Number-Phase Wigner Representation for Scalable Stochastic Simulations of Controlled Quantum Systems
NNumber-Phase Wigner Representation for Scalable Stochastic Simulations ofControlled Quantum Systems
M. R. Hush, A. R. R. Carvalho, and J. J. Hope
Department of Quantum Science, Research School of Physics and Engineering,The Australian National University, ACT 0200, Australia (Dated: November 10, 2018)Simulation of conditional master equations is important to describe systems under continuousmeasurement and for the design of control strategies in quantum systems. For large bosonic sys-tems, such as BEC and atom lasers, full quantum field simulations must rely on scalable stochasticmethods whose convergence time is restricted by the use of representations based on coherent states.Here we show that typical measurements on atom-optical systems have a common form that allowsfor an efficient simulation using the number-phase Wigner (NPW) phase-space representation. Wedemonstrate that a stochastic method based on the NPW can converge over an order of magni-tude longer and more precisely than its coherent equivalent. This opens the possibility of realisticsimulations of controlled multi-mode quantum systems.
PACS numbers: 42.50.Lc,05.10.Gg,03.75.Gg,03.75.Kk,02.70.-c
Exciting advances in physics have led to a boom of re-search into technologies that exploit fundamental quan-tum properties. Such quantum technologies now encom-pass more than lasers and superconductors. Indeed, thereare applications to precision metrology [1, 2], quantuminformation processing and quantum cryptography [3, 4].A key feature of quantum technologies is that they re-quire the precise creation, measurement and control of in-dividual quantum systems. In particular, measurement-based feedback control has shown promise as an effectiveand robust technique for controlling quantum systems.The first experiments [5, 6] and much theoretical work[7–12] on feedback control of quantum systems have beenapplied to relatively low-dimensional systems. This let-ter describes a technique for efficient simulation of largebosonic conditional quantum systems that is more thanan order of magnitude more precise and converges for sig-nificantly longer timescales than previous methods, andthat scales logarithmically with the size of the Hilbertspace.A large bosonic system of particular interest toquantum science is the Bose-Einstein condensate.Measurement-based feedback control of BECs and atomlasers was first investigated in a single-mode model,where a continuous number measurement was used toreduce the interaction-induced phase diffusion that lim-its single-mode atom laser linewidth [13]. It was thenshown that position measurement and feedback on a sin-gle trapped atom could bring it to the ground state [9],but the proposed measurement scheme was not suitablefor large atomic clouds such as a condensate. A multi-mode quantum field model of a condensate measured byan existing experimental technique (phase-contrast imag-ing) was then produced, but it could only be solved usinga semiclassical approximation [11, 12]. Analysis of thelinewidth of a multimode atom laser undergoing feedbackwill require a viable stochastic method for conditional quantum states that can deal with both high nonlinear-ities and number-like measurements. The number-phaseWigner function method fulfills both of these require-ments.The most effective methods for dynamic simulation ofhigh-dimensional bosonic quantum systems are stochas-tic techniques based on phase-space representations [14,15]. Each stochastic method is derived from a specificphase-space representation, which is akin to the choiceof a basis for the Hilbert space. Naively, these tech-niques require memory and computational resources thatscale logarithmically with the size of the Hilbert space.Practically, the overall computational efficiency is sys-tem dependent, and strongly depends on how well theunderlying phase-space representation matches the nat-ural basis for the quantum system under consideration.The most commonly used stochastic simulation meth-ods are based on phase-space representations that useGaussian states. These methods have enabled the sim-ulation of quantum optical [15, 16], atomic [13, 17, 18],and fermionic quantum fields [19]. In particular, stochas-tic methods have been used extensively in the field ofquantum-atom optics, where dilute atomic gases canbe cooled to produce BECs and atom lasers [20–22].The two most successful varieties are based on posi-tive P (P + ) and truncated Wigner (TW) representa-tions. P + is an exact technique, but requires a dou-bling of the phase space that often leads to instabili-ties [23]. Truncated Wigner is an approximate techniquethat typically has significantly longer convergence timesthan P + . However it makes an uncontrolled approxima-tion [24], and may therefore converge to incorrect so-lutions. Both of these