1 00:00:00,050 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,010 Commons license. 3 00:00:04,010 --> 00:00:06,350 Your support will help MIT OpenCourseWare 4 00:00:06,350 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,202 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,202 --> 00:00:17,827 at ocw.mit.edu. 8 00:00:21,752 --> 00:00:22,835 PROFESSOR: Good afternoon. 9 00:00:27,330 --> 00:00:31,060 So in the last week of this semester, 10 00:00:31,060 --> 00:00:36,625 we will be finishing up the chapter on coherence. 11 00:00:39,920 --> 00:00:42,540 What we want to continue to explore today 12 00:00:42,540 --> 00:00:45,620 is the presence of a dark state. 13 00:00:45,620 --> 00:00:48,370 If you have a three level system, and that's 14 00:00:48,370 --> 00:00:50,630 what we went through last week, we 15 00:00:50,630 --> 00:00:56,800 will always find one state which is dark, 16 00:00:56,800 --> 00:01:00,080 which means we fully illuminate the atom with laser light, 17 00:01:00,080 --> 00:01:02,240 but there is one state which is not 18 00:01:02,240 --> 00:01:05,498 coupled to the excited state which does not scatter light, 19 00:01:05,498 --> 00:01:06,415 and therefore is dark. 20 00:01:09,980 --> 00:01:13,730 I showed you that for very general conditions, if you have 21 00:01:13,730 --> 00:01:17,640 two laser beams exciting the two states, 22 00:01:17,640 --> 00:01:21,070 there will always be a coherent superposition of the two 23 00:01:21,070 --> 00:01:25,260 states, which is the dark state. 24 00:01:25,260 --> 00:01:29,840 The dark state is the novel feature of three level systems, 25 00:01:29,840 --> 00:01:33,800 and I want to show you different perspectives of it. 26 00:01:33,800 --> 00:01:37,650 We started out by talking about just the existence 27 00:01:37,650 --> 00:01:39,330 of the dark state. 28 00:01:39,330 --> 00:01:44,160 Then we talked about dark state transfer. 29 00:01:44,160 --> 00:01:47,250 You can regard, for detuned light, the dark state 30 00:01:47,250 --> 00:01:50,460 as the lowest energy state of the system. 31 00:01:50,460 --> 00:01:53,250 And then an adiabatic theorem tells you 32 00:01:53,250 --> 00:01:56,670 that you can keep particles in the dark state 33 00:01:56,670 --> 00:01:59,000 even if you change the laser parameters 34 00:01:59,000 --> 00:02:02,300 and change what state is now the dark state. 35 00:02:02,300 --> 00:02:05,700 This is the basis of coherent population transfer, 36 00:02:05,700 --> 00:02:09,250 or the famous STIRAP method. 37 00:02:09,250 --> 00:02:11,950 There is another aspect of the dark state 38 00:02:11,950 --> 00:02:16,170 which gives us a possibility of lasing without inversion. 39 00:02:16,170 --> 00:02:20,160 As I reminded you, lasing has a threshold, 40 00:02:20,160 --> 00:02:23,310 which is inversion that you need more atoms in the excited state 41 00:02:23,310 --> 00:02:27,510 than in the ground state because the atoms in the ground state 42 00:02:27,510 --> 00:02:31,450 absorb the laser light and you want a net gain. 43 00:02:31,450 --> 00:02:33,500 However, if you have a dark state, 44 00:02:33,500 --> 00:02:36,460 you have a situation where the atoms do not absorb the laser 45 00:02:36,460 --> 00:02:42,260 light, and therefore, the conditions for net gain 46 00:02:42,260 --> 00:02:43,270 have changed. 47 00:02:43,270 --> 00:02:46,650 And indeed, lasing without inversion becomes possible. 48 00:02:49,540 --> 00:03:00,660 I started to explain to you the way 49 00:03:00,660 --> 00:03:03,970 how lasing without inversion could come about. 50 00:03:03,970 --> 00:03:07,040 It relies on the fact that in a three level system, 51 00:03:07,040 --> 00:03:10,230 absorption destructively interferes 52 00:03:10,230 --> 00:03:14,050 but stimulated emission does not destructively interfere. 53 00:03:14,050 --> 00:03:16,300 And therefore, you can have lasing 54 00:03:16,300 --> 00:03:20,670 without inversion in such three level systems. 55 00:03:20,670 --> 00:03:29,090 I showed you one possible realization, 56 00:03:29,090 --> 00:03:33,080 and this is about hydrogen in a DC electric field. 57 00:03:33,080 --> 00:03:37,860 If we mix the 2S and 2P states with an electric field, 58 00:03:37,860 --> 00:03:41,060 we have this three level structure. 59 00:03:41,060 --> 00:03:45,380 And for a laser tuned right in the middle between the two 60 00:03:45,380 --> 00:03:50,720 states, the two amplitudes for excitation cancel exactly, 61 00:03:50,720 --> 00:03:52,840 and therefore, you have a zero absorption feature. 62 00:03:56,630 --> 00:03:59,760 However, if you put now a little bit of population into, 63 00:03:59,760 --> 00:04:03,450 let's say, the upper state, this upper state 64 00:04:03,450 --> 00:04:06,150 in the wings of its profile has still 65 00:04:06,150 --> 00:04:09,190 gained for stimulated emission, and what we get 66 00:04:09,190 --> 00:04:12,270 is lasing without inversion. 67 00:04:12,270 --> 00:04:15,530 Now you say, OK, but that's a hydrogen atom. 68 00:04:15,530 --> 00:04:18,880 Which atom is really degenerate between two levels, 69 00:04:18,880 --> 00:04:23,180 and you can split regeneracy with a static electric field. 70 00:04:23,180 --> 00:04:27,030 Well, you're already an expert at this point in the course. 71 00:04:27,030 --> 00:04:31,500 If the 2S and 2P state are widely separated, 72 00:04:31,500 --> 00:04:36,990 well, you add a photon, and the photon, 73 00:04:36,990 --> 00:04:40,360 which is in resonance with the 2S and 2P state, 74 00:04:40,360 --> 00:04:42,770 creates, in the dressed atom picture, degeneracy. 75 00:04:47,905 --> 00:04:49,780 Maybe I should have shown the P state higher. 76 00:04:49,780 --> 00:04:53,250 The 2S state with one more photon and the 2P 77 00:04:53,250 --> 00:04:56,300 state with one photon less have the same energy, 78 00:04:56,300 --> 00:04:59,590 and then you create exactly this situation. 79 00:04:59,590 --> 00:05:03,215 So therefore, the way you can realize 80 00:05:03,215 --> 00:05:06,160 that in atoms other than hydrogen 81 00:05:06,160 --> 00:05:19,915 is use an AC electric field to mix S and P states. 82 00:05:22,820 --> 00:05:25,090 And I'll show you in five minutes a little bit more 83 00:05:25,090 --> 00:05:29,268 in detail what I mean by that. 84 00:05:32,860 --> 00:05:38,470 There is a trivial realization which 85 00:05:38,470 --> 00:05:41,760 I want to mention for lasing without inversion, 86 00:05:41,760 --> 00:05:48,330 and this would be if you have a three level 87 00:05:48,330 --> 00:05:54,900 system with an excited state, two levels, g 88 00:05:54,900 --> 00:05:59,065 and f in the ground state. 89 00:06:07,280 --> 00:06:19,180 You may have an inversion for the e to g transition. 90 00:06:19,180 --> 00:06:23,410 And therefore, you can get lasing 91 00:06:23,410 --> 00:06:26,570 because the population in this state f is not coupled. 92 00:06:26,570 --> 00:06:34,100 This is pretty trivial, but the more subtle part, of course, 93 00:06:34,100 --> 00:06:41,820 is that we can realize it using a driven system using a control 94 00:06:41,820 --> 00:06:49,290 laser by creating the same situation with the bright 95 00:06:49,290 --> 00:06:55,850 and the dark state, and population in the dark state 96 00:06:55,850 --> 00:07:00,880 is hidden from the light and does not absorb light. 97 00:07:00,880 --> 00:07:05,280 Let me just indicate that in both those states, 98 00:07:05,280 --> 00:07:10,885 we would have no absorption, dark state. 99 00:07:18,990 --> 00:07:22,450 Those examples may raise the question whether whenever 100 00:07:22,450 --> 00:07:25,140 you have lasing without inversion, 101 00:07:25,140 --> 00:07:33,360 if you can find a basis where you have inversion again 102 00:07:33,360 --> 00:07:35,970 between the two levels which are relevant 103 00:07:35,970 --> 00:07:39,330 and the extra population is just hidden. 104 00:07:39,330 --> 00:07:41,300 I want to make two comments about it. 105 00:07:41,300 --> 00:07:42,920 This question is sometimes discussed 106 00:07:42,920 --> 00:07:46,740 in the literature, sometimes in a semi-controversial way. 107 00:07:46,740 --> 00:07:48,150 There are two comments about it. 108 00:07:48,150 --> 00:07:53,170 One is when you start dressing up 109 00:07:53,170 --> 00:07:56,680 your system with laser beams, you have strong control lasers. 110 00:07:56,680 --> 00:07:59,800 You have two lasers, omega 1, omega 2. 111 00:07:59,800 --> 00:08:01,990 One is often a strong control laser 112 00:08:01,990 --> 00:08:03,940 and the other one is the weak laser 113 00:08:03,940 --> 00:08:06,990 where we want to have lasing. 114 00:08:06,990 --> 00:08:11,880 You have actually a time dependent system driven 115 00:08:11,880 --> 00:08:16,000 by time dependent fields, and once you have a time dependent 116 00:08:16,000 --> 00:08:20,150 system, it's no longer clear what the eigenstates are, 117 00:08:20,150 --> 00:08:24,280 what the populations are, and what are the coherences. 118 00:08:24,280 --> 00:08:27,760 There is no longer a unique way to distinguish 119 00:08:27,760 --> 00:08:30,770 what are the eigenstates because every state is, so to speak, 120 00:08:30,770 --> 00:08:33,789 time dependent almost by definition. 121 00:08:33,789 --> 00:08:35,720 On the other hand, I think the example 122 00:08:35,720 --> 00:08:38,350 I gave you with atomic hydrogen where you just 123 00:08:38,350 --> 00:08:40,970 a little bit of mixing with an electric field 124 00:08:40,970 --> 00:08:43,669 is an example where you genuinely 125 00:08:43,669 --> 00:08:47,380 have less population in the excited state 126 00:08:47,380 --> 00:08:48,910 than you have in the ground state, 127 00:08:48,910 --> 00:08:52,090 and at least the equation tells you, even without inversion, 128 00:08:52,090 --> 00:08:54,160 you have a net gain. 129 00:08:54,160 --> 00:08:56,890 So my own understanding of that situation 130 00:08:56,890 --> 00:09:00,150 is that in many situations, you can actually reduce it 131 00:09:00,150 --> 00:09:05,970 to a simple picture where you have simply hidden population 132 00:09:05,970 --> 00:09:15,080 in a dark state, but without any sort of unnatural definitions, 133 00:09:15,080 --> 00:09:18,350 you may not find that in some other systems. 