Multi-time correlations in the positive-P, Q, and doubled phase-space representations

Piotr Deuar

Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

A number of physically intuitive results for the in quantum information science (see [58] and [129] for calculation of multi-time correlations in phase- recent reviews) and for the simulation of large-scale space representations of quantum mechanics are quantum dynamics. Negativity of Wigner, P, and other obtained. They relate time-dependent stochastic phase-space quasi-distributions is a major criterion for samples to multi-time observables, and rely on quantumness and closely related to contextuality and the presence of derivative-free operator identi- nonlocality in quantum mechanics [58, 65]. The inabil- ties. In particular, expressions for time-ordered ity to interpret the Glauber-Sudarshan P [67, 132] and normal-ordered observables in the positive-P Wigner distributions in terms of a classical probabil- distribution are derived which replace Heisen- ity density is the fundamental benchmark for quan- berg operators with the bare time-dependent tum light [88, 128]. Quasi-probability representations stochastic variables, confirming extension of ear- arise naturally when looking for hidden variable de- lier such results for the Glauber-Sudarshan P. scriptions and an ontological model of quantum theory Analogous expressions are found for the anti- [16, 58, 93, 127]. Wigner and P-distribution negativ- normal-ordered case of the doubled phase-space ity are considered a resource for quantum computation Q representation, along with conversion rules [8, 58] and closely related to magic state distillation among doubled phase-space s-ordered represen- and quantum advantage [138]; the Glauber-Sudarshan tations. The latter are then shown to be read- P distribution defines a hierarchy of nonclassicalities ily exploited to further calculate anti-normal and nonclassicality witnesses [88, 139] and can be ex- and mixed-ordered multi-time observables in the perimentally reconstructed [3, 83]. Moreover, P, Q and positive-P, Wigner, and doubled-Wigner repre- Wigner distributions and have been used for simulations sentations. Which mixed-order observables are of models important for quantum information topics amenable and which are not is indicated, and such as open spin-qubit systems [136] and boson sam- explicit tallies are given up to 4th order. Over- pling [53, 112, 113]. all, the theory of quantum multi-time observ- For simulation of quantum mechanics, the prime ad- ables in phase-space representations is extended, vantage of the phase-space approach is that its com- allowing non-perturbative treatment of many putational cost typically grows only linearly with sys- cases. The accuracy, usability, and scalability of tem size even in interacting systems of many particles. the results to large systems is demonstrated us- Therefore it provides a route to the non-perturbative ing stochastic simulations of the unconventional treatment of the quantum dynamics of large systems. photon blockade system and a related Bose- It has been used for solving and simulating a multitude Hubbard chain. In addition, a robust but simple of problems in various physical fields: e.g. quantum op- algorithm for integration of stochastic equations arXiv:2011.10107v2 [quant-ph] 6 May 2021 tics [21, 28, 30, 32, 52, 54, 84, 91, 110], ultracold atoms for phase-space samples is provided. [22, 39, 40, 49, 74, 82, 94, 95, 101, 102, 108, 125, 131, 134, 144], fermionic systems [6,7, 27, 29], spin systems [11, 106, 107], nuclear physics [137], dissipative systems 1 Introduction in condensed matter [26, 45, 143], or cosmology [111]. The vast majority of such calculations so far have con- Phase-space representations of quantum mechanics such sidered only equal time correlations and observables. as the Wigner, P, positive-P, Q and related approaches are a powerful tool for the study and understanding of Multi-time correlations are also important for an- quantum mechanics [14, 50, 62, 93, 118]. Their use in swering many physical questions which cannot neces- recent times has been directed particularly as a tool sarily be dealt with by monitoring the time depen- dence of equal time observables [13, 62, 92, 118, 139]. Piotr Deuar: [email protected] For example, the determination of lifetimes in an equi-

1 librium or stationary state, response functions, out-of- Therefore, after basic background in Sec.2, the first time-order correlations (OTOCs), or finding the time order of business in this paper in Sec.3 is to extend the resolution required to observe a transient phenomenon. long known Glauber-Sudarshan P results on time- and However their calculation in the phase-space framework normal-ordered multi-time correlations to the positive- has been restricted because few general results have P representation. This is sort of an obvious extension, been available. What is known is largely restricted to but was missing from the literature, and allows complete time- and normal-ordered correlations in the Glauber- quantum calculations. It also lets us explain the ins and Sudarshan P representation[5, 62], time-symmetric ones outs and set the stage for the less obvious and broader in the truncated Wigner representation [13, 115], or the extensions that follow: corresponding linear response corrections for closed sys- The first one of those in Sec. 4.1 considers general tems. The Glauber-Sudarshan P representation results representations and reformulates the case of Q repre- provide a particularly intuitive framework, simply re- sentations in an operational form that can be extended † placing Heisenberg operators ba (t) and ba(t) with time- to doubled phase-space. This gives stochastic access to † dependent phase-space variables in normal-ordered ob- anti-normal ordered observables such as a(t1)a (t2). It † b b servables such as ba (t1)ba(t2). They also apply to open is then shown in Sec.5 how the Gaussian convolu- systems [2, 62]. Such ordering corresponds to general tion relationship between Wigner, P, and Q distribu- photon counting measurements [67, 68, 81]. Still, while tions allows one to easily evaluate anti-normal ordered scaling with system size is excellent, both above ap- correlations in Wigner and P. This is extended also proaches usually end up being only approximate be- to a wide range of mixed ordered observables such as † † cause one is forced to omit part of the full quantum me- ba (t1)ba(t2)ba (t3)ba(t4) in Sec. 5.4, and further extended chanics in their numerical implementations [62]. Hence, to cover doubled phase-space representations in Sec.6. a broader application of full quantum phase-space meth- In Sec.7, the use and accuracy of this approach is ods to multi-time observables would be advantageous. demonstrated on the numerical example of the uncon- This is especially so, since recent years have shown a ventional photon blockade, a system that may have ap- lot of interest in such quantities, be it in the study of plication to the creation of single-photon sources. Fi- nonclassicality in the time dimension [9, 89, 104, 119], nally, AppendixA gives details of a robust algorithm for time crystals [55, 135, 142, 145], quantum technologies integrating the resulting phase-space stochastic equa- [78, 96, 103] where time correlations and time resolu- tions, a matter that has been somewhat neglected in tion are crucial, methods development [86, 115], and the literature to date. the OTOCs that have recently found to be important for the study of quantum scrambling [15, 64, 133], quan- tum chaos [98, 124] and many-body localisation [56, 70]. 2 Background Therefore, this paper sets out to extend the infrastruc- ture available for multi-time observables with phase- 2.1 Positive-P representation space methods. Consider an M-mode system (modes 1,...,j,...,M) The positive-P representation [50] does not suffer with bosonic annihilation operators baj for the jth mode. from the limitations of the Glauber-Sudarshan P (seen The density operator of the system is written in terms of e.g. in Sec. 7.3) due to its anchoring in doubled phase bosonic coherent states of complex amplitude αj defined space. This allows it to incorporate the full quantum relative to the vacuum: mechanics for most Hamiltonians, including all two- 2 † −|αj | /2 αj a body interactions. Importantly, as we shall see, the same αj j = e e bj vac , (1) | i | i applies for its differently ordered analogues like the doubled-Wigner and doubled-Q. While the positive-P using [50]: is known to be limited to short times for closed sys- Z 4M tems due to a noise amplification instability [38, 66], ρb = d λ P+(λ) Λ(b λ) (2a) this abates in dissipative systems [45, 66]. Recent work ∗ has shown that a very broad range of driven dissipa- O αj j βj j Λ(b λ) = | ∗i h | . (2b) tive Bose-Hubbard models can be simulated with the β j αj j j h j | | i positive-P into and beyond the stationary state [45]. This covers systems of current interest such as micropil- ∗ The kernel Λb involves “ket” αj j and “bra” βj j states lars, transmon qubits, and includes strong quantum ef- described by separate and independent| i variables.h | Let us fects such two-photon interference in the unconventional define the container variable photon blockade [10, 96], making its extension quite timely. λ = α , . . . , α , β , . . . , β = λ (3) { 1 M 1 M } { µ}

2 as a shorthand to hold the full configuration informa- n 2, in which case this is a Fokker-Planck equation max ¬ tion. Individual complex variables λµ can be indexed (FPE): by µ = 1,..., 2M. The positive-P distribution P+ is ( 2 ) guaranteed real and non-negative [50]. This is essential ∂P+ X ∂ X ∂ Dµν (λ) for the utility of the method, which lies in its ability = [ Aµ(λ)] + P+. ∂t ∂λµ − ∂λµ∂λν 2 to represent full quantum mechanics using stochastic µ µν (8) trajectories of the samples λ of the distribution P . + An exception occurs if irreducible higher order partial Notably, the size of these samples scales merely linearly derivatives appear, e.g. due to explicit three-body inter- with M, allowing for the simulation of the full quan- actions in the Hamiltonian. Most first principles models tum dynamics of systems with even millions of modes use only two-particle interactions, though. [39, 82]. Standard methods convert an FPE like (8) to the fol- The distribution (2) can be compared to the simpler lowing Ito stochastic equations of the samples [62]: and more widely known Glauber-Sudarshan P represen- tation [67, 132], which uses a single set of coherent state dλµ X = A (λ) + B (λ)ξ (t) (9) amplitudes and ties the “bra” and ”ket” amplitudes to dt µ µσ σ be equal. It is written σ Z where 2M O X ρ = d α P (α) αj j αj j, (4) D = B B (10) b | i h | µν µσ νσ j σ T where the multimode coherent amplitude is α = i.e. D = BB in matrix notation, and ξσ(t) are inde- pendent real white noises with variances α1, . . . , αM . This representation gives a well behaved representation{ } of only a subset of possible quantum ξ (t)ξ (t0) = δ(t t0)δ . (11) states [20, 128], though a very useful one that includes h µ ν istoch − µν Gaussian density operators [4]. The notation indicates a stochastic average over For full computational utility of the positive-P stoch the samples.h·i The equations (9) give us quantum me- method, all quantum mechanical actions should be able chanical evolution in terms of the samples, which be- to be written in terms of the samples λ. Central to this comes ever more exact as the number of samples grows. are the following identities: Quantum expectation values at a given time are evalu- ated via bajΛb = αjΛb, (5a)   † † † ∂ aj aj ak1 akM = βj1 βjN αk1 αkM stoch. bajΛb = βj + Λb, (5b) hb 1 ··· b N b ··· b i h ··· ··· i ∂αj (12)  ∂  The equivalence can be written more explicitly in terms Λbbaj = αj + Λb, (5c) of individual samples λ(u) labelled by u = 1,..., as: ∂βj S S † † † Λba = βjΛb. (5d) a a ak ak (13) bj hbj1 ··· bjN b 1 ··· b M i 1 P (u) (u) (u) (u) A general master equation for the density matrix = limS→∞ u βj1 βjN αk αk . S ··· 1 ··· M written in Lindblad form with reservoir operators R bn The mode labels in the above can be in any combina- is tion, provided all annihilation operators are to the right of all creation operators (which is called “normal order- ∂ρb i h i X h † † † i = H,b ρb + 2RbnρbRbn RbnRbnρb ρbRbnRbn . ing”). Any single-time operator can be expressed as a ∂t −~ − − n sum of normally ordered terms like (12). (6) This can be transformed by standard methods [62], using identities (5), to a partial differential equation 2.2 Multi-time averages in open systems (PDE) for P of the following general form: + The situation is much more complicated when the op-

(nmax 2M n ! ) erators in the expectation value are not evaluated all ∂P+ X X Y ∂ at the same time. The root of the difficulty is that = Fµ1,...,µn (λ) P+, ∂t ∂λµk n=1 µ1,...,µk k=1 multi-time commutation relations usually depend non- (7) trivially on the full system dynamics, and a reduction with maximum differential order nmax and coefficients of arbitrary multi-time operators to a normal-ordered F that depend on the details of (6). Most cases have form is not generally possible.

