1 00:00:00,050 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,000 under a Creative Commons license. 3 00:00:04,000 --> 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 --> 00:00:17,183 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,183 --> 00:00:17,807 at ocw.mit.edu. 8 00:00:20,435 --> 00:00:22,230 PROFESSOR: So, good afternoon. 9 00:00:24,740 --> 00:00:31,120 Our subject for all of today and some part of Monday 10 00:00:31,120 --> 00:00:34,640 is coherence in free-level systems. 11 00:00:34,640 --> 00:00:37,180 And what I want to show you are there 12 00:00:37,180 --> 00:00:41,690 are certain fundamentally new and, to some extent, 13 00:00:41,690 --> 00:00:45,210 surprising processes which are possible when 14 00:00:45,210 --> 00:00:48,100 you have three levels and not only two. 15 00:00:48,100 --> 00:00:49,960 Because when you have three levels, 16 00:00:49,960 --> 00:00:52,360 you have one excited state which is 17 00:00:52,360 --> 00:00:55,190 coupled to two ground states. 18 00:00:55,190 --> 00:00:59,270 And that gives us the possibility of interference. 19 00:00:59,270 --> 00:01:01,790 And actually, all the special things 20 00:01:01,790 --> 00:01:03,760 I will tell you about three-level system 21 00:01:03,760 --> 00:01:05,730 are related to interference. 22 00:01:05,730 --> 00:01:09,645 Certain phenomena will constructively interfere. 23 00:01:09,645 --> 00:01:12,290 Other will destructively interfere. 24 00:01:12,290 --> 00:01:25,006 And that leads to a number of important effects. 25 00:01:25,006 --> 00:01:26,630 The first effect you want to talk about 26 00:01:26,630 --> 00:01:28,870 is the possibility of dark states, 27 00:01:28,870 --> 00:01:31,520 that there are superposition states which 28 00:01:31,520 --> 00:01:35,180 are bombarded with two laser beams, and they're not excited. 29 00:01:35,180 --> 00:01:39,040 Because the amplitudes to go to the excited state destructively 30 00:01:39,040 --> 00:01:41,140 interfere. 31 00:01:41,140 --> 00:01:43,820 When you have a state which is dark, 32 00:01:43,820 --> 00:01:47,750 which cannot absorb the light, then you already understand 33 00:01:47,750 --> 00:01:51,140 the possibility of doing lasing without inversion. 34 00:01:51,140 --> 00:01:54,570 Because if a lot of the population is in a dark state, 35 00:01:54,570 --> 00:01:57,660 you don't have to invert excited and ground state. 36 00:01:57,660 --> 00:02:01,160 Because the dark state doesn't count. 37 00:02:01,160 --> 00:02:03,570 Lasing without inversion is a little bit more intricate 38 00:02:03,570 --> 00:02:05,830 than this, but just kind of the idea, 39 00:02:05,830 --> 00:02:08,330 the notion of a dark state helps you already 40 00:02:08,330 --> 00:02:16,360 to anticipate why something like lasing with inversion 41 00:02:16,360 --> 00:02:19,010 may be possible. 42 00:02:19,010 --> 00:02:23,120 The dark state is only dark in a very narrow frequency 43 00:02:23,120 --> 00:02:25,070 interval of the two lasers. 44 00:02:25,070 --> 00:02:27,360 So if you're a very narrow resonance, 45 00:02:27,360 --> 00:02:28,910 then you have a very narrow resonance 46 00:02:28,910 --> 00:02:30,490 where something goes dark. 47 00:02:30,490 --> 00:02:33,420 But if something goes dark, it's not absorbing. 48 00:02:33,420 --> 00:02:35,170 And therefore you have electromagnetically 49 00:02:35,170 --> 00:02:38,650 induced transparency, which is a very, very sharp feature 50 00:02:38,650 --> 00:02:40,740 in frequency space. 51 00:02:40,740 --> 00:02:44,170 And when you have very, very sharp features in frequency, 52 00:02:44,170 --> 00:02:47,370 you can change the group velocity of light, 53 00:02:47,370 --> 00:02:52,540 because if transmission and absorption index of refraction 54 00:02:52,540 --> 00:02:54,490 changes a functional frequency very, 55 00:02:54,490 --> 00:02:58,220 very sharply, you have huge effects on propagation lights. 56 00:02:58,220 --> 00:03:02,390 So kind of all of these different subjects which 57 00:03:02,390 --> 00:03:05,880 I want to go through are different perspectives, 58 00:03:05,880 --> 00:03:08,120 but you will find similar physics. 59 00:03:08,120 --> 00:03:09,910 It's just we look at interference. 60 00:03:09,910 --> 00:03:14,760 We look at dark states in various ways. 61 00:03:14,760 --> 00:03:19,440 And each of them has created a cottage industry 62 00:03:19,440 --> 00:03:22,570 and a subfield of itself. 63 00:03:22,570 --> 00:03:30,280 So I started to remind you that you're 64 00:03:30,280 --> 00:03:35,760 very familiar with the dark state in the situation where 65 00:03:35,760 --> 00:03:37,530 we have optical pumping. 66 00:03:37,530 --> 00:03:40,850 If you have two ground states in an excited state, 67 00:03:40,850 --> 00:03:44,430 and you switch only one laser, sooner or later you 68 00:03:44,430 --> 00:03:47,690 will just pump all the atoms into the state f 69 00:03:47,690 --> 00:03:52,520 and the state f is dark, and vice versa. 70 00:03:52,520 --> 00:03:57,490 So what I want to show you now is that even if both lasers are 71 00:03:57,490 --> 00:04:04,080 on, then no matter in what kind of state or superposition state 72 00:04:04,080 --> 00:04:05,710 you have the atoms, the atoms see 73 00:04:05,710 --> 00:04:08,240 light, which is ready to excite them. 74 00:04:08,240 --> 00:04:11,280 But there is-- and this is what I want to show you now-- 75 00:04:11,280 --> 00:04:15,560 a coherent superposition state, which is still the dark state. 76 00:04:18,339 --> 00:04:23,340 So the last thing I did on Monday, I wrote down for you 77 00:04:23,340 --> 00:04:26,690 our Hamiltonian, which is nothing 78 00:04:26,690 --> 00:04:29,840 else in the well-known dipole Hamiltonian in the rotating 79 00:04:29,840 --> 00:04:31,650 wave approximation. 80 00:04:31,650 --> 00:04:35,210 And what is important now is that we have pathways 81 00:04:35,210 --> 00:04:38,440 with Rabi frequency omega 1 and omega 2. 82 00:04:38,440 --> 00:04:40,880 And we have one laser fields. 83 00:04:40,880 --> 00:04:42,660 The photons are created, annihilated 84 00:04:42,660 --> 00:04:45,680 by the a operator, and the second laser field, 85 00:04:45,680 --> 00:04:49,080 where the operator is C and C dega. 86 00:04:49,080 --> 00:04:56,150 So there are two ways how we go to the excited state. 87 00:04:56,150 --> 00:04:59,700 We can couple from the ground state f to the excited state 88 00:04:59,700 --> 00:05:01,950 and from the ground state to g to the excited state. 89 00:05:13,690 --> 00:05:15,290 So when you shine on the light, atoms 90 00:05:15,290 --> 00:05:18,050 may be in some initial state. 91 00:05:18,050 --> 00:05:22,320 They will sort of spontaneously emit but scatter light. 92 00:05:22,320 --> 00:05:27,260 But then after some transient, the state 93 00:05:27,260 --> 00:05:30,920 which is stable against further illumination, is a dark state. 94 00:05:30,920 --> 00:05:32,720 That's like optical pumping. 95 00:05:32,720 --> 00:05:36,710 So you will optically pump into the following state, 96 00:05:36,710 --> 00:05:38,390 which I write down for you. 97 00:05:38,390 --> 00:05:42,940 And then we can inspect that indeed it is a dark state. 98 00:05:42,940 --> 00:05:48,150 It is a coherent superposition of the two ground states. 99 00:05:48,150 --> 00:05:53,880 And the coefficient involves the Rabi frequency, 100 00:05:53,880 --> 00:05:58,070 omega 2 of laser 2, omega 1 of laser 1. 101 00:05:58,070 --> 00:06:05,250 And we just normalize it by taking the quadrature some 102 00:06:05,250 --> 00:06:06,470 of the two Rabi frequencies. 103 00:06:10,800 --> 00:06:19,530 So this state is indeed a dark state. 104 00:06:19,530 --> 00:06:31,420 So let me show that to you by looking 105 00:06:31,420 --> 00:06:35,740 what happens when we expose it to this operator. 106 00:06:35,740 --> 00:06:38,170 This operator v in the Hamiltonian 107 00:06:38,170 --> 00:06:43,230 is the operator which couples the atoms to light, which 108 00:06:43,230 --> 00:06:46,010 couples the atom to the excited state 109 00:06:46,010 --> 00:06:50,830 and this causes spontaneous emission. 110 00:06:50,830 --> 00:07:00,430 So what I want to show you that is a dark state, 111 00:07:00,430 --> 00:07:05,110 since if you take this operator, apply it to the dark state 112 00:07:05,110 --> 00:07:09,690 and ask, is there any possibility 113 00:07:09,690 --> 00:07:23,530 how you can excite, how you can excite to the excited state? 114 00:07:23,530 --> 00:07:26,780 And the answer is, it will be zero. 115 00:07:26,780 --> 00:07:40,930 The dark state has a component on-- the dark state 116 00:07:40,930 --> 00:07:49,590 has one component in f, and it has one component in g. 117 00:07:57,097 --> 00:07:58,180 How did I want to do that? 118 00:08:16,434 --> 00:08:18,100 Sometimes it's easier to say it in words 119 00:08:18,100 --> 00:08:19,558 than to write down a long equation. 120 00:08:22,640 --> 00:08:27,620 The light atom interaction couples the component 121 00:08:27,620 --> 00:08:30,630 of the ground state to the excited state 122 00:08:30,630 --> 00:08:36,330 with Rabi frequency omega 1 and the state f with omega 2. 123 00:08:36,330 --> 00:08:41,549 However, in the construction of the dark state, 124 00:08:41,549 --> 00:08:45,030 I've made sure that the amplitude of the component g 125 00:08:45,030 --> 00:08:46,760 is omega 2. 126 00:08:46,760 --> 00:08:49,900 So therefore when we apply this operator, 127 00:08:49,900 --> 00:08:53,370 we have omega 1 from the operator and omega 2 here. 128 00:08:53,370 --> 00:08:56,860 And here we also get omega 1 and omega 2. 129 00:08:56,860 --> 00:09:00,430 And this minus sign means the two effects cancel. 130 00:09:00,430 --> 00:09:12,780 So therefore, I've shown you that even 131 00:09:12,780 --> 00:09:17,550 though in its full frame with second quantization, photon 132 00:09:17,550 --> 00:09:21,130 operators and all this, this coupling is identically 133 00:09:21,130 --> 00:09:23,250 0 and therefore we have a dark state. 134 00:09:26,080 --> 00:09:31,910 So this phenomenon of having a dark state 135 00:09:31,910 --> 00:09:37,530 and populating the dark state by light scattering, 136 00:09:37,530 --> 00:09:39,890 you have spontaneous emission into the dark state 137 00:09:39,890 --> 00:09:43,620 until the scattering light staggering stops and no matter 138 00:09:43,620 --> 00:09:45,060 what the initial state of the atom 139 00:09:45,060 --> 00:09:49,070 bar where the dark state is now 100% populated. 140 00:09:49,070 --> 00:09:51,660 This is called coherent population trapping. 141 00:10:11,530 --> 00:10:13,980 Let me mention that this coherent population 142 00:10:13,980 --> 00:10:17,520 trapping can be some good thing if you want to pump atoms 143 00:10:17,520 --> 00:10:20,110 into a certain state and then they stop scattering light. 144 00:10:20,110 --> 00:10:22,080 They're not heated up anymore. 145 00:10:22,080 --> 00:10:25,210 But sometimes you want the atoms to scatter light. 146 00:10:25,210 --> 00:10:27,990 For instance, in current laser cooling experiments 147 00:10:27,990 --> 00:10:30,870 with molecules, there is a problem 148 00:10:30,870 --> 00:10:35,820 that the multiplicity of the angular momentum states, which 149 00:10:35,820 --> 00:10:37,820 are suitable for laser cooling in molecules 150 00:10:37,820 --> 00:10:40,380 is such that you always have a dark state. 151 00:10:40,380 --> 00:10:44,240 So if you irradiate the atoms with laser light, 152 00:10:44,240 --> 00:10:46,220 they will not be on a cycling transition. 153 00:10:46,220 --> 00:10:48,130 The cycling transition will stop. 154 00:10:48,130 --> 00:10:50,220 So you have to play additional tricks. 155 00:10:50,220 --> 00:10:53,330 One is to apply a transverse magnetic field 156 00:10:53,330 --> 00:10:57,200 that the axis of the laser and the quantization axis given 157 00:10:57,200 --> 00:10:59,910 by the magnetic field are in two different directions. 