A Lindblad Model of Quantum Brownian Motion

A Lindblad model of quantum Brownian motion

Aniello Lampo,1, ∗ Soon Hoe Lim,2 Jan Wehr,2 Pietro Massignan,1 and Maciej Lewenstein1, 3 1ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA 3ICREA – Instituci´oCatalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, E-08010 Barcelona, Spain (Dated: December 16, 2016) The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.

PACS numbers: 05.40.-a,03.65.Yz,72.70.+m,03.75.Gg

I. INTRODUCTION default choice for evaluating decoherence and dissipation processes, and in general it provides a way to treat quan- A physical system interacting with the environment is titatively the effects experienced by an open system due referred to as open [1]. In reality, such an interaction to the interaction with the environment [27–30]. Here- is unavoidable, so every physical system is affected by after, we will refer to the central particle studied in the the presence of its surroundings, leading to dissipation, model as the Brownian particle. thermalization, and, in the quantum case, decoherence The main tool for the investigation of the dynamics of [2,3]. the system is the master equation (ME), which describes During the recent few years, interest in open quantum the evolution of the reduced density matrix of the Brown- systems increased because of several reasons. On the one ian particle, obtained by taking the trace over the degrees hand, both decoherence and dissipation generally con- of freedom of the bath. The ME allows to compute var- stitute the main obstacle to the realization of quantum ious physical quantities, such as the time scales of deco- computers and other quantum devices. On the other herence and dissipation processes, as well as the average hand, recently there has been a series of very interesting values of variables like position and momentum. Widely proposals to exploit the interaction with the environment used in the literature is the so-called Born-Markov mas- to dissipatively engineer challenging states of matter [4– ter equation (BMME) [1,3]. The latter, however, does 7], and of works where the engineering of environments not always preserve positivity of the density matrix, lead- paved the way to the creation of entanglement and su- ing to violations of the Heisenberg uncertainty principle perpositions of quantum states [8–10]. (HUP), i.e., σX σP ≥ ~/2, especially at very low temper- Moreover, open quantum system techniques are often atures [31, 32]. Here σX and σP are the standard devi- adopted to investigate problems lying at the core of the ation of the position and momentum, respectively. The foundations of quantum mechanics. Here, one of the MEs in the so-called Lindblad form preserve the positiv- unsolved issues regards the problem of the quantum-to- ity of the density operator at all times [1,3, 33], and this classical transition, i.e. the question how do classical fea- in turn guarantees that the HUP is always satisfied. A tures we experience in the macroscopic world arise from brief, self-contained demonstration of the latter is given the underlying quantum phenomena [3, 11–13]. Most in the Appendix. Various ways of addressing this diffi- of the theories addressing the emergence of the classical culty have been put forward [34–43]. In this paper we world deem it a consequence of the coupling of quantum add a term to the BMME, that vanishes in the classical systems with the environment [14–20]. limit, bringing the equation to the Lindblad form and, In this work we focus on an ubiquitous model of open in particular, ensuring that the HUP is always satisfied arXiv:1604.06033v2 [quant-ph] 15 Dec 2016 quantum system, the quantum Brownian motion (QBM), [33]. We study the dynamics of the obtained equation, which describes the dynamics of a particle (playing the in particular its stationary state. role of the open system) coupled with a thermal bath An important purpose of the current paper is to inves- made up by a large number of bosonic oscillators (the tigate models of QBM more general than those usually environment) [21–26]. QBM is in many situations the studied in the literature. Usually, the particle-bath coupling is assumed to be linear in both the coordinates of the particle and the bath oscillators, and for definiteness in the following we will refer to this specific case with ∗ [email protected] the name linear QBM. We are here also interested in a 2 more general case, where the coupling is still a linear in which: function of the positions of the oscillators of the bath, but depends nonlinearly on the position of the Brownian Pˆ2 mΩ2Xˆ 2 Hˆ = + , (2) particle. This situation arises when dealing with inhomo- S 2m 2 geneous environments, in which damping and diffusion 2 2 2 X pˆk mkωkxˆk vary in space. An immediate application of this gener- HÊ = + , 2mk 2 alization concerns the physical behavior of an impurity k embedded in an ultra cold gas. In this case spatial inho- X HÎ = gkxˆkX,ˆ mogeneities are due to the presence of trapping potentials k and, possibly, stray fields [44, 45]. Here, we will study in detail the case when the coupling depends quadratically where m is the mass of the Brownian particle, Ω is the on the position of the test particle, and we will refer to frequency of the harmonic potential trapping it, mk and this case with the name quadratic QBM. th ωk are the mass and the frequency of the k oscillator The paper is organized as follows. In Sec.II we con- of the environment, and gk are the bath-particle cou- sider QBM with a linear coupling. We first introduce the pling constants. In this Section, the interaction term HÎ BMME, and briefly discuss the lack of positivity preser- depends linearly on the positions of both the Brownian vation mentioned above. We then add a term to obtain particle and the oscillators of the environment. We refer a LME according to the procedure proposed in [38], and to this model as a linear QBM. rewrite it in the Wigner function representation. The The Hamiltonian is the starting point to derive the Wