4. Non-classical light, squeezing, Part 2
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Ketterle, Wolfgang
In this lecture, the professor discussed Homodyne detection and teleportation of light.
 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. Well, we are in the middle of something really interesting. We are talking about squeezing. We are talking about non-classical light. And today I sort of want to wrap it up. I think it will be really an exciting class. But before I continue with the material, I want to address a question which actually came up in discussions with several of the students. And this is, I realize that some people said, OK, everything makes sense. But what are we plotting? What is really squeezed? Are we squeezing in the spatial domain? Are we squeezing in the temporal domain? So the plots look wonderful with these ellipses and the circles. But what is it really we are doing here? So let me address that. First of all, we are talking about, as we are talking about a single harmonic oscillator. We showed that the Maxwell's equations can be reduced to a bunch of harmonic oscillator equation, one for each mode. And now we are talking today and in the previous few classes about one single mode, about one harmonic oscillator. And the harmonic oscillator has canonically variables of momentum and position. But this is just to make a connection for you with something you have learned. What we are talking about is a single harmonic oscillator, which is one single mode of the electromagnetic field. So maybe let me draw a cartoon for that. So let's assume we have a cavity. We have an electromagnetic wave. There is propagation. There is e to the ikz. There is transverse confinement. Maybe there is a Gaussian, e to the minus x squared plus y squared over some beam waves parameter. All that is simply the spatial mode. And we just take that for given, because we are not solving the spatial differential equation. All we are doing it, we are looking at this one mode. And the two degrees of freedom is that this mode can have a certain number of photons. It's the amplitude. And the second one is, you can say, the temporal phase. It can be a cosine omega t. It can be a sine omega t. It can be a superposition. But whatever we are talking is in this mode. There is nothing happening in the spatial domain. We're just asking what is the oscillation in this mode. The whole mode does what it should. It has a pre-factor, which is the amplitude. And it has a temporal factor, which we factor out. And this is what we are talking about. Let me be a little bit more specific and say that when we are plotting things, we are plotting the q representation, the phase-based representation of the statistical operator, which is simply describing this single mode of the harmonic oscillator. And by performing the diagonal matrix elements, we obtain the q distribution. In that case, we have the vacuum. We have a displaced vacuum, which is in coherent state. And our axes are, from the very definition, we are in the complex plane with a real part and imaginary part of alpha. However, we can also define a Wigner distribution, which is another phase-based distribution. It's almost the same as the q distribution. It's just this little bit smeared out by h bar because of some commutators, but nothing you have to worry about. In that case, the projection of the w function on the vertical axis, on the y axis, is the momentum wave function squared. On the x axis, on the x axis, it is the spatial wave function. It's squared of the harmonic oscillator. So therefore, we may sometimes think when we have a distribution here, we project it and we see what is the momentum distribution or what is the spatial distribution of the mechanical harmonic oscillator, which is analogous, which is equivalent to the one mode of the electromagnetic field we are using. I know it may help you to some extent to think about the p and q, but it may also be misleading because it gives you the impression something is moving with the momentum p in real space. Let me therefore emphasize what are the normalized forms of p and q. If I do the symmetric and anti-symmetric combination of the annihilation and creation operator, i over square root 2, I call those a1 and a2, and they are nothing else than q and p normalized by the characteristic momentum or spatial coordinate of the harmonic oscillator. So what is important here is that a1 and a2, forget about pq now, they are equivalent. But for the electromagnetic field, a1 and a2 have a very direct interpretation. They are called the two quadrature operators. And what I mean by that becomes clear when I use the Heisenberg representation for the electric field. And I'm here using the formula which is given in the book of Weisblath in page 175. Some copies of the book have been, some pages copied from this book have been posted on the website. So we have our normalization factor, which is the electric field related to the electric field of a single photon. We have the polarization factor. But now we have an expression which involves the quadrature operators a1 and a2. Just to be specific, we are not in a cavity here. Therefore, we have propagating waves cosine kR sine kR. But you can also immediately use similar expression for the case of a cavity. Let us specify at r equals 0. And then we realize what the two quadrature operators are. A2 is the operator which creates and annihilates an electromagnetic field, so to speak photons, which have an electric field which oscillates as cosine omega t. And a1 is the quadrature operator for the sine omega t component. So therefore, if you simply analyze the electric field, what is cosine omega 2 is related to the a2 quadrature operator, the sine omega 2 oscillation is related to the a1 quadrature operator. So life would be easier, but more boring if you could create a pure cosine or pure sine oscillation of the electromagnetic field. But you can't, because a1 and a2 do not commute. And there is an uncertainty relation that delta a1 times delta a2 is larger or equal than 1 half. And we have the equal sign for coherent states alpha. So therefore, if we look at the electric field, everything moves around periodically in r, because it's a traveling wave in t, because it's an oscillating wave. So let's not confuse things. Let's simply peak r equals 0. We've already done that. But now let's peak t equals 0. At t equals 0, the sine omega t is 0. And therefore, the distribution for a2, the expectation value, and the variance for the quadrature operator, or a2, can be simply read off by looking at the electric field. So in other words, at t equals 0, the electric field, which is obtained by projecting our quasi probabilities on the y-axis, gives the expectation value for a1 and the variance delta a1 squared. Somebody says, 2 at t equals 0, yes. Yes. And now, if you want to see what the other quadrature component is, well, we just wait a quarter period until the sine, which was 0, is maximum, and the cosine omega t is 0. So therefore, at pi over 2 omega, using the projection on the y-axis, gives us a1 and delta a1 squared. Or alternatively, we don't need to wait. We can do t equals 0, and we can project onto the x-axis. So therefore, let me just draw a few things into this diagram. If you had a classical motion, which would simply be cosine omega t, then that would mean we would just have a, if you had a motion which were only cosine omega t, yes, it would be a point on the y-axis. However, classically, we can never have something which is just cosine omega t. The point has to be blurred into a circle. This is a coherent state. So this coherent state has now, let me just call that one for the sake of the argument, the point would be the classical oscillator. It has just cosine omega t. Everything is dataministic. No uncertainty, no nothing. Of course, it means that in the time evolution, it goes in a circle. But this is what everything does in a harmonic oscillator when time evolves due to the e to the i omega t factor. So let's not get confused with that. Let's just look at t equals 0. And for traveling wave r equals 0, the classical oscillator is one point. But now we have a spread here. This says that quantum mechanically, the amplitude of the cosine omega t term is not arbitrarily fast. It's not arbitrarily sharp. They are fluctuations. And in addition, we have widths in this direction, which we project onto the x-axis. And this tells us what the distribution in our ensemble, in our quantum mechanical ensemble, is for the amplitudes of the sine omega t motion. So the best we can do is quantum mechanically, if you want to have something which is really just cosine omega t, we have to squeeze it that the cosine omega t amplitude is now extremely sharp. But the sine omega t amplitude in the ensemble is completely smeared out. So this is what we are talking about. Now, what I think has confused some of you is what I thought was a wonderful example, the classical squeezing experiment. I mean, these are visuals which will be in your head forever when you saw Professor Britschard with a circular pendulum he's just pulling, and then the circle squeezes into an ellipse. And it seems that something is here squeezed in real space. But this is actually wrong. How you should have looked at the experiment, and I made a comment about it, but maybe not emphatically enough, you should have really thought about a single pendulum. And this single pendulum, if it has an arbitrary phase, is in a superposition of sine omega t and cosine omega t. And if you pull on the string, if you shorten and lengthen the pendulum at sine 2 omega t, you will amplify the pre-factor in front of sine omega t, and you will exponentially de-amplify the factor in front of cosine omega t. So therefore, what will happen if this pendulum oscillates, and let me say with a phase, well, sine omega t plus delta, if delta is 90 degrees, cosine, if delta is 0, it's sine. And let's say this pendulum oscillates at 45 degrees, sine omega t plus 45 degrees. If you now parametrically drive it, the squeezing action would now mean that you, let's just make a sine convention, you de-amplify the cosine, you amplify the sine, and after a while, instead of oscillating with sine omega t plus 45 degree, it will oscillate with an amplified amplitude at sine omega t. This is what you have done. And this is the mechanical analogy. There is, of course, no squeezing in any way, because in a classical pendulum, we start with one definite value, if you prepare the system well, and then we just change the motion. We amplify, we pick out a phase, and that's what we are doing. Now, there are two ways how classical squeezing can come in. One is, if the motion of the pendulum is, maybe there is an uncertainty. Maybe Professor Britscher did experiments with an ion trap, and actually 20 years ago, he published a Fuss-Refletter on classical squeezing. You think, how can you publish a Fuss-Refletter on classical squeezing? Well, he had the world's most accurate ion trap, measuring atomic masses with 10 and 11 digits precision. And what was actually one limiting factor was, for Kelvin, the thermal distribution of harmonic oscillator modes. And so what he had is he didn't have just one clean amplitude. The sine omega t amplitudes had a spread because of the thermal distribution he started from. And so what he then did is, by simply classical squeezing, by doing it classically with the ion trap, exactly what he did with a pendulum, drive it with sine 2 omega t, he could now take this classical distribution of, this is a classical distribution. In one axis, it's a distribution of amplitudes of the cosine motion. And here, it's the classical distribution of the amplitudes of the sine motion. And he was squeezing it into this direction. So he had a narrower definition of the coefficient for the cosine omega t motion. And as I will tell you today is, you can now do a homodyne measurement, which is reducing the noise. So essentially, he prepared, quote, unquote, effectively a colder ensemble by squeezing the uncertainty in the cosine pre-factor at the expense of increasing the pre-factor, the uncertainty, the variance in the pre-factor of the sine omega t motion. Finally, you all saw something visually. You saw how a circular motion became a linear motion. So what was going on here? Well, I mentioned to you that the circular pendulum actually has two modes. These are two modes of the harmonic oscillator. And I'm not talking about two modes of the harmonic oscillator. Everything we're discussing here is about one mode of the harmonic oscillator. The circular motion of the pendulum was sort of just a nice visualization trick that if the pendulum moves in a circle, you have a degenerate harmonic oscillator. One is excited with sine omega t. The other one is excited with cosine omega t. And instead of doing two experiments, if you would start with sine omega t