1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,238 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,238 --> 00:00:17,863 at ocw.mit.edu. 8 00:00:21,130 --> 00:00:24,470 PROFESSOR: Good afternoon, everybody. 9 00:00:24,470 --> 00:00:26,590 Let me just remind you where we are. 10 00:00:26,590 --> 00:00:29,120 We're describing interaction between 11 00:00:29,120 --> 00:00:31,820 the electromagnetic field and atoms. 12 00:00:31,820 --> 00:00:36,620 And we had formulated an exact approach 13 00:00:36,620 --> 00:00:39,730 using the time evolution operators and diagrams. 14 00:00:39,730 --> 00:00:42,200 And well, I think we understood what 15 00:00:42,200 --> 00:00:45,830 it means when atoms in the ground state emit 16 00:00:45,830 --> 00:00:48,730 photons which are virtually absorbed and all that. 17 00:00:48,730 --> 00:00:52,960 So we figured out what is really inside this formalism 18 00:00:52,960 --> 00:00:55,450 and what are the processes. 19 00:00:55,450 --> 00:00:57,710 What we want to continue discussing today 20 00:00:57,710 --> 00:01:02,230 is one problem which you often have such approaches. 21 00:01:02,230 --> 00:01:05,010 And this is the problem of resonance. 22 00:01:05,010 --> 00:01:07,160 If you have a perturbative treatment, 23 00:01:07,160 --> 00:01:09,600 even if you carry to infinite order, 24 00:01:09,600 --> 00:01:12,860 you have, formally, divergences if you 25 00:01:12,860 --> 00:01:15,450 have resonant interaction. 26 00:01:15,450 --> 00:01:18,000 Because the ground state and the photon 27 00:01:18,000 --> 00:01:21,170 has exactly the same energy as the excited state. 28 00:01:21,170 --> 00:01:25,660 And that means if you write down the perturbative expansion, 29 00:01:25,660 --> 00:01:27,650 you have a 0 in the denominator. 30 00:01:27,650 --> 00:01:29,730 You have a divergence. 31 00:01:29,730 --> 00:01:36,430 And I reminded you that in a phenomenological way, 32 00:01:36,430 --> 00:01:40,570 you've seen that this problem can be "fixed" 33 00:01:40,570 --> 00:01:43,840 by adding an imaginary part to the energy level 34 00:01:43,840 --> 00:01:46,340 just saying, well, the excited state 35 00:01:46,340 --> 00:01:50,330 couples by spontaneous emission to the radiation field. 36 00:01:50,330 --> 00:01:53,330 And the level has [INAUDIBLE]. 37 00:01:53,330 --> 00:01:56,490 Well, but you know, sometimes putting an imaginary part 38 00:01:56,490 --> 00:01:58,072 into Schrodinger's equation means 39 00:01:58,072 --> 00:01:59,655 it's no longer unitary time evolution. 40 00:01:59,655 --> 00:02:01,350 It has its problem. 41 00:02:01,350 --> 00:02:06,950 But anyway, we want to now look deeper into it. 42 00:02:06,950 --> 00:02:09,030 I want to show you what are the tools 43 00:02:09,030 --> 00:02:12,840 to treat those infinity source divergences 44 00:02:12,840 --> 00:02:16,450 in a consistent and a systematic way. 45 00:02:16,450 --> 00:02:22,580 And one hint how we have to do it comes by simply taking 46 00:02:22,580 --> 00:02:27,200 this energy denominator and expanding it in gamma, simply 47 00:02:27,200 --> 00:02:29,420 a Taylor expansion in gamma. 48 00:02:29,420 --> 00:02:32,990 And then, we realize gamma is often 49 00:02:32,990 --> 00:02:36,100 calculated in second order Fermi's golden rule. 50 00:02:36,100 --> 00:02:39,000 But since you have here all orders n, 51 00:02:39,000 --> 00:02:42,400 that tells us that doing something here 52 00:02:42,400 --> 00:02:46,292 probably means infinite orders in a perturbation series. 53 00:02:46,292 --> 00:02:48,000 And that's what I want to show you today. 54 00:02:48,000 --> 00:02:52,710 I want to show you that I can go beyond this result. 55 00:02:52,710 --> 00:02:56,330 But I can reproduce this result by going 56 00:02:56,330 --> 00:02:58,920 to infinite order in perturbation theory. 57 00:02:58,920 --> 00:03:03,750 And it means to sum up an infinite number of diagrams. 58 00:03:03,750 --> 00:03:05,930 Who of you have actually seen those kind 59 00:03:05,930 --> 00:03:09,080 of diagrammatic tricks and summation? 60 00:03:09,080 --> 00:03:11,660 A few, OK. 61 00:03:11,660 --> 00:03:13,220 So it's maybe nice to see it again. 62 00:03:13,220 --> 00:03:16,110 But for those who haven't seen it, 63 00:03:16,110 --> 00:03:18,030 welcome to the magic of diagrams. 64 00:03:18,030 --> 00:03:20,200 I learned it from those examples. 65 00:03:20,200 --> 00:03:21,115 And I really like it. 66 00:03:21,115 --> 00:03:26,050 It's a very elegant way to combine equations 67 00:03:26,050 --> 00:03:28,470 with graphical manipulations. 68 00:03:28,470 --> 00:03:34,510 So that's our agenda for at least the first part of today. 69 00:03:34,510 --> 00:03:41,320 And OK, we want to understand the time 70 00:03:41,320 --> 00:03:42,770 evolution of this system. 71 00:03:42,770 --> 00:03:46,060 Our tool is a time evolution operator. 72 00:03:46,060 --> 00:03:47,880 And at the end of the class on Monday, 73 00:03:47,880 --> 00:03:51,600 I told you, well, let's simplify things. 74 00:03:51,600 --> 00:03:55,900 Let's get rid of those temporal integrations 75 00:03:55,900 --> 00:03:59,760 and multiple integrals by simply doing a Fourier transform. 76 00:03:59,760 --> 00:04:02,820 Because a Fourier transform turns an integral 77 00:04:02,820 --> 00:04:06,000 or convolution into a product. 78 00:04:06,000 --> 00:04:10,020 And so therefore, we introduced the Fourier transform, 79 00:04:10,020 --> 00:04:14,620 or the Laplace transform, of the time evolution operator. 80 00:04:14,620 --> 00:04:21,250 And this iterative equation where we get the nth order 81 00:04:21,250 --> 00:04:23,510 by plugging the n minus first order 82 00:04:23,510 --> 00:04:28,250 on the right hand side, this iterative equation 83 00:04:28,250 --> 00:04:31,950 turns now into a simpler algebraic iterative 84 00:04:31,950 --> 00:04:34,980 equation for the Fourier transform. 85 00:04:34,980 --> 00:04:38,040 So this is now the starting point for our discussion today. 86 00:04:38,040 --> 00:04:41,880 We want to calculate the Fourier transform of the time evolution 87 00:04:41,880 --> 00:04:43,350 operator to infinite orders. 88 00:04:46,850 --> 00:04:55,760 So let me-- well, of course, any questions before I continue? 89 00:04:59,670 --> 00:05:05,410 So unfortunately, [INAUDIBLE]. 90 00:05:05,410 --> 00:05:08,690 We should copy this equation. 91 00:05:08,690 --> 00:05:09,730 Because we need it. 92 00:05:15,380 --> 00:05:17,645 So OK, we want to take this equation. 93 00:05:25,130 --> 00:05:27,775 And now we want to iterate it. 94 00:05:36,820 --> 00:05:43,730 So the resolving G in 0's order is G0. 95 00:05:43,730 --> 00:05:49,420 Now plug G0 into the right hand side of the equation, 96 00:05:49,420 --> 00:05:52,030 and you get the first order, which is G0VG0. 97 00:05:55,300 --> 00:05:58,130 Now plug that into the end of the iterative equation, 98 00:05:58,130 --> 00:06:00,720 and you get the second order, G0VG0VG0. 99 00:06:05,250 --> 00:06:07,160 I think you've got the idea. 100 00:06:07,160 --> 00:06:14,300 It's almost like a geometric series-- 101 00:06:14,300 --> 00:06:17,220 well, a geometric series with operators. 102 00:06:17,220 --> 00:06:18,933 But this is already sort of a hint. 103 00:06:18,933 --> 00:06:23,890 A geometric series can be summed up to infinity rather easy. 104 00:06:27,720 --> 00:06:36,396 So we have to introduce now the eigenfunctions 105 00:06:36,396 --> 00:06:39,660 of the unperturbed operator. 106 00:06:39,660 --> 00:06:43,810 So these are, let's say, the ground state, excited state, 107 00:06:43,810 --> 00:06:46,770 next excited state of your favorite atom. 108 00:06:46,770 --> 00:06:57,010 And they have eigenenergies Ek-- nope. 109 00:07:09,550 --> 00:07:17,640 So if you are now expressing the equation above in the basis 110 00:07:17,640 --> 00:07:20,230 kl-- so in other words, this is an operator, 111 00:07:20,230 --> 00:07:22,090 and we want to know the matrix element 112 00:07:22,090 --> 00:07:24,760 between eigenfunctions k and l. 113 00:07:29,720 --> 00:07:33,750 The other thing we need is that G0. 114 00:07:33,750 --> 00:07:39,120 Remember the Fourier transform-- and I 115 00:07:39,120 --> 00:07:42,990 gave you sort a mini derivation here-- is just 1 116 00:07:42,990 --> 00:07:54,900 over energy minus H. And if you apply that now to the operator 117 00:07:54,900 --> 00:08:02,480 G0, so G0 is nothing like-- is given by that. 118 00:08:02,480 --> 00:08:06,055 So therefore, if you write now this equation 119 00:08:06,055 --> 00:08:11,940 as matrix elements, the first part, the G0, 120 00:08:11,940 --> 00:08:17,490 gives us 1 minus 1 over Z minus Ek. 121 00:08:17,490 --> 00:08:20,470 And since we are diagonal in H0, it's delta kl. 122 00:08:23,840 --> 00:08:27,700 And well, now you see the structure 123 00:08:27,700 --> 00:08:37,250 if you sum over intermediate states, 124 00:08:37,250 --> 00:08:39,264 or if you introduce intermediate states. 125 00:08:42,650 --> 00:08:46,700 So this is how we have to write it. 126 00:08:46,700 --> 00:08:50,635 So this is just writing it down in basis function 127 00:08:50,635 --> 00:08:53,770 of the unperturbed operator H0. 128 00:08:53,770 --> 00:09:01,360 But now we can formulate the problem we are encountering 129 00:09:01,360 --> 00:09:05,010 and we want to solve. 130 00:09:05,010 --> 00:09:10,030 Namely, we have the problem with one state. 131 00:09:10,030 --> 00:09:14,340 Our problem is the excited state b. 132 00:09:14,340 --> 00:09:15,980 And we have a resonant excitation 133 00:09:15,980 --> 00:09:17,930 from the count state a. 134 00:09:20,460 --> 00:09:32,640 So this is a discrete eigenstate of the unperturbed Hamiltonian 135 00:09:32,640 --> 00:09:39,080 with energy Eb. 136 00:09:39,080 --> 00:09:55,780 And therefore, we have terms, and actually divergent terms, 137 00:09:55,780 --> 00:10:02,840 which are 1 over Z minus Eb. 138 00:10:08,800 --> 00:10:11,380 And just to make the connection, this 139 00:10:11,380 --> 00:10:15,170 is sort of the description that book Z is in the complex plane 140 00:10:15,170 --> 00:10:17,080 and can have imaginary values. 141 00:10:17,080 --> 00:10:20,410 Sometimes, scattering and evolution equations 142 00:10:20,410 --> 00:10:24,960 are better formulated when you do it in the complex plane. 143 00:10:24,960 --> 00:10:26,740 It doesn't really matter for us here. 144 00:10:26,740 --> 00:10:30,000 Just remember, Z is the energy. 145 00:10:30,000 --> 00:10:32,030 And it is the initial energy. 