3 To describe what is meant at the operator level it is Then, first helpful to introduce the two-sided evolution opera- ˘ Ab (t )Ab (t ) AbN (tN )Bb (s )Bb (s ) BbM(sM) tor V (t1, t2) such that evolution by the master equation h 1 1 2 2 ··· 1 1 2 2 ··· i (6) can be summarised as h = Tr F˘RV˘ (τR, τR− )F˘R− V˘ (τR− , τR− ) 1 1 1 2 ··· ˘ i ρb(t2) = V (t2, t1)ρb(t1). (14) F˘2V˘ (τ2, τ1)F˘1ρ(τ1)) . (20) ··· b Notably, the evolution operator has the semigroup prop- erty [130] V˘ (t3, t2)V˘ (t2, t1) = V˘ (t3, t1). We will use 2.3 Multi-time correlations in the Glauber- the convention that two-sided operators, indicated by Sudarshan P representation a breve˘, act on everything to their right. Hence Time-ordered correlations for the Glauber-Sudarshan P n o representation and a range of other single-phase space V˘ AB = V˘ AB = VA˘ B. (15) { } 6 representations like the Husimi Q were first studied by Agarwal and Wolf in a Greens function framework [1] Let us also define the time-ordered form via extended to the case of open systems [2] and for Hamil- tonian systems in a formal integral form [5]. Later, Gar- Ab (t )Ab (t ) AbN (tN )Bb (s )Bb (s ) BbM(sM) h 1 1 2 2 ··· 1 1 2 2 ··· i diner [62] used a different more operational approach (16) to derive equivalent expressions for normally and time- where the Abj(t) and Bbj(t) are Heisenberg picture oper- ordered operator averages in the Glauber-Sudarshan P ators, and the times obey representation. These have an intuitive form similar to the single-time stochastic expression (12), as follows: t t ... tN (17a) 1 ¬ 2 ¬ ¬ a† t a† t a s a s s1 s2 ... sM. (17b) bp1 ( 1) bpN ( N )bq1 ( 1) bqM ( M) = (21) ­ ­ ­ h ···∗ ∗ ··· i αp1 (t1) αpN (tN )αq1 (s1) αqM (sM) stoch There are operators Ab with times labelled t , . . . , tN h ··· ··· i N 1 increasing to the right, inward, and operators Bb with provided the times are ordered according to (17). The M times labelled s1, . . . , sM increasing to the left, also in- requirement that the operators in (21) be normally or- ward. The location of the inner “meeting point” is ar- dered is an additional constraint on top of time-ordering bitrary, and the number of Ab or Bb operators can also (17), but one that leads to all operators sorted as they be zero. No particular constraints are imposed on the occur in photo-counting theory [81]. It covers a very Ab and Bb operators, except that they should be single large subset of the potentially physically interesting cor- time quantities. These operators refer to measurements relations. The Gardiner approach is more amenable to made at the respective times τr, when the density oper- extension to doubled phase space and will be used in ator was ρb(τr). Between measurements the state evolves what follows. according to the master equation (6) (i.e. (14)). This ordering can be contrasted with the It has been shown [62] that multi-time correlations time-symmetric ordering for which straight- that correspond to sequences of measurements can al- forward truncated Wigner correspondences ways be written in the above form (16). Therefore this for closed system evolution were found in time ordering is not an arbitrary one, and not particu- [13, 115, 118]. Examples of time-symmetric or- larly restrictive in itself. It has also been shown that a dered quantities are 1  † †  and 2 ba (t2)ba(t1) + ba(t1)ba (t2) general time-ordered correlation function (16) obeying 1  † † † 4 ba(t1)ba(t2)ba (t3) + ba(t2)ba (t3)ba(t1) + ba(t1)ba (t3)ba(t2) (17) can be written in a form that uses the V˘ . To do †  +ba (t3)ba(t2)ba(t1) . so it is necessary to be careful about operator ordering. Following [62], let us order all the times tp and sq in the correlation in sequence from earliest to latest. Let 3 Time ordered moments in the us then rename them so that τr positive-P representation

τ1 τ2 τR−1 τR (18) ¬ ¬ · · · ¬ ¬ 3.1 Normally ordered observables with = + . We also define the corresponding To derive an expression like (21) for the positive-P R N M ˘ two-sided operators Fr representation, we will follow Gardiner’s approach [62] ( that was previously used for the Glauber-Sudarshan P. ˘ ρbAbp if τr = tp Smaller steps will, however, be taken here to draw at- Frρb = (19) Bbqρb if τr = sq tention to a few subtleties that will be necessary later.

4 3.1.1 First order correlation function later interpretation as part of a joint probability, while the kernel is expressed in the variables λ that ac- First consider the correlation Λb company later times, ready for application of the next (1) 0 † 0 operator identity. G (t , t) = baj(t )bak(t) . (22) h i Notice that there are no particularly stringent as- † Comparing to (20), we can identify Ab1 = baj, Bb1 = sumptions about P+ for (27) itself to apply. For ex- 0 bak, and the times t1 = t , s1 = t. For this low order ample, (27) applies equally well if one replaces P+ with correlation, the time ordering (17) sets no additional some complex distribution function Pe. This point will conditions. There are two possibilities for τ1, depending soon be useful. However, there was an assumption that on whether time t or t0 is later. Consider first t0 t, so the propagator is well behaved. This is certainly 0 ­ P that τ1 = t, τ2 = t . Using (20) and (19) we have that true if the PDE was of a Fokker-Planck form (8), and h i therefore is always justified if we have an exact map- a†(t0)a (t) = Tr V˘ (t0, t) a (t)ρ(t) a†(t0) (23) hbj bk i {bk b } bj ping of the master equation (6) to stochastic equations h i (9). However, in some other cases of third/higher order = Tr a†(t0)V˘ (t0, t) a (t)ρ(t) (24) bj {bk b } terms in the PDE, it might not. We will not be con- The 2nd line follows from the cyclic property of traces. cerned with such cases here. The is kept for now for clarity. We can see that the Now in (26), V˘ is acting on a distribution Pe(λ, t) = {·} evolution operator from t to t0 acts to the right on all αkP+(λ, t). Using (27) we get the quantities at time t, while the later-time operator h † i † (1) 0 R 4M R 4M 0 0 G = Tr aj(t ) d λ d λ (λ, t λ, t)Pe(λ, t) Λ(b λ) baj(t ) acts only on the evolved quantities to its right. b P | h i This makes intuitive physical sense. In the second case † 0 R 4M R 4M 0 0 0 = Tr aj(t ) d λ d λ αk (λ, t λ, t)P+(λ, t) Λ(b λ) . of t < t, we have τ1 = t , τ2 = t, and get that b P | h n oi (28) † 0 ˘ 0 0 † 0 baj(t )bak(t) = Tr bak(t)V (t, t ) ρb(t )baj(t ) . (25) h i The two-way operator acting on the right that was V˘ , This again has the intuitive form of the operator V˘ act- has now been gotten rid of, by virtue of being incor- ing to the right on all the earlier-time quantities. porated in the propagator . Therefore, the remaining † 0 P Take the first case with t t. Expressing the density operator baj can now be shifted to the right due to the matrix in (24) in the positive-P­ representation (2a), cyclic property of the trace, and then processed via (5d) as so:  Z  (1) † 0 0 4M G = Tr a (t )V˘ (t , t) d λ P (λ, t) a (t)Λ(b λ) R M R M 0 h † 0 i bj + bk G(1) = d4 λ d4 λ (λ, t λ, t)Pe(λ, t) Tr Λ(b λ)a (t ) P | bj  Z  h i † 0 0 4M R 4M R 4M 0 = Tr a (t )V˘ (t , t) d λ α P (λ, t) Λ(b λ) . (26) = d λ βj d λ αk (λ, t λ, t)P+(λ, t) Tr Λ(b λ) bj k + P | = R d4M λ β R d4M λ α (λ, t0 λ, t)P (λ, t). (29a) The 2nd line follows from application of (5a). We cannot j kP | + do the same for † 0 yet, because the kernel finds h i baj(t ) Λb The last line follows from Tr Λb = 1, which is pre-set ˘ itself inside the prior action of the V operator. by the definition (2b). To deal with this, consider now the action of the evo- The quantity P+ is the just the joint probability lution operator V˘ on distributions P+. i.e. the action of P the PDE (7). If we define the conditional distribution 0 0 P (λ, t ; λ, t) = (λ, t λ, t)P+(λ, t) (30) (λ, t0 λ, t) as the solution of this PDE at time t0 t P | P | ­ starting from the initial condition δ4M (λ λ), i.e. the of having configuration λ at time t and configuration λ “propagator”, then it can be used to formally− write at time t0. (Provided is well behaved, positive, real, P Z  as mentioned before, which is the case for any model 0 4M V˘ (t , t) d λ P+(λ, t)Λ(b λ) fully described by an FPE (8)). Therefore, Z Z  Z Z 0 4M 4M 4M (1) 4M 4M 0 = V˘ (t , t) d λ d λ δ (λ λ) P (λ, t)Λ(b λ) G = d λ d λ βjα P (λ, t ; λ, t). (31) − + k Z Z = d4M λ d4M λ (λ, t0 λ, t)P (λ, t)Λ(b λ). (27) At this stage we can identify probability with stochas- P | + tic realisations. The noises ξµ(t) introduced during time This now contains no more two-sided operators. evolution are independent from each other, indepen- Through this convolution, P+ is expressed in λ vari- dent at each time step, independent for each sam- ables which accompany the earliest time t to aid for ple’s trajectory. They are also independent of any other