158 00:10:59,910 --> 00:11:02,980 And then once the atom is pumped into a dark state, 159 00:11:02,980 --> 00:11:06,670 it will actually precess, Zeeman precess out of the dark state. 160 00:11:06,670 --> 00:11:09,960 Or magneto-optical traps for molecules 161 00:11:09,960 --> 00:11:14,130 use a magnetic field gradient, which is rapidly switched, 162 00:11:14,130 --> 00:11:16,450 because when you are in a dark state for one 163 00:11:16,450 --> 00:11:20,090 value of the magnetic field, you just change the sign quickly. 164 00:11:20,090 --> 00:11:22,350 And then the atoms are no longer in the dark state 165 00:11:22,350 --> 00:11:24,630 for the new situation you've created. 166 00:11:24,630 --> 00:11:28,110 So in other words, to have these wonderful dark state, 167 00:11:28,110 --> 00:11:30,640 which is a major intellectual accomplishment 168 00:11:30,640 --> 00:11:34,680 and eventually was exploited in many experiments, 169 00:11:34,680 --> 00:11:37,630 for a number of experiments it's a real nuisance. 170 00:11:37,630 --> 00:11:40,050 And you have to find ways around it 171 00:11:40,050 --> 00:11:44,577 to make sure that the atoms are not stopping scattering light 172 00:11:44,577 --> 00:11:45,910 and don't stop scattering light. 173 00:11:49,010 --> 00:11:52,250 I gave you the simple example that the dark state 174 00:11:52,250 --> 00:11:56,130 exists for resonant excitation. 175 00:11:56,130 --> 00:12:01,910 But as you can immediately show, the dark state 176 00:12:01,910 --> 00:12:08,455 also exists for detuning of resonance. 177 00:12:19,330 --> 00:12:25,285 The only thing which is important as long as the two 178 00:12:25,285 --> 00:12:31,850 lasers fulfill the two photon resonance condition. 179 00:12:41,980 --> 00:12:45,460 So you want the difference between the two lasers 180 00:12:45,460 --> 00:12:50,070 is the energy difference between the two states. 181 00:12:50,070 --> 00:12:57,190 Also remember if you have two lasers which 182 00:12:57,190 --> 00:13:05,340 are very different in power-- so let's say one Rabi frequency is 183 00:13:05,340 --> 00:13:07,500 much smaller than the other one. 184 00:13:12,210 --> 00:13:20,840 So then the dark state, the superposition 185 00:13:20,840 --> 00:13:22,710 which is the dark state, the dark state 186 00:13:22,710 --> 00:13:27,761 is now predominantly which one? 187 00:13:27,761 --> 00:13:29,260 The state with the stronger coupling 188 00:13:29,260 --> 00:13:30,426 or with the weaker coupling? 189 00:13:39,090 --> 00:13:42,850 Well, it's the state with the weaker coupling. 190 00:13:42,850 --> 00:13:45,010 The dark state has always a larger amplitude 191 00:13:45,010 --> 00:13:47,550 in the state for which the coupling is weaker. 192 00:13:47,550 --> 00:13:51,290 And that of course immediately makes a connection 193 00:13:51,290 --> 00:13:53,110 to optical pumping. 194 00:13:53,110 --> 00:13:57,350 For instance, if omega 1 the Rabi frequency is 0, 195 00:13:57,350 --> 00:14:00,250 then the dark state is what we started out 196 00:14:00,250 --> 00:14:03,140 with a trivial example, is a state g. 197 00:14:03,140 --> 00:14:05,010 So you see that actually what I taught you 198 00:14:05,010 --> 00:14:06,890 about coherent population trapping 199 00:14:06,890 --> 00:14:09,920 is nothing else than extension of the concept 200 00:14:09,920 --> 00:14:12,770 of optical pumping. 201 00:14:12,770 --> 00:14:18,630 OK, so we've talked about the dark state 202 00:14:18,630 --> 00:14:24,410 as a superposition of state g and f. 203 00:14:24,410 --> 00:14:26,540 But if you have a two-dimensional Hilbert 204 00:14:26,540 --> 00:14:30,590 space given by two ground states g and f, 205 00:14:30,590 --> 00:14:35,050 we can now find a new basis. 206 00:14:35,050 --> 00:14:38,540 We have defined the dark state. 207 00:14:38,540 --> 00:14:43,260 But now we can get the orthogonal state, 208 00:14:43,260 --> 00:14:44,925 which is the bright state. 209 00:14:44,925 --> 00:14:52,200 It has a plus sign here and it has the omega 1 and omega 2, 210 00:14:52,200 --> 00:14:55,080 the Rabi frequencies reversed. 211 00:14:55,080 --> 00:14:56,490 So this state is now orthogonal. 212 00:15:03,420 --> 00:15:15,710 And you can now visualize the preparation of the dark state 213 00:15:15,710 --> 00:15:16,210 as follows. 214 00:15:18,890 --> 00:15:21,660 We have the dark state. 215 00:15:21,660 --> 00:15:23,900 We have the bright state. 216 00:15:23,900 --> 00:15:26,085 The bright state, as you can immediately verify, 217 00:15:26,085 --> 00:15:31,390 is strongly coupled by the laser field to the excited state. 218 00:15:31,390 --> 00:15:34,230 So we have an excited state. 219 00:15:34,230 --> 00:15:38,340 The laser fields-- well, its two laser fields strongly 220 00:15:38,340 --> 00:15:42,850 couple the bright state to the excited state. 221 00:15:42,850 --> 00:15:48,600 And then spontaneous emission-- two photon lights scattering-- 222 00:15:48,600 --> 00:15:52,330 can take you back to either of the two states. 223 00:15:52,330 --> 00:16:02,890 But now you see that the concept of optical pumping, 224 00:16:02,890 --> 00:16:05,700 which I started out with a trivial example, 225 00:16:05,700 --> 00:16:08,470 is now applying in the new basis to the bright 226 00:16:08,470 --> 00:16:09,830 and to the dark state. 227 00:16:09,830 --> 00:16:12,920 And what happens is you would, by optical pumping, 228 00:16:12,920 --> 00:16:15,622 populate the dark state and completely pump out 229 00:16:15,622 --> 00:16:16,330 the bright state. 230 00:16:19,370 --> 00:16:23,420 Just as a side remark, there are lots of subtleties. 231 00:16:23,420 --> 00:16:27,270 Most of them you will probably see by yourself 232 00:16:27,270 --> 00:16:29,960 by just inspecting the results. 233 00:16:29,960 --> 00:16:31,800 But if you may ask yourself, what 234 00:16:31,800 --> 00:16:35,930 happens when the Raman resonance is not 235 00:16:35,930 --> 00:16:38,320 made, when the difference between the two lasers 236 00:16:38,320 --> 00:16:40,990 is not exactly the difference between the two? 237 00:16:40,990 --> 00:16:44,310 Well, then what you have is you have a superposition 238 00:16:44,310 --> 00:16:45,850 of ground and excited state. 239 00:16:45,850 --> 00:16:50,510 So we have the two ground states g and f. 240 00:16:50,510 --> 00:16:56,600 But the two laser fields have a phase 241 00:16:56,600 --> 00:17:00,340 which evolves different from the phase between g and f. 242 00:17:00,340 --> 00:17:05,510 And suddenly, what you have is relative-- I mean, 243 00:17:05,510 --> 00:17:09,079 everything is relative to the laser field, so to speak. 244 00:17:09,079 --> 00:17:11,740 This plus sign turns into a minus sign. 245 00:17:11,740 --> 00:17:14,930 So therefore, if you have pumped into the dark state, 246 00:17:14,930 --> 00:17:18,550 but now the frequency difference of the two lasers 247 00:17:18,550 --> 00:17:21,329 is slightly different from the frequency difference between g 248 00:17:21,329 --> 00:17:24,140 and f, this mean in essence that as time 249 00:17:24,140 --> 00:17:28,470 goes by, with the frequency, with the detuning away 250 00:17:28,470 --> 00:17:32,230 from the two photon resonance, the dark state will now 251 00:17:32,230 --> 00:17:35,570 precess into the bright state. 252 00:17:35,570 --> 00:17:38,570 And it is only for the Raman resonance, 253 00:17:38,570 --> 00:17:42,580 when it is exactly met, that you have a long-lived dark state, 254 00:17:42,580 --> 00:17:45,530 a dark state which is a 2 eigenstate of this Hamiltonian. 255 00:17:45,530 --> 00:17:46,030 Yes? 256 00:17:46,030 --> 00:17:48,054 AUDIENCE: Would the precession also 257 00:17:48,054 --> 00:17:49,759 change omega 1 into omega 2? 258 00:17:56,577 --> 00:17:58,910 PROFESSOR: No, we have defined omega 1 and omega 2 259 00:17:58,910 --> 00:18:01,310 as the Rabi frequencies of the two lasers. 260 00:18:01,310 --> 00:18:04,250 Omega 1 is a Rabi frequency of the laser which 261 00:18:04,250 --> 00:18:07,710 talks to, I think, the state g. 262 00:18:07,710 --> 00:18:09,475 And omega two is the Rabi frequency 263 00:18:09,475 --> 00:18:11,141 of the laser which talks to the state f. 264 00:18:11,141 --> 00:18:13,200 So this doesn't change. 265 00:18:13,200 --> 00:18:16,960 I'm just sort of telling you how you may want to think about it, 266 00:18:16,960 --> 00:18:20,270 that if you put in all of the temporal phase factors, 267 00:18:20,270 --> 00:18:23,450 on resonance, at least in some dressed atom picture, 268 00:18:23,450 --> 00:18:25,530 all the phase factors are 0. 269 00:18:25,530 --> 00:18:29,120 Because the e to the i omega t of the atomic wave function 270 00:18:29,120 --> 00:18:31,840 is compensated by the light atom coupling. 271 00:18:31,840 --> 00:18:34,170 And therefore phase factors just disappear, 272 00:18:34,170 --> 00:18:36,030 and the state is stationary. 273 00:18:36,030 --> 00:18:38,730 But if you write down what happens as a function of time 274 00:18:38,730 --> 00:18:42,030 to the state, and you couple it as a function of time 275 00:18:42,030 --> 00:18:44,520 to the laser field, you will find out 276 00:18:44,520 --> 00:18:47,240 that there is an evolving relative phase 277 00:18:47,240 --> 00:18:50,240 between the light field and the atomic state. 278 00:18:50,240 --> 00:18:53,600 And the essence of that is that with these, there 279 00:18:53,600 --> 00:18:57,060 will be a beat node at which the dark state becomes bright 280 00:18:57,060 --> 00:18:58,759 and the bright state becomes dark. 281 00:19:05,060 --> 00:19:05,560 OK. 282 00:19:08,480 --> 00:19:15,460 One important application of the dark state 283 00:19:15,460 --> 00:19:18,940 is the STIRAP technique. 284 00:19:18,940 --> 00:19:25,715 STIRAP is adiabetic population transfer. 285 00:19:43,690 --> 00:19:48,470 So STIRAP stands for stimulated-- what is the r? 286 00:19:52,380 --> 00:19:54,455 Stimulated Rapid Adiabetic Passage? 287 00:19:54,455 --> 00:19:56,163 AUDIENCE: I've seen both rapid and Raman. 288 00:19:59,360 --> 00:20:01,170 PROFESSOR: So we have two choices. 289 00:20:01,170 --> 00:20:03,290 It's both Raman and rapid. 290 00:20:03,290 --> 00:20:06,170 A is adiabetic, p is passage. 291 00:20:06,170 --> 00:20:08,420 And STI is stimulated, because it's 292 00:20:08,420 --> 00:20:12,070 a coherent stimulated process. 293 00:20:12,070 --> 00:20:13,570 The idea is the following. 294 00:20:13,570 --> 00:20:15,860 With this concept of a dark state, 295 00:20:15,860 --> 00:20:18,900 I can immediately explain to you how, 296 00:20:18,900 --> 00:20:22,310 by changing the intensities of the laser beams, 297 00:20:22,310 --> 00:20:26,410 you can make an adiabetic transfer from one state 298 00:20:26,410 --> 00:20:28,570 g to the other state f. 299 00:20:28,570 --> 00:20:32,010 And this is important because in many atomic physics 300 00:20:32,010 --> 00:20:34,670 experiments, you start with one state 301 00:20:34,670 --> 00:20:37,120 and then you want to prepare another state. 302 00:20:37,120 --> 00:20:41,220 You always have the option of having some suitable pi pulse 303 00:20:41,220 --> 00:20:42,820 and going from one state to the next. 304 00:20:42,820 --> 00:20:45,610 But the pi pulse has to be exactly pi. 305 00:20:45,610 --> 00:20:47,440 You have to be careful what the intensity 306 00:20:47,440 --> 00:20:49,420 and duration of your pulse is. 307 00:20:49,420 --> 00:20:52,780 But if you can adiabetically go from one stage 308 00:20:52,780 --> 00:20:54,650 to the other one, this is terribly 309 00:20:54,650 --> 00:20:58,280 robust against pretty much everything. 310 00:20:58,280 --> 00:21:01,400 And I want to now use the concept of the dark state 311 00:21:01,400 --> 00:21:06,270 to tell you how this population transfer works. 312 00:21:06,270 --> 00:21:08,830 But before I even get into any explanation, 313 00:21:08,830 --> 00:21:11,860 the picture is the following. 