and you parametrically drive it, you amplify it, you start with cosine omega t, you could bring the pendulum to a stop. But instead of doing two experiments, Professor Bridger just did one and showed that the sine omega t motion became larger and the orthogonal cosine omega t motion shrunk. And therefore, you saw that the circular motion, which was a superposition of sine and cosine, became pure sine motion. But the fact that there was something we could see in a spatial domain was simply due to the fact that we had kind of two experiments at one, two versions of the same harmonic oscillator, one in x and one in y. And then when we did the experiment, we saw something visually in the spatial domain. So that's why we saw squeezing in the spatial domain. But you should really think about it. What the whole action is, is it's an interplay of de-amplifying prefactors of cosine, amplifying prefactors of sine. And if the prefactor has a distribution, by de-amplifying it, you also shrink the widths of the distribution. And this is what we call squeezing. Yes, Timo? Two for question. So the operators A1 and A2 here, right? You use those instead of A and A dagger, because you used cosine and sine rather than e to the i omega t and e to the minus i. Because they should contain in the dagger, right? Let me go back to the definition. They are actually exactly. All right. They correspond exactly to position and momentum of the mechanical harmonic oscillator. So if that makes sense. The other thing is from the technical speaking, would we call the last goal that field e, cos k r minus omega t as maximum squeeze? If it only has a cos component? Well, not if I use it. Depends how I define squeezing. So you would now give a definition of squeezing, which says that the variance in A1 is now unequal to the variance in A2. So the classical oscillator is a point. It has zero variance in A1, zero variance in A2. But as I said, you can actually apply all that to a classical oscillator if you add technical noise or thermal noise. Then your system is prepared not with a sharp value, but with a distribution, which is simply maybe a Boltzmann distribution due to the preparation. So we call it squeezing when the noise in the amplitude of the sine motion is not equal to the noise in the amplitude of the cosine motion. Some people say it a little bit narrower. They reply squeezing to the situation that we are at. We are uncertainty limited, and then we squeeze. But of course, you can say you can always reduce the noise in your system by just preparing the system, by cooling the system, by selecting the system through measurements until you reach the quantum limit. So you can get a smaller delta A1, a smaller delta A2 without squeezing by just better preparation or by selecting your ensemble. So squeezing in a narrower sense only makes sense when you hit the limit what quantum mechanics allows you. And now you want to distribute the variance unequally between A1 and A2 because then you can get something in delta A1 or delta A2, which is better than 1 over square root 2. And this is now really quantum mechanically squeezed. But there are both definitions. Classical squeezing exists. It's just not as common as quantum mechanical squeezing. Other questions? Yes? I'm a little confused when you said that in the classical squeezing, you are attenuating one amplitude and you are amplifying the other amplitude. So in this picture, then, shouldn't we have the ellipse come down from the circle on the y-axis? OK. You're just not just changing delta A, but also changing A itself. Let me just get a sketchbed here. So this is A2. This is A1. A2 is for the cosine omega t and A1 for the sine omega t. So just to be specific. So what do you want to prepare? You want to prepare a harmonic oscillator, which is just sine omega t. This is a point here. If you are now parametrically, so this has a value at t equals 0, our A1 is S0. So if you are now squeezing your classical harmonic oscillator, you would have a situation where A1 of t is S0 times e to the plus or minus rt, depending whether you do the parametric drive at sine 2 omega t or cosine 2 omega t. So therefore, what would happen is this point will be amplified. It would just move out on the x-axis. So this would be for the plus sign. Or for the other case, you would damp the motion to 0. And this is the minus sign here. OK. And you are also changing the variance. Well, a point does not have variance. But for an example. So if you want to build up variance, you need, let's say, three points. One is the average value. One is the left outlier. One is the right outlier. And what happens is now, as you amplify the motion, you would also amplify, magnify the distance between the points. And if you deamplify it with the minus sign, the distance between the points would shrink because all the three points converge to 0. So pretty much what I just told you for the three point ensemble, you can now use it and construct any initial condition you want and see what happens due to squeezing. Other questions? Good. So the question now is how to measure squeezing, how to take advantage of squeezing. So the situation we are facing is the following. Let's assume we have done a nice squeezing job. And that means that we have a sharp value. We have created a narrow distribution of the cosine omega 2 coefficients. So the cosine omega 2 motion is rather sharp. But we also know that the electric field itself is sharp at this moment. But since this ellipse rotates, the electric field will have enormous uncertainty a quarter period later. So if we want to take advantage that we have squeezed the electromagnetic field, there are now a couple of ideas which we can use. One is we could just measure the electric field stoposcopically. We would just make a setup where we look at the system in a quantum measurement process. Only we only measure the electric field when the ellipse is like this or is like this. And therefore, we have a sharp value of the electric field. But instead of doing a stoposcopic measurement, we can do something else. Remember, we have a distribution of cosine omega t and sine omega t. And we have a distribution of coefficients c and s. And we know we are interested in the cosine omega t. So how to pick that out has actually been solved in early radios. You do a homodyne detection. In other words, you take a reference oscillator, which is strong. B of t is B naught times cosine omega naught plus delta. And if you mix, maybe before I use the word mixing, I should simply multiply. If you multiply the two signals, your signal you are interested in, or at least you're interested in one component, you multiply it with your huge local oscillator. And then you integrate over time. Then, of course, when you pick the phase delta to be 0, cosine omega t times cosine omega p t gives cosine square omega t. It averages to 1 half. So you have a time average. Whereas cosine omega t times sine omega t averages to 0. So for delta equals 0, you project out the cosine component. And for delta equals 90 degree, you pick out the sine component. So therefore, you can have a measurement. It's a phase sensitive measurement by multiplying your signal with a local oscillator, where you are only sensitive to the component you have squeezed. And therefore, your measurement uncertainty has now been reduced by the squeezing factor. This omega should be the frequency of the laser. Omega has, yes, actually, homodyne means we use the same frequency. Heterodyne would mean we use two different frequencies. But I'm not talking about that. So we have to use exactly the same frequency here. So this reference also should also need the laser to make sure that. Yes. So to address your question, what usually happens is you start with one laser in those experiment. You frequency double the laser. If you want to do some squeezing, you remember that we need a parametric oscillator, where one energetic photon gives us two photons. So what you do is you start with a laser, often it's a YET laser at 1064 nanometer. You frequency double it to a green laser. The green laser pumps your parametric oscillator. And then you get through down conversion. You get squeezed light at 1064. But this is because you first double the laser, and then you break the photon into two pieces, has exactly the same frequency as your laser you started with. And this laser is then the local oscillator, or the reference clock, for your whole experiment. So everything in your experiment, the doubled laser, the parametrically down converted beams, everything is related to the single one laser you started with, and everything is phase coherent. So that's how usually the experiment is done. Before I tell you what we're doing quantum mechanically, let me just also get another question out of the system, which I've been asked several times. People ask me, well, the problem is that kind of the ellipse rotates like this. Isn't there a way, now I need my hand, that we can have an ellipse rotating like this? I mean, that would be great, but this is sort of unnatural because the harmonic oscillator does that. So if you wanted to do that, you need an operator, which is really at every cycle of the electromagnetic field, is when the light wants to sort of do this. No, you always push it back, and this is impractical. You need really an oscillator which would completely change the quadrature components of your harmonic oscillator in every single cycle of the electromagnetic field. But what homodyne detection is, instead of now forcing the light to stay aligned to sort of do this, which is very unnatural, we allow the light to freely evolve. But we have now an observer, our local oscillator, which is rotating synchronously with the ellipse. So we have a local oscillator, which is cosine omega t. It does so to speak exactly what the ellipse is doing. So in that sense, the local oscillator allows us now to observe the ellipse always from its narrow side because the local oscillator is rotating. But the mathematics is pretty much the Fourier transform. The mathematics is the Fourier transform. The physics is the physics of a log-in detector. OK, now the only question remains is, how do we mix? How do we get a product of our signal and the strong local oscillator? In an old radio, it's done by an element, maybe a diode, which has a nonlinear circuit. If you drive a nonlinear element with two input sources, you get something which involves the product of the two. OK, so the principle of homodyne detection is now that we want to mix light at a beam splitter. So the device which does all that for us is, after using so many words, I would say, is disgustingly simple, it's really a half-silver mirror. We'll talk about beam splitters a lot here because beam splitters perform wonderful unitary transformations and we will exploit them for many purposes. For the purpose of this lecture, I simply assume that we have a 50-50 beam splitter. So what I'm telling you now about the beam splitter will be generalized either later today or in the lecture on Wednesday. So a beam splitter has two input ports and two output ports. We have light impinging on the beam splitter. We call those modes A and B. And after the beam splitter, we have two output modes. And let me call them A0 and B0. And what we measure is the output of the beam splitter. So we have two photo detectors and we measure the output. OK, the output modes A0 and B0 are simply obtained by taking the input modes and propagating them. We have two modes, AB. You can say what goes vertical is A. We call it A0. Here it becomes B0. And what we have to do is we have to transform the operator A0. Let me just make a comment. Sometimes in the beam splitter, you want to think a quantum state comes and is transformed. But instead of transforming the state of a photon, I can also transform the operator which creates the photon. One is the Heisenberg picture. One is the Schrödinger picture. So right now I use the Heisenberg picture. I have two operators, AB. They transform into, before the beam splitter is there, I have two operators A0 and B0. Afterwards, I have two other operators. I have operators AB. Afterwards, I have operators A0 and B0. And if you do the time evolution, it's a unitary transformation. And for operators, we have to multiply the operators from the left and right hand side with the unitary operator u and u dagger. And we really talk about the beam splitter in its full beauty in a short while. It will be a special case of what we discuss later, but I think it's pretty obvious that the 50-50 beam splitter is simply creating two modes. One are the symmetric combination and one are the anti-symmetric combination. OK, so in our homodyne detector, we are now measuring the number of photons in the mode A0. We measure with the upper detector the number of photons in the mode B0. But what we are then doing is we run it through a different circuit. We