146 00:10:32,030 --> 00:10:34,340 And the initial energy is if it's 147 00:10:34,340 --> 00:10:37,240 a ground state and a resonant photon, we have a problem. 148 00:10:37,240 --> 00:10:41,260 Because the denominator is 0. 149 00:10:41,260 --> 00:10:47,300 So in other words, for resonant excitation, 150 00:10:47,300 --> 00:10:53,260 we are interested in the case that Z 151 00:10:53,260 --> 00:10:56,500 is on the order of this close to Eb. 152 00:11:04,050 --> 00:11:09,750 OK, so what we want to do now is-- 153 00:11:09,750 --> 00:11:12,160 and this is the basic idea. 154 00:11:12,160 --> 00:11:18,420 By looking at this formally, exact calculation, 155 00:11:18,420 --> 00:11:23,410 we say the difficult parts are those where we have this energy 156 00:11:23,410 --> 00:11:24,970 denominator, which vanishes. 157 00:11:24,970 --> 00:11:26,940 The other parts are simple. 158 00:11:26,940 --> 00:11:28,610 They don't have any divergences. 159 00:11:28,610 --> 00:11:30,820 They are not resonant and such. 160 00:11:30,820 --> 00:11:33,710 So what we want to do is now, in some way, 161 00:11:33,710 --> 00:11:36,570 we want to sort of give special treatment, 162 00:11:36,570 --> 00:11:39,570 factor out the problematic terms. 163 00:11:39,570 --> 00:11:41,610 And the rest is easy. 164 00:11:41,610 --> 00:11:44,360 And for the easy part, which has no divergence, 165 00:11:44,360 --> 00:11:46,320 we can make any kind of approximation 166 00:11:46,320 --> 00:11:50,250 we want without altering the physics. 167 00:11:50,250 --> 00:11:53,490 But the resonant part, this needs special attention. 168 00:11:53,490 --> 00:11:56,239 Because if I treat it literally in those expressions, 169 00:11:56,239 --> 00:11:57,780 they don't make sense mathematically. 170 00:11:57,780 --> 00:11:59,615 Because they cause infinities. 171 00:12:04,910 --> 00:12:08,660 I could continue to do it completely algebraically. 172 00:12:08,660 --> 00:12:13,380 But because I think it's just beautiful method, 173 00:12:13,380 --> 00:12:21,390 I want to look at this equation and write it down in symbols. 174 00:12:21,390 --> 00:12:35,290 So we want to arrive at a diagrammatic representation 175 00:12:35,290 --> 00:12:41,640 for this matrix element of the resolvent Gbb. 176 00:12:41,640 --> 00:12:51,230 And I use symbols where the circle means the interaction. 177 00:12:51,230 --> 00:12:56,440 The straight line stands for the term which is problematic, 178 00:12:56,440 --> 00:13:00,720 which has resonant-- the resonant interaction has 179 00:13:00,720 --> 00:13:03,410 the divergent denominator. 180 00:13:03,410 --> 00:13:13,600 And then, I'll use a dashed line for all other terms, 181 00:13:13,600 --> 00:13:18,270 all other intermediate states, where i is not b, 182 00:13:18,270 --> 00:13:20,480 and therefore there is not a problem. 183 00:13:20,480 --> 00:13:21,270 It's not resonant. 184 00:13:21,270 --> 00:13:22,395 We don't have a divergence. 185 00:13:25,830 --> 00:13:31,350 So I'm just sort of-- you see the structure of the sum. 186 00:13:31,350 --> 00:13:38,655 We sort of propagate here. 187 00:13:38,655 --> 00:13:41,420 We make a transition from l to k. 188 00:13:41,420 --> 00:13:42,990 And then, we propagate here. 189 00:13:42,990 --> 00:13:46,160 So in this kind of order, you start in this state, 190 00:13:46,160 --> 00:13:49,127 you have a vertex, you go to the other state, you have a vertex, 191 00:13:49,127 --> 00:13:50,960 you go to the next state, you have a vertex. 192 00:13:50,960 --> 00:13:52,170 This is how the system works. 193 00:13:52,170 --> 00:13:56,280 This is sort of what quantum mechanics does for you. 194 00:13:56,280 --> 00:14:01,420 And what we're going to do is, this is an algebraic equation. 195 00:14:01,420 --> 00:14:05,000 And now we want to order them in the following way. 196 00:14:05,000 --> 00:14:10,830 We want to figure out which of those expressions 197 00:14:10,830 --> 00:14:16,710 include this problematic term exactly twice or three times 198 00:14:16,710 --> 00:14:18,010 or four times. 199 00:14:18,010 --> 00:14:23,170 So we regroup those infinite sums, these algebraic terms, 200 00:14:23,170 --> 00:14:25,550 in such a way that we say, OK, which 201 00:14:25,550 --> 00:14:28,970 one has the occurrence of this once, twice, three times, 202 00:14:28,970 --> 00:14:29,810 four times? 203 00:14:29,810 --> 00:14:34,350 So we regroup the terms, and then we see what we can do. 204 00:14:34,350 --> 00:14:37,460 Now, I've picked the matrix element Gbb. 205 00:14:37,460 --> 00:14:41,020 So that means over here and over here 206 00:14:41,020 --> 00:14:44,130 we start out in the state b. 207 00:14:50,670 --> 00:14:59,490 So if I write down all terms which 208 00:14:59,490 --> 00:15:10,070 contain this resonant term twice, 209 00:15:10,070 --> 00:15:13,040 well, we start with a resonant term. 210 00:15:13,040 --> 00:15:15,320 And we have to end with a resonant term, 211 00:15:15,320 --> 00:15:16,910 because we have the matrix element 212 00:15:16,910 --> 00:15:19,790 Gbb, which I'm focusing now on. 213 00:15:19,790 --> 00:15:22,625 And so there can be one vertex. 214 00:15:25,210 --> 00:15:27,320 We can start with this term. 215 00:15:29,840 --> 00:15:33,110 And now we can do something else in between. 216 00:15:33,110 --> 00:15:36,290 But we are not allowed to go back to the state b. 217 00:15:36,290 --> 00:15:38,330 Because we are only looking at terms 218 00:15:38,330 --> 00:15:41,720 where the solid line appears twice. 219 00:15:41,720 --> 00:15:44,930 So therefore, this can only be a dashed line, another state. 220 00:15:47,950 --> 00:15:51,720 We can go through two vertices. 221 00:15:51,720 --> 00:15:57,150 But it can only include dashed lines in between. 222 00:16:01,620 --> 00:16:03,980 So let me use another symbol for it. 223 00:16:03,980 --> 00:16:08,120 So we start with that state. 224 00:16:08,120 --> 00:16:10,260 We end with a state b. 225 00:16:10,260 --> 00:16:14,830 But in between, we can sort of once, twice, three times 226 00:16:14,830 --> 00:16:16,750 go through other intermediate states. 227 00:16:16,750 --> 00:16:19,950 But we never are allowed to create another divergence. 228 00:16:19,950 --> 00:16:22,470 And this infinite sum over all other 229 00:16:22,470 --> 00:16:25,660 states-- I don't know how to calculate it yet. 230 00:16:25,660 --> 00:16:28,580 But I'll just call it a square box. 231 00:16:28,580 --> 00:16:31,510 So in other words, what is sort of in lowest order, 232 00:16:31,510 --> 00:16:35,560 the operator V, with these other infinity terms, 233 00:16:35,560 --> 00:16:40,210 I symbolize that kind of much more complicated vertex, which 234 00:16:40,210 --> 00:16:47,650 includes an infinite sum, by a square and not by a circle. 235 00:16:47,650 --> 00:16:51,360 So diagrammatically-- feel free to write it down 236 00:16:51,360 --> 00:16:52,170 mathematically. 237 00:16:52,170 --> 00:16:53,890 It's obvious how to do it. 238 00:16:53,890 --> 00:17:06,490 The square box is the circle plus all terms like this. 239 00:17:12,829 --> 00:17:16,359 Let's go back to equations. 240 00:17:16,359 --> 00:17:20,480 We call this the function Rb of Z. 241 00:17:20,480 --> 00:17:25,310 And what I've shown in diagrams above 242 00:17:25,310 --> 00:17:30,510 is nothing else than the circle Vbb, the matrix element 243 00:17:30,510 --> 00:17:34,010 of the interaction. 244 00:17:34,010 --> 00:17:37,980 But now we have sums where we're not 245 00:17:37,980 --> 00:17:40,830 allowed to go through the resonant state. 246 00:17:40,830 --> 00:17:44,445 We have to go from b to an intermediate state. 247 00:17:49,100 --> 00:17:52,590 We propagate in the intermediate state. 248 00:17:52,590 --> 00:17:55,860 And then, we have to go back. 249 00:17:55,860 --> 00:18:01,820 And it's clear how to go to higher terms. 250 00:18:01,820 --> 00:18:04,720 So we've just defined this function R 251 00:18:04,720 --> 00:18:10,554 by focusing on two occurrences of the straight line. 252 00:18:10,554 --> 00:18:15,060 Well, let's look at higher order terms. 253 00:18:15,060 --> 00:18:26,750 What happens when Z minus Eb, the divergent term, 254 00:18:26,750 --> 00:18:30,140 comes to the power n? 255 00:18:30,140 --> 00:18:33,160 Well, we just dealt with n equals 2. 256 00:18:33,160 --> 00:18:35,365 Let's now look at n equals 3. 257 00:18:38,260 --> 00:18:39,032 Colin. 258 00:18:39,032 --> 00:18:41,000 AUDIENCE: I'm trying to find the definition. 259 00:18:41,000 --> 00:18:45,024 Is the box a T-matrix or the S-matrix? 260 00:18:45,024 --> 00:18:48,300 What's the definition of the [INAUDIBLE]. 261 00:18:48,300 --> 00:18:49,430 The S is-- 262 00:18:49,430 --> 00:19:01,920 PROFESSOR: No, the T-matrix is actually 263 00:19:01,920 --> 00:19:06,150 the matrix, the relevant matrix, of the time evolution operator. 264 00:19:06,150 --> 00:19:09,210 And if you factor out the delta function for the energy shell, 265 00:19:09,210 --> 00:19:10,880 you get the S-matrix. 266 00:19:10,880 --> 00:19:13,580 But we're not talking about a time evolution operator here. 267 00:19:13,580 --> 00:19:15,620 We're talking about the Fourier transform. 268 00:19:15,620 --> 00:19:17,620 And we're looking at the resolvent, which 269 00:19:17,620 --> 00:19:22,960 is the function G. And now, we've introduced a function R. 270 00:19:22,960 --> 00:19:25,080 I actually tried to look up-- I wanted 271 00:19:25,080 --> 00:19:26,840 to use the correct word in class. 272 00:19:26,840 --> 00:19:28,110 I couldn't find the name. 273 00:19:28,110 --> 00:19:30,270 In the book, it's just called the function R. 274 00:19:30,270 --> 00:19:33,660 So it is the function R. And the function R turns out 275 00:19:33,660 --> 00:19:38,350 to be the kernel of the resolvent G. But none of it 276 00:19:38,350 --> 00:19:39,720 is the S-matrix and T-matrix. 277 00:19:39,720 --> 00:19:40,550 It is related. 278 00:19:40,550 --> 00:19:43,310 Because if you do the inverse Fourier transform from G, 279 00:19:43,310 --> 00:19:46,217 we go back to U. And then, we have, 280 00:19:46,217 --> 00:19:47,300 I would say, the T-matrix. 281 00:19:49,880 --> 00:19:53,560 AUDIENCE: This is the self energy? 282 00:19:53,560 --> 00:19:57,730 PROFESSOR: We will find that R has a real and imaginary part. 283 00:19:57,730 --> 00:19:59,800 One is a self energy, and the other one 284 00:19:59,800 --> 00:20:02,110 is a decay rate, has an imaginary part. 285 00:20:02,110 --> 00:20:04,830 But the real part will be the self energy. 286 00:20:04,830 --> 00:20:07,350 Yes, we connect a lot of passwords 287 00:20:07,350 --> 00:20:09,320 you may have heard here and there. 