1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 00:00:06,880 Your support will help MIT OpenCourseWare continue 4 00:00:06,880 --> 00:00:10,740 to offer high quality educational resources for free. 5 00:00:10,740 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,258 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,258 --> 00:00:17,883 at ocw.mit.edu. 8 00:00:21,337 --> 00:00:22,420 PROFESSOR: Good afternoon. 9 00:00:27,190 --> 00:00:29,200 Usually, my first question is do you 10 00:00:29,200 --> 00:00:33,310 have any questions about the last lecture? 11 00:00:33,310 --> 00:00:37,370 We did this fly over over 100 equations 12 00:00:37,370 --> 00:00:40,940 to go from first principles to the [INAUDIBLE] Hamiltonian. 13 00:00:43,820 --> 00:00:46,600 I've put up here for you the overview of this course. 14 00:00:46,600 --> 00:00:48,850 I won't go through it again. 15 00:00:48,850 --> 00:00:52,550 But I want to sort of make you aware that we started 16 00:00:52,550 --> 00:00:57,700 with the fundamental Hamiltonian for light atom interactions. 17 00:00:57,700 --> 00:00:59,990 But now we take a step back. 18 00:00:59,990 --> 00:01:04,060 In the next few lectures, we simply talk about light. 19 00:01:04,060 --> 00:01:05,960 We talk about photons. 20 00:01:05,960 --> 00:01:09,530 We talk about the quantum nature of photons. 21 00:01:09,530 --> 00:01:13,530 And then, after understanding photons in more depth, 22 00:01:13,530 --> 00:01:17,819 we come back and look at more depths at atom photon 23 00:01:17,819 --> 00:01:18,360 interactions. 24 00:01:21,750 --> 00:01:30,190 Let me just give you a pictorial summary of what I've just said. 25 00:01:30,190 --> 00:01:36,770 AMO physics is about atoms and photons. 26 00:01:36,770 --> 00:01:43,650 And in a cartoon like way, I can symbolize atoms 27 00:01:43,650 --> 00:01:45,725 by two level systems. 28 00:01:48,570 --> 00:01:50,980 Or sometimes, we have a two level system 29 00:01:50,980 --> 00:01:53,460 which is coupled by tunneling that's also 30 00:01:53,460 --> 00:01:56,650 equivalent to a two level system. 31 00:01:56,650 --> 00:02:06,580 And then we have photons, two state systems. 32 00:02:06,580 --> 00:02:07,990 And photons are different. 33 00:02:07,990 --> 00:02:13,200 Photons are not two state systems. 34 00:02:13,200 --> 00:02:15,240 They are actually harmonic oscillators. 35 00:02:18,660 --> 00:02:23,770 Usually, electromagnetic waves. 36 00:02:23,770 --> 00:02:27,720 But Hamiltonians for photons can be 37 00:02:27,720 --> 00:02:30,830 and have been realized by replacing 38 00:02:30,830 --> 00:02:35,470 the harmonic oscillator with the photon with vibrations, 39 00:02:35,470 --> 00:02:37,910 for example, of ions in an ion trap 40 00:02:37,910 --> 00:02:41,520 or micromechanical oscillators. 41 00:02:41,520 --> 00:02:50,350 So cartoon picture again is photons in a cavity. 42 00:02:55,460 --> 00:02:59,615 But this is just another harmonic oscillator. 43 00:03:04,480 --> 00:03:08,380 So we have the atoms. 44 00:03:08,380 --> 00:03:13,920 And most of what you should know about atoms 45 00:03:13,920 --> 00:03:17,060 has been discussed in 8.421. 46 00:03:17,060 --> 00:03:21,000 For those of you who will take 8.421 next year, 47 00:03:21,000 --> 00:03:24,070 we'll have a lot of fun talking about the structure of atoms. 48 00:03:28,370 --> 00:03:33,530 Photons, well, I would hope a lot 49 00:03:33,530 --> 00:03:36,490 of it, at least the harmonic oscillator aspects, 50 00:03:36,490 --> 00:03:41,520 you took in 8.05 or in 8.321. 51 00:03:41,520 --> 00:03:49,200 And actually, starting today in this unit, 52 00:03:49,200 --> 00:03:52,800 we want to talk about the quantum age of light. 53 00:03:55,800 --> 00:03:59,280 So you can say 8.421, we had a deep look at the atoms. 54 00:03:59,280 --> 00:04:00,990 We have now look at the photons. 55 00:04:00,990 --> 00:04:10,400 But eventually, what is very important is the interactions 56 00:04:10,400 --> 00:04:12,180 between the atoms and the photons. 57 00:04:14,940 --> 00:04:15,960 That's not all. 58 00:04:19,120 --> 00:04:22,210 This is sort of a very well defined system. 59 00:04:22,210 --> 00:04:29,180 A two level system and one mode of the harmonic oscillator. 60 00:04:29,180 --> 00:04:31,570 But it will become very important-- 61 00:04:31,570 --> 00:04:33,740 and that's what we'll do in a few weeks-- 62 00:04:33,740 --> 00:04:37,060 to discuss what happens when this system-, 63 00:04:37,060 --> 00:04:39,900 the system above the dotted line, 64 00:04:39,900 --> 00:04:46,840 becomes an open system and we interact with the reservoir. 65 00:04:46,840 --> 00:04:51,450 The reservoir will be relevant for relaxation. 66 00:04:51,450 --> 00:04:54,640 Atoms can decohere, can lose their decoherence. 67 00:04:54,640 --> 00:04:57,280 Photons will decay or decohere. 68 00:04:57,280 --> 00:04:59,840 Entanglement will get lost. 69 00:04:59,840 --> 00:05:03,300 And this phenomenon of relaxation 70 00:05:03,300 --> 00:05:07,570 requires the formalism of the master equation. 71 00:05:07,570 --> 00:05:09,480 And we will find the master equation 72 00:05:09,480 --> 00:05:12,700 in the form of the optical block equation. 73 00:05:12,700 --> 00:05:17,840 If you want me to add in cartoon picture of the reservoir, 74 00:05:17,840 --> 00:05:24,420 I would add many, many harmonic oscillators. 75 00:05:30,630 --> 00:05:41,340 So reservoir is characterized by having many modes 76 00:05:41,340 --> 00:05:45,500 and having a capacity to absorb energy 77 00:05:45,500 --> 00:05:48,150 without changing its state. 78 00:05:48,150 --> 00:05:52,750 So this is another kind of illustrated summary 79 00:05:52,750 --> 00:05:57,090 of what we will be doing in the next few weeks. 80 00:05:57,090 --> 00:06:00,830 And today, we want to talk about the quantum nature of light. 81 00:06:07,650 --> 00:06:13,640 We're actually talking mainly about already 82 00:06:13,640 --> 00:06:16,670 a very special form of light, namely 83 00:06:16,670 --> 00:06:19,600 monochromatic light in a single mode 84 00:06:19,600 --> 00:06:22,384 of the harmonic oscillator. 85 00:06:22,384 --> 00:06:25,220 And there is still a lot to be discussed 86 00:06:25,220 --> 00:06:28,370 about the statistics, about squeezed states, 87 00:06:28,370 --> 00:06:31,470 about coherent states. 88 00:06:31,470 --> 00:06:38,110 We want to understand how the quantum state of light 89 00:06:38,110 --> 00:06:41,649 is modified, is transformed, by propagation, 90 00:06:41,649 --> 00:06:43,940 or how it's modified when the light runs through a beam 91 00:06:43,940 --> 00:06:45,380 splitter. 92 00:06:45,380 --> 00:06:47,810 And all of this is already relevant for just 93 00:06:47,810 --> 00:06:49,380 a single mode. 94 00:06:49,380 --> 00:06:54,000 And then very soon, we'll also talk about two modes of light. 95 00:06:54,000 --> 00:06:56,370 Because we need two modes if we want 96 00:06:56,370 --> 00:07:01,340 to talk about entanglement of two modes. 97 00:07:01,340 --> 00:07:03,870 I hope you will realize in this discussion 98 00:07:03,870 --> 00:07:07,140 that the quantum nature of light is more 99 00:07:07,140 --> 00:07:10,140 than light is a wave in the form of electromagnetic wave, 100 00:07:10,140 --> 00:07:13,190 or light is a particle in the form of photons. 101 00:07:13,190 --> 00:07:15,060 Things go much, much deeper. 102 00:07:17,680 --> 00:07:25,530 So the unit on quantum light will actually 103 00:07:25,530 --> 00:07:29,023 evolve in six stages. 104 00:07:36,320 --> 00:07:49,540 Today, we will be talking about photons and statistics, 105 00:07:49,540 --> 00:07:54,740 fluctuation, coherent states. 106 00:07:54,740 --> 00:08:00,910 On next week, on Monday, we want to talk 107 00:08:00,910 --> 00:08:09,650 about non-classical light, states 108 00:08:09,650 --> 00:08:15,340 which have no classical analogy in squeezed states. 109 00:08:15,340 --> 00:08:18,720 Eventually, we want to focus on single photons. 110 00:08:22,960 --> 00:08:26,795 We will then talk about entangled photons. 111 00:08:33,240 --> 00:08:39,460 The fifth part is on interferometry, or more 112 00:08:39,460 --> 00:08:40,554 generally on metrology. 113 00:08:43,130 --> 00:08:45,820 We will talk about the short noise 114 00:08:45,820 --> 00:08:48,290 limit and the fundamental Heisenberg 115 00:08:48,290 --> 00:08:52,690 limit of performing quantum measurements. 116 00:08:52,690 --> 00:08:59,770 And the last one is now opening up the light to interactions 117 00:08:59,770 --> 00:09:01,600 with atoms. 118 00:09:01,600 --> 00:09:03,630 So we will talk about interaction 119 00:09:03,630 --> 00:09:09,600 of one photon and one atom in this specific situation 120 00:09:09,600 --> 00:09:12,700 of cavity QED. 121 00:09:12,700 --> 00:09:15,610 So this is an overview over the next, I don't know, 122 00:09:15,610 --> 00:09:19,910 three, four, five, six lectures. 123 00:09:19,910 --> 00:09:43,760 Today, we start with a very, very short review 124 00:09:43,760 --> 00:09:45,710 of the simple harmonic oscillator, just 125 00:09:45,710 --> 00:09:47,195 to set the stage. 126 00:09:51,150 --> 00:09:53,290 I then want to show you that in three minutes, 127 00:09:53,290 --> 00:09:56,430 I can discuss thermal light and the Planck Law 128 00:09:56,430 --> 00:09:58,700 by just writing down one or two equations 129 00:09:58,700 --> 00:10:02,440 in the formalism we have chosen. 130 00:10:02,440 --> 00:10:03,730 So we cover thermal states. 131 00:10:08,530 --> 00:10:10,640 When we do laser physics, when we 132 00:10:10,640 --> 00:10:16,640 have light atom in the action, we always need coherent states. 133 00:10:16,640 --> 00:10:19,077 And we talk about quasi-probability 134 00:10:19,077 --> 00:10:19,660 distributions. 135 00:10:25,010 --> 00:10:30,410 We then want to understand fluctuations, noise, 136 00:10:30,410 --> 00:10:35,200 and second order correlation functions. 137 00:10:35,200 --> 00:10:38,600 Probably this will spill over into the next class. 138 00:10:38,600 --> 00:10:43,060 We want to address some properties 139 00:10:43,060 --> 00:10:45,960 of the single photon. 140 00:10:45,960 --> 00:10:50,120 Which, as you will see, it seems a simple system, 141 00:10:50,120 --> 00:10:53,610 but it is the most non-classical system you can have, 142 00:10:53,610 --> 00:10:54,625 a single photon. 143 00:10:54,625 --> 00:10:58,480 But we'll come to that. 144 00:10:58,480 --> 00:11:04,490 Let's start with very, very short review 145 00:11:04,490 --> 00:11:06,470 of the simple harmonic oscillator. 146 00:11:06,470 --> 00:11:10,820 It just also connects me with the last lecture. 147 00:11:10,820 --> 00:11:16,990 So how do we describe light in a single mode? 148 00:11:16,990 --> 00:11:20,350 And single mode means it is perfectly monochromatic. 149 00:11:25,100 --> 00:11:30,740 Well, we know that-- and that's what 150 00:11:30,740 --> 00:11:34,200 we did last class-- that we have the Hamiltonian 151 00:11:34,200 --> 00:11:39,550 for the electromagnetic field, which is the quantum 152 00:11:39,550 --> 00:11:49,600 version of the electromagnetic energy, B square E square. 