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 --> 00:00:17,200 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,200 --> 00:00:17,825 at ocw.mit.edu. 8 00:00:25,830 --> 00:00:28,080 PROFESSOR: Good afternoon. 9 00:00:28,080 --> 00:00:31,480 So we are still in a discussion of atoms 10 00:00:31,480 --> 00:00:33,740 in external magnetic fields. 11 00:00:33,740 --> 00:00:38,840 And we are working our way up from simple static fields 12 00:00:38,840 --> 00:00:41,190 to time dependent fields. 13 00:00:41,190 --> 00:00:43,850 Last week and the week before, we 14 00:00:43,850 --> 00:00:47,760 covered external magnetic fields, Zeeman shifts, 15 00:00:47,760 --> 00:00:53,660 different coupling limits, strong field, and weak field. 16 00:00:53,660 --> 00:00:56,970 Last class on Friday, we talked about what 17 00:00:56,970 --> 00:01:01,030 happens when you put atoms into DC electric fields. 18 00:01:01,030 --> 00:01:04,250 So what we did was simple. 19 00:01:04,250 --> 00:01:07,980 Lowest order, which means in this case, second order 20 00:01:07,980 --> 00:01:09,340 perturbation theory. 21 00:01:09,340 --> 00:01:12,720 And we derived an explicit expression 22 00:01:12,720 --> 00:01:15,350 for the polarizability, alpha. 23 00:01:15,350 --> 00:01:19,300 And this polarizability tells us how energy levels 24 00:01:19,300 --> 00:01:23,900 are shifted quadratically with the electric field. 25 00:01:23,900 --> 00:01:28,060 I put some emphasis in talking about what 26 00:01:28,060 --> 00:01:34,210 is inside perturbation theory and identified with you that, 27 00:01:34,210 --> 00:01:36,350 yes, if you have an electric field, 28 00:01:36,350 --> 00:01:38,370 we have electrostatic energy. 29 00:01:38,370 --> 00:01:41,370 But in order to polarize the atoms, 30 00:01:41,370 --> 00:01:43,950 we have to create an internal energy. 31 00:01:43,950 --> 00:01:46,410 In a concrete example, if we have an s state, 32 00:01:46,410 --> 00:01:50,180 we have to mix a p state to create a dipole moment. 33 00:01:50,180 --> 00:01:52,160 And this costs energy. 34 00:01:52,160 --> 00:01:56,280 Exactly in the same way as when you have a spring 35 00:01:56,280 --> 00:01:58,890 and pull on the spring with gravity, 36 00:01:58,890 --> 00:02:00,630 you gain gravitational energy. 37 00:02:00,630 --> 00:02:03,645 But you have two pay exactly half of it 38 00:02:03,645 --> 00:02:06,590 to create internal energy in your spring. 39 00:02:06,590 --> 00:02:09,820 And this is actually the reason for this factor of 1/2 40 00:02:09,820 --> 00:02:12,055 as we discussed in great length. 41 00:02:15,570 --> 00:02:20,680 Question is do you have any questions about that part? 42 00:02:20,680 --> 00:02:23,150 Because we want to go to the next level. 43 00:02:26,230 --> 00:02:27,125 Perturbation theory? 44 00:02:30,760 --> 00:02:33,360 DC polorizability? 45 00:02:33,360 --> 00:02:36,350 So anyway, the menu for today is we 46 00:02:36,350 --> 00:02:40,180 have done perturbation theory here for weak electric fields. 47 00:02:40,180 --> 00:02:44,570 But the question came already up in class, is it really valid? 48 00:02:44,570 --> 00:02:46,470 Or for what regime is it invalid? 49 00:02:46,470 --> 00:02:48,700 So today I want to talk to you briefly 50 00:02:48,700 --> 00:02:52,550 what happens when we go beyond perturbation theory. 51 00:02:52,550 --> 00:03:02,810 When we go beyond the quadratic Stark effect. 52 00:03:02,810 --> 00:03:04,540 And that leads us to a discussion 53 00:03:04,540 --> 00:03:08,400 on stability of atoms in strong electric field and field 54 00:03:08,400 --> 00:03:10,560 ionization. 55 00:03:10,560 --> 00:03:12,970 I like to sort of feature it because it allows 56 00:03:12,970 --> 00:03:16,650 me to tell you something about the peculiar properties 57 00:03:16,650 --> 00:03:18,550 of Rydberg atoms. 58 00:03:18,550 --> 00:03:21,730 And also the ionization of Rydberg 59 00:03:21,730 --> 00:03:24,420 atoms through electric field. 60 00:03:24,420 --> 00:03:28,050 This is how people in our field create cold plasmas. 61 00:03:28,050 --> 00:03:32,730 And it's also a way to do a very sensitive detection of atoms. 62 00:03:32,730 --> 00:03:37,110 So what I'm telling you today is interesting for its own sake, 63 00:03:37,110 --> 00:03:43,020 but also because it's an important tool for manipulating 64 00:03:43,020 --> 00:03:46,900 atoms, creating plasmas, or sensitive detection. 65 00:03:46,900 --> 00:03:49,850 So this will probably only take 10 or 20 minutes. 66 00:03:49,850 --> 00:03:53,580 And then we want to go form DC electric fields 67 00:03:53,580 --> 00:03:55,530 to AC electric fields. 68 00:03:55,530 --> 00:03:59,570 So we then discuss the AC polarizability. 69 00:03:59,570 --> 00:04:03,730 And, well, that will take us from perturbation theory 70 00:04:03,730 --> 00:04:07,280 in a time independent way what we have done now for DC fields 71 00:04:07,280 --> 00:04:10,160 to time dependent perturbation theory. 72 00:04:10,160 --> 00:04:13,470 So all the topics are rather basic aspects 73 00:04:13,470 --> 00:04:15,020 of quantum physics. 74 00:04:15,020 --> 00:04:19,250 But as usual, I try to give you some special perspectives 75 00:04:19,250 --> 00:04:20,560 from the atomic physics side. 76 00:04:23,300 --> 00:04:36,550 So in perturbation theory, we have a mixture of other states. 77 00:04:36,550 --> 00:04:41,725 And this said mixture is done with the matrix element. 78 00:04:47,110 --> 00:04:49,650 And in perturbation theory, we always 79 00:04:49,650 --> 00:04:52,430 have an energy denominator. 80 00:04:52,430 --> 00:04:55,510 Intermediate state to ground state. 81 00:04:55,510 --> 00:04:59,740 And if the electric field is smaller than this value, 82 00:04:59,740 --> 00:05:04,880 then we have mixture of other states into the ground state. 83 00:05:04,880 --> 00:05:07,850 So usually when we estimate the validity of perturbation 84 00:05:07,850 --> 00:05:15,590 theory, we look for the state which 85 00:05:15,590 --> 00:05:17,270 is closest to the ground state. 86 00:05:17,270 --> 00:05:20,320 For which the energy denominator is smallest. 87 00:05:20,320 --> 00:05:27,510 So therefore, i here is the nearest state, but, and this 88 00:05:27,510 --> 00:05:29,090 is important, of opposite parity. 89 00:05:35,410 --> 00:05:40,060 Otherwise, because of the parity selection rule, 90 00:05:40,060 --> 00:05:42,672 the matrix element would be 0 and electric field 91 00:05:42,672 --> 00:05:44,460 does nothing. 92 00:05:44,460 --> 00:05:47,685 I made a comment on Friday, but let me do it again. 93 00:05:47,685 --> 00:05:50,800 That, of course, means that when we apply electric field 94 00:05:50,800 --> 00:05:52,890 to our favorite atoms, we don't have 95 00:05:52,890 --> 00:05:57,660 to worry about the other hyperfine states. 96 00:05:57,660 --> 00:06:00,840 The other hyperfine states have the same spatial wave function. 97 00:06:00,840 --> 00:06:02,260 Have the same parity. 98 00:06:02,260 --> 00:06:05,770 So we are really here talking about the first excited state. 99 00:06:05,770 --> 00:06:07,630 And a concrete example for those of you 100 00:06:07,630 --> 00:06:11,420 who work with alkalis with an s ground 101 00:06:11,420 --> 00:06:13,930 state, the relevant energy scale here 102 00:06:13,930 --> 00:06:18,460 is the excitation energy to the first p state. 103 00:06:18,460 --> 00:06:25,910 So let's just estimate it for a single electron atom. 104 00:06:25,910 --> 00:06:34,220 And well, this is a hydrogenic estimate. 105 00:06:34,220 --> 00:06:44,890 The excitation energy to the first excited state. 106 00:06:44,890 --> 00:06:48,610 1s to 2p is about 1 Rydberg. 107 00:06:48,610 --> 00:06:51,170 And the Rydberg is nothing, or Hartree, 108 00:06:51,170 --> 00:06:54,720 and that's nothing else than e square over a0. 109 00:06:54,720 --> 00:06:57,800 It's electric potential of two charges separated 110 00:06:57,800 --> 00:07:00,160 by Bohr radius. 111 00:07:00,160 --> 00:07:08,560 And if we estimate that the matrix element 112 00:07:08,560 --> 00:07:12,830 for strong transition is on the order of the Bohr radius, 113 00:07:12,830 --> 00:07:15,330 there are no other length scales in the problem, 114 00:07:15,330 --> 00:07:20,750 we find that the value for the electric field in atomic units 115 00:07:20,750 --> 00:07:25,990 is the charge divided by the Bohr radius squared. 116 00:07:25,990 --> 00:07:28,070 And this is really high. 117 00:07:28,070 --> 00:07:36,350 It is on the order of 5 times 10 to the 9 volt per centimeter. 118 00:07:36,350 --> 00:07:47,190 And this is 1,000 times larger than laboratory 119 00:07:47,190 --> 00:07:48,100 electric fields. 120 00:07:56,130 --> 00:07:59,400 Those fields would just create sparking along the electrodes. 121 00:07:59,400 --> 00:08:01,820 You cannot apply such high electric fields 122 00:08:01,820 --> 00:08:03,390 in a laboratory. 123 00:08:03,390 --> 00:08:05,020 So, therefore, nothing to worry about. 124 00:08:05,020 --> 00:08:07,670 If you have ground state atoms, the Stark effect 125 00:08:07,670 --> 00:08:10,974 in perturbation theory is all you need. 126 00:08:10,974 --> 00:08:12,890 Actually, to be a little bit more precise when 127 00:08:12,890 --> 00:08:18,740 I used the Rydberg and a0, I made a little bit overestimate. 128 00:08:18,740 --> 00:08:21,820 So that the typically the critical electric field 129 00:08:21,820 --> 00:08:25,050 which would cause a breakdown of perturbation theory 130 00:08:25,050 --> 00:08:28,590 for the ground state is around 10 131 00:08:28,590 --> 00:08:31,780 to the 9 volt per centimeter. 132 00:08:31,780 --> 00:08:38,340 OK, so we are safe when we talk about ground states. 133 00:08:38,340 --> 00:08:43,830 But once we got to the excited state, we have degeneracy. 134 00:08:43,830 --> 00:08:46,320 p states have a three fold degeneracy 135 00:08:46,320 --> 00:08:48,720 and they are states with opposite parity. 136 00:08:48,720 --> 00:08:50,450 So we can't have mixing there. 137 00:08:50,450 --> 00:08:54,670 And actually as we will see for excited states, 138 00:08:54,670 --> 00:08:57,420 we have already a breakdown of perturbation theory 139 00:08:57,420 --> 00:09:00,050 at very, very small electric field. 140 00:09:02,660 --> 00:09:06,380 So let us discuss hydrogenic orbits 141 00:09:06,380 --> 00:09:10,180 with principal quantum number, n. 142 00:09:10,180 --> 00:09:17,810 And if you estimate what is the size of the matrix element? 143 00:09:17,810 --> 00:09:20,040 Well, it's not just a0. 144 00:09:20,040 --> 00:09:24,140 There is a scaling with n which is n square. 145 00:09:24,140 --> 00:09:28,210 The matrix element in higher and higher excited state scales 146 00:09:28,210 --> 00:09:28,990 with n square. 147 00:09:33,150 --> 00:09:37,230 Well, how does the energy separation scale? 148 00:09:37,230 --> 00:09:39,530 Well, let's not discuss hydrogen here. 149 00:09:39,530 --> 00:09:42,074 Because in hydrogen, energy levels are degenerate. 150 00:09:42,074 --> 00:09:43,740 And we would immediately get a breakdown 151 00:09:43,740 --> 00:09:45,500 of perturbation theory. 152 00:09:45,500 --> 00:09:49,810 Let's rather formulate it for general atoms. 153 00:09:49,810 --> 00:09:56,200 And we had this nice discussion about the quantum defect. 154 00:09:56,200 --> 00:10:05,587 So if we compare the energy of two l states, they scale as 1 155 00:10:05,587 --> 00:10:06,170 over n square. 156 00:10:09,780 --> 00:10:22,500 But for different l states, we have different quantum defects. 157 00:10:22,500 --> 00:10:24,860 Delta l plus 1. 