314 00:21:11,860 --> 00:21:17,520 The dark state-- if you change laser parameters, 315 00:21:17,520 --> 00:21:20,300 you can change which state is the dark state. 316 00:21:20,300 --> 00:21:23,620 I gave you the simple example if one laser beam is on, 317 00:21:23,620 --> 00:21:25,740 the state g is the dark state. 318 00:21:25,740 --> 00:21:29,310 If the other laser beam is on, the state f is the dark state. 319 00:21:29,310 --> 00:21:33,200 So if you switch one laser beam off and the other one on, 320 00:21:33,200 --> 00:21:36,090 you've changed the definition of the dark state. 321 00:21:36,090 --> 00:21:40,140 And as I want to show you is the dark state, if we can set it up 322 00:21:40,140 --> 00:21:42,620 in the situation that the dark state is the absolute ground 323 00:21:42,620 --> 00:21:45,430 state of the system, and you know when you change parameters 324 00:21:45,430 --> 00:21:47,940 of your Hamiltonian, adiabeticity 325 00:21:47,940 --> 00:21:51,660 tells you that you always stay in the ground state. 326 00:21:51,660 --> 00:21:53,800 See now, even without any equations 327 00:21:53,800 --> 00:21:56,950 you understand what adiabetic population transfer is. 328 00:21:56,950 --> 00:22:00,570 But let's work it out. 329 00:22:00,570 --> 00:22:09,250 So we want to understand-- and this is our adiabetic process-- 330 00:22:09,250 --> 00:22:15,950 what happens when the laser parameters, omega 1 and omega 331 00:22:15,950 --> 00:22:17,885 2, change slowly. 332 00:22:23,790 --> 00:22:28,950 So for pedagogical reasons, I will 333 00:22:28,950 --> 00:22:30,850 drop all the assumptions I'm making now, 334 00:22:30,850 --> 00:22:32,930 but let me make them. 335 00:22:32,930 --> 00:22:39,310 So let's discuss degenerate states first. 336 00:22:44,400 --> 00:22:48,010 So g and f are degenerate. 337 00:22:48,010 --> 00:22:51,930 And let me also discuss blue-detuned light. 338 00:22:51,930 --> 00:22:53,345 And you will see in a moment why. 339 00:22:57,190 --> 00:23:08,760 So that would mean that-- and let's set up the situation. 340 00:23:08,760 --> 00:23:11,830 We have the state g, f, and e. 341 00:23:17,470 --> 00:23:23,230 We have a green laser beam and an orange laser beam. 342 00:23:23,230 --> 00:23:26,850 And we know already, from what we have just learned, 343 00:23:26,850 --> 00:23:32,950 that this should be transformed into-- 344 00:23:32,950 --> 00:23:34,740 and let me just be more specific here. 345 00:23:37,420 --> 00:23:43,720 I said I want to use blue-detuned light. 346 00:23:43,720 --> 00:23:50,850 So now we do what we just learned. 347 00:23:50,850 --> 00:23:56,100 We express things as the dark state and the bright state. 348 00:23:56,100 --> 00:23:58,730 And here we have the excited state. 349 00:23:58,730 --> 00:24:02,330 And the beauty of degeneracy is that dark state 350 00:24:02,330 --> 00:24:04,080 and bright state are superposition states, 351 00:24:04,080 --> 00:24:05,220 but since they're superposition states 352 00:24:05,220 --> 00:24:06,750 of two states with the same energy, 353 00:24:06,750 --> 00:24:09,710 I can draw them without twisting my hand 354 00:24:09,710 --> 00:24:13,330 or doing sort of a double line. 355 00:24:13,330 --> 00:24:16,040 Anyway, but now what happens is the following. 356 00:24:16,040 --> 00:24:19,630 The bright state is coupled to the excited 357 00:24:19,630 --> 00:24:23,270 state by the two laser fields. 358 00:24:23,270 --> 00:24:28,370 And the question is, what is the energy 359 00:24:28,370 --> 00:24:30,580 of the bright state relative to the dark state? 360 00:24:35,980 --> 00:24:38,660 Well, if I have a blue-detuned laser beam, 361 00:24:38,660 --> 00:24:44,690 I get an ac-Stark shift, which increases the energy. 362 00:24:44,690 --> 00:24:50,730 So therefore, the state which sees the light, which 363 00:24:50,730 --> 00:24:52,990 scatters photon-- scattering photon 364 00:24:52,990 --> 00:24:55,610 means there is an excited state admixture. 365 00:24:55,610 --> 00:24:59,130 An excited state admixture means there's an ac-Stark effect. 366 00:24:59,130 --> 00:25:00,820 And the ac-Stark effect is positive. 367 00:25:03,570 --> 00:25:08,310 So therefore, the dark state is-- the uncoupled state 368 00:25:08,310 --> 00:25:11,560 is the lowest state in the ground state manifold. 369 00:25:17,560 --> 00:25:19,520 And even if you would not generalize it 370 00:25:19,520 --> 00:25:23,810 to multiple levels and all that, if there is one dark state, 371 00:25:23,810 --> 00:25:26,460 it's not upshifted by the ac-Stark effect. 372 00:25:26,460 --> 00:25:28,765 Whereas every state which sees the light 373 00:25:28,765 --> 00:25:31,750 is upshifted by the ac-Stark effect 374 00:25:31,750 --> 00:25:33,400 which provides a blue shift. 375 00:25:33,400 --> 00:25:36,270 So the dark state is the lowest quantum state. 376 00:25:41,100 --> 00:25:46,250 So therefore, we can now use the general concept 377 00:25:46,250 --> 00:25:49,670 in quantum mechanics of adiabetic state transfer 378 00:25:49,670 --> 00:25:57,600 to achieve a perfect 100% transfer from one ground 379 00:25:57,600 --> 00:26:10,316 state to the other one by tailoring as a function of time 380 00:26:10,316 --> 00:26:13,480 our laser fields as follows. 381 00:26:13,480 --> 00:26:23,160 We start out in a situation where the only beam which is on 382 00:26:23,160 --> 00:26:34,260 is-- I want to change that to orange-- is the orange laser. 383 00:26:34,260 --> 00:26:38,850 And when the orange laser is on, then the dark state is g. 384 00:26:41,760 --> 00:26:45,390 Then we ramp down the orange laser. 385 00:26:45,390 --> 00:26:48,470 But we ramp up the green laser. 386 00:26:51,550 --> 00:26:55,340 And we have the situation that initially g 387 00:26:55,340 --> 00:26:57,750 and eventually f is the dark state. 388 00:27:03,140 --> 00:27:07,120 So this is a picture behind the rapid adiabetic passage 389 00:27:07,120 --> 00:27:11,270 from state g to f. 390 00:27:13,860 --> 00:27:18,090 I want to point out that the application of the two laser 391 00:27:18,090 --> 00:27:21,770 pulse, first orange and then green 392 00:27:21,770 --> 00:27:25,490 is called the counter-intuitive sequence. 393 00:27:25,490 --> 00:27:28,030 Because if you ask yourself the problem, 394 00:27:28,030 --> 00:27:30,470 you're starting out in state g and you 395 00:27:30,470 --> 00:27:33,480 want to go over to state f, which laser 396 00:27:33,480 --> 00:27:36,810 would you first switch on if you're in state g? 397 00:27:36,810 --> 00:27:38,950 Well, if you want to get out of state g, 398 00:27:38,950 --> 00:27:40,660 you would say first the green laser 399 00:27:40,660 --> 00:27:44,820 to go up and then the orange laser to go down. 400 00:27:44,820 --> 00:27:47,450 So what I just said, first the green and then the orange 401 00:27:47,450 --> 00:27:49,640 is called the intuitive sequence. 402 00:27:49,640 --> 00:27:52,590 But STIRAP works with the counter-intuitive sequence. 403 00:28:10,450 --> 00:28:12,520 OK. 404 00:28:12,520 --> 00:28:13,778 Any questions? 405 00:28:13,778 --> 00:28:14,278 Yes? 406 00:28:14,278 --> 00:28:18,150 AUDIENCE: So what's the intuition we have with this? 407 00:28:18,150 --> 00:28:20,560 PROFESSOR: Well, I gave you the intuition. 408 00:28:20,560 --> 00:28:22,720 I said, you want to have a dark state. 409 00:28:22,720 --> 00:28:24,480 The dark state is the ground state, 410 00:28:24,480 --> 00:28:26,270 the lowest state of the manifold. 411 00:28:26,270 --> 00:28:28,860 So you want to first make sure that g is the ground state. 412 00:28:28,860 --> 00:28:31,200 And then you turn opposite your Hamiltonian, 413 00:28:31,200 --> 00:28:33,770 and eventually f is the ground state. 414 00:28:33,770 --> 00:28:35,330 And this is exactly the situation. 415 00:28:35,330 --> 00:28:37,580 f is the ground state with the green light on. 416 00:28:37,580 --> 00:28:40,730 I've given you the intuitive picture. 417 00:28:40,730 --> 00:28:42,605 Now I will actually come back to that. 418 00:28:42,605 --> 00:28:51,650 There is the kind of-- it's called 419 00:28:51,650 --> 00:28:54,850 the magic of the adiabetic state transfer 420 00:28:54,850 --> 00:28:59,190 that if you always stay in the dark state, 421 00:28:59,190 --> 00:29:02,010 if you always stay in the dark state, 422 00:29:02,010 --> 00:29:05,090 you never populate the excited state. 423 00:29:05,090 --> 00:29:07,520 So it looks like magic that you can 424 00:29:07,520 --> 00:29:09,730 go from the ground state to the final state 425 00:29:09,730 --> 00:29:12,430 without having ever any population in the excited 426 00:29:12,430 --> 00:29:13,750 state. 427 00:29:13,750 --> 00:29:19,100 And of course, can that be true? 428 00:29:19,100 --> 00:29:21,960 No, because there is no coupling direct. 429 00:29:21,960 --> 00:29:25,805 So somehow in this picture of adiabetic transfer always 430 00:29:25,805 --> 00:29:29,430 staying in the dark and never populating the excited state, 431 00:29:29,430 --> 00:29:31,630 maybe we have over-emphasized something. 432 00:29:31,630 --> 00:29:33,670 We need a little bit of excited state, 433 00:29:33,670 --> 00:29:37,180 otherwise we cannot go there, because our Hamiltonian does 434 00:29:37,180 --> 00:29:39,780 not allow the direct path from g to f. 435 00:29:39,780 --> 00:29:41,990 So a lot of people think that STIRAP 436 00:29:41,990 --> 00:29:44,180 is a method where you can go from g to f 437 00:29:44,180 --> 00:29:45,860 without going through the excited state. 438 00:29:45,860 --> 00:29:48,040 I have many, many people talking about it. 439 00:29:48,040 --> 00:29:51,170 So I want to discuss it in the next few minutes 440 00:29:51,170 --> 00:29:53,630 and to some extent demystify it. 441 00:29:53,630 --> 00:29:58,710 But since you're asking me-- now in a black and white picture, 442 00:29:58,710 --> 00:30:00,900 STIRAP is the way. 443 00:30:00,900 --> 00:30:02,490 I'm correcting myself in a moment, 444 00:30:02,490 --> 00:30:05,150 but STIRAP is a way where you can go directly 445 00:30:05,150 --> 00:30:07,540 without going through the excited state. 446 00:30:07,540 --> 00:30:10,840 And this is done by the counter-intuitive sequence 447 00:30:10,840 --> 00:30:13,640 where you just want to keep atoms in the dark state, 448 00:30:13,640 --> 00:30:17,720 whereas the intuitive sequence is really based on the fact you 449 00:30:17,720 --> 00:30:21,230 want to take population here, put population here and the 450 00:30:21,230 --> 00:30:22,150 stimulate it down. 451 00:30:22,150 --> 00:30:26,040 And this is not what STIRAP is about. 452 00:30:26,040 --> 00:30:26,910 Other questions? 453 00:30:26,910 --> 00:30:27,410 Yes? 454 00:30:27,410 --> 00:30:29,860 AUDIENCE: So if there is very strong coupling 455 00:30:29,860 --> 00:30:34,720 to a fourth state from the excited state, because you need 456 00:30:34,720 --> 00:30:37,660 to-- the population needs to travel through the excited 457 00:30:37,660 --> 00:30:41,075 state in STIRAP, it's possible to spoil the whole process. 458 00:30:41,075 --> 00:30:42,450 PROFESSOR: We will talk about it. 459 00:30:42,450 --> 00:30:45,520 And actually in the next minute, I want to sort of simply 460 00:30:45,520 --> 00:30:50,410 calculate how many photons are emitted during the passage. 461 00:30:50,410 --> 00:30:53,610 And if you now say the photons have a branching ratio, 462 00:30:53,610 --> 00:30:56,030 this could be a branching ratio to you fourth state. 463 00:30:56,030 --> 00:31:00,080 But by just calculating what is the time integrated population 464 00:31:00,080 --> 00:31:02,990 in the excited state multiplied with gamma, 465 00:31:02,990 --> 00:31:05,120 this is the amount of photons scattered. 466 00:31:05,120 --> 00:31:07,820 And you can say if you try to do a STIRAP with the bosons 467 00:31:07,820 --> 00:31:13,300 then condensate, every time you scatter, you heat up, 468 00:31:13,300 --> 00:31:15,860 you reach momentum states outside the condensate. 