 Today Christina is here. She's in the back. You can recognize her in case you haven't filled in your reimbursement forms you can do so today and There is going to be a photograph of everyone here and also of the online participants So that will happen just before lunch. So I'll remind you again 12 but we'll go out on the terrace and take the picture and then people who are online can can have their picture taken by the People here Then there is a conference dinner tonight at seven and we're just going to meet at the restaurant and the restaurant is just Just here basically Right on this side of the Institute. You you go out you go towards the sea and it's there on your Just next good So then the the chair of this session is Yes, hello, so I'm you put you from our CA Sackley and the first speaker Will be a Thomas Stefan virally from Shahbrook and more real and time domain quantum microwave Please Hi, so can you hear me? Yeah So good morning everyone My name is Stefan Virally Probably you don't know me. It's normal But I've been working. I'm risked. I'm now affiliated with Polytechnique Montréal, but I used to be working at Université Sherbrooke and And most of the the work that's been done and that they will present here comes from a Sherbrook So it used to work, but it doesn't know so okay So I'm basically a spokesperson for a group of real scientists So the most important I guess for this project is Bertrand Rollet who you probably know So he's at Université Sherbrooke working on noise in microwave microwave circuits and Most of the work has been done really by two PhD students Jean Olivier Simoneau who is now working for a Quantum company because there's a huge amount of new quantum companies in Sherbrooke that that That I just spawned in the in the in the last couple of years and Simoneau-Boldig-Bauduin is The current PhD student on the project So the slides a few slides that I will show you are coming from here Now I'm at Polytechnique Montréal. I'm working with Denis Sileski Maybe some of you know him is in ultra-fast photonics and we have a very good PhD student also who's working on it, but so Everything that I will talk about Basically, we discussed all of us on these things and it's the fruit of a lot of a long time of reflection so the motivation for this talk and also to try to motivate you to to listen to the talk up to the end the idea is to try to To work now quantum optics or quantum photonics or quantum microwaves Depending on the support we're working in with very short pulses and The good thing is in microwaves is Because we're so low in in frequency. We can really probe those short pulses relatively well So Because because these systems are very useful Because we're going to be able to probe fast dynamics stuff like that We want to understand these These pulses in in a quantum in a quantum way Also QED is usually cast in space and time So we really want to be able to to to have a good description of what happens to light when it interacts in small volumes of of space and time and And also as you will see Because we started working on this time domain Which is a bit different point of view from whatever Exists in general in the in the literature. We already have found some surprises and and there are probably more to come So this is basically what we started with Bertrand it started with Bertrand and I having a conversation about about this Bertrand says said to me, you know, I can send a pulse that's as As short as I want in a in a microwave line and And And I want to understand how it works in in terms of quantum for instance We will see usually the Hamiltonian is is is shown as to be h equals h bar omega times n And what is the omega that's associated with the pulse? That's so short that it has a huge amount of frequencies available And we'll see that the answer in the very beginning the answer seemed to be very very easy We just take an average or something like that, but it doesn't absolutely nothing to do With that in reality So let's start with that in books We have this But if I forget the one half of We have this relation if you if you open a quantum optics book and you and you Go a few pages you will find this relationship and books are a very dangerous thing because in the in in the end you will you will end up really believing that it's true and And it's not and and and it took me a lot of time to figure out that it's not that it's not true It's true in in some sense in some settings When third quantization is done in the frequency domain And that's usually the case of all the books that you encounter in the literature still from current energy to To Fox or whatever you will find That first quantization is done in the frequency domain or in the case space if you want and And you end up with this kind of of relation But so the energy is in this case is proportional to the number of photons The reason why it's true is because you have of course given a very definite Frequency to your photons. So of course you can say that if I measure a photon in this mode It will have the energy H bar omega, but in reality Of course photons are not quanta of energy. We all know that and and there is no real relationship between The Hamiltonian and the number of photons if I take something that's very broad my Hamiltonian is going to be some of the H bar omega n of omega and and of the total number of photons is just the integral of All the ends at all omega n's and they are not proportional So this idea that we have all that h and n are proportional isn't it just not true It's it's true only in very very specific circumstances and the circumstances do not apply to the type of research that we do with very broad pulses So What gives them The first thing is We know that energy is basically the square of the electromagnetic magnetic field so we do e squared plus b squared basically and we have the energy of of our En field But the number of photons Are something else they are the square of another field which I call the photonic field, but you know And it's and and it carries Information the same way the EM field does But it's not the same one and we'll see the relationship between the two but So that that's where we started we started with so what is this? weird thing photonic field that really gives me the the photonic field will give me the The the probability of seeing a click on a detector and The electromagnetic field will give me the probability of measuring some some type some energy in the in the M field oops so How do we how do we relate the two