1 00:00:00,090 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:17,815 at ocw.mit.edu. 8 00:00:26,225 --> 00:00:30,920 PROFESSOR: Then let me just-- I gave you 9 00:00:30,920 --> 00:00:34,540 two things about the last class, one 10 00:00:34,540 --> 00:00:42,880 is the question about the electron and factors of 2 11 00:00:42,880 --> 00:00:45,160 and what precesses at what frequency. 12 00:00:49,160 --> 00:00:53,230 Think about it, I mean, write down for yourself 13 00:00:53,230 --> 00:00:55,530 what is a magnetic moment of an electron? 14 00:00:55,530 --> 00:00:59,610 What is a magnetic moment of a classical particles which 15 00:00:59,610 --> 00:01:02,570 is one unit of angular momentum and try to sort of re 16 00:01:02,570 --> 00:01:03,530 derive it. 17 00:01:03,530 --> 00:01:08,300 The way how I did it is deceptively simple. 18 00:01:08,300 --> 00:01:11,000 But whenever you think about it, you 19 00:01:11,000 --> 00:01:13,660 will get confused about some factors. 20 00:01:13,660 --> 00:01:18,410 But this is, of course, where I showed you here 21 00:01:18,410 --> 00:01:23,540 is the quantum mechanical mystery about the g-factor of 2 22 00:01:23,540 --> 00:01:25,680 that you have an electron which has one more 23 00:01:25,680 --> 00:01:28,620 magneton of magnetic moment, and you have a classical particle 24 00:01:28,620 --> 00:01:30,180 which has one more magneton. 25 00:01:30,180 --> 00:01:33,840 The energy levels of up and down what are the same. 26 00:01:33,840 --> 00:01:37,350 The energy is 1.4 megahertz per Gauss. 27 00:01:37,350 --> 00:01:40,740 But the precession of frequency of the electron 28 00:01:40,740 --> 00:01:43,420 is 2 times faster than the precession frequency 29 00:01:43,420 --> 00:01:45,440 of the classical particle, and this 30 00:01:45,440 --> 00:01:49,260 is what the g-factor of 2, which comes out of the Dirac equation 31 00:01:49,260 --> 00:01:51,130 means. 32 00:01:51,130 --> 00:01:53,880 And at least one intuitive explanation 33 00:01:53,880 --> 00:01:57,440 I can give you is a frequency which is observed 34 00:01:57,440 --> 00:02:00,070 as a precession frequency in the laboratory 35 00:02:00,070 --> 00:02:03,420 is the beat frequency between two neighboring levels. 36 00:02:03,420 --> 00:02:05,014 And here you have a level in between. 37 00:02:05,014 --> 00:02:07,180 So the beat frequency between two neighboring levels 38 00:02:07,180 --> 00:02:09,664 is 1.4 megahertz, and here the beat frequency 39 00:02:09,664 --> 00:02:12,210 is 2.8 megahertz. 40 00:02:12,210 --> 00:02:13,910 And you will actually see that if you 41 00:02:13,910 --> 00:02:16,660 take a classic system which has much, 42 00:02:16,660 --> 00:02:20,260 much larger angular momentum, maybe L equals 10. 43 00:02:20,260 --> 00:02:21,790 It will have this level structure, 44 00:02:21,790 --> 00:02:26,690 but it will go from minus 10 to plus 10. 45 00:02:26,690 --> 00:02:31,020 But the precession frequency is the beat frequency 46 00:02:31,020 --> 00:02:33,840 between two levels. 47 00:02:33,840 --> 00:02:37,460 Or if you don't like the word beat frequency, 48 00:02:37,460 --> 00:02:40,370 it precesses and you can increase the precession. 49 00:02:42,789 --> 00:02:44,580 So if you're not increasing the impression, 50 00:02:44,580 --> 00:02:47,550 you can increase the energy of the precessing system 51 00:02:47,550 --> 00:02:51,250 by driving it at the resonance frequency which 52 00:02:51,250 --> 00:02:52,850 is a precession frequency. 53 00:02:52,850 --> 00:02:55,930 And the resonance is always the energy difference 54 00:02:55,930 --> 00:02:57,220 between two levels. 55 00:02:57,220 --> 00:03:00,300 So no matter how many levels you have in a classical system 56 00:03:00,300 --> 00:03:05,640 with a g-factor of 1, you're always driving it step by step, 57 00:03:05,640 --> 00:03:09,410 and each step is the resonance and this 58 00:03:09,410 --> 00:03:11,820 is the precession frequency of the system. 59 00:03:14,430 --> 00:03:16,250 Questions about it? 60 00:03:16,250 --> 00:03:20,380 I know people get confused all the time, 61 00:03:20,380 --> 00:03:22,170 but just think about it. 62 00:03:22,170 --> 00:03:24,160 I think I gave you all the different angles 63 00:03:24,160 --> 00:03:28,130 you can use to look at it. 64 00:03:28,130 --> 00:03:29,210 Questions about that? 65 00:03:31,990 --> 00:03:33,680 OK. 66 00:03:33,680 --> 00:03:40,490 Next thing I just want to point out is rotation. 67 00:03:40,490 --> 00:03:45,360 Yes, we are spending quite some time in class 68 00:03:45,360 --> 00:03:48,670 just to figure out that something is rotating. 69 00:03:48,670 --> 00:03:53,420 So what we figured out here is that if you 70 00:03:53,420 --> 00:04:03,550 go to a rotating frame, if you have angular momentum, 71 00:04:03,550 --> 00:04:07,500 and we go to a rotating frame, that that simply 72 00:04:07,500 --> 00:04:11,570 means that the effect of a rotating frame 73 00:04:11,570 --> 00:04:18,720 is to simply add a fictitious magnetic fields 74 00:04:18,720 --> 00:04:19,930 to the real magnetic field. 75 00:04:23,700 --> 00:04:27,390 Well, that's very handy. 76 00:04:27,390 --> 00:04:29,610 For instance, we know then immediately 77 00:04:29,610 --> 00:04:33,660 if you have a real magnetic field 78 00:04:33,660 --> 00:04:35,840 and we pick the fictitious magnetic field 79 00:04:35,840 --> 00:04:37,850 in such a way that the real magnetic field 80 00:04:37,850 --> 00:04:39,215 that the total field cancels it. 81 00:04:39,215 --> 00:04:42,870 It cancels the total magnetic field, well, 82 00:04:42,870 --> 00:04:46,740 then we have a system viewed from the rotating frame 83 00:04:46,740 --> 00:04:48,780 where the effective field is 0. 84 00:04:48,780 --> 00:04:51,210 Well, the spin is zero field does nothing 85 00:04:51,210 --> 00:04:53,750 and then we know when you go back into the lab frame, 86 00:04:53,750 --> 00:04:55,450 all this spin is doing it rotates. 87 00:04:55,450 --> 00:04:58,230 It's an exact solution. 88 00:04:58,230 --> 00:05:01,670 I want to use it today again but in a different way, 89 00:05:01,670 --> 00:05:04,290 but I want to just point out today 90 00:05:04,290 --> 00:05:07,120 we will not picking the fictitious field to make 91 00:05:07,120 --> 00:05:08,990 the magnetic field zero. 92 00:05:08,990 --> 00:05:11,890 Today we are talking about what happens 93 00:05:11,890 --> 00:05:15,620 to a spin in a rotating field. 94 00:05:15,620 --> 00:05:16,466 Complicated. 95 00:05:16,466 --> 00:05:20,680 A time-dependent problem, time-dependent Hamiltonian. 96 00:05:20,680 --> 00:05:23,610 But if you go in a rotating frame, if you rotate 97 00:05:23,610 --> 00:05:26,980 with the field, then in the rotating frame 98 00:05:26,980 --> 00:05:29,280 it becomes a time-independent problem 99 00:05:29,280 --> 00:05:31,260 which we can immediately solve. 100 00:05:31,260 --> 00:05:34,490 So today we use the same transformation again, 101 00:05:34,490 --> 00:05:38,650 but we pick our frequency not to cancel some static field. 102 00:05:38,650 --> 00:05:41,850 We pick our frequency to co-rotate 103 00:05:41,850 --> 00:05:46,590 with an external rotating field that in our rotating frame now 104 00:05:46,590 --> 00:05:47,704 everything is stationary. 105 00:05:47,704 --> 00:05:49,370 So that's what you're going to do today. 106 00:05:53,110 --> 00:05:55,550 And finally just to give you an outlook, 107 00:05:55,550 --> 00:05:58,120 just try to sort of make you aware that often what we're 108 00:05:58,120 --> 00:06:01,050 doing is the same thing in a different angle. 109 00:06:01,050 --> 00:06:02,930 So this is one way to deal with rotation. 110 00:06:02,930 --> 00:06:07,590 I'll give you an exact solution by going to rotating frame. 111 00:06:07,590 --> 00:06:12,960 I will later show you today that quantum mechanically 112 00:06:12,960 --> 00:06:17,300 the solution to the Heisenberg equation of motion for angular 113 00:06:17,300 --> 00:06:20,290 momentum or magnetic moment in a magnetic field 114 00:06:20,290 --> 00:06:22,370 is exactly a rotation. 115 00:06:22,370 --> 00:06:25,350 So we'll again show you that a quantum mechanical solution, 116 00:06:25,350 --> 00:06:29,530 the solution of a time-dependent problem is exactly rotation. 117 00:06:29,530 --> 00:06:36,740 And later on when we use the spin-1/2 Hamiltonian and we 118 00:06:36,740 --> 00:06:40,250 write down the wave function, we solve sometimes the wave 119 00:06:40,250 --> 00:06:42,540 function by transforming the wave function. 120 00:06:42,540 --> 00:06:45,840 And this will be, again, the rotating frame transformation. 121 00:06:45,840 --> 00:06:49,400 It's not always called like this, but it's always the same. 122 00:06:49,400 --> 00:06:52,250 You go to some form of rotating frame, 123 00:06:52,250 --> 00:06:55,210 and we'll do that in three different ways. 124 00:06:55,210 --> 00:06:58,920 This is the first way just as a general classical physics 125 00:06:58,920 --> 00:07:01,780 transformation to rotating frame. 126 00:07:01,780 --> 00:07:04,600 We will do it again for the expectation 127 00:07:04,600 --> 00:07:07,340 value as a solution of Heisenberg's equation 128 00:07:07,340 --> 00:07:10,800 of motion, and then we do it again 129 00:07:10,800 --> 00:07:14,025 when we transform the wave function 130 00:07:14,025 --> 00:07:15,700 with a unitary transformation. 131 00:07:18,590 --> 00:07:23,580 A lot of time for simple rotations, but its good thing. 132 00:07:23,580 --> 00:07:25,080 It really provides a lot of insight. 133 00:07:27,670 --> 00:07:31,050 Anyway, this is more sort of an outlook over today's lecture 134 00:07:31,050 --> 00:07:34,200 and part of Tuesday's lecture. 135 00:07:34,200 --> 00:07:36,890 Any question about the summary and the outlook? 136 00:07:42,090 --> 00:07:46,130 Because before I really come back to the rotating frame, 137 00:07:46,130 --> 00:07:52,380 I quickly want to do something I couldn't. 138 00:07:52,380 --> 00:07:56,900 I ran out of time on Monday. 139 00:07:56,900 --> 00:08:00,150 We talked so much about harmonic oscillators, the precision 140 00:08:00,150 --> 00:08:01,900 at which we can determine the frequency 141 00:08:01,900 --> 00:08:05,320 of the harmonic oscillator, and I really 142 00:08:05,320 --> 00:08:06,760 want to give you examples. 143 00:08:06,760 --> 00:08:12,200 I want give you two outstanding examples for atomic clocks. 144 00:08:17,270 --> 00:08:21,850 And the two extreme examples are, well, 145 00:08:21,850 --> 00:08:24,420 the two best atomic clocks in the world. 146 00:08:24,420 --> 00:08:32,020 One is the cesium atom with a fountain clock. 147 00:08:32,020 --> 00:08:36,030 Well, we'll talk about it later, but some of you 148 00:08:36,030 --> 00:08:39,299 know that the cesium atom is hyper fine structure. 149 00:08:39,299 --> 00:08:42,539 In one state, the electron and nucleus spin are parallel. 