469 00:31:15,860 --> 00:31:18,450 And those momentum states are your fourth state. 470 00:31:18,450 --> 00:31:21,280 So actually, by discussing how much light scattering happens 471 00:31:21,280 --> 00:31:23,910 in this transfer, I will pretty much 472 00:31:23,910 --> 00:31:26,570 talk indirectly about how much population 473 00:31:26,570 --> 00:31:28,480 is going into a fourth state. 474 00:31:28,480 --> 00:31:30,300 So that's what you want to understand now-- 475 00:31:30,300 --> 00:31:33,800 how true is the magic of the dark state transfer where 476 00:31:33,800 --> 00:31:35,920 we can go from one state to the next 477 00:31:35,920 --> 00:31:38,421 without ever going through the excited state. 478 00:31:38,421 --> 00:31:38,920 Yes? 479 00:31:38,920 --> 00:31:42,536 AUDIENCE: So are the two laser frequencies the same for that? 480 00:31:42,536 --> 00:31:43,370 PROFESSOR: Yes. 481 00:31:43,370 --> 00:31:47,460 I actually assumed that-- I come to that in a second-- 482 00:31:47,460 --> 00:31:49,820 that the two states are degenerate. 483 00:31:49,820 --> 00:31:53,010 And for all this dark state busyness, 484 00:31:53,010 --> 00:31:55,090 we have to fulfill the Raman resonance. 485 00:31:55,090 --> 00:31:57,420 So if the two states are degenerate, 486 00:31:57,420 --> 00:31:59,166 then the two laser beams. 487 00:31:59,166 --> 00:32:00,902 AUDIENCE: So then what's the meaning 488 00:32:00,902 --> 00:32:02,142 of increasing one of them? 489 00:32:02,142 --> 00:32:04,126 We are satisfied-- 490 00:32:04,126 --> 00:32:06,630 PROFESSOR: OK. 491 00:32:06,630 --> 00:32:11,770 One possibility is that this is an m equals 0 state. 492 00:32:11,770 --> 00:32:13,460 This is an m equals minus 1 state. 493 00:32:13,460 --> 00:32:17,260 This is m equals plus 1. 494 00:32:17,260 --> 00:32:20,360 And this is sigma plus. 495 00:32:20,360 --> 00:32:23,010 And this is sigma minus radiation. 496 00:32:23,010 --> 00:32:25,590 So yes, and this was actually very 497 00:32:25,590 --> 00:32:28,590 important for the whole discussion of three level 498 00:32:28,590 --> 00:32:32,530 coherence that this level can only talk to the green laser, 499 00:32:32,530 --> 00:32:34,820 this level can only talk with the orange laser. 500 00:32:34,820 --> 00:32:36,830 And we have two ways to accomplish it. 501 00:32:36,830 --> 00:32:39,530 One is by frequency difference. 502 00:32:39,530 --> 00:32:42,060 But if you don't have a frequency difference, 503 00:32:42,060 --> 00:32:46,270 we need polarization selection rules. 504 00:32:46,270 --> 00:32:47,083 Yes? 505 00:32:47,083 --> 00:32:50,618 AUDIENCE: It's sort of a silly question. 506 00:32:50,618 --> 00:32:54,320 In this particular case, if I have all my atoms in g, 507 00:32:54,320 --> 00:32:56,188 can I turn on the green laser? 508 00:32:56,188 --> 00:32:59,152 Can I optically pump everything to f? 509 00:32:59,152 --> 00:33:00,640 Right? 510 00:33:00,640 --> 00:33:03,657 And so in this particular case, is there 511 00:33:03,657 --> 00:33:06,115 an advantage of doing STIRAP as opposed to optical pumping? 512 00:33:15,810 --> 00:33:19,410 PROFESSOR: Yes there is, and we'll talk about that. 513 00:33:19,410 --> 00:33:24,080 There is a difference between transferring population or only 514 00:33:24,080 --> 00:33:26,490 transferring an amplitude. 515 00:33:26,490 --> 00:33:31,370 And so it depends what you want. 516 00:33:31,370 --> 00:33:33,430 If all you want is to get the atoms from g 517 00:33:33,430 --> 00:33:36,490 to f and you don't care how they are heated up, so that's OK. 518 00:33:36,490 --> 00:33:40,000 But the STIRAP process is a stimulated process, 519 00:33:40,000 --> 00:33:45,330 so you have momentum k and here you have the momentum k2. 520 00:33:45,330 --> 00:33:48,290 So you take the atoms from an initial state 521 00:33:48,290 --> 00:33:52,080 and put them in a very well-defined momentum state. 522 00:33:52,080 --> 00:33:55,560 So a PEC in one state after STIRAP transfer 523 00:33:55,560 --> 00:33:57,590 is in another state, maybe if you 524 00:33:57,590 --> 00:34:00,130 have the lasers in a counter-propagating way 525 00:34:00,130 --> 00:34:01,810 back to zero momentum. 526 00:34:01,810 --> 00:34:03,480 Whereas if you do optical pumping, which 527 00:34:03,480 --> 00:34:05,810 involves spontaneous emission, then you always 528 00:34:05,810 --> 00:34:07,110 have dissipation. 529 00:34:07,110 --> 00:34:08,620 You create entropy. 530 00:34:08,620 --> 00:34:12,679 And because spontaneous emission goes into many modes, 531 00:34:12,679 --> 00:34:17,100 you have many different recoils, and eventually your final state 532 00:34:17,100 --> 00:34:18,909 is no longer a single state. 533 00:34:18,909 --> 00:34:22,760 It has many different momentum states involved. 534 00:34:22,760 --> 00:34:26,520 So that's why we've been talking here about a coherent transfer, 535 00:34:26,520 --> 00:34:28,380 sort of assuming that optical pumping 536 00:34:28,380 --> 00:34:31,340 with spontaneous emission and the many modes involved 537 00:34:31,340 --> 00:34:33,100 is not what you want because of heating. 538 00:34:35,690 --> 00:34:40,289 OK, so lots of interesting questions. 539 00:34:47,380 --> 00:34:57,520 First, let's now drop all the assumptions we've made. 540 00:34:57,520 --> 00:35:00,040 So what happens when we change from blue detuning 541 00:35:00,040 --> 00:35:00,945 to red detuning? 542 00:35:15,570 --> 00:35:17,535 Well, for blue detuning, we use the argument 543 00:35:17,535 --> 00:35:20,350 that adiabetically we stay in the ground state, 544 00:35:20,350 --> 00:35:22,250 but for red detuning it's a higher state, 545 00:35:22,250 --> 00:35:24,900 and we adiabetically stay in the higher state. 546 00:35:24,900 --> 00:35:25,590 Same thing. 547 00:35:28,550 --> 00:35:30,570 What happens if f and g are not degenerate? 548 00:35:49,141 --> 00:35:52,650 Well, the argument is the following. 549 00:35:52,650 --> 00:35:55,600 It's the same result, and I can quickly show that to you 550 00:35:55,600 --> 00:36:03,060 by saying use the dressed state basis. 551 00:36:03,060 --> 00:36:18,280 You have the state g with Na and Nb photons in laser beams. 552 00:36:18,280 --> 00:36:19,200 Let me be consistent. 553 00:36:19,200 --> 00:36:22,870 I call the operators c and Na and Nc. 554 00:36:22,870 --> 00:36:26,950 And if you now look at the state f, 555 00:36:26,950 --> 00:36:33,230 which has one photon less here and one photon more here, 556 00:36:33,230 --> 00:36:34,880 those two states are degenerate. 557 00:36:39,290 --> 00:36:41,440 So I was just showing the situation 558 00:36:41,440 --> 00:36:44,185 if g and f are not degenerate, here 559 00:36:44,185 --> 00:36:46,220 we have a laser beam with Na photons. 560 00:36:46,220 --> 00:36:50,120 Here we have a laser beam with Nc photons. 561 00:36:50,120 --> 00:36:52,390 If we include the photons in our description, 562 00:36:52,390 --> 00:36:54,590 we have a degeneracy. 563 00:36:54,590 --> 00:36:56,090 And that's what matters. 564 00:36:56,090 --> 00:36:58,180 But what is relevant of course is 565 00:36:58,180 --> 00:37:01,450 you have this degeneracy only when the energy 566 00:37:01,450 --> 00:37:03,980 difference between g and f is the energy frequency 567 00:37:03,980 --> 00:37:05,720 difference between the two laser beams. 568 00:37:05,720 --> 00:37:08,410 So it is essential, and you see that again here, 569 00:37:08,410 --> 00:37:10,059 that the two photon Raman resonance 570 00:37:10,059 --> 00:37:11,100 condition has to be made. 571 00:37:19,430 --> 00:37:23,030 So we've talked all the assumptions so we know you can, 572 00:37:23,030 --> 00:37:24,620 in a general system you can transfer 573 00:37:24,620 --> 00:37:25,960 between two hyperfine states. 574 00:37:25,960 --> 00:37:27,835 You can use red or blue detuned laser. 575 00:37:27,835 --> 00:37:28,710 Everything is robust. 576 00:37:32,880 --> 00:37:33,520 OK. 577 00:37:33,520 --> 00:37:43,315 So STIRAP goes by the name dark state transfer. 578 00:37:47,010 --> 00:37:51,280 And what I want to address now is, 579 00:37:51,280 --> 00:37:53,650 what is the magic which is going on? 580 00:37:53,650 --> 00:38:01,040 How can we transfer from state g to state f 581 00:38:01,040 --> 00:38:02,870 without going to the excited state? 582 00:38:05,930 --> 00:38:07,660 And the answer is-- I gave it already 583 00:38:07,660 --> 00:38:11,850 to you-- it's not possible, because the only couplings we 584 00:38:11,850 --> 00:38:14,700 have are the two couplings which I've shown. 585 00:38:18,060 --> 00:38:22,680 But-- and this is sort of the gist you should carry away. 586 00:38:22,680 --> 00:38:29,980 Since the whole transfer is a coherent process-- 587 00:38:29,980 --> 00:38:32,800 and I will be more exact in a moment-- 588 00:38:32,800 --> 00:38:34,340 we are not building up population 589 00:38:34,340 --> 00:38:36,030 in the excited state. 590 00:38:36,030 --> 00:38:38,550 We're just building up a small amplitude. 591 00:38:38,550 --> 00:38:40,925 And the population is the small amplitude squared. 592 00:38:40,925 --> 00:38:43,520 It's, so to speak, epsilon squared. 593 00:38:43,520 --> 00:38:46,860 And so you win one power of epsilon, 594 00:38:46,860 --> 00:38:49,040 one power of the small amplitude, 595 00:38:49,040 --> 00:38:51,850 if you do a coherent transfer, as compared 596 00:38:51,850 --> 00:38:54,050 to incoherent transfer, which relies 597 00:38:54,050 --> 00:38:56,450 on shuffling over population. 598 00:38:56,450 --> 00:39:00,005 And I want to show you now how this pans out. 599 00:39:00,005 --> 00:39:00,505 Jenny? 600 00:39:00,505 --> 00:39:05,170 AUDIENCE: But in the limit of infinitely adiabetic, 601 00:39:05,170 --> 00:39:11,880 infinitely slow, you still would have the transfer happen 602 00:39:11,880 --> 00:39:15,240 without ever going into the excited state, right? 603 00:39:15,240 --> 00:39:20,530 PROFESSOR: Well, I will show you that the integrated population 604 00:39:20,530 --> 00:39:23,850 in the excited state goes as one over the transfer time. 605 00:39:23,850 --> 00:39:26,880 So if you allow an infinite amount for the transfer, 606 00:39:26,880 --> 00:39:30,630 you can make the population [INAUDIBLE] slow. 607 00:39:30,630 --> 00:39:32,510 But ultimately, what I also want to show 608 00:39:32,510 --> 00:39:36,010 you is that how much you put into the excited state 609 00:39:36,010 --> 00:39:39,810 depends-- and I think this is a very satisfying result 610 00:39:39,810 --> 00:39:42,050 for me-- depends on your resources. 611 00:39:42,050 --> 00:39:43,746 It depends on your laser power. 612 00:39:43,746 --> 00:39:45,120 if you have infinite laser power, 613 00:39:45,120 --> 00:39:48,180 you can keep the population infinitesimally small. 614 00:39:48,180 --> 00:39:50,090 And if you have infinite time. 615 00:39:50,090 --> 00:39:52,510 But if you go to the lab and do an experiment and say, 616 00:39:52,510 --> 00:39:54,360 I want to be done in 24 hours, well 617 00:39:54,360 --> 00:39:56,200 this defines your time budget. 618 00:39:56,200 --> 00:39:58,580 And the strongest laser you have in the laboratory 619 00:39:58,580 --> 00:40:00,160 defines your laser power budget. 620 00:40:00,160 --> 00:40:03,410 And given those two resources, there is a small amount. 621 00:40:03,410 --> 00:40:05,470 And I want to give you an estimate for that. 622 00:40:05,470 --> 00:40:08,060 And then we have to discuss, is that small amount, 623 00:40:08,060 --> 00:40:10,180 which we get in STIRAP, does that really 624 00:40:10,180 --> 00:40:14,180 stand out over other methods to transfer population? 625 00:40:14,180 --> 00:40:15,665 And the answer will be interesting. 626 00:40:20,220 --> 00:40:22,130 So let me just say, no. 627 00:40:25,740 --> 00:40:30,510 Excited state is needed as a stepping stone. 