want to cancel certain noise, as you will see in a moment. This is why we obtain the different signal, which we call I minus. So let's now calculate what I minus is. Well, the number of photons in the mode A0 is A0 dagger A0. That's the operator for the intensity of the number of photons. We subtract B0 dagger B0. And as you can immediately see, is that because of the beam splitter, this is now involving a product of A and B, AB dagger plus A dagger B. So in other words, when you ask yourself, how can we remember, I said we pick out the cosine omega t component by multiplying our signal with a local oscillator? And you would say, how do you multiply two modes of the electromagnetic field? Well, just send it to a beam splitter. In a beam splitter, you create the sum of them. But then your photo detector is a square law detector. You measure the electric field squared, or you have to measure the operator A dagger A. And now you get a cross term between AB dagger and A dagger B. So this is how we multiply two operators, how we get a signal which is proportional to A or A dagger times B dagger. By the way, if you wouldn't take the difference, we get terms of A, A dagger, B, B dagger, just the mode A or the mode B by themselves. And we want to get rid of them. And by taking the difference between the two photo detectors, those parts of the signal are coming mode and are subtracted out. So therefore, this is called balanced homodyne. Because we have a balanced beam splitter, we measure two signals, and then we take the difference of the two signals. OK. There's one more thing we have to add. And then we find we understand the balanced homodyne detection. We want to exploit that our mode B, remember, we want to measure mode A. A is squeezed. A has quantum properties. A has only single photons. And B is just a trick to project out the cosine omega t. And we do that by choosing for B a strong coherent state, described by a coherent state parameter beta. Of course, whenever we have a strong coherent state, that means that we can replace B by beta and B dagger by beta star. This is sort of the classical field limit of a strong coherent state. And now we can ask, what is our difference signal i minus? Well, the coherent state depends on the phase angle theta. So it will be phase sensitive. It will depend on the angle theta. But now there is one more thing. And that is, if you use a stronger coherent state, of course, both of our outputs go up in proportion to beta. So therefore, we want to do a normalization by dividing by beta. So this is now our normalized output. So this is now simply, I'm looking at this. So this is our operator for the signal i minus. And it is A. B has been replaced by, B dagger has been replaced by B star. That gives us A times E to the, because of the complex conjugation, minus i theta, plus A dagger times E to the i theta divided by 2. So therefore, if we choose the phase to be 0, we measure the symmetric combination A plus A dagger over 2. And this is, just comparing nodes, simply the A1 quadrature operator divided by square root 2. And if we put a phase shifter, just a dispersive element into the strong local oscillator and shift the phase by pi over 2 by delaying the light pulse by a quarter wavelength, what we are now projecting out is the other quadrature component. So therefore, using a beam splitter in the local oscillator, we can now measure the expectation value for A1 and for A2. I've posted a few papers on the website which show some of the pioneering experiments where people did exactly that. So then when they observed, let's say, they observed the quadrature component which was squeezed, they usually did it for squeezed vacuum. So A1 and A2 had zero expectation value. But the interesting part is how much noise was there. If you don't squeeze, you find that your normalized noise simply corresponds to the classical photon shot noise. But if you are squeezed, you find that one quadrature component has smaller noise than shot noise. So you detect n photons in your different signal, and your noise is less than square root n, whereas in the other quadrature component is larger. So you can look up on the website papers which where people slowly varied the phase of the local oscillator. And you see this is shot noise. How the noise is below shot noise exceeds shot noise. It's below shot noise and exceeds shot noise. And this was the first evidence for the generation of squeezed light. Questions? OK, so we have discussed the detection of squeezed light using balanced homodyne detector. And balanced means that the beam splitter was 50-50. So now we are ready to discuss the unbalanced case. So let's get our unbalanced beam splitter. And just for the sake of the argument, let's say it has a really good transmission of t. t is a transmission coefficient. t squared tells you what fraction of the power is transmitted. And let's just assume for the sake of the argument, 99% are transmitted. So therefore, if we start with our signal a, which may be an interesting squeezed quantum state, a number state, you name it, a naught is pretty much the same as a. We haven't lost so much. But now we use our local oscillator, which is very strong. Only a very small fraction of it will be reflected. But we can always compensate for the small reflection by making b even stronger. Let's see what we get. So the mode a naught, the output mode, is now the transmission coefficient times a. a and a, the operators are sort of amplitudes. So we have to use, if you have a 99% beam splitter, we have to take the square root of 0.99. This is the transmission coefficient. And now we have the transmission coefficient for the strong local oscillator, which we will again approximate by its eigenvalue, coherent state eigenvalue. And the reflection coefficient is 1 minus t squared. So if we make the approximation that t is approximately 1, we can take it out of the parentheses. And we obtain that. If t is close to 1, we can neglect it. And what we see now is what we have obtained is actually the original quantum state a, or I should say the operator, I'm in the Heisenberg picture. a is the mode operator for the input of the unbalanced homodyne detector. And what we have simply done is we have displaced the operator. We have displaced the mode operator a. And the displacement operator has an argument of 1 minus t squared beta. And this is the Hermitian conjugate, 1 minus t squared beta. So what I've shown you here is that a local oscillator and a beam splitter is the realization or implementation of the displacement operator. So in the limit that t goes to unity, we are not losing anything of our quantum state. But by reflecting in the amplitude of a strong coherent state, we simply take our quantum state and we displace it in this two-dimensional plane. So that's what this beam splitter does for us. Any questions? In the next few classes, we really take advantage of those elements. We know now that the displacement operator is just a beam splitter. We know when we have squeezed some light by using balanced homodyne detection, we can just do measurement of one quadrature component of the other. So what I hope for those of you who haven't heard about it, what you take away from that is that these are extremely simple elements. But by combining them, we can really realize very sophisticated schemes of quantum optics. To some extent, when I heard about for the first time, I think the mathematics look so fancy, I couldn't believe that such simple elements can actually realize what those operators describe. So I gave you here the example for the displacement operator. When I learned about the beam splitter and its underlying physics, there was one thing which really fascinated me. And this is the most simple element you can think. I mean, what is simpler than a beam splitter? A beam splitter has two inputs, two outputs. The simplest optical element is just an attenuator. Put in a window in your laser beam, and you lose 4% of your power per surface. Or put in just a little bit of dirty optics, and you lose a few percent. So what I want to discuss with you now is what is really an attenuator quantum mechanically? Well, if classically, the attenuator would do the following. An attenuator is a device which has a transmission, which is a transmission coefficient square, which is smaller than unity. And in a classical system, if you have a coherent state, you would simply assume that the coherent state gets multiplied with a transmission coefficient. In other words, that you have your original state, described by this phase or alpha. And then the action of the attenuator would simply be to scale everything down by a transmission factor of alpha. So the picture you should have is the following. You have a coherent state. It gets attenuated by the transmission coefficient. But if your fluctuations about the coherent state, also the fluctuations get attenuated, because everything gets attenuated by this attenuator. So if you look at that, you should immediately say, no, this is quantum mechanically forbidden. Because the coherent state, with a minimum uncertainty state, this shaded area cannot be smaller than 1 half. But here it has become smaller. So what I've shown to you here is a violation of quantum mechanics. It would actually mean, let me just give you the example. It would mean that if you had a, the coherent state is quantum limited. And if you calculate what is the fluctuations in the photo number, it's just shot noise. So just to give you a sort of simple intuitive example, if you have 10,000 photons plus minus 100, this is shot noise, square root n. If you could now attenuate it by a factor of 100, and you would go from 10,000 plus minus 100 to 100 plus minus 1, that's much better than the shot noise. I mean, this is what I'm just telling you. What a simple-minded attenuator would do, and you would immediately say, that's too good to be true. I cannot get sub-shot noise light. So what is wrong? Impossible, not allowed. Well, we have just tried to formulate something intuitively, and we have to be careful. Well, we know already one way how we can attenuate an input beam, and maybe we should go back to the situation and analyze it. We know that we can attenuate it with a beam splitter, and this beam splitter has a transmission coefficient of t. And then we get our transmitted coherent light. There is something getting reflected. But now we realize that this beam splitter is not just a device which has one input, it has another input. And you may say, well, I don't care about the other input. I don't want to use it. Well, if you don't want to use the input, it has the vacuum state. So therefore, if you would realize the attenuator with a beam splitter, it would mean that in addition, and this is what the math really shows, in addition to the attenuated coherent state, which mathematically is also attenuating the fluctuations, you have to add something which is the reflection coefficient times the vacuum state, which is this. And if you now correctly do the math, if you add the two together, you get a coherent state, which has an amplitude of t alpha, but has the correct Poissonian fluctuation, is again a minimum uncertainty state. So the disc of your attenuated state has exactly the same area as the unattenuated state. So I've shown you the physical part. I have shown you the graphical solution. The math is very simple, but I really want you to do the math yourself. This is a new homework problem we designed to illustrate the physics. But what it tells you is the following. If you take a neutral density filter out of the lab drawer and say, this is not a beam splitter, this is an attenuator, sorry, you cannot simply attenuate a quantum mechanical mode. This is not a unitary time evolution. What your attenuator does is, without you knowing it, it couples the electromagnetic wave to whatever, to the heat bath of the black paint, which is in your attenuator. I don't even want to describe it. But you're not circumventing the limitation of a beam splitter. Whenever you attenuate, whenever you have a laser beam and it undergoes losses, when you send a laser beam through the atmosphere and it undergoes some losses by, who knows, really scattering through the air or something like this, that means you get an attenuated coherent state, but you couple in the fluctuations of the vacuum, and this establishes shot noise now at a lower level of intensity. So your attenuator is not a single state device. Disappation, attenuation, really means you connect with other parts of Hilbert space. You cannot attenuate in a small part of Hilbert space. This is impossible. This is not a unitary time evolution. So what I've just told you has traumatic consequences for any form of non-classical or squeezed light. And that's the following. Most of you are experimentalists, and you know that when you run a laser at one or two watts, you send it through