150 00:08:42,539 --> 00:08:44,375 In the other state, they are antiparallel, 151 00:08:44,375 --> 00:08:48,560 and the transition frequency is 10 gigahertz, well, 152 00:08:48,560 --> 00:08:51,650 9.something, but for the matter of this discussion, 153 00:08:51,650 --> 00:08:54,640 it's 10 gigahertz. 154 00:08:54,640 --> 00:09:00,620 And the definition of time, the definition for 1 second 155 00:09:00,620 --> 00:09:06,110 is in terms of so-and-so many cycles of this transition. 156 00:09:06,110 --> 00:09:09,380 So this transition has this frequency, 157 00:09:09,380 --> 00:09:13,000 and for decades, this frequency was 158 00:09:13,000 --> 00:09:14,590 determined in an atomic beam. 159 00:09:14,590 --> 00:09:17,110 You have an atomic cesium beam, and you interrogate it 160 00:09:17,110 --> 00:09:19,000 with microwave fields. 161 00:09:19,000 --> 00:09:23,200 But now with cold atoms, we can achieve much, much longer 162 00:09:23,200 --> 00:09:27,320 interrogation times by almost completely eliminating 163 00:09:27,320 --> 00:09:28,620 the atomic motion. 164 00:09:28,620 --> 00:09:30,400 In the current experiment, is that you 165 00:09:30,400 --> 00:09:33,330 have a cloud [? lace ?] are cooled 166 00:09:33,330 --> 00:09:34,930 to micro Kelvin temperature. 167 00:09:34,930 --> 00:09:37,300 You launch it into a fountain. 168 00:09:37,300 --> 00:09:42,500 The cloud goes up and goes down, and you interrogate it twice 169 00:09:42,500 --> 00:09:44,940 and the interrogation time is 1 second. 170 00:09:44,940 --> 00:09:47,910 And this interrogation time is no longer as in the convention 171 00:09:47,910 --> 00:09:51,890 atomic clocks limited by the thermal velocity of the atom, 172 00:09:51,890 --> 00:09:53,670 it's limited by gravity. 173 00:09:53,670 --> 00:09:57,420 If you want to increase a time to 10 seconds, 174 00:09:57,420 --> 00:10:00,360 you need a 100-meter tower and nobody wants to build that. 175 00:10:00,360 --> 00:10:03,780 I mean, it would be a really big atomic clock. 176 00:10:03,780 --> 00:10:10,050 So therefore you usually deal with an interrogation time 177 00:10:10,050 --> 00:10:13,700 on the order of 1 second. 178 00:10:13,700 --> 00:10:18,970 So we know that based on Fourier's theorem, 179 00:10:18,970 --> 00:10:22,120 delta is a factor of 2 pi, but the line widths, 180 00:10:22,120 --> 00:10:24,840 if you would recall now the spectrum, 181 00:10:24,840 --> 00:10:28,410 the line widths, delta omega, would be that's a factor of 2, 182 00:10:28,410 --> 00:10:31,670 but it would be 1 over 1 second. 183 00:10:31,670 --> 00:10:37,970 And therefore the fractional accuracy is 10 to the 11, 184 00:10:37,970 --> 00:10:41,960 is on the order of 10 to the minus 11. 185 00:10:41,960 --> 00:10:48,930 However, the accuracy of the base cesium fountains 186 00:10:48,930 --> 00:10:54,620 is now 10 to the minus 16. 187 00:10:54,620 --> 00:10:59,290 So people are able to split the line to 1 part in 100,000, 188 00:10:59,290 --> 00:11:04,630 which requires exquisite knowledge, 189 00:11:04,630 --> 00:11:06,400 exquisite knowledge of the line shape 190 00:11:06,400 --> 00:11:09,680 and also systematic effects, but well, an atomic clock 191 00:11:09,680 --> 00:11:12,180 is a piece of art. 192 00:11:12,180 --> 00:11:17,930 And I just want to emphasize that I sometimes 193 00:11:17,930 --> 00:11:20,830 feel when I talk about the uncertainty at which you can 194 00:11:20,830 --> 00:11:25,135 measure a classical oscillator or such is almost trivial, 195 00:11:25,135 --> 00:11:27,070 and you should know all about it. 196 00:11:27,070 --> 00:11:31,590 But I can just say without looking at anybody that I just 197 00:11:31,590 --> 00:11:34,270 had a lunch discussion with some of my graduate student 198 00:11:34,270 --> 00:11:36,810 and we talked about laser stabilization. 199 00:11:36,810 --> 00:11:41,320 And one graduate student asked, but if the natural line widths 200 00:11:41,320 --> 00:11:44,610 of a transition is 10 megahertz, can we stabilized a laser 201 00:11:44,610 --> 00:11:47,070 to better than a megahertz? 202 00:11:47,070 --> 00:11:48,240 Of course, we can. 203 00:11:48,240 --> 00:11:50,940 I mean, here we have a natural line width of 10 204 00:11:50,940 --> 00:11:53,780 to the minus 11, but we can get an a accuracy 205 00:11:53,780 --> 00:11:57,710 of a signal, which is 10 to the minus 16. 206 00:11:57,710 --> 00:12:00,810 So a microwave oscillator, which is the microwave accrual 207 00:12:00,810 --> 00:12:05,240 end of a laser can now be locked to the cesium transition 208 00:12:05,240 --> 00:12:08,600 with a precision of 10 to the minus 16, 209 00:12:08,600 --> 00:12:12,560 100,000 times better than the line width. 210 00:12:12,560 --> 00:12:17,065 Let me give you a second example. 211 00:12:19,630 --> 00:12:21,220 Just noticing the red color. 212 00:12:21,220 --> 00:12:23,790 This is red. 213 00:12:23,790 --> 00:12:27,042 I wanted to highlight it. 214 00:12:27,042 --> 00:12:28,500 There's probably nothing we can do. 215 00:12:28,500 --> 00:12:32,650 It seems the projector is not showing the color. 216 00:12:32,650 --> 00:12:35,160 Well, in the lecture post, the lecture notes, 217 00:12:35,160 --> 00:12:38,680 this will be bright red, but maybe I 218 00:12:38,680 --> 00:12:41,550 should use more blue and green and yellow 219 00:12:41,550 --> 00:12:42,760 for highlighting it today. 220 00:12:47,050 --> 00:12:49,650 That's really odd. 221 00:12:49,650 --> 00:12:55,530 Yellow-- OK. 222 00:12:55,530 --> 00:12:59,160 The red is completely missing. 223 00:12:59,160 --> 00:13:08,210 The other example I want to give you is the strontium-- yeah, 224 00:13:08,210 --> 00:13:10,570 it's blue-- the strontium optical clock. 225 00:13:14,440 --> 00:13:18,500 And there was a really nice paper 226 00:13:18,500 --> 00:13:32,550 in Nature just a few weeks ago, and well, here 227 00:13:32,550 --> 00:13:37,110 is a level diagram of atomic strontium 228 00:13:37,110 --> 00:13:40,740 what happens is there is a very fast transition s 229 00:13:40,740 --> 00:13:43,770 to p transition for laser cooling and trapping 230 00:13:43,770 --> 00:13:47,850 and all that, but then there is a very, very slow transition 231 00:13:47,850 --> 00:13:51,460 [? forbidden ?] to a triplet state, which is meta stable. 232 00:13:51,460 --> 00:13:54,070 And this transition, those states 233 00:13:54,070 --> 00:13:57,030 have a very long lifetime, and therefore this transition 234 00:13:57,030 --> 00:14:00,210 is extremely level. 235 00:14:00,210 --> 00:14:05,380 So what is state of the art for the strontium clock? 236 00:14:05,380 --> 00:14:13,220 Well, in the experiment, they use an interrogation time. 237 00:14:17,750 --> 00:14:23,200 So they observe the atom for delta t, which 238 00:14:23,200 --> 00:14:27,540 is on the order of 160 milliseconds 239 00:14:27,540 --> 00:14:31,980 and putting in the 2 pis in the right place that would mean 240 00:14:31,980 --> 00:14:35,920 the frequency resolution is 1 Hertz. 241 00:14:35,920 --> 00:14:38,890 And you see that on the left-hand side when 242 00:14:38,890 --> 00:14:43,260 we record the resonance, the blue one is about 1 Hertz 243 00:14:43,260 --> 00:14:44,900 and there is something broader. 244 00:14:44,900 --> 00:14:46,690 This is when they are not actively 245 00:14:46,690 --> 00:14:48,830 feeding back the magnetic field. 246 00:14:48,830 --> 00:14:53,000 So for this fantastic position, you have to control everything, 247 00:14:53,000 --> 00:14:54,910 but the blue line is sort of what 248 00:14:54,910 --> 00:14:56,320 we record as a clock transition. 249 00:14:56,320 --> 00:14:58,310 It's about 1 Hertz. 250 00:14:58,310 --> 00:15:06,030 Now, compared to a cesium clock, the big advantage 251 00:15:06,030 --> 00:15:13,420 is that the strontium clock operates in the optical domain, 252 00:15:13,420 --> 00:15:18,404 and this is a frequency at 5 times 10 to the 14 Hertz. 253 00:15:18,404 --> 00:15:20,320 So we're seeing if the frequency is much, much 254 00:15:20,320 --> 00:15:23,910 faster even if you have a shorter interrogation time, 255 00:15:23,910 --> 00:15:27,310 your relative accuracy is better. 256 00:15:27,310 --> 00:15:33,250 And here the Q value the mu over delta mu 257 00:15:33,250 --> 00:15:37,460 is on the order of 10 times 10 to the minus 15. 258 00:15:37,460 --> 00:15:38,190 Fantastic. 259 00:15:38,190 --> 00:15:50,800 15 orders of magnitude, and they are splitting the line, not as 260 00:15:50,800 --> 00:15:54,800 extreme as the cesium atomic clock by a factor of 100,000, 261 00:15:54,800 --> 00:15:58,310 by a factor which is on the order of 300. 262 00:15:58,310 --> 00:16:05,800 And with that, they have an accuracy, which is now really 263 00:16:05,800 --> 00:16:08,080 the record for the base performance 264 00:16:08,080 --> 00:16:15,255 of any atomic clock, which is 6 times 10 to the minus 18. 265 00:16:19,140 --> 00:16:22,260 Now, this is now in the optical domain. 266 00:16:22,260 --> 00:16:26,320 You may wonder about the laser, how stable is the laser. 267 00:16:26,320 --> 00:16:29,680 The laser is stabilized to an optical activity, 268 00:16:29,680 --> 00:16:34,270 but the laser because of thermal fluctuations, not 269 00:16:34,270 --> 00:16:36,910 terminal fluctuations, thermal fluctuations 270 00:16:36,910 --> 00:16:44,560 in the mirror because of thermal noise 271 00:16:44,560 --> 00:16:48,660 is limited in the short term to 10 to the minus 16. 272 00:16:53,770 --> 00:16:57,440 So the short term stability between 1 and 1,000 273 00:16:57,440 --> 00:17:00,250 second is 10 to the minus 16. 274 00:17:00,250 --> 00:17:05,109 So they use a laser which has a stability of 10 275 00:17:05,109 --> 00:17:07,260 to the minus 16. 276 00:17:07,260 --> 00:17:12,089 Every 1.3 seconds, they take a data point. 277 00:17:12,089 --> 00:17:16,089 Each data point or the spectral widths 278 00:17:16,089 --> 00:17:23,180 is 2 times 10 to the-- the laser is 10 to the 16. 279 00:17:23,180 --> 00:17:26,020 The line we record is 10 to the 15, 280 00:17:26,020 --> 00:17:28,810 but seen since the thermal noise is completely random 281 00:17:28,810 --> 00:17:31,834 and by averaging it, they can determine the line center 282 00:17:31,834 --> 00:17:33,000 to better than 10 to the 17. 283 00:17:35,610 --> 00:17:39,340 So I think this just illustrates the precision at the which 284 00:17:39,340 --> 00:17:42,490 you can observe the harmonic oscillator, 285 00:17:42,490 --> 00:17:44,620 and what I like about this example, 286 00:17:44,620 --> 00:17:49,160 it shows you that both the transition you are recording 287 00:17:49,160 --> 00:17:53,580 and the laser itself-- 10 to the 15, 10 to the 16-- 288 00:17:53,580 --> 00:17:57,449 are worse than the final precision of the measurement, 289 00:17:57,449 --> 00:17:58,865 which is better than 10 to the 17. 