288 00:20:09,320 --> 00:20:13,480 OK, the question is, do we have to now define 289 00:20:13,480 --> 00:20:14,575 pentagons and hexagons? 290 00:20:14,575 --> 00:20:18,500 Do we have to find more and more symbols 291 00:20:18,500 --> 00:20:20,530 for more and more complicated sums? 292 00:20:20,530 --> 00:20:22,820 But the nice thing is no. 293 00:20:22,820 --> 00:20:29,770 Because n equals 3 means we have to start in state b. 294 00:20:29,770 --> 00:20:31,890 We have to stand in state b. 295 00:20:31,890 --> 00:20:34,490 And one time in the time evolution, 296 00:20:34,490 --> 00:20:36,710 we can go through state b. 297 00:20:36,710 --> 00:20:40,450 And now, between that, we can go from here to there 298 00:20:40,450 --> 00:20:43,580 with any combination of states you want. 299 00:20:43,580 --> 00:20:46,730 But one thing is not allowed-- to involve the state b. 300 00:20:46,730 --> 00:20:50,650 Because we are focusing on three occurrences of the state b. 301 00:20:50,650 --> 00:20:54,000 And everything else other than the state b 302 00:20:54,000 --> 00:20:55,350 has already a symbol. 303 00:20:55,350 --> 00:20:58,020 It is the square symbol. 304 00:20:58,020 --> 00:21:05,010 So this is the exact representation for n equals 3. 305 00:21:05,010 --> 00:21:13,710 And the contribution to the resolvent 306 00:21:13,710 --> 00:21:16,870 G, the Fourier transform of the time evolution operator, 307 00:21:16,870 --> 00:21:22,890 is-- well, we have factored out three occurrences of the state 308 00:21:22,890 --> 00:21:23,390 b. 309 00:21:26,030 --> 00:21:30,500 And then, we have three occurrences of the state b. 310 00:21:30,500 --> 00:21:32,450 We need two square boxes. 311 00:21:32,450 --> 00:21:35,010 But the square box has already a name 312 00:21:35,010 --> 00:21:38,090 and an algebraic definition. 313 00:21:38,090 --> 00:21:43,580 So this is nothing else than-- I think 314 00:21:43,580 --> 00:21:48,120 it's called the kernel Rb squared. 315 00:21:51,320 --> 00:21:53,290 OK, I've shown you n equals 2. 316 00:21:53,290 --> 00:21:55,430 I've shown you n equals 3. 317 00:21:55,430 --> 00:22:00,070 I assume it's now absolutely clear how to continue. 318 00:22:00,070 --> 00:22:03,980 It just involves more and more power. 319 00:22:03,980 --> 00:22:09,310 So the lowest order is that. 320 00:22:09,310 --> 00:22:15,570 And whenever we ask what happens when 321 00:22:15,570 --> 00:22:21,690 we allow more appearances of the state b, for each of them, 322 00:22:21,690 --> 00:22:24,210 we obtain another square box. 323 00:22:30,790 --> 00:22:35,310 So by looking at the terms which are bothersome 324 00:22:35,310 --> 00:22:39,420 and regrouping the infinite terms 325 00:22:39,420 --> 00:22:44,370 according to one occurrence, two occurrence, three 326 00:22:44,370 --> 00:22:46,920 occurrence of this divergent denominator, 327 00:22:46,920 --> 00:22:53,810 we have now found an exact expression for Gb of Z. 328 00:22:53,810 --> 00:22:56,830 And since this is now an algebraic equation 329 00:22:56,830 --> 00:23:01,800 with a geometric series, we can write it exactly 330 00:23:01,800 --> 00:23:18,420 as this minus Rb of Z. 331 00:23:18,420 --> 00:23:22,220 Well, like with every exact result, 332 00:23:22,220 --> 00:23:24,140 you have to ask, what is the use for it? 333 00:23:24,140 --> 00:23:28,640 Because we started out with U, which 334 00:23:28,640 --> 00:23:31,180 we couldn't calculate with Fourier transform. 335 00:23:31,180 --> 00:23:33,400 We had G, which we couldn't calculate. 336 00:23:33,400 --> 00:23:36,900 And now we've expressed G in R, which of course we cannot 337 00:23:36,900 --> 00:23:38,950 calculate exactly. 338 00:23:38,950 --> 00:23:41,860 But there is an importance. 339 00:23:41,860 --> 00:23:45,935 We have made progress for the following reasons. 340 00:23:55,850 --> 00:24:06,570 Namely, the first is that those resonant terms, which 341 00:24:06,570 --> 00:24:10,120 appears in the time evolution whenever the system goes back 342 00:24:10,120 --> 00:24:14,650 to the state b, is now fully accounted for. 343 00:24:17,750 --> 00:24:19,640 We have sort of factored those terms out. 344 00:24:19,640 --> 00:24:22,330 We've given them special treatment. 345 00:24:22,330 --> 00:24:26,480 And therefore, and this is the main result, the exposition 346 00:24:26,480 --> 00:24:29,210 which now is the non-trivial expression, the function 347 00:24:29,210 --> 00:24:32,160 of the kernel, has no divergences. 348 00:24:42,640 --> 00:24:51,700 And therefore, because there is no critical part to it, 349 00:24:51,700 --> 00:25:07,880 rather simple approximations can be made 350 00:25:07,880 --> 00:25:12,400 and lead to physically meaningful results. 351 00:25:12,400 --> 00:25:14,530 That's an idea you may see often in physics. 352 00:25:14,530 --> 00:25:17,040 You have a theory whether it's something complicated, 353 00:25:17,040 --> 00:25:19,710 non-perturbative divergence. 354 00:25:19,710 --> 00:25:23,130 But you just rewrite the theory, transform the equations 355 00:25:23,130 --> 00:25:27,740 in such a way that structure of the equations 356 00:25:27,740 --> 00:25:30,400 now accounts for the physics behind it. 357 00:25:30,400 --> 00:25:33,630 And the part which has to be calculated now 358 00:25:33,630 --> 00:25:36,390 can be calculated with crude approximation. 359 00:25:36,390 --> 00:25:38,950 And you still get the correct physics out of it. 360 00:25:38,950 --> 00:25:41,320 The structure of the equation accounts for the physics. 361 00:25:41,320 --> 00:25:44,420 And the numerical part has become very harmless. 362 00:25:47,700 --> 00:25:51,050 But what it does is even if you do now a lowest order 363 00:25:51,050 --> 00:25:54,830 approximation to the function R, even 364 00:25:54,830 --> 00:26:03,360 those very simple approximations correspond 365 00:26:03,360 --> 00:26:11,666 to an infinite number of terms in the original expansion. 366 00:26:24,050 --> 00:26:28,160 Or in other words, the message is, you have an expression. 367 00:26:28,160 --> 00:26:31,460 And if you do a perturbative expansion-- well, 368 00:26:31,460 --> 00:26:34,230 maybe you should do some form of perturbative expansion 369 00:26:34,230 --> 00:26:36,230 not to the whole expression. 370 00:26:36,230 --> 00:26:38,450 You should do it to some denominator 371 00:26:38,450 --> 00:26:40,280 or to some part of the denominator. 372 00:26:40,280 --> 00:26:44,590 Because then, the perturbative expansion 373 00:26:44,590 --> 00:26:48,775 involves no divergent terms and can be performed. 374 00:26:53,350 --> 00:26:57,660 So therefore, we are now in a position 375 00:26:57,660 --> 00:27:11,580 to make approximations to the function R. 376 00:27:11,580 --> 00:27:17,730 And the simplest approximation which we can do 377 00:27:17,730 --> 00:27:23,030 is we can just try to see if we can get away 378 00:27:23,030 --> 00:27:25,380 with very low order. 379 00:27:25,380 --> 00:27:29,690 It's maybe not getting into any divergences here. 380 00:27:29,690 --> 00:27:34,940 And let me call this now the triangle. 381 00:27:34,940 --> 00:27:37,540 So the circle was the naked interaction. 382 00:27:37,540 --> 00:27:40,260 The square would be the exact function 383 00:27:40,260 --> 00:27:42,810 if you sum up those interactions to all orders. 384 00:27:42,810 --> 00:27:46,990 And the triangle is now, well, the step in between. 385 00:27:46,990 --> 00:27:50,108 We hope to get away with a triangle. 386 00:27:50,108 --> 00:27:52,610 So that would mean the following, 387 00:27:52,610 --> 00:28:06,130 that the exact formulation involved-- let me get black-- 388 00:28:06,130 --> 00:28:07,005 had a propagator. 389 00:28:07,005 --> 00:28:14,480 The state b has to go through multiple squares. 390 00:28:14,480 --> 00:28:16,470 This is how it propagates. 391 00:28:26,840 --> 00:28:30,810 This would be the exact result after the resummation 392 00:28:30,810 --> 00:28:32,300 we have done. 393 00:28:32,300 --> 00:28:43,140 And an approximate result is now that the squares 394 00:28:43,140 --> 00:28:45,915 are replaced by triangles. 395 00:28:49,060 --> 00:28:50,670 So that's pretty neat. 396 00:28:50,670 --> 00:28:55,330 But the importance comes now when I pull things together, 397 00:28:55,330 --> 00:28:58,810 I want to show you what we have exactly 398 00:28:58,810 --> 00:29:03,720 done for treating an atom in the excited state 399 00:29:03,720 --> 00:29:06,440 and for treating light scattering. 400 00:29:06,440 --> 00:29:08,540 So in other words, that's our result. 401 00:29:08,540 --> 00:29:10,050 We have derived it. 402 00:29:10,050 --> 00:29:20,420 I just go now and apply to an excited atomic state. 403 00:29:25,530 --> 00:29:27,820 So the state we are interested in 404 00:29:27,820 --> 00:29:31,850 is the atomic state b and no photons. 405 00:29:34,810 --> 00:29:46,640 And the property of the atomic state 406 00:29:46,640 --> 00:29:52,250 is obtained when we know the function Gb of Z. 407 00:29:52,250 --> 00:29:57,280 Just to tell you what it means, this is the time evolution 408 00:29:57,280 --> 00:30:02,920 operator when we Fourier transform, 409 00:30:02,920 --> 00:30:12,540 an inverse Fourier or Laplace transform, which is a contour 410 00:30:12,540 --> 00:30:16,360 integral in the complex plane, a generalization of the Fourier 411 00:30:16,360 --> 00:30:17,800 transform. 412 00:30:17,800 --> 00:30:21,940 This would take us then to the time evolution 413 00:30:21,940 --> 00:30:25,720 for the atom in state b. 414 00:30:25,720 --> 00:30:29,540 So this is now the diagram matrix element of the T-matrix. 415 00:30:40,090 --> 00:30:48,269 And this is the matrix element between state B0B0 and the time 416 00:30:48,269 --> 00:30:49,060 evolution operator. 417 00:30:53,510 --> 00:30:56,130 So we are calculating, of course, 418 00:30:56,130 --> 00:30:59,970 the Fourier transform of the time evolution of the state 419 00:30:59,970 --> 00:31:05,520 b by the Fourier transform through the resolvent G. 420 00:31:05,520 --> 00:31:08,950 And all the work we have done with our diagrams 421 00:31:08,950 --> 00:31:13,500 means instead of calculating G, we 422 00:31:13,500 --> 00:31:20,790 are calculating the kernel of Z. And if you 423 00:31:20,790 --> 00:31:32,030 use the lowest second order approach, then 424 00:31:32,030 --> 00:31:48,600 diagrammatically-- just give me one second, photon was emitted. 425 00:32:09,890 --> 00:32:17,390 Yeah, sorry, so the process we have considered 426 00:32:17,390 --> 00:32:19,700 is that we go to second order. 