153 00:11:49,600 --> 00:11:55,730 And from that, we derived the quantized Hamiltonian. 154 00:11:55,730 --> 00:12:00,130 We introduced, just as a reminder, vector potentials, 155 00:12:00,130 --> 00:12:02,840 normal modes, and the normal mode operators 156 00:12:02,840 --> 00:12:05,530 became the a daggers. 157 00:12:05,530 --> 00:12:09,640 And in second quantization, this is the Hamiltonian 158 00:12:09,640 --> 00:12:12,460 for light, which is of course the Hamiltonian 159 00:12:12,460 --> 00:12:13,460 for harmonic oscillator. 160 00:12:16,440 --> 00:12:22,570 What I need today is I have to rewrite the harmonic 161 00:12:22,570 --> 00:12:23,680 oscillator. 162 00:12:23,680 --> 00:12:27,770 And we have to look at two variables-- 163 00:12:27,770 --> 00:12:30,980 the momentum and the precision. 164 00:12:30,980 --> 00:12:33,490 Actually, the momentum will later, 165 00:12:33,490 --> 00:12:35,140 I will show you that the momentum 166 00:12:35,140 --> 00:12:37,650 of this harmonic oscillator is actually 167 00:12:37,650 --> 00:12:39,860 the operator of the electric field. 168 00:12:39,860 --> 00:12:43,630 So we need that in a second. 169 00:12:43,630 --> 00:12:46,640 And of course, I assume you know from your knowledge 170 00:12:46,640 --> 00:12:51,560 of the harmonic oscillator that precision and momentum are 171 00:12:51,560 --> 00:12:55,750 linear combinations, symmetric and anti-symmetric, of a dagger 172 00:12:55,750 --> 00:12:58,110 a. 173 00:12:58,110 --> 00:13:02,810 And the unit is the prefactor square root 174 00:13:02,810 --> 00:13:04,880 of h bar omega over 2. 175 00:13:04,880 --> 00:13:10,800 Or here it is h bar over 2 omega. 176 00:13:10,800 --> 00:13:17,240 And of course, in precision and momentum operators, 177 00:13:17,240 --> 00:13:22,080 the Hamiltonian is simply p square, 178 00:13:22,080 --> 00:13:25,760 should remind you of kinetic energy. 179 00:13:25,760 --> 00:13:30,100 Omega square, Q square should remind you of potential energy. 180 00:13:33,330 --> 00:13:38,000 When we discuss light, we want to discuss the electric field. 181 00:13:40,900 --> 00:13:44,140 I had shown you in one of those 100 182 00:13:44,140 --> 00:13:54,950 equations the operator for the electric field. 183 00:13:58,090 --> 00:14:02,090 The operator for the electric field 184 00:14:02,090 --> 00:14:05,470 was, of course, related to the vector potential. 185 00:14:05,470 --> 00:14:08,290 And the vector potential, which is a normal mode expansion. 186 00:14:08,290 --> 00:14:10,250 And then we put everything backward. 187 00:14:10,250 --> 00:14:13,820 And we found that the electric field operator can actually 188 00:14:13,820 --> 00:14:14,780 be written. 189 00:14:14,780 --> 00:14:17,450 Have to fill in a few things, but essentially, 190 00:14:17,450 --> 00:14:21,990 it's a minus a dagger. 191 00:14:21,990 --> 00:14:26,580 And a minus a dagger is nothing else 192 00:14:26,580 --> 00:14:30,940 than the momentum of the harmonic oscillator. 193 00:14:30,940 --> 00:14:33,620 So here, the momentum has a special role, 194 00:14:33,620 --> 00:14:37,570 because it is the operator of the electric field. 195 00:14:37,570 --> 00:14:39,900 If you have light in a single mode, 196 00:14:39,900 --> 00:14:42,420 it has a harmonic oscillator representation. 197 00:14:42,420 --> 00:14:45,360 And the operator for the electric field 198 00:14:45,360 --> 00:14:48,110 is-- I put in a few bells and whistles in a moment-- 199 00:14:48,110 --> 00:14:51,050 but is in essence the momentum operator 200 00:14:51,050 --> 00:14:53,590 of the mechanical and analogous harmonic oscillator. 201 00:14:57,350 --> 00:15:02,560 OK but the electric field has a polarization. 202 00:15:02,560 --> 00:15:03,810 We need that. 203 00:15:10,640 --> 00:15:16,590 We have the photon energy and the quantization volume. 204 00:15:16,590 --> 00:15:20,300 So what I'm writing down here is actually 205 00:15:20,300 --> 00:15:23,460 the electric field, this square root, 206 00:15:23,460 --> 00:15:26,370 is the electric field strings of a single photon 207 00:15:26,370 --> 00:15:30,600 in the quantized volume of our cavity. 208 00:15:30,600 --> 00:15:32,820 So v is the quantization volume. 209 00:15:39,350 --> 00:15:45,200 Then if you have a single mode of the electromagnetic field, 210 00:15:45,200 --> 00:15:50,030 well, in contrast to a harmonic oscillator, 211 00:15:50,030 --> 00:15:55,780 we have a prorogation kr plus omega t. 212 00:15:55,780 --> 00:15:58,440 And here we have e to the minus i. 213 00:16:01,020 --> 00:16:02,460 I hope I get the signs wrong here. 214 00:16:02,460 --> 00:16:03,720 Minus omega t. 215 00:16:07,340 --> 00:16:09,760 But, OK, we will mainly be looking 216 00:16:09,760 --> 00:16:11,390 at an atom at the origin. 217 00:16:11,390 --> 00:16:13,960 That's what we did in the dipole approximation. 218 00:16:13,960 --> 00:16:17,630 So when I said that the electric field operator corresponds 219 00:16:17,630 --> 00:16:24,000 to the momentum, I have to add now that it is at t equals 0 220 00:16:24,000 --> 00:16:25,070 and at r equals 0. 221 00:16:30,010 --> 00:16:34,410 We'll come back to that in a few moments. 222 00:16:34,410 --> 00:16:38,270 OK, since I've given you the operator-- 223 00:16:38,270 --> 00:16:42,150 let me just put the operator symbol on it-- 224 00:16:42,150 --> 00:16:45,010 the operator of the electric field, 225 00:16:45,010 --> 00:16:49,740 I have to say now one word about Heisenberg operators 226 00:16:49,740 --> 00:16:52,020 versus Schrodinger operators. 227 00:16:52,020 --> 00:16:54,200 I assume you're all familiar with the two 228 00:16:54,200 --> 00:16:57,080 different representations of quantum mechanics. 229 00:16:57,080 --> 00:16:59,660 We often use the Schrodinger representation 230 00:16:59,660 --> 00:17:02,160 where the operators are time independent. 231 00:17:02,160 --> 00:17:04,750 But in the Heisenberg representation, 232 00:17:04,750 --> 00:17:08,310 the wave function, the state vector, is time independent. 233 00:17:08,310 --> 00:17:13,050 And the time dependence of the evolution of the quantum system 234 00:17:13,050 --> 00:17:16,240 leads to a time dependence of the operator. 235 00:17:16,240 --> 00:17:19,420 So what I've written down for you here 236 00:17:19,420 --> 00:17:27,280 is actually the Heisenberg operator, 237 00:17:27,280 --> 00:17:29,450 because it has a time revolution. 238 00:17:29,450 --> 00:17:34,590 But for a single mode, monochromatic electromagnetic 239 00:17:34,590 --> 00:17:37,820 field, the time evolution is just e to the i omega t. 240 00:17:40,850 --> 00:17:49,600 So if you want to obtain the Schrodinger operator, 241 00:17:49,600 --> 00:17:52,060 you have to eliminate the time dependence 242 00:17:52,060 --> 00:17:55,560 by setting t equals 0. 243 00:17:55,560 --> 00:18:00,030 At t equals 0, by definition, the Schrodinger operator 244 00:18:00,030 --> 00:18:03,230 and the Heisenberg operator are the same. 245 00:18:03,230 --> 00:18:05,260 But then, in the Schrodinger picture, 246 00:18:05,260 --> 00:18:08,270 the quantum state evolves and the operator stays put. 247 00:18:08,270 --> 00:18:11,820 In the Heisenberg picture, the quantum state stays constant 248 00:18:11,820 --> 00:18:14,230 and the operator evolves. 249 00:18:14,230 --> 00:18:17,520 So I'm usually writing down for you 250 00:18:17,520 --> 00:18:19,330 expressions for Heisenberg operators. 251 00:18:19,330 --> 00:18:22,677 And if you simply set t equals 0 in this expression, 252 00:18:22,677 --> 00:18:24,093 you have the Schrodinger operator. 253 00:18:28,260 --> 00:18:30,170 Questions? 254 00:18:30,170 --> 00:18:30,710 Yes. 255 00:18:30,710 --> 00:18:33,043 AUDIENCE: What happens when there's a factor of two that 256 00:18:33,043 --> 00:18:35,395 shows up in the electric field of a single photon, 257 00:18:35,395 --> 00:18:36,880 like the [INAUDIBLE]? 258 00:18:39,850 --> 00:18:42,572 I thought it was like-- [INAUDIBLE] it's 259 00:18:42,572 --> 00:18:44,320 just a factor of two [INAUDIBLE]. 260 00:18:44,320 --> 00:18:47,070 PROFESSOR: Yes, you have to be careful with factors of two. 261 00:18:47,070 --> 00:18:48,242 I think I got it right here. 262 00:18:48,242 --> 00:18:49,575 There is no extra factor of two. 263 00:18:53,670 --> 00:18:58,410 But the question is, OK, whenever you write down 264 00:18:58,410 --> 00:19:00,330 equations for the electric field, 265 00:19:00,330 --> 00:19:05,020 you want to sort of use a and a daggers photon number, 266 00:19:05,020 --> 00:19:08,560 or create photon numbers, so they are dimensionless. 267 00:19:08,560 --> 00:19:11,820 So you always need a prefactor, which 268 00:19:11,820 --> 00:19:14,390 has a dimension of the electric field. 269 00:19:14,390 --> 00:19:20,110 And for the quantisation of light, you have two choices. 270 00:19:20,110 --> 00:19:23,400 One is I can take the electric field associated 271 00:19:23,400 --> 00:19:24,890 with the single photon. 272 00:19:24,890 --> 00:19:26,160 That's one option. 273 00:19:26,160 --> 00:19:29,440 The other option is I take as a prefactor 274 00:19:29,440 --> 00:19:34,990 the electric field in the ground state, which you can say 275 00:19:34,990 --> 00:19:37,750 is the zero point fluctuations of the electric field 276 00:19:37,750 --> 00:19:39,960 and the two differ by a factor of two, 277 00:19:39,960 --> 00:19:43,320 because the energy of the vacuum field 278 00:19:43,320 --> 00:19:47,130 is 1/2 h bar omega, whereas 1 [INAUDIBLE] equals h bar omega. 279 00:19:47,130 --> 00:19:51,395 So that's one reason why you may sometimes find factors of two 280 00:19:51,395 --> 00:19:54,190 in some textbooks or not in others. 281 00:19:54,190 --> 00:19:56,417 But I think-- I checked. 282 00:19:56,417 --> 00:19:58,250 I think, as far as I know, it is consistent. 283 00:19:58,250 --> 00:20:01,238 AUDIENCE: So these operators you've written down for e, 284 00:20:01,238 --> 00:20:04,420 this is the Heisenberg operator with the time frames right now? 285 00:20:04,420 --> 00:20:06,170 PROFESSOR: This is the Heisenberg operator 286 00:20:06,170 --> 00:20:08,580 for the electric field. 287 00:20:08,580 --> 00:20:09,520 But it differs. 288 00:20:09,520 --> 00:20:14,710 I mean, at that point, don't-- there's nothing really to worry 289 00:20:14,710 --> 00:20:16,440 about or get concerned about. 290 00:20:16,440 --> 00:20:18,245 The question is, for monochromatic light, 291 00:20:18,245 --> 00:20:22,550 do we put the e to the i omega t into the state or the operator? 292 00:20:22,550 --> 00:20:24,550 It's really just an e to the i omega 293 00:20:24,550 --> 00:20:29,617 t factor has to be put either on this state or on the operator. 294 00:20:29,617 --> 00:20:30,700 These are the two choices. 295 00:20:35,980 --> 00:20:40,910 Actually, my advice to you is don't even 296 00:20:40,910 --> 00:20:43,880 develop an intuitive conceptual understanding. 