628 00:40:40,990 --> 00:40:47,420 I want to give you a quantitative touch 629 00:40:47,420 --> 00:40:49,680 without doing too much math. 630 00:40:49,680 --> 00:40:56,070 So let me just write down Schrodinger's equation. 631 00:40:56,070 --> 00:40:59,570 We have the three amplitudes, the three states involved, 632 00:40:59,570 --> 00:41:02,240 e, g, f. 633 00:41:02,240 --> 00:41:04,870 Schrodinger's equation gives us the time derivative. 634 00:41:09,250 --> 00:41:12,470 c dot f. 635 00:41:12,470 --> 00:41:16,700 Since we assumed in the dressed state picture, 636 00:41:16,700 --> 00:41:19,860 we can assume everything is degenerate. 637 00:41:19,860 --> 00:41:21,620 But now we have couplings. 638 00:41:21,620 --> 00:41:23,310 And the only couplings are-- and this 639 00:41:23,310 --> 00:41:26,140 is what I was emphasizing-- between ground and excited 640 00:41:26,140 --> 00:41:33,820 state with the Rabi frequency omega 1 and between the state f 641 00:41:33,820 --> 00:41:38,700 and the excited state with Rabi frequency omega 2. 642 00:41:47,340 --> 00:41:50,420 So the important part is-- I can write it down, 643 00:41:50,420 --> 00:41:53,640 but the third equation is the one I want to focus, 644 00:41:53,640 --> 00:41:59,000 that you can only build up population in the final state 645 00:41:59,000 --> 00:42:02,930 by transferring with a Rabi frequency omega 646 00:42:02,930 --> 00:42:06,940 two population from the excited state. 647 00:42:06,940 --> 00:42:09,150 So let me just assume that Rabi frequency omega 648 00:42:09,150 --> 00:42:13,240 1 equals omega 2 equals omega Rabi. 649 00:42:13,240 --> 00:42:17,250 Let's have a symmetric situation so I can now completely focus 650 00:42:17,250 --> 00:42:20,150 how can you get from the excited state to the final state, 651 00:42:20,150 --> 00:42:21,750 but how you get from the ground state, 652 00:42:21,750 --> 00:42:25,240 from the state g to the excited state is fairly symmetric here. 653 00:42:25,240 --> 00:42:32,100 OK, so what we want is we want that at the end of the transfer 654 00:42:32,100 --> 00:42:36,200 time, t transfer, the final state 655 00:42:36,200 --> 00:42:38,050 amplitude is on the order of 1. 656 00:42:38,050 --> 00:42:41,010 This would be 100% percent transfer. 657 00:42:41,010 --> 00:42:44,060 So therefore, just looking at the differential equation 658 00:42:44,060 --> 00:42:56,360 above, that means that this here has 659 00:42:56,360 --> 00:43:04,200 to be 1 divided by the transfer time. 660 00:43:04,200 --> 00:43:08,990 So therefore, we find that for this adiabetic transfer, 661 00:43:08,990 --> 00:43:11,920 the excited state amplitude has to be-- 662 00:43:11,920 --> 00:43:13,900 and I neglect factors on the order of 2 663 00:43:13,900 --> 00:43:19,220 now-- has to be 1 over the Rabi frequency times the transfer 664 00:43:19,220 --> 00:43:21,910 time. 665 00:43:21,910 --> 00:43:26,110 So the probability to be in the excited state 666 00:43:26,110 --> 00:43:28,005 is the amplitude squared. 667 00:43:30,890 --> 00:43:33,870 And if you are asking, which I think is a practical question 668 00:43:33,870 --> 00:43:37,420 and I suggest to use this as a figure of merit, 669 00:43:37,420 --> 00:43:43,880 when we discuss how well, how perfectly we have transferred 670 00:43:43,880 --> 00:43:46,960 population, we can now ask, what is 671 00:43:46,960 --> 00:43:51,500 the probability of spontaneous emission? 672 00:43:54,640 --> 00:44:00,160 P spontaneous is the probability to be in the excited state 673 00:44:00,160 --> 00:44:03,870 times gamma integrated over time. 674 00:44:07,460 --> 00:44:11,130 So what we obtain for that is yes, 675 00:44:11,130 --> 00:44:14,040 it's proportionate to gamma. 676 00:44:14,040 --> 00:44:17,590 But we have the excited state amplitude squared. 677 00:44:17,590 --> 00:44:20,490 We've multiplied with the transfer time. 678 00:44:20,490 --> 00:44:24,990 So therefore, what we have in the denominator 679 00:44:24,990 --> 00:44:27,090 is the laser power or the Rabi frequency 680 00:44:27,090 --> 00:44:30,627 squared times the transfer time. 681 00:44:35,030 --> 00:44:39,360 So therefore, this probability to go 682 00:44:39,360 --> 00:44:42,040 to a spontaneous emission, which is also the integrated 683 00:44:42,040 --> 00:44:48,320 probability of having populated the excited state, 684 00:44:48,320 --> 00:44:55,010 it goes to 0 in the case that you have infinite laser power 685 00:44:55,010 --> 00:45:00,018 or that you're infinitely patient. 686 00:45:08,000 --> 00:45:13,260 So one question is, how long can the transfer time be? 687 00:45:13,260 --> 00:45:19,530 And this is actually setting an limit to it. 688 00:45:19,530 --> 00:45:21,240 We're talking about a coherent transfer. 689 00:45:23,950 --> 00:45:27,410 So the transfer time has to be smaller 690 00:45:27,410 --> 00:45:31,250 than the coherence time of your system. 691 00:45:31,250 --> 00:45:33,250 For instance, we do that all the time in my lab. 692 00:45:33,250 --> 00:45:35,900 If you go from on sublevel to another sublevel, 693 00:45:35,900 --> 00:45:39,210 and we have magnetic field noise, if the magnetic field 694 00:45:39,210 --> 00:45:42,480 noise is destroying the phase relationship between the two 695 00:45:42,480 --> 00:45:45,260 hyperfine states, then the coherent transfer 696 00:45:45,260 --> 00:45:47,705 gets interrupted, and then eventually we 697 00:45:47,705 --> 00:45:49,455 are no longer talking about the amplitude. 698 00:45:49,455 --> 00:45:54,260 We are talking about population in the excited state. 699 00:45:54,260 --> 00:46:00,240 OK, so we have this result, which at least demystifies what 700 00:46:00,240 --> 00:46:02,680 many people say-- actually I've heard 701 00:46:02,680 --> 00:46:05,980 truly experienced researchers in the field to say, 702 00:46:05,980 --> 00:46:09,340 we can do the transfer without ever populating 703 00:46:09,340 --> 00:46:10,800 the excited state. 704 00:46:10,800 --> 00:46:13,780 So the truth is what I just derived for you. 705 00:46:13,780 --> 00:46:16,680 But now we want to compare it to what 706 00:46:16,680 --> 00:46:19,745 would happen if you use an incoherent picture. 707 00:46:25,640 --> 00:46:27,950 If you use an incoherent picture, 708 00:46:27,950 --> 00:46:30,990 let's assume we would use a pi pulse, 709 00:46:30,990 --> 00:46:33,650 put all the population in the excited state, 710 00:46:33,650 --> 00:46:35,250 and then we would stimulate it down. 711 00:46:43,680 --> 00:46:45,980 So we have a pi pulse. 712 00:46:45,980 --> 00:46:47,870 So we build our population. 713 00:46:47,870 --> 00:46:49,768 And then we use a second pi pulse. 714 00:46:52,640 --> 00:46:59,280 Well, we would then have about unity population in the excited 715 00:46:59,280 --> 00:47:00,742 state. 716 00:47:00,742 --> 00:47:05,950 It would emit light at a rate gamma. 717 00:47:05,950 --> 00:47:10,790 And it would take the time of the inverse Rabi frequency, 718 00:47:10,790 --> 00:47:13,320 the Rabi period, to stimulate it down. 719 00:47:16,920 --> 00:47:19,280 Now you can say, what happens if-- maybe that's 720 00:47:19,280 --> 00:47:22,750 dumb to have a pi pulse and put all the population in, take 721 00:47:22,750 --> 00:47:26,781 a pi over 20 pulse, put a little bit population in, stimulate it 722 00:47:26,781 --> 00:47:27,280 down. 723 00:47:27,280 --> 00:47:29,980 The next batch, the next batch, the next batch, but you 724 00:47:29,980 --> 00:47:33,790 get exactly the same result, because each population which 725 00:47:33,790 --> 00:47:35,790 you've put into the excited state 726 00:47:35,790 --> 00:47:40,040 had to wait for a Rabi period to be stimulated down. 727 00:47:40,040 --> 00:47:42,220 So therefore in the incoherent picture, 728 00:47:42,220 --> 00:47:45,690 if you sort of transfer batches of population, 729 00:47:45,690 --> 00:47:49,970 you realize that the probability for spontaneous emission 730 00:47:49,970 --> 00:47:53,340 or for populating a four state is independent of the transfer 731 00:47:53,340 --> 00:47:57,110 time, because it cancels out. 732 00:47:57,110 --> 00:47:59,770 If you do small pieces of population, 733 00:47:59,770 --> 00:48:01,900 you have a smaller rate of emission, 734 00:48:01,900 --> 00:48:04,330 but you integrate for longer and the two temporal factors 735 00:48:04,330 --> 00:48:05,450 cancel out. 736 00:48:05,450 --> 00:48:08,610 And now you realize that the coherent transfer 737 00:48:08,610 --> 00:48:13,234 is more favorable by another power of omega Rabi times t 738 00:48:13,234 --> 00:48:14,400 transfer in the denominator. 739 00:48:17,550 --> 00:48:19,080 So that's why people prefer STIRAP. 740 00:48:23,745 --> 00:48:24,620 Questions about that? 741 00:48:34,790 --> 00:48:37,200 Can you suggest another method how 742 00:48:37,200 --> 00:48:42,840 we could transfer population from one state to the next? 743 00:48:42,840 --> 00:48:44,520 I gave you two examples. 744 00:48:44,520 --> 00:48:49,640 One was this magic, or not so magic, dark state transfer. 745 00:48:49,640 --> 00:48:51,430 The other one is a method where we 746 00:48:51,430 --> 00:48:54,800 put population, stimulated it down, next batch of population, 747 00:48:54,800 --> 00:48:57,420 stimulate it down. 748 00:48:57,420 --> 00:49:02,125 What is another process which we can do to transfer population? 749 00:49:02,125 --> 00:49:03,520 AUDIENCE: Landau-Zener? 750 00:49:03,520 --> 00:49:04,270 PROFESSOR: Pardon? 751 00:49:04,270 --> 00:49:05,330 AUDIENCE: Landau-Zener? 752 00:49:05,330 --> 00:49:06,330 PROFESSOR: Landau-Zener? 753 00:49:08,450 --> 00:49:10,970 Actually, the STIRAP is a Landau-Zener. 754 00:49:10,970 --> 00:49:13,370 The dark state is always the lowest state. 755 00:49:13,370 --> 00:49:15,290 There is a crossing sort of in between, 756 00:49:15,290 --> 00:49:16,760 but it's an avoided crossing. 757 00:49:16,760 --> 00:49:20,170 Actually, in a way, you may want to think about it. 758 00:49:20,170 --> 00:49:22,870 The STIRAP is actually a Landau-Zener transfer 759 00:49:22,870 --> 00:49:24,894 between where the lower state is a dark state, 760 00:49:24,894 --> 00:49:26,310 the upper state is a bright state. 761 00:49:26,310 --> 00:49:26,810 Will? 762 00:49:26,810 --> 00:49:28,294 AUDIENCE: I still have a question. 763 00:49:28,294 --> 00:49:39,454 So Landau-Zener [INAUDIBLE] But the STIRAP scheme 764 00:49:39,454 --> 00:49:41,864 is sensitive to the run-on resonance. 765 00:49:41,864 --> 00:49:43,030 PROFESSOR: Absolutely it is. 766 00:49:43,030 --> 00:49:46,949 AUDIENCE: So how robust is it against small violations 767 00:49:46,949 --> 00:49:47,448 [INAUDIBLE]? 768 00:49:52,327 --> 00:49:55,100 PROFESSOR: Well, I gave you already the hint 769 00:49:55,100 --> 00:49:58,760 when I discussed that if you are detuned from the Raman 770 00:49:58,760 --> 00:50:02,250 resonance, this detuning, at the detuning frequency 771 00:50:02,250 --> 00:50:04,460 would be between dark and bright state. 772 00:50:04,460 --> 00:50:07,020 So this picture what is dark and what is bright 773 00:50:07,020 --> 00:50:08,900 is scrambled up by that. 774 00:50:08,900 --> 00:50:11,400 So in other words, as long as you're 775 00:50:11,400 --> 00:50:16,400 inverse detuning is smaller than the transfer time, you're OK. 776 00:50:16,400 --> 00:50:18,280 Because the dark state is the dark state. 777 00:50:18,280 --> 00:50:21,210 But if during the transfer time you turn the dark state 778 00:50:21,210 --> 00:50:22,955 into a bright state, you've completely 779 00:50:22,955 --> 00:50:23,830 messed up the scheme. 780 00:50:27,270 --> 00:50:31,180 So in other words, I know people in this hallway, in my group 781 00:50:31,180 --> 00:50:34,270 want to STIRAP transfer between molecules. 782 00:50:34,270 --> 00:50:36,420 And we have to stabilize our lasers. 783 00:50:36,420 --> 00:50:39,210 So we have automatically little bit laser fluctuations. 