290 00:18:03,920 --> 00:18:07,330 Make sense, but yes, that's what it is. 291 00:18:07,330 --> 00:18:08,420 Questions about that? 292 00:18:20,350 --> 00:18:20,850 Yes, Collin. 293 00:18:20,850 --> 00:18:24,561 AUDIENCE: What was the accuracy of the [INAUDIBLE] result? 294 00:18:27,998 --> 00:18:31,707 Was it better than 6 times 10 to the minus 18? 295 00:18:31,707 --> 00:18:32,290 PROFESSOR: No. 296 00:18:32,290 --> 00:18:34,724 This 6 times 10 to the minus 18 is really 297 00:18:34,724 --> 00:18:35,682 the [INAUDIBLE] record. 298 00:18:35,682 --> 00:18:38,944 AUDIENCE: Is that the [? atomic ?] clock? 299 00:18:38,944 --> 00:18:42,350 PROFESSOR: There was an aluminum ion clock, which had, 300 00:18:42,350 --> 00:18:44,469 I think, 10 the minus 17 precision. 301 00:18:44,469 --> 00:18:47,010 But I think it's 6 times 10 to the minus 18 is close to that. 302 00:18:47,010 --> 00:18:49,166 So they are close or they are better now 303 00:18:49,166 --> 00:18:54,160 than the aluminum clock, but the aluminum clock is a single ion. 304 00:18:54,160 --> 00:18:56,370 You have to ever reach for much, much longer time 305 00:18:56,370 --> 00:18:58,310 to get this precision. 306 00:18:58,310 --> 00:19:01,600 So that's a big advantage for the strontium clock, which 307 00:19:01,600 --> 00:19:04,900 is many, many atoms in an optical lattice, 308 00:19:04,900 --> 00:19:07,570 and they say that they have improved on the best 309 00:19:07,570 --> 00:19:10,980 previous lattice clock by a factor of 20. 310 00:19:10,980 --> 00:19:14,230 But I think the improvement effect of 20, 311 00:19:14,230 --> 00:19:18,380 you always have to distinguish between sensitivity 312 00:19:18,380 --> 00:19:19,740 and absolute precision. 313 00:19:19,740 --> 00:19:21,776 In the absolute precision, you also 314 00:19:21,776 --> 00:19:24,090 have to control all systematic effects. 315 00:19:24,090 --> 00:19:26,110 And the big step here which was really 316 00:19:26,110 --> 00:19:29,420 boosting the absolute precision was they completely 317 00:19:29,420 --> 00:19:32,760 controlled the black body environment. 318 00:19:32,760 --> 00:19:35,830 So you really have to know with high precision what 319 00:19:35,830 --> 00:19:39,610 is the effective temperature of the black body radiation 320 00:19:39,610 --> 00:19:42,340 because at the 10 to the minus 18 level, 321 00:19:42,340 --> 00:19:44,860 it causes a shift of the atomic resonance. 322 00:19:47,890 --> 00:19:51,270 We talk later about it, but it's the ac Stark effect 323 00:19:51,270 --> 00:19:53,430 of the black body radiation, which 324 00:19:53,430 --> 00:19:57,177 becomes an important systematic at that level of precision. 325 00:19:57,177 --> 00:19:59,760 I don't know exactly the number of the previous atomic clocks, 326 00:19:59,760 --> 00:20:02,280 but this is sort of now the gold standard 327 00:20:02,280 --> 00:20:04,300 of frequency methodology. 328 00:20:10,760 --> 00:20:11,490 OK. 329 00:20:11,490 --> 00:20:12,250 Let's go. 330 00:20:16,290 --> 00:20:18,610 Yes, what you have here is you have 331 00:20:18,610 --> 00:20:21,140 sort of classical harmonic oscillator, 332 00:20:21,140 --> 00:20:23,130 but the classical harmonic oscillator 333 00:20:23,130 --> 00:20:27,290 is based on doing measurements on seamless strontium 334 00:20:27,290 --> 00:20:29,890 atoms and single cesium atoms. 335 00:20:29,890 --> 00:20:32,860 So you see that the accuracy, and we had a discussion on it 336 00:20:32,860 --> 00:20:35,720 last class, the accuracy at which you can observe 337 00:20:35,720 --> 00:20:37,320 a quantum mechanical oscillator. 338 00:20:37,320 --> 00:20:41,000 In a classical oscillator, it's pretty much the same. 339 00:20:41,000 --> 00:20:44,610 It's the same kind of-- if you have a good signal to noise, 340 00:20:44,610 --> 00:20:50,047 you can improve your precision for the line center 341 00:20:50,047 --> 00:20:50,630 substantially. 342 00:20:53,230 --> 00:20:57,830 From those results, and exciting atomic clocks, 343 00:20:57,830 --> 00:21:00,480 let's go back to classical physics. 344 00:21:00,480 --> 00:21:06,410 So we want to go back to our classical magnetic moment 345 00:21:06,410 --> 00:21:11,920 and understand the motion of it. 346 00:21:11,920 --> 00:21:15,300 But in addition to what we discussed so far, 347 00:21:15,300 --> 00:21:17,945 a stationary magnetic field, we now 348 00:21:17,945 --> 00:21:20,280 want to add to it a rotating magnetic field. 349 00:21:38,430 --> 00:21:50,860 So the situation is that-- just use 350 00:21:50,860 --> 00:21:56,270 another color-- we have a magnetic moment. 351 00:21:56,270 --> 00:22:04,690 We assume it's classical, and we have a magnetic field, 352 00:22:04,690 --> 00:22:10,980 which we assume points along the z-axis. 353 00:22:10,980 --> 00:22:13,410 And then, of course, we know that 354 00:22:13,410 --> 00:22:16,530 from our previous discussion that the spin 355 00:22:16,530 --> 00:22:18,420 undergoes precession. 356 00:22:18,420 --> 00:22:23,130 It's sort of precesses around the magnetic field 357 00:22:23,130 --> 00:22:24,140 at the Larmor frequency. 358 00:22:27,700 --> 00:22:46,070 And now, let's assume we have-- the yellow shows up. 359 00:22:46,070 --> 00:22:48,570 We have a rotating field. 360 00:22:48,570 --> 00:22:52,580 We add a rotating field eight B1, 361 00:22:52,580 --> 00:22:55,550 and just to keep things simple, we 362 00:22:55,550 --> 00:23:00,330 want to assume that the rotating field rotates 363 00:23:00,330 --> 00:23:03,280 at the same frequency as a magnetic moment, 364 00:23:03,280 --> 00:23:06,310 so we are on a resonance. 365 00:23:06,310 --> 00:23:13,430 Because what happens now is we can simply 366 00:23:13,430 --> 00:23:16,690 do a transformation to the rotating frame. 367 00:23:19,920 --> 00:23:26,390 The rotating frame is now at the Larmor frequency 368 00:23:26,390 --> 00:23:31,730 and we have just learned that in the rotating frame, of course, 369 00:23:31,730 --> 00:23:38,120 in the rotating frame the rotating field stands still, 370 00:23:38,120 --> 00:23:43,190 so it becomes a static field which 371 00:23:43,190 --> 00:23:45,705 points in the x direction. 372 00:23:52,850 --> 00:23:58,620 And we have the static field in the z direction, 373 00:23:58,620 --> 00:24:01,280 but now that's what I just reviewed. 374 00:24:01,280 --> 00:24:06,840 We have a fictitious magnetic field, 375 00:24:06,840 --> 00:24:09,560 which comes from the transformation to the rotating 376 00:24:09,560 --> 00:24:14,290 frame and on resonance, at the Larmor frequency, 377 00:24:14,290 --> 00:24:17,085 this is exactly the negative of B naught. 378 00:24:20,660 --> 00:24:27,660 So in other words, we started out 379 00:24:27,660 --> 00:24:33,160 to expose a magnetic moment to a time-dependent field. 380 00:24:38,810 --> 00:24:54,010 So we had a rotating field cosine omega Lt sine omega Lt. 381 00:24:58,460 --> 00:25:04,790 But in the rotating frame, this becomes now the x prime x, 382 00:25:04,790 --> 00:25:06,800 and it's stationary in the rotating frame. 383 00:25:10,940 --> 00:25:15,360 So this was the field in the lab frame. 384 00:25:15,360 --> 00:25:31,770 In the rotating frame, we have an effective field, 385 00:25:31,770 --> 00:25:34,590 which is the field in the lab frame 386 00:25:34,590 --> 00:25:40,850 minus the fictitious field, and the fictitious field 387 00:25:40,850 --> 00:25:42,240 was given by that. 388 00:25:42,240 --> 00:25:45,750 It just cancels the B naught component. 389 00:25:45,750 --> 00:25:51,150 And in the rotating frame, this vector and this vector 390 00:25:51,150 --> 00:25:55,250 cancels and we are just left with a field of strengths 391 00:25:55,250 --> 00:26:03,930 B1, which points in the x-axis, or actually 392 00:26:03,930 --> 00:26:08,410 the x-axis in the rotating frame is what I call x prime. 393 00:26:08,410 --> 00:26:11,240 So therefore, we have a very simple problem 394 00:26:11,240 --> 00:26:13,140 that in the rotating frame. 395 00:26:13,140 --> 00:26:18,200 We have a static field of value B1. 396 00:26:22,839 --> 00:26:24,630 And the good thing is, we know already what 397 00:26:24,630 --> 00:26:28,240 a magnetic moment does with a static field. 398 00:26:28,240 --> 00:26:30,140 The magnetic moment is just precessing 399 00:26:30,140 --> 00:26:33,190 around the static magnetic field. 400 00:26:33,190 --> 00:26:35,850 So therefore, our solution is now 401 00:26:35,850 --> 00:26:39,020 we have transformed the time-dependent field 402 00:26:39,020 --> 00:26:53,040 and now we know that mu precesses around this field, 403 00:26:53,040 --> 00:26:57,240 and the precession frequency is the Rabi frequency, which 404 00:26:57,240 --> 00:26:58,970 we discussed previously. 405 00:27:04,700 --> 00:27:10,310 The Rabi frequency is the gyromagnetic ratio times B1. 406 00:27:10,310 --> 00:27:13,080 So therefore, if we would start out 407 00:27:13,080 --> 00:27:19,670 with a magnetic moment aligned with the z-axis. 408 00:27:19,670 --> 00:27:25,200 If we would wait half a Rabi's cycle, 409 00:27:25,200 --> 00:27:28,885 the magnetic moment would now be inverted. 410 00:27:32,440 --> 00:27:34,800 So the situation is we have a magnetic moment which 411 00:27:34,800 --> 00:27:38,010 points in the z-axis with a field in the z-axis. 412 00:27:38,010 --> 00:27:41,440 But we expose it to a rotating field 413 00:27:41,440 --> 00:27:43,270 which rotates in the xy plane. 414 00:27:47,080 --> 00:27:50,240 So in the rotating frame, we have a stationary field 415 00:27:50,240 --> 00:27:52,970 which points in x. 416 00:27:52,970 --> 00:27:56,010 In this rotating frame, the spin is simply 417 00:27:56,010 --> 00:28:00,030 precessing around what is now a steady field, 418 00:28:00,030 --> 00:28:04,550 and after half a Ravi cycle, the spin points down. 419 00:28:04,550 --> 00:28:07,680 So therefore we know-- going back to lab frame-- 420 00:28:07,680 --> 00:28:12,660 that this rotating field has caused in quantum mechanics 421 00:28:12,660 --> 00:28:15,770 I would say a spin flip, a full reversal 422 00:28:15,770 --> 00:28:19,290 of the magnetic moment, and this is 423 00:28:19,290 --> 00:28:23,750 what we call the pi pulse-- it has already 424 00:28:23,750 --> 00:28:29,810 rotated the spin by pi-- or we call it a spin flip, 425 00:28:29,810 --> 00:28:32,030 but it's a completely classical system. 426 00:28:37,760 --> 00:28:38,480 Any questions? 427 00:28:48,430 --> 00:28:51,660 Well, then I would question for you. 428 00:28:51,660 --> 00:28:58,180 I've discussed with you the case that the rotating field rotates 429 00:28:58,180 --> 00:29:01,660 at the Larmor frequency. 430 00:29:01,660 --> 00:29:13,530 But now I want to discuss the case that the rotating field is 431 00:29:13,530 --> 00:29:16,680 not at the Larmor frequency, it's 432 00:29:16,680 --> 00:29:30,520 at the frequency omega 1, which is faster 433 00:29:30,520 --> 00:29:31,625 than the Larmor frequency. 