fields? So I will I will now go to the microwave domain and in the microwave domain The the main thing that we can measure from the EM field is is the voltage but it has the same structure as as The electric field or the magnetic field for instance and the important thing is that inside of the Of the expression of the voltage there's this pesky square root of h bar omega term that exists and that multiplies all the frequencies Whereas in the photonic field of the field that we really want to represent the number of photons There's this this factor doesn't appear. It's just straightforward lead of the quadrature of the of the field without without any without any Dimension So how do we go from here to here? So we have this square root of h bar omega So the best thing to do is to divide by the square root of h bar omega and That's exactly what we can do actually So when we do that basically what we do is we we have a transform We have a convolution in time domain With a one something that is the the Fourier transform on one over h bar omega and it happens to be one over T a square root of t basically So from the time domain perspective we can transform the voltage Into things that resemble the quadrature of the of the photonic field that will give us real information about photons How how we would measure clicks on detectors The big problem is there's those are non local obviously and non causal relations So it's a it's a bit weird. There's a there's a non-causal relation between the photonic field and the EM field Which means that we basically need information from the past and from the future But it's not the first time we encounter this kind of problem in in electrodynamics But if we if we implement this This relation we can we can measure Clicks the the the equivalent of clicks on detectors one of these this one Commutes with V at all times and that's very important because what we can do is what we can register a trace of V over time and then we can apply this Transform and if we have a Vigna function that is that is Rotationally symmetric for instance we only need one quadrature to be able to measure to do a full tomography of the of the of the state and we can from V obtain Q one of the quadratures and Check whether we have the good relations for Photon numbers when we when we do that So just this was just a slide about the non-causal relation between V and and and and the quadratures of the photonic field Interestingly when we when we do something just a simple pulse like that The the probability of finding a photon is exactly maximum when the voltage is equal to zero But so we wanted to test this this thing and we decided to do it on a total junction Which is basically the you know, it's the job of of Bertrand to Look at these things Tunnel junction is really what what? Makes him tick So when we base a tunnel junction with a DC and AC signal We have three waves mixing taking place so we can for instance Send a 12 gigahertz AC signal and we can And we can measure pairs of photons at let's say two two at six gigahertz or one at four and one at eight Or we can go down to two and ten We're just limited by the by this the the setup that we have and When we do that we can Collect the traces and apply the voltage transforms that I showed you before and And and then we we can check that we have the real the the good relationships for For Photons So just the setup the idea is that you can we can send both DC and AC on on the tunnel junction here And then we send everything back to a very very fast acquisition called 32 giga samples per second and we have a 12 gigahertz Bend width so we can basically measure everything Well, not the low frequencies because we're limited by our amplifiers and Here is the result. So let's say I send a 12 gigahertz pulse on to my on my junction and as I said, I will have pairs at six gigahertz I will have pairs that are that span four and eight for instance and And so I should be able to if I if I'm clever and if I decide to filter correctly my signal I should be able to to recover The equivalent of single mode squeezing so pairs that appear pairs of photon that appear all the time in in all the in all the Frequencies that I want so for instance if I go at six gigahertz It's already there. Just if I have a narrow band filter around six gigahertz I should see pairs of photons and that's what I see here So there's a there's a small contribution for a thermal state at 65 milik a but other than that the Fano factor at Zero should be close. It's it's it's not one clearly not one. It's close to two and And the reason is that we were generating pairs of photons Now we can do that at six gigahertz But if we do that at eight gigahertz, which is you know, we should have a normal Thermal state because if you take just one mode Of us of a two-mode squeeze state you will have a thermal state So the the Fano factor should be one and that's what we recover now We could go to four and eight gigahertz. So what we do is we have a filter a numerical digital filter that Goes that by Goshen around four and eight and we do the same thing on the trace with that That that's a simple filter and we recover the statistics of pairs of photons because we have now a mode a full mode that is both four and eight gigahertz and That will generate pairs of photons if we do that for with five and eight For instance, which is which does not corresponds to pairs that should be created at the same time We don't we recover the thermal the thermal statistics So we've shown that and and of course I Don't show you here because we have a problem in the curve But there's no there's no such thing as the same Relationships if we do if we do that with just V if we if we use V squared as a proxy for the number of photons We don't recover these these relationships. We have to implement the The transforms that I talked about So let's go to to another subject the quadratures in the time domain Because that was the question that we asked for for our saves a lot of for a long time We discovered that they are Hilbert transform of one another which is not surprising When you think about it, but in the very beginning it wasn't clear to us So the idea is if you have a Casio monochromatic System you have two quadratures you have beside you have basically the cosine and the sign and if you do if you go in the frequency domain Basically, you have the cosine is just two real parts and the sign is Two imaginary parts one positive the other one negative on the on the other side If if if we want