1 00:00:00,050 --> 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 00:00:20,450 --> 00:00:24,190 PROFESSOR: So over the last lecture, 9 00:00:24,190 --> 00:00:28,070 we have talked about coherence within an atom, 10 00:00:28,070 --> 00:00:32,880 coherence between two levels, coherence between three levels. 11 00:00:32,880 --> 00:00:35,020 And today, in the last class, we want 12 00:00:35,020 --> 00:00:40,270 to talk about coherence between atoms. 13 00:00:40,270 --> 00:00:44,380 So this is now, I think, for the first time 14 00:00:44,380 --> 00:00:47,550 in this course that we really have more than one atom. 15 00:00:47,550 --> 00:00:50,550 Well, maybe we discussed some collision or broadening, 16 00:00:50,550 --> 00:00:56,320 or we discussed [INAUDIBLE] interaction between two atoms. 17 00:00:56,320 --> 00:01:00,530 But usually, atomic physics is one atom at a time. 18 00:01:00,530 --> 00:01:05,269 But now we want to understand one important phenomenon which 19 00:01:05,269 --> 00:01:07,810 happens when we have many atoms. 20 00:01:07,810 --> 00:01:10,700 And the phenomenon is called superradiance. 21 00:01:10,700 --> 00:01:13,380 So I left something good for the end. 22 00:01:13,380 --> 00:01:16,800 And superradiance has in common the word 23 00:01:16,800 --> 00:01:20,800 super with superconductivity and superfluidity. 24 00:01:20,800 --> 00:01:25,240 And it really represents that the atom, many atoms 25 00:01:25,240 --> 00:01:26,600 act together. 26 00:01:26,600 --> 00:01:29,300 And the word super also means coherence 27 00:01:29,300 --> 00:01:34,120 among atoms-- superfluidity and superconductivity 28 00:01:34,120 --> 00:01:37,550 have macroscopic wave function where all the atoms, the meta 29 00:01:37,550 --> 00:01:39,830 waves are coherent. 30 00:01:39,830 --> 00:01:43,770 The phenomenon of superradiance, as we will see, 31 00:01:43,770 --> 00:01:46,660 has not so much to do with coherent atoms. 32 00:01:46,660 --> 00:01:48,740 It has more to do with coherent photons. 33 00:01:48,740 --> 00:01:53,730 So it's more-- some people regard superradiance as a laser 34 00:01:53,730 --> 00:01:55,850 without mirrors. 35 00:01:55,850 --> 00:01:59,250 But you'll see where the story leads us to. 36 00:01:59,250 --> 00:02:05,620 So just to set the stage-- for many atoms, 37 00:02:05,620 --> 00:02:11,770 we should first talk about single atoms. 38 00:02:11,770 --> 00:02:14,280 And all that is described in this landmark paper 39 00:02:14,280 --> 00:02:18,580 by [INAUDIBLE] in 1954, which is posted on our website. 40 00:02:18,580 --> 00:02:22,710 So if you have a single atom prepared in the excited state, 41 00:02:22,710 --> 00:02:25,060 it decays to the ground state, and we 42 00:02:25,060 --> 00:02:28,170 want to characterize the system by emission rate 43 00:02:28,170 --> 00:02:30,850 as a function of time. 44 00:02:30,850 --> 00:02:38,450 So the emission rate initially is gamma, the natural language 45 00:02:38,450 --> 00:02:39,887 of the excited state. 46 00:02:39,887 --> 00:02:41,470 And then, of course, the emission rate 47 00:02:41,470 --> 00:02:45,330 decays because we don't have any atoms left. 48 00:02:45,330 --> 00:02:49,000 Similarly, the probability to be in the ground state 49 00:02:49,000 --> 00:02:55,410 is zero initially, and then with an exponential approach, 50 00:02:55,410 --> 00:02:57,820 it eventually goes to unity after a while, 51 00:02:57,820 --> 00:03:00,897 after we have only atoms in the gamma state. 52 00:03:08,670 --> 00:03:11,050 So this is rather straightforward. 53 00:03:11,050 --> 00:03:13,840 But now we want to bring in a second atom. 54 00:03:13,840 --> 00:03:17,830 And I'm asking, what happens when 55 00:03:17,830 --> 00:03:20,660 we have not the one atom, but two atoms? 56 00:03:20,660 --> 00:03:21,830 One is in the ground state. 57 00:03:21,830 --> 00:03:23,390 One is excited. 58 00:03:23,390 --> 00:03:25,965 So pretty much what we have added to the original situation 59 00:03:25,965 --> 00:03:29,320 with one excited atom was we've brought in one 60 00:03:29,320 --> 00:03:35,870 ground state atom, which naively you would think does nothing. 61 00:03:35,870 --> 00:03:38,450 But that's not the case. 62 00:03:38,450 --> 00:03:46,910 What happens is-- and I assume just for review-- 63 00:03:46,910 --> 00:03:48,930 we will drop the assumption later, 64 00:03:48,930 --> 00:03:51,130 but we assume for now that all the atoms 65 00:03:51,130 --> 00:03:53,550 are within one optical wavelength. 66 00:03:53,550 --> 00:03:56,160 What we then realize is for two atoms-- 67 00:03:56,160 --> 00:03:59,530 and I will show you that in its full beauty-- 68 00:03:59,530 --> 00:04:02,810 that the initial rate of light which comes out of the system 69 00:04:02,810 --> 00:04:03,660 is the same. 70 00:04:03,660 --> 00:04:08,660 So the extra ground state atom does not 71 00:04:08,660 --> 00:04:17,950 change the initial emission rate, but it goes down faster. 72 00:04:17,950 --> 00:04:23,010 And if we ask what is the probability that the atom is 73 00:04:23,010 --> 00:04:27,520 in the ground state, we find that it's only one half. 74 00:04:27,520 --> 00:04:31,120 So in other words, normalized [INAUDIBLE] system, 75 00:04:31,120 --> 00:04:35,520 we have a ground and excited state atom, and what comes out 76 00:04:35,520 --> 00:04:37,500 is only half a photon. 77 00:04:37,500 --> 00:04:41,250 Half of the atoms do not decay. 78 00:04:41,250 --> 00:04:45,730 So it's not the same rate and the same decay. 79 00:04:45,730 --> 00:04:47,544 Something profoundly has happened. 80 00:04:47,544 --> 00:04:49,210 And this is what you want to understand. 81 00:04:52,200 --> 00:04:53,820 So let me give you the correct answer. 82 00:04:57,560 --> 00:05:00,720 The rate of emission is a function 83 00:05:00,720 --> 00:05:02,490 of time for this situation. 84 00:05:02,490 --> 00:05:07,780 We start out with gamma, but then the emission decays, not 85 00:05:07,780 --> 00:05:11,030 with gamma but with 2 gamma. 86 00:05:11,030 --> 00:05:19,470 And the probability that both atoms are in the ground state-- 87 00:05:19,470 --> 00:05:23,560 or that the second atom, so to speak, is in the ground state-- 88 00:05:23,560 --> 00:05:27,910 will only asymptotically go to 1/2. 89 00:05:27,910 --> 00:05:31,820 And it does so exponentially-- but again, 90 00:05:31,820 --> 00:05:37,050 with the time constant, which is two times faster 91 00:05:37,050 --> 00:05:40,150 than for the single-atom system. 92 00:05:40,150 --> 00:05:57,970 So we have the same initial emission rate, but only 93 00:05:57,970 --> 00:06:02,260 probability of 1/2 to emit at all. 94 00:06:07,970 --> 00:06:15,920 So in order to understand it, we have 95 00:06:15,920 --> 00:06:22,350 to look at an atom in the excited state and atom 96 00:06:22,350 --> 00:06:23,730 in the ground state. 97 00:06:23,730 --> 00:06:26,680 And we want to write down the wave function 98 00:06:26,680 --> 00:06:29,910 as a superposition of a symmetrized and antisymmetrized 99 00:06:29,910 --> 00:06:30,780 wave function. 100 00:06:43,530 --> 00:06:47,610 I should tell you, I'm going very slowly for two atoms. 101 00:06:47,610 --> 00:06:51,330 And then once I've introduced the concept for two atoms, 102 00:06:51,330 --> 00:06:55,090 with a few pen strokes, we can immediately discuss in atoms. 103 00:06:55,090 --> 00:06:57,470 So all the phsyics, all the understanding 104 00:06:57,470 --> 00:06:59,601 what goes on in superradiance is already 105 00:06:59,601 --> 00:07:00,600 displayed for two atoms. 106 00:07:03,250 --> 00:07:04,930 So we want to have a superposition 107 00:07:04,930 --> 00:07:09,490 of symmetric and antisymmetric wave function. 108 00:07:09,490 --> 00:07:12,150 The symmetric one is a normalized wave function 109 00:07:12,150 --> 00:07:14,190 which is ge plus eg. 110 00:07:17,240 --> 00:07:25,160 And we call that the superradiant wave function, 111 00:07:25,160 --> 00:07:28,050 for reasons which will become clear in a moment. 112 00:07:28,050 --> 00:07:30,260 And if you have a minus sign here, 113 00:07:30,260 --> 00:07:35,980 the antisymmetric combination, we 114 00:07:35,980 --> 00:07:40,420 call this subradiant wave function. 115 00:07:40,420 --> 00:07:44,620 Now, what happens is, we have to consider-- 116 00:07:44,620 --> 00:07:46,662 so we have symmetrized the wave function. 117 00:07:46,662 --> 00:07:48,120 Well, I didn't really tell you why, 118 00:07:48,120 --> 00:07:50,450 but it's always good to symmetrize. 119 00:07:50,450 --> 00:07:52,740 Symmetry is, if you can use it, something good. 120 00:07:52,740 --> 00:07:58,740 And the reason why I symmetrized it is because I want 121 00:07:58,740 --> 00:08:00,240 look at the interaction Hamiltonian. 122 00:08:05,090 --> 00:08:08,060 And if I look at the interaction Hamiltonian-- the one 123 00:08:08,060 --> 00:08:11,005 we have seen many, many times but now for two atoms-- 124 00:08:11,005 --> 00:08:14,780 we will immediately realize that this interaction Hamiltonian is 125 00:08:14,780 --> 00:08:16,520 symmetric. 126 00:08:16,520 --> 00:08:19,686 So therefore, if the Hamiltonian is symmetric, 127 00:08:19,686 --> 00:08:21,060 it's a really good starting point 128 00:08:21,060 --> 00:08:23,615 to have the wave function for the atoms expanded 129 00:08:23,615 --> 00:08:25,870 in a symmetric basis. 130 00:08:25,870 --> 00:08:30,220 And since I want to emphasize that the whole story I'm 131 00:08:30,220 --> 00:08:32,870 telling you today has nothing to do 132 00:08:32,870 --> 00:08:41,350 with the kind of second quantization-- 133 00:08:41,350 --> 00:08:42,730 it is about spontaneous emission, 134 00:08:42,730 --> 00:08:44,550 but it's not involving any subtlety 135 00:08:44,550 --> 00:08:47,520 of spontaneous emission and field 136 00:08:47,520 --> 00:08:54,140 quantization-- I want to write down the interaction 137 00:08:54,140 --> 00:08:58,070 Hamiltonian both in a classical and a quantum mechanical way. 138 00:08:58,070 --> 00:09:03,540 In the classical way, we have the dipole moment d1. 139 00:09:03,540 --> 00:09:06,310 We have the dipole moment d2. 140 00:09:06,310 --> 00:09:12,660 And the atoms talk to the electric field at position RNT. 141 00:09:16,950 --> 00:09:20,490 And now you realize where some of the assumptions 142 00:09:20,490 --> 00:09:23,820 are important, since the atoms are localized to 143 00:09:23,820 --> 00:09:25,730 within a wavelength, they rarely talk 144 00:09:25,730 --> 00:09:26,890 to the same electric field. 145 00:09:26,890 --> 00:09:29,650 There are no phase factors. 146 00:09:29,650 --> 00:09:33,210 In about 55 minutes or so, we introduce phase factors 147 00:09:33,210 --> 00:09:34,450 for extended samples. 148 00:09:34,450 --> 00:09:36,330 But for now, we don't. 149 00:09:36,330 --> 00:09:39,340 And therefore, what the atoms couple with 150 00:09:39,340 --> 00:09:41,350 is with a dipole moment, which is 151 00:09:41,350 --> 00:09:42,985 a sum of the two dipole moments. 152 00:09:51,510 --> 00:09:56,160 So this is classical or semi-classical. 153 00:10:00,800 --> 00:10:02,470 So what enters in the Hamiltonian 154 00:10:02,470 --> 00:10:05,750 is only the sum of the operators for the two atoms. 155 00:10:05,750 --> 00:10:09,017 And the same happens in the QED Hamiltonian. 156 00:10:09,017 --> 00:10:11,100 And actually, I will get a little bit more mileage 157 00:10:11,100 --> 00:10:16,050 out of the QED Hamiltonian, as you will see in a moment. 