1 00:00:00,050 --> 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 --> 00:00:17,202 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,202 --> 00:00:17,827 at ocw.mit.edu. 8 00:00:20,465 --> 00:00:21,930 PROFESSOR: All right. 9 00:00:21,930 --> 00:00:26,860 Then let's go back to our discussion of what 10 00:00:26,860 --> 00:00:34,050 happens to classical and quantum mechanical magnetic moments 11 00:00:34,050 --> 00:00:40,200 when they are exposed to magnetic fields. 12 00:00:40,200 --> 00:00:48,830 I just want to remind you of what we did last class 13 00:00:48,830 --> 00:00:57,450 and what we want to wrap up today. 14 00:00:57,450 --> 00:01:00,790 This is rapid adiabatic passage. 15 00:01:00,790 --> 00:01:05,650 I mentioned to you and explained it to you 16 00:01:05,650 --> 00:01:09,390 that rapid adiabatic passage is a powerful way 17 00:01:09,390 --> 00:01:12,430 to manipulate a classical and quantum system, 18 00:01:12,430 --> 00:01:16,170 and what we discussed is that when a spin points in the up 19 00:01:16,170 --> 00:01:19,810 direction and you sweep the resonance 20 00:01:19,810 --> 00:01:22,610 of an oscillating magnetic field-- the frequency 21 00:01:22,610 --> 00:01:26,400 of an oscillating magnetic field, through the resonance, 22 00:01:26,400 --> 00:01:31,920 you create an effective magnetic field in the moving frame which 23 00:01:31,920 --> 00:01:33,410 will rotate. 24 00:01:33,410 --> 00:01:37,080 And the atom, when the change is done adiabatically, 25 00:01:37,080 --> 00:01:41,080 will follow the rotation and therefore invert the spin. 26 00:01:41,080 --> 00:01:43,890 So it's a perfect, very robust method 27 00:01:43,890 --> 00:01:48,905 to invert population in spin systems. 28 00:01:52,920 --> 00:01:57,390 What I want to pick up today is the question, 29 00:01:57,390 --> 00:01:59,120 how slow is adiabatic. 30 00:01:59,120 --> 00:02:03,040 We have to fulfill an adiabatic condition 31 00:02:03,040 --> 00:02:06,470 and we have already an idea already 32 00:02:06,470 --> 00:02:07,940 of what the adiabatic condition is 33 00:02:07,940 --> 00:02:10,490 but now we want to derive it. 34 00:02:10,490 --> 00:02:13,090 That we had this picture of a spin 35 00:02:13,090 --> 00:02:14,950 which is rapidly precessing. 36 00:02:14,950 --> 00:02:21,060 It always precesses around the direction 37 00:02:21,060 --> 00:02:25,700 of the effective magnetic field, and the condition adiabadicity 38 00:02:25,700 --> 00:02:30,190 is that the rotation of the effective magnetic field 39 00:02:30,190 --> 00:02:33,060 has to be much slower than the precession. 40 00:02:33,060 --> 00:02:35,430 Let me just make it clear by a counter example. 41 00:02:35,430 --> 00:02:38,310 If the atom precesses around the magnetic field 42 00:02:38,310 --> 00:02:41,330 and the magnetic field would suddenly jump, 43 00:02:41,330 --> 00:02:43,320 then the atom would now start precessing 44 00:02:43,320 --> 00:02:46,160 about the new direction of the magnetic field 45 00:02:46,160 --> 00:02:48,870 and it would have completely changed its angle 46 00:02:48,870 --> 00:02:50,560 relative to the magnetic field. 47 00:02:50,560 --> 00:02:53,660 It would have lost its alignment with the magnetic field. 48 00:02:53,660 --> 00:02:56,040 So you clearly see that the condition is 49 00:02:56,040 --> 00:02:59,570 the direction of the magnetic field must not jump, 50 00:02:59,570 --> 00:03:02,330 and the only other time scheme is the frequency of the Larmor 51 00:03:02,330 --> 00:03:05,510 precession, so our condition for adiabadicity 52 00:03:05,510 --> 00:03:11,100 is the rotation of the effective magnetic field 53 00:03:11,100 --> 00:03:13,445 has to be slow compared to the precession frequency. 54 00:03:18,330 --> 00:03:24,597 And so we want to now derive from that condition 55 00:03:24,597 --> 00:03:25,930 the conditions for adiabadicity. 56 00:03:28,586 --> 00:03:31,610 And just as an outlook to make it interesting, 57 00:03:31,610 --> 00:03:35,150 what I will derive for you in the classical picture 58 00:03:35,150 --> 00:03:37,260 is now something which you will later 59 00:03:37,260 --> 00:03:40,280 encounter as the Landau-Zener parameter. 60 00:03:40,280 --> 00:03:44,120 But Landau-Zener sweeps, we talk about it later today 61 00:03:44,120 --> 00:03:47,030 is a quantum mechanical version of rapid adiabatic passage, 62 00:03:47,030 --> 00:03:48,990 but we now get classically a result, 63 00:03:48,990 --> 00:03:51,720 which will feature the Landau-Zener parameter. 64 00:03:54,750 --> 00:04:04,000 So with that, let us write down what we want to look at. 65 00:04:04,000 --> 00:04:13,180 It is adiabatic condition, and to write it down in words 66 00:04:13,180 --> 00:04:19,350 is that the Larmor frequency, omega L, which 67 00:04:19,350 --> 00:04:24,120 is given by the effective magnetic field, 68 00:04:24,120 --> 00:04:30,970 has to be much larger than theta dot. 69 00:04:30,970 --> 00:04:34,140 Now things in general are rather complicated. 70 00:04:34,140 --> 00:04:36,240 If we are far away from resonance, 71 00:04:36,240 --> 00:04:39,110 you change the frequency, but the effective field 72 00:04:39,110 --> 00:04:40,550 is not changing a lot. 73 00:04:40,550 --> 00:04:45,050 The critical moment is really when we have the real field, we 74 00:04:45,050 --> 00:04:48,760 add the fictitious field and that cause a rotation. 75 00:04:48,760 --> 00:04:53,120 The critical moment is when we are near resonance. 76 00:04:53,120 --> 00:04:56,540 So in other words, we have to fulfill an inequality. 77 00:04:56,540 --> 00:04:59,360 The left side has to be larger than the right side, 78 00:04:59,360 --> 00:05:05,910 but the left side is actually smallest 79 00:05:05,910 --> 00:05:08,670 in the vicinity of the resonance, 80 00:05:08,670 --> 00:05:12,340 and the angle theta dot is actually 81 00:05:12,340 --> 00:05:14,820 largest near resonance. 82 00:05:14,820 --> 00:05:17,475 That's when it sort of quickly goes to 90 degrees. 83 00:05:20,750 --> 00:05:25,420 So therefore, if we want to find the condition of adiabadicity, 84 00:05:25,420 --> 00:05:28,210 we can derive it by looking at the region 85 00:05:28,210 --> 00:05:29,105 around the resonance. 86 00:05:32,200 --> 00:05:40,490 So the effective magnetic field is the real field 87 00:05:40,490 --> 00:05:47,200 minus the fictitious field caused 88 00:05:47,200 --> 00:05:50,520 by the rotation by the transformation 89 00:05:50,520 --> 00:05:51,735 into the rotating frame. 90 00:05:54,740 --> 00:06:04,410 So we have the magnetic field at an angle theta with respect 91 00:06:04,410 --> 00:06:04,993 to the z-axis. 92 00:06:10,480 --> 00:06:14,750 I've just written down the z component for you, 93 00:06:14,750 --> 00:06:20,580 and the transverse component is-- so this is this component 94 00:06:20,580 --> 00:06:23,521 and the transverse component is the amplitude of our drive 95 00:06:23,521 --> 00:06:24,020 field B1. 96 00:06:27,010 --> 00:06:29,035 So we can just read it form the diagram. 97 00:06:34,070 --> 00:06:39,060 The resonance data is 90 degrees, 98 00:06:39,060 --> 00:06:43,450 and the correction angle is whatever 99 00:06:43,450 --> 00:06:49,420 we have of the effective z field over B1, 100 00:06:49,420 --> 00:06:54,080 and that means that the derivative, the angle theta 101 00:06:54,080 --> 00:06:58,220 dot, the angular velocity at which the magnetic field 102 00:06:58,220 --> 00:07:04,350 rotates, there's a time derivative because we 103 00:07:04,350 --> 00:07:06,180 sweep the frequency. 104 00:07:06,180 --> 00:07:10,260 So therefore, theta dot is nothing else 105 00:07:10,260 --> 00:07:16,960 than omega dot, the sweep rate of the frequency divided 106 00:07:16,960 --> 00:07:18,120 by gamma B1. 107 00:07:22,680 --> 00:07:29,975 But gamma B1 is nothing else than the Rabi frequency. 108 00:07:34,500 --> 00:07:40,650 And on resonance, the Larmor frequency is just a Rabi 109 00:07:40,650 --> 00:07:46,220 frequency, because on resonance-- sorry for repeating 110 00:07:46,220 --> 00:07:51,240 myself-- the fictitious field has canceled the bias field, 111 00:07:51,240 --> 00:07:54,450 and the only field left is the rotating field, 112 00:07:54,450 --> 00:07:58,510 but the rotating field is the Rabi frequency 113 00:07:58,510 --> 00:08:00,780 with a gamma factor. 114 00:08:00,780 --> 00:08:07,520 So therefore, we have the adiabatic condition 115 00:08:07,520 --> 00:08:10,490 that omega dot over omega Rabi has 116 00:08:10,490 --> 00:08:17,180 to be smaller than omega Rabi or to say it inverts the change 117 00:08:17,180 --> 00:08:20,820 delta omega of your drive frequency, 118 00:08:20,820 --> 00:08:27,330 the change delta omega in one Ravi period 119 00:08:27,330 --> 00:08:29,370 has to be smaller than the Rabi frequency. 120 00:08:34,409 --> 00:08:37,080 So omega has units of frequency. 121 00:08:37,080 --> 00:08:41,350 Omega dot is a derivative that's in units of frequency squared, 122 00:08:41,350 --> 00:08:44,120 and this is to be smaller than the Rabi frequency squared. 123 00:08:59,250 --> 00:09:01,280 You find that actually quite often if you 124 00:09:01,280 --> 00:09:05,550 do an adiabatic change of your trap frequency, 125 00:09:05,550 --> 00:09:09,610 things are adiabatic as long as the change of the trap 126 00:09:09,610 --> 00:09:13,250 frequency in one period of the trap frequency 127 00:09:13,250 --> 00:09:15,210 is smaller than the trap frequency, 128 00:09:15,210 --> 00:09:18,240 and you find something else that the derivative of your trap 129 00:09:18,240 --> 00:09:21,760 frequency has to be smaller than the trap frequency squared. 130 00:09:21,760 --> 00:09:23,310 So these are adiabatic conditions, 131 00:09:23,310 --> 00:09:27,430 when you tighten up magnetic or optical confinement for atoms. 