18 00:01:02,450 --> 00:01:05,310 a Hamiltonian with the semiclassical electric field. 19 00:01:05,310 --> 00:01:08,130 So if you don't drive the system with an electric field, 20 00:01:08,130 --> 00:01:11,090 you cannot stimulate the emission of photons. 21 00:01:11,090 --> 00:01:14,200 But we know this is not what happens. 22 00:01:14,200 --> 00:01:16,630 Photons are emitted into empty space, 23 00:01:16,630 --> 00:01:18,690 photons are emitted into a vacuum. 24 00:01:18,690 --> 00:01:22,340 And for that we needed a quantized description 25 00:01:22,340 --> 00:01:23,940 of the electromagnetic field. 26 00:01:23,940 --> 00:01:26,120 We did field quantization, and we 27 00:01:26,120 --> 00:01:29,170 have now our quantized Hamiltonian. 28 00:01:29,170 --> 00:01:33,210 And on Monday I started to discuss 29 00:01:33,210 --> 00:01:35,285 what is sort of the paradigmatic situation, 30 00:01:35,285 --> 00:01:39,030 the paradigmatic example, for how you should think 31 00:01:39,030 --> 00:01:41,910 about the vacuum and how you should think about emission 32 00:01:41,910 --> 00:01:43,710 of photons into the vacuum. 33 00:01:43,710 --> 00:01:46,880 And these are the vacuum Rabi oscillation 34 00:01:46,880 --> 00:01:49,780 described by the Jaynes-Cummings model. 35 00:01:49,780 --> 00:01:52,635 So the situation which I have in mind, 36 00:01:52,635 --> 00:01:54,010 or which you should have in mind, 37 00:01:54,010 --> 00:01:57,370 is an idealized situation, but it 38 00:01:57,370 --> 00:02:00,450 has been realized experimentally. 39 00:02:00,450 --> 00:02:04,130 And some of those idealized experiments 40 00:02:04,130 --> 00:02:06,620 were recognized with the Nobel Prize 41 00:02:06,620 --> 00:02:08,310 research of Haroche and Dave Wineland. 42 00:02:08,310 --> 00:02:11,039 So the situation is we have an atom, 43 00:02:11,039 --> 00:02:15,610 but it only talks to one mode of the electromagnetic field, 44 00:02:15,610 --> 00:02:17,340 and we make sure that the atom only 45 00:02:17,340 --> 00:02:20,580 talks to one mode of the electromagnetic field 46 00:02:20,580 --> 00:02:22,980 not by eliminating other modes; they exist. 47 00:02:22,980 --> 00:02:26,460 I mean, an atom can emit upwards and downwards. 48 00:02:26,460 --> 00:02:32,340 But we surround it with a cavity which has such a small mode 49 00:02:32,340 --> 00:02:37,460 volume, it has such a small volume, that the single photon 50 00:02:37,460 --> 00:02:40,620 Rabi frequency is huge, and therefore 51 00:02:40,620 --> 00:02:44,410 the emission into this one single mode 52 00:02:44,410 --> 00:02:47,970 dominates over the emission into all other modes. 53 00:02:50,510 --> 00:02:54,410 So this is a condition that the single photon Rabi frequency 54 00:02:54,410 --> 00:02:56,830 has to be larger than gamma. 55 00:02:56,830 --> 00:02:58,900 And, of course, we also have to make sure 56 00:02:58,900 --> 00:03:02,840 that the system is idealized so the loss of photons because 57 00:03:02,840 --> 00:03:05,240 of losses in the mirror, or finite reflectivity 58 00:03:05,240 --> 00:03:08,060 in the mirror, also has to be smaller. 59 00:03:08,060 --> 00:03:11,200 So that means for several Rabi periods 60 00:03:11,200 --> 00:03:15,600 we have a system which has only two parts, a two-level atom 61 00:03:15,600 --> 00:03:17,740 and one single mode of the cavity. 62 00:03:20,520 --> 00:03:23,810 So that's the system we have in mind, 63 00:03:23,810 --> 00:03:26,580 and we discussed the Hamiltonian. 64 00:03:30,070 --> 00:03:34,320 We saw that the Hilbert space of the atom 65 00:03:34,320 --> 00:03:38,020 is excited in ground state, the Hilbert space of the photons 66 00:03:38,020 --> 00:03:42,157 is spent by the [? flux ?] states, 67 00:03:42,157 --> 00:03:44,740 but what happens is-- so there's an infinite number of states, 68 00:03:44,740 --> 00:03:47,770 because of the infinite number of states of the photon field-- 69 00:03:47,770 --> 00:03:53,710 but what happens is the Hamiltonian couples only 70 00:03:53,710 --> 00:03:57,540 an excited state with n photons to a ground state with n 71 00:03:57,540 --> 00:03:59,080 plus 1 photons. 72 00:03:59,080 --> 00:04:03,890 So the whole Hilbert space is segmented now 73 00:04:03,890 --> 00:04:09,250 into just pairs of states labeled by the index n. 74 00:04:09,250 --> 00:04:15,320 So after so much work, we are back to a two-level system. 75 00:04:15,320 --> 00:04:19,810 And here is our two-level Hamiltonian. 76 00:04:19,810 --> 00:04:22,350 And, well, a two-level system does 77 00:04:22,350 --> 00:04:24,120 oscillations between the two levels. 78 00:04:24,120 --> 00:04:26,220 Rabi oscillations, no surprise. 79 00:04:26,220 --> 00:04:28,910 And this is what I want to discuss now. 80 00:04:28,910 --> 00:04:36,060 But the new feature is that these are really, well, 81 00:04:36,060 --> 00:04:38,950 these are now really two levels. 82 00:04:38,950 --> 00:04:41,910 Each of them is the combined state 83 00:04:41,910 --> 00:04:44,620 of the atom and the quantized radiation field. 84 00:04:44,620 --> 00:04:48,950 So now we have included in our two-level description 85 00:04:48,950 --> 00:04:51,950 the quantum state of the electromagnetic field. 86 00:04:56,910 --> 00:05:08,930 So first you should realize that this Hamiltonian is absolutely 87 00:05:08,930 --> 00:05:13,540 identical to spn 1/2 in magnetic fields. 88 00:05:20,330 --> 00:05:27,490 And you can recognize by [? comparing ?] 89 00:05:27,490 --> 00:05:29,740 this Hamiltonian, this matrix, to the matrices 90 00:05:29,740 --> 00:05:33,880 we discussed for spn 1/2 in the magnetic field, 91 00:05:33,880 --> 00:05:39,720 that this corresponds to the situation 92 00:05:39,720 --> 00:05:44,540 where this spn 1/2 had a transverse field 93 00:05:44,540 --> 00:05:49,520 in the x direction which caused a precession from spn 94 00:05:49,520 --> 00:05:51,310 up to spn down. 95 00:05:51,310 --> 00:05:54,930 And this x component of the field 96 00:05:54,930 --> 00:05:58,700 corresponds now to the single photon Rabi frequency times 97 00:05:58,700 --> 00:05:59,880 n plus 1. 98 00:05:59,880 --> 00:06:02,780 That's the off diagonal matrix element in this matrix. 99 00:06:07,050 --> 00:06:08,750 The thing which we have to discuss, 100 00:06:08,750 --> 00:06:12,330 and I will focus later, is that it depends on n. 101 00:06:12,330 --> 00:06:15,970 So for each pairs of state labeled by n, the photon 102 00:06:15,970 --> 00:06:21,530 number, we have a different off diagonal matrix element. 103 00:06:21,530 --> 00:06:27,600 But let's discuss first the most important and simplest case. 104 00:06:27,600 --> 00:06:32,640 Let's assume we are on resonance, 105 00:06:32,640 --> 00:06:42,520 and we want to assume that we have a vacuum. 106 00:06:42,520 --> 00:06:51,070 Then our Hamiltonian is simply this. 107 00:06:51,070 --> 00:06:56,690 And when we prepare the system in an initial state, which 108 00:06:56,690 --> 00:07:06,250 is an excited state with no photon in the vacuum, 109 00:07:06,250 --> 00:07:11,180 then we'll have oscillations to the ground 110 00:07:11,180 --> 00:07:12,120 state with one photon. 111 00:07:15,360 --> 00:07:17,740 These oscillations are exactly the oscillations 112 00:07:17,740 --> 00:07:19,262 we saw on the spn 1/2 system. 113 00:07:19,262 --> 00:07:20,470 We can just map the solution. 114 00:07:20,470 --> 00:07:23,640 I'm not really writing anything here. 115 00:07:23,640 --> 00:07:28,750 So what we obtain is the famous vacuum Rabi oscillations. 116 00:07:31,274 --> 00:07:38,220 where the probability to be in the excited state 117 00:07:38,220 --> 00:07:46,100 oscillates with the single photon Rabi frequency omega 1. 118 00:07:46,100 --> 00:07:48,940 I think there's a little bit of an ambiguity in language. 119 00:07:48,940 --> 00:07:50,985 Is it the single photon Rabi frequency? 120 00:07:50,985 --> 00:07:53,700 or is it the vacuum Rabi frequency? 121 00:07:53,700 --> 00:07:56,580 Because there's always the question about plus 122 00:07:56,580 --> 00:07:59,010 minus one photon because we start in the excited 123 00:07:59,010 --> 00:08:01,170 state without photon so you want to say 124 00:08:01,170 --> 00:08:02,980 it's a vacuum Rabi frequency. 125 00:08:02,980 --> 00:08:05,850 But then you have the ground state with one photon, 126 00:08:05,850 --> 00:08:08,040 and this photon is reabsorbed and then 127 00:08:08,040 --> 00:08:10,840 you may want to call it the one photon Rabi frequency. 128 00:08:10,840 --> 00:08:15,060 So I leave it to you, but it's called vacuum Rabi oscillation 129 00:08:15,060 --> 00:08:18,710 and this Rabi frequency is usually referred to as the one 130 00:08:18,710 --> 00:08:21,860 photon Rabi frequency because we obtained the Rabi 131 00:08:21,860 --> 00:08:24,889 frequency by calculating the electric field 132 00:08:24,889 --> 00:08:25,680 of a single photon. 133 00:08:29,200 --> 00:08:34,140 So the Rabi oscillations which we are observing now 134 00:08:34,140 --> 00:08:40,130 correspond to the periodic spontaneous emission 135 00:08:40,130 --> 00:08:43,370 and re-absorption of the same photon. 