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 --> 00:00:17,237 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,237 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:21,940 --> 00:00:28,640 PROFESSOR: OK, so let's start to discuss ion traps. 9 00:00:28,640 --> 00:00:35,190 Now, in many situations, ions are just charged neutral atoms. 10 00:00:35,190 --> 00:00:37,300 And if the charge doesn't matter, 11 00:00:37,300 --> 00:00:41,440 you can do very similar physics with ions and neutral atoms. 12 00:00:41,440 --> 00:00:46,220 However, the charge makes major differences 13 00:00:46,220 --> 00:00:50,460 when it comes to confining and manipulating particles. 14 00:00:50,460 --> 00:00:59,410 So when I compare neutrals to ions, 15 00:00:59,410 --> 00:01:02,660 the forces which we exert on neutral atoms 16 00:01:02,660 --> 00:01:10,225 are magnetic forces, optical forces. 17 00:01:13,760 --> 00:01:19,930 And typically, the traps we can create 18 00:01:19,930 --> 00:01:22,680 have a depth of about 1 kelvin. 19 00:01:22,680 --> 00:01:25,140 Actually, you just need superconducting magnets. 20 00:01:25,140 --> 00:01:30,000 The magnetic traps we build here at MIT have millikelvin depths. 21 00:01:30,000 --> 00:01:34,390 So the trap depth is often on the order 22 00:01:34,390 --> 00:01:38,610 of microkelvin, millikelvin, and kelvin at the high scale. 23 00:01:38,610 --> 00:01:41,580 Well, this is very different in ions. 24 00:01:41,580 --> 00:01:43,835 In ions, you have the Coulomb force. 25 00:01:49,040 --> 00:01:59,080 If you take electric fields of 100 volts per millimeter, 26 00:01:59,080 --> 00:02:03,595 you can create confinement forces for 10,000 kelvin. 27 00:02:03,595 --> 00:02:06,785 So you have no problem in confining room temperature 28 00:02:06,785 --> 00:02:07,285 ions. 29 00:02:10,840 --> 00:02:19,880 When it comes to-- so this is trapping and confinement. 30 00:02:19,880 --> 00:02:28,565 If you talk about cooling, well, our workhorses 31 00:02:28,565 --> 00:02:32,500 are Doppler cooling and evaporative cooling. 32 00:02:35,630 --> 00:02:37,865 You can actually cool ions by evaporation. 33 00:02:37,865 --> 00:02:41,400 You can also cool ions by evaporative cooling. 34 00:02:41,400 --> 00:02:45,930 But what has been very common and special to ions 35 00:02:45,930 --> 00:02:50,870 is the ability to resolve sidebands, sidebands 36 00:02:50,870 --> 00:02:56,090 of the vibration and motion in the harmonic oscillator trap. 37 00:02:56,090 --> 00:02:58,030 And in problem set number 10, you 38 00:02:58,030 --> 00:03:03,090 have looked through the physics of sideband cooling. 39 00:03:03,090 --> 00:03:06,790 However, sideband cooling is now also coming back 40 00:03:06,790 --> 00:03:08,050 to neutral atoms. 41 00:03:08,050 --> 00:03:11,350 For instance, if you have neutral atoms in a very tightly 42 00:03:11,350 --> 00:03:13,850 confining dipole trap, you can resolve sidebands 43 00:03:13,850 --> 00:03:15,520 and use those for cooling. 44 00:03:18,190 --> 00:03:19,695 OK, what is the science? 45 00:03:22,860 --> 00:03:33,520 Well, a driving force for both ions and neutral atoms 46 00:03:33,520 --> 00:03:35,020 has been clocks. 47 00:03:35,020 --> 00:03:37,490 If atoms are cold, you can use them 48 00:03:37,490 --> 00:03:41,630 to have very accurate ways of keeping time. 49 00:03:41,630 --> 00:03:43,610 And I think currently, there is a race 50 00:03:43,610 --> 00:03:47,990 between the based neutral atom atomic clocks, 51 00:03:47,990 --> 00:03:50,270 neutral [INAUDIBLE] optical lattices, 52 00:03:50,270 --> 00:03:58,130 and ion traps in particular, the aluminum ion at Boulder. 53 00:03:58,130 --> 00:04:00,930 So we have clocks. 54 00:04:00,930 --> 00:04:07,190 In ions-- and this is what I want to talk a little bit more 55 00:04:07,190 --> 00:04:12,620 about today-- a major driving force was quantum information. 56 00:04:12,620 --> 00:04:15,140 You have two ions and get quantum gates between them, 57 00:04:15,140 --> 00:04:17,579 or you have eight ions and can have now 58 00:04:17,579 --> 00:04:22,019 eight qubits in a prototypical quantum computer. 59 00:04:22,019 --> 00:04:25,340 So quantum information science and quantum algorithms 60 00:04:25,340 --> 00:04:26,890 was quite important. 61 00:04:26,890 --> 00:04:29,210 But I would say some of the physics 62 00:04:29,210 --> 00:04:33,399 is at least discussed and partially pursued 63 00:04:33,399 --> 00:04:34,690 with atoms in optical lattices. 64 00:04:37,310 --> 00:04:40,610 We have talked a lot about, for neutral atoms, 65 00:04:40,610 --> 00:04:45,320 quantum simulations, studying super fluidity, 66 00:04:45,320 --> 00:04:48,500 studying phase transitions, magnetic properties 67 00:04:48,500 --> 00:04:50,110 of many body systems. 68 00:04:50,110 --> 00:04:54,920 But there are more efforts to do that with ions. 69 00:04:54,920 --> 00:04:56,810 Chris Monroe's group in Maryland is 70 00:04:56,810 --> 00:05:00,700 studying antiferromagnetic ordering of a spin chain. 71 00:05:00,700 --> 00:05:06,060 And the spin chain consists of a string of ions in an ion trap. 72 00:05:06,060 --> 00:05:08,340 So see actually on the science side 73 00:05:08,340 --> 00:05:14,250 that all three-- clocks, precision measurements, quantum 74 00:05:14,250 --> 00:05:17,780 information, qubits, quantum algorithms, and quantum 75 00:05:17,780 --> 00:05:19,670 simulation of many body systems-- 76 00:05:19,670 --> 00:05:22,810 are done with both neutral atoms and ions. 77 00:05:22,810 --> 00:05:25,340 So these are now two very different platforms. 78 00:05:29,340 --> 00:05:33,340 So I could spend a long time teaching 79 00:05:33,340 --> 00:05:35,650 about special aspects of ions. 80 00:05:35,650 --> 00:05:36,970 How do you do a trap? 81 00:05:36,970 --> 00:05:37,845 How do you cool ions? 82 00:05:37,845 --> 00:05:41,090 How do you do quantum state control? 83 00:05:41,090 --> 00:05:45,480 Because this is our last lecture, 84 00:05:45,480 --> 00:05:48,620 I decided to focus on two aspects. 85 00:05:48,620 --> 00:05:56,510 One is, I want to talk about a special way of trapping ions. 86 00:05:56,510 --> 00:06:00,620 Because there is one difference for ions. 87 00:06:00,620 --> 00:06:05,960 Ions, until recently, have not been 88 00:06:05,960 --> 00:06:08,260 trapped in stationary traps. 