434 00:29:37,850 --> 00:29:41,150 And the question is now what will 435 00:29:41,150 --> 00:29:44,590 happen to the spin or the magnetic moment. 436 00:29:44,590 --> 00:29:48,810 I explained to you that on a resonance, 437 00:29:48,810 --> 00:29:54,690 the magnetic moment was just flipping over, rotating 438 00:29:54,690 --> 00:29:59,040 precessing at the Rabi frequency. 439 00:29:59,040 --> 00:30:02,350 And I want you to think about it for a moment 440 00:30:02,350 --> 00:30:05,870 and then decide if we go off resonant, 441 00:30:05,870 --> 00:30:08,660 if we drive this system away from the resonance 442 00:30:08,660 --> 00:30:11,320 with a rotating field which is faster than the Larmor 443 00:30:11,320 --> 00:30:18,260 frequency, what is now the oscillation 444 00:30:18,260 --> 00:30:23,575 frequency of the magnetic moment. 445 00:30:31,940 --> 00:30:44,860 And so the choices are, is it larger, smaller, or the same? 446 00:30:44,860 --> 00:30:51,560 And this was, of course, compared to the Rabi frequency. 447 00:30:58,780 --> 00:31:03,860 So I've explained to you that the spin flip, the Rabi 448 00:31:03,860 --> 00:31:07,000 flopping, or the pi pulse-- and this picture 449 00:31:07,000 --> 00:31:12,520 was at the magnetic moment-- does Rabi flopping rotates 450 00:31:12,520 --> 00:31:18,260 plus z minus, z plus, z minus, z at the Rabi frequency. 451 00:31:18,260 --> 00:31:20,930 But now we drive the system faster 452 00:31:20,930 --> 00:31:23,230 than the Larmor frequency and the question 453 00:31:23,230 --> 00:31:26,410 is, is whatever this magnetic moment does, 454 00:31:26,410 --> 00:31:28,657 is it faster, slower, or does it always 455 00:31:28,657 --> 00:31:29,865 happen at the Rabi frequency? 456 00:31:43,070 --> 00:31:45,220 Larger. 457 00:31:45,220 --> 00:31:45,720 Good. 458 00:31:48,910 --> 00:31:53,630 Well, let me then immediately add another twist. 459 00:31:53,630 --> 00:31:56,890 What would happen-- let's ask the same question, 460 00:31:56,890 --> 00:32:00,315 but now we are driving it at a lower frequency. 461 00:32:03,692 --> 00:32:04,686 The yellow. 462 00:32:09,670 --> 00:32:11,090 This projector has a problem. 463 00:32:14,000 --> 00:32:20,280 So now same question as before, but instead 464 00:32:20,280 --> 00:32:23,980 of driving the system with a faster rotating field, faster 465 00:32:23,980 --> 00:32:27,170 than the Larmor frequency, we are driving it 466 00:32:27,170 --> 00:32:29,330 with a smaller frequency. 467 00:32:29,330 --> 00:32:32,770 Is now the response of the magnetic moment, 468 00:32:32,770 --> 00:32:36,730 the effective precession frequency, larger, smaller, 469 00:32:36,730 --> 00:32:40,430 or the same as the Rabi frequency as the resonant case? 470 00:32:51,340 --> 00:32:52,190 OK, good. 471 00:32:52,190 --> 00:32:56,320 So that means I can go very quickly about the explanation. 472 00:32:56,320 --> 00:32:57,210 It's correct. 473 00:32:57,210 --> 00:33:00,450 Whenever you are off resonant, this system 474 00:33:00,450 --> 00:33:04,970 precesses faster, so let me summarize 475 00:33:04,970 --> 00:33:08,420 what probably is obvious to all of you. 476 00:33:08,420 --> 00:33:13,110 What happens when we have an off-resonant rotating field. 477 00:33:17,940 --> 00:33:20,870 When we have an off-resonant rotating field, 478 00:33:20,870 --> 00:33:25,370 we go to the rotating frame, but of course, the rotating frame 479 00:33:25,370 --> 00:33:29,420 we go to is now not rotating at the Larmor frequency 480 00:33:29,420 --> 00:33:31,620 because the purpose of going to rotating frame 481 00:33:31,620 --> 00:33:33,860 is get rid of the time dependence of the rotating 482 00:33:33,860 --> 00:33:34,540 field. 483 00:33:34,540 --> 00:33:37,070 So we go to rotating frame which rotates 484 00:33:37,070 --> 00:33:40,820 at the frequency of the rotating field. 485 00:33:40,820 --> 00:33:56,515 So if the rotating field rotates at a frequency omega, 486 00:33:56,515 --> 00:34:01,210 we have a fictitious field, which is omega over gamma. 487 00:34:01,210 --> 00:34:02,715 Gamma is the gyromagnetic ratio. 488 00:34:06,570 --> 00:34:11,750 For the resonant case, we were just completely canceling 489 00:34:11,750 --> 00:34:14,030 the static field in the z direction, 490 00:34:14,030 --> 00:34:16,750 but for the off-resonant case, when 491 00:34:16,750 --> 00:34:21,480 omega is larger or smaller in both cases, 492 00:34:21,480 --> 00:34:26,690 this one here is no longer 0. 493 00:34:26,690 --> 00:34:33,139 And our total effective field is now 494 00:34:33,139 --> 00:34:42,630 the quadrature sum of what we have in the z direction, 495 00:34:42,630 --> 00:34:46,360 and what we have in the x direction or x prime direction, 496 00:34:46,360 --> 00:34:46,889 this is B1. 497 00:34:49,780 --> 00:34:54,429 So in the z direction, we have the static field 498 00:34:54,429 --> 00:35:02,960 minus the fictitious field and then the two are added up 499 00:35:02,960 --> 00:35:06,320 On resonance, the angle theta is 90 degrees, 500 00:35:06,320 --> 00:35:09,480 but for the off-resonant case, the angle 501 00:35:09,480 --> 00:35:18,720 is different given by the simple geometric result. 502 00:35:18,720 --> 00:35:28,910 And the effective field is the quadrature sum of B1 squared 503 00:35:28,910 --> 00:35:50,120 plus B naught minus-- and the fact 504 00:35:50,120 --> 00:35:54,270 is this adds something to the effective field in the rotating 505 00:35:54,270 --> 00:35:57,490 frame whether we drive it above or below resonance. 506 00:35:57,490 --> 00:36:10,070 So therefore the magnetic moment precesses 507 00:36:10,070 --> 00:36:18,500 at what is called the generalized Rabi 508 00:36:18,500 --> 00:36:34,200 frequency, which I now call-- this would be red-- let's use 509 00:36:34,200 --> 00:36:41,420 green instead-- the generalized Rabi frequency, which 510 00:36:41,420 --> 00:36:50,620 is gamma times B effective, and this 511 00:36:50,620 --> 00:36:59,830 is the quadrature sum of the detuning 512 00:36:59,830 --> 00:37:07,310 plus the capital letter omega, omega R, 513 00:37:07,310 --> 00:37:10,120 is the Rabi frequency at resonance, 514 00:37:10,120 --> 00:37:13,440 and this is nothing else than a measure for the drive 515 00:37:13,440 --> 00:37:17,130 field for the strengths of the drive field B1 in frequencies. 516 00:37:19,640 --> 00:37:22,820 So therefore, the generalized Rabi frequency 517 00:37:22,820 --> 00:37:27,110 is the resonant Rabi frequency added in quadrature 518 00:37:27,110 --> 00:37:28,330 with the detuning squared. 519 00:37:40,460 --> 00:37:41,515 Any questions? 520 00:37:47,770 --> 00:37:55,140 So because it's an exact result and I like the result Rabi 521 00:37:55,140 --> 00:37:58,140 flopping at the generalized Rabi frequency, 522 00:37:58,140 --> 00:38:01,160 I want to derive it for you. 523 00:38:01,160 --> 00:38:06,430 So I want to figure out what is the dynamic of a spin which 524 00:38:06,430 --> 00:38:13,400 is originally aligned, and now it 525 00:38:13,400 --> 00:38:18,830 undergoes-- it is driven by the rotating field. 526 00:38:18,830 --> 00:38:21,770 Remember, the resonant case was very simple. 527 00:38:21,770 --> 00:38:24,160 The spin was just doing Rabi flopping. 528 00:38:24,160 --> 00:38:27,540 It was fully inverted, came back, and just 529 00:38:27,540 --> 00:38:30,980 did this at that Rabi frequency. 530 00:38:30,980 --> 00:38:33,880 Now we know that in the off-resonant case, 531 00:38:33,880 --> 00:38:36,690 there will be an effective magnetic field 532 00:38:36,690 --> 00:38:39,740 and it will precess at a faster frequency, which 533 00:38:39,740 --> 00:38:46,620 is a generalized Rabi frequency, but 534 00:38:46,620 --> 00:38:51,940 since the effective magnetic field is not transverse, 535 00:38:51,940 --> 00:38:59,440 it has a z component, the spin will never fully invert. 536 00:38:59,440 --> 00:39:02,180 So geometrically, it's very easy. 537 00:39:02,180 --> 00:39:06,820 We start out with a magnetic moment at zero time, 538 00:39:06,820 --> 00:39:10,530 and I can immediately draw to you the complete solution. 539 00:39:10,530 --> 00:39:18,710 The complete solution is that in the rotating frame, this sort 540 00:39:18,710 --> 00:39:26,130 of precesses around the effective magnetic field. 541 00:39:26,130 --> 00:39:27,780 This is the solution. 542 00:39:27,780 --> 00:39:29,900 But I just wanted to do is because it takes me 543 00:39:29,900 --> 00:39:32,690 three or four minutes, I want to read from this graph, 544 00:39:32,690 --> 00:39:37,110 from this drawing, one of two trigonometric identities 545 00:39:37,110 --> 00:39:40,980 and derive for you the explicit expression, 546 00:39:40,980 --> 00:39:42,850 what is the value of the magnetic moment 547 00:39:42,850 --> 00:39:45,940 as a function of time. 548 00:39:45,940 --> 00:39:49,980 But it's clear from that it will have a maximum value. 549 00:39:49,980 --> 00:39:53,000 It precesses around the tilt direction, 550 00:39:53,000 --> 00:39:56,020 and when it's over there, it has a minimum value, 551 00:39:56,020 --> 00:39:59,520 but it will never completely invert. 552 00:39:59,520 --> 00:40:02,400 Well, quantum mechanically, if you drive a system not 553 00:40:02,400 --> 00:40:06,100 on resonance, you cannot completely invert 554 00:40:06,100 --> 00:40:10,220 the population, but we'll come to the later. 555 00:40:10,220 --> 00:40:11,750 So what do I need? 556 00:40:11,750 --> 00:40:19,675 Well, the spin is moving here on a circle. 557 00:40:24,110 --> 00:40:25,910 I need a few angles. 558 00:40:30,240 --> 00:40:34,210 So let's say the spin was here at one time. 559 00:40:34,210 --> 00:40:38,030 At another time, it is there, and that 560 00:40:38,030 --> 00:40:44,740 would mean that on the circle, it has moved an angle phi. 561 00:40:47,350 --> 00:40:53,300 The tilt angle between the spin and the magnetic field 562 00:40:53,300 --> 00:40:56,650 is what I call theta. 563 00:40:56,650 --> 00:41:06,080 And the angle between the initial magnetic moment 564 00:41:06,080 --> 00:41:12,810 and the magnetic moment at time, t, is what I call alpha. 565 00:41:15,122 --> 00:41:16,955 Yeah, these other the three relevant angles. 566 00:41:22,860 --> 00:41:28,060 The tip of the magnetic moment goes in a circle, 567 00:41:28,060 --> 00:41:41,490 and this circle has a radius, which is mu times sine theta. 568 00:41:41,490 --> 00:41:48,600 Sine theta is nothing else than the rotating magnetic field 569 00:41:48,600 --> 00:41:51,890 over the effective magnetic field, which 570 00:41:51,890 --> 00:41:58,680 is nothing else than the resonant Rabi frequency divided 571 00:41:58,680 --> 00:42:00,165 by the generalized Rabi frequency. 572 00:42:13,920 --> 00:42:16,110 I said what I want to determinant 573 00:42:16,110 --> 00:42:24,850 is the magnetic moment in the z direction as a function 574 00:42:24,850 --> 00:42:30,500 of time, and for that, I defined the angle cosine alpha. 