132 00:09:27,430 --> 00:09:29,450 So this is very, very genetic. 133 00:09:29,450 --> 00:09:32,090 The small rate of the frequency you change 134 00:09:32,090 --> 00:09:34,910 has to be smaller than the relevant frequency squared. 135 00:09:37,930 --> 00:09:41,020 As I said, we come back to that when 136 00:09:41,020 --> 00:09:45,610 we do the quantized treatment of rapid adiabatic passage 137 00:09:45,610 --> 00:09:48,130 and we encounter that combination 138 00:09:48,130 --> 00:09:51,010 in the Landau-Zener parameter, which 139 00:09:51,010 --> 00:09:55,530 takes us to our next topic. 140 00:09:55,530 --> 00:10:01,760 We want to now talk about quantized spin 141 00:10:01,760 --> 00:10:02,870 in a magnetic field. 142 00:10:12,549 --> 00:10:14,590 And one of the first things I will be telling you 143 00:10:14,590 --> 00:10:16,760 is that everything we have learned 144 00:10:16,760 --> 00:10:20,410 about the classical magnetic moment you don't have 145 00:10:20,410 --> 00:10:23,080 to unlearn or re-learn, it exactly 146 00:10:23,080 --> 00:10:26,020 applies to the quantum spin. 147 00:10:29,750 --> 00:10:34,600 Before we look at the Hamiltonian 148 00:10:34,600 --> 00:10:41,530 and the standard Hamiltonian for two-level system, 149 00:10:41,530 --> 00:10:45,450 let us first look at something more general, which 150 00:10:45,450 --> 00:10:47,655 is Heisenberg equations of motion. 151 00:10:52,580 --> 00:10:57,570 So we want to write down the differential equation, 152 00:10:57,570 --> 00:11:01,290 the equation of motion for expectation values. 153 00:11:10,300 --> 00:11:15,950 So for an atom in a magnetic field, 154 00:11:15,950 --> 00:11:20,470 our Hamiltonian is simply the same Hamiltonian, 155 00:11:20,470 --> 00:11:24,370 which involves the gyromagnetic ratio, the angular momentum 156 00:11:24,370 --> 00:11:28,120 operator, and the magnetic field. 157 00:11:28,120 --> 00:11:34,220 And you know from quantum mechanics 158 00:11:34,220 --> 00:11:47,760 that Heisenberg's equation of motion for any operator 159 00:11:47,760 --> 00:11:53,980 are simply that the time derivative of the operator 160 00:11:53,980 --> 00:11:58,410 equals the commutator with the Hamiltonian. 161 00:12:02,190 --> 00:12:05,770 In some cases, if you have an explicit time derivative 162 00:12:05,770 --> 00:12:09,230 of the operator, you have to edit, 163 00:12:09,230 --> 00:12:11,390 but we are talking here about the angular momentum 164 00:12:11,390 --> 00:12:13,510 operator which has no explicit time dependents. 165 00:12:16,180 --> 00:12:22,410 So we are interested in the operator which 166 00:12:22,410 --> 00:12:30,370 describes the magnetic moment, but the magnetic moment 167 00:12:30,370 --> 00:12:32,920 is nothing else than the gyromagnetic ratio 168 00:12:32,920 --> 00:12:38,510 times the angular momentum. 169 00:12:38,510 --> 00:12:42,880 So therefore, the relevant commutator 170 00:12:42,880 --> 00:12:51,820 is the commutator of the Hamiltonian with the angular 171 00:12:51,820 --> 00:12:54,100 momentum operator. 172 00:12:54,100 --> 00:12:57,350 Just to remind you, the Hamiltonian 173 00:12:57,350 --> 00:13:01,530 was proportional to z. 174 00:13:01,530 --> 00:13:09,310 So what we are talking about is commutators, not surprisingly, 175 00:13:09,310 --> 00:13:14,620 between the angular momentum operators 176 00:13:14,620 --> 00:13:25,180 and those commutators are just this cyclic commutation 177 00:13:25,180 --> 00:13:29,090 involving the epsilon tensor. 178 00:13:29,090 --> 00:13:36,100 So if you just put that in component by component, 179 00:13:36,100 --> 00:13:43,260 you realize immediately that the operator equation for the time 180 00:13:43,260 --> 00:13:47,930 derivative of the magnetic moment 181 00:13:47,930 --> 00:13:55,170 is nothing else than the cross product, the vector product, 182 00:13:55,170 --> 00:13:57,820 of the operator of the magnetic moment 183 00:13:57,820 --> 00:13:59,920 times the magnetic field. 184 00:14:03,640 --> 00:14:11,400 To commence, this is exact, but also it 185 00:14:11,400 --> 00:14:15,470 looks exactly like the classical result. 186 00:14:22,060 --> 00:14:28,130 Well, you would say it's an operator equation. 187 00:14:28,130 --> 00:14:31,610 Usually operator equation some of them 188 00:14:31,610 --> 00:14:34,620 are pretty useless because you can't calculate the operators, 189 00:14:34,620 --> 00:14:37,670 but in this case, we can immediately 190 00:14:37,670 --> 00:14:40,730 take the expectation value, so we 191 00:14:40,730 --> 00:14:44,170 can get some immediately meaningfully equation 192 00:14:44,170 --> 00:14:53,900 namely that the expectation value follows 193 00:14:53,900 --> 00:15:02,030 the same equation, and this tells us 194 00:15:02,030 --> 00:15:04,310 that whenever we have a quantum system, 195 00:15:04,310 --> 00:15:07,390 it has a magnetic moment. 196 00:15:07,390 --> 00:15:10,680 In an applied external magnetic field, 197 00:15:10,680 --> 00:15:15,360 the result is simply rotation, precession, and this is exact. 198 00:15:15,360 --> 00:15:17,210 It's not a classical approximation. 199 00:15:17,210 --> 00:15:19,365 It's an exact result for quantum mechanics. 200 00:15:22,890 --> 00:15:30,180 So the way how we derived it makes it obvious that 201 00:15:30,180 --> 00:15:37,710 it's an exact result, which is valid not only for spin-1/2, 202 00:15:37,710 --> 00:15:39,980 but it is valid for any spin. 203 00:15:39,980 --> 00:15:44,600 If you have a magnetic moment corresponding to a spin of 10 H 204 00:15:44,600 --> 00:15:49,420 bar, this spin follows the same equation of motion as spin-1/2. 205 00:15:55,150 --> 00:15:59,450 Of course, a special case is valid for spin-1/2, 206 00:15:59,450 --> 00:16:04,200 but spin-1/2 is isomorphous to a two-level system. 207 00:16:04,200 --> 00:16:08,140 Any two-level system can be regarded as spin-1/2 system, 208 00:16:08,140 --> 00:16:13,110 therefore, this geometric interpretation that 209 00:16:13,110 --> 00:16:17,250 the dynamics of the quantum system is just a precession 210 00:16:17,250 --> 00:16:21,115 rigorously, exactly applies to any two-level system. 211 00:16:33,560 --> 00:16:38,880 It's also valid, and this will be relevant for atoms, 212 00:16:38,880 --> 00:16:41,830 if we have composite angular momentum. 213 00:16:48,010 --> 00:16:51,500 For instance, we will encounter the total angular momentum 214 00:16:51,500 --> 00:16:55,050 F of an atom which has components 215 00:16:55,050 --> 00:16:56,970 from the orbital motion of the electron, 216 00:16:56,970 --> 00:17:00,450 the spin of the electron, and the spin of the nucleus. 217 00:17:00,450 --> 00:17:02,890 But if we have such an angular momentum, 218 00:17:02,890 --> 00:17:07,450 F, the creation of motion is it will precess around 219 00:17:07,450 --> 00:17:10,720 a magnetic field. 220 00:17:10,720 --> 00:17:26,810 Well the small print here is, unless the B field is so strong 221 00:17:26,810 --> 00:17:29,580 that it de-couples the components 222 00:17:29,580 --> 00:17:39,490 or that it breaks up the coupling of the different parts 223 00:17:39,490 --> 00:17:40,540 of the angular momentum. 224 00:17:46,020 --> 00:17:48,330 In other words, we have simply assumed here 225 00:17:48,330 --> 00:17:52,890 that magnetic moment is gamma times angular momentum, 226 00:17:52,890 --> 00:17:55,630 and that requires that the angular momentum are 227 00:17:55,630 --> 00:17:57,126 coupled in a certain way. 228 00:17:57,126 --> 00:17:58,500 If we don't fully understand what 229 00:17:58,500 --> 00:18:00,850 coupling of angular momentum is, we really 230 00:18:00,850 --> 00:18:03,480 talk about that when we talk about atomic structure. 231 00:18:03,480 --> 00:18:07,520 So as long as the spin state coupled to one total spin, 232 00:18:07,520 --> 00:18:09,480 this total spin will just precess. 233 00:18:14,920 --> 00:18:19,340 This picture of precession will also 234 00:18:19,340 --> 00:18:37,590 be valid for a system of N two-level systems 235 00:18:37,590 --> 00:18:44,200 coupled to an external field, and this 236 00:18:44,200 --> 00:18:52,550 will be the example of Dicke superrradiance, which 237 00:18:52,550 --> 00:18:54,830 we will discussed towards the end of the course. 238 00:19:05,500 --> 00:19:08,260 So very simple result, but very powerful, 239 00:19:08,260 --> 00:19:10,540 and this is your permission whenever 240 00:19:10,540 --> 00:19:13,530 you encounter any of the systems to see 241 00:19:13,530 --> 00:19:15,700 a vector precessing in your head. 242 00:19:15,700 --> 00:19:16,490 This is exact. 243 00:19:28,690 --> 00:19:31,025 So we've talked about Heisenberg equation of motion 244 00:19:31,025 --> 00:19:33,150 for general spin, but-- you have a question, Nancy? 245 00:19:33,150 --> 00:19:36,935 AUDIENCE: What did you mean by N two-level systems here? 246 00:19:36,935 --> 00:19:39,350 Are we talking about coherent systems 247 00:19:39,350 --> 00:19:42,731 or non-coherent systems? 248 00:19:42,731 --> 00:19:47,350 PROFESSOR: We talk about in two-level systems 249 00:19:47,350 --> 00:19:52,100 and to be more specific the coupling comes because they all 250 00:19:52,100 --> 00:19:54,320 talk to the same magnetic field. 251 00:19:54,320 --> 00:19:58,270 So we have in two-level systems connected 252 00:19:58,270 --> 00:20:01,710 to the modes of the electromagnetic field. 253 00:20:01,710 --> 00:20:03,440 We start with the symmetric state. 254 00:20:03,440 --> 00:20:05,250 The coupling is symmetric and that 255 00:20:05,250 --> 00:20:09,310 preserves the symmetry of the atomic state. 256 00:20:09,310 --> 00:20:14,920 In other words, we will have a situation where the angular 257 00:20:14,920 --> 00:20:19,110 momentum is the maximum angular momentum we 258 00:20:19,110 --> 00:20:22,420 can get in two-level system, and the dynamics 259 00:20:22,420 --> 00:20:25,330 of this two-level system, the description of Dicke 260 00:20:25,330 --> 00:20:29,410 superradiance has the geometric visualization 261 00:20:29,410 --> 00:20:30,720 of this precessing motion. 