158 00:10:24,860 --> 00:10:26,520 And here we have delta l. 159 00:10:31,770 --> 00:10:35,420 So, therefore, doing an expansion in m 160 00:10:35,420 --> 00:10:40,820 which we assume to be large, we find that the energy difference 161 00:10:40,820 --> 00:10:46,920 is proportional to the difference between the quantum 162 00:10:46,920 --> 00:10:48,480 defects for the two states we want 163 00:10:48,480 --> 00:10:51,490 to mix with the electric field. 164 00:10:51,490 --> 00:10:56,360 And then again, the scaling with the principal quantum number 165 00:10:56,360 --> 00:10:57,270 is n cube. 166 00:11:00,960 --> 00:11:06,470 So, therefore, we find for the critical field 167 00:11:06,470 --> 00:11:10,240 using the criterion I mentioned above. 168 00:11:10,240 --> 00:11:12,060 We take the energy splitting. 169 00:11:12,060 --> 00:11:17,420 The energy the denominator which appears in perturbation theory. 170 00:11:17,420 --> 00:11:21,895 We divide by the value of the matrix element. 171 00:11:25,590 --> 00:11:36,940 Well, we had the Rydberg constant or two times 172 00:11:36,940 --> 00:11:45,150 the Rydberg constant is nothing else than b squared over a0. 173 00:11:45,150 --> 00:11:49,190 The matrix element was 1 over a0. 174 00:11:49,190 --> 00:11:56,280 And then we have the difference between quantum defects. 175 00:11:56,280 --> 00:11:59,310 And now-- and this makes it really so dramatic. 176 00:11:59,310 --> 00:12:02,730 We had an n square scaling of the matrix element. 177 00:12:02,730 --> 00:12:06,970 And we have an into the minus 3 scaling of energy differences. 178 00:12:06,970 --> 00:12:13,560 So that means the critical field scales by n to the 5. 179 00:12:13,560 --> 00:12:16,920 Go to an excited state with n equals 10. 180 00:12:16,920 --> 00:12:18,665 And the breakdown of perturbation theory 181 00:12:18,665 --> 00:12:22,110 happens 100,000 times earlier. 182 00:12:22,110 --> 00:12:25,280 So some of the scaling in atomic physics 183 00:12:25,280 --> 00:12:27,365 is very, very dramatic when you go 184 00:12:27,365 --> 00:12:29,540 to more highly excited state. 185 00:12:32,370 --> 00:12:44,650 If you threw in that quantum defects become very small, 186 00:12:44,650 --> 00:12:47,810 once you go beyond s and p states. 187 00:12:47,810 --> 00:12:51,260 The higher states just don't penetrate into the nucleus. 188 00:12:51,260 --> 00:12:58,010 So if l is larger than 2, if you have more complicated atoms, 189 00:12:58,010 --> 00:13:01,740 you may add the angular momentum of the core here. 190 00:13:01,740 --> 00:13:05,490 But so if you put n to the 5 scaling and the small quantum 191 00:13:05,490 --> 00:13:10,780 defect together, you find that critical electric fields 192 00:13:10,780 --> 00:13:16,490 are smaller than one volt per centimeter 193 00:13:16,490 --> 00:13:22,340 already for principal quantum numbers as low as 7. 194 00:13:22,340 --> 00:13:28,610 So that means bring a 1.5 volt battery close to your atom 195 00:13:28,610 --> 00:13:30,080 and you drive it crazy. 196 00:13:30,080 --> 00:13:31,835 You drive it out of perturbation theory. 197 00:13:36,810 --> 00:13:39,790 So what we have is the following. 198 00:13:39,790 --> 00:13:41,620 We have, of course, the structure 199 00:13:41,620 --> 00:13:43,120 of atoms in excited field. 200 00:13:43,120 --> 00:13:45,910 Here is the electric field. 201 00:13:45,910 --> 00:13:50,810 And let me just pick three n values. 202 00:13:50,810 --> 00:13:53,660 18, 19, 20. 203 00:13:53,660 --> 00:13:56,500 And now the structure, of course, 204 00:13:56,500 --> 00:14:04,630 is that by the criterion I actually 205 00:14:04,630 --> 00:14:07,740 gave you was the criterion for the application of perturbation 206 00:14:07,740 --> 00:14:08,790 theory. 207 00:14:08,790 --> 00:14:13,640 If the energy levels are smaller than the matrix element, 208 00:14:13,640 --> 00:14:16,940 you have to rediagonalize between those levels. 209 00:14:16,940 --> 00:14:20,710 And that gives you then not a quadratic, but a linear effect. 210 00:14:20,710 --> 00:14:23,160 So, therefore, the structure is here 211 00:14:23,160 --> 00:14:31,071 that you have a region where you have strong l mixing. 212 00:14:31,071 --> 00:14:32,820 So you have to use degenerate perturbation 213 00:14:32,820 --> 00:14:35,550 theory for the different l states. 214 00:14:35,550 --> 00:14:40,450 But the many folds in n, the principle quantum number, 215 00:14:40,450 --> 00:14:43,640 are still well separated. 216 00:14:43,640 --> 00:14:49,100 But then, eventually, when you go further, 217 00:14:49,100 --> 00:14:53,100 you have a region which is called n mixing. 218 00:14:53,100 --> 00:14:55,150 So now the electric field is really 219 00:14:55,150 --> 00:14:58,900 completely rediagonalizing your states with different quantum 220 00:14:58,900 --> 00:14:59,710 numbers, n. 221 00:15:04,450 --> 00:15:07,500 So the result of this discussion is 222 00:15:07,500 --> 00:15:12,210 that highly excited states of atoms 223 00:15:12,210 --> 00:15:15,985 behave very differently from ground state atoms. 224 00:15:15,985 --> 00:15:18,880 And n to the 5 scaling is sensitivity to volt 225 00:15:18,880 --> 00:15:20,070 per centimeter. 226 00:15:20,070 --> 00:15:22,680 Level mixing all over the place. 227 00:15:22,680 --> 00:15:28,530 And that's why for those highly excited states, 228 00:15:28,530 --> 00:15:32,730 people have coined the word Rydberg atoms or Rydberg 229 00:15:32,730 --> 00:15:33,440 matter. 230 00:15:33,440 --> 00:15:36,930 That means atoms with higher principal quantum numbers. 231 00:15:36,930 --> 00:15:45,310 And the study of Rydberg atoms was pioneered, well, 232 00:15:45,310 --> 00:15:49,610 the early pioneering work by our own, Dan Kleppner. 233 00:15:49,610 --> 00:15:54,370 And then Herbert Walther in Munich 234 00:15:54,370 --> 00:15:57,900 who happened to be my Ph.D. Advisor. 235 00:15:57,900 --> 00:16:01,330 And finally, Serge Haroche, who was recognized 236 00:16:01,330 --> 00:16:04,510 with the last Nobel Prize together with Dave Wineland. 237 00:16:10,160 --> 00:16:13,020 So this is not just theory. 238 00:16:13,020 --> 00:16:21,600 What I am showing to you here is spectroscopy 239 00:16:21,600 --> 00:16:24,410 done at MIT by Dan Kleppner. 240 00:16:24,410 --> 00:16:27,990 So what is done here is from the ground state, 241 00:16:27,990 --> 00:16:31,970 they excite to an excited state. 242 00:16:31,970 --> 00:16:37,400 And whenever you hit an excited state, you see a signal. 243 00:16:37,400 --> 00:16:43,250 Let's focus on the upper part. 244 00:16:46,020 --> 00:16:49,980 So if at a given electric field, you scan the laser, 245 00:16:49,980 --> 00:16:51,520 you get one of those traces. 246 00:16:51,520 --> 00:16:54,640 You find peaks, peaks, peaks. 247 00:16:54,640 --> 00:16:58,510 And those peaks correspond to the different and many folds 248 00:16:58,510 --> 00:17:00,880 with strong Stark mixing. 249 00:17:00,880 --> 00:17:03,860 And eventually when you go to somewhat higher field, 250 00:17:03,860 --> 00:17:05,980 you have states all over the place. 251 00:17:05,980 --> 00:17:10,130 And this is the regime where you have done n mixing. 252 00:17:10,130 --> 00:17:13,640 So in the '70s and '80s, there was really, 253 00:17:13,640 --> 00:17:16,859 those experiments obtained a clear understanding 254 00:17:16,859 --> 00:17:20,680 and description of atoms in, well, 255 00:17:20,680 --> 00:17:22,400 I would say high electric fields. 256 00:17:22,400 --> 00:17:23,900 But the fields were not so high. 257 00:17:23,900 --> 00:17:26,185 It was just the atoms were very sensitive that already 258 00:17:26,185 --> 00:17:27,560 is at lower electric fields, they 259 00:17:27,560 --> 00:17:29,920 reached what is regarded as the high-field limit. 260 00:17:34,790 --> 00:17:39,990 Now the question is when we recorded signal, 261 00:17:39,990 --> 00:17:42,930 suddenly the traces stop. 262 00:17:42,930 --> 00:17:46,520 And that means the electric field is now so high 263 00:17:46,520 --> 00:17:49,980 that the atom no longer has a stable state. 264 00:17:49,980 --> 00:17:52,860 The electric field is so high that it literally rips 265 00:17:52,860 --> 00:17:55,740 the electron away from the atom. 266 00:17:55,740 --> 00:17:59,580 And if you go to higher states, the electric field 267 00:17:59,580 --> 00:18:01,900 where this happens is lower. 268 00:18:01,900 --> 00:18:04,440 This is the process of field ionization 269 00:18:04,440 --> 00:18:07,891 and that's what we want to discuss next. 270 00:18:07,891 --> 00:18:08,390 Question? 271 00:18:08,390 --> 00:18:09,687 AUDIENCE: Um, for what atoms? 272 00:18:09,687 --> 00:18:12,020 PROFESSOR: Those studies were actually done for lithium. 273 00:18:14,690 --> 00:18:16,050 It's actually peculiar. 274 00:18:16,050 --> 00:18:18,030 Dan Kleppner really liked hydrogen. 275 00:18:18,030 --> 00:18:19,940 Dan Kleppner is the person who tried 276 00:18:19,940 --> 00:18:22,570 to do almost all experiments with hydrogen. 277 00:18:22,570 --> 00:18:24,630 The famous BEC experiment. 278 00:18:24,630 --> 00:18:27,820 He also had the Rydberg experiment 279 00:18:27,820 --> 00:18:31,125 which was just in building 26 where Vladan Vuletic teaches 280 00:18:31,125 --> 00:18:33,110 now, his labs. 281 00:18:33,110 --> 00:18:34,780 This is where spectroscopy of hydrogen 282 00:18:34,780 --> 00:18:37,120 were done with the goal of a precision 283 00:18:37,120 --> 00:18:39,440 measurement of the Rydberg constant. 284 00:18:39,440 --> 00:18:42,700 So we excited hydrogen to some of those high levels. 285 00:18:42,700 --> 00:18:45,900 But as probably the experts know, 286 00:18:45,900 --> 00:18:50,080 the hydrogen atom is the hardest atom to work with. 287 00:18:50,080 --> 00:18:52,790 Because you need [INAUDIBLE] and alpha. 288 00:18:52,790 --> 00:18:56,280 You have this huge gap to the first excited state. 289 00:18:56,280 --> 00:18:58,180 And that's why if you can get away, 290 00:18:58,180 --> 00:19:00,090 you try to work with other atoms. 291 00:19:00,090 --> 00:19:02,615 And in those experiments, those people 292 00:19:02,615 --> 00:19:04,710 worked with the lithium atom. 293 00:19:04,710 --> 00:19:07,760 So the lithium atom has a quantum defect 294 00:19:07,760 --> 00:19:10,710 which will actually contrast to hydrogen where the quantum 295 00:19:10,710 --> 00:19:11,900 defect is 0. 296 00:19:11,900 --> 00:19:14,450 And this will actually be very, very important 297 00:19:14,450 --> 00:19:18,334 for field ionization as I want to discuss now. 298 00:19:18,334 --> 00:19:19,000 Other questions? 299 00:19:23,280 --> 00:19:30,720 OK, so at a given electric field, states, so to speak, 300 00:19:30,720 --> 00:19:32,580 just disappear. 301 00:19:32,580 --> 00:19:34,720 They're no longer stable. 302 00:19:34,720 --> 00:19:37,583 And this is the process which is called field ionization. 303 00:19:49,450 --> 00:19:56,990 So the phenomenon is that sufficiently strong 304 00:19:56,990 --> 00:20:03,950 electric fields ionize the atom. 305 00:20:08,420 --> 00:20:11,150 And whenever there is a simple model 306 00:20:11,150 --> 00:20:15,270 and I can give you an analytic answer, I try to do that. 307 00:20:15,270 --> 00:20:17,690 Because I feel a lot of our intuition 308 00:20:17,690 --> 00:20:20,760 is shaped by understanding simple models. 309 00:20:20,760 --> 00:20:24,370 And the simplest model for field ionization 310 00:20:24,370 --> 00:20:27,520 is just the classical model by calculating 311 00:20:27,520 --> 00:20:31,570 what is the settle point in the combined potential. 312 00:20:31,570 --> 00:20:33,550 The combined potential of the nucleus 313 00:20:33,550 --> 00:20:38,320 which is a Coulomb potential and the external magnetic field. 