methods, along with all othercoherent-state based representations, experience difficul-ties dealing with large number-conserving nonlinearities,as the underlying Gaussian basis becomes inappropriate.Such large number-conserving nonlinearities are typically a r X i v : . [ qu a n t - ph ] M a y the dominant energies in confined cold atomic systemsby a couple of orders of magnitude. Recently, we in-troduced a new stochastic method based on a number-phase Wigner (NPW) representation [25], that providesa non-approximate method for simulating large number-conserving nonlinearities. It was found that this dramat-ically improved the convergence of simulations of thesehighly nonlinear systems.Modelling highly nonlinear systems undergoing contin-uous monitoring and feedback requires the simulation ofa conditional quantum state. In a recent paper [26], weextended stochastic simulation techniques to apply to aclass of conditional quantum systems. Continuous mea-surement of a quantum system can have a dramatic effecton its dynamics. In fact the choice of measurement caneven be used as a controlling mechanism by itself [27]. Itis therefore unsurprising that the appropriate choice ofphase-space representation is heavily influenced by thechoice of measurement, as it may drive the conditionedsystem towards a state that is simpler to describe in aparticular representation. Also, not all measurementsautomatically produce a usable method. Our previouspaper demonstrated how to unravel a particular form ofstochastic Fokker-Planck equation (SFPE) [26], and it isonly possible to generate SFPEs of this form with par-ticular combinations of measurement schemes and phase-space representations. In particular, methods based oncoherent state representations are badly suited to mea-surements involving number-like observables rather thanquadrature-like observables, which are prevalent in atomoptics. In this letter, we show that the NPW representa-tion produces dramatically superior results than coherentstate based representations for these calculations.A common quantum atom-optical system under mon-itoring is governed by the conditional master equation d ˆ ρ = − i [ ˆ H, ˆ ρ ] dt + (cid:88) i D [ ˆ L i ]ˆ ρ dt + (cid:88) i H [ ˆ L i ]ˆ ρ dW i . (1)where dW is an Ito Wiener increment; D [ˆ c ]ˆ ρ = ˆ c ˆ ρ ˆ c † − (ˆ c † ˆ cρ + ρ ˆ c † ˆ c ); H [ˆ c ] = ˆ cρ + ρ ˆ c † − Tr[ˆ cρ + ρ ˆ c † ]ˆ ρ ; ˆ H isthe Hamiltonian and contains the contributions from ki-netic, potential and many-body interaction energies; andˆ L i = (cid:82) d x ˆ ψ † ( x ) L i ( x ) ˆ ψ ( x ) is the measurement operatorwhere L i ( x ) ∈ L are the measured density momentsof the multimode object. The only restriction we haveapplied to the measurement operators ˆ L i is the orderof the field operators which we note are ‘number like’algebraically. This form is the lowest order number-conserving interaction possible for a multimode system.Measurements that are lower order with respect to thefield operators will not conserve number and may besuited to traditional coherent-state representations, butin many cases these systems may be treated using an-alytic techniques like the Kalman filter, making simula-tion less important. Number-conserving measurementsare quite common in engineered monitoring of BECs [11, 12, 28–33], as any phase-sensitive measurement re-quires the existence of an atomic local oscillator to useas a phase standard. Thus, the efficient simulation ofEq. (1) will be relevant to a wide variety of atom-opticsystems, including all those involving current experimen-tal detection schemes.To compare the performance of the NPW represen-tation to coherent methods when simulating conditionalmaster equations of the form Eq. (1) we require a veri-fiable solution for comparison. Unfortunately, Gaussiananalytic techniques commonly applied in quantum con-trol are not appropriate for Eq. (1). This can be under-stood by noting the measurement operator ˆ L i is secondorder with respect to the field operators, which generatesnon-quadratic terms. Thus Gaussian analytic techniquessuch as the Kalman filter are not guaranteed to be exact[34], and we are forced to integrate the master equationdirectly to generate a benchmark for comparison. Thisrestricts us to looking at single mode systems, as directintegration is not scalable to multimode systems. Thesingle mode problem that is algebraically equivalent tothe multimode Eq. (1) is d ˆ ρ = γ D [ˆ a † ˆ a ]ˆ ρ dt + γ C [ˆ a † ˆ a ]ˆ ρ dt + √ γ H [ˆ a † ˆ a ] ◦ dW. (2)where ◦ dW is a StratonovichWiener increment and C [ c ] ρ = − (cid:0) ˆ c ρ + 2ˆ cρ ˆ c † + ρ (ˆ c † ) − Tr[ˆ c ρ + 2ˆ c ˆ ρ ˆ c † + ˆ ρ (ˆ c † ) ] ρ (cid:1) + (cid:0) ˆ c ˆ ρ + ˆ ρ ˆ c † − Tr[ˆ c ˆ ρ + ˆ ρ ˆ c † ]ˆ ρ (cid:1) Tr[ˆ c ˆ ρ + ˆ ρ ˆ c † ] is the Stratonovichcorrection superoperator. This master equation is of asystem undergoing continuous collopse under a numbermeasurement.The scalability