262 00:20:33,510 --> 00:20:35,880 I know I'm not explaining it exactly. 263 00:20:35,880 --> 00:20:39,230 I want to sort of whet your appetite for what comes later 264 00:20:39,230 --> 00:20:42,839 and also sort of prep you that some of the simple pictures 265 00:20:42,839 --> 00:20:44,380 will really carry through the course. 266 00:20:48,180 --> 00:20:51,670 OK, so this is for very general spin. 267 00:20:51,670 --> 00:20:56,810 Let's now talk about features of, yes, 268 00:20:56,810 --> 00:21:01,370 the most important spin for us, namely the two-level system, 269 00:21:01,370 --> 00:21:02,680 which is spin-1/2. 270 00:21:16,440 --> 00:21:27,530 Well, the most generic system is an electron 271 00:21:27,530 --> 00:21:38,170 in a magnetic field B times ez. 272 00:21:41,060 --> 00:21:46,465 And well, why don't we start with a clicker question. 273 00:21:51,940 --> 00:21:54,540 So the question is, what is the level structure 274 00:21:54,540 --> 00:21:57,610 of the electron in the magnetic field? 275 00:21:57,610 --> 00:22:01,970 It's a two-level system, and you have two options 276 00:22:01,970 --> 00:22:06,940 A and B. One option is the upper state is spin up. 277 00:22:06,940 --> 00:22:17,740 The ground state is spin down or the opposite. 278 00:22:17,740 --> 00:22:24,800 So in other words, and tell me whether an electron 279 00:22:24,800 --> 00:22:27,620 is in the lower state when the speed is up 280 00:22:27,620 --> 00:22:29,250 or when the spin is down. 281 00:22:38,560 --> 00:22:40,440 You would say it's a stupid definition, 282 00:22:40,440 --> 00:22:42,730 but we talk all the time about an electron 283 00:22:42,730 --> 00:22:44,960 is in the spin down state or the spin 284 00:22:44,960 --> 00:22:47,900 up state, which is the lower energy state of the electron. 285 00:22:52,390 --> 00:22:54,720 So I think by exchanging the better recent number 286 00:22:54,720 --> 00:22:57,730 of responses has considerably gone up. 287 00:23:04,291 --> 00:23:04,790 OK. 288 00:23:10,030 --> 00:23:14,770 The answer is the electron is in the ground state. 289 00:23:14,770 --> 00:23:24,190 Spin down is the lowest state for the electron. 290 00:23:28,340 --> 00:23:31,890 To say it in words, of course, a compass needle 291 00:23:31,890 --> 00:23:34,430 wants to be aligned with the magnetic field. 292 00:23:34,430 --> 00:23:41,450 So you want the magnetic moment to be aligned 293 00:23:41,450 --> 00:23:43,330 with the magnetic field, and that 294 00:23:43,330 --> 00:23:45,340 means the magnetic moment has to point 295 00:23:45,340 --> 00:23:49,300 in the plus e direction, which up but the electron has 296 00:23:49,300 --> 00:23:52,500 negative charge, the gamma factor is negative, 297 00:23:52,500 --> 00:23:56,170 and that's why for the electron, the vector of the spin 298 00:23:56,170 --> 00:24:00,360 and the vector of the magnetic moment are opposite. 299 00:24:00,360 --> 00:24:02,700 So just try to find some main-mode technical thing. 300 00:24:02,700 --> 00:24:04,380 The electron lives in the basement. 301 00:24:04,380 --> 00:24:06,630 It wants to be down, spin down. 302 00:24:06,630 --> 00:24:10,170 This is the lowest state for the electron. 303 00:24:10,170 --> 00:24:13,270 However, if you have a system which 304 00:24:13,270 --> 00:24:16,500 has a positive gyromagnetic ratio, which would correspond 305 00:24:16,500 --> 00:24:26,820 to, well, positive charge, nucleus if it has somewhat 306 00:24:26,820 --> 00:24:31,590 normal magnetic moment, then in that case, the spin up state 307 00:24:31,590 --> 00:24:33,870 is less energetic than the spin down state. 308 00:24:38,440 --> 00:24:39,920 So let me just write that down. 309 00:24:39,920 --> 00:24:45,200 So the correct answer is this one, 310 00:24:45,200 --> 00:24:50,470 and it involves that gamma is negative. 311 00:24:50,470 --> 00:24:52,580 The gamma is a gyromagnetic ratio, 312 00:24:52,580 --> 00:24:55,270 the ratio between magnetic moment and angular momentum, 313 00:24:55,270 --> 00:24:57,065 and for negative charges, it's negative. 314 00:25:01,280 --> 00:25:04,400 For positive gamma, the situation is inverted. 315 00:25:07,770 --> 00:25:10,850 So let's just use for a moment the result 316 00:25:10,850 --> 00:25:13,860 we got from Heisenberg's equation of motion. 317 00:25:13,860 --> 00:25:16,150 We know the classical result. 318 00:25:18,960 --> 00:25:21,710 We have already derived for the classic spin, 319 00:25:21,710 --> 00:25:24,270 the classical result for the expectation 320 00:25:24,270 --> 00:25:25,620 value of the magnetic moment. 321 00:25:28,540 --> 00:25:31,420 But now I want to sort of relate it to something quantum 322 00:25:31,420 --> 00:25:34,920 mechanical because we know that the classical solution equals 323 00:25:34,920 --> 00:25:37,660 the quantum mechanical solution. 324 00:25:37,660 --> 00:25:45,330 So if you have a two-level system, 325 00:25:45,330 --> 00:25:48,900 the z component of the magnetic moment 326 00:25:48,900 --> 00:25:57,450 is the difference between spin down and spin up. 327 00:26:01,750 --> 00:26:04,860 And because of conservation probability, p up and p down 328 00:26:04,860 --> 00:26:11,180 is unity, we can also write that as 2 times-- 329 00:26:11,180 --> 00:26:13,880 let me now introduce e for excited state, 330 00:26:13,880 --> 00:26:16,750 just I know it's hard to keep track of spin up, spin down. 331 00:26:16,750 --> 00:26:19,450 I want to make sure that I mean now the excited state, 332 00:26:19,450 --> 00:26:23,850 so the excited state for the electron is spin up. 333 00:26:23,850 --> 00:26:41,550 So we have this condition, so therefore, the excited state 334 00:26:41,550 --> 00:26:48,070 fraction of a two-level system is related to the expectation 335 00:26:48,070 --> 00:26:53,150 value of the magnetic moment in that wave, 336 00:26:53,150 --> 00:26:56,320 and now we want to use the classical result we derived. 337 00:27:01,540 --> 00:27:05,550 We derived the classical result when 338 00:27:05,550 --> 00:27:21,170 for t equals 0 all the spins were in the ground state, 339 00:27:21,170 --> 00:27:26,150 and by using the result we derived previously, 340 00:27:26,150 --> 00:27:28,930 we have the 1/2 from the previous line 341 00:27:28,930 --> 00:27:33,400 and then the magnetic moment, mu z. 342 00:27:33,400 --> 00:27:41,510 We found an expression which involved the Rabi frequency, 343 00:27:41,510 --> 00:27:55,108 and the off resonant Rabi frequency 344 00:27:55,108 --> 00:28:02,290 are sine square generalized Rabi frequency times time over 2. 345 00:28:05,380 --> 00:28:10,200 So the two factors of 1/2 and 1/2 346 00:28:10,200 --> 00:28:17,230 cancel, and what we find now for the quantum mechanical system 347 00:28:17,230 --> 00:28:19,930 using Heisenberg's equation of motion 348 00:28:19,930 --> 00:28:23,560 is that if you prepare a system initially 349 00:28:23,560 --> 00:28:27,290 in the ground state, the excited state 350 00:28:27,290 --> 00:28:33,460 probability, the fraction, the excited state oscillates 351 00:28:33,460 --> 00:28:37,300 with a Rabi frequency, and this is, 352 00:28:37,300 --> 00:28:41,660 I think, the second time in this course and not the last time 353 00:28:41,660 --> 00:28:44,150 that we see the Rabi, that we obtain, 354 00:28:44,150 --> 00:28:45,620 the Rabi transition probability. 355 00:28:55,350 --> 00:28:56,300 But let's go further. 356 00:28:56,300 --> 00:28:58,400 We have now discussed the classical spin. 357 00:28:58,400 --> 00:29:01,540 We have sort of done classical quantum correspondence 358 00:29:01,540 --> 00:29:03,730 with Heisenberg's equation of motion. 359 00:29:03,730 --> 00:29:09,490 We know that this in general, it implies Rabi oscillations, 360 00:29:09,490 --> 00:29:15,200 but now we want to go deeper into the quantum domain 361 00:29:15,200 --> 00:29:19,440 by talking about the spin-1/2 Hamiltonian. 362 00:29:25,760 --> 00:29:29,880 So in other words, we want to go beyond expectation values. 363 00:29:34,480 --> 00:29:37,490 We want to talk about the wave function itself. 364 00:29:41,350 --> 00:29:49,200 So the Hamiltonian, which we will 365 00:29:49,200 --> 00:29:53,250 use for major parts in this course, 366 00:29:53,250 --> 00:29:56,519 it's one of the fundamental Hamiltonians in physics. 367 00:29:56,519 --> 00:29:58,310 Of course, there's the harmonic oscillator. 368 00:29:58,310 --> 00:30:00,630 There is a hydrogen atom, but then there's 369 00:30:00,630 --> 00:30:06,000 this Hamiltonian, which is a two-level system with splitting 370 00:30:06,000 --> 00:30:07,790 omega naught. 371 00:30:07,790 --> 00:30:17,170 And then we make, which is often a simplification, 372 00:30:17,170 --> 00:30:26,270 where we use a pure exponential, so a single frequency 373 00:30:26,270 --> 00:30:30,890 in complex notation where the complex exponential e 374 00:30:30,890 --> 00:30:34,170 to the i omega t is the drive term. 375 00:30:37,610 --> 00:30:40,100 So this is one of the simplest Hamiltonian 376 00:30:40,100 --> 00:30:41,230 for this kind of system. 377 00:30:41,230 --> 00:30:46,130 It's a two-level system with a splitting, and now it is driven 378 00:30:46,130 --> 00:30:49,780 and the simplest drive term is not cosine omega t or sine 379 00:30:49,780 --> 00:30:51,810 omega t as we will see. 380 00:30:51,810 --> 00:30:55,430 The simplest drive term is e to the i omega t. 381 00:30:55,430 --> 00:30:58,870 So this is now our Hamiltonian. 382 00:30:58,870 --> 00:31:02,740 and since it is so important, let 383 00:31:02,740 --> 00:31:14,580 me ask you a clicker question whether this Hamiltonian can 384 00:31:14,580 --> 00:31:32,775 be exactly realized in nature or it is an approximation. 