5 random variables used to produce the initial ensem- Working similarly, using just the 1st and 4th identities ble λ(1),..., λ(S) (t) that samples P (λ, t). Therefore, in (5), one readily but somewhat cumbersomely finds { } + the evolved configuration λ(u)(t0) at time t0 depends that the stochastic estimator for the general time-and- only on mutually independent random variables that normal-ordered correlation function is consist of λ(u) and the noise history of the th tra- (t) u † † a (t ) a (tN )a (s ) a (sM) jectory. As a result, the combination of initial config- hbp1 1 ··· bpN bq1 1 ··· bqM i (u) (u) 0 uration λ (t) and the evolved configuration λ (t ), = βp1 (t1) βpN (tN )αq1 (s1) αqM (sM) stoch. (39) together form an unbiased sample of the joint distri- h ··· ··· i bution P (λ, t0; λ, t). We arrive then at the final result Here, of course (17) must hold, and the stochastic aver- that aging is over products constructed using values from the evolution of a single sample, as in (34). It con- (1) 0 † 0 0 firms the suspicion and intuition that the behaviour of G (t , t) = baj(t )bak(t) = βj(t )αk(t) stoch. (32) h i h i the positive-P representation in this regard should be 0 The procedure for the case t < t gives a result analo- similar to the earlier expression (21), for the Glauber- gous to (31): Sudarshan P. Z Z G(1) = d4M λ d4M λ β α P (λ, t; λ, t0), (33) j k 3.2 Other ordering in the positive-P which once again leads to (32). Now to see the limitations of this scheme, consider the It is especially important to note that the quantity to anti-normally (but time-ordered) ordered correlation be averaged comes from the time evolution of individual † sample trajectories. Explicitly: = a (t0)a (t) (40) A hbj bk i S with 0 . The first point to make is that we † 0 1 X (u) 0 (u) t > t a (t )ak(t) = lim β (t )α (t). (34) hbj b i S→∞ j k cannot rearrange this to a normal-ordered form like u=1 † 0 S a (t)aj(t ) + δjk and then use (39) because generally hbk b i i This allows for very efficient calculations. a (t0), a† (t) = δ when t = t0. Instead it is some bj bk 6 jk 6 time-dependent operator. Now applying (20), (40) can 3.1.2 Higher order correlations be written

Other time and normal ordered correlations follow a 0 † h 0 0 n † oi 00 0 = a (t )a (t) = Tr a (t )V˘ (t , t) a (t)ρ(t) .(41) similar pattern. For example, when t t t, A hbj bk i bj bk b ­ ­ 00 0 baj(t )bak(t )bal(t) (35) Upon expansion, we will need to act on Λb using the h h i n oi † 00 ˘ 00 0 0 ˘ 0 2nd identity in (5), (5b), to convert a to variable form. = Tr baj(t )V (t , t ) bak(t )V (t , t) bal(t)ρb(t) . bk { } Thus In this case, following a similar procedure to before, one " ( can act alternately with (5a) on the kernel to extract a 0 ˘ 0 = Tr baj(t )V (t , t) factor of α, and (27) to convert the evolution operators A to propagators. One finds Z   )# 4M ∂ Z d λ P+(λ, t) βk + Λ(b λ) . (42) a (t00)a (t0)a (t) = d4M λd4M λd4M λ α α α ∂αk hbj bk bl i j k l 00 0 0 (λ, t λ, t ) (λ, t λ, t)P+(λ, t). (36) This is not of a form amenable to (27). We can soldier on ×P | P | applying integration by parts and assuming negligible 00 0 Since the times are ordered t t t, and the con- boundary terms to obtain ditional probabilities follow from­ the­ evolution of the FPE, λ are parent variables of the λ and so on, and " ( = Tr a (t0)V˘ (t0, t) (43) the product of conditional probabilities is just the joint A bj probability. Hence (36) is )# Z  ∂P (λ, t) Z d4M λ β P (λ, t) + Λ(b λ) . d4M λd4M λd4M λ α α α P λ, t00 λ, t0 λ, t , k + − ∂α j k l ( ; ; ) (37) k

and The matter of whether boundary terms can be discarded has been studied in depth [23, 34, 37, 66, 85, 126]. The a (t00)a (t0)a (t) = α (t00)α (t0)α (t) . (38) summary is that one can determine operationally in a hbj bk bl i h j k l istoch

6 stochastic simulation whether boundary terms are neg- Notably, while the trace of the symmetric form in the ligible or not. If deemed negligible, then integration by Wigner representation corresponds to α 2 with no cor- h i | | parts is justified. In (43) we can now identify a distri- 1 † †
2 rections: Tr a a + aa ΛbWig = α , this does not ∂ 2 b b bb | | bution Pe(λ, t) = [βk ]P+(λ, t) to act on with (27). 2 ∂αk remove derivatives in the corresponding identity . The Doing so gives: − best that appears to be achievable in this way is " ZZ 0 4M 4M † †  2  = Tr aj(t ) d λ d λ (44) ba ba + baba 2 1 ∂ A b ΛbWig = α + ΛbWig. (46) 2 | | 4 ∂α∂α∗ #  ∂P (λ, t)  λ, t0 λ, t β P λ, t + λ . Hence, the path-integral and time-symmetric approach ( ) k +( ) Λ(b ) P | − ∂αk [13] is more suited to the Wigner representation. Also the phase-space representations developed for spin sys- 0 The second operator baj(t ) can now act on Λb from the tems [11, 100, 107], contain derivatives for all identities. left. One obtains One notable exception is the Q representation, which ZZ admits derivative-free identities similar to (5) for anti- = d4M λ d4M λ (45) A normally ordered operators, and so is a good candidate for convenient phase-space expressions. Time-ordered  ∂P (λ, t)  (λ, t0 λ, t)α β P (λ, t) + . anti-normal operators occur for example in the theory of j k + ∂α P | − k photon detectors that operate via emission rather than While this is formally acceptable (given those negligible absorption of photons [99]. Formal integral expressions boundary terms), and could be used for some analytic for this kind of correlation were also provided in [5]. work in small systems such as demonstrated in [62], Below, the operational stochastic expressions are found using the Gardiner approach. unfortunately the derivative of P+ is not amenable to interpretation in terms of stochastic samples. At least not the direct samples we are investigating here. It may 4.1 The case of the Q representation be partially treatable using the quantum jump and re- sponse theory approach previously applied to truncated The Husimi Q representation [75] is defined as Wigner [13, 115, 118], which is a topic for another time. 1 However – in Sections5 and6, a different direct way α α α Q( ) = M ρb (47) to evaluate anti-normal ordered observables such as = π h | | i 0 † A aj(t )a (t) will be demonstrated. and is positive for any . Due to the Q distribution being hb bk i ρb defined in this explicit way, rather than the implicit form (2), observable expressions in the Q distribution 4 Other phase-space representations have traditionally been found by simply expanding the trace: In the derivations of Sec. 3.1, one can see that the cru- cial aspect for obtaining a stochastically useful expres- h i 1 Z Tr Ob ρ = d2M α α Ob ρ α , (48) sion is to use only those identities which do not contain b πM h | b| i derivatives1. This suggests that convenient expressions for multi-time correlations similar to (39) will be ob- and applying the eigenvalue equation for coherent states tainable whenever the operators in the correlation can a α = α α . (49) be converted to phase-space variables without resorting bj| i j| i to identities with derivatives. For anti-normal ordered correlations at equal times, the However, it happens that such identities without cyclic property of traces gives a a a† a† = bj1 bjN bk1 bkM derivatives are not particularly abundant in other h i h ··· ··· i phase-space representations. For example, the Wigner Tr a† a† ρ a a which immediately leads to bk1 bkM b bj1 bjN representation [105, 118, 141] has derivatives in all iden- ··· ··· tities, as does its dimension-doubled analogue [71, 116]. † † ∗ ∗ aj aj a a = αj αj α α . hb 1 ··· b N bk1 ··· bkM i h 1 ··· N k1 ··· kM istoch 1 (50) Strictly speaking, a small exception to this appears if the However, this traditional approach fails with time- derivative appears only at the final time when the only remain- ing operator is Λ since Tr[ ∂ Λ] = ∂ Tr[Λ] = 0 removes any ordered correlations. Whatever way one orders the op- b ∂λµ b ∂λµ b awkward terms. Such a case can, however, also be treated by an erators, working this way on expression (20) will lead identity without any derivatives after using the cyclic property of 2 the trace at the right step. Here, ΛbWig = Λbs from (53) with s = 0.