297 00:20:43,880 --> 00:20:45,520 And whether you are in the Schrodinger, 298 00:20:45,520 --> 00:20:48,490 try to sort of see the structure of the operator. 299 00:20:48,490 --> 00:20:51,190 I sometimes like the omega t, because it shows you 300 00:20:51,190 --> 00:20:53,070 how the electric field evolves. 301 00:20:53,070 --> 00:20:54,690 And you see that explicitly. 302 00:20:54,690 --> 00:20:58,290 But in the end, it's only when you do real calculation 303 00:20:58,290 --> 00:21:01,370 that you have to think hard, OK, which picture am I using now? 304 00:21:01,370 --> 00:21:03,790 Which representation am I using now? 305 00:21:03,790 --> 00:21:06,262 I'm probably not 100% rigorous in this course 306 00:21:06,262 --> 00:21:08,720 when I write down quantum states and operators, whether I'm 307 00:21:08,720 --> 00:21:10,410 always in the Schrodinger picture 308 00:21:10,410 --> 00:21:12,070 or in the Heisenberg picture. 309 00:21:12,070 --> 00:21:14,800 But by simply looking, does it have a time dependence? 310 00:21:14,800 --> 00:21:15,300 Yes. 311 00:21:15,300 --> 00:21:16,760 It's a Heisenberg picture. 312 00:21:16,760 --> 00:21:21,030 If I write down a quantum state, which has a time dependence, 313 00:21:21,030 --> 00:21:24,160 then I've chosen the Schrodinger representation. 314 00:21:24,160 --> 00:21:27,000 It's as easy as that. 315 00:21:27,000 --> 00:21:27,674 There was other? 316 00:21:27,674 --> 00:21:28,173 Yeah. 317 00:21:28,173 --> 00:21:29,839 AUDIENCE: There's a sign wrong up there, 318 00:21:29,839 --> 00:21:34,054 because both a and a dagger goes into the i omega 319 00:21:34,054 --> 00:21:35,548 t as you have written down. 320 00:21:35,548 --> 00:21:39,895 So they should be kind of rotating right. 321 00:21:39,895 --> 00:21:41,145 PROFESSOR: Give me one second. 322 00:22:11,700 --> 00:22:17,070 Actually, I just wanted to say I was careful with my notes, 323 00:22:17,070 --> 00:22:19,417 but I didn't carefully look at my notes. 324 00:22:19,417 --> 00:22:20,750 This is what I have in my notes. 325 00:22:20,750 --> 00:22:21,710 Does that satisfy you? 326 00:22:21,710 --> 00:22:22,501 AUDIENCE: Yeah. 327 00:22:22,501 --> 00:22:23,334 PROFESSOR: OK, good. 328 00:22:27,780 --> 00:22:34,365 OK so we want to talk about quantum states of light. 329 00:22:37,650 --> 00:22:42,110 The simplest eigenstates are, of course, 330 00:22:42,110 --> 00:22:45,910 the harmonic oscillator eigenstates. 331 00:22:45,910 --> 00:22:49,050 But, as I've already mentioned, and we'll 332 00:22:49,050 --> 00:22:52,380 elaborate on it further, these are actually 333 00:22:52,380 --> 00:22:54,030 non-classical states. 334 00:22:54,030 --> 00:22:55,970 So quite often, we don't want those states. 335 00:22:55,970 --> 00:22:57,960 We want coherent states. 336 00:22:57,960 --> 00:23:05,840 But we need those number states said as one representation. 337 00:23:05,840 --> 00:23:07,740 So let me just introduce them. 338 00:23:14,900 --> 00:23:20,410 So we'll label by n the states with n photons. 339 00:23:23,190 --> 00:23:29,855 And the energy of those states is n plus 1/2 h bar omega. 340 00:23:33,167 --> 00:23:34,250 The states are normalized. 341 00:23:38,580 --> 00:23:42,370 Sometimes, out of laziness, because it doesn't matter, 342 00:23:42,370 --> 00:23:45,240 we can eliminate the 1/2. 343 00:23:45,240 --> 00:23:49,020 It just makes some equations more compact. 344 00:23:49,020 --> 00:24:11,700 And if I do that, I can write the Hamiltonian like this. 345 00:24:15,250 --> 00:24:18,820 You are familiar, of course, with commutators 346 00:24:18,820 --> 00:24:20,680 of a and a daggers. 347 00:24:20,680 --> 00:24:26,170 And the only non-trivial matrix element of a and a dagger 348 00:24:26,170 --> 00:24:33,880 that it raises and lowers the photon number by one. 349 00:24:37,330 --> 00:24:42,490 OK, this was a short review of the simple harmonic oscillator 350 00:24:42,490 --> 00:24:49,320 as it applies to a single mode of the electromagnetic field. 351 00:24:49,320 --> 00:24:56,590 Now, just as a short example, in subsection two, 352 00:24:56,590 --> 00:24:59,490 we want to discuss the thermal state. 353 00:24:59,490 --> 00:25:01,530 And the thermal state is nothing else 354 00:25:01,530 --> 00:25:04,960 than photons in a black body cavity. 355 00:25:04,960 --> 00:25:08,770 We have a photon field, which is in equilibrium 356 00:25:08,770 --> 00:25:10,290 at temperature t. 357 00:25:16,630 --> 00:25:22,390 So if you use the canonical ensemble, 358 00:25:22,390 --> 00:25:26,120 our statistical operator is given 359 00:25:26,120 --> 00:25:30,610 by this well known expression. 360 00:25:30,610 --> 00:25:34,317 And if I use-- I mean that's all. 361 00:25:34,317 --> 00:25:35,900 I mean, isn't it wonderful once you're 362 00:25:35,900 --> 00:25:38,010 trained in statistical mechanics? 363 00:25:38,010 --> 00:25:40,740 This is all you have to know about a thermal state of light? 364 00:25:40,740 --> 00:25:42,200 That's it. 365 00:25:42,200 --> 00:25:44,370 This is the Planck Law. 366 00:25:44,370 --> 00:25:48,299 All we have to do is take the Hamiltonian, 367 00:25:48,299 --> 00:25:49,840 which we have written down, upstairs. 368 00:25:54,160 --> 00:25:56,810 So this is e to the minus n photons. 369 00:25:56,810 --> 00:26:01,420 n times h bar omega is the energy over kt 370 00:26:01,420 --> 00:26:03,130 divided by the partition function. 371 00:26:08,040 --> 00:26:12,000 This is the density matrix, a statistical operator 372 00:26:12,000 --> 00:26:14,635 for single mode light at thermal equilibrium. 373 00:26:19,320 --> 00:26:24,230 The partition function is very simple, 374 00:26:24,230 --> 00:26:32,390 because this sum over Boltzmann factors this 375 00:26:32,390 --> 00:26:36,340 is a geometric series, which can be simply 376 00:26:36,340 --> 00:26:44,830 summed up in this way-- divided by kt. 377 00:26:44,830 --> 00:26:56,870 So therefore, this expression here 378 00:26:56,870 --> 00:27:08,420 can be written as-- the density matrix in thermal equilibrium 379 00:27:08,420 --> 00:27:11,560 can be written in the number bases. 380 00:27:11,560 --> 00:27:14,260 And it's diagonal. 381 00:27:14,260 --> 00:27:17,910 And the prefactor is the probability 382 00:27:17,910 --> 00:27:21,520 to have n photons populated in thermal equilibrium. 383 00:27:26,130 --> 00:27:33,590 And the probability for n photons is given here. 384 00:27:33,590 --> 00:27:36,610 We've evaluated the partition function. 385 00:27:36,610 --> 00:27:39,140 So then if you put it together, we simply 386 00:27:39,140 --> 00:27:44,750 get this-- you can call it Boltz-Einesten factor, which 387 00:27:44,750 --> 00:27:48,730 tells us what is the probability to find 388 00:27:48,730 --> 00:27:52,930 an harmonic oscillator with n [INAUDIBLE]. 389 00:27:52,930 --> 00:27:57,415 And this is the expression-- e to the minus h bar omega. 390 00:28:00,550 --> 00:28:14,910 So this result takes care of many situations-- 391 00:28:14,910 --> 00:28:17,200 actually, almost all situations-- 392 00:28:17,200 --> 00:28:20,350 you find in every day life unless you 393 00:28:20,350 --> 00:28:22,350 switch on a laser pointer. 394 00:28:22,350 --> 00:28:31,260 So this describes-- and resistive filament. 395 00:28:37,570 --> 00:28:40,545 It describes the light generated by gas discharges. 396 00:28:44,610 --> 00:28:48,980 It describes the light of the sun. 397 00:28:48,980 --> 00:28:55,010 And this kind of thermal light, because it 398 00:28:55,010 --> 00:28:59,800 has the-- I mean, light in thermal equilibrium 399 00:28:59,800 --> 00:29:05,350 maximizing the entropy under the given boundary condition, 400 00:29:05,350 --> 00:29:07,080 it's also called chaotic light. 401 00:29:11,360 --> 00:29:13,470 It's just another name for thermal light. 402 00:29:18,650 --> 00:29:24,120 Well, we should now lean back and take the attitude, 403 00:29:24,120 --> 00:29:27,310 if I know the statistical operator of an ensemble, 404 00:29:27,310 --> 00:29:29,800 I know everything about it. 405 00:29:29,800 --> 00:29:32,600 And every quantity I'm interested in 406 00:29:32,600 --> 00:29:35,420 can be calculated. 407 00:29:35,420 --> 00:29:40,520 So just as an example, we know what 408 00:29:40,520 --> 00:29:42,760 is a probability to find n photons. 409 00:29:46,730 --> 00:29:51,570 But now we want to ask what is the energy density, 410 00:29:51,570 --> 00:29:55,990 or what is the mean excitation number. 411 00:29:55,990 --> 00:30:00,010 What is the average of n? 412 00:30:00,010 --> 00:30:05,630 So for that, we have to take the number operator, a dagger a, 413 00:30:05,630 --> 00:30:08,250 and perform the trace with the product 414 00:30:08,250 --> 00:30:10,730 of the statistical operator. 415 00:30:10,730 --> 00:30:14,710 And this is, of course, nothing else when 416 00:30:14,710 --> 00:30:17,920 we look for the probability to find n photons. 417 00:30:17,920 --> 00:30:22,960 And this probability is weighted with n. 418 00:30:22,960 --> 00:30:27,235 And what we get is the very simple result. 419 00:30:30,700 --> 00:30:32,700 It's a Bose-Einstein distribution law 420 00:30:32,700 --> 00:30:35,720 with a chemical potential of 0, which 421 00:30:35,720 --> 00:30:37,610 is the familiar distribution. 422 00:30:43,300 --> 00:30:52,530 We can now use the quantity n to rewrite our previous expression 423 00:30:52,530 --> 00:31:00,300 for the probability to find n photons in thermal equilibrium. 424 00:31:00,300 --> 00:31:04,310 It is n bar to the power n divided 425 00:31:04,310 --> 00:31:09,470 by 1 over n bar times n plus 1. 426 00:31:09,470 --> 00:31:12,450 So all this should be very familiar to you. 427 00:31:12,450 --> 00:31:16,020 And I'm not doing the final step, 428 00:31:16,020 --> 00:31:23,080 but I could immediately get Planck's Law now. 429 00:31:23,080 --> 00:31:26,550 All I have to use it sum up now. 430 00:31:26,550 --> 00:31:29,500 This is a single mode of light, single monochromatic light. 431 00:31:29,500 --> 00:31:31,900 All I have to do is to get Planck's law is 432 00:31:31,900 --> 00:31:35,470 to sum overall modes using the density of modes. 433 00:31:47,844 --> 00:31:48,385 Any question? 434 00:31:54,040 --> 00:31:57,040 We will not be talking a lot in this course 435 00:31:57,040 --> 00:32:01,870 about chaotic and thermal light, because we 436 00:32:01,870 --> 00:32:05,450 are mainly interested in the purer form 437 00:32:05,450 --> 00:32:07,340 how atom interacts with light. 438 00:32:07,340 --> 00:32:11,080 And this will be when light is in a more specific state, 439 00:32:11,080 --> 00:32:14,200 not in a thermal ensemble, in a more specific state 440 00:32:14,200 --> 00:32:16,850 which are number states or coherent states. 441 00:32:16,850 --> 00:32:19,810 But nevertheless, I feel if I don't tell you 442 00:32:19,810 --> 00:32:21,950 a little bit about ordinary light, 443 00:32:21,950 --> 00:32:25,610 you will not be able to appreciate quantum light. 444 00:32:25,610 --> 00:32:28,540 So I want to use this example now of a thermal distribution 445 00:32:28,540 --> 00:32:33,500 of light and write down for you in the next 90 seconds 446 00:32:33,500 --> 00:32:35,630 expressions for the fluctuations. 