WEBVTT 00:00:00.090 --> 00:00:01.770 The following content is provided 00:00:01.770 --> 00:00:04.010 under a Creative Commons license. 00:00:04.010 --> 00:00:06.860 Your support will help MIT OpenCourseWare continue 00:00:06.860 --> 00:00:10.720 to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.190 --> 00:00:17.815 at ocw.mit.edu. 00:00:26.225 --> 00:00:30.920 PROFESSOR: Then let me just-- I gave you 00:00:30.920 --> 00:00:34.540 two things about the last class, one 00:00:34.540 --> 00:00:42.880 is the question about the electron and factors of 2 00:00:42.880 --> 00:00:45.160 and what precesses at what frequency. 00:00:49.160 --> 00:00:53.230 Think about it, I mean, write down for yourself 00:00:53.230 --> 00:00:55.530 what is a magnetic moment of an electron? 00:00:55.530 --> 00:00:59.610 What is a magnetic moment of a classical particles which 00:00:59.610 --> 00:01:02.570 is one unit of angular momentum and try to sort of re 00:01:02.570 --> 00:01:03.530 derive it. 00:01:03.530 --> 00:01:08.300 The way how I did it is deceptively simple. 00:01:08.300 --> 00:01:11.000 But whenever you think about it, you 00:01:11.000 --> 00:01:13.660 will get confused about some factors. 00:01:13.660 --> 00:01:18.410 But this is, of course, where I showed you here 00:01:18.410 --> 00:01:23.540 is the quantum mechanical mystery about the g-factor of 2 00:01:23.540 --> 00:01:25.680 that you have an electron which has one more 00:01:25.680 --> 00:01:28.620 magneton of magnetic moment, and you have a classical particle 00:01:28.620 --> 00:01:30.180 which has one more magneton. 00:01:30.180 --> 00:01:33.840 The energy levels of up and down what are the same. 00:01:33.840 --> 00:01:37.350 The energy is 1.4 megahertz per Gauss. 00:01:37.350 --> 00:01:40.740 But the precession of frequency of the electron 00:01:40.740 --> 00:01:43.420 is 2 times faster than the precession frequency 00:01:43.420 --> 00:01:45.440 of the classical particle, and this 00:01:45.440 --> 00:01:49.260 is what the g-factor of 2, which comes out of the Dirac equation 00:01:49.260 --> 00:01:51.130 means. 00:01:51.130 --> 00:01:53.880 And at least one intuitive explanation 00:01:53.880 --> 00:01:57.440 I can give you is a frequency which is observed 00:01:57.440 --> 00:02:00.070 as a precession frequency in the laboratory 00:02:00.070 --> 00:02:03.420 is the beat frequency between two neighboring levels. 00:02:03.420 --> 00:02:05.014 And here you have a level in between. 00:02:05.014 --> 00:02:07.180 So the beat frequency between two neighboring levels 00:02:07.180 --> 00:02:09.664 is 1.4 megahertz, and here the beat frequency 00:02:09.664 --> 00:02:12.210 is 2.8 megahertz. 00:02:12.210 --> 00:02:13.910 And you will actually see that if you 00:02:13.910 --> 00:02:16.660 take a classic system which has much, 00:02:16.660 --> 00:02:20.260 much larger angular momentum, maybe L equals 10. 00:02:20.260 --> 00:02:21.790 It will have this level structure, 00:02:21.790 --> 00:02:26.690 but it will go from minus 10 to plus 10. 00:02:26.690 --> 00:02:31.020 But the precession frequency is the beat frequency 00:02:31.020 --> 00:02:33.840 between two levels. 00:02:33.840 --> 00:02:37.460 Or if you don't like the word beat frequency, 00:02:37.460 --> 00:02:40.370 it precesses and you can increase the precession. 00:02:42.789 --> 00:02:44.580 So if you're not increasing the impression, 00:02:44.580 --> 00:02:47.550 you can increase the energy of the precessing system 00:02:47.550 --> 00:02:51.250 by driving it at the resonance frequency which 00:02:51.250 --> 00:02:52.850 is a precession frequency. 00:02:52.850 --> 00:02:55.930 And the resonance is always the energy difference 00:02:55.930 --> 00:02:57.220 between two levels. 00:02:57.220 --> 00:03:00.300 So no matter how many levels you have in a classical system 00:03:00.300 --> 00:03:05.640 with a g-factor of 1, you're always driving it step by step, 00:03:05.640 --> 00:03:09.410 and each step is the resonance and this 00:03:09.410 --> 00:03:11.820 is the precession frequency of the system. 00:03:14.430 --> 00:03:16.250 Questions about it? 00:03:16.250 --> 00:03:20.380 I know people get confused all the time, 00:03:20.380 --> 00:03:22.170 but just think about it. 00:03:22.170 --> 00:03:24.160 I think I gave you all the different angles 00:03:24.160 --> 00:03:28.130 you can use to look at it. 00:03:28.130 --> 00:03:29.210 Questions about that? 00:03:31.990 --> 00:03:33.680 OK. 00:03:33.680 --> 00:03:40.490 Next thing I just want to point out is rotation. 00:03:40.490 --> 00:03:45.360 Yes, we are spending quite some time in class 00:03:45.360 --> 00:03:48.670 just to figure out that something is rotating. 00:03:48.670 --> 00:03:53.420 So what we figured out here is that if you 00:03:53.420 --> 00:04:03.550 go to a rotating frame, if you have angular momentum, 00:04:03.550 --> 00:04:07.500 and we go to a rotating frame, that that simply 00:04:07.500 --> 00:04:11.570 means that the effect of a rotating frame 00:04:11.570 --> 00:04:18.720 is to simply add a fictitious magnetic fields 00:04:18.720 --> 00:04:19.930 to the real magnetic field. 00:04:23.700 --> 00:04:27.390 Well, that's very handy. 00:04:27.390 --> 00:04:29.610 For instance, we know then immediately 00:04:29.610 --> 00:04:33.660 if you have a real magnetic field 00:04:33.660 --> 00:04:35.840 and we pick the fictitious magnetic field 00:04:35.840 --> 00:04:37.850 in such a way that the real magnetic field 00:04:37.850 --> 00:04:39.215 that the total field cancels it. 00:04:39.215 --> 00:04:42.870 It cancels the total magnetic field, well, 00:04:42.870 --> 00:04:46.740 then we have a system viewed from the rotating frame 00:04:46.740 --> 00:04:48.780 where the effective field is 0. 00:04:48.780 --> 00:04:51.210 Well, the spin is zero field does nothing 00:04:51.210 --> 00:04:53.750 and then we know when you go back into the lab frame, 00:04:53.750 --> 00:04:55.450 all this spin is doing it rotates. 00:04:55.450 --> 00:04:58.230 It's an exact solution. 00:04:58.230 --> 00:05:01.670 I want to use it today again but in a different way, 00:05:01.670 --> 00:05:04.290 but I want to just point out today 00:05:04.290 --> 00:05:07.120 we will not picking the fictitious field to make 00:05:07.120 --> 00:05:08.990 the magnetic field zero. 00:05:08.990 --> 00:05:11.890 Today we are talking about what happens 00:05:11.890 --> 00:05:15.620 to a spin in a rotating field. 00:05:15.620 --> 00:05:16.466 Complicated. 00:05:16.466 --> 00:05:20.680 A time-dependent problem, time-dependent Hamiltonian. 00:05:20.680 --> 00:05:23.610 But if you go in a rotating frame, if you rotate 00:05:23.610 --> 00:05:26.980 with the field, then in the rotating frame 00:05:26.980 --> 00:05:29.280 it becomes a time-independent problem 00:05:29.280 --> 00:05:31.260 which we can immediately solve. 00:05:31.260 --> 00:05:34.490 So today we use the same transformation again, 00:05:34.490 --> 00:05:38.650 but we pick our frequency not to cancel some static field. 00:05:38.650 --> 00:05:41.850 We pick our frequency to co-rotate 00:05:41.850 --> 00:05:46.590 with an external rotating field that in our rotating frame now 00:05:46.590 --> 00:05:47.704 everything is stationary. 00:05:47.704 --> 00:05:49.370 So that's what you're going to do today. 00:05:53.110 --> 00:05:55.550 And finally just to give you an outlook, 00:05:55.550 --> 00:05:58.120 just try to sort of make you aware that often what we're 00:05:58.120 --> 00:06:01.050 doing is the same thing in a different angle. 00:06:01.050 --> 00:06:02.930 So this is one way to deal with rotation. 00:06:02.930 --> 00:06:07.590 I'll give you an exact solution by going to rotating frame. 00:06:07.590 --> 00:06:12.960 I will later show you today that quantum mechanically 00:06:12.960 --> 00:06:17.300 the solution to the Heisenberg equation of motion for angular 00:06:17.300 --> 00:06:20.290 momentum or magnetic moment in a magnetic field 00:06:20.290 --> 00:06:22.370 is exactly a rotation. 00:06:22.370 --> 00:06:25.350 So we'll again show you that a quantum mechanical solution, 00:06:25.350 --> 00:06:29.530 the solution of a time-dependent problem is exactly rotation. 00:06:29.530 --> 00:06:36.740 And later on when we use the spin-1/2 Hamiltonian and we 00:06:36.740 --> 00:06:40.250 write down the wave function, we solve sometimes the wave 00:06:40.250 --> 00:06:42.540 function by transforming the wave function. 00:06:42.540 --> 00:06:45.840 And this will be, again, the rotating frame transformation. 00:06:45.840 --> 00:06:49.400 It's not always called like this, but it's always the same. 00:06:49.400 --> 00:06:52.250 You go to some form of rotating frame, 00:06:52.250 --> 00:06:55.210 and we'll do that in three different ways. 00:06:55.210 --> 00:06:58.920 This is the first way just as a general classical physics 00:06:58.920 --> 00:07:01.780 transformation to rotating frame. 00:07:01.780 --> 00:07:04.600 We will do it again for the expectation 00:07:04.600 --> 00:07:07.340 value as a solution of Heisenberg's equation 00:07:07.340 --> 00:07:10.800 of motion, and then we do it again 00:07:10.800 --> 00:07:14.025 when we transform the wave function 00:07:14.025 --> 00:07:15.700 with a unitary transformation. 00:07:18.590 --> 00:07:23.580 A lot of time for simple rotations, but its good thing. 00:07:23.580 --> 00:07:25.080 It really provides a lot of insight. 00:07:27.670 --> 00:07:31.050 Anyway, this is more sort of an outlook over today's lecture 00:07:31.050 --> 00:07:34.200 and part of Tuesday's lecture. 00:07:34.200 --> 00:07:36.890 Any question about the summary and the outlook? 00:07:42.090 --> 00:07:46.130 Because before I really come back to the rotating frame, 00:07:46.130 --> 00:07:52.380 I quickly want to do something I couldn't. 00:07:52.380 --> 00:07:56.900 I ran out of time on Monday. 00:07:56.900 --> 00:08:00.150 We talked so much about harmonic oscillators, the precision 00:08:00.150 --> 00:08:01.900 at which we can determine the frequency 00:08:01.900 --> 00:08:05.320 of the harmonic oscillator, and I really 00:08:05.320 --> 00:08:06.760 want to give you examples. 00:08:06.760 --> 00:08:12.200 I want give you two outstanding examples for atomic clocks. 00:08:17.270 --> 00:08:21.850 And the two extreme examples are, well, 00:08:21.850 --> 00:08:24.420 the two best atomic clocks in the world. 