136 00:08:43,370 --> 00:08:46,430 There's only one photon which is spontaneously 137 00:08:46,430 --> 00:08:51,450 emitted and reabsorbed in a completely reversible 138 00:08:51,450 --> 00:08:54,560 coherent way, and the time evolution is unitary. 139 00:09:12,350 --> 00:09:22,390 So it's a periodic spontaneous emission 140 00:09:22,390 --> 00:09:25,180 and re-absorption of the same photon. 141 00:09:38,630 --> 00:09:40,595 This has been experimentally observed. 142 00:09:43,220 --> 00:09:45,280 Actually, let me back up. 143 00:09:45,280 --> 00:09:51,875 Experiments are done in the microwave regime. 144 00:09:55,320 --> 00:09:58,460 The leading groups are, well, in the older days, 145 00:09:58,460 --> 00:10:00,030 Dan Kleppner, Herbert [? Weidner, ?] 146 00:10:00,030 --> 00:10:02,000 and Serge Haroche. 147 00:10:02,000 --> 00:10:04,830 And this involves Rydberg atoms. 148 00:10:04,830 --> 00:10:13,685 Rydberg atoms in superconducting high q cavities. 149 00:10:17,550 --> 00:10:20,150 And those Rydberg atoms, because things 150 00:10:20,150 --> 00:10:22,910 scale with n and n squared, the principal quantum number, 151 00:10:22,910 --> 00:10:24,860 have a fantastically strong coupling 152 00:10:24,860 --> 00:10:27,120 to the electromagnetic field. 153 00:10:27,120 --> 00:10:29,130 And there is a homework assignment 154 00:10:29,130 --> 00:10:32,580 on Rydberg atoms in such cavities. 155 00:10:32,580 --> 00:10:35,660 The other example is in the optical domain. 156 00:10:43,970 --> 00:10:48,470 And this really involves the D line of alkali atoms. 157 00:10:48,470 --> 00:10:50,490 You drive them on the D line. 158 00:10:50,490 --> 00:10:53,470 rubidium and caesium are often used, 159 00:10:53,470 --> 00:10:56,490 and the work is enabled by the development 160 00:10:56,490 --> 00:10:59,700 of so-called supermirrors which have an extremely 161 00:10:59,700 --> 00:11:05,360 high reflectivity, and you can realize an excellent q factor. 162 00:11:05,360 --> 00:11:09,560 And the leaders in this field are Jeff Kimble and Gerhard 163 00:11:09,560 --> 00:11:12,720 Rempe at the Max Planck Institute. 164 00:11:12,720 --> 00:11:17,230 So let me just discuss an example 165 00:11:17,230 --> 00:11:19,660 taken from the optical domain. 166 00:11:19,660 --> 00:11:28,130 So the generic situation is that you 167 00:11:28,130 --> 00:11:35,410 have two mirrors which define a single mode cavity. 168 00:11:35,410 --> 00:11:40,120 Usually, you have a stream of atoms. 169 00:11:40,120 --> 00:11:42,390 Traditionally in atomic beams, then 170 00:11:42,390 --> 00:11:46,230 in some experiments in slowed atomic beams, more recently 171 00:11:46,230 --> 00:11:49,260 in atoms which are falling out off the mode, 172 00:11:49,260 --> 00:11:52,870 and only very recently single atoms with the help 173 00:11:52,870 --> 00:11:56,550 of other laser beams that are trapped inside the cavity. 174 00:11:56,550 --> 00:11:58,590 So they are streamed in such a way 175 00:11:58,590 --> 00:12:03,630 that only one or a few atoms are in the mode volume interact 176 00:12:03,630 --> 00:12:06,800 with a single mode of the cavity at a given time. 177 00:12:06,800 --> 00:12:12,270 And then you want to figure out what is now happening, 178 00:12:12,270 --> 00:12:17,790 and you have the probe laser, you send it through the cavity, 179 00:12:17,790 --> 00:12:23,260 and then you record the transmission with a photodiode. 180 00:12:23,260 --> 00:12:23,982 Yanosh? 181 00:12:23,982 --> 00:12:28,320 AUDIENCE: What is the mirror made of for the [INAUDIBLE]? 182 00:12:28,320 --> 00:12:31,320 PROFESSOR: The mirror is made of a glass substrate, 183 00:12:31,320 --> 00:12:33,500 but then you would [INAUDIBLE] the coating. 184 00:12:33,500 --> 00:12:36,110 And the mastering is really to put coatings 185 00:12:36,110 --> 00:12:39,310 on which are very pure, but then also I 186 00:12:39,310 --> 00:12:43,150 think using ion sputtering, you make 187 00:12:43,150 --> 00:12:45,750 sure that the coating is extremely smooth 188 00:12:45,750 --> 00:12:48,930 and does not have any surface irregularities which 189 00:12:48,930 --> 00:12:52,130 would scatter a tiny fraction of the light. 190 00:12:52,130 --> 00:12:55,030 I know there are some people in Ike's group and [INAUDIBLE] 191 00:12:55,030 --> 00:12:56,760 group who work with high q mirrors. 192 00:12:56,760 --> 00:13:00,280 What is a typical example for the reflectivity? 193 00:13:00,280 --> 00:13:02,146 Or the q factor you can reach? 194 00:13:02,146 --> 00:13:07,006 AUDIENCE: 5 [INAUDIBLE] and a finesse of, maybe, 500,000. 195 00:13:07,006 --> 00:13:09,440 And they're called superpolished mirrors. 196 00:13:09,440 --> 00:13:12,110 PROFESSOR: So finesse of about a million, and that 197 00:13:12,110 --> 00:13:16,480 means the mirrors have 99.9999% reflectivity. 198 00:13:16,480 --> 00:13:20,140 And the superpolishing, I think, that was the last step. 199 00:13:20,140 --> 00:13:22,100 People had controlled the materials, 200 00:13:22,100 --> 00:13:24,810 but then they found ways to make a super polish 201 00:13:24,810 --> 00:13:27,830 and avoid these even one part per million 202 00:13:27,830 --> 00:13:31,184 scattering by surface roughness. 203 00:13:33,741 --> 00:13:34,240 OK. 204 00:13:34,240 --> 00:13:37,590 So if you do that experiment, what would you expect? 205 00:13:37,590 --> 00:13:40,250 Well, it's a [INAUDIBLE] experiment 206 00:13:40,250 --> 00:13:46,440 so if you would scan the probe laser, 207 00:13:46,440 --> 00:13:48,580 and there is nothing in it, what you 208 00:13:48,580 --> 00:13:54,610 would expect is you would just expect a transmission 209 00:13:54,610 --> 00:13:57,180 peak at the cavity resonance. 210 00:13:57,180 --> 00:13:59,270 And if you tune much further, you 211 00:13:59,270 --> 00:14:03,030 get the next peak at the free spectral range. 212 00:14:03,030 --> 00:14:05,400 Let me just indicate that. 213 00:14:05,400 --> 00:14:09,230 So this is a case for 0 atoms in the cavity. 214 00:14:09,230 --> 00:14:11,820 If you put 1 photon in the cavity, 215 00:14:11,820 --> 00:14:14,770 you no longer-- sorry, 1 atom in the cavity, 216 00:14:14,770 --> 00:14:17,620 you're no longer probing a cavity, 217 00:14:17,620 --> 00:14:19,790 you're really probing a system, which 218 00:14:19,790 --> 00:14:21,510 is no longer the cavity by itself. 219 00:14:21,510 --> 00:14:23,390 It's an atom-photon system. 220 00:14:23,390 --> 00:14:25,110 It's a couple system. 221 00:14:25,110 --> 00:14:28,580 And we know it's described by our two-by-two Hamiltonian, 222 00:14:28,580 --> 00:14:32,060 and this Hamiltonian has two solutions. 223 00:14:32,060 --> 00:14:36,670 And the two solutions are split by the one photon Rabi 224 00:14:36,670 --> 00:14:37,770 frequency. 225 00:14:37,770 --> 00:14:40,550 So the two eigenvalues of our Hamiltonian 226 00:14:40,550 --> 00:14:46,100 are at plus minus omega 1 photon. 227 00:14:46,100 --> 00:14:50,730 So therefore, for n equals 1, we have a situation 228 00:14:50,730 --> 00:14:56,520 that we have two peaks split by the single photon Rabi 229 00:14:56,520 --> 00:14:58,190 frequency. 230 00:14:58,190 --> 00:15:00,540 Of course, I have assumed that great care has 231 00:15:00,540 --> 00:15:04,570 been spent to make sure that the cavity resonance is right 232 00:15:04,570 --> 00:15:07,290 where the atomic resonance is. 233 00:15:07,290 --> 00:15:10,370 So this is now for 1. 234 00:15:10,370 --> 00:15:15,840 If you have 10 atoms, remember the two-by-two Hamiltonian 235 00:15:15,840 --> 00:15:19,920 looks the same, but it has the square root n plus 1 factor. 236 00:15:19,920 --> 00:15:24,150 So neglecting the 1 roughly when we have 10 atoms in the cavity, 237 00:15:24,150 --> 00:15:27,980 it's square root 10 larger Rabi frequency. 238 00:15:27,980 --> 00:15:30,680 And therefore, we would expect that we 239 00:15:30,680 --> 00:15:34,390 have now a splitting of the two modes, which 240 00:15:34,390 --> 00:15:37,110 is square root 10 plus 1 larger. 241 00:15:43,110 --> 00:15:46,220 Actually, I didn't-- sorry, I have to collect myself now. 242 00:15:50,538 --> 00:15:54,200 I showed you that the Rabi frequency 243 00:15:54,200 --> 00:15:58,670 scales with square root n plus 1 in the photon field. 244 00:15:58,670 --> 00:16:00,880 But you should realize that everything 245 00:16:00,880 --> 00:16:02,950 is [? isometric ?] between photons and atoms. 246 00:16:02,950 --> 00:16:06,180 It's the complete coupling between photons and atoms. 247 00:16:06,180 --> 00:16:10,110 And if you would now look-- but I don't want to do it now-- 248 00:16:10,110 --> 00:16:13,530 if you would now look what happens if several atoms are 249 00:16:13,530 --> 00:16:16,000 present in the mode volume, you would also 250 00:16:16,000 --> 00:16:21,450 get a scaling which is n plus 1 in the atom number, 251 00:16:21,450 --> 00:16:23,840 because the atom coupled coherently. 252 00:16:23,840 --> 00:16:25,910 It is actually an effect of super-radiance, 253 00:16:25,910 --> 00:16:27,380 which we'll discussed later. 254 00:16:27,380 --> 00:16:29,250 So just take my word. 255 00:16:29,250 --> 00:16:32,170 You have the same scaling with the atom number. 