314 00:20:38,320 --> 00:20:40,880 So many features of the experiment 315 00:20:40,880 --> 00:20:43,880 can be understood by this simple three line derivation. 316 00:20:46,460 --> 00:20:48,920 So we have a potential. 317 00:20:48,920 --> 00:20:56,240 One part of it is the Coulomb potential. 318 00:20:56,240 --> 00:20:59,620 And focusing on one spatial direction here. 319 00:20:59,620 --> 00:21:09,655 And then, in addition, we apply an electric field. 320 00:21:13,460 --> 00:21:16,785 And the electric field creates a linear potential. 321 00:21:19,360 --> 00:21:24,970 And if I take the sum of the two, well, at large distances, 322 00:21:24,970 --> 00:21:27,860 see, it's the electric field which dominates. 323 00:21:27,860 --> 00:21:30,320 Then the Coulomb potential takes over. 324 00:21:33,350 --> 00:21:35,800 So that's how it looks like. 325 00:21:35,800 --> 00:21:42,190 So now we have the situation if we would put in atoms 326 00:21:42,190 --> 00:21:45,620 and we would look at the energy eigenvalues. 327 00:21:45,620 --> 00:21:52,100 At this point, this is the maximum excited state 328 00:21:52,100 --> 00:21:54,800 in the atom which is still stable. 329 00:21:54,800 --> 00:21:58,400 So what I want to derive for you is what determines 330 00:21:58,400 --> 00:22:01,630 the stability is simply the settle point. 331 00:22:01,630 --> 00:22:06,170 When the binding energy of the excited state for which we 332 00:22:06,170 --> 00:22:09,990 use the Rydberg formula is not stronger 333 00:22:09,990 --> 00:22:12,680 than the position of the settle point, 334 00:22:12,680 --> 00:22:17,140 the atom becomes unstable it and becomes field ionized. 335 00:22:17,140 --> 00:22:18,900 And we'll discuss a little bit later 336 00:22:18,900 --> 00:22:21,510 if this really applies to real atoms. 337 00:22:21,510 --> 00:22:25,820 The quick answer is for lithium and all the other atoms, 338 00:22:25,820 --> 00:22:26,810 it applies. 339 00:22:26,810 --> 00:22:28,630 For hydrogen, it doesn't. 340 00:22:28,630 --> 00:22:30,624 Because hydrogen has too many symmetries. 341 00:22:30,624 --> 00:22:31,790 Too many exact degeneracies. 342 00:22:34,730 --> 00:22:46,050 OK, so the total potential is the Coulomb potential 343 00:22:46,050 --> 00:22:49,396 plus the electric potential. 344 00:22:56,250 --> 00:23:02,210 What we need is the position of the settle point. 345 00:23:02,210 --> 00:23:07,200 Where we have a maximum in this one. 346 00:23:07,200 --> 00:23:08,570 This is a one dimensional cut. 347 00:23:08,570 --> 00:23:12,810 And this one dimensional cut has a maximum at this position. 348 00:23:12,810 --> 00:23:16,610 And by taking the derivative of the total potential, 349 00:23:16,610 --> 00:23:21,530 you immediately find this to be of that value. 350 00:23:21,530 --> 00:23:25,490 And now what we are calculating next 351 00:23:25,490 --> 00:23:30,455 is what is the potential energy at this point. 352 00:23:33,260 --> 00:23:42,160 And, well, this is just copy from the notes, e to the 3/2. 353 00:23:45,020 --> 00:23:47,140 And now what we want to do is we want 354 00:23:47,140 --> 00:23:51,290 to postulate that for field ionization, 355 00:23:51,290 --> 00:23:56,970 this should be able to the binding energy of the electron. 356 00:23:56,970 --> 00:24:01,620 Which is nothing else than the Rydberg 357 00:24:01,620 --> 00:24:03,160 constant divided by n square. 358 00:24:10,900 --> 00:24:14,160 OK, now here, we have the square root 359 00:24:14,160 --> 00:24:20,140 of the electric field from this calculation. 360 00:24:20,140 --> 00:24:25,530 So that means the critical electric field will scale as 1 361 00:24:25,530 --> 00:24:26,690 over n to the 4. 362 00:24:29,740 --> 00:24:32,930 And this is a famous scaling which 363 00:24:32,930 --> 00:24:35,900 can be found in many textbooks. 364 00:24:35,900 --> 00:24:43,070 That the critical electric field for ionization equals, 365 00:24:43,070 --> 00:24:45,080 and now, that's the beauty of atomic units, 366 00:24:45,080 --> 00:24:50,090 it is 1/16, n to the 4. 367 00:24:50,090 --> 00:24:54,670 Beautiful formula derived from the settle point criteria. 368 00:24:54,670 --> 00:24:58,070 Of course, what I mean here is, and if you do the derivation, 369 00:24:58,070 --> 00:24:59,215 it's in atomic units. 370 00:25:03,130 --> 00:25:08,460 Which means in units of the atomic unit 371 00:25:08,460 --> 00:25:12,390 of the electric field which is e over the Bohr radius squared. 372 00:25:19,250 --> 00:25:21,650 So it's a simple model. 373 00:25:21,650 --> 00:25:23,150 It's an analytic result. 374 00:25:23,150 --> 00:25:25,980 The question is is it valid? 375 00:25:25,980 --> 00:25:27,560 Does it make any sense? 376 00:25:27,560 --> 00:25:33,579 And the answer is yes, but in a quantum mechanical problem, 377 00:25:33,579 --> 00:25:35,620 you would actually solve Schroedinger's equation, 378 00:25:35,620 --> 00:25:36,750 such a potential. 379 00:25:36,750 --> 00:25:39,510 But then, the onset of field ionization 380 00:25:39,510 --> 00:25:44,060 comes when tunnelling becomes possible for this barrier. 381 00:25:44,060 --> 00:25:46,510 But it is the nature of tunnelling 382 00:25:46,510 --> 00:25:50,190 that if you're a little bit too low, tunneling is negligible. 383 00:25:50,190 --> 00:25:53,730 You may have ionization rates of 1 per millisecond or so. 384 00:25:53,730 --> 00:25:55,880 And if you just go a little bit closer, 385 00:25:55,880 --> 00:25:58,710 it becomes exponentially larger. 386 00:25:58,710 --> 00:26:01,750 So, therefore, this scaling is very, very accurate. 387 00:26:01,750 --> 00:26:04,630 Because the transition where you go from weak tunneling, 388 00:26:04,630 --> 00:26:07,470 to strong tunneling, to spilling over the barrier, 389 00:26:07,470 --> 00:26:09,530 it's a very narrow range of electric fields. 390 00:26:13,520 --> 00:26:16,460 But, yes, people have looked at it in great details 391 00:26:16,460 --> 00:26:24,220 and have calculated corrections due to tunneling. 392 00:26:24,220 --> 00:26:29,890 So these are quantum corrections to the classical threshold 393 00:26:29,890 --> 00:26:30,890 which I just calculated. 394 00:26:39,120 --> 00:27:02,370 But now in hydrogen, a lot of n, l mixing matrix elements, 395 00:27:02,370 --> 00:27:05,000 matrix elements due to the electric field 396 00:27:05,000 --> 00:27:07,910 between an l states vanish. 397 00:27:07,910 --> 00:27:13,990 Hydrogen is just too pure, too precise. 398 00:27:13,990 --> 00:27:16,670 There is actually parabolic quantum numbers 399 00:27:16,670 --> 00:27:18,330 where you can exactly diagonalize 400 00:27:18,330 --> 00:27:20,040 hydrogen electric fields. 401 00:27:20,040 --> 00:27:23,860 And you find some stable states which do not decay. 402 00:27:23,860 --> 00:27:26,360 And they are above the classical threshold 403 00:27:26,360 --> 00:27:31,840 we have just calculated. 404 00:27:31,840 --> 00:27:35,440 So as Dan Kleppner would have said, the simplest of all atoms 405 00:27:35,440 --> 00:27:39,050 is the most complicated when it comes to field ionization. 406 00:27:39,050 --> 00:27:46,670 Because it has a lot of stable states 407 00:27:46,670 --> 00:27:47,940 above the classical barrier. 408 00:28:01,550 --> 00:28:05,410 So you can sort of envision that there 409 00:28:05,410 --> 00:28:09,060 will be orbits which are just confined to this region. 410 00:28:09,060 --> 00:28:13,180 And the electron never samples the settle point. 411 00:28:13,180 --> 00:28:19,210 And if you look at it this diagram on the wiki, 412 00:28:19,210 --> 00:28:21,930 these are actually calculations for hydrogen 413 00:28:21,930 --> 00:28:24,740 which include ionization rates. 414 00:28:24,740 --> 00:28:27,680 You will find that the states which 415 00:28:27,680 --> 00:28:30,600 are the ones which go down, which 416 00:28:30,600 --> 00:28:33,730 are on the downhill side of the electric field, 417 00:28:33,730 --> 00:28:38,870 you see always here marked an onset for ionization. 418 00:28:38,870 --> 00:28:40,980 And then you see a rapid increase 419 00:28:40,980 --> 00:28:42,610 in the ionization rate. 420 00:28:42,610 --> 00:28:45,160 But you also see those hydrogenic state 421 00:28:45,160 --> 00:28:49,590 which go upward in energy and they refuse to ionize. 422 00:28:49,590 --> 00:28:52,410 Because of the symmetry of parabolic coordinates 423 00:28:52,410 --> 00:28:55,190 and the things I've mentioned. 424 00:28:55,190 --> 00:28:58,020 Anyway, it's too especially to spend more time here 425 00:28:58,020 --> 00:29:02,590 in class on it, but I just think you should at least know 426 00:29:02,590 --> 00:29:07,890 qualitatively what is different for hydrogen. 427 00:29:11,870 --> 00:29:14,170 OK, so that's what I wanted to tell you 428 00:29:14,170 --> 00:29:21,220 about high electric fields and field ionization in principle. 429 00:29:21,220 --> 00:29:25,820 Let me now briefly mention important applications 430 00:29:25,820 --> 00:29:27,700 of field ionization. 431 00:29:39,440 --> 00:29:55,070 One is close to 100% detection efficiency for atoms. 432 00:29:55,070 --> 00:29:57,680 If you want to detect single atoms, 433 00:29:57,680 --> 00:30:00,500 for instance, you'll have a krypton sample 434 00:30:00,500 --> 00:30:03,950 and you want to find a rare isotope of krypton 435 00:30:03,950 --> 00:30:06,720 for dating the material. 436 00:30:06,720 --> 00:30:08,710 You need an extremely high sensitivity 437 00:30:08,710 --> 00:30:12,000 and you may just have a few single atoms in the sample. 438 00:30:12,000 --> 00:30:19,680 One way to do it would be that you excite the atom maybe 439 00:30:19,680 --> 00:30:22,640 through an intermediate state, to Rydberg state. 440 00:30:30,210 --> 00:30:33,380 And then by just applying a few volt per centimeter, 441 00:30:33,380 --> 00:30:34,460 you get an ion. 442 00:30:34,460 --> 00:30:38,380 And ions can be counted by particle detectors. 443 00:30:38,380 --> 00:30:41,320 You can accelerate the ion, smash it into a surface 444 00:30:41,320 --> 00:30:46,390 and count the particles with close to 100% efficiency. 445 00:30:46,390 --> 00:30:51,130 And this is one of the most sensitive detection schemes. 446 00:30:51,130 --> 00:30:54,680 I remember in the aftermath of Chernobyl, 447 00:30:54,680 --> 00:30:56,890 there was an interest in detection schemes 448 00:30:56,890 --> 00:30:58,180 for radioactive strontium. 449 00:30:58,180 --> 00:31:00,450 And on the wiki, I give you a reference 450 00:31:00,450 --> 00:31:04,070 where some people developed this resonance ionization 451 00:31:04,070 --> 00:31:07,480 spectroscopy for some atomic isotopes. 452 00:31:07,480 --> 00:31:10,760 Which unfortunately appeared more frequently 453 00:31:10,760 --> 00:31:13,030 after the Chernobyl disaster. 454 00:31:13,030 --> 00:31:15,700 And they developed a method based 455 00:31:15,700 --> 00:31:18,180 on excitation to Rydberg states. 456 00:31:18,180 --> 00:31:21,020 Which was more sensitive than other methods. 457 00:31:21,020 --> 00:31:25,295 You may ask why don't you ionize it with a laser? 458 00:31:25,295 --> 00:31:28,930 Well, the fact is, you can photoionize it with a laser. 459 00:31:28,930 --> 00:31:32,460 It's another alternative, but it takes much more laser power. 460 00:31:32,460 --> 00:31:34,880 Because if you excite into the continuum, 461 00:31:34,880 --> 00:31:37,090 the matrix element is much smaller. 