89 00:06:08,260 --> 00:06:12,550 They've only been trapped in dynamic traps. 90 00:06:12,550 --> 00:06:17,020 So you cannot trap ions by simply putting an electric 91 00:06:17,020 --> 00:06:18,660 potential around it. 92 00:06:18,660 --> 00:06:22,650 Because the divergence of the electric field is 0. 93 00:06:22,650 --> 00:06:26,050 And that means there's always one direction in space 94 00:06:26,050 --> 00:06:28,830 where the ions are dragged out. 95 00:06:28,830 --> 00:06:32,530 You cannot have an inward force from all directions. 96 00:06:32,530 --> 00:06:37,210 So therefore, when you want to use electric fields, 97 00:06:37,210 --> 00:06:39,760 you can only use dynamic traps. 98 00:06:43,310 --> 00:06:49,000 Well, for the experts I should say, 99 00:06:49,000 --> 00:06:53,530 this does not apply for the optical dipole trap. 100 00:06:53,530 --> 00:06:56,170 We know how we can create trapping potentials 101 00:06:56,170 --> 00:06:58,370 for neutral atoms using the optical dipole trap. 102 00:06:58,370 --> 00:07:03,480 And very recently, a group was able to trap ions 103 00:07:03,480 --> 00:07:08,800 not with electric and magnetic fields provided by electrodes 104 00:07:08,800 --> 00:07:12,129 or radio frequency, but by inner focused laser beams. 105 00:07:12,129 --> 00:07:13,420 So it's a very new development. 106 00:07:13,420 --> 00:07:15,860 But it uses exactly the mechanism 107 00:07:15,860 --> 00:07:19,870 which has been used for neutral atoms. 108 00:07:19,870 --> 00:07:22,310 So when it comes to dynamic traps, 109 00:07:22,310 --> 00:07:25,190 the standard two traps which have 110 00:07:25,190 --> 00:07:31,030 been used for ion trapping-- one is the Penning trap, 111 00:07:31,030 --> 00:07:33,870 which I do not want to describe in detail. 112 00:07:33,870 --> 00:07:39,520 The Penning trap uses a quadrupole configuration 113 00:07:39,520 --> 00:07:44,910 of two positive charges. 114 00:07:44,910 --> 00:07:48,000 And then you have a ring with negative charge. 115 00:07:48,000 --> 00:07:54,400 That means that an ion is confined here 116 00:07:54,400 --> 00:08:02,160 but would see anti-confinement in the outward direction. 117 00:08:02,160 --> 00:08:05,090 Now, what you do with the anti-confinement? 118 00:08:05,090 --> 00:08:13,170 Well, the solution is that you add a magnetic field. 119 00:08:13,170 --> 00:08:15,620 And what happens now is the particle 120 00:08:15,620 --> 00:08:17,230 is well-confined axially. 121 00:08:17,230 --> 00:08:19,480 But radially, when it wants to move out, 122 00:08:19,480 --> 00:08:22,820 it's turned into a cyclotron orbit, 123 00:08:22,820 --> 00:08:24,640 and therefore stays trapped. 124 00:08:24,640 --> 00:08:29,930 So this is the principal of Penning traps 125 00:08:29,930 --> 00:08:34,030 where the confinement in the axial direction 126 00:08:34,030 --> 00:08:43,840 is electrostatic, and the confinement in x, y 127 00:08:43,840 --> 00:08:46,390 is through cyclotron motion. 128 00:08:54,570 --> 00:08:59,830 OK, it's wonderful physics to describe 129 00:08:59,830 --> 00:09:02,210 the motion of ions in this trap. 130 00:09:02,210 --> 00:09:04,770 It's the combination of an axial motion, 131 00:09:04,770 --> 00:09:07,630 cyclotron motion, magnetron motion. 132 00:09:07,630 --> 00:09:09,390 It's very interesting physics. 133 00:09:09,390 --> 00:09:12,292 And for many years, Dave Pritchard here at MIT 134 00:09:12,292 --> 00:09:17,880 used such Penning traps to measure the masses, determine 135 00:09:17,880 --> 00:09:20,800 the masses, of ions with the highest precision ever 136 00:09:20,800 --> 00:09:22,100 possible. 137 00:09:22,100 --> 00:09:24,900 MIT still has the world record of the most precise 138 00:09:24,900 --> 00:09:27,940 mass measurements at the 10 to the minus 11 level. 139 00:09:27,940 --> 00:09:31,920 And this was done in Penning traps. 140 00:09:31,920 --> 00:09:39,320 But I want to talk more today about RF traps, or Paul traps, 141 00:09:39,320 --> 00:09:42,980 because the are used in the majority of experiments. 142 00:09:42,980 --> 00:09:46,150 And they're used in pretty much all experiments 143 00:09:46,150 --> 00:09:51,700 on quantum information processing with ions. 144 00:09:51,700 --> 00:09:56,870 So the neat thing-- and I try to teach you 145 00:09:56,870 --> 00:10:01,120 concepts in physics-- the interesting thing about Paul 146 00:10:01,120 --> 00:10:05,880 traps or RF traps is that you alternate at a radio 147 00:10:05,880 --> 00:10:11,650 frequency between confinement and anti-confinement. 148 00:10:11,650 --> 00:10:14,430 You can say you have a settle point potential, which 149 00:10:14,430 --> 00:10:18,320 is confining in x and anti-confining in y. 150 00:10:18,320 --> 00:10:20,820 But then, you switch x and y. 151 00:10:20,820 --> 00:10:24,790 So therefore, the situation is just imagine you're an ion, 152 00:10:24,790 --> 00:10:29,590 and you feel half of the time a confining harmonic oscillator 153 00:10:29,590 --> 00:10:33,070 and half of the time an anti-confining harmonic 154 00:10:33,070 --> 00:10:35,080 oscillator. 155 00:10:35,080 --> 00:10:37,530 The question I have for you, and this is a general physics 156 00:10:37,530 --> 00:10:41,610 question, well, is the sum of a confining 157 00:10:41,610 --> 00:10:47,005 and an anti-confining potential of the same strength give 0, 158 00:10:47,005 --> 00:10:49,850 or does it give something which in the end is confining? 159 00:10:56,090 --> 00:11:00,470 So if you were in a potential, and the potential switches 160 00:11:00,470 --> 00:11:03,860 permanently form parabolic upward to parabolic 161 00:11:03,860 --> 00:11:09,780 downward, at any given position, the average force is 0. 162 00:11:09,780 --> 00:11:12,662 Because in one phase, you have a force to the left. 163 00:11:12,662 --> 00:11:14,370 In the other case, you have the opposite. 