256 00:16:32,170 --> 00:16:35,870 But I have to give you my word now, because in the experiment 257 00:16:35,870 --> 00:16:37,005 this is what people varied. 258 00:16:40,810 --> 00:16:44,280 AUDIENCE: If they had varied power or number of photons 259 00:16:44,280 --> 00:16:47,380 instead, like, we couldn't have drawn these same diagrams, 260 00:16:47,380 --> 00:16:47,880 right? 261 00:16:47,880 --> 00:16:52,906 Because then the top part of the Lorentzian changes. 262 00:16:52,906 --> 00:16:57,840 If you're changing the photon number then Lorentzians change. 263 00:16:57,840 --> 00:16:58,790 PROFESSOR: Say again? 264 00:16:58,790 --> 00:17:00,920 AUDIENCE: So, like, right now, yes, we 265 00:17:00,920 --> 00:17:02,760 are varying the number of atoms so we 266 00:17:02,760 --> 00:17:04,930 can talk about the splitting. 267 00:17:04,930 --> 00:17:07,773 But if we were varying the power or the number of photons 268 00:17:07,773 --> 00:17:09,942 instead, then each of the Lorentzians, 269 00:17:09,942 --> 00:17:11,250 their height would change. 270 00:17:15,839 --> 00:17:17,806 How would you draw this observation 271 00:17:17,806 --> 00:17:19,574 if you were changing the photon numbers? 272 00:17:19,574 --> 00:17:21,960 PROFESSOR: You know, I don't want to go into line shape. 273 00:17:21,960 --> 00:17:24,410 I would probably be a Lorentzian. 274 00:17:24,410 --> 00:17:25,829 I mean, all I want to discuss here 275 00:17:25,829 --> 00:17:29,880 is that we have a two-by-two Hamiltonian, which is split. 276 00:17:29,880 --> 00:17:35,380 And if we have one atom and one photon, 277 00:17:35,380 --> 00:17:38,270 it is split by the single photon Rabi frequency. 278 00:17:38,270 --> 00:17:43,610 If we have one atom and 10 photons, 279 00:17:43,610 --> 00:17:48,560 the atom can of course absorb and emit only one. 280 00:17:48,560 --> 00:17:50,820 As I derived on the previous page, 281 00:17:50,820 --> 00:17:53,650 we would have now a Rabi splitting, which is square root 282 00:17:53,650 --> 00:17:57,060 n plus 1, n being the number of photons. 283 00:17:57,060 --> 00:18:00,290 But if you would start in an empty cavity 284 00:18:00,290 --> 00:18:03,240 with 10 atoms in the excited state, 285 00:18:03,240 --> 00:18:05,600 because all the atoms are identical, 286 00:18:05,600 --> 00:18:08,410 they would spontaneously emit together, 287 00:18:08,410 --> 00:18:10,880 and then you would have 10 atoms in the ground state, 288 00:18:10,880 --> 00:18:13,050 and then you would have 10 photons. 289 00:18:13,050 --> 00:18:14,690 And so maybe this helps you. 290 00:18:14,690 --> 00:18:17,230 If you start with 10 atoms in the excited state, 291 00:18:17,230 --> 00:18:18,520 they do everything together. 292 00:18:18,520 --> 00:18:21,240 If you have 10 atoms in the ground state with 10 photons, 293 00:18:21,240 --> 00:18:23,380 and now you have 10 photons and it's clear 294 00:18:23,380 --> 00:18:25,880 the 10 photons lead to a Rabi frequency, 295 00:18:25,880 --> 00:18:28,760 which is proportional to the square root of 10 296 00:18:28,760 --> 00:18:31,000 or to the square root of 11. 297 00:18:31,000 --> 00:18:34,640 So therefore, what you will observe 298 00:18:34,640 --> 00:18:38,520 is you will now observe a splitting of the single mode 299 00:18:38,520 --> 00:18:44,329 of the cavity which goes by the square root of n plus 1. 300 00:18:44,329 --> 00:18:46,870 I don't want to discuss the line shape and the [? strings, ?] 301 00:18:46,870 --> 00:18:51,010 I just want to sort of discuss, in a way, the eigenvalues 302 00:18:51,010 --> 00:18:53,660 of the Hamiltonian, and the eigenvalues 303 00:18:53,660 --> 00:18:55,662 are the positions of the transmission 304 00:18:55,662 --> 00:18:56,495 peaks with a cavity. 305 00:18:59,720 --> 00:19:02,605 And that has been observed. 306 00:19:07,460 --> 00:19:09,490 I mentioned the two leaders of the field 307 00:19:09,490 --> 00:19:11,030 are Gerhard Rempe and Jeff Kimble. 308 00:19:11,030 --> 00:19:16,090 Well, Gerhard Rempe, he did his Ph.D. 309 00:19:16,090 --> 00:19:17,680 In the same group at the same time 310 00:19:17,680 --> 00:19:19,560 as I did so I know him very well. 311 00:19:19,560 --> 00:19:21,964 Then he went and did post doc work with Jeff Kimble, 312 00:19:21,964 --> 00:19:24,130 and now is the Director of the Max Planck Institute. 313 00:19:24,130 --> 00:19:27,120 He has the world leading group in cavity QED. 314 00:19:27,120 --> 00:19:29,710 But this is sort of here the two leaders have 315 00:19:29,710 --> 00:19:35,590 a joint paper, which is the first observation of the vacuum 316 00:19:35,590 --> 00:19:38,090 Rabi splitting in an optical cavity. 317 00:19:38,090 --> 00:19:41,970 Of course, you can easily observe it 318 00:19:41,970 --> 00:19:44,960 if you have a strong atomic bean with many atoms, 319 00:19:44,960 --> 00:19:46,900 because then you have a good signal. 320 00:19:46,900 --> 00:19:51,390 And secondly, the splitting is large and easily resolved. 321 00:19:51,390 --> 00:19:55,120 So what they managed to do is they managed to throttle down 322 00:19:55,120 --> 00:19:59,470 the atomic beam that fewer and fewer atoms at the given time 323 00:19:59,470 --> 00:20:01,190 were in the cavity. 324 00:20:01,190 --> 00:20:07,970 And eventually they came down to the limit of one atom. 325 00:20:07,970 --> 00:20:10,380 That was an historic experiment. 326 00:20:10,380 --> 00:20:13,870 Of course, it's not perfect in the sense 327 00:20:13,870 --> 00:20:18,640 that you do not see the deep cut between the two peaks simply 328 00:20:18,640 --> 00:20:21,910 because, when on average you have one atom in the cavity, 329 00:20:21,910 --> 00:20:24,150 sometimes you have to atom in the cavity, 330 00:20:24,150 --> 00:20:27,520 and then you have a peak in the middle. 331 00:20:27,520 --> 00:20:29,590 So those experiments in those days 332 00:20:29,590 --> 00:20:31,900 were done only with average atom numbers 333 00:20:31,900 --> 00:20:35,440 and not with trapped atoms where you know for sure there's 334 00:20:35,440 --> 00:20:37,044 exactly one atom in the cavity. 335 00:20:51,701 --> 00:20:52,200 OK. 336 00:20:58,810 --> 00:21:02,500 So I don't show you an experiment, 337 00:21:02,500 --> 00:21:11,090 but let me just state that this sort of single photon Rabi 338 00:21:11,090 --> 00:21:15,115 flopping has been observed. 339 00:21:24,360 --> 00:21:28,340 You start with the cavity in the vacuum field, 340 00:21:28,340 --> 00:21:31,010 and you sort of see this oscillation 341 00:21:31,010 --> 00:21:34,140 to the ground state with one photon. 342 00:21:34,140 --> 00:21:40,170 But what I want to discuss now is the situation 343 00:21:40,170 --> 00:21:44,500 that we are not starting with an empty cavity. 344 00:21:44,500 --> 00:21:48,140 We are starting with a coherent field. 345 00:21:48,140 --> 00:21:50,880 You can also start with a thermal field 346 00:21:50,880 --> 00:21:54,820 so there are different experiments you can do. 347 00:21:54,820 --> 00:21:56,260 What would we expect now? 348 00:22:00,430 --> 00:22:12,370 So now the initial photon state is not 349 00:22:12,370 --> 00:22:20,490 the vacuum state, but the thermal state. 350 00:22:20,490 --> 00:22:22,660 If you have a microwave cavity and you heat it 351 00:22:22,660 --> 00:22:27,340 up a little bit, you have to cool it down to below 1 Kelvin. 352 00:22:27,340 --> 00:22:30,970 People use either helium-free chrio stats or dilution 353 00:22:30,970 --> 00:22:33,640 refrigerators, but if you warm it up a little bit, 354 00:22:33,640 --> 00:22:36,180 you have a few microwave photons in the cavity. 355 00:22:36,180 --> 00:22:39,780 Or, that's even more controlled, you can make the cavity ice 356 00:22:39,780 --> 00:22:43,700 cold, but then you inject a few photons from your synthesizer 357 00:22:43,700 --> 00:22:45,690 into it-- from your microwave generator-- 358 00:22:45,690 --> 00:22:48,680 and then you have a weak coherent field. 359 00:22:48,680 --> 00:22:51,470 But a thermal state or a coherent state. 360 00:23:13,300 --> 00:23:16,780 So what we then have is, OK, we would expect now 361 00:23:16,780 --> 00:23:23,290 a Rabi oscillation; however, the frequency for the Rabi flopping 362 00:23:23,290 --> 00:23:27,330 is now proportional to n plus 1. 363 00:23:27,330 --> 00:23:30,240 And we have our photon field in the superposition 364 00:23:30,240 --> 00:23:31,280 of flux states. 365 00:23:36,220 --> 00:23:45,410 So the fact that we have a superposition state implies now 366 00:23:45,410 --> 00:23:58,540 that the Rabi oscillations have a different oscillation 367 00:23:58,540 --> 00:24:07,970 frequency for the different [? tablets ?] of states labeled 368 00:24:07,970 --> 00:24:09,720 by n. 369 00:24:09,720 --> 00:24:11,690 And that leads to a dephasing. 370 00:24:14,790 --> 00:24:18,685 So that would mean that if you would look at the probability 371 00:24:18,685 --> 00:24:23,140 to be in the excited state-- just think about it. 372 00:24:23,140 --> 00:24:26,210 You have a wave function where the atom starts in the excited 373 00:24:26,210 --> 00:24:29,520 state, and the photon field is in a superposition. 