00:08:24.420 --> 00:08:32.020 One is the cesium atom with a fountain clock. 00:08:32.020 --> 00:08:36.030 Well, we'll talk about it later, but some of you 00:08:36.030 --> 00:08:39.299 know that the cesium atom is hyper fine structure. 00:08:39.299 --> 00:08:42.539 In one state, the electron and nucleus spin are parallel. 00:08:42.539 --> 00:08:44.375 In the other state, they are antiparallel, 00:08:44.375 --> 00:08:48.560 and the transition frequency is 10 gigahertz, well, 00:08:48.560 --> 00:08:51.650 9.something, but for the matter of this discussion, 00:08:51.650 --> 00:08:54.640 it's 10 gigahertz. 00:08:54.640 --> 00:09:00.620 And the definition of time, the definition for 1 second 00:09:00.620 --> 00:09:06.110 is in terms of so-and-so many cycles of this transition. 00:09:06.110 --> 00:09:09.380 So this transition has this frequency, 00:09:09.380 --> 00:09:13.000 and for decades, this frequency was 00:09:13.000 --> 00:09:14.590 determined in an atomic beam. 00:09:14.590 --> 00:09:17.110 You have an atomic cesium beam, and you interrogate it 00:09:17.110 --> 00:09:19.000 with microwave fields. 00:09:19.000 --> 00:09:23.200 But now with cold atoms, we can achieve much, much longer 00:09:23.200 --> 00:09:27.320 interrogation times by almost completely eliminating 00:09:27.320 --> 00:09:28.620 the atomic motion. 00:09:28.620 --> 00:09:30.400 In the current experiment, is that you 00:09:30.400 --> 00:09:33.330 have a cloud [? lace ?] are cooled 00:09:33.330 --> 00:09:34.930 to micro Kelvin temperature. 00:09:34.930 --> 00:09:37.300 You launch it into a fountain. 00:09:37.300 --> 00:09:42.500 The cloud goes up and goes down, and you interrogate it twice 00:09:42.500 --> 00:09:44.940 and the interrogation time is 1 second. 00:09:44.940 --> 00:09:47.910 And this interrogation time is no longer as in the convention 00:09:47.910 --> 00:09:51.890 atomic clocks limited by the thermal velocity of the atom, 00:09:51.890 --> 00:09:53.670 it's limited by gravity. 00:09:53.670 --> 00:09:57.420 If you want to increase a time to 10 seconds, 00:09:57.420 --> 00:10:00.360 you need a 100-meter tower and nobody wants to build that. 00:10:00.360 --> 00:10:03.780 I mean, it would be a really big atomic clock. 00:10:03.780 --> 00:10:10.050 So therefore you usually deal with an interrogation time 00:10:10.050 --> 00:10:13.700 on the order of 1 second. 00:10:13.700 --> 00:10:18.970 So we know that based on Fourier's theorem, 00:10:18.970 --> 00:10:22.120 delta is a factor of 2 pi, but the line widths, 00:10:22.120 --> 00:10:24.840 if you would recall now the spectrum, 00:10:24.840 --> 00:10:28.410 the line widths, delta omega, would be that's a factor of 2, 00:10:28.410 --> 00:10:31.670 but it would be 1 over 1 second. 00:10:31.670 --> 00:10:37.970 And therefore the fractional accuracy is 10 to the 11, 00:10:37.970 --> 00:10:41.960 is on the order of 10 to the minus 11. 00:10:41.960 --> 00:10:48.930 However, the accuracy of the base cesium fountains 00:10:48.930 --> 00:10:54.620 is now 10 to the minus 16. 00:10:54.620 --> 00:10:59.290 So people are able to split the line to 1 part in 100,000, 00:10:59.290 --> 00:11:04.630 which requires exquisite knowledge, 00:11:04.630 --> 00:11:06.400 exquisite knowledge of the line shape 00:11:06.400 --> 00:11:09.680 and also systematic effects, but well, an atomic clock 00:11:09.680 --> 00:11:12.180 is a piece of art. 00:11:12.180 --> 00:11:17.930 And I just want to emphasize that I sometimes 00:11:17.930 --> 00:11:20.830 feel when I talk about the uncertainty at which you can 00:11:20.830 --> 00:11:25.135 measure a classical oscillator or such is almost trivial, 00:11:25.135 --> 00:11:27.070 and you should know all about it. 00:11:27.070 --> 00:11:31.590 But I can just say without looking at anybody that I just 00:11:31.590 --> 00:11:34.270 had a lunch discussion with some of my graduate student 00:11:34.270 --> 00:11:36.810 and we talked about laser stabilization. 00:11:36.810 --> 00:11:41.320 And one graduate student asked, but if the natural line widths 00:11:41.320 --> 00:11:44.610 of a transition is 10 megahertz, can we stabilized a laser 00:11:44.610 --> 00:11:47.070 to better than a megahertz? 00:11:47.070 --> 00:11:48.240 Of course, we can. 00:11:48.240 --> 00:11:50.940 I mean, here we have a natural line width of 10 00:11:50.940 --> 00:11:53.780 to the minus 11, but we can get an a accuracy 00:11:53.780 --> 00:11:57.710 of a signal, which is 10 to the minus 16. 00:11:57.710 --> 00:12:00.810 So a microwave oscillator, which is the microwave accrual 00:12:00.810 --> 00:12:05.240 end of a laser can now be locked to the cesium transition 00:12:05.240 --> 00:12:08.600 with a precision of 10 to the minus 16, 00:12:08.600 --> 00:12:12.560 100,000 times better than the line width. 00:12:12.560 --> 00:12:17.065 Let me give you a second example. 00:12:19.630 --> 00:12:21.220 Just noticing the red color. 00:12:21.220 --> 00:12:23.790 This is red. 00:12:23.790 --> 00:12:27.042 I wanted to highlight it. 00:12:27.042 --> 00:12:28.500 There's probably nothing we can do. 00:12:28.500 --> 00:12:32.650 It seems the projector is not showing the color. 00:12:32.650 --> 00:12:35.160 Well, in the lecture post, the lecture notes, 00:12:35.160 --> 00:12:38.680 this will be bright red, but maybe I 00:12:38.680 --> 00:12:41.550 should use more blue and green and yellow 00:12:41.550 --> 00:12:42.760 for highlighting it today. 00:12:47.050 --> 00:12:49.650 That's really odd. 00:12:49.650 --> 00:12:55.530 Yellow-- OK. 00:12:55.530 --> 00:12:59.160 The red is completely missing. 00:12:59.160 --> 00:13:08.210 The other example I want to give you is the strontium-- yeah, 00:13:08.210 --> 00:13:10.570 it's blue-- the strontium optical clock. 00:13:14.440 --> 00:13:18.500 And there was a really nice paper 00:13:18.500 --> 00:13:32.550 in Nature just a few weeks ago, and well, here 00:13:32.550 --> 00:13:37.110 is a level diagram of atomic strontium 00:13:37.110 --> 00:13:40.740 what happens is there is a very fast transition s 00:13:40.740 --> 00:13:43.770 to p transition for laser cooling and trapping 00:13:43.770 --> 00:13:47.850 and all that, but then there is a very, very slow transition 00:13:47.850 --> 00:13:51.460 [? forbidden ?] to a triplet state, which is meta stable. 00:13:51.460 --> 00:13:54.070 And this transition, those states 00:13:54.070 --> 00:13:57.030 have a very long lifetime, and therefore this transition 00:13:57.030 --> 00:14:00.210 is extremely level. 00:14:00.210 --> 00:14:05.380 So what is state of the art for the strontium clock? 00:14:05.380 --> 00:14:13.220 Well, in the experiment, they use an interrogation time. 00:14:17.750 --> 00:14:23.200 So they observe the atom for delta t, which 00:14:23.200 --> 00:14:27.540 is on the order of 160 milliseconds 00:14:27.540 --> 00:14:31.980 and putting in the 2 pis in the right place that would mean 00:14:31.980 --> 00:14:35.920 the frequency resolution is 1 Hertz. 00:14:35.920 --> 00:14:38.890 And you see that on the left-hand side when 00:14:38.890 --> 00:14:43.260 we record the resonance, the blue one is about 1 Hertz 00:14:43.260 --> 00:14:44.900 and there is something broader. 00:14:44.900 --> 00:14:46.690 This is when they are not actively 00:14:46.690 --> 00:14:48.830 feeding back the magnetic field. 00:14:48.830 --> 00:14:53.000 So for this fantastic position, you have to control everything, 00:14:53.000 --> 00:14:54.910 but the blue line is sort of what 00:14:54.910 --> 00:14:56.320 we record as a clock transition. 00:14:56.320 --> 00:14:58.310 It's about 1 Hertz. 00:14:58.310 --> 00:15:06.030 Now, compared to a cesium clock, the big advantage 00:15:06.030 --> 00:15:13.420 is that the strontium clock operates in the optical domain, 00:15:13.420 --> 00:15:18.404 and this is a frequency at 5 times 10 to the 14 Hertz. 00:15:18.404 --> 00:15:20.320 So we're seeing if the frequency is much, much 00:15:20.320 --> 00:15:23.910 faster even if you have a shorter interrogation time, 00:15:23.910 --> 00:15:27.310 your relative accuracy is better. 00:15:27.310 --> 00:15:33.250 And here the Q value the mu over delta mu 00:15:33.250 --> 00:15:37.460 is on the order of 10 times 10 to the minus 15. 00:15:37.460 --> 00:15:38.190 Fantastic. 00:15:38.190 --> 00:15:50.800 15 orders of magnitude, and they are splitting the line, not as 00:15:50.800 --> 00:15:54.800 extreme as the cesium atomic clock by a factor of 100,000, 00:15:54.800 --> 00:15:58.310 by a factor which is on the order of 300. 00:15:58.310 --> 00:16:05.800 And with that, they have an accuracy, which is now really 00:16:05.800 --> 00:16:08.080 the record for the base performance 00:16:08.080 --> 00:16:15.255 of any atomic clock, which is 6 times 10 to the minus 18. 00:16:19.140 --> 00:16:22.260 Now, this is now in the optical domain. 00:16:22.260 --> 00:16:26.320 You may wonder about the laser, how stable is the laser. 00:16:26.320 --> 00:16:29.680 The laser is stabilized to an optical activity, 00:16:29.680 --> 00:16:34.270 but the laser because of thermal fluctuations, not 00:16:34.270 --> 00:16:36.910 terminal fluctuations, thermal fluctuations 00:16:36.910 --> 00:16:44.560 in the mirror because of thermal noise 00:16:44.560 --> 00:16:48.660 is limited in the short term to 10 to the minus 16. 00:16:53.770 --> 00:16:57.440 So the short term stability between 1 and 1,000 00:16:57.440 --> 00:17:00.250 second is 10 to the minus 16. 00:17:00.250 --> 00:17:05.109 So they use a laser which has a stability of 10 00:17:05.109 --> 00:17:07.260 to the minus 16. 00:17:07.260 --> 00:17:12.089 Every 1.3 seconds, they take a data point. 00:17:12.089 --> 00:17:16.089 Each data point or the spectral widths 00:17:16.089 --> 00:17:23.180 is 2 times 10 to the-- the laser is 10 to the 16. 00:17:23.180 --> 00:17:26.020 The line we record is 10 to the 15, 00:17:26.020 --> 00:17:28.810 but seen since the thermal noise is completely random 00:17:28.810 --> 00:17:31.834 and by averaging it, they can determine the line center 00:17:31.834 --> 00:17:33.000 to better than 10 to the 17. 00:17:35.610 --> 00:17:39.340 So I think this just illustrates the precision at the which 00:17:39.340 --> 00:17:42.490 you can observe the harmonic oscillator, 00:17:42.490 --> 00:17:44.620 and what I like about this example, 00:17:44.620 --> 00:17:49.160 it shows you that both the transition you are recording 00:17:49.160 --> 00:17:53.580 and the laser itself-- 10 to the 15, 10 to the 16-- 00:17:53.580 --> 00:17:57.449 are worse than the final precision of the measurement, 00:17:57.449 --> 00:17:58.865 which is better than 10 to the 17. 