462 00:31:37,090 --> 00:31:40,710 And often, if you want to have 100% ionization probability 463 00:31:40,710 --> 00:31:45,170 to go into the continuum, you need such high laser power 464 00:31:45,170 --> 00:31:49,750 that you may get some background of resonant ionization 465 00:31:49,750 --> 00:31:52,160 of other elements and such. 466 00:31:52,160 --> 00:31:54,760 So Rydberg atoms is really the smart way to go. 467 00:31:54,760 --> 00:31:59,280 You go to an almost bound to an almost unbound electron. 468 00:31:59,280 --> 00:32:01,170 And then it's just the electric field 469 00:32:01,170 --> 00:32:06,655 which causes the final act of ionization. 470 00:32:11,590 --> 00:32:17,710 I also want to briefly mention that the famous experiments 471 00:32:17,710 --> 00:32:22,380 on Rydberg atoms by Herbert Walther and Serge 472 00:32:22,380 --> 00:32:27,520 Haroche and collaborators would not have been possible 473 00:32:27,520 --> 00:32:28,690 without field ionization. 474 00:32:31,990 --> 00:32:38,370 I give you this one reference, but I'm 475 00:32:38,370 --> 00:32:42,160 sure you'll find more in the actually very, 476 00:32:42,160 --> 00:32:44,910 very nicely written Nobel lecture of Serge Haroche. 477 00:32:44,910 --> 00:32:47,300 I just read it a few weeks ago and it's 478 00:32:47,300 --> 00:32:51,480 a delight to read how he exposes the field. 479 00:32:51,480 --> 00:32:58,690 So they did QED experiments by having microwave transitions 480 00:32:58,690 --> 00:33:02,020 between atoms in two highly excited states. 481 00:33:02,020 --> 00:33:05,640 Let's say with principal quantum number 50 and 51. 482 00:33:05,640 --> 00:33:10,320 So that's now conveniently in the microwave regime. 483 00:33:10,320 --> 00:33:21,070 Then those atoms are passed through a cavity. 484 00:33:21,070 --> 00:33:26,410 And in this cavity, single atoms interact with single photons. 485 00:33:26,410 --> 00:33:29,840 And they have done beautiful quantum non-demolition 486 00:33:29,840 --> 00:33:31,440 experiments of single photons. 487 00:33:31,440 --> 00:33:37,830 I mean that's really wonderful, state of the art experiments. 488 00:33:37,830 --> 00:33:40,430 Back to Rydberg atoms and field ionization. 489 00:33:40,430 --> 00:33:42,740 Eventually, the read out of those experiments 490 00:33:42,740 --> 00:33:46,570 was you prepare atoms in the state 50 or 51. 491 00:33:46,570 --> 00:33:49,600 And afterwards, if they have absorbed or emitted a photon, 492 00:33:49,600 --> 00:33:51,640 they should be in a different state. 493 00:33:51,640 --> 00:33:55,500 So you were interested in a very high detection efficiency which 494 00:33:55,500 --> 00:33:59,660 could distinguish between 50 and 51. 495 00:33:59,660 --> 00:34:03,950 And of course there is a way to distinguish that. 496 00:34:03,950 --> 00:34:07,230 And this is because of n to the 4. 497 00:34:07,230 --> 00:34:09,400 You first apply an electric field 498 00:34:09,400 --> 00:34:12,810 which can only field ionized 51. 499 00:34:12,810 --> 00:34:14,750 And then you allow the atoms to propagate 500 00:34:14,750 --> 00:34:17,239 into a slightly higher electric field. 501 00:34:17,239 --> 00:34:19,850 And then the 50s are ionized. 502 00:34:19,850 --> 00:34:23,530 So the standard experiment is that you 503 00:34:23,530 --> 00:34:29,065 pass those atoms between two field plates. 504 00:34:31,889 --> 00:34:39,920 And by putting an angle between them that indicates 505 00:34:39,920 --> 00:34:43,469 the voltage is increasing. 506 00:34:43,469 --> 00:34:47,390 And then you have two little holes. 507 00:34:47,390 --> 00:34:52,310 You have some channel-drawn particle detectors. 508 00:34:52,310 --> 00:34:59,200 And the first detector will detect the 51s 509 00:34:59,200 --> 00:35:02,850 and the second detector will detect the lower lying states. 510 00:35:02,850 --> 00:35:06,690 So you can detect every atom with high probability, 511 00:35:06,690 --> 00:35:09,300 but also in a state selective way. 512 00:35:15,590 --> 00:35:21,930 So this way of doing state selective field ionization 513 00:35:21,930 --> 00:35:25,370 based on the discussion we had earlier, this 514 00:35:25,370 --> 00:35:29,590 is sort of the method of choice for experiments 515 00:35:29,590 --> 00:35:31,080 involving Rydberg atoms. 516 00:35:36,960 --> 00:35:41,680 Any questions about atoms in the electric fields? 517 00:35:48,230 --> 00:35:50,320 Well, then. 518 00:35:54,650 --> 00:35:59,010 Frequently we add time dependence. 519 00:35:59,010 --> 00:36:03,690 So what we should do next is atoms 520 00:36:03,690 --> 00:36:07,790 in oscillating electric fields. 521 00:36:07,790 --> 00:36:10,320 It's also a good way to review what we have done. 522 00:36:10,320 --> 00:36:12,190 Because the first thing to do now 523 00:36:12,190 --> 00:36:16,160 is we calculate the polarizability for AC fields. 524 00:36:16,160 --> 00:36:18,360 We calculate the AC Stark effect. 525 00:36:18,360 --> 00:36:21,830 And, of course, if in the AC Stark effect we said omega 2, 0 526 00:36:21,830 --> 00:36:24,260 will retrieve the DC Stark effect. 527 00:36:24,260 --> 00:36:26,460 So in a way, what I'm doing for you now is 528 00:36:26,460 --> 00:36:29,010 I'm using time dependent perturbation theory 529 00:36:29,010 --> 00:36:30,680 to obtain a new result. 530 00:36:30,680 --> 00:36:33,400 But it will reproduce the result of time 531 00:36:33,400 --> 00:36:38,250 independent perturbation theory which we have just discussed. 532 00:36:38,250 --> 00:36:48,450 All right, so atoms in oscillating electric fields. 533 00:36:56,490 --> 00:36:58,330 Of course the next step is, and this 534 00:36:58,330 --> 00:37:00,490 is where we are working towards, we 535 00:37:00,490 --> 00:37:04,214 will in the next few lectures starting on Wednesday. 536 00:37:04,214 --> 00:37:06,630 Well, then there is Spring Break, but in the next lectures 537 00:37:06,630 --> 00:37:09,140 develop a deep understanding what 538 00:37:09,140 --> 00:37:12,120 happens when atoms interact with light. 539 00:37:12,120 --> 00:37:14,800 And oscillating electric fields is already 540 00:37:14,800 --> 00:37:16,230 pretty close to light. 541 00:37:16,230 --> 00:37:17,760 And I want to actually also show you 542 00:37:17,760 --> 00:37:21,830 that we capture already a lot of the phenomena which 543 00:37:21,830 --> 00:37:25,175 happen with light except for a full understanding 544 00:37:25,175 --> 00:37:28,220 of spontaneous emission. 545 00:37:28,220 --> 00:37:32,170 So we pretty much when we use an oscillating electric field, 546 00:37:32,170 --> 00:37:35,310 we allow the atom to interact with just one 547 00:37:35,310 --> 00:37:37,310 mode of the electromagnetic field. 548 00:37:37,310 --> 00:37:39,740 Which is filled with a coherent state. 549 00:37:39,740 --> 00:37:41,580 And this is so classical that we don't even 550 00:37:41,580 --> 00:37:42,760 need field quantization. 551 00:37:42,760 --> 00:37:45,050 We just use a classical electric field. 552 00:37:45,050 --> 00:37:49,570 And this already gives us the interaction of atoms with light 553 00:37:49,570 --> 00:37:51,570 except for spontaneous emission which 554 00:37:51,570 --> 00:37:53,810 involves all the other modes. 555 00:37:53,810 --> 00:37:55,790 So that's what we do later. 556 00:37:55,790 --> 00:37:59,300 But today, we just do the same classical description 557 00:37:59,300 --> 00:38:03,680 of an atom in an oscillating electric field. 558 00:38:03,680 --> 00:38:09,120 And this is the theory of the AC Stark effect. 559 00:38:09,120 --> 00:38:14,650 So all we do is application of time dependent perturbation 560 00:38:14,650 --> 00:38:15,150 theory. 561 00:38:22,530 --> 00:38:25,546 So our electric field is now time dependent. 562 00:38:28,070 --> 00:38:32,740 It has a value epsilon, polarization e hat, 563 00:38:32,740 --> 00:38:33,490 and it oscillates. 564 00:38:36,390 --> 00:38:41,120 Our perturbation Hamiltonian is exactly 565 00:38:41,120 --> 00:38:45,890 the same dipole Hamiltonian we had before for the DC Stark 566 00:38:45,890 --> 00:38:47,220 effect. 567 00:38:47,220 --> 00:38:49,170 But now it's a time dependent one. 568 00:38:52,940 --> 00:39:03,430 And it will be useful to break up the oscillating term in e 569 00:39:03,430 --> 00:39:06,740 to the i omega, t and e to the minus i omega t. 570 00:39:10,670 --> 00:39:14,320 I don't want to bore you with perturbation theory in quantum 571 00:39:14,320 --> 00:39:21,570 mechanics because you've all seen it in 805 or 806. 572 00:39:21,570 --> 00:39:23,570 I just want to jump to the result. 573 00:39:23,570 --> 00:39:26,120 You find more details about it on the wiki. 574 00:39:26,120 --> 00:39:30,530 But all you do is you parametrize your wave function, 575 00:39:30,530 --> 00:39:35,560 you expand it into eigenstates with amplitude an. 576 00:39:41,130 --> 00:39:44,820 And then you put it into the Schroedinger equation 577 00:39:44,820 --> 00:39:51,140 and assume that for short times the atom 578 00:39:51,140 --> 00:39:52,260 is in the ground state. 579 00:39:52,260 --> 00:39:54,700 The amplitude of the ground state is 1. 580 00:39:54,700 --> 00:39:57,150 And the amplitude of the excited state 581 00:39:57,150 --> 00:39:59,810 is infinitesimal that you can use the lowest order 582 00:39:59,810 --> 00:40:02,240 perturbation. 583 00:40:02,240 --> 00:40:05,720 This immediately gives you the first order result 584 00:40:05,720 --> 00:40:10,150 for the amplitude in an excited state, k. 585 00:40:10,150 --> 00:40:13,870 It only comes about because your initial state, the ground 586 00:40:13,870 --> 00:40:20,760 state, is coupled by the matrix element to the excited state. 587 00:40:20,760 --> 00:40:23,190 It's linear in the applied electric field. 588 00:40:26,590 --> 00:40:29,750 So what you do is you have the Schroedinger equation 589 00:40:29,750 --> 00:40:33,410 and you integrate it from times 0 to time 1. 590 00:40:33,410 --> 00:40:39,490 And since you have e to the i omega, t and e 591 00:40:39,490 --> 00:40:47,460 to the minus i omega t, you get two time dependent terms. 592 00:40:47,460 --> 00:40:48,080 e to the i. 593 00:40:53,140 --> 00:40:59,850 So what appears now is we have the frequencies omega 594 00:40:59,850 --> 00:41:02,990 n of the excited state. 595 00:41:02,990 --> 00:41:09,690 And now when we couple the excited state, k, 596 00:41:09,690 --> 00:41:11,620 to the ground state, what appears 597 00:41:11,620 --> 00:41:14,160 is the frequency difference between the two. 598 00:41:14,160 --> 00:41:18,650 That's pretty much the excitation gap. 599 00:41:18,650 --> 00:41:24,360 And we have a time dependent oscillation at omega. 600 00:41:24,360 --> 00:41:29,160 And then we have, of course, the same term 601 00:41:29,160 --> 00:41:32,810 where we flip the sign from omega-- 602 00:41:32,810 --> 00:41:34,120 let me write that more clearly. 603 00:41:34,120 --> 00:41:44,210 Where we flip the sign from omega to minus omega. 604 00:41:44,210 --> 00:41:45,390 So that's the second term. 605 00:41:51,260 --> 00:41:54,880 By integrating an exponential function with respect to time, 606 00:41:54,880 --> 00:42:00,150 we get an energy and a frequency denominator which is this one. 607 00:42:00,150 --> 00:42:02,710 So this is really just straight forward, 608 00:42:02,710 --> 00:42:05,310 most basic plain vanilla application 609 00:42:05,310 --> 00:42:07,082 of perturbation theory. 610 00:42:07,082 --> 00:42:09,290 The only thing I want to discuss because it sometimes 611 00:42:09,290 --> 00:42:12,940 confuses people is that we integrate 612 00:42:12,940 --> 00:42:16,550 from times 0 to the finite time. 613 00:42:16,550 --> 00:42:21,450 And then you integrate, you get contributions 614 00:42:21,450 --> 00:42:23,280 from the upper integration limit. 