1 00:00:00,890 --> 00:00:04,430 Finally in this video we're going to discuss a very interesting class of 2 00:00:04,430 --> 00:00:08,324 quantum phenomenon that appear in the presence of slow, or so called adiabatic 3 00:00:08,324 --> 00:00:14,964 time-dependent perturbations. And such adiabatic perturbations give 4 00:00:14,964 --> 00:00:20,670 rise to a very elegant mathematical structure. 5 00:00:20,670 --> 00:00:24,828 And, in particular, this so called geometric, Berry Phase that we're going 6 00:00:24,828 --> 00:00:29,733 to derive. so the this Barry Phase was discovered 7 00:00:29,733 --> 00:00:36,380 theoretically by Sir Michael Barry. And this paper, published in 1984 in the 8 00:00:36,380 --> 00:00:40,800 Proceedings of the Royal Society of London. 9 00:00:40,800 --> 00:00:45,350 And this is a really very well written and elegant paper, and actually even 10 00:00:45,350 --> 00:00:50,110 though it's a research-level paper which in all likelihood may lead to a Nobel 11 00:00:50,110 --> 00:00:54,870 Prize for Sir Michael Berry you are well-equipped to actually understand 12 00:00:54,870 --> 00:01:01,816 everything in this paper. So now, both the derivation and the final 13 00:01:01,816 --> 00:01:05,346 results. because it requires nothing but the basic 14 00:01:05,346 --> 00:01:09,580 knowledge of single particle quantum mechanics and Schrodinger equation. 15 00:01:09,580 --> 00:01:13,927 So we're going to reproduce part of the derivation in today's lecture, in this 16 00:01:13,927 --> 00:01:18,412 part but of course if you want to know more details I would like to refer you to 17 00:01:18,412 --> 00:01:25,140 the original paper which again is very well written. 18 00:01:25,140 --> 00:01:29,820 So the problem that Sir Michael Berry considered was well, in retrospect a very 19 00:01:29,820 --> 00:01:33,668 simple problem. So basically it was the canonical 20 00:01:33,668 --> 00:01:37,320 Schrödinger equation with some time-dependent Hamiltonian. 21 00:01:37,320 --> 00:01:43,870 And he assumed two things. he looked into adiabatic perturbation. 22 00:01:43,870 --> 00:01:53,680 That is to say that So the time-dependent in h of t is, is assumed to be slow. 23 00:01:53,680 --> 00:01:59,760 what it means precisely, we're going to define in the next few slides, but at 24 00:01:59,760 --> 00:02:07,370 this stage that's just leave it at that. So we have some very slowly changing 25 00:02:07,370 --> 00:02:12,807 Hamiltonian. And also let's assume that the 26 00:02:12,807 --> 00:02:20,720 Hamiltonian returns to itself after a certain period let me call it capital T. 27 00:02:20,720 --> 00:02:25,370 So basically, we have a periodic in time slow perturbation. 28 00:02:25,370 --> 00:02:30,426 And the question that he asked is what happens when the wave function As we get 29 00:02:30,426 --> 00:02:36,479 to this moment of time, capital T. So what is the wave function at the final 30 00:02:36,479 --> 00:02:40,460 moment of time? So and, it turned out that, as we will 31 00:02:40,460 --> 00:02:45,035 see, so that this wave function has a very interesting contribution sort of 32 00:02:45,035 --> 00:02:51,130 topilogical, or geometric contributions. There are various sort of descriptions of 33 00:02:51,130 --> 00:02:56,990 it that appear in the literature. that was completely counterintuitive, and 34 00:02:56,990 --> 00:03:00,880 that takes some time to digest. even after a derivation. 35 00:03:02,220 --> 00:03:06,084 So now let me formulate the problem that he solved more precisely and proceed to 36 00:03:06,084 --> 00:03:12,041 the actual solution of the problem. And to the formulation of these Theory of 37 00:03:12,041 --> 00:03:16,726 the Berry phase. So, let's imagine that we have a 38 00:03:16,726 --> 00:03:21,310 Hamiltonian that depends on a parameter lambda, okay? 39 00:03:21,310 --> 00:03:25,506 So this could be a, anything. For example, it can be magnetic field or 40 00:03:25,506 --> 00:03:29,820 it can be some parameters of a harmonic oscillator. 