optics and shutters and optical fibers. How much do you get at the end of your experiment? Not even 50%. So whenever you create light and then you want to do something, you lose some of the light. And let's now assume that you have done what was really breakthrough and scientific headlines within your lifetime. You have generated squeezed light. And now you want to use the squeezed light, shine it on atoms, and do a precision measurement, which is better than the standard quantum limit, because you have squeezed the ellipse, and you want to now exploit the sharpness of the ellipse. What happens to your aspect ratio of the ellipse when your beam is attenuated? So let me just discuss it with you graphically. So let's assume we have a squeezed state, and we send it through an optical fiber. The result is you will never send a squeezed state through an optical fiber, but I want you to realize that. So let's assume we have our squeezed state symbolized by that. Let's use some red color for the state. And now in the time evolution, we have some Rayleigh scattering, some fiber absorption. But you know already, the absorption is in reality a beam splitter, which couples in the vacuum. So we have now some finite transmission coefficient. And that would mean, which is the bad news, that your ellipse gets shrunk, it shrinks, by this transmission factor t. But that's already bad. You lose some of your power. But the really bad thing is that with a multiplied, with the reflection coefficient, you couple in the vacuum. Oops, I want to change to red. And as a result, since the noise in the vacuum is equal in both quadrature components, you've worked so hard to squeeze it to make it asymmetric. But what you get now is, well, from your ellipse, you get something which is, well, much more egg-shaped now. So in other words, I mean, you can write it down with operators. But once you understand what is going on, you immediately realize that losses reduce the squeezing. And this is a challenge to all experiments using squeezed or non-classical light. And you see how the scaling works. If you have squeezed your ellipse by a factor of 100, even 1% of the vacuum will spoil your squeezing. So the more you have squeezed, the more non-classical the light is, the more valuable it is, the more sensitive it is to even very small losses. Questions? OK, good. Now we have 10 more minutes. And what I want to do now is I want to show you that the language we have used, the methods we have introduced, can now be used to something which is really cool, teleportation. So I want to show you how balanced homodyne detection, quadrature measurement, and displacement operator can be put together to generalize a scheme which is a scheme for teleporting quantum states. So this is an application of squeezing, homodyning, and all that, teleportation of light. Let me just illustrate what the problem is. I know we have only 10 minutes, and I've decided not to write down all the math for you. The math, I went actually to the Wikipedia page and corrected a few mistakes in the equations and added a few explanations. I think you can just sit down and read it by yourself. What I want to explain to you here is the physical concepts behind teleportation and the crucial steps to realize teleportation. So first, what is teleportation? Well, teleportation has a center and a receiver which in quantum information science are called Alice and Bob. And teleportation means Alice has a quantum state psi. And she wants to send this quantum state psi to Bob. In other words, you have maybe some squeezed light. You have something, a quantum state psi, which is interesting. And the question is, how can Bob create an identical copy of this quantum state psi that he can walk away with the same quantum state? Before you appreciate teleportation, you have to realize what the problem is. And the problem is really in fundamental properties of quantum systems, fundamental limitations what you can do with a quantum system. First, I have to tell you what is allowed and what not. In this game, teleportation means, of course, you will not take your atom and your photons in the state psi. Just propagate them to Bob and Bob. Bob has them. That's trivial. I can send any quantum state to you by transmitting an atom or a laser beam to you. And you have the same quantum state I had earlier. This is not teleportation. This is trivial propagation. What is meant is that we don't have a quantum channel. I'm not allowed to send my quantum state to you. But in teleportation, I can do a measurement on the quantum state, call you up through a classical communication channel, and tell you the result of my measurement is such and such. And then you would say, well, if I have a spin system and I measure spin up, I call you and say, my measurement was spin up. And you create a spin up system. Isn't that teleportation? Well, the answer is no, because maybe the state I had was a superposition of spin up and down. And what I measured is only spin up. And by telling you it's spin up, you would never create a superposition state. You would just create a spin up state. So the fact is that if I do a measurement on my quantum state and report my results to you, you have insufficient information, because a projective measurement on a quantum state inevitably leads to loss of information. A measurement on a single quantum state does not create enough information to recreate the quantum state. Of course, there is an obvious solution. When we want to obtain information about a quantum state, we often, in quantum mechanics, we have to do many measurements. And we can take a spin state. We can measure what is its x, what its y, what its c component of the spin. We can completely characterize the spin state. And then we know everything about it. But the problem is we have only one quantum state. And we can't do repeated measurements. Then you would say, well, the next thing is, why don't you just take your quantum state and xerox it, make many, many copies? And then you have an ensemble. And you can take as many measurements you want. You can do an x measurement, p measurement. When you can reconstruct the complete wave function, you can measure with an x basis. You can measure with a momentum basis. You can collect all information. But the problem is, otherwise the whole teleportation would not be an issue at all, there is no cloning theorem. You cannot duplicate a quantum state. So if you have an atom in a certain quantum state and another atom, it is quantum mechanically forbidden. There is no unitary transformation, no way of creating a situation that you have one unknown quantum state plus another atom or another light beam. And after some interaction, you have two times the same quantum state. You cannot clone. So therefore, based on all that, we cannot clone the quantum state, we are only left with the one copy of the quantum state which Alice has. Alice is not allowed to send it to Bob. Maybe Bob is on the other side of the ocean or on another planet. All what Alice can do is she can do one measurement and tell Bob, this is my measurement. So this is the problem of teleportation. How is that possible? Well, the way how I put it, it seems impossible. But there's a way out of it. So maybe let me just write down what I said. So the goal is now Alice performs measurement, reports result to Bob. And now Bob will recreate the quantum state. What I've just said is to perform a single measurement is not enough information. So this is not enough. So we need one more resource. And the resource which is now used to realize quantum teleportation is that you take some entangled system, or, and this is why I talk about it today, or squeezed light, which is also does the same thing. It's a form of entanglement. I told you that when we generate squeezed light with a parametric down conversion, the parametric down conversion takes a green photon and creates two identical infrared photons. Until now, we have discussed that those identical infrared photons go into the same mode which is squeezed. But now a slight extension of this concept would mean in parametric down conversion, you squeeze something. But one photon goes to Alice, one photon goes to Bob. So now Alice and Bob have an additional resource. They sort of own each of them half of the two twin brothers which are the photons created in the parametric down conversion process. And that will work. So the idea is the following. To create those twin beams of photons is simply done with an optical parametric oscillator. There is one extension which is described on the wick here. I won't have time to explain it to you. But it's a two-mode OPO. It puts the two identical photons not in the same beam as we did before. We had the a dagger squared a square operator. They go into two different modes. And now there's, you know, the magic of beams paid us by now. But now two beams come out. One goes to Alice, one goes to Bob. And now if Alice would take her input state, this unknown state which is handed to her by Victor, somebody else who participates in the game. So now we know already if Alice would perform a measurement, the quantum state would be destroyed. The result of the measurement, let's talk about spin 1 1 half would only be one spin projection. It's not enough to reproduce a state. But what she is doing is she uses again the magic of the beam splitter. So one of those twin brother photons is now mixed with the unknown quantum state at the beam splitter, and the output of the beam splitter is now entering the balanced homodyne detection, which we just discussed. So the output of this beam splitter and the two outputs, both of them is now becoming part of a balanced homodyne measurement. And you see the ingredients. So this and this will be measured. The other input for the balanced homodyne is a strong local oscillator. And the phase of the local oscillator is chosen in one case that you measure the x or a 1 quadrature component or the p or the a 2 component. So now what Alice has done is by using this balanced homodyne, she has, with this local oscillator, has actually now performed two measurements. The quantum state is destroyed. But here she gets an x value, and here she gets a p value. So how the magic works out, it's really just a few lines of mathematics now, is these were sort of twin brothers. But they were, it wasn't clear in which quantum state the twin brothers are. So if you write it down, and I can show you the formula in a moment, these are twin brothers. But those twin brothers, this is brother 1, this is brother 2, are in a linear super, are in sort of a continuum of states. And when Alice would measure that this twin brother is in state big x, that would be a projective measurement. And Bob's twin brother would now be certainty also be in the state x. So therefore, what happens is the measurement of x and p is now producing, is now through the measurement process, putting the other photon or the other beam, the other twin brother, into a specific quantum state. And if you look through a few lines of math, the magic is that the quantum state, which is now here with Bob, turns out to be a displaced copy, a displaced copy of the original state. And the displacement depends on x and p. So if Alice now tells Bob, hey, I measured x and p. And Bob is now programming his displacement operator. Remember, displacement operator is nothing else than a beam splitter, an unbalanced beam splitter, with a huge local oscillator as an input. So if Bob is now setting up his displacement operator based on and makes a displacement which depends on x and p, he can take this other twin brother, shift it back, and he will exactly regenerate the quantum state, which Alice had. So this is now how a quantum state can be transmitted without having any quantum channel for transmission. You're not propagating the quantum state. You use classical communication, but the resource you use is some EPR pairs or squeeze light. It's a few lines of equations, but I don't have time to go through it. They are really annotated in a way that I think it will be an enjoyable reading for you. Any questions? Time is over. OK, reminder for those who came late. We have today. This week we have three classes, Monday, Wednesday, Friday. Have a good afternoon.