164 00:11:14,370 --> 00:11:16,155 You have the same force but to the right 165 00:11:16,155 --> 00:11:19,270 with the opposite sign. 166 00:11:19,270 --> 00:11:22,570 But what's going on here? 167 00:11:22,570 --> 00:11:23,302 Yes. 168 00:11:23,302 --> 00:11:26,176 AUDIENCE: So I can't answer this mathematically or even 169 00:11:26,176 --> 00:11:26,676 physically. 170 00:11:26,676 --> 00:11:29,568 But I just remember this very cool example 171 00:11:29,568 --> 00:11:33,135 that someone showed me once where they took a saddle shape 172 00:11:33,135 --> 00:11:35,570 in space, put a ball in the middle, 173 00:11:35,570 --> 00:11:37,383 and then you spin the saddle. 174 00:11:37,383 --> 00:11:39,466 And as soon as the ball tries to fall to one side, 175 00:11:39,466 --> 00:11:41,414 the saddle will spin over and pick it up. 176 00:11:41,414 --> 00:11:43,414 So I know the answer has to be, it is confining. 177 00:11:43,414 --> 00:11:44,970 But I can't explain why. 178 00:11:44,970 --> 00:11:47,420 PROFESSOR: OK, Colin? 179 00:11:47,420 --> 00:11:49,440 AUDIENCE: Could it possibly be like a threshold? 180 00:11:49,440 --> 00:11:51,830 Because if you consider spinning it sort of slowly, 181 00:11:51,830 --> 00:11:53,962 then the particle can just sort of roll off. 182 00:11:53,962 --> 00:11:55,045 Maybe there's a threshold. 183 00:11:55,045 --> 00:11:56,628 PROFESSOR: Yeah, there is a threshold. 184 00:11:56,628 --> 00:11:58,620 I mean, if there's a confining potential 185 00:11:58,620 --> 00:12:00,585 for a long, long time, and then there's 186 00:12:00,585 --> 00:12:03,095 an anti-confining potential, and I just fall down 187 00:12:03,095 --> 00:12:06,840 the anti-confining potential, and I disappear to infinity, 188 00:12:06,840 --> 00:12:08,750 the confining part doesn't help me. 189 00:12:08,750 --> 00:12:11,890 So there will definitely be a threshold in time. 190 00:12:11,890 --> 00:12:15,420 We have to switch between confining and anti-confining 191 00:12:15,420 --> 00:12:16,480 sufficiently rapidly. 192 00:12:16,480 --> 00:12:16,980 Yes. 193 00:12:16,980 --> 00:12:18,646 AUDIENCE: Does it have something to do 194 00:12:18,646 --> 00:12:21,980 with fictitious force [INAUDIBLE]? 195 00:12:21,980 --> 00:12:25,890 PROFESSOR: No, but this would have been a possibility. 196 00:12:25,890 --> 00:12:27,110 Maybe there is a formalism. 197 00:12:27,110 --> 00:12:29,890 But the standard formalism does not use the rotating frame. 198 00:12:29,890 --> 00:12:30,390 Let me-- 199 00:12:30,390 --> 00:12:32,431 AUDIENCE: You could think of the classical analog 200 00:12:32,431 --> 00:12:35,440 where you have a bead on a wire, and you have a stable position 201 00:12:35,440 --> 00:12:39,960 at sort of a finite angle only at certain frequencies. 202 00:12:39,960 --> 00:12:43,720 PROFESSOR: Yes, I'm not sure. 203 00:12:43,720 --> 00:12:49,280 Anyway, you all-- I think everybody here works 204 00:12:49,280 --> 00:12:51,650 with optics or knows what optics is. 205 00:12:51,650 --> 00:12:54,070 So why don't I discuss the following. 206 00:12:54,070 --> 00:12:57,540 If you take a positive lens and a negative lens 207 00:12:57,540 --> 00:13:03,700 with equal strength, what happens now to your laser beam? 208 00:13:03,700 --> 00:13:05,980 You send a laser beam through, it 209 00:13:05,980 --> 00:13:11,790 gets an inward bend by the convex lens, an outward bend 210 00:13:11,790 --> 00:13:13,260 by the concave lens. 211 00:13:13,260 --> 00:13:16,450 Is the net effect nothing, or is it focusing? 212 00:13:22,690 --> 00:13:26,950 Well, it depends. 213 00:13:26,950 --> 00:13:29,650 What happens is the following. 214 00:13:29,650 --> 00:13:38,110 If the lenses are put in close proximity, 215 00:13:38,110 --> 00:13:44,680 the two bending actions on the rays just completely cancel. 216 00:13:44,680 --> 00:13:47,470 But now we can do the following. 217 00:13:47,470 --> 00:13:55,040 We can put the two lenses further apart. 218 00:13:58,940 --> 00:14:01,110 And now the following happens. 219 00:14:01,110 --> 00:14:04,890 We have a laser beam which is bent inward. 220 00:14:04,890 --> 00:14:07,990 And you probably all know the lens maker's equation, 221 00:14:07,990 --> 00:14:10,150 that the further out the rays are, 222 00:14:10,150 --> 00:14:13,670 the more they have to be bent inward to come to a focus. 223 00:14:13,670 --> 00:14:17,950 So what happens is those beams are now bent inward. 224 00:14:17,950 --> 00:14:20,960 But when they reach the other lens, 225 00:14:20,960 --> 00:14:23,290 they are now bent outward somewhat. 226 00:14:23,290 --> 00:14:25,460 I'm not very accurate in my drawing. 227 00:14:25,460 --> 00:14:27,750 But the angle by which they're bent 228 00:14:27,750 --> 00:14:30,700 inward is proportional to the separation here. 229 00:14:30,700 --> 00:14:32,805 The angle by which they're bent outward 230 00:14:32,805 --> 00:14:34,660 is proportional to the separation there. 231 00:14:34,660 --> 00:14:38,470 So the inward bend is stronger than the outward bend. 232 00:14:38,470 --> 00:14:40,130 So therefore what I'm telling you 233 00:14:40,130 --> 00:14:43,760 is convex and concave, with a certain distance 234 00:14:43,760 --> 00:14:45,755 between the two, results in focusing. 235 00:14:48,490 --> 00:14:50,780 Now maybe you can think about it. 236 00:14:50,780 --> 00:14:54,565 What happens if I reverse convex and concave? 237 00:14:56,901 --> 00:14:57,776 AUDIENCE: [INAUDIBLE] 238 00:15:00,670 --> 00:15:02,550 PROFESSOR: Is it now still focusing, 239 00:15:02,550 --> 00:15:04,767 or is it the opposite of the example above? 240 00:15:09,440 --> 00:15:17,356 Now the first lens bends the beam outward, yes? 241 00:15:17,356 --> 00:15:20,920 And the second lens bends it inward. 242 00:15:20,920 --> 00:15:24,930 But since we reach the second lens further out, 243 00:15:24,930 --> 00:15:29,640 the invert kick is stronger than the outward bend initially. 244 00:15:29,640 --> 00:15:34,330 So therefore, when you have positive and negative lenses 245 00:15:34,330 --> 00:15:38,950 of equal strength, no matter in what order you apply it, 246 00:15:38,950 --> 00:15:40,525 you always get an inward force. 247 00:15:43,780 --> 00:15:46,150 So this should actually explain-- 248 00:15:48,705 --> 00:15:51,556 AUDIENCE: As long as they're within the focal length. 