374 00:24:29,520 --> 00:24:32,720 So now you have a two-component wave function 375 00:24:32,720 --> 00:24:35,940 which has different parts, and each part 376 00:24:35,940 --> 00:24:39,480 has a specific Rabi frequency. 377 00:24:39,480 --> 00:24:43,944 So you would have oscillations. 378 00:24:43,944 --> 00:24:45,610 Let's say there is a certain probability 379 00:24:45,610 --> 00:24:47,600 that the cavity is in the vacuum, 380 00:24:47,600 --> 00:24:51,080 and then that means that there is a component which oscillates 381 00:24:51,080 --> 00:24:53,980 at the vacuum Rabi oscillation frequency. 382 00:24:53,980 --> 00:24:57,890 But if you have a component in your coherent 383 00:24:57,890 --> 00:25:01,020 or thermal state which has two photons in it, 384 00:25:01,020 --> 00:25:06,690 then you have Rabi oscillations which are faster. 385 00:25:06,690 --> 00:25:09,370 And now you have to superimpose them all. 386 00:25:09,370 --> 00:25:11,580 And if you all superimpose them, and you 387 00:25:11,580 --> 00:25:16,290 find that very soon there is a damping and maybe 388 00:25:16,290 --> 00:25:18,360 a little bit of vigor, but you see 389 00:25:18,360 --> 00:25:27,403 at damping of the population in the excited state. 390 00:25:30,520 --> 00:25:32,790 Q [? 2 ?] dephasing. 391 00:25:32,790 --> 00:25:33,720 I'm just hesitating. 392 00:25:33,720 --> 00:25:36,290 I think I took this plot out of my notes, 393 00:25:36,290 --> 00:25:39,050 but I would expect now the damping should actually 394 00:25:39,050 --> 00:25:43,040 lead to a probability to be in the excited state of 1/2. 395 00:25:43,040 --> 00:25:46,720 So let me just try to correct that. 396 00:25:49,330 --> 00:25:51,576 So there is a little bit of oscillation, 397 00:25:51,576 --> 00:25:53,075 but then there is sort of a damping. 398 00:25:58,070 --> 00:26:12,710 And eventually, if you have only a small number 399 00:26:12,710 --> 00:26:15,060 of photon states, then there will 400 00:26:15,060 --> 00:26:18,850 be a time where you have sort of at least 401 00:26:18,850 --> 00:26:20,930 a partial commensurability. 402 00:26:20,930 --> 00:26:22,580 You have maybe five frequencies. 403 00:26:22,580 --> 00:26:25,690 You know, square root 5, square root 4, square root 3, 404 00:26:25,690 --> 00:26:26,870 square root 2. 405 00:26:26,870 --> 00:26:29,020 But then there is sort of a time where 406 00:26:29,020 --> 00:26:32,050 all these different frequencies have done an integer 407 00:26:32,050 --> 00:26:34,430 number of oscillations each, and then you 408 00:26:34,430 --> 00:26:35,800 get what is called a revival. 409 00:26:39,470 --> 00:26:43,430 And if you go to a large photon number, 410 00:26:43,430 --> 00:26:47,120 you have square root 100, square root 99, square root 88, 411 00:26:47,120 --> 00:26:49,930 the revival will happen at a later and later time 412 00:26:49,930 --> 00:26:54,400 and eventually at infinite times if you use a microscopic field. 413 00:26:54,400 --> 00:26:59,030 But for small coherent states, or thermal states, 414 00:26:59,030 --> 00:27:01,300 which only involve a few photons, 415 00:27:01,300 --> 00:27:02,970 you will get a revival phenomenon. 416 00:27:08,070 --> 00:27:12,890 And this has indeed been observed. 417 00:27:17,920 --> 00:27:23,660 This was actually the PhD thesis of Gerhard Rempe, 418 00:27:23,660 --> 00:27:28,930 and it shows the probability in the excited state. 419 00:27:28,930 --> 00:27:32,420 They had previously observed the Rabi oscillations 420 00:27:32,420 --> 00:27:35,880 at early times, but now the experiment 421 00:27:35,880 --> 00:27:39,620 had to be adjusted, I think by using slower atoms, 422 00:27:39,620 --> 00:27:41,660 to observe the longer time. 423 00:27:41,660 --> 00:27:46,800 And here, well, 1987 for the first time revivals 424 00:27:46,800 --> 00:27:47,735 have been seen. 425 00:27:54,500 --> 00:27:57,110 Let me dwell on that, or first are there 426 00:27:57,110 --> 00:28:01,610 any questions about what happens now? 427 00:28:01,610 --> 00:28:05,770 Atoms in the cavity to Rabi oscillations? 428 00:28:05,770 --> 00:28:09,756 And if the photon field is a superposition 429 00:28:09,756 --> 00:28:15,510 of only a few states due to this pseudo commensurability, 430 00:28:15,510 --> 00:28:17,400 you find times where you have revivals. 431 00:28:22,530 --> 00:28:26,620 I just worked out something this morning 432 00:28:26,620 --> 00:28:30,310 which I think is nice, because it will highlight 433 00:28:30,310 --> 00:28:33,330 how you should think about spontaneous emission. 434 00:28:33,330 --> 00:28:34,615 So let me discuss. 435 00:28:38,020 --> 00:28:39,520 It doesn't really matter, but I want 436 00:28:39,520 --> 00:28:45,260 to give you a specific example that we have a coherent state. 437 00:28:45,260 --> 00:28:47,770 A lot of you know what a coherent photon state is. 438 00:28:47,770 --> 00:28:49,580 For those who don't, it doesn't really 439 00:28:49,580 --> 00:28:51,810 matter for what I want to explain, and recover 440 00:28:51,810 --> 00:28:53,680 that in [? 8.4.22. ?] 441 00:28:53,680 --> 00:28:58,970 But if you have a laser or if you have a microwave generator, 442 00:28:58,970 --> 00:29:04,270 what comes out is a field which has a normalized amplitude 443 00:29:04,270 --> 00:29:09,980 of alpha, but your field is in a superposition 444 00:29:09,980 --> 00:29:11,135 state or [? flux ?] states. 445 00:29:16,430 --> 00:29:20,480 With these prefactors, I just wanted to give you an example. 446 00:29:20,480 --> 00:29:22,740 What I really just mean is that we 447 00:29:22,740 --> 00:29:26,370 have a coherent superposition of number states. 448 00:29:26,370 --> 00:29:28,710 We have prepared that. 449 00:29:28,710 --> 00:29:32,630 So now we have one atom in the excited state, 450 00:29:32,630 --> 00:29:36,520 it enters the cavity which has been prepared 451 00:29:36,520 --> 00:29:39,820 with the short pulse for a laser or microwave 452 00:29:39,820 --> 00:29:42,710 synthesizer in these state alpha. 453 00:29:46,610 --> 00:29:53,970 And now we want to discuss-- so this is at t equals 0-- 454 00:29:53,970 --> 00:29:58,950 and now I want to discuss what happens as a function of time. 455 00:29:58,950 --> 00:30:06,450 Well, we know that if you have one tablet, 456 00:30:06,450 --> 00:30:16,000 n, we have Rabi oscillations between the atoms 457 00:30:16,000 --> 00:30:19,630 in the excited state, and we have n photons. 458 00:30:19,630 --> 00:30:22,640 Or it has emitted the photon, and then we 459 00:30:22,640 --> 00:30:24,945 have n plus 1 photon and the cavity. 460 00:30:36,140 --> 00:30:45,600 But now, we have a superposition state, 461 00:30:45,600 --> 00:30:47,835 and we have amplitudes an. 462 00:30:58,140 --> 00:31:01,100 So I mean, that's what we get. 463 00:31:01,100 --> 00:31:03,340 And this includes everything. 464 00:31:03,340 --> 00:31:06,420 It includes everything a two-level atom 465 00:31:06,420 --> 00:31:08,860 does in a single mode of a cavity. 466 00:31:08,860 --> 00:31:12,150 And this is spontaneous emission, stimulated emission, 467 00:31:12,150 --> 00:31:13,980 and reabsorption. 468 00:31:13,980 --> 00:31:18,650 But I want to use that now to discuss 469 00:31:18,650 --> 00:31:24,425 with you the misconceptions about spontaneous emission. 470 00:31:27,545 --> 00:31:28,045 Colin? 471 00:31:28,045 --> 00:31:29,824 AUDIENCE: We're talking about just spontaneous emissions 472 00:31:29,824 --> 00:31:30,800 into the cavity? 473 00:31:34,587 --> 00:31:35,170 PROFESSOR: OK. 474 00:31:35,170 --> 00:31:37,850 I've singled out a single mode. 475 00:31:37,850 --> 00:31:42,980 But what happens is-- and you're just two minutes, 30 seconds, 476 00:31:42,980 --> 00:31:47,800 ahead of me-- that we had discussed vacuum Rabi 477 00:31:47,800 --> 00:31:49,760 oscillations or Rabi oscillations when 478 00:31:49,760 --> 00:31:51,720 we have n photons in the cavity. 479 00:31:51,720 --> 00:31:55,020 This was our two-level system, our Hamiltonian, and all 480 00:31:55,020 --> 00:31:59,340 we get is Rabi oscillations with the Rabi frequency omega n. 481 00:31:59,340 --> 00:32:02,180 And now we have to sort of do averaging. 482 00:32:02,180 --> 00:32:06,070 I'm now discussing that we have a coherent superposition 483 00:32:06,070 --> 00:32:07,610 of number states. 484 00:32:07,610 --> 00:32:10,920 Let's say, a pulse of coherent radiation, a coherent state, 485 00:32:10,920 --> 00:32:13,260 and this is what we get. 486 00:32:13,260 --> 00:32:18,370 You can now, if you want, put in a [? zillion ?] of other modes, 487 00:32:18,370 --> 00:32:21,370 have another sum over all the other modes you want. 488 00:32:21,370 --> 00:32:24,020 So I'm just doing the first step in discussing 489 00:32:24,020 --> 00:32:28,090 with you what will happen, but adding more and more 490 00:32:28,090 --> 00:32:31,270 modes will actually not change the structure of the answer 491 00:32:31,270 --> 00:32:34,650 and will be, of course, quantitatively a mess 492 00:32:34,650 --> 00:32:38,180 but conceptually not more complicated. 493 00:32:38,180 --> 00:32:40,100 So I want you to really look at that 494 00:32:40,100 --> 00:32:43,100 and realize where is the spontaneity 495 00:32:43,100 --> 00:32:46,120 of spontaneous emission. 