41 00:03:29,820 --> 00:03:34,520 let's say, the frequency or something else, whatever you want to think about. 42 00:03:34,520 --> 00:03:38,230 So let's have with some we have some set of parameters lambda And we combine those 43 00:03:38,230 --> 00:03:43,188 parameter in a vector so in principle it could be more than one parameter. 44 00:03:43,188 --> 00:03:46,898 And we just construct a, sort of a vector in the parameter space which does not 45 00:03:46,898 --> 00:03:50,184 have to be three-dimensional but let me just for sake of simplicity, let me 46 00:03:50,184 --> 00:03:55,410 imagine that they have a three-dimensional parameter space. 47 00:03:55,410 --> 00:04:00,265 We will say this is going to be lambda x or lambda one, lambda two, lambda three. 48 00:04:00,265 --> 00:04:04,780 Okay. And let me consider the station where 49 00:04:04,780 --> 00:04:08,540 this parameter lambda is changing in, in time. 50 00:04:08,540 --> 00:04:12,820 So, for example, again, it can be a magnetic field that is changing. 51 00:04:12,820 --> 00:04:18,571 And also, let me consider the situation where the after a certain period of time 52 00:04:18,571 --> 00:04:24,010 capital T, so the parameter. Returns to the original point. 53 00:04:24,010 --> 00:04:27,400 So if this is my parameter space. So, basically what I'm talking about. 54 00:04:27,400 --> 00:04:31,960 I'm talking about a slow evolution of this lambda such that it starts at a 55 00:04:31,960 --> 00:04:38,440 certain point and returns to the same point after, time capital T. 56 00:04:38,440 --> 00:04:43,137 So, and as I advertised in the previous slide, so what I want to know Is what 57 00:04:43,137 --> 00:04:48,982 happens with the wave function? So, I have some initial condition for the 58 00:04:48,982 --> 00:04:51,190 wave function. And, 59 00:04:51,190 --> 00:04:57,150 So, the Hamiltonian, after these period. Capital T returns to itself. 60 00:04:57,150 --> 00:05:00,750 So, but the wave function, does not necessarily have to return to itself. 61 00:05:00,750 --> 00:05:04,150 So the question that we're interest in is what happens? 62 00:05:04,150 --> 00:05:08,084 With this wave function. As a first step in this derivation, let 63 00:05:08,084 --> 00:05:12,690 me define precisely what I mean by slow, or, adiabatic perturbation. 64 00:05:13,750 --> 00:05:19,870 So what does it actually mean, that the Hamiltonian is changing slowly with time? 65 00:05:19,870 --> 00:05:24,575 And to provide this definition let me formulate an eigenvalue problem. 66 00:05:24,575 --> 00:05:27,231 Instantaneous eigenvalue problem for the Hamiltonian. 67 00:05:27,231 --> 00:05:30,559 So the Hamiltonian itself is changing with time so there is no reason for us to 68 00:05:30,559 --> 00:05:33,670 write the stationary Schrodinger equation. 69 00:05:33,670 --> 00:05:36,580 But never the less mathematically it is well defined. 70 00:05:36,580 --> 00:05:40,466 So, we can consider at any moment of time t, we can consider an eigenvalue problem 71 00:05:40,466 --> 00:05:45,440 for this h of t. And we can find a set of eigenfunctions. 72 00:05:45,440 --> 00:05:52,240 And that the responding eigen values which I denote as e sub n of t. 73 00:05:52,240 --> 00:05:57,060 So, for the sake of simplicity let me actually assume that the 74 00:05:57,060 --> 00:06:01,320 Spectrum of the Hamiltonian is discrete. So I have just a set of discrete levels. 75 00:06:01,320 --> 00:06:06,630 So n equals 0, n equals 1, et cetera. Also, let me assume that the initial 76 00:06:06,630 --> 00:06:11,186 condition for the wave function, psi at t equals 0, is one of the eigenstates of my 77 00:06:11,186 --> 00:06:17,258 Hamiltonian at t equals 0. So, for the sake of concreteness, let me 78 00:06:17,258 --> 00:06:22,020 actually assume that this is the ground state. 