00:18:03.920 --> 00:18:07.330 Make sense, but yes, that's what it is. 00:18:07.330 --> 00:18:08.420 Questions about that? 00:18:20.350 --> 00:18:20.850 Yes, Collin. 00:18:20.850 --> 00:18:24.561 AUDIENCE: What was the accuracy of the [INAUDIBLE] result? 00:18:27.998 --> 00:18:31.707 Was it better than 6 times 10 to the minus 18? 00:18:31.707 --> 00:18:32.290 PROFESSOR: No. 00:18:32.290 --> 00:18:34.724 This 6 times 10 to the minus 18 is really 00:18:34.724 --> 00:18:35.682 the [INAUDIBLE] record. 00:18:35.682 --> 00:18:38.944 AUDIENCE: Is that the [? atomic ?] clock? 00:18:38.944 --> 00:18:42.350 PROFESSOR: There was an aluminum ion clock, which had, 00:18:42.350 --> 00:18:44.469 I think, 10 the minus 17 precision. 00:18:44.469 --> 00:18:47.010 But I think it's 6 times 10 to the minus 18 is close to that. 00:18:47.010 --> 00:18:49.166 So they are close or they are better now 00:18:49.166 --> 00:18:54.160 than the aluminum clock, but the aluminum clock is a single ion. 00:18:54.160 --> 00:18:56.370 You have to ever reach for much, much longer time 00:18:56.370 --> 00:18:58.310 to get this precision. 00:18:58.310 --> 00:19:01.600 So that's a big advantage for the strontium clock, which 00:19:01.600 --> 00:19:04.900 is many, many atoms in an optical lattice, 00:19:04.900 --> 00:19:07.570 and they say that they have improved on the best 00:19:07.570 --> 00:19:10.980 previous lattice clock by a factor of 20. 00:19:10.980 --> 00:19:14.230 But I think the improvement effect of 20, 00:19:14.230 --> 00:19:18.380 you always have to distinguish between sensitivity 00:19:18.380 --> 00:19:19.740 and absolute precision. 00:19:19.740 --> 00:19:21.776 In the absolute precision, you also 00:19:21.776 --> 00:19:24.090 have to control all systematic effects. 00:19:24.090 --> 00:19:26.110 And the big step here which was really 00:19:26.110 --> 00:19:29.420 boosting the absolute precision was they completely 00:19:29.420 --> 00:19:32.760 controlled the black body environment. 00:19:32.760 --> 00:19:35.830 So you really have to know with high precision what 00:19:35.830 --> 00:19:39.610 is the effective temperature of the black body radiation 00:19:39.610 --> 00:19:42.340 because at the 10 to the minus 18 level, 00:19:42.340 --> 00:19:44.860 it causes a shift of the atomic resonance. 00:19:47.890 --> 00:19:51.270 We talk later about it, but it's the ac Stark effect 00:19:51.270 --> 00:19:53.430 of the black body radiation, which 00:19:53.430 --> 00:19:57.177 becomes an important systematic at that level of precision. 00:19:57.177 --> 00:19:59.760 I don't know exactly the number of the previous atomic clocks, 00:19:59.760 --> 00:20:02.280 but this is sort of now the gold standard 00:20:02.280 --> 00:20:04.300 of frequency methodology. 00:20:10.760 --> 00:20:11.490 OK. 00:20:11.490 --> 00:20:12.250 Let's go. 00:20:16.290 --> 00:20:18.610 Yes, what you have here is you have 00:20:18.610 --> 00:20:21.140 sort of classical harmonic oscillator, 00:20:21.140 --> 00:20:23.130 but the classical harmonic oscillator 00:20:23.130 --> 00:20:27.290 is based on doing measurements on seamless strontium 00:20:27.290 --> 00:20:29.890 atoms and single cesium atoms. 00:20:29.890 --> 00:20:32.860 So you see that the accuracy, and we had a discussion on it 00:20:32.860 --> 00:20:35.720 last class, the accuracy at which you can observe 00:20:35.720 --> 00:20:37.320 a quantum mechanical oscillator. 00:20:37.320 --> 00:20:41.000 In a classical oscillator, it's pretty much the same. 00:20:41.000 --> 00:20:44.610 It's the same kind of-- if you have a good signal to noise, 00:20:44.610 --> 00:20:50.047 you can improve your precision for the line center 00:20:50.047 --> 00:20:50.630 substantially. 00:20:53.230 --> 00:20:57.830 From those results, and exciting atomic clocks, 00:20:57.830 --> 00:21:00.480 let's go back to classical physics. 00:21:00.480 --> 00:21:06.410 So we want to go back to our classical magnetic moment 00:21:06.410 --> 00:21:11.920 and understand the motion of it. 00:21:11.920 --> 00:21:15.300 But in addition to what we discussed so far, 00:21:15.300 --> 00:21:17.945 a stationary magnetic field, we now 00:21:17.945 --> 00:21:20.280 want to add to it a rotating magnetic field. 00:21:38.430 --> 00:21:50.860 So the situation is that-- just use 00:21:50.860 --> 00:21:56.270 another color-- we have a magnetic moment. 00:21:56.270 --> 00:22:04.690 We assume it's classical, and we have a magnetic field, 00:22:04.690 --> 00:22:10.980 which we assume points along the z-axis. 00:22:10.980 --> 00:22:13.410 And then, of course, we know that 00:22:13.410 --> 00:22:16.530 from our previous discussion that the spin 00:22:16.530 --> 00:22:18.420 undergoes precession. 00:22:18.420 --> 00:22:23.130 It's sort of precesses around the magnetic field 00:22:23.130 --> 00:22:24.140 at the Larmor frequency. 00:22:27.700 --> 00:22:46.070 And now, let's assume we have-- the yellow shows up. 00:22:46.070 --> 00:22:48.570 We have a rotating field. 00:22:48.570 --> 00:22:52.580 We add a rotating field eight B1, 00:22:52.580 --> 00:22:55.550 and just to keep things simple, we 00:22:55.550 --> 00:23:00.330 want to assume that the rotating field rotates 00:23:00.330 --> 00:23:03.280 at the same frequency as a magnetic moment, 00:23:03.280 --> 00:23:06.310 so we are on a resonance. 00:23:06.310 --> 00:23:13.430 Because what happens now is we can simply 00:23:13.430 --> 00:23:16.690 do a transformation to the rotating frame. 00:23:19.920 --> 00:23:26.390 The rotating frame is now at the Larmor frequency 00:23:26.390 --> 00:23:31.730 and we have just learned that in the rotating frame, of course, 00:23:31.730 --> 00:23:38.120 in the rotating frame the rotating field stands still, 00:23:38.120 --> 00:23:43.190 so it becomes a static field which 00:23:43.190 --> 00:23:45.705 points in the x direction. 00:23:52.850 --> 00:23:58.620 And we have the static field in the z direction, 00:23:58.620 --> 00:24:01.280 but now that's what I just reviewed. 00:24:01.280 --> 00:24:06.840 We have a fictitious magnetic field, 00:24:06.840 --> 00:24:09.560 which comes from the transformation to the rotating 00:24:09.560 --> 00:24:14.290 frame and on resonance, at the Larmor frequency, 00:24:14.290 --> 00:24:17.085 this is exactly the negative of B naught. 00:24:20.660 --> 00:24:27.660 So in other words, we started out 00:24:27.660 --> 00:24:33.160 to expose a magnetic moment to a time-dependent field. 00:24:38.810 --> 00:24:54.010 So we had a rotating field cosine omega Lt sine omega Lt. 00:24:58.460 --> 00:25:04.790 But in the rotating frame, this becomes now the x prime x, 00:25:04.790 --> 00:25:06.800 and it's stationary in the rotating frame. 00:25:10.940 --> 00:25:15.360 So this was the field in the lab frame. 00:25:15.360 --> 00:25:31.770 In the rotating frame, we have an effective field, 00:25:31.770 --> 00:25:34.590 which is the field in the lab frame 00:25:34.590 --> 00:25:40.850 minus the fictitious field, and the fictitious field 00:25:40.850 --> 00:25:42.240 was given by that. 00:25:42.240 --> 00:25:45.750 It just cancels the B naught component. 00:25:45.750 --> 00:25:51.150 And in the rotating frame, this vector and this vector 00:25:51.150 --> 00:25:55.250 cancels and we are just left with a field of strengths 00:25:55.250 --> 00:26:03.930 B1, which points in the x-axis, or actually 00:26:03.930 --> 00:26:08.410 the x-axis in the rotating frame is what I call x prime. 00:26:08.410 --> 00:26:11.240 So therefore, we have a very simple problem 00:26:11.240 --> 00:26:13.140 that in the rotating frame. 00:26:13.140 --> 00:26:18.200 We have a static field of value B1. 00:26:22.839 --> 00:26:24.630 And the good thing is, we know already what 00:26:24.630 --> 00:26:28.240 a magnetic moment does with a static field. 00:26:28.240 --> 00:26:30.140 The magnetic moment is just precessing 00:26:30.140 --> 00:26:33.190 around the static magnetic field. 00:26:33.190 --> 00:26:35.850 So therefore, our solution is now 00:26:35.850 --> 00:26:39.020 we have transformed the time-dependent field 00:26:39.020 --> 00:26:53.040 and now we know that mu precesses around this field, 00:26:53.040 --> 00:26:57.240 and the precession frequency is the Rabi frequency, which 00:26:57.240 --> 00:26:58.970 we discussed previously. 00:27:04.700 --> 00:27:10.310 The Rabi frequency is the gyromagnetic ratio times B1. 00:27:10.310 --> 00:27:13.080 So therefore, if we would start out 00:27:13.080 --> 00:27:19.670 with a magnetic moment aligned with the z-axis. 00:27:19.670 --> 00:27:25.200 If we would wait half a Rabi's cycle, 00:27:25.200 --> 00:27:28.885 the magnetic moment would now be inverted. 00:27:32.440 --> 00:27:34.800 So the situation is we have a magnetic moment which 00:27:34.800 --> 00:27:38.010 points in the z-axis with a field in the z-axis. 00:27:38.010 --> 00:27:41.440 But we expose it to a rotating field 00:27:41.440 --> 00:27:43.270 which rotates in the xy plane. 00:27:47.080 --> 00:27:50.240 So in the rotating frame, we have a stationary field 00:27:50.240 --> 00:27:52.970 which points in x. 00:27:52.970 --> 00:27:56.010 In this rotating frame, the spin is simply 00:27:56.010 --> 00:28:00.030 precessing around what is now a steady field, 00:28:00.030 --> 00:28:04.550 and after half a Ravi cycle, the spin points down. 00:28:04.550 --> 00:28:07.680 So therefore we know-- going back to lab frame-- 00:28:07.680 --> 00:28:12.660 that this rotating field has caused in quantum mechanics 00:28:12.660 --> 00:28:15.770 I would say a spin flip, a full reversal 00:28:15.770 --> 00:28:19.290 of the magnetic moment, and this is 00:28:19.290 --> 00:28:23.750 what we call the pi pulse-- it has already 00:28:23.750 --> 00:28:29.810 rotated the spin by pi-- or we call it a spin flip, 00:28:29.810 --> 00:28:32.030 but it's a completely classical system. 00:28:37.760 --> 00:28:38.480 Any questions? 00:28:48.430 --> 00:28:51.660 Well, then I would question for you. 00:28:51.660 --> 00:28:58.180 I've discussed with you the case that the rotating field rotates 00:28:58.180 --> 00:29:01.660 at the Larmor frequency. 