134 00:09:21,850 --> 00:09:24,535 Questions about lasing without inversion? 135 00:09:24,535 --> 00:09:25,946 Nancy? 136 00:09:25,946 --> 00:09:27,938 STUDENT: Is lasing without inversion 137 00:09:27,938 --> 00:09:32,420 important in lab, as opposed to lasing with inversion, 138 00:09:32,420 --> 00:09:35,906 or is it more about teaching us [INAUDIBLE]?? 139 00:09:38,910 --> 00:09:41,190 PROFESSOR: Well, lasing without inversion 140 00:09:41,190 --> 00:09:43,880 has definitely been touted as a way 141 00:09:43,880 --> 00:09:47,130 to get lasing deeper in the UV, to get 142 00:09:47,130 --> 00:09:53,120 lasing for very blue transitions because when 143 00:09:53,120 --> 00:09:54,980 you want to create inversion-- 144 00:09:54,980 --> 00:09:57,500 this actually has been the problem in creating 145 00:09:57,500 --> 00:10:00,170 x-ray lasers in atomic systems. 146 00:10:00,170 --> 00:10:04,890 If you have larger and larger energy separation, 147 00:10:04,890 --> 00:10:10,890 spontaneous emission scales with omega [? cube. ?] 148 00:10:10,890 --> 00:10:13,790 And so therefore, it becomes harder and harder 149 00:10:13,790 --> 00:10:16,350 to fulfill the ordinary gain equation. 150 00:10:16,350 --> 00:10:18,798 STUDENT: So in those cases, even these last two 151 00:10:18,798 --> 00:10:21,926 methods [INAUDIBLE] because [INAUDIBLE]?? 152 00:10:25,650 --> 00:10:27,900 PROFESSOR: Well, lasing without inversion 153 00:10:27,900 --> 00:10:30,460 alleviates the requirement to build a laser, 154 00:10:30,460 --> 00:10:32,505 and so people have discussed that where 155 00:10:32,505 --> 00:10:36,630 it's really hard to create a laser in the conventional way, 156 00:10:36,630 --> 00:10:39,770 deep, deep in the blue of the UV or in the x-ray regime, 157 00:10:39,770 --> 00:10:42,360 that lasing without inversion may help. 158 00:10:42,360 --> 00:10:45,490 I'm not aware that any practical development 159 00:10:45,490 --> 00:10:49,610 has emerged from that because there is a price to be paid, 160 00:10:49,610 --> 00:10:52,010 and that is usually in the form of coherence. 161 00:10:52,010 --> 00:10:54,300 You need a certain degree of coherence 162 00:10:54,300 --> 00:10:56,370 in your system to be able to do that. 163 00:10:59,500 --> 00:11:04,960 It's an idea which is powerful, but as far as I know, 164 00:11:04,960 --> 00:11:08,950 there was no killer application of it. 165 00:11:08,950 --> 00:11:11,480 The importance of dark states is definitely 166 00:11:11,480 --> 00:11:16,320 in slow light, manipulation, quantum computation, 167 00:11:16,320 --> 00:11:18,720 and concepts of storing light, and this is 168 00:11:18,720 --> 00:11:21,650 what we want to discuss next. 169 00:11:21,650 --> 00:11:23,940 Actually not next, but after next. 170 00:11:23,940 --> 00:11:27,260 What I first want to discuss is another aspect 171 00:11:27,260 --> 00:11:31,610 of the dark state, and this goes by the name 172 00:11:31,610 --> 00:11:36,885 EIT, electromagnetically induced transparency. 173 00:12:04,410 --> 00:12:10,320 I can introduce this topic by a question to Radio Yerevan, 174 00:12:10,320 --> 00:12:13,570 is it possible to send a laser beam through a brick wall? 175 00:12:13,570 --> 00:12:16,990 And the answer of Radio Yerevan, if you know the joke, 176 00:12:16,990 --> 00:12:20,725 is always, in principle yes, but you need another very powerful 177 00:12:20,725 --> 00:12:22,270 laser. 178 00:12:22,270 --> 00:12:24,430 So in an incoherent way, of course, 179 00:12:24,430 --> 00:12:27,550 a very powerful laser can drill a hole into the wall 180 00:12:27,550 --> 00:12:29,800 and then the next laser can go through the wall 181 00:12:29,800 --> 00:12:32,740 without absorption, but you can be smarter. 182 00:12:32,740 --> 00:12:36,080 If the very powerful laser through coherence 183 00:12:36,080 --> 00:12:41,230 puts all the atoms in the brick in the beam pass of the laser 184 00:12:41,230 --> 00:12:43,680 into a coherent superposition state, 185 00:12:43,680 --> 00:12:45,740 and then they become a dark state, 186 00:12:45,740 --> 00:12:49,108 then your laser can go through a brick wall. 187 00:12:56,080 --> 00:13:03,895 So can a laser beam penetrate an optically thick medium? 188 00:13:13,610 --> 00:13:21,285 And the answer is yes, with the help of another laser. 189 00:13:27,480 --> 00:13:30,480 Original ideas along those lines were formulated by Steve 190 00:13:30,480 --> 00:13:33,310 Harris, who has really pioneered this field, 191 00:13:33,310 --> 00:13:38,750 and he first considered special auto-ionizing excited states 192 00:13:38,750 --> 00:13:40,160 which couple to-- 193 00:13:40,160 --> 00:13:43,370 there were two pathways of coupling into the continuum, 194 00:13:43,370 --> 00:13:46,230 but later work has shown that it can 195 00:13:46,230 --> 00:13:50,380 be realized in a lambda system. 196 00:13:50,380 --> 00:13:52,985 Let me talk about this conceptionally simpler 197 00:13:52,985 --> 00:13:59,495 realization in a three level lambda system. 198 00:14:05,040 --> 00:14:12,470 Let's assume we have again our normal three level system 199 00:14:12,470 --> 00:14:18,273 gf and an excited state which has a width, gamma. 200 00:14:21,514 --> 00:14:32,170 And we want to send a probe laser through a dense medium, 201 00:14:32,170 --> 00:14:35,470 and it would be completely absorbed by the resonance 202 00:14:35,470 --> 00:14:38,380 to the excited state. 203 00:14:38,380 --> 00:14:57,400 But now we can have a strong coupling laser 204 00:14:57,400 --> 00:15:01,990 with a rapid frequency, omega c, and so if we drive 205 00:15:01,990 --> 00:15:08,610 the system very strongly, we can create a situation where 206 00:15:08,610 --> 00:15:12,730 the coupling laser does, if it's strong enough, complete 207 00:15:12,730 --> 00:15:16,890 mixing between the excited state and the ground state, 208 00:15:16,890 --> 00:15:19,640 and that means if you have two levels which are completely 209 00:15:19,640 --> 00:15:24,850 mixed, they are split by the energy or the frequency 210 00:15:24,850 --> 00:15:26,630 of the coupling. 211 00:15:26,630 --> 00:15:32,800 So in other words, what we obtain is we 212 00:15:32,800 --> 00:15:39,420 have now two states, e plus f and e minus f. 213 00:15:42,330 --> 00:15:44,500 You know the how you should read is the following. 214 00:15:44,500 --> 00:15:47,990 You can just assume for a second the state 215 00:15:47,990 --> 00:15:52,070 f will degenerate with gamma and then I put in a very strong mix 216 00:15:52,070 --> 00:15:53,470 in between the two. 217 00:15:53,470 --> 00:15:56,500 This is exactly the example we had with hydrogen 218 00:15:56,500 --> 00:15:58,600 in an electric field, and then we 219 00:15:58,600 --> 00:16:02,770 get two states which have both width gamma over 2. 220 00:16:02,770 --> 00:16:05,630 They are strongly mixed, and the splitting 221 00:16:05,630 --> 00:16:09,730 is nothing else than the matrix element of the electric field. 222 00:16:09,730 --> 00:16:13,150 But if you don't have two degenerate states 223 00:16:13,150 --> 00:16:16,980 and we add a photon, then the photon-- 224 00:16:16,980 --> 00:16:19,820 I mentioned it again and again in addressed atom picture-- 225 00:16:19,820 --> 00:16:21,520 creates a degeneracy. 226 00:16:21,520 --> 00:16:24,090 You can just add a photon, draw a dashed line, 227 00:16:24,090 --> 00:16:26,340 and this is your virtual state if you want. 228 00:16:26,340 --> 00:16:28,480 Or if you look at Schrodinger's equation, 229 00:16:28,480 --> 00:16:30,240 you have something which oscillates 230 00:16:30,240 --> 00:16:32,460 at the frequency of the state, f, 231 00:16:32,460 --> 00:16:35,930 but now you multiply it with an electric field which oscillates 232 00:16:35,930 --> 00:16:37,920 at the resonance frequency, and then 233 00:16:37,920 --> 00:16:39,380 you have something which oscillates 234 00:16:39,380 --> 00:16:41,430 at the sum of the two frequencies, 235 00:16:41,430 --> 00:16:44,560 and this is exactly what I indicated with a dashed line. 236 00:16:44,560 --> 00:16:47,310 So that's how you create, so to speak, 237 00:16:47,310 --> 00:16:50,610 a degeneracy by using the frequency of the laser 238 00:16:50,610 --> 00:16:52,970 to overcome the energy splitting, 239 00:16:52,970 --> 00:16:57,280 and the result is that you have created exactly 240 00:16:57,280 --> 00:17:01,220 this excited state level structure. 241 00:17:01,220 --> 00:17:04,609 And if you now look at the ground state, 242 00:17:04,609 --> 00:17:17,329 and our photon is tuned right between those two continua, 243 00:17:17,329 --> 00:17:18,654 then we have a dark resonance. 244 00:17:23,910 --> 00:17:25,520 And in order to accomplish this, I'm 245 00:17:25,520 --> 00:17:27,369 not going into any calculations here, 246 00:17:27,369 --> 00:17:29,950 but you need a sufficiently strong coupling 247 00:17:29,950 --> 00:17:31,720 laser who can accomplish that. 248 00:17:31,720 --> 00:17:33,270 For instance, coupling laser which 249 00:17:33,270 --> 00:17:37,430 is stronger than the spontaneous emission rate or decoherence 250 00:17:37,430 --> 00:17:38,970 rate gamma in the excited state. 251 00:17:45,730 --> 00:18:04,140 So if we now scan the probe laser 252 00:18:04,140 --> 00:18:21,990 and we look for transmission, let 253 00:18:21,990 --> 00:18:28,620 me assume for simplicity that there is no relaxation 254 00:18:28,620 --> 00:18:32,480 between the two ground states. 