79 00:06:22,020 --> 00:06:26,066 So n equals 0 at t equals 0. So by the way, it's not necessarily 80 00:06:26,066 --> 00:06:29,360 really for the main conclusions of the derivation, but for the sake of 81 00:06:29,360 --> 00:06:34,808 completeness, let me just be specific. Okay, now let us recall what we discussed 82 00:06:34,808 --> 00:06:38,702 actually in the previous lecture, in the previous part of this lecture where we 83 00:06:38,702 --> 00:06:44,780 dealt with the opposite case of fast or sudden perturbations. 84 00:06:44,780 --> 00:06:48,608 So there, we know that if we change the Hamiltonian very fast, the state, the 85 00:06:48,608 --> 00:06:51,972 initial state, which in this case is the ground state, would be sort of 86 00:06:51,972 --> 00:06:57,707 redistributed all over the place. so the particle after the quench, or 87 00:06:57,707 --> 00:07:01,542 after a fast sudden change, would exist in all possible eigenstates of the new 88 00:07:01,542 --> 00:07:06,351 Hamiltonian. And sometimes it's natural to define a 89 00:07:06,351 --> 00:07:12,160 small perterbation as the opposite to the sudden perturbation. 90 00:07:12,160 --> 00:07:16,896 That is to say that, we will define a small perterbation aside that it does not 91 00:07:16,896 --> 00:07:22,270 induce transitions. So, basically what happens here is, let's 92 00:07:22,270 --> 00:07:25,700 say this is the parameter lambda. It's equal to 0. 93 00:07:25,700 --> 00:07:30,070 And we have our initial state, the ground state, with n equals 0. 94 00:07:30,070 --> 00:07:34,480 And, as I change my lambda, so, with time. 95 00:07:34,480 --> 00:07:39,167 So, these energy levels start to move. So, the energy levels really are this 96 00:07:39,167 --> 00:07:43,690 Solutions to this, eigenvalue problem. But the Hamiltonian was changing. 97 00:07:43,690 --> 00:07:47,810 And so the, energy levels are moving. And the principal. 98 00:07:47,810 --> 00:07:51,340 Also, the explicit, wave functions are changing. 99 00:07:51,340 --> 00:07:53,930 So, and what I can imagine happening is that. 100 00:07:53,930 --> 00:07:57,240 The wave, the particle, which initially, was a. 101 00:07:57,240 --> 00:08:01,464 sort of confined to the ground state canon principal propagate to all other 102 00:08:01,464 --> 00:08:04,850 states. And this indeed will, will happen if the 103 00:08:04,850 --> 00:08:09,950 change in Hamiltonian is fast enough. But if this is slow, and this will give 104 00:08:09,950 --> 00:08:14,525 the definition of slow, the the state of the system would remain in the 105 00:08:14,525 --> 00:08:20,160 instantaneous ground state of my hamiltonian. 106 00:08:20,160 --> 00:08:23,778 And also in this case, if I start form this point in the parameter space and 107 00:08:23,778 --> 00:08:27,396 return to the same point in the parameter space so that at the end of this periodic 108 00:08:27,396 --> 00:08:31,068 evolution basically I require then that the system, that the system would remain 109 00:08:31,068 --> 00:08:36,544 in the old ground state. And this will be my definition of a slow 110 00:08:36,544 --> 00:08:40,380 perturbation. So to mathematically define a precise 111 00:08:40,380 --> 00:08:44,340 condition for this to happen requires a little bit of work and that provide this 112 00:08:44,340 --> 00:08:49,620 attenuation in the supplementary material in the notes. 113 00:08:49,620 --> 00:08:53,845 And if you go through the attenuation, you will see that the condition for this 114 00:08:53,845 --> 00:08:58,330 to happen for the condition for having no transitions Is that the derivative of the 115 00:08:58,330 --> 00:09:02,360 Hamiltonian, which is an operator, and the matrix element of this d H d t 116 00:09:02,360 --> 00:09:06,780 between two states, let's say this ground state and for example the first excited 117 00:09:06,780 --> 00:09:15,710 state, this must be much smaller than the level spacing, the distance. 118 00:09:15,710 --> 00:09:20,964 between these energy levels divided by the typical time well, the concentric 119 00:09:20,964 --> 00:09:26,690 time at which the change in the Hamiltonian occurs. 