00:29:01.660 --> 00:29:13.530 But now I want to discuss the case that the rotating field is 00:29:13.530 --> 00:29:16.680 not at the Larmor frequency, it's 00:29:16.680 --> 00:29:30.520 at the frequency omega 1, which is faster 00:29:30.520 --> 00:29:31.625 than the Larmor frequency. 00:29:37.850 --> 00:29:41.150 And the question is now what will 00:29:41.150 --> 00:29:44.590 happen to the spin or the magnetic moment. 00:29:44.590 --> 00:29:48.810 I explained to you that on a resonance, 00:29:48.810 --> 00:29:54.690 the magnetic moment was just flipping over, rotating 00:29:54.690 --> 00:29:59.040 precessing at the Rabi frequency. 00:29:59.040 --> 00:30:02.350 And I want you to think about it for a moment 00:30:02.350 --> 00:30:05.870 and then decide if we go off resonant, 00:30:05.870 --> 00:30:08.660 if we drive this system away from the resonance 00:30:08.660 --> 00:30:11.320 with a rotating field which is faster than the Larmor 00:30:11.320 --> 00:30:18.260 frequency, what is now the oscillation 00:30:18.260 --> 00:30:23.575 frequency of the magnetic moment. 00:30:31.940 --> 00:30:44.860 And so the choices are, is it larger, smaller, or the same? 00:30:44.860 --> 00:30:51.560 And this was, of course, compared to the Rabi frequency. 00:30:58.780 --> 00:31:03.860 So I've explained to you that the spin flip, the Rabi 00:31:03.860 --> 00:31:07.000 flopping, or the pi pulse-- and this picture 00:31:07.000 --> 00:31:12.520 was at the magnetic moment-- does Rabi flopping rotates 00:31:12.520 --> 00:31:18.260 plus z minus, z plus, z minus, z at the Rabi frequency. 00:31:18.260 --> 00:31:20.930 But now we drive the system faster 00:31:20.930 --> 00:31:23.230 than the Larmor frequency and the question 00:31:23.230 --> 00:31:26.410 is, is whatever this magnetic moment does, 00:31:26.410 --> 00:31:28.657 is it faster, slower, or does it always 00:31:28.657 --> 00:31:29.865 happen at the Rabi frequency? 00:31:43.070 --> 00:31:45.220 Larger. 00:31:45.220 --> 00:31:45.720 Good. 00:31:48.910 --> 00:31:53.630 Well, let me then immediately add another twist. 00:31:53.630 --> 00:31:56.890 What would happen-- let's ask the same question, 00:31:56.890 --> 00:32:00.315 but now we are driving it at a lower frequency. 00:32:03.692 --> 00:32:04.686 The yellow. 00:32:09.670 --> 00:32:11.090 This projector has a problem. 00:32:14.000 --> 00:32:20.280 So now same question as before, but instead 00:32:20.280 --> 00:32:23.980 of driving the system with a faster rotating field, faster 00:32:23.980 --> 00:32:27.170 than the Larmor frequency, we are driving it 00:32:27.170 --> 00:32:29.330 with a smaller frequency. 00:32:29.330 --> 00:32:32.770 Is now the response of the magnetic moment, 00:32:32.770 --> 00:32:36.730 the effective precession frequency, larger, smaller, 00:32:36.730 --> 00:32:40.430 or the same as the Rabi frequency as the resonant case? 00:32:51.340 --> 00:32:52.190 OK, good. 00:32:52.190 --> 00:32:56.320 So that means I can go very quickly about the explanation. 00:32:56.320 --> 00:32:57.210 It's correct. 00:32:57.210 --> 00:33:00.450 Whenever you are off resonant, this system 00:33:00.450 --> 00:33:04.970 precesses faster, so let me summarize 00:33:04.970 --> 00:33:08.420 what probably is obvious to all of you. 00:33:08.420 --> 00:33:13.110 What happens when we have an off-resonant rotating field. 00:33:17.940 --> 00:33:20.870 When we have an off-resonant rotating field, 00:33:20.870 --> 00:33:25.370 we go to the rotating frame, but of course, the rotating frame 00:33:25.370 --> 00:33:29.420 we go to is now not rotating at the Larmor frequency 00:33:29.420 --> 00:33:31.620 because the purpose of going to rotating frame 00:33:31.620 --> 00:33:33.860 is get rid of the time dependence of the rotating 00:33:33.860 --> 00:33:34.540 field. 00:33:34.540 --> 00:33:37.070 So we go to rotating frame which rotates 00:33:37.070 --> 00:33:40.820 at the frequency of the rotating field. 00:33:40.820 --> 00:33:56.515 So if the rotating field rotates at a frequency omega, 00:33:56.515 --> 00:34:01.210 we have a fictitious field, which is omega over gamma. 00:34:01.210 --> 00:34:02.715 Gamma is the gyromagnetic ratio. 00:34:06.570 --> 00:34:11.750 For the resonant case, we were just completely canceling 00:34:11.750 --> 00:34:14.030 the static field in the z direction, 00:34:14.030 --> 00:34:16.750 but for the off-resonant case, when 00:34:16.750 --> 00:34:21.480 omega is larger or smaller in both cases, 00:34:21.480 --> 00:34:26.690 this one here is no longer 0. 00:34:26.690 --> 00:34:33.139 And our total effective field is now 00:34:33.139 --> 00:34:42.630 the quadrature sum of what we have in the z direction, 00:34:42.630 --> 00:34:46.360 and what we have in the x direction or x prime direction, 00:34:46.360 --> 00:34:46.889 this is B1. 00:34:49.780 --> 00:34:54.429 So in the z direction, we have the static field 00:34:54.429 --> 00:35:02.960 minus the fictitious field and then the two are added up 00:35:02.960 --> 00:35:06.320 On resonance, the angle theta is 90 degrees, 00:35:06.320 --> 00:35:09.480 but for the off-resonant case, the angle 00:35:09.480 --> 00:35:18.720 is different given by the simple geometric result. 00:35:18.720 --> 00:35:28.910 And the effective field is the quadrature sum of B1 squared 00:35:28.910 --> 00:35:50.120 plus B naught minus-- and the fact 00:35:50.120 --> 00:35:54.270 is this adds something to the effective field in the rotating 00:35:54.270 --> 00:35:57.490 frame whether we drive it above or below resonance. 00:35:57.490 --> 00:36:10.070 So therefore the magnetic moment precesses 00:36:10.070 --> 00:36:18.500 at what is called the generalized Rabi 00:36:18.500 --> 00:36:34.200 frequency, which I now call-- this would be red-- let's use 00:36:34.200 --> 00:36:41.420 green instead-- the generalized Rabi frequency, which 00:36:41.420 --> 00:36:50.620 is gamma times B effective, and this 00:36:50.620 --> 00:36:59.830 is the quadrature sum of the detuning 00:36:59.830 --> 00:37:07.310 plus the capital letter omega, omega R, 00:37:07.310 --> 00:37:10.120 is the Rabi frequency at resonance, 00:37:10.120 --> 00:37:13.440 and this is nothing else than a measure for the drive 00:37:13.440 --> 00:37:17.130 field for the strengths of the drive field B1 in frequencies. 00:37:19.640 --> 00:37:22.820 So therefore, the generalized Rabi frequency 00:37:22.820 --> 00:37:27.110 is the resonant Rabi frequency added in quadrature 00:37:27.110 --> 00:37:28.330 with the detuning squared. 00:37:40.460 --> 00:37:41.515 Any questions? 00:37:47.770 --> 00:37:55.140 So because it's an exact result and I like the result Rabi 00:37:55.140 --> 00:37:58.140 flopping at the generalized Rabi frequency, 00:37:58.140 --> 00:38:01.160 I want to derive it for you. 00:38:01.160 --> 00:38:06.430 So I want to figure out what is the dynamic of a spin which 00:38:06.430 --> 00:38:13.400 is originally aligned, and now it 00:38:13.400 --> 00:38:18.830 undergoes-- it is driven by the rotating field. 00:38:18.830 --> 00:38:21.770 Remember, the resonant case was very simple. 00:38:21.770 --> 00:38:24.160 The spin was just doing Rabi flopping. 00:38:24.160 --> 00:38:27.540 It was fully inverted, came back, and just 00:38:27.540 --> 00:38:30.980 did this at that Rabi frequency. 00:38:30.980 --> 00:38:33.880 Now we know that in the off-resonant case, 00:38:33.880 --> 00:38:36.690 there will be an effective magnetic field 00:38:36.690 --> 00:38:39.740 and it will precess at a faster frequency, which 00:38:39.740 --> 00:38:46.620 is a generalized Rabi frequency, but 00:38:46.620 --> 00:38:51.940 since the effective magnetic field is not transverse, 00:38:51.940 --> 00:38:59.440 it has a z component, the spin will never fully invert. 00:38:59.440 --> 00:39:02.180 So geometrically, it's very easy. 00:39:02.180 --> 00:39:06.820 We start out with a magnetic moment at zero time, 00:39:06.820 --> 00:39:10.530 and I can immediately draw to you the complete solution. 00:39:10.530 --> 00:39:18.710 The complete solution is that in the rotating frame, this sort 00:39:18.710 --> 00:39:26.130 of precesses around the effective magnetic field. 00:39:26.130 --> 00:39:27.780 This is the solution. 00:39:27.780 --> 00:39:29.900 But I just wanted to do is because it takes me 00:39:29.900 --> 00:39:32.690 three or four minutes, I want to read from this graph, 00:39:32.690 --> 00:39:37.110 from this drawing, one of two trigonometric identities 00:39:37.110 --> 00:39:40.980 and derive for you the explicit expression, 00:39:40.980 --> 00:39:42.850 what is the value of the magnetic moment 00:39:42.850 --> 00:39:45.940 as a function of time. 00:39:45.940 --> 00:39:49.980 But it's clear from that it will have a maximum value. 00:39:49.980 --> 00:39:53.000 It precesses around the tilt direction, 00:39:53.000 --> 00:39:56.020 and when it's over there, it has a minimum value, 00:39:56.020 --> 00:39:59.520 but it will never completely invert. 00:39:59.520 --> 00:40:02.400 Well, quantum mechanically, if you drive a system not 00:40:02.400 --> 00:40:06.100 on resonance, you cannot completely invert 00:40:06.100 --> 00:40:10.220 the population, but we'll come to the later. 00:40:10.220 --> 00:40:11.750 So what do I need? 00:40:11.750 --> 00:40:19.675 Well, the spin is moving here on a circle. 00:40:24.110 --> 00:40:25.910 I need a few angles. 00:40:30.240 --> 00:40:34.210 So let's say the spin was here at one time. 00:40:34.210 --> 00:40:38.030 At another time, it is there, and that 00:40:38.030 --> 00:40:44.740 would mean that on the circle, it has moved an angle phi. 00:40:47.350 --> 00:40:53.300 The tilt angle between the spin and the magnetic field 00:40:53.300 --> 00:40:56.650 is what I call theta. 00:40:56.650 --> 00:41:06.080 And the angle between the initial magnetic moment 00:41:06.080 --> 00:41:12.810 and the magnetic moment at time, t, is what I call alpha. 00:41:15.122 --> 00:41:16.955 Yeah, these other the three relevant angles. 00:41:22.860 --> 00:41:28.060 The tip of the magnetic moment goes in a circle, 00:41:28.060 --> 00:41:41.490 and this circle has a radius, which is mu times sine theta. 00:41:41.490 --> 00:41:48.600 Sine theta is nothing else than the rotating magnetic field 00:41:48.600 --> 00:41:51.890 over the effective magnetic field, which 00:41:51.890 --> 00:41:58.680 is nothing else than the resonant Rabi frequency divided 00:41:58.680 --> 00:42:00.165 by the generalized Rabi frequency. 