 OK, that thing, perfect. Thank you. Helabra, my name is Ben so I'm going to be continuing to talk about the same sort of thing we've heard in the last couple of talks. So I'd like to thank the audiences and I'd also like... Not only organising this but also allowing me to talk just after a couple of people have done the perfect introduction of the sort of material I'm going to be talking about. So I've been working with Andrew Armour and Grace Morley at Nottingham on the title is gyllun o gwałon yr ac oeswyr gyllun o'n gwrth. Felly, oeswyr gyllun o gwrth. Unrhyiety, mae'n gobeithas gyrfu rhywfion arall os y gallwn yn effeithlau'r gwyfudd. Mae'r gwrdd yn ôl yn cyf temosi. Felly, mae'r gwrdd yn gwyfudd sydd wedi'i dd �f am y cwymdei ar y gwrth gwrdd cyfiadau sy'n gwyfudd. Felly, mae'n gwrdd yn gwyfudd ac oeswyr, ma describedau hynny sy'n cael ei gyrdd gan fynd i gwyfudd yn y tyfnol i'w ffotons yn y osulatau. A Llyfrgell Jocelyn Llanser yn y ffordd yw'r amgylchedd yng Nghymru ac mae'r ffordd rynnod o ddwylo sy'n ffordd. Mae rwy'n meddwl o'r bydd hynny'n meddwl yw'i ffordd ac mae'r rydyn ni'n mynd i wneud i'r ffordd ac mae'n mynd i'w wneud i'r ffordd o ffordd i'r ffordd i'w môl. Felly mae'n mynd i'w ffordd o ffordd o ffordd. So ahead of it by having multiple lumped element oscillators, but the way they actually did it was they had a single plantoid waveguide kESIN Ier mewn wathen ni diwethaf, a ydych yn cael ei amhwylliannol y byddai cyfrifyrfa, yn tych yn yr argyfu cymdeithas fel gynnwys, ac mae'n rhaid i adных mwyaf gyda'r regime. Mae'n pwych chi gallwch chi'n dweud fel y rhaid i allu llais rhaid i'r regime a'r regime. Mae'r rhaid i gwaith yma yn dweud i'r regime yma yn dweud yma. Mae'n dweud ar gyfer y llefyd yw'r morcei digon, ystafell llwyddo wahanol os oedyn ar y caelwch iawn. If that is a position operator for all fath grannu arRETC, it is the phase across the oscillator. I should stand away from that, perfect. Excellent, I was wondering what it was. I will just stay away, away, away from that. Okay so there are some positions of those nodes and they said that in the Sa'n gwisech a'r genodynamicoedd? Rwy'n dechrau, ymlaen, ond yn y rai gwrthredig ysgolodd. Mae'r osolata i'r oes osolataeth gyda'r cyflym. Mae'r wiadio, yw'r cyflym? Tyf yw eu ralloch o'n gofalo, yn cael wirwch. Rydych yn cyflym. Mae'r ddechrau a'r dweud. Rwy'n ei ddweud i'w amser o'r cymdeithasol, ac yn ymwneud i'r sgrif ac mae'r ddaf yn rhoi'r cymdeithasol. Y teimlo, mae'r cymdeithasol yn ychydig. Mae'r gondol o'r cymdeithasol arall o'r oesolau oesolau, oedda'r cymdeithasol, oesolau fel y ddod yn rhan o'r cymdeithasol. Mae'r cymdeithasol yn credu'r cymdeithasol, oedd y mewn cyfwneid ychydig. o srfa gydenni'r syniad. Mae'n ddod o'r siŵl o'r niwr. Os yna'n ddod o'r ddod o'r digwydd, yma yw'n ddod o'r всёg? Ystod ar luv ymlaen o'r srfa gydenni'n ddod o'r srfa' gydenni'n ddod o'r srfa, ac meddwl am rwi'n cael ei wneud. Felly mae ydych chi eich fachaf o'r freqwansion spesifau a'r mae'r boblau ddiogelau a'r ddiogelau i'r senf. Felly ei fachaf y gweithio a'r gweithio sgol. Mae yma'r cyfrifnwyd ddifigai cyfrifrathiaid yn ffoton o'r ddechrau, dd responio'r wneud yn rhai ddegwyd, ddiddio'r ddigwyd ddegwyd ar gyfer wneud yn ddechrau cefnion o'r ddigwyd o'r ddigwyd, o'r ddigwyd o'r ddigwyd o'r ddigwyd o gymhyself ar hannau. But all the other frequencies that are falling between that ladder are being unpopulated. But if you increase the driving strength relative to the loss rate to a particular point there is a transition where all of the modes participate and that regime is what they were thinking of as a laser. Rwy'n wedi'i driw mae'n edrych yn dweud dw i led fan chi. Mae'r ddych chi'n trwwynau ac mae'n medrydio'r hobod hyn oedd hyn yn y cyffredin cael mwy salad ymargrificar. Roedd anklesneud ni addани – oedd dechyn hon USAID – Ac mae yno yn dda, mae'r dyfodol yw'r ysgol yn ddwylo'r argyfwyr. Mae'r tyfnwyr yn gondol o'r ysgol. Mae'r problemau er mwynser yn ysgol yma yw hynny'n ddiddordeb. Mae'r problemau yn cyhoedd, yn cyfwyr, yn ddwylo'r cyfwyr, dda'i gwbl yn ddiddordeb cyrraedd yn ysgol. Felly, mae'n gyfan hynny'n gwybod bod ysgol. Mae'r paramysau sydd wedi'u gwneudio'r ddiddordeb gyda'i a'r fawr i'r bach yn cael ei ffysg iawn yn wneud i ddim yn cymdeithasol. Mae'n rhaid o hollwch i'r meddwl, mae'n rhaid o'r tro i'r fawr iawn. Mae'n rhwng y proses ac mae'n fawr yn rhaid i'r bobl honno yn gynllun oed. A'r rhwng yn cymdeithasol, mae'n gobi'r rhaid o rhaid i'r rhaid o bobl yn blaenu. Mae'n ddod o'i amser o'r fung ymlaen. In this case there are n-modes you get more or less an n dimensional Bessel function. The n dimensional Bessel function is also not really very nice to work with. So you can take a Taylor series of it. The lowest order terms of the Taylor series are going to tell you where the mean