427 00:32:19,700 --> 00:32:31,335 We can go through an intermediate state, 428 00:32:31,335 --> 00:32:34,540 can say, absorbing it and meeting a virtual photon. 429 00:32:34,540 --> 00:32:36,150 That's what two vertices mean. 430 00:32:36,150 --> 00:32:38,390 At the first vertex, the photon appears. 431 00:32:38,390 --> 00:32:42,890 At the second vertex, the photon disappears. 432 00:32:42,890 --> 00:32:49,000 So that means now that we, with this approximation, 433 00:32:49,000 --> 00:32:54,890 approximate Gb of Z, the Fourier transform, the resolvents, 434 00:32:54,890 --> 00:32:56,990 in the following way. 435 00:32:56,990 --> 00:33:00,570 We let b just propagate, which is problematic 436 00:33:00,570 --> 00:33:02,850 because this has divergences. 437 00:33:02,850 --> 00:33:07,440 But we are allowing now the state b 438 00:33:07,440 --> 00:33:15,190 to go through intermediate states a. 439 00:33:15,190 --> 00:33:26,070 And remember, since this appears in the denominator, that 440 00:33:26,070 --> 00:33:29,850 means for G, just make a Taylor expansion in R, 441 00:33:29,850 --> 00:33:34,770 that it has this process to all orders. 442 00:33:34,770 --> 00:33:41,320 So that propagator, the sort of propagation G, 443 00:33:41,320 --> 00:33:54,970 involves now all possibilities that we 444 00:33:54,970 --> 00:33:57,580 have to end up in state b again. 445 00:33:57,580 --> 00:34:01,020 But we can go through intermediate states 446 00:34:01,020 --> 00:34:08,449 a or a prime as often as we want. 447 00:34:08,449 --> 00:34:11,130 Or actually, we have summed it up. 448 00:34:11,130 --> 00:34:14,040 Our results contain those processes to infinite order. 449 00:34:20,120 --> 00:34:22,159 So the question is, what have we neglected? 450 00:34:22,159 --> 00:34:23,510 I mean, that looks like a lot. 451 00:34:23,510 --> 00:34:27,290 We allow the state b to emit a photon, go to another state. 452 00:34:27,290 --> 00:34:29,110 It's reabsorbed and such. 453 00:34:29,110 --> 00:34:31,200 So the question is now, what is neglected? 454 00:34:35,929 --> 00:34:40,690 So what we neglect is actually a lot. 455 00:34:40,690 --> 00:34:45,150 In this lowest order approximation for the function 456 00:34:45,150 --> 00:34:51,179 R, we approximated R by second order. 457 00:34:51,179 --> 00:34:52,650 We have only two vertices. 458 00:34:52,650 --> 00:34:57,640 So what we neglect are all processes 459 00:34:57,640 --> 00:35:01,050 where we have not just two vertices 460 00:35:01,050 --> 00:35:02,850 and one intermediate states, where 461 00:35:02,850 --> 00:35:16,370 we have several intermediate states between two 462 00:35:16,370 --> 00:35:23,850 occurrences of the state B0. 463 00:35:23,850 --> 00:35:28,420 Or diagrammatically, what we have neglected 464 00:35:28,420 --> 00:35:30,445 is-- let me just give you an example. 465 00:35:34,500 --> 00:35:38,430 So what we have neglected is we have to always start 466 00:35:38,430 --> 00:35:42,270 with a state b, and we have to end with a state b. 467 00:35:42,270 --> 00:35:45,280 But now the idea was that-- one, two, 468 00:35:45,280 --> 00:35:47,780 let me just go through four vertices. 469 00:35:47,780 --> 00:35:50,545 The approximation we have done-- you 470 00:35:50,545 --> 00:35:52,920 have to sort of look through that maybe after the lecture 471 00:35:52,920 --> 00:35:54,730 and see that that's what we've really done, 472 00:35:54,730 --> 00:36:01,380 but trust me-- is that when the system goes away from the state 473 00:36:01,380 --> 00:36:03,940 b into an intermediate state a, at 474 00:36:03,940 --> 00:36:08,350 the next vertex it has to go back to the state b. 475 00:36:08,350 --> 00:36:13,190 So what we have not included is processes 476 00:36:13,190 --> 00:36:18,980 where we go through states a, a prime, a double prime, 477 00:36:18,980 --> 00:36:22,500 and then eventually we go back to the state a. 478 00:36:22,500 --> 00:36:28,700 Or we have not included processes where we scatter, 479 00:36:28,700 --> 00:36:29,790 absorb. 480 00:36:29,790 --> 00:36:31,090 This is a state a. 481 00:36:31,090 --> 00:36:32,930 But then, we don't go back to b. 482 00:36:32,930 --> 00:36:34,640 We go to a prime. 483 00:36:34,640 --> 00:36:38,020 Then, we scatter again. 484 00:36:38,020 --> 00:36:39,685 Here we are in state a double prime. 485 00:36:39,685 --> 00:36:41,610 And then, we are back in b. 486 00:36:41,610 --> 00:36:45,620 So in other words, we have said whenever something happens, 487 00:36:45,620 --> 00:36:49,640 and we go away from state b, the next vertex 488 00:36:49,640 --> 00:36:51,610 has to go back to state b. 489 00:36:51,610 --> 00:36:54,890 This is the nature of the lowest order approximation. 490 00:36:54,890 --> 00:36:59,960 We have included to infinite order 491 00:36:59,960 --> 00:37:02,840 all processes where the state b emits a photon, 492 00:37:02,840 --> 00:37:04,810 reabsorbs it, emits and reabsorbs it. 493 00:37:04,810 --> 00:37:14,280 But the system cannot go sort of two steps away from the state 494 00:37:14,280 --> 00:37:14,780 b. 495 00:37:14,780 --> 00:37:16,650 It's only one step, and then go back. 496 00:37:16,650 --> 00:37:18,316 This is the nature of the approximation. 497 00:37:28,040 --> 00:37:37,020 OK, so our result for this kernel, 498 00:37:37,020 --> 00:37:52,945 which describes the state b, is we have-- just go back, 499 00:37:52,945 --> 00:37:53,820 let you see it again. 500 00:38:01,290 --> 00:38:05,190 I'm writing now for you down the equation for the triangle, 501 00:38:05,190 --> 00:38:08,380 which is an interaction V, an intermediate state, 502 00:38:08,380 --> 00:38:14,390 and another interaction V, which is nothing else than Fermi's 503 00:38:14,390 --> 00:38:21,690 golden rule where we-- well, with a little twist-- have 504 00:38:21,690 --> 00:38:24,330 the initial state. 505 00:38:24,330 --> 00:38:28,020 The dipole interaction or the [? p.a ?] interaction 506 00:38:28,020 --> 00:38:31,040 takes us to an intermediate state 507 00:38:31,040 --> 00:38:35,420 with a photon with [INAUDIBLE] and polarization epsilon. 508 00:38:38,330 --> 00:38:41,060 We propagate in the intermediate state Ea. 509 00:38:45,970 --> 00:38:50,410 And now we have to go back. 510 00:38:50,410 --> 00:38:52,609 But we have to go back with the same matrix element. 511 00:38:52,609 --> 00:38:54,400 So therefore the matrix element is squared. 512 00:39:00,790 --> 00:39:10,420 And we have a double sum. 513 00:39:10,420 --> 00:39:14,480 We sum over all possible states of the photon. 514 00:39:14,480 --> 00:39:17,230 And we can sum over all intermediate atomic states. 515 00:39:24,260 --> 00:39:26,400 Yes, so this is what we have done. 516 00:39:31,370 --> 00:39:37,540 This expression has in general a real part and imaginary part. 517 00:39:41,650 --> 00:39:48,410 It's a function of the initial energy E. 518 00:39:48,410 --> 00:39:50,150 And it has an imaginary part. 519 00:40:03,650 --> 00:40:09,460 Yes, let me now interpret it in two ways. 520 00:40:09,460 --> 00:40:11,850 But before I do that, are there any questions? 521 00:40:11,850 --> 00:40:12,406 Yes. 522 00:40:12,406 --> 00:40:13,376 AUDIENCE: Just mathematically, how do you 523 00:40:13,376 --> 00:40:14,792 get an imaginary part out of this? 524 00:40:14,792 --> 00:40:16,894 It's all real components, because we 525 00:40:16,894 --> 00:40:20,062 have magnitude squared divided by, presumably, 526 00:40:20,062 --> 00:40:25,660 the energies real and so on. 527 00:40:25,660 --> 00:40:27,680 PROFESSOR: OK, this is now something 528 00:40:27,680 --> 00:40:30,570 to do that we have resonant terms. 529 00:40:30,570 --> 00:40:44,910 And what we often do is we add an infinitesimal eta, 530 00:40:44,910 --> 00:40:47,540 and then let eta go to 0. 531 00:40:47,540 --> 00:40:53,060 And it's the same if you have the function 1 over x. 532 00:40:53,060 --> 00:40:58,280 And look at a real part and imaginary part. 533 00:40:58,280 --> 00:41:01,460 It needs a little bit of correct treatment 534 00:41:01,460 --> 00:41:03,175 of functions in the complex plane. 535 00:41:06,520 --> 00:41:07,770 Let me actually write it down. 536 00:41:07,770 --> 00:41:08,686 Then it becomes clear. 537 00:41:08,686 --> 00:41:11,120 But yes, thanks for the question. 538 00:41:11,120 --> 00:41:14,625 So I always said, we do a Fourier transform, 539 00:41:14,625 --> 00:41:17,060 but it's a Laplace transform. 540 00:41:17,060 --> 00:41:19,600 We do a Fourier transform at real energy. 541 00:41:19,600 --> 00:41:21,805 But if you do an integration along when 542 00:41:21,805 --> 00:41:23,690 you do the Fourier transform, you 543 00:41:23,690 --> 00:41:25,530 have to integrate over omega. 544 00:41:25,530 --> 00:41:28,620 But the function we integrate, the time evolution operator, 545 00:41:28,620 --> 00:41:30,160 has poles in omega. 546 00:41:30,160 --> 00:41:32,150 So we can't just Fourier transform. 547 00:41:32,150 --> 00:41:34,570 Because we go right through the poles. 548 00:41:34,570 --> 00:41:38,210 But what we can do is we can add an imaginary part plus or minus 549 00:41:38,210 --> 00:41:39,040 eta. 550 00:41:39,040 --> 00:41:41,480 And we can just go around the poles. 551 00:41:41,480 --> 00:41:43,990 And then, it becomes mathematically meaningful. 552 00:41:43,990 --> 00:41:46,790 And what we're doing here is-- but I'm not really 553 00:41:46,790 --> 00:41:49,651 explaining it mathematically-- we 554 00:41:49,651 --> 00:41:50,900 have played those tricks here. 555 00:41:53,470 --> 00:41:58,860 But I hope it becomes clear if I say 556 00:41:58,860 --> 00:42:01,740 what the real and imaginary parts are. 557 00:42:01,740 --> 00:42:13,140 So the real part is this matrix element 558 00:42:13,140 --> 00:42:29,260 squared, but double sum. 559 00:42:29,260 --> 00:42:34,350 But what we use is the principle part of it, 560 00:42:34,350 --> 00:42:38,080 which is well defined in the theory of complex functions. 561 00:42:38,080 --> 00:42:43,519 And it's divergent, but you take a certain symmetric 562 00:42:43,519 --> 00:42:44,310 limiting procedure. 563 00:42:44,310 --> 00:42:48,790 So you have to interpret that as-- you 564 00:42:48,790 --> 00:42:51,340 have to introduce a limiting procedure to make sure 565 00:42:51,340 --> 00:42:54,120 that the divergence cancels out. 566 00:42:54,120 --> 00:42:58,250 And the imaginary part, value of the imaginary part of something 567 00:42:58,250 --> 00:43:05,230 which diverges-- and if you treat the eta correctly, 568 00:43:05,230 --> 00:43:09,910 let the eta go to 0, you will realize that the imaginary part 569 00:43:09,910 --> 00:43:11,260 turns into a delta function. 