7 to a form which can be verified by equating the LHS and RHS " when Tb(0, s) is expanded in number states. Therefore, 1 Z − d2M α α F˘ (51) in the limit s 1 corresponding to the Q representa- πM h | R → − tion with Λb−1 = ΛbQ, # n  o " # V˘ (τ , τ − ) F˘ − V˘ (τ − , τ − ) F˘ ρ(τ ) α R R 1 R 1 R 1 R 2 ··· 1 b 1 | i ∂ bajΛbQ = αj ∗ ΛbQ, (56a) − ∂αj ˘ in which it is the latest time operator FR(τR) that † ∗ acts on the outer coherent states. One can convert bajΛbQ = αj ΛbQ, (56b) F˘ to phase-space variables using (49), and move the R ΛbQbaj = αjΛbQ, (56c) state vectors α or α closer to the density matrix   † ∂ and the form (|47i). However,h | in the next putative step, Λb a = α∗ Λb . (56d) Qbj j − ∂α Q there is no clear way to convert the evolution opera- j ˘ tor V (τR, τR−1) to phase space form. Moreover, when We see then that multi-time correlations in which only there is phase-space diffusion, the propagator is well- † P the orderings bajΛbQ and ΛbQbaj appear could have the behaved only in the forward time direction, so there capacity to correspond to simple stochastic expressions. is no way to act with on the outer state vectors and P This implies anti-normal ordered moments since by the variables which correspond to later times than the inner cyclic property of traces ones.  † †  † † Therefore, we proceed in a non-traditional way, us- Tr a a ρ a a = a a a a . bq1 ··· bqM bbp1 ··· bpN hbp1 ··· bpN bq1 ··· bqM i ing an implicit form similar to what was done for the (57) positive-P distribution in Sec. 3.1. The Q representation The implicit form (53) for ρb allows us to proceed in is the s 1 limiting case of the family of s-ordered → − a similar fashion to Sec. 3.1. Consider then the anti- representations Ws(α) studied by Cahill and Glauber normal and time-ordered correlation [19, 20] (the Glauber-Sudarshan P and Wigner corre- spond to s = 1 and s = 0, respectively). All these dis- 0 a t a t a† s a† s = bp1 ( 1) bpN ( N )bq1 ( 1) bqM ( M) (58) tributions can be written using coherent displacement A h ··· ··· i operators with times obeying (17). A potentially awkward † ∗ issue is that the kernel Λ is not very well αj a −αj aj Y bs Dbj(αj) = e bj b , ; Db(α) = Dbj(αj), (52) bounded in the s 1 limit, with β Λbs(α) β = j  → − h | | i 2 exp 2 α β 2/ε where s = ε 1. To gauge whether ε − | − | − in an implicit form similar to (2): this is a problem, we will work using infinitesimal ε > 0 Z in which case 2M ρb = d α Ws(α)Λbs(α) (53a) Λb Λb + (ε). (59) s → Q O Y † Λbs(α) = Dbj(αj)Tbj(0, s)Dbj( αj), (53b) Using the form (20) on (57) with now B = a and A = a, − − b b j one obtains with the base operator [20] Z " ( 0 2M α ˘ ˘ † = (ε) + d Tr FRV (τR, τR−1) (60) a aj A O 2 s 1bj b Tbj(0, s) = − . (54) )# − 1 + s 1 + s n  o F˘ V˘ (τ , τ ) F˘ V˘ (τ , τ ) W (α, τ )F˘ Λb (α) . h i ··· 3 3 2 2 2 1 s 1 1 Q One has Tr Λbs = 1 and the operator identities " # The part in the inner brackets will be either 1 s ∂ {} ajΛbs = αj − Λbs, (55a) b − 2 ∂α∗ j W (α, t )Λb a if t sM s 1 Qbp1 1 ¬ (61)   W α, s a† otherwise. † ∗ 1 + s ∂ s( M)bqM ΛbQ bajΛbs = αj + Λbs, (55b) 2 ∂αj " # Both cases allow us to use derivative-free identities (56c) 1 + s ∂ or (56b), to replace (61) with P (α, τ )Λ where Λbsbaj = αj + ∗ Λbs, (55c) es 1 bQ 2 ∂αj   1 s ∂ Pes = αp1 Ws(α, sM) if t1 sM a† α∗ , ¬ (62) Λbsbj = j − Λbs (55d) ∗ α otherwise. − 2 ∂αj Pes = αqM Ws( , t1)

8 Now assuming an acceptable propagator (α, τ α, τ ) convolution of the more normally-ordered distribution Ps 2| 1 exists for the evolution of the s-ordered distribution Ws, Ws0 , and reflects the well known property that Q dis- use of (27) leads to tributions (s = 1) are broader and more smoothed than Wigner ( − ), which are in turn broader than Z Z " ( s = 0 0 2M 2M Glauber-Sudarshan P distributions (s = 1) for the same = (ε) + d α d α Tr F˘RV˘ (τR, τR−1) A O state. Importantly for us here, this means that if we )# have samples α of a more normally ordered distribu- n o 0 F˘3V˘ (τ3, τ2) s(α, τ2 α, τ1)Pes(α, τ1)F˘2ΛbQ , (63) tion, we can easily also obtain samples α of the less ··· P | normally ordered distributions simply by adding Gaus- in which the objects related to the first time inter- sian noise. The prescription is val have been fully converted to phase-space quantities. r 0 s0 s Proceeding in this fashion with increasing time for the α = αj + − ζj (67) j 2 remaining a and a† operators, one arrives at b b for each mode j, with each ζ a complex random variable  Z Z Z j 0 2M 2M 2M of variance 1: = d α(τ1) d α(τ2) d α(τR) A ··· ∗ ∗ ∗ ζj stoch = 0 ; ζjζk stoch = 0 ; ζj ζk stoch = 1. (68) αp1 (t1) αpN (tN )αq1 (s1) αqM (sM) h i h i h i ··· ··· × In particular, converting samples of a P distribution to s(α(τR), τR α(τR−1), τR−1) s(α(τ2), τ2 α(τ1), τ1) P | ···P  | samples of Q requires Ws(α(τ1), τ1) + (ε). (64) 0 × O αj = αj + ζj. (69)

Variables α, α etc were relabelled to α(τR), α(τR−1). Converting samples of a Wigner distribution (if it is In the limit ε 0 that we are considering, W (α) nonnegative to begin with) to samples of Q can be done → s → Q(α), which is real non-negative. To be consistent, the with ζj propagator − must also be real nonnegative. When 0 Wig 1 αj = αj + . (70) the masterP equation (6) is faithfully reproduced by √2 a Fokker-Planck equation for Q(α), this will be the case. Then, the s ... sWs factors can be interpreted 5.2 Evaluation of anti-normal ordered moments similarly to (30)P as theP joint probability of samples starting from P and Wigner representations α(τ ), α(τ ),... at successive times. With that, we ar- 1 2 Therefore, if one has samples α of a P or Wigner rep- rive at the hoped for result that a time ordered (as (17)) (t0) resentation up to a time (e.g. from a prior stochastic and anti-normally ordered correlation is evaluated as t0 evolution), the prescription (67) can be used to con- a t a t a† s a† s vert them to samples α0 of the Q representation at bp1 ( 1) bpN ( N )bq1 ( 1) bqM ( M) (t0) h ··· ···∗ ∗i that time. The noises ζj are generated just once at this = αp1 (t1) αpN (tN )αq (s1) αq (sM) stoch (65) h ··· 1 ··· M i time. Subsequent evolution according to the Q stochas- using Q representation samples αj. Stochastic averag- tic equations then leads to Q samples α0(t) at later ing is over products constructed using values from the times t > t0. These can be directly used in expression evolution of a single sample, like in (34). (65) to evaluate anti-normal ordered multi-time observ- ables. This is potentially a little less convenient that remain- 5 Evaluation by conversion to the Q rep- ing in one distribution throughout because the conver- resentation sion time t0 has to be chosen before starting a simu- lation, and the anti-normal ordered operators cannot 5.1 Conversion between samples of s-ordered extend to times before t0. It does preserve the principal distributions advantages of phase-space simulation, though: intuitive and computationally tractable expressions for observ- The s-ordered distributions Ws introduced in (53) are ables, stochasticity, gentle scaling with system size. The mutually related by [20] evolution equations (9) are usually of similar form in all 0 Ws(α ) (66) s-ordered representations, apart from simplification at M special s values.  2  Z d2M α  2 α0 α 2  = exp | − | W (α). One cannot convert the other way with this procedure s s πM − s s s0 0 − 0 − (e.g. from Wigner to P), so normally ordered multi-time in the sense that Ws and Ws0 represent the same quan- correlations cannot be extracted this way from samples tum density matrix ρb. When s0 > s, this is a Gaussian of Wigner or Q distributions.

9 5.3 Mixed-order moments we have The above discussion suggests a way that some mixed- Z " ( )# 00 2M 0 † ˘ 0 0 order multi-time correlations that contain both normal = d α Tr aq V (s1, t2) Pe(α , t2) ΛbQ(α ) . A b 1 and anti-normal factors could be evaluated. Suppose early time operators (t t ) are normally ordered (can (76) 0 Now using (27), and taking care with variable labels: be evaluated using the P¬ distribution), while later time ZZ operators (t > t0) are anti-normally ordered (can be 00 2M 0 2M 0 0 0 0 = d α d α − (α , s α , t )Pe(α , t ) evaluated using the Q representation). A switch ac- A P 1 1| 2 2 cording to (69) could be made at t0 after evaluating h i Tr a† Λb (α0) . (77) any normally-ordered factors, and the Q distribution × bq1 Q evolved and used at later times to obtain the remaining After the variable change caused by (27), now inner factors with t > t0. Let us check in detail whether this is feasible. α0 = α + ζ. (78) Consider the time-ordered but neither anti- or nor- mally ordered correlation Notice also the 1 label on the latest propagator in (77), − 00 = a† (t )a (t )a† (s )a (s ) indicating that it is according to the Q representation A hbp1 1 bp2 2 bq1 1 bq2 2 i Z   equations in the time interval (t2, s2]. Finally, applying d2M α a† V˘ s , t (56b), one arrives at: = Tr bq1 ( 1 2) (71)  Z −|ζ|2 ! n  † o 00 2M 2M 2M 2M e 2M 0 V˘ (t , s ) a V˘ (s , t ) P (α, t )Λb (α)a a = d α d α d α d ζ d α 2 2 bq2 2 1 1 1 bp1 bp2 A πM ∗ 0∗ 0 whose times satisfy (17), and t1 s2 t2 < s1 (for in- α α (αp2 + ζp2 )α −1(α , s1 α + ζ, t2) q2 p1 q1 stance). The initial expansion of ρ¬is made¬ in the P rep- × P | b 1(α, t2 α, s2) 1(α, s2 α, t1)P (α, t1). (79) resentation, where P (α) = W1(α) and Λb1(α) = α α . ×P | P | The first two operators a† , a , and first two V˘ convert| ih | This encodes the following sequence of operations: bp1 bq2 easily to phase-space variables via (5a) and (5d), giving 1. Start with initial samples α at t1. 2. Propagate P representation equations to s2 obtain- ZZZ " ( ing samples α. 00 d2M α d2M α d2M α a† V˘ s , t 3. Propagate P representation equations to t obtain- = Tr bq1 ( 1 2) (72) 2 A ing samples α. )# 4. Add Gaussian noise as per (78), to get samples α0. α α α α α ∗ α 1( , t2 , s2) 1( , s2 , t1)P ( , t1) α αp1 Λb1( )ap2 . 5. Propagate Q representation equations to s1 obtain- P | P | q2 b ing samples α0. Further work by this route is now closed because Along the way, samples are collected to use in the final Λb1bap2 involves (5c) and derivatives. Instead, another result stochastic average. The s factors in (79) together with the Gaussian factor in theP top line form the joint prob- that follows from from [19] allows us to convert kernels: 0 0 ability P (α , s1; α , t2; α, s2; α, t1). Therefore, the final 00 Λbs(α) (73) estimator for the observable is A  M Z 2M 0  0 2  2 d α 2 α α 0 00 † † α = ap (t1)ap2 (t2)aq (s1)aq2 (s2) = (80) = M exp | − | Λbs0 ( ). b 1 b b 1 b s s0 π − s s0 A h ∗ 0 0∗ i − − α (t1)α (t2)α (s1)αq2 (s2) stoch. h p1 p2 q1 i Taking s = 1 and s0 = 1, to move to a Q representa- tion kernel, one finds − Primed variables α0 are samples of the Q distribution, while un-primed ones α are samples of the P distribu- Z d2M ζ α  ζ 2 α0 tion. This all matches intuitively with the evolution and Λb1( ) = M exp ΛbQ( ). (74) π −| | the expectation that anti-normally ordered elements where ζ = α0 α = ζ has exactly the proper- will use Q distribution samples and normally-ordered − { j} ties of the noise in (68), and α0 = α + ζ is given by elements samples of the P distribution. (69). After substituting (74) into (72), applying (56c) to α0 a , and defining the distribution ΛbQ( )bp2 5.4 Most general ordering case 2M Z α 0 2ZZ 0 0 d −|α −α| 2M 2M The widest generalisation of this procedure to other Pe(α , t2) = αp e d α d α (75) 2 πM (time-ordered) cases is as follows: If there is an early ∗ 1(α, t2 α, s2) 1(α, s2 α, t1)P (α, t1) α αp . time set of normally ordered operators, on either side P | P | q2 1