385 00:31:44,710 --> 00:31:51,550 For instance, that you always have cosine omega t as a drive, 386 00:31:51,550 --> 00:31:53,880 and cosine omega t is e to the plus i omega t 387 00:31:53,880 --> 00:31:55,410 and e to the minus i omega t. 388 00:31:55,410 --> 00:31:57,700 And then maybe with the rotating wave approximation, 389 00:31:57,700 --> 00:31:59,430 you throw away a term. 390 00:31:59,430 --> 00:32:01,500 So I sort of wanted to ask you, is this 391 00:32:01,500 --> 00:32:05,620 an idealization that we have a coupling which 392 00:32:05,620 --> 00:32:10,240 is a simple, complex exponential, this nature always 393 00:32:10,240 --> 00:32:15,600 more complicated or is there a simple way 394 00:32:15,600 --> 00:32:17,470 to realize this Hamiltonian in nature? 395 00:32:20,420 --> 00:32:21,850 So what do you think? 396 00:32:36,510 --> 00:32:37,980 Stop display. 397 00:32:44,080 --> 00:32:45,470 What's funny about that? 398 00:32:45,470 --> 00:32:46,853 AUDIENCE: Equal to. 399 00:32:46,853 --> 00:32:48,700 AUDIENCE: There are two [INAUDIBLE]. 400 00:32:48,700 --> 00:32:50,700 PROFESSOR: Oh, they couldn't make up their mind. 401 00:32:54,140 --> 00:32:57,660 The system should reject those votes. 402 00:32:57,660 --> 00:33:02,290 Anyway, it means most of you anticipate what I want 403 00:33:02,290 --> 00:33:05,950 to derive to you that we can actually exactly get this 404 00:33:05,950 --> 00:33:11,210 Hamiltonian and indeed, this is the Hamiltonian I will derive 405 00:33:11,210 --> 00:33:14,510 for you in the next few minutes, which is the Hamiltonian 406 00:33:14,510 --> 00:33:21,960 of spin-1/2 in a magnetic field coupled to a rotating magnetic 407 00:33:21,960 --> 00:33:23,720 field. 408 00:33:23,720 --> 00:33:26,210 So we start out with fields which 409 00:33:26,210 --> 00:33:29,700 are real, real field, no imaginary numbers, 410 00:33:29,700 --> 00:33:30,940 no complex numbers. 411 00:33:30,940 --> 00:33:34,410 These are real fields, but when we write down how the real 412 00:33:34,410 --> 00:33:39,706 fields couple to spin-1/2, we get e to the i omega t and e 413 00:33:39,706 --> 00:33:42,025 to the minus omega t without any approximation. 414 00:33:46,330 --> 00:33:52,430 So since at least 80% of you know the result, 415 00:33:52,430 --> 00:33:58,770 just regard it as an exercise to introduce how we spin matrices 416 00:33:58,770 --> 00:34:02,850 in a nice way, also this will help you 417 00:34:02,850 --> 00:34:06,770 how to do pre-set number 1. 418 00:34:06,770 --> 00:34:12,750 So the Hamiltonian is the spin coupled to a magnetic field, 419 00:34:12,750 --> 00:34:18,130 and if we express that by angular momentum operators, 420 00:34:18,130 --> 00:34:20,949 the gyromagnetic ratio it involves the operator 421 00:34:20,949 --> 00:34:22,603 for the spin in the z direction. 422 00:34:25,170 --> 00:34:30,920 So the two-level system has a splitting of H bar omega, 423 00:34:30,920 --> 00:34:35,260 so it's plus 1/2 minus 1/2 H bar omega, 424 00:34:35,260 --> 00:34:37,540 and that means the diagonal part, 425 00:34:37,540 --> 00:34:47,969 the non-driven part is simply given by the Pauli spin matrix 426 00:34:47,969 --> 00:34:55,719 sigma z and omega naught is the energy splitting, 427 00:34:55,719 --> 00:35:00,220 which is proportional to the applied magnetic field. 428 00:35:00,220 --> 00:35:04,580 And up-down, and excited and ground 429 00:35:04,580 --> 00:35:15,280 are the eigenstates of this Hamiltonian with energies 430 00:35:15,280 --> 00:35:19,290 plus minus H bar omega naught over 2. 431 00:35:22,150 --> 00:35:31,450 So this is the same Hamiltonian, but now add a real rotating B 432 00:35:31,450 --> 00:35:35,520 field B1. 433 00:35:40,510 --> 00:35:46,950 So the drive Hamiltonian H1 is the same magnetic moment 434 00:35:46,950 --> 00:35:49,935 but now coupled to a time-dependent rotating field. 435 00:35:52,770 --> 00:35:54,440 The amplitude of the rotating field 436 00:35:54,440 --> 00:35:58,670 is the Rabi frequency divided by gamma, 437 00:35:58,670 --> 00:36:09,970 and we assume that the field is rotating in the xy plane. 438 00:36:09,970 --> 00:36:18,960 So it's ex ey cosine omega t sine omega t. 439 00:36:18,960 --> 00:36:22,066 And now I put in two minus signs here for convenience. 440 00:36:26,860 --> 00:36:31,180 If you want, I've just shifted-- it's just a definition. 441 00:36:31,180 --> 00:36:34,250 I've changed the definition of the amplitude by a minus sign. 442 00:36:41,130 --> 00:36:53,900 So this is the Rabi frequency, and the magnetic moment 443 00:36:53,900 --> 00:36:59,040 divided by gamma is nothing than the spin. 444 00:36:59,040 --> 00:37:03,440 Magnetic moment is gamma times the spin, 445 00:37:03,440 --> 00:37:19,220 so therefore, we are back to the spin operators 446 00:37:19,220 --> 00:37:27,210 and the spin operators, if I factor out 1/2 H bar, 447 00:37:27,210 --> 00:37:33,325 are now the Pauli spin matrices sigma x and sigma y. 448 00:37:40,720 --> 00:37:48,970 And if you look at those spin matrices, 449 00:37:48,970 --> 00:37:54,500 then you'll realize that we go complex in our Hamiltonian, 450 00:37:54,500 --> 00:37:58,930 not because we have approximated a real field cosine omega 451 00:37:58,930 --> 00:38:03,290 t by some e to the i omega t, but because when 452 00:38:03,290 --> 00:38:06,950 we have a rotating field and we write down it in Pauli spin 453 00:38:06,950 --> 00:38:12,035 matrices, we get imaginary units form the sigma y spin matrix. 454 00:38:17,800 --> 00:38:24,060 So that means we have now for this system 455 00:38:24,060 --> 00:38:28,940 rewritten the coupling term took you to the rotating field H1 456 00:38:28,940 --> 00:38:38,720 as 0 0 e to the plus i omega t e to the minus i omega t. 457 00:38:38,720 --> 00:38:48,720 And therefore, the Hamiltonian is 458 00:38:48,720 --> 00:38:54,800 the famous two-level Hamiltonian with omega naught 459 00:38:54,800 --> 00:39:02,190 and the Rabi frequency, which I wrote down 460 00:39:02,190 --> 00:39:07,660 at the beginning of this chapter. 461 00:39:12,270 --> 00:39:21,030 So we'll leave that here, but we will use it even more in 8.421. 462 00:39:21,030 --> 00:39:25,810 This is the famous dressed atom Hamiltonian. 463 00:39:29,016 --> 00:39:32,590 It is the starting point to calculate eigenstates 464 00:39:32,590 --> 00:39:36,960 and eigenvalues, not just in perturbation theory. 465 00:39:36,960 --> 00:39:43,550 You can go to oscillating fields at arbitrary strengths, 466 00:39:43,550 --> 00:39:47,510 so you can solve exactly in the dressed atom picture using 467 00:39:47,510 --> 00:39:51,480 this Hamiltonian, the problem of a two-level system 468 00:39:51,480 --> 00:39:54,530 plus one mode of the electromagnetic field 469 00:39:54,530 --> 00:39:56,430 no matter what the drive term, what 470 00:39:56,430 --> 00:39:58,620 the strengths of the electromagnetic field 471 00:39:58,620 --> 00:39:59,610 in the spin mode is. 472 00:40:04,060 --> 00:40:11,460 So this describes the two-level system 473 00:40:11,460 --> 00:40:23,300 plus one mode of the electromagnetic field 474 00:40:23,300 --> 00:40:24,870 with arbitrary strengths. 475 00:40:34,210 --> 00:40:39,350 And as I said, we talk about some things here 476 00:40:39,350 --> 00:40:50,500 but others I explored in 8.422. 477 00:40:50,500 --> 00:40:53,220 Questions? 478 00:40:53,220 --> 00:40:53,855 Yes, Will. 479 00:40:53,855 --> 00:40:57,990 AUDIENCE: We refer to the eigenstates and eigenenergies 480 00:40:57,990 --> 00:41:01,410 of this Hamiltonian as dressed in states in the same way 481 00:41:01,410 --> 00:41:04,924 as we refer to address states in a fully quantized fiction? 482 00:41:04,924 --> 00:41:10,492 Do we refer to both cases as dressed states? 483 00:41:10,492 --> 00:41:12,360 PROFESSOR: Yes. 484 00:41:12,360 --> 00:41:13,460 That's a good question. 485 00:41:13,460 --> 00:41:16,260 The question is now, what are the dressed states, and Will, 486 00:41:16,260 --> 00:41:18,066 I think you are referring that there 487 00:41:18,066 --> 00:41:23,480 are two ways to talk about the coupling of a two-level system 488 00:41:23,480 --> 00:41:25,570 to one mode of the electromagnetic field. 489 00:41:25,570 --> 00:41:28,100 It is this same classical picture 490 00:41:28,100 --> 00:41:33,220 where we introduce and let me say an analog amplitude 491 00:41:33,220 --> 00:41:35,730 of the electromagnetic field which drives it. 492 00:41:35,730 --> 00:41:38,190 And then there is a fully quantized picture 493 00:41:38,190 --> 00:41:40,930 where you first quantize electromagnetic field 494 00:41:40,930 --> 00:41:44,920 and you couple to photon number states. 495 00:41:44,920 --> 00:41:48,110 Actually the beauty of it that the two solutions 496 00:41:48,110 --> 00:41:51,060 are exactly the same. 497 00:41:51,060 --> 00:42:01,490 So in other words, how to say, if you couple an atom 498 00:42:01,490 --> 00:42:04,490 to one mode of the electromagnetic field. 499 00:42:04,490 --> 00:42:06,780 We have two ways how we can solve it. 500 00:42:06,780 --> 00:42:13,860 One is, we introduce a coherent electromagnetic field, 501 00:42:13,860 --> 00:42:17,940 and there is an exact unitary transformation which tells us 502 00:42:17,940 --> 00:42:21,130 if we have the quantized field in a coherent state, 503 00:42:21,130 --> 00:42:23,540 we can do unitary transformation, 504 00:42:23,540 --> 00:42:26,490 and what we get is exactly this Hamiltonian. 505 00:42:26,490 --> 00:42:29,690 So therefore, this is also-- you may not recognize it-- 506 00:42:29,690 --> 00:42:32,740 this is actually the quantum description 507 00:42:32,740 --> 00:42:37,780 of the electromagnetic field when it is in a coherent state. 