 The following content is provided under a Creative  Commons license.  Your support will help MIT OpenCourseWare continue to  offer high quality educational resources for free.  To make a donation or view additional materials from  hundreds of MIT courses, visit MIT OpenCourseWare at  ocw.mit.edu.  Well, we are in the middle of something really interesting.  We are talking about squeezing.  We are talking about non-classical light.  And today I sort of want to wrap it up.  I think it will be really an exciting class.  But before I continue with the material, I want to address  a question which actually came up in discussions with  several of the students.  And this is, I realize that some people said, OK,  everything makes sense.  But what are we plotting?  What is really squeezed?  Are we squeezing in the spatial domain?  Are we squeezing in the temporal domain?  So the plots look wonderful with these ellipses and the  circles.  But what is it really we are doing here?  So let me address that.  First of all, we are talking about,  as we are talking about a single harmonic oscillator.  We showed that the Maxwell's equations can be reduced to a  bunch of harmonic oscillator equation, one for each mode.  And now we are talking today and in the previous few classes  about one single mode, about one harmonic oscillator.  And the harmonic oscillator has canonically variables of  momentum and position.  But this is just to make a connection for you with  something you have learned.  What we are talking about is a single harmonic oscillator,  which is one single mode of the electromagnetic field.  So maybe let me draw a cartoon for that.  So let's assume we have a cavity.  We have an electromagnetic wave.  There is propagation.  There is e to the ikz.  There is transverse confinement.  Maybe there is a Gaussian, e to the minus x squared plus y  squared over some beam waves parameter.  All that is simply the spatial mode.  And we just take that for given, because we are not  solving the spatial differential equation.  All we are doing it, we are looking at this one mode.  And the two degrees of freedom is that this mode can have a  certain number of photons.  It's the amplitude.  And the second one is, you can say, the temporal phase.  It can be a cosine omega t.  It can be a sine omega t.  It can be a superposition.  But whatever we are talking is in this mode.  There is nothing happening in the spatial domain.  We're just asking what is the oscillation in this mode.  The whole mode does what it should.  It has a pre-factor, which is the amplitude.  And it has a temporal factor, which we factor out.  And this is what we are talking about.  Let me be a little bit more specific and say that when  we are plotting things, we are plotting the q representation,  the phase-based representation of the statistical operator,  which is simply describing this single mode of the harmonic  oscillator.  And by performing the diagonal matrix elements,  we obtain the q distribution.  In that case, we have the vacuum.  We have a displaced vacuum, which is in coherent state.  And our axes are, from the very definition,  we are in the complex plane with a real part  and imaginary part of alpha.  However, we can also define a Wigner distribution, which  is another phase-based distribution.  It's almost the same as the q distribution.  It's just this little bit smeared out  by h bar because of some commutators,  but nothing you have to worry about.  In that case, the projection of the w function  on the vertical axis, on the y axis,  is the momentum wave function squared.  On the x axis, on the x axis, it is the spatial wave function.  It's squared of the harmonic oscillator.  So therefore, we may sometimes think  when we have a distribution here, we project it  and we see what is the momentum distribution  or what is the spatial distribution  of the mechanical harmonic oscillator, which  is analogous, which is equivalent to the one  mode of the electromagnetic field we are using.  I know it may help you to some extent  to think about the p and q, but it may also be misleading  because it gives you the impression something  is moving with the momentum p in real space.  Let me therefore emphasize what are the normalized forms  of p and q.  If I do the symmetric and anti-symmetric combination  of the annihilation and creation operator,  i over square root 2, I call those a1 and a2, and they are  nothing else than q and p normalized  by the characteristic momentum or spatial coordinate  of the harmonic oscillator.  So what is important here is that a1 and a2,  forget about pq now, they are equivalent.  But for the electromagnetic field, a1 and a2  have a very direct interpretation.  They are called the two quadrature operators.  And what I mean by that becomes clear when  I use the Heisenberg representation  for the electric field.  And I'm here using the formula which  is given in the book of Weisblath in page 175.  Some copies of the book have been,  some pages copied from this book have been posted on the website.  So we have our normalization factor,  which is the electric field related  to the electric field of a single photon.  We have the polarization factor.  But now we have an expression which  involves the quadrature operators a1 and a2.  Just to be specific, we are not in a cavity here.  Therefore, we have propagating waves cosine kR sine kR.  But you can also immediately use similar expression  for the case of a cavity.  Let us specify at r equals 0.  And then we realize what the two quadrature operators are.  A2 is the operator which creates and annihilates  an electromagnetic field, so to speak photons, which  have an electric field which oscillates as cosine omega t.  And a1 is the quadrature operator  for the sine omega t component.  So therefore, if you simply analyze the electric field,  what is cosine omega 2 is related to the a2 quadrature  operator, the sine omega 2 oscillation  is related to the a1 quadrature operator.  So life would be easier, but more boring  if you could create a pure cosine or pure sine  oscillation of the electromagnetic field.  But you can't, because a1 and a2 do not commute.  And there is an uncertainty relation  that delta a1 times delta a2 is larger or equal than 1 half.  And we have the equal sign for coherent states alpha.  So therefore, if we look at the electric field,  everything moves around periodically in r,  because it's a traveling wave in t,  because it's an oscillating wave.  So let's not confuse things.  Let's simply peak r equals 0.  We've already done that.  But now let's peak t equals 0.  At t equals 0, the sine omega t is 0.  And therefore, the distribution for a2,  the expectation value, and the variance for the quadrature  operator, or a2, can be simply read off  by looking at the electric field.  So in other words, at t equals 0,  the electric field, which is obtained  by projecting our quasi probabilities on the y-axis,  gives the expectation value for a1 and the variance delta a1  squared.  Somebody says, 2 at t equals 0, yes.  Yes.  And now, if you want to see what the other quadrature  component is, well, we just wait a quarter period  until the sine, which was 0, is maximum,  and the cosine omega t is 0.  So therefore, at pi over 2 omega,  using the projection on the y-axis,  gives us a1 and delta a1 squared.  Or alternatively, we don't need to wait.  We can do t equals 0, and we can project onto the x-axis.  So therefore, let me just draw a few things into this diagram.  If you had a classical motion, which would simply be cosine  omega t, then that would mean we would just have a,  if you had a motion which were only cosine omega t,  yes, it would be a point on the y-axis.  However, classically, we can never  have something which is just cosine omega t.  The point has to be blurred into a circle.  This is a coherent state.  So this coherent state has now, let me just call that one  for the sake of the argument, the point  would be the classical oscillator.  It has just cosine omega t.  Everything is dataministic.  No uncertainty, no nothing.  Of course, it means that in the time evolution,  it goes in a circle.  But this is what everything does in a harmonic oscillator  when time evolves due to the e to the i omega t factor.  So let's not get confused with that.  Let's just look at t equals 0.  And for traveling wave r equals 0,  the classical oscillator is one point.  But now we have a spread here.  This says that quantum mechanically,  the amplitude of the cosine omega t term  is not arbitrarily fast.  It's not arbitrarily sharp.  They are fluctuations.  And in addition, we have widths in this direction, which  we project onto the x-axis.  And this tells us what the distribution in our ensemble,  in our quantum mechanical ensemble,  is for the amplitudes of the sine omega t motion.  So the best we can do is quantum mechanically,  if you want to have something which is really just cosine  omega t, we have to squeeze it that the cosine omega t  amplitude is now extremely sharp.  But the sine omega t amplitude in the ensemble  is completely smeared out.  So this is what we are talking about.  Now, what I think has confused some of you  is what I thought was a wonderful example,  the classical squeezing experiment.  I mean, these are visuals which will be in your head forever  when you saw Professor Britschard with a circular pendulum  he's just pulling, and then the circle squeezes into an ellipse.  And it seems that something is here squeezed in real space.  But this is actually wrong.  How you should have looked at the experiment,  and I made a comment about it, but maybe not emphatically  enough, you should have really thought about a single pendulum.  And this single pendulum, if it has an arbitrary phase,  is in a superposition of sine omega t and cosine omega t.  And if you pull on the string, if you shorten and lengthen  the pendulum at sine 2 omega t, you  will amplify the pre-factor in front of sine omega t,  and you will exponentially de-amplify the factor  in front of cosine omega t.  So therefore, what will happen if this pendulum oscillates,  and let me say with a phase, well, sine omega t plus delta,  if delta is 90 degrees, cosine, if delta is 0, it's sine.  And let's say this pendulum oscillates at 45 degrees,  sine omega t plus 45 degrees.  If you now parametrically drive it,  the squeezing action would now mean that you, let's just  make a sine convention, you de-amplify the cosine,  you amplify the sine, and after a while,  instead of oscillating with sine omega t plus 45 degree,  it will oscillate with an amplified amplitude  at sine omega t.  This is what you have done.  And this is the mechanical analogy.  There is, of course, no squeezing in any way,  because in a classical pendulum, we  start with one definite value, if you prepare the system well,  and then we just change the motion.  We amplify, we pick out a phase, and that's what we are doing.  Now, there are two ways how classical squeezing can come in.  One is, if the motion of the pendulum is,  maybe there is an uncertainty.  Maybe Professor Britscher did experiments  with an ion trap, and actually 20 years ago,  he published a Fuss-Refletter on classical squeezing.  You think, how can you publish a Fuss-Refletter  on classical squeezing?  Well, he had the world's most accurate ion trap,  measuring atomic masses with 10 and 11 digits precision.  And what was actually one limiting factor  was, for Kelvin, the thermal distribution  of harmonic oscillator modes.  And so what he had is he didn't have just one clean amplitude.  The sine omega t amplitudes had a spread  because of the thermal distribution he started from.  And so what he then did is, by simply classical squeezing,  by doing it classically with the ion trap, exactly what  he did with a pendulum, drive it with sine 2 omega t,  he could now take this classical distribution of,  this is a classical distribution.  In one axis, it's a distribution of amplitudes  of the cosine motion.  And here, it's the classical distribution  of the amplitudes of the sine motion.  And he was squeezing it into this direction.  So he had a narrower definition of the coefficient  for the cosine omega t motion.  And as I will tell you today is, you  can now do a homodyne measurement, which  is reducing the noise.  So essentially, he prepared, quote,  unquote, effectively a colder ensemble  by squeezing the uncertainty in the cosine pre-factor  at the expense of increasing the pre-factor,  the uncertainty, the variance in the pre-factor  of the sine omega t motion.  Finally, you all saw something visually.  You saw how a circular motion became a linear motion.  So what was going on here?  Well, I mentioned to you that the circular pendulum actually  has two modes.  These are two modes of the harmonic oscillator.  And I'm not talking about two modes of the harmonic oscillator.  Everything we're discussing here is about one mode  of the harmonic oscillator.  The circular motion of the pendulum  was sort of just a nice visualization trick  that if the pendulum moves in a circle,  you have a degenerate harmonic oscillator.  One is excited with sine omega t.  The other one is excited with cosine omega t.  And instead of doing two experiments,  if you would start with sine omega t and you parametrically  drive it, you amplify it, you start with cosine omega t,  you could bring the pendulum to a stop.  But instead of doing two experiments,  Professor Bridger just did one and showed  that the sine omega t motion became larger  and the orthogonal cosine omega t motion shrunk.  