10 of the correlation, it can be dealt with by sampling the Order 1st 2nd 3rd P distribution according to the replacements (number of operators) order order order a α ; a† α∗. (81) Total permutations 2 12 104 bj → j bj → j single time correlations 2 4 8 Once this avenue becomes exhausted, one adds noise via (69) to convert α’s to Q distribution samples α0. If multi-time accessible the remaining later time inner factors are anti-normally with P representation – 4 22 ordered, they can then be dealt with using the replace- additional accessible ment with Q representation – 4 22 a α0 ; a† α0∗. (82) bj → j bj → j additional accessible The above covers both fully normal and fully anti- with mixed order (Sec. 5.4) – – 22 normal ordered products as special cases. On the other Total doable 2 12 74 hand, the case of an early time anti-normally ordered time ordered not doable – – 6 block and a later time normally ordered block contain- ing several times is not amenable to this approach be- Not time ordered, not doable – – 24 cause one cannot stochastically convert Q samples to P Table 1: A tally of ba, ba† product permutations that can/cannot samples. be evaluated with the various approaches discussed. The gen- A large number of cases can also be reduced to a form eral form considered is A(ta)B(tb)C(tc) , where A, B, C can h b b b i b b b amenable to this procedure by the use of the commuta- be either of ba or ba† (same mode), and the time arguments † can take up to three distinct times t < t < t . Permuta- tor [ba, ba ] = 1 for equal time factors. Also, simulations 1 2 3 starting in a Wigner representation can be used for eval- tions with the same time topology (e.g. Ab(t1)Bb(t1)Cb(t2) and uation of outer symmetric ordered parts, and then a Ab(t2)Bb(t2)Cb(t3)) are counted only once. switch can be made to the Q representation via (70) to evaluate any remaining inner anti-normal ordered parts.

5.5 Correlation function coverage time ordered four-operator products of this form can Table1 counts the number of distinct 1, 2, and 3 an- be evaluated, including atypical combinations such as nihilation/creation operator products that can be eval- a(0)a†(τ)a†(τ)a(0) , but very many require the Q rep- hb b b b i uated using the various methodology described above. resentation. (72 out of the 160 accessible ones require All one and two-factor combinations can be evaluated the use of the mid-simulation switching to the Q repre- (though 4/12 of the latter require use of the Q rep- sentation described in Sec. 5.4). The only correlations resentation). For third order correlations, almost all that are inaccessible are the non time ordered ones such time-ordered cases can be evaluated (74 out of 80). as e.g. a†(τ)a†(0)a(0)a(τ) . hb b b b i The majority require use of the Q representation or a shift from a P to Q representation as described in The change of distribution can introduce additional Sec. 5.4 to work. There are 6 exceptions that cannot restrictions. In particular, Wigner and Q distributions † † † † be evaluated: ba (t2)ba (t3)ba (t1) , ba(t2)ba (t3)ba(t1) , have a tendency to involve higher order derivatives in h † i h † i ba(t1)ba(t3)ba(t2) , ba(t1)ba (t3)ba(t2) , ba(t3)ba(t2)ba (t1) , the PDE (7) than P distributions, so that a standard h † † i h i h i ba (t3)ba(t2)ba (t1) , where t1 < t2 < t3 is assumed. diffusion process no longer captures the full quantum Theseh all share thei feature that the earliest factor al- dynamics. An example are two-photon losses with oper- 2 ready requires the Q representation, while the next op- ators Rb = ba for which P distributions produce only 2nd erator in time requires the P representation to which order derivatives but Q distributions 4th order terms, one cannot return. There are also a number of correla- and Wigner distributions 3rd order ones. This precludes tions that are not time ordered, for which the earliest fully accurate calculation of correlation functions such † † ∗ 0 0 0∗ time t1 is on the middle operator, and these are not as ba (0)ba(t)ba(t)ba (0) = α (0)α (t)α (t)α (0) stoch possible to evaluate according to the schemes presented requiringh Q evolutioni afterh the changeover, buti not here. cases where the change to a Q distribution is only The greatest interest in multi-time correlations usu- needed at the final time such as a†(0)a(t)a†(t)a(0) = hb b b b i ally concerns those involving two times (say t = 0 α∗(0)α0(t)α0∗(t)α(0) . Stochastic techniques for h istoch and t = τ > 0). Examples are counting correlations dealing with higher order PDE terms have been investi- † † like ba (0)ba (τ)ba(τ)ba(0) , and pair correlations such as gated [79, 80, 109, 117] particularly in [48] for doubled † h † i ba (0)ba (0)ba(τ)ba(τ) . The case count for these is sum- phase space, though attempts to date have shown strong marisedh by Table2.i A we can see, all 160 kinds of time and stability limitations.

11 h i + Order 2nd 3rd 4th where necessary. The usual properties Tr Λbs = 1 and (number of operators) order order order db−1(α, β) = db( α, β) continue to apply. The operator Total permutations 12 56 240 identities are now− − single time correlations 4 8 16 multi-time accessible   + 1 s ∂ + with P representation 4 14 36 bajΛbs = αj − Λbs , (85a) − 2 ∂βj additional accessible   † + 1 + s ∂ + with Q representation 4 14 36 bajΛbs = βj + Λbs , (85b) 2 ∂αj additional accessible   + 1 + s ∂ + with mixed order (Sec. 5.4) – 12 72 Λbs baj = αj + Λbs , (85c) 2 ∂βj Total doable 12 48 160   + † 1 s ∂ + time ordered not doable – – – Λbs baj = βj − Λbs . (85d) − 2 ∂αj Not time ordered, not doable – 8 80 The s 1 limit of all the above gives the positive-P rep- Table 2: A tally of ba, ba† products involving up to four operators, → evaluated at one of two times. The general form considered is resentation, s = 0 the doubled-Wigner representation of [71], and the limit s 1 a doubled phase space Ab(ta)Bb(tb)Cb(tc)Db(td) , where A,b B,b C,b Db can be either of a h i b analogue to the Q representation→ − (“doubled-Q”). Eqs. or ba† (same mode), and the time arguments can take up to two distinct times t = 0 and t = τ > 0. (85) are equivalent to the correspondences found in [31]. An advantage of the doubled-Q and doubled Wigner representations relative to their single-phase space ana- 6 Q representations and s-ordering in logues is that all 2nd order derivative terms in the PDE doubled phase space can be made positive-definite and converted fully to stochastic equations via the same analytic kernel trick To use the mechanisms described in Sec.5 in the dou- as for the positive-P representation. For example, spon- bled phase-space representations that give more com- taneous emission in a Schwinger boson system need not plete coverage of quantum mechanics, we need also to be amputated like in [73] for the Wigner representation. consider the doubled phase space Q representation and The kernel transform between different orderings in how to switch to it from the positive-P. doubled phase-space is found to be

+ α, β 6.1 Doubled phase space s-ordered representa- Λbs ( ) (86)  2 MZ d2M ζ  2 ζ 2  tions + α ζ, β ζ∗ , = M exp | | Λbs0 ( + + ) s s0 π −s s0 The s-ordered representations were first generalised to − − doubled phase space by de Oliveira [31]. Later studies which is easily verified with the help of [35, 71, 116] used a different normalisation that is closer Z 1 2 to the original single-phase space formulation (52) of 1 2 − s|γ| Tbj(0, s) = d γe 2 Dbj(γ) (87) Cahill and Glauber [19]. It will also be used here. The − π explicit expansion of the density matrix was given in from [19], the easy to show identities [35] as3:

1 ∗ ∗ Z (αγ −α γ) 4M + + Dbj(α)Dbj(γ) = Dbj(α + γ)e 2 (88a) ρb = d λ Ws (λ)Λbs (λ) (83a) 1 0 0 0 0 0 0 αβ −βα 2 ( ) + Y dbj(α, β)dbj(α , β ) = dbj(α + α , β + β )e ,(88b) Λb (λ) = dbj(αj, βj)Tbj(0, s)dbj( αj, βj), (83b) s − − − j and Gaussian integrals. The distribution transform where the displacement-like operator db is + Ws (α, β) (89) α a†−β a ∗  MZ 2M  2  jbj jbj 2 d ζ 2 ζ dbj(αj, βj) = e ; dbj(α, α ) = Dbj(α), (84) W + α ζ, β ζ∗ . = M exp | | s0 ( + + ) s0 s π −s0 s obtained by the replacement α∗ β in (52). To distin- − − guish from single phase space, the→ superscript + is used is readily found by equating two (83a) expansions of ρb which have different s, and applying (86) to the one 3In the supplemental material therein. with higher s (= s0).