158 00:10:16,050 --> 00:10:24,490 Because with the QED Hamiltonian we describe the atomic system-- 159 00:10:24,490 --> 00:10:27,260 so at first atom one-- with the raising 160 00:10:27,260 --> 00:10:30,500 and lowering operator with the atoms interacting 161 00:10:30,500 --> 00:10:33,710 with a and a [INAUDIBLE]. 162 00:10:33,710 --> 00:10:38,370 And then I have to add the term where the index one and two are 163 00:10:38,370 --> 00:10:38,870 exchanged. 164 00:10:42,910 --> 00:10:44,780 So we are introducing here-- that's 165 00:10:44,780 --> 00:10:49,020 convenient for two-level atom-- the spin notation sigma plus 166 00:10:49,020 --> 00:10:52,820 and sigma minus are the raising and lowering operator 167 00:10:52,820 --> 00:10:56,150 which flip the atom from the ground to the excited state 168 00:10:56,150 --> 00:10:58,520 and vice versa. 169 00:10:58,520 --> 00:11:01,170 But the important part now is-- and this 170 00:11:01,170 --> 00:11:06,370 is where, actually, everything comes from in superradiance-- 171 00:11:06,370 --> 00:11:12,790 that the coupling involves not the individual spins, 172 00:11:12,790 --> 00:11:16,940 little sigma plus, sigma 1 and sigma 2-- 173 00:11:16,940 --> 00:11:27,650 it only involves the sum of the individuals. 174 00:11:27,650 --> 00:11:33,280 i equals 122, and later we extend the sum to n. 175 00:11:36,900 --> 00:11:41,630 So therefore, what matters for the interaction 176 00:11:41,630 --> 00:11:46,550 of the atoms with the electromagnetic field 177 00:11:46,550 --> 00:11:51,210 is the sum of all the atomic spin operators. 178 00:11:51,210 --> 00:11:56,240 And the sum is, of course, symmetric against exchange. 179 00:11:56,240 --> 00:11:59,950 So therefore, when we are asking what 180 00:11:59,950 --> 00:12:06,160 is the coupling of the state which I called 181 00:12:06,160 --> 00:12:10,810 the superradiant state, the one where we had symmetrized eg 182 00:12:10,810 --> 00:12:16,770 plus ge, or we ask, what is the coupling 183 00:12:16,770 --> 00:12:25,830 of the subradiant state to-- well, the state where 184 00:12:25,830 --> 00:12:28,960 both atoms are in the ground state. 185 00:12:28,960 --> 00:12:31,610 Well, now we can use symmetry. 186 00:12:31,610 --> 00:12:33,370 The left-hand side is symmetric. 187 00:12:33,370 --> 00:12:35,170 The operator is symmetric. 188 00:12:35,170 --> 00:12:37,980 And now only the symmetric state will couple. 189 00:12:37,980 --> 00:12:40,690 The antisymmetric state will not couple. 190 00:12:40,690 --> 00:12:46,830 So therefore, the subradiant state, eg minus ge, 191 00:12:46,830 --> 00:12:47,986 cannot decay. 192 00:12:47,986 --> 00:12:49,360 That's why we call it subradiant. 193 00:12:52,020 --> 00:12:55,275 I think a better word would be non-radiant, 194 00:12:55,275 --> 00:12:58,760 but non is definitely subradiant. 195 00:12:58,760 --> 00:13:02,800 And for the matrix element eg and ge, 196 00:13:02,800 --> 00:13:06,670 we find that we have actually an enhancement of the coupling 197 00:13:06,670 --> 00:13:08,090 by a factor of square root 2. 198 00:13:17,960 --> 00:13:21,480 So now we pretty much know what we have to do. 199 00:13:24,240 --> 00:13:33,000 You want to use the symmetry of-- let's assume we consider 200 00:13:33,000 --> 00:13:37,020 ground and excited state of each atom as spin 1/2. 201 00:13:37,020 --> 00:13:41,510 But now we want to look at the total spin, the total pseudo 202 00:13:41,510 --> 00:13:43,730 angular momentum of the two atoms, 203 00:13:43,730 --> 00:13:46,120 and later we extend it to n atoms. 204 00:13:46,120 --> 00:13:48,700 So we want to use now the power of the angular momentum 205 00:13:48,700 --> 00:13:50,950 description. 206 00:13:50,950 --> 00:13:52,620 And that goes like follows. 207 00:13:52,620 --> 00:13:55,980 We have four states of two atoms. 208 00:13:55,980 --> 00:14:05,930 And this is gg, ge, eg, and ee. 209 00:14:05,930 --> 00:14:10,410 And if I denote with ground state spin 210 00:14:10,410 --> 00:14:19,920 down, excited state spin up, I'm talking about 2 211 00:14:19,920 --> 00:14:23,960 spin 1/2 states. 212 00:14:27,540 --> 00:14:38,580 And 2 spin 1/2 states can couple to s 213 00:14:38,580 --> 00:14:43,100 equals 1, total s equals 1, and total spin s equals 0. 214 00:14:48,300 --> 00:14:51,820 And that's what I've done here. 215 00:14:51,820 --> 00:14:56,940 I've arranged the states ee, the symmetric superradiant state, 216 00:14:56,940 --> 00:14:59,775 the ground state, and the subradiant state. 217 00:15:02,780 --> 00:15:04,700 A variation energy level diagram-- 218 00:15:04,700 --> 00:15:07,660 here we have 0 excitation energy, 219 00:15:07,660 --> 00:15:09,470 here we have 1 excitation energy, 220 00:15:09,470 --> 00:15:12,520 and here we have two excitations energies of the atom. 221 00:15:12,520 --> 00:15:16,080 But I've also labeled now the spin labels 222 00:15:16,080 --> 00:15:18,580 for the combined system. 223 00:15:18,580 --> 00:15:23,790 Those symmetrized states correspond to a spin equals 1. 224 00:15:23,790 --> 00:15:27,350 It's a triplet letter with three different magnetic quantum 225 00:15:27,350 --> 00:15:28,590 numbers. 226 00:15:28,590 --> 00:15:31,420 m equals plus 1 means everything is highly excited. 227 00:15:31,420 --> 00:15:35,550 m equals minus 1 means we are in the total ground state. 228 00:15:35,550 --> 00:15:39,340 And here we have the simulate state, 229 00:15:39,340 --> 00:15:45,060 which is the antisymmetric state or the subradiant state. 230 00:15:45,060 --> 00:15:54,660 And our interaction Hamiltonian is the total spin plus minus. 231 00:15:54,660 --> 00:15:57,540 It is the raising and the lowering operator. 232 00:15:57,540 --> 00:15:59,680 And you know that the raising and lowering 233 00:15:59,680 --> 00:16:03,910 operator for the spin is only making transitions 234 00:16:03,910 --> 00:16:06,910 within a manifold of total s. 235 00:16:06,910 --> 00:16:10,940 It just changes the end quantum number by plus minus 1. 236 00:16:10,940 --> 00:16:14,750 So the Hamiltonian cannot do anything to the simulate state, 237 00:16:14,750 --> 00:16:17,260 because there is no other simulate state to couple. 238 00:16:17,260 --> 00:16:21,100 But within the triplet manifold, the sigma plus sigma 239 00:16:21,100 --> 00:16:24,140 minus operator is creating transitions 240 00:16:24,140 --> 00:16:26,850 between the different end states. 241 00:16:26,850 --> 00:16:30,030 And the coupling constant, which for an individual atom 242 00:16:30,030 --> 00:16:36,670 was little g is now factor of square root 2 enhanced. 243 00:16:36,670 --> 00:16:39,230 And we will see in a few minutes, that for n atoms, 244 00:16:39,230 --> 00:16:41,220 it's square root n enhanced. 245 00:16:41,220 --> 00:16:45,340 And if any speak, that's where the word super in superradiance 246 00:16:45,340 --> 00:16:45,860 comes from. 247 00:16:50,760 --> 00:16:54,920 Yeah, actually let me just quickly add 248 00:16:54,920 --> 00:16:58,410 the diagram for the single atom. 249 00:16:58,410 --> 00:17:03,010 The single atom has only an excited state, a ground state. 250 00:17:03,010 --> 00:17:06,480 It corresponds to s equals 1/2. 251 00:17:06,480 --> 00:17:11,369 And we have magnetic quantum numbers off plus 1/2 252 00:17:11,369 --> 00:17:13,990 and minus 1/2. 253 00:17:13,990 --> 00:17:18,980 And the coupling due to the light atom interaction 254 00:17:18,980 --> 00:17:20,579 goes with the coupling constant g. 255 00:17:34,180 --> 00:17:37,010 So the key message we have learned here 256 00:17:37,010 --> 00:17:44,810 is that when we have several atoms 257 00:17:44,810 --> 00:17:47,680 within an optical wavelength, we should 258 00:17:47,680 --> 00:17:50,590 use for their description symmetrized 259 00:17:50,590 --> 00:17:52,270 and antisymmetrized states. 260 00:17:52,270 --> 00:17:56,840 Or when we generalize to more than two atoms, 261 00:17:56,840 --> 00:18:00,930 we should just add the total angular momenta 262 00:18:00,930 --> 00:18:04,390 by treating each atom as pseudo spin 1/2. 263 00:18:04,390 --> 00:18:07,040 And it is this angular classification 264 00:18:07,040 --> 00:18:11,410 which tells us how the radiation proceeds. 265 00:18:11,410 --> 00:18:20,110 Because the coupling to the electromagnetic field 266 00:18:20,110 --> 00:18:24,820 is only involving the lowering and raising 267 00:18:24,820 --> 00:18:31,770 operators for the total spin. 268 00:18:31,770 --> 00:18:38,620 And this only acts on a manifold where 269 00:18:38,620 --> 00:18:45,820 the total spin s is conserved. 270 00:18:45,820 --> 00:18:49,450 And what we get is transitions with delta in plus minus 1. 271 00:18:53,870 --> 00:18:58,980 So the question, have those effects been observed? 272 00:18:58,980 --> 00:19:00,210 Yes, they have, actually. 273 00:19:00,210 --> 00:19:02,540 And they're important for a lot of research. 274 00:19:02,540 --> 00:19:07,460 But just for two atoms, the simplest observation 275 00:19:07,460 --> 00:19:09,320 is when you take two atoms-- let's 276 00:19:09,320 --> 00:19:12,120 say two sodium atoms-- bring them very close, 277 00:19:12,120 --> 00:19:14,590 and you form a sodium 2 molecule. 278 00:19:14,590 --> 00:19:21,160 And to some extent, in four states where the molecule are 279 00:19:21,160 --> 00:19:23,220 binding is not completely changing 280 00:19:23,220 --> 00:19:27,850 the electronic structure, we can regard the sodium 2 molecule 281 00:19:27,850 --> 00:19:30,400 as consisting of two sodium atoms. 282 00:19:30,400 --> 00:19:33,350 And indeed, if you do spectroscopy of the sodium 2 283 00:19:33,350 --> 00:19:36,390 molecule, you find some molecular states 284 00:19:36,390 --> 00:19:40,080 which are very long lived, like the subradiant states, which 285 00:19:40,080 --> 00:19:42,030 do not radiate at all, but then you 286 00:19:42,030 --> 00:19:45,970 find states which have a spontaneous emission rate which 287 00:19:45,970 --> 00:19:50,070 is about two times faster than the spontaneous emission rate. 288 00:19:50,070 --> 00:19:52,590 So you find that you can understand 289 00:19:52,590 --> 00:19:55,710 some radiative properties of molecules 290 00:19:55,710 --> 00:20:00,060 by assuming that they are related to the sub 291 00:20:00,060 --> 00:20:02,459 and superradiant state of the two atoms 292 00:20:02,459 --> 00:20:03,500 which form this molecule. 293 00:20:08,920 --> 00:20:14,810 So an example here is sodium 2 molecule. 294 00:20:21,270 --> 00:20:31,150 A state where the gamma molecule is approximately 2 times gamma 295 00:20:31,150 --> 00:20:34,570 sodium, or other states where it's very small. 296 00:20:38,220 --> 00:20:38,720 OK. 297 00:20:38,720 --> 00:20:44,790 Now we understand the basic four of superradiance in two atoms. 298 00:20:44,790 --> 00:20:52,485 And therefore, we can now generalize it to end particles. 299 00:21:02,550 --> 00:21:06,910 But before I use the spin algebra to describe 300 00:21:06,910 --> 00:21:13,720 end particles, I want to glean some intuition where we just 301 00:21:13,720 --> 00:21:16,070 consider-- and this takes us back 302 00:21:16,070 --> 00:21:19,020 to the beginning of the course-- where we consider 303 00:21:19,020 --> 00:21:22,000 end spins in a magnetic field. 304 00:21:30,890 --> 00:21:34,085 And I really invite you to think now completely classically. 