508 00:42:37,780 --> 00:42:41,830 The other option is, we use the dressed atom picture maybe 509 00:42:41,830 --> 00:42:45,800 following some work of [INAUDIBLE] and others 510 00:42:45,800 --> 00:42:49,320 where we assume this single mode of electromagnetic field 511 00:42:49,320 --> 00:42:53,665 has in photons, and then we solve it 512 00:42:53,665 --> 00:42:57,450 for this photon number state. 513 00:42:57,450 --> 00:42:59,860 So in other words, these are the two ways 514 00:42:59,860 --> 00:43:02,450 how we can relatively, easily treat the problem. 515 00:43:02,450 --> 00:43:05,970 Either we assume the quantum field is in a coherence state 516 00:43:05,970 --> 00:43:08,460 or it's in a flux state. 517 00:43:08,460 --> 00:43:12,090 But since the dressed atom picture in the standard way 518 00:43:12,090 --> 00:43:16,270 assumes that the photon number, N, is large, 519 00:43:16,270 --> 00:43:19,330 there is a correspondence that in the limit of N of N 520 00:43:19,330 --> 00:43:21,840 being large, the flux state description 521 00:43:21,840 --> 00:43:24,280 and the coherent state description fully agree. 522 00:43:28,320 --> 00:43:30,830 And you pick what you want. 523 00:43:30,830 --> 00:43:33,240 If you introduce the electromagnetic field 524 00:43:33,240 --> 00:43:35,220 explicitly with it's quantum state, 525 00:43:35,220 --> 00:43:38,460 you get the dressed atom picture as a solution of a time 526 00:43:38,460 --> 00:43:41,550 independent problem, whereas here with a coherent state 527 00:43:41,550 --> 00:43:45,380 description, the coherent state oscillates, cosine omega t, 528 00:43:45,380 --> 00:43:47,890 with a time-dependent problem. 529 00:43:47,890 --> 00:43:49,570 And actually I should say whenever 530 00:43:49,570 --> 00:43:52,520 I get confused in one picture, I look in the other picture 531 00:43:52,520 --> 00:43:55,410 and it becomes clear. 532 00:43:55,410 --> 00:43:59,510 I generally prefer where we have N photons, 533 00:43:59,510 --> 00:44:01,830 it's because we can discuss everything 534 00:44:01,830 --> 00:44:06,280 in a time-independent way, but for certain intuitive aspects, 535 00:44:06,280 --> 00:44:10,010 this is also variable, so in the end, you have to learn both. 536 00:44:10,010 --> 00:44:12,334 And in your homework, you will actually 537 00:44:12,334 --> 00:44:14,500 write down the general solution for this Hamiltonian 538 00:44:14,500 --> 00:44:17,210 as an exercise. 539 00:44:17,210 --> 00:44:17,710 Nancy. 540 00:44:17,710 --> 00:44:21,370 AUDIENCE: I think I'm confused a little bit. 541 00:44:21,370 --> 00:44:24,885 So in the flux state picture, the dressed states 542 00:44:24,885 --> 00:44:30,115 can be exactly part of an independent matter of coupling 543 00:44:30,115 --> 00:44:33,490 between a lesser photon on any excited state. 544 00:44:33,490 --> 00:44:38,627 So like we can write eN as 1 and g or something like that. 545 00:44:38,627 --> 00:44:39,670 PROFESSOR: Yeah. 546 00:44:39,670 --> 00:44:42,950 You couple a photon field with N photons and energy 547 00:44:42,950 --> 00:44:46,390 in H bar omega to N minus 1 H bar omega. 548 00:44:46,390 --> 00:44:51,890 AUDIENCE: But in this one, is there like a direct photon 549 00:44:51,890 --> 00:44:54,390 number thing, because we haven't quantized the field yet? 550 00:44:54,390 --> 00:44:59,890 So what do the dressed states mean at this point? 551 00:44:59,890 --> 00:45:06,320 PROFESSOR: Well, the fact is that if you start out 552 00:45:06,320 --> 00:45:10,470 with a coherent state, your photon field is not in photons, 553 00:45:10,470 --> 00:45:12,600 it's a laser beam. 554 00:45:12,600 --> 00:45:15,380 The laser beam or the coherent state 555 00:45:15,380 --> 00:45:18,970 is in a quantized description, a superposition of many flux 556 00:45:18,970 --> 00:45:20,040 states. 557 00:45:20,040 --> 00:45:23,850 So therefore, the number of photons in a coherent state 558 00:45:23,850 --> 00:45:27,530 fluctuates or has a large Plutonian statistics, 559 00:45:27,530 --> 00:45:30,770 and if you take one photon out or not, 560 00:45:30,770 --> 00:45:32,260 it doesn't make a big difference. 561 00:45:32,260 --> 00:45:34,150 For instance, for those of you who 562 00:45:34,150 --> 00:45:38,450 know how the coherent state is state defined, 563 00:45:38,450 --> 00:45:42,100 the coherent state is defined as when 564 00:45:42,100 --> 00:45:45,630 you act on the coherent state with an annihilation operator, 565 00:45:45,630 --> 00:45:48,730 you get the eigenvalue times the coherent state. 566 00:45:48,730 --> 00:45:51,350 So that tells you you have a fully-quantized description 567 00:45:51,350 --> 00:45:54,100 of your laser in terms of a coherent state. 568 00:45:54,100 --> 00:45:56,530 You take one photon out and what you get? 569 00:45:56,530 --> 00:45:58,100 the same state back. 570 00:45:58,100 --> 00:46:01,910 And this may immediately justify that what we write down here 571 00:46:01,910 --> 00:46:05,295 is simply the coherent state with its amplitude 572 00:46:05,295 --> 00:46:06,920 and the amplitude of the coherent state 573 00:46:06,920 --> 00:46:10,420 would be B1, the amplitude of the drive field. 574 00:46:10,420 --> 00:46:12,450 And we don't really need other states 575 00:46:12,450 --> 00:46:14,360 because a coherent state has the property. 576 00:46:14,360 --> 00:46:17,462 You to take a photon out and you still have the same state. 577 00:46:17,462 --> 00:46:18,920 So therefore, we don't have to keep 578 00:46:18,920 --> 00:46:20,410 track of the coherent state. 579 00:46:20,410 --> 00:46:23,500 It's there all the time. 580 00:46:23,500 --> 00:46:25,580 But what I'm saying can be formulated 581 00:46:25,580 --> 00:46:28,949 more exactly when we use the appropriate formulas. 582 00:46:28,949 --> 00:46:30,240 But this is sort of the bridge. 583 00:46:30,240 --> 00:46:33,570 That's why we do not have to keep track of the photon state. 584 00:46:33,570 --> 00:46:38,754 It's because the coherent state has those wonderful properties. 585 00:46:38,754 --> 00:46:39,420 Other questions? 586 00:46:44,710 --> 00:46:47,149 OK. 587 00:46:47,149 --> 00:46:48,565 So this is the famous Hamiltonian. 588 00:46:52,420 --> 00:46:55,450 And of course, if it's the famous Hamiltonian, 589 00:46:55,450 --> 00:46:58,070 we want to solve it. 590 00:47:02,640 --> 00:47:13,520 As I said, the general solution is left to the homework, 591 00:47:13,520 --> 00:47:16,710 but I want to sort of show you parts 592 00:47:16,710 --> 00:47:19,190 of the solution to tell a story. 593 00:47:19,190 --> 00:47:22,550 And the question is, well, how do we solve this Hamiltonian? 594 00:47:25,100 --> 00:47:27,690 The answer is, we do exactly what we 595 00:47:27,690 --> 00:47:29,670 did in the classical problem. 596 00:47:29,670 --> 00:47:33,000 We transform to the rotating frame. 597 00:47:38,090 --> 00:47:41,960 In other words, this Hamiltonian is best 598 00:47:41,960 --> 00:47:48,150 solved by doing-- you can actually solve it directly. 599 00:47:48,150 --> 00:47:51,040 You can just put in a tri wave function and solve it. 600 00:47:51,040 --> 00:47:55,360 But I want to sort of bring out the big idea here 601 00:47:55,360 --> 00:47:57,920 which is analogous to what we have done 602 00:47:57,920 --> 00:47:59,950 in the last few classes, namely we 603 00:47:59,950 --> 00:48:03,180 have involved rotating frames. 604 00:48:03,180 --> 00:48:09,570 So what solves this Hamiltonian is a unitary transformation, 605 00:48:09,570 --> 00:48:17,040 and the unitary transformation is this one. 606 00:48:27,120 --> 00:48:32,402 And so this unitary transformation, 607 00:48:32,402 --> 00:48:37,710 let me first write it down, it transforms the Hamiltonian 608 00:48:37,710 --> 00:48:44,250 to the time independent one. 609 00:48:44,250 --> 00:48:48,010 We have now time independent of diagonal matrix elements. 610 00:48:52,260 --> 00:48:56,100 Our diagonal matrix element have changed. 611 00:48:56,100 --> 00:49:02,030 Delta is now the detuning of the tri frequency from the energy 612 00:49:02,030 --> 00:49:04,440 splitting of the two-level system. 613 00:49:04,440 --> 00:49:07,340 In particular, when we are on resonance, 614 00:49:07,340 --> 00:49:09,880 the diagonal matrix elements have disappeared. 615 00:49:09,880 --> 00:49:21,460 This is the result of the unitary transformation, 616 00:49:21,460 --> 00:49:30,240 and let me just show you this transformation over here. 617 00:49:30,240 --> 00:49:37,730 Can be actually written as an operator involving 618 00:49:37,730 --> 00:49:42,620 the z component of the magnetic field. 619 00:49:42,620 --> 00:49:45,770 And what I just wrote down for you is actually 620 00:49:45,770 --> 00:49:48,930 the quantum mechanical operator, the rotation 621 00:49:48,930 --> 00:49:57,295 operator for performing at rotation around the z-axis. 622 00:50:00,300 --> 00:50:06,070 So by selecting the rotation angle to be omega t, 623 00:50:06,070 --> 00:50:10,290 that's how I can generate the unitary transformation, 624 00:50:10,290 --> 00:50:16,201 and this unitary transformation makes the Hamiltonian time 625 00:50:16,201 --> 00:50:16,700 independent. 626 00:50:23,180 --> 00:50:26,510 So in other words, everything is in the classical system. 627 00:50:26,510 --> 00:50:29,262 We just go to a frame which rotates with a [INAUDIBLE], 628 00:50:29,262 --> 00:50:30,970 and we find the time-independent problem. 629 00:50:38,840 --> 00:50:46,470 So now this Hamiltonian can be easily solved. 