 Okay, I'm assuming everyone can hear me. Let me begin by extending my gratitude to the conference organizers for giving me the opportunity to speak. I am Onola Voray. I'm currently a postdoc in Michele Kastner's group at the National Institute for Theoretical Physics in Steyl and Bosch, which is affiliated to Steyl and Bosch University in South Africa. The title of my talk, I've modified it a little bit from the title that was announced, is Many Body Quantum Kinetics in the Phase Space, Simulating Spin Dynamics by the BBGKY trajectories of sampled phase points. Before I begin, let me espouse a little bit on this acronym. BBGKY is an acronym of five physicists. Nikolai Bogolyubov, Max Born, Herbert Green, John Kirkwood and Jacques Yvonne. Yvonne is underlined because chronologically he was first here. The BBGKY trajectory or the BBGKY hierarchy in many body physics refers to a set of equations describing the dynamics of interacting particles with respect to s less than n particle distributions. If you have n interacting particles, the BBGKY hierarchy gives you a set of coupled ordinary differential equations involving all possible s particle distributions where s is less than or equal to n. The equation for the s particle correlations in this hierarchy includes the s plus one particle correlations, thus forming coupled chains of equations of increasing order. So this is basically all of us. Whoops, for some reason the pictures have all vanished. I apologize for that. Our group at Stellenbosch is myself, another former postdoc, Dr. Lorenzo Pucci and of course Mikhail himself. We have an experimental collaboration for benchmarking and for validating our method with the Instituto de Fisica de Casar Carlos in Brazil with Professor Roman Paquilar who is here as well, as well as one of his graduate students, Thiago. We extend acknowledgments to Robin Kaiser, who was here up until today from Nice, as well as on the truncated Wigner approximation method that was largely developed and formulated by Dr. Johann Schruckenmeier from Ana Maria Race Theory Group at the University of Colorado at Boulder and some experimental data that we're benchmarking against came from John Bollinger's group at NIST. So let me introduce what I'm going to do in this talk is to present a novel method for simulating quantum many-body spin dynamics. This method involves the truncated Wigner approximation applied discreetly in the spin phase space and extended in accuracy with the quantum kinetics of two-point correlations. Now, in principle, this method is completely general. In practice, it's applicable to a large variety of both integrable and non-integrable many-body systems, especially to long-range interactions as well as to systems high larger than one dimension, especially in systems where matrix product states or DMRG methods begin to fail and with the computational resources at our disposal, we can apply it to systems of 10 to the 2 to 10 to the 3 particles. Our motivation for developing this method is to study quantum spin systems with long-range interactions out of equilibrium with a special focus on closed quantum systems that model ongoing experiments involving long-range interactions. There has been quite a bit of talk at this conference about Rydberg atoms, so I won't go into those and analogously polar molecules. The experiment that we're focusing on for our data for benchmarking our method comes from an experiment involving ultra-cold ions in a penning trap in 2D from the group of John Bollinger, and they've published their details here in this nature paper. There's a poster out back that also cites this experiment as a basis for their work. It's basically ultra-cold ions in a penning trap that's in a Wigner crystal, that's triangular, and by inducing, using a travelling wave to induce transverse excitations, they can effectively model easing-like interactions in the long range where the power-law decay between the site-to-site is tunable from zero essentially mean field all-to-all interactions all the way to dipoles 3. In addition to closed quantum systems, we're also interested in looking at open quantum systems that model experimental decoherence. For instance, in this penning trap experiment, it essentially only remains closed up to a certain point in time after which interaction with the external environment starts to induce quantum decoherence. We want to generalize, I'm currently mostly going to present it for closed quantum systems, but we're trying to generalize it to open quantum systems to model the decoherence, and we're also interested in looking at periodically driven systems, including open quantum systems that are being periodically driven, such as light scattering dipoles with detuning, and I'll go into that at the end a little bit. So let me first explain what I mean by the discrete truncated Wigner approximation and do so by providing a brief review of what Wigner functions are. Now Eugene Wigner, all the way back in the 1930s, had shown that in quantum mechanics, the state of any system can be completely represented by a quasi-probability Wigner distribution function in a phase space. So here I've shown it notionally for a boson in a continuum, but more generally, if you have any Hermitian operator like omega hat here, you can uniquely define what's called a vial transform, which is basically just the trace of the operator with respect to a phase point operator here, which is defined on a phase space spanned by position and momentum eigenvalues, which are for bosons in a continuum or continuous. And the phase point operator here is basically the operator that generates a coherent state, a Gaussian-Glauber coherent state, at a point, well, this should be Q actually, but Qp. So for every point in the phase space, there's a coherent state, this generates the coherent state, and tracing any