570 00:43:19,200 --> 00:43:27,110 So what we get is 2pi over H bar matrix 571 00:43:27,110 --> 00:43:30,930 element squared times the delta function. 572 00:43:30,930 --> 00:43:36,500 So that's something which you have seen. 573 00:43:36,500 --> 00:43:41,625 So the imaginary part gets us Fermi's golden rule. 574 00:43:45,600 --> 00:43:52,400 And the real part has actually-- remember 575 00:43:52,400 --> 00:43:54,800 when we discussed the AC Stark shift. 576 00:43:54,800 --> 00:43:58,600 The AC Stark shift has a 1 over [INAUDIBLE] dependence. 577 00:43:58,600 --> 00:44:00,950 And you recognize that here. 578 00:44:00,950 --> 00:44:04,970 So this is actually nothing else than the AC Stark 579 00:44:04,970 --> 00:44:16,190 shift not due to a laser beam, but due to one photon per mode. 580 00:44:16,190 --> 00:44:18,710 Because we started with an atom in an excited state. 581 00:44:18,710 --> 00:44:20,910 It can emit a photon in any mode. 582 00:44:20,910 --> 00:44:23,980 And this photon is now creating an AC Stark shift. 583 00:44:23,980 --> 00:44:26,920 And this is mathematically the expression. 584 00:44:26,920 --> 00:44:32,910 And such AC Stark shifts which appear as self energies, 585 00:44:32,910 --> 00:44:36,300 as energy shifts created by the state, this 586 00:44:36,300 --> 00:44:41,170 is nothing else than the famous Lamb shift. 587 00:44:41,170 --> 00:44:42,708 So that's what we get out here. 588 00:44:48,450 --> 00:44:58,780 I have already-- do I in this sum? 589 00:45:03,240 --> 00:45:07,950 What we have here is we have this function R 590 00:45:07,950 --> 00:45:12,720 in the real and imaginary part, which depends on the energy E. 591 00:45:12,720 --> 00:45:16,420 But remember, we worked so hard with diagrams to make sure 592 00:45:16,420 --> 00:45:20,670 that the triangle-- first the square, and then the triangle, 593 00:45:20,670 --> 00:45:22,500 and this is what we calculate here-- 594 00:45:22,500 --> 00:45:26,190 has no resonant structure at the energy Eb. 595 00:45:26,190 --> 00:45:32,270 So therefore, we can neglect the energy dependence of that 596 00:45:32,270 --> 00:45:38,975 and simply replace the argument E by the energy 597 00:45:38,975 --> 00:45:44,130 we are interested in, namely energies close to Eb. 598 00:45:44,130 --> 00:45:50,860 So in other words, when we had the function, the resolvent Gb 599 00:45:50,860 --> 00:45:56,540 of Z, all the dependence on energy came from this kernel. 600 00:45:56,540 --> 00:45:59,902 But this kernel is now so well-behaved, there 601 00:45:59,902 --> 00:46:02,680 are no resonant terms, that we can neglect its energy 602 00:46:02,680 --> 00:46:03,180 dependence. 603 00:46:06,020 --> 00:46:09,210 This actually has a name which we will encounter later 604 00:46:09,210 --> 00:46:12,320 when we discuss master equation and optical Bloch equations. 605 00:46:15,030 --> 00:46:22,990 So this replace, neglect E and set, or replace 606 00:46:22,990 --> 00:46:25,630 the dependence by E, by taking the value 607 00:46:25,630 --> 00:46:34,160 at Eb, this corresponds to the Markov approximation. 608 00:46:38,790 --> 00:46:41,300 The Markov approximation often means 609 00:46:41,300 --> 00:46:47,420 that some relaxation time or some response of a system 610 00:46:47,420 --> 00:46:50,350 is replaced by delta function. 611 00:46:50,350 --> 00:46:52,400 Well, we've done the same here. 612 00:46:52,400 --> 00:46:58,240 Because if you replace some temporal response function 613 00:46:58,240 --> 00:47:01,830 by a delta function, that means the Fourier transform 614 00:47:01,830 --> 00:47:03,710 becomes constant. 615 00:47:03,710 --> 00:47:06,180 And by neglecting the energy dependence, 616 00:47:06,180 --> 00:47:08,430 we are now saying everything is constant as a function 617 00:47:08,430 --> 00:47:09,059 of energy. 618 00:47:09,059 --> 00:47:10,600 And that means in the temporal domain 619 00:47:10,600 --> 00:47:12,047 that we have a delta function. 620 00:47:12,047 --> 00:47:13,380 I don't want to go further here. 621 00:47:13,380 --> 00:47:16,340 But when we talk about the master equation, 622 00:47:16,340 --> 00:47:19,480 we will also make a Markov approximation 623 00:47:19,480 --> 00:47:22,280 later on, but then in the temporal domain. 624 00:47:22,280 --> 00:47:26,020 And the two are equivalent here. 625 00:47:26,020 --> 00:47:33,850 So what we've got now is we found 626 00:47:33,850 --> 00:47:40,340 a solution for the Fourier transform of the time evolution 627 00:47:40,340 --> 00:47:50,330 operator, which initially had a divergence at energy b. 628 00:47:50,330 --> 00:47:52,950 And this was the problem we are facing. 629 00:47:52,950 --> 00:47:57,220 But by now, calculating the function R, 630 00:47:57,220 --> 00:48:05,080 we have a correction, which is a radiative shift, which 631 00:48:05,080 --> 00:48:07,330 comes from the real part. 632 00:48:07,330 --> 00:48:14,620 And we obtained, as promised, the imaginary part, 633 00:48:14,620 --> 00:48:18,750 which we can approximate by Fermi's golden rule. 634 00:48:18,750 --> 00:48:25,700 If we now Fourier transform back and obtain the time evolution 635 00:48:25,700 --> 00:48:37,360 of this state, it no longer evolves with the energy Eb. 636 00:48:37,360 --> 00:48:44,480 It has a shifted energy by this self energy. 637 00:48:44,480 --> 00:48:54,350 And this is called the radiative shift. 638 00:48:54,350 --> 00:49:03,060 But in addition, because of the imaginary part, 639 00:49:03,060 --> 00:49:05,585 it has now an exponential decay. 640 00:49:10,340 --> 00:49:14,010 And you should now-- well, this is what we may have expected. 641 00:49:14,010 --> 00:49:15,810 But there are two things you should learn. 642 00:49:15,810 --> 00:49:21,960 The first thing is that the exponential decay 643 00:49:21,960 --> 00:49:27,410 would be different if we had not made the Markov approximation. 644 00:49:27,410 --> 00:49:36,900 If we had kept a dependence of this imaginary part on energy, 645 00:49:36,900 --> 00:49:38,840 the Fourier transform would not have simply 646 00:49:38,840 --> 00:49:40,570 given us an exponential. 647 00:49:40,570 --> 00:49:43,900 So therefore, the exponential decay 648 00:49:43,900 --> 00:49:48,510 involves an approximation that the R function 649 00:49:48,510 --> 00:49:51,240 has no energy dependence. 650 00:49:51,240 --> 00:49:56,130 And you would say, well, is that really possible? 651 00:49:56,130 --> 00:49:58,690 If you have an atom, and it decays, 652 00:49:58,690 --> 00:50:04,790 or you have the atom in state b, at very, very short times 653 00:50:04,790 --> 00:50:07,840 you need Fourier transform elements 654 00:50:07,840 --> 00:50:10,870 of large amounts of energy. 655 00:50:10,870 --> 00:50:13,470 So maybe for the first femtosecond for the time 656 00:50:13,470 --> 00:50:18,320 evolution of an excited state, you need whatever, 657 00:50:18,320 --> 00:50:21,110 a whole X-ray spectrum of energies. 658 00:50:21,110 --> 00:50:24,740 And it's obvious that the properties of this expression 659 00:50:24,740 --> 00:50:27,030 where you sum over all states, something 660 00:50:27,030 --> 00:50:31,190 will happen when you go past the normal excitation 661 00:50:31,190 --> 00:50:34,090 energy or the ionization energy of the atom. 662 00:50:34,090 --> 00:50:36,100 So what you can immediately read form here 663 00:50:36,100 --> 00:50:39,940 is that exponential decay is a simple approximation. 664 00:50:39,940 --> 00:50:41,330 It works very well. 665 00:50:41,330 --> 00:50:43,910 But at very early times, it will break down. 666 00:50:43,910 --> 00:50:47,900 Because then, the energy dependence matters. 667 00:50:47,900 --> 00:50:51,400 But the longer you wait-- if you wait a few nanoseconds, 668 00:50:51,400 --> 00:50:55,120 the Fourier transform, the relevant part of the Fourier 669 00:50:55,120 --> 00:51:03,850 transform, is only a small energy or frequency interval 670 00:51:03,850 --> 00:51:06,050 around the resonance energy. 671 00:51:06,050 --> 00:51:09,220 And then, the density of states of your photon field 672 00:51:09,220 --> 00:51:11,250 is pretty much constant around here. 673 00:51:11,250 --> 00:51:14,230 And then, this approximation is excellent. 674 00:51:14,230 --> 00:51:16,510 So I hope what you have learned from the treatment 675 00:51:16,510 --> 00:51:19,970 is number one, where the exponential decay comes from. 676 00:51:19,970 --> 00:51:22,280 Of course you knew that from coupling to all modes, 677 00:51:22,280 --> 00:51:31,140 but that the approximation which leads to exponential decay 678 00:51:31,140 --> 00:51:33,870 would also involve the density of states 679 00:51:33,870 --> 00:51:36,300 and the density of modes is constant, 680 00:51:36,300 --> 00:51:39,380 which is excellent for certain times 681 00:51:39,380 --> 00:51:41,400 but which is of course violated at early times. 682 00:51:46,070 --> 00:51:54,150 And finally, if we hadn't done the infinite summation 683 00:51:54,150 --> 00:52:02,080 of diagrams, if we had done a perturbative expansion, 684 00:52:02,080 --> 00:52:08,530 we would have never obtained exponential decay. 685 00:52:12,640 --> 00:52:15,410 We would have obtained some polynominal decay. 686 00:52:31,870 --> 00:52:32,770 Questions about that? 687 00:52:36,340 --> 00:52:36,840 Yes. 688 00:52:36,840 --> 00:52:39,049 AUDIENCE: What is the polynominal decay? 689 00:52:43,040 --> 00:52:44,670 PROFESSOR: Some power law. 690 00:52:47,830 --> 00:52:58,120 1 minus the time-- as far as I know, 691 00:52:58,120 --> 00:53:01,450 it would just involve powers of n. 692 00:53:01,450 --> 00:53:06,080 If you do lowest order perturbation theory, 693 00:53:06,080 --> 00:53:08,610 instead of getting an exponential decay, 694 00:53:08,610 --> 00:53:10,680 you would just get a linear slope. 695 00:53:10,680 --> 00:53:12,000 That's perturbation theory. 696 00:53:12,000 --> 00:53:14,710 And if you fix it, I think you get quadratic terms. 697 00:53:14,710 --> 00:53:16,980 So it's the sum of power loss. 698 00:53:20,930 --> 00:53:23,000 Of course, an exponential function 699 00:53:23,000 --> 00:53:25,120 has a Taylor expansion, which is an infinite sum 700 00:53:25,120 --> 00:53:27,390 of polynominal terms. 701 00:53:27,390 --> 00:53:31,077 And therefore, we need infinite order to get the exponential. 702 00:53:31,077 --> 00:53:32,910 So it's not really profound what I'm saying. 703 00:53:32,910 --> 00:53:35,451 It's pretty much an exponential function is non-perturbative. 704 00:53:37,734 --> 00:53:38,400 Other questions? 