255 00:18:32,480 --> 00:18:55,550 If the coupling laser strength is 0, 256 00:18:55,550 --> 00:19:01,290 then we have a broad feature, which 257 00:19:01,290 --> 00:19:03,830 is simply the single photon absorption of the probe light. 258 00:19:08,320 --> 00:19:14,200 If we have an infinitesimal coupling laser strength, 259 00:19:14,200 --> 00:19:17,340 we get a very, very sharp feature, 260 00:19:17,340 --> 00:19:22,390 and if the coupling laser is stronger and stronger, 261 00:19:22,390 --> 00:19:27,710 we get a window of transparency, and the width is given. 262 00:19:27,710 --> 00:19:32,020 The width delta omega of this central feature 263 00:19:32,020 --> 00:19:35,130 is given by the Rabi frequency of the coupling laser. 264 00:19:39,930 --> 00:19:45,340 So what are possible applications? 265 00:20:04,030 --> 00:20:09,930 One is you can design non-linear materials. 266 00:20:12,640 --> 00:20:17,860 Usually, when you want to have frequency conversion processes 267 00:20:17,860 --> 00:20:21,110 or optical switches at one laser beam 268 00:20:21,110 --> 00:20:23,460 affects another laser beam, you want 269 00:20:23,460 --> 00:20:26,170 a very strong non-linear response of your medium, 270 00:20:26,170 --> 00:20:29,230 you usually get that if you go near resonance, 271 00:20:29,230 --> 00:20:31,790 but near resonance, you have strong absorption. 272 00:20:31,790 --> 00:20:36,150 But now, using the concept of EIT, you can have both. 273 00:20:36,150 --> 00:20:38,570 You can have the strong non-linear response 274 00:20:38,570 --> 00:20:42,190 of your medium near resonance but you suppress absorption, 275 00:20:42,190 --> 00:20:44,690 you get a window of transparency using EIT. 276 00:20:51,130 --> 00:20:58,246 So you can have near resonant materials without absorption. 277 00:21:02,230 --> 00:21:04,280 Again, in principle, I'm not aware 278 00:21:04,280 --> 00:21:07,260 that this has really taken off in a bigger time. 279 00:21:07,260 --> 00:21:11,470 One would be if you want to do very sensitive spectroscopy. 280 00:21:11,470 --> 00:21:13,790 Assume you want to measure one isotope which 281 00:21:13,790 --> 00:21:17,370 has a tiny little abundance but you 282 00:21:17,370 --> 00:21:19,990 have to observe it against the background of a very 283 00:21:19,990 --> 00:21:21,530 strong isotope. 284 00:21:21,530 --> 00:21:25,080 If you could switch off the absorption of the background, 285 00:21:25,080 --> 00:21:27,210 the absorption of the strong isotope, 286 00:21:27,210 --> 00:21:30,330 you could still see a small amount of a trace isotope 287 00:21:30,330 --> 00:21:32,310 in the presence of a strong absorption line. 288 00:21:36,260 --> 00:21:43,206 So sensitive detection of trace elements. 289 00:21:55,780 --> 00:21:59,600 Questions about that? 290 00:21:59,600 --> 00:22:01,800 I've given you qualitative pictures 291 00:22:01,800 --> 00:22:05,470 for lasing without inversion and for EIT. 292 00:22:05,470 --> 00:22:12,250 I want to give it a little bit more quantitative touch, 293 00:22:12,250 --> 00:22:14,860 not by going through to the optical Bloch equations which 294 00:22:14,860 --> 00:22:18,390 would be necessary to describe all features of it, 295 00:22:18,390 --> 00:22:23,260 but at least I want to give you one picture where 296 00:22:23,260 --> 00:22:25,630 you can derive and discuss things 297 00:22:25,630 --> 00:22:29,710 in a more quantitative way, and this is the eigenstate picture. 298 00:22:33,180 --> 00:22:36,400 I've also done here what I've said several times, 299 00:22:36,400 --> 00:22:44,200 that when we have splittings between the levels, 300 00:22:44,200 --> 00:22:47,100 we can actually focus on what really matters, 301 00:22:47,100 --> 00:22:51,870 namely the detunings, by absorbing the laser 302 00:22:51,870 --> 00:22:55,190 frequency into the definition of our levels, 303 00:22:55,190 --> 00:22:56,680 and that's what I've done here. 304 00:22:56,680 --> 00:23:02,750 Instead of using levels g, f, and e, I have levels with g, f, 305 00:23:02,750 --> 00:23:06,450 e, but this is the photon number in laser field one 306 00:23:06,450 --> 00:23:09,950 and the photon number in laser field two. 307 00:23:09,950 --> 00:23:13,580 So if all of the photons, the laser field one and laser 308 00:23:13,580 --> 00:23:16,730 field two, would be resonance with their respective 309 00:23:16,730 --> 00:23:19,565 transitions, then all those three levels 310 00:23:19,565 --> 00:23:21,860 would be degenerate. 311 00:23:21,860 --> 00:23:24,220 But now in the three level system, 312 00:23:24,220 --> 00:23:27,090 they're not degenerate only because we 313 00:23:27,090 --> 00:23:32,340 have a relative detuning delta, detuning small delta 314 00:23:32,340 --> 00:23:37,350 from the Raman resonance, and we have a detuning peak delta, 315 00:23:37,350 --> 00:23:40,700 which is sort the common detuning of the Raman laser 316 00:23:40,700 --> 00:23:42,360 from the excited state levels. 317 00:23:45,130 --> 00:23:50,760 So therefore, if I did define the Rabi frequencies, 318 00:23:50,760 --> 00:23:53,500 the Rabi frequencies over 2 are gain coefficient, 319 00:23:53,500 --> 00:23:55,910 and the Rabi frequencies are proportional 320 00:23:55,910 --> 00:23:57,450 to the electric field. 321 00:23:57,450 --> 00:24:00,340 Therefore, they scale with the photon numbers n 322 00:24:00,340 --> 00:24:02,640 and m in the two laser beams. 323 00:24:10,190 --> 00:24:15,580 If I do that, I have a really very simple Hamiltonian. 324 00:24:15,580 --> 00:24:18,645 On the diagonal, we simply have the detuning 325 00:24:18,645 --> 00:24:22,440 of both laser beams form the excited state. 326 00:24:22,440 --> 00:24:26,730 Here we have the Raman detuning, and we have two couplings 327 00:24:26,730 --> 00:24:28,130 to the excited state. 328 00:24:28,130 --> 00:24:31,920 One is laser field one, g1, with n photons, 329 00:24:31,920 --> 00:24:34,430 and the other one is laser field g2 with m photons. 330 00:24:37,270 --> 00:24:39,420 Any questions about? 331 00:24:39,420 --> 00:24:43,070 Just setting up the simple equations 332 00:24:43,070 --> 00:24:46,320 which we have done a few times. 333 00:24:46,320 --> 00:24:49,470 Let me focus first on the simple case 334 00:24:49,470 --> 00:24:54,330 that everything is on resonance. 335 00:24:57,250 --> 00:25:01,870 Then, if everything is on resonance, 336 00:25:01,870 --> 00:25:06,950 we have the structure which I've shown here. 337 00:25:06,950 --> 00:25:09,450 You can sort of say you have three levels, which 338 00:25:09,450 --> 00:25:13,480 are degenerate without any Rabi frequency, 339 00:25:13,480 --> 00:25:15,120 because the detunings are all zero, 340 00:25:15,120 --> 00:25:16,880 or all the diagonal is zero. 341 00:25:16,880 --> 00:25:20,240 And then the off diagonal matrix elements, the Rabi frequencies, 342 00:25:20,240 --> 00:25:23,930 are just spreading the three levels apart, 343 00:25:23,930 --> 00:25:27,370 and the general structure of this matrix 344 00:25:27,370 --> 00:25:29,815 is that in the middle, you always 345 00:25:29,815 --> 00:25:34,800 have a state which is just a superposition of g and f. 346 00:25:34,800 --> 00:25:36,050 So it's a dark state. 347 00:25:36,050 --> 00:25:39,510 It has no contribution in the excited state. 348 00:25:39,510 --> 00:25:42,755 And the two outer states have equal contribution 349 00:25:42,755 --> 00:25:45,090 of the excited state. 350 00:25:45,090 --> 00:25:47,965 So the excited state has been distributed over the two 351 00:25:47,965 --> 00:25:50,610 outer states, and the widths of those levels 352 00:25:50,610 --> 00:25:53,700 is therefore gamma over 2. 353 00:25:53,700 --> 00:25:57,940 And we know when we had two levels 354 00:25:57,940 --> 00:26:01,930 and we were driving them with a Rabi frequency resonantly, 355 00:26:01,930 --> 00:26:05,590 we had splittings which were just 356 00:26:05,590 --> 00:26:10,530 given by the Rabi frequency, and now 357 00:26:10,530 --> 00:26:14,370 the splitting between the outer level and the dark state 358 00:26:14,370 --> 00:26:18,023 is the quadrature sum of the two Rabi frequencies. 359 00:26:18,023 --> 00:26:19,940 PROFESSOR: So that's a very general structure. 360 00:26:23,840 --> 00:26:28,430 I want to go back to the situation 361 00:26:28,430 --> 00:26:32,040 where one laser is a probe laser. 362 00:26:32,040 --> 00:26:33,960 Let's assume this is our laser beam which 363 00:26:33,960 --> 00:26:36,080 wants to go through the brick wall, 364 00:26:36,080 --> 00:26:40,700 and the other laser has to prepare the system. 365 00:26:40,700 --> 00:26:45,920 Let me discuss the limit where the photon number becomes very 366 00:26:45,920 --> 00:26:50,850 small in laser one, the photon number n goes to unity, 367 00:26:50,850 --> 00:26:53,530 and this is much smaller than m. 368 00:26:53,530 --> 00:26:58,090 So we have the situation of a weak probe field. 369 00:27:01,350 --> 00:27:09,330 If we have this limit, then the dark state 370 00:27:09,330 --> 00:27:14,750 has much, much more amplitude in state g, 371 00:27:14,750 --> 00:27:21,500 and the state g is almost decoupled. 372 00:27:21,500 --> 00:27:22,720 It's in a trivial way. 373 00:27:22,720 --> 00:27:25,290 The dark state and the laser beam 374 00:27:25,290 --> 00:27:29,140 is very strongly mixing the state f and the excited state. 375 00:27:34,130 --> 00:27:38,740 So in this limit, we have a nice physical situation. 376 00:27:42,880 --> 00:27:45,190 I should actually point out that you 377 00:27:45,190 --> 00:27:48,740 can solve most of those situations for three level 378 00:27:48,740 --> 00:27:50,180 systems analytically. 379 00:27:50,180 --> 00:27:53,440 It's just those explanations get long and are not 380 00:27:53,440 --> 00:27:54,810 very transparent. 381 00:27:54,810 --> 00:27:57,780 So what I'm trying here in the classroom, 382 00:27:57,780 --> 00:27:59,820 I try to pick certain examples-- 383 00:27:59,820 --> 00:28:02,050 weak probe field, resonance-- 384 00:28:02,050 --> 00:28:06,270 where we can easily understand the new features which 385 00:28:06,270 --> 00:28:08,050 happen in the system. 