00:42:13.920 --> 00:42:16.110 I said what I want to determinant 00:42:16.110 --> 00:42:24.850 is the magnetic moment in the z direction as a function 00:42:24.850 --> 00:42:30.500 of time, and for that, I defined the angle cosine alpha. 00:42:37.870 --> 00:42:43.080 So the way how I derive it the easiest way, its geometry, 00:42:43.080 --> 00:42:45.930 its three dimension, and triangles, and all that. 00:42:45.930 --> 00:42:53.410 But the best way how I can describe it for you, 00:42:53.410 --> 00:42:58.360 let me introduce this auxiliary line, which connects the tip 00:42:58.360 --> 00:43:02.780 of the magnetic moment at time t equals 0 and at time t. 00:43:02.780 --> 00:43:06.480 And I call the length of this line A. 00:43:06.480 --> 00:43:13.230 And I want to derive-- I want to have now two triangles where 00:43:13.230 --> 00:43:16.880 one side is A, determine A in two different ways, 00:43:16.880 --> 00:43:19.550 combine the equation, we are done. 00:43:19.550 --> 00:43:26.400 So the first way is that we have the magnetic moment 00:43:26.400 --> 00:43:31.160 at time 0, the magnetic moment at time, t. 00:43:31.160 --> 00:43:34.880 We said the angle is alpha, and the two tips 00:43:34.880 --> 00:43:50.640 are connected by A, and you know in every triangle you have this 00:43:50.640 --> 00:43:56.010 would Pythagoras, but in the general case, 00:43:56.010 --> 00:43:58.980 this is valid for general triangle. 00:43:58.980 --> 00:44:03.600 So applying that to this triangle, we have a square. 00:44:03.600 --> 00:44:07.210 b square is mu square. c square is mu square, 00:44:07.210 --> 00:44:10.470 so we get 2 mu square. 00:44:10.470 --> 00:44:17.780 And then this term 2 ab is 2 mu times mu times cosine alpha, 00:44:17.780 --> 00:44:21.930 so therefore, this is 1 minus cosine alpha. 00:44:21.930 --> 00:44:23.820 So we've taken care of the first triangle. 00:44:31.000 --> 00:44:34.560 I hope the drawing is not completely confusing 00:44:34.560 --> 00:44:36.450 at this point, but why don't we just 00:44:36.450 --> 00:44:41.380 look at the drawing looking down the effective field. 00:44:41.380 --> 00:44:43.500 We look down at the effective field 00:44:43.500 --> 00:44:48.050 and then we see the circle where the magnetic moment precesses, 00:44:48.050 --> 00:44:52.990 and the radius of the circle I've already given to you. 00:44:52.990 --> 00:44:58.410 So now we want to look down the effective magnetic moment. 00:44:58.410 --> 00:45:01.800 We see the circle. 00:45:01.800 --> 00:45:04.170 We want to lock down the effective magnetic field. 00:45:04.170 --> 00:45:07.495 We see the circular where the magnetic moment precesses. 00:45:12.320 --> 00:45:14.730 It's has precessed form here to there. 00:45:14.730 --> 00:45:16.350 We connect this line. 00:45:16.350 --> 00:45:25.700 This was our A, and the angle at which the magnetic moment 00:45:25.700 --> 00:45:28.730 has precesses is phi. 00:45:28.730 --> 00:45:35.810 And the radius as we derived before was mu times sine theta. 00:45:35.810 --> 00:45:40.250 So now just using the same equation for this triangle, 00:45:40.250 --> 00:45:47.170 we find that A square is 2 mu square sine square theta 00:45:47.170 --> 00:45:52.040 times 1 minus cosine phi. 00:45:54.570 --> 00:46:01.490 And for cosine phi, I want to use the trig identity 00:46:01.490 --> 00:46:06.820 and express it by half the angle. 00:46:06.820 --> 00:46:08.610 All right, now we are done. 00:46:08.610 --> 00:46:10.225 We're pretty much looking at this-- 00:46:10.225 --> 00:46:14.360 the drawing is clear precession around a tilt angle, 00:46:14.360 --> 00:46:16.510 and we're just doing geometry here. 00:46:16.510 --> 00:46:20.540 And we have now two expressions for A square. 00:46:20.540 --> 00:46:25.930 We can set the two expression equal 00:46:25.930 --> 00:46:34.140 and solve for the unknown, which is cosine over [? phi. ?] 00:46:34.140 --> 00:46:37.320 And with that, we find the cosine alpha 00:46:37.320 --> 00:46:47.375 is 1 minus 2 sine square theta sine square phi over 2. 00:46:50.560 --> 00:46:58.590 And the purpose of this exercise was that cosine alpha tells us 00:46:58.590 --> 00:47:01.750 the tilt angle of the magnetic moment away form 00:47:01.750 --> 00:47:05.000 the vertical axis, so therefore, we 00:47:05.000 --> 00:47:07.450 have done what we wanted to do. 00:47:10.480 --> 00:47:15.190 We know the z component of the magnetic moment 00:47:15.190 --> 00:47:24.970 as a function of time is this times 1 00:47:24.970 --> 00:47:34.415 minus generalized Rabi frequency squared times sine square. 00:47:37.710 --> 00:47:42.550 And now we know the precession phi, the precession at which 00:47:42.550 --> 00:47:46.280 the keep of the magnetic moment moves in a circle, 00:47:46.280 --> 00:47:49.640 we discussed it already before, and you gave the correct answer 00:47:49.640 --> 00:47:51.720 with the clicker, is this generalized 00:47:51.720 --> 00:47:56.070 Rabi frequency omega Rabi. 00:47:59.840 --> 00:48:04.060 So that's a nice formula, but before we lean back and look 00:48:04.060 --> 00:48:08.200 at it, let me just do one tiny step. 00:48:13.490 --> 00:48:16.380 Based on our quantum mechanical background, 00:48:16.380 --> 00:48:37.560 we can now define that the probability that the spin has 00:48:37.560 --> 00:48:45.820 been flipped is the relative difference 00:48:45.820 --> 00:48:51.230 in the z component appropriately normalized. 00:48:51.230 --> 00:48:54.170 This is just the normalized change in the z component, 00:48:54.170 --> 00:48:58.271 and if I call that the probability that, 00:48:58.271 --> 00:49:00.820 that which is on the left-hand side, 00:49:00.820 --> 00:49:07.110 then I find that the probability for this classically expression 00:49:07.110 --> 00:49:10.840 for which expresses how much the z component of the magnetic 00:49:10.840 --> 00:49:25.660 moment has changed, and this is exactly the celebrated formula 00:49:25.660 --> 00:49:28.445 for spin flips in a spin-1/2 system. 00:49:42.330 --> 00:49:48.370 So we have derived exactly the solution 00:49:48.370 --> 00:49:54.080 for the motion for the precession of a magnetic moment 00:49:54.080 --> 00:49:59.030 in a magnetic field plus the rotating magnetic field, 00:49:59.030 --> 00:50:02.410 and what we found is, we found that the magnetic moment 00:50:02.410 --> 00:50:06.610 precesses at the generalized Rabi frequency. 00:50:06.610 --> 00:50:09.890 And as I will show you next week, 00:50:09.890 --> 00:50:12.450 this result is exactly the same as 00:50:12.450 --> 00:50:16.650 for quantum mechanical expectation values. 00:50:16.650 --> 00:50:17.553 Question, yeah? 00:50:17.553 --> 00:50:20.692 AUDIENCE: So what if we have magnetic field oscillating just 00:50:20.692 --> 00:50:24.315 in one direction instead of rotation magnetic fields? 00:50:24.315 --> 00:50:27.190 PROFESSOR: Oh, that's a much more complicated problem. 00:50:27.190 --> 00:50:30.620 What happens when we have the magnetic field which 00:50:30.620 --> 00:50:33.130 is linearly polarized, which is only 00:50:33.130 --> 00:50:35.190 oscillating in one direction? 00:50:35.190 --> 00:50:40.030 Well, light or a vector which is oscillating linearly 00:50:40.030 --> 00:50:44.240 in one direction can be regarded as a superposition 00:50:44.240 --> 00:50:47.400 of a left and right rotating field. 00:50:47.400 --> 00:50:50.460 In other words, if you superimpose left-handed 00:50:50.460 --> 00:50:53.030 and right-handed circular polarized light, 00:50:53.030 --> 00:50:56.290 the sum of the two is linearly polarized light. 00:50:56.290 --> 00:50:59.070 So now we have actually the situation 00:50:59.070 --> 00:51:02.327 that linearly polarized magnetic field-- that's 00:51:02.327 --> 00:51:03.660 what we usually do when the lab. 00:51:03.660 --> 00:51:05.100 I mean, we have coils. 00:51:05.100 --> 00:51:08.430 We connect them to synthesizer and the field 00:51:08.430 --> 00:51:09.810 is not going in a circle. 00:51:09.810 --> 00:51:11.080 It's going back and forth. 00:51:11.080 --> 00:51:12.590 It's linearly polarized. 00:51:12.590 --> 00:51:16.910 This corresponds to a magnetic field 00:51:16.910 --> 00:51:20.730 which corresponds to two magnetic field, one 00:51:20.730 --> 00:51:24.460 rotates left and one rotates right. 00:51:24.460 --> 00:51:26.875 But the problem is if you now do a transformation 00:51:26.875 --> 00:51:29.230 in the rotating frame, do we want 00:51:29.230 --> 00:51:32.970 to rotate omega to the left or omega to the right? 00:51:32.970 --> 00:51:36.870 So what we can do is, we can pick our rotating frame 00:51:36.870 --> 00:51:39.830 and we are now in the rotating frame. 00:51:39.830 --> 00:51:43.280 One of the rotating fields has become time independent, 00:51:43.280 --> 00:51:45.780 the other rotates now at 2 omega. 00:51:48.540 --> 00:51:52.040 And at that point, we need the celebrated rotating wave 00:51:52.040 --> 00:51:55.770 approximation that we keep the one term [? via ?] rectified 00:51:55.770 --> 00:51:58.120 and the other one at two omega rotates 00:51:58.120 --> 00:52:01.430 so rapidly that we say these rapid oscillations do nothing 00:52:01.430 --> 00:52:03.760 and we discard it. 00:52:03.760 --> 00:52:05.440 We'll discussed it later in this course. 00:52:07.970 --> 00:52:11.340 But the gist is, if you have linearly-polarized 00:52:11.340 --> 00:52:13.940 light, linearly-polarized magnetic fields, 00:52:13.940 --> 00:52:17.110 we usually have to do an additional approximation, 00:52:17.110 --> 00:52:18.950 the rotating wave approximation. 00:52:18.950 --> 00:52:22.090 And since the rotating wave approximation is done always 00:52:22.090 --> 00:52:25.030 in almost any treatment, any paper you can find, 00:52:25.030 --> 00:52:27.350 we think it's always necessary. 00:52:27.350 --> 00:52:30.580 But what I've shown to you is, when we have a rotating field, 00:52:30.580 --> 00:52:32.330 we don't need any approximation. 00:52:32.330 --> 00:52:37.780 The transformation in the rotating frame is exact, 00:52:37.780 --> 00:52:41.170 but that's the beauty of it the when we assume rotating frames, 00:52:41.170 --> 00:52:44.540 we can hold on to exact solutions for longer and only 00:52:44.540 --> 00:52:48.450 later then discuss what happens when we introduce 00:52:48.450 --> 00:52:51.210 linearly-polarized magnetic fields. 