447 00:32:35,630 --> 00:32:37,380 So you should know a little bit about what 448 00:32:37,380 --> 00:32:40,260 is the statistic of thermal light. 449 00:32:40,260 --> 00:32:46,569 If you find n photons in a black cavity and you look again, 450 00:32:46,569 --> 00:32:49,110 it's the standard deviation of the photon number square root. 451 00:32:49,110 --> 00:32:51,500 And is it Poissonian, or is it much larger? 452 00:32:51,500 --> 00:32:54,700 So you should know about that, because it will then 453 00:32:54,700 --> 00:32:57,550 make a difference when we talk about the purer 454 00:32:57,550 --> 00:33:01,190 states of light, coherent states, and number 455 00:33:01,190 --> 00:33:02,800 states, squeeze states, and all that. 456 00:33:06,550 --> 00:33:11,750 OK, so we want to look now at fluctuations. 457 00:33:11,750 --> 00:33:16,496 Let me just first mention that we have normalized correctly 458 00:33:16,496 --> 00:33:17,246 the probabilities. 459 00:33:24,830 --> 00:33:26,660 Just one second. 460 00:33:26,660 --> 00:33:30,440 What we are interested now is the fluctuations. 461 00:33:30,440 --> 00:33:33,350 So we want to know the variance. 462 00:33:33,350 --> 00:33:39,100 We want to know what is delta n squared. 463 00:33:39,100 --> 00:33:42,860 And delta n squared is of course n 464 00:33:42,860 --> 00:33:46,155 square average minus n average squared. 465 00:33:49,170 --> 00:33:56,205 And we want to find out what is that for thermal light, 466 00:33:56,205 --> 00:33:58,300 for chaotic light. 467 00:33:58,300 --> 00:34:02,050 So what we need now is we need an expression. 468 00:34:02,050 --> 00:34:06,370 We have already an expression up there for n average. 469 00:34:06,370 --> 00:34:08,680 And now we need an expression for n square average. 470 00:34:17,500 --> 00:34:26,540 We can get that as follows. 471 00:34:26,540 --> 00:34:45,719 If we take the probability, multiply it with n-- n minus 1. 472 00:34:45,719 --> 00:34:46,640 Sum over. 473 00:34:57,330 --> 00:34:58,330 So I have a brain cramp. 474 00:34:58,330 --> 00:35:02,640 This equation is in my notes, but I don't-- maybe step back-- 475 00:35:02,640 --> 00:35:04,660 I don't see where it comes from. 476 00:35:04,660 --> 00:35:06,160 But from this expression, of course, 477 00:35:06,160 --> 00:35:09,020 we have now an expression for n average squared. 478 00:35:11,970 --> 00:35:18,890 n squared-- the sum over it. 479 00:35:29,840 --> 00:35:30,440 Just a second. 480 00:35:37,699 --> 00:35:54,954 n average-- n thermal-- n average squared. 481 00:35:57,912 --> 00:36:00,399 Anyway, I don't want to waste class time. 482 00:36:00,399 --> 00:36:02,190 It made sense to me when I wrote the notes. 483 00:36:02,190 --> 00:36:05,360 I just can't reproduce the argument. 484 00:36:05,360 --> 00:36:08,630 So this should be obvious. 485 00:36:08,630 --> 00:36:10,640 It's not obvious to me right now. 486 00:36:10,640 --> 00:36:12,640 But let's just take it and run with it. 487 00:36:15,400 --> 00:36:19,240 The left hand side is n square minus n 488 00:36:19,240 --> 00:36:21,220 multiplied with the probability. 489 00:36:30,860 --> 00:36:32,110 So let me just take a note. 490 00:36:35,680 --> 00:36:38,370 I will show you where it comes from on Monday. 491 00:36:38,370 --> 00:36:41,352 But assuming that it's true, and I 492 00:36:41,352 --> 00:36:47,960 know it is true, that actually contains 493 00:36:47,960 --> 00:37:04,525 n square average minus n average is 2n average squared. 494 00:37:11,460 --> 00:37:15,200 So therefore, we have now our expression 495 00:37:15,200 --> 00:37:19,120 for the average of n square, which 496 00:37:19,120 --> 00:37:27,360 is 2n average squared plus n average. 497 00:37:27,360 --> 00:37:39,860 And therefore, for thermal light, for chaotic light, 498 00:37:39,860 --> 00:37:48,210 the variance, delta n square, is n average square 499 00:37:48,210 --> 00:37:49,510 plus n average. 500 00:37:52,740 --> 00:37:59,810 So the question is if this is our variance-- 501 00:37:59,810 --> 00:38:05,680 the variance of thermal light, of chaotic light, is that. 502 00:38:05,680 --> 00:38:07,700 What you should see immediately is 503 00:38:07,700 --> 00:38:11,590 that it is much more fluctuating then Poissonian statistics. 504 00:38:17,110 --> 00:38:28,800 Well, if you compare now with Poissonian statistics, 505 00:38:28,800 --> 00:38:36,050 Poissonian statistics would have fluctuations 506 00:38:36,050 --> 00:38:41,840 where, well, the RMS fluctuations are square root n. 507 00:38:41,840 --> 00:38:43,540 The variance is just n average. 508 00:38:46,550 --> 00:38:50,670 So thermal light is super Poissonian, 509 00:38:50,670 --> 00:38:55,209 has much, much stronger fluctuations than Poissonian 510 00:38:55,209 --> 00:38:55,750 distribution. 511 00:39:02,360 --> 00:39:06,140 OK, one question for you. 512 00:39:06,140 --> 00:39:10,435 If you look at the distribution of photons-- and we 513 00:39:10,435 --> 00:39:13,390 have written it down here. 514 00:39:13,390 --> 00:39:14,400 Let me just scroll up. 515 00:39:18,280 --> 00:39:22,150 Well, you probably don't have an auto-plot function 516 00:39:22,150 --> 00:39:23,150 in your brain. 517 00:39:23,150 --> 00:39:26,890 But if you plot this function, and I'm 518 00:39:26,890 --> 00:39:31,720 asking you what is the most probable photon number 519 00:39:31,720 --> 00:39:34,510 you will find for light in thermal equilibrium? 520 00:39:39,210 --> 00:39:40,150 AUDIENCE: Zero. 521 00:39:40,150 --> 00:39:42,850 PROFESSOR: Zero. 522 00:39:42,850 --> 00:39:45,880 It is a broad Gaussian centered at zero. 523 00:39:45,880 --> 00:39:47,840 So you have a broad Gaussian. 524 00:39:47,840 --> 00:39:49,810 The average photon number is related 525 00:39:49,810 --> 00:39:52,130 to the width of the Gaussian. 526 00:39:52,130 --> 00:39:55,590 But the nature of thermal light is 527 00:39:55,590 --> 00:39:59,080 that whenever you look into a black body cavity 528 00:39:59,080 --> 00:40:03,970 and ask, what is the most probable photon 529 00:40:03,970 --> 00:40:08,260 number for any mode, it is actually zero. 530 00:40:08,260 --> 00:40:11,410 No matter how hot the cavity is and how many 531 00:40:11,410 --> 00:40:14,650 photons you have in it, the photon number distribution 532 00:40:14,650 --> 00:40:21,800 is always a broad Gaussian with a maximum at zero. 533 00:40:21,800 --> 00:40:23,790 We'll come back to that later When 534 00:40:23,790 --> 00:40:25,860 we talk about coherent states. 535 00:40:25,860 --> 00:40:27,920 Then you will appreciate the difference. 536 00:40:27,920 --> 00:40:31,219 But I will be referring back to the result we have just 537 00:40:31,219 --> 00:40:33,010 derived for the photon number distribution. 538 00:40:38,810 --> 00:40:40,030 Let me just take a note. 539 00:40:42,770 --> 00:40:48,800 The most probably n is n equals 0. 540 00:40:53,900 --> 00:41:04,790 OK, I hope with that-- and this is III, the third subsection-- 541 00:41:04,790 --> 00:41:13,355 you can really appreciate the properties of coherent states. 542 00:41:18,130 --> 00:41:26,820 Coherent states were introduced by Roy Glauber in 1963. 543 00:41:26,820 --> 00:41:30,530 It's actually something you should wonder about. 544 00:41:30,530 --> 00:41:32,730 Coherent states seem so natural. 545 00:41:32,730 --> 00:41:34,620 I mean, we teach it in all courses. 546 00:41:34,620 --> 00:41:37,490 And when we think about harmonic oscillator, 547 00:41:37,490 --> 00:41:40,510 we often think in terms of coherent states. 548 00:41:40,510 --> 00:41:43,560 But they were not immediately used 549 00:41:43,560 --> 00:41:45,900 and invented when quantum mechanics 550 00:41:45,900 --> 00:41:47,710 was invented in the '30s. 551 00:41:47,710 --> 00:41:50,330 It really took somebody like Roy Glauber 552 00:41:50,330 --> 00:41:51,890 to invent coherent states. 553 00:41:51,890 --> 00:41:53,340 And that happened in the '60s. 554 00:41:58,320 --> 00:42:00,490 Maybe the reason why people didn't immediately 555 00:42:00,490 --> 00:42:04,090 jump at coherent states is because coherent states are not 556 00:42:04,090 --> 00:42:07,460 eigenstates of an Hamiltonian. 557 00:42:07,460 --> 00:42:09,340 The definition of a coherent state 558 00:42:09,340 --> 00:42:17,250 is actually an eigenstate, not of an Hamiltonian. 559 00:42:21,280 --> 00:42:24,250 It's an eigenstate of the annihilation operator. 560 00:42:30,334 --> 00:42:32,750 And of course, the annihilation operator is non-Hermitian. 561 00:42:32,750 --> 00:42:37,530 So it's a strange operator to look for eigenstates. 562 00:42:37,530 --> 00:42:40,800 But this is what defines the coherent state, 563 00:42:40,800 --> 00:42:46,650 namely that e acting on a coherent state alpha 564 00:42:46,650 --> 00:42:48,970 equals alpha times alpha. 565 00:42:51,730 --> 00:42:54,590 I'm using here the standard notation 566 00:42:54,590 --> 00:43:00,300 that alpha is both the label for the coherent state 567 00:43:00,300 --> 00:43:03,560 written down here, and is also the eigenvalue 568 00:43:03,560 --> 00:43:05,960 of the coherent state when acted upon 569 00:43:05,960 --> 00:43:07,890 with the annihilation operator. 570 00:43:13,920 --> 00:43:17,140 We normalize those states. 571 00:43:17,140 --> 00:43:20,690 And since we're talking about eigenvalues 572 00:43:20,690 --> 00:43:24,090 of a non-Hermitian operator, alpha 573 00:43:24,090 --> 00:43:25,760 can be any complex number. 574 00:43:37,380 --> 00:43:43,340 OK, so now we have introduced a second set of basis functions 575 00:43:43,340 --> 00:43:44,390 for light. 576 00:43:44,390 --> 00:43:48,170 Remember, we had the basis functions n, the eigenfunctions 577 00:43:48,170 --> 00:43:50,340 of the harmonic oscillator, and now we've 578 00:43:50,340 --> 00:43:52,927 defined a new set of eigenfunctions, 579 00:43:52,927 --> 00:43:54,010 the coherent state alphas. 580 00:43:56,720 --> 00:43:58,770 Of course, the next obvious question 581 00:43:58,770 --> 00:44:01,265 is how are the two representations connected? 582 00:44:04,540 --> 00:44:08,490 So what we want to do now is we want to figure out 583 00:44:08,490 --> 00:44:13,880 how do we write a coherent state alpha in the n basis 584 00:44:13,880 --> 00:44:16,640 in the basis of eigenfunctions of the harmonic oscillator? 585 00:44:20,900 --> 00:44:24,470 So we want to write-- this is just-- 586 00:44:24,470 --> 00:44:31,000 want to expand the coherent state into eigenfunctions n. 587 00:44:31,000 --> 00:44:33,480 So the coefficients we want to calculate 588 00:44:33,480 --> 00:44:38,630 are those matrix elements which are defined as cn. 589 00:44:46,190 --> 00:44:48,410 So what we have to use is in order 590 00:44:48,410 --> 00:44:50,500 to find this representation, we have 591 00:44:50,500 --> 00:45:00,600 to use the fact that alpha is an eigenstate of the annihilation 592 00:45:00,600 --> 00:45:02,000 operator. 593 00:45:02,000 --> 00:45:06,180 So we know that this is alpha times alpha. 