630 00:50:55,470 --> 00:51:00,020 And you will find as a special case 631 00:51:00,020 --> 00:51:02,750 when you start with an amplitude, 632 00:51:02,750 --> 00:51:05,510 initially you start in the ground state, 633 00:51:05,510 --> 00:51:10,580 then the excited state amplitude square 634 00:51:10,580 --> 00:51:17,750 is the Rabi oscillation, something 635 00:51:17,750 --> 00:51:32,140 we discussed 40 minutes ago, but before, we 636 00:51:32,140 --> 00:51:33,810 got it from the classical quantum 637 00:51:33,810 --> 00:51:36,430 mechanical correspondence using the Heisenberg equation 638 00:51:36,430 --> 00:51:41,600 of motion and here it comes out by explicitly solving 639 00:51:41,600 --> 00:51:44,518 for the wave function for the dressed Hamiltonian. 640 00:51:56,021 --> 00:51:56,520 Questions? 641 00:52:13,550 --> 00:52:26,370 I want to say a few words now about rapid adiabatic passage, 642 00:52:26,370 --> 00:52:31,700 but this time by emphasizing the quantum mechanical aspects. 643 00:52:36,910 --> 00:52:42,079 In other words, we have a clear understanding 644 00:52:42,079 --> 00:52:43,120 what happens classically. 645 00:52:43,120 --> 00:52:45,070 We have a clear understanding what 646 00:52:45,070 --> 00:52:48,070 happens in the adiabatic limit, but I just 647 00:52:48,070 --> 00:52:50,670 want to sort of in the next 10 minutes 648 00:52:50,670 --> 00:52:52,870 use what we have already learned, 649 00:52:52,870 --> 00:52:56,210 combine it with the quantum mechanical Hamiltonian 650 00:52:56,210 --> 00:53:00,950 and tell you that, well, when you are not fully adiabatic, 651 00:53:00,950 --> 00:53:04,130 you actually have transition probabilities between the two 652 00:53:04,130 --> 00:53:05,160 states. 653 00:53:05,160 --> 00:53:11,530 So I want to sort of bring in the concept of transition 654 00:53:11,530 --> 00:53:19,040 probabilities to the case of-- what 655 00:53:19,040 --> 00:53:21,362 I want to say is rapid adiabatic passage when it's 656 00:53:21,362 --> 00:53:23,820 no longer adiabatic, but what this just means when we sweep 657 00:53:23,820 --> 00:53:28,080 the frequency and we're not in the adiabatic limit. 658 00:53:28,080 --> 00:53:35,400 So how do we describe it quantum mechanically? 659 00:53:35,400 --> 00:53:46,076 We start out with a Hamiltonian, which has-- we use our rotating 660 00:53:46,076 --> 00:53:49,510 framework for convenience that allows to write down 661 00:53:49,510 --> 00:53:56,330 exactly the same Hamiltonian in time-independent picture. 662 00:53:56,330 --> 00:54:03,680 So the Hamiltonian has two parts, a diagonal part 663 00:54:03,680 --> 00:54:09,330 and an off-diagonal part. 664 00:54:14,350 --> 00:54:24,930 So if I show the energy as a function of detuning delta-- 665 00:54:24,930 --> 00:54:29,095 well, maybe I should times 2 over H bar, 666 00:54:29,095 --> 00:54:35,190 just normalize it so then it becomes just the straight line 667 00:54:35,190 --> 00:54:38,150 at 45 degree y equals x. 668 00:54:38,150 --> 00:54:48,530 So the unperturbed-- the Hamiltonian without drive 669 00:54:48,530 --> 00:54:52,455 has a level crossing at detuning 0. 670 00:54:55,770 --> 00:54:58,808 Then we add to it the drive term. 671 00:55:05,640 --> 00:55:09,140 Well, let me just write down, not 672 00:55:09,140 --> 00:55:15,950 the drive term but the full Hamiltonian. 673 00:55:15,950 --> 00:55:20,280 So the full Hamiltonian with the addition of the drive term 674 00:55:20,280 --> 00:55:23,530 has delta minus delta and now it has 675 00:55:23,530 --> 00:55:27,370 the coupling with the Rabi frequency. 676 00:55:27,370 --> 00:55:31,660 That means that on resonance, the degeneracy between the two 677 00:55:31,660 --> 00:55:35,840 levels is split by the Rabi frequency, 678 00:55:35,840 --> 00:55:38,200 and if I now show you the energy eigen 679 00:55:38,200 --> 00:55:42,580 levels of this two-by-two Hamiltonian, 680 00:55:42,580 --> 00:55:47,650 it will asymptotically coincide with a dashed line 681 00:55:47,650 --> 00:55:51,130 through this and through that. 682 00:55:54,090 --> 00:55:57,440 So in other words, I'm just reminding you 683 00:55:57,440 --> 00:56:02,250 that a non-diagonal matrix element turns a crossing 684 00:56:02,250 --> 00:56:03,520 into an avoided crossing. 685 00:56:12,640 --> 00:56:18,800 So when we take the frequency omega 686 00:56:18,800 --> 00:56:27,120 and we sweep the detuning, so we change delta 687 00:56:27,120 --> 00:56:36,790 and do a sweep of the frequency omega at a rate omega dot, 688 00:56:36,790 --> 00:56:40,960 then we sweep through the resonance and in one limit, 689 00:56:40,960 --> 00:56:46,470 we have rapid adiabatic passage or in general, we 690 00:56:46,470 --> 00:56:49,850 realize the Landau-Zener problem of a sweep 691 00:56:49,850 --> 00:56:52,970 through an avoided crossing. 692 00:56:52,970 --> 00:56:56,010 So what I'm formulating here is it's 693 00:56:56,010 --> 00:56:58,670 the so-called Landau-Zener crossing 694 00:56:58,670 --> 00:57:01,480 or the Landau-Zener problem, which 695 00:57:01,480 --> 00:57:07,430 is the quantum mechanical description of you take 696 00:57:07,430 --> 00:57:12,370 a system by changing an external parameter. 697 00:57:12,370 --> 00:57:17,300 Here, we sweep the frequency of the rotating field, 698 00:57:17,300 --> 00:57:20,310 but by changing the external parameter, 699 00:57:20,310 --> 00:57:24,418 we sweep the system through the avoided crossing. 700 00:57:33,880 --> 00:57:36,600 And it has the two limiting cases 701 00:57:36,600 --> 00:57:41,620 that when we go through this crossing very, very slowly, 702 00:57:41,620 --> 00:57:44,640 the adiabatic field then tells us we 703 00:57:44,640 --> 00:57:48,590 stay on one of these adiabatic solid curves, 704 00:57:48,590 --> 00:57:51,850 and this is the case of rapid adiabatic passage, which 705 00:57:51,850 --> 00:57:54,350 we discuss in the classical limit. 706 00:57:54,350 --> 00:57:57,090 But it is also the other solution 707 00:57:57,090 --> 00:58:00,510 if you would sweep through it very, very fast, 708 00:58:00,510 --> 00:58:03,830 you're in the diabatic limit, you follow the dashed line 709 00:58:03,830 --> 00:58:06,654 and you start up here and you wind up there. 710 00:58:15,810 --> 00:58:17,310 The Landau-Zener problem is actually 711 00:58:17,310 --> 00:58:20,790 a problem which you find it in all it text books, 712 00:58:20,790 --> 00:58:22,590 but to the best of my knowledge, there 713 00:58:22,590 --> 00:58:25,580 is no simple, elementary derivation 714 00:58:25,580 --> 00:58:27,480 which I could give you in a few minutes. 715 00:58:27,480 --> 00:58:29,750 And if the mathematical problem is 716 00:58:29,750 --> 00:58:34,720 a nice, mathematical demonstration 717 00:58:34,720 --> 00:58:41,430 of an exact, solvable model, but to my knowledge explicitly 718 00:58:41,430 --> 00:58:43,940 deriving it is not providing additional insight. 719 00:58:43,940 --> 00:58:47,260 It's one of the cases where the result is 720 00:58:47,260 --> 00:58:51,210 more insightful and much simpler than the derivation. 721 00:58:51,210 --> 00:58:53,180 So what I want to give you is, therefore, 722 00:58:53,180 --> 00:58:54,540 simply the textbook result. 723 00:59:00,650 --> 00:59:05,590 So in the adiabatic limit, you stay on the solid line. 724 00:59:05,590 --> 00:59:12,390 If you do not cross the avoided crossing very, very slowly, 725 00:59:12,390 --> 00:59:16,610 you'll have a non-adiabatic probability 726 00:59:16,610 --> 00:59:21,080 to jump from one level to the other one. 727 00:59:21,080 --> 00:59:26,660 And this non-adiabatic probability 728 00:59:26,660 --> 00:59:32,360 is expressed as an exponential function which 729 00:59:32,360 --> 00:59:35,680 involves the Landau-Zener parameter. 730 00:59:35,680 --> 00:59:46,290 And the Landau-Zener parameter in this exact solution 731 00:59:46,290 --> 00:59:56,070 is omega Rabi squared times this new rate, d omega 732 00:59:56,070 --> 01:00:00,048 dt or d delta dt minus 1. 733 01:00:04,780 --> 01:00:08,070 This square should go outside the brackets, 734 01:00:08,070 --> 01:00:11,200 so therefore, what we find is from the exact solution 735 01:00:11,200 --> 01:00:16,190 that the Landau-Zener parameter is a quarter times-- 736 01:00:16,190 --> 01:00:20,175 and this should now look familiar, 737 01:00:20,175 --> 01:00:22,890 the omega Ravi frequencies squared over omega dot. 738 01:00:26,160 --> 01:00:30,610 And when we discussed the limit of adiabaticity classically, 739 01:00:30,610 --> 01:00:33,910 I hope you remember I gave you the argument by looking 740 01:00:33,910 --> 01:00:39,520 at the adiabatic condition that the adiabatic case requires 741 01:00:39,520 --> 01:00:43,090 omega dot to be much smaller than omega Rabi squared. 742 01:00:43,090 --> 01:00:46,050 So here very naturally what appears in the quantum 743 01:00:46,050 --> 01:00:49,500 mechanical problem is just the ratio of the two quantities 744 01:00:49,500 --> 01:00:54,930 we compared when we looked for the limit of adiabaticity. 745 01:00:54,930 --> 01:01:03,980 So therefore, the probability for a non-adiabatic transition 746 01:01:03,980 --> 01:01:08,525 is simply involving this ratio omega Rabi squared 747 01:01:08,525 --> 01:01:09,830 over omega dot. 748 01:01:15,240 --> 01:01:17,780 So in other words, we know already 749 01:01:17,780 --> 01:01:19,530 from the classical argument, but here we 750 01:01:19,530 --> 01:01:29,657 confirm it, adiabaticity require that this inequality is met. 751 01:01:45,260 --> 01:01:52,300 OK, I could stop here, but since we 752 01:01:52,300 --> 01:01:59,220 are using sort of diabatic sweeps in the laboratory, 753 01:01:59,220 --> 01:02:03,950 as long as I've been involved in doing quote "atom science," 754 01:02:03,950 --> 01:02:09,330 I want to sort of go one step further and teach you 755 01:02:09,330 --> 01:02:11,440 a little bit more about this formula 756 01:02:11,440 --> 01:02:14,970 and try to provide insight, and often 757 01:02:14,970 --> 01:02:19,120 insight is also provided when you apply perturbation theory. 