305 00:21:36,545 --> 00:21:38,670 We'll describe it quantum mechanically in a moment. 306 00:21:38,670 --> 00:21:41,280 But I've often said in this course, 307 00:21:41,280 --> 00:21:43,950 if in doubt, if you have a classical description 308 00:21:43,950 --> 00:21:47,830 and a quantum mechanical and they seem to contradict, 309 00:21:47,830 --> 00:21:50,600 usually there is more truth in the classical description. 310 00:21:50,600 --> 00:21:52,630 It's so much easier to fool yourself 311 00:21:52,630 --> 00:21:55,040 with the formalism of quantum mechanics. 312 00:21:55,040 --> 00:22:02,310 So let's take end spins in a magnetic field 313 00:22:02,310 --> 00:22:04,300 and ask what happens. 314 00:22:04,300 --> 00:22:07,260 So we have end spins. 315 00:22:07,260 --> 00:22:08,670 So these are now real spins. 316 00:22:08,670 --> 00:22:10,180 They have a real magnetic moment. 317 00:22:10,180 --> 00:22:12,580 These are tiny little bar magnets. 318 00:22:12,580 --> 00:22:17,620 And we do pi over 2 pulse. 319 00:22:17,620 --> 00:22:23,490 And after we've done a pi over 2 products, 320 00:22:23,490 --> 00:22:26,960 the spins are aligned like this. 321 00:22:26,960 --> 00:22:30,130 Let's assume we had our magnetic field. 322 00:22:30,130 --> 00:22:34,020 And now what happens is these spins 323 00:22:34,020 --> 00:22:44,030 will precess at the line of frequency. 324 00:22:49,630 --> 00:22:52,200 So now you have your end spins. 325 00:22:52,200 --> 00:22:53,820 They precess together. 326 00:22:53,820 --> 00:22:58,120 And if you have a magnetic moment which oscillates, 327 00:22:58,120 --> 00:23:01,430 the classical equation of electromagnetism 328 00:23:01,430 --> 00:23:06,270 tells you that you have now a system which radiates. 329 00:23:11,460 --> 00:23:18,970 But compared to a single atom, the dipole moment 330 00:23:18,970 --> 00:23:25,423 is now n times the single atom dipole moment. 331 00:23:36,750 --> 00:23:42,750 So therefore, what do we expect for the radiated power? 332 00:23:47,450 --> 00:23:51,260 Well, if the electromagnetic radiation 333 00:23:51,260 --> 00:23:54,450 by an oscillating electric or an oscillating magnetic dipole 334 00:23:54,450 --> 00:23:57,890 moment scales with a dipole moment squared, 335 00:23:57,890 --> 00:24:02,810 therefore, we would expect that the power radiated 336 00:24:02,810 --> 00:24:07,960 is proportional to n squared. 337 00:24:07,960 --> 00:24:11,780 And that means I have to take the perfect of n [INAUDIBLE]. 338 00:24:11,780 --> 00:24:20,450 This means this is n times higher than if you assume 339 00:24:20,450 --> 00:24:27,520 you have n individual particles, and each of them 340 00:24:27,520 --> 00:24:30,250 emits electromagnetic radiation. 341 00:24:30,250 --> 00:24:32,560 what I'm telling you is if you scatter 342 00:24:32,560 --> 00:24:38,300 n spins through your laboratory, you excite them. 343 00:24:38,300 --> 00:24:40,980 Pi over 2 pulse, they radiate. 344 00:24:40,980 --> 00:24:44,160 They radiate a power which is proportion to n. 345 00:24:44,160 --> 00:24:47,400 But if you put them all together, localize them better 346 00:24:47,400 --> 00:24:50,150 than the wavelengths, their radiated power 347 00:24:50,150 --> 00:24:52,880 is proportional to n square, which is an n times 348 00:24:52,880 --> 00:24:53,380 enhancement. 349 00:24:57,740 --> 00:25:06,290 So the way how I put it for n spins-- 350 00:25:06,290 --> 00:25:12,410 and this is a situation of nuclear magnetic resonance-- 351 00:25:12,410 --> 00:25:18,110 this is the completely natural picture. 352 00:25:18,110 --> 00:25:22,110 But if I would have asked you the question-- let's 353 00:25:22,110 --> 00:25:25,690 take n atoms which are excited and put them close together, 354 00:25:25,690 --> 00:25:29,000 you say, well, each atom does spontaneous emission, 355 00:25:29,000 --> 00:25:34,100 and if you have n atoms, we get n times c intensity 356 00:25:34,100 --> 00:25:36,100 you would have gotten a different result. 357 00:25:36,100 --> 00:25:41,370 So we are so accustomed to look at spins in NMR 358 00:25:41,370 --> 00:25:47,000 as a coherent system, look that all the spins add up 359 00:25:47,000 --> 00:25:50,430 to one giant antenna, to one giant oscillating dipole 360 00:25:50,430 --> 00:25:53,960 moment, whereas for atoms, we are so much used 361 00:25:53,960 --> 00:25:56,700 to saying each atom is its own particle and thus 362 00:25:56,700 --> 00:25:57,810 its own thing. 363 00:26:03,380 --> 00:26:15,220 So for n excited atoms, they are usually 364 00:26:15,220 --> 00:26:16,400 regarded as independent. 365 00:26:19,580 --> 00:26:27,570 However-- and this is the message of today-- 366 00:26:27,570 --> 00:26:29,150 there shouldn't be a difference. 367 00:26:29,150 --> 00:26:32,383 All 2 level systems are equivalent. 368 00:26:43,620 --> 00:26:48,320 Side remark-- for NMR in spins, it is much, much easier 369 00:26:48,320 --> 00:26:51,310 to observe the effect, because the condition 370 00:26:51,310 --> 00:26:54,410 that all the spins are localized within one wavelength 371 00:26:54,410 --> 00:26:57,050 is always fulfilled if the wavelength is 372 00:26:57,050 --> 00:26:58,750 meter or kilometers. 373 00:26:58,750 --> 00:27:00,710 But if you have atoms which radiate 374 00:27:00,710 --> 00:27:05,490 at the optical wavelengths, this condition becomes nontrivial. 375 00:27:05,490 --> 00:27:09,120 That is partially responsible for the misconception 376 00:27:09,120 --> 00:27:11,340 that you treat the two-level system which 377 00:27:11,340 --> 00:27:13,493 is a spin in your head differently 378 00:27:13,493 --> 00:27:15,284 from the two-level system which is an atom. 379 00:27:25,380 --> 00:27:34,130 So the important difference here is lambda. 380 00:27:38,660 --> 00:27:41,320 And we have to compare it with a sample size. 381 00:27:41,320 --> 00:27:46,240 And usually, the sample size is much larger 382 00:27:46,240 --> 00:27:51,710 in the optical domain, and is much, much smaller 383 00:27:51,710 --> 00:27:55,360 in the NMR domain. 384 00:27:55,360 --> 00:27:58,640 However-- and that's what we'll see 385 00:27:58,640 --> 00:28:02,060 during the remainder of this class-- 386 00:28:02,060 --> 00:28:05,870 some of the dramatic consequences of superradiance 387 00:28:05,870 --> 00:28:09,400 will even survive under suitable conditions 388 00:28:09,400 --> 00:28:11,210 in the extended samples. 389 00:28:11,210 --> 00:28:14,560 So when we have samples of excited atoms much, 390 00:28:14,560 --> 00:28:16,370 much larger than the optical wavelengths, 391 00:28:16,370 --> 00:28:18,510 we can still observe superradiance. 392 00:28:18,510 --> 00:28:20,960 So therefore, for pedagogical reasons, 393 00:28:20,960 --> 00:28:24,380 I first complete the focus on the case 394 00:28:24,380 --> 00:28:26,730 that everything is tightly localized. 395 00:28:26,730 --> 00:28:28,800 We derive some interesting equations, 396 00:28:28,800 --> 00:28:30,590 and then we see how they are modified 397 00:28:30,590 --> 00:28:32,160 when we go to extended samples. 398 00:28:35,170 --> 00:28:39,850 But I want to say, the intuition from spin systems, 399 00:28:39,850 --> 00:28:42,265 the intuition from classical precession 400 00:28:42,265 --> 00:28:45,820 and nuclear magnetic resonance, will help us 401 00:28:45,820 --> 00:28:48,720 what happens for electronically excited atoms. 402 00:28:48,720 --> 00:28:51,460 So we want to use this other spin 1/2 403 00:28:51,460 --> 00:28:54,678 system as a powerful analogy to guide us. 404 00:28:58,550 --> 00:29:04,330 So before I start with the angular momentum formalism, 405 00:29:04,330 --> 00:29:15,450 I want to emphasize that what are the ingredients here. 406 00:29:15,450 --> 00:29:25,490 Well, we're talking about coherence-- coherent radiation, 407 00:29:25,490 --> 00:29:29,250 coherence between atoms-- and we'll talk about radiation. 408 00:29:29,250 --> 00:29:33,160 And the important part here is the following. 409 00:29:33,160 --> 00:29:38,360 That when we talk about radiation, 410 00:29:38,360 --> 00:29:50,230 we have the situation that all atoms interact 411 00:29:50,230 --> 00:29:51,730 with a common radiation field. 412 00:29:57,550 --> 00:30:01,120 In other words, all the spins, all the atoms 413 00:30:01,120 --> 00:30:04,373 have to emit their photons into the same mode 414 00:30:04,373 --> 00:30:06,460 of the electromagnetic field. 415 00:30:06,460 --> 00:30:09,340 And therefore, you may be right in some limit 416 00:30:09,340 --> 00:30:12,850 that the atoms are independent, but not the photons they emit. 417 00:30:12,850 --> 00:30:15,530 They go into the same mode. 418 00:30:15,530 --> 00:30:30,563 And therefore, the emitted photons cannot be treated 419 00:30:30,563 --> 00:30:31,146 independently. 420 00:30:43,980 --> 00:30:45,610 And that's why the classical picture 421 00:30:45,610 --> 00:30:47,210 is so powerful for that. 422 00:30:47,210 --> 00:30:49,500 Because in the classic picture, we 423 00:30:49,500 --> 00:30:52,710 do a coherent summation of the field amplitudes. 424 00:30:52,710 --> 00:30:55,240 So we have constructive interference. 425 00:30:55,240 --> 00:30:57,960 The superposition principle of field amplitudes 426 00:30:57,960 --> 00:31:03,710 build into our equations and deeply engraved in our brains. 427 00:31:03,710 --> 00:31:06,380 And that's why when we use classical arguments, 428 00:31:06,380 --> 00:31:10,820 we automatically account for that the photons interfere, 429 00:31:10,820 --> 00:31:13,245 that the photons are emitted into the same mode 430 00:31:13,245 --> 00:31:16,610 of the electromagnetic field. 431 00:31:16,610 --> 00:31:21,640 And eventually, this leads to the phenomenon 432 00:31:21,640 --> 00:31:27,870 that we have coherence and enhancement when 433 00:31:27,870 --> 00:31:39,480 we look at spontaneous emission for n atoms 434 00:31:39,480 --> 00:31:40,980 which are sufficiently localized. 435 00:31:49,560 --> 00:31:59,330 So let me also discuss what we have assumed here. 436 00:31:59,330 --> 00:32:02,560 Number one is, we have assumed we 437 00:32:02,560 --> 00:32:06,740 have a localization of the sample 438 00:32:06,740 --> 00:32:10,330 smaller than the optical wavelengths. 439 00:32:10,330 --> 00:32:12,340 The other thing-- and this is really important-- 440 00:32:12,340 --> 00:32:15,530 we are talking here about a collective phenomenon 441 00:32:15,530 --> 00:32:20,555 where n atoms act together and do something. 442 00:32:20,555 --> 00:32:22,430 They develop the phenomenon of superradiance. 443 00:32:22,430 --> 00:32:25,336 They decay much, much faster than any individual atom could 444 00:32:25,336 --> 00:32:26,490 do by itself. 445 00:32:26,490 --> 00:32:30,720 But nevertheless, we have not assumed-- or we have actually 446 00:32:30,720 --> 00:32:33,120 excluded in our description-- that there 447 00:32:33,120 --> 00:32:35,470 is any direct interaction between the atoms. 448 00:32:35,470 --> 00:32:37,890 The atoms have no [INAUDIBLE] interaction. 449 00:32:37,890 --> 00:32:39,500 They're not forming molecules. 450 00:32:39,500 --> 00:32:43,420 They're not part of a solid with shared electrons. 451 00:32:43,420 --> 00:32:46,720 The atoms are, in that sense, non-interacting. 452 00:32:46,720 --> 00:32:49,560 And therefore, in a way, as long as 453 00:32:49,560 --> 00:32:51,040 they are just atoms, independent. 