594 00:45:06,180 --> 00:45:14,190 But we can also let the annihilation operator a, 595 00:45:14,190 --> 00:45:18,920 we can let the annihilation operator a act on this state 596 00:45:18,920 --> 00:45:21,980 and in this sum. 597 00:45:21,980 --> 00:45:26,740 And the annihilation operator, of course, 598 00:45:26,740 --> 00:45:29,630 does what it is supposed to do. 599 00:45:29,630 --> 00:45:31,695 It lowers the photon number by one. 600 00:45:36,230 --> 00:45:39,130 And we get the matrix element square root n. 601 00:45:41,770 --> 00:45:48,530 And then by just changing the summation index by one, 602 00:45:48,530 --> 00:45:54,440 we can write it as n times cn plus 1 times the square root 603 00:45:54,440 --> 00:45:57,250 of n plus 1. 604 00:45:57,250 --> 00:46:00,319 So this is when the annihilation operator acts on the sum. 605 00:46:00,319 --> 00:46:01,860 And now, I'll continue with the lower 606 00:46:01,860 --> 00:46:10,060 line, which is alpha cn times n. 607 00:46:10,060 --> 00:46:13,070 So what I've found now is I've found two ways 608 00:46:13,070 --> 00:46:17,550 to hide the same state, namely a acting on alpha. 609 00:46:17,550 --> 00:46:21,240 And therefore, those coefficients 610 00:46:21,240 --> 00:46:22,530 have to be the same. 611 00:46:25,930 --> 00:46:28,360 And what I just indicated in blue 612 00:46:28,360 --> 00:46:38,765 is a recursion relation, which expresses cn plus 1 by cn. 613 00:46:46,940 --> 00:46:54,140 So every time you go from cn to cn plus 1, 614 00:46:54,140 --> 00:46:57,363 you multiply with the square root of n plus 1. 615 00:47:01,310 --> 00:47:05,300 So if you stare at it for a moment, 616 00:47:05,300 --> 00:47:09,010 you find that cn is related to c0. 617 00:47:21,990 --> 00:47:41,040 Yes, cn plus 1 by alpha n over n factorial. 618 00:47:41,040 --> 00:47:42,830 So of course, we get only the relation 619 00:47:42,830 --> 00:47:44,450 between the coefficients. 620 00:47:44,450 --> 00:47:52,560 But if you throw in that we want to normalize the state, 621 00:47:52,560 --> 00:47:55,880 this determines c0. 622 00:47:55,880 --> 00:48:00,470 And then, we get as a final result 623 00:48:00,470 --> 00:48:06,490 the normalized representation of the coherent state 624 00:48:06,490 --> 00:48:07,790 in terms of number states. 625 00:48:19,110 --> 00:48:22,800 Excuse me, I think the square root is missing here, 626 00:48:22,800 --> 00:48:26,600 because it appears here times n. 627 00:48:31,020 --> 00:48:37,370 OK, so what we have found here is now 628 00:48:37,370 --> 00:48:42,910 how coherent states are written down in the number state basis. 629 00:48:53,080 --> 00:48:56,120 So now we know coherent states. 630 00:48:56,120 --> 00:48:58,500 And we can ask a number of questions. 631 00:48:58,500 --> 00:49:03,640 So what is this? 632 00:49:03,640 --> 00:49:05,320 So what did we define? 633 00:49:05,320 --> 00:49:07,644 And what have we now derived? 634 00:49:07,644 --> 00:49:09,185 Let's look at some of the properties. 635 00:49:12,340 --> 00:49:18,150 The first one is that if you look at this expression, 636 00:49:18,150 --> 00:49:21,550 this is the amplitude that will have 637 00:49:21,550 --> 00:49:25,410 n photons in a coherent state. 638 00:49:25,410 --> 00:49:27,560 The square of the amplitude is the probability. 639 00:49:30,150 --> 00:49:34,490 So the probability to find n photons 640 00:49:34,490 --> 00:49:44,140 is the square of the amplitude above. 641 00:49:46,660 --> 00:49:49,630 So this is that. 642 00:49:49,630 --> 00:49:51,870 And this is now a Poissonian distribution. 643 00:49:59,290 --> 00:50:03,340 So it's immediately clear that those coherent states 644 00:50:03,340 --> 00:50:06,220 have much, much smaller fluctuations in the photon 645 00:50:06,220 --> 00:50:09,050 number than the thermal state we discussed before. 646 00:50:09,050 --> 00:50:12,170 Because this is Poissonian, whereas the thermal state 647 00:50:12,170 --> 00:50:14,630 had a variance which was much, much larger than Poissonian. 648 00:50:19,080 --> 00:50:22,160 OK, that's number one. 649 00:50:22,160 --> 00:50:30,110 Number two is we would like to know what is the photon number. 650 00:50:30,110 --> 00:50:33,900 So we want to know what is the expectation value for n. 651 00:50:36,410 --> 00:50:39,280 That's actually very easy to find. 652 00:50:39,280 --> 00:50:43,460 You could, of course, get the average number of photons 653 00:50:43,460 --> 00:50:49,520 in by evaluating the expansion of the coherent state in terms 654 00:50:49,520 --> 00:50:51,670 of number states. 655 00:50:51,670 --> 00:50:55,190 But sometimes, you should also try to maybe directly evaluate 656 00:50:55,190 --> 00:50:56,650 the coherent state. 657 00:50:56,650 --> 00:51:01,166 And the photon number operator is a dagger a. 658 00:51:01,166 --> 00:51:03,260 And now you see here we are lucky, 659 00:51:03,260 --> 00:51:07,400 because a acting on alpha is just getting alpha. 660 00:51:07,400 --> 00:51:10,760 But a dagger acting on alpha on the left hand side 661 00:51:10,760 --> 00:51:14,990 gives us alpha star, the complex conjugate. 662 00:51:14,990 --> 00:51:18,010 So what we obtain here is simply the absolute value 663 00:51:18,010 --> 00:51:19,490 of alpha square. 664 00:51:19,490 --> 00:51:22,360 And this is our expression for the average photon number. 665 00:51:35,500 --> 00:51:39,060 I have already mentioned that the variance must 666 00:51:39,060 --> 00:51:44,220 be-- the variance must be n, because it's a Poissonian 667 00:51:44,220 --> 00:51:45,830 distribution. 668 00:51:45,830 --> 00:51:51,400 But since it is so nice to have creation annihilation 669 00:51:51,400 --> 00:51:54,290 operators acting on coherent states, 670 00:51:54,290 --> 00:51:57,390 let's just calculate-- let's verify 671 00:51:57,390 --> 00:52:00,220 that this distribution-- we know that distribution 672 00:52:00,220 --> 00:52:02,870 is Poissonian, but let's verify directly 673 00:52:02,870 --> 00:52:06,330 that the variance of the photon number in the coherent state 674 00:52:06,330 --> 00:52:08,660 is n, is Poissonian. 675 00:52:08,660 --> 00:52:11,950 If you need the variance, we need an average. 676 00:52:11,950 --> 00:52:13,760 And we need n square average. 677 00:52:13,760 --> 00:52:15,670 Let's just calculate n square average. 678 00:52:21,080 --> 00:52:24,120 So we need now the operator for n square, 679 00:52:24,120 --> 00:52:28,630 which is a dagger a, a dagger a. 680 00:52:28,630 --> 00:52:31,580 We know already if those two operators 681 00:52:31,580 --> 00:52:37,350 act on the [INAUDIBLE], that this just 682 00:52:37,350 --> 00:52:40,630 gives us alpha star alpha. 683 00:52:43,500 --> 00:52:46,190 And then we are left with a a dagger. 684 00:52:49,320 --> 00:52:51,530 Now, a dagger acts on alpha. 685 00:52:51,530 --> 00:52:54,410 We don't know what a dagger is acting on alpha. 686 00:52:54,410 --> 00:52:57,945 We know that alpha is an eigenstate of a, but not 687 00:52:57,945 --> 00:52:59,180 of a dagger. 688 00:52:59,180 --> 00:53:04,060 So therefore, we want to change the order of a and a dagger, 689 00:53:04,060 --> 00:53:04,855 use a commutator. 690 00:53:08,600 --> 00:53:12,720 And then of course, a dagger can act here. 691 00:53:12,720 --> 00:53:14,520 A can act here. 692 00:53:14,520 --> 00:53:17,820 And we just get another factor of alpha. 693 00:53:17,820 --> 00:53:20,930 So therefore, in a very straightforward way, 694 00:53:20,930 --> 00:53:23,880 we find now that n square is alpha 695 00:53:23,880 --> 00:53:30,450 square times alpha square plus 1. 696 00:53:30,450 --> 00:53:33,650 And using the result for the mean photon number, 697 00:53:33,650 --> 00:53:38,980 this is nothing else than n bar, 1 plus n bar. 698 00:53:47,430 --> 00:53:52,400 So if we now put it together, we want to know the variance. 699 00:53:52,400 --> 00:53:55,370 The variance is n square average, which we've just 700 00:53:55,370 --> 00:53:59,000 calculated, minus n average squared. 701 00:53:59,000 --> 00:54:03,130 And hooray, we find n bar. 702 00:54:03,130 --> 00:54:09,910 And we verify again that the fluctuations are Poissonian. 703 00:54:21,360 --> 00:54:24,350 What else can we ask? 704 00:54:24,350 --> 00:54:28,580 We've asked about the statistics of fluctuation. 705 00:54:28,580 --> 00:54:31,900 We've asked about the probability to find photons. 706 00:54:31,900 --> 00:54:33,900 Well, what else do you want to know about light? 707 00:54:37,040 --> 00:54:37,940 Electric field. 708 00:54:41,740 --> 00:54:44,789 Oh, that's something I forgot to ask. 709 00:54:44,789 --> 00:54:46,080 I will come back to that later. 710 00:54:46,080 --> 00:54:48,680 But just sort of to connect it, what 711 00:54:48,680 --> 00:54:52,240 is the average electric field for thermal light? 712 00:54:57,280 --> 00:54:58,556 Zero. 713 00:54:58,556 --> 00:55:01,140 And what else can it be? 714 00:55:01,140 --> 00:55:06,710 Because any value of the electric field, 715 00:55:06,710 --> 00:55:09,830 whether the electric field is positive or negative, 716 00:55:09,830 --> 00:55:15,190 requires that the-- I say it loosely-- 717 00:55:15,190 --> 00:55:16,840 that the photon has a phase. 718 00:55:16,840 --> 00:55:20,310 The phase will define whether it's positive or negative. 719 00:55:20,310 --> 00:55:25,060 But all different phases of the photon field, all different-- 720 00:55:25,060 --> 00:55:26,700 what I mean now is coherent states, 721 00:55:26,700 --> 00:55:29,410 but have different phases-- they are all degenerate. 722 00:55:29,410 --> 00:55:35,870 So in a thermal ensemble, the states are only 723 00:55:35,870 --> 00:55:38,150 populated due to their energy, they're 724 00:55:38,150 --> 00:55:40,800 populated by Boltzmann factors. 725 00:55:40,800 --> 00:55:43,370 And therefore, there is no distinction 726 00:55:43,370 --> 00:55:45,850 what the phase of the state are. 727 00:55:45,850 --> 00:55:49,990 So therefore, we haven't defined yet what the phase of a photon 728 00:55:49,990 --> 00:55:53,630 is, so therefore I'm a little bit guarded with my language. 729 00:55:53,630 --> 00:55:57,600 But in other words, whatever the state with the phase 730 00:55:57,600 --> 00:56:01,960 is in a thermal ensemble, you have all phases simultaneously. 731 00:56:01,960 --> 00:56:05,050 And therefore, you never have an expectation value of e. 732 00:56:07,630 --> 00:56:11,860 But of course, we had an average number of photons. 733 00:56:11,860 --> 00:56:14,940 And that means we have an expectation value of e square, 734 00:56:14,940 --> 00:56:17,340 because the expectation value of e square 735 00:56:17,340 --> 00:56:19,460 is the expectation value of the energy. 736 00:56:19,460 --> 00:56:22,494 The state has energy. 737 00:56:22,494 --> 00:56:24,160 But it doesn't have an expectation value 738 00:56:24,160 --> 00:56:27,130 for the electric field. 739 00:56:27,130 --> 00:56:30,050 OK, so maybe with that motivation, 740 00:56:30,050 --> 00:56:33,770 we want to figure out now does this coherent state, which 741 00:56:33,770 --> 00:56:37,520 we have constructed, does that have now 742 00:56:37,520 --> 00:56:40,170 an expectation value for the electric field? 