249 00:15:51,556 --> 00:15:53,730 Could you imagine if I had positive and negative 250 00:15:53,730 --> 00:15:55,941 that was larger than the focal length away? 251 00:15:55,941 --> 00:15:58,960 PROFESSOR: Yeah, everything within reason 252 00:15:58,960 --> 00:16:01,550 here-- there will be a stability diagram. 253 00:16:01,550 --> 00:16:04,090 But I want to first make it very plausible what happens. 254 00:16:04,090 --> 00:16:07,710 And just to address the question which I raised before, 255 00:16:07,710 --> 00:16:10,190 if I'm in a potential, and I get an equal amount 256 00:16:10,190 --> 00:16:14,650 of inward and outward force, shouldn't that average to 0? 257 00:16:14,650 --> 00:16:16,840 The question is how you average. 258 00:16:16,840 --> 00:16:22,600 If I would keep the distance from the x as constant, 259 00:16:22,600 --> 00:16:26,050 I would say here as a function of-- let me call it x. 260 00:16:26,050 --> 00:16:29,170 The bending angle as a function of x is positive. 261 00:16:29,170 --> 00:16:32,730 The bending angle at the same x is negative. 262 00:16:32,730 --> 00:16:37,800 So if I would do the average at constant x, I would get 0. 263 00:16:37,800 --> 00:16:41,430 But what I do is I do the average following 264 00:16:41,430 --> 00:16:44,380 the trajectory of the beam. 265 00:16:44,380 --> 00:16:46,390 So in other words, if I'm sitting 266 00:16:46,390 --> 00:16:49,565 on this potential, which switches from attractive 267 00:16:49,565 --> 00:16:53,350 to repulsive, if I would be here at my position, 268 00:16:53,350 --> 00:16:56,000 the average force is 0. 269 00:16:56,000 --> 00:17:01,240 But when the potential is repulsive, in 1/2 cycle 270 00:17:01,240 --> 00:17:02,710 I'm pushed outwards. 271 00:17:02,710 --> 00:17:05,829 And further out, the attractive force 272 00:17:05,829 --> 00:17:08,020 is now stronger than it was before. 273 00:17:08,020 --> 00:17:13,160 And therefore, if you average the force at my position, 274 00:17:13,160 --> 00:17:15,460 you will always find an inward force. 275 00:17:15,460 --> 00:17:18,530 But if you average the force at a fixed location, 276 00:17:18,530 --> 00:17:22,450 it averages out to 0. 277 00:17:22,450 --> 00:17:29,230 So this is exactly how RF ion traps work. 278 00:17:29,230 --> 00:17:44,790 And well, the rest are just equations to describe that. 279 00:17:44,790 --> 00:17:46,690 So I think we have to make a choice. 280 00:17:46,690 --> 00:17:48,920 I could derive for you the equations, how 281 00:17:48,920 --> 00:17:53,720 you average over the rapid oscillations of the potential, 282 00:17:53,720 --> 00:17:55,680 and you get a harmonic oscillator potential, 283 00:17:55,680 --> 00:17:57,440 and this is how you trap ions. 284 00:17:57,440 --> 00:18:00,630 But it's pretty much showing to you 285 00:18:00,630 --> 00:18:04,210 how the concept I've just explained to you 286 00:18:04,210 --> 00:18:06,840 is reflected in equations. 287 00:18:06,840 --> 00:18:09,495 Or the second option we have that takes about 20 minutes 288 00:18:09,495 --> 00:18:12,680 is to explain to you how quantum computation is 289 00:18:12,680 --> 00:18:15,290 done with two ions, in particular 290 00:18:15,290 --> 00:18:19,859 how the famous Zoller-Cirac gate works. 291 00:18:19,859 --> 00:18:21,150 Do you have a clear preference? 292 00:18:21,150 --> 00:18:23,102 AUDIENCE: Second one. 293 00:18:23,102 --> 00:18:30,880 PROFESSOR: OK, fine, but there is one piece 294 00:18:30,880 --> 00:18:33,250 of mathematics you should understand. 295 00:18:33,250 --> 00:18:35,150 Really, and this is the only essence 296 00:18:35,150 --> 00:18:36,930 of the mathematical derivation. 297 00:18:36,930 --> 00:18:40,680 If I'm standing here, and I feel the alternating force, 298 00:18:40,680 --> 00:18:44,780 it's a cosine omega t, and it averages to 0. 299 00:18:44,780 --> 00:18:48,020 But if the force drives me, it shakes me. 300 00:18:48,020 --> 00:18:49,740 This is micromotion. 301 00:18:49,740 --> 00:18:51,875 Like the beam here, if you go through many lasers, 302 00:18:51,875 --> 00:18:54,080 the beam goes up and down, up and down. 303 00:18:54,080 --> 00:18:55,320 This is micromotion. 304 00:18:55,320 --> 00:18:58,606 The micromotion is at cosine omega t. 305 00:18:58,606 --> 00:19:01,850 And the net attractive force comes 306 00:19:01,850 --> 00:19:05,760 because the force which oscillates at cosine omega t, 307 00:19:05,760 --> 00:19:08,500 and I'm oscillating at cosine omega t, 308 00:19:08,500 --> 00:19:11,930 then you get cosine squared omega t. 309 00:19:11,930 --> 00:19:15,240 And this gives you the net force which 310 00:19:15,240 --> 00:19:17,750 provides the net attractive force. 311 00:19:17,750 --> 00:19:22,620 And it's the cosine square term averaged over the micromotion, 312 00:19:22,620 --> 00:19:25,920 which is one half, which is responsible for the trapping 313 00:19:25,920 --> 00:19:28,490 potential in ion traps. 314 00:19:28,490 --> 00:19:31,530 Anyway, you know the physics very well. 315 00:19:31,530 --> 00:19:34,470 You understand where the math comes form in that it's really 316 00:19:34,470 --> 00:19:35,906 just the product of the two. 317 00:19:35,906 --> 00:19:36,780 It's a rectification. 318 00:19:36,780 --> 00:19:38,730 It's an alternating force. 319 00:19:38,730 --> 00:19:41,470 But since the system itself, the particle itself, 320 00:19:41,470 --> 00:19:44,980 is driven at the micromotion, there's rectification going on, 321 00:19:44,980 --> 00:19:46,970 which leads to the cosine squared term. 322 00:19:49,790 --> 00:19:50,500 [? Timor, ?] yes? 323 00:19:50,500 --> 00:19:51,760 AUDIENCE: Two quick questions. 324 00:19:51,760 --> 00:19:54,343 So I think ion trappers always complain about the micromotion. 325 00:19:54,343 --> 00:19:57,160 But in fact, they need it for trapping 326 00:19:57,160 --> 00:20:00,940 to work-- is pretty much what's going on here, right? 327 00:20:00,940 --> 00:20:03,351 PROFESSOR: Yes, actually, the one interesting side 328 00:20:03,351 --> 00:20:09,270 of the mathematics is the micromotion is kinetic energy. 329 00:20:09,270 --> 00:20:11,050 The harmonic trapping potential is 330 00:20:11,050 --> 00:20:13,910 nothing else than the micromotion. 