something that is very short so here I I will show you a frequency come because I Think it's easier to see so you can have a very Wide spectrum you can have very different when you can have very strange relations between the Phase relations between the frequencies, but the idea is that the the the the quadrature is the Hilbert transform so the Basically the orange and the blue lines are 90 degrees from from one another and I have this neat animation to try and show To give you an idea of what quadratures are in the in the in the time domain so now if we want to go full time so we decide that we We go to the opposite direction of frequency with just time We realize that energy was an observable for for frequency domain and time of arrival of photons is a is a Observable for for the time domain and I call that Graphene single photon detector because it has You know it has a wide band going from zero to infinity So it's an ideal detector with infinite bandwidth should be able to to recover this These times of arrival so so if we have the the Hamiltonian which is equal to The sum of h bar omega a dagger omega a omega a dagger omega a omega We can think about something like theta which would be an average sum of The time of arrival of our of our photons and it would be the basically the jewel of the Hamiltonian in the in the time domain and interestingly There's a nice Commutation relation between the two And that was surprising because we thought there was no mutation relation between Hamiltonian and time There's a good argument for that but this commutation relation Is not as problematic as the one that people think about which usually doesn't include the end here so So this is this is a valid commutation Relation and I find it neat because I think it shows very well that What we measure and the the number of photons is really a number of action Now if we if I want to go beyond what we've done and do a full tomography of the field So we I want to measure up both quadrature's for instance in the frequency domain. I know that I can do Homodyne detection with a local oscillator So I send a local oscillator. I I split it in two. There's one part that has a pi over two shift the other part I leave alone and then I mix that On beam splitter with with my signal and I get and I recover the the two quadrature's I Can also inverse the signal and the LO so I can apply the pi over two shift on the on the signal It's it's it's going to give us the same thing now in time domain. It's more difficult One of the things that I do that we do with Bertrand. We measure we measure voltages so when we measure voltages we have We measure one point and we can do the same thing with electro optic sampling in in in optics basically sending a Dirac and Sampling the field at one time the problem with and and the way to to recover the To recover the quadrature's is to do the river transform But as I said, it's non-causal. So it's not obvious how do you do that? There is a device that's called a hybrid that we've talked about two days ago That actually does this but it has to be very wide-bend to be able to do that I'm not going to discuss this the last thing I wanted to show you is Something that you've you've heard about a lot You you when you have a coherent state you can see nice figures in the literature that that look like this but this this figure is a is an actual Simulation of what would happen in time domain? It's not like the figures that we've seen in the literature where basically those wiggles would be would corresponds to measuring the Changing the phase Of of our measurement here. Let's say I have a sine wave and it's a coherent state if I if I put it on a on a On a oscilloscope I should see exactly this this type of trace if I have a squeeze state For instance, we all know that we have this kind of here The state is squeezed in in amplitude. So we will recover this kind of thing But I just wanted to show you this because I I think it's beautiful It's the it's the cat state because I always Asked myself what is a cat state because it's a it's a plus alpha and a minus alpha and I couldn't So it looks like it should be zero all the time But it's it's far from being zero all the time the the signal that we are looking for in Second quantification quantization is really in the noise and that's the specialty of Bertrand That's it. I didn't want to Go further than this Okay questions Thank you for the nice talk. Maybe it's a stupid question But it's a formality when when you go from a frequency domain to 10 domain you need to do this I mean integral with I mean the frequency range is going from infinite to Minus infinity in a way, but it is not true. I mean you have a Width on a window of frequencies. So I mean in a way you have also restrictions for your I mean formalism You're absolutely right. So That's why I went full-time only after I showed you some some some actual Measurements that we did so if you go full-time you have this binding with the the teta and the and The time of arrival that you can measure but you in order to do that you have to have infinite bandwidth as you said And in general you're in between you're you're so but the idea is that we want to have tools To treat that in between instead of just saying okay, it's it's almost monochromatic so there's an omega Average omega and then we treat the stuff just as if it was monochromatic We wanted to be able to have a full description in time or in in frequency and And and that's why we're doing it like that You mentioned that you have this a causal connection between the you know the voltage and the quadratures and maybe I mean You start with definitions in terms of Fourier transforms, which are a cough is an a causal transform Did you think about or reconsidered? I mean could you do definite start with a Laplace transform and then get a causal theory? I've never thought of that so I It it's possible and that you mentioned it But I've never looked at it Yeah here, so whenever you open a single processing book you you learn about the the window function of The FFT or the Fourier transform engine on so maybe it's a dumb question, but how is this different from a window function? I mean you're the fact that you need to take the integral properly No It's not just a window. What would be exactly the same thing as the window function there I mean basically the yeah, yeah, so taking a very well or dear I mean the as far as understand but maybe it's a dumb question But as far as understand the point of the window function in signal