743 00:56:52,730 --> 00:56:57,330 So there is one thing you of course 744 00:56:57,330 --> 00:56:59,330 know about this electric field. 745 00:56:59,330 --> 00:57:01,760 Whenever we have an electric field, 746 00:57:01,760 --> 00:57:03,655 it needs an amplitude and a phase. 747 00:57:08,580 --> 00:57:12,300 And what I just mentioned to you is that in a thermal ensemble, 748 00:57:12,300 --> 00:57:16,680 we were just looking at the partition function and such. 749 00:57:16,680 --> 00:57:18,480 We didn't even mention a phase. 750 00:57:18,480 --> 00:57:22,300 There was not any complex number in any of the equations 751 00:57:22,300 --> 00:57:24,000 we wrote down about thermal light. 752 00:57:24,000 --> 00:57:28,180 No complex number, no phase, means no electric field. 753 00:57:28,180 --> 00:57:32,100 But this is different now for the coherent state. 754 00:57:32,100 --> 00:57:37,780 Because I mentioned that alpha in general can be complex, 755 00:57:37,780 --> 00:57:42,130 the annihilation operator can have complex eigenvalues. 756 00:57:42,130 --> 00:57:45,080 So therefore, in the coherent state-- 757 00:57:45,080 --> 00:57:48,610 because it is characterized by a complex number-- 758 00:57:48,610 --> 00:57:53,630 we have now the possibility to include a phase. 759 00:57:56,980 --> 00:57:59,510 And what I want to show you is, in the next few minutes, 760 00:57:59,510 --> 00:58:03,050 is that yes indeed, the coherent state 761 00:58:03,050 --> 00:58:05,670 has an expectation value of the electric field. 762 00:58:11,840 --> 00:58:16,750 Instead of simply using the equation for the electric field 763 00:58:16,750 --> 00:58:20,620 operator-- remember electric field operator was a minus 764 00:58:20,620 --> 00:58:28,140 a dagger-- I want to introduce something else which 765 00:58:28,140 --> 00:58:30,320 will turn out to be very, very useful. 766 00:58:30,320 --> 00:58:34,960 And one of the application of this concept 767 00:58:34,960 --> 00:58:39,452 is to show you what the electric field is. 768 00:58:39,452 --> 00:58:46,010 This is sort of a visual representation, 769 00:58:46,010 --> 00:58:50,830 a visual representation of the states 770 00:58:50,830 --> 00:58:54,080 of the electromagnetic field. 771 00:58:54,080 --> 00:58:58,340 So I want to sort of give you, in a two dimensional diagram, 772 00:58:58,340 --> 00:59:03,010 a visual representation of thermal states, 773 00:59:03,010 --> 00:59:05,330 coherent states, number states. 774 00:59:05,330 --> 00:59:07,050 And this visual representation will 775 00:59:07,050 --> 00:59:11,830 make it immediately obvious what the electric field is. 776 00:59:11,830 --> 00:59:19,220 So the visual representation will 777 00:59:19,220 --> 00:59:21,730 represent what are called quasi-probabilities. 778 00:59:27,830 --> 00:59:30,740 So I have to define for you now those quasi-probabilities. 779 00:59:36,490 --> 00:59:44,710 So I denote the quasi-probability by Q. 780 00:59:44,710 --> 00:59:49,090 And I want to know what is the quasi-probability Q 781 00:59:49,090 --> 00:59:53,350 for any statistical operator which describes a light. 782 00:59:53,350 --> 00:59:56,470 It could be the operator for thermal light. 783 00:59:56,470 --> 01:00:01,050 It could be statistical operator also includes, of course, 784 01:00:01,050 --> 01:00:01,850 pure states. 785 01:00:01,850 --> 01:00:04,050 And the pure state could be a coherence state. 786 01:00:04,050 --> 01:00:07,710 So this is a way to define our quasi-probabilities 787 01:00:07,710 --> 01:00:12,920 for any statistical operator which is describing the light. 788 01:00:12,920 --> 01:00:19,030 And at the quasi-probabilities, well, the probability for what? 789 01:00:19,030 --> 01:00:22,490 Well, the probabilities for alpha. 790 01:00:22,490 --> 01:00:26,620 And alpha is related to the coherent state. 791 01:00:26,620 --> 01:00:31,820 So in other words, we say a statistical operator 792 01:00:31,820 --> 01:00:37,340 has a quasi-probability with a coherent state alpha by simply 793 01:00:37,340 --> 01:00:40,360 calculating this diagonal matrix element 794 01:00:40,360 --> 01:00:42,380 of the statistical operator with alpha. 795 01:00:47,020 --> 01:00:51,630 Well, that's an abstract definition, but it's exact. 796 01:00:51,630 --> 01:00:56,110 What we should now do is I want to show you some examples. 797 01:00:56,110 --> 01:00:58,230 And then you will actually see that it's actually 798 01:00:58,230 --> 01:00:59,188 a wonderful definition. 799 01:01:02,030 --> 01:01:06,310 So my first example is let's look 800 01:01:06,310 --> 01:01:10,180 at the quasi-probability for the vacuum state. 801 01:01:10,180 --> 01:01:14,160 So if the statistical operator is simply 802 01:01:14,160 --> 01:01:16,580 the ground state of the harmonic oscillator, 803 01:01:16,580 --> 01:01:18,755 no photon, no nothing, just empty vacuum. 804 01:01:21,700 --> 01:01:27,970 So the quasi-probability for the statistical operator 805 01:01:27,970 --> 01:01:30,300 is now a pure state. 806 01:01:30,300 --> 01:01:33,150 So I can say, what is a quasi-probability of the vacuum 807 01:01:33,150 --> 01:01:35,270 state to be in state alpha? 808 01:01:38,410 --> 01:01:40,950 Well, following the definition above, 809 01:01:40,950 --> 01:01:45,580 it's nothing else than the matrix element, the overlap, 810 01:01:45,580 --> 01:01:48,210 between the vacuum and the coherent state alpha. 811 01:01:52,560 --> 01:01:55,010 I've given you a path, the representation 812 01:01:55,010 --> 01:01:59,730 of the coherent state alpha in terms of number states. 813 01:01:59,730 --> 01:02:01,730 And all you have to do is look now 814 01:02:01,730 --> 01:02:06,550 upstairs, what is the amplitude that this representation 815 01:02:06,550 --> 01:02:07,900 includes a vacuum state. 816 01:02:07,900 --> 01:02:12,380 So it's just read of the amplitude c0 817 01:02:12,380 --> 01:02:14,220 from the state above. 818 01:02:14,220 --> 01:02:20,290 And this was simply e to the minus alpha squared. 819 01:02:20,290 --> 01:02:29,110 So therefore, if I-- those quasi-probabilities 820 01:02:29,110 --> 01:02:32,440 can be plotted in a two dimensional plane, 821 01:02:32,440 --> 01:02:35,700 because it's a function of alpha. 822 01:02:35,700 --> 01:02:40,730 So if I use the real part of alpha, the imaginary part 823 01:02:40,730 --> 01:02:47,720 of alpha, now I can plot the quasi-probability Q. 824 01:02:47,720 --> 01:02:55,130 And since it's hard for me to do a 3-D plot on the tablet, 825 01:02:55,130 --> 01:02:57,550 I just sort of shade the region. 826 01:02:57,550 --> 01:02:59,917 Black means the amplitude is large. 827 01:02:59,917 --> 01:03:01,000 And now it's sort of here. 828 01:03:01,000 --> 01:03:02,940 It means the amplitude falls off. 829 01:03:02,940 --> 01:03:06,700 So what I'm plotting here is a Gaussian, 830 01:03:06,700 --> 01:03:17,700 a Gaussian with the widths of [INAUDIBLE] 831 01:03:17,700 --> 01:03:19,197 with the width of 1/2. 832 01:03:19,197 --> 01:03:21,280 I think in the standard deviation of the Gaussian, 833 01:03:21,280 --> 01:03:23,179 you have a factor of 2. 834 01:03:23,179 --> 01:03:25,220 Anyway, it's on the order of-- the Gaussian width 835 01:03:25,220 --> 01:03:26,190 on the order of [INAUDIBLE] here, 836 01:03:26,190 --> 01:03:27,606 I think it's 1 over square root 2, 837 01:03:27,606 --> 01:03:30,416 or whatever the definition of the standard deviation is. 838 01:03:30,416 --> 01:03:31,540 So here we have a Gaussian. 839 01:03:34,230 --> 01:03:34,870 Any question? 840 01:03:42,090 --> 01:03:45,530 So for pure state, we're simply asking 841 01:03:45,530 --> 01:03:48,720 what is the amplitude of the purse state 842 01:03:48,720 --> 01:03:50,782 to overlap to be in a coherent state. 843 01:03:50,782 --> 01:03:52,240 And what we are plotting is nothing 844 01:03:52,240 --> 01:03:56,990 else than the probability that this state 845 01:03:56,990 --> 01:04:00,126 is a coherent state with value alpha. 846 01:04:00,126 --> 01:04:01,212 You had a question? 847 01:04:01,212 --> 01:04:02,128 AUDIENCE: [INAUDIBLE]. 848 01:04:08,300 --> 01:04:13,070 PROFESSOR: The next example, and we have already 849 01:04:13,070 --> 01:04:17,610 prepared for that, is the thermal state. 850 01:04:17,610 --> 01:04:24,560 Example number two is the thermal state. 851 01:04:24,560 --> 01:04:28,420 So we want to know what is the quasi-probability 852 01:04:28,420 --> 01:04:33,480 for the thermal state as a function of alpha. 853 01:04:38,880 --> 01:04:42,350 The statistical operator for the thermal states, we had derived 854 01:04:42,350 --> 01:04:47,450 [INAUDIBLE] was the probability to find n photons. 855 01:04:47,450 --> 01:04:50,120 And in the number representation, 856 01:04:50,120 --> 01:04:54,050 the statistical operator is diagonal. 857 01:04:54,050 --> 01:04:59,960 And then the quasi-probabilities are nothing else, 858 01:04:59,960 --> 01:05:09,760 following the definition above, then those probabilities times 859 01:05:09,760 --> 01:05:15,020 the matrix element, the overlap between alpha and n. 860 01:05:15,020 --> 01:05:21,490 And if you look up what we had derived for the probability pn, 861 01:05:21,490 --> 01:05:24,520 you find a very simple result. 862 01:05:24,520 --> 01:05:32,280 What we obtain is again a Gaussian. 863 01:05:39,650 --> 01:05:46,200 It is also centered at the origin. 864 01:05:46,200 --> 01:05:49,070 So therefore, since I didn't label my axis, 865 01:05:49,070 --> 01:05:51,657 it looks exactly as a vacuum state. 866 01:05:51,657 --> 01:05:53,740 It's just that the Gaussian is much, much broader. 867 01:05:57,700 --> 01:06:08,560 And I will come to that, but an intuitive reading 868 01:06:08,560 --> 01:06:11,280 of this Gaussian is, well, this Gaussian 869 01:06:11,280 --> 01:06:13,680 has a peak at the origin. 870 01:06:13,680 --> 01:06:17,640 At the origin, this is where alpha equals 0. 871 01:06:17,640 --> 01:06:20,210 And this is directly related to the question I asked you 872 01:06:20,210 --> 01:06:25,520 before, that the photon number probability is 873 01:06:25,520 --> 01:06:26,860 peaked at n equals 0. 874 01:06:30,030 --> 01:06:34,000 But I hope this will become even clearer when 875 01:06:34,000 --> 01:06:44,720 we move on to the third example, which is now 876 01:06:44,720 --> 01:06:50,480 Q of alpha of a coherent state. 877 01:07:00,870 --> 01:07:04,200 Maybe just before I even discuss the quasi-probability 878 01:07:04,200 --> 01:07:10,980 of a coherent state, I have to address one thing up front. 879 01:07:10,980 --> 01:07:15,740 What would you expect the quasi-probability 880 01:07:15,740 --> 01:07:18,891 of a coherent state to be? 881 01:07:18,891 --> 01:07:20,382 AUDIENCE: The Gaussian [INAUDIBLE]. 882 01:07:23,870 --> 01:07:24,930 PROFESSOR: Pretty good. 883 01:07:24,930 --> 01:07:29,800 But I said before, the quasi-probability 884 01:07:29,800 --> 01:07:35,360 is-- let me just show you how it was defined. 