And therefore, you saw that the circular motion,  which was a superposition of sine and cosine, became  pure sine motion.  But the fact that there was something  we could see in a spatial domain was simply  due to the fact that we had kind of two experiments at one,  two versions of the same harmonic oscillator, one  in x and one in y.  And then when we did the experiment,  we saw something visually in the spatial domain.  So that's why we saw squeezing in the spatial domain.  But you should really think about it.  What the whole action is, is it's  an interplay of de-amplifying prefactors of cosine,  amplifying prefactors of sine.  And if the prefactor has a distribution,  by de-amplifying it, you also shrink  the widths of the distribution.  And this is what we call squeezing.  Yes, Timo?  Two for question.  So the operators A1 and A2 here, right?  You use those instead of A and A dagger,  because you used cosine and sine rather than e to the i omega t  and e to the minus i.  Because they should contain in the dagger, right?  Let me go back to the definition.  They are actually exactly.  All right.  They correspond exactly to position and momentum  of the mechanical harmonic oscillator.  So if that makes sense.  The other thing is from the technical speaking,  would we call the last goal that field e,  cos k r minus omega t as maximum squeeze?  If it only has a cos component?  Well, not if I use it.  Depends how I define squeezing.  So you would now give a definition of squeezing,  which says that the variance in A1 is now  unequal to the variance in A2.  So the classical oscillator is a point.  It has zero variance in A1, zero variance in A2.  But as I said, you can actually apply all that  to a classical oscillator if you add technical noise  or thermal noise.  Then your system is prepared not with a sharp value,  but with a distribution, which is simply maybe a Boltzmann  distribution due to the preparation.  So we call it squeezing when the noise in the amplitude  of the sine motion is not equal to the noise  in the amplitude of the cosine motion.  Some people say it a little bit narrower.  They reply squeezing to the situation that we are at.  We are uncertainty limited, and then we squeeze.  But of course, you can say you can always  reduce the noise in your system by just  preparing the system, by cooling the system,  by selecting the system through measurements  until you reach the quantum limit.  So you can get a smaller delta A1, a smaller delta A2  without squeezing by just better preparation  or by selecting your ensemble.  So squeezing in a narrower sense only  makes sense when you hit the limit what  quantum mechanics allows you.  And now you want to distribute the variance unequally  between A1 and A2 because then you  can get something in delta A1 or delta A2, which  is better than 1 over square root 2.  And this is now really quantum mechanically squeezed.  But there are both definitions.  Classical squeezing exists.  It's just not as common as quantum mechanical squeezing.  Other questions?  Yes?  I'm a little confused when you said that in the classical  squeezing, you are attenuating one amplitude  and you are amplifying the other amplitude.  So in this picture, then, shouldn't we  have the ellipse come down from the circle on the y-axis?  OK.  You're just not just changing delta A,  but also changing A itself.  Let me just get a sketchbed here.  So this is A2.  This is A1.  A2 is for the cosine omega t and A1 for the sine omega t.  So just to be specific.  So what do you want to prepare?  You want to prepare a harmonic oscillator,  which is just sine omega t.  This is a point here.  If you are now parametrically, so this  has a value at t equals 0, our A1 is S0.  So if you are now squeezing your classical harmonic oscillator,  you would have a situation where A1 of t is S0 times e  to the plus or minus rt, depending  whether you do the parametric drive at sine 2 omega t  or cosine 2 omega t.  So therefore, what would happen is this point  will be amplified.  It would just move out on the x-axis.  So this would be for the plus sign.  Or for the other case, you would damp the motion to 0.  And this is the minus sign here.  OK.  And you are also changing the variance.  Well, a point does not have variance.  But for an example.  So if you want to build up variance,  you need, let's say, three points.  One is the average value.  One is the left outlier.  One is the right outlier.  And what happens is now, as you amplify the motion,  you would also amplify, magnify the distance  between the points.  And if you deamplify it with the minus sign,  the distance between the points  would shrink because all the three points converge to 0.  So pretty much what I just told you for the three point  ensemble, you can now use it and construct  any initial condition you want and see what happens due to squeezing.  Other questions?  Good.  So the question now is how to measure squeezing,  how to take advantage of squeezing.  So the situation we are facing is the following.  Let's assume we have done a nice squeezing job.  And that means that we have a sharp value.  We have created a narrow distribution  of the cosine omega 2 coefficients.  So the cosine omega 2 motion is rather sharp.  But we also know that the electric field itself  is sharp at this moment.  But since this ellipse rotates, the electric field  will have enormous uncertainty a quarter period later.  So if we want to take advantage that we have squeezed  the electromagnetic field, there are now  a couple of ideas which we can use.  One is we could just measure the electric field  stoposcopically.  We would just make a setup where we look at the system  in a quantum measurement process.  Only we only measure the electric field when the ellipse is  like this or is like this.  And therefore, we have a sharp value of the electric field.  But instead of doing a stoposcopic measurement,  we can do something else.  Remember, we have a distribution of cosine omega t  and sine omega t.  And we have a distribution of coefficients c and s.  And we know we are interested in the cosine omega t.  So how to pick that out has actually  been solved in early radios.  You do a homodyne detection.  In other words, you take a reference oscillator,  which is strong.  B of t is B naught times cosine omega naught plus delta.  And if you mix, maybe before I use the word mixing,  I should simply multiply.  If you multiply the two signals,  your signal you are interested in,  or at least you're interested in one component,  you multiply it with your huge local oscillator.  And then you integrate over time.  Then, of course, when you pick the phase delta to be 0,  cosine omega t times cosine omega p t  gives cosine square omega t.  It averages to 1 half.  So you have a time average.  Whereas cosine omega t times sine omega t averages to 0.  So for delta equals 0, you project out the cosine component.  And for delta equals 90 degree, you  pick out the sine component.  So therefore, you can have a measurement.  It's a phase sensitive measurement  by multiplying your signal with a local oscillator,  where you are only sensitive to the component you have squeezed.  And therefore, your measurement uncertainty  has now been reduced by the squeezing factor.  This omega should be the frequency of the laser.  Omega has, yes, actually, homodyne means  we use the same frequency.  Heterodyne would mean we use two different frequencies.  But I'm not talking about that.  So we have to use exactly the same frequency here.  So this reference also should also  need the laser to make sure that.  Yes.  So to address your question, what usually happens  is you start with one laser in those experiment.  You frequency double the laser.  If you want to do some squeezing,  you remember that we need a parametric oscillator, where  one energetic photon gives us two photons.  So what you do is you start with a laser,  often it's a YET laser at 1064 nanometer.  You frequency double it to a green laser.  The green laser pumps your parametric oscillator.  And then you get through down conversion.  You get squeezed light at 1064.  But this is because you first double the laser,  and then you break the photon into two pieces,  has exactly the same frequency as your laser you started with.  And this laser is then the local oscillator,  or the reference clock, for your whole experiment.  So everything in your experiment, the doubled laser,  the parametrically down converted beams,  everything is related to the single one laser  you started with, and everything is phase coherent.  So that's how usually the experiment is done.  Before I tell you what we're doing quantum mechanically,  let me just also get another question out  of the system, which I've been asked several times.  People ask me, well, the problem is  that kind of the ellipse rotates like this.  Isn't there a way, now I need my hand,  that we can have an ellipse rotating like this?  I mean, that would be great, but this  is sort of unnatural because the harmonic oscillator does that.  So if you wanted to do that, you need an operator, which  is really at every cycle of the electromagnetic field,  is when the light wants to sort of do this.  No, you always push it back, and this is impractical.  You need really an oscillator which  would completely change the quadrature components  of your harmonic oscillator in every single cycle  of the electromagnetic field.  But what homodyne detection is, instead of now forcing  the light to stay aligned to sort of do this, which  is very unnatural, we allow the light to freely evolve.  But we have now an observer, our local oscillator,  which is rotating synchronously with the ellipse.  So we have a local oscillator, which is cosine omega t.  It does so to speak exactly what the ellipse is doing.  So in that sense, the local oscillator  allows us now to observe the ellipse always  from its narrow side because the local oscillator is  rotating.  But the mathematics is pretty much the Fourier transform.  The mathematics is the Fourier transform.  The physics is the physics of a log-in detector.  OK, now the only question remains is, how do we mix?  How do we get a product of our signal  and the strong local oscillator?  In an old radio, it's done by an element, maybe a diode,  which has a nonlinear circuit.  If you drive a nonlinear element with two input sources,  you get something which involves the product of the two.  OK, so the principle of homodyne detection  is now that we want to mix light at a beam splitter.  So the device which does all that for us is,  after using so many words, I would say, is disgustingly  simple, it's really a half-silver mirror.  We'll talk about beam splitters a lot here  because beam splitters perform wonderful unitary transformations  and we will exploit them for many purposes.  For the purpose of this lecture, I simply  assume that we have a 50-50 beam splitter.  So what I'm telling you now about the beam splitter  will be generalized either later today  or in the lecture on Wednesday.  So a beam splitter has two input ports and two output ports.  We have light impinging on the beam splitter.  We call those modes A and B. And after the beam splitter,  we have two output modes.  And let me call them A0 and B0.  And what we measure is the output of the beam splitter.  So we have two photo detectors and we measure the output.  OK, the output modes A0 and B0 are simply  obtained by taking the input modes and propagating them.  We have two modes, AB.  You can say what goes vertical is A. We call it A0.  Here it becomes B0.  And what we have to do is we have to transform  the operator A0.  Let me just make a comment.  Sometimes in the beam splitter, you  want to think a quantum state comes and is transformed.  But instead of transforming the state of a photon,  I can also transform the operator which creates the photon.  One is the Heisenberg picture.  One is the Schrödinger picture.  So right now I use the Heisenberg picture.  I have two operators, AB.  They transform into, before the beam splitter is there,  I have two operators A0 and B0.  Afterwards, I have two other operators.  I have operators AB.  Afterwards, I have operators A0 and B0.  And if you do the time evolution,  it's a unitary transformation.  And for operators, we have to multiply the operators  from the left and right hand side  with the unitary operator u and u dagger.  And we really talk about the beam splitter  in its full beauty in a short while.  It will be a special case of what we discuss later,  but I think it's pretty obvious that the 50-50 beam splitter  is simply creating two modes.  One are the symmetric combination  and one are the anti-symmetric combination.  OK, so in our homodyne detector, we  are now measuring the number of photons in the mode A0.  We measure with the upper detector  the number of photons in the mode B0.  But what we are then doing is we run it  through a different circuit.  