386 00:28:08,050 --> 00:28:10,680 So the situation we have now prepared 387 00:28:10,680 --> 00:28:15,300 is we have our dark state, which is level g, 388 00:28:15,300 --> 00:28:19,050 we said we have only one photon in our probe beam, 389 00:28:19,050 --> 00:28:23,050 and we have lots and lots of photons in the coupling laser 390 00:28:23,050 --> 00:28:30,630 which couples the other ground state, f, to the excited state, 391 00:28:30,630 --> 00:28:31,130 e. 392 00:28:35,680 --> 00:28:40,880 So we have the structure that we have now two states which 393 00:28:40,880 --> 00:28:44,580 have half of the widths of the excited state, 394 00:28:44,580 --> 00:28:48,310 and they are both bright. 395 00:28:48,310 --> 00:28:53,750 We can call one the bright state 1 396 00:28:53,750 --> 00:28:56,610 and the other the bright state 2. 397 00:28:59,940 --> 00:29:04,200 The splitting is related to the Rabi frequency. 398 00:29:04,200 --> 00:29:06,058 Let me just call it delta bar. 399 00:29:14,390 --> 00:29:17,630 This is now the level structure which we have, 400 00:29:17,630 --> 00:29:21,100 and what I want to now emphasize in this picture 401 00:29:21,100 --> 00:29:22,543 is the phenomenon of interference. 402 00:29:26,260 --> 00:29:29,030 We've talked about interference of amplitudes, 403 00:29:29,030 --> 00:29:31,470 but now I want to take the system 404 00:29:31,470 --> 00:29:35,530 and show you how we get now interference 405 00:29:35,530 --> 00:29:39,740 when we send one probe photon through the system. 406 00:29:42,580 --> 00:29:45,540 What we can now formulate is a scattering problem. 407 00:29:49,640 --> 00:29:58,250 We have one photon in our probe beam in a special mode, 408 00:29:58,250 --> 00:30:08,230 but then, all the other modes are unoccupied. 409 00:30:08,230 --> 00:30:12,630 And when we are asking, does the probe photon get absorbed? 410 00:30:12,630 --> 00:30:15,990 Does the weak laser beam get stuck in the brick wall? 411 00:30:15,990 --> 00:30:19,460 We are actually are asking if it is possible 412 00:30:19,460 --> 00:30:23,380 that we scatter the photon out of the mode 413 00:30:23,380 --> 00:30:25,880 and it gets absorbed. 414 00:30:25,880 --> 00:30:27,700 But you know absorption is actually always 415 00:30:27,700 --> 00:30:31,360 a two photon process involves spontaneous emission, 416 00:30:31,360 --> 00:30:35,130 and we have emitted the photon into another mode. 417 00:30:41,780 --> 00:30:47,950 The scattering problem is a two photon process, 418 00:30:47,950 --> 00:31:06,980 and the matrix element needs an intermediate state, 419 00:31:06,980 --> 00:31:07,830 but now we have two. 420 00:31:16,790 --> 00:31:21,760 We start with one photon, we have the light atom coupling, 421 00:31:21,760 --> 00:31:25,600 we can go through bright state 1. 422 00:31:25,600 --> 00:31:29,660 From bright state 1, we have the light atom coupling again, 423 00:31:29,660 --> 00:31:32,520 and eventually, we go back to the ground state 424 00:31:32,520 --> 00:31:33,560 without four photon. 425 00:31:36,710 --> 00:31:38,845 And here we have a detuning. 426 00:31:38,845 --> 00:31:41,260 Let's assume we are halfway detuned 427 00:31:41,260 --> 00:31:45,135 between the two bright states. 428 00:31:47,730 --> 00:31:50,470 And then we have a second amplitude, 429 00:31:50,470 --> 00:31:51,770 and it's indistinguishable. 430 00:31:51,770 --> 00:31:54,650 We have a Feynman double slit experiment, 431 00:31:54,650 --> 00:32:01,320 and everything here is the same except that we 432 00:32:01,320 --> 00:32:04,950 scatter through bright state 2. 433 00:32:12,550 --> 00:32:15,150 This matrix element, when it vanishes, 434 00:32:15,150 --> 00:32:18,375 this is now the condition of electromagnetically 435 00:32:18,375 --> 00:32:19,250 induced transparency. 436 00:32:25,360 --> 00:32:27,710 But we want to now understand what happens 437 00:32:27,710 --> 00:32:29,780 when we detune the probe laser. 438 00:32:29,780 --> 00:32:33,170 So we have set up the system with a strong control laser, 439 00:32:33,170 --> 00:32:39,060 we have completely mixed the ground state f with the excited 440 00:32:39,060 --> 00:32:42,535 state, and now we want to ask, can a weak probe laser 441 00:32:42,535 --> 00:32:46,760 go through the brick wall How much of the probe laser 442 00:32:46,760 --> 00:32:47,680 is absorbed? 443 00:32:47,680 --> 00:32:51,790 So what we want to understand now, 444 00:32:51,790 --> 00:32:54,320 and this is a new feature I want to discuss now, 445 00:32:54,320 --> 00:33:04,020 what happens when we detune the probe laser by delta? 446 00:33:04,020 --> 00:33:09,570 Well, it's clear, and I just wanted to show you the formula. 447 00:33:09,570 --> 00:33:17,310 We had two detunings here, and if we detune the probe laser 448 00:33:17,310 --> 00:33:26,420 by delta, that would mean now that those denominators are 449 00:33:26,420 --> 00:33:32,890 no longer opposite but equal. 450 00:33:32,890 --> 00:33:34,940 In one case, we add delta. 451 00:33:34,940 --> 00:33:38,840 In the other case, we subtract delta. 452 00:33:38,840 --> 00:33:41,550 And therefore, we have no longer the cancellation 453 00:33:41,550 --> 00:33:43,870 of the two amplitudes and we have 454 00:33:43,870 --> 00:33:45,415 the scattering of the photon. 455 00:33:51,060 --> 00:33:53,380 This is sort of the framework, and I 456 00:33:53,380 --> 00:33:55,483 want to show you now several examples. 457 00:33:58,880 --> 00:34:02,503 I want to show you examples of a probe absorption spectra. 458 00:34:02,503 --> 00:34:04,170 It's more a little bit of show and tell. 459 00:34:04,170 --> 00:34:09,010 I want to show you the result if you would work that out. 460 00:34:09,010 --> 00:34:12,889 It's pretty interesting. 461 00:34:12,889 --> 00:34:19,170 So first, I want to discuss the case where the coupling 462 00:34:19,170 --> 00:34:20,989 laser is really in resonance. 463 00:34:41,360 --> 00:34:43,860 We are near the one photon resonance. 464 00:34:51,820 --> 00:34:58,970 Let me assume first that the Rabi frequency of the coupling 465 00:34:58,970 --> 00:35:01,540 laser is much, much larger than gamma. 466 00:35:04,450 --> 00:35:17,330 Then if you look at the probe transmission, 467 00:35:17,330 --> 00:35:19,640 we have the situation that we know 468 00:35:19,640 --> 00:35:21,610 when we are right here in the middle, 469 00:35:21,610 --> 00:35:23,870 we have a window of transparency, 470 00:35:23,870 --> 00:35:30,630 but if we detune by the Rabi frequency over two, 471 00:35:30,630 --> 00:35:34,525 we hit the bright state 1, and if we detune in the opposite, 472 00:35:34,525 --> 00:35:37,170 we hit the bright state 2. 473 00:35:37,170 --> 00:35:40,475 And the splitting in this situation, delta bar, 474 00:35:40,475 --> 00:35:45,660 is nothing else than the Rabi frequency omega 2. 475 00:35:45,660 --> 00:35:49,660 So what we have is we have the detuning in between, 476 00:35:49,660 --> 00:35:53,605 and we know already here is our special point, 477 00:35:53,605 --> 00:35:54,980 and that's what we have discussed 478 00:35:54,980 --> 00:35:59,020 for long, electromagnetically induced transparency. 479 00:35:59,020 --> 00:36:03,700 The two bright states are at plus and minus 480 00:36:03,700 --> 00:36:07,910 half the Rabi frequency of the coupling laser. 481 00:36:07,910 --> 00:36:12,080 We know that we have very strong absorption here. 482 00:36:16,110 --> 00:36:19,335 What you get is sort of broad feature. 483 00:36:22,970 --> 00:36:27,180 But this was a situation when we drive the system very strongly. 484 00:36:29,980 --> 00:36:35,680 I now want to discuss the case that the coupling laser is 485 00:36:35,680 --> 00:36:36,770 much weaker than gamma. 486 00:36:40,700 --> 00:36:45,880 Then the splitting between the two bright states 487 00:36:45,880 --> 00:36:48,890 was the Rabi frequency, but the width is gamma, 488 00:36:48,890 --> 00:36:51,970 so then the two bright states pretty much merge 489 00:36:51,970 --> 00:36:56,380 into one continuum feature of width gamma. 490 00:36:56,380 --> 00:37:17,010 For that situation, we have a broad feature of absorption, 491 00:37:17,010 --> 00:37:19,510 which is on the order of gamma, or if you 492 00:37:19,510 --> 00:37:21,180 have an opaque medium, of course, 493 00:37:21,180 --> 00:37:23,190 you put the absorption coefficient 494 00:37:23,190 --> 00:37:26,995 into the exponent of an exponential function 495 00:37:26,995 --> 00:37:30,480 and you get a blackout which is wider than gamma. 496 00:37:30,480 --> 00:37:38,520 But then we have our phenomenon of EIT, 497 00:37:38,520 --> 00:37:43,872 but the width of this feature is now much smaller than gamma. 498 00:37:48,210 --> 00:37:53,490 So what we have here in those two situations 499 00:37:53,490 --> 00:37:57,150 where the lasers are close to resonance with the excited 500 00:37:57,150 --> 00:38:01,710 level, we have the situation that the strong absorption 501 00:38:01,710 --> 00:38:04,910 feature due to the bright states, 502 00:38:04,910 --> 00:38:08,540 either one strong feature of width gamma or two features 503 00:38:08,540 --> 00:38:12,260 here, those broad, similar photon absorption features, 504 00:38:12,260 --> 00:38:17,230 they really overlap with our window of transparency. 505 00:38:17,230 --> 00:38:21,420 And what I find very insightful is now to discuss the situation 506 00:38:21,420 --> 00:38:25,480 where we separate the two, and I want to show you what happens. 507 00:38:25,480 --> 00:38:27,040 It gives a very interesting profile. 508 00:38:30,820 --> 00:38:42,140 So what we want to discuss now are the famous Fano profiles, 509 00:38:42,140 --> 00:38:45,715 and I want to discuss two photon absorption features. 510 00:38:55,910 --> 00:39:01,140 We have discussed the case where the one photon detuning. 