885 01:07:35,360 --> 01:07:38,110 Here was my definition of the quasi-probability. 886 01:07:38,110 --> 01:07:41,140 And you can see the quasi-probability 887 01:07:41,140 --> 01:07:47,550 is the diagram matrix element of the statistical operator 888 01:07:47,550 --> 01:07:49,380 with a coherent state. 889 01:07:49,380 --> 01:07:55,180 But if our statistical operator is a coherent state, what would 890 01:07:55,180 --> 01:07:57,120 you expect this probability to be? 891 01:07:57,120 --> 01:07:58,170 Just naively. 892 01:07:58,170 --> 01:07:59,400 AUDIENCE: One. 893 01:07:59,400 --> 01:08:01,270 PROFESSOR: One or a delta function. 894 01:08:01,270 --> 01:08:04,190 I mean, it's sort of-- we ask, we 895 01:08:04,190 --> 01:08:08,320 ask-- if you expand this statistical operator 896 01:08:08,320 --> 01:08:10,650 in coherent states, what do we get? 897 01:08:10,650 --> 01:08:13,740 But if this operator is a coherent state, 898 01:08:13,740 --> 01:08:16,479 let's say it's a coherent state beta, you would expect, 899 01:08:16,479 --> 01:08:19,840 well, then we only get something non-vanishing if alpha 900 01:08:19,840 --> 01:08:21,029 equals beta. 901 01:08:21,029 --> 01:08:23,670 So you would naively expect that what you get 902 01:08:23,670 --> 01:08:25,210 is a delta function. 903 01:08:25,210 --> 01:08:28,710 The coherent state beta has a quasi-probability Q 904 01:08:28,710 --> 01:08:31,180 of alpha, which is peaked at beta. 905 01:08:31,180 --> 01:08:34,310 But it's peaked as a delta function. 906 01:08:34,310 --> 01:08:37,100 Now, I'm calculating it for you right now, 907 01:08:37,100 --> 01:08:40,620 and the result is this is not the case. 908 01:08:40,620 --> 01:08:43,290 The reason is-- and I will show that 909 01:08:43,290 --> 01:08:47,529 to you-- is that the coherent states are not 910 01:08:47,529 --> 01:08:49,529 your ordinary basis function. 911 01:08:49,529 --> 01:08:54,220 The coherent state forms a basis which is over complete. 912 01:08:54,220 --> 01:08:58,930 So we have sort of more coherent states than necessary. 913 01:08:58,930 --> 01:09:00,540 There is some redundancy. 914 01:09:00,540 --> 01:09:03,260 And therefore, the coherent state alpha 915 01:09:03,260 --> 01:09:08,185 and the coherent state beta, if there's only 916 01:09:08,185 --> 01:09:11,620 a small difference between alpha and beta, they have overlap. 917 01:09:11,620 --> 01:09:14,779 Coherent states which differ by only a little bit 918 01:09:14,779 --> 01:09:20,160 in their eigenvalue, they're not orthogonal. 919 01:09:20,160 --> 01:09:21,770 So this is a complication. 920 01:09:21,770 --> 01:09:23,750 It's one of the complications we have 921 01:09:23,750 --> 01:09:26,979 to deal with when we want to describe coherent states 922 01:09:26,979 --> 01:09:30,455 and when we want to describe laser lights. 923 01:09:30,455 --> 01:09:31,930 But this is one of the properties. 924 01:09:31,930 --> 01:09:33,950 So therefore, when I'm now discussing 925 01:09:33,950 --> 01:09:36,990 with you is what are the quasi-probabilities 926 01:09:36,990 --> 01:09:40,520 of a coherent state, it will not be a delta function. 927 01:09:46,740 --> 01:09:52,040 But we should simply follow the definition, 928 01:09:52,040 --> 01:09:55,120 and everything will fall into place. 929 01:09:55,120 --> 01:10:02,420 So our statistical operator for a pure coherent state 930 01:10:02,420 --> 01:10:06,730 is, of course-- this is a statistical operator 931 01:10:06,730 --> 01:10:09,040 with pure state beta. 932 01:10:09,040 --> 01:10:14,200 And the quasi-probability for the coherent state beta 933 01:10:14,200 --> 01:10:24,505 to be an alpha is this matrix element squared. 934 01:10:27,580 --> 01:10:30,370 And you can calculate that, for instance, 935 01:10:30,370 --> 01:10:36,610 by just putting in alpha, alpha and beta, 936 01:10:36,610 --> 01:10:39,580 the expansion of alpha and beta in number states. 937 01:10:39,580 --> 01:10:44,100 So just have to solve the integral and calculate it. 938 01:10:44,100 --> 01:10:50,870 And what you find is not a delta function, 939 01:10:50,870 --> 01:10:53,290 but a decaying exponential. 940 01:10:53,290 --> 01:10:58,260 So you can say, as long as alpha and beta do not 941 01:10:58,260 --> 01:11:03,280 differ by more than one, there is substantial overlap. 942 01:11:03,280 --> 01:11:06,740 So in other words, what we get for this quasi-probability 943 01:11:06,740 --> 01:11:12,600 is something which is centered at beta. 944 01:11:15,260 --> 01:11:16,550 But it is again a Gaussian. 945 01:11:20,840 --> 01:11:28,390 So the important message here is coherent states 946 01:11:28,390 --> 01:11:30,400 are not orthogonal to each other. 947 01:11:37,130 --> 01:11:39,195 They form an over complete basis. 948 01:11:43,000 --> 01:11:45,930 Maybe as a side remark, it may come as a surprise 949 01:11:45,930 --> 01:11:49,760 to you, because you have been trained too much in basis 950 01:11:49,760 --> 01:11:54,490 states of emission operators, of energy eigenstates. 951 01:11:54,490 --> 01:11:58,470 And those energy eigenstates form a complete basis. 952 01:11:58,470 --> 01:12:00,900 The coherent states are eigenfunctions 953 01:12:00,900 --> 01:12:03,870 of very strange operator, a non-emission annihilation 954 01:12:03,870 --> 01:12:05,450 operator. 955 01:12:05,450 --> 01:12:09,190 So therefore, the few things you took for granted 956 01:12:09,190 --> 01:12:10,190 do not apply here. 957 01:12:15,110 --> 01:12:23,820 So we have our nice diagram with-- [INAUDIBLE] diagram 958 01:12:23,820 --> 01:12:26,050 for quasi-probabilities. 959 01:12:26,050 --> 01:12:31,660 So now if you have-- that was the real part of alpha. 960 01:12:31,660 --> 01:12:34,690 That was the imaginary part of alpha. 961 01:12:34,690 --> 01:12:39,020 So if you take that as a complex plane, 962 01:12:39,020 --> 01:12:44,770 and I'm drawing now as a phase of the complex number 963 01:12:44,770 --> 01:12:48,870 beta, which is the eigenvalue of the coherent state beta, 964 01:12:48,870 --> 01:12:58,910 then the quasi-probability is centered near beta. 965 01:12:58,910 --> 01:13:00,120 And it's a decaying Gaussian. 966 01:13:04,850 --> 01:13:07,470 I'm not showing it in here explicitly, 967 01:13:07,470 --> 01:13:09,490 but it is simply the vacuum state 968 01:13:09,490 --> 01:13:12,375 we discussed before displaced by beta. 969 01:13:22,760 --> 01:13:25,510 My fourth example, just to show you. 970 01:13:25,510 --> 01:13:28,250 So far, I've shown you that everything is Gaussian. 971 01:13:28,250 --> 01:13:30,910 Thermal light is a broad Gaussian, coherent state 972 01:13:30,910 --> 01:13:35,061 is a narrow Gaussian, the vacuum state is a narrow Gaussian. 973 01:13:35,061 --> 01:13:36,560 Let me just show you something which 974 01:13:36,560 --> 01:13:40,790 is non-Gaussian, which is the number state. 975 01:13:40,790 --> 01:13:44,890 If you put into the definition of the quasi-probability 976 01:13:44,890 --> 01:13:49,360 the number state, and you calculate the quasi-probability 977 01:13:49,360 --> 01:14:00,160 as a function of alpha, then you find that what you get 978 01:14:00,160 --> 01:14:04,260 is actually not a Gaussian. 979 01:14:04,260 --> 01:14:07,790 What you get is a ring, a ring with a certain width. 980 01:14:10,810 --> 01:14:15,880 So this is the representation of a number state. 981 01:14:18,740 --> 01:14:29,940 And the radius of this ring is proportional to the square root 982 01:14:29,940 --> 01:14:33,920 of the number of photons in a number state. 983 01:14:33,920 --> 01:14:38,490 We'll come back to photon states in much more detail very soon. 984 01:14:38,490 --> 01:14:40,950 But I just wanted to show you that at this point. 985 01:14:40,950 --> 01:14:43,950 So quasi-probabilities are not always Gaussian. 986 01:14:43,950 --> 01:14:46,230 They really depict something, they 987 01:14:46,230 --> 01:14:49,255 show us something important about-- they 988 01:14:49,255 --> 01:14:51,760 show us important differences about the quantum 989 01:14:51,760 --> 01:14:52,450 states of light. 990 01:15:01,750 --> 01:15:02,990 Any questions? 991 01:15:02,990 --> 01:15:03,796 Yes. 992 01:15:03,796 --> 01:15:07,198 AUDIENCE: [INAUDIBLE] a delta function [INAUDIBLE] radius, 993 01:15:07,198 --> 01:15:09,640 [INAUDIBLE] little past [INAUDIBLE]? 994 01:15:09,640 --> 01:15:12,690 PROFESSOR: Can you hold the question for three minutes? 995 01:15:12,690 --> 01:15:14,470 OK. 996 01:15:14,470 --> 01:15:17,780 They quick answer is it's not. 997 01:15:17,780 --> 01:15:20,720 But in three minutes, I want to tell you 998 01:15:20,720 --> 01:15:23,000 that there are three possible definitions 999 01:15:23,000 --> 01:15:26,270 of quasi-probabilities-- the Q function, to W function, 1000 01:15:26,270 --> 01:15:27,590 and the P function. 1001 01:15:27,590 --> 01:15:29,720 In one of the functions, it's a delta function. 1002 01:15:29,720 --> 01:15:31,540 In the Q function, it's not. 1003 01:15:31,540 --> 01:15:32,934 But we'll come to that. 1004 01:15:35,556 --> 01:15:37,305 But let's for now stick to the Q function. 1005 01:15:43,210 --> 01:15:48,750 The next thing we want to put in is the time dependence. 1006 01:15:48,750 --> 01:15:54,322 I want to show you that when we have quantum states of light-- 1007 01:15:54,322 --> 01:15:56,530 and right now, we've just shown the quasi-probability 1008 01:15:56,530 --> 01:15:59,900 distribution at a snapshot of t equals 0. 1009 01:15:59,900 --> 01:16:02,820 What happens as a function of time? 1010 01:16:02,820 --> 01:16:04,690 What I want to show you is-- and this 1011 01:16:04,690 --> 01:16:09,620 makes those quasi-probabilities also nice and intuitive-- 1012 01:16:09,620 --> 01:16:13,290 that as a function of time, the quasi-probability distribution 1013 01:16:13,290 --> 01:16:18,260 simply rotates with an angular frequency of omega. 1014 01:16:18,260 --> 01:16:21,450 So let me show that to you. 1015 01:16:21,450 --> 01:16:27,600 We know that we have a coherent state. 1016 01:16:27,600 --> 01:16:29,120 And what we want to understand is 1017 01:16:29,120 --> 01:16:35,420 what happens when we act on the coherent state with the time 1018 01:16:35,420 --> 01:16:37,790 propagation operator, which is the Hamiltonian 1019 01:16:37,790 --> 01:16:41,260 in the exponent. 1020 01:16:41,260 --> 01:16:47,270 Well, in order to evaluate the left hand side, 1021 01:16:47,270 --> 01:16:55,060 we can simply take the expansion of coherent states 1022 01:16:55,060 --> 01:17:01,000 into in a number basis, because we 1023 01:17:01,000 --> 01:17:10,030 know that the time evolution of a number state 1024 01:17:10,030 --> 01:17:20,020 is simply the energy of the number state 1025 01:17:20,020 --> 01:17:22,260 with which it's given by the energy of the number 1026 01:17:22,260 --> 01:17:24,040 state, which is n times h bar omega. 