120 00:09:26,690 --> 00:09:31,100 So this is the definition if you want, if you want or, the con, con, constraint 121 00:09:31,100 --> 00:09:37,290 that we have to satisfy in order for the perturbation to be slow. 122 00:09:37,290 --> 00:09:43,641 And one thing that we that we notice here is that, of course, if e n minus e of 0 123 00:09:43,641 --> 00:09:48,310 vanishes. That is, if the levels cross, or if they 124 00:09:48,310 --> 00:09:52,120 come very close to one another. So this condition cannot be satisfied 125 00:09:52,120 --> 00:09:56,360 because when necessary we'll induce transitions between these states. 126 00:09:56,360 --> 00:10:01,190 But if the gaps don't close. If the level spacing remains large enough 127 00:10:01,190 --> 00:10:05,150 and if the change in the connectonium is slow enough so we can satisfy this 128 00:10:05,150 --> 00:10:09,039 condition. An do good approximation, we can assume 129 00:10:09,039 --> 00:10:12,940 that the system remains in the ground state during the revolution. 130 00:10:14,190 --> 00:10:17,728 So, now we are at the position to actually derive this geometric berry 131 00:10:17,728 --> 00:10:20,934 phase. And what it entails is essentially 132 00:10:20,934 --> 00:10:25,460 solving the Schrodinger equation, the time dependent Schrodinger equation under 133 00:10:25,460 --> 00:10:31,610 the assumption of this slow perturbation that does not induce the transitions. 134 00:10:31,610 --> 00:10:37,590 So, this is my Schrodinger equation. I'm just writing it down again. 135 00:10:37,590 --> 00:10:45,670 And here I'm basically requiring that psi of t remains proportional to the ground 136 00:10:45,670 --> 00:10:51,227 state. And the only thing I can have here is a 137 00:10:51,227 --> 00:10:52,800 phase. Right? 138 00:10:52,800 --> 00:10:57,408 So, because, well, the total number of particles s conserved, so therefore I 139 00:10:57,408 --> 00:11:02,664 cannot have here any, coefficient whose absolute value is larger than 1 because 140 00:11:02,664 --> 00:11:07,488 it would imply that the probability to find a the particle in this instantaneous 141 00:11:07,488 --> 00:11:15,230 ground is larger than 1, and it doesn't make any sense. 142 00:11:15,230 --> 00:11:18,025 So, and I also require that well, within my approximation, there are no 143 00:11:18,025 --> 00:11:21,160 transitions, so the particle remains in the ground state. 144 00:11:21,160 --> 00:11:25,580 So there's basically only, only one possibility for this to happen. 145 00:11:25,580 --> 00:11:30,216 if this coefficient relating the wave function at time t, and the instantaneous 146 00:11:30,216 --> 00:11:36,205 ground state is a pure phase, either for i phi of t, or if i is a real function. 147 00:11:36,205 --> 00:11:42,590 So under these assumptions, what remains is is to find this one phase. 148 00:11:42,590 --> 00:11:45,419 Everything else we know so this is really the remaining unknown in our [INAUDIBLE] 149 00:11:45,419 --> 00:11:49,372 time-dependent problem. And if we plug in this form of the 150 00:11:49,372 --> 00:11:54,286 solution back into the Schrodinger equation and recall by definition that 151 00:11:54,286 --> 00:12:01,680 this 0, t is the instantaneous eigenstate of the Hamiltonian. 152 00:12:01,680 --> 00:12:06,183 So simply get, so basically combining these 2 things will lead to the 153 00:12:06,183 --> 00:12:12,358 Schrodinger equation in this form. So it will be I H D over D T of phi of T, 154 00:12:12,358 --> 00:12:20,530 in this form is equal to the energy of teh ground state which is time dependant. 155 00:12:20,530 --> 00:12:23,003 And this is because energy is moving around. 156 00:12:23,003 --> 00:12:26,789 si of t. Okay? 157 00:12:26,789 --> 00:12:31,920 And well so this is no longer an operator actually. 158 00:12:31,920 --> 00:12:36,470 This is just a function and naively you may sort of assume there exists a simple 159 00:12:36,470 --> 00:12:41,620 solution to this problem. where if I would be equal to sort of a 160 00:12:41,620 --> 00:12:46,450 generalization of usual quantum phase that appears even in the time-independent 161 00:12:46,450 --> 00:12:51,122 problems. So we could naively just write it as 162 00:12:51,122 --> 00:12:57,040 minus 1 over h bar, an integral from zero to t, E naught of t prime, d t prime. 163 00:12:57,040 --> 00:13:02,570 So again, if the Hamiltonian were not to depend, on time, that is, in this picture 164 00:13:02,570 --> 00:13:08,780 would mean that basically Lambda's a point which stays there. 165 00:13:08,780 --> 00:13:13,068 So I would just get the quantum phase being e to the power i minus i over h 166 00:13:13,068 --> 00:13:17,292 bar, energy times time, which is the usual quantum phase that appears in 167 00:13:17,292 --> 00:13:23,417 stationary quantum mechanics. So here, instead, I may just go ahead and 168 00:13:23,417 --> 00:13:28,119 integrate from 0 to t. Well it turns out however that this is 169 00:13:28,119 --> 00:13:31,215 not entirely correct. And there exists an important 170 00:13:31,215 --> 00:13:35,440 contribution which appears in well, to this quantum fix, and let me write this 171 00:13:35,440 --> 00:13:40,476 contribution as gamma. And this addtional phase gamma that we 172 00:13:40,476 --> 00:13:44,800 will derive in a second is exactly what is called the Berry phase. 173 00:13:44,800 --> 00:13:48,008 So this is my, our main quantity of interest. 174 00:13:48,008 --> 00:13:51,019 While the first part is, is sort of the usual dynamical phase. 175 00:13:51,019 --> 00:13:59,014 It's actually called dynamical phase, let me, you note it as Theta with the 176 00:13:59,014 --> 00:14:07,750 subscript of D, and this is called dynamical phase. 177 00:14:09,910 --> 00:14:13,514 If we plug in now, back, you know, basically, this expression for the phase 178 00:14:13,514 --> 00:14:17,224 back into the Schrodinger equation, we're going to get in the left hand side, we're 179 00:14:17,224 --> 00:14:24,231 going to get the following term. So there's going to be derivative pf e to 180 00:14:24,231 --> 00:14:31,000 the power i theta, e to the power i gamma. 181 00:14:31,000 --> 00:14:35,818 And there will be the instantaneous ground state, 0 t, and the right hand 182 00:14:35,818 --> 00:14:43,101 side will remain, will remain the same. So in here, we have the identity which is 183 00:14:43,101 --> 00:14:46,614 going to act on. Three terms so they are going to, they 184 00:14:46,614 --> 00:14:49,670 are going to be three contributions in the left hand side. 185 00:14:49,670 --> 00:14:52,866 So the first contribution essentially, by construction, is going to cancel out the 186 00:14:52,866 --> 00:14:56,369 right hand side. So if we differentiate this exponential 187 00:14:56,369 --> 00:15:03,265 so let me just write it down. It's going to be i, h bar times i theta d 188 00:15:03,265 --> 00:15:07,700 dot. Times psi. 189 00:15:07,700 --> 00:15:14,286 And this theta d dot is this the derivative of this integral, which will 190 00:15:14,286 --> 00:15:21,550 just pull out the value of the energy. So this whole thing is going to be equal 191 00:15:21,550 --> 00:15:25,816 to the energy. as a function of time, which will be 192 00:15:25,816 --> 00:15:29,630 cancelled by the same term in the right hand side. 193 00:15:29,630 --> 00:15:32,690 So, there's going to be also another term here. 194 00:15:32,690 --> 00:15:37,010 This is going to be i h bar i gamma dot times psi. 195 00:15:37,010 --> 00:15:42,446 And finally, there's going to be a derivative of the wave function itself. 196 00:15:42,446 --> 00:15:48,333 So the eigenstates are changing. So the Hamiltonian is changing and so are 197 00:15:48,333 --> 00:15:54,589 the the eigenstate, so therefore there's going to be another term so let me write 198 00:15:54,589 --> 00:16:01,680 a plus i h bar e to the power i theta plus gamma. 199 00:16:01,680 --> 00:16:05,586 Here we're going to have the derivative of this of this guy and the right-hand 200 00:16:05,586 --> 00:16:10,440 side is e naught times psi. Which as I already mentioned, is going to 201 00:16:10,440 --> 00:16:14,853 be cancelled precisely by the first term in the left hand side. 202 00:16:14,853 --> 00:16:18,950 That's why we introduced this dynamical phase in the first place. 