511 00:39:01,140 --> 00:39:15,630 We want to go now to a large one photon detuning delta 2, 512 00:39:15,630 --> 00:39:19,410 and the new feature now is that this 513 00:39:19,410 --> 00:39:27,340 will separate the window of electromagnetically 514 00:39:27,340 --> 00:39:30,695 induced transparency from the broad absorption features. 515 00:39:41,660 --> 00:39:45,045 Let me just draw a diagram of the states. 516 00:39:56,130 --> 00:40:02,180 As usual, we have our two states in a lambda transition, g 517 00:40:02,180 --> 00:40:04,460 and f. 518 00:40:04,460 --> 00:40:08,230 We have this continuum of the excited state, 519 00:40:08,230 --> 00:40:12,140 but now, and this is what often is done in the experiment, 520 00:40:12,140 --> 00:40:15,930 you're not using Raman lasers which are in resonance 521 00:40:15,930 --> 00:40:17,580 with the excited state. 522 00:40:17,580 --> 00:40:21,050 You're using Raman lasers-- 523 00:40:21,050 --> 00:40:24,560 here is the strong coupling lasers-- 524 00:40:24,560 --> 00:40:25,480 which are far detuned. 525 00:40:28,370 --> 00:40:34,395 So here we have a detuning for laser two called gamma 2. 526 00:40:34,395 --> 00:40:36,820 The Rabi frequency is omega 2. 527 00:40:40,660 --> 00:40:47,823 Here we have a weak probe laser omega 1, and the detuning. 528 00:40:52,110 --> 00:40:56,670 Let's call it gamma 1. 529 00:40:56,670 --> 00:41:01,370 So in order to keep the situation simple, 530 00:41:01,370 --> 00:41:03,400 we use a weak probe laser. 531 00:41:03,400 --> 00:41:05,980 Omega 1 is small. 532 00:41:05,980 --> 00:41:09,620 And our Raman detuning delta is the difference 533 00:41:09,620 --> 00:41:12,350 between the two single photons detuning, 534 00:41:12,350 --> 00:41:14,866 capital delta 2 minus delta 1. 535 00:41:18,440 --> 00:41:29,770 For these situations, there are nice analytic expressions, 536 00:41:29,770 --> 00:41:33,020 and together with the class notes, 537 00:41:33,020 --> 00:41:38,330 I will post a wonderfully clear paper 538 00:41:38,330 --> 00:41:42,755 by [? Lunis ?] and [INAUDIBLE] where the two authors discuss 539 00:41:42,755 --> 00:41:43,380 this situation. 540 00:41:46,500 --> 00:41:51,410 Let's just figure out what are the features in the system. 541 00:41:57,020 --> 00:42:05,700 Let's just go through different situations. 542 00:42:05,700 --> 00:42:08,980 One is if the Raman detuning is zero, 543 00:42:08,980 --> 00:42:10,570 we should always get the phenomenon 544 00:42:10,570 --> 00:42:12,620 of electromagnetically induced transparency. 545 00:42:23,620 --> 00:42:29,050 If we don't have any coupling, then 546 00:42:29,050 --> 00:42:32,310 we simply have a two level system, 547 00:42:32,310 --> 00:42:36,070 and if we tune omega 1 into resonance, 548 00:42:36,070 --> 00:42:37,810 we get simple single photon absorption. 549 00:42:47,950 --> 00:42:50,470 So if we look at the system of the three levels 550 00:42:50,470 --> 00:42:57,370 and we are asking what are now the relevant processes, 551 00:42:57,370 --> 00:43:02,700 one limit is, of course, the trivial limit 552 00:43:02,700 --> 00:43:15,960 that we have single photon absorption. 553 00:43:18,810 --> 00:43:20,825 This is, of course, a trivial case. 554 00:43:20,825 --> 00:43:23,970 Let's now go to the more interesting case 555 00:43:23,970 --> 00:43:25,385 that we have a coupling laser. 556 00:43:33,130 --> 00:43:33,980 What happens now? 557 00:43:39,200 --> 00:43:42,320 We have our two states. 558 00:43:42,320 --> 00:43:51,020 The excited state is coupled with the laser 559 00:43:51,020 --> 00:43:55,360 and Rabi frequency omega 2. 560 00:43:55,360 --> 00:44:02,490 Now we know that the laser omega 2, if it is not on resonance, 561 00:44:02,490 --> 00:44:04,560 will give us an AC Stark shift. 562 00:44:07,990 --> 00:44:11,340 This AC Stark shift, we've gone through that several times. 563 00:44:11,340 --> 00:44:15,090 It's the matrix element or Rabi frequency squared 564 00:44:15,090 --> 00:44:16,190 divided by the detuning. 565 00:44:39,520 --> 00:44:44,550 If we now bring in the probe laser, 566 00:44:44,550 --> 00:44:46,280 what are the features we expect? 567 00:44:49,360 --> 00:44:51,330 Well, there are two features. 568 00:44:51,330 --> 00:44:55,540 One is we've just discussed the trivial case above. 569 00:44:55,540 --> 00:44:59,030 If we tune the probe laser into resonance with the excited 570 00:44:59,030 --> 00:45:01,860 state, we have single photon absorption. 571 00:45:01,860 --> 00:45:03,220 We get a broad feature. 572 00:45:03,220 --> 00:45:05,770 It's almost like in the two level system. 573 00:45:05,770 --> 00:45:09,410 But in contrast to the case I just discussed above, 574 00:45:09,410 --> 00:45:14,370 the excited state level e has now an AC Stark shift 575 00:45:14,370 --> 00:45:17,700 so the resonance is shifted. 576 00:45:17,700 --> 00:45:20,500 That's now becoming a four photon process 577 00:45:20,500 --> 00:45:22,700 because we need two photons going 578 00:45:22,700 --> 00:45:25,450 up and down with a coupling laser 579 00:45:25,450 --> 00:45:27,670 to create the AC Stark shift. 580 00:45:27,670 --> 00:45:30,270 And now we have a laser from the probe beam 581 00:45:30,270 --> 00:45:33,640 and the photon is scattered, so it's a four photon process 582 00:45:33,640 --> 00:45:36,050 and it will give us a broad resonance which 583 00:45:36,050 --> 00:45:39,450 is now AC stack shifted. 584 00:45:39,450 --> 00:45:43,040 But in addition, we have a resonance, 585 00:45:43,040 --> 00:45:46,730 which is the Raman resonance. 586 00:45:46,730 --> 00:45:52,840 When the Raman detuning is zero, then we 587 00:45:52,840 --> 00:45:56,420 absorb from the probe laser and we 588 00:45:56,420 --> 00:45:59,040 emit in a stimulated way with the coupling laser, 589 00:45:59,040 --> 00:46:03,540 and we have a stimulated two photon transition. 590 00:46:03,540 --> 00:46:06,670 Now, what is the width of this stimulated two photon 591 00:46:06,670 --> 00:46:07,170 transition? 592 00:46:09,830 --> 00:46:14,830 Well, we go from a stable ground state to-- 593 00:46:14,830 --> 00:46:17,480 I wanted to say another stable ground state, 594 00:46:17,480 --> 00:46:20,850 but this other stable ground state 595 00:46:20,850 --> 00:46:22,250 is now scattering photons. 596 00:46:24,860 --> 00:46:33,680 So because of the presence of a strong coupling laser, 597 00:46:33,680 --> 00:46:38,035 you have broadened this level f by photon scattering. 598 00:46:38,035 --> 00:46:40,150 You interrupt the coherent time evolution 599 00:46:40,150 --> 00:46:44,110 by scattering photons, and the photon scattering 600 00:46:44,110 --> 00:46:49,140 happens in perturbation theory by the amplitude to be 601 00:46:49,140 --> 00:47:02,860 in the excited state squared times gamma. 602 00:47:06,690 --> 00:47:08,960 The scattering rate, gamma scattering, 603 00:47:08,960 --> 00:47:14,960 is Rabi frequency divided by detuning. 604 00:47:14,960 --> 00:47:17,670 This is the amplitude to be in the excited state. 605 00:47:21,720 --> 00:47:26,760 We square that, and then we multiply with gamma, 606 00:47:26,760 --> 00:47:28,322 and if I can trust my notes, it's 607 00:47:28,322 --> 00:47:30,700 a factor of two, which I don't want to discuss further. 608 00:47:33,220 --> 00:47:35,110 This is a quantitative argument. 609 00:47:35,110 --> 00:47:37,500 The analytic expressions are in the reference 610 00:47:37,500 --> 00:47:39,060 I've given to you. 611 00:47:39,060 --> 00:47:42,170 So the situation which we have right now 612 00:47:42,170 --> 00:47:47,470 can be in a very powerful way summarized as follows. 613 00:47:47,470 --> 00:47:53,010 We have our ground state, we have 614 00:47:53,010 --> 00:47:56,610 two continua we can couple to. 615 00:47:56,610 --> 00:48:00,290 We can couple to the excited state which 616 00:48:00,290 --> 00:48:05,430 has a width, gamma, through a single photon, 617 00:48:05,430 --> 00:48:07,890 but there is the AC Stark shift. 618 00:48:07,890 --> 00:48:12,310 Or we can couple through a two photon Raman transition 619 00:48:12,310 --> 00:48:22,050 to the state f, but the width is much, much smaller 620 00:48:22,050 --> 00:48:24,960 because it is only the scattering rate due 621 00:48:24,960 --> 00:48:29,710 to the off resonant coupling laser. 622 00:48:29,710 --> 00:48:33,010 Let me just write that down because I said a lot of things. 623 00:48:33,010 --> 00:48:41,520 So g couples now. 624 00:48:41,520 --> 00:48:44,080 When we detune the probe laser, we 625 00:48:44,080 --> 00:48:46,270 can be in resonance with this feature 626 00:48:46,270 --> 00:48:52,670 and we can be in resonance with that feature 627 00:48:52,670 --> 00:49:07,990 to a narrow and wide excited state. 628 00:49:07,990 --> 00:49:11,070 One excited state, of course, is the state f, 629 00:49:11,070 --> 00:49:14,550 but the coupling laser puts some character of the excited 630 00:49:14,550 --> 00:49:17,290 states into the state f. 631 00:49:17,290 --> 00:49:21,040 And the most important thing now is the following. 632 00:49:21,040 --> 00:49:25,570 This is the theme I've emphasized again and again when 633 00:49:25,570 --> 00:49:28,480 we discussed three level systems. 634 00:49:28,480 --> 00:49:31,740 Those two states have a width, and the width 635 00:49:31,740 --> 00:49:34,760 means they spontaneously emit light, 636 00:49:34,760 --> 00:49:37,910 but they emit light into the same continuum. 637 00:49:37,910 --> 00:49:40,530 So if you start in the ground state, 638 00:49:40,530 --> 00:49:44,360 your probe laser has a photon and the photon gets scattered. 639 00:49:44,360 --> 00:49:48,230 You do not know through which channel it has been scattered. 640 00:49:48,230 --> 00:49:51,110 So in general, and this is as far 641 00:49:51,110 --> 00:49:55,220 as I want to push it in this class, in three level system, 642 00:49:55,220 --> 00:49:59,730 we have now the interference between those two continua. 