of stochastic techniques for solving con-ditional quantum dynamics has already been demon-strated in [26], but we aim to investigate the effectof choosing different representations. We use masterequation Eq. (2) to compare the performance of leadingcoherent-based scalable stochastic methods to the NPWrepresentation. The convergence of these techniques iscompared to a direct integration of the master equation.We start our analysis with the coherent state basedrepresentations P + and TW. The success of these tech-niques have been primarily concerned with Hamiltonianand decoherence evolution of BEC and quantum-opticalsystems. Starting with P + , we now investigate the ap-plicability of these techniques on a conditional masterequation. Using the correspondences in [15] we can con-vert the master equation (2) to d P ( α ) = (cid:8) γ (cid:2) ∂ α α (cid:0) | α | − E P (cid:2) | α | (cid:3)(cid:1) + ∂ α ∗ α ∗ (cid:0) | α | − E P (cid:2) | α | (cid:3)(cid:1) − ∂ α α − ∂ α ∗ ( α ∗ ) − (cid:0) | α | + | α | − E P (cid:2) | α | (cid:3) − E P (cid:2) | α | (cid:3)(cid:1) +4 E P (cid:2) | α | (cid:3) (cid:0) | α | − E P (cid:2) | α | (cid:3)(cid:1)(cid:3) dt + √ γ [ − ∂ α α − ∂ α ∗ α ∗ +2 (cid:0) | α | − E P (cid:2) | α | (cid:3)(cid:1)(cid:3) ◦ dW (cid:9) P ( α ) , (3)where P ( α ) is the P-representation quasi-probabilitydistribution which reproduces normally ordered mo-ments of the master equation (2). Here E Q [ f ( x )] ≡ (cid:82) d x (cid:48) f ( x (cid:48) ) Q ( x (cid:48) ) is our notation for taking the expecta-tion values of a function f ( x ) with respect to the quasi-probability distribution Q ( x ).We immediately note thatthis equation contains non-positive definite diffusion thatmust be simulated by doubling the phase space. Thus P + techniques are required. This representation can then beunravelled into the following set of stochastic equations: dα = − γα ( βα − E f [ βα ]) dt + √ γα ◦ ( idV + idV + dW ); dβ = − γβ ( βα − E f [ βα ]) dt + √ γβ ◦ ( − idV + idV + dW ); dω = − γω (cid:0) βα + β α − βα E f [ βα ] (cid:1) + 2 √ γωβα ◦ dW. (4)Where dV and dV are the set of ‘fictitious noises’that are averaged over to obtain the weighted averages E f [ f ( x )] ≡ (cid:80) i ω i f ( x i ) / (cid:80) i ω i . For more details on thetechniques used to unravel equation (3) into (4) and howto simulate them see [26]. Eqs. (4) will be used to bench-mark the unravelling of Eq. (2) using coherent-basedmethods.We continue our analysis with the TW representa-tion. Using the operator correspondences given in [15],we can write the master equation for the Wigner quasi-probability distribution W ( α ) as d W ( α ) = (cid:26) γ (cid:20) − ∂ α ∗ ∂ α − ∂ α α − ∂ α ∗ ( α ∗ ) + ∂ α ∗ ∂ α (cid:0) | α | − E W (cid:2) | α | (cid:3)(cid:1) − (cid:0) | α | − E W (cid:2) | α | (cid:3)(cid:1) + 4 E W (cid:2) | α | (cid:3) (cid:0) | α | − E W (cid:2) | α | (cid:3)(cid:1)(cid:3) dt + (cid:20) − ∂ α ∗ ∂ α + 2 (cid:0) | α | − E W (cid:2) | α | (cid:3)(cid:1)(cid:21) ◦ dW (cid:27) W ( α ) . (5)Note that the first term contains higher order derivativesand a truncation is required in order to obtain a stochas-tic unravelling of Eq. (5). Note also that a P + style ex-tension of the phase space would be required to simulatethe diffusion in the conditioning term. Traditionally theWigner representation is guaranteed to produce strictlypositive-definite diffusion [15], but this is under the as-sumption that the calculus increment is positive as is thecase with dt . Unfortunately this assumption does nothold with the dW increment. A new ‘positive’ Wignerrepresentation could be derived by analogy to the P + rep-resentation, but the higher order terms would still need tobe truncated. This would make this hypothetical repre-sentation both approximate and doubled in phase space,which would make it unlikely to compete with P + . Thusit is not worthy of further investigation. Finally we consider the number-phase Wigner repre-sentation. The NPW was first derived in [25] and wasused in the simulation of large nonlinear equations. Wenow consider its applicability for use on conditioned largeatom-optic systems. Using the operator correspondencesgiven in [25] we get the following equation d N ( n, φ ) = (cid:26) γ (cid:20) ∂ φ − n − E N (cid:2) n (cid:3) )+4 E N [ n ] ( n − E N [ n ]) (cid:105) dt +2 √ γ [ n − E N [ n ]] ◦ dW (cid:111) N ( n, φ ) , (6)where N ( n, φ ) is the NPW representation that producesa complete set of moments of the master equation asoutlined in [25]. We next unravel Eq. (6) using [26] to dn = 0; dφ = √ γdV ; dω = γω ( − n + 4 E f [ n ] n ) dt + √ γωn ◦ dW. (7)Note we did not need to apply any truncations or dou-ble the phase space. The simplicity of the equations (7)compared to (4) show how an appropriate choice of rep-resentation, NPW in this case, can greatly reduce thecomplexity of the evolution, just as an appropriate choiceof basis can simplify analysis of other