615 00:42:23,280 --> 00:42:28,000 And from the lower integration limit. 616 00:42:28,000 --> 00:42:35,650 And this contribution at the lower integration limit 617 00:42:35,650 --> 00:42:40,950 is actually a transient. 618 00:42:40,950 --> 00:42:43,090 It is at the frequency. 619 00:42:49,530 --> 00:42:51,410 It doesn't depend on omega. 620 00:42:51,410 --> 00:42:54,590 So it's a beat node between the ground and excited state. 621 00:42:54,590 --> 00:42:56,650 I could say at frequency omega k, 622 00:42:56,650 --> 00:43:01,020 but then it's together with the ground state, it's omega kg. 623 00:43:01,020 --> 00:43:06,450 And this is due to if you switch on a perturbation, 624 00:43:06,450 --> 00:43:08,390 it's like you suddenly switch on the drive 625 00:43:08,390 --> 00:43:09,580 of an harmonic oscillator. 626 00:43:09,580 --> 00:43:10,940 And you have some ringing. 627 00:43:10,940 --> 00:43:16,216 You have a transient at the natural frequency 628 00:43:16,216 --> 00:43:17,340 of the harmonic oscillator. 629 00:43:17,340 --> 00:43:18,780 It has nothing to do with the drive. 630 00:43:18,780 --> 00:43:19,835 It's just a sudden onset. 631 00:43:22,650 --> 00:43:27,320 It's transient and frequent due to the sudden switch on. 632 00:43:31,410 --> 00:43:33,580 So like any transient, we haven't 633 00:43:33,580 --> 00:43:34,742 included damping in here. 634 00:43:34,742 --> 00:43:36,200 We don't have spontaneous emission. 635 00:43:36,200 --> 00:43:37,240 Everything is undamped. 636 00:43:37,240 --> 00:43:40,640 But eventually, all those transients 637 00:43:40,640 --> 00:43:43,560 will damp out with time. 638 00:43:43,560 --> 00:43:46,360 And as we should have known also from the beginning, 639 00:43:46,360 --> 00:43:50,890 when we drive a system, when we switch on a perturbation, 640 00:43:50,890 --> 00:43:52,670 just think about an harmonic oscillator. 641 00:43:52,670 --> 00:43:55,460 You have a response at the harmonic oscillator frequency 642 00:43:55,460 --> 00:43:57,520 which is always transient in nature. 643 00:43:57,520 --> 00:44:00,650 And then you have a response at the dry frequency. 644 00:44:00,650 --> 00:44:02,520 And we are interested, of course, 645 00:44:02,520 --> 00:44:14,760 only in the driven response The driven response 646 00:44:14,760 --> 00:44:17,610 is at frequency plus minus omega. 647 00:44:17,610 --> 00:44:19,290 That's how we drive it. 648 00:44:19,290 --> 00:44:21,380 But now I have to be careful since I'm 649 00:44:21,380 --> 00:44:27,050 looking at the amplitude and I factor out the time dependence 650 00:44:27,050 --> 00:44:32,540 of the time dependence of the wave function, the time 651 00:44:32,540 --> 00:44:34,560 dependence of the eigenstates. 652 00:44:34,560 --> 00:44:38,240 I'm now looking for drive terms in this expression 653 00:44:38,240 --> 00:44:41,940 for the amplitude which are at frequency omega, 654 00:44:41,940 --> 00:44:50,570 but modified by the ground. 655 00:44:50,570 --> 00:44:52,760 Or the frequency of the ground state. 656 00:44:52,760 --> 00:44:55,180 But anyway, what I mean is the relevant term 657 00:44:55,180 --> 00:44:57,330 is the one which depends on omega. 658 00:44:57,330 --> 00:45:01,230 And in the following discussion, I simply drop the minus 1 659 00:45:01,230 --> 00:45:02,740 because it's a transient. 660 00:45:02,740 --> 00:45:06,440 If you would switch on your time dependent electric field 661 00:45:06,440 --> 00:45:08,715 in a smoother way, this term would disappear. 662 00:45:13,100 --> 00:45:16,750 OK, let's now be specific. 663 00:45:16,750 --> 00:45:21,540 Let's assume the electric field points in the z direction. 664 00:45:21,540 --> 00:45:33,670 For an isotropic medium, the dipole moment, 665 00:45:33,670 --> 00:45:37,406 the time dependent dipole moment which we induce 666 00:45:37,406 --> 00:45:42,660 is also pointing in the z direction. 667 00:45:42,660 --> 00:45:48,940 And so we want to calculate now what is the dipole moment which 668 00:45:48,940 --> 00:45:53,610 is created by the drive term? 669 00:45:53,610 --> 00:45:58,610 By the driven electric Field 670 00:45:58,610 --> 00:46:04,940 And for that, we simply use the perturbation theory 671 00:46:04,940 --> 00:46:06,820 we have just applied. 672 00:46:06,820 --> 00:46:12,640 We take the ground state and its first order correction. 673 00:46:12,640 --> 00:46:16,445 And calculate the expectation value of the dipole moment. 674 00:46:20,220 --> 00:46:23,820 In the line at the top, we have the first order correction 675 00:46:23,820 --> 00:46:26,850 to the ground state wave function. 676 00:46:26,850 --> 00:46:29,600 And so we just plug it in. 677 00:46:29,600 --> 00:46:34,330 And what we obtain is result where 678 00:46:34,330 --> 00:46:39,930 we have the matrix elements squared. 679 00:46:39,930 --> 00:46:42,270 Remember, we do first order perturbation theory 680 00:46:42,270 --> 00:46:44,660 which is one occurrence of the matrix element, 681 00:46:44,660 --> 00:46:46,640 but now we take a second matrix element 682 00:46:46,640 --> 00:46:49,550 because we're interested in the dipole operator. 683 00:46:49,550 --> 00:46:54,085 So this gives us now a sum over matrix element squared. 684 00:46:57,450 --> 00:47:01,890 We have e to the plus i omega t and e to the minus i omega t. 685 00:47:01,890 --> 00:47:04,905 This means we get 2 times the real part of this expression. 686 00:47:08,000 --> 00:47:13,890 And the time dependent term is e to the i omega t. 687 00:47:20,130 --> 00:47:23,900 And then we have the term with plus and minus omega. 688 00:47:27,941 --> 00:47:28,440 Yes. 689 00:47:35,400 --> 00:47:38,530 And so, most importantly, everything 690 00:47:38,530 --> 00:47:40,340 is driven by the electric field. 691 00:47:49,340 --> 00:47:57,500 We can now just to write the result in an easier way, 692 00:47:57,500 --> 00:48:01,420 we can use this result with omega minus omega. 693 00:48:01,420 --> 00:48:07,840 And the real part and write that. 694 00:48:07,840 --> 00:48:13,980 We can write that as 2 times omega k, g. 695 00:48:13,980 --> 00:48:23,720 Omega k, g squared minus omega squared, times cosine omega t, 696 00:48:23,720 --> 00:48:26,500 times the electric field. 697 00:48:26,500 --> 00:48:30,410 And now finally, we have the matrix element. 698 00:48:30,410 --> 00:48:37,020 We have integrated the e to the i omega t function and such. 699 00:48:37,020 --> 00:48:41,350 So what we have here now is the time dependent electric field. 700 00:48:41,350 --> 00:48:45,710 And what we have here is the factor 701 00:48:45,710 --> 00:48:48,510 by which we multiply the electric field 702 00:48:48,510 --> 00:48:50,550 to obtain the bipole moment. 703 00:48:50,550 --> 00:48:52,820 And this is the definition of the now 704 00:48:52,820 --> 00:48:55,260 time dependent or frequency dependent polarizability. 705 00:48:59,260 --> 00:49:00,260 AUDIENCE: Excuse me. 706 00:49:00,260 --> 00:49:01,012 PROFESSOR: Yes? 707 00:49:01,012 --> 00:49:03,964 AUDIENCE: Why are we only getting the cosine and not 708 00:49:03,964 --> 00:49:05,440 [INAUDIBLE]? 709 00:49:05,440 --> 00:49:11,836 You only multiply by the other terms to get the [INAUDIBLE]? 710 00:49:15,280 --> 00:49:16,756 In terms of the cosine. 711 00:49:21,347 --> 00:49:23,180 PROFESSOR: I haven't done the math yesterday 712 00:49:23,180 --> 00:49:24,740 when I prepared for the class. 713 00:49:24,740 --> 00:49:27,530 I did it a while ago when I wrote those notes, 714 00:49:27,530 --> 00:49:36,370 but you know, one comment, I know you don't want to hear it, 715 00:49:36,370 --> 00:49:37,650 but it's a following. 716 00:49:37,650 --> 00:49:40,010 This system has no dissipation at all. 717 00:49:40,010 --> 00:49:42,490 And when I drive it, I will always 718 00:49:42,490 --> 00:49:48,740 get a response which is in phase with the drive. 719 00:49:48,740 --> 00:49:51,030 It can be cosine omega t, or minus cosine omega t. 720 00:49:51,030 --> 00:49:53,120 You cannot get a quadrature at this point. 721 00:49:53,120 --> 00:49:57,350 So if you find I've made a mistake, and you would say, 722 00:49:57,350 --> 00:49:59,900 there is a sine, omega term, I've made a mistake. 723 00:49:59,900 --> 00:50:03,220 I know for physical reasons, I cannot get a phase shift. 724 00:50:03,220 --> 00:50:06,180 You only get a phase shift in the response of a system 725 00:50:06,180 --> 00:50:08,956 to drive when you have dissipation. 726 00:50:08,956 --> 00:50:10,980 AUDIENCE: OK. 727 00:50:10,980 --> 00:50:12,864 I'll have to talk with you. 728 00:50:12,864 --> 00:50:14,280 PROFESSOR: But why don't-- I mean, 729 00:50:14,280 --> 00:50:15,990 I know the result is correct. 730 00:50:15,990 --> 00:50:18,360 And this is just, I hate to spend class time 731 00:50:18,360 --> 00:50:23,000 in trying to figure out if I've omitted one term. 732 00:50:23,000 --> 00:50:24,000 AUDIENCE: The real part. 733 00:50:24,000 --> 00:50:28,772 PROFESSOR: But the real part what I've probably done is-- 734 00:50:28,772 --> 00:50:30,730 AUDIENCE: I think that's probably what you did. 735 00:50:30,730 --> 00:50:31,229 Yeah. 736 00:50:31,229 --> 00:50:32,689 Thank you. 737 00:50:32,689 --> 00:50:34,230 PROFESSOR: I think that's what I did. 738 00:50:34,230 --> 00:50:37,460 So let me, therefore, also say we have now 739 00:50:37,460 --> 00:50:40,520 included the real part of it. 740 00:50:40,520 --> 00:50:41,040 Yes. 741 00:50:41,040 --> 00:50:43,240 OK, thank you. 742 00:50:43,240 --> 00:50:56,760 OK, this part here can actually-- 743 00:50:56,760 --> 00:50:58,190 that's how we often report. 744 00:50:58,190 --> 00:51:01,340 And that's how you often find in textbook the result, 745 00:51:01,340 --> 00:51:04,920 the frequency dependence of the AC polarizability. 746 00:51:04,920 --> 00:51:19,470 But I like to rewrite it now in a different way which 747 00:51:19,470 --> 00:51:20,915 is identical. 748 00:51:23,880 --> 00:51:31,340 And it shows now that there are two contributions. 749 00:51:31,340 --> 00:51:40,680 And I will discuss them in a moment. 750 00:51:40,680 --> 00:51:46,725 But those two contributions, one has in the denominator, 751 00:51:46,725 --> 00:51:50,080 let's assume we excite the system close to resonance. 752 00:51:50,080 --> 00:51:52,200 Omega is close to omega kg. 753 00:51:52,200 --> 00:51:54,720 Then one term is much, much closer. 754 00:51:54,720 --> 00:51:57,980 It's to a near resonant excitation. 755 00:51:57,980 --> 00:52:01,040 And from our discussion about rotating frames, 756 00:52:01,040 --> 00:52:05,100 we have all of this near resonant excitation corresponds 757 00:52:05,100 --> 00:52:08,310 to a corotating term. 758 00:52:08,310 --> 00:52:11,134 And the other one corresponds to a counter rotating term. 759 00:52:14,850 --> 00:52:16,770 We've not assumed any rotating fields here, 760 00:52:16,770 --> 00:52:21,000 but we find those terms with the same mathematical signature. 761 00:52:21,000 --> 00:52:23,520 And I will discuss that little bit later. 762 00:52:23,520 --> 00:52:27,570 But the physics is between the corotating and counterrotating 763 00:52:27,570 --> 00:52:28,070 term. 764 00:52:28,070 --> 00:52:35,190 And it's the corotating term which 765 00:52:35,190 --> 00:52:37,799 is the term which [INAUDIBLE] the so-called rotating 766 00:52:37,799 --> 00:52:38,590 wave approximation. 767 00:52:42,590 --> 00:52:45,050 I just want to identify those two terms 768 00:52:45,050 --> 00:52:50,720 and let's hold the thought for until we have the discussion. 