705 00:53:43,060 --> 00:53:47,120 So let me wrap up this chapter. 706 00:53:47,120 --> 00:53:50,570 What we have discussed here is-- I haven't really 707 00:53:50,570 --> 00:53:52,940 discussed resonant scattering. 708 00:53:52,940 --> 00:53:56,600 I've now focused on the function Gb. 709 00:53:56,600 --> 00:54:02,510 I focused on what happens to the state b. 710 00:54:02,510 --> 00:54:07,140 But this is-- and this is what I want to show you now-- 711 00:54:07,140 --> 00:54:11,780 the only element we need to discuss resonant scattering. 712 00:54:15,240 --> 00:54:19,270 So when we have an atom in the ground state, 713 00:54:19,270 --> 00:54:31,530 and a photon comes along, and it takes the atom to the excited 714 00:54:31,530 --> 00:54:45,250 state b, then we go back to the same state-- 715 00:54:45,250 --> 00:54:48,290 could be also another state-- by emitting a photon k 716 00:54:48,290 --> 00:54:49,560 prime epsilon prime. 717 00:54:52,470 --> 00:54:57,860 And the relevant matrix element of the time evolution operator, 718 00:54:57,860 --> 00:55:08,860 which is the T-matrix, involves now the matrix element, 719 00:55:08,860 --> 00:55:22,590 the initial energy minus the intermediate energy. 720 00:55:22,590 --> 00:55:26,300 And the critical part is really-- 721 00:55:26,300 --> 00:55:27,900 you can do it mathematically. 722 00:55:27,900 --> 00:55:30,090 I just show it here as a summary. 723 00:55:30,090 --> 00:55:33,210 The critical part is really the propagation of the state 724 00:55:33,210 --> 00:55:35,300 b, which is problematic. 725 00:55:35,300 --> 00:55:38,840 But we have now learned, and it transfers exactly 726 00:55:38,840 --> 00:55:41,560 to the light scattering problem, that we 727 00:55:41,560 --> 00:55:50,870 have to include now radiative shifts and an imaginary part 728 00:55:50,870 --> 00:55:58,330 for the decay to the time evolution. 729 00:55:58,330 --> 00:56:09,110 And that means that this diagram here for light scattering 730 00:56:09,110 --> 00:56:19,590 has been-- we have added other terms to it. 731 00:56:19,590 --> 00:56:21,640 And the other terms are, of course, 732 00:56:21,640 --> 00:56:28,780 that when we scatter light of an excited state like this, 733 00:56:28,780 --> 00:56:36,080 the excited state can sort of emit photons and reabsorb them. 734 00:56:36,080 --> 00:56:45,480 And it can do that to-- so we go to that state. 735 00:56:49,270 --> 00:56:53,600 It can do that to infinite order. 736 00:56:53,600 --> 00:56:56,250 So in other words, for any problem now 737 00:56:56,250 --> 00:56:59,550 which involves the excited state, 738 00:56:59,550 --> 00:57:05,240 we replace the 0's order propagation of the state b. 739 00:57:05,240 --> 00:57:08,430 And mathematically, it means we replace this function 740 00:57:08,430 --> 00:57:15,260 by the resolvent, which we have calculated 741 00:57:15,260 --> 00:57:24,960 by doing an approximation to the kernel R. Questions? 742 00:57:24,960 --> 00:57:29,211 AUDIENCE: Question, [INAUDIBLE]? 743 00:57:29,211 --> 00:57:31,175 PROFESSOR: Pardon? 744 00:57:31,175 --> 00:57:32,648 AUDIENCE: Is this [INAUDIBLE]? 745 00:57:38,135 --> 00:57:39,510 PROFESSOR: Yeah, OK, what happens 746 00:57:39,510 --> 00:57:57,120 is if you're off-resonant, you don't have a problem. 747 00:57:57,120 --> 00:58:01,710 This extra term delta and gamma, the radiative shift 748 00:58:01,710 --> 00:58:04,940 and the line widths only matter when 749 00:58:04,940 --> 00:58:07,530 the black term is close to 0. 750 00:58:07,530 --> 00:58:12,100 If you have a large detuning delta here, 751 00:58:12,100 --> 00:58:15,900 then the small shift in the line widths don't matter. 752 00:58:15,900 --> 00:58:20,550 So everything we have done by correcting 753 00:58:20,550 --> 00:58:22,760 the naked propagation of the state 754 00:58:22,760 --> 00:58:26,470 b by the correct propagation with this infinite emission 755 00:58:26,470 --> 00:58:29,280 and reabsorption of virtual photons, 756 00:58:29,280 --> 00:58:33,750 this is only needed if the denominator is 0. 757 00:58:33,750 --> 00:58:36,600 And then, we have to figure out what else happens. 758 00:58:36,600 --> 00:58:38,630 And what else happens is obtained 759 00:58:38,630 --> 00:58:40,940 in higher order with this non-perturbative treatment. 760 00:58:48,579 --> 00:58:49,245 Other questions? 761 00:58:54,540 --> 00:59:09,680 OK, 20 minutes left. 762 00:59:17,720 --> 00:59:18,980 So we now change gears. 763 00:59:18,980 --> 00:59:22,280 We move onto the optical Bloch equation. 764 00:59:22,280 --> 00:59:27,360 But let me give you one summary on this chapter of diagrams. 765 00:59:27,360 --> 00:59:31,020 Until maybe 10, 15 years ago here at MIT, 766 00:59:31,020 --> 00:59:32,790 we were not teaching that. 767 00:59:32,790 --> 00:59:35,520 And I felt often in discussion with students 768 00:59:35,520 --> 00:59:39,840 that a little bit more of a complete picture behind atom 769 00:59:39,840 --> 00:59:43,180 photon processes is needed. 770 00:59:43,180 --> 00:59:48,140 What I summarized could of course cover a whole semester 771 00:59:48,140 --> 00:59:50,780 course in QED and how to do calculation. 772 00:59:50,780 --> 00:59:56,250 And if you're interested in mathematical rigor, 773 00:59:56,250 --> 00:59:59,180 the green book, Atom-Photon Interactions, 774 00:59:59,180 --> 01:00:02,520 is pretty rigorous and still very physical. 775 01:00:02,520 --> 01:00:04,970 But on the other hand, many of you 776 01:00:04,970 --> 01:00:06,740 experimentalists, I think you should 777 01:00:06,740 --> 01:00:09,740 have sort of this picture behind it, what really happens, 778 01:00:09,740 --> 01:00:12,700 what kind of emission processes are 779 01:00:12,700 --> 01:00:14,757 responsible for which effect. 780 01:00:14,757 --> 01:00:16,215 And at least this is sort of for me 781 01:00:16,215 --> 01:00:20,480 a take-home message which I hope you can enjoy even 782 01:00:20,480 --> 01:00:27,990 without mathematical rigor, that the fact that the excited 783 01:00:27,990 --> 01:00:32,490 state has [INAUDIBLE] really comes from an infinite number 784 01:00:32,490 --> 01:00:37,250 of absorption, of emission and reabsorption processes. 785 01:00:37,250 --> 01:00:41,650 So you should maybe think about when 786 01:00:41,650 --> 01:00:43,870 you take an atom to the excited state 787 01:00:43,870 --> 01:00:47,220 that the excited state does more than just absorb the photon, 788 01:00:47,220 --> 01:00:50,090 the atom is excited, and then it emits. 789 01:00:50,090 --> 01:00:53,230 The real nature of this state is that it 790 01:00:53,230 --> 01:00:54,800 couples to many, many modes. 791 01:00:54,800 --> 01:00:57,180 It emits photons and reabsorbs them. 792 01:00:57,180 --> 01:01:01,820 And you can often neglect that in the simple description 793 01:01:01,820 --> 01:01:03,110 of your experiment. 794 01:01:03,110 --> 01:01:09,360 But if you take certain expressions seriously, 795 01:01:09,360 --> 01:01:11,320 they would have divergences. 796 01:01:11,320 --> 01:01:15,170 And that's what we discussed without this infinite number 797 01:01:15,170 --> 01:01:17,692 of processes which happen. 798 01:01:17,692 --> 01:01:19,900 Of course, yes, the whole other regime which I should 799 01:01:19,900 --> 01:01:23,030 mention-- and this is when you can completely 800 01:01:23,030 --> 01:01:24,840 neglect the coupling to many modes. 801 01:01:24,840 --> 01:01:27,880 If you do Rabi oscillation with resonant interaction, 802 01:01:27,880 --> 01:01:29,300 you don't need all that. 803 01:01:29,300 --> 01:01:32,640 Because then, you're really looking at discrete states. 804 01:01:32,640 --> 01:01:34,970 So it really also depends what you want to describe. 805 01:01:34,970 --> 01:01:37,040 If you do cavity QED with an atom, 806 01:01:37,040 --> 01:01:38,540 you have the Jaynes-Cummings model. 807 01:01:38,540 --> 01:01:41,570 And you have an exact solution. 808 01:01:41,570 --> 01:01:43,630 Then you have a similar mode problem. 809 01:01:43,630 --> 01:01:46,030 But here we have discussed what happens 810 01:01:46,030 --> 01:01:48,540 if you want to scatter light in free space. 811 01:01:48,540 --> 01:01:51,609 And then, you have to deal with the divergence of the excited 812 01:01:51,609 --> 01:01:52,109 state. 813 01:01:56,420 --> 01:02:16,420 OK, the next chapter is called Derivation of the Optical Bloch 814 01:02:16,420 --> 01:02:16,920 Equation. 815 01:02:27,550 --> 01:02:36,660 And yes, what we really need for a number of phenomenon 816 01:02:36,660 --> 01:02:41,800 in AMO physics for laser cooling, light forces, and much 817 01:02:41,800 --> 01:02:45,850 more are the optical Bloch equations. 818 01:02:45,850 --> 01:02:48,660 But what I try in this chapter-- to give you 819 01:02:48,660 --> 01:02:53,300 sort of the fundamental story, the profound story, 820 01:02:53,300 --> 01:02:56,800 the fundamental insight behind the optical Bloch equations. 821 01:02:56,800 --> 01:03:01,450 Because what the optical Bloch equations are is the following. 822 01:03:01,450 --> 01:03:04,530 We want to describe a quantum system. 823 01:03:04,530 --> 01:03:09,570 But the quantum system is coupled to the environment. 824 01:03:09,570 --> 01:03:12,380 And that means we have dissipation. 825 01:03:12,380 --> 01:03:17,240 And this is something which is not easily dealt 826 01:03:17,240 --> 01:03:19,410 with in simple quantum physics. 827 01:03:19,410 --> 01:03:24,200 Because a simply quantum system undergoes unitary time 828 01:03:24,200 --> 01:03:26,930 evolution described by a Hamiltonian. 829 01:03:26,930 --> 01:03:31,800 And unitary time evolution does not allow entropy to increase, 830 01:03:31,800 --> 01:03:34,610 does not allow energies to be exchanged. 831 01:03:34,610 --> 01:03:37,570 So a lot of things which happen in our experiments 832 01:03:37,570 --> 01:03:41,580 and in daily life come because the system we are looking at, 833 01:03:41,580 --> 01:03:44,760 it follows Schrodinger's equation. 834 01:03:44,760 --> 01:03:48,020 But it is coupled to a much bigger system. 835 01:03:48,020 --> 01:03:51,060 And so in this section, what I want 836 01:03:51,060 --> 01:03:53,860 to address at the most fundamental limit is, what 837 01:03:53,860 --> 01:03:58,400 is the step where we go from reversible equation, 838 01:03:58,400 --> 01:04:01,880 unitary time evolution, to something which is called 839 01:04:01,880 --> 01:04:04,990 relaxation, which is dissipative, 840 01:04:04,990 --> 01:04:07,330 where entropy is increased? 