454 00:33:10,560 --> 00:33:18,990 Finally-- and I want you to think about it-- 455 00:33:18,990 --> 00:33:21,710 you can think about already for two atoms 456 00:33:21,710 --> 00:33:24,300 before we generalize it to n atoms. 457 00:33:24,300 --> 00:33:26,370 Think about it. 458 00:33:26,370 --> 00:33:32,610 What was really the assumption about the atoms? 459 00:33:32,610 --> 00:33:36,910 Do the atoms have to be bosons to be in this symmetric state? 460 00:33:39,850 --> 00:33:42,466 Can they be fermions? 461 00:33:42,466 --> 00:33:45,600 Or can they be even distinguishable particles? 462 00:33:50,230 --> 00:33:52,880 If the two atoms where one would be a sodium atom 463 00:33:52,880 --> 00:33:55,740 and one would be a rubidium atom-- but let's just 464 00:33:55,740 --> 00:33:59,560 say we live in a world where sodium and rubidium atoms emit 465 00:33:59,560 --> 00:34:02,040 exactly the same color of light. 466 00:34:02,040 --> 00:34:05,330 Would we have been sub and superradiant state 467 00:34:05,330 --> 00:34:08,710 for two atoms, one of which is sodium and one of which 468 00:34:08,710 --> 00:34:09,210 is rubidium? 469 00:34:24,909 --> 00:34:25,409 Yes? 470 00:34:25,409 --> 00:34:26,361 AUDIENCE: [INAUDIBLE]. 471 00:34:33,929 --> 00:34:34,720 PROFESSOR: Exactly. 472 00:34:34,720 --> 00:34:37,120 AUDIENCE: [INAUDIBLE]. 473 00:34:37,120 --> 00:34:39,909 PROFESSOR: Yes. 474 00:34:39,909 --> 00:34:41,429 And that confuses many people. 475 00:34:41,429 --> 00:34:44,020 It is the indistinguishability of the photons 476 00:34:44,020 --> 00:34:45,440 they have emitted. 477 00:34:45,440 --> 00:34:48,400 It is the common mode where the photons are emitted. 478 00:34:48,400 --> 00:34:53,110 The atoms can be distinguished. 479 00:34:53,110 --> 00:34:57,820 Also, we've made the assumption that the atoms are localized to 480 00:34:57,820 --> 00:35:05,160 within an area which is much smaller than lambda. 481 00:35:05,160 --> 00:35:08,530 But you could imagine you have a solid state matrix 482 00:35:08,530 --> 00:35:11,660 and you have one atom here, one atom there, 483 00:35:11,660 --> 00:35:14,510 and you can go with a microscope and distinguish them. 484 00:35:14,510 --> 00:35:16,890 So therefore, the moment you can distinguish them 485 00:35:16,890 --> 00:35:18,900 because they are pinned down in a lattice-- 486 00:35:18,900 --> 00:35:22,180 or if you don't like a lattice, take two microscopic ion traps 487 00:35:22,180 --> 00:35:24,380 a few nanometers apart, and you tightly 488 00:35:24,380 --> 00:35:26,720 hold onto two ions-- it doesn't matter 489 00:35:26,720 --> 00:35:28,910 whether they are bosons or fermions. 490 00:35:28,910 --> 00:35:31,740 It only matters whether your bosons or fermions 491 00:35:31,740 --> 00:35:34,520 when the atomic wave functions overlap 492 00:35:34,520 --> 00:35:35,810 and you have to symmatrize it. 493 00:35:35,810 --> 00:35:40,130 As long as you have two atoms which are spatially separated, 494 00:35:40,130 --> 00:35:44,250 it doesn't matter whether they are bosons or fermions. 495 00:35:44,250 --> 00:35:47,550 And that also means they can be completely different atoms. 496 00:35:47,550 --> 00:35:49,010 You can already call the Boson A, 497 00:35:49,010 --> 00:35:52,300 Boson B. Now you can call it sodium and rubidium, 498 00:35:52,300 --> 00:35:55,940 and they can have different numbers of nucleus. 499 00:35:55,940 --> 00:35:58,980 They can be different numbers of neutrons in the nucleus. 500 00:35:58,980 --> 00:36:01,490 It could be different isotopes of the same atom. 501 00:36:01,490 --> 00:36:06,230 The whole collective phenomenon comes 502 00:36:06,230 --> 00:36:08,986 when they emit a photon into the same mode. 503 00:36:39,550 --> 00:36:40,250 OK. 504 00:36:40,250 --> 00:36:49,129 So now we want to treat to a formula 505 00:36:49,129 --> 00:36:50,295 treatment for end particles. 506 00:36:53,680 --> 00:36:58,950 So we have now the individual pseudo spins one half. 507 00:36:58,950 --> 00:37:02,460 We perform now with some overall end particles. 508 00:37:02,460 --> 00:37:05,350 We get the total spin s. 509 00:37:05,350 --> 00:37:07,440 The total spin s quantum number has 510 00:37:07,440 --> 00:37:09,830 to be smaller or equal to n over 2, 511 00:37:09,830 --> 00:37:12,335 because we have m spin 1/2 systems. 512 00:37:16,060 --> 00:37:24,100 The end quantum number is 1/2 times 513 00:37:24,100 --> 00:37:27,990 the difference of the atoms which 514 00:37:27,990 --> 00:37:30,495 are in the excited state minus the atoms which 515 00:37:30,495 --> 00:37:32,270 are in the ground state. 516 00:37:32,270 --> 00:37:34,250 And this is, of course, trivial. 517 00:37:34,250 --> 00:37:36,210 Trivially must be smaller than s. 518 00:37:38,790 --> 00:37:45,520 Because m is this z component of s. 519 00:37:45,520 --> 00:37:50,990 And we are now describing the system 520 00:37:50,990 --> 00:37:56,030 by the eigenstates s and n of the collective spin. 521 00:38:00,380 --> 00:38:12,270 So that means we have the following situation. 522 00:38:12,270 --> 00:38:18,470 We have a manifold-- we want to show now all the energy levels. 523 00:38:18,470 --> 00:38:22,920 We have a manifold which has a maximum spin n over 2. 524 00:38:22,920 --> 00:38:26,790 The next manifold has n over 2 minus 1. 525 00:38:26,790 --> 00:38:32,150 And the last one has-- let's assume 526 00:38:32,150 --> 00:38:35,480 we have an odd number of particles-- x equals 1/2. 527 00:38:35,480 --> 00:38:42,700 So here, we have now n energy levels. 528 00:38:42,700 --> 00:38:46,582 We can go from all the n atoms excited 529 00:38:46,582 --> 00:38:50,970 to all the n atoms being de-excited. 530 00:38:50,970 --> 00:38:58,545 In the following manifold, we have s is one less. 531 00:39:01,110 --> 00:39:07,410 And therefore, we have a letter of states 532 00:39:07,410 --> 00:39:09,030 which is a little bit shorter. 533 00:39:09,030 --> 00:39:16,360 And eventually, for s equals 1/2, 534 00:39:16,360 --> 00:39:19,305 we have only two components plus 1/2 and minus 1/2. 535 00:39:25,210 --> 00:39:34,900 So those levels interact with the electromagnetic field. 536 00:39:37,470 --> 00:39:40,490 The operator of the electromagnetic field, 537 00:39:40,490 --> 00:39:43,300 we have already derived that, involves 538 00:39:43,300 --> 00:39:47,400 the sum of all of the little sigma pluses, sigma i pluses, 539 00:39:47,400 --> 00:39:53,960 and we call the sum of all of them s plus and s minus. 540 00:39:53,960 --> 00:40:04,500 And the matrix element is now for spontaneous emission. 541 00:40:04,500 --> 00:40:07,370 You have a state with sm. 542 00:40:07,370 --> 00:40:13,900 s minus is the lowering operator for the n particle system, 543 00:40:13,900 --> 00:40:17,090 so it goes from a state with a certain number 544 00:40:17,090 --> 00:40:20,600 of atomic excitations to one excitation less, 545 00:40:20,600 --> 00:40:25,150 and that means this is the act of emitting spontaneously 546 00:40:25,150 --> 00:40:27,400 one photon. 547 00:40:27,400 --> 00:40:32,960 The operator s minus stays within the s manifold, 548 00:40:32,960 --> 00:40:36,560 so we state in the same letter, which 549 00:40:36,560 --> 00:40:39,350 is characterized by the quantum number s. 550 00:40:39,350 --> 00:40:42,230 But we lower the end quantum number by one. 551 00:40:42,230 --> 00:40:44,000 The end quantum number is a measure 552 00:40:44,000 --> 00:40:47,130 of the number of excitations. 553 00:40:47,130 --> 00:40:51,680 And we know from general spin algebra 554 00:40:51,680 --> 00:41:02,020 that this matrix element is s minus m plus 1 times s plus n. 555 00:41:10,490 --> 00:41:12,020 There are, of course, pre-factors 556 00:41:12,020 --> 00:41:15,770 like the dipole matrix element of a single atom. 557 00:41:15,770 --> 00:41:20,890 But I always want to normalize things to a single atom. 558 00:41:20,890 --> 00:41:25,900 And by just using the square root, 559 00:41:25,900 --> 00:41:30,470 if you have a single particle, which is in the s to m 560 00:41:30,470 --> 00:41:36,260 equals 1/2 state, then you see that it this square root 561 00:41:36,260 --> 00:41:38,370 is just 1. 562 00:41:38,370 --> 00:41:42,940 So therefore, for when I discuss now the relative strengths 563 00:41:42,940 --> 00:41:46,690 of the transitions between those eigenstates, 564 00:41:46,690 --> 00:41:49,060 I've always normalized to a single particle. 565 00:41:49,060 --> 00:41:52,370 For a single particle, the transition matrix element is 1. 566 00:42:03,900 --> 00:42:04,980 OK. 567 00:42:04,980 --> 00:42:09,620 So therefore, what we want to discuss now is, 568 00:42:09,620 --> 00:42:13,140 we want to discuss the rate, which 569 00:42:13,140 --> 00:42:16,720 is the matrix element squared, or the intensity 570 00:42:16,720 --> 00:42:31,140 of the observed radiation relative to a single particle. 571 00:42:38,100 --> 00:42:42,180 So the intensity-- and this is what we are talking about-- 572 00:42:42,180 --> 00:42:45,910 is now the square root of the square root, or the square 573 00:42:45,910 --> 00:42:53,974 of the square root, which is s minus 1 plus m times s plus m. 574 00:42:56,820 --> 00:43:00,870 Pretty much, this is the complete description 575 00:43:00,870 --> 00:43:04,660 of superradiance for strongly localized atoms. 576 00:43:04,660 --> 00:43:07,250 It's all in this one formula. 577 00:43:07,250 --> 00:43:10,120 Once we learned how to classify the states, 578 00:43:10,120 --> 00:43:12,550 we can just borrow all the results 579 00:43:12,550 --> 00:43:17,590 from angular momentum, addition, and angular momentum operators. 580 00:43:17,590 --> 00:43:24,120 So I want to use this formula for the intensity 581 00:43:24,120 --> 00:43:31,190 and look at which is the most superradiant state, the state 582 00:43:31,190 --> 00:43:33,780 where all the particles are symmetric. 583 00:43:33,780 --> 00:43:36,220 And this is a state with s equals 584 00:43:36,220 --> 00:43:41,380 n where the spin is n over 2. 585 00:43:41,380 --> 00:43:48,670 So I'm looking now at the letter of m states, 586 00:43:48,670 --> 00:43:52,350 and I want to figure out what happens. 587 00:43:52,350 --> 00:43:55,810 So the maximum m state is m equals s, 588 00:43:55,810 --> 00:43:59,460 all the atoms are excited. 589 00:43:59,460 --> 00:44:05,650 And now the first photon gets emitted. 590 00:44:05,650 --> 00:44:19,780 So just put s equals m equals n half into the formula 591 00:44:19,780 --> 00:44:26,390 for the intensity, and you find that the intensity gives us 592 00:44:26,390 --> 00:44:28,173 this expression is just n. 593 00:44:31,490 --> 00:44:36,790 So we have n excited atoms, and they initially 594 00:44:36,790 --> 00:44:41,130 emit with an intensity which is n. 595 00:44:41,130 --> 00:44:45,630 And this is the same as for n completely independent atoms. 596 00:44:50,640 --> 00:44:56,240 So nothing really special to write home about. 597 00:44:56,240 --> 00:44:59,570 But now we should go further down the ladder, 598 00:44:59,570 --> 00:45:05,050 and let's look at the state which has m equals zero. 599 00:45:05,050 --> 00:45:16,320 Well, then the intensity of the matrix element 600 00:45:16,320 --> 00:45:20,950 squared for the transition, which goes to m equals zero, 601 00:45:20,950 --> 00:45:25,820 has an intensity which is n over 2. 