processing is to avoid this Echoes other things because it restricts yourself to the proper frequency range. So basically you don't have any problem anymore so How is this? different or no, but let's say you have a very wide signal so I have a I have those pairs at four and eight gigahertz. So they have their widely separated or two and ten They're very widely separated. Yes, I can apply a double window function as you as you say and that's what I That's what I get. That's the type of stuff that we do We apply those two window functions if you want to to go to two gaussians around two and ten But when we do that and we have the right relationship phase relationship between those two windows It's not like they're separated They have a phase relation between the two and it has to be the right phase relation between the two and when we do that we can recover the quadrature of of the of our signal and We can show that we have it basically it becomes a full mode the full mode of the electromagnetic field but with but with photons at four and eight so they're in a superposition of four and eight Yes, it's a two mode quiz state sure, but no, it's not a two modes quiz state It becomes like a single mode quiz state But it's a single mode that is Delocalized in frequency domain at four and eight. It's not the same thing as I have my two modes and I and I Okay, so maybe you guess I'm missing something. Maybe we will discuss that at the coffee break. Thanks So I guess we're all asking the same thing because we're trying to wrap our head around this non-causal thing This is the most provocative thing you brought up. So it generates questions. Maybe even on purpose It's really nice to think about this fundamental issue. So The photon is is really defined by the mode in which you measure Yes, and you're determining that by whatever window function you set up and actually Our intuition is usually more sort of like a wavelet story and and you want to have In the Vigna representation sort of like what's called a Feynman basis Or maybe a tiling of face base in terms of Gaussian states each one taking up roughly h bar Yes, and and that's sort of the the intuitive coarse-graining. We like to think about when we want to say it's localized symmetrically both in space and in time So Essentially, I guess I'm repeating all the questions that have been here before is when you do that sort of localization that to that degree that's symmetric basis because The Fourier transform puts all the weight in time in terms of complete decal delocalization in time and and Dirac localization in frequency, but the fundamental basis is the symmetric one Yes, and it's an overcomplete basis because they overlap Yes, but still it's it's the intuitive one and the question is have you looked at that one maybe as well I guess it's a complimentary Yeah, so so You're absolutely right that it's it's the right basis I think to to To explore these things so what that means is that there are some very important Relationships between the the frequencies and they basically have this it's like a cosine as you it's like a cosine they have the right relationship between the different frequencies and that's the basis that you're talking about and That's the one that we use for for our measurements Okay, thank you. Hi So if I understood well, you're essentially explaining continuous variable or quantum optics and You just choose one way of representing the signals in time domain, which is just complementary to describing them in frequency domain So the the only difference is first quantization the first quantization step is different. Yes I mean, there are trillions of recipes into dealing with a continuous variable One two maptics in frequency domain. Of course, it doesn't look like localized states. It's the tenor and ones and I mean At some point you have a finite bandwidth. So it's just a point of view. Yes, and when you insist on the weird Distribution of squeezing of course it depends on into which temporal mode or you're projecting it and I mean It is very well known from the optical community of continuous variables and We even have a very good tool as soon as it is Gaussian here with a squeeze state How already looking at all the covariance matrices in frequency domain you can not only Look at the local squeezing in frequency domain, but look at what would be the the temporal modes of them I mean all these tools as far as I know are already known. Yes. So Yes, what's your question? Okay. No, I was just missing the No, no, but it's I mean, I mean I'm interested in the time domain picture. I want to understand it. So It's then all these temporal modes appear directly from the diagonalization of the covariant matrices I mean, of course, it can have very weird shapes. It will depend on how you tailor your policies. I understand I understand that it's it's not intuitive, but I Understand but still I that's I wanted to have a different point of view I want to try to understand what happens in time domain and I think at least to me when when I do that Even though they are already very well known things that I can do to to do exactly the same thing I'm more interested in in something new just to try to understand Not new physics, but a different point of view that maybe will bring something Okay, let's take your last question Hi, so can you just explain a bit more for the cat state? I didn't understand so in the time domain what you would measure and In ultimately how it's different than what you would do conventionally No, well, so it's just that this this is this is something that you should be able to replicate with, you know Just having a cat state and and doing the tomography as you rotate along the the phase Basically doing a full tomography This is just but this is what I would see in time It's the same thing, but it's what I would see in time if I generate generated a cat state in a in a very narrow band Frequencies if I generate a cat state in this and I just put it on on on an oscilloscope That's what I would see and that's just something that I was It's multi-valued I'm sorry. I mean it seems like it's multi-valued. I mean Yeah, yeah, yeah, but that's that's well, so on average is zero But that's that's the the beauty of quantum basically this the signal is the noise, right? So what you see is that everything is always zero on average But there's noise that is not distributed in time in always in the same fashion So I find that fascinating but Okay, thank you very much and thank you for having time for discussion