575 00:42:37,870 --> 00:42:43,080 So the way how I derive it the easiest way, its geometry, 576 00:42:43,080 --> 00:42:45,930 its three dimension, and triangles, and all that. 577 00:42:45,930 --> 00:42:53,410 But the best way how I can describe it for you, 578 00:42:53,410 --> 00:42:58,360 let me introduce this auxiliary line, which connects the tip 579 00:42:58,360 --> 00:43:02,780 of the magnetic moment at time t equals 0 and at time t. 580 00:43:02,780 --> 00:43:06,480 And I call the length of this line A. 581 00:43:06,480 --> 00:43:13,230 And I want to derive-- I want to have now two triangles where 582 00:43:13,230 --> 00:43:16,880 one side is A, determine A in two different ways, 583 00:43:16,880 --> 00:43:19,550 combine the equation, we are done. 584 00:43:19,550 --> 00:43:26,400 So the first way is that we have the magnetic moment 585 00:43:26,400 --> 00:43:31,160 at time 0, the magnetic moment at time, t. 586 00:43:31,160 --> 00:43:34,880 We said the angle is alpha, and the two tips 587 00:43:34,880 --> 00:43:50,640 are connected by A, and you know in every triangle you have this 588 00:43:50,640 --> 00:43:56,010 would Pythagoras, but in the general case, 589 00:43:56,010 --> 00:43:58,980 this is valid for general triangle. 590 00:43:58,980 --> 00:44:03,600 So applying that to this triangle, we have a square. 591 00:44:03,600 --> 00:44:07,210 b square is mu square. c square is mu square, 592 00:44:07,210 --> 00:44:10,470 so we get 2 mu square. 593 00:44:10,470 --> 00:44:17,780 And then this term 2 ab is 2 mu times mu times cosine alpha, 594 00:44:17,780 --> 00:44:21,930 so therefore, this is 1 minus cosine alpha. 595 00:44:21,930 --> 00:44:23,820 So we've taken care of the first triangle. 596 00:44:31,000 --> 00:44:34,560 I hope the drawing is not completely confusing 597 00:44:34,560 --> 00:44:36,450 at this point, but why don't we just 598 00:44:36,450 --> 00:44:41,380 look at the drawing looking down the effective field. 599 00:44:41,380 --> 00:44:43,500 We look down at the effective field 600 00:44:43,500 --> 00:44:48,050 and then we see the circle where the magnetic moment precesses, 601 00:44:48,050 --> 00:44:52,990 and the radius of the circle I've already given to you. 602 00:44:52,990 --> 00:44:58,410 So now we want to look down the effective magnetic moment. 603 00:44:58,410 --> 00:45:01,800 We see the circle. 604 00:45:01,800 --> 00:45:04,170 We want to lock down the effective magnetic field. 605 00:45:04,170 --> 00:45:07,495 We see the circular where the magnetic moment precesses. 606 00:45:12,320 --> 00:45:14,730 It's has precessed form here to there. 607 00:45:14,730 --> 00:45:16,350 We connect this line. 608 00:45:16,350 --> 00:45:25,700 This was our A, and the angle at which the magnetic moment 609 00:45:25,700 --> 00:45:28,730 has precesses is phi. 610 00:45:28,730 --> 00:45:35,810 And the radius as we derived before was mu times sine theta. 611 00:45:35,810 --> 00:45:40,250 So now just using the same equation for this triangle, 612 00:45:40,250 --> 00:45:47,170 we find that A square is 2 mu square sine square theta 613 00:45:47,170 --> 00:45:52,040 times 1 minus cosine phi. 614 00:45:54,570 --> 00:46:01,490 And for cosine phi, I want to use the trig identity 615 00:46:01,490 --> 00:46:06,820 and express it by half the angle. 616 00:46:06,820 --> 00:46:08,610 All right, now we are done. 617 00:46:08,610 --> 00:46:10,225 We're pretty much looking at this-- 618 00:46:10,225 --> 00:46:14,360 the drawing is clear precession around a tilt angle, 619 00:46:14,360 --> 00:46:16,510 and we're just doing geometry here. 620 00:46:16,510 --> 00:46:20,540 And we have now two expressions for A square. 621 00:46:20,540 --> 00:46:25,930 We can set the two expression equal 622 00:46:25,930 --> 00:46:34,140 and solve for the unknown, which is cosine over [? phi. ?] 623 00:46:34,140 --> 00:46:37,320 And with that, we find the cosine alpha 624 00:46:37,320 --> 00:46:47,375 is 1 minus 2 sine square theta sine square phi over 2. 625 00:46:50,560 --> 00:46:58,590 And the purpose of this exercise was that cosine alpha tells us 626 00:46:58,590 --> 00:47:01,750 the tilt angle of the magnetic moment away form 627 00:47:01,750 --> 00:47:05,000 the vertical axis, so therefore, we 628 00:47:05,000 --> 00:47:07,450 have done what we wanted to do. 629 00:47:10,480 --> 00:47:15,190 We know the z component of the magnetic moment 630 00:47:15,190 --> 00:47:24,970 as a function of time is this times 1 631 00:47:24,970 --> 00:47:34,415 minus generalized Rabi frequency squared times sine square. 632 00:47:37,710 --> 00:47:42,550 And now we know the precession phi, the precession at which 633 00:47:42,550 --> 00:47:46,280 the keep of the magnetic moment moves in a circle, 634 00:47:46,280 --> 00:47:49,640 we discussed it already before, and you gave the correct answer 635 00:47:49,640 --> 00:47:51,720 with the clicker, is this generalized 636 00:47:51,720 --> 00:47:56,070 Rabi frequency omega Rabi. 637 00:47:59,840 --> 00:48:04,060 So that's a nice formula, but before we lean back and look 638 00:48:04,060 --> 00:48:08,200 at it, let me just do one tiny step. 639 00:48:13,490 --> 00:48:16,380 Based on our quantum mechanical background, 640 00:48:16,380 --> 00:48:37,560 we can now define that the probability that the spin has 641 00:48:37,560 --> 00:48:45,820 been flipped is the relative difference 642 00:48:45,820 --> 00:48:51,230 in the z component appropriately normalized. 643 00:48:51,230 --> 00:48:54,170 This is just the normalized change in the z component, 644 00:48:54,170 --> 00:48:58,271 and if I call that the probability that, 645 00:48:58,271 --> 00:49:00,820 that which is on the left-hand side, 646 00:49:00,820 --> 00:49:07,110 then I find that the probability for this classically expression 647 00:49:07,110 --> 00:49:10,840 for which expresses how much the z component of the magnetic 648 00:49:10,840 --> 00:49:25,660 moment has changed, and this is exactly the celebrated formula 649 00:49:25,660 --> 00:49:28,445 for spin flips in a spin-1/2 system. 650 00:49:42,330 --> 00:49:48,370 So we have derived exactly the solution 651 00:49:48,370 --> 00:49:54,080 for the motion for the precession of a magnetic moment 652 00:49:54,080 --> 00:49:59,030 in a magnetic field plus the rotating magnetic field, 653 00:49:59,030 --> 00:50:02,410 and what we found is, we found that the magnetic moment 654 00:50:02,410 --> 00:50:06,610 precesses at the generalized Rabi frequency. 655 00:50:06,610 --> 00:50:09,890 And as I will show you next week, 656 00:50:09,890 --> 00:50:12,450 this result is exactly the same as 657 00:50:12,450 --> 00:50:16,650 for quantum mechanical expectation values. 658 00:50:16,650 --> 00:50:17,553 Question, yeah? 659 00:50:17,553 --> 00:50:20,692 AUDIENCE: So what if we have magnetic field oscillating just 660 00:50:20,692 --> 00:50:24,315 in one direction instead of rotation magnetic fields? 661 00:50:24,315 --> 00:50:27,190 PROFESSOR: Oh, that's a much more complicated problem. 662 00:50:27,190 --> 00:50:30,620 What happens when we have the magnetic field which 663 00:50:30,620 --> 00:50:33,130 is linearly polarized, which is only 664 00:50:33,130 --> 00:50:35,190 oscillating in one direction? 665 00:50:35,190 --> 00:50:40,030 Well, light or a vector which is oscillating linearly 666 00:50:40,030 --> 00:50:44,240 in one direction can be regarded as a superposition 667 00:50:44,240 --> 00:50:47,400 of a left and right rotating field. 668 00:50:47,400 --> 00:50:50,460 In other words, if you superimpose left-handed 669 00:50:50,460 --> 00:50:53,030 and right-handed circular polarized light, 670 00:50:53,030 --> 00:50:56,290 the sum of the two is linearly polarized light. 671 00:50:56,290 --> 00:50:59,070 So now we have actually the situation 672 00:50:59,070 --> 00:51:02,327 that linearly polarized magnetic field-- that's 673 00:51:02,327 --> 00:51:03,660 what we usually do when the lab. 674 00:51:03,660 --> 00:51:05,100 I mean, we have coils. 675 00:51:05,100 --> 00:51:08,430 We connect them to synthesizer and the field 676 00:51:08,430 --> 00:51:09,810 is not going in a circle. 677 00:51:09,810 --> 00:51:11,080 It's going back and forth. 678 00:51:11,080 --> 00:51:12,590 It's linearly polarized. 679 00:51:12,590 --> 00:51:16,910 This corresponds to a magnetic field 680 00:51:16,910 --> 00:51:20,730 which corresponds to two magnetic field, one 681 00:51:20,730 --> 00:51:24,460 rotates left and one rotates right. 682 00:51:24,460 --> 00:51:26,875 But the problem is if you now do a transformation 683 00:51:26,875 --> 00:51:29,230 in the rotating frame, do we want 684 00:51:29,230 --> 00:51:32,970 to rotate omega to the left or omega to the right? 685 00:51:32,970 --> 00:51:36,870 So what we can do is, we can pick our rotating frame 686 00:51:36,870 --> 00:51:39,830 and we are now in the rotating frame. 687 00:51:39,830 --> 00:51:43,280 One of the rotating fields has become time independent, 688 00:51:43,280 --> 00:51:45,780 the other rotates now at 2 omega. 689 00:51:48,540 --> 00:51:52,040 And at that point, we need the celebrated rotating wave 690 00:51:52,040 --> 00:51:55,770 approximation that we keep the one term [? via ?] rectified 691 00:51:55,770 --> 00:51:58,120 and the other one at two omega rotates 692 00:51:58,120 --> 00:52:01,430 so rapidly that we say these rapid oscillations do nothing 693 00:52:01,430 --> 00:52:03,760 and we discard it. 694 00:52:03,760 --> 00:52:05,440 We'll discussed it later in this course. 695 00:52:07,970 --> 00:52:11,340 But the gist is, if you have linearly-polarized 696 00:52:11,340 --> 00:52:13,940 light, linearly-polarized magnetic fields, 697 00:52:13,940 --> 00:52:17,110 we usually have to do an additional approximation, 698 00:52:17,110 --> 00:52:18,950 the rotating wave approximation. 699 00:52:18,950 --> 00:52:22,090 And since the rotating wave approximation is done always 700 00:52:22,090 --> 00:52:25,030 in almost any treatment, any paper you can find, 701 00:52:25,030 --> 00:52:27,350 we think it's always necessary. 702 00:52:27,350 --> 00:52:30,580 But what I've shown to you is, when we have a rotating field, 703 00:52:30,580 --> 00:52:32,330 we don't need any approximation. 704 00:52:32,330 --> 00:52:37,780 The transformation in the rotating frame is exact, 705 00:52:37,780 --> 00:52:41,170 but that's the beauty of it the when we assume rotating frames, 706 00:52:41,170 --> 00:52:44,540 we can hold on to exact solutions for longer and only 707 00:52:44,540 --> 00:52:48,450 later then discuss what happens when we introduce 708 00:52:48,450 --> 00:52:51,210 linearly-polarized magnetic fields. 709 00:52:55,370 --> 00:52:57,310 But that's something we definitely 710 00:52:57,310 --> 00:53:00,870 do later in its full beauty. 