496 00:32:46,120 --> 00:32:49,681 Where do you see any form of randomness associated 497 00:32:49,681 --> 00:32:51,555 with spontaneous emission in this expression? 498 00:33:02,000 --> 00:33:03,530 I don't see it. 499 00:33:03,530 --> 00:33:08,140 This is a wave function, and this time evolution is unitary. 500 00:33:13,780 --> 00:33:19,680 Everything is deterministic, and depending now 501 00:33:19,680 --> 00:33:23,060 how we choose our coefficient, there is even a revival. 502 00:33:23,060 --> 00:33:25,340 It's not dissipative that a photon is spontaneously 503 00:33:25,340 --> 00:33:26,660 emitted, and it's done. 504 00:33:26,660 --> 00:33:28,495 We saw in the single photon Rabi oscillation 505 00:33:28,495 --> 00:33:32,450 it can be reabsorbed, we saw in a slightly more complicated 506 00:33:32,450 --> 00:33:36,550 situation that there are at least partial revivals, 507 00:33:36,550 --> 00:33:39,680 and it now depends how long we wait 508 00:33:39,680 --> 00:33:41,920 whether revivals will take place or whether they 509 00:33:41,920 --> 00:33:44,550 will be complete revivals or partial revivals. 510 00:33:44,550 --> 00:33:48,520 But we don't need a revival in a coherent evolution, 511 00:33:48,520 --> 00:33:51,190 the coherent evolution can just go to a complicated wave 512 00:33:51,190 --> 00:33:55,620 function and it's still a single coherent wave function 513 00:33:55,620 --> 00:34:02,100 fully deterministically obtained form the Hamilton operator. 514 00:34:02,100 --> 00:34:06,180 Sometimes it pops into our eyes through a reversible 515 00:34:06,180 --> 00:34:09,321 oscillation or through revival, but we don't need that. 516 00:34:19,750 --> 00:34:23,570 So let me write it down but then explain you something. 517 00:34:23,570 --> 00:34:26,719 So it's unitary. 518 00:34:26,719 --> 00:34:31,820 There is no spontaneity at all. 519 00:34:35,000 --> 00:34:50,500 However, eventually we want to retrieve the classical limit. 520 00:34:50,500 --> 00:34:53,980 So if we would go to this situation 521 00:34:53,980 --> 00:34:58,560 that the average photon number is much smaller than 1, 522 00:34:58,560 --> 00:35:06,550 then the fluctuation in the photon field 523 00:35:06,550 --> 00:35:08,905 around the mean number are very small. 524 00:35:08,905 --> 00:35:13,520 For the coherent state the fluctuations are square root n. 525 00:35:13,520 --> 00:35:23,880 And then, we retrieve the limit of semiclassical Rabi 526 00:35:23,880 --> 00:35:30,700 flopping with the Rabi frequency omega 527 00:35:30,700 --> 00:35:41,310 r, which is-- I'm not consistent here with lower and uppercase 528 00:35:41,310 --> 00:35:41,860 [INAUDIBLE]. 529 00:35:41,860 --> 00:35:45,070 So it's uppercase or lowercase omega n, 530 00:35:45,070 --> 00:35:48,980 and this is square root n times the single photon Rabi 531 00:35:48,980 --> 00:35:49,970 frequency. 532 00:35:49,970 --> 00:35:54,270 And, of course, for a large number of photons, 533 00:35:54,270 --> 00:35:59,654 we can always make the approximation 534 00:35:59,654 --> 00:36:01,070 that we do not have to distinguish 535 00:36:01,070 --> 00:36:02,285 between n and n plus 1. 536 00:36:06,230 --> 00:36:09,200 So this is the ultimate limit if we 537 00:36:09,200 --> 00:36:12,580 would work in the limit of large photon numbers. 538 00:36:12,580 --> 00:36:16,280 So the way how you should look at it is the following. 539 00:36:16,280 --> 00:36:19,810 This system undergoes a time evolution 540 00:36:19,810 --> 00:36:22,360 to a state which is rather complicated. 541 00:36:22,360 --> 00:36:25,990 But if you make the number n large, 542 00:36:25,990 --> 00:36:31,450 this becomes approximately a state 543 00:36:31,450 --> 00:36:33,340 where you have simply-- you know what 544 00:36:33,340 --> 00:36:34,840 the rate of the semiclassical limit? 545 00:36:34,840 --> 00:36:37,680 In the semiclassical limit, we have a constant laser beam 546 00:36:37,680 --> 00:36:40,090 with constant electric field amplitude, e, 547 00:36:40,090 --> 00:36:42,130 and then we have driven Rabi oscillations 548 00:36:42,130 --> 00:36:45,800 between ground and excited state. 549 00:36:45,800 --> 00:36:48,900 So therefore, I don't want to show you mathematically, 550 00:36:48,900 --> 00:36:52,770 but in the limit of large n, you can approximate 551 00:36:52,770 --> 00:36:55,570 this complicated entangled wave function 552 00:36:55,570 --> 00:36:58,470 by the product of Rabi oscillations 553 00:36:58,470 --> 00:37:03,180 between ground and excited state times a coherent photon field. 554 00:37:07,040 --> 00:37:11,240 And the correction between what I just said 555 00:37:11,240 --> 00:37:17,470 and this complicated wave function is like 1 over n, 556 00:37:17,470 --> 00:37:20,900 because it's sort of a 1 over n approximation 557 00:37:20,900 --> 00:37:22,810 where we have neglected terms which 558 00:37:22,810 --> 00:37:27,040 the relative importance of them is 1 over n. 559 00:37:27,040 --> 00:37:31,000 So therefore, there are people who will say 560 00:37:31,000 --> 00:37:33,380 and who will tell you when we have 561 00:37:33,380 --> 00:37:36,470 an interaction of an atom with a coherent state, 562 00:37:36,470 --> 00:37:41,010 and let's just think in the number of n being large, 563 00:37:41,010 --> 00:37:46,260 that n times out of n plus 1, we have a coherent state. 564 00:37:46,260 --> 00:37:48,110 The atom does Rabi oscillation and what 565 00:37:48,110 --> 00:37:51,410 it does is it just emits photon into the coherent field 566 00:37:51,410 --> 00:37:54,760 and takes it back, like in semiclassical physics. 567 00:37:54,760 --> 00:38:00,030 But in one case out of n cases, or the rate 568 00:38:00,030 --> 00:38:03,302 1 over n of the rate of the wave function is sort of fuzzy. 569 00:38:03,302 --> 00:38:05,010 It's not a coherent state; it's something 570 00:38:05,010 --> 00:38:06,840 much more complicated. 571 00:38:06,840 --> 00:38:11,160 And if you do not keep track of this complicated nature 572 00:38:11,160 --> 00:38:15,192 of the wave function and just do some simple measurement by, 573 00:38:15,192 --> 00:38:16,650 let's say, just measuring the phase 574 00:38:16,650 --> 00:38:19,195 of the electromagnetic wave by projecting 575 00:38:19,195 --> 00:38:21,970 onto a coherent state, then you would 576 00:38:21,970 --> 00:38:26,750 find that with the probability of n 577 00:38:26,750 --> 00:38:29,590 the system was just staying in a coherent state. 578 00:38:29,590 --> 00:38:32,770 And with a probability which is one part out of n, 579 00:38:32,770 --> 00:38:37,315 something else has happened, and your detector cannot capture 580 00:38:37,315 --> 00:38:39,680 the entanglement of that state. 581 00:38:39,680 --> 00:38:42,550 And this last part is what some people 582 00:38:42,550 --> 00:38:44,370 associate with spontaneous emission. 583 00:38:47,050 --> 00:38:48,080 I don't know. 584 00:38:48,080 --> 00:38:51,920 That's my view where the spontaneity in this process is. 585 00:38:51,920 --> 00:38:54,650 It's not a spontaneity in the time evolution. 586 00:38:54,650 --> 00:38:57,290 It's more a spontaneity if you do not 587 00:38:57,290 --> 00:39:00,820 care to detect this complexity, but map it back 588 00:39:00,820 --> 00:39:02,500 to a coherent state. 589 00:39:02,500 --> 00:39:05,880 And then with a precision which is 1 over n, 590 00:39:05,880 --> 00:39:08,530 you retrieve the semiclassical limit, 591 00:39:08,530 --> 00:39:11,050 but the difference between the semiclassical limit 592 00:39:11,050 --> 00:39:14,600 and the entangled wave function, this is what some people say 593 00:39:14,600 --> 00:39:17,980 is spontaneous because it's not captured by a single picture. 594 00:39:24,240 --> 00:39:27,210 I'm actually expecting some people to disagree with me, 595 00:39:27,210 --> 00:39:29,800 but this is sort of my view, what 596 00:39:29,800 --> 00:39:32,989 I'm sort of learning from the simple examples I've 597 00:39:32,989 --> 00:39:33,530 given to you. 598 00:39:50,640 --> 00:39:53,050 Since Ike is an expert on it, maybe, Ike, 599 00:39:53,050 --> 00:39:56,480 can I ask you the question is there actually a simple way 600 00:39:56,480 --> 00:40:00,470 to show that if I go to a large n limit 601 00:40:00,470 --> 00:40:06,460 that you can sort of really show that n parts out of n plus 1 602 00:40:06,460 --> 00:40:10,250 is really described fully by the semiclassical limit 603 00:40:10,250 --> 00:40:13,110 and there is only a 1 over n fraction 604 00:40:13,110 --> 00:40:16,270 where we have to look at the more complicated wave function? 605 00:40:16,270 --> 00:40:18,520 AUDIENCE: I don't think there's a simple way to do it, 606 00:40:18,520 --> 00:40:20,970 but one can look at the equivalence 607 00:40:20,970 --> 00:40:24,615 of a [? and a factor ?] state, and they're only 608 00:40:24,615 --> 00:40:26,675 different by one photon number. 609 00:40:26,675 --> 00:40:27,925 PROFESSOR: It's sort of clear. 610 00:40:27,925 --> 00:40:32,800 I mean, everything is if you approximate n by n plus 1. 