We want to cancel certain noise, as you will see in a moment.  This is why we obtain the different signal,  which we call I minus.  So let's now calculate what I minus is.  Well, the number of photons in the mode A0  is A0 dagger A0.  That's the operator for the intensity  of the number of photons.  We subtract B0 dagger B0.  And as you can immediately see, is  that because of the beam splitter,  this is now involving a product of A and B, AB dagger  plus A dagger B. So in other words, when you ask yourself,  how can we remember, I said we pick out  the cosine omega t component by multiplying our signal  with a local oscillator?  And you would say, how do you multiply two modes  of the electromagnetic field?  Well, just send it to a beam splitter.  In a beam splitter, you create the sum of them.  But then your photo detector is a square law detector.  You measure the electric field squared,  or you have to measure the operator A dagger A.  And now you get a cross term between AB dagger and A dagger  B. So this is how we multiply two operators,  how we get a signal which is proportional to A or A dagger  times B dagger.  By the way, if you wouldn't take the difference,  we get terms of A, A dagger, B, B dagger, just the mode A  or the mode B by themselves.  And we want to get rid of them.  And by taking the difference between the two photo detectors,  those parts of the signal are coming mode  and are subtracted out.  So therefore, this is called balanced homodyne.  Because we have a balanced beam splitter,  we measure two signals, and then we  take the difference of the two signals.  OK.  There's one more thing we have to add.  And then we find we understand the balanced homodyne  detection.  We want to exploit that our mode B,  remember, we want to measure mode A. A is squeezed.  A has quantum properties.  A has only single photons.  And B is just a trick to project out the cosine omega t.  And we do that by choosing for B a strong coherent state,  described by a coherent state parameter beta.  Of course, whenever we have a strong coherent state,  that means that we can replace B by beta and B dagger  by beta star.  This is sort of the classical field  limit of a strong coherent state.  And now we can ask, what is our difference signal i minus?  Well, the coherent state depends on the phase angle theta.  So it will be phase sensitive.  It will depend on the angle theta.  But now there is one more thing.  And that is, if you use a stronger coherent state,  of course, both of our outputs go up in proportion to beta.  So therefore, we want to do a normalization  by dividing by beta.  So this is now our normalized output.  So this is now simply, I'm looking at this.  So this is our operator for the signal i minus.  And it is A. B has been replaced by,  B dagger has been replaced by B star.  That gives us A times E to the, because of the complex  conjugation, minus i theta, plus A dagger times  E to the i theta divided by 2.  So therefore, if we choose the phase to be 0,  we measure the symmetric combination A plus A  dagger over 2. And this is, just comparing nodes, simply  the A1 quadrature operator divided by square root 2.  And if we put a phase shifter, just a dispersive element  into the strong local oscillator and shift the phase  by pi over 2 by delaying the light pulse by a quarter  wavelength, what we are now projecting out  is the other quadrature component.  So therefore, using a beam splitter in the local oscillator,  we can now measure the expectation value for A1 and for A2.  I've posted a few papers on the website  which show some of the pioneering experiments  where people did exactly that.  So then when they observed, let's say,  they observed the quadrature component which was squeezed,  they usually did it for squeezed vacuum.  So A1 and A2 had zero expectation value.  But the interesting part is how much noise was there.  If you don't squeeze, you find that your normalized noise  simply corresponds to the classical photon shot noise.  But if you are squeezed, you find  that one quadrature component has smaller noise than shot  noise.  So you detect n photons in your different signal,  and your noise is less than square root n,  whereas in the other quadrature component is larger.  So you can look up on the website papers which  where people slowly varied the phase of the local oscillator.  And you see this is shot noise.  How the noise is below shot noise exceeds shot noise.  It's below shot noise and exceeds shot noise.  And this was the first evidence for the generation  of squeezed light.  Questions?  OK, so we have discussed the detection of squeezed light  using balanced homodyne detector.  And balanced means that the beam splitter was 50-50.  So now we are ready to discuss the unbalanced case.  So let's get our unbalanced beam splitter.  And just for the sake of the argument,  let's say it has a really good transmission of t.  t is a transmission coefficient.  t squared tells you what fraction of the power is  transmitted.  And let's just assume for the sake of the argument,  99% are transmitted.  So therefore, if we start with our signal a, which  may be an interesting squeezed quantum state,  a number state, you name it, a naught  is pretty much the same as a.  We haven't lost so much.  But now we use our local oscillator, which is very strong.  Only a very small fraction of it will be reflected.  But we can always compensate for the small reflection  by making b even stronger.  Let's see what we get.  So the mode a naught, the output mode,  is now the transmission coefficient times a.  a and a, the operators are sort of amplitudes.  So we have to use, if you have a 99% beam splitter,  we have to take the square root of 0.99.  This is the transmission coefficient.  And now we have the transmission coefficient  for the strong local oscillator, which  we will again approximate by its eigenvalue,  coherent state eigenvalue.  And the reflection coefficient is 1 minus t squared.  So if we make the approximation that t is approximately 1,  we can take it out of the parentheses.  And we obtain that.  If t is close to 1, we can neglect it.  And what we see now is what we have obtained  is actually the original quantum state a,  or I should say the operator, I'm in the Heisenberg picture.  a is the mode operator for the input  of the unbalanced homodyne detector.  And what we have simply done is we have displaced the operator.  We have displaced the mode operator a.  And the displacement operator has an argument of 1 minus t  squared beta.  And this is the Hermitian conjugate, 1 minus t squared beta.  So what I've shown you here is that a local oscillator  and a beam splitter is the realization or implementation  of the displacement operator.  So in the limit that t goes to unity,  we are not losing anything of our quantum state.  But by reflecting in the amplitude  of a strong coherent state, we simply take our quantum state  and we displace it in this two-dimensional plane.  So that's what this beam splitter does for us.  Any questions?  In the next few classes, we really  take advantage of those elements.  We know now that the displacement operator  is just a beam splitter.  We know when we have squeezed some light  by using balanced homodyne detection,  we can just do measurement of one quadrature component  of the other.  So what I hope for those of you who haven't heard about it,  what you take away from that is that these are extremely  simple elements.  But by combining them, we can really  realize very sophisticated schemes of quantum optics.  To some extent, when I heard about for the first time,  I think the mathematics look so fancy,  I couldn't believe that such simple elements can actually  realize what those operators describe.  So I gave you here the example for the displacement operator.  When I learned about the beam splitter  and its underlying physics, there  was one thing which really fascinated me.  And this is the most simple element you can think.  I mean, what is simpler than a beam splitter?  A beam splitter has two inputs, two outputs.  The simplest optical element is just an attenuator.  Put in a window in your laser beam,  and you lose 4% of your power per surface.  Or put in just a little bit of dirty optics,  and you lose a few percent.  So what I want to discuss with you now  is what is really an attenuator quantum mechanically?  Well, if classically, the attenuator would do the following.  An attenuator is a device which has a transmission, which  is a transmission coefficient square, which  is smaller than unity.  And in a classical system, if you have a coherent state,  you would simply assume that the coherent state gets  multiplied with a transmission coefficient.  In other words, that you have your original state,  described by this phase or alpha.  And then the action of the attenuator  would simply be to scale everything down  by a transmission factor of alpha.  So the picture you should have is the following.  You have a coherent state.  It gets attenuated by the transmission coefficient.  But if your fluctuations about the coherent state,  also the fluctuations get attenuated,  because everything gets attenuated by this attenuator.  So if you look at that, you should immediately say, no,  this is quantum mechanically forbidden.  Because the coherent state, with a minimum uncertainty state,  this shaded area cannot be smaller than 1 half.  But here it has become smaller.  So what I've shown to you here is a violation of quantum  mechanics.  It would actually mean, let me just give you the example.  It would mean that if you had a,  the coherent state is quantum limited.  And if you calculate what is the fluctuations in the photo  number, it's just shot noise.  So just to give you a sort of simple intuitive example,  if you have 10,000 photons plus minus 100,  this is shot noise, square root n.  If you could now attenuate it by a factor of 100,  and you would go from 10,000 plus minus 100 to 100 plus  minus 1, that's much better than the shot noise.  I mean, this is what I'm just telling you.  What a simple-minded attenuator would do,  and you would immediately say, that's too good to be true.  I cannot get sub-shot noise light.  So what is wrong?  Impossible, not allowed.  Well, we have just tried to formulate something  intuitively, and we have to be careful.  Well, we know already one way how we can attenuate an input  beam, and maybe we should go back to the situation  and analyze it.  We know that we can attenuate it with a beam splitter,  and this beam splitter has a transmission coefficient of t.  And then we get our transmitted coherent light.  There is something getting reflected.  But now we realize that this beam splitter is not  just a device which has one input, it has another input.  And you may say, well, I don't care about the other input.  I don't want to use it.  Well, if you don't want to use the input,  it has the vacuum state.  So therefore, if you would realize the attenuator  with a beam splitter, it would mean that in addition,  and this is what the math really shows,  in addition to the attenuated coherent state, which  mathematically is also attenuating the fluctuations,  you have to add something which is the reflection coefficient  times the vacuum state, which is this.  And if you now correctly do the math,  if you add the two together, you get a coherent state, which  has an amplitude of t alpha, but has the correct Poissonian  fluctuation, is again a minimum uncertainty state.  So the disc of your attenuated state  has exactly the same area as the unattenuated state.  So I've shown you the physical part.  I have shown you the graphical solution.  The math is very simple, but I really  want you to do the math yourself.  This is a new homework problem we designed  to illustrate the physics.  But what it tells you is the following.  If you take a neutral density filter out of the lab drawer  and say, this is not a beam splitter,  this is an attenuator, sorry, you cannot simply attenuate  a quantum mechanical mode.  This is not a unitary time evolution.  What your attenuator does is, without you knowing it,  it couples the electromagnetic wave  to whatever, to the heat bath of the black paint, which  is in your attenuator.  I don't even want to describe it.  But you're not circumventing the limitation of a beam splitter.  Whenever you attenuate, whenever you have a laser beam  and it undergoes losses, when you send a laser beam  through the atmosphere and it undergoes some losses by,  who knows, really scattering through the air  or something like this, that means  you get an attenuated coherent state,  but you couple in the fluctuations of the vacuum,  and this establishes shot noise now  at a lower level of intensity.  So your attenuator is not a single state device.  Disappation, attenuation, really means  you connect with other parts of Hilbert space.  You cannot attenuate in a small part of Hilbert space.  