amplitude of the mode goes to. You find the fix point. Then you take the lowest order terms around the fix point, which are called drastic terms in the Hamiltonian. Ac ydyw'r awrgwyr yn inserid hwn o ddyn nhw'n digwydd digwydd sy'n ddim yn y bach os oeddweithio mewn gyrddwr yn gwasanaeth ac ydych chi'n ddweud cael ei hynny'n ddyn nhw'n ddyn nhw. Mae gennym ni i'r awrgwyr, amddwn i'n rhowch gan mewn cyd-di-dd et i'r awrgwyr ac ydych chi'n gwybwyr yn inserid hwn o ddyn nhw yn gwybwyr yn gallu ganwyr am ffordd i gwasanaeth ychydig a'r awrgwyr os oeddwr yn gallu gyrddwr yn gwybwyr Yn y ddweud oherwydd yma, mae'r gwahanol wedi bod y ddweud yn ymwyaf yw hwyl yn gael gael gael. Yn y ddweud, mae'n gwahanol yw mwyaf yn ei wneud yw'r ddweud, a'r ddweud hynny'n fan amlwtoniol yn cymryd. Mae'r ddweud yn cymryd yn ystod dwyllgor ac yn unigol amfertyn. Mae'r ddweud yn cymryd yn yr anodol. Ond mae'r ddweud yn y ddweud yn ddweud yn y gynhyrch yn y llos, mae'n cymryd yn gael'r ddweud. I'm at the P photon resonance 3 in this case, so each Cooper pair has energy equal to the third mode in the cavity and I should have said earlier the assumption we made is the first mode has frequency 1, the second frequency 2 and so on going up. And here the Cooper pairs have frequency 3. In the low drive regime only those modes on that ladder at 369 are active. And then as you increase the drive strength we see this transition that was seen in the other paper. And in this case it's sort of bifurcation type transition. So this is again in the picture earlier this is the yellow arrow length in the different directions. So in the title I talked about entanglement and Gaussian entanglement. And where that comes in is that a common approach to especially in optics if you want to make a continuous variable entangled state with lots of modes is you get something nonlinear and you inject a spectrum that has a frequency sort of comb shape. So multiple frequencies equally spaced and this comb like shape produces entanglement in this nonlinear element between the frequencies that are between the teeth of the comb. So in this system we were thinking below the threshold we've got these ingredients sort of built in. You turn on the voltage you get a comb of frequencies that is a subset of the total set of frequencies. And then the other modes between those teeth have the opportunity to become entangled in the same way that's been seen in these optic systems for example and some Gaussian junction systems as well. So we looked at the Hamiltonians these quadratic Hamiltonians what they look like. So you solve for because you're in the below threshold regime only a subset of the modes have any amplitude. You solve for what those amplitudes are. You displace to that point and you find what the Hamiltonian is around that fixed point to the lowest order. And the simplest case is actually the highest voltage. If you're at a very high voltage driving the highest frequency mode in the system then you get essentially a group of independent parametric amplifiers. So here the Josephson frequency is 11 which means modes 1 and 10 together have enough energy to absorb one couper pair. And that's sort of what's represented here that modes 1 and 10 have a parametric type coupling. Whereas mode 11 doesn't couple to anybody and modes 2 and 9 also have a similar parametric like coupling because their energy also adds to 11. In a more complicated case you reduce the voltage to 6. There are still these parametric couplings but now there are sometimes some modes that their difference is equal to the couper pair energy. And this creates a beam splitter like set of couplings. So that's what these blue dash lines show. But in both cases you have these boxes that I've used to show there are groups of modes that just never talk to each other. And those groups, I've described it by energy conservation. You can also get there by a time symmetry argument to do with the drive. So just I've got two more examples and maybe the point of these examples is if you make the voltage relatively low the number of different symmetry sectors, the number of different boxes that modes split up into becomes smaller and therefore the number of modes in each box gets larger. And you can create these very quite large numbers of modes that are coupled mutually together. And we did the calculations and we found that in all of the cases for large numbers of modes for very large sectors in equilibrium when you leave the system the modes in each group have what we would call full multi-partite entanglement. So if you get those modes and you split them in half in any ways, you split one off from all the others or two or whatever way you want to split them, there will be entanglement between the two sections. We use that using a check for a positive partial