784 00:50:39,210 --> 00:50:41,630 But if you do the transfer in a millisecond, 785 00:50:41,630 --> 00:50:44,560 it is sufficient to have the laser frequency under control 786 00:50:44,560 --> 00:50:46,220 at the kilohertz level. 787 00:50:46,220 --> 00:50:48,440 So that's a requirement. 788 00:50:48,440 --> 00:50:50,320 But back to my question. 789 00:50:50,320 --> 00:50:52,365 We have discussed sort of an incoherent transfer 790 00:50:52,365 --> 00:50:53,920 and we've discussed STIRAP. 791 00:50:53,920 --> 00:50:55,500 Can you suggest another method how 792 00:50:55,500 --> 00:50:58,351 you can go from one count state to the next? 793 00:51:08,091 --> 00:51:10,039 AUDIENCE: We talked about optical pumping. 794 00:51:10,039 --> 00:51:12,470 PROFESSOR: Optical pumping? 795 00:51:12,470 --> 00:51:14,580 Yes, but optical pumping means we 796 00:51:14,580 --> 00:51:16,120 really go with spontaneous emission. 797 00:51:16,120 --> 00:51:18,780 And then we have 100% spontaneous emission. 798 00:51:18,780 --> 00:51:22,330 So we want a process where we go from one momentum state 799 00:51:22,330 --> 00:51:23,490 to another momentum state. 800 00:51:28,360 --> 00:51:29,040 Well-- 801 00:51:29,926 --> 00:51:31,144 AUDIENCE: [INAUDIBLE]. 802 00:51:31,144 --> 00:51:32,940 PROFESSOR: Yes. 803 00:51:32,940 --> 00:51:34,939 Optical pumping is a two-photon process. 804 00:51:34,939 --> 00:51:36,980 Actually, everything's a two-photon process here. 805 00:51:36,980 --> 00:51:40,480 But optical pumping is actually a process 806 00:51:40,480 --> 00:51:43,000 where we have a Raman to photon transition. 807 00:51:43,000 --> 00:51:44,670 The first leg is stimulated. 808 00:51:44,670 --> 00:51:46,440 The second one is spontaneous. 809 00:51:46,440 --> 00:51:48,770 But I would suggest now what [INAUDIBLE] said, 810 00:51:48,770 --> 00:51:51,240 let's have a two photon Raman transition. 811 00:52:01,210 --> 00:52:11,010 So we want to go from state g to state f with two photon Raman 812 00:52:11,010 --> 00:52:12,900 transition. 813 00:52:12,900 --> 00:52:14,770 Let's assume we have a detuning delta. 814 00:52:18,180 --> 00:52:21,420 So now, do you have any expectations 815 00:52:21,420 --> 00:52:25,130 whether these-- we can do a pi pulse. 816 00:52:25,130 --> 00:52:28,925 Two photon Raman transition is a two photon Rabi frequency. 817 00:52:28,925 --> 00:52:30,710 I'll remind you in a second of it. 818 00:52:30,710 --> 00:52:32,310 And if you do a pi pulse in that, 819 00:52:32,310 --> 00:52:34,290 we have half a Rabi oscillation, which 820 00:52:34,290 --> 00:52:38,840 takes us from cg to f and we have 100% transfer. 821 00:52:38,840 --> 00:52:42,390 So do you have an expectation if this Raman 822 00:52:42,390 --> 00:52:47,450 process how will it turn out to be in our figure of merit, 823 00:52:47,450 --> 00:52:50,560 how many photons are emitted from the excited 824 00:52:50,560 --> 00:52:52,500 state during the transfer? 825 00:52:52,500 --> 00:52:56,180 Will it be the same as STIRAP? 826 00:52:56,180 --> 00:52:59,830 Will it be the same as in population transfer? 827 00:52:59,830 --> 00:53:03,265 Will it be something else? 828 00:53:03,265 --> 00:53:06,151 AUDIENCE: You have to scale. 829 00:53:06,151 --> 00:53:08,660 PROFESSOR: It will scale with the detuning, 830 00:53:08,660 --> 00:53:12,650 but-- OK, what I will do is actually-- I will in a moment 831 00:53:12,650 --> 00:53:13,650 say, OK. 832 00:53:13,650 --> 00:53:14,930 We have a certain laser. 833 00:53:14,930 --> 00:53:18,840 What I'm going to say? 834 00:53:18,840 --> 00:53:21,260 We have to compare apples with apples. 835 00:53:21,260 --> 00:53:23,860 And here we have the detuning as a parameter. 836 00:53:23,860 --> 00:53:25,640 And what I want to replace is I will-- 837 00:53:25,640 --> 00:53:29,090 what will replace the detuning by the time it 838 00:53:29,090 --> 00:53:32,450 takes to do the pi pulse. 839 00:53:32,450 --> 00:53:35,670 So the detuning is determined by the condition 840 00:53:35,670 --> 00:53:39,430 that we want to do the transfer in the same time as the STIRAP. 841 00:53:39,430 --> 00:53:42,210 So this will eliminate the detuning. 842 00:53:42,210 --> 00:53:44,330 So we want to use the same resources. 843 00:53:44,330 --> 00:53:47,100 Our resources are time and laser power. 844 00:53:47,100 --> 00:53:49,520 And we want to use our resources of time and laser power 845 00:53:49,520 --> 00:53:53,120 in the same way for the two photon transition. 846 00:53:53,120 --> 00:53:58,380 So the question is what happens to-- what 847 00:53:58,380 --> 00:54:01,760 is the answer to the question, how much excited state is 848 00:54:01,760 --> 00:54:07,226 involved in the two photon Raman process as compared to STIRAP? 849 00:54:07,226 --> 00:54:08,350 Do you have an expectation? 850 00:54:13,910 --> 00:54:18,060 Who of you is doing STIRAP in the laboratory? 851 00:54:18,060 --> 00:54:21,080 A few people, yeah. 852 00:54:21,080 --> 00:54:25,170 Actually, I know that 99-- my observation 853 00:54:25,170 --> 00:54:28,850 is that 99% of the people say STIRAP is special 854 00:54:28,850 --> 00:54:31,870 because it's a dark state transfer. 855 00:54:31,870 --> 00:54:35,770 And when I started to confront famous people in the field 856 00:54:35,770 --> 00:54:39,039 and say the two photon Raman process does exactly the same, 857 00:54:39,039 --> 00:54:40,330 they were completely surprised. 858 00:54:43,150 --> 00:54:45,590 Anyway I want to show you in one equation 859 00:54:45,590 --> 00:54:47,680 that the two photon Raman process has 860 00:54:47,680 --> 00:54:51,580 exactly the same integrated population in the excited 861 00:54:51,580 --> 00:54:56,520 state, and therefore it is as good-- the two photon Raman 862 00:54:56,520 --> 00:55:01,740 process performs exactly as well as the so-called dark state 863 00:55:01,740 --> 00:55:05,350 transfer, where you never go through the excited state. 864 00:55:05,350 --> 00:55:07,870 And this for me completely demystifies it, 865 00:55:07,870 --> 00:55:09,520 because the two photon Raman process 866 00:55:09,520 --> 00:55:13,220 has in common with the STIRAP that everything is coherent. 867 00:55:13,220 --> 00:55:14,820 And if everything is coherent, it 868 00:55:14,820 --> 00:55:18,410 means we are not building a population. 869 00:55:18,410 --> 00:55:23,480 We are only building up amplitude in the excited state. 870 00:55:23,480 --> 00:55:25,330 So let me just show you in one equation 871 00:55:25,330 --> 00:55:28,070 that indeed it works out in that way. 872 00:55:28,070 --> 00:55:32,520 It's also a nice way to quickly recapitulate 873 00:55:32,520 --> 00:55:34,640 what we learned in the previous chapter 874 00:55:34,640 --> 00:55:37,330 about two photon transitions. 875 00:55:37,330 --> 00:55:40,440 So the two photon Rabi frequency is nothing else 876 00:55:40,440 --> 00:55:43,280 than the product of the two Rabi frequencies divided 877 00:55:43,280 --> 00:55:45,270 by the detuning. 878 00:55:45,270 --> 00:55:49,810 And of course as-- if you use studied larger and larger 879 00:55:49,810 --> 00:55:52,600 detuning as you said, everything will slow down. 880 00:55:52,600 --> 00:55:57,610 But we know that the transfer time 881 00:55:57,610 --> 00:56:03,120 will be the inverse of the two photon Rabi frequency. 882 00:56:03,120 --> 00:56:06,310 And this is delta over omega square. 883 00:56:06,310 --> 00:56:10,260 So therefore, we will eliminate delta from our equations 884 00:56:10,260 --> 00:56:16,540 by replacing it by the time it will 885 00:56:16,540 --> 00:56:22,290 take to perform a pi pulse between the two ground states. 886 00:56:22,290 --> 00:56:27,860 OK, so what is the probability to be in the excited state? 887 00:56:27,860 --> 00:56:30,330 Well, we've done the theory of the ac-Stark effect. 888 00:56:30,330 --> 00:56:32,870 We have beaten perturbation theory to death. 889 00:56:32,870 --> 00:56:35,120 You know the first order admixture 890 00:56:35,120 --> 00:56:38,465 is omega Rabi over delta and the probability 891 00:56:38,465 --> 00:56:41,610 is omega Rabi squared over delta squared. 892 00:56:41,610 --> 00:56:46,620 So therefore, the probability to spontaneously emit a photon 893 00:56:46,620 --> 00:56:51,820 is omega Rabi squared over delta squared times gamma times t 894 00:56:51,820 --> 00:56:54,780 transfer. 895 00:56:54,780 --> 00:57:00,430 And this is nothing else than gamma divided by omega Rabi 896 00:57:00,430 --> 00:57:04,210 squared times t transfer. 897 00:57:04,210 --> 00:57:07,020 So if you use a Raman pulse to transfer the population 898 00:57:07,020 --> 00:57:11,610 coherently, the integrated population 899 00:57:11,610 --> 00:57:14,410 of the excited state, which is a measure for heating and measure 900 00:57:14,410 --> 00:57:17,220 for winding up in the wrong state, emission 901 00:57:17,220 --> 00:57:20,410 to a false state is inversely proportional to your laser 902 00:57:20,410 --> 00:57:23,840 power and inversely proportional to the time 903 00:57:23,840 --> 00:57:26,940 you can afford to do the transfer. 904 00:57:26,940 --> 00:57:30,300 Exactly the same as for STIRAP. 905 00:57:30,300 --> 00:57:32,890 And the limit to the transfer time in both cases 906 00:57:32,890 --> 00:57:34,503 is set by the coherence time. 907 00:57:38,450 --> 00:57:44,140 Try to tell people who do STIRAP that a two photon Raman 908 00:57:44,140 --> 00:57:46,290 pulse would do exactly the same as STIRAP, 909 00:57:46,290 --> 00:57:48,900 at least in populating the excited state. 910 00:57:48,900 --> 00:57:50,210 They will be surprised. 911 00:57:50,210 --> 00:57:53,460 Because here, you explicitly go through the excited state. 912 00:57:53,460 --> 00:57:55,570 We don't have a counter-intuitive sequence. 913 00:57:55,570 --> 00:57:58,800 We switch on both lasers simultaneously. 914 00:57:58,800 --> 00:58:02,560 But I think this for me illustrates or completely 915 00:58:02,560 --> 00:58:06,030 demystifies what people have said about the dark state 916 00:58:06,030 --> 00:58:07,670 transfer and the STIRAP process. 917 00:58:12,140 --> 00:58:13,410 Any question? 918 00:58:13,410 --> 00:58:13,910 Cody? 919 00:58:13,910 --> 00:58:16,544 AUDIENCE: Why do people use STIRAP? 920 00:58:16,544 --> 00:58:17,915 What's the advantage? 921 00:58:17,915 --> 00:58:22,440 PROFESSOR: Well, why are you using Landau-Zener sweeps when 922 00:58:22,440 --> 00:58:24,650 you go from one hyperfine state to the next, 923 00:58:24,650 --> 00:58:27,599 and why don't you use a pi pulse with your f generator, 924 00:58:27,599 --> 00:58:28,640 because it's more robust. 925 00:58:28,640 --> 00:58:29,634 AUDIENCE: [INAUDIBLE]. 926 00:58:34,350 --> 00:58:35,350 As you said [INAUDIBLE]. 927 00:58:39,688 --> 00:58:43,090 PROFESSOR: But similar, if you detune-- 928 00:58:43,090 --> 00:58:47,430 if you detune your-- the condition is the same. 929 00:58:47,430 --> 00:58:50,540 If you have your two Raman laser and you detune them, 930 00:58:50,540 --> 00:58:53,720 you no longer have resonant Rabi oscillations. 931 00:58:53,720 --> 00:58:56,170 You have off-resonant Rabi oscillations. 932 00:58:56,170 --> 00:59:00,040 And you will find if you want to do the transfer in half 933 00:59:00,040 --> 00:59:04,560 a period, you have to make sure that you don't 934 00:59:04,560 --> 00:59:06,840 get one cycle of the beat node or one 935 00:59:06,840 --> 00:59:09,270 cycle of the detuning in your transfer time. 936 00:59:09,270 --> 00:59:13,290 So the condition for the frequency stability for the two 937 00:59:13,290 --> 00:59:15,860 photon Raman and the STIRAP I think 938 00:59:15,860 --> 00:59:18,970 are pretty exactly identical. 939 00:59:18,970 --> 00:59:20,890 So I think it's pretty much the robustness. 