Hermitian operator with respect to that phase point at that point gives you the value of a distribution at that point. This distribution is shown as the vial distribution, and a special case, the vial distribution for the density matrix is just the Wigner function. So if you invert this result using trace properties of the phase point operators, you can get the density matrix written as an integral of the Wigner function with respect to the phase point operators. Now, the reason why, I mean, one of the, and there's, of course, here, there's a more modern review article by Anatoliy Polkovnikov from the Anals of Physics that gives a much more detailed overview of Wigner functions. Here are some properties of the Wigner function that we'll be needing. The Wigner function is bounded. It's quasi-probability, which is not necessarily non-negative. It can get negative, but it's always bounded. The way I've defined it here is bounded between minus 2 and 2, although it's always real. It's normal, integrated over the entire phase space, it's unity. It has traceality properties. If you take two Wigner functions and you integrate them, you'll get the trace of the corresponding density matrices, and most importantly, it's the projection property, which means that if you integrate this along between any two, between a strip that is delineated between any two lines, let's say AQ plus BP is C1 and AQ plus BP is C2, then what you get is the probability that the observable AQ hat plus BP hat has a measurement between C1 and C2. Now, by utilizing this property and inverting it using the trace properties of the phase point operators, we can map these properties to analogous properties of the phase point operators themselves. So the phase point operator is normal, so it traces unity. It has traceality properties, two phase point operators trace to delta functions, and integrating a phase point operator on that strip gives you an operator that projects this to this interval. This will be useful because when we're going to discretize this for discrete spins, we're going to get expressions for the phase point operators by utilizing these properties that have been established for the continuous case. Now, the reason why I gave you an overview of the Wigner function is that if we look at quantum dynamics in the Wigner representation, here I have the von Neumann dynamics of density operator here for a closed quantum system, Hamiltonian system. If I take while transforms on both the sides, on the left hand side, I have the time derivative of the Wigner distribution. On the right hand side, I have the while transform of this commutator, which, when you work it out, gives you the moll bracket between the while transforms of the constituent operators in the commutator. So if rho here, this becomes W, the Wigner function, and the while symbol of a normal-ordered Hamiltonian actually just becomes a classical Hamiltonian. So this is a moll bracket. The moll bracket is defined as thus so. The sine of this, which means that it's the sine series power, and these arrows means they act on this operator and the right-facing arrows means they act on this operator. You can clearly see, writing it in this form, is that if you just look at the lowest order term here, sine theta is theta, get rid of the sine, you can clearly see that this is just the Poisson bracket. So that means that in this expansion, the lowest order contribution is just the classical dynamics. This motivates the truncated Wigner approximation that basically involves a sampling biased by this Wigner distribution of the phase points followed by the classical evolution. So here I've shown it notionally. This is basically just the Bosonic-Fox state at the zeroth level. It's just a Gaussian. In the Wigner space, it's also a Gaussian. So we try to get some quantum physics from our approximation by sampling phase points with this probability distribution. Here we're assuming that it is a probability distribution. It doesn't go negative, which for this state it doesn't. And so we're going to sample more points along the maximum here and fewer points away from the bell curve. And then with each point, we're going to just evolve it classically. So it's the lowest order in this expansion. And observables in the truncated Wigner approximation is obtained from classically evolving Vile symbols. So the expectation value of an observable exactly is given by the trace of the density matrix and the corresponding operator, which is exactly given by the integral of the full Wigner function continued in time, according to this dynamics, times the Vile symbol of the operator also continued in time, which we will approximate by an arithmetic mean of the Vile symbol of the sample points that we've sampled. So this is actually, this is the continuous truncated Wigner approximation. It's been in use for a very long time. There is a reference here that provides more details on it. A lot of people in chemistry, as well as in atomic and molecular physics, have used this on single particle problems. We're now going to look at how to discretize it. So now let's just look at a single spin half. And this was essentially shown in this seminal paper by William Booters all the way back in 1987 and has been generalized to many body systems by Johann Schock and Meyer and the folks at the Anna Maria Race Theory Group which they published on this PRX last year. So we'll start from just a single spin half. So a single spin half has canonically conjugate degrees of freedom that are linear combinations of the Pauli matrices. They have two discrete eigenvalues each. Let's denote them by up and down. So as a result, sigma x, sigma y, let's say we have four discrete phase points which I'm labeling by the letter alpha and I'm labeling the four discrete phase points now by simply 0, 0, 0, 1, 1, 0 and 1, 1, which