00:52:55.370 --> 00:52:57.310 But that's something we definitely 00:52:57.310 --> 00:53:00.870 do later in its full beauty. 00:53:00.870 --> 00:53:02.190 15 minutes. 00:53:07.260 --> 00:53:07.760 OK. 00:53:07.760 --> 00:53:12.792 There's one thing I want to do about classical spins 00:53:12.792 --> 00:53:14.750 and then we do with the full quantum treatment. 00:53:18.220 --> 00:53:21.800 And this aspect of classical spins 00:53:21.800 --> 00:53:24.720 is called rapid adiabatic passage. 00:53:46.080 --> 00:53:50.340 So we want to add one more piece to our discussion. 00:53:50.340 --> 00:53:55.360 So far we have assumed a static field and a rotating field, 00:53:55.360 --> 00:53:58.040 which rotates at one frequency. 00:53:58.040 --> 00:54:01.240 But now we want to change the frequency of the rotating 00:54:01.240 --> 00:54:02.280 field. 00:54:02.280 --> 00:54:06.300 So we have our magnetic moment in a static magnetic field, 00:54:06.300 --> 00:54:08.380 which is our quantization axis. 00:54:08.380 --> 00:54:10.260 And now we have a rotating field, 00:54:10.260 --> 00:54:13.640 but we increase the frequency of the rotating field from slow 00:54:13.640 --> 00:54:16.670 to fast and ask what happens. 00:54:16.670 --> 00:54:20.460 And the result is that by increasing the frequency, 00:54:20.460 --> 00:54:23.230 sweeping the frequency through the resonance, 00:54:23.230 --> 00:54:25.510 we can do something very useful. 00:54:25.510 --> 00:54:27.890 We can invert the spin. 00:54:27.890 --> 00:54:32.220 We can turn over the magnetic moment in a very robust way, 00:54:32.220 --> 00:54:35.550 and this is the concept of rapid adiabatic passage. 00:54:35.550 --> 00:54:38.550 A lot of you may be familiar with the concept 00:54:38.550 --> 00:54:40.920 a Landau-Zener transition, what I'm 00:54:40.920 --> 00:54:44.790 telling you what is exactly the classical counterpart 00:54:44.790 --> 00:54:46.700 of the Landau-Zener transition. 00:54:46.700 --> 00:54:49.770 actually in many cases when it comes to spin physics, 00:54:49.770 --> 00:54:53.030 the classical physics and the quantum physics is the same. 00:54:53.030 --> 00:54:56.430 So that's why I want to discuss rapid adiabatic passage 00:54:56.430 --> 00:55:03.970 and Landau-Zener physics first in the classical environment. 00:55:03.970 --> 00:55:09.610 So rapid adiabatic passage is a technique 00:55:09.610 --> 00:55:20.800 for inverting turning around spins or magnetic moments 00:55:20.800 --> 00:55:28.550 by sweeping the frequency of your drive field 00:55:28.550 --> 00:55:29.980 across the resonance. 00:55:36.160 --> 00:55:48.875 So adiabatic means that this frequency sweep has to be slow. 00:55:53.800 --> 00:55:59.905 Slow is slow compared to the Larmor frequency. 00:56:04.810 --> 00:56:08.080 We get any given moment the magnetic moment precesses 00:56:08.080 --> 00:56:14.360 at the Larmor frequency, which is the gyromagnetic ratio 00:56:14.360 --> 00:56:17.030 times g-factor magnetic field, so it's 00:56:17.030 --> 00:56:18.970 sort of quasi stationary. 00:56:18.970 --> 00:56:22.450 The spin precesses around the effective magnetic field 00:56:22.450 --> 00:56:25.700 and we want to change the frequency of the drive 00:56:25.700 --> 00:56:29.820 slow compared to that motion. 00:56:29.820 --> 00:56:31.940 Well, the word rapid adiabatic passage 00:56:31.940 --> 00:56:35.890 has the word adiabatic, which means slow but also rapid. 00:56:41.140 --> 00:56:44.270 The word rapid means we have to be rapid 00:56:44.270 --> 00:56:47.540 compared to relaxation processes, which we do not 00:56:47.540 --> 00:56:50.860 discuss here in an idealized environment. 00:56:50.860 --> 00:56:53.030 For instance, if you do rapid adiabatic passage 00:56:53.030 --> 00:56:55.950 in the environment where the atoms can collide, 00:56:55.950 --> 00:56:59.570 rapid means you have to do it fast enough before you have 00:56:59.570 --> 00:57:04.400 decoherence in two collisions, so slow compared to the Larmor 00:57:04.400 --> 00:57:08.370 frequency and rapid compared to all the things 00:57:08.370 --> 00:57:12.900 I'm not mentioning here-- rapid compared 00:57:12.900 --> 00:57:17.270 to decoherence and relaxation processes. 00:57:31.910 --> 00:57:38.940 I will not set up differential equation and solve them. 00:57:38.940 --> 00:57:47.120 I want to give you the intuitive picture of what goes on, 00:57:47.120 --> 00:57:53.330 but then also derive what is sort 00:57:53.330 --> 00:57:56.329 of the criterion for adiabadicity, 00:57:56.329 --> 00:57:57.370 which we have to fulfill. 00:58:07.270 --> 00:58:08.630 So what are our ingredients? 00:58:08.630 --> 00:58:11.450 We have a magnetic moment, mu. 00:58:15.300 --> 00:58:18.310 We have a static field B naught. 00:58:20.920 --> 00:58:32.750 We have a drive field B1, which rotates at a frequency omega. 00:58:32.750 --> 00:58:41.930 And we assume that the rotating field 00:58:41.930 --> 00:58:45.310 is smaller than the static field. 00:58:45.310 --> 00:58:48.220 It's not absolutely necessary, but you 00:58:48.220 --> 00:58:50.060 apply a big static field and then you 00:58:50.060 --> 00:58:51.210 have a repetitive drive. 00:58:51.210 --> 00:58:53.150 That's a standard situation. 00:58:53.150 --> 00:58:56.640 Or we always want to have quantization axis, which 00:58:56.640 --> 00:58:58.840 is given with the z-axis as defined 00:58:58.840 --> 00:59:01.600 by the static magnetic field, but it's only 00:59:01.600 --> 00:59:03.820 defined by the static magnetic field 00:59:03.820 --> 00:59:05.870 if the transverse field is not much 00:59:05.870 --> 00:59:07.670 larger than the static field, otherwise, 00:59:07.670 --> 00:59:09.978 we are talking about a somewhat different problem. 00:59:12.820 --> 00:59:17.710 And let me just assume to be specific we later 00:59:17.710 --> 00:59:20.100 discuss what happens if it's not the case. 00:59:20.100 --> 00:59:25.020 We assume that we start with a frequency omega, which 00:59:25.020 --> 00:59:27.457 is much, much smaller than the Larmor frequency, 00:59:27.457 --> 00:59:29.040 much, much smaller than the resonance. 00:59:36.710 --> 00:59:41.150 So what does that mean for our effective magnetic field? 00:59:43.990 --> 00:59:46.470 Let me just sketch it for you. 00:59:46.470 --> 00:59:48.770 Remember our effective magnetic field, 00:59:48.770 --> 00:59:50.860 we have a field B naught. 00:59:50.860 --> 00:59:54.660 We have a drive field B1, which we assume is smaller. 00:59:54.660 --> 00:59:57.410 But then when we go to the rotating frame, 00:59:57.410 --> 01:00:01.260 we have a fictitious field, but if the frequency omega 01:00:01.260 --> 01:00:04.340 is below the Larmor frequency, this fictitious field 01:00:04.340 --> 01:00:05.970 is very small. 01:00:05.970 --> 01:00:08.660 So therefore, we start out in a situation 01:00:08.660 --> 01:00:11.850 where the effective field is pretty much pointing 01:00:11.850 --> 01:00:13.001 along the z direction. 01:00:18.100 --> 01:00:23.750 So just to remind you, so we have a situation 01:00:23.750 --> 01:00:33.450 where the effective field is just at a tiny angle, 01:00:33.450 --> 01:00:42.994 and if we started out with our magnetic moment aligned in z, 01:00:42.994 --> 01:00:46.290 and we assume B1 is really perturbative, pretty 01:00:46.290 --> 01:00:50.170 much the magnetic moment is very tightly coupled. 01:00:50.170 --> 01:00:54.040 It has a very small precession angle, or if you want, 01:00:54.040 --> 01:00:56.570 if you take the magnetic field B1 to be perturbative, 01:00:56.570 --> 01:00:59.440 you can say the magnetic moment is 01:00:59.440 --> 01:01:03.390 aligned with the effective field. 01:01:03.390 --> 01:01:06.260 That's the limit that the cone angle of precession 01:01:06.260 --> 01:01:07.040 is very small. 01:01:17.860 --> 01:01:19.600 So let we write that down. 01:01:19.600 --> 01:01:23.720 This is what's omega much, much smaller 01:01:23.720 --> 01:01:26.270 than the Larmor frequency. 01:01:39.210 --> 01:01:42.100 So now we want to turn up the knob 01:01:42.100 --> 01:01:43.700 on the frequency of the drive. 01:01:43.700 --> 01:01:49.200 We want to rotate the drive field B1 faster and faster. 01:01:49.200 --> 01:01:53.850 And the picture you should have is-- I just wiped it away, 01:01:53.850 --> 01:01:58.590 but what that means is, the effective field, 01:01:58.590 --> 01:02:02.380 the fictitious field is no longer and longer component 01:02:02.380 --> 01:02:05.800 and on resonance, the fictitious magnetic field 01:02:05.800 --> 01:02:08.420 will cancel B naught. 01:02:08.420 --> 01:02:10.370 So in other words, at this point, 01:02:10.370 --> 01:02:13.710 when B naught is canceled, the effective field 01:02:13.710 --> 01:02:19.100 has only the B1 component in the x direction. 01:02:19.100 --> 01:02:25.110 So therefore, when be go to delta equal 0, 01:02:25.110 --> 01:02:32.690 the effective field is only B1 and points in the x direction. 01:02:32.690 --> 01:02:36.510 So what we have done is by changing the frequency, 01:02:36.510 --> 01:02:39.600 by ramping up the frequency to resonance, 01:02:39.600 --> 01:02:42.430 we have tilted the effective magnetic field 01:02:42.430 --> 01:02:47.070 from the z direction into the x direction. 01:02:47.070 --> 01:02:49.230 And at any given moment, I mean, we 01:02:49.230 --> 01:02:52.170 know what the exact solution of the magnetic moment is. 01:02:52.170 --> 01:02:54.860 At any given moment, the magnetic moment 01:02:54.860 --> 01:02:59.120 precesses around the effective magnetic field. 01:02:59.120 --> 01:03:01.710 And if the precession is very fast 01:03:01.710 --> 01:03:05.070 and the effective magnetic field is slowly rotating, 01:03:05.070 --> 01:03:08.930 the magnetic moment is just following. 01:03:08.930 --> 01:03:12.310 So therefore, at this point when we are at resonance, 01:03:12.310 --> 01:03:17.560 we have tilted the magnetic moment by 90 degrees. 01:03:17.560 --> 01:03:19.010 Just one second. 01:03:19.010 --> 01:03:24.100 And if we go with a frequency much higher 01:03:24.100 --> 01:03:31.910 than the Larmor frequency, then our fictitious magnetic field 01:03:31.910 --> 01:03:35.610 is much larger than B naught and therefore, 01:03:35.610 --> 01:03:42.410 the effective magnetic field points now 01:03:42.410 --> 01:03:43.470 in the minus z direction. 