331 00:20:13,910 --> 00:20:16,360 In other words, if people show in an ion trap 332 00:20:16,360 --> 00:20:18,670 that you have a harmonic oscillator, 333 00:20:18,670 --> 00:20:21,600 the harmonic oscillator potential is high here. 334 00:20:21,600 --> 00:20:24,950 But mathematically, this harmonic oscillator potential 335 00:20:24,950 --> 00:20:27,520 is the kinetic energy of the micromotion. 336 00:20:27,520 --> 00:20:30,070 In other words, if you look at the micromotion, 337 00:20:30,070 --> 00:20:31,630 it has a kinetic energy. 338 00:20:31,630 --> 00:20:35,090 The kinetic energy appears as the potential energy 339 00:20:35,090 --> 00:20:39,100 in this slow secular motion. 340 00:20:39,100 --> 00:20:42,000 So in other words, the potential energy for the trap 341 00:20:42,000 --> 00:20:44,860 is the kinetic energy of the micromotion. 342 00:20:44,860 --> 00:20:46,820 So you can't avoid it. 343 00:20:46,820 --> 00:20:50,364 This is responsible for the restoring force. 344 00:20:50,364 --> 00:20:52,155 AUDIENCE: Is it a really conservative trap, 345 00:20:52,155 --> 00:20:55,717 or can I have an out of phase component 346 00:20:55,717 --> 00:20:57,457 to the micromotion response? 347 00:20:57,457 --> 00:20:59,942 Is there like a sine part to the response? 348 00:21:04,415 --> 00:21:07,880 PROFESSOR: Well, the secular motion is conservative. 349 00:21:07,880 --> 00:21:08,720 You're now asking-- 350 00:21:08,720 --> 00:21:11,220 AUDIENCE: I'm trying to think if there would be a reason why 351 00:21:11,220 --> 00:21:14,134 my response, my dipole response, would 352 00:21:14,134 --> 00:21:17,536 have a different [INAUDIBLE]. 353 00:21:17,536 --> 00:21:19,480 Is there a reason? 354 00:21:25,820 --> 00:21:30,060 PROFESSOR: Well, to the best of my knowledge, 355 00:21:30,060 --> 00:21:32,050 everything is wonderfully conservative 356 00:21:32,050 --> 00:21:34,210 if you have a separation of timescale. 357 00:21:34,210 --> 00:21:36,290 If you want to sort of get something 358 00:21:36,290 --> 00:21:39,380 strange and non-conservative out of the micromotion, 359 00:21:39,380 --> 00:21:41,625 you have to violate the adiabatic separation 360 00:21:41,625 --> 00:21:44,660 of timescales. 361 00:21:44,660 --> 00:21:48,280 There is a possibility that if particles move quickly 362 00:21:48,280 --> 00:21:50,040 through the potential, they're not 363 00:21:50,040 --> 00:21:52,580 following the micromotion adiabatically. 364 00:21:52,580 --> 00:21:54,575 And then there is a collision where micromotion 365 00:21:54,575 --> 00:21:56,270 is released as collisional energy. 366 00:21:56,270 --> 00:22:03,055 But on the similar particle level, 367 00:22:03,055 --> 00:22:04,430 it's a pretty good approximation. 368 00:22:07,800 --> 00:22:14,200 OK, we want to talk about quantum computation with ions. 369 00:22:14,200 --> 00:22:16,080 [INAUDIBLE] has a wonderful write-up 370 00:22:16,080 --> 00:22:19,770 on the wiki on the basic concept of quantum computation 371 00:22:19,770 --> 00:22:24,070 and how it applies to ions. 372 00:22:24,070 --> 00:22:28,250 But let me immediately jump to qubit operations. 373 00:22:28,250 --> 00:22:32,180 You've heard many, many times that a general quantum 374 00:22:32,180 --> 00:22:36,150 algorithm can be constructed out of single qubit 375 00:22:36,150 --> 00:22:40,420 rotations and two qubit gates. 376 00:22:40,420 --> 00:22:43,150 Single qubit rotations are just oscillations 377 00:22:43,150 --> 00:22:44,390 on the Bloch sphere. 378 00:22:44,390 --> 00:22:48,070 If you excite a particle, move it around on the Bloch sphere, 379 00:22:48,070 --> 00:22:51,510 this is how you can rotate a single qubit from ground state 380 00:22:51,510 --> 00:22:55,220 to excited state, any possible superposition state. 381 00:22:55,220 --> 00:22:58,750 So the physics of single atoms, Rabi oscillation, 382 00:22:58,750 --> 00:23:01,440 rotation on the Bloch sphere, this is all you 383 00:23:01,440 --> 00:23:03,950 have to know about single qubit operations. 384 00:23:03,950 --> 00:23:08,640 The interesting thing is how do we create two qubit operations? 385 00:23:08,640 --> 00:23:10,710 And two qubit operation would mean 386 00:23:10,710 --> 00:23:13,920 you want to do something to one ion here depending 387 00:23:13,920 --> 00:23:16,040 what the state of the other ion is. 388 00:23:16,040 --> 00:23:18,150 And this is much more subtle. 389 00:23:18,150 --> 00:23:20,020 So what I want to explain to you now 390 00:23:20,020 --> 00:23:25,440 is the famous Cirac-Zoller gate, which was really 391 00:23:25,440 --> 00:23:30,350 a pioneering paper which has thousands of citations now. 392 00:23:30,350 --> 00:23:33,490 Because it showed that quantum computation 393 00:23:33,490 --> 00:23:36,240 can be experimentally realized in a rather straightforward 394 00:23:36,240 --> 00:23:36,740 way. 395 00:23:40,500 --> 00:23:49,340 So it's a fundamental theorem of quantum information processing 396 00:23:49,340 --> 00:23:53,155 that if you have single qubit operations, and one 397 00:23:53,155 --> 00:23:56,400 to qubit gate, which is a CNOT gate, that this 398 00:23:56,400 --> 00:23:59,310 is a universal set of quantum gates, 399 00:23:59,310 --> 00:24:03,220 any arbitrary quantum algorithm can be constructed out 400 00:24:03,220 --> 00:24:05,410 of those gate operations. 401 00:24:05,410 --> 00:24:08,780 So let me just remind you what the CNOT gate is. 402 00:24:08,780 --> 00:24:12,580 We have qubits ion one, ion two. 403 00:24:12,580 --> 00:24:14,870 They can be in 0 or 1. 404 00:24:14,870 --> 00:24:17,460 These can be two different hyperfine states. 405 00:24:17,460 --> 00:24:20,920 Or in the case of the calcium ion, it can be the ground state 406 00:24:20,920 --> 00:24:24,670 or metastable excited state-- just two different states 407 00:24:24,670 --> 00:24:28,610 of ion one, two different states of ion two. 408 00:24:28,610 --> 00:24:31,580 The CNOT gate would now say the following. 409 00:24:31,580 --> 00:24:35,060 The first ion is the control bit. 410 00:24:35,060 --> 00:24:38,850 And whenever the control bit is 1, 411 00:24:38,850 --> 00:24:41,820 it flips the qubit, the second qubit. 