602 00:45:25,820 --> 00:45:29,540 I will just look. s is n over 2, and m is zero. 603 00:45:29,540 --> 00:45:31,300 So we have the question whether we 604 00:45:31,300 --> 00:45:37,020 have odd or even number of particles, 605 00:45:37,020 --> 00:45:38,410 but it doesn't really matter. 606 00:45:38,410 --> 00:45:42,850 What dominates is always the big factor n over 2. 607 00:45:42,850 --> 00:45:46,850 So what we find out is that we have 608 00:45:46,850 --> 00:45:50,680 an enhancement, huge enhancement over independent atom, 609 00:45:50,680 --> 00:45:53,830 because this intensity goes with n squared, 610 00:45:53,830 --> 00:45:56,560 and this proportionality to n squared, 611 00:45:56,560 --> 00:45:58,060 this is a hallmark of superradiance. 612 00:46:00,710 --> 00:46:06,980 So this is what is characteristic 613 00:46:06,980 --> 00:46:09,040 for superradiance. 614 00:46:09,040 --> 00:46:17,920 We have an n times enhancement relative to the a singular 615 00:46:17,920 --> 00:46:19,680 atom. 616 00:46:19,680 --> 00:46:37,440 So this is one important aspect. 617 00:46:37,440 --> 00:46:41,860 Now, in the classical picture, that 618 00:46:41,860 --> 00:46:43,420 should come very naturally. 619 00:46:43,420 --> 00:46:45,610 If you have all the spins aligned 620 00:46:45,610 --> 00:46:48,460 and they start the [INAUDIBLE] procession, 621 00:46:48,460 --> 00:46:51,470 there is not a lot of oscillating dipole moment. 622 00:46:51,470 --> 00:46:55,460 But when half of the spins are de-excited, 623 00:46:55,460 --> 00:46:57,410 they are now in the XY plane. 624 00:46:57,410 --> 00:47:02,310 Now you have this giant antenna which oscillates and radiates. 625 00:47:02,310 --> 00:47:04,230 So it's clear that at the beginning, 626 00:47:04,230 --> 00:47:07,310 the effect is less pronounced, and if you're halfway 627 00:47:07,310 --> 00:47:10,225 down the Bloch sphere, then you would expect this n times 628 00:47:10,225 --> 00:47:10,725 enhancement. 629 00:47:16,190 --> 00:47:18,610 But now let's go further down the ladder 630 00:47:18,610 --> 00:47:24,530 and ask what happens when we arrive at the end. 631 00:47:24,530 --> 00:47:31,330 So I'm asking now, what is the intensity when 632 00:47:31,330 --> 00:47:36,150 the last photon gets emitted. 633 00:47:39,090 --> 00:47:42,570 There is only one excitation in the system. 634 00:47:42,570 --> 00:47:47,510 And the answer is, it's not one like an independent atom. 635 00:47:47,510 --> 00:47:52,950 If you inspect the square root expression, you find it's n. 636 00:47:52,950 --> 00:47:55,870 So we have one excitation in the system, 637 00:47:55,870 --> 00:47:57,270 but it's completely symmetrized. 638 00:48:02,750 --> 00:48:05,320 And therefore, we have an n times enhancement. 639 00:48:05,320 --> 00:48:11,410 And I want to show you where it comes about. 640 00:48:11,410 --> 00:48:13,750 So there's only one particle excited. 641 00:48:17,570 --> 00:48:20,735 And here, we have an n times enhancement. 642 00:48:24,880 --> 00:48:28,630 By the way, the states with the classification s and m 643 00:48:28,630 --> 00:48:31,070 are called the [INAUDIBLE] state. 644 00:48:31,070 --> 00:48:34,350 And this state here, which has a single excitation 645 00:48:34,350 --> 00:48:37,830 but it radiates n times faster than a single atom, 646 00:48:37,830 --> 00:48:39,800 is a very special [INAUDIBLE] state. 647 00:48:39,800 --> 00:48:43,580 And there is currently an effort in Professor [INAUDIBLE] lab 648 00:48:43,580 --> 00:48:47,530 to realize in a very well-controlled way 649 00:48:47,530 --> 00:48:50,790 this special [INAUDIBLE] state in the laboratory. 650 00:48:50,790 --> 00:48:53,340 There are non-classical states, because they're not 651 00:48:53,340 --> 00:48:57,790 behaving as you would maybe naturally assume 652 00:48:57,790 --> 00:49:00,930 a system with a single excitation to behave. 653 00:49:06,140 --> 00:49:11,290 So let's maybe try to shed some light on it. 654 00:49:11,290 --> 00:49:14,490 One way how you can intuitively understand superradiance 655 00:49:14,490 --> 00:49:16,810 is really with a classical antenna picture 656 00:49:16,810 --> 00:49:20,170 that you have n spins which form a microscopic dipole 657 00:49:20,170 --> 00:49:22,690 moment which oscillates. 658 00:49:22,690 --> 00:49:25,860 And this is a very nice picture to understand the n times 659 00:49:25,860 --> 00:49:29,230 enhancement when we have half of the atoms 660 00:49:29,230 --> 00:49:31,690 excited and the other half de-excited. 661 00:49:31,690 --> 00:49:33,380 Let me now give you a nice argument 662 00:49:33,380 --> 00:49:38,290 which explains why a single excitation in this system 663 00:49:38,290 --> 00:49:41,620 now leads to an n times enhanced decay. 664 00:49:44,400 --> 00:49:47,860 The situation is that the initial state 665 00:49:47,860 --> 00:49:53,090 for this last photon is, we have an excited atom, 666 00:49:53,090 --> 00:49:56,640 and all the atoms are in the ground state. 667 00:49:56,640 --> 00:50:02,420 However, we could also have in this nomenclature 668 00:50:02,420 --> 00:50:03,765 the second atom excited. 669 00:50:09,460 --> 00:50:15,070 Or we could have the third one excited, and so on. 670 00:50:15,070 --> 00:50:19,110 So therefore, what we have is-- because we 671 00:50:19,110 --> 00:50:25,780 are in the left-most [INAUDIBLE] which has the maximum s spin 672 00:50:25,780 --> 00:50:28,060 quantum number of n over 2, that means 673 00:50:28,060 --> 00:50:30,260 everything is fully symmetrized. 674 00:50:30,260 --> 00:50:32,370 So therefore, we have to fully symmetrize 675 00:50:32,370 --> 00:50:35,800 by summing over the n possibilities. 676 00:50:35,800 --> 00:50:44,420 And our final state is, of course, the fully symmetrized 677 00:50:44,420 --> 00:50:46,040 [INAUDIBLE] state. 678 00:50:46,040 --> 00:50:49,110 And now you realize that you have 679 00:50:49,110 --> 00:50:55,510 a coherent summation over-- you have n contributions. 680 00:51:00,480 --> 00:51:09,380 So therefore, the matrix element has n contributions 681 00:51:09,380 --> 00:51:10,860 compared to single atom. 682 00:51:10,860 --> 00:51:13,730 The normalization is only square root n. 683 00:51:13,730 --> 00:51:21,640 So therefore, the matrix element is square root n times larger 684 00:51:21,640 --> 00:51:24,710 than for an individual atom. 685 00:51:30,380 --> 00:51:34,080 So by simply having one atom excited and n 686 00:51:34,080 --> 00:51:38,360 minus 1 atom not excited, but if you now 687 00:51:38,360 --> 00:51:40,580 have the fully symmetrized state, 688 00:51:40,580 --> 00:51:46,350 you don't know for fundamental reasons which atom is excited. 689 00:51:46,350 --> 00:51:49,010 You have a superposition state where the excitation 690 00:51:49,010 --> 00:51:51,140 can be with either of the atom. 691 00:51:51,140 --> 00:51:54,550 This state, which has a similar quantum of excitation, 692 00:51:54,550 --> 00:51:57,528 radiates n times faster than a single atom would. 693 00:52:05,510 --> 00:52:08,350 Let me make a side remark. 694 00:52:13,470 --> 00:52:19,070 Maybe some of you remember when we went to [INAUDIBLE] QED, 695 00:52:19,070 --> 00:52:22,390 we had just proudly quantized the electromagnetic field, 696 00:52:22,390 --> 00:52:25,530 and we discussed the vacuum Rabi splitting. 697 00:52:25,530 --> 00:52:29,060 And I told you that the vacuum Rabi splitting 698 00:52:29,060 --> 00:52:37,420 is if the cavity is not empty but is filled with n atoms, 699 00:52:37,420 --> 00:52:42,790 because of the matrix element of the a dagger operator, 700 00:52:42,790 --> 00:52:46,240 you get an enhancement of the vacuum Rabi splitting, which 701 00:52:46,240 --> 00:52:49,010 is square root n, then photon number. 702 00:52:49,010 --> 00:52:52,110 But then I showed you the important first observation, 703 00:52:52,110 --> 00:52:55,410 the pioneering research at Cal Tech by JF Kimball and Gerhard 704 00:52:55,410 --> 00:52:58,570 Rempe, and they didn't vary the photon number. 705 00:52:58,570 --> 00:53:00,810 They varied the atom number. 706 00:53:00,810 --> 00:53:03,650 And when they had more flux in the atomic beam, 707 00:53:03,650 --> 00:53:06,840 the Rabi splitting became larger and larger. 708 00:53:06,840 --> 00:53:10,320 Well, we've just learned that when 709 00:53:10,320 --> 00:53:19,630 we have n atoms, that the matrix element for emitting the photon 710 00:53:19,630 --> 00:53:22,020 is square root n times enhanced. 711 00:53:22,020 --> 00:53:25,730 So if you put n atoms in a cavity filled with little n 712 00:53:25,730 --> 00:53:30,140 photons, the Rabi splitting between the two modes 713 00:53:30,140 --> 00:53:33,080 has square root n plus 1 in the photon number 714 00:53:33,080 --> 00:53:37,200 and square root big n in the atom number. 715 00:53:37,200 --> 00:53:40,470 So the effect I've shown you in the demonstration of the vacuum 716 00:53:40,470 --> 00:53:44,160 Rabi splitting is this scaling with the atom number. 717 00:53:44,160 --> 00:53:47,772 This actually can be understood as a superradiant effect. 718 00:54:02,550 --> 00:54:04,970 OK. 719 00:54:04,970 --> 00:54:07,910 So that's pretty much what I wanted 720 00:54:07,910 --> 00:54:14,240 to tell you about the basic phenomenon of superradiance. 721 00:54:14,240 --> 00:54:15,830 Now I want to discuss two more things. 722 00:54:15,830 --> 00:54:18,596 One is superradiance in an extended sample. 723 00:54:18,596 --> 00:54:20,290 Ad we have time for that. 724 00:54:20,290 --> 00:54:25,190 But I also want to discuss with you the question. 725 00:54:25,190 --> 00:54:27,230 Let's assume we have the same system, 726 00:54:27,230 --> 00:54:30,480 and we just convinced ourselves, yes, it's superradiant. 727 00:54:30,480 --> 00:54:33,157 Photons are emitted n times faster. 728 00:54:36,020 --> 00:54:41,900 Now, what would you think will happen when we are not 729 00:54:41,900 --> 00:54:44,060 looking for spontaneous emission, 730 00:54:44,060 --> 00:54:46,100 but we shine a laser light on it, 731 00:54:46,100 --> 00:54:49,750 and we are asking for induced emission? 732 00:54:49,750 --> 00:54:52,987 Or the other way around-- we ask-- 733 00:54:52,987 --> 00:54:54,820 and you know that from Einstein's treatment, 734 00:54:54,820 --> 00:54:56,610 it's completely recipocal-- where 735 00:54:56,610 --> 00:55:01,253 we are asking the question, what happens to the absorption 736 00:55:01,253 --> 00:55:01,753 process? 737 00:55:06,690 --> 00:55:13,530 So is a stimulated emission process or an absorption 738 00:55:13,530 --> 00:55:17,685 process, are they also enhanced n times? 739 00:55:21,329 --> 00:55:21,870 I don't know. 740 00:55:21,870 --> 00:55:26,307 Do you have any opinions about that? 741 00:55:41,714 --> 00:55:42,708 AUDIENCE: [INAUDIBLE]. 742 00:55:56,540 --> 00:55:58,680 PROFESSOR: Yes. 743 00:55:58,680 --> 00:56:00,520 It's a subtle way of counting. 744 00:56:00,520 --> 00:56:03,710 I've shown you that certain matrix element-- especially 745 00:56:03,710 --> 00:56:09,080 the matrix element when the spin is in the middle, 746 00:56:09,080 --> 00:56:14,420 is at 90 degrees-- that we have matrix elements which 747 00:56:14,420 --> 00:56:16,470 are n times enhanced. 748 00:56:16,470 --> 00:56:20,270 And of course, if you ask for absorption or stimulated 749 00:56:20,270 --> 00:56:23,740 emission, we are talking to a system which has an n times 750 00:56:23,740 --> 00:56:25,760 enhanced matrix element. 