769 00:52:50,720 --> 00:52:55,740 What I first want to say is the limiting case. 770 00:52:55,740 --> 00:52:59,630 We have not made any assumptions about frequency. 771 00:52:59,630 --> 00:53:07,380 When we let omega go to 0, we obtain the DC result. 772 00:53:12,310 --> 00:53:14,900 It is important to point out that when we have the DC 773 00:53:14,900 --> 00:53:19,170 result, we can only get to correct results because we have 774 00:53:19,170 --> 00:53:25,720 equal contributions from co and counterrotating terms. 775 00:53:33,380 --> 00:53:37,650 So that's sort of a question one could ask. 776 00:53:37,650 --> 00:53:40,850 You know, which mistake do you do for the DC polarizability 777 00:53:40,850 --> 00:53:43,740 when you do the rotating wave approximation? 778 00:53:43,740 --> 00:53:46,300 Well, you miss out on exactly 50% of it. 779 00:53:46,300 --> 00:53:48,215 Because both terms become equally important. 780 00:53:52,350 --> 00:53:58,750 I have deliberately focused here on the calculation 781 00:53:58,750 --> 00:54:01,380 of the dipole moment. 782 00:54:01,380 --> 00:54:05,640 Because the dipole, I simply calculated the dipole moment 783 00:54:05,640 --> 00:54:08,150 as being proportion to the electric field. 784 00:54:08,150 --> 00:54:13,570 And the coefficient in front of it is alpha the polarizability. 785 00:54:13,570 --> 00:54:18,260 You may remember that when we calculated 786 00:54:18,260 --> 00:54:20,490 the effect of a static electric field, 787 00:54:20,490 --> 00:54:25,480 we looked for the DC Stark shift for the shift of energy levels. 788 00:54:25,480 --> 00:54:30,125 We can now discuss also the AC Stark shift. 789 00:54:33,750 --> 00:54:39,890 Which is a shift of the energy levels due to the time 790 00:54:39,890 --> 00:54:41,220 dependent field. 791 00:54:41,220 --> 00:54:43,880 But I have to say you have to be a little bit careful. 792 00:54:43,880 --> 00:54:46,700 And sometimes when I looked at equations like this, 793 00:54:46,700 --> 00:54:49,250 there is a moment of confusion what you want to actually, 794 00:54:49,250 --> 00:54:50,890 what the question is. 795 00:54:50,890 --> 00:54:53,980 Because the wave function is now a time dependent wave function. 796 00:54:53,980 --> 00:54:55,420 It's a driven system. 797 00:54:55,420 --> 00:54:57,705 It's no longer your time independent Schroedinger 798 00:54:57,705 --> 00:54:59,550 equation and you are asked, what is 799 00:54:59,550 --> 00:55:02,210 the shift in the value of eigenvalues? 800 00:55:02,210 --> 00:55:05,700 So the AC Stark shift here is now 801 00:55:05,700 --> 00:55:10,710 given by the frequency dependent polarizability. 802 00:55:10,710 --> 00:55:14,540 And then, and I know some textbooks do it right away. 803 00:55:14,540 --> 00:55:17,520 And at the end of the day, it may confuse you. 804 00:55:17,520 --> 00:55:21,390 It uses an average of e square. 805 00:55:21,390 --> 00:55:24,060 So in other words, if you have an electric field which 806 00:55:24,060 --> 00:55:29,390 is cosine omega t, and you calculate what is the AC Stark 807 00:55:29,390 --> 00:55:32,930 shift, you get another factor of 1/2. 808 00:55:32,930 --> 00:55:37,700 Because cosine square omega t time average is 1/2. 809 00:55:37,700 --> 00:55:41,190 So anyway, just think about that. 810 00:55:41,190 --> 00:55:44,237 It's one of those factors of 1/2 which is confusing. 811 00:55:44,237 --> 00:55:45,320 Will, you have a question? 812 00:55:45,320 --> 00:55:47,570 AUDIENCE: So when we take omega goes 813 00:55:47,570 --> 00:55:49,820 to 0 from our previous results, are we still justified 814 00:55:49,820 --> 00:55:52,320 in neglecting the transient term? 815 00:55:56,820 --> 00:55:58,320 PROFESSOR: Yes. 816 00:55:58,320 --> 00:55:59,136 But why? 817 00:56:12,260 --> 00:56:14,790 What will happen is the transient term 818 00:56:14,790 --> 00:56:18,910 is really a term which has time dependence. 819 00:56:18,910 --> 00:56:21,600 And even if omega is 0, just the step function 820 00:56:21,600 --> 00:56:26,380 of switching it on creates an oscillation 821 00:56:26,380 --> 00:56:30,790 in the atom at a frequency which is omega excited 822 00:56:30,790 --> 00:56:33,240 state minus omega ground state. 823 00:56:33,240 --> 00:56:34,830 You may think about it like this. 824 00:56:34,830 --> 00:56:36,530 I give you more the intuitive answer. 825 00:56:36,530 --> 00:56:39,630 Take an atoms and put it in electric field. 826 00:56:39,630 --> 00:56:42,790 If you gradually switch on the electric field, 827 00:56:42,790 --> 00:56:48,680 you create a dipole moment by mixing at 0 frequency 828 00:56:48,680 --> 00:56:50,650 a p state into the s state. 829 00:56:50,650 --> 00:56:53,890 And that displaces the electron from the origin. 830 00:56:53,890 --> 00:56:57,030 But if you suddenly switch on the electric field, 831 00:56:57,030 --> 00:57:00,520 you actually create a response of the atoms which 832 00:57:00,520 --> 00:57:04,500 has a beat node between the excitation frequency of the p 833 00:57:04,500 --> 00:57:07,110 state and the excitation frequency of the s state. 834 00:57:07,110 --> 00:57:11,260 And what you regard is the DC response of the atom 835 00:57:11,260 --> 00:57:15,650 is everything except for this transient term. 836 00:57:15,650 --> 00:57:17,940 However, and this tells you maybe something 837 00:57:17,940 --> 00:57:20,380 about the different formulas in quantum mechanics 838 00:57:20,380 --> 00:57:22,835 when we talked about the time independent perturbation 839 00:57:22,835 --> 00:57:23,720 theory. 840 00:57:23,720 --> 00:57:25,780 We never worried about the switch 841 00:57:25,780 --> 00:57:29,020 on because we just did time independent perturbation theory 842 00:57:29,020 --> 00:57:32,650 and we sort of assumed that the perturbation term had already 843 00:57:32,650 --> 00:57:35,370 existed from the beginning of the universe. 844 00:57:35,370 --> 00:57:38,910 So it's not that we excluded the term. 845 00:57:38,910 --> 00:57:41,450 We formulated the theory in such a way 846 00:57:41,450 --> 00:57:44,210 that the term just didn't appear. 847 00:57:44,210 --> 00:57:47,270 But if you switch on a DC field, you should actually 848 00:57:47,270 --> 00:57:48,770 if you want an accurate description, 849 00:57:48,770 --> 00:57:51,390 do time dependent perturbation theory, 850 00:57:51,390 --> 00:57:54,280 you get the transient term even for DC field. 851 00:57:54,280 --> 00:57:57,055 And then you discuss it away the way I did. 852 00:58:01,420 --> 00:58:04,640 OK, so if you want, these are the textbook results. 853 00:58:04,640 --> 00:58:08,520 We could stop here. 854 00:58:08,520 --> 00:58:13,513 But I want to add three points to the discussion. 855 00:58:20,272 --> 00:58:21,980 You can also see at this point, we really 856 00:58:21,980 --> 00:58:24,520 understand that AC Stark shift theory 857 00:58:24,520 --> 00:58:28,062 as you find it in generic quantum mechanics textbooks. 858 00:58:28,062 --> 00:58:30,020 And now I want to give you a little bit of sort 859 00:58:30,020 --> 00:58:37,840 of extra insight based on my knowledge of atomic physics. 860 00:58:37,840 --> 00:58:40,600 So there are three points I would like to discuss here. 861 00:58:44,420 --> 00:58:51,890 The first one is the relation to the dressed-atom picture. 862 00:58:56,700 --> 00:59:03,210 The second one is I want to parametrize the results 863 00:59:03,210 --> 00:59:06,520 for the polarizability using the concept of oscillator 864 00:59:06,520 --> 00:59:07,020 strengths. 865 00:59:09,700 --> 00:59:13,860 And thirdly, I want to tell you how 866 00:59:13,860 --> 00:59:18,020 you can, already at this point, take our result for the AC 867 00:59:18,020 --> 00:59:23,110 polarizability and calculate how do atoms absorb light 868 00:59:23,110 --> 00:59:28,140 and what is the dispersive phase shift which atoms generate 869 00:59:28,140 --> 00:59:30,040 when they're exposed to light. 870 00:59:30,040 --> 00:59:31,920 Or in other words, I want to show you 871 00:59:31,920 --> 00:59:34,390 that based on this simple result, 872 00:59:34,390 --> 00:59:37,400 we have pretty much already all the information we need 873 00:59:37,400 --> 00:59:40,700 to understand how absorption imaging and dispersive 874 00:59:40,700 --> 00:59:43,130 imaging is done in the laboratory. 875 00:59:43,130 --> 00:59:49,960 So these are the three directions 876 00:59:49,960 --> 00:59:54,320 I would like to take it. 877 00:59:54,320 --> 01:00:05,966 So let's start with number one. 878 01:00:05,966 --> 01:00:07,340 The relation to the dressed-atom. 879 01:00:15,720 --> 01:00:19,030 So what I want to show you now is that a result we obtained 880 01:00:19,030 --> 01:00:21,450 in time dependent perturbation theory, 881 01:00:21,450 --> 01:00:24,390 we could've actually obtained in time independent perturbation 882 01:00:24,390 --> 01:00:28,380 theory by not using a coherent electric field which 883 01:00:28,380 --> 01:00:29,300 oscillates. 884 01:00:29,300 --> 01:00:32,600 But just assuming that they're stationary Fock states 885 01:00:32,600 --> 01:00:33,590 of photons. 886 01:00:33,590 --> 01:00:35,800 And this is actually the dressed-atom picture. 887 01:00:35,800 --> 01:00:38,770 I know I'm throwing now a lot of sort of lingo at you. 888 01:00:38,770 --> 01:00:40,325 It's actually very, very trivial. 889 01:00:40,325 --> 01:00:42,950 But I want to show you that you have a result where if you just 890 01:00:42,950 --> 01:00:45,730 open your eyes, you see actually the dressed-atom shining 891 01:00:45,730 --> 01:00:47,300 through. 892 01:00:47,300 --> 01:00:51,390 So what we had is we had an energy shift here. 893 01:00:51,390 --> 01:00:54,800 Which was let me just summarize what we have derived. 894 01:00:54,800 --> 01:01:00,170 Was this result with a polarizability 895 01:01:00,170 --> 01:01:02,350 which we derived. 896 01:01:02,350 --> 01:01:06,206 And let me rewrite the results from above. 897 01:01:20,810 --> 01:01:23,430 So this is why I made the remark about the averaging 898 01:01:23,430 --> 01:01:25,810 of the electric field. 899 01:01:25,810 --> 01:01:30,020 If you combine it with, as you will see, 900 01:01:30,020 --> 01:01:33,690 Rabi frequencies and dressed-atom picture, 901 01:01:33,690 --> 01:01:36,040 you'll need the amplitude of the electric field. 902 01:01:36,040 --> 01:01:39,170 And formulated in the amplitude of the electric field, 903 01:01:39,170 --> 01:01:40,750 you have a one quarter. 904 01:01:40,750 --> 01:01:42,350 And this is not a mistake. 905 01:01:42,350 --> 01:01:44,590 And you can really trace it down to the time 906 01:01:44,590 --> 01:01:47,440 average of the cosine term. 907 01:01:47,440 --> 01:01:58,930 OK, so this is now-- I'm really copying from the previous page. 908 01:01:58,930 --> 01:02:04,890 We had the difference frequency between state 1 and state 2. 909 01:02:04,890 --> 01:02:07,610 So I'm simply assuming that we couple two states. 910 01:02:07,610 --> 01:02:12,410 Now an s and p state, If you want. 911 01:02:12,410 --> 01:02:14,680 We have a matrix element which is 912 01:02:14,680 --> 01:02:17,200 the matrix element of the position operator, 913 01:02:17,200 --> 01:02:22,350 z, between state 1 and state 2 squared. 914 01:02:22,350 --> 01:02:30,010 We have an energy denominator which was this one. 915 01:02:30,010 --> 01:02:35,000 And we have-- so I'm just rewriting the previous result. 916 01:02:35,000 --> 01:02:39,290 But now I usually hate matrix elements 917 01:02:39,290 --> 01:02:41,110 when they appear in an equation. 918 01:02:41,110 --> 01:02:44,570 I mean, who knows matrix elements. 