643 00:49:59,730 --> 00:50:01,740 One is narrow and one is wide. 644 00:50:12,100 --> 00:50:14,460 Let me just write that down because this is important, 645 00:50:14,460 --> 00:50:26,255 but both excited states couple to the same continuum. 646 00:50:29,200 --> 00:50:31,610 And by continuum, I mean the vacuum 647 00:50:31,610 --> 00:50:36,940 of all empty states where photons can be scattered, 648 00:50:36,940 --> 00:50:38,940 and this is the condition for interference. 649 00:50:46,240 --> 00:50:48,510 And this is, of course, what gives rise 650 00:50:48,510 --> 00:50:50,910 to electromagnetically induced transparency. 651 00:50:54,565 --> 00:50:55,940 I'll take your questions, but let 652 00:50:55,940 --> 00:50:58,020 me first give you a drawing which 653 00:50:58,020 --> 00:51:05,110 may illustrate, or summarize, what I've just said. 654 00:51:05,110 --> 00:51:10,490 So what we try to understand is what happens 655 00:51:10,490 --> 00:51:14,530 when we detune the probe laser. 656 00:51:14,530 --> 00:51:18,950 Until now, we had always the EIT feature, the EIT window, 657 00:51:18,950 --> 00:51:22,300 was completely overlapping with the one photon resonance, 658 00:51:22,300 --> 00:51:27,910 but now, because the coupling laser has 659 00:51:27,910 --> 00:51:38,205 a detuning of gamma 2, we have to use with the probe laser. 660 00:51:57,270 --> 00:52:00,090 I have to trace back how the detunings are defined. 661 00:52:03,400 --> 00:52:09,930 The Raman detuning was big delta 2 minus big delta 1. 662 00:52:09,930 --> 00:52:13,810 If the detuning delta is chosen to be delta 2, 663 00:52:13,810 --> 00:52:16,670 we have the simple situation that we go from the ground 664 00:52:16,670 --> 00:52:23,520 state right to the excited state, 665 00:52:23,520 --> 00:52:29,380 and we have the feature which has a width of gamma. 666 00:52:29,380 --> 00:52:34,469 This is what you would call the single photon resonance. 667 00:52:34,469 --> 00:52:35,886 It is the single photon resonance. 668 00:52:38,830 --> 00:52:44,290 It is due by resonantly coupling into this continuum, 669 00:52:44,290 --> 00:52:47,360 and the only feature of the coupling laser 670 00:52:47,360 --> 00:52:51,820 is that there's an AC Stark shift to it. 671 00:52:51,820 --> 00:52:55,090 Single normal resonance, almost like in a two level system, 672 00:52:55,090 --> 00:52:58,220 but the only addition is the AC Stark effect. 673 00:53:18,800 --> 00:53:24,280 Now we have a second feature, which can be very sharp. 674 00:53:33,610 --> 00:53:40,380 This is when we do the two photon Raman process 675 00:53:40,380 --> 00:53:41,640 into the other ground state. 676 00:53:47,580 --> 00:53:52,480 Due to photon scattering, this resonance 677 00:53:52,480 --> 00:54:00,060 has a width of gamma scattering, and the position 678 00:54:00,060 --> 00:54:05,750 is not at zero at the naked Raman resonance. 679 00:54:05,750 --> 00:54:09,770 It also has an AC Stark shift because the coupling laser 680 00:54:09,770 --> 00:54:13,060 does an AC Stark shift to both the excited and the ground 681 00:54:13,060 --> 00:54:14,550 state. 682 00:54:14,550 --> 00:54:16,610 The coupling laser, the AC Stark shift, 683 00:54:16,610 --> 00:54:20,120 pushes ground and excited state in opposite directions. 684 00:54:20,120 --> 00:54:24,540 So therefore, we find that at the Rabi frequency 685 00:54:24,540 --> 00:54:28,067 of the coupling laser divided by gamma over 2. 686 00:54:33,770 --> 00:54:42,870 The name of this feature is the two photon Raman resonance 687 00:54:42,870 --> 00:54:53,080 plus the AC Stark shift, which actually means 688 00:54:53,080 --> 00:54:54,205 it's a four photon process. 689 00:55:00,570 --> 00:55:04,958 The question is now, where is the electromagnetically induced 690 00:55:04,958 --> 00:55:05,500 transparency? 691 00:55:11,160 --> 00:55:14,780 We have introduced now specifically the absorption 692 00:55:14,780 --> 00:55:15,580 feature. 693 00:55:15,580 --> 00:55:18,780 We have identified a one photon absorption feature 694 00:55:18,780 --> 00:55:21,370 plus AC Stark shift, a two photon absorption 695 00:55:21,370 --> 00:55:25,330 feature plus AC Stark shift, but where is electromagnetically 696 00:55:25,330 --> 00:55:26,330 introduced transparency? 697 00:55:31,340 --> 00:55:33,810 Here. 698 00:55:33,810 --> 00:55:36,700 Electromagnetically induced transparency 699 00:55:36,700 --> 00:55:38,860 is always at delta equals 0. 700 00:55:38,860 --> 00:55:46,850 You have to fulfill the Raman resonance, 701 00:55:46,850 --> 00:55:51,250 and this resonance is not affected by any AC Stark shift. 702 00:55:51,250 --> 00:55:53,880 It's always at delta equals 0. 703 00:55:53,880 --> 00:55:57,610 So therefore, what that means is you have two absorption 704 00:55:57,610 --> 00:56:01,210 features, and you would think these are two Lorentzians, 705 00:56:01,210 --> 00:56:16,843 but they interfere and they go to exactly 0 at delta equals 0, 706 00:56:16,843 --> 00:56:18,010 and this is our EIT feature. 707 00:56:22,150 --> 00:56:28,540 So there is an interference effect between a broad feature 708 00:56:28,540 --> 00:56:32,140 and a narrow feature, and this is 709 00:56:32,140 --> 00:56:37,200 found in many different parts of spectroscopy, 710 00:56:37,200 --> 00:56:39,270 but you also find it in nuclear physics 711 00:56:39,270 --> 00:56:41,960 whenever you have a narrow feature embedded 712 00:56:41,960 --> 00:56:43,480 into continuum. 713 00:56:43,480 --> 00:56:47,880 What we have here is a narrow feature and a broader feature, 714 00:56:47,880 --> 00:56:49,590 but a narrow feature and something 715 00:56:49,590 --> 00:56:53,360 which is broader or continuum is called a Fano resonance. 716 00:56:58,090 --> 00:57:00,660 You actually have the same situation 717 00:57:00,660 --> 00:57:04,877 when you look at scattering of atoms. 718 00:57:04,877 --> 00:57:06,960 Many of you are familiar with Feshbach resonances. 719 00:57:06,960 --> 00:57:10,720 Well, a lot of people call it Fano Feshbach resonance, 720 00:57:10,720 --> 00:57:15,040 and what you have in a Fano Feshbach resonance is two atoms 721 00:57:15,040 --> 00:57:16,810 can scatter off each other. 722 00:57:16,810 --> 00:57:18,650 This is the continuum. 723 00:57:18,650 --> 00:57:20,820 This would be your broad feature. 724 00:57:20,820 --> 00:57:24,450 But then they can also scatter through a molecular state, 725 00:57:24,450 --> 00:57:26,730 and this is a narrow feature. 726 00:57:26,730 --> 00:57:29,770 And what we've identified now for electromagnetically 727 00:57:29,770 --> 00:57:32,810 induced transparency are the two features. 728 00:57:32,810 --> 00:57:34,670 One is the single photon absorption, 729 00:57:34,670 --> 00:57:36,390 one is the Ramon resonance. 730 00:57:36,390 --> 00:57:39,400 But in general, the concept is much more general. 731 00:57:39,400 --> 00:57:43,870 You have a narrow feature, you have a broader feature. 732 00:57:43,870 --> 00:57:46,110 It's responsible for scattering two atoms 733 00:57:46,110 --> 00:57:48,510 or it's responsible for scattering a photon, 734 00:57:48,510 --> 00:57:51,300 and once the photon or the atoms have been scattered, 735 00:57:51,300 --> 00:57:54,430 you have no way of telling which intermediate state was 736 00:57:54,430 --> 00:57:55,310 involved. 737 00:57:55,310 --> 00:57:57,630 And therefore, you get interference. 738 00:57:57,630 --> 00:58:00,930 And what I just said, what is EIT for light 739 00:58:00,930 --> 00:58:03,780 is the zero crossing of a scattering length 740 00:58:03,780 --> 00:58:06,470 that the atoms do not scatter off each other 741 00:58:06,470 --> 00:58:10,360 because the two different processes completely 742 00:58:10,360 --> 00:58:11,560 destructively interfere. 743 00:58:15,210 --> 00:58:19,350 So what I've shown here is two Lorentzian, two absorption 744 00:58:19,350 --> 00:58:21,570 features. 745 00:58:21,570 --> 00:58:30,720 Let me know re-plot it and plot the index of refraction 746 00:58:30,720 --> 00:58:31,610 minus 1. 747 00:58:35,930 --> 00:58:40,590 Here was EIT, zero detuning. 748 00:58:40,590 --> 00:58:44,340 We have a sharp feature here at the two photon resonance, 749 00:58:44,340 --> 00:58:48,430 we have a broad feature here at the single photon resonance, 750 00:58:48,430 --> 00:58:56,670 and if I now transform the Lorentzian 751 00:58:56,670 --> 00:58:58,870 into a dispersive feature. 752 00:58:58,870 --> 00:59:03,860 I use freehand, so this is a dispersive feature 753 00:59:03,860 --> 00:59:06,680 for the broad transition. 754 00:59:06,680 --> 00:59:09,510 The narrow transition has a much, much sharper 755 00:59:09,510 --> 00:59:14,360 dispersive feature, and the important part 756 00:59:14,360 --> 00:59:28,800 is now at the EIT, at the detuning delta 757 00:59:28,800 --> 00:59:31,250 is zero where we have electromagnetically 758 00:59:31,250 --> 00:59:35,370 induced transparency, we have an index of refraction 759 00:59:35,370 --> 00:59:41,750 which is exactly one because it's a dark state, 760 00:59:41,750 --> 00:59:44,810 you have no absorption, you have no light scattering, 761 00:59:44,810 --> 00:59:47,170 you have no reaction to the light. 762 00:59:47,170 --> 00:59:50,560 And therefore, the index of refraction of the material 763 00:59:50,560 --> 00:59:53,174 is like the index of refraction of the vacuum. 764 00:59:55,920 --> 00:59:57,830 So you have n equals 1. 765 00:59:57,830 --> 01:00:02,760 It looks like a vacuum in terms of index of refraction. 766 01:00:02,760 --> 01:00:06,440 It looks like a vacuum because you have no absorption. 767 01:00:06,440 --> 01:00:12,200 But what you have is you have a large derivative 768 01:00:12,200 --> 01:00:15,890 of the index of refraction with the frequency detuning, 769 01:00:15,890 --> 01:00:18,880 and that affects, and that's what I want to tell you now, 770 01:00:18,880 --> 01:00:20,776 the group velocity of light. 