758 01:02:19,120 --> 01:02:22,120 So I know the adiabatic case is very simple, 759 01:02:22,120 --> 01:02:24,830 but I want to look at the diabatic case 760 01:02:24,830 --> 01:02:27,620 and then look at transition probabilities 761 01:02:27,620 --> 01:02:29,240 in a perturbative way. 762 01:02:29,240 --> 01:02:34,100 This is actually the way how we often transfer population 763 01:02:34,100 --> 01:02:34,850 in the laboratory. 764 01:02:42,870 --> 01:02:48,030 So I want to understand better the way how we transfer 765 01:02:48,030 --> 01:02:52,000 population from one curve to the other one. 766 01:02:52,000 --> 01:02:57,040 So if we do a fast sweep, we call it diabatic. 767 01:03:02,040 --> 01:03:10,240 So in other words, if we have this crossing 768 01:03:10,240 --> 01:03:15,740 and we go really fast, well, what happens 769 01:03:15,740 --> 01:03:19,100 is this is the crossing between spin up and spin down. 770 01:03:19,100 --> 01:03:22,160 If you go much, much faster than the Rabi frequency, 771 01:03:22,160 --> 01:03:27,050 the spin has no opportunity to change its orientation. 772 01:03:27,050 --> 01:03:33,130 So therefore, the wave function, the spin has to stay up or down 773 01:03:33,130 --> 01:03:36,015 and that means the system just goes straight 774 01:03:36,015 --> 01:03:37,800 through the crossing. 775 01:03:37,800 --> 01:03:41,790 Because spin up has positive slopes, 776 01:03:41,790 --> 01:03:43,830 spin down has negative slope. 777 01:03:43,830 --> 01:03:48,360 Being adiabatic, staying on this lower adiabatic curve 778 01:03:48,360 --> 01:03:50,460 would actually require the system 779 01:03:50,460 --> 01:03:54,160 to go from spin up in this part of the adiabatic curve 780 01:03:54,160 --> 01:03:56,600 to spin down in the other part. 781 01:03:56,600 --> 01:04:00,120 And to flip a spin cannot be done faster than the Rabi 782 01:04:00,120 --> 01:04:05,430 frequency, so if you sweep fast, that's what's happening. 783 01:04:05,430 --> 01:04:08,830 So we have two trivial limits, one is the adiabatic limit 784 01:04:08,830 --> 01:04:11,770 or just the adiabatic curve and nothing happens. 785 01:04:11,770 --> 01:04:14,660 The other limit is the infinitely-fast limit 786 01:04:14,660 --> 01:04:17,380 and nothing happens again when we 787 01:04:17,380 --> 01:04:21,170 look at the diabatic basis, which is spin up and spin down. 788 01:04:21,170 --> 01:04:28,270 But now let's be almost adiabatic, 789 01:04:28,270 --> 01:04:30,080 and this is a problem which we really 790 01:04:30,080 --> 01:04:34,330 want to understand physically and intuitively because that 791 01:04:34,330 --> 01:04:40,530 means the system spins the small timelier resonance, 792 01:04:40,530 --> 01:04:51,036 and there is a small probability to make a transition. 793 01:04:56,880 --> 01:04:58,960 So if you had one of your hyperfine states, 794 01:04:58,960 --> 01:05:01,140 pick your favorite hyperfine state, 795 01:05:01,140 --> 01:05:03,230 you'll rapidly sweep the frequency. 796 01:05:03,230 --> 01:05:06,650 You will find, unless you sweep it infinitely fast 797 01:05:06,650 --> 01:05:10,220 that there's a small probability in the other hyperfine state. 798 01:05:10,220 --> 01:05:12,280 And that's what you want to calculate now, 799 01:05:12,280 --> 01:05:15,220 and I want you to understand how would you 800 01:05:15,220 --> 01:05:17,390 estimate and calculate the small probability. 801 01:05:23,390 --> 01:05:33,230 So let's now estimate the result, namely 802 01:05:33,230 --> 01:05:35,670 for the small probability in perturbation theory. 803 01:05:43,820 --> 01:05:47,750 And actually what I'm calculating for you here is, 804 01:05:47,750 --> 01:05:50,110 if you've used evaporation-- I know half of the class 805 01:05:50,110 --> 01:05:53,170 is doing that-- if you apply an i F field, 806 01:05:53,170 --> 01:05:54,630 you don't have to make it so strong 807 01:05:54,630 --> 01:05:56,150 that you and the adiabatic limit. 808 01:05:56,150 --> 01:05:57,920 You are exactly in this limit. 809 01:05:57,920 --> 01:06:00,900 The atom will slosh several times 810 01:06:00,900 --> 01:06:04,160 through the resonance in an almost diabatic way, 811 01:06:04,160 --> 01:06:06,840 but there is a finite spin flip probability 812 01:06:06,840 --> 01:06:09,160 and that's how you evaporate atoms. 813 01:06:09,160 --> 01:06:11,630 And I want you now to fully understand 814 01:06:11,630 --> 01:06:15,750 the derivation, what is the probability of ejecting atoms 815 01:06:15,750 --> 01:06:19,570 in the almost diabatic limit with i F spin flips. 816 01:06:19,570 --> 01:06:24,520 That's a limit where 90% of the BEC experiments operate. 817 01:06:27,880 --> 01:06:30,640 So I hope everyone realize it's an important question 818 01:06:30,640 --> 01:06:32,290 and also I hope everybody understands 819 01:06:32,290 --> 01:06:36,450 the question because now I have bigger questions for you. 820 01:06:36,450 --> 01:06:40,250 The first question which I will ask you, 821 01:06:40,250 --> 01:06:48,460 should we calculate that transition probability 822 01:06:48,460 --> 01:06:54,020 by using perturbation theory for an incoherent transition 823 01:06:54,020 --> 01:06:56,490 or for coherent transition? 824 01:06:56,490 --> 01:06:59,150 Let me just explain you what I mean and then I 825 01:06:59,150 --> 01:07:01,860 ask you for your opinion. 826 01:07:01,860 --> 01:07:05,700 Coherently, we simply say in perturbation theory, 827 01:07:05,700 --> 01:07:10,860 we start with our population in state 1. 828 01:07:10,860 --> 01:07:14,910 We have to do the coupling Hamiltonian time 829 01:07:14,910 --> 01:07:18,300 dependence of the population in state 2 830 01:07:18,300 --> 01:07:26,000 and that means if we integrate this equation for a short time, 831 01:07:26,000 --> 01:07:29,060 we find an amplitude a2. 832 01:07:29,060 --> 01:07:31,400 And the probability to be in the state 833 01:07:31,400 --> 01:07:35,140 2, which is the amplitude squared, 834 01:07:35,140 --> 01:07:38,860 is proportional to the Rabi frequency squared 835 01:07:38,860 --> 01:07:41,370 times the effective time squared, 836 01:07:41,370 --> 01:07:45,320 the effective time of tribing the system. 837 01:07:45,320 --> 01:07:48,160 Coherent processes are always quadratic in time. 838 01:07:51,500 --> 01:07:55,610 If we do it incoherently, well, the way 839 01:07:55,610 --> 01:07:57,860 how we describe incoherent processes 840 01:07:57,860 --> 01:08:03,450 are Fermi's golden rule, which we've all seen. 841 01:08:03,450 --> 01:08:12,510 And the probability in Fermi's golden rule is very different. 842 01:08:12,510 --> 01:08:15,600 Well, it is proportional to the Rabi frequency squared, 843 01:08:15,600 --> 01:08:18,250 to the matrix element squared, but Fermi's golden rule 844 01:08:18,250 --> 01:08:22,340 gives us a constant rate, and for constant rate, 845 01:08:22,340 --> 01:08:25,180 the probability is rate times time. 846 01:08:25,180 --> 01:08:30,660 So now it is linear in time and then 847 01:08:30,660 --> 01:08:37,290 because of the delta function in Fermi's golden rule-- 848 01:08:37,290 --> 01:08:40,890 I'm missing a symbol so I use gamma here. 849 01:08:40,890 --> 01:08:43,540 It has nothing to do with the Landau-Zener probability. 850 01:08:43,540 --> 01:08:45,534 This is just the density of states. 851 01:08:49,890 --> 01:08:51,899 So I hope you know now what is the difference 852 01:08:51,899 --> 01:08:54,109 between coherent or incoherent. 853 01:08:54,109 --> 01:08:57,979 The most important part is that things 854 01:08:57,979 --> 01:09:01,014 are linear in time for an incoherent pulses rate equation 855 01:09:01,014 --> 01:09:03,897 and at least for small times quadratic in time 856 01:09:03,897 --> 01:09:04,730 for coherent pulses. 857 01:09:08,290 --> 01:09:15,330 So now we come to this process where 858 01:09:15,330 --> 01:09:20,470 we take atoms from spin up to spin down. 859 01:09:20,470 --> 01:09:25,750 We evaporate with a weak, course, almost adiabatic 860 01:09:25,750 --> 01:09:29,560 with weaker f drive, so we are closer to the diabatic limit. 861 01:09:29,560 --> 01:09:33,670 And so if you think about this problem, 862 01:09:33,670 --> 01:09:41,590 I want you to tell me if this process, 863 01:09:41,590 --> 01:09:47,170 the perturbative transition close to the diabatic case 864 01:09:47,170 --> 01:09:51,399 is that should we use when we apply perturbation 865 01:09:51,399 --> 01:10:03,810 theory, the coherent picture or the incoherent picture. 866 01:10:03,810 --> 01:10:07,450 In other words, is the dynamics of the quantum system, when 867 01:10:07,450 --> 01:10:13,855 we go relatively quickly for the Landau-Zener crossing, 868 01:10:13,855 --> 01:10:16,630 is that a coherent or an incoherent process? 869 01:10:24,250 --> 01:10:27,217 I could see where it doesn't matter, but it does matter. 870 01:10:36,970 --> 01:10:40,656 So I think it's an open question. 871 01:10:40,656 --> 01:10:41,780 Let me give you the answer. 872 01:10:44,370 --> 01:10:47,940 It is coherent, and you can see it in the following way. 873 01:10:47,940 --> 01:10:50,736 What is the source of incoherent here? 874 01:10:50,736 --> 01:10:52,710 We have a Hamiltonian. 875 01:10:52,710 --> 01:10:55,520 The Hilbert space is by two-by-two. 876 01:10:55,520 --> 01:10:58,470 There is no coherence which can be lost. 877 01:10:58,470 --> 01:11:01,700 There is no spontaneous emission to other states. 878 01:11:01,700 --> 01:11:04,000 There is no reservoir. 879 01:11:04,000 --> 01:11:06,510 We don't have a small system which couples to a bigger 880 01:11:06,510 --> 01:11:08,810 system, and then the small system-- 881 01:11:08,810 --> 01:11:12,680 we do that on Wednesday-- has to-- tomorrow, 882 01:11:12,680 --> 01:11:15,500 Wednesday-- has to be described by density matrix. 