711 00:53:00,870 --> 00:53:02,190 15 minutes. 712 00:53:07,260 --> 00:53:07,760 OK. 713 00:53:07,760 --> 00:53:12,792 There's one thing I want to do about classical spins 714 00:53:12,792 --> 00:53:14,750 and then we do with the full quantum treatment. 715 00:53:18,220 --> 00:53:21,800 And this aspect of classical spins 716 00:53:21,800 --> 00:53:24,720 is called rapid adiabatic passage. 717 00:53:46,080 --> 00:53:50,340 So we want to add one more piece to our discussion. 718 00:53:50,340 --> 00:53:55,360 So far we have assumed a static field and a rotating field, 719 00:53:55,360 --> 00:53:58,040 which rotates at one frequency. 720 00:53:58,040 --> 00:54:01,240 But now we want to change the frequency of the rotating 721 00:54:01,240 --> 00:54:02,280 field. 722 00:54:02,280 --> 00:54:06,300 So we have our magnetic moment in a static magnetic field, 723 00:54:06,300 --> 00:54:08,380 which is our quantization axis. 724 00:54:08,380 --> 00:54:10,260 And now we have a rotating field, 725 00:54:10,260 --> 00:54:13,640 but we increase the frequency of the rotating field from slow 726 00:54:13,640 --> 00:54:16,670 to fast and ask what happens. 727 00:54:16,670 --> 00:54:20,460 And the result is that by increasing the frequency, 728 00:54:20,460 --> 00:54:23,230 sweeping the frequency through the resonance, 729 00:54:23,230 --> 00:54:25,510 we can do something very useful. 730 00:54:25,510 --> 00:54:27,890 We can invert the spin. 731 00:54:27,890 --> 00:54:32,220 We can turn over the magnetic moment in a very robust way, 732 00:54:32,220 --> 00:54:35,550 and this is the concept of rapid adiabatic passage. 733 00:54:35,550 --> 00:54:38,550 A lot of you may be familiar with the concept 734 00:54:38,550 --> 00:54:40,920 a Landau-Zener transition, what I'm 735 00:54:40,920 --> 00:54:44,790 telling you what is exactly the classical counterpart 736 00:54:44,790 --> 00:54:46,700 of the Landau-Zener transition. 737 00:54:46,700 --> 00:54:49,770 actually in many cases when it comes to spin physics, 738 00:54:49,770 --> 00:54:53,030 the classical physics and the quantum physics is the same. 739 00:54:53,030 --> 00:54:56,430 So that's why I want to discuss rapid adiabatic passage 740 00:54:56,430 --> 00:55:03,970 and Landau-Zener physics first in the classical environment. 741 00:55:03,970 --> 00:55:09,610 So rapid adiabatic passage is a technique 742 00:55:09,610 --> 00:55:20,800 for inverting turning around spins or magnetic moments 743 00:55:20,800 --> 00:55:28,550 by sweeping the frequency of your drive field 744 00:55:28,550 --> 00:55:29,980 across the resonance. 745 00:55:36,160 --> 00:55:48,875 So adiabatic means that this frequency sweep has to be slow. 746 00:55:53,800 --> 00:55:59,905 Slow is slow compared to the Larmor frequency. 747 00:56:04,810 --> 00:56:08,080 We get any given moment the magnetic moment precesses 748 00:56:08,080 --> 00:56:14,360 at the Larmor frequency, which is the gyromagnetic ratio 749 00:56:14,360 --> 00:56:17,030 times g-factor magnetic field, so it's 750 00:56:17,030 --> 00:56:18,970 sort of quasi stationary. 751 00:56:18,970 --> 00:56:22,450 The spin precesses around the effective magnetic field 752 00:56:22,450 --> 00:56:25,700 and we want to change the frequency of the drive 753 00:56:25,700 --> 00:56:29,820 slow compared to that motion. 754 00:56:29,820 --> 00:56:31,940 Well, the word rapid adiabatic passage 755 00:56:31,940 --> 00:56:35,890 has the word adiabatic, which means slow but also rapid. 756 00:56:41,140 --> 00:56:44,270 The word rapid means we have to be rapid 757 00:56:44,270 --> 00:56:47,540 compared to relaxation processes, which we do not 758 00:56:47,540 --> 00:56:50,860 discuss here in an idealized environment. 759 00:56:50,860 --> 00:56:53,030 For instance, if you do rapid adiabatic passage 760 00:56:53,030 --> 00:56:55,950 in the environment where the atoms can collide, 761 00:56:55,950 --> 00:56:59,570 rapid means you have to do it fast enough before you have 762 00:56:59,570 --> 00:57:04,400 decoherence in two collisions, so slow compared to the Larmor 763 00:57:04,400 --> 00:57:08,370 frequency and rapid compared to all the things 764 00:57:08,370 --> 00:57:12,900 I'm not mentioning here-- rapid compared 765 00:57:12,900 --> 00:57:17,270 to decoherence and relaxation processes. 766 00:57:31,910 --> 00:57:38,940 I will not set up differential equation and solve them. 767 00:57:38,940 --> 00:57:47,120 I want to give you the intuitive picture of what goes on, 768 00:57:47,120 --> 00:57:53,330 but then also derive what is sort 769 00:57:53,330 --> 00:57:56,329 of the criterion for adiabadicity, 770 00:57:56,329 --> 00:57:57,370 which we have to fulfill. 771 00:58:07,270 --> 00:58:08,630 So what are our ingredients? 772 00:58:08,630 --> 00:58:11,450 We have a magnetic moment, mu. 773 00:58:15,300 --> 00:58:18,310 We have a static field B naught. 774 00:58:20,920 --> 00:58:32,750 We have a drive field B1, which rotates at a frequency omega. 775 00:58:32,750 --> 00:58:41,930 And we assume that the rotating field 776 00:58:41,930 --> 00:58:45,310 is smaller than the static field. 777 00:58:45,310 --> 00:58:48,220 It's not absolutely necessary, but you 778 00:58:48,220 --> 00:58:50,060 apply a big static field and then you 779 00:58:50,060 --> 00:58:51,210 have a repetitive drive. 780 00:58:51,210 --> 00:58:53,150 That's a standard situation. 781 00:58:53,150 --> 00:58:56,640 Or we always want to have quantization axis, which 782 00:58:56,640 --> 00:58:58,840 is given with the z-axis as defined 783 00:58:58,840 --> 00:59:01,600 by the static magnetic field, but it's only 784 00:59:01,600 --> 00:59:03,820 defined by the static magnetic field 785 00:59:03,820 --> 00:59:05,870 if the transverse field is not much 786 00:59:05,870 --> 00:59:07,670 larger than the static field, otherwise, 787 00:59:07,670 --> 00:59:09,978 we are talking about a somewhat different problem. 788 00:59:12,820 --> 00:59:17,710 And let me just assume to be specific we later 789 00:59:17,710 --> 00:59:20,100 discuss what happens if it's not the case. 790 00:59:20,100 --> 00:59:25,020 We assume that we start with a frequency omega, which 791 00:59:25,020 --> 00:59:27,457 is much, much smaller than the Larmor frequency, 792 00:59:27,457 --> 00:59:29,040 much, much smaller than the resonance. 793 00:59:36,710 --> 00:59:41,150 So what does that mean for our effective magnetic field? 794 00:59:43,990 --> 00:59:46,470 Let me just sketch it for you. 795 00:59:46,470 --> 00:59:48,770 Remember our effective magnetic field, 796 00:59:48,770 --> 00:59:50,860 we have a field B naught. 797 00:59:50,860 --> 00:59:54,660 We have a drive field B1, which we assume is smaller. 798 00:59:54,660 --> 00:59:57,410 But then when we go to the rotating frame, 799 00:59:57,410 --> 01:00:01,260 we have a fictitious field, but if the frequency omega 800 01:00:01,260 --> 01:00:04,340 is below the Larmor frequency, this fictitious field 801 01:00:04,340 --> 01:00:05,970 is very small. 802 01:00:05,970 --> 01:00:08,660 So therefore, we start out in a situation 803 01:00:08,660 --> 01:00:11,850 where the effective field is pretty much pointing 804 01:00:11,850 --> 01:00:13,001 along the z direction. 805 01:00:18,100 --> 01:00:23,750 So just to remind you, so we have a situation 806 01:00:23,750 --> 01:00:33,450 where the effective field is just at a tiny angle, 807 01:00:33,450 --> 01:00:42,994 and if we started out with our magnetic moment aligned in z, 808 01:00:42,994 --> 01:00:46,290 and we assume B1 is really perturbative, pretty 809 01:00:46,290 --> 01:00:50,170 much the magnetic moment is very tightly coupled. 810 01:00:50,170 --> 01:00:54,040 It has a very small precession angle, or if you want, 811 01:00:54,040 --> 01:00:56,570 if you take the magnetic field B1 to be perturbative, 812 01:00:56,570 --> 01:00:59,440 you can say the magnetic moment is 813 01:00:59,440 --> 01:01:03,390 aligned with the effective field. 814 01:01:03,390 --> 01:01:06,260 That's the limit that the cone angle of precession 815 01:01:06,260 --> 01:01:07,040 is very small. 816 01:01:17,860 --> 01:01:19,600 So let we write that down. 817 01:01:19,600 --> 01:01:23,720 This is what's omega much, much smaller 818 01:01:23,720 --> 01:01:26,270 than the Larmor frequency. 819 01:01:39,210 --> 01:01:42,100 So now we want to turn up the knob 820 01:01:42,100 --> 01:01:43,700 on the frequency of the drive. 821 01:01:43,700 --> 01:01:49,200 We want to rotate the drive field B1 faster and faster. 822 01:01:49,200 --> 01:01:53,850 And the picture you should have is-- I just wiped it away, 823 01:01:53,850 --> 01:01:58,590 but what that means is, the effective field, 824 01:01:58,590 --> 01:02:02,380 the fictitious field is no longer and longer component 825 01:02:02,380 --> 01:02:05,800 and on resonance, the fictitious magnetic field 826 01:02:05,800 --> 01:02:08,420 will cancel B naught. 827 01:02:08,420 --> 01:02:10,370 So in other words, at this point, 828 01:02:10,370 --> 01:02:13,710 when B naught is canceled, the effective field 829 01:02:13,710 --> 01:02:19,100 has only the B1 component in the x direction. 830 01:02:19,100 --> 01:02:25,110 So therefore, when be go to delta equal 0, 831 01:02:25,110 --> 01:02:32,690 the effective field is only B1 and points in the x direction. 832 01:02:32,690 --> 01:02:36,510 So what we have done is by changing the frequency, 833 01:02:36,510 --> 01:02:39,600 by ramping up the frequency to resonance, 834 01:02:39,600 --> 01:02:42,430 we have tilted the effective magnetic field 835 01:02:42,430 --> 01:02:47,070 from the z direction into the x direction. 836 01:02:47,070 --> 01:02:49,230 And at any given moment, I mean, we 837 01:02:49,230 --> 01:02:52,170 know what the exact solution of the magnetic moment is. 838 01:02:52,170 --> 01:02:54,860 At any given moment, the magnetic moment 839 01:02:54,860 --> 01:02:59,120 precesses around the effective magnetic field. 840 01:02:59,120 --> 01:03:01,710 And if the precession is very fast 841 01:03:01,710 --> 01:03:05,070 and the effective magnetic field is slowly rotating, 842 01:03:05,070 --> 01:03:08,930 the magnetic moment is just following. 843 01:03:08,930 --> 01:03:12,310 So therefore, at this point when we are at resonance, 844 01:03:12,310 --> 01:03:17,560 we have tilted the magnetic moment by 90 degrees. 845 01:03:17,560 --> 01:03:19,010 Just one second. 846 01:03:19,010 --> 01:03:24,100 And if we go with a frequency much higher 847 01:03:24,100 --> 01:03:31,910 than the Larmor frequency, then our fictitious magnetic field 848 01:03:31,910 --> 01:03:35,610 is much larger than B naught and therefore, 849 01:03:35,610 --> 01:03:42,410 the effective magnetic field points now 850 01:03:42,410 --> 01:03:43,470 in the minus z direction. 