611 00:40:32,800 --> 00:40:34,790 If you don't care about the small difference, 612 00:40:34,790 --> 00:40:37,010 everything falls into place and is simple. 613 00:40:37,010 --> 00:40:38,605 But I was just wondering if one could 614 00:40:38,605 --> 00:40:43,530 show sort of in a more direct or more intuitive or maybe 615 00:40:43,530 --> 00:40:47,810 more quantitatively what is really 616 00:40:47,810 --> 00:40:51,920 the extra part beyond stimulated emission 617 00:40:51,920 --> 00:40:54,220 absorption into the coherent state. 618 00:40:54,220 --> 00:40:55,690 So what sort of really the nature 619 00:40:55,690 --> 00:41:00,214 of what people call the spontaneously emitted photon? 620 00:41:00,214 --> 00:41:04,885 AUDIENCE: I think that I don't the question, because I still 621 00:41:04,885 --> 00:41:07,507 argue that it's purely [? unitary ?] evolution even-- 622 00:41:07,507 --> 00:41:08,090 PROFESSOR: OK. 623 00:41:08,090 --> 00:41:10,038 AUDIENCE: For that system, and therefore, it's 624 00:41:10,038 --> 00:41:11,986 purely [INAUDIBLE] and nothing spontaneous 625 00:41:11,986 --> 00:41:14,910 is happening at all. 626 00:41:14,910 --> 00:41:16,931 PROFESSOR: OK. 627 00:41:16,931 --> 00:41:17,430 All right. 628 00:41:17,430 --> 00:41:17,930 Good. 629 00:41:17,930 --> 00:41:20,050 OK. 630 00:41:20,050 --> 00:41:21,020 Fine. 631 00:41:21,020 --> 00:41:21,910 What is next? 632 00:41:21,910 --> 00:41:30,210 I think this finishes our discussion on vacuum Rabi 633 00:41:30,210 --> 00:41:31,850 oscillations and revivals. 634 00:41:36,020 --> 00:41:38,240 I have now two topics in light atom 635 00:41:38,240 --> 00:41:43,240 interaction which you may not find in many textbooks, 636 00:41:43,240 --> 00:41:47,410 but it's my experience that they're really relevant. 637 00:41:47,410 --> 00:41:49,400 One is very conceptual. 638 00:41:49,400 --> 00:41:52,000 It's about the rotating wave approximation. 639 00:41:52,000 --> 00:41:56,140 And the other one is just the opposite, very technical. 640 00:41:56,140 --> 00:41:58,390 It's not really a new concept, but this 641 00:41:58,390 --> 00:42:03,490 is about saturation intensities and cross-section 642 00:42:03,490 --> 00:42:05,600 of an atom for absorption. 643 00:42:05,600 --> 00:42:08,730 The last things, cross-section for absorption and saturation 644 00:42:08,730 --> 00:42:11,320 intensity, that's what you need when 645 00:42:11,320 --> 00:42:13,440 you talk to atoms in the laboratory. 646 00:42:13,440 --> 00:42:17,360 These are the quantities in which we think intuitively 647 00:42:17,360 --> 00:42:19,290 about light atom interaction. 648 00:42:19,290 --> 00:42:21,890 So it's not involving any concept. 649 00:42:21,890 --> 00:42:26,940 I want to spend 20 minutes in introducing for you saturation, 650 00:42:26,940 --> 00:42:29,800 saturation parameter, cross-section, 651 00:42:29,800 --> 00:42:32,480 what's different between monochromatic light 652 00:42:32,480 --> 00:42:33,910 and broadband light. 653 00:42:33,910 --> 00:42:36,380 But before I do that, I have a few minutes 654 00:42:36,380 --> 00:42:37,900 on the rotating wave approximation. 655 00:42:45,340 --> 00:42:57,010 So let's call it rotating wave approximation revisited. 656 00:42:57,010 --> 00:42:58,670 Again, rotating wave approximation. 657 00:43:02,550 --> 00:43:05,490 And what I want to discuss can be discussed in 658 00:43:05,490 --> 00:43:08,550 the fully quantized picture, but also 659 00:43:08,550 --> 00:43:10,070 in the semiclassical picture. 660 00:43:17,130 --> 00:43:22,950 In the fully quantized picture, just a reminder, 661 00:43:22,950 --> 00:43:31,020 what we discussed earlier was that when 662 00:43:31,020 --> 00:43:36,770 we have the atomic raising and lowering 663 00:43:36,770 --> 00:43:41,380 operator and the photonic raising and blowing operator, 664 00:43:41,380 --> 00:43:44,870 we got four terms. 665 00:43:44,870 --> 00:43:46,930 And two of the terms are co-rotating, 666 00:43:46,930 --> 00:43:48,045 two are counter-rotating. 667 00:43:51,760 --> 00:43:54,340 But I can get exactly the same number 668 00:43:54,340 --> 00:43:58,080 of four terms in this semiclassical picture, 669 00:43:58,080 --> 00:44:00,670 and I want you to see both. 670 00:44:00,670 --> 00:44:05,540 But in the quantized picture, it's actually easier, 671 00:44:05,540 --> 00:44:07,650 because when you see a and [? a dega, ?] you 672 00:44:07,650 --> 00:44:10,520 know immediately one is absorption one is emission. 673 00:44:10,520 --> 00:44:12,792 So therefore let me explain to you 674 00:44:12,792 --> 00:44:14,750 what I want to tell you about the rotating wave 675 00:44:14,750 --> 00:44:17,960 approximation using the semiclassical picture, 676 00:44:17,960 --> 00:44:19,450 because then you immediately know 677 00:44:19,450 --> 00:44:21,158 how to apply it to the quantized picture. 678 00:44:29,280 --> 00:44:32,160 So what I want to bring in the here in addition to what we 679 00:44:32,160 --> 00:44:34,630 have discussed about light atom interaction, 680 00:44:34,630 --> 00:44:38,130 we had sort of a dipole Hamiltonian, 681 00:44:38,130 --> 00:44:42,850 is the fact that we have circular polarized light, 682 00:44:42,850 --> 00:44:46,440 left-handed and right-handed light, 683 00:44:46,440 --> 00:44:49,470 and I want to sort of use that and combine it 684 00:44:49,470 --> 00:44:53,510 with angular momentum selection roles, which as you remember 685 00:44:53,510 --> 00:44:56,690 we discussed after our discussion of dipole, 686 00:44:56,690 --> 00:45:00,040 quadrupole, and magnetic dipole positions. 687 00:45:00,040 --> 00:45:02,130 So I put now all those parts together 688 00:45:02,130 --> 00:45:04,835 and revisit the rotating wave approximation. 689 00:45:10,740 --> 00:45:14,100 So what I hope for is it tells you a little bit 690 00:45:14,100 --> 00:45:18,400 how selection roles, angular momentum, 691 00:45:18,400 --> 00:45:22,670 circular polarization, and semiclassical field 692 00:45:22,670 --> 00:45:25,275 which rotate in one direction how they are all connected. 693 00:45:29,260 --> 00:45:33,760 So I have to set up the situation 694 00:45:33,760 --> 00:45:47,690 by saying that we use as a quantization axis the direction 695 00:45:47,690 --> 00:45:52,590 k, which is either the direction of the propagation 696 00:45:52,590 --> 00:45:55,352 of the light beam or, in general, 697 00:45:55,352 --> 00:45:56,810 it's orthogonal to the polarization 698 00:45:56,810 --> 00:46:01,170 of the electric and magnetic field. 699 00:46:01,170 --> 00:46:06,660 And I can talk about an electric field driving 700 00:46:06,660 --> 00:46:08,480 an electric dipole transition. 701 00:46:08,480 --> 00:46:10,950 I can talk about a magnetic field 702 00:46:10,950 --> 00:46:12,710 driving a magnetic transition. 703 00:46:12,710 --> 00:46:16,160 It doesn't really matter. 704 00:46:16,160 --> 00:46:18,680 I will use [? Bsc ?] amplitude, but you can also 705 00:46:18,680 --> 00:46:21,360 immediately think electric dipole, 706 00:46:21,360 --> 00:46:23,910 and this field is linearly polarized. 707 00:46:28,970 --> 00:46:32,840 But I want to immediately decompose this field 708 00:46:32,840 --> 00:46:37,660 into right-handed and left-handed field. 709 00:46:37,660 --> 00:46:39,910 Or a linearly polarized field can 710 00:46:39,910 --> 00:46:43,300 be regarded as a superposition of a field which circulates 711 00:46:43,300 --> 00:46:46,830 this way plus one which circulates the other way. 712 00:46:46,830 --> 00:46:49,770 And ultimately, the message we will see 713 00:46:49,770 --> 00:46:53,440 is that if you have linearly polarized light, 714 00:46:53,440 --> 00:46:55,820 we always get counter-rotating term, we always 715 00:46:55,820 --> 00:46:57,690 have a [INAUDIBLE] shift and such. 716 00:46:57,690 --> 00:47:02,020 But if you use rotating fields or circularly polarized light, 717 00:47:02,020 --> 00:47:05,450 selection roles may actually lead to the result 718 00:47:05,450 --> 00:47:08,080 that there is no counter-rotating term at all. 719 00:47:08,080 --> 00:47:10,780 So this is eventually what I'm aiming for, 720 00:47:10,780 --> 00:47:14,235 and this will be the final point of the discussion. 721 00:47:17,770 --> 00:47:27,610 So the field which rotates in the right-handed direction 722 00:47:27,610 --> 00:47:31,110 where the rotating field is a superposition of x and y, 723 00:47:31,110 --> 00:47:32,780 or i and j. 724 00:47:32,780 --> 00:47:35,950 And one the rotates has a cosine omega t 725 00:47:35,950 --> 00:47:39,340 and one has sine omega t. 726 00:47:39,340 --> 00:47:42,334 I don't need to write down the left-handed part, 727 00:47:42,334 --> 00:47:43,750 because there's just a minus sign. 728 00:47:46,580 --> 00:47:50,440 Or this will actually become very handy. 729 00:47:50,440 --> 00:47:53,460 I do all the discussion for the right-handed part, 730 00:47:53,460 --> 00:47:56,800 but I can always obtain the expression 731 00:47:56,800 --> 00:48:02,120 for the left-handed part by replacing omega by minus omega. 