01:03:49.070 --> 01:03:52.750 So the idea is that as long as this rotation 01:03:52.750 --> 01:03:57.200 of the effective field from being plus z in the x direction 01:03:57.200 --> 01:04:01.850 and into minus z is slow enough, the rapid precession is 01:04:01.850 --> 01:04:05.750 locking, is keeping the magnetic moment aligned 01:04:05.750 --> 01:04:07.970 with the effective magnetic field 01:04:07.970 --> 01:04:12.800 and we have just a handle we invert. 01:04:12.800 --> 01:04:14.550 We move around the magnetic moment. 01:04:17.180 --> 01:04:26.780 So in the adiabatic limit, the spin precesses tightly, 01:04:26.780 --> 01:04:33.880 and by tightly I mean the angle theta is small around B ef, B 01:04:33.880 --> 01:04:44.910 effective, and follows the direction 01:04:44.910 --> 01:04:48.990 of the effective magnetic field. 01:04:48.990 --> 01:04:51.910 So we are rotating an effective field. 01:04:51.910 --> 01:04:54.160 We are rotating the magnetic moment, 01:04:54.160 --> 01:04:57.230 but we're not rotating anything in the laboratory. 01:04:57.230 --> 01:04:59.025 The only thing we are doing is, we 01:04:59.025 --> 01:05:01.610 are changing the frequency of the rotating magnetic field. 01:05:11.530 --> 01:05:12.040 Questions? 01:05:12.040 --> 01:05:12.540 Jenny. 01:05:12.540 --> 01:05:13.666 AUDIENCE: Oh, yeah. 01:05:13.666 --> 01:05:15.230 I was thinking, can you also do this 01:05:15.230 --> 01:05:17.198 by keeping the frequency the same 01:05:17.198 --> 01:05:19.166 and just ramping up the strength? 01:05:19.166 --> 01:05:23.594 Like say, put it at the-- make the B1 01:05:23.594 --> 01:05:26.546 equal to the frequency of that Larmor frequency 01:05:26.546 --> 01:05:29.498 and just start from 0 and ramp up? 01:05:29.498 --> 01:05:30.430 PROFESSOR: Yes. 01:05:30.430 --> 01:05:34.620 I mean, the essence here is that the effective magnetic field 01:05:34.620 --> 01:05:37.640 is [INAUDIBLE] resonance, and what you are suggesting 01:05:37.640 --> 01:05:40.120 is if the frequency is constant, that would 01:05:40.120 --> 01:05:43.090 mean the fictitious magnetic field is constant. 01:05:43.090 --> 01:05:46.060 But if you were given a fictitious magnetic field 01:05:46.060 --> 01:05:49.830 and we a huge field B naught, the effective magnetic field 01:05:49.830 --> 01:05:51.800 points out, points up. 01:05:51.800 --> 01:05:54.800 But if we now make this static field B naught smaller 01:05:54.800 --> 01:05:57.110 and smaller, both through resonance 01:05:57.110 --> 01:05:59.710 and make it even smaller, we have also 01:05:59.710 --> 01:06:02.680 done an inversion of the effective magnetic field. 01:06:02.680 --> 01:06:05.220 The result is the same. 01:06:05.220 --> 01:06:07.180 Actually, sometimes in the laboratory, 01:06:07.180 --> 01:06:09.610 if you have a synthesizer which is not easily 01:06:09.610 --> 01:06:12.530 computer controlled, we do an analog sweep 01:06:12.530 --> 01:06:14.980 of the magnetic field, so we actually 01:06:14.980 --> 01:06:18.910 change the Larmor frequency of the atom instead 01:06:18.910 --> 01:06:20.580 of changing the drive frequency. 01:06:20.580 --> 01:06:22.290 What really matters in the whole business 01:06:22.290 --> 01:06:24.480 is the relative frequency between the two. 01:06:24.480 --> 01:06:29.400 AUDIENCE: Is it necessarily true that the spin resistance 01:06:29.400 --> 01:06:31.860 is tightly around the effective fields? 01:06:31.860 --> 01:06:35.796 Isn't it at resonance the radius of the circle in precession 01:06:35.796 --> 01:06:38.748 is equal to mu? 01:06:38.748 --> 01:06:42.438 Is it my [INAUDIBLE] before we found 01:06:42.438 --> 01:06:44.160 the radius which was mu sine theta. 01:06:44.160 --> 01:06:49.160 Then sine theta cosine Omega R-- 01:06:49.160 --> 01:06:49.895 PROFESSOR: Yes. 01:06:49.895 --> 01:06:50.978 AUDIENCE: --divided into-- 01:06:50.978 --> 01:06:52.776 PROFESSOR: But the way is we start-- 01:06:52.776 --> 01:06:54.150 the way we've started it up here. 01:06:54.150 --> 01:06:59.820 Now if we are far away from resonance, 01:06:59.820 --> 01:07:04.390 the angle theta is infinitesimally small. 01:07:04.390 --> 01:07:05.500 Something confuses you. 01:07:05.500 --> 01:07:06.125 AUDIENCE: Yeah. 01:07:06.125 --> 01:07:09.702 I mean, it's just [INAUDIBLE]. 01:07:09.702 --> 01:07:11.160 PROFESSOR: See, what we have solved 01:07:11.160 --> 01:07:14.730 before, we have done something else before. 01:07:14.730 --> 01:07:16.550 What we have discussed before is, 01:07:16.550 --> 01:07:19.140 that we have a spin which is aligned 01:07:19.140 --> 01:07:23.520 and then we looked at the-- you can say the transient solution, 01:07:23.520 --> 01:07:27.120 we switched suddenly on the rotating field. 01:07:27.120 --> 01:07:29.090 So that may sort of described that. 01:07:29.090 --> 01:07:31.410 We have a magnetic field, which is in the z direction. 01:07:31.410 --> 01:07:32.330 Obviously we don't. 01:07:32.330 --> 01:07:34.370 Our spin is aligned in the z direction. 01:07:34.370 --> 01:07:38.290 And if you then suddenly switch on a rotating field, 01:07:38.290 --> 01:07:40.290 that could mean you have suddenly 01:07:40.290 --> 01:07:42.800 created an effective field, which is tilted. 01:07:42.800 --> 01:07:46.020 So then by the sudden switch on of your drive, 01:07:46.020 --> 01:07:49.680 you have an angle theta and the magnetic moment precesses 01:07:49.680 --> 01:07:51.990 with the precession angle theta. 01:07:51.990 --> 01:07:53.900 But what we are here doing is, we 01:07:53.900 --> 01:07:59.060 are ever so slightly changing the angle theta 01:07:59.060 --> 01:08:01.169 and then the spin stays aligned. 01:08:01.169 --> 01:08:02.085 That's the difference. 01:08:08.740 --> 01:08:18.750 So before I give you another example 01:08:18.750 --> 01:08:25.704 and discuss the limit of adiabadicity, 01:08:25.704 --> 01:08:27.120 let's just have a quicker question 01:08:27.120 --> 01:08:29.620 to make sure that everybody follows. 01:08:29.620 --> 01:08:37.479 So OK, what I think we have at least understood 01:08:37.479 --> 01:08:40.410 in the adiabatic limit is, that when 01:08:40.410 --> 01:08:49.710 we start with a magnetic moment, which is aligned, 01:08:49.710 --> 01:08:56.960 and now we do a sweep, we start with a small rotating frequency 01:08:56.960 --> 01:08:58.819 and we make it large. 01:08:58.819 --> 01:09:03.590 We sweep from low to high frequency through resonance, 01:09:03.590 --> 01:09:10.180 then we can invert the magnetic moment. 01:09:10.180 --> 01:09:20.420 My question for you now is, what happens 01:09:20.420 --> 01:09:25.330 when we start with a spin but we switch 01:09:25.330 --> 01:09:29.930 on a drive, which is a buff resonance 01:09:29.930 --> 01:09:32.960 and we sweep it with small frequencies? 01:09:32.960 --> 01:09:37.020 What will now happen when we do the opposite sweep? 01:09:37.020 --> 01:09:44.619 Will the spin just stay, will it be flipped, 01:09:44.619 --> 01:09:47.659 or will it sort of get diffused or get disoriented? 01:09:50.250 --> 01:09:55.250 So this is answer A, this is answer B, and this is answer C. 01:09:55.250 --> 01:09:57.020 So what happens? 01:09:57.020 --> 01:09:59.310 I have assumed in already discussion 01:09:59.310 --> 01:10:02.890 before that we start with a very low frequency where 01:10:02.890 --> 01:10:05.920 the fictitious magnetic field is negligible. 01:10:05.920 --> 01:10:08.070 And then we change the frequency, 01:10:08.070 --> 01:10:09.580 we sweep the frequency. 01:10:09.580 --> 01:10:12.860 It cause the Larmor frequency to very high frequency. 01:10:12.860 --> 01:10:15.460 My question now is, what happens when 01:10:15.460 --> 01:10:18.740 we reverse the frequency of the sweep? 01:10:37.445 --> 01:10:37.945 OK. 01:10:40.870 --> 01:10:43.440 There is some distribution. 01:10:43.440 --> 01:10:47.570 The correct answer is indeed B, we flip the spin. 01:10:47.570 --> 01:10:51.510 What happens is the following. 01:10:51.510 --> 01:10:55.050 The spin is in the z direction, but at high frequency, 01:10:55.050 --> 01:10:57.850 the fictitious magnetic field is very large. 01:10:57.850 --> 01:11:00.440 So therefore, the effective magnetic field 01:11:00.440 --> 01:11:03.500 is now a large effective magnetic field 01:11:03.500 --> 01:11:06.540 pointing in the minus z direction. 01:11:06.540 --> 01:11:10.390 So now we have a situation where the spin tightly precesses 01:11:10.390 --> 01:11:12.980 around the magnetic effective magnetic field 01:11:12.980 --> 01:11:17.400 at an angle of 180 degrees, not 0 degrees, but 180 degrees. 01:11:17.400 --> 01:11:23.410 And now, when we change the detuning, 01:11:23.410 --> 01:11:26.250 the effective magnetic field tilts, 01:11:26.250 --> 01:11:29.580 but the spin keeps on precessing at 180 degrees 01:11:29.580 --> 01:11:33.070 and eventually when we sweep through the resonance, 01:11:33.070 --> 01:11:36.890 the effective magnetic field at low frequency of the rotating 01:11:36.890 --> 01:11:39.710 field becomes the real magnetic field is now 01:11:39.710 --> 01:11:43.020 pointing in the z direction and the spin 01:11:43.020 --> 01:11:46.150 has followed the 180 degree rotation. 01:11:46.150 --> 01:11:48.570 So in other words, it does not matter 01:11:48.570 --> 01:11:51.930 whether you go from low to high or from high to low frequency. 01:11:51.930 --> 01:11:55.230 Whenever you go through the resonance, you flip the spin. 01:12:02.100 --> 01:12:05.150 Let's close that Let's see. 01:12:05.150 --> 01:12:05.916 It is that. 01:12:14.970 --> 01:12:22.250 So the more generalized answer is 01:12:22.250 --> 01:12:38.130 that the rapid adiabatic passage always swaps the spin state 01:12:38.130 --> 01:12:39.890 no matter which way you sweep. 01:12:39.890 --> 01:12:43.240 When you start in spin up, you wind up in spin down. 01:12:43.240 --> 01:12:46.050 When you start in spin down, you wind up 01:12:46.050 --> 01:12:48.320 in spin up because what you're doing 01:12:48.320 --> 01:12:53.280 is you're inverting the direction 01:12:53.280 --> 01:12:55.950 of the effective magnetic field and the spin just follows. 