851 01:03:49,070 --> 01:03:52,750 So the idea is that as long as this rotation 852 01:03:52,750 --> 01:03:57,200 of the effective field from being plus z in the x direction 853 01:03:57,200 --> 01:04:01,850 and into minus z is slow enough, the rapid precession is 854 01:04:01,850 --> 01:04:05,750 locking, is keeping the magnetic moment aligned 855 01:04:05,750 --> 01:04:07,970 with the effective magnetic field 856 01:04:07,970 --> 01:04:12,800 and we have just a handle we invert. 857 01:04:12,800 --> 01:04:14,550 We move around the magnetic moment. 858 01:04:17,180 --> 01:04:26,780 So in the adiabatic limit, the spin precesses tightly, 859 01:04:26,780 --> 01:04:33,880 and by tightly I mean the angle theta is small around B ef, B 860 01:04:33,880 --> 01:04:44,910 effective, and follows the direction 861 01:04:44,910 --> 01:04:48,990 of the effective magnetic field. 862 01:04:48,990 --> 01:04:51,910 So we are rotating an effective field. 863 01:04:51,910 --> 01:04:54,160 We are rotating the magnetic moment, 864 01:04:54,160 --> 01:04:57,230 but we're not rotating anything in the laboratory. 865 01:04:57,230 --> 01:04:59,025 The only thing we are doing is, we 866 01:04:59,025 --> 01:05:01,610 are changing the frequency of the rotating magnetic field. 867 01:05:11,530 --> 01:05:12,040 Questions? 868 01:05:12,040 --> 01:05:12,540 Jenny. 869 01:05:12,540 --> 01:05:13,666 AUDIENCE: Oh, yeah. 870 01:05:13,666 --> 01:05:15,230 I was thinking, can you also do this 871 01:05:15,230 --> 01:05:17,198 by keeping the frequency the same 872 01:05:17,198 --> 01:05:19,166 and just ramping up the strength? 873 01:05:19,166 --> 01:05:23,594 Like say, put it at the-- make the B1 874 01:05:23,594 --> 01:05:26,546 equal to the frequency of that Larmor frequency 875 01:05:26,546 --> 01:05:29,498 and just start from 0 and ramp up? 876 01:05:29,498 --> 01:05:30,430 PROFESSOR: Yes. 877 01:05:30,430 --> 01:05:34,620 I mean, the essence here is that the effective magnetic field 878 01:05:34,620 --> 01:05:37,640 is [INAUDIBLE] resonance, and what you are suggesting 879 01:05:37,640 --> 01:05:40,120 is if the frequency is constant, that would 880 01:05:40,120 --> 01:05:43,090 mean the fictitious magnetic field is constant. 881 01:05:43,090 --> 01:05:46,060 But if you were given a fictitious magnetic field 882 01:05:46,060 --> 01:05:49,830 and we a huge field B naught, the effective magnetic field 883 01:05:49,830 --> 01:05:51,800 points out, points up. 884 01:05:51,800 --> 01:05:54,800 But if we now make this static field B naught smaller 885 01:05:54,800 --> 01:05:57,110 and smaller, both through resonance 886 01:05:57,110 --> 01:05:59,710 and make it even smaller, we have also 887 01:05:59,710 --> 01:06:02,680 done an inversion of the effective magnetic field. 888 01:06:02,680 --> 01:06:05,220 The result is the same. 889 01:06:05,220 --> 01:06:07,180 Actually, sometimes in the laboratory, 890 01:06:07,180 --> 01:06:09,610 if you have a synthesizer which is not easily 891 01:06:09,610 --> 01:06:12,530 computer controlled, we do an analog sweep 892 01:06:12,530 --> 01:06:14,980 of the magnetic field, so we actually 893 01:06:14,980 --> 01:06:18,910 change the Larmor frequency of the atom instead 894 01:06:18,910 --> 01:06:20,580 of changing the drive frequency. 895 01:06:20,580 --> 01:06:22,290 What really matters in the whole business 896 01:06:22,290 --> 01:06:24,480 is the relative frequency between the two. 897 01:06:24,480 --> 01:06:29,400 AUDIENCE: Is it necessarily true that the spin resistance 898 01:06:29,400 --> 01:06:31,860 is tightly around the effective fields? 899 01:06:31,860 --> 01:06:35,796 Isn't it at resonance the radius of the circle in precession 900 01:06:35,796 --> 01:06:38,748 is equal to mu? 901 01:06:38,748 --> 01:06:42,438 Is it my [INAUDIBLE] before we found 902 01:06:42,438 --> 01:06:44,160 the radius which was mu sine theta. 903 01:06:44,160 --> 01:06:49,160 Then sine theta cosine Omega R-- 904 01:06:49,160 --> 01:06:49,895 PROFESSOR: Yes. 905 01:06:49,895 --> 01:06:50,978 AUDIENCE: --divided into-- 906 01:06:50,978 --> 01:06:52,776 PROFESSOR: But the way is we start-- 907 01:06:52,776 --> 01:06:54,150 the way we've started it up here. 908 01:06:54,150 --> 01:06:59,820 Now if we are far away from resonance, 909 01:06:59,820 --> 01:07:04,390 the angle theta is infinitesimally small. 910 01:07:04,390 --> 01:07:05,500 Something confuses you. 911 01:07:05,500 --> 01:07:06,125 AUDIENCE: Yeah. 912 01:07:06,125 --> 01:07:09,702 I mean, it's just [INAUDIBLE]. 913 01:07:09,702 --> 01:07:11,160 PROFESSOR: See, what we have solved 914 01:07:11,160 --> 01:07:14,730 before, we have done something else before. 915 01:07:14,730 --> 01:07:16,550 What we have discussed before is, 916 01:07:16,550 --> 01:07:19,140 that we have a spin which is aligned 917 01:07:19,140 --> 01:07:23,520 and then we looked at the-- you can say the transient solution, 918 01:07:23,520 --> 01:07:27,120 we switched suddenly on the rotating field. 919 01:07:27,120 --> 01:07:29,090 So that may sort of described that. 920 01:07:29,090 --> 01:07:31,410 We have a magnetic field, which is in the z direction. 921 01:07:31,410 --> 01:07:32,330 Obviously we don't. 922 01:07:32,330 --> 01:07:34,370 Our spin is aligned in the z direction. 923 01:07:34,370 --> 01:07:38,290 And if you then suddenly switch on a rotating field, 924 01:07:38,290 --> 01:07:40,290 that could mean you have suddenly 925 01:07:40,290 --> 01:07:42,800 created an effective field, which is tilted. 926 01:07:42,800 --> 01:07:46,020 So then by the sudden switch on of your drive, 927 01:07:46,020 --> 01:07:49,680 you have an angle theta and the magnetic moment precesses 928 01:07:49,680 --> 01:07:51,990 with the precession angle theta. 929 01:07:51,990 --> 01:07:53,900 But what we are here doing is, we 930 01:07:53,900 --> 01:07:59,060 are ever so slightly changing the angle theta 931 01:07:59,060 --> 01:08:01,169 and then the spin stays aligned. 932 01:08:01,169 --> 01:08:02,085 That's the difference. 933 01:08:08,740 --> 01:08:18,750 So before I give you another example 934 01:08:18,750 --> 01:08:25,704 and discuss the limit of adiabadicity, 935 01:08:25,704 --> 01:08:27,120 let's just have a quicker question 936 01:08:27,120 --> 01:08:29,620 to make sure that everybody follows. 937 01:08:29,620 --> 01:08:37,479 So OK, what I think we have at least understood 938 01:08:37,479 --> 01:08:40,410 in the adiabatic limit is, that when 939 01:08:40,410 --> 01:08:49,710 we start with a magnetic moment, which is aligned, 940 01:08:49,710 --> 01:08:56,960 and now we do a sweep, we start with a small rotating frequency 941 01:08:56,960 --> 01:08:58,819 and we make it large. 942 01:08:58,819 --> 01:09:03,590 We sweep from low to high frequency through resonance, 943 01:09:03,590 --> 01:09:10,180 then we can invert the magnetic moment. 944 01:09:10,180 --> 01:09:20,420 My question for you now is, what happens 945 01:09:20,420 --> 01:09:25,330 when we start with a spin but we switch 946 01:09:25,330 --> 01:09:29,930 on a drive, which is a buff resonance 947 01:09:29,930 --> 01:09:32,960 and we sweep it with small frequencies? 948 01:09:32,960 --> 01:09:37,020 What will now happen when we do the opposite sweep? 949 01:09:37,020 --> 01:09:44,619 Will the spin just stay, will it be flipped, 950 01:09:44,619 --> 01:09:47,659 or will it sort of get diffused or get disoriented? 951 01:09:50,250 --> 01:09:55,250 So this is answer A, this is answer B, and this is answer C. 952 01:09:55,250 --> 01:09:57,020 So what happens? 953 01:09:57,020 --> 01:09:59,310 I have assumed in already discussion 954 01:09:59,310 --> 01:10:02,890 before that we start with a very low frequency where 955 01:10:02,890 --> 01:10:05,920 the fictitious magnetic field is negligible. 956 01:10:05,920 --> 01:10:08,070 And then we change the frequency, 957 01:10:08,070 --> 01:10:09,580 we sweep the frequency. 958 01:10:09,580 --> 01:10:12,860 It cause the Larmor frequency to very high frequency. 959 01:10:12,860 --> 01:10:15,460 My question now is, what happens when 960 01:10:15,460 --> 01:10:18,740 we reverse the frequency of the sweep? 961 01:10:37,445 --> 01:10:37,945 OK. 962 01:10:40,870 --> 01:10:43,440 There is some distribution. 963 01:10:43,440 --> 01:10:47,570 The correct answer is indeed B, we flip the spin. 964 01:10:47,570 --> 01:10:51,510 What happens is the following. 965 01:10:51,510 --> 01:10:55,050 The spin is in the z direction, but at high frequency, 966 01:10:55,050 --> 01:10:57,850 the fictitious magnetic field is very large. 967 01:10:57,850 --> 01:11:00,440 So therefore, the effective magnetic field 968 01:11:00,440 --> 01:11:03,500 is now a large effective magnetic field 969 01:11:03,500 --> 01:11:06,540 pointing in the minus z direction. 970 01:11:06,540 --> 01:11:10,390 So now we have a situation where the spin tightly precesses 971 01:11:10,390 --> 01:11:12,980 around the magnetic effective magnetic field 972 01:11:12,980 --> 01:11:17,400 at an angle of 180 degrees, not 0 degrees, but 180 degrees. 973 01:11:17,400 --> 01:11:23,410 And now, when we change the detuning, 974 01:11:23,410 --> 01:11:26,250 the effective magnetic field tilts, 975 01:11:26,250 --> 01:11:29,580 but the spin keeps on precessing at 180 degrees 976 01:11:29,580 --> 01:11:33,070 and eventually when we sweep through the resonance, 977 01:11:33,070 --> 01:11:36,890 the effective magnetic field at low frequency of the rotating 978 01:11:36,890 --> 01:11:39,710 field becomes the real magnetic field is now 979 01:11:39,710 --> 01:11:43,020 pointing in the z direction and the spin 980 01:11:43,020 --> 01:11:46,150 has followed the 180 degree rotation. 981 01:11:46,150 --> 01:11:48,570 So in other words, it does not matter 982 01:11:48,570 --> 01:11:51,930 whether you go from low to high or from high to low frequency. 983 01:11:51,930 --> 01:11:55,230 Whenever you go through the resonance, you flip the spin. 984 01:12:02,100 --> 01:12:05,150 Let's close that Let's see. 985 01:12:05,150 --> 01:12:05,916 It is that. 986 01:12:14,970 --> 01:12:22,250 So the more generalized answer is 987 01:12:22,250 --> 01:12:38,130 that the rapid adiabatic passage always swaps the spin state 988 01:12:38,130 --> 01:12:39,890 no matter which way you sweep. 989 01:12:39,890 --> 01:12:43,240 When you start in spin up, you wind up in spin down. 990 01:12:43,240 --> 01:12:46,050 When you start in spin down, you wind up 991 01:12:46,050 --> 01:12:48,320 in spin up because what you're doing 992 01:12:48,320 --> 01:12:53,280 is you're inverting the direction 993 01:12:53,280 --> 01:12:55,950 of the effective magnetic field and the spin just follows. 994 01:12:58,820 --> 01:13:01,330 So therefore it goes both ways. 995 01:13:01,330 --> 01:13:04,030 It applies to-- you can say to the ground state 996 01:13:04,030 --> 01:13:07,300 with the lowest energy state, it applies to the highest energy 997 01:13:07,300 --> 01:13:11,640 state, and as such, it is actually 998 01:13:11,640 --> 01:13:17,320 a swap operation like the pi over 2 pulse. 