919 01:02:44,570 --> 01:02:46,745 What is the relevant thing when we 920 01:02:46,745 --> 01:02:49,990 couple two different states is the Rabi frequency. 921 01:02:49,990 --> 01:02:52,540 Frequency units is what we want. 922 01:02:52,540 --> 01:02:56,320 So, therefore, I have prepared the formula 923 01:02:56,320 --> 01:03:02,890 that I can take the matrix element 924 01:03:02,890 --> 01:03:05,620 with the electric field. 925 01:03:05,620 --> 01:03:09,830 And this is nothing else than the Rabi frequency squared. 926 01:03:09,830 --> 01:03:12,430 Or actually, one is measured as energy units. 927 01:03:12,430 --> 01:03:14,020 The other is frequency units. 928 01:03:14,020 --> 01:03:16,690 So there is an h bar square. 929 01:03:16,690 --> 01:03:24,650 So, therefore, I have now written this result 930 01:03:24,650 --> 01:03:28,120 in what I think is a more physically insightful way 931 01:03:28,120 --> 01:03:31,960 by explicitly identifying the Rabi frequency which 932 01:03:31,960 --> 01:03:34,950 couples ground and excited states. 933 01:03:34,950 --> 01:03:39,460 And I also want to separate, want 934 01:03:39,460 --> 01:03:43,080 to introduce the detuning of the time 935 01:03:43,080 --> 01:03:46,016 dependent oscillating electric field from resonance. 936 01:03:58,990 --> 01:04:01,560 And then I obtain this result. 937 01:04:01,560 --> 01:04:05,630 Doesn't it look so much simpler than what we had before? 938 01:04:05,630 --> 01:04:10,310 And it has a lot of physics we can discuss now. 939 01:04:10,310 --> 01:04:18,100 One over delta is sort of like an AC Stark 940 01:04:18,100 --> 01:04:21,540 effect in one limit. 941 01:04:21,540 --> 01:04:25,460 It's a far-off resonant case of an optical trapping potential. 942 01:04:25,460 --> 01:04:27,220 So this formula has a lot of insight 943 01:04:27,220 --> 01:04:28,600 which I want to provide now. 944 01:04:31,510 --> 01:04:35,960 The second part I give you the name and the interpretation 945 01:04:35,960 --> 01:04:37,930 will become obvious in a moment. 946 01:04:37,930 --> 01:04:41,500 Is the important Bloch-Siegert shift. 947 01:04:41,500 --> 01:04:45,600 It is the AC Stark shift due to the counterrotating term. 948 01:04:51,490 --> 01:04:56,530 So what I'm motivating here is just don't get confused. 949 01:04:56,530 --> 01:04:58,340 What I write down is very simple. 950 01:04:58,340 --> 01:05:02,097 And I sometimes use advanced language for those of you 951 01:05:02,097 --> 01:05:03,430 who have heard those buzz words. 952 01:05:03,430 --> 01:05:06,180 But what I really mean is what I want to discuss 953 01:05:06,180 --> 01:05:10,086 and you to follow are the simple steps we do here. 954 01:05:10,086 --> 01:05:11,460 So anyway, what I've just done is 955 01:05:11,460 --> 01:05:14,550 I've rewritten the result from the previous page 956 01:05:14,550 --> 01:05:17,110 by just introducing what I suggest 957 01:05:17,110 --> 01:05:20,690 as more physically appealing symbols. 958 01:05:20,690 --> 01:05:28,080 And now I want to remind you that this result for the AC 959 01:05:28,080 --> 01:05:36,870 Stark effect doesn't it look very similar not to a result, 960 01:05:36,870 --> 01:05:40,537 to the standard result of time independent perturbation 961 01:05:40,537 --> 01:05:41,037 theory? 962 01:05:53,970 --> 01:05:55,890 And, of course, you remember that in time 963 01:05:55,890 --> 01:05:58,320 independent perturbation theory, you 964 01:05:58,320 --> 01:06:03,360 get an energy shift which is the square of a matrix element 965 01:06:03,360 --> 01:06:04,250 divided by detuning. 966 01:06:07,010 --> 01:06:12,730 So it seems when we inspect our result for the AC Stark 967 01:06:12,730 --> 01:06:16,390 effect which came from time dependent perturbation theory, 968 01:06:16,390 --> 01:06:21,320 that this result here has actually-- if we map it 969 01:06:21,320 --> 01:06:27,200 on time independent perturbation theory, it has two terms. 970 01:06:27,200 --> 01:06:29,710 Both coupled by the Rabi frequency. 971 01:06:29,710 --> 01:06:32,610 But one has a detuning of delta. 972 01:06:32,610 --> 01:06:43,970 And the other has a detuning of minus 2 omega, minus delta. 973 01:06:43,970 --> 01:06:48,160 So it seems that the result for the AC Stark shift 974 01:06:48,160 --> 01:06:54,370 can be completely understood by a mixture of not one, 975 01:06:54,370 --> 01:06:59,340 but two different states with different detunings. 976 01:06:59,340 --> 01:07:04,150 And this is exactly what we will do 977 01:07:04,150 --> 01:07:06,810 in 8.422 in the dressed-atom picture 978 01:07:06,810 --> 01:07:16,600 when we have quantized the electromagnetic field. 979 01:07:16,600 --> 01:07:17,970 In other words, we have photons. 980 01:07:22,550 --> 01:07:25,810 Because then what we have is the following. 981 01:07:25,810 --> 01:07:32,720 We have the ground state with n photons, n gamma. 982 01:07:32,720 --> 01:07:36,300 Well, there are sort of n quanta in the system. 983 01:07:36,300 --> 01:07:37,720 n photons. 984 01:07:37,720 --> 01:07:41,580 But what we can do is we can now have one quantum of excitation 985 01:07:41,580 --> 01:07:43,730 with the atom. 986 01:07:43,730 --> 01:07:45,580 And n minus 1 photon. 987 01:07:45,580 --> 01:07:48,980 So it's almost like absorbing a photon. 988 01:07:48,980 --> 01:07:56,120 And this state has a detuning of delta. 989 01:08:03,750 --> 01:08:09,040 But then we can also consider an excited state. 990 01:08:09,040 --> 01:08:11,900 In other words, here, we connect to the excited state 991 01:08:11,900 --> 01:08:13,260 by absorbing a photon. 992 01:08:13,260 --> 01:08:17,729 But we can also, we talk about it more later, 993 01:08:17,729 --> 01:08:20,540 we can also connect to the excited state 994 01:08:20,540 --> 01:08:21,709 by emitting a photon. 995 01:08:24,779 --> 01:08:29,310 So this state has now not one quantum of excitation, 996 01:08:29,310 --> 01:08:30,794 there's three quanta of excitation. 997 01:08:30,794 --> 01:08:33,689 One in the atom and two in the photon. 998 01:08:33,689 --> 01:08:38,600 So, therefore, its detuning is now much, much larger. 999 01:08:38,600 --> 01:08:40,930 Actually, if we're on resonant, the detuning 1000 01:08:40,930 --> 01:08:43,080 would be just 2 omega. 1001 01:08:43,080 --> 01:08:47,710 But if you are detuned, there is the delta. 1002 01:08:47,710 --> 01:08:52,420 So in other words, we can just say 1003 01:08:52,420 --> 01:08:55,434 if it would do time independent-- 1004 01:08:55,434 --> 01:08:56,350 I'm not doing it here. 1005 01:08:56,350 --> 01:08:57,850 And I leave kind of all the beauty 1006 01:08:57,850 --> 01:09:00,840 to when we discuss the dressed-atom picture 1007 01:09:00,840 --> 01:09:02,560 in its full-fledged version. 1008 01:09:02,560 --> 01:09:06,430 But all I'm telling you is that the result for the AC Stark 1009 01:09:06,430 --> 01:09:09,899 shift looks like type independent perturbation theory 1010 01:09:09,899 --> 01:09:12,140 with those two detunings. 1011 01:09:12,140 --> 01:09:15,609 And I'm now offering you the physical picture behind it 1012 01:09:15,609 --> 01:09:18,830 by saying, look, when we have the ground state with n 1013 01:09:18,830 --> 01:09:21,460 photons, and we have those two other states, 1014 01:09:21,460 --> 01:09:25,439 they have exactly the detuning which our results suggests. 1015 01:09:25,439 --> 01:09:29,290 And yes, indeed, if you look at the many folds for those three 1016 01:09:29,290 --> 01:09:33,620 states and we would simply do time independent perturbation 1017 01:09:33,620 --> 01:09:37,210 theory, we would find exactly the same frequency shifts, 1018 01:09:37,210 --> 01:09:39,609 the AC Stark shift as we just obtained 1019 01:09:39,609 --> 01:09:42,720 in a time dependent picture. 1020 01:09:42,720 --> 01:09:45,140 So in other words, what I'm telling you is there 1021 01:09:45,140 --> 01:09:48,069 are two ways to obtain the AC Stark shift. 1022 01:09:48,069 --> 01:09:51,510 One is you do time dependent perturbation theory assuming 1023 01:09:51,510 --> 01:09:53,420 oscillating electric field. 1024 01:09:53,420 --> 01:09:56,010 And that would mean in the quantized language 1025 01:09:56,010 --> 01:10:00,110 you assume the electromagnetic field is in a coherent state. 1026 01:10:00,110 --> 01:10:03,280 Alternatively, you can quantize the electric field 1027 01:10:03,280 --> 01:10:04,940 and introduce Fock states. 1028 01:10:04,940 --> 01:10:10,470 And then because Fock states are type independent, 1029 01:10:10,470 --> 01:10:14,170 then these are the eigenstates of the electromagnetic field 1030 01:10:14,170 --> 01:10:15,360 in a cavity. 1031 01:10:15,360 --> 01:10:18,324 And now you obtain the same atomic level shift 1032 01:10:18,324 --> 01:10:19,990 in time independent perturbation theory. 1033 01:10:29,070 --> 01:10:37,785 So in other words, we can have photon number states. 1034 01:10:40,570 --> 01:10:51,090 And do time independent perturbation theory. 1035 01:10:51,090 --> 01:10:57,160 Or alternatively, we can use a semi-classical electric field. 1036 01:10:59,680 --> 01:11:05,460 Which means we have a classical electric field. 1037 01:11:05,460 --> 01:11:09,399 And then we can treat it in time dependent. 1038 01:11:16,890 --> 01:11:22,330 All the textbooks generally use the latter approach. 1039 01:11:22,330 --> 01:11:25,390 Because it uses a same classical electric field. 1040 01:11:25,390 --> 01:11:29,960 But I can tell you, I strongly prefer the first approach. 1041 01:11:29,960 --> 01:11:31,625 Because in the first approach, you 1042 01:11:31,625 --> 01:11:33,520 have no problems whatsoever. 1043 01:11:33,520 --> 01:11:35,980 What is the time dependent wave function? 1044 01:11:35,980 --> 01:11:39,530 What is an energy level shift when you have a driven system? 1045 01:11:42,130 --> 01:11:44,960 In a time independent way, everything just 1046 01:11:44,960 --> 01:11:48,650 is simple and right there in a simple way. 1047 01:11:48,650 --> 01:11:51,600 But it's two different physical regimes. 1048 01:11:54,101 --> 01:11:54,600 Questions? 1049 01:11:59,720 --> 01:12:03,932 OK, second point of discussion is the concept 1050 01:12:03,932 --> 01:12:05,098 of the oscillator strengths. 1051 01:12:14,870 --> 01:12:16,820 So what I'm teaching in the next five minutes 1052 01:12:16,820 --> 01:12:19,750 is so old-fashioned that I sometimes 1053 01:12:19,750 --> 01:12:22,102 wonder should I still teach it or not. 1054 01:12:22,102 --> 01:12:25,320 On the other hand, you find it in all the textbooks. 1055 01:12:25,320 --> 01:12:27,820 You also want to understand a little bit the tradition. 1056 01:12:27,820 --> 01:12:30,540 And at least I'm giving you some motivation 1057 01:12:30,540 --> 01:12:35,010 to learn about it that if I parametrize the matrix 1058 01:12:35,010 --> 01:12:37,840 element with an oscillators strength. 1059 01:12:37,840 --> 01:12:40,760 And most of your atoms, most of the alkali atoms 1060 01:12:40,760 --> 01:12:42,530 have an oscillator strength for the s 1061 01:12:42,530 --> 01:12:44,170 to p transition for the D-lines. 1062 01:12:44,170 --> 01:12:46,100 Which is unity. 1063 01:12:46,100 --> 01:12:49,810 You can actually write down what is the matrix element. 1064 01:12:49,810 --> 01:12:53,410 What is the spontaneous lifetime of your atom without knowing 1065 01:12:53,410 --> 01:12:55,510 anything about atomic structure. 1066 01:12:55,510 --> 01:12:58,960 Just memorizing that f equals 1, the oscillator strength 1067 01:12:58,960 --> 01:13:02,150 is 1 is pretty much all you have to know about your atom. 1068 01:13:02,150 --> 01:13:04,740 And the rest, the only other thing you have to know 1069 01:13:04,740 --> 01:13:07,580 is what is the resonant frequency of your laser? 