751 00:56:25,760 --> 00:56:29,510 And you would say, things go n times faster. 752 00:56:29,510 --> 00:56:33,000 Why don't you hold this thought for a moment? 753 00:56:33,000 --> 00:56:34,766 Now let me just wear another hat and let 754 00:56:34,766 --> 00:56:38,410 me say that we have assumed that we have n independent 755 00:56:38,410 --> 00:56:40,740 spins that are closely next to each other, 756 00:56:40,740 --> 00:56:42,880 but they're not interacting. 757 00:56:42,880 --> 00:56:50,110 And now I take these n spins, and for stimulated emission 758 00:56:50,110 --> 00:56:53,670 and absorption, we can just use a picture of Rabi oscillation. 759 00:56:53,670 --> 00:56:55,570 On the first Rabi cycle, we emit. 760 00:56:55,570 --> 00:56:57,990 On the next Rabi cycle, we absorb. 761 00:56:57,990 --> 00:57:00,030 So if I take now-- and why don't you 762 00:57:00,030 --> 00:57:03,972 think not about these pseudo spins electronically, 763 00:57:03,972 --> 00:57:05,430 atoms with [INAUDIBLE] excitation-- 764 00:57:05,430 --> 00:57:08,350 just think of real spins which have a magnetic moment, 765 00:57:08,350 --> 00:57:10,750 and you drive them to the magnetic field. 766 00:57:10,750 --> 00:57:12,890 So now you have your n little spins. 767 00:57:12,890 --> 00:57:14,840 You apply a magnetic field to them, 768 00:57:14,840 --> 00:57:16,760 time-dependent magnetic field, and 769 00:57:16,760 --> 00:57:18,860 the time-dependent magnetic field 770 00:57:18,860 --> 00:57:23,070 is now driving the spin in Rabi oscillations. 771 00:57:23,070 --> 00:57:27,630 And the external drive field talks to one spin, 772 00:57:27,630 --> 00:57:30,040 talks to the next, talks to all of them. 773 00:57:30,040 --> 00:57:33,920 But each of the spins does exactly 774 00:57:33,920 --> 00:57:37,830 the same Rabi oscillation it would do if all the other atoms 775 00:57:37,830 --> 00:57:40,840 where not present. 776 00:57:40,840 --> 00:57:44,480 So the picture is, you have an external field. 777 00:57:44,480 --> 00:57:47,020 All the atoms couple to the external field. 778 00:57:47,020 --> 00:57:49,980 But the coupling of each atom to this external field 779 00:57:49,980 --> 00:57:52,500 is exactly the same as of a single atom 780 00:57:52,500 --> 00:57:53,990 coupling to the external field. 781 00:57:53,990 --> 00:57:56,890 And the Rabi frequency for each atom 782 00:57:56,890 --> 00:57:59,230 is exactly the Rabi frequency you 783 00:57:59,230 --> 00:58:01,990 would get for a single atom. 784 00:58:01,990 --> 00:58:03,800 So therefore, based on this picture, 785 00:58:03,800 --> 00:58:06,750 I would expect that I have my end spins, 786 00:58:06,750 --> 00:58:09,180 and these can now be real spins with a magnetic moment, 787 00:58:09,180 --> 00:58:12,590 or can be atoms in the electronically excited state. 788 00:58:12,590 --> 00:58:15,490 I coherently drive them with a drive field, 789 00:58:15,490 --> 00:58:17,680 and they will do Rabi oscillations. 790 00:58:17,680 --> 00:58:20,140 But the frequency or Rabi oscillations 791 00:58:20,140 --> 00:58:23,950 will be the same as for a single atom. 792 00:58:23,950 --> 00:58:24,450 OK. 793 00:58:24,450 --> 00:58:26,890 We've just held the thought that their matrix 794 00:58:26,890 --> 00:58:30,910 elements in the Dickey states, which are square root 795 00:58:30,910 --> 00:58:34,270 n times larger, and the Dickey state 796 00:58:34,270 --> 00:58:36,680 seem to suggest something to us which 797 00:58:36,680 --> 00:58:38,950 would say there should be an enhancement, 798 00:58:38,950 --> 00:58:42,470 whereas the analysis in independent atoms, 799 00:58:42,470 --> 00:58:48,210 which are driven by an external field, also seems compelling. 800 00:58:48,210 --> 00:58:51,570 So now we have to reconcile the two approaches. 801 00:58:51,570 --> 00:58:53,920 Is the question clear? 802 00:58:53,920 --> 00:58:56,980 We want to figure out-- we have matrix elements in the Dickey 803 00:58:56,980 --> 00:59:00,010 state which suggest enhancement, but the simple picture 804 00:59:00,010 --> 00:59:02,960 of n independent atoms driven by an external field 805 00:59:02,960 --> 00:59:06,910 would say there is no enhancement. 806 00:59:06,910 --> 00:59:07,700 OK. 807 00:59:07,700 --> 00:59:13,440 So let me just write down more formally what I explained. 808 00:59:13,440 --> 00:59:17,100 When we have an initial state, which is all the atoms 809 00:59:17,100 --> 00:59:24,910 are in the ground state to the power n, 810 00:59:24,910 --> 00:59:29,720 and the state n would evolve when 811 00:59:29,720 --> 00:59:39,000 it is driven in a state phi of t, 812 00:59:39,000 --> 00:59:49,200 so now the exact wave function for our n particles 813 00:59:49,200 --> 00:59:52,360 is nothing else than the time-dependent solution 814 00:59:52,360 --> 00:59:56,680 of Schrodinger's equation for the single particle. 815 00:59:56,680 --> 01:00:08,550 So this is single particle to the power n. 816 01:00:12,870 --> 01:00:16,380 So therefore, this is pretty much a mathematical proof 817 01:00:16,380 --> 01:00:22,450 unless I've made a mistake, which I haven't. 818 01:00:22,450 --> 01:00:34,580 So it takes exactly half a single atom Rabi period 819 01:00:34,580 --> 01:00:48,930 to completely invert the population exactly 820 01:00:48,930 --> 01:00:50,759 s for a single atom. 821 01:00:56,990 --> 01:01:03,630 So that's the result. 822 01:01:03,630 --> 01:01:18,010 However, if you describe the system by Dickie states, 823 01:01:18,010 --> 01:01:27,040 you have matrix element which are matrix elements which 824 01:01:27,040 --> 01:01:28,250 are proportional to n. 825 01:01:30,910 --> 01:01:35,280 However, if you want to-- I've described it just 826 01:01:35,280 --> 01:01:36,390 as a two-level system. 827 01:01:36,390 --> 01:01:38,040 Each atom does Rabi oscillation. 828 01:01:38,040 --> 01:01:42,220 I've said, OK, the system of n atoms 829 01:01:42,220 --> 01:01:44,860 is just n individual systems. 830 01:01:44,860 --> 01:01:49,160 But if you insist to describe it as a collective spin, 831 01:01:49,160 --> 01:01:50,970 then we have the Dickie states. 832 01:01:50,970 --> 01:01:52,620 Then we have the n times enhancement 833 01:01:52,620 --> 01:01:53,940 of the matrix element. 834 01:01:53,940 --> 01:01:57,490 But then we also have to go through n states. 835 01:01:57,490 --> 01:02:00,470 So we have n steps in the Dickie letter ladder. 836 01:02:04,160 --> 01:02:07,820 And one can say now-- and this is the exact argument-- 837 01:02:07,820 --> 01:02:09,680 you have n steps. 838 01:02:09,680 --> 01:02:11,990 You take each step n times faster, 839 01:02:11,990 --> 01:02:16,040 but the total time is the same. 840 01:02:16,040 --> 01:02:20,210 So n times 1 over n is 1. 841 01:02:20,210 --> 01:02:21,440 OK. 842 01:02:21,440 --> 01:02:28,360 But now when we talk about spontaneous emission, 843 01:02:28,360 --> 01:02:31,220 we are not driving the system with an external field. 844 01:02:31,220 --> 01:02:34,380 It's really driven by the system itself, which emits photons 845 01:02:34,380 --> 01:02:36,070 into the empty mode. 846 01:02:36,070 --> 01:02:42,840 Spontaneous emission, each step is 847 01:02:42,840 --> 01:02:48,800 proportional to the matrix element squared, 848 01:02:48,800 --> 01:02:50,530 because we're talking about [INAUDIBLE] 849 01:02:50,530 --> 01:02:52,060 spontaneous emission. 850 01:02:52,060 --> 01:02:55,990 So this is proportional to n square. 851 01:02:55,990 --> 01:03:00,500 And if you say that we have n steps, 852 01:03:00,500 --> 01:03:06,980 well, then we have n squared over n. 853 01:03:06,980 --> 01:03:10,060 Then we have a speed up. 854 01:03:10,060 --> 01:03:13,730 Each step is n squared faster. 855 01:03:13,730 --> 01:03:17,095 We divide by n, and we get the superradiant speed 856 01:03:17,095 --> 01:03:18,700 of which is n. 857 01:03:18,700 --> 01:03:22,590 So superradiance is something that you observe in spontaneous 858 01:03:22,590 --> 01:03:26,840 emission, but you cannot absorb it in a driven system. 859 01:03:26,840 --> 01:03:28,340 Because in a driven system, you can 860 01:03:28,340 --> 01:03:30,630 say you have a classical external field. 861 01:03:30,630 --> 01:03:33,090 And this external talks to one atom, 862 01:03:33,090 --> 01:03:35,460 to n atoms exactly in the same way. 863 01:03:35,460 --> 01:03:38,935 It is really the interference of spontaneously emitted photons 864 01:03:38,935 --> 01:03:40,560 which is at the heart of superradiance. 865 01:03:49,560 --> 01:03:52,860 As a side remark, we are talking here 866 01:03:52,860 --> 01:03:58,780 about coherent effect, which is n times enhanced. 867 01:03:58,780 --> 01:04:01,610 And you can actually regard that as a kind 868 01:04:01,610 --> 01:04:06,510 of bosonic enhancement in the emission of photons, 869 01:04:06,510 --> 01:04:10,070 because the photons are bosons. 870 01:04:10,070 --> 01:04:13,360 When Bose-Einstein condensation was discovered 871 01:04:13,360 --> 01:04:17,000 and people were thinking about basic experiments, of course, 872 01:04:17,000 --> 01:04:19,120 one thing which was on our mind is, 873 01:04:19,120 --> 01:04:21,580 we wanted to show that there are processes 874 01:04:21,580 --> 01:04:25,310 in the Bose-Einstein condensate we are n times enhanced. 875 01:04:25,310 --> 01:04:27,110 For fermions, they would be suppressed. 876 01:04:27,110 --> 01:04:28,590 This is the flip side. 877 01:04:28,590 --> 01:04:30,710 Big enhancements for bosons, complete suppression 878 01:04:30,710 --> 01:04:32,160 for fermions. 879 01:04:32,160 --> 01:04:35,730 And we found that, for instance, the formation of the condensate 880 01:04:35,730 --> 01:04:37,890 had an n times enhancement. 881 01:04:37,890 --> 01:04:40,670 There was a stimulation factor. 882 01:04:40,670 --> 01:04:44,160 But we also thought you should actually-- there may 883 01:04:44,160 --> 01:04:48,370 be ways where you can observe suppression of light scattering 884 01:04:48,370 --> 01:04:50,930 or enhancement of light scattering. 885 01:04:50,930 --> 01:04:54,190 But we thought about it with two laser beam [INAUDIBLE] 886 01:04:54,190 --> 01:04:56,540 scattering, and the idea seemed compelling. 887 01:04:56,540 --> 01:04:58,100 And then we said, no, wait a moment. 888 01:04:58,100 --> 01:05:02,000 If you use laser beams, everything is stimulated. 889 01:05:02,000 --> 01:05:06,190 You can observe bosonic enhancement 890 01:05:06,190 --> 01:05:08,470 and fermeonic suppression only when 891 01:05:08,470 --> 01:05:10,270 you have spontaneous events. 892 01:05:10,270 --> 01:05:13,150 If you drive it in a unitary time evolution, 893 01:05:13,150 --> 01:05:16,580 you will not be able to see quantum statistical suppression 894 01:05:16,580 --> 01:05:17,750 or enhancement. 895 01:05:17,750 --> 01:05:19,940 And the same thing as we have seen here-- 896 01:05:19,940 --> 01:05:22,440 when you have a stimulated system, 897 01:05:22,440 --> 01:05:26,240 everything is undergoing a unitary time evolution, 898 01:05:26,240 --> 01:05:28,570 and the unitary time evolution for n atoms 899 01:05:28,570 --> 01:05:30,460 is the same as for a single atom. 900 01:05:30,460 --> 01:05:34,360 You need the element of spontaneous emission. 