412 00:24:41,820 --> 00:24:48,580 If the control bit is 0, it does nothing to the second ion. 413 00:24:48,580 --> 00:24:55,750 So therefore, the table which shows how the quantum 414 00:24:55,750 --> 00:25:00,530 states perform is 1, 1, 0, 0, 1, 1. 415 00:25:00,530 --> 00:25:04,180 It just shows that I'm writing down 416 00:25:04,180 --> 00:25:05,520 the matrix in these spaces. 417 00:25:05,520 --> 00:25:10,350 It's a truth table 0, 0, 0, 1, 1, 0, 1, 1. 418 00:25:10,350 --> 00:25:14,290 And it shows that when you have 1, 1 at the input, 419 00:25:14,290 --> 00:25:16,460 you get 1, 0 at the output. 420 00:25:16,460 --> 00:25:21,070 If you have 1, 0 at the input, you get 1, 1 at the output. 421 00:25:21,070 --> 00:25:26,780 Well, the two qubit gate I want to discuss, 422 00:25:26,780 --> 00:25:32,690 which can be realized in the simplest possible way 423 00:25:32,690 --> 00:25:38,330 with the Cirac-Zoller algorithm, is the controlled phase qubit, 424 00:25:38,330 --> 00:25:41,730 which has a truth table of 1, 1, 1, minus 1. 425 00:25:41,730 --> 00:25:50,920 And it means it simply changes the face of the two 426 00:25:50,920 --> 00:25:53,710 qubit states when both qubits are 1. 427 00:25:57,920 --> 00:26:04,910 Now, the CNOT gate can be constructed out 428 00:26:04,910 --> 00:26:11,900 of the controlled phase gate by having two single qubit 429 00:26:11,900 --> 00:26:14,820 rotations done with the Hadamard gate. 430 00:26:14,820 --> 00:26:19,650 The Hadamard gate is taking the bases 1 and 2 and creates 1 431 00:26:19,650 --> 00:26:22,860 plus 2, 1 minus 2, the symmetric and anti-symmetric 432 00:26:22,860 --> 00:26:24,460 superposition state. 433 00:26:24,460 --> 00:26:29,890 So therefore, the CPHASE gate is S general as the CNOT gate. 434 00:26:29,890 --> 00:26:33,110 Because one can be used to construct the other. 435 00:26:39,780 --> 00:26:45,690 So how do we create the Cirac-Zoller gate? 436 00:26:45,690 --> 00:26:47,750 So how do we get now the controlled phase 437 00:26:47,750 --> 00:26:50,670 gate out of trapped ions? 438 00:26:50,670 --> 00:26:52,920 So the effect is the following. 439 00:26:52,920 --> 00:26:59,880 Remember, if we have two ions, we 440 00:26:59,880 --> 00:27:05,050 want to change the phase to minus 1 of the state 441 00:27:05,050 --> 00:27:10,200 only if the two ions are qubit up. 442 00:27:10,200 --> 00:27:14,130 So in other words, we want to do 180 degree rotation of the wave 443 00:27:14,130 --> 00:27:18,000 function, but only if both ions are up. 444 00:27:18,000 --> 00:27:19,880 And the question is, how do we do that? 445 00:27:19,880 --> 00:27:24,340 How does one ion feel what the other ion is doing? 446 00:27:24,340 --> 00:27:26,750 And let me explain that now. 447 00:27:26,750 --> 00:27:29,970 The critical element in the Cirac-Zoller case 448 00:27:29,970 --> 00:27:32,890 is a center of mass motion. 449 00:27:32,890 --> 00:27:39,370 We want at some point-- say when this ion is spin up, 450 00:27:39,370 --> 00:27:42,930 we swap the quantum information from the ion 451 00:27:42,930 --> 00:27:44,740 to the center of mass motion. 452 00:27:44,740 --> 00:27:48,200 In other words, when this ion is up, we use laser pulses. 453 00:27:48,200 --> 00:27:49,660 I will explain that to you. 454 00:27:49,660 --> 00:27:52,270 Then all the ions are moving together. 455 00:27:52,270 --> 00:27:55,630 And then, this ion, because they're all moving, 456 00:27:55,630 --> 00:27:59,970 knows that this ion was spin up before. 457 00:27:59,970 --> 00:28:06,470 So the center of mass motion, the common dipole oscillation 458 00:28:06,470 --> 00:28:11,550 of all ions together in the trap, this is the data bus. 459 00:28:11,550 --> 00:28:13,900 So this is the key idea. 460 00:28:13,900 --> 00:28:22,280 But in order to explain it to you, I have to go in two steps. 461 00:28:22,280 --> 00:28:25,540 So let's just take that as a genetic level structure. 462 00:28:25,540 --> 00:28:27,590 We have an S state. 463 00:28:27,590 --> 00:28:29,690 And it has two different hyperfine states, 464 00:28:29,690 --> 00:28:30,940 spin up and spin down. 465 00:28:30,940 --> 00:28:33,450 And these are our two qubits. 466 00:28:33,450 --> 00:28:36,690 There may be a P state, which we have used for laser cooling. 467 00:28:36,690 --> 00:28:38,910 But everything is in the ground state of motion 468 00:28:38,910 --> 00:28:40,970 using sideband cooling, which you've 469 00:28:40,970 --> 00:28:43,030 studied in your homework. 470 00:28:43,030 --> 00:28:45,380 And what I want to show you first 471 00:28:45,380 --> 00:28:49,250 is how we can create a phase gate. 472 00:28:49,250 --> 00:28:51,730 But I'm only focusing on ion one. 473 00:28:51,730 --> 00:28:53,530 So hold the thought. 474 00:28:53,530 --> 00:28:57,710 Ion two will come in a few moments, will come back. 475 00:28:57,710 --> 00:29:00,490 So what we can do is the following. 476 00:29:00,490 --> 00:29:04,240 Each qubit state has vibrational structure-- ground state, 477 00:29:04,240 --> 00:29:07,240 excited state without center of mass motion. 478 00:29:07,240 --> 00:29:10,180 But then, as a function of vibrational quantum number 479 00:29:10,180 --> 00:29:11,625 for the center of mass motion, we 480 00:29:11,625 --> 00:29:15,810 have g1, g2, e1, e2, and such. 481 00:29:15,810 --> 00:29:20,040 And what we can now do is, let's assume 482 00:29:20,040 --> 00:29:22,660 we have another state, a D state, which 483 00:29:22,660 --> 00:29:24,760 is an auxiliary state here. 484 00:29:24,760 --> 00:29:28,040 Fortunately, ions and atoms have many states. 485 00:29:28,040 --> 00:29:35,980 And we can drive a laser transition from S 486 00:29:35,980 --> 00:29:41,880 to D, which is only in resonance with the first vibrational 487 00:29:41,880 --> 00:29:45,600 excited state to the D state. 488 00:29:45,600 --> 00:29:48,690 So we use a [INAUDIBLE] sideband here, 489 00:29:48,690 --> 00:29:51,940 which only drives the transition from e1 490 00:29:51,940 --> 00:29:57,630 to the non-vibrational state of the auxiliary state. 491 00:29:57,630 --> 00:30:02,380 And that means automatically that this laser does nothing 492 00:30:02,380 --> 00:30:06,090 to e0, does nothing to g0, does nothing to g1. 