732 00:48:06,370 --> 00:48:09,500 Which will mean that some emission process 733 00:48:09,500 --> 00:48:12,420 by the right-handed part will be an absorption 734 00:48:12,420 --> 00:48:14,400 process by the left-hand part. 735 00:48:14,400 --> 00:48:15,900 Be we'll see. 736 00:48:15,900 --> 00:48:18,410 You can change angular momentum by plus 1 737 00:48:18,410 --> 00:48:21,104 by absorbing a right-handed photon or-- you'll see. 738 00:48:21,104 --> 00:48:21,770 We'll get there. 739 00:48:21,770 --> 00:48:23,660 So anyway, those signs will become important. 740 00:48:26,510 --> 00:48:29,110 Let me now take the above expression 741 00:48:29,110 --> 00:48:37,950 for the right-handed part and replace cosine omega t 742 00:48:37,950 --> 00:48:41,670 and sine omega t by e to the i omega t. 743 00:48:51,950 --> 00:48:57,600 So this was the i component, this was the j component. 744 00:48:57,600 --> 00:48:58,850 We divide by 2. 745 00:49:03,890 --> 00:49:06,790 Just to avoid confusion, I want to emphasize 746 00:49:06,790 --> 00:49:09,210 I've started with a real field. 747 00:49:09,210 --> 00:49:13,570 So I'm not using, as you often do in e and m complex field 748 00:49:13,570 --> 00:49:16,240 and the real fields are the real part, 749 00:49:16,240 --> 00:49:18,570 I have not started out by adding, 750 00:49:18,570 --> 00:49:20,700 you know, imaginary parts to the field. 751 00:49:20,700 --> 00:49:23,470 I've started out with a linearly polarized field 752 00:49:23,470 --> 00:49:25,800 in the x direction cosine omega t, 753 00:49:25,800 --> 00:49:28,600 and I've decomposed it into two real fields. 754 00:49:28,600 --> 00:49:31,300 One is right-handed, one is left-handed. 755 00:49:31,300 --> 00:49:33,710 Complex numbers only come because I 756 00:49:33,710 --> 00:49:37,270 want to use a complex exponential to replace 757 00:49:37,270 --> 00:49:38,670 cosine omega t and sine omega t. 758 00:49:41,930 --> 00:49:45,460 We are almost done with the decomposition of the field. 759 00:49:45,460 --> 00:49:49,670 I just wanted to-- we have now four terms, 760 00:49:49,670 --> 00:49:53,320 and I want to [? recoup ?] them. 761 00:49:53,320 --> 00:49:57,670 i minus imaginary unit j. 762 00:49:57,670 --> 00:50:03,030 i plus imaginary unit j. 763 00:50:03,030 --> 00:50:07,040 This is e to the i omega t. 764 00:50:07,040 --> 00:50:10,980 And this is e to the minus i omega t. 765 00:50:14,340 --> 00:50:17,200 So what have we done? 766 00:50:17,200 --> 00:50:20,370 Well, we've just started with linearly repolarized light, 767 00:50:20,370 --> 00:50:23,430 and I've rewritten the expression twice, 768 00:50:23,430 --> 00:50:27,160 and now we are looking only at one of the circular components, 769 00:50:27,160 --> 00:50:30,980 and in the end what we have is four terms. 770 00:50:30,980 --> 00:50:35,320 Well, that's also what we had in the fully quantized 771 00:50:35,320 --> 00:50:39,330 Hamiltonian, and we now want to identify what those four terms. 772 00:50:39,330 --> 00:50:43,360 Two will be co-rotating, two will be counter-rotating, 773 00:50:43,360 --> 00:50:46,110 but it's very helpful to analyze those terms. 774 00:50:50,340 --> 00:50:57,650 But there are two things we have to look at now. 775 00:50:57,650 --> 00:51:00,480 One is we have an e to the i omega t. 776 00:51:03,740 --> 00:51:06,240 And, well, it's probably a sign convention, 777 00:51:06,240 --> 00:51:09,170 but trust me, if you put that in the Schrodinger equation, 778 00:51:09,170 --> 00:51:13,330 it mean that you increase the energy of the atom 779 00:51:13,330 --> 00:51:15,430 if you drive it with e to the i omega t. 780 00:51:15,430 --> 00:51:18,130 You take it from a ground state to an excited state, which 781 00:51:18,130 --> 00:51:21,370 differs in frequency by omega, and therefore this 782 00:51:21,370 --> 00:51:24,130 means you have increased the energy of the system, 783 00:51:24,130 --> 00:51:26,660 and this corresponds to absorption. 784 00:51:26,660 --> 00:51:31,550 Whereas this one here means we take an atom 785 00:51:31,550 --> 00:51:34,010 from an excited state to a ground state, 786 00:51:34,010 --> 00:51:36,480 and this is the situation of stimulated emission. 787 00:51:39,880 --> 00:51:43,110 Remember, in selection roles we take our field 788 00:51:43,110 --> 00:51:46,160 and we multiply it with a dipole moment, 789 00:51:46,160 --> 00:51:48,550 electric, magnetic dipole moment whatever. 790 00:51:48,550 --> 00:51:53,580 But now we want to use also the spherical tensor decomposition 791 00:51:53,580 --> 00:51:54,940 of those dipole moments. 792 00:51:54,940 --> 00:51:57,980 It's a complicated word, but what 793 00:51:57,980 --> 00:52:03,710 it means is those terms are dotted with the dipole moment, 794 00:52:03,710 --> 00:52:08,940 and if you do it now component-by-component, 795 00:52:08,940 --> 00:52:15,780 we retrieve selection roles because this peaks out the-- 796 00:52:15,780 --> 00:52:18,890 let me just write it down-- the x plus y 797 00:52:18,890 --> 00:52:25,860 tensor component of the matrix element. 798 00:52:25,860 --> 00:52:31,990 And this corresponds to delta m equals plus 1. 799 00:52:31,990 --> 00:52:34,430 We change the angular momentum by one unit, 800 00:52:34,430 --> 00:52:38,115 and of course this term is then delta m equals minus 1. 801 00:52:45,080 --> 00:52:47,030 So we have done the work. 802 00:52:47,030 --> 00:52:50,480 What I want to do now is just map those terms into 803 00:52:50,480 --> 00:52:52,110 and energy level diagram. 804 00:52:52,110 --> 00:52:55,040 I like sort of pictorial representations, 805 00:52:55,040 --> 00:53:00,430 and each term becomes now a graphical [INAUDIBLE]. 806 00:53:00,430 --> 00:53:07,180 So let us assume we have a system, hydrogen is to p state. 807 00:53:07,180 --> 00:53:11,520 But let's say generally we go from a j equals 0 to j 808 00:53:11,520 --> 00:53:17,555 equals 1 state, which has three components. 809 00:53:20,170 --> 00:53:29,750 Now, I have set it up in such a way that-- oops, 810 00:53:29,750 --> 00:53:31,165 we need a little bit extra space. 811 00:53:35,190 --> 00:53:39,370 I've set it up in such a way that the states here, this 812 00:53:39,370 --> 00:53:45,510 is m equals 0, this is m equals plus 1, 813 00:53:45,510 --> 00:53:49,490 and this is m equals minus 1. 814 00:53:49,490 --> 00:53:53,560 So therefore-- let me just use color 815 00:53:53,560 --> 00:54:10,360 coding now-- this one here is delta m equals plus 1 816 00:54:10,360 --> 00:54:17,040 so this one always moves to the right. 817 00:54:17,040 --> 00:54:21,100 It changes angular momentum by 1 so it can always 818 00:54:21,100 --> 00:54:27,130 move to the right, whereas the other one, delta m 819 00:54:27,130 --> 00:54:30,280 equals minus 1, moves to the bank. 820 00:54:30,280 --> 00:54:34,760 Absorption is e to the i omega t, always moves up. 821 00:54:34,760 --> 00:54:36,860 And stimulated emission moves down. 822 00:54:58,120 --> 00:55:07,230 So with that what happens is this term here 823 00:55:07,230 --> 00:55:11,120 transfers one unit of angular momentum and energy. 824 00:55:11,120 --> 00:55:15,270 So that would mean this term goes up here. 825 00:55:15,270 --> 00:55:19,950 It could go up here, if [? there were a ?] state. 826 00:55:19,950 --> 00:55:30,130 The other term-- let me use a green color-- is driving 827 00:55:30,130 --> 00:55:32,065 the process in the opposite direction. 828 00:55:35,410 --> 00:55:40,890 But now we have to also consider that you can go down here, 829 00:55:40,890 --> 00:55:44,750 and you can go down to a virtual state. 830 00:55:47,470 --> 00:55:49,510 A virtual state is just something 831 00:55:49,510 --> 00:55:52,300 which has the same wave function as a state, 832 00:55:52,300 --> 00:55:55,170 it just has an e to the i omega t, which 833 00:55:55,170 --> 00:55:57,079 is not-- it's a driven system. 834 00:55:57,079 --> 00:55:57,620 You drive it. 835 00:55:57,620 --> 00:55:58,890 You [INAUDIBLE] a state. 836 00:55:58,890 --> 00:56:01,130 You [INAUDIBLE] a state at the drive frequency, 837 00:56:01,130 --> 00:56:04,160 and it just means, in this case, this state 838 00:56:04,160 --> 00:56:09,350 has an oscillation e to the i omega t, which is very, very 839 00:56:09,350 --> 00:56:12,560 different from what a state which is populated would have, 840 00:56:12,560 --> 00:56:15,720 and this is what we call a virtual state. 841 00:56:15,720 --> 00:56:19,670 So in other words, what is possible is 842 00:56:19,670 --> 00:56:24,490 we have our three states, plus minus 1 and 0, 843 00:56:24,490 --> 00:56:27,300 but this is the spatial wave function 844 00:56:27,300 --> 00:56:30,030 including angular momentum. 845 00:56:30,030 --> 00:56:33,840 But we can now drive it by plus omega and minus omega, 846 00:56:33,840 --> 00:56:37,240 and therefore we can have it as virtual states 847 00:56:37,240 --> 00:56:40,190 pretty much at any energy we want. 848 00:56:40,190 --> 00:56:43,550 But this process here is not possible, 849 00:56:43,550 --> 00:56:48,490 because this would require to go to a state which has m 850 00:56:48,490 --> 00:56:51,660 equals 2, which does not exist. 851 00:56:51,660 --> 00:56:54,030 So now what I've shown here is if we 852 00:56:54,030 --> 00:56:56,770 would stock in the m equals 0 state, 853 00:56:56,770 --> 00:57:00,670 I've shown you the four terms, two are co-rotating 854 00:57:00,670 --> 00:57:02,480 and two are counter-rotating. 