771 01:00:25,210 --> 01:00:28,170 So anyway, this is maybe as far as I want to push it, 772 01:00:28,170 --> 01:00:30,780 and I was actually wondering if this is a little bit too 773 01:00:30,780 --> 01:00:32,780 complicated to present in class. 774 01:00:32,780 --> 01:00:35,160 But on the other hand, I think it sort of also 775 01:00:35,160 --> 01:00:37,000 wraps up the course. 776 01:00:37,000 --> 01:00:39,230 We have a three level system, and we 777 01:00:39,230 --> 01:00:43,812 find a lot of things we have studied separately before-- 778 01:00:43,812 --> 01:00:47,910 two photon Raman feature, single photon light scattering-- 779 01:00:47,910 --> 01:00:50,360 but now they act together and they 780 01:00:50,360 --> 01:00:53,520 interfere and have this additional feature 781 01:00:53,520 --> 01:00:56,670 of electromagnetically induced transparency. 782 01:00:56,670 --> 01:00:57,338 Colin? 783 01:00:57,338 --> 01:01:02,218 STUDENT: So the way you drew the level diagram, the f, I think, 784 01:01:02,218 --> 01:01:03,682 [INAUDIBLE] state [INAUDIBLE]? 785 01:01:09,315 --> 01:01:09,940 PROFESSOR: Yes. 786 01:01:09,940 --> 01:01:22,550 Actually, I emphasized that we have an AC Stark shift, 787 01:01:22,550 --> 01:01:25,085 and what I didn't say when I discussed it here 788 01:01:25,085 --> 01:01:28,610 that the AC Stark shift pushes this level down 789 01:01:28,610 --> 01:01:30,820 and pushes the other level up. 790 01:01:30,820 --> 01:01:34,070 But since we are talking about a very broad resonance 791 01:01:34,070 --> 01:01:37,150 in the excited state, for all practical matters, 792 01:01:37,150 --> 01:01:39,095 the AC Stark shift doesn't matter, 793 01:01:39,095 --> 01:01:41,010 whereas for the narrow Raman resonance, 794 01:01:41,010 --> 01:01:43,760 the AC Stark shift is important. 795 01:01:43,760 --> 01:01:46,210 STUDENT: Also, you showed on the plot 796 01:01:46,210 --> 01:01:52,760 that the shift of the excited state [INAUDIBLE].. 797 01:01:52,760 --> 01:01:54,370 PROFESSOR: Sorry. 798 01:01:54,370 --> 01:01:55,090 Thanks. 799 01:01:55,090 --> 01:01:57,880 So the AC Stark shift for that detuning 800 01:01:57,880 --> 01:02:03,820 would shift this level up and would shift this level down. 801 01:02:03,820 --> 01:02:05,540 What's the second question? 802 01:02:05,540 --> 01:02:08,480 STUDENT: You drew on the plot-- 803 01:02:08,480 --> 01:02:10,536 PROFESSOR: This one? 804 01:02:10,536 --> 01:02:12,012 STUDENT: That one. 805 01:02:12,012 --> 01:02:14,472 That the shift of the excited state 806 01:02:14,472 --> 01:02:18,900 was much higher than the shift of the ground state. 807 01:02:21,853 --> 01:02:23,395 PROFESSOR: You mean those two shifts? 808 01:02:23,395 --> 01:02:23,978 STUDENT: Yeah. 809 01:02:29,135 --> 01:02:31,010 PROFESSOR: We have to now do the bookkeeping. 810 01:02:33,870 --> 01:02:37,470 We have assumed that the coupling laser has 811 01:02:37,470 --> 01:02:39,400 a large detuning, the coupling laser 812 01:02:39,400 --> 01:02:42,510 is very far away from resonance. 813 01:02:42,510 --> 01:02:45,400 And if you want to hit the excited state, 814 01:02:45,400 --> 01:02:48,990 we know we need a Raman resonance, which is delta 2, 815 01:02:48,990 --> 01:02:52,580 but the Raman resonance, capital delta 2, 816 01:02:52,580 --> 01:02:55,670 means that we are smack on the single photon 817 01:02:55,670 --> 01:02:57,500 resonance for the probe laser. 818 01:02:57,500 --> 01:03:00,405 So this feature is pretty much we take the ground state 819 01:03:00,405 --> 01:03:05,010 and go exactly to the excited state with a single photon. 820 01:03:05,010 --> 01:03:06,650 There is an AC Stark shift involved 821 01:03:06,650 --> 01:03:10,370 but it's not relevant here, whereas the other feature is 822 01:03:10,370 --> 01:03:13,680 the two photon Raman feature. 823 01:03:13,680 --> 01:03:16,520 And the one thing I wanted to point out in this context 824 01:03:16,520 --> 01:03:21,390 is that there is actually a small energy 825 01:03:21,390 --> 01:03:25,820 splitting between the two photon Raman feature by the AC Stark 826 01:03:25,820 --> 01:03:28,510 effect, whereas the EIT feature always 827 01:03:28,510 --> 01:03:31,190 happens at Raman resonance delta equals 0. 828 01:03:38,500 --> 01:03:42,140 Because, just to emphasize that, delta 829 01:03:42,140 --> 01:03:46,570 equals 0 is really you induce a coherence between g and f. 830 01:03:46,570 --> 01:03:48,490 It's a dark state, and when you have 831 01:03:48,490 --> 01:03:51,390 a dark state in that situation, you 832 01:03:51,390 --> 01:03:53,000 don't have an AC Stark effect. 833 01:03:53,000 --> 01:03:56,470 So the EIT feature is at delta equals 0, 834 01:03:56,470 --> 01:03:59,870 whereas the photon scattering features, they suffer, 835 01:03:59,870 --> 01:04:01,870 or they experience-- 836 01:04:01,870 --> 01:04:04,140 it may not be negative to suffer an AC Stark shift, 837 01:04:04,140 --> 01:04:05,868 but they experience an AC Stark shift. 838 01:04:09,360 --> 01:04:10,140 Further questions? 839 01:04:10,140 --> 01:04:10,670 Yes? 840 01:04:10,670 --> 01:04:13,000 STUDENT: Can we somehow relate this 841 01:04:13,000 --> 01:04:16,740 to Doppler free spectroscopy? 842 01:04:16,740 --> 01:04:20,990 PROFESSOR: Can we relate that to Doppler free spectroscopy? 843 01:04:20,990 --> 01:04:24,760 Actually, I don't think so because I 844 01:04:24,760 --> 01:04:28,402 would say for the whole discussion here, 845 01:04:28,402 --> 01:04:30,860 let's assume we have an atom which has infinite mass, which 846 01:04:30,860 --> 01:04:32,300 is not moving at all. 847 01:04:32,300 --> 01:04:36,310 We're really talking about internal coherences. 848 01:04:36,310 --> 01:04:41,540 However, and that's where it becomes related, 849 01:04:41,540 --> 01:04:44,860 if you look at very, very narrow features 850 01:04:44,860 --> 01:04:48,320 as a function of detuning then, of course, Doppler shifts 851 01:04:48,320 --> 01:04:51,900 play a role, and if you have very, very, very 852 01:04:51,900 --> 01:04:56,710 narrow features, you become sensitive to very, very, very 853 01:04:56,710 --> 01:04:59,290 small velocities, and therefore, you 854 01:04:59,290 --> 01:05:01,950 have an opportunity to cool. 855 01:05:01,950 --> 01:05:05,080 So if you can distinguish spectroscopically 856 01:05:05,080 --> 01:05:07,770 an atom which moves a tiny little bit and an atom which 857 01:05:07,770 --> 01:05:10,780 stands still, if you can, by a narrow line, 858 01:05:10,780 --> 01:05:12,920 distinguish the two, then you can actually 859 01:05:12,920 --> 01:05:15,630 laser cool this atom. 860 01:05:15,630 --> 01:05:21,170 The EIT feature can give you extremely high resolution. 861 01:05:21,170 --> 01:05:23,090 I'm not discussing it here. 862 01:05:23,090 --> 01:05:25,600 I've discussed the phenomenon of EIT, which is 863 01:05:25,600 --> 01:05:27,590 coherent population trapping. 864 01:05:27,590 --> 01:05:34,810 But there is an extension, which is VSCPT, Velocity Selective 865 01:05:34,810 --> 01:05:39,880 Coherent Population Trapping, and VSCPT was a powerful method 866 01:05:39,880 --> 01:05:43,300 to cool atoms below the recoil limit, 867 01:05:43,300 --> 01:05:48,070 but I've not connected anything of coherent population trapping 868 01:05:48,070 --> 01:05:49,080 with the Doppler shift. 869 01:05:49,080 --> 01:05:52,300 So for all this discussion, please 870 01:05:52,300 --> 01:05:56,223 assume the atom is not moving. 871 01:05:56,223 --> 01:05:56,890 Other questions? 872 01:06:01,870 --> 01:06:10,810 Then let me just say a few words about the fact 873 01:06:10,810 --> 01:06:18,850 that we have a large derivative of the index of refraction, 874 01:06:18,850 --> 01:06:33,350 and this, of course, is used for generating 875 01:06:33,350 --> 01:06:34,670 what is called slow light. 876 01:06:37,410 --> 01:06:47,000 The group velocity of light is the speed of light, 877 01:06:47,000 --> 01:06:49,290 but then it has a denominator which 878 01:06:49,290 --> 01:06:52,002 is the derivative of the index of refraction 879 01:06:52,002 --> 01:06:53,085 with respect to frequency. 880 01:07:01,770 --> 01:07:05,240 Towards the end of the last century, 881 01:07:05,240 --> 01:07:10,110 there were predictions that electromagnetically induced 882 01:07:10,110 --> 01:07:18,370 transparency would give you very sharp features which can 883 01:07:18,370 --> 01:07:20,980 be used for very slow light. 884 01:07:25,610 --> 01:07:30,770 And what eventually triggered major developments in the field 885 01:07:30,770 --> 01:07:38,230 was this landmark paper by Lene Hau where 886 01:07:38,230 --> 01:07:41,410 she used the Bose-Einstein condensate to eliminate 887 01:07:41,410 --> 01:07:43,290 all kinds of Doppler broadening. 888 01:07:43,290 --> 01:07:45,290 There are other tricks how you can eliminate it, 889 01:07:45,290 --> 01:07:47,260 but this was the most powerful way 890 01:07:47,260 --> 01:07:49,850 to just take a Bose-Einstein condensate where 891 01:07:49,850 --> 01:08:00,880 atoms have no thermal velocity and, in this research, 892 01:08:00,880 --> 01:08:03,970 she was able to show that light propagated 893 01:08:03,970 --> 01:08:05,680 at the speed of a bicycle. 894 01:08:05,680 --> 01:08:09,020 So it was a dramatic reduction of the speed of light, 895 01:08:09,020 --> 01:08:11,740 and this showed the true potential of EIT. 896 01:08:11,740 --> 01:08:13,240 There have been other demonstrations 897 01:08:13,240 --> 01:08:15,940 before where light has been slowed by a factor of 100 898 01:08:15,940 --> 01:08:18,340 or a few hundred, but eventually, 899 01:08:18,340 --> 01:08:21,740 combining that with a Doppler-less feature 900 01:08:21,740 --> 01:08:25,010 because the atoms don't move in a BEC 901 01:08:25,010 --> 01:08:26,790 created a dramatic effect. 