203 00:16:18,950 --> 00:16:23,250 And now, what remains is simply to balance out these 2 terms. 204 00:16:23,250 --> 00:16:25,540 And as you can see, a lot of things cancel out. 205 00:16:25,540 --> 00:16:29,800 So the Planc constant cancels out, so the phase factors. 206 00:16:29,800 --> 00:16:35,410 In this term are going to be canceled by the corresponding phase factors in this 207 00:16:35,410 --> 00:16:39,540 wave function. So this goes away. 208 00:16:39,540 --> 00:16:43,840 And I times I is equal to minus one, and so what you can write is the following. 209 00:16:43,840 --> 00:16:48,366 So let's say on the left hand side we're going to have a gamma dot the wave 210 00:16:48,366 --> 00:16:53,111 function, the ground state wave function is equal, basically, to this guy, i d 211 00:16:53,111 --> 00:16:59,591 over dt 0t. So finally, what we can do here, we can 212 00:16:59,591 --> 00:17:06,780 take advantage of the fact that we have normalized states. 213 00:17:06,780 --> 00:17:10,182 So the number of particles is conserved, and therefore, and hold up, and our 214 00:17:10,182 --> 00:17:13,908 particles exist in this ground state, therefore, this bracket product of this 215 00:17:13,908 --> 00:17:18,360 instantaneous ground state with itself is equal to one. 216 00:17:18,360 --> 00:17:22,264 And so if we multiply this equation from the left by a broad vector of zero t, so 217 00:17:22,264 --> 00:17:26,990 we're going to have simply one in the left hand side. 218 00:17:26,990 --> 00:17:32,490 And we're going to have you know, zero t g over d t zero t in the right hand side. 219 00:17:32,490 --> 00:17:38,855 So therefore, what we can do, we can write the, the Berry phase, well, this 220 00:17:38,855 --> 00:17:47,360 Berry phase gamma at the time T as an integral from 0 to capital T. 221 00:17:47,360 --> 00:17:51,018 So this is the time that it takes the particle, well, the, the system, to 222 00:17:51,018 --> 00:17:59,490 return to the original Hamiltonian. And under this integral we're going to 223 00:17:59,490 --> 00:18:07,770 have zero T, D over D T, zero T, D T. So it's an integral over time. 224 00:18:07,770 --> 00:18:13,670 So we get this interesting term which is the Berry phase. 225 00:18:13,670 --> 00:18:16,918 And and as we shall discuss in the last slide, so this term has a very 226 00:18:16,918 --> 00:18:22,270 interesting, geometric rotation, that's why it's called a geometric phase. 227 00:18:23,770 --> 00:18:28,326 To see this let us recall the definition of these eigenstates, zero T, so those 228 00:18:28,326 --> 00:18:33,250 are the instantaneous eigenstates of the Hamiltonian. 229 00:18:33,250 --> 00:18:37,154 But, per our assumption, so the Hamiltonian depends on time only through 230 00:18:37,154 --> 00:18:41,790 the parameter lambda, this extra parameter lambda. 231 00:18:41,790 --> 00:18:45,572 So instead of writing this guise as being parametrized by t, we can as well 232 00:18:45,572 --> 00:18:51,160 parametrize them by this parameter lambda, in this parameter space here. 233 00:18:51,160 --> 00:18:56,788 And if we do so, we can transform this integral over time into an integral over 234 00:18:56,788 --> 00:19:03,310 lambda in this parameter space. And this would require a very curious, as 235 00:19:03,310 --> 00:19:08,500 I said, geometric interpretation. So almost the last step here is to 236 00:19:08,500 --> 00:19:12,040 perform a change of variables and to go from an integral over time, to an 237 00:19:12,040 --> 00:19:17,200 integral over Landa. And this can be done by simply writing. 238 00:19:17,200 --> 00:19:23,530 So here, this dd, d over dt can be written, so this is the identity. 239 00:19:23,530 --> 00:19:29,090 It can be written as d lambda over dt d over d lambda. 