883 01:11:15,500 --> 01:11:17,670 We have the none of the physics which 884 01:11:17,670 --> 01:11:20,010 would give arise to incoherent physics. 885 01:11:20,010 --> 01:11:20,900 It is coherent. 886 01:11:25,000 --> 01:11:26,690 But maybe I'm oversimplifying. 887 01:11:26,690 --> 01:11:29,140 Is somebody who said it's incoherent 888 01:11:29,140 --> 01:11:31,959 who wants to maybe press me harder and tell me 889 01:11:31,959 --> 01:11:33,250 why you think it is incoherent. 890 01:11:39,760 --> 01:11:41,770 Well, one possibility is-- and this 891 01:11:41,770 --> 01:11:44,600 is why we often use Landau-Zener sweeps in the lab. 892 01:11:44,600 --> 01:11:47,570 We have fluctuations of the resonance frequency. 893 01:11:47,570 --> 01:11:50,004 And when we go and sweep through it, 894 01:11:50,004 --> 01:11:53,400 we don't know exactly when we hit the resonance. 895 01:11:53,400 --> 01:11:56,060 And if you would take an ensemble of systems 896 01:11:56,060 --> 01:11:59,020 and you go through the resonance at different times, 897 01:11:59,020 --> 01:12:01,640 you will get an ensemble of wave functions, which 898 01:12:01,640 --> 01:12:04,860 has different phase factors in it, and in the end, 899 01:12:04,860 --> 01:12:08,530 you will actually need a density matrix to describe it. 900 01:12:08,530 --> 01:12:10,860 But this is now an experimental imperfection 901 01:12:10,860 --> 01:12:13,200 which I haven't assumed here. 902 01:12:13,200 --> 01:12:16,070 So in other words, what you should do is the following. 903 01:12:16,070 --> 01:12:18,660 Whenever you sweep through Landau-Zener crossing, 904 01:12:18,660 --> 01:12:21,120 you start with the ground state and what you get out 905 01:12:21,120 --> 01:12:23,890 is a superposition of ground and excited state. 906 01:12:23,890 --> 01:12:28,720 And the Hamiltonian determines absolutely every aspect 907 01:12:28,720 --> 01:12:31,890 of the amplitude in the ground and the excited state including 908 01:12:31,890 --> 01:12:33,480 all phase factors. 909 01:12:33,480 --> 01:12:37,965 In other words, if you do a Landau-Zener crossing in a way 910 01:12:37,965 --> 01:12:40,720 that you prepare 50-50% of the atoms, 911 01:12:40,720 --> 01:12:45,270 they are always face coherent and you can use this process 912 01:12:45,270 --> 01:12:48,250 as a beam splitter in an atomic thermometer. 913 01:12:48,250 --> 01:12:49,170 Fully coherent. 914 01:12:56,230 --> 01:13:09,470 We have five minutes, so if we assume now, 915 01:13:09,470 --> 01:13:15,770 I hope we all agree that the amplitude is obtained 916 01:13:15,770 --> 01:13:19,640 in this coherent wave, then, of course, the question is, 917 01:13:19,640 --> 01:13:22,450 but what is the effective time? 918 01:13:22,450 --> 01:13:25,414 When we sweep the resonance, we are far away. 919 01:13:25,414 --> 01:13:26,080 Nothing happens. 920 01:13:26,080 --> 01:13:26,820 Nothing happens. 921 01:13:26,820 --> 01:13:28,010 Nothing happens. 922 01:13:28,010 --> 01:13:30,620 We go through the resonance, everything happens. 923 01:13:30,620 --> 01:13:31,300 Nothing happens. 924 01:13:31,300 --> 01:13:31,966 Nothing happens. 925 01:13:31,966 --> 01:13:32,760 Nothing happens. 926 01:13:32,760 --> 01:13:36,930 So this is the effective time when the wave function really 927 01:13:36,930 --> 01:13:39,660 changes and we create the coherent and mixture 928 01:13:39,660 --> 01:13:42,030 in the second state. 929 01:13:42,030 --> 01:13:45,170 And it is this effective times squared 930 01:13:45,170 --> 01:13:51,670 which determines what happens, what our transmission 931 01:13:51,670 --> 01:13:53,830 amplitude is. 932 01:13:53,830 --> 01:13:59,480 So therefore, the question is, what 933 01:13:59,480 --> 01:14:02,170 is the effective time in the Landau-Zener crossing? 934 01:14:04,940 --> 01:14:08,630 I can give you three choices. 935 01:14:08,630 --> 01:14:15,900 One is, the effective time is, well, we change omega 936 01:14:15,900 --> 01:14:20,270 and the effective time is, how long, what 937 01:14:20,270 --> 01:14:23,490 is the time until we have a detuning, which 938 01:14:23,490 --> 01:14:25,200 is equal to the Rabi frequency. 939 01:14:29,520 --> 01:14:32,870 Another possibility is that the effective time where 940 01:14:32,870 --> 01:14:35,320 we coherently drive the system is 941 01:14:35,320 --> 01:14:39,370 1 over the Rabi frequency, just the Rabi period. 942 01:14:39,370 --> 01:14:44,365 Or another choice, how I can construct time out 943 01:14:44,365 --> 01:14:50,240 of the two frequencies where it's omega Rabi and omega dot. 944 01:14:50,240 --> 01:14:52,010 These are our two elements. 945 01:14:52,010 --> 01:14:55,890 So another possibility how I can get time 946 01:14:55,890 --> 01:15:00,680 is, omega dot is frequency squared, 947 01:15:00,680 --> 01:15:04,290 and the square root of 1 over omega dot at least 948 01:15:04,290 --> 01:15:09,040 fulfills the dimensional criterion that this is time. 949 01:15:09,040 --> 01:15:13,640 So my question for you is, what do you think 950 01:15:13,640 --> 01:15:15,610 is the effective time during which 951 01:15:15,610 --> 01:15:19,230 we drive the system coherently? 952 01:15:19,230 --> 01:15:22,360 I have to tell you before I made up the problem, 953 01:15:22,360 --> 01:15:24,550 I do not know the answer. 954 01:15:24,550 --> 01:15:26,720 But I can also tell you that there 955 01:15:26,720 --> 01:15:28,490 is only one answer which is correct. 956 01:15:36,920 --> 01:15:37,600 All right. 957 01:15:45,080 --> 01:15:52,110 I have to say I expected A to be the correct answer, 958 01:15:52,110 --> 01:15:56,040 but I convinced myself it's only B. 959 01:15:56,040 --> 01:16:02,600 And the answer is the following, and I know I have to stop, 960 01:16:02,600 --> 01:16:04,880 but I only need three more lines. 961 01:16:04,880 --> 01:16:09,300 The effective time is not the time until you are detuned, 962 01:16:09,300 --> 01:16:14,770 the effective time is which I can call the dephasing 963 01:16:14,770 --> 01:16:21,600 time, the time during which everything is coherent. 964 01:16:21,600 --> 01:16:29,110 What happens is, we change the frequency delta omega, 965 01:16:29,110 --> 01:16:33,810 and the delta omega is, of course, 966 01:16:33,810 --> 01:16:37,610 omega dot times delta t. 967 01:16:37,610 --> 01:16:42,050 So if we change the frequency by sweeping it in such a way 968 01:16:42,050 --> 01:16:46,120 that we are detuning and now with that detuning, 969 01:16:46,120 --> 01:16:50,950 if we let the system evolve, we would get a phase shift of pi. 970 01:16:50,950 --> 01:16:55,630 That's sort of the maximum where everything adds up coherently. 971 01:16:55,630 --> 01:16:58,560 If we would go longer in time, the frequency 972 01:16:58,560 --> 01:17:00,660 has changed to the point that what 973 01:17:00,660 --> 01:17:05,460 we add to the amplitude of the other state 974 01:17:05,460 --> 01:17:09,266 is no longer in phase to what we have added before. 975 01:17:09,266 --> 01:17:10,960 You can sort of look at it like this. 976 01:17:10,960 --> 01:17:12,999 You have a little bit of phase a2, You build up. 977 01:17:12,999 --> 01:17:13,540 You build up. 978 01:17:13,540 --> 01:17:16,600 You build up by adding amplitude with the same phase. 979 01:17:16,600 --> 01:17:18,290 But now you are sweeping, and this 980 01:17:18,290 --> 01:17:23,720 is the criterion where you start to pile up 981 01:17:23,720 --> 01:17:25,220 things with the wrong phase and then 982 01:17:25,220 --> 01:17:26,904 the phase eventually becomes randomized 983 01:17:26,904 --> 01:17:28,570 and you're not effectively contributing. 984 01:17:32,580 --> 01:17:35,970 This equation defines the window delta 985 01:17:35,970 --> 01:17:41,180 t during which we effectively add substantial amplitude 986 01:17:41,180 --> 01:17:45,550 in the second state, and it involves delta t squared, 987 01:17:45,550 --> 01:17:49,335 so neglecting factors of unity, the result of this 988 01:17:49,335 --> 01:17:58,640 is that, indeed, delta t is 1 over omega dot plus. 989 01:17:58,640 --> 01:18:04,630 And indeed, if we say the probability for coherent drive 990 01:18:04,630 --> 01:18:10,150 is Rabi frequency squared times delta t squared, what we now 991 01:18:10,150 --> 01:18:15,480 obtain is the Rabi frequency squared over omega dot, 992 01:18:15,480 --> 01:18:19,560 and this is our Landau-Zener parameter. 993 01:18:19,560 --> 01:18:33,240 So in other words, if we check with the exact result, 994 01:18:33,240 --> 01:18:41,040 the Landau-Zener probability, 1 minus p non-adiabatic-- 995 01:18:41,040 --> 01:18:44,240 the Landau-Zener problem is e to the minus 2 pi 996 01:18:44,240 --> 01:18:46,900 over gamma or gamma pi over 2 over the Landau-Zener 997 01:18:46,900 --> 01:18:51,980 parameter, if I do an expansion of the Landau-Zener parameter 998 01:18:51,980 --> 01:18:54,880 for a small value of the exponent, 999 01:18:54,880 --> 01:18:59,950 the exact result is 2 pi times gamma, 1000 01:18:59,950 --> 01:19:03,880 and this is indeed proportional to omega Rabi squared 1001 01:19:03,880 --> 01:19:05,870 over omega dot. 1002 01:19:05,870 --> 01:19:07,590 So in other words, what I've shown 1003 01:19:07,590 --> 01:19:10,510 you is that coherent time evolution 1004 01:19:10,510 --> 01:19:15,590 with this weird effective time, what I motivated physically, 1005 01:19:15,590 --> 01:19:18,580 exactly that produce the limit of small gamma 1006 01:19:18,580 --> 01:19:20,790 from the exact result for the Landau-Zener crossing. 1007 01:19:23,420 --> 01:19:24,870 OK. 1008 01:19:24,870 --> 01:19:25,930 I know time is over. 1009 01:19:25,930 --> 01:19:26,810 Any questions? 1010 01:19:30,130 --> 01:19:30,630 OK. 1011 01:19:30,630 --> 01:19:33,650 So today was officially our Monday class, 1012 01:19:33,650 --> 01:19:38,514 so we meet again tomorrow, the same place, the same time.