12 6.2 Non normally-ordered correlations in the 7 Test example: unconventional photon positive-P representation blockade The ideas from Sec.5 can be used to treat non normally- ordered correlations in the positive-P representation, 7.1 The system which – unlike the Glauber-Sudarshan P – is applicable Consider the two-site Bose-Hubbard Hamiltonian for all quantum states and systems. With the tools for   the doubled s-ordered phase space described in Sec. 6.1, X † U † Hb = a ∆ + a aj aj bj − 2 bjb b derivation of the expressions for mixed-time expectation j=1,2 values follows the same path as in Secs. 5.1 and 5.3, but h i h i +F a† + a J a†a + a†a (95) some care needs to be taken to incorporate the doubled b1 b1 − b1b2 b2b1 phase-space. We obtain that: with standard annihilation operators a for modes j us- (1) To shift samples from a more to a less normally bj ing units = m = 1. This describes two sites with ordered doubled-phase space representation, one adds ~ (real) tunnelling J, local on-site interaction constant U, the following noise: detuning ∆, and a coherent drive F (real) only on the first site. There is also a decay process with rate that r r γ 0 s0 s 0 s0 s ∗ is treated by describing the evolution of the system with α = αj + − ζj ; β = βj + − ζ . (90) j 2 j 2 j the master equation

∗ Notably – the same noise for α and β , which was ∂ρb h i γN X h † † † i = i H,b ρ + 2a ρaj aja ρ ρ ajaj not obvious a priori. The prefactor is for positive- ∂t − b 2 bj bb − b bj b− b b b 1 j P to “doubled”-Q, and 1/√2 for positive-P to doubled- γ(N + 1) X h † † † i Wigner. + 2ajρa a ajρ ρa aj . (96) 2 b bbj − bjb b− bbjb (2) Anti-normal ordered mixed-time correlations are j evaluated as To test the performance a bit more beyond the standard

† † model, we have also addded a thermal bath with mean a (t ) a (tN )a (s ) a (sM) bp1 1 bpN bq1 1 bqM occupation N. Such a system can be realised using e.g. h 0 ··· 0 0··· 0 i = α (t ) α (tN )β (s ) β (sM) . (91) micropillars [69, 123] or transmon qubits [72, 122]. The h p1 1 ··· pN q1 1 ··· qM istoch nontrivial feature here is a two-boson destructive in- where α0 and β0 are doubled-Q representation samples, terference effect between photons injected by the drive, and other photons that have been previously injected, possibly created via (90) (with s0 s = 2) from initial positive-P samples, and later evolved− via the appropri- tunnelled to site 2 and then back, returning with a rel- ate doubled-Q representation evolution equations. ative phase of π [10]. The result of this is that in the (3) For mixed ordering, the procedure in Sec. 5.4 fol- steady state only single photons can be present at the lows with the same structure, except that the correspon- driven site 1, providing possibly an avenue to create dences are single-photon sources [96]. The lack of double occupa- tion is evidenced in a single-time two body correlation †2 2 † 2 function g11 = a a / a a1 , which is very close to a α ; a† β . (92) hb1 b1i hb1b i bj → j bj → j zero. However, for practical application, it is of partic- ular interest to find out how large a time mismatch be- in the positive-P and tween measured photons can be accommodated without (2) significantly increasing g11 from this low level. If it is 0 † 0 baj αj ; baj βj. (93) too short, then the single-photon source will have a too → → short active time for practical applications. Therefore in the doubled-Q. For example, the expression for the correlation function of particular interest is the correlation 00 (80) using samples starting in the A † † positive-P is ba1(t)ba1(t + τ)ba1(t + τ)ba1(t) g11(τ) = h 2 i (97) n1 00 = β (t )α0 (t )β0 (s )α (s ) (94) A h p1 1 p2 2 q1 1 q2 2 istoch in the stationary state, with delay time τ. The mean † occupation is n1 = ba1ba1 . (4) The correlation tallies of Sec. 5.5 apply without The Ito stochastich evolutioni equations in the positive- change to the doubled phase space representations. P(s = 1) and doubled-Q (s = 1) representations are −

13 † where nj(t) = ba (t)ba(t) = Re αj(t)βj(t) stoch is the mean occupationh of site ji. In theh positive-Pi representa- dαj h γ i = iU(αjβj + s 1) + √ isU ξj(t) αj iFj tion the correlation (101) is calculated via dt − − − 2 − − q X sp 1−s
i H αk + γ N + ηj(t) (98a) Re α1(t0)β1(t0)αj(t0 + τ)βj(t0 + τ) − jk 2 g (τ) = h istoch . k 1,j n1(t0)nj(t0 + τ) dβj h γ i (102) = iU(αjβj + s 1) + √isU ξej(t) βj + iFj dt − − 2 We can take the real parts above, because the imaginary q X sp −s
∗ parts must converge to zero in the limit. This +i H β + γ N + 1 η (t). (98b) jk k 2 j behaviour is verified in the simulations.S → ∞ k These were derived in [45] for the positive-P. Here the 2.5 matrix elements of the one-particle Hamiltonian are Hsp = ∆, Hsp = H = J, the drives are F = F , 2 jj − 12 21 − 1 F2 = 0, while ξj and ξej are delta-correlated independent ) 1.5 white real noises with variance τ ( 1 , 1

0 0 g ξ (t)ξ 0 (t ) = δ 0 δ(t t ) (99) 1 h j j i jj − N and ηj are similarly correlated independent complex 0.5 = 0 noises: 0 ∗ 0 0 0 0 1 2 3 4 5 6 η (t)η 0 (t ) = δ 0 δ(t t ); η (t)η 0 (t ) = 0. h j j i jj − h j j i (100) τ Appendix A.3 gives some detail on generation of the 1.2 noise. In some cases, calculations with these stochastic 1 equations can already be faster than brute force calcu- lations directly with the density matrix ρb in a suitably 0.8 truncated number state basis. For large systems, they ) τ

( 0.6 are the only tractable way to access full quantum me- 1 8 , N = 5 10− 1

g × chanics. AppendixA gives details of an integration algo- 0.4 rithm that is robust to the multiplicative noise appear- 8 ing in (98), and was used for the simulations reported 0.2 N = 10− N = 0 here and elsewhere [35, 36, 41–45, 82, 107, 120, 134]. 0

0 0.2 0.4 0.6 0.8 7.2 Two-time photon-photon correlations τ Consider first the strong antibunching parameters stud- Figure 1: Time duration of antibunching in the unconven- ied in [10] and [45]: U = 0.0856, J = 3, ∆ = 0.275, tional photon blockade system, F = 0.01. Top: clean system − γ = 1, F = 0.01. We will study the correlations pri- (N = 0), strong antibunching. Bottom: degradation with back- marily in the stationary state. To this end, a positive-P ground thermal intensity N > 0. Black dashed lines: direct so- simulation is initialised in the vacuum α = β = 0, and lution of the master equation. Triple lines show 1σ statistical uncertainty. evolved up to t = 30. The stationary state is attained after t & 15, and we calculate multi time correlation functions from times t0 = 20 to t0 + τ using the avail- Fig.1 (top) shows the multi-time local photon cor- able samples. ( = 216 in all cases). Uncertainty in pre- relation (101) at site 1 (i.e. j = 1) for the clean S dictions is calculated using sub-ensemble averaging, as case (N = 0), and verifies that the result perfectly explained in Appendix A.4. These errors are shown as matches the exact brute force solution. The desired anti- triple lines in all the plots. correlation dip is seen around τ = 0, along with char- The two-time photon-photon correlations between acteristic oscillations out to delay times of about τ = 5. one photon measured at site 1 at time t0 and the other The bottom panel shows how the anti-correlation dip at site j after a delay time of τ are degrades when the system is linked to a particle reser- voir (growing N). Notably the dip timescale does not † † ba1(t0)baj(t0 + τ)baj(t0 + τ)ba1(t0) change appreciably as the minimum correlation rises. A g1,j(τ) = h i, (101) n1(t0)nj(t0 + τ) 10% remnant correlation which might still be acceptable

14 −8 0.12 for applications is found when N = 10 = 0.026n1. This sets a limit on how much background photon reser- 0.1 voir is acceptable. 0.08

2 8 2 5 N = 5 10− N = 2 10− × × 1 0.06 1.5 1.5 n ) ) τ τ ( (

1 1 1 1 0.04 , , 1 1 g 0.5 g 0.5 0.02 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 τ τ 0 5 10 15 20 25 30 Figure 2: Breakdown of the Glauber-Sudarshan P representa- t tion: shown in pink; Positive-P simulation: yellow; exact: black Figure 3: Estimation in positive-P and doubled-Q simulations: dashed. F = 0.01. occupation of mode 1, when F = 3, after switching from the positive-P to the doubled-Q representation at t = t0 = 20.

7.3 Breakdown of the Glauber-Sudarshan P predictions for the parameters in Fig.1 because Q dis- tributions have a width of (1) even in vacuum. This This system also shows the breakdown of the single- O phase-space P representation very clearly. The evolution is a certain limitation to the Q distribution approach. The difference in sampling accuracy can be nicely seen equation is (98a) with s = 1, but it is a correct represen- tation of the quantum FPE (8) only when the diffusion in Fig.3 which shows the mean and uncertainty of n1 during the simulation used to generate Figs.4-5. The matrix Dµν in the FPE has all nonnegative eigenvalues [62]. Here, the diffusion matrix for real, imaginary parts samples are switched from positive-P to doubled-Q at t = t = 20. of has elements U 2 γN , 0 αj = αjr +iαji Djr,jr = 2 Im(αj )+ 2 D = U Im(α2)+ γN , D = D = U Re(α2). ji,ji − 2 j 2 jr,ji ji,jr − 2 j F = 3 with eigenvalues γN U 2. These only be- 6 λj,± = 2 2 αj ± | |2 come non-negative once γN > U αj , i.e. γN & Unj. The question then is: for what parameters| | does the evo- a lution remain well described? When N = 0 The anti- 4 G bunched mode has mean occupation n = 3.87 10−7, 1 × naively suggesting N 5 10−8 = 1.5Un /γ to already ∼ × 1 be a value for which the description is good. However, 2 we can see in Fig.2 that it does not give correct results Re b at all. This is because the problem lies in the nr. 2 mode G −5 −5 with n = 1.07 10 . Taking a far larger N = 2 10 anti-normal correlations 2 × × 0 (in which case γN 10Un1, 7Un2) gives almost correct ∼ Im b results with the Glauber-Sudarshan P (though still not G fully), but of course antibunching in mode 1 is long gone -2 for such a relatively high thermal noise level. 0 2 4 6 8 This is an indication that skimping on full quantum τ effects by trying approximate semiclassical methods is Figure 4: Anti-normal ordered moments. Strongly pumped case not a good strategy for this kind of system. (F = 3), mode 2. Shown are a and the real and imaginary 1 G parts of b as per (103). Exact results: dashed black. G 7.4 Differently ordered correlations Arguably the primary practical interest lies in cases To test how the prescriptions developed in Secs. 4.1 with four factors and two times – corresponding roughly and5 work, we use a different driving, F = 3, which to either detection of single particles at two times, or generates larger occupations (in the stationary state creation and destruction of pairs. Fig.4 shows the simu- n1 0.043 and n2 0.98) and as a result more in- lation of some anti-normally ordered correlations of this teresting≈ anti-normal≈ and mixed-order correlations. The kind (not normalised), evaluated using the doubled-Q doubled-Q simulations are also too noisy to get useful representation:

15 3 but with a longer chain of 32 sites, which is already be- F = 3 yond or at the limit of the capabilities of alternative methods such as brute force or corner space renormali- 2 c sation [24, 59]. The tunnelling term in (95) now becomes G h i J P31 a†a + a† a , and a driving of F = 3 re- − j=1 bjbj+1 bj+1bj mains only at the j = 1 site. While the stationary state 1 does not show any time-dependent change in expecta- tion values, we should expect to still see signatures of Re Gd transport in its multi-time correlations as a function of distance and delay time if the method is good.

mixed order0 correlations Im Gd 1.01 -1 0 2 4 6 8 1 τ

Figure 5: Moments that are neither normal nor anti-normal. ) 0.99 τ

( j = 32 F = 3 like in Fig.4. Shown are c and the real and imaginary ,j

G 1 parts of d as per (104). Exact results: dashed black lines. g 0.98 G j = 21 j = 11 0.97 j = 1 = a (t )a (t + τ)a†(t + τ)a†(t ) a b2 0 b2 0 b2 0 b2 0 0.96 G h 0 0 0 0 i = Re α2(t0)α2(t0 + τ)β2(t0 + τ)β2(t0) stoch (103) h † i 0 2 4 6 8 10 = a (t + τ)2[a (t )]2 = α0 (t + τ)2β0 (t )2 Gb hb2 0 b2 0 i h 2 0 2 0 istoch τ The latter b is an anomalous correlation that has Figure 6: Spreading of correlations in a 32 site chain, in the both real andG imaginary components. Similarly, Fig.5 stationary state. F = 3. The correlation is given by (101). shows simulations of a number of mixed-order correla- tions which absolutely require a swapping from positive- Fig.6 shows such a calculation in the 32 site sys- P to doubled-Q representation using the procedure of tem, plotting the normalised photon-photon correlation Sec. 5.4: g1,j(τ) with spatial separation j 1 and delay time τ, as defined in (101). The correlations− are evaluated † † from t = 20. Despite this being the stationary state, = a (t + τ)a (t + τ)a (t )a (t ) 0 Gc hb2 0 b2 0 b2 0 b2 0 i a very clear anti-correlation signal can be seen mov- = Re α0 (t + τ)β0 (t + τ)β0 (t )α (t ) h 2 0 2 0 2 0 2 0 istoch ing steadily with delay time. Its speed is approximately † 2 2J, twice the tunnelling rate. Correlation waves often d = ba2(t0)[ba2(t0 + τ)] ba2(t0) G h 0 0 i travel at twice the characteristic speed for single parti- = α (t )β (t + τ)2α (t ) (104) h 2 0 2 0 2 0 istoch cles [25, 97, 114], so this is not unexpected. However, In all cases, primed variables are evaluated in the the speed is clearly not related to the superfluid speed p doubled-Q representation, un-primed in the positive-P. of sound, here Unj, which is far lower. Notably, in both figures the stochastic simulations ∼ perfectly and very accurately agree with brute force cal- culations using the density matrix. This is despite the 8 Summary regime being one which is very poorly treated by ap- proximate semiclassical methods (occupations are (1) The known framework for evaluation of multi-time ob- or smaller). This is strong evidence that the intuitiveO ex- servables from the work of Gardiner in the P represen- pressions and approach laid out in the previous sections tation [62] has been extended here to include firstly: is appropriate. the much more widely applicable positive-P represen- tation (39); Secondly the Q (65), doubled-Q and other single and double phase space s-ordered representations; 7.5 Correlation dynamics in the stationary state Thirdly: other orderings such as anti-normal and mixed- As a larger-sized example, consider the same Hamilto- ordered observable products (Sec.5, and especially (80) nian and master equation as (95) and (96) (take N = 0), and the algorithm of Sec. 5.4). These results allow the

16 evaluation of a very wide range of quantum multi-time Some practical elements that are usually skipped over observables in bosonic systems (classified in Tables.1-2) in the literature are also pointed out. Example text- in a way that contains the full quantum mechanics, and books for the broad topic of stochastic integration in- is scalable to large systems. Systems with localised in- clude [76, 87]. teractions require computational effort proportional to M or M log M with the number of modes or sites, M. A.1 Integration algorithm While out-of-time correlations (OTOCs) are not di- rectly accessible through the mechanism outlined here, Let us introduce the following notation. The Ito stochas- time reversal schemes like those used in experiment [63] tic equations for a set of variables ~v with elements vj or theory [15, 46, 133] could be attempted by chang- indexed by j will be written ing the signs of constants in the equation of motion. dv When combined with the techniques developed here, j = (~v, t) = A (~v, t) + X (~v, t) (105) dt Dj j j this would give access to a wide range of information about quantum chaos [15, 60, 98], many body locali- where X are noise terms with zero mean X = 0, j h jistoch sation [56, 70], and quantum phase transitions [124] in whereas Aj are deterministic “drift terms” that contain larger systems than were accessible to date [15, 63]. no explicit noise contribution, apart from the statistical Along the way, a number of additional results were spread of the variables vj themselves. At the level of obtained regarding conversion formulae between differ- the Ito equations (105) we also require that the noise 0 0 ent orderings in the doubled-phase space representa- is not correlated in time Xj(t)Xj0 (t ) δ(t t ), h istoch ∝ − tions ((86) and (89)) and their stochastic samples (90). i.e. proportional to Wiener increments. For example, in A clear case of the breakdown of the Glauber-Sudarshan (98): P representation was seen in Fig.2. Also, in Sec.7 s   some results regarding the correlation functions (Figs.4 1 s Xα = αj√ isU ξj + γ N + − ηj, (106a) and5), susceptibility to background thermal density j − 2 (Fig.1) and signal speed (Fig.6) in the unconventional s   photon blockade system were obtained. √ 1 s ∗ Xβj = βj isU ξej + γ N + − ηj . (106b) Finally, AppendixA details a convenient algorithm 2 for simulation of phase-space stochastic equations, an For various reasons, the standard mathematical envi- item that has been hard to find in the literature in the ronments and numerical packages tend not to perform past. well when integrating stochastic equations4, especially ones that contain variable-dependent diffusion coeffi- cients such as in (106) or the prototypical Kubo oscil- Acknowledgments lator [90] with multiplicative noise. Phase space meth- ods that treat the full quantum dynamics also typically I am grateful to Michał Matuszewski, Marzena Szy- contain such terms. Once one leaves the simplest Ito- mańska, and Alex Ferrier for discussions on topics Euler algorithm which requires very small step sizes, leading to this paper, and to Wouter Vestraelen for two of the recurring issues are that advanced integra- discussions of numerics. This research was supported tion algorithms usually assume sufficiently continuous by the National Science Centre (Poland) grant No. derivatives (completely violated by white noise) or try 2018/31/B/ST2/01871. to adapt the timestep in real time (which can introduce systematic errors when noise is involved). A third dif- ficulty is the need for autocorrelation corrections (seen A Numerical techniques for stochastic Sec. A.2) which become much more difficult to derive simulation of phase-space trajectories as the complexity of the algorithm grows. The semi-implicit midpoint approach [47, 126] has This appendix details how the stochastic equations for been shown to be robust against spurious numerical in- the examples in Sec.7 were integrated, and describes stability as carefully analysed in [51, 140], but remains a robust and general algorithm that is particularly ap- simple enough for any autocorrelation corrections to be plicable to stochastic trajectories generated by phase- readily calculated. space descriptions of quantum systems. The bulk of the Suppose we are to advance time by ∆t, starting from (0) 0 method is based on the semi-implicit midstep algorithm variables vj at time t to vj at t + ∆t. Define a con- described in [51, 140], adding some modified propaga- stituent substep or propagator that advances times by τ tors as per [44] which allow for more efficient treatment of exponential and dominant deterministic processes. 4A notable exception is the XMDS package [33].

17 h i and variables to vstep ~v(0), (~v (0), t), τ which depends and solve the equation for the E and R parts j j D in order on the starting variables, the form of the deriva- dvj E R tive, and the timestep τ. The simplest choice for this = j vj + j (111) propagator is the Ito-Euler form dτ D D h i as if the coefficients were constant [44]. This gives the vstep v(0), (~v (0), t), τ = v(0) + (~v (0), t) τ (107) j j D j D full propagator which just uses the values at the beginning of the step h i vstep v(0), (~v (0), t), τ = for evaluating the derivative, as per the definition of Ito j j D stochastic calculus. R (0) (0) DE (0)  DE (0)  j (~v , t) The semi-implicit midpoint algorithm can be viewed v eτ j (~v ,t) + eτ j (~v ,t) 1 D j − E(~v (0), t) as a container procedure that calculates the final vari- Dj ables v0 via a series of iterations [140] + L(~v (0), t) τ. (112) j Dj h i v(1) = vstep v(0), (~v (0), t), ∆t/2 , Then, the leading processes are often integrated exactly, j j j D h i with only higher order corrections needed to be worked v(2) = vstep v(0), (~v (1), t), ∆t/2 , j j j D on by the midpoint algorithm. This is particularly effi- (108) cient when the stochastic terms are a perturbation on ··· h i the dominant exponential deterministic evolution. It is v(n) = vstep v(0), (~v (n−1), t), ∆t/2 , j j j D usually optimal to place all non-exponential parts into h i the “R” part, but some special cases can require avoid- v0 (t + ∆t) = vstep v(0), (~v (n), t + ∆t/2), ∆t . (109) j j j D ance of the nonlinear solution [44]. Both A drift and X The numbered iterations always start from the initial noise parts can enter the coefficients as one chooses. point, but iteratively find a self-consistent estimate of In the case of the calculations in this paper using (98), the midpoint value of the derivatives. The last itera- E γ sp E tion (109) uses this midpoint value and midstep time = iU(αjβj + s 1) + √ isU ξj iH + C , Dαj − − − 2 − − jj αj to make the final advancement of the variables. Usu- s   ally, a single midpoint iteration (i.e. n = 1) is suffi- R X sp 1 s = iFj i H αk + γ N + − ηj, cient. When using the basic Ito-Euler propagator (107) Dαj − − jk 2 k=6 j at time argument t + ∆t/2 in (109), the final step is ac- γ 2 E √ sp E curate to (∆t ) in the deterministic parts A , and the βj = iU(αjβj + s 1) + isU ξej + iHjj + Cβj , O j D − − 2 step’s mean to order (∆t) in the noise terms Xj [51], s O   provided the autocorrelation correction is incorporated R X sp 1 s ∗ = iFj + i H βk + γ N + − η , into step, as explained in Sec. A.2. This is so-called Dβj jk 2 j vj k=6 j “weak” stochastic convergence to (∆t) in the terminol- (113) ogy of [51] because the accuracy forO single trajectories is √ ( ∆t). “Strong” stochastic convergence to (∆t) for and L = 0 were used for the propagator (112) and O O Dj single trajectories requires a more involved and more combined with a single iteration (n = 1) of the mid- inconvenient autocorrelation correction than that pre- point algorithm (108). Note the presence of the auto- sented in Sec. A.2. Nevertheless, the “weak” level of ac- correlation corrections curacy in the noise term is actually quite high already. For example, even 4th or 9th order Runge-Kutta imple- E isU C = (114a) mentations made available are generally only accurate αj 2 to √ in this regard [33]. isU ( ∆t) CE = (114b) Moreover,O in practice, one can often easily but sub- βj − 2 stantially improve on the Euler step (107). For example, as explained in Sec. A.2. if ∆, γ, or U are large in (98), the evolution is primar- Importantly – the form of the corrections (114) above ily exponential. Another typical case is in continuum arises when the noises ξ, η etc. appearing in the X models when the kinetic energy or external potential j are calculated just once at the beginning of the entire energy contributions are dominant and trivial to inte- procedure for making the full t step. Other noise input grate [44]. A pragmatic and flexible propagator choice is ∆ can change the expressions (114). to separate the evolution into parts that will be treated The actual timestep that must be used is con- exponentially (E), strictly linearly (L) and the rest (R): ∆t strained by the need for the coefficients to change Dj (~v) = E(~v) v + R(~v) + L(~v), (110) little over a single timestep ∆t. A good practical rule of Dj Dj j Dj Dj