1070 01:13:07,580 --> 01:13:09,030 780 nanometer? 1071 01:13:09,030 --> 01:13:10,506 589 nanometer? 1072 01:13:10,506 --> 01:13:12,180 671 nanometer? 1073 01:13:12,180 --> 01:13:15,420 So the modern motivation for this old-fashioned concept 1074 01:13:15,420 --> 01:13:18,390 is for simple atoms where the oscillator strength 1075 01:13:18,390 --> 01:13:21,980 is close to 1, this is probably the parametrization 1076 01:13:21,980 --> 01:13:24,810 you want to use because you can forget 1077 01:13:24,810 --> 01:13:28,080 about the atomic structure. 1078 01:13:28,080 --> 01:13:34,580 So but the derivation would go as follows. 1079 01:13:34,580 --> 01:13:38,220 I want to compare our result. 1080 01:13:38,220 --> 01:13:41,880 How an atom responds to an electric field. 1081 01:13:41,880 --> 01:13:44,620 I want to compare this result to the result 1082 01:13:44,620 --> 01:13:48,290 of a classical oscillator. 1083 01:13:48,290 --> 01:14:03,490 So compare our result for the AC polarizability 1084 01:14:03,490 --> 01:14:08,690 to a classical harmonic oscillator. 1085 01:14:08,690 --> 01:14:11,360 So I assume this classical harmonic oscillator 1086 01:14:11,360 --> 01:14:19,540 has a charge, a mass term, and a frequency. 1087 01:14:19,540 --> 01:14:24,400 And both the atom and the classical harmonic oscillator 1088 01:14:24,400 --> 01:14:29,230 are driven by the time dependent electric field. 1089 01:14:29,230 --> 01:14:34,860 Which we have already parametrized by cosine omega t. 1090 01:14:34,860 --> 01:14:35,360 OK. 1091 01:14:35,360 --> 01:14:41,320 If you look at the classical harmonic oscillator, 1092 01:14:41,320 --> 01:14:53,800 you find that the-- you drive it at omega 1093 01:14:53,800 --> 01:14:57,220 and ask what is it time dependent dipole moment? 1094 01:15:00,060 --> 01:15:02,930 It's driven by the cosine term. 1095 01:15:02,930 --> 01:15:06,170 And what I mean, of course, is dipole moment 1096 01:15:06,170 --> 01:15:07,610 of a classical harmonic oscillator 1097 01:15:07,610 --> 01:15:12,154 is nothing else than charge times displacement. 1098 01:15:12,154 --> 01:15:14,880 And, well, if you spend one minute 1099 01:15:14,880 --> 01:15:18,780 and solve the equation for the driven harmonic oscillator, 1100 01:15:18,780 --> 01:15:23,480 you find that the response, the amplitude, the steady state 1101 01:15:23,480 --> 01:15:27,050 amplitude, [? zk ?] of the harmonic oscillator 1102 01:15:27,050 --> 01:15:28,660 is cosine omega t. 1103 01:15:28,660 --> 01:15:32,582 Times a prefactor which I'm writing down now. 1104 01:15:39,810 --> 01:15:41,305 There is this resonant behavior. 1105 01:15:44,690 --> 01:15:47,195 So that's the response of the classical harmonic oscillator. 1106 01:15:50,450 --> 01:15:56,190 And yeah. 1107 01:15:56,190 --> 01:15:58,730 So this is just classical harmonic oscillator physics. 1108 01:15:58,730 --> 01:16:04,270 And I now want to define a quantity which 1109 01:16:04,270 --> 01:16:09,234 I call the oscillator strength of the atom. 1110 01:16:09,234 --> 01:16:11,900 So I'm just jumping now from the harmonic oscillator to the atom 1111 01:16:11,900 --> 01:16:13,320 and then I combine a the two. 1112 01:16:13,320 --> 01:16:16,750 And the oscillator strength is nothing else 1113 01:16:16,750 --> 01:16:21,220 than a parametrization of the matrix element 1114 01:16:21,220 --> 01:16:23,395 with between different states. 1115 01:16:25,970 --> 01:16:28,210 But it's dimensionless. 1116 01:16:28,210 --> 01:16:31,390 And it's made dimensionless by using the mass. 1117 01:16:31,390 --> 01:16:33,960 By using h-bar. 1118 01:16:33,960 --> 01:16:35,798 And by using the transition frequency. 1119 01:16:39,980 --> 01:16:54,430 So the atomic-- let me just make sure we take care of it. 1120 01:16:54,430 --> 01:16:58,020 This is the result for the classical harmonic oscillator. 1121 01:16:58,020 --> 01:17:00,720 And this is now the result for the atom 1122 01:17:00,720 --> 01:17:07,480 that we have found already they the result for the atom before. 1123 01:17:07,480 --> 01:17:20,920 And now I'm rewriting it simply by expressing 1124 01:17:20,920 --> 01:17:25,470 the matrix element square by the oscillator strength. 1125 01:17:38,590 --> 01:17:45,230 And this here is just another expression 1126 01:17:45,230 --> 01:17:49,100 for the polarizability alpha. 1127 01:17:49,100 --> 01:17:52,190 Well, let's now compare the result 1128 01:17:52,190 --> 01:17:55,370 of a quantum mechanical atom exactly described 1129 01:17:55,370 --> 01:17:57,480 by time dependent perturbation theory 1130 01:17:57,480 --> 01:17:59,530 to the result of a classical harmonic oscillator. 1131 01:18:02,300 --> 01:18:05,540 The frequency structure is the same. 1132 01:18:05,540 --> 01:18:11,210 So if I would know say we have an ensemble 1133 01:18:11,210 --> 01:18:19,320 of harmonic oscillators, and the harmonic oscillators 1134 01:18:19,320 --> 01:18:24,200 may have different frequencies and different charges. 1135 01:18:24,200 --> 01:18:38,010 Then I have made those formulas exactly equal. 1136 01:18:38,010 --> 01:18:42,590 And I can now formulate that the atom reacts 1137 01:18:42,590 --> 01:18:45,770 to an electric field to time dependent electric field. 1138 01:18:45,770 --> 01:18:49,890 Exactly as an ensemble of classical oscillators 1139 01:18:49,890 --> 01:18:52,470 with effective charge. 1140 01:18:52,470 --> 01:18:55,600 If I would say I have an ensemble of oscillators 1141 01:18:55,600 --> 01:19:06,350 with effective charge, then the response 1142 01:19:06,350 --> 01:19:08,530 of the atom and the response of the ensemble 1143 01:19:08,530 --> 01:19:11,018 of classical oscillators is absolutely identical. 1144 01:19:27,130 --> 01:19:41,880 So the atom response is a set classical oscillators 1145 01:19:41,880 --> 01:19:48,670 with an effective charge which is given here. 1146 01:19:48,670 --> 01:19:51,510 So, therefore, you don't have to go further 1147 01:19:51,510 --> 01:19:54,500 if you want to have any intuition how an atom reacts 1148 01:19:54,500 --> 01:19:55,310 to light. 1149 01:19:55,310 --> 01:19:57,190 The classical harmonic oscillator 1150 01:19:57,190 --> 01:19:59,280 is not an approximation. 1151 01:19:59,280 --> 01:20:02,650 It is exact. 1152 01:20:02,650 --> 01:20:12,640 So that result is relevant because it 1153 01:20:12,640 --> 01:20:18,520 allows us to clearly formulate the classical correspondence. 1154 01:20:23,190 --> 01:20:26,770 The second thing is as you can easily 1155 01:20:26,770 --> 01:20:30,190 show with basic commutator algebra, 1156 01:20:30,190 --> 01:20:33,140 there is the Thomas Kuhn sum rule 1157 01:20:33,140 --> 01:20:36,130 which is discussed in all texts in quantum physics 1158 01:20:36,130 --> 01:20:43,020 which says that the sum over all oscillator strengths is 1. 1159 01:20:43,020 --> 01:20:46,340 So, therefore, we know if we have transitions 1160 01:20:46,340 --> 01:20:49,220 from the ground state to different states, 1161 01:20:49,220 --> 01:20:52,000 the sum of all the oscillator strengths to all the states 1162 01:20:52,000 --> 01:20:53,508 can only be 1. 1163 01:20:57,240 --> 01:21:02,830 And another advantage of the formulation with oscillator 1164 01:21:02,830 --> 01:21:06,330 strength is that it is a dimensionless unit. 1165 01:21:10,190 --> 01:21:14,470 It's a dimensionless parameter which tells us 1166 01:21:14,470 --> 01:21:21,330 how the atom corresponds to an external electromagnetic field. 1167 01:21:21,330 --> 01:21:23,270 I just need two or three more minutes 1168 01:21:23,270 --> 01:21:26,760 to show you what that means. 1169 01:21:26,760 --> 01:21:32,010 If you have hydrogen, the 1s to 2p transition 1170 01:21:32,010 --> 01:21:34,010 is the strongest transition. 1171 01:21:34,010 --> 01:21:36,290 And it has a matrix element which 1172 01:21:36,290 --> 01:21:42,610 corresponds to an oscillator strength of about 0.4. 1173 01:21:42,610 --> 01:21:48,170 So the rest comes from more highly excited states. 1174 01:21:48,170 --> 01:22:00,220 However, for alkali atoms, the D-line, the s to p transition 1175 01:22:00,220 --> 01:22:03,180 has an oscillator strength. 1176 01:22:03,180 --> 01:22:05,360 I didn't write down the second digit, 1177 01:22:05,360 --> 01:22:09,780 but it's with excellent approximation, 1.98 1178 01:22:09,780 --> 01:22:10,330 or something. 1179 01:22:10,330 --> 01:22:13,970 So not just qualitatively, almost quantitatively, 1180 01:22:13,970 --> 01:22:19,490 you capture the response from the atom by saying f equals 1. 1181 01:22:19,490 --> 01:22:23,410 So if you use for the alkali atoms f equals 1, 1182 01:22:23,410 --> 01:22:33,090 then simply the transition frequency of the D-line 1183 01:22:33,090 --> 01:22:37,170 gives you the polarizability alpha. 1184 01:22:37,170 --> 01:22:40,850 And as we will see later, because we haven't introduced 1185 01:22:40,850 --> 01:22:49,230 it is, but it will also give you gamma the natural line beats. 1186 01:22:49,230 --> 01:22:51,830 Because all the coupling to the electromagnetic field 1187 01:22:51,830 --> 01:22:55,580 to an external field is really captured by saying what 1188 01:22:55,580 --> 01:22:56,970 the matrix element is. 1189 01:22:56,970 --> 01:22:59,720 And f equals 1 is nothing else than saying 1190 01:22:59,720 --> 01:23:02,880 the matrix element is such and such. 1191 01:23:02,880 --> 01:23:07,360 And, indeed, if I now use the definition of the oscillator 1192 01:23:07,360 --> 01:23:11,820 strength in reverse, the matrix element 1193 01:23:11,820 --> 01:23:14,690 between two transitions, between two states 1194 01:23:14,690 --> 01:23:19,060 is the matrix element squared is the oscillator length 1195 01:23:19,060 --> 01:23:20,860 times one half. 1196 01:23:20,860 --> 01:23:23,370 And if you just go back and look at the formula, 1197 01:23:23,370 --> 01:23:29,670 you find that this is now dimensionless. 1198 01:23:29,670 --> 01:23:32,930 So we need now because the left hand side is the length square, 1199 01:23:32,930 --> 01:23:34,440 we need two lengths. 1200 01:23:34,440 --> 01:23:42,550 One is the Compton wave length which is h-bar over 2 times 1201 01:23:42,550 --> 01:23:44,690 the mass of the electron. 1202 01:23:44,690 --> 01:23:49,960 And lambda bar is the transition wave lengths, 1203 01:23:49,960 --> 01:23:53,870 lambda divided by 2 pi. 1204 01:23:53,870 --> 01:23:58,020 So, therefore, I haven't found it anywhere in textbooks, 1205 01:23:58,020 --> 01:24:02,280 but this is my sort of summary of this. 1206 01:24:02,280 --> 01:24:06,550 If you have a strong transition, and strong 1207 01:24:06,550 --> 01:24:12,760 means that the oscillator length is close to 1, 1208 01:24:12,760 --> 01:24:17,430 then the matrix element for the transition 1209 01:24:17,430 --> 01:24:25,160 is approximately the geometric mean 1210 01:24:25,160 --> 01:24:28,080 of the Compton wave length of the electron. 1211 01:24:31,260 --> 01:24:36,680 And the reduced wave lengths of the resonant transition. 1212 01:24:36,680 --> 01:24:38,160 So now if you want to know what is 1213 01:24:38,160 --> 01:24:42,190 the matrix element for the d-line of rubidium, 1214 01:24:42,190 --> 01:24:45,560 take the wave length of 780 nanometer. 1215 01:24:45,560 --> 01:24:47,650 Take the Compton wave lengths of the electron 1216 01:24:47,650 --> 01:24:50,800 and you get an accurate expression. 1217 01:24:50,800 --> 01:24:51,997 I know time is over. 1218 01:24:51,997 --> 01:24:52,580 Any questions? 1219 01:24:56,480 --> 01:24:58,330 All right.