493 00:30:06,090 --> 00:30:11,730 So if you have this Hilbert space of g0, g1, e0, e1, 494 00:30:11,730 --> 00:30:13,580 the only transition which is effected here, 495 00:30:13,580 --> 00:30:17,034 or the only state which is effected here, is the e1 state. 496 00:30:19,700 --> 00:30:21,240 OK, I'm talking about a single ion. 497 00:30:21,240 --> 00:30:23,615 I'm just talking about talking to a single ion right now. 498 00:30:23,615 --> 00:30:24,540 It's really simple. 499 00:30:24,540 --> 00:30:27,690 So if we switch on this laser, what happens 500 00:30:27,690 --> 00:30:33,980 is we simply induce Rabi oscillations-- simple physics. 501 00:30:33,980 --> 00:30:40,240 The Rabi oscillation, however, after one Rabi period 502 00:30:40,240 --> 00:30:45,230 has changed the phase of the initial state by minus 1. 503 00:30:45,230 --> 00:30:47,880 If you carefully look what Rabi oscillations are, 504 00:30:47,880 --> 00:30:50,560 the wave function goes up, down, up, down. 505 00:30:50,560 --> 00:30:53,920 But we usually discuss what happens to the probability. 506 00:30:53,920 --> 00:30:55,880 And this is shown in green. 507 00:30:55,880 --> 00:31:00,070 So when we are back in the original state, 508 00:31:00,070 --> 00:31:04,500 we have changed the phase of the wave function by minus 1. 509 00:31:04,500 --> 00:31:07,410 So let's just assume that we switch on this laser 510 00:31:07,410 --> 00:31:14,211 to this auxiliary state, and we have a-- well, 511 00:31:14,211 --> 00:31:17,390 it's a 2 pi pulse in population. 512 00:31:17,390 --> 00:31:20,290 But it makes a phase shift of pi to the wave function, 513 00:31:20,290 --> 00:31:22,990 which is minus 1. 514 00:31:22,990 --> 00:31:24,920 So what we have accomplished now is 515 00:31:24,920 --> 00:31:28,520 a phase gate not between two ions. 516 00:31:28,520 --> 00:31:33,350 But we have a phase gate which, in this Hilbert space, 517 00:31:33,350 --> 00:31:39,850 between the qubit up, qubit down-- 518 00:31:39,850 --> 00:31:42,590 I call it ground and excited state. 519 00:31:42,590 --> 00:31:46,890 But the other qubit, so to speak, 520 00:31:46,890 --> 00:31:50,600 is the center of mass motion 0, 1, 0, 1. 521 00:31:50,600 --> 00:31:54,790 So this is the truth table of the phase gate. 522 00:31:54,790 --> 00:32:02,490 And so technically, we have now done a local phase gate 523 00:32:02,490 --> 00:32:05,540 with one ion where the two qubits are 524 00:32:05,540 --> 00:32:07,740 ground and excited state of the ion. 525 00:32:07,740 --> 00:32:09,740 This is our qubit we want to work with. 526 00:32:09,740 --> 00:32:14,200 But then, the second qubit to realize the phase gate 527 00:32:14,200 --> 00:32:17,065 was 0, 1 of the center of mass vibration. 528 00:32:20,020 --> 00:32:23,470 But once we have realized that, we 529 00:32:23,470 --> 00:32:31,040 can now immediately take this idea 530 00:32:31,040 --> 00:32:35,470 and get a phase gate between two ions. 531 00:32:35,470 --> 00:32:41,540 Because the center of mass vibration is everywhere. 532 00:32:41,540 --> 00:32:46,165 So therefore, what we can do is this was ion one. 533 00:32:49,200 --> 00:32:51,490 We can now-- and this is the missing step 534 00:32:51,490 --> 00:32:54,890 I want to show you-- go to ion two. 535 00:32:54,890 --> 00:33:00,590 And we can swap the qubit information-- ground, excited, 536 00:33:00,590 --> 00:33:01,860 spin up, spin down. 537 00:33:01,860 --> 00:33:05,680 We can swap that with the center of mass motion. 538 00:33:05,680 --> 00:33:10,600 So therefore, what we can do is, when the second ion is spin up, 539 00:33:10,600 --> 00:33:13,940 we do a swapping of the information 540 00:33:13,940 --> 00:33:16,790 from spin up, spin down to center of mass motion 541 00:33:16,790 --> 00:33:18,930 being 0 or 1. 542 00:33:18,930 --> 00:33:21,340 So the idea is the following. 543 00:33:21,340 --> 00:33:24,890 Ion two should control what ion one is doing. 544 00:33:24,890 --> 00:33:27,950 So we first focus on ion two. 545 00:33:27,950 --> 00:33:33,660 We swap the qubit of ion two into the center of mass motion. 546 00:33:33,660 --> 00:33:37,210 Then, we do what I just said, the controlled phase 547 00:33:37,210 --> 00:33:41,910 gate for ion one with the center of mass motion. 548 00:33:41,910 --> 00:33:48,230 And after this is done, we write the center of mass motion back 549 00:33:48,230 --> 00:33:49,480 into ion two. 550 00:33:49,480 --> 00:33:51,090 And then we have nothing to ion two. 551 00:33:51,090 --> 00:33:53,070 Ion two was just putting its information 552 00:33:53,070 --> 00:33:54,700 into the center of mass motion. 553 00:33:54,700 --> 00:33:58,140 It helped ion one to do the controlled phase gate. 554 00:33:58,140 --> 00:34:00,320 And then, we swapped the information back. 555 00:34:00,320 --> 00:34:02,030 So this is the sequence. 556 00:34:02,030 --> 00:34:05,530 After preparing the center of mass, the first of two lasers 557 00:34:05,530 --> 00:34:12,040 cool everything in the center of mass motion into the 0 state. 558 00:34:12,040 --> 00:34:15,850 Then we swap from ion two the information 559 00:34:15,850 --> 00:34:19,360 to this phonon state center of mass vibration. 560 00:34:19,360 --> 00:34:24,375 Then we do the controlled phase gate on ion one. 561 00:34:24,375 --> 00:34:26,000 And then, we swap the information back. 562 00:34:32,340 --> 00:34:35,020 So what I should show you-- you may have seen it. 563 00:34:35,020 --> 00:34:37,150 But what I should show you explicitly-- 564 00:34:37,150 --> 00:34:41,710 how do we swap the information ge 565 00:34:41,710 --> 00:34:44,290 from the internal state of ion two? 566 00:34:44,290 --> 00:34:48,159 How do we write this information into the data bus, which 567 00:34:48,159 --> 00:34:50,750 is the phonon, which is the center 568 00:34:50,750 --> 00:34:53,719 of mass vibration in the trap? 569 00:34:53,719 --> 00:34:57,500 Well, we again use [INAUDIBLE] side pulse. 570 00:34:57,500 --> 00:35:01,100 We do a pi pulse driving this transition. 571 00:35:01,100 --> 00:35:03,575 And just look how beautifully the math works out. 572 00:35:03,575 --> 00:35:05,910 We have an arbitrary qubit state, 573 00:35:05,910 --> 00:35:08,940 a superposition of ground and excited. 