01:12:58.820 --> 01:13:01.330 So therefore it goes both ways. 01:13:01.330 --> 01:13:04.030 It applies to-- you can say to the ground state 01:13:04.030 --> 01:13:07.300 with the lowest energy state, it applies to the highest energy 01:13:07.300 --> 01:13:11.640 state, and as such, it is actually 01:13:11.640 --> 01:13:17.320 a swap operation like the pi over 2 pulse. 01:13:17.320 --> 01:13:20.950 When you have a pi over 2 pulse, you have a spin 01:13:20.950 --> 01:13:24.410 and you just pulse on a rotating magnetic field, which 01:13:24.410 --> 01:13:28.260 in the rotating flame means you have an x field. 01:13:28.260 --> 01:13:32.170 When the spin was up at the Rabi frequency, rotates down. 01:13:32.170 --> 01:13:36.300 When the spin was down at the Rabi frequency, it rotates up. 01:13:36.300 --> 01:13:40.410 And quantum mechanically it just means you take the population 01:13:40.410 --> 01:13:44.910 in two states and your unitary time evolution 01:13:44.910 --> 01:13:45.950 is a swap operation. 01:13:48.540 --> 01:13:51.670 And the classical counterpart is what we just discussed, 01:13:51.670 --> 01:13:54.670 the rapid adiabatic passage. 01:13:54.670 --> 01:13:58.470 However, there is one big experimental advantage 01:13:58.470 --> 01:14:03.320 of doing rapid adiabatic passage over pi over 2 pulse. 01:14:03.320 --> 01:14:08.120 And I think many of you use rapid adiabatic passage 01:14:08.120 --> 01:14:09.250 in the lab. 01:14:09.250 --> 01:14:11.760 Remember, if you give yourself the drop, 01:14:11.760 --> 01:14:17.730 you want to transfer the magnetic moment form spin up 01:14:17.730 --> 01:14:19.870 to spin down. 01:14:19.870 --> 01:14:22.502 If you want to do it with the pi pulse-- 01:14:22.502 --> 01:14:25.910 if you want to do it with the-- I said pi over 2 pulse, 01:14:25.910 --> 01:14:27.057 I meant pi pulse. 01:14:33.740 --> 01:14:36.860 The only way how with a pulse you 01:14:36.860 --> 01:14:42.460 can rotate the magnetic moment by 180 degrees, if you 01:14:42.460 --> 01:14:46.650 are exactly on resonance, where the fictitious field cancels 01:14:46.650 --> 01:14:50.900 the static field and what you have is in the rotating frame 01:14:50.900 --> 01:14:52.920 is only a field in x. 01:14:52.920 --> 01:14:55.940 And then you can rotate around the x-axis. 01:14:55.940 --> 01:15:01.100 But if for some reason you're not exactly on resonance, then 01:15:01.100 --> 01:15:04.090 the fictitious field does not cancel the static field, 01:15:04.090 --> 01:15:07.370 and your effective magnetic field is at an angle. 01:15:07.370 --> 01:15:10.120 And if you rotate an effective magnetic field, 01:15:10.120 --> 01:15:12.990 which is not in the x-axis but at an angle, 01:15:12.990 --> 01:15:16.790 you cannot do a full inversion of the magnetic moment. 01:15:16.790 --> 01:15:20.540 So in other words, if you want to use a pi pulse to flip over 01:15:20.540 --> 01:15:24.300 a spin, you have to pulse on your drive exactly 01:15:24.300 --> 01:15:25.970 on resonance. 01:15:25.970 --> 01:15:29.690 And if you have MBN magnetic fields which drift by a few 01:15:29.690 --> 01:15:32.250 milligals and you're not sure where the resonance is, 01:15:32.250 --> 01:15:35.330 you cannot do a perfect pi pulse. 01:15:35.330 --> 01:15:40.160 But with the sweep, you just have to sweep from point A 01:15:40.160 --> 01:15:43.340 to point B, and if you know you cross the resonance, 01:15:43.340 --> 01:15:47.750 you have a perfect inversion of the magnetic moment, which 01:15:47.750 --> 01:15:49.726 is robust against frequency drifts. 01:15:59.446 --> 01:16:04.360 Of course, you have to do a longer sweep. 01:16:04.360 --> 01:16:06.610 The pi pulse is the shortest possible wave. 01:16:06.610 --> 01:16:09.396 If you heat the cloud on resonance, you first sweep, 01:16:09.396 --> 01:16:10.770 nothing happens, nothing happens, 01:16:10.770 --> 01:16:13.394 and you go through resonance and just view further and further, 01:16:13.394 --> 01:16:14.840 you waste some of your time. 01:16:14.840 --> 01:16:18.230 So you pay a price for it, but often we want precision 01:16:18.230 --> 01:16:20.470 and we have the time to do it and then 01:16:20.470 --> 01:16:22.970 the job is done by rapid adiabatic passage. 01:16:27.880 --> 01:16:30.050 Let me just mention one more thing and then I stop. 01:16:33.570 --> 01:16:42.004 I have discussed the physics of keeping-- 01:16:42.004 --> 01:16:43.420 with rapid adiabatic passage, I've 01:16:43.420 --> 01:16:48.140 discussed with you the physics that rapid precession keeps 01:16:48.140 --> 01:16:52.750 a magnetic moment aligned with an effective magnetic field. 01:16:52.750 --> 01:16:57.410 Let me now discuss the same phenomenon 01:16:57.410 --> 01:16:58.910 but in a very different environment. 01:17:03.720 --> 01:17:11.170 And this is a similar process happen in a magnetic trap. 01:17:11.170 --> 01:17:19.110 In a magnetic trap, we don't have any drive 01:17:19.110 --> 01:17:23.730 or we have a time-dependent field. 01:17:23.730 --> 01:17:27.450 But what happens is, we have a magnetic field, 01:17:27.450 --> 01:17:30.950 but the atoms move through the atom trap. 01:17:30.950 --> 01:17:35.720 So the atom sees a changing magnetic field. 01:17:35.720 --> 01:17:39.350 And in many of our experiments here at MIT, 01:17:39.350 --> 01:17:43.290 we use a quadrupolar field. 01:17:43.290 --> 01:17:47.210 We've discussed some of these aspects in 8.422, 01:17:47.210 --> 01:17:50.780 so a quadrupolar field, the field 01:17:50.780 --> 01:17:56.080 has to be homogeneous in order to provide trapping forces, 01:17:56.080 --> 01:18:00.320 so we often use quadrupolar fields, a lot of advantages. 01:18:00.320 --> 01:18:03.240 That's how we build the tightest magnetic traps. 01:18:03.240 --> 01:18:07.000 But what happens now is, if an atom moves 01:18:07.000 --> 01:18:12.520 along this projector, it moves up in the laboratory, 01:18:12.520 --> 01:18:16.830 here t let's say the atom is in spin up. 01:18:16.830 --> 01:18:18.710 Here it's now a aligned. 01:18:21.670 --> 01:18:24.660 The magnetic field is opposite to the spin 01:18:24.660 --> 01:18:28.390 and now it moves up, and now the magnetic field up here 01:18:28.390 --> 01:18:30.090 is pointing up. 01:18:30.090 --> 01:18:34.100 The physics I explain to you rapid adiabatic passage means 01:18:34.100 --> 01:18:36.400 that the rapid precession off this spin 01:18:36.400 --> 01:18:40.730 means that as the atom moves, the spin stays 01:18:40.730 --> 01:18:43.050 aligned with the magnetic field. 01:18:43.050 --> 01:18:46.700 So you find the same physics here 01:18:46.700 --> 01:18:49.615 in a different environment, but the mathematical description 01:18:49.615 --> 01:18:50.894 is the same. 01:18:53.786 --> 01:18:57.380 Of course, and that's my last comment for today, 01:18:57.380 --> 01:18:59.790 if you have a spheric quadrupole trap 01:18:59.790 --> 01:19:04.150 and you go right through the origin, 01:19:04.150 --> 01:19:06.970 you're out of luck because here, the atom 01:19:06.970 --> 01:19:09.040 sees the magnetic field is down. 01:19:09.040 --> 01:19:10.506 It gets smaller, the magnetic field 01:19:10.506 --> 01:19:11.630 gets smaller, gets smaller. 01:19:11.630 --> 01:19:13.270 The magnet field gets zero. 01:19:13.270 --> 01:19:16.600 Oopsy, the magnetic field points in the other direction. 01:19:16.600 --> 01:19:18.050 And there was no warning. 01:19:18.050 --> 01:19:22.600 The magnetic field has jumped from 0 degree to 180 degree. 01:19:22.600 --> 01:19:27.260 There was never, ever any transverse field around which 01:19:27.260 --> 01:19:30.360 the atom could precess and change its orientation. 01:19:30.360 --> 01:19:33.580 So therefore, when an atom is aligned with a magnetic field 01:19:33.580 --> 01:19:36.590 moves through the origin, oopsy, it's anti aligned. 01:19:36.590 --> 01:19:40.280 It has lost its orientation with respect to the magnetic field, 01:19:40.280 --> 01:19:43.610 and this is the breakdown of rapid adiabatic passage 01:19:43.610 --> 01:19:45.230 because there's no adiabadicity. 01:19:45.230 --> 01:19:47.770 It is not an adiabatic change of the direction 01:19:47.770 --> 01:19:50.210 of the magnetic field, it's a sudden field. 01:19:50.210 --> 01:19:52.310 And the consequences are bad. 01:19:52.310 --> 01:19:54.580 You lose your atoms from the magnetic trap. 01:19:54.580 --> 01:19:56.207 It's called Majorana losses. 01:19:56.207 --> 01:19:58.040 A lot of people know what I'm talking about, 01:19:58.040 --> 01:20:01.200 but I'm not explaining it in detail. 01:20:01.200 --> 01:20:02.020 Time is over. 01:20:02.020 --> 01:20:07.760 Any question about what I've discussed? 01:20:07.760 --> 01:20:08.639 Yes. 01:20:08.639 --> 01:20:09.637 AUDIENCE: [INAUDIBLE]. 01:20:20.116 --> 01:20:22.262 PROFESSOR: You said the weights of the-- 01:20:22.262 --> 01:20:23.750 AUDIENCE: Yeah, so [INAUDIBLE]. 01:20:31.686 --> 01:20:34.690 PROFESSOR: I would say-- the question is about the frequency 01:20:34.690 --> 01:20:37.950 weights and if you switch on the frequency drive, 01:20:37.950 --> 01:20:39.610 it's not a delta function. 01:20:39.610 --> 01:20:41.270 If we are far away from resonance, 01:20:41.270 --> 01:20:43.570 it doesn't really matter. 01:20:43.570 --> 01:20:47.160 That is, of course, a criterion that the effective weights 01:20:47.160 --> 01:20:54.690 of the frequency should not be comparable to the detuning. 01:20:54.690 --> 01:20:57.259 So then you switch it on, but you're so far away 01:20:57.259 --> 01:20:58.800 from resonance that it doesn't matter 01:20:58.800 --> 01:21:01.410 if you have a small weights, the effective detuning is still 01:21:01.410 --> 01:21:02.120 large. 01:21:02.120 --> 01:21:03.980 You scan for resonance. 01:21:03.980 --> 01:21:06.210 But these are sort of the boundary condition. 01:21:06.210 --> 01:21:08.860 The exact solution in the Landau-Zener problem, 01:21:08.860 --> 01:21:11.830 for instance, assumes you go from minus infinity 01:21:11.830 --> 01:21:13.990 to plus infinity in the detuning. 01:21:13.990 --> 01:21:15.250 Nobody does that. 01:21:15.250 --> 01:21:20.220 So we're discussing sort of sweep finite duration effects, 01:21:20.220 --> 01:21:26.550 but usually we are pretty close to the idealized assumption. 01:21:26.550 --> 01:21:29.100 OK, enjoy the Monday holiday. 01:21:29.100 --> 01:21:32.410 We meet on Tuesday, and Tuesday is 01:21:32.410 --> 01:21:35.830 in building 37 in our standard classroom.