This is impossible.  This is not a unitary time evolution.  So what I've just told you has traumatic consequences  for any form of non-classical or squeezed light.  And that's the following.  Most of you are experimentalists, and you  know that when you run a laser at one or two watts,  you send it through optics and shutters and optical fibers.  How much do you get at the end of your experiment?  Not even 50%.  So whenever you create light and then you  want to do something, you lose some of the light.  And let's now assume that you have  done what was really breakthrough and scientific  headlines within your lifetime.  You have generated squeezed light.  And now you want to use the squeezed light, shine it on atoms,  and do a precision measurement, which  is better than the standard quantum limit, because you  have squeezed the ellipse, and you want to now  exploit the sharpness of the ellipse.  What happens to your aspect ratio of the ellipse  when your beam is attenuated?  So let me just discuss it with you graphically.  So let's assume we have a squeezed state,  and we send it through an optical fiber.  The result is you will never send a squeezed state  through an optical fiber, but I want you to realize that.  So let's assume we have our squeezed state  symbolized by that.  Let's use some red color for the state.  And now in the time evolution, we  have some Rayleigh scattering, some fiber absorption.  But you know already, the absorption  is in reality a beam splitter, which couples in the vacuum.  So we have now some finite transmission coefficient.  And that would mean, which is the bad news,  that your ellipse gets shrunk, it shrinks, by this transmission  factor t.  But that's already bad.  You lose some of your power.  But the really bad thing is that with a multiplied,  with the reflection coefficient, you  couple in the vacuum.  Oops, I want to change to red.  And as a result, since the noise in the vacuum  is equal in both quadrature components,  you've worked so hard to squeeze it to make it asymmetric.  But what you get now is, well, from your ellipse,  you get something which is, well, much more egg-shaped now.  So in other words, I mean, you can write it down with operators.  But once you understand what is going on,  you immediately realize that losses reduce the squeezing.  And this is a challenge to all experiments using  squeezed or non-classical light.  And you see how the scaling works.  If you have squeezed your ellipse by a factor of 100,  even 1% of the vacuum will spoil your squeezing.  So the more you have squeezed, the more non-classical  the light is, the more valuable it is,  the more sensitive it is to even very small losses.  Questions?  OK, good.  Now we have 10 more minutes.  And what I want to do now is I want  to show you that the language we have used,  the methods we have introduced, can now  be used to something which is really cool, teleportation.  So I want to show you how balanced homodyne detection,  quadrature measurement, and displacement operator  can be put together to generalize a scheme which  is a scheme for teleporting quantum states.  So this is an application of squeezing, homodyning,  and all that, teleportation of light.  Let me just illustrate what the problem is.  I know we have only 10 minutes, and I've decided not  to write down all the math for you.  The math, I went actually to the Wikipedia page  and corrected a few mistakes in the equations  and added a few explanations.  I think you can just sit down and read it by yourself.  What I want to explain to you here  is the physical concepts behind teleportation  and the crucial steps to realize teleportation.  So first, what is teleportation?  Well, teleportation has a center and a receiver  which in quantum information science  are called Alice and Bob.  And teleportation means Alice has a quantum state psi.  And she wants to send this quantum state psi to Bob.  In other words, you have maybe some squeezed light.  You have something, a quantum state psi, which is interesting.  And the question is, how can Bob create  an identical copy of this quantum state psi  that he can walk away with the same quantum state?  Before you appreciate teleportation,  you have to realize what the problem is.  And the problem is really in fundamental properties  of quantum systems, fundamental limitations  what you can do with a quantum system.  First, I have to tell you what is allowed and what not.  In this game, teleportation means, of course,  you will not take your atom and your photons in the state psi.  Just propagate them to Bob and Bob.  Bob has them.  That's trivial.  I can send any quantum state to you  by transmitting an atom or a laser beam to you.  And you have the same quantum state I had earlier.  This is not teleportation.  This is trivial propagation.  What is meant is that we don't have a quantum channel.  I'm not allowed to send my quantum state to you.  But in teleportation, I can do a measurement  on the quantum state, call you up  through a classical communication channel,  and tell you the result of my measurement is such and such.  And then you would say, well, if I have a spin system  and I measure spin up, I call you and say,  my measurement was spin up.  And you create a spin up system.  Isn't that teleportation?  Well, the answer is no, because maybe the state I had  was a superposition of spin up and down.  And what I measured is only spin up.  And by telling you it's spin up,  you would never create a superposition state.  You would just create a spin up state.  So the fact is that if I do a measurement  on my quantum state and report my results to you,  you have insufficient information,  because a projective measurement on a quantum state  inevitably leads to loss of information.  A measurement on a single quantum state  does not create enough information  to recreate the quantum state.  Of course, there is an obvious solution.  When we want to obtain information about a quantum  state, we often, in quantum mechanics,  we have to do many measurements.  And we can take a spin state.  We can measure what is its x, what its y,  what its c component of the spin.  We can completely characterize the spin state.  And then we know everything about it.  But the problem is we have only one quantum state.  And we can't do repeated measurements.  Then you would say, well, the next thing is,  why don't you just take your quantum state and xerox it,  make many, many copies?  And then you have an ensemble.  And you can take as many measurements you want.  You can do an x measurement, p measurement.  When you can reconstruct the complete wave function,  you can measure with an x basis.  You can measure with a momentum basis.  You can collect all information.  But the problem is, otherwise the whole teleportation would  not be an issue at all, there is no cloning theorem.  You cannot duplicate a quantum state.  So if you have an atom in a certain quantum state  and another atom, it is quantum mechanically forbidden.  There is no unitary transformation,  no way of creating a situation that you  have one unknown quantum state plus another atom  or another light beam.  And after some interaction, you have two times  the same quantum state.  You cannot clone.  So therefore, based on all that, we cannot clone the quantum  state, we are only left with the one copy of the quantum  state which Alice has.  Alice is not allowed to send it to Bob.  Maybe Bob is on the other side of the ocean  or on another planet.  All what Alice can do is she can do one measurement  and tell Bob, this is my measurement.  So this is the problem of teleportation.  How is that possible?  Well, the way how I put it, it seems impossible.  But there's a way out of it.  So maybe let me just write down what I said.  So the goal is now Alice performs measurement,  reports result to Bob.  And now Bob will recreate the quantum state.  What I've just said is to perform  a single measurement is not enough information.  So this is not enough.  So we need one more resource.  And the resource which is now used to realize quantum teleportation  is that you take some entangled system,  or, and this is why I talk about it today,  or squeezed light, which is also does the same thing.  It's a form of entanglement.  I told you that when we generate squeezed light  with a parametric down conversion,  the parametric down conversion takes a green photon  and creates two identical infrared photons.  Until now, we have discussed that those identical infrared  photons go into the same mode which is squeezed.  But now a slight extension of this concept  would mean in parametric down conversion,  you squeeze something.  But one photon goes to Alice, one photon goes to Bob.  So now Alice and Bob have an additional resource.  They sort of own each of them half of the two twin brothers  which are the photons created in the parametric down  conversion process.  And that will work.  So the idea is the following.  To create those twin beams of photons  is simply done with an optical parametric oscillator.  There is one extension which is described on the wick here.  I won't have time to explain it to you.  But it's a two-mode OPO.  It puts the two identical photons not in the same beam  as we did before.  We had the a dagger squared a square operator.  They go into two different modes.  And now there's, you know, the magic of beams  paid us by now.  But now two beams come out.  One goes to Alice, one goes to Bob.  And now if Alice would take her input state,  this unknown state which is handed to her by Victor,  somebody else who participates in the game.  So now we know already if Alice would perform a measurement,  the quantum state would be destroyed.  The result of the measurement, let's talk about spin 1  1 half would only be one spin projection.  It's not enough to reproduce a state.  But what she is doing is she uses again  the magic of the beam splitter.  So one of those twin brother photons  is now mixed with the unknown quantum state at the beam  splitter, and the output of the beam splitter  is now entering the balanced homodyne detection, which  we just discussed.  So the output of this beam splitter and the two outputs,  both of them is now becoming part of a balanced homodyne  measurement.  And you see the ingredients.  So this and this will be measured.  The other input for the balanced homodyne  is a strong local oscillator.  And the phase of the local oscillator  is chosen in one case that you measure the x or a 1  quadrature component or the p or the a 2 component.  So now what Alice has done is by using this balanced homodyne,  she has, with this local oscillator,  has actually now performed two measurements.  The quantum state is destroyed.  But here she gets an x value, and here she gets a p value.  So how the magic works out, it's really just a few lines  of mathematics now, is these were sort of twin brothers.  But they were, it wasn't clear in which  quantum state the twin brothers are.  So if you write it down, and I can show you  the formula in a moment, these are twin brothers.  But those twin brothers, this is brother 1, this is brother 2,  are in a linear super, are in sort of a continuum of states.  And when Alice would measure that this twin brother is  in state big x, that would be a projective measurement.  And Bob's twin brother would now be certainty also  be in the state x.  So therefore, what happens is the measurement of x and p  is now producing, is now through the measurement process,  putting the other photon or the other beam,  the other twin brother, into a specific quantum state.  And if you look through a few lines of math,  the magic is that the quantum state, which is now here with Bob,  turns out to be a displaced copy, a displaced copy  of the original state.  And the displacement depends on x and p.  So if Alice now tells Bob, hey, I measured x and p.  And Bob is now programming his displacement operator.  Remember, displacement operator is nothing else  than a beam splitter, an unbalanced beam splitter,  with a huge local oscillator as an input.  So if Bob is now setting up his displacement operator based  on and makes a displacement which depends on x and p,  he can take this other twin brother, shift it back,  and he will exactly regenerate the quantum state, which  Alice had.  So this is now how a quantum state  can be transmitted without having any quantum channel  for transmission.  You're not propagating the quantum state.  You use classical communication,  but the resource you use is some EPR pairs or squeeze light.  It's a few lines of equations, but I  don't have time to go through it.  They are really annotated in a way that I think  it will be an enjoyable reading for you.  Any questions?  Time is over.  OK, reminder for those who came late.  We have today.  This week we have three classes, Monday, Wednesday, Friday.  Have a good afternoon.
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