I can do without loss of generality. Now, of course, if I can define phase point operators that have unit trace like so here, then I can define a Wigner function consisting of four values, one for each phase point as a trace of the density matrix for the spin half times the phase point operator. Of course, these r alphas, these r's are different three vectors, r's different three r's for each alpha and the r alphas are not unique but they're all formally equivalent and I choose r alphas as a sampling scheme to satisfy the projection operator condition that I had shown in the previous slide for the continuous case. So I apply them, I get equations and then I get non-unique solutions for r and I can choose one and use them. So this can be easily generalized to many body systems. So now I have a whole bunch of spins, they may be interacting. We have four discrete phase points for each spin. So for n spins, we have a total of four to the n. We can, assuming let's say that we're starting from a disentangled state that's a product state, then the Wigner function for the full system is just given by the product of the constituent Wigner functions and irrespective of the initial condition at t is equal to zero, the phase point operator is given by direct product of the constituent phase point operator. So if I, let's say sample a few points from here, here denoted by red, for each point I have an alpha. So if I sample from all of them, I have an array of alphas, n array of alphas. I have four to the n such combinations. So I sample a small number of those and then for each such sample phase points in the truncated, discrete truncated Wigner approximation I do a classical evolution. And then this classical evolution approximates this exact evolution here which is shown in the Heisenberg representation. The Heisenberg representation of course the Wigner function stays the same and the phase point operators which were originally decoupled here now evolve in time and may couple to a full many body operators because of the Hamiltonian. And we get observable just like in the continuous truncated Wigner case as an arithmetic mean of the wild symbols. And here notionally I've demonstrated this is basically the classical, this is a Poisson bracket, this is basically the classical evolution for general spin Hamiltonian. So of course we can clearly see that this method has certain quantitative shortcomings which is that suppose I have an initial state that may be correlated or may not be correlated, the quantum physics from those correlations will manifest themselves here because the correlations will reflect upon the Wigner function. So when we sample from the Wigner function with that probability distribution we'll get all the quantum effects that are associated with those correlations. However let's say we have just two sample points and we're now evolving the classical dynamics these may be correlated and when we sample these two points we're going to sample them with a probability distribution that reflects those correlations when they're evolving in time these are classical trajectories they don't see each other. These are essentially just single particle expectation value trajectories and so individual trajectories don't see each other which means that the discrete truncated Wigner approximation incorporates dynamical fluctuations only on the level of the initial state and therefore in reality accounts for them only for short times. Longer times they won't account quantitatively so what we've done is that we've improved upon this method using the BBGKY hierarchy and our results are published we put our results out there on this PRB this year so the idea is that we recast the exact dynamics of the sampled face points as a BBGKY hierarchy so we've sampled a bunch of points so we have a face point operator a unique face point operator characterized by our sample each face point operator shows an exact quantum evolution we're going to approximate that by adding a BBGKY hierarchy both of the single point observables recovering truncated Wigner approximation and add to that classical dynamics of local correlations to higher order the BBGKY hierarchy offers us a way offers us a prescription to do this we just do it to two particle correlation orders we can go to higher orders as well but we truncate out connected correlations to third order and beyond and this yields significantly better approximations to the dynamics of each Wigner sampled face points especially in strongly correlated systems and furthermore because we're using BBGKY it also makes our method highly conducive to the study of long range interactions this has been discussed at some length in earlier talks as well and there's a reference here that has also been cited in several talks of BBGKY in the context of the earlier talks the BBGKY actually offers a prescription for deriving Vlasov dynamics and going beyond that in classical cases and we're just doing the quantum we're just doing the quantum version and here in this diagram we can see that from the earlier diagrams in addition to the single point trajectories and their classical evolution we add to that, coupled to these the classical evolution of correlations there are of course many many more NC2 correlations but I've just shown it here notionally so I'm not going to give you the entire maths for this because it's too complicated I'm just going to give you an overview of the ingredients that lead that allow us to formulate this hierarchy for each sampled face point here shown in this superscript here we have the exact von Neumann dynamics of the face point here shown like this what we do is first we define reduced one particle and two particle operators by taking partial traces on both the sides so we trace out over all but one particle and do it for each particle we have n single particle