574 00:35:08,940 --> 00:35:11,630 And the center of mass motion was 575 00:35:11,630 --> 00:35:14,730 laser cooled to the ground state. 576 00:35:14,730 --> 00:35:17,900 So let me just multiply that out. 577 00:35:17,900 --> 00:35:23,470 So we have coefficient a times g0 plus coefficient b times e0. 578 00:35:23,470 --> 00:35:30,530 If we now do a pi pulse, we transfer the population from e0 579 00:35:30,530 --> 00:35:31,790 into g1. 580 00:35:31,790 --> 00:35:34,180 And this is what I've done here. 581 00:35:34,180 --> 00:35:38,256 But now you see I can factor out the g. 582 00:35:38,256 --> 00:35:42,460 And what I have now is I have the same qubit information 583 00:35:42,460 --> 00:35:46,700 with coefficients e and b, but no longer in the internal state 584 00:35:46,700 --> 00:35:50,960 of the ion but in the phonon state 585 00:35:50,960 --> 00:35:55,130 in the qubit, which is the center of mass motion. 586 00:35:55,130 --> 00:35:58,950 So the idea here is you can have an arbitrary number of ions. 587 00:35:58,950 --> 00:36:01,250 And now you want to have a quantum gate. 588 00:36:01,250 --> 00:36:03,970 What you do with a quantum gate, you say, 589 00:36:03,970 --> 00:36:07,800 I want to now take this ion here and have a quantum 590 00:36:07,800 --> 00:36:10,530 gate with 10 ions further down. 591 00:36:10,530 --> 00:36:12,840 So you first go to your first ion, 592 00:36:12,840 --> 00:36:16,580 transfer the information of this qubit into the data bus, 593 00:36:16,580 --> 00:36:19,470 into the center of mass motion. 594 00:36:19,470 --> 00:36:22,630 And now with a local operation here, 595 00:36:22,630 --> 00:36:28,890 you can do something to this ion conditionally on the motion, 596 00:36:28,890 --> 00:36:30,320 the center of mass motion. 597 00:36:30,320 --> 00:36:33,890 But the center of mass motion reflects the ion 598 00:36:33,890 --> 00:36:35,770 over there, in which state it was. 599 00:36:35,770 --> 00:36:37,660 And after you have done the two qubit 600 00:36:37,660 --> 00:36:40,210 operation between those two ions, 601 00:36:40,210 --> 00:36:42,220 you take the center of mass motion, 602 00:36:42,220 --> 00:36:44,120 put it back into this ion. 603 00:36:44,120 --> 00:36:46,230 And at the end of the day what you have done 604 00:36:46,230 --> 00:36:49,460 is you have done a two qubit operation. 605 00:36:49,460 --> 00:36:52,580 You've changed the state of this ion. 606 00:36:52,580 --> 00:36:55,840 But all the other ions have remained the same. 607 00:36:55,840 --> 00:36:58,840 And with that, you can now, in a chain of ion, 608 00:36:58,840 --> 00:37:01,295 implement arbitrary two qubit operations. 609 00:37:05,580 --> 00:37:08,150 I think I said it clearer in words here. 610 00:37:08,150 --> 00:37:10,190 It is in a little bit more formal way 611 00:37:10,190 --> 00:37:12,740 the steps you will do to the wave function. 612 00:37:12,740 --> 00:37:13,950 You initialize. 613 00:37:13,950 --> 00:37:16,430 Your transfer the information to the data bus, 614 00:37:16,430 --> 00:37:18,940 you execute the controlled phase gate, 615 00:37:18,940 --> 00:37:21,380 and you put the information from the data bus 616 00:37:21,380 --> 00:37:23,470 back into the control ion. 617 00:37:23,470 --> 00:37:26,660 And now your data bus is free, is back again 618 00:37:26,660 --> 00:37:30,870 to the initial value and ready for the next operation. 619 00:37:30,870 --> 00:37:33,595 I should mention that this was pioneering paper. 620 00:37:36,140 --> 00:37:39,700 Right now, this quantum gate is not the most popular one. 621 00:37:39,700 --> 00:37:42,620 Because it has severe requirements. 622 00:37:42,620 --> 00:37:47,250 One is you really have to cool to the absolute ground state. 623 00:37:47,250 --> 00:37:48,860 The center of mass motion has to be 624 00:37:48,860 --> 00:37:50,930 cooled to the absolute ground state. 625 00:37:50,930 --> 00:37:52,920 There are other quantum gates which 626 00:37:52,920 --> 00:37:56,120 do not require such extreme cooling. 627 00:37:56,120 --> 00:38:01,020 And secondly, the center of mass motion, because all particles 628 00:38:01,020 --> 00:38:05,620 move together, is one of the slowest vibrational motions 629 00:38:05,620 --> 00:38:06,870 in the system. 630 00:38:06,870 --> 00:38:08,880 And if you want to scale up the system 631 00:38:08,880 --> 00:38:13,090 to more and more particles, the speed of the gate operation 632 00:38:13,090 --> 00:38:15,750 has to be slower than the vibrational period 633 00:38:15,750 --> 00:38:17,850 of the center of mass motion. 634 00:38:17,850 --> 00:38:21,340 So these are two disadvantages. 635 00:38:21,340 --> 00:38:27,740 But there are different gates, like geometric gates, 636 00:38:27,740 --> 00:38:31,560 like Sorensen gates. 637 00:38:31,560 --> 00:38:34,620 There are other possibilities to involve the same quantum 638 00:38:34,620 --> 00:38:36,820 logic with ions. 639 00:38:36,820 --> 00:38:39,860 But the one of Cirac-Zoller really 640 00:38:39,860 --> 00:38:41,830 stands out by it's pedagogy. 641 00:38:41,830 --> 00:38:43,980 It's really a wonderful way to see 642 00:38:43,980 --> 00:38:47,080 how we take the information, put it in a data bus, 643 00:38:47,080 --> 00:38:50,840 do the controlled phase operation, 644 00:38:50,840 --> 00:38:56,030 and then proceed to the next two qubit gate. 645 00:38:56,030 --> 00:39:01,055 OK, so that's the end of ions, the end of this class. 646 00:39:01,055 --> 00:39:05,740 Do you have any questions about the last topic? 647 00:39:09,360 --> 00:39:10,840 If not, let me say thanks. 648 00:39:10,840 --> 00:39:12,780 You were a fabulous class. 649 00:39:12,780 --> 00:39:14,990 I really enjoyed the classroom atmosphere. 650 00:39:14,990 --> 00:39:17,410 I enjoyed all the questions, whether it 651 00:39:17,410 --> 00:39:19,280 was after class or during class. 652 00:39:19,280 --> 00:39:22,150 And, well, I'm happy that most of you 653 00:39:22,150 --> 00:39:25,720 I will see around for a number of years. 654 00:39:25,720 --> 00:39:27,290 All right. 655 00:39:27,290 --> 00:39:30,940 [APPLAUSE]