1027 01:17:31,780 --> 01:17:41,030 So now if you look for a second at this expression, 1028 01:17:41,030 --> 01:17:47,086 you realize that we can absorb the time evolution 1029 01:17:47,086 --> 01:17:55,610 if you redefine alpha to become alpha times e to the i omega t. 1030 01:17:55,610 --> 01:18:00,320 So in other words, the time evolution of the coherent state 1031 01:18:00,320 --> 01:18:06,040 alpha preserves the character as a coherent state. 1032 01:18:06,040 --> 01:18:09,290 It just means we get an incoherent state 1033 01:18:09,290 --> 01:18:13,600 whose eigenvalue is now alpha times e to the i omega t. 1034 01:18:19,830 --> 01:18:24,560 So that means the following. 1035 01:18:24,560 --> 01:18:28,870 If you want to look at the time evolution in terms 1036 01:18:28,870 --> 01:18:36,260 of quasi-probabilities, we have our diagram 1037 01:18:36,260 --> 01:18:41,930 with the real part of alpha, with the imaginary part 1038 01:18:41,930 --> 01:18:42,600 of alpha. 1039 01:18:46,110 --> 01:18:51,020 And let's assume we had a coherent state, which 1040 01:18:51,020 --> 01:18:54,190 happened-- this is sort of now my circle. 1041 01:18:54,190 --> 01:18:56,840 It's, you can say, a high contrast 1042 01:18:56,840 --> 01:18:59,120 representation of the Gaussian. 1043 01:18:59,120 --> 01:19:03,480 So this is the quasi-probability of the initial state alpha. 1044 01:19:03,480 --> 01:19:07,780 But as time propagates, it just gets multiplied with e 1045 01:19:07,780 --> 01:19:09,280 to the i omega t. 1046 01:19:09,280 --> 01:19:17,280 That means when the time evolution is just 1047 01:19:17,280 --> 01:19:21,610 displacing the state over there, and as time goes by, 1048 01:19:21,610 --> 01:19:25,530 the coherent state is just moving in a circle. 1049 01:19:25,530 --> 01:19:28,987 And after one full period of omega, 1050 01:19:28,987 --> 01:19:30,195 we are back where we started. 1051 01:19:48,340 --> 01:19:56,210 Now let me give you now an intuitive picture 1052 01:19:56,210 --> 01:20:00,310 what the electric field is. 1053 01:20:00,310 --> 01:20:03,520 I will be a little bit more exact in two or three minutes. 1054 01:20:03,520 --> 01:20:06,610 So we have a diagram of the real part of alpha 1055 01:20:06,610 --> 01:20:08,816 and the imaginary part of alpha. 1056 01:20:11,720 --> 01:20:21,146 For a harmonic oscillator, I can also label it as x and p. 1057 01:20:21,146 --> 01:20:23,660 You know, in harmonic oscillator, 1058 01:20:23,660 --> 01:20:26,990 when you can think about it classically, 1059 01:20:26,990 --> 01:20:28,990 if you have some initial distribution 1060 01:20:28,990 --> 01:20:31,750 of a mechanical oscillator in x and p, 1061 01:20:31,750 --> 01:20:34,640 what the system actually does is it simply 1062 01:20:34,640 --> 01:20:37,880 rotates on a circle in phase space. 1063 01:20:37,880 --> 01:20:40,760 The classical phase space is x and p. 1064 01:20:40,760 --> 01:20:42,925 And I haven't really told you that I 1065 01:20:42,925 --> 01:20:49,690 was-- I didn't want to get lost in complex definitions. 1066 01:20:49,690 --> 01:20:52,120 I defined for you the quasi-probability, 1067 01:20:52,120 --> 01:20:57,670 but the quasi-probability is a generalization 1068 01:20:57,670 --> 01:21:01,240 of the classical phase space function. 1069 01:21:01,240 --> 01:21:04,600 Phase space function is a function of p and x. 1070 01:21:04,600 --> 01:21:08,110 So take my word for a second, and allow 1071 01:21:08,110 --> 01:21:13,800 me to identify the real axis with x and the imaginary axis 1072 01:21:13,800 --> 01:21:15,910 with p. 1073 01:21:15,910 --> 01:21:21,910 Well, then p is the electric field. 1074 01:21:21,910 --> 01:21:26,940 So now, what you should visually take from those pictures 1075 01:21:26,940 --> 01:21:31,770 is that if you want to know what the electric field is, 1076 01:21:31,770 --> 01:21:35,120 you just sort of project this fuzzy 1077 01:21:35,120 --> 01:21:38,760 ball on the imaginary axis. 1078 01:21:38,760 --> 01:21:44,300 So in other words, when I ask you 1079 01:21:44,300 --> 01:21:51,640 what is now the electric field as a function of time, at t 1080 01:21:51,640 --> 01:21:59,830 equals 0, I project this onto the p axis. 1081 01:21:59,830 --> 01:22:05,860 And what I get is something which is centered at 0. 1082 01:22:11,100 --> 01:22:13,990 But it has also a certain fuzziness. 1083 01:22:13,990 --> 01:22:16,880 The fuzziness is given by the size of the disk, 1084 01:22:16,880 --> 01:22:19,760 or by the width of the Gaussian. 1085 01:22:19,760 --> 01:22:23,150 And if I now use this picture as a function of time, 1086 01:22:23,150 --> 01:22:28,490 as this fuzzy ball rotates around, 1087 01:22:28,490 --> 01:22:35,080 the electric field goes up and down in one cycle. 1088 01:22:39,250 --> 01:22:41,140 And the fuzziness moves with it. 1089 01:22:44,970 --> 01:22:53,510 So the fuzziness here for the electric field is related-- 1090 01:22:53,510 --> 01:22:57,770 and that's what we want to discuss next week-- 1091 01:22:57,770 --> 01:23:00,840 is related to short noise. 1092 01:23:00,840 --> 01:23:04,360 I also want to tell you that the coherent state is, 1093 01:23:04,360 --> 01:23:08,020 to some extent, the best possible way 1094 01:23:08,020 --> 01:23:10,010 to define an electric field. 1095 01:23:10,010 --> 01:23:12,600 It's a minimum uncertainty state. 1096 01:23:12,600 --> 01:23:14,840 I want to show you that the coherent state is 1097 01:23:14,840 --> 01:23:17,000 sort of at the minimum of Heisenberg's Uncertainty 1098 01:23:17,000 --> 01:23:19,260 Relation. 1099 01:23:19,260 --> 01:23:21,790 So I think if you take this picture, 1100 01:23:21,790 --> 01:23:25,010 you will immediately realize that you 1101 01:23:25,010 --> 01:23:28,550 draw the quasi-probability, you project on the vertical axis. 1102 01:23:28,550 --> 01:23:32,080 And you get the electric field that tells you immediately 1103 01:23:32,080 --> 01:23:33,920 if you have a thermal state, which 1104 01:23:33,920 --> 01:23:35,660 is a Gaussian centered here. 1105 01:23:35,660 --> 01:23:36,820 You project it. 1106 01:23:36,820 --> 01:23:39,920 You get an electric field 0. 1107 01:23:39,920 --> 01:23:43,170 It has an enormous fuzziness, but it's 0. 1108 01:23:43,170 --> 01:23:48,970 And this picture that everything rotates with angular frequency 1109 01:23:48,970 --> 01:23:51,620 omega also tells you, as expected, 1110 01:23:51,620 --> 01:23:54,620 that in a thermal state, it's already 1111 01:23:54,620 --> 01:23:57,040 a circularly symmetric distribution. 1112 01:23:57,040 --> 01:23:59,998 When it rotates, nothing changes. 1113 01:23:59,998 --> 01:24:02,742 AUDIENCE: [INAUDIBLE], couldn't you 1114 01:24:02,742 --> 01:24:04,988 find, eigenstates for that operator itself, 1115 01:24:04,988 --> 01:24:06,984 and then get rid of this fuzziness? 1116 01:24:22,952 --> 01:24:24,948 AUDIENCE: Could it just be, like, 1117 01:24:24,948 --> 01:24:29,938 eigenstates of the harmonic oscillator, so not 1118 01:24:29,938 --> 01:24:31,435 stationary at all? 1119 01:24:36,470 --> 01:24:39,710 PROFESSOR: I'm a little bit confused now. 1120 01:24:39,710 --> 01:24:44,010 Actually, two people mind if I teach a few minutes longer? 1121 01:24:44,010 --> 01:24:47,050 I can just finish the-- I want to have one, 1122 01:24:47,050 --> 01:24:47,940 I have one footnote. 1123 01:24:47,940 --> 01:24:49,820 I really want to sort of give it to you, 1124 01:24:49,820 --> 01:24:52,622 because I think it will help some of you. 1125 01:24:52,622 --> 01:24:53,122 Pardon? 1126 01:24:53,122 --> 01:24:55,110 AUDIENCE: We have a class starting at 2:30. 1127 01:24:55,110 --> 01:24:56,460 PROFESSOR: There's a class starting at 2:30? 1128 01:24:56,460 --> 01:24:57,290 AUDIENCE: Yes. 1129 01:24:57,290 --> 01:24:58,945 PROFESSOR: Then I have to wrap up. 1130 01:24:58,945 --> 01:24:59,445 Sorry. 1131 01:25:02,650 --> 01:25:04,580 Then let me just tell you what I will 1132 01:25:04,580 --> 01:25:06,620 do at the beginning of the next class. 1133 01:25:06,620 --> 01:25:08,940 What I've told you here with the projection 1134 01:25:08,940 --> 01:25:12,710 of the electric field, this is not 1135 01:25:12,710 --> 01:25:17,610 100% rigorous for the quasi-probabilities Q. 1136 01:25:17,610 --> 01:25:19,930 Due to the non-commutativity of operators, 1137 01:25:19,930 --> 01:25:24,850 when we generalize phase space function into quantum 1138 01:25:24,850 --> 01:25:28,730 mechanics, we can order operators in a symmetric way, 1139 01:25:28,730 --> 01:25:31,000 in a normal way, in an anti- normal way. 1140 01:25:31,000 --> 01:25:33,770 And therefore, there are three different quasi-probabilities. 1141 01:25:33,770 --> 01:25:37,400 The Q probability is the easiest to define. 1142 01:25:37,400 --> 01:25:39,170 It comes in very handy. 1143 01:25:39,170 --> 01:25:41,670 But what I showed you with the projection 1144 01:25:41,670 --> 01:25:44,840 does not rigorously apply to the Q distribution. 1145 01:25:44,840 --> 01:25:48,500 It applies to the p or w distribution. 1146 01:25:48,500 --> 01:25:51,720 But the difference is, unless you really go into subtleties, 1147 01:25:51,720 --> 01:25:52,780 are minor. 1148 01:25:52,780 --> 01:25:55,440 So intuitively, what I did is correct. 1149 01:25:55,440 --> 01:25:57,790 And to answer your question, an eigenstate 1150 01:25:57,790 --> 01:26:01,340 of the electric field, I have a little bit problems with that. 1151 01:26:01,340 --> 01:26:04,060 Because if you take this projection, what you need 1152 01:26:04,060 --> 01:26:07,450 is you need something which is very, very narrow. 1153 01:26:07,450 --> 01:26:10,930 But since this area is in Heisenberg uncertainty 1154 01:26:10,930 --> 01:26:15,880 limited area, it means delta x delta p equals h bar over 2. 1155 01:26:15,880 --> 01:26:18,490 The only way how you can create this electric field 1156 01:26:18,490 --> 01:26:21,020 is you can completely squeeze it. 1157 01:26:21,020 --> 01:26:24,540 So to the best of my knowledge, an electric field eigenstate 1158 01:26:24,540 --> 01:26:27,800 would be a completely strongly squeezed state. 1159 01:26:27,800 --> 01:26:29,830 We'll talk about that next week. 1160 01:26:29,830 --> 01:26:31,690 But then you realize, if you've completed 1161 01:26:31,690 --> 01:26:36,650 squeezed your state after a quarter period, 1162 01:26:36,650 --> 01:26:42,560 this infinitely squeezed ellipse is now standing upside down. 1163 01:26:42,560 --> 01:26:45,090 And after a quarter period your electric field 1164 01:26:45,090 --> 01:26:46,860 is completely uncertain. 1165 01:26:46,860 --> 01:26:49,520 So I think you can define an operator, 1166 01:26:49,520 --> 01:26:51,620 you can find a state which has a sharp value 1167 01:26:51,620 --> 01:26:54,320 of the electric field at one moment of time. 1168 01:26:54,320 --> 01:26:56,440 But then it rotates around. 1169 01:26:56,440 --> 01:26:59,060 And what was a highly certain electric field 1170 01:26:59,060 --> 01:27:02,840 becomes highly uncertain later. 1171 01:27:02,840 --> 01:27:04,950 OK, enough things to discuss. 1172 01:27:04,950 --> 01:27:07,480 We'll go on next week.