940 00:59:20,890 --> 00:59:25,070 And you have to decide what you need and what you want. 941 00:59:25,070 --> 00:59:28,120 For instance, if you have laser beam and the laser beam has 942 00:59:28,120 --> 00:59:31,660 an inhomogeneous profile, you cannot meet the pi pulse 943 00:59:31,660 --> 00:59:34,350 condition for the atoms in the middle and at the edge 944 00:59:34,350 --> 00:59:35,400 of your laser beam. 945 00:59:35,400 --> 00:59:39,060 But in STIRAP you just provide plenty of extra power 946 00:59:39,060 --> 00:59:43,140 and then everything is robust against the laser beam profile. 947 00:59:43,140 --> 00:59:45,832 So there are clearly advantages like that. 948 00:59:45,832 --> 00:59:48,390 But all what I point out is the advantage 949 00:59:48,390 --> 00:59:51,660 is not the population of the excited state. 950 00:59:51,660 --> 00:59:54,300 It's the same. 951 00:59:54,300 --> 00:59:54,800 Questions? 952 01:00:01,650 --> 01:00:02,150 OK. 953 01:00:04,780 --> 01:00:09,550 So this is dark state and STIRAP. 954 01:00:09,550 --> 01:00:20,290 Our next topic is lasing or gain without inversion. 955 01:00:33,360 --> 01:00:43,100 So in other words, I want to show you 956 01:00:43,100 --> 01:00:47,200 what was actually rather recent accomplishment 957 01:00:47,200 --> 01:00:58,090 that this common belief that you need population in the excited 958 01:00:58,090 --> 01:01:00,980 state, which is larger in the ground state, 959 01:01:00,980 --> 01:01:12,680 for optical gain and lasing is not true. 960 01:01:12,680 --> 01:01:14,730 Well, it's true for two level system, 961 01:01:14,730 --> 01:01:17,460 but it's not true when you have more than two levels. 962 01:01:17,460 --> 01:01:21,300 And I discuss it here for the case of a three-level system. 963 01:01:21,300 --> 01:01:25,515 And this is really insight which came as recent as in the '80s. 964 01:01:28,540 --> 01:01:34,310 One reference is the person who pioneered a lot of work 965 01:01:34,310 --> 01:01:37,310 on three-level systems and provided a lot of insight 966 01:01:37,310 --> 01:01:40,640 was Steve Harris at Stanford. 967 01:01:40,640 --> 01:01:47,520 And this was a paper in the late '80s which 968 01:01:47,520 --> 01:01:49,600 enunciated the possibility of having 969 01:01:49,600 --> 01:01:51,460 lasing without inversion. 970 01:01:51,460 --> 01:01:53,890 And when you are a graduate student like me, 971 01:01:53,890 --> 01:01:58,290 who was educated earlier, you say, gee whiz. 972 01:01:58,290 --> 01:01:59,910 That's not what I learned in class. 973 01:01:59,910 --> 01:02:02,100 But of course it's often the small sprint. 974 01:02:02,100 --> 01:02:05,130 It's the assumption for two-level system. 975 01:02:05,130 --> 01:02:10,760 So the secret behind lasing without inversion 976 01:02:10,760 --> 01:02:20,534 is that you can have cancellation of absorption 977 01:02:20,534 --> 01:02:21,200 by interference. 978 01:02:26,220 --> 01:02:30,930 In other words, this belief that you need inversion 979 01:02:30,930 --> 01:02:34,620 comes that you want that the stimulated emission is 980 01:02:34,620 --> 01:02:36,830 stronger than the absorption. 981 01:02:36,830 --> 01:02:40,060 But if you have a three-level system, you have to be careful. 982 01:02:40,060 --> 01:02:42,170 We have interference effect. 983 01:02:42,170 --> 01:02:45,910 And what I want to show you is that we can create a situation 984 01:02:45,910 --> 01:02:49,580 where we have destructive interference for absorption, 985 01:02:49,580 --> 01:02:51,740 but we don't have destructive interference 986 01:02:51,740 --> 01:02:53,820 for stimulated emission. 987 01:02:53,820 --> 01:02:59,320 So therefore, we can now afford to have more population 988 01:02:59,320 --> 01:03:03,390 in the ground state, because the absorption out of the ground 989 01:03:03,390 --> 01:03:06,810 state does not have to be suppressed by minimizing 990 01:03:06,810 --> 01:03:08,700 the population in the ground state. 991 01:03:08,700 --> 01:03:12,420 We have an interference effect, which suppresses absorption 992 01:03:12,420 --> 01:03:14,131 from the ground state population. 993 01:03:14,131 --> 01:03:15,880 And therefore, the ground state population 994 01:03:15,880 --> 01:03:17,950 can be larger than in a two-level system. 995 01:03:22,950 --> 01:03:26,030 I could simply tell you you can hide population 996 01:03:26,030 --> 01:03:28,200 in the dark state, end of discussion, 997 01:03:28,200 --> 01:03:29,880 and you get the basic idea. 998 01:03:29,880 --> 01:03:32,400 But the concept of lasing without inversion 999 01:03:32,400 --> 01:03:34,380 is richer than that. 1000 01:03:34,380 --> 01:03:37,330 It's connected to the dark state, but there is more to it. 1001 01:03:37,330 --> 01:03:40,690 So that's why I want to spend the rest of today's class 1002 01:03:40,690 --> 01:03:42,490 in discussing this concept. 1003 01:03:55,870 --> 01:04:03,100 So I want to discuss it in the simplest possible scheme. 1004 01:04:07,290 --> 01:04:12,355 So I want to discuss first the simple system. 1005 01:04:15,310 --> 01:04:17,880 In the end, this is not the system 1006 01:04:17,880 --> 01:04:20,670 in which it has been accomplished. 1007 01:04:20,670 --> 01:04:23,470 But I think what you realize is I draw you 1008 01:04:23,470 --> 01:04:24,830 up a very simple scheme. 1009 01:04:24,830 --> 01:04:27,700 I make all assumptions, and there is maybe 1010 01:04:27,700 --> 01:04:31,100 no atomic, no suitable atom where you can realize it. 1011 01:04:31,100 --> 01:04:34,420 But then I will do what I've done so often in this class. 1012 01:04:34,420 --> 01:04:37,510 I turn around and say, but if you go to a tri-state basis, 1013 01:04:37,510 --> 01:04:39,300 we can exactly engineer that. 1014 01:04:39,300 --> 01:04:42,140 So in other words, if you need a certain level scheme 1015 01:04:42,140 --> 01:04:45,510 and you don't get it, you can just take another level, 1016 01:04:45,510 --> 01:04:48,450 put it there with a strong laser as a virtual level, 1017 01:04:48,450 --> 01:04:50,320 and you get the level scheme you want. 1018 01:04:50,320 --> 01:04:52,860 And in a dress state description, 1019 01:04:52,860 --> 01:04:55,620 you have exactly the idealized level 1020 01:04:55,620 --> 01:04:58,050 scheme I'm presenting to you. 1021 01:04:58,050 --> 01:05:00,330 So therefore with that justification, 1022 01:05:00,330 --> 01:05:05,400 let me just try to give you the simple scheme where 1023 01:05:05,400 --> 01:05:11,512 you can understand this absorption 1024 01:05:11,512 --> 01:05:12,720 cancellation by interference. 1025 01:05:17,340 --> 01:05:19,820 I mean, later in the end, we want 1026 01:05:19,820 --> 01:05:22,140 to use lambda systems with two stable ground 1027 01:05:22,140 --> 01:05:23,660 state, two hyperfine states. 1028 01:05:23,660 --> 01:05:25,440 That's how all the research is done. 1029 01:05:25,440 --> 01:05:28,005 But now I assume we have a V system. 1030 01:05:30,900 --> 01:05:37,080 So we have two excited states and we have one ground state. 1031 01:05:37,080 --> 01:05:47,170 And I want to assume that by coupling through radiation, 1032 01:05:47,170 --> 01:05:51,650 spontaneous emission, those excited states 1033 01:05:51,650 --> 01:05:56,320 couple to the same continuum. 1034 01:06:08,110 --> 01:06:10,330 I want to say that this is an important assumption. 1035 01:06:10,330 --> 01:06:13,910 If you have two states, hyperfine states, 1036 01:06:13,910 --> 01:06:16,080 n equals plus 1, n equals minus 1, 1037 01:06:16,080 --> 01:06:18,490 which would emit photons of different circular 1038 01:06:18,490 --> 01:06:20,450 polarization, no. 1039 01:06:20,450 --> 01:06:22,480 They are not coupling to the same continuum. 1040 01:06:22,480 --> 01:06:25,620 They would be distinguishable, and then certain interference 1041 01:06:25,620 --> 01:06:27,430 effects will not happen. 1042 01:06:27,430 --> 01:06:30,660 So eventually, the magic of quantum interference 1043 01:06:30,660 --> 01:06:33,230 comes if it is absolutely indistinguishable 1044 01:06:33,230 --> 01:06:35,240 which excited state has emitted the photon. 1045 01:06:35,240 --> 01:06:39,650 So we have to really be-- set up something like this. 1046 01:06:39,650 --> 01:06:48,330 So it must be fundamentally impossible to distinguish 1047 01:06:48,330 --> 01:06:52,150 photons emitted through one state or the other. 1048 01:06:52,150 --> 01:06:56,810 So we know already if we treat the system, 1049 01:06:56,810 --> 01:06:59,840 we couple the atoms to the continuum. 1050 01:06:59,840 --> 01:07:02,500 That means in the end, that those states 1051 01:07:02,500 --> 01:07:03,925 acquire a certain width. 1052 01:07:06,500 --> 01:07:14,510 And yes, this is a situation we want to discuss. 1053 01:07:14,510 --> 01:07:21,180 So what we have set up now is a situation 1054 01:07:21,180 --> 01:07:26,950 where we have, with these two states, two 1055 01:07:26,950 --> 01:07:36,280 indistinguishable paths to scatter photons. 1056 01:07:39,800 --> 01:07:44,450 So let's say we take the ground state 1057 01:07:44,450 --> 01:07:45,980 and we shine laser light on it. 1058 01:07:45,980 --> 01:07:47,260 It's a two photon process. 1059 01:07:47,260 --> 01:07:51,510 A photon goes in, a photon comes out. 1060 01:07:51,510 --> 01:07:55,100 And we can now go through excited state 1, 1061 01:07:55,100 --> 01:07:58,300 through excited state out then a photon comes out 1062 01:07:58,300 --> 01:08:03,480 and eventually we have now the coupled through this two photon 1063 01:08:03,480 --> 01:08:06,085 process the ground state to continuum of modes. 1064 01:08:08,800 --> 01:08:11,910 But those two processes to go through e1 and e2 1065 01:08:11,910 --> 01:08:13,640 are indistinguishable. 1066 01:08:13,640 --> 01:08:15,890 So I will give you immediately an expression 1067 01:08:15,890 --> 01:08:19,204 where in perturbation theory, we are not adding intensities. 1068 01:08:19,204 --> 01:08:21,050 We are adding the two amplitudes. 1069 01:08:21,050 --> 01:08:24,649 It's like a double-slit experiment. 1070 01:08:24,649 --> 01:08:31,300 Or in other words, the situation I want to discuss now 1071 01:08:31,300 --> 01:08:33,170 is the following. 1072 01:08:33,170 --> 01:08:36,330 We have all of the population in the ground state. 1073 01:08:39,330 --> 01:08:41,790 We have the two excited states. 1074 01:08:41,790 --> 01:08:45,000 And our laser is tuned in between. 1075 01:08:45,000 --> 01:08:46,319 And now we can scatter light. 1076 01:08:46,319 --> 01:08:48,510 We've talked about really scattering and such. 1077 01:08:48,510 --> 01:08:51,470 And you know that for an infinitely heavy atom, 1078 01:08:51,470 --> 01:08:55,359 the emitted radiation is a delta function at the incident light. 1079 01:08:55,359 --> 01:08:57,010 And you will not know, you can never 1080 01:08:57,010 --> 01:09:01,779 know which excited state was involved in the scattering. 1081 01:09:01,779 --> 01:09:06,180 So we assume that we have excited 1082 01:09:06,180 --> 01:09:09,180 state-- which one is the lower one? 1083 01:09:09,180 --> 01:09:10,510 I think e1. 1084 01:09:10,510 --> 01:09:12,550 This is e2. 1085 01:09:12,550 --> 01:09:16,250 And our laser beam is detuned. 1086 01:09:16,250 --> 01:09:20,810 We have a detuning of delta 1 and delta 2 respectively. 1087 01:09:20,810 --> 01:09:32,040 So now we have to add amplitudes when we do perturbation theory. 1088 01:09:32,040 --> 01:09:42,901 So in second order perturbation theory, 1089 01:09:42,901 --> 01:09:45,040 for the light scattering, if you want 1090 01:09:45,040 --> 01:09:49,170 to derive Fermi's golden rule as we have done a number of times 1091 01:09:49,170 --> 01:09:53,790 here, you remember, the critical part-- 1092 01:09:53,790 --> 01:09:56,800 I'm not writing down the whole expression-- the critical part 1093 01:09:56,800 --> 01:10:01,110 is sort of a product of two matrix element or two photon 1094 01:10:01,110 --> 01:10:04,940 matrix element which takes us from the ground state 1095 01:10:04,940 --> 01:10:08,740 through the light atom coupling to the excited state. 