841 01:04:07,330 --> 01:04:10,500 And the step, of course, is we go from pure states 842 01:04:10,500 --> 01:04:11,910 to statistical operators. 843 01:04:11,910 --> 01:04:13,950 We describe the big system. 844 01:04:13,950 --> 01:04:16,100 But then, we focus on the small system. 845 01:04:16,100 --> 01:04:19,580 And I want to show you, first with a simple example, 846 01:04:19,580 --> 01:04:23,070 but next week in more generality, 847 01:04:23,070 --> 01:04:27,610 how this completely changes the nature 848 01:04:27,610 --> 01:04:29,650 of the description of the small system. 849 01:04:29,650 --> 01:04:33,150 A small system which I just described 850 01:04:33,150 --> 01:04:36,410 by Schrodinger's equation, now follows the density matrix 851 01:04:36,410 --> 01:04:41,960 equation, has relaxation, the entropy increases, and such. 852 01:04:41,960 --> 01:04:47,880 So to maybe set the stage, one of the simplest systems 853 01:04:47,880 --> 01:04:51,890 we can imagine is the Jaynes-Cummings model, 854 01:04:51,890 --> 01:04:59,480 that we have an atom in a cavity interacting 855 01:04:59,480 --> 01:05:02,930 with one mode of the radiation field. 856 01:05:02,930 --> 01:05:07,650 And in this cause in part one, we've dealt with it 857 01:05:07,650 --> 01:05:10,250 and looked at vacuum Robi oscillation and a few really 858 01:05:10,250 --> 01:05:12,040 neat things. 859 01:05:12,040 --> 01:05:16,210 But what happens is that the system is an open quantum 860 01:05:16,210 --> 01:05:17,950 system. 861 01:05:17,950 --> 01:05:21,560 You can have spontaneous emission. 862 01:05:21,560 --> 01:05:28,390 And if your mirrors are not 100.00% reflectivity, 863 01:05:28,390 --> 01:05:30,490 some light leaks out. 864 01:05:30,490 --> 01:05:35,135 So in other words, we have coupling to the environment. 865 01:05:45,200 --> 01:05:51,470 Of course, we can say we simply describe it. 866 01:05:51,470 --> 01:05:56,040 We have our atoms, maybe the one mode 867 01:05:56,040 --> 01:06:04,750 plus the one mode of the electromagnetic field. 868 01:06:04,750 --> 01:06:15,070 And then, we have the environment, 869 01:06:15,070 --> 01:06:17,780 which may consist of photons which are leaking out 870 01:06:17,780 --> 01:06:20,900 and photons which are spontaneously emitted. 871 01:06:20,900 --> 01:06:23,700 So this is our system. 872 01:06:23,700 --> 01:06:29,390 And of course, if you write down the total Hamiltonian 873 01:06:29,390 --> 01:06:33,630 and do the time evolution, something 874 01:06:33,630 --> 01:06:36,970 will come out which, in general, is 875 01:06:36,970 --> 01:06:40,200 very complicated, very entangled. 876 01:06:40,200 --> 01:06:41,610 The atom is entangled. 877 01:06:41,610 --> 01:06:43,320 The atom moving to the left side is 878 01:06:43,320 --> 01:06:46,840 entangled with a photon which was emitted to the right side. 879 01:06:46,840 --> 01:06:48,890 And the recall of the photons push the atom. 880 01:06:48,890 --> 01:06:52,160 So you have to know, to keep track of all the photons, which 881 01:06:52,160 --> 01:06:55,140 have been scattered in the lifetime of an atom. 882 01:06:55,140 --> 01:06:56,620 So you can do that. 883 01:06:56,620 --> 01:06:59,240 And you do that by propagating the system 884 01:06:59,240 --> 01:07:02,130 with its total Hamiltonian. 885 01:07:02,130 --> 01:07:10,120 But often, what we do is we simply 886 01:07:10,120 --> 01:07:12,370 put the photons in a trash can. 887 01:07:12,370 --> 01:07:13,080 We trash them. 888 01:07:13,080 --> 01:07:14,910 We're not interested in what the photons are doing. 889 01:07:14,910 --> 01:07:16,297 We're not keeping track of them. 890 01:07:16,297 --> 01:07:18,380 Actually, they hit the wall of our vacuum chamber, 891 01:07:18,380 --> 01:07:22,160 and we couldn't even keep track of them. 892 01:07:22,160 --> 01:07:24,380 The vacuum chamber has taken care of them. 893 01:07:24,380 --> 01:07:26,820 So all that we are interested in-- how do we now 894 01:07:26,820 --> 01:07:29,240 describe the atomic system? 895 01:07:29,240 --> 01:07:31,660 What, after the time evolution, is now 896 01:07:31,660 --> 01:07:35,630 the density matrix of the system? 897 01:07:35,630 --> 01:07:39,615 Of course, if we use the full description, 898 01:07:39,615 --> 01:07:41,490 we know the initial state of the environment. 899 01:07:41,490 --> 01:07:43,470 We know the initial state or our system. 900 01:07:43,470 --> 01:07:47,230 We propagate it exactly with the correct time evolution, 901 01:07:47,230 --> 01:07:49,260 we get everything. 902 01:07:49,260 --> 01:07:52,220 And then, we could reduce by doing a partial trace. 903 01:07:52,220 --> 01:07:55,020 We could reduce the description. 904 01:07:55,020 --> 01:07:57,710 What is now the probabilistic description 905 01:07:57,710 --> 01:08:00,270 for density matrix of the atom? 906 01:08:00,270 --> 01:08:02,620 But this is rather complicated. 907 01:08:02,620 --> 01:08:08,080 What we want to do is we want to do it as a derivation. 908 01:08:08,080 --> 01:08:11,290 But in the end, we want to have a formulation which simply 909 01:08:11,290 --> 01:08:16,160 tells us, what is the atomic density 910 01:08:16,160 --> 01:08:25,439 matrix as a function of the initial atomic density matrix? 911 01:08:25,439 --> 01:08:29,109 So in other words, all that happens with the environment-- 912 01:08:29,109 --> 01:08:31,939 that there's initial state, that it gets entangled, 913 01:08:31,939 --> 01:08:36,189 and then we neglect maybe not keeping track of the photons-- 914 01:08:36,189 --> 01:08:39,300 we're not interested in all of that. 915 01:08:39,300 --> 01:08:43,920 We really want to focus on what happens to the atom. 916 01:08:43,920 --> 01:08:46,779 How does an initial state of the atom 917 01:08:46,779 --> 01:08:49,670 propagate into a final state? 918 01:08:49,670 --> 01:08:52,630 And this is done by optical Bloch equations. 919 01:08:52,630 --> 01:08:55,120 This is done by the master equation. 920 01:08:55,120 --> 01:08:57,800 So the master equation, you can see, 921 01:08:57,800 --> 01:09:00,620 focuses on the relevant part of the system, maybe just 922 01:09:00,620 --> 01:09:01,859 a single atom. 923 01:09:01,859 --> 01:09:04,529 And we can neglect the million photons 924 01:09:04,529 --> 01:09:05,870 which have been emitted. 925 01:09:05,870 --> 01:09:08,029 But those million photons which have 926 01:09:08,029 --> 01:09:09,850 been emitted into the environment, 927 01:09:09,850 --> 01:09:12,500 they change, of course. 928 01:09:12,500 --> 01:09:16,380 They change this density matrix of the atom. 929 01:09:16,380 --> 01:09:20,569 And if you find a description, the master equation 930 01:09:20,569 --> 01:09:24,970 is including with extra terms what those photons have done. 931 01:09:24,970 --> 01:09:26,710 Maybe this sounds very abstract. 932 01:09:26,710 --> 01:09:29,729 But in the end, you will find that maybe photons which 933 01:09:29,729 --> 01:09:32,080 emitted produce some damping. 934 01:09:32,080 --> 01:09:35,500 Or if you put atoms in molasses, the atomic motion 935 01:09:35,500 --> 01:09:37,420 comes to a standstill. 936 01:09:37,420 --> 01:09:39,090 So in other words, we want to develop 937 01:09:39,090 --> 01:09:42,859 a systematic approximation scheme for the master equation 938 01:09:42,859 --> 01:09:46,240 whether effect of all these many, many degrees of freedom 939 01:09:46,240 --> 01:09:51,379 can maybe be simply expressed by a few damping terms. 940 01:09:51,379 --> 01:09:52,170 So that's the idea. 941 01:10:00,000 --> 01:10:36,180 So yes, you know already one example of a master equation. 942 01:10:36,180 --> 01:10:46,320 And these are Einstein's rate equations, 943 01:10:46,320 --> 01:10:51,440 which we have discussed in the first part of the course. 944 01:10:51,440 --> 01:10:55,830 If you have a two level system which 945 01:10:55,830 --> 01:10:59,150 is coupled to the radiation field, 946 01:10:59,150 --> 01:11:02,930 then we obtained equations that the rate 947 01:11:02,930 --> 01:11:05,920 of change of the ground state population 948 01:11:05,920 --> 01:11:09,730 is related to the excited state population 949 01:11:09,730 --> 01:11:12,990 through spontaneous emission described by the Einstein e 950 01:11:12,990 --> 01:11:15,020 coefficient. 951 01:11:15,020 --> 01:11:18,130 And if you have a spectral density 952 01:11:18,130 --> 01:11:28,290 of the electromagnetic field, it causes stimulated emission. 953 01:11:28,290 --> 01:11:39,556 And it causes absorption described by the Einstein b 954 01:11:39,556 --> 01:11:40,055 coefficient. 955 01:11:44,200 --> 01:11:47,830 And you have a similar equation for the excited state. 956 01:11:47,830 --> 01:11:57,890 So this is clearly the semi-classical limit 957 01:11:57,890 --> 01:12:00,400 of what we want to accomplish. 958 01:12:00,400 --> 01:12:01,500 But we want to know more. 959 01:12:01,500 --> 01:12:04,360 We really want to know not just, what 960 01:12:04,360 --> 01:12:06,460 is the rate equation for the atom? 961 01:12:06,460 --> 01:12:11,560 We want to know, you can see, the wave function. 962 01:12:11,560 --> 01:12:14,160 Well, wave function slash statistical operator-- 963 01:12:14,160 --> 01:12:15,700 the statistical operator contains 964 01:12:15,700 --> 01:12:17,790 all information of the wave function. 965 01:12:17,790 --> 01:12:20,397 And if the wave function is a pure state, 966 01:12:20,397 --> 01:12:21,980 it has a certain statistical operator. 967 01:12:21,980 --> 01:12:24,670 And we can find the pure state. 968 01:12:24,670 --> 01:12:26,050 So that's what we want to do. 969 01:12:26,050 --> 01:12:28,650 But sort of just genetically, we really 970 01:12:28,650 --> 01:12:33,410 want to find the full quantum time evolution. 971 01:12:33,410 --> 01:12:36,810 And now I just want to express that. 972 01:12:36,810 --> 01:12:38,510 We have to be careful. 973 01:12:38,510 --> 01:12:42,650 The time evolution as a Hamiltonian, 974 01:12:42,650 --> 01:12:45,350 if you now bring in the environment, 975 01:12:45,350 --> 01:12:52,920 cannot be simply included by adding an imaginary term. 976 01:12:52,920 --> 01:12:59,355 This here violates the unitary time evolution. 977 01:13:07,420 --> 01:13:12,400 In other words, when we find an equation for the quantum 978 01:13:12,400 --> 01:13:16,930 system, how it evolves, it's not that everything 979 01:13:16,930 --> 01:13:19,460 you think which phenomenologically works will 980 01:13:19,460 --> 01:13:20,160 work. 981 01:13:20,160 --> 01:13:23,030 It has to be consistent with the loss of quantum physics. 