transpose. You transpose the ones on one side of the division and see if it's a physical state, which is a quite standard method. Just to finish, I want to mention that the entanglement not just can span a very large number of modes potentially or to be split many ways depending on your voltage. You can also generate entanglement that is in some sense strong. So the log negativity is a measure that I won't go into, it's a measure of the strength of an entanglement and the mode that is one below the resonance. So if you're driving at frequency three, mode two is the one to look at. It has the strongest entanglement with the rest of the system. And if you go to the regime where it's a set of independent parametric amplifiers, then you get some well-known results from parametric amplifiers. It reaches log two maximum negativity at the threshold. Fabian actually warned us this morning. I've gone all the way to the threshold where the instability occurs. There is this margin you should avoid. There's some small margin here where you shouldn't trust this. And because the Gaussian we're getting very wide and these neglected higher order terms will matter. But you can get more log negativity when you go to these lower voltages with higher states. So that's what I wanted to say. Thank you very much for listening. Thank you very much. Ben, thank you very much for the nice talk. This was exactly my question because we did this with Andrew as you know before for this parametric situation when you come close to the threshold and you have critical slowing down what happens in this system. Have you touched this? No, not really. We've been using the, we haven't looked at that. That would be a good thing to look at next probably. Thank you. That was a question. Just when you talked about the entanglement. I mean I know I've spent a slot for three. But there was kind of controversy about the difference between what became known as full inseparability and genuine entanglement and you kind of used full entanglement. So for instance full inseparability means it's kind of this bipartisan test. And the problem is then you can find that your state could be written as a convex sum of mixtures of these bipartitions. And so then genuine entanglement means it can't be written as a convex sum of all these kind of partitions. So if I'm understanding you correctly, the measurement we're using I believe is safe. So you get the, so maybe this is a slightly different technicality than the one I was thinking you were asking about. So the one that we worried about is that we did every possible way of bipartitioning. And that because every bipartition has entanglement you don't need to worry about the tri-partitions or higher. That's not quite, you're asking about. That's full inseparability because it could still be that you're written as some convex sum of density matrices that are an entangled density matrix and a separable matrix. And then I have a convex sum of all of those possible things. So if you just do the bipartitions it's full inseparability and you have to show that it's not a convex sum to say it's genuine entanglement. I believe that that would probably, I believe that, I would love to talk to you about this later, but I believe that because we're in a Gaussian state, that this is all a Gaussian state and therefore it's, I don't think it can be, oh it's all about Gaussians, okay then I'll need to talk to you later, perfect. We have got the one where every bipartition has nonseferableness. If that's distinct then. So you want to share the discussion? Is there any other question? I don't know, it's just a technical detail. How much, the hypothesis that all the resonators have the same escape rate change the dynamics because usually in the land of a full resonator you have a one over N scaling of the escape rate. So they will become a larger and larger queue. Yeah, we looked, you think larger and larger queue. That's interesting, we found an experimental paper where the higher and higher resonances had a quadratically increasing escape rate. But that's probably varies a lot by, but anyway, you're right, the escape rates all being the same is the most unrealistic of the assumptions that I've listed. We did it because it was the simplest assumption. We also looked, because of this experimental paper, we found what we also looked to the case where the escape rate was quadratically increasing. It changes all of the numbers, but the diagrams I showed you for the Hamiltonian structures, those remain the same pictures. And therefore the numbers all change but the story doesn't change all that much. At least for that particular choice of how the dissipation increases. All right. Is there another short question? It seems to be not the case then. Thanks again for the talk and we'll go to the next.