999 01:13:17,320 --> 01:13:20,950 When you have a pi over 2 pulse, you have a spin 1000 01:13:20,950 --> 01:13:24,410 and you just pulse on a rotating magnetic field, which 1001 01:13:24,410 --> 01:13:28,260 in the rotating flame means you have an x field. 1002 01:13:28,260 --> 01:13:32,170 When the spin was up at the Rabi frequency, rotates down. 1003 01:13:32,170 --> 01:13:36,300 When the spin was down at the Rabi frequency, it rotates up. 1004 01:13:36,300 --> 01:13:40,410 And quantum mechanically it just means you take the population 1005 01:13:40,410 --> 01:13:44,910 in two states and your unitary time evolution 1006 01:13:44,910 --> 01:13:45,950 is a swap operation. 1007 01:13:48,540 --> 01:13:51,670 And the classical counterpart is what we just discussed, 1008 01:13:51,670 --> 01:13:54,670 the rapid adiabatic passage. 1009 01:13:54,670 --> 01:13:58,470 However, there is one big experimental advantage 1010 01:13:58,470 --> 01:14:03,320 of doing rapid adiabatic passage over pi over 2 pulse. 1011 01:14:03,320 --> 01:14:08,120 And I think many of you use rapid adiabatic passage 1012 01:14:08,120 --> 01:14:09,250 in the lab. 1013 01:14:09,250 --> 01:14:11,760 Remember, if you give yourself the drop, 1014 01:14:11,760 --> 01:14:17,730 you want to transfer the magnetic moment form spin up 1015 01:14:17,730 --> 01:14:19,870 to spin down. 1016 01:14:19,870 --> 01:14:22,502 If you want to do it with the pi pulse-- 1017 01:14:22,502 --> 01:14:25,910 if you want to do it with the-- I said pi over 2 pulse, 1018 01:14:25,910 --> 01:14:27,057 I meant pi pulse. 1019 01:14:33,740 --> 01:14:36,860 The only way how with a pulse you 1020 01:14:36,860 --> 01:14:42,460 can rotate the magnetic moment by 180 degrees, if you 1021 01:14:42,460 --> 01:14:46,650 are exactly on resonance, where the fictitious field cancels 1022 01:14:46,650 --> 01:14:50,900 the static field and what you have is in the rotating frame 1023 01:14:50,900 --> 01:14:52,920 is only a field in x. 1024 01:14:52,920 --> 01:14:55,940 And then you can rotate around the x-axis. 1025 01:14:55,940 --> 01:15:01,100 But if for some reason you're not exactly on resonance, then 1026 01:15:01,100 --> 01:15:04,090 the fictitious field does not cancel the static field, 1027 01:15:04,090 --> 01:15:07,370 and your effective magnetic field is at an angle. 1028 01:15:07,370 --> 01:15:10,120 And if you rotate an effective magnetic field, 1029 01:15:10,120 --> 01:15:12,990 which is not in the x-axis but at an angle, 1030 01:15:12,990 --> 01:15:16,790 you cannot do a full inversion of the magnetic moment. 1031 01:15:16,790 --> 01:15:20,540 So in other words, if you want to use a pi pulse to flip over 1032 01:15:20,540 --> 01:15:24,300 a spin, you have to pulse on your drive exactly 1033 01:15:24,300 --> 01:15:25,970 on resonance. 1034 01:15:25,970 --> 01:15:29,690 And if you have MBN magnetic fields which drift by a few 1035 01:15:29,690 --> 01:15:32,250 milligals and you're not sure where the resonance is, 1036 01:15:32,250 --> 01:15:35,330 you cannot do a perfect pi pulse. 1037 01:15:35,330 --> 01:15:40,160 But with the sweep, you just have to sweep from point A 1038 01:15:40,160 --> 01:15:43,340 to point B, and if you know you cross the resonance, 1039 01:15:43,340 --> 01:15:47,750 you have a perfect inversion of the magnetic moment, which 1040 01:15:47,750 --> 01:15:49,726 is robust against frequency drifts. 1041 01:15:59,446 --> 01:16:04,360 Of course, you have to do a longer sweep. 1042 01:16:04,360 --> 01:16:06,610 The pi pulse is the shortest possible wave. 1043 01:16:06,610 --> 01:16:09,396 If you heat the cloud on resonance, you first sweep, 1044 01:16:09,396 --> 01:16:10,770 nothing happens, nothing happens, 1045 01:16:10,770 --> 01:16:13,394 and you go through resonance and just view further and further, 1046 01:16:13,394 --> 01:16:14,840 you waste some of your time. 1047 01:16:14,840 --> 01:16:18,230 So you pay a price for it, but often we want precision 1048 01:16:18,230 --> 01:16:20,470 and we have the time to do it and then 1049 01:16:20,470 --> 01:16:22,970 the job is done by rapid adiabatic passage. 1050 01:16:27,880 --> 01:16:30,050 Let me just mention one more thing and then I stop. 1051 01:16:33,570 --> 01:16:42,004 I have discussed the physics of keeping-- 1052 01:16:42,004 --> 01:16:43,420 with rapid adiabatic passage, I've 1053 01:16:43,420 --> 01:16:48,140 discussed with you the physics that rapid precession keeps 1054 01:16:48,140 --> 01:16:52,750 a magnetic moment aligned with an effective magnetic field. 1055 01:16:52,750 --> 01:16:57,410 Let me now discuss the same phenomenon 1056 01:16:57,410 --> 01:16:58,910 but in a very different environment. 1057 01:17:03,720 --> 01:17:11,170 And this is a similar process happen in a magnetic trap. 1058 01:17:11,170 --> 01:17:19,110 In a magnetic trap, we don't have any drive 1059 01:17:19,110 --> 01:17:23,730 or we have a time-dependent field. 1060 01:17:23,730 --> 01:17:27,450 But what happens is, we have a magnetic field, 1061 01:17:27,450 --> 01:17:30,950 but the atoms move through the atom trap. 1062 01:17:30,950 --> 01:17:35,720 So the atom sees a changing magnetic field. 1063 01:17:35,720 --> 01:17:39,350 And in many of our experiments here at MIT, 1064 01:17:39,350 --> 01:17:43,290 we use a quadrupolar field. 1065 01:17:43,290 --> 01:17:47,210 We've discussed some of these aspects in 8.422, 1066 01:17:47,210 --> 01:17:50,780 so a quadrupolar field, the field 1067 01:17:50,780 --> 01:17:56,080 has to be homogeneous in order to provide trapping forces, 1068 01:17:56,080 --> 01:18:00,320 so we often use quadrupolar fields, a lot of advantages. 1069 01:18:00,320 --> 01:18:03,240 That's how we build the tightest magnetic traps. 1070 01:18:03,240 --> 01:18:07,000 But what happens now is, if an atom moves 1071 01:18:07,000 --> 01:18:12,520 along this projector, it moves up in the laboratory, 1072 01:18:12,520 --> 01:18:16,830 here t let's say the atom is in spin up. 1073 01:18:16,830 --> 01:18:18,710 Here it's now a aligned. 1074 01:18:21,670 --> 01:18:24,660 The magnetic field is opposite to the spin 1075 01:18:24,660 --> 01:18:28,390 and now it moves up, and now the magnetic field up here 1076 01:18:28,390 --> 01:18:30,090 is pointing up. 1077 01:18:30,090 --> 01:18:34,100 The physics I explain to you rapid adiabatic passage means 1078 01:18:34,100 --> 01:18:36,400 that the rapid precession off this spin 1079 01:18:36,400 --> 01:18:40,730 means that as the atom moves, the spin stays 1080 01:18:40,730 --> 01:18:43,050 aligned with the magnetic field. 1081 01:18:43,050 --> 01:18:46,700 So you find the same physics here 1082 01:18:46,700 --> 01:18:49,615 in a different environment, but the mathematical description 1083 01:18:49,615 --> 01:18:50,894 is the same. 1084 01:18:53,786 --> 01:18:57,380 Of course, and that's my last comment for today, 1085 01:18:57,380 --> 01:18:59,790 if you have a spheric quadrupole trap 1086 01:18:59,790 --> 01:19:04,150 and you go right through the origin, 1087 01:19:04,150 --> 01:19:06,970 you're out of luck because here, the atom 1088 01:19:06,970 --> 01:19:09,040 sees the magnetic field is down. 1089 01:19:09,040 --> 01:19:10,506 It gets smaller, the magnetic field 1090 01:19:10,506 --> 01:19:11,630 gets smaller, gets smaller. 1091 01:19:11,630 --> 01:19:13,270 The magnet field gets zero. 1092 01:19:13,270 --> 01:19:16,600 Oopsy, the magnetic field points in the other direction. 1093 01:19:16,600 --> 01:19:18,050 And there was no warning. 1094 01:19:18,050 --> 01:19:22,600 The magnetic field has jumped from 0 degree to 180 degree. 1095 01:19:22,600 --> 01:19:27,260 There was never, ever any transverse field around which 1096 01:19:27,260 --> 01:19:30,360 the atom could precess and change its orientation. 1097 01:19:30,360 --> 01:19:33,580 So therefore, when an atom is aligned with a magnetic field 1098 01:19:33,580 --> 01:19:36,590 moves through the origin, oopsy, it's anti aligned. 1099 01:19:36,590 --> 01:19:40,280 It has lost its orientation with respect to the magnetic field, 1100 01:19:40,280 --> 01:19:43,610 and this is the breakdown of rapid adiabatic passage 1101 01:19:43,610 --> 01:19:45,230 because there's no adiabadicity. 1102 01:19:45,230 --> 01:19:47,770 It is not an adiabatic change of the direction 1103 01:19:47,770 --> 01:19:50,210 of the magnetic field, it's a sudden field. 1104 01:19:50,210 --> 01:19:52,310 And the consequences are bad. 1105 01:19:52,310 --> 01:19:54,580 You lose your atoms from the magnetic trap. 1106 01:19:54,580 --> 01:19:56,207 It's called Majorana losses. 1107 01:19:56,207 --> 01:19:58,040 A lot of people know what I'm talking about, 1108 01:19:58,040 --> 01:20:01,200 but I'm not explaining it in detail. 1109 01:20:01,200 --> 01:20:02,020 Time is over. 1110 01:20:02,020 --> 01:20:07,760 Any question about what I've discussed? 1111 01:20:07,760 --> 01:20:08,639 Yes. 1112 01:20:08,639 --> 01:20:09,637 AUDIENCE: [INAUDIBLE]. 1113 01:20:20,116 --> 01:20:22,262 PROFESSOR: You said the weights of the-- 1114 01:20:22,262 --> 01:20:23,750 AUDIENCE: Yeah, so [INAUDIBLE]. 1115 01:20:31,686 --> 01:20:34,690 PROFESSOR: I would say-- the question is about the frequency 1116 01:20:34,690 --> 01:20:37,950 weights and if you switch on the frequency drive, 1117 01:20:37,950 --> 01:20:39,610 it's not a delta function. 1118 01:20:39,610 --> 01:20:41,270 If we are far away from resonance, 1119 01:20:41,270 --> 01:20:43,570 it doesn't really matter. 1120 01:20:43,570 --> 01:20:47,160 That is, of course, a criterion that the effective weights 1121 01:20:47,160 --> 01:20:54,690 of the frequency should not be comparable to the detuning. 1122 01:20:54,690 --> 01:20:57,259 So then you switch it on, but you're so far away 1123 01:20:57,259 --> 01:20:58,800 from resonance that it doesn't matter 1124 01:20:58,800 --> 01:21:01,410 if you have a small weights, the effective detuning is still 1125 01:21:01,410 --> 01:21:02,120 large. 1126 01:21:02,120 --> 01:21:03,980 You scan for resonance. 1127 01:21:03,980 --> 01:21:06,210 But these are sort of the boundary condition. 1128 01:21:06,210 --> 01:21:08,860 The exact solution in the Landau-Zener problem, 1129 01:21:08,860 --> 01:21:11,830 for instance, assumes you go from minus infinity 1130 01:21:11,830 --> 01:21:13,990 to plus infinity in the detuning. 1131 01:21:13,990 --> 01:21:15,250 Nobody does that. 1132 01:21:15,250 --> 01:21:20,220 So we're discussing sort of sweep finite duration effects, 1133 01:21:20,220 --> 01:21:26,550 but usually we are pretty close to the idealized assumption. 1134 01:21:26,550 --> 01:21:29,100 OK, enjoy the Monday holiday. 1135 01:21:29,100 --> 01:21:32,410 We meet on Tuesday, and Tuesday is 1136 01:21:32,410 --> 01:21:35,830 in building 37 in our standard classroom.