902 01:08:31,850 --> 01:08:38,430 So we have now two ways how we can get 903 01:08:38,430 --> 01:08:40,660 a large derivative, dn d omega. 904 01:08:40,660 --> 01:08:43,819 I've discussed here the general case 905 01:08:43,819 --> 01:08:47,800 that we have a narrow feature, a broad feature, because we 906 01:08:47,800 --> 01:08:55,750 have the coupling light detuned from the excited state, 907 01:08:55,750 --> 01:09:07,810 but let me just point out that even if you have the coupling 908 01:09:07,810 --> 01:09:15,680 light on resonance, depends what you really want, 909 01:09:15,680 --> 01:09:23,410 but you can get an even stronger feature in the index 910 01:09:23,410 --> 01:09:24,804 of refraction versus frequency. 911 01:09:31,270 --> 01:09:34,174 This is now the situation where we have the strong absorption 912 01:09:34,174 --> 01:09:40,670 feature, but then we have the EIT window. 913 01:09:40,670 --> 01:09:44,760 So we have this superposition of a positive Lorentzian 914 01:09:44,760 --> 01:09:48,109 and a negative Lorentzian, and if I now 915 01:09:48,109 --> 01:09:55,490 run it through my Kramers-Kronig calculator 916 01:09:55,490 --> 01:09:57,700 and I take the dispersive shape, I 917 01:09:57,700 --> 01:10:01,780 can sort of do it for the broad feature in this way 918 01:10:01,780 --> 01:10:05,460 and for the narrow feature in this way, 919 01:10:05,460 --> 01:10:07,760 and now you have to add up the two. 920 01:10:07,760 --> 01:10:12,270 And what you realize is at this point, 921 01:10:12,270 --> 01:10:21,390 you have a huge dn d omega. 922 01:10:21,390 --> 01:10:24,340 What I'm plotting here is on the left side, the absorption 923 01:10:24,340 --> 01:10:28,290 of the Lorentzian, and you can regard this sharp notch 924 01:10:28,290 --> 01:10:30,040 as a second Lorentzian. 925 01:10:30,040 --> 01:10:33,440 So you have the positive Lorentzian, negative 926 01:10:33,440 --> 01:10:36,190 Lorentzian, and then you take the dispersive features 927 01:10:36,190 --> 01:10:38,680 and you add them up with the correct sign. 928 01:10:38,680 --> 01:10:44,180 So whether you're realizing now for quite a general situation 929 01:10:44,180 --> 01:10:48,030 where you have single photon detuning, which I discussed 930 01:10:48,030 --> 01:10:51,160 before, or whether you're on the single photon resonance, 931 01:10:51,160 --> 01:10:55,550 you can have extremely sharp features. 932 01:11:01,310 --> 01:11:03,630 So now you can take it to the next level. 933 01:11:03,630 --> 01:11:06,970 You have a light pulse which enters a medium, 934 01:11:06,970 --> 01:11:12,780 and now the light pulse slowly moves through the medium. 935 01:11:12,780 --> 01:11:17,760 But while the light pulse moves through the medium, 936 01:11:17,760 --> 01:11:20,894 you reduce the strength of the coupling laser. 937 01:11:24,212 --> 01:11:25,160 What happens? 938 01:11:28,490 --> 01:11:32,380 So you do now an adiabatic change of your system. 939 01:11:32,380 --> 01:11:40,840 You do an adiabatic change of the control field 940 01:11:40,840 --> 01:11:55,020 omega 2 while the probe pulse is in the medium. 941 01:11:55,020 --> 01:12:04,430 Well, that means that under idealized assumptions which 942 01:12:04,430 --> 01:12:06,810 we've discussed, this feature gets narrower 943 01:12:06,810 --> 01:12:08,800 and narrower and narrower. 944 01:12:08,800 --> 01:12:13,850 If omega 2 goes to 0, the strength of the control field, 945 01:12:13,850 --> 01:12:17,250 this feature becomes infinitesimally narrow. 946 01:12:17,250 --> 01:12:21,480 And therefore, this feature becomes infinitely sharp, 947 01:12:21,480 --> 01:12:27,440 and that means that the group velocity goes to zero. 948 01:12:31,210 --> 01:12:35,950 This is now in the popular press, 949 01:12:35,950 --> 01:12:44,510 it's called stopped light or frozen light 950 01:12:44,510 --> 01:12:46,960 because the light has come to a standstill. 951 01:12:50,540 --> 01:12:54,450 What really happens is the following. 952 01:12:54,450 --> 01:13:06,480 We have our coupling laser, omega 2, 953 01:13:06,480 --> 01:13:10,600 and we have our probe laser, omega 1. 954 01:13:10,600 --> 01:13:19,370 When we do what I just said is that omega 2 goes through zero, 955 01:13:19,370 --> 01:13:26,840 then the dark state originally, for very strong omega 2, 956 01:13:26,840 --> 01:13:33,590 remember the dark state was g? 957 01:13:33,590 --> 01:13:35,800 But now if we let omega 2 go to zero, 958 01:13:35,800 --> 01:13:37,280 the dark state will become f. 959 01:13:43,420 --> 01:13:50,990 And that means that in a way, every photon in the probe pulse 960 01:13:50,990 --> 01:13:55,820 has now pumped an atom from g through two photon Raman 961 01:13:55,820 --> 01:13:57,940 process into f. 962 01:13:57,940 --> 01:14:00,650 So therefore, what it means to stop light 963 01:14:00,650 --> 01:14:05,710 or to freeze light means simply that the photons of the laser 964 01:14:05,710 --> 01:14:11,860 have turned into an atomic excitation where the excitation 965 01:14:11,860 --> 01:14:14,270 is now the state, f. 966 01:14:14,270 --> 01:14:17,560 In other words, you have written the photon. 967 01:14:17,560 --> 01:14:20,570 The photon has now put the atom into a different hyperfine 968 01:14:20,570 --> 01:14:21,070 state. 969 01:14:23,840 --> 01:14:26,750 So if this is done adiabatically, 970 01:14:26,750 --> 01:14:30,400 and I can't do full justice in this course, 971 01:14:30,400 --> 01:14:35,420 but this means that the light is coherently 972 01:14:35,420 --> 01:14:52,760 converted into the atomic-- 973 01:14:55,600 --> 01:14:58,580 when I say "coherently" say "into population," I 974 01:14:58,580 --> 01:15:01,940 mean all the quantum phases, everything which 975 01:15:01,940 --> 01:15:06,470 was in the quantum nature of the light has now been converted, 976 01:15:06,470 --> 01:15:08,450 has been written into the state, f. 977 01:15:12,290 --> 01:15:17,720 This is often called, because g and f are hyperfine states, 978 01:15:17,720 --> 01:15:20,880 this means that you have coherently 979 01:15:20,880 --> 01:15:24,870 converted the photon or the electromagnetic wave 980 01:15:24,870 --> 01:15:30,030 in the probe beam into a spin wave or a magnon. 981 01:15:37,480 --> 01:15:40,910 Anyway, I just want to show you the analogy. 982 01:15:40,910 --> 01:15:45,440 The fact that you can put the quantum information of light 983 01:15:45,440 --> 01:15:47,930 into an atomic state and back and forth, 984 01:15:47,930 --> 01:15:52,140 we've discussed that when we had the situation of cavity QED. 985 01:15:52,140 --> 01:15:55,840 We prepared a superposition of ground and excited state 986 01:15:55,840 --> 01:15:59,640 and exactly the same quantum state which we had in the atom 987 01:15:59,640 --> 01:16:02,950 we later found in the cavity as a superposition 988 01:16:02,950 --> 01:16:06,250 of the zero photon and the one photon state. 989 01:16:06,250 --> 01:16:10,390 So from those general concepts, it should be clear to you 990 01:16:10,390 --> 01:16:13,130 that it is possible to coherently 991 01:16:13,130 --> 01:16:17,330 transform a quantum state from light to atoms 992 01:16:17,330 --> 01:16:19,950 and back to light. 993 01:16:19,950 --> 01:16:22,300 And here you see the different realization. 994 01:16:22,300 --> 01:16:28,530 We have a quantum state of the photon in the probe laser, 995 01:16:28,530 --> 01:16:31,780 and we can now describe the excitations 996 01:16:31,780 --> 01:16:37,340 in the system in a parametrized way. 997 01:16:40,170 --> 01:16:43,650 What it means is for the strong probe laser, 998 01:16:43,650 --> 01:16:48,120 for the strong coupling laser, the excitation in the system 999 01:16:48,120 --> 01:16:51,550 travels as a photon in the field one, 1000 01:16:51,550 --> 01:16:56,810 but when you reduce the coupling in the coupling laser, 1001 01:16:56,810 --> 01:17:01,650 the excitation becomes less and less photon-like. 1002 01:17:01,650 --> 01:17:06,520 It becomes more and more magnon-like, spin wave like. 1003 01:17:06,520 --> 01:17:09,230 And the moment you reduce the power in the coupling laser 1004 01:17:09,230 --> 01:17:12,370 to zero, what used to be an excitation 1005 01:17:12,370 --> 01:17:16,330 in the electromagnetic field has now been termed adiabatically 1006 01:17:16,330 --> 01:17:20,015 into a spin excitation. 1007 01:17:20,015 --> 01:17:23,660 Coherence has been written into the hyperfine states 1008 01:17:23,660 --> 01:17:25,629 or your atoms, g and f. 1009 01:17:31,620 --> 01:17:34,900 All this is done coherently, and therefore reversibly. 1010 01:17:39,630 --> 01:17:50,500 You can read out the information by simply time reversing 1011 01:17:50,500 --> 01:17:52,480 the process. 1012 01:17:52,480 --> 01:17:55,570 You ramp up again the coupling laser, 1013 01:17:55,570 --> 01:18:00,170 and that adiabatically turns the spin excitation back 1014 01:18:00,170 --> 01:18:02,580 into an excitation of the electromagnetic field. 1015 01:18:10,750 --> 01:18:11,520 Any questions? 1016 01:18:20,830 --> 01:18:22,600 Well, we have three minutes left, 1017 01:18:22,600 --> 01:18:25,770 but I'm not getting started with superradiance. 1018 01:18:25,770 --> 01:18:28,070 We have one more topic left, and this 1019 01:18:28,070 --> 01:18:31,610 will be the topic on Wednesday, decay superradiance, 1020 01:18:31,610 --> 01:18:36,620 and this is when we discuss the phenomenon of coherence where 1021 01:18:36,620 --> 01:18:42,100 we have coherence not only in one atom between two or three 1022 01:18:42,100 --> 01:18:42,890 levels. 1023 01:18:42,890 --> 01:18:45,040 We will then discuss on Wednesday 1024 01:18:45,040 --> 01:18:48,280 if we have coherence between many atoms, 1025 01:18:48,280 --> 01:18:51,480 and this at the heart of superradiance. 1026 01:18:51,480 --> 01:18:51,980 OK. 1027 01:18:51,980 --> 01:18:54,130 See you on Wednesday.