1096 01:10:08,740 --> 01:10:14,360 And then from the excited state why are the coupling eventually 1097 01:10:14,360 --> 01:10:17,880 back to the ground state, which is now a continuum of modes. 1098 01:10:17,880 --> 01:10:20,730 And we sum over all possible modes. 1099 01:10:20,730 --> 01:10:26,750 And what we have here is the detuning denominator. 1100 01:10:26,750 --> 01:10:31,950 Well, if you want to do a little bit better, 1101 01:10:31,950 --> 01:10:34,200 I'm just mentioning here for completeness, 1102 01:10:34,200 --> 01:10:37,655 we want to add the imaginary part. 1103 01:10:40,750 --> 01:10:44,050 That's more placeholder for a command I want to make later. 1104 01:10:44,050 --> 01:10:45,580 It's not really essential. 1105 01:10:45,580 --> 01:10:48,110 But the new thing I want to discuss now 1106 01:10:48,110 --> 01:10:53,750 is that since we have two excited states and not one, 1107 01:10:53,750 --> 01:10:57,301 we have to sum over the two excited states, 1108 01:10:57,301 --> 01:10:58,300 and it's a coherent sum. 1109 01:11:04,400 --> 01:11:18,370 So if I detune between-- if my laser detuning is tuned 1110 01:11:18,370 --> 01:11:21,820 in between, then delta 1 is positive 1111 01:11:21,820 --> 01:11:24,090 and delta 2 is negative. 1112 01:11:24,090 --> 01:11:28,650 So therefore, we do a sum here with opposite signs. 1113 01:11:28,650 --> 01:11:32,580 And if I can neglect the gamma, assuming the gamma 1114 01:11:32,580 --> 01:11:35,180 is small or much smaller than the detuning, 1115 01:11:35,180 --> 01:11:41,030 then I'm just adding up two numbers with opposite sign. 1116 01:11:41,030 --> 01:11:46,140 And depending now on the matrix element, if they're identical 1117 01:11:46,140 --> 01:11:49,260 I get a cancellation if I'm detuning half between. 1118 01:11:49,260 --> 01:11:51,650 But even if the matrix element are different, 1119 01:11:51,650 --> 01:11:55,500 I will always find a detuning where the sum is 0. 1120 01:11:58,700 --> 01:12:03,550 So to the extent that I can neglect the imaginary part, 1121 01:12:03,550 --> 01:12:08,080 the gamma in the denominator, I have now this situation 1122 01:12:08,080 --> 01:12:12,660 that this is 0. 1123 01:12:12,660 --> 01:12:28,910 It vanishes for a certain laser frequency, omega 0, 1124 01:12:28,910 --> 01:12:35,152 tuned between the levels e1 and e2. 1125 01:12:39,110 --> 01:12:48,910 And what I've assumed here is that the detunings are larger 1126 01:12:48,910 --> 01:12:53,700 than the decay widths, and therefore I can neglect that. 1127 01:12:53,700 --> 01:12:54,920 Any questions about that? 1128 01:12:57,730 --> 01:12:58,230 So OK. 1129 01:12:58,230 --> 01:13:00,170 Very trivially, two excited states 1130 01:13:00,170 --> 01:13:01,660 which are indistinguishable. 1131 01:13:01,660 --> 01:13:03,540 It's the modern version of Feynman's 1132 01:13:03,540 --> 01:13:04,950 double-slit experiment. 1133 01:13:04,950 --> 01:13:09,850 We add up the amplitudes and by necessity, we get 0. 1134 01:13:09,850 --> 01:13:14,870 But that sounds so trivial, but it took a Steve Harris, 1135 01:13:14,870 --> 01:13:19,080 some genius person to invent it, to realize 1136 01:13:19,080 --> 01:13:28,680 that when we have light at omega 0, it would be absorbed by e1. 1137 01:13:28,680 --> 01:13:30,910 It would be absorbed by e2. 1138 01:13:30,910 --> 01:13:35,845 But it is not absorbed in the situation 1139 01:13:35,845 --> 01:13:39,860 we have with e1 and e2. 1140 01:13:39,860 --> 01:13:46,270 But now we have to make the connection that the statement 1141 01:13:46,270 --> 01:13:50,670 that we need inversion for lasing was related 1142 01:13:50,670 --> 01:13:54,210 to absorption, but now we have cancelled absorption. 1143 01:13:54,210 --> 01:13:56,850 So now what we want to do is we want 1144 01:13:56,850 --> 01:14:00,090 to create a situation where we have lasing. 1145 01:14:00,090 --> 01:14:03,400 We want to put population into the excited state. 1146 01:14:06,670 --> 01:14:11,630 But the population in this excited state 1147 01:14:11,630 --> 01:14:14,940 will be much, much smaller than the population in the ground 1148 01:14:14,940 --> 01:14:15,510 state. 1149 01:14:15,510 --> 01:14:18,680 So you're not even close to inversion. 1150 01:14:18,680 --> 01:14:20,660 So let me draw the diagram. 1151 01:14:20,660 --> 01:14:22,600 We have lots and lots and lots and lots 1152 01:14:22,600 --> 01:14:27,300 of population in the ground state. 1153 01:14:27,300 --> 01:14:34,800 We have 0 population in e1. 1154 01:14:34,800 --> 01:14:44,100 And in e2, we just have a little bit. 1155 01:14:44,100 --> 01:14:48,610 And now you would say, well, you don't have inversion. 1156 01:14:48,610 --> 01:14:50,660 How can you get lasing? 1157 01:14:50,660 --> 01:15:00,620 But if we have now a cavity at frequency omega 0, 1158 01:15:00,620 --> 01:15:09,550 then you can say the Lorentzian profile of the excited state e2 1159 01:15:09,550 --> 01:15:11,730 overlaps with omega 0. 1160 01:15:11,730 --> 01:15:15,850 So at least you have some possibility for the excited 1161 01:15:15,850 --> 01:15:20,510 state e2 to decay into the cavity mode. 1162 01:15:20,510 --> 01:15:24,220 And therefore, a very weak field in the cavity mode 1163 01:15:24,220 --> 01:15:27,260 will be amplified by stimulated emission 1164 01:15:27,260 --> 01:15:29,800 without suffering any absorption. 1165 01:15:29,800 --> 01:15:32,490 So therefore it will be amplified and we have gain. 1166 01:15:37,670 --> 01:15:41,310 So if you have a cavity at omega 0, 1167 01:15:41,310 --> 01:15:52,778 we have gain because we have some stimulated emission 1168 01:15:52,778 --> 01:15:56,178 at omega 0 but no absorption. 1169 01:16:01,400 --> 01:16:03,600 Of course, if you would tune your cavity 1170 01:16:03,600 --> 01:16:06,540 to be in a resonance with e2, you 1171 01:16:06,540 --> 01:16:09,000 would have more stimulated emission, 1172 01:16:09,000 --> 01:16:10,720 but you would have much more absorption 1173 01:16:10,720 --> 01:16:14,470 and you would not be able to get a net gain for a small probe 1174 01:16:14,470 --> 01:16:14,970 field. 1175 01:16:14,970 --> 01:16:16,675 So you would not have lasing. 1176 01:16:20,070 --> 01:16:22,970 So you can say that what we have achieved here 1177 01:16:22,970 --> 01:16:28,660 in a three-level system, we have accomplished 1178 01:16:28,660 --> 01:16:32,810 destructive interference for absorption. 1179 01:16:32,810 --> 01:16:36,340 But the way how we've put population just in e2, 1180 01:16:36,340 --> 01:16:39,270 we have not any destructive interference 1181 01:16:39,270 --> 01:16:40,860 for the stimulated gain. 1182 01:16:40,860 --> 01:16:43,420 So we have tweaked on the lasing equation, 1183 01:16:43,420 --> 01:16:45,730 on the laser equation the absorption part, 1184 01:16:45,730 --> 01:16:48,345 but not the stimulated emission part by interference 1185 01:16:48,345 --> 01:16:49,626 in a three-level system. 1186 01:16:56,140 --> 01:16:58,258 So how can this be realized? 1187 01:17:06,970 --> 01:17:11,310 Let me just give you the first possible realization 1188 01:17:11,310 --> 01:17:15,210 and then I think we should stop. 1189 01:17:15,210 --> 01:17:17,380 We could, in principle, realize it 1190 01:17:17,380 --> 01:17:21,370 with hydrogen and a DC electric field. 1191 01:17:24,680 --> 01:17:30,830 Remember, hydrogen has the 1s level, has the 2s level, 1192 01:17:30,830 --> 01:17:33,970 and the 2p state. 1193 01:17:33,970 --> 01:17:38,260 And you all had a nice, little nice homework assignment 1194 01:17:38,260 --> 01:17:44,320 that when you have a DC field, you actually couple 2s and 2p. 1195 01:17:44,320 --> 01:17:52,050 And if your DC field is stronger than the Lamb shift, 1196 01:17:52,050 --> 01:17:58,780 you actually create a superposition 1197 01:17:58,780 --> 01:18:01,910 of 2s and 2p states. 1198 01:18:01,910 --> 01:18:08,427 And now, if you tune your laser right in between, 1199 01:18:08,427 --> 01:18:09,510 you have a dark resonance. 1200 01:18:13,900 --> 01:18:19,230 So the situation is if I would now 1201 01:18:19,230 --> 01:18:24,960 plot the absorption versus frequency, 1202 01:18:24,960 --> 01:18:30,020 you would find that if you're in resonance with the upper state 1203 01:18:30,020 --> 01:18:32,750 you absorb, but then even in between, 1204 01:18:32,750 --> 01:18:37,790 there is this notch where you have zero absorption. 1205 01:18:37,790 --> 01:18:39,650 So for this simple case with hydrogen 1206 01:18:39,650 --> 01:18:42,690 in a DC electric field, the dark resonance 1207 01:18:42,690 --> 01:18:47,180 occurs when you have equal detuning. 1208 01:18:47,180 --> 01:18:50,450 You have an equal detuning for the atom with respect 1209 01:18:50,450 --> 01:18:52,290 to the two states. 1210 01:18:52,290 --> 01:18:57,770 And I like to sort of connect it to classical physics. 1211 01:18:57,770 --> 01:19:01,050 You know that absorption is related. 1212 01:19:01,050 --> 01:19:04,416 When you absorb, you have light scattering. 1213 01:19:04,416 --> 01:19:05,790 The photons are not disappearing. 1214 01:19:05,790 --> 01:19:07,470 They're just scattered into other modes. 1215 01:19:07,470 --> 01:19:09,250 That's what absorption is about. 1216 01:19:09,250 --> 01:19:11,850 And classically, light scattering 1217 01:19:11,850 --> 01:19:15,300 comes because you have an oscillating dipole moment. 1218 01:19:15,300 --> 01:19:18,780 But what happens is if you have this coherent superposition 1219 01:19:18,780 --> 01:19:22,590 and you're right in between, you have a positive detuning 1220 01:19:22,590 --> 01:19:25,430 with one harmonic oscillator and a negative detuning 1221 01:19:25,430 --> 01:19:27,430 with the other harmonic oscillator. 1222 01:19:27,430 --> 01:19:32,030 So if you drive an harmonic oscillator-- and this is one 1223 01:19:32,030 --> 01:19:34,780 and this is the other-- with positive detuning, 1224 01:19:34,780 --> 01:19:37,340 with red detuning, the dipole moment 1225 01:19:37,340 --> 01:19:39,460 is in phase with the electric field. 1226 01:19:39,460 --> 01:19:41,830 In the other case, it's out of phase. 1227 01:19:41,830 --> 01:19:44,460 So now you have the two excited states 1228 01:19:44,460 --> 01:19:48,800 and you're driving two equal but opposite dipole moments 1229 01:19:48,800 --> 01:19:51,560 in those two states. 1230 01:19:51,560 --> 01:19:55,590 And therefore, the two dipole moments add up to 0. 1231 01:19:55,590 --> 01:19:58,235 Therefore the laser, just using our understanding 1232 01:19:58,235 --> 01:20:00,610 of a simple harmonic oscillator, is not 1233 01:20:00,610 --> 01:20:02,230 creating any dipole moment. 1234 01:20:02,230 --> 01:20:04,110 Therefore, we're not scattering any light 1235 01:20:04,110 --> 01:20:07,540 and therefore we're not absorbing any light. 1236 01:20:07,540 --> 01:20:10,950 So this is how you can now realize in the hydrogen atom 1237 01:20:10,950 --> 01:20:14,540 using DC magnetic field and creating this level structure, 1238 01:20:14,540 --> 01:20:17,960 you can create lasing without inversion. 1239 01:20:17,960 --> 01:20:21,646 But I will tell you on Wednesday-- no, 1240 01:20:21,646 --> 01:20:23,020 today is Wednesday-- next Monday, 1241 01:20:23,020 --> 01:20:25,390 next week that we don't have to deal 1242 01:20:25,390 --> 01:20:27,270 with the difficulties of hydrogen atoms 1243 01:20:27,270 --> 01:20:28,400 in the laboratory. 1244 01:20:28,400 --> 01:20:30,610 We can just use our favorite alkalize 1245 01:20:30,610 --> 01:20:33,450 and dress the alkalize up with laser beams 1246 01:20:33,450 --> 01:20:36,280 and eventually create the same situation 1247 01:20:36,280 --> 01:20:38,770 with an atom which is experimentally 1248 01:20:38,770 --> 01:20:41,420 much simpler to address. 1249 01:20:41,420 --> 01:20:42,085 Any questions? 1250 01:20:45,130 --> 01:20:47,380 OK, one more week to go. 1251 01:20:47,380 --> 01:20:49,530 See you on Monday.