operators that are reduced from here on the right hand side because the Hamiltonian may contain two body operators like hopping and so on if we take partial trace we're actually going to get both reduced one body operators and reduced two body operators so to complete the dynamical so now we have to take two particle traces so here we define them as we take partial trace over all but the i th and j th side and do it for all combinations of i and j to get reduced two body operators and then when we take the same partial trace on the right hand side we're now going to get because of the Hamiltonian reduced one body, two body and three body so then to complete the dynamics we take partial trace of reduced three body, a four body and so on and so forth all the way to how many atoms there are n bodies of course if we do it for all of them we'll get the exact dynamics but now this hierarchy offers us a prescription to select the dominant contribution of the correlations, two body correlations which is one of the things that we're interested in looking at we can do that by rewriting the reduced operators as cluster expansions shown here for the two body case and three body case and truncating out equating these and all the higher order correlations to zero this gives us a fairly accurate estimate when plugged into the dynamics for the evolution of two body correlations and so of course I wrote them all in operators if we have a general kind of Hamiltonian that involves one body terms and two body terms and all we do now is we write out the reduced operators in the Pauli basis shown here schematically for the single particle reduced one body operators and two body operators and compare coefficients on either side of each equation to get a chain of coupled ODE's of coefficients coefficients of A which will involve functions of A as well as functions of these C's and higher order terms and so on and so forth if we truncate the hierarchy at the first level by equating these and all higher order connected correlations to zero this is basically just the classical dynamics of the truncated Wigner approximation if we truncated lower here by equating the third order and higher order connected correlations to zero we get the dominant contribution of the correlations that were lost in truncated Wigner so the only thing that we do is we do truncated Wigner we sample it and instead of evolving just the classical trajectories we now evolve this BBGKY hierarchy dynamics truncated to this order so here are some results here we benchmark these track methods as well as against old fashioned truncated Wigner let's start with the bottom case this is basically a bunch of XX dipoles so if cubic interactions here at t is equal to zero we are starting from a fully polarized spin state with all the spins in x direction in blue we have truncated Wigner in black here we have exact dynamics for 100 spins we've done here DMRG this is in one dimension and in red we have truncated Wigner with BBGKY now you can clearly see that truncated Wigner for single this is just the expectation value of SX summed over all sites per site we can see that this is truncated Wigner is better than just bare classical dynamics because you can clearly see from this example that if all the spins are pointing in the x direction that's the classical ground state so classically you won't see any dynamics of course quantum mechanically you will see dynamics and although these are essentially classical trajectories you do see dynamics here in the sampling because initially while all the spins are in the x direction the Wigner function however has a uniform distribution of y and z spins that average out to zero in the classical case so when we sample from it we sample uniformly a bunch of y and z z spins between minus one and one so the y and z spins are not zero so each sample point is not in the classical ground state and so they evolve and it is their average effect that produces this dynamics to the exact dynamics if we add the BBGKY correction that I had described previously we get excellent agreement up to experimentally accessible time scales here this one this is in units of h bar over j here well j here and here we are looking at correlations this is basically spin squeezing this is an entanglement witness that's basically at each time the minimum variance of the spin operator in a direction perpendicular to the spin vector qualitatively quantitatively we get far better agreement for correlations here sorry okay so again we've also shown this for the easing model we can see excellent agreement both for mean field all to all interactions as well as dipolar interactions and here we have the triangular lattice from John Bollinger's group that I had shown previously we've got it a transverse field to it as a proof of concept we've done spin squeezing in two dimensions we're waiting for experimental data to benchmark this against and finally as an outlook we're interested in seeing whether we can extend this to longer times to do theoretical investigations whether this reflects whether we can get universality in non-equilibrium systems like the Kibble-Zurek mechanism as well as a lot of other things like quantum ergodicity, Lieb Robinson bounds and so on a lot of this has been discussed in earlier talks and of course we're also looking at this for dissipative open quantum systems in the Markov limit this is a collaboration that is ongoing with Romain Roba as well as Tiago and finally well let's skip this and finally also modeling experimental decoherence these are Raman scattered and Rayleigh scattered processes that bring decoherence in the Bollinger experiment and other spin systems and we can add these as non-herbition lindblad corrections and we're trying to extend our BBGKY prescription and a truncated Wigner prescription to model these systems