quantum problems.We can now compare the performance of the NPW rep-resentation to P + by integrating equations (4) and (7),respectively.The simulations comparing the P + , NPW and a directintegration of the master equation are shown in Fig. 1.The number-phase Wigner representation converges forthe longest time interval. In fact it converges until a com-plete collapse into the correct number state. As this isthe steady state of the equation we expect the number-phase representation to converge indefinitely. Not onlyis the NPW more accurate it is also significantly moreprecise. The increase in precision in turn improves theaccuracy of the evolution, as simulation of the condi-tional master equation uses an estimate of the observ-able Tr[ˆ c ˆ ρ + ˆ ρ ˆ c † ], thus lack of precision results in a lackof accuracy in the long term. This dynamic instabilityis not seen in non-conditional master equation evolution,and makes the precision of stochastic techniques consid-erably more important in these problems.The results show that the NPW-based simulationsare significantly better than the simulations based oncoherent-state representations for the conditional masterequation described in Eq. (2). The simulations promiseto be stable enough to consider long-term behaviour ofsystems and model the effects of feedback strategies. Thishigh level of convergence can be explained by noting thesuitability of the basis underlying the representation tothe measurement eigenstates. Importantly, due to prac-tical difficulties in producing stable atomic local oscil-lators, any current detection scheme used in quantum N u m be r N u m be r A cc u r a cy / P r e c i s i on (a)(b) FIG. 1. (Color online) (a) Plot of number versus time (indimensionless units) for a single measurement run, compar-ing the direct integration of the master equation to the NPWand P + methods. Simulations had an initial condition of a co-herent state with an amplitude of 10. Numerical integrationof the master equation (2) is plotted with a solid green line,the NPW (7) is plotted with a dashed red line and P + (4)is plotted with a dot-dashed blue line. Uncertainty in bothstochastic methods is plotted with dotted lines. The P + be-comes divergent around t = 0 .
15, and is not plotted beyondthis point. A close-up view of this divergence is shown in theinset. Part (b) shows the accuracy, defined as the differencebetween each stochastic result and the master equation so-lution, and the precision, defined as the standard deviationof the averages, for each stochastic method on a logarithmicscale. The NPW accuracy is described with a solid red line,the NPW precision is a dashed red line, the P + accuracy is adash-dotted blue line, and the P + precision is a dotted blueline. The NPW is considerably more precise and is convergentfor at least an order of magnitude longer than the competingP + representation. The numerical integration was performedby using the open source software package XMDS [35]. gases is of a form that is suited to the NPW method.The NPW method described in this letter is also theonly simulation tool that is deterministic for the strongnumber-conserving nonlinearities that are present in suchsystems, and is therefore the only suitable candidate forsimulating conditional states of ultracold atomic gasesfor feedback or state estimation.This research was supported under Australian Re-search Council’s Discovery Projects funding scheme(project number DP0556073) and the Australian Re-search Council Centre of Excellence for Quantum-Atom Optics (ACQAO). We acknowledge the use of CPU timeat the National Computational Infrastructure NationalFacility and thank Graham Dennis for his help with sim-ulations. [1] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev.Lett. , 010401 (2006).[2] D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton,J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, andD. J. Wineland, Science , 1476 (2004).[3] M. A. Nielsen and I. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,2000).[4] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev.Mod. Phys. , 145 (2002).[5] H. Mabuchi and A. C. Doherty, Science , 1372 (2002).[6] D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya,and S. Habib, Phys. Rev. Lett. , 223004 (2004).[7] V. Belavkin, Automatica and Remote Control , 178(1983).[8] H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. ,548 (1993).[9] A. C. Doherty and K. Jacobs, Phys. Rev. A , 2700(1999).[10] J. K. S. Ramon van Handel and H. Mabuchi, Journalof Optics B: Quantum and Semiclassical Optics , S179(2005).[11] S. S. Szigeti, M. R. Hush, A. R. R. Carvalho, and J. J.Hope, Phys. Rev. A , 013614 (2009).[12] S. S. Szigeti, M. R. Hush, A. R. R. Carvalho, and J. J.Hope, Phys. Rev. A , 043632 (2010).[13] H. M. Wiseman and L. K. Thomsen, Phys. Rev. Lett. , 1143 (2001).[14] C. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 1983).[15] C. Gardiner,