18 thumb is to require one needs D E (ν) (0) 3/2 (~v, t) ∆t ∆vj stoch = Aj(~v )∆t + (∆t) D 0.1, (115) h i stoch O (0) . v D (0) (0) 2E 3/2 j ∆vj∆vk stoch = Xj(~v )Xk(~v )∆t + (∆t) . h i stoch O for each term (ν) in = P (ν). One caveat is that (117) D D ν D one should take care not to estimate (~v (0), t) on the D In orders of ∆t, the drift is Aj∆t (∆t) and the noise run using noise or even variable values from individ- ∼ O X ∆t (√∆t). The time-dependence of coefficients ual trajectories. Doing so can, and often does, intro- j ∼ O duce autocorrelations between noise values and time A and X does not enter at this order of ∆t. A simple Ito- Euler timestep v(0) v(0) + A (~v (0)) ∆t + X (~v (0)) ∆t step lengths, leading to systematic errors. For treat- j → j j j ing noise terms, it is useful to use the RMS estimate meets both conditions (117). p 2 However, the midstep algorithm with an Ito-Euler Xj Xj √∆t based on expected mean vari- ≈ h| | i ∝ propagator (107) can introduce an extra correlation of able values vj for timestep estimates. Finally, forh completenessi and general applicability be- order ∆t by virtue of using the same noise for both the yond the examples of Sec.7, it can be advantageous to halfstep of ∆t/2 and then for the final full step. The treat widely differing processes using a split-step ap- first halfstep advance using (107) produces proach [140]. In particular, the treatment of kinetic en- (1) (1) (0) (0) ∆t (0) ∆t ∆v = v v = Aj(~v ) + Xj(~v ) , ergy and tunnelling is much bettered for many-mode j j − j 2 2 systems. A split step allows to take advantage of the (√∆t) (118) fact that kinetic energy and tunnelling are diagonal in k- ∼ O 1 space and can be done exactly there. k-space is reached Note that the Xj themselves are ( √ ), which is im- O ∆t using a discrete Fast Fourier Transform that has com- portant below. The next step in (108) involves coeffi- (1) (1) putational cost of only M log M for an M-site system cients Aj(~v ) and Xj(~v ). Taylor expanding around [61]. The split-step is applied in three parts: ~v (0), (i) First evolve terms diagonal in k-space for ∆t/2; (ii) then change variables to x-space and evolve the re- (1) (0) X dXj (0) (1) √ Xj(~v ) = Xj(~v ) + (~v )∆vk + ( ∆t), dvk O maining terms there for ∆t; k (iii) finally return to k-space for another k-space evolu- (119) tion of ∆t/2. and analogously for Aj. Using (118) and discarding For cases like (98) in which the main mean-field evolu- terms of subleading order (√∆t) in the new coeffi- tion has a Gross-Pitaevskii form, the split step method cients one finds O is symplectic (conserves energy) and has been shown to (1) (0) have an accuracy of (∆t)2 for the long time solution Xj(~v ) = Xj(~v ) O X dXj ∆t [77] provided that the latest available copy of the field is ~v (0) X ~v (0) √ t , (0) + ( ) j( ) + ( ∆ ) input as ~v after each Fourier transform [57]. Within dvk 2 O k each k- or x-space split step, the midpoint method (108) A (~v (1)) = A (~v (0)) + (√∆t). (120) can employed if the coefficients of the substep are j j O D (1) variable-dependent, or otherwise if the coefficients are Notice the appearance of a term in Xj(~v ) of the (0) constant just a plain propagator substep (112) is done. same order as the original coefficients Xj(~v ). Assum- Thus, one has the split-step algorithm as a container ing first just one n = 1 iteration, the final step (109) for midpoint iterations, which themselves are contain- becomes ers for the base propagators. Long-range interactions 0 (0) (0) (0) (eg. dipole-dipole) can also often be treated efficiently v = vj + Aj(~v )∆t + Xj(~v ) ∆t (121) 2 through Fourier transforms [144]. X dXj (0) (0) (∆t) 3/2 + (~v ) Xj(~v ) + (∆t) . Sumarizing, the algorithm used for the simulations of dvk 2 O Sec.7 consists of (108), (109), (112)–(114). k It turns out that iterations of n > 1 do not affect the ex- A.2 Timestep autocorrelation correction pression (121) at (∆t). The new term in (121) breaks the equivalence ofO the algorithm (117) because it is of Realisations of Ito stochastic equations (105) must at the same order as the drift Aj∆t. the least produce the required average and noise vari- To restore equivalence, one can add a correction Cj ance to lowest order. That is, writing to the drift used in the algorithm as per

∆v = v0 (t + ∆t) v(0), (116) A A + C . (122) j j − j j → j j

19 in either the “strongly convergent” form limiting distribution for each ∆t step and avoids a po- tential need to reduce timestep further to capture the strong ∆t X dXj right noise distribution. For real noises ξ, one generates Cj = (~v) Xj(~v) (123) − 2 dvk k fresh Gaussian random variables of variance 1/∆t for each j at the beginning of each timestep. Generating or the “weakly convergent” averaged form Cj = new ones for each time step ensures the δ(t t0) limit Cstrong . For the case of Ito-Euler propagator in − j stoch as ∆t 0. The Box-Muller algorithm [17] is a simple the midstep algorithm, way to→ obtain two such independent Gaussian variables * + from two uniformly distributed noises, whereas a vari- Strat ∆t X dXj ety of more efficient though less transparent algorithms Cj = Cj = (~v) Xj(~v) . (124) − 2 dvk are also known [12, 18]. In our case Box-Muller was k stoch used, while the underlying uniformly distributed noise The strong forms (123) have different values for each was generated using the SFMT fast Mersenne twister stochastic realization [51] and obtain (∆t) timestep method [121] 5. Complex noise η is simply constructed O accuracy for the noise terms, while the weak forms use as η = (ξr +iξi)/√2 using two real Gaussian noises ξi,r. a pre-calculated mean (124). The weak variant is more When using efficient random number generation such commonly used in practice and sufficient to obtain over- as [121] it turns out that the creation of Gaussian noises all stable and accurate integration. It is what is applied from uniform ones is the most computationally costly in (114) and the calculations of Sec.7. step. The Brent algorithm [18] alleviates this apprecia- The weak autocorrelation correction Cj has often bly, while a simple alternative route applicable at least been dubbed a “Stratonovich correction” because the for stochastic simulations – due to their central limit form (124) is identical to the correction used to move properties – is to produce binomially distributed noise between Ito and Stratonovich stochastic calculus. How- instead. One adds nB uniform noises rn on [0, 1] as per q ever, this is a misnomer because the match is merely a 12 PnB 1 ξ (rn ). In practice nB = 4 or nB = 3 coincidence that occurs when an Ito-Euler propagator ≈ nB ∆t n=1 − 2 is used with the midpoint algorithm. For the case of is already sufficient. the partly exponential propagator (112), the same pro- cedure as above leads to the following, different, form A.4 Error estimation of the autocorrelation corrections. When n in the 1 Statistical uncertainty in quantities calculated from midstep algorithm: ­ the simulations can be robustly estimatedQ via sub- ensemble averaging [140] even when the distribution of Strat ∆t E L Cj = C + X (~v)X (~v) , (125) j 2 j j stoch samples is unknown. The full ensemble of, say , re- alisations is divided into a smaller number u ofS sub- or when one uses no midstepping (n ): = 0 ensembles. Auxiliary subensemble means (i=1,...,u) are calculated for each sub-ensemble individually.Q Then by ∆t C XE ~v XE ~v v XE ~v XR ~v . the central limit theorem, the 1σ uncertainty in the full j = j ( ) j ( ) j + j ( ) j ( ) stoch − 2 ensemble prediction is (126) As it happens, in the case of the equations (98), the cor- s   rections are the same whether using an Ito-Euler prop- var (i) ∆ = Q . (127) agator ((114)) or the partially exponential one (125)- Q u 1 (126), but that is not always the case. − In practice u √ is useful. In the simulations reported . S A.3 Noise implementation in this paper, u = 32 was used. A separate issue is testing the time discretization ac- Since the underlying stochastic equations are defined curacy in stochastic equations. In a simple implemen- in the infinitesimal limit ∆t 0, in principle any tation where noise is generated sequentially, changing implementation of the noise that→ has zero mean and timestep changes also the noise history, making com- satisfies the variance conditions (99) and (100) in the parison of single trajectories with different ∆t useless ∆t 0 limit is correct. By the central limit theo- for determination of accuracy. Yet, comparing the en- rem,→ after several timesteps the effective distribution semble means for different timesteps can be onerous in of the sum of the noises will always converge to a Gaus- large systems. A solution is to compare two runs of a sian one. In practice, however, it is usually desirable single realisation with identical underlying noise sources to use explicitly Gaussian distributed noises in the im- plementation. Doing so already accurately depicts the 5http://www.math.sci.hiroshima-u.ac.jp/m-mat/MT/SFMT/

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