901 01:05:34,360 --> 01:05:39,450 So I'm not proving it to you, but I'm just 902 01:05:39,450 --> 01:05:42,080 making as a remark-- what we have seen here, 903 01:05:42,080 --> 01:05:44,100 that the superradiance only shows up 904 01:05:44,100 --> 01:05:46,660 in spontaneous emission and not when we drive 905 01:05:46,660 --> 01:05:50,400 the system-- a driven system is a unitary evolution. 906 01:05:50,400 --> 01:05:55,240 And the same conclusion which we just got for superradiance 907 01:05:55,240 --> 01:05:59,070 also applies if you want to observe fermionic suppression 908 01:05:59,070 --> 01:06:02,440 or pulsonic enhancement in quantum gases. 909 01:06:02,440 --> 01:06:04,485 It needs an element of spontaneous scattering 910 01:06:04,485 --> 01:06:06,229 or spontaneous emission. 911 01:06:06,229 --> 01:06:07,696 Yes? 912 01:06:07,696 --> 01:06:11,119 AUDIENCE: If we think of it in terms of interference 913 01:06:11,119 --> 01:06:14,053 of photons, how does that tie in here? 914 01:06:14,053 --> 01:06:17,965 Because if the stimulated [INAUDIBLE] photons are still 915 01:06:17,965 --> 01:06:22,366 interfering, then you can get emission [INAUDIBLE]. 916 01:06:26,278 --> 01:06:28,720 PROFFESOR: The quick answer is, you 917 01:06:28,720 --> 01:06:32,790 have a classical field which you use for-- you have a laser 918 01:06:32,790 --> 01:06:35,050 field for stimulate emission for absorption. 919 01:06:35,050 --> 01:06:37,180 There are so many photons in the laser field 920 01:06:37,180 --> 01:06:40,710 that the few photons which your system emits do not matter. 921 01:06:40,710 --> 01:06:42,840 They are really talking to a classical field, 922 01:06:42,840 --> 01:06:46,730 and it doesn't matter whether the other n minus 1 atoms 923 01:06:46,730 --> 01:06:49,560 have emitted a photon, because you have zillions of photons 924 01:06:49,560 --> 01:06:51,225 in your laser field, and they determine 925 01:06:51,225 --> 01:06:52,486 the dynamics of the system. 926 01:07:01,960 --> 01:07:04,500 OK. 927 01:07:04,500 --> 01:07:07,290 Super radiance would not be as important 928 01:07:07,290 --> 01:07:14,790 as it is if it could not observed in extended samples. 929 01:07:14,790 --> 01:07:18,040 So now I want to use the last 10 minutes 930 01:07:18,040 --> 01:07:24,050 to show you what is kept and what 931 01:07:24,050 --> 01:07:27,270 has to be dropped when we talk about extended samples. 932 01:07:39,390 --> 01:07:45,100 So let's-- it doesn't really matter, 933 01:07:45,100 --> 01:07:49,252 but for pedagogical reasons, let's assume we have 934 01:07:49,252 --> 01:07:52,480 an extended sample which is much, 935 01:07:52,480 --> 01:07:58,160 much longer than the optical wavelengths along the long axis 936 01:07:58,160 --> 01:08:01,350 of the cigar and smaller along the short axis. 937 01:08:01,350 --> 01:08:02,930 The second condition that the cigar 938 01:08:02,930 --> 01:08:05,610 is smaller than along the long axis does not really matter. 939 01:08:30,640 --> 01:08:32,810 No, I'm not making this assumption. 940 01:08:32,810 --> 01:08:36,550 So it's a cigar, and let's just assume 941 01:08:36,550 --> 01:08:43,819 that everything is-- just saw a contradiction in my notes. 942 01:08:43,819 --> 01:08:46,430 So anyway, we have now a system which 943 01:08:46,430 --> 01:08:53,490 is-- let's have a Cuban cigar, a really thick cigar. 944 01:08:53,490 --> 01:08:56,490 And this is now our extended sample. 945 01:08:56,490 --> 01:09:02,689 And what I need is, I need the cross section of the sample A. 946 01:09:02,689 --> 01:09:08,840 And let's assume the length is l. 947 01:09:12,779 --> 01:09:14,090 This is diameter d. 948 01:09:14,090 --> 01:09:18,200 It's a cigar much, much larger than here. 949 01:09:18,200 --> 01:09:22,970 And yes, we are talking about superradiance, 950 01:09:22,970 --> 01:09:25,010 we are talking about spontaneous emission. 951 01:09:25,010 --> 01:09:29,109 But if you see a long cigar with excited atoms, 952 01:09:29,109 --> 01:09:32,390 you think immediately about lasing action. 953 01:09:32,390 --> 01:09:35,870 The photon is emitted is amplified along the path. 954 01:09:35,870 --> 01:09:38,720 And of course, the preferred direction 955 01:09:38,720 --> 01:09:41,810 where you would expect the maximum effect to happen 956 01:09:41,810 --> 01:09:44,540 is when the light is emitted along the long axis 957 01:09:44,540 --> 01:09:45,680 of the cigar. 958 01:09:45,680 --> 01:09:57,180 So you want to consider now preferential modes 959 01:09:57,180 --> 01:09:58,786 along the x-axis. 960 01:10:01,780 --> 01:10:10,200 So if you now assume that you have many atoms, 961 01:10:10,200 --> 01:10:14,650 and they emit light, if an atom here and here would emit light 962 01:10:14,650 --> 01:10:17,380 in this direction, it may constructively interfere. 963 01:10:17,380 --> 01:10:22,090 But in another direction, it will destructively interfere. 964 01:10:22,090 --> 01:10:28,480 But let us now consider what is the solid angle into which all 965 01:10:28,480 --> 01:10:31,550 of the atoms can coherently emit. 966 01:10:31,550 --> 01:10:38,760 Well, you know that from classical optics, the emission 967 01:10:38,760 --> 01:10:46,970 into a solid angle of lambda square over a can be coherent. 968 01:10:53,250 --> 01:10:55,670 Sort of similar to when you have a double slit 969 01:10:55,670 --> 01:10:59,016 and you ask, over what angle do the two slits emit inface. 970 01:10:59,016 --> 01:11:01,140 You get a bright fringe, and you get a dark fringe, 971 01:11:01,140 --> 01:11:02,750 you get the next bright fringe. 972 01:11:02,750 --> 01:11:06,090 The coherence, the angle over which the pass lengths 973 01:11:06,090 --> 01:11:10,360 differences do not add up to more than lambda. 974 01:11:10,360 --> 01:11:15,060 It's the deflection-limited angle which for a beam of size 975 01:11:15,060 --> 01:11:17,270 d is lambda over d. 976 01:11:17,270 --> 01:11:19,090 And if you take it to the second dimension, 977 01:11:19,090 --> 01:11:21,460 the solid angle is lambda squared over d squared. 978 01:11:21,460 --> 01:11:23,970 So that's what I'm talking about. 979 01:11:23,970 --> 01:11:28,260 So if you would give all the atoms in your assemble 980 01:11:28,260 --> 01:11:30,200 just the right phase that they are 981 01:11:30,200 --> 01:11:32,840 coherent to emit into the x-axis, 982 01:11:32,840 --> 01:11:36,270 they will also coherently emit into a small, solid angle, 983 01:11:36,270 --> 01:11:38,190 and the solid angle is given by this number. 984 01:11:40,980 --> 01:11:44,630 So the just of it is-- and I will not completely 985 01:11:44,630 --> 01:11:47,150 prove it to you, but I just want to give you a taste-- 986 01:11:47,150 --> 01:11:53,680 is that therefore, we still have a superradiant enhancement. 987 01:11:53,680 --> 01:11:57,690 We know the superradiant enhancement previously 988 01:11:57,690 --> 01:12:02,000 when we had the localized system was n. 989 01:12:02,000 --> 01:12:06,520 But now we have the n atoms act together, 990 01:12:06,520 --> 01:12:10,080 but they're not acting together for emission into 4 pi. 991 01:12:10,080 --> 01:12:12,790 They are acting together for emission into the solid angle. 992 01:12:17,360 --> 01:12:22,330 And if I write the big n as density 993 01:12:22,330 --> 01:12:28,950 n times l times a squared, the a squared cancels out 994 01:12:28,950 --> 01:12:32,280 and I get n lambda square l. 995 01:12:32,280 --> 01:12:36,800 And if you remember that the cross section of an atom 996 01:12:36,800 --> 01:12:39,410 was lambda square for absorption, 997 01:12:39,410 --> 01:12:42,410 if the atom is excited, the cross section 998 01:12:42,410 --> 01:12:48,100 for amplification of light for stimulated emission 999 01:12:48,100 --> 01:12:50,040 is also lambda squared. 1000 01:12:50,040 --> 01:12:53,065 So lambda squared is the gain cross section. 1001 01:12:56,110 --> 01:13:00,050 And what we find now as a superradiant enhancement factor 1002 01:13:00,050 --> 01:13:02,830 is nothing else like something which reminds us 1003 01:13:02,830 --> 01:13:06,370 of a laser, which reminds us of optical gain. 1004 01:13:06,370 --> 01:13:08,870 And actually, the lasing phenomenon 1005 01:13:08,870 --> 01:13:13,490 in superradiance in extended samples has a lot of analogies. 1006 01:13:13,490 --> 01:13:15,982 In some limits, it's even identical. 1007 01:13:15,982 --> 01:13:17,940 When we are talking about spontaneous emission, 1008 01:13:17,940 --> 01:13:19,814 we are not talking about stimulated emission. 1009 01:13:19,814 --> 01:13:23,270 But if you have a system which is in some excited state 1010 01:13:23,270 --> 01:13:25,420 superradiant Dickey states, and we 1011 01:13:25,420 --> 01:13:28,040 are asking what are the spontaneously emitted 1012 01:13:28,040 --> 01:13:33,430 photons coming out, to say different atoms 1013 01:13:33,430 --> 01:13:35,700 emit into the same mode, and now you 1014 01:13:35,700 --> 01:13:38,880 have to add up the feeds coherently, 1015 01:13:38,880 --> 01:13:40,890 this is a language we have used so far. 1016 01:13:40,890 --> 01:13:45,030 Or if you use a language atom emits a photon and this photon 1017 01:13:45,030 --> 01:13:50,920 gets amplified on its way out, those two language strongly 1018 01:13:50,920 --> 01:13:53,460 overlap or in some limits are even identical. 1019 01:13:53,460 --> 01:13:57,020 So the amplification of a photon on its way 1020 01:13:57,020 --> 01:14:01,590 out, this is behind superradiance. 1021 01:14:01,590 --> 01:14:05,440 But when the localize the atoms to lessen the wavelengths, 1022 01:14:05,440 --> 01:14:08,240 well, the atoms pretty much emit as a whole 1023 01:14:08,240 --> 01:14:12,070 and there is no pass lengths of the size 1024 01:14:12,070 --> 01:14:14,420 of the optical wavelengths where you can say the photon 1025 01:14:14,420 --> 01:14:16,030 propagates, gets amplified. 1026 01:14:16,030 --> 01:14:18,560 So we have looked at just what comes out of it. 1027 01:14:18,560 --> 01:14:20,550 But in extended sample, you could even 1028 01:14:20,550 --> 01:14:22,200 address this situation. 1029 01:14:22,200 --> 01:14:25,830 How do the photons get amplified, magnified, 1030 01:14:25,830 --> 01:14:30,590 augmented when they travel from the center to the edge? 1031 01:14:30,590 --> 01:14:33,010 So you could actually ask, what is the light intensity 1032 01:14:33,010 --> 01:14:35,400 as a function of the position within the cigar? 1033 01:14:35,400 --> 01:14:39,070 For localized samples, you can't. 1034 01:14:39,070 --> 01:14:40,730 So let me just write that down. 1035 01:14:40,730 --> 01:15:04,410 So this is analogous to optical amplification 1036 01:15:04,410 --> 01:15:06,620 in an elongated, inverted medium. 1037 01:15:16,680 --> 01:15:19,300 OK. 1038 01:15:19,300 --> 01:15:33,340 So you can formally describe that. 1039 01:15:33,340 --> 01:15:45,150 You can now define new Dickey states with respect 1040 01:15:45,150 --> 01:15:50,800 to the preferred mode. 1041 01:15:50,800 --> 01:15:54,930 And the preferred mode is the mode in the x direction. 1042 01:15:54,930 --> 01:15:59,060 So what I've done is-- remember, we have those atoms. 1043 01:15:59,060 --> 01:16:04,640 Those atoms are now sitting at different positions, x1 and x2. 1044 01:16:04,640 --> 01:16:07,220 And if I define Dickey states which 1045 01:16:07,220 --> 01:16:10,510 have phase factors into the ik x1, 1046 01:16:10,510 --> 01:16:16,710 into the ik x2, if now this atom emits a photon 1047 01:16:16,710 --> 01:16:19,530 and this atom emits a photon, well, 1048 01:16:19,530 --> 01:16:24,250 the second photon is x2 minus x1 ahead of this photon 1049 01:16:24,250 --> 01:16:29,600 if you think of those atoms sitting aligned in a string. 