982 01:13:23,030 --> 01:13:28,690 In other words, when we find an equation for the statistical-- 983 01:13:28,690 --> 01:13:33,690 if you find an equation which describes the atomic system, 984 01:13:33,690 --> 01:13:36,610 it will be a requirement that a density matrix 985 01:13:36,610 --> 01:13:38,500 turns into a density matrix. 986 01:13:38,500 --> 01:13:41,510 So certain structure has to be obeyed. 987 01:13:41,510 --> 01:13:44,250 And that is actually extremely restrictive. 988 01:13:44,250 --> 01:13:46,530 And our derivation of the master equation 989 01:13:46,530 --> 01:13:52,190 will actually show what kind of operators applied 990 01:13:52,190 --> 01:13:56,335 to the atomic density matrix are consistent, are 991 01:13:56,335 --> 01:13:58,620 quantum mechanically consistent. 992 01:13:58,620 --> 01:14:01,020 This is actually something-- well, 993 01:14:01,020 --> 01:14:03,370 we're not doing a lot of equations or work today, 994 01:14:03,370 --> 01:14:08,070 so let me rather be a little bit chatty. 995 01:14:08,070 --> 01:14:12,470 This is actually something which is the frontier of our field, 996 01:14:12,470 --> 01:14:15,910 both in theory in ion traps and with neutral atoms, 997 01:14:15,910 --> 01:14:18,870 that we have some evolution of an atomic system 998 01:14:18,870 --> 01:14:21,880 by coupling it to the environment. 999 01:14:21,880 --> 01:14:25,380 Well, the usual environment you can think of 1000 01:14:25,380 --> 01:14:28,350 is just taking photons away. 1001 01:14:28,350 --> 01:14:31,750 It gives us a damping term of the excited state. 1002 01:14:31,750 --> 01:14:33,650 But now you can ask the question, 1003 01:14:33,650 --> 01:14:36,330 can you construct an environment which 1004 01:14:36,330 --> 01:14:38,840 has some degrees of freedom with laser fields, RF 1005 01:14:38,840 --> 01:14:39,760 fields, and such? 1006 01:14:39,760 --> 01:14:43,660 You call it the environment, and this environment 1007 01:14:43,660 --> 01:14:46,710 does something really fancy. 1008 01:14:46,710 --> 01:14:48,310 And the system comes in equilibrium 1009 01:14:48,310 --> 01:14:50,300 with the environment. 1010 01:14:50,300 --> 01:14:52,970 But could you engineer the environment 1011 01:14:52,970 --> 01:14:55,620 that the system comes into equilibrium 1012 01:14:55,620 --> 01:14:58,870 with the environment, and it is in a fancy super fluid 1013 01:14:58,870 --> 01:15:01,960 or fancy entangled state? 1014 01:15:01,960 --> 01:15:05,230 So can you engineer the environment in such a way 1015 01:15:05,230 --> 01:15:09,480 that it does something really fancy to your system? 1016 01:15:09,480 --> 01:15:11,770 Well, you can dream of it. 1017 01:15:11,770 --> 01:15:15,865 But you dreams are restricted by the mathematical structure 1018 01:15:15,865 --> 01:15:19,000 of all possible master equations in the world. 1019 01:15:19,000 --> 01:15:23,710 Because the environment cannot do everything for you. 1020 01:15:23,710 --> 01:15:31,370 The environment can only do for you 1021 01:15:31,370 --> 01:15:35,690 what can come out of all possible Hamiltonians. 1022 01:15:35,690 --> 01:15:38,440 And the fact that the total system 1023 01:15:38,440 --> 01:15:42,060 evolves in a unitary way with the total Hamiltonian 1024 01:15:42,060 --> 01:15:45,320 of the system, this is really restricting. 1025 01:15:45,320 --> 01:15:48,660 This operator sort of stands for all possible master equations. 1026 01:15:48,660 --> 01:15:50,710 It's restricting the master equation 1027 01:15:50,710 --> 01:15:53,830 for your atomic density matrix. 1028 01:15:53,830 --> 01:15:55,693 There was just-- I think was it this year 1029 01:15:55,693 --> 01:15:59,940 or last year-- a nice science and nature paper 1030 01:15:59,940 --> 01:16:03,300 by [INAUDIBLE] where they engineered 1031 01:16:03,300 --> 01:16:08,040 the environment around ions in an ion trap. 1032 01:16:08,040 --> 01:16:10,670 And that stabilized the ion in the state, 1033 01:16:10,670 --> 01:16:13,760 which was an unusual state, not what the normal environment 1034 01:16:13,760 --> 01:16:15,400 would have done. 1035 01:16:15,400 --> 01:16:18,590 So anyway, what I will be telling 1036 01:16:18,590 --> 01:16:21,200 you is also sort of relevant to understand 1037 01:16:21,200 --> 01:16:24,020 this sort of frontier in our field which 1038 01:16:24,020 --> 01:16:25,510 is called environment engineering. 1039 01:16:37,890 --> 01:16:46,020 OK, density matrix-- good, five more minutes. 1040 01:16:52,940 --> 01:17:08,500 So what we have is we have a system. 1041 01:17:08,500 --> 01:17:11,420 And we have this environment. 1042 01:17:11,420 --> 01:17:16,350 And what we are exchanging with the environment 1043 01:17:16,350 --> 01:17:20,760 is both energy, but also entropy. 1044 01:17:24,350 --> 01:17:32,500 And so when we transfer energy or heat, 1045 01:17:32,500 --> 01:17:36,660 there is a corresponding change in energy. 1046 01:17:36,660 --> 01:17:41,450 And it's a general property of all quantum systems. 1047 01:17:41,450 --> 01:17:44,610 It's a consequence of the fluctuation dissipation 1048 01:17:44,610 --> 01:17:47,250 principle, the fluctuation dissipation theorem, 1049 01:17:47,250 --> 01:17:51,140 that you cannot have any form of relaxation without noise. 1050 01:18:10,920 --> 01:18:17,830 So for instance, when we discuss optical molasses-- 1051 01:18:17,830 --> 01:18:21,170 we do that in a few weeks, where the atomic motion is damped-- 1052 01:18:21,170 --> 01:18:23,290 well, you have a damping term. 1053 01:18:23,290 --> 01:18:26,050 It seems it brings some motion through friction 1054 01:18:26,050 --> 01:18:27,540 to a standstill. 1055 01:18:27,540 --> 01:18:29,600 But we know by general principles 1056 01:18:29,600 --> 01:18:32,220 that damping is not possible without noise. 1057 01:18:32,220 --> 01:18:35,400 So when somebody sells you a wonderful damping scheme which 1058 01:18:35,400 --> 01:18:37,460 damps the motion, you should always ask, 1059 01:18:37,460 --> 01:18:38,510 but there must be noise. 1060 01:18:38,510 --> 01:18:40,150 What is the ultimate noise? 1061 01:18:40,150 --> 01:18:42,650 And it's fundamental that it is there. 1062 01:18:42,650 --> 01:18:46,400 So therefore, our derivation of the master equation 1063 01:18:46,400 --> 01:18:49,750 will also display that, that we do not 1064 01:18:49,750 --> 01:18:53,870 get any form of damping without at least the fundamental 1065 01:18:53,870 --> 01:18:56,250 quantum noise. 1066 01:18:56,250 --> 01:19:08,930 So what we need is we need a description of the quantum 1067 01:19:08,930 --> 01:19:13,360 noise, which comes from coupling to the environment. 1068 01:19:22,530 --> 01:19:28,830 The tool which we use for that is the density matrix. 1069 01:19:34,200 --> 01:19:36,700 I assume everybody here is familiar with the density 1070 01:19:36,700 --> 01:19:38,560 matrix. 1071 01:19:38,560 --> 01:19:41,750 The atomic lecture notes on the wiki 1072 01:19:41,750 --> 01:19:46,170 have a small tutorial on the density matrix 1073 01:19:46,170 --> 01:19:51,380 if you want to freshen up your knowledge about density matrix, 1074 01:19:51,380 --> 01:19:52,620 maybe read about it. 1075 01:19:52,620 --> 01:19:55,070 I don't want to cover it in class. 1076 01:20:03,160 --> 01:20:06,740 The one thing we need, and I just want to remind you, 1077 01:20:06,740 --> 01:20:16,470 is for all density matrices, you can always unravel it. 1078 01:20:19,250 --> 01:20:22,360 The density matrix can be written 1079 01:20:22,360 --> 01:20:30,005 as a probabilistic sum over states. 1080 01:20:34,400 --> 01:20:39,040 This will actually play a major role. 1081 01:20:39,040 --> 01:20:43,165 We will make certain models for damping. 1082 01:20:43,165 --> 01:20:45,210 And it's really beautiful. 1083 01:20:45,210 --> 01:20:49,150 On Monday, I will give you the beam splitter model 1084 01:20:49,150 --> 01:20:50,505 for the optical Bloch equation. 1085 01:20:50,505 --> 01:20:51,500 I really like it. 1086 01:20:51,500 --> 01:20:54,080 Because it's a microscopic model. 1087 01:20:54,080 --> 01:20:58,440 And it shows you a lot of fundamental principles. 1088 01:20:58,440 --> 01:21:01,320 But the important part is whenever 1089 01:21:01,320 --> 01:21:05,215 we have a way to construct a density matrix by saying, 1090 01:21:05,215 --> 01:21:07,420 we have certain quantum states k, 1091 01:21:07,420 --> 01:21:10,600 and we just add them up probabilistically, 1092 01:21:10,600 --> 01:21:13,730 this kind of microscopic interpretation of the density 1093 01:21:13,730 --> 01:21:15,740 matrix is called unravelling. 1094 01:21:15,740 --> 01:21:19,770 It's sort of writing it as a specific diagonal sum 1095 01:21:19,770 --> 01:21:20,800 over states. 1096 01:21:20,800 --> 01:21:26,860 But those unravellings are not unique. 1097 01:21:26,860 --> 01:21:28,745 They describe one possibility. 1098 01:21:28,745 --> 01:21:31,660 But there are other possibilities. 1099 01:21:31,660 --> 01:21:42,240 And the one example which I can give you 1100 01:21:42,240 --> 01:21:45,540 is that if you have a density matrix like this, 1101 01:21:45,540 --> 01:21:52,820 you can write the density matrix as this form suggests, 1102 01:21:52,820 --> 01:22:02,240 as a probabilistic sum of being in the state 0, 0, 1103 01:22:02,240 --> 01:22:09,350 or in the state 1, 1 with probability 1/4 and 3/4. 1104 01:22:09,350 --> 01:22:17,070 But you can also write it as being with equal probability 1105 01:22:17,070 --> 01:22:30,130 in two states a and b where the states a and b 1106 01:22:30,130 --> 01:22:37,420 are superpositions of state 0 and 1. 1107 01:22:37,420 --> 01:22:49,390 And you can see by inspection that this will do the trick. 1108 01:22:49,390 --> 01:22:52,020 So we'll talk a lot about unravelling of the density 1109 01:22:52,020 --> 01:22:52,520 matrix. 1110 01:22:52,520 --> 01:22:53,936 That's why I want to say up front, 1111 01:22:53,936 --> 01:22:57,385 that the same density matrix can be thought 1112 01:22:57,385 --> 01:23:02,710 of as being created by different processes. 1113 01:23:02,710 --> 01:23:06,150 But this actually makes it even more powerful. 1114 01:23:06,150 --> 01:23:08,810 Because we have a unified description, or even 1115 01:23:08,810 --> 01:23:11,505 an identical description, for different microscopic 1116 01:23:11,505 --> 01:23:12,005 processes. 1117 01:23:15,120 --> 01:23:16,835 OK, any last questions? 1118 01:23:19,605 --> 01:23:23,200 Well then, let's enjoy the open house with incoming graduate 1119 01:23:23,200 --> 01:23:25,940 students, and I'll see you on Monday.