240 00:19:29,090 --> 00:19:33,680 Okay? And on the other hand, when this guy is 241 00:19:33,680 --> 00:19:39,320 going to multiply. This differential, so we can simply 242 00:19:39,320 --> 00:19:46,777 replace this product by d lambda. And so another way to write this this 243 00:19:46,777 --> 00:19:52,636 Barry phase, so let me present it here, so gamma can be written as an integral 244 00:19:52,636 --> 00:19:58,774 and now this integral will go actually uhhhm around this loop in this bramoshir 245 00:19:58,774 --> 00:20:06,076 space. So it's going to be closed integral so 246 00:20:06,076 --> 00:20:12,186 lets save this contour, we'll call it C, so it's going to be basically encircling 247 00:20:12,186 --> 00:20:21,170 the C, and here were going to have zero d over d lambda 0 lambda d lambda. 248 00:20:21,170 --> 00:20:23,266 Okay. So this is another way to write 249 00:20:23,266 --> 00:20:26,566 essentially the same quantity, but the time is completely gone from our 250 00:20:26,566 --> 00:20:30,251 description even though we started with time-dependent problem so here time is 251 00:20:30,251 --> 00:20:36,421 not essential anymore. So, another thing we can do, we can 252 00:20:36,421 --> 00:20:40,447 introduce a new notation, let me call it A a differentiation A is the function of 253 00:20:40,447 --> 00:20:45,550 Lambda, which is by definition is going to be this guy. 254 00:20:50,710 --> 00:20:56,910 And so in this notations we can, we can see that the, 255 00:20:56,910 --> 00:21:01,785 A vary phase is nothing but a circulation of this potential if you want, A of 256 00:21:01,785 --> 00:21:07,267 lambda, around this contour. Okay, so another thing that we can do 257 00:21:07,267 --> 00:21:10,144 here. We can take advantage of certain 258 00:21:10,144 --> 00:21:16,571 identities from the vector analysis. Hopefully familiar to some of you who 259 00:21:16,571 --> 00:21:21,350 have studied the theory of electromagnetism, namely we can define 260 00:21:21,350 --> 00:21:26,291 the curl of this vector A, so let's define the curl A Curl with this respect 261 00:21:26,291 --> 00:21:32,621 to this lambda. And let me call it b of lambda. 262 00:21:32,621 --> 00:21:39,792 And if we do so this berry phase gamma can be rewritten as a flux of this I will 263 00:21:39,792 --> 00:21:46,155 call it, fictitious magnetic field, through the area enclosed by this loop 264 00:21:46,155 --> 00:21:56,810 that corresponds to this quantum evolution, okay. 265 00:21:56,810 --> 00:22:00,626 So essentially well it's very complicated but, so those of you who again who are 266 00:22:00,626 --> 00:22:04,030 familiar with the theory of electromagnetism. 267 00:22:04,030 --> 00:22:08,580 Now remember, that to describe magnetic field in real space is often times 268 00:22:08,580 --> 00:22:12,640 convenient to introduce a vector potential such that the curl of this 269 00:22:12,640 --> 00:22:19,674 vector potential is the magnetic field. And the, so here, instead of the real 270 00:22:19,674 --> 00:22:23,444 space have this sort of fictitious parameter space, which does not have to 271 00:22:23,444 --> 00:22:26,650 be a real space. It can be something else. 272 00:22:26,650 --> 00:22:29,250 Can be just be a set of parameters in our problem. 273 00:22:29,250 --> 00:22:32,955 And what we found is that the quantum phase that will be required by the wave 274 00:22:32,955 --> 00:22:38,062 function as a result of this evolution. It's essentially a flux of some 275 00:22:38,062 --> 00:22:41,380 fictitious magnetic field that exists in this model. 276 00:22:41,380 --> 00:22:44,238 Okay? So, by the way, notice that even though 277 00:22:44,238 --> 00:22:49,788 we started with a time-dependent problem, this b of lambda, and the, corresponding 278 00:22:49,788 --> 00:22:55,700 vector potential a can be calculated by themselves. 279 00:22:55,700 --> 00:22:59,264 So if we just have a, Hamiltonian, which depends on Lambda, whether or not it 280 00:22:59,264 --> 00:23:02,882 depends on time is a separate matter, we can just go ahead and calculate these 281 00:23:02,882 --> 00:23:09,400 properties. And so, what we may think about is that. 282 00:23:09,400 --> 00:23:12,951 When we have a quantum system, which is parameterized by some parameter lambda, 283 00:23:12,951 --> 00:23:16,449 so there exists an internal and in some sense fictitious magnetic field in this 284 00:23:16,449 --> 00:23:21,240 parameter space. And so it exists whether or not we 285 00:23:21,240 --> 00:23:25,320 perform a quantum evolution, but if we do perform quantum evolution and if we have 286 00:23:25,320 --> 00:23:30,210 this sort of periodic Time depend on perturbation. 