855 00:57:02,480 --> 00:57:05,020 If you neglect this virtual state which 856 00:57:05,020 --> 00:57:09,530 has a detuning of about 2 omega, or 2 resonance 857 00:57:09,530 --> 00:57:12,070 frequency of the atom, this is the rotating wave 858 00:57:12,070 --> 00:57:13,730 approximation. 859 00:57:13,730 --> 00:57:16,850 One term is responsible for absorption; 860 00:57:16,850 --> 00:57:21,360 the other term is responsible for a stimulated emission. 861 00:57:21,360 --> 00:57:24,640 But if I don't make the rotating wave approximation, 862 00:57:24,640 --> 00:57:25,875 I have those two extra terms. 863 00:57:36,130 --> 00:57:39,830 So this is only the right-handed light, 864 00:57:39,830 --> 00:57:42,400 and I want to sort of play a little bit with this concept. 865 00:57:42,400 --> 00:57:45,110 If I would take the left-handed light, 866 00:57:45,110 --> 00:57:47,690 I would add sort of four more arrows. 867 00:57:47,690 --> 00:57:51,330 Two more here and two more here. 868 00:57:51,330 --> 00:57:52,870 But let's just keep the situation 869 00:57:52,870 --> 00:57:54,260 as simple as possible. 870 00:57:54,260 --> 00:57:56,177 But I really sort of like that you 871 00:57:56,177 --> 00:57:58,770 write down right-handed, left-handed side, 872 00:57:58,770 --> 00:58:00,970 decompose it into its components, 873 00:58:00,970 --> 00:58:04,410 and each component is now in this diagram connected 874 00:58:04,410 --> 00:58:07,900 to an arrow where one direction is angular momentum, 875 00:58:07,900 --> 00:58:09,020 the other one is energy. 876 00:58:12,250 --> 00:58:26,050 So let me now talk about other energy diagrams. 877 00:58:26,050 --> 00:58:28,550 And this will lead to the answer. 878 00:58:28,550 --> 00:58:35,430 Well, can we create a situation where 879 00:58:35,430 --> 00:58:39,310 we have only two terms, which would be the simplest 880 00:58:39,310 --> 00:58:41,430 two-level system, can be directly realized 881 00:58:41,430 --> 00:58:46,090 without any rotating wave approximation 882 00:58:46,090 --> 00:58:48,770 a two-level system? 883 00:58:48,770 --> 00:58:52,035 So if we had two levels, which have only m 884 00:58:52,035 --> 00:58:56,640 equals 0 and m equals 1. 885 00:58:56,640 --> 00:58:58,670 So this would be the situation I just 886 00:58:58,670 --> 00:59:00,140 discussed with those two levels. 887 00:59:09,240 --> 00:59:14,200 So the only way how I can fit in this arrow is this one, 888 00:59:14,200 --> 00:59:18,130 and the diagonally downward arrow is that. 889 00:59:18,130 --> 00:59:22,095 So in this case, rotating wave approximation 890 00:59:22,095 --> 00:59:24,600 is not an approximation, it is exact. 891 00:59:27,250 --> 00:59:30,430 But some purists will actually say, hey, 892 00:59:30,430 --> 00:59:33,240 you can never realize that when you have an n 893 00:59:33,240 --> 00:59:36,350 equals plus 1 state. 894 00:59:36,350 --> 00:59:39,490 Then you always have an m equals minus 1 state. 895 00:59:39,490 --> 00:59:41,950 And then you have a virtual state down there, 896 00:59:41,950 --> 00:59:44,350 and then you get two more terms, which 897 00:59:44,350 --> 00:59:48,610 are the counter-rotating terms which I just showed above. 898 00:59:48,610 --> 00:59:51,430 So whole I would say if you have a neutron star which 899 00:59:51,430 --> 00:59:53,890 makes an infinitely high magnetic field, 900 00:59:53,890 --> 00:59:56,045 you can have a huge [INAUDIBLE] splitting between m 901 00:59:56,045 --> 00:59:58,240 equals plus 1 and m equals minus 1 902 00:59:58,240 --> 01:00:01,240 and completely move one of the angular momentum 903 01:00:01,240 --> 01:00:02,990 states out of the picture. 904 01:00:02,990 --> 01:00:06,330 But, of course, in the rotating wave approximation 905 01:00:06,330 --> 01:00:09,760 we are neglecting off resonant terms at 2 omega, omega 906 01:00:09,760 --> 01:00:12,052 being an electronic excitation energy, 907 01:00:12,052 --> 01:00:13,760 so I'm really talking about Zeeman shifts 908 01:00:13,760 --> 01:00:16,070 here to eliminate the other state which 909 01:00:16,070 --> 01:00:18,890 may be comparable to electronic energies. 910 01:00:18,890 --> 01:00:22,820 So in principle, I can say this is my Hilbert space, 911 01:00:22,820 --> 01:00:25,195 and in this Hilbert space no rotating wave approximations 912 01:00:25,195 --> 01:00:26,500 is needed. 913 01:00:26,500 --> 01:00:30,100 But it's maybe an artificial Hilbert space. 914 01:00:30,100 --> 01:00:32,860 When I had a discussion with other people, 915 01:00:32,860 --> 01:00:39,470 we came up with the possibility of some forbidden transition. 916 01:00:39,470 --> 01:00:43,240 If you go from a doublet s to a doublet s state 917 01:00:43,240 --> 01:00:47,560 so all you have is a spin system which 918 01:00:47,560 --> 01:00:54,540 has 1/2 angular momentum plus 1/2 minus 1/2. 919 01:00:54,540 --> 01:01:00,840 And then you realize that the only way how 920 01:01:00,840 --> 01:01:07,060 you can fit in the orange arrow is 921 01:01:07,060 --> 01:01:10,620 in this way, and the green arrow in this way. 922 01:01:10,620 --> 01:01:12,470 So here you would have a situation 923 01:01:12,470 --> 01:01:15,110 where the rotating wave approximation is exact. 924 01:01:15,110 --> 01:01:17,720 But, of course, it's not an electric dipole transition; 925 01:01:17,720 --> 01:01:24,542 it's some sort of weaker conversation, which 926 01:01:24,542 --> 01:01:25,250 may be forbidden. 927 01:01:32,110 --> 01:01:33,990 I need two more minutes. 928 01:01:33,990 --> 01:01:38,940 I have discussed the case where we have quantized 929 01:01:38,940 --> 01:01:42,630 along a direction, I called it the k direction, 930 01:01:42,630 --> 01:01:45,420 and the polarization of the electromagnetic field 931 01:01:45,420 --> 01:01:48,610 was [? i and j ?] was perpendicular to it. 932 01:01:48,610 --> 01:01:55,530 So let me now discuss a case where we quantize along 933 01:01:55,530 --> 01:01:59,010 the polarization of the electromagnetic field, 934 01:01:59,010 --> 01:02:01,980 and you remember from our discussion on selection roles 935 01:02:01,980 --> 01:02:04,440 that this is pi light. 936 01:02:04,440 --> 01:02:09,260 So in this case, our magnetic or electric field 937 01:02:09,260 --> 01:02:12,690 is polarized along the i direction, 938 01:02:12,690 --> 01:02:18,970 and the real cosine omega t gets decomposed into e to the plus 939 01:02:18,970 --> 01:02:20,750 e to the minus i omega t. 940 01:02:20,750 --> 01:02:24,580 And we know already one term is absorption one is emission. 941 01:02:24,580 --> 01:02:37,020 And now, if I take my j equals 0 to j equals 1 system, 942 01:02:37,020 --> 01:02:42,230 pi light has a selection role of delta m equals 0. 943 01:02:42,230 --> 01:02:47,870 So now I have an arrow, which I want to be orange, 944 01:02:47,870 --> 01:02:48,950 which goes up. 945 01:02:48,950 --> 01:02:54,240 And a green arrow-- this is a great program. 946 01:02:54,240 --> 01:02:57,270 The only thing is you have to be very careful when 947 01:02:57,270 --> 01:02:59,130 you change color and press carefully. 948 01:02:59,130 --> 01:03:02,060 That's why sometimes the colors are not doing what I want. 949 01:03:02,060 --> 01:03:03,500 But here is green. 950 01:03:03,500 --> 01:03:10,710 But now, of course, with linearly polarized light 951 01:03:10,710 --> 01:03:13,870 we can always go down to a virtual state. 952 01:03:13,870 --> 01:03:15,480 We have now four terms. 953 01:03:15,480 --> 01:03:23,330 Two are rotating, two are counter-rotating. 954 01:03:23,330 --> 01:03:26,400 So therefore the quick conclusion 955 01:03:26,400 --> 01:03:32,440 of the last ten minutes is that there is the possibility 956 01:03:32,440 --> 01:03:40,550 that counter-rotating terms can be 957 01:03:40,550 --> 01:03:58,130 0 for sigma plus sigma minus light due to angular momentum 958 01:03:58,130 --> 01:03:58,945 selection roles. 959 01:04:01,620 --> 01:04:06,890 But what we have also learned is if you have the m plus 1 state 960 01:04:06,890 --> 01:04:11,230 in there's an m minus state, if you have circularly polarized 961 01:04:11,230 --> 01:04:15,330 light and we drive a transition between two m states, 962 01:04:15,330 --> 01:04:18,320 the counter-rotating term does not 963 01:04:18,320 --> 01:04:21,490 come from the same set of two states, m equals 1. 964 01:04:21,490 --> 01:04:23,580 It involves m equals minus 1. 965 01:04:23,580 --> 01:04:28,140 So it's the other state which is maybe degenerate or only 966 01:04:28,140 --> 01:04:30,570 slightly [? split ?] by a magnetic field which 967 01:04:30,570 --> 01:04:34,200 is responsible for the counter-rotating terms. 968 01:04:34,200 --> 01:04:36,870 Anyway we have talked so much about rotating wave 969 01:04:36,870 --> 01:04:38,680 approximation and those terms, I just 970 01:04:38,680 --> 01:04:41,670 wanted to show you how it is modified 971 01:04:41,670 --> 01:04:46,490 if you use degeneracy p states and angular momentum. 972 01:04:46,490 --> 01:04:48,560 Any question? 973 01:04:48,560 --> 01:04:50,110 OK.