1050 01:16:29,600 --> 01:16:34,950 But the phase vector is exactly canceling the propagation phase 1051 01:16:34,950 --> 01:16:38,380 for the first atom in such a way that if you are now 1052 01:16:38,380 --> 01:16:41,870 coupling these states to the electromagnetic field, 1053 01:16:41,870 --> 01:16:44,370 the phase factors of the electromagnetic field 1054 01:16:44,370 --> 01:16:47,330 in the mode cancel with those phase factors, 1055 01:16:47,330 --> 01:16:52,940 and you again have the situation that each state here 1056 01:16:52,940 --> 01:16:55,890 has an equal amplitude for emission. 1057 01:16:55,890 --> 01:16:59,000 So now you have n possible contributions, 1058 01:16:59,000 --> 01:17:01,460 and the normalization is 1 over square root n, 1059 01:17:01,460 --> 01:17:03,070 and everything falls into place. 1060 01:17:05,620 --> 01:17:08,950 And you can define that for two excited atoms with two phase 1061 01:17:08,950 --> 01:17:10,120 factors and so on. 1062 01:17:10,120 --> 01:17:14,610 So you can use immediately the same formalism. 1063 01:17:17,250 --> 01:17:47,420 And what happens is those phase factors for the interaction 1064 01:17:47,420 --> 01:17:55,290 Hamiltonian-- and our interaction 1065 01:17:55,290 --> 01:17:57,250 Hamiltonian is now different. 1066 01:17:57,250 --> 01:18:00,420 It is di. 1067 01:18:00,420 --> 01:18:02,940 And now in an extended sample, we 1068 01:18:02,940 --> 01:18:06,430 have to keep track of the precision of the atom. 1069 01:18:09,060 --> 01:18:14,380 So for the coupling to the mode in the x direction, 1070 01:18:14,380 --> 01:18:17,820 we have those phase factors. 1071 01:18:28,060 --> 01:18:31,275 So all the phase factors cancel. 1072 01:18:31,275 --> 01:18:32,650 And actually, I'm not telling you 1073 01:18:32,650 --> 01:18:36,360 whether this is plus or minus in order to cancel. 1074 01:18:36,360 --> 01:18:38,260 You pick the sign that they all cancel, 1075 01:18:38,260 --> 01:18:41,560 and then you have superradiance. 1076 01:18:41,560 --> 01:18:43,560 You have fully constructive interference. 1077 01:19:05,010 --> 01:19:07,790 Yes. 1078 01:19:07,790 --> 01:19:11,670 So all this looks now the same as superradiance, 1079 01:19:11,670 --> 01:19:13,990 but there are also things which are different, 1080 01:19:13,990 --> 01:19:15,500 and this is the following. 1081 01:19:15,500 --> 01:19:19,150 If the atom would emit now photons, not in the preferred 1082 01:19:19,150 --> 01:19:27,100 mode, then-- remember, we had the Dickey ladder. 1083 01:19:27,100 --> 01:19:30,420 We had the most superradiant ladder, little bit 1084 01:19:30,420 --> 01:19:32,660 less superradiant ladder, and eventually we 1085 01:19:32,660 --> 01:19:35,440 had the subradiant ladder in order 1086 01:19:35,440 --> 01:19:41,200 of smaller and smaller total quantum number s. 1087 01:19:41,200 --> 01:19:44,300 Emission in the preferred mode stays in each letter, 1088 01:19:44,300 --> 01:19:46,590 and we have the superradiant cascade. 1089 01:19:46,590 --> 01:19:49,530 But emission into other modes is now 1090 01:19:49,530 --> 01:19:52,260 coupling different s states. 1091 01:19:52,260 --> 01:19:57,000 Because the operator or the phase factor into the IKR 1092 01:19:57,000 --> 01:19:59,530 has broken to complete permutation symmetry 1093 01:19:59,530 --> 01:20:00,340 between the sides. 1094 01:20:00,340 --> 01:20:01,780 We have changed the symmetry. 1095 01:20:01,780 --> 01:20:03,980 We have not the completely symmetric sum. 1096 01:20:03,980 --> 01:20:06,440 We have a symmetric summation with phase factors. 1097 01:20:06,440 --> 01:20:10,170 So therefore, the phenomenon is somewhat different. 1098 01:20:10,170 --> 01:20:14,730 But we still have superradiant cascade for the preferred mode. 1099 01:20:19,100 --> 01:20:28,480 And the result is that we have an enhancement 1100 01:20:28,480 --> 01:20:31,610 for the most symmetric for the superradiant 1101 01:20:31,610 --> 01:20:35,570 states, which is given by that. 1102 01:20:38,600 --> 01:20:41,950 And this is nothing else than the resonant optical density 1103 01:20:41,950 --> 01:20:45,010 of your center. 1104 01:20:45,010 --> 01:20:49,030 So in experiments-- many of them go on in [INAUDIBLE] lab, 1105 01:20:49,030 --> 01:20:53,160 where he uses collective spin and the storage 1106 01:20:53,160 --> 01:20:56,830 of single photons in n atoms, the figure 1107 01:20:56,830 --> 01:21:00,790 of [INAUDIBLE] of his samples is always the optical density, 1108 01:21:00,790 --> 01:21:05,716 the number of atoms times lambda squared times the lengths. 1109 01:21:10,380 --> 01:21:14,370 Finally-- and sorry for keeping you three more 1110 01:21:14,370 --> 01:21:19,240 minutes-- the form of superradiance 1111 01:21:19,240 --> 01:21:22,935 which is very important is Raman superradiance. 1112 01:21:28,160 --> 01:21:30,070 We don't have an excited state where 1113 01:21:30,070 --> 01:21:34,390 we put a lot of excitations on, because the excited state would 1114 01:21:34,390 --> 01:21:36,160 be very short lived. 1115 01:21:36,160 --> 01:21:47,020 So what we instead do is, we have Rabi frequency omega 1. 1116 01:21:47,020 --> 01:21:51,150 We have a large [INAUDIBLE] delta. 1117 01:21:51,150 --> 01:21:54,180 And then the spontaneous emission 1118 01:21:54,180 --> 01:21:57,460 with the coupling constant g takes us down 1119 01:21:57,460 --> 01:22:00,420 to the excited state. 1120 01:22:00,420 --> 01:22:05,270 In the case that the Rabi frequency is much, much smaller 1121 01:22:05,270 --> 01:22:13,350 than delta, we can eliminate the excited state 1122 01:22:13,350 --> 01:22:15,010 from the description. 1123 01:22:15,010 --> 01:22:18,540 And what we obtain is now a system 1124 01:22:18,540 --> 01:22:20,435 which has an excited state. 1125 01:22:23,250 --> 01:22:27,470 The widths of this excited state-- 1126 01:22:27,470 --> 01:22:34,330 this is pretty much the virtual state here-- is the scattering 1127 01:22:34,330 --> 01:22:40,200 rate which is the probability to excite the atom 1128 01:22:40,200 --> 01:22:42,650 is Rabi frequencied over detuning squared. 1129 01:22:42,650 --> 01:22:44,550 That's just perturbation theory. 1130 01:22:44,550 --> 01:22:49,120 And then we multiply with gamma or gamma over 2. 1131 01:22:49,120 --> 01:22:51,300 So this is the rate of spontaneous emission out 1132 01:22:51,300 --> 01:22:54,750 of the virtual state. 1133 01:22:54,750 --> 01:23:02,250 And from this virtual state, we go now to the ground state. 1134 01:23:02,250 --> 01:23:06,210 And the Rabi frequency, or the coupling 1135 01:23:06,210 --> 01:23:09,550 for this virtual state, is the original coupling 1136 01:23:09,550 --> 01:23:12,140 g between ground and excited state, 1137 01:23:12,140 --> 01:23:15,100 but now pro-rated by the amplitude 1138 01:23:15,100 --> 01:23:19,741 that we have mixed the excited state into the virtual state. 1139 01:23:23,220 --> 01:23:28,980 So therefore, we have now obtained a superradiant system. 1140 01:23:28,980 --> 01:23:31,690 And for instance, we did experiments 1141 01:23:31,690 --> 01:23:34,540 which became classic now because they are conceptually so 1142 01:23:34,540 --> 01:23:37,210 clear-- we took a Bose-Einstein condensate, 1143 01:23:37,210 --> 01:23:40,760 we switched on one strong of resonant laser beam, 1144 01:23:40,760 --> 01:23:44,070 and then we had a system which was 100% inverted, 1145 01:23:44,070 --> 01:23:47,320 because we had no atoms into the final state. 1146 01:23:47,320 --> 01:23:49,910 The final state is a Bose-Einstein condensate 1147 01:23:49,910 --> 01:23:51,999 but with a recoil kick. 1148 01:23:51,999 --> 01:23:53,915 So by just having a Bose-Einstein condensation 1149 01:23:53,915 --> 01:23:58,250 and shining this laser light on it, we had now in this picture 1150 01:23:58,250 --> 01:24:03,950 a 100% inverted system, which is the ideal realization 1151 01:24:03,950 --> 01:24:06,310 of a fully inverted Dicky state. 1152 01:24:06,310 --> 01:24:08,300 Everything is completely symmetric, 1153 01:24:08,300 --> 01:24:11,255 and then we observed superradiant emission 1154 01:24:11,255 --> 01:24:12,380 of light pulses. 1155 01:24:16,420 --> 01:24:16,920 OK. 1156 01:24:20,000 --> 01:24:24,370 So this has been the important realize-- 1157 01:24:24,370 --> 01:24:30,380 so important experiments have been done via BECs in my group 1158 01:24:30,380 --> 01:24:36,350 and with cold atoms in with laser code samples 1159 01:24:36,350 --> 01:24:37,045 in [INAUDIBLE]. 1160 01:24:45,070 --> 01:24:48,110 So why is superradiance so important? 1161 01:24:48,110 --> 01:24:54,220 And this my last statement for this class 1162 01:24:54,220 --> 01:24:55,970 and for the semester. 1163 01:24:55,970 --> 01:25:02,990 So if you have extended sample superradiance, 1164 01:25:02,990 --> 01:25:06,570 those samples are no longer coupling 1165 01:25:06,570 --> 01:25:10,940 to the electromagnetic field with the coupling constant g. 1166 01:25:10,940 --> 01:25:15,050 The coupling constant g is now multiplied 1167 01:25:15,050 --> 01:25:19,130 by the optical density of your sample. 1168 01:25:19,130 --> 01:25:24,850 And there is a lot of interest for current research 1169 01:25:24,850 --> 01:25:28,000 for quantum computation, manipulation of photon states, 1170 01:25:28,000 --> 01:25:31,810 and all that, to do cavity QED. 1171 01:25:31,810 --> 01:25:35,450 And in cavity QED, we try to have 1172 01:25:35,450 --> 01:25:38,220 very good mirrors, very small mode volume 1173 01:25:38,220 --> 01:25:39,845 to have a very, very large g. 1174 01:25:44,600 --> 01:25:47,720 But this large g which we achieve in a cavity, 1175 01:25:47,720 --> 01:25:51,540 if you put many atoms in it, gets 1176 01:25:51,540 --> 01:25:53,730 enhanced by the optical density. 1177 01:25:53,730 --> 01:25:57,790 So the cavity enhancement and the superradiant enhancement 1178 01:25:57,790 --> 01:25:59,360 is multiplicative. 1179 01:25:59,360 --> 01:26:03,110 And often, it's very favorable for single photon manipulation 1180 01:26:03,110 --> 01:26:03,930 if you do Bose. 1181 01:26:03,930 --> 01:26:08,750 You getting enhancement form the cavity and enhancement 1182 01:26:08,750 --> 01:26:10,180 due to superradiance. 1183 01:26:10,180 --> 01:26:12,810 And the person who has really pioneered work 1184 01:26:12,810 --> 01:26:17,370 along this direction is [INAUDIBLE] here at MIT. 1185 01:26:17,370 --> 01:26:20,020 Anyway, yes, with five minutes delay, 1186 01:26:20,020 --> 01:26:21,740 I finish the chapter on superradiance. 1187 01:26:24,700 --> 01:26:26,420 Well, that's the end of this course. 1188 01:26:26,420 --> 01:26:30,250 Let me thank you for your active participation. 1189 01:26:30,250 --> 01:26:33,410 Sometimes as a lecturer, you learn as much as the students. 1190 01:26:33,410 --> 01:26:36,570 And I think partially based on your questions and discussions, 1191 01:26:36,570 --> 01:26:37,640 this is really true. 1192 01:26:37,640 --> 01:26:40,020 I've learned a few new aspects of atomic physics. 1193 01:26:40,020 --> 01:26:41,660 I hope you have learned something, too. 1194 01:26:41,660 --> 01:26:44,710 And good luck in the future.