287 00:23:30,210 --> 00:23:34,618 So the quantum phase that would be acquired by the particle is going to be 288 00:23:34,618 --> 00:23:39,786 basically the flux of this magnetic field if you want, through this area enclosed 289 00:23:39,786 --> 00:23:44,932 by this loop. And that's why it's called the geometric 290 00:23:44,932 --> 00:23:49,300 phase, sometimes it's also called the pological phase. 291 00:23:49,300 --> 00:23:52,717 It's a very non-trivial concept, so I mean, first of all it was not discovered 292 00:23:52,717 --> 00:23:58,100 in the early days of quantum mechanics. It was discovered only in the 80s. 293 00:23:58,100 --> 00:24:01,460 And furthermore, only now, and this is basically the subject of current 294 00:24:01,460 --> 00:24:06,610 research, we're starting to understand the true importance of this Berry phase. 295 00:24:06,610 --> 00:24:10,657 And, one of the recent discoveries that you might have heard about Is so called 296 00:24:10,657 --> 00:24:14,780 topological insulators. Something I didn't have time to talk 297 00:24:14,780 --> 00:24:18,989 about these quite amazing discoveries, but I just wanted to mention them in 298 00:24:18,989 --> 00:24:23,320 passing to sort of emphasize that this is not just a pure mathematical construct, 299 00:24:23,320 --> 00:24:27,041 which is sort of elegant by itself, but also does have Important physical 300 00:24:27,041 --> 00:24:32,888 applications. And, this is the subject of current 301 00:24:32,888 --> 00:24:35,903 research. So, the final thing I'm going to mention 302 00:24:35,903 --> 00:24:40,313 here sort of concluding this technical discussion, is that unlike the magnetic 303 00:24:40,313 --> 00:24:45,310 field in real space, which does not have sources or sinks. 304 00:24:45,310 --> 00:24:48,926 So, there are no monopoles as we know. So, the divergence of the magnetic field 305 00:24:48,926 --> 00:24:51,938 is 0. So this fictitious magnetic field 306 00:24:51,938 --> 00:24:57,113 actually it can have sources. And this sources are, are, well some sort 307 00:24:57,113 --> 00:25:01,066 of monopoles in this parameter space correspond to the degeneracies in this 308 00:25:01,066 --> 00:25:05,915 spectrum of the problem. So what it actually means is and this is 309 00:25:05,915 --> 00:25:11,080 one of the results eh, understood by Berry in his original paper. 310 00:25:11,080 --> 00:25:15,364 Is that if we have some, value, some special values of parameter lambda, let's 311 00:25:15,364 --> 00:25:19,774 say, equal lambda star, such that, there exist, two, there may exist two wave 312 00:25:19,774 --> 00:25:26,404 functions with the same energy. So basically, the energy levels cross So 313 00:25:26,404 --> 00:25:31,268 then these points, let's say this point, serve as a source of the monopoles of 314 00:25:31,268 --> 00:25:36,730 this magnetic field. So in this parameter is best. 315 00:25:36,730 --> 00:25:41,810 This is also a very interesting result. Well, it's a very interesting subject. 316 00:25:41,810 --> 00:25:45,430 I would definitely talk more about it. It's a very exciting field. 317 00:25:45,430 --> 00:25:49,600 But I feel that I probably should stop here. 318 00:25:49,600 --> 00:25:53,050 Because I've been talking for twenty-five minutes now on this last part of the last 319 00:25:53,050 --> 00:25:56,542 lecture. And I just would like to thank you very 320 00:25:56,542 --> 00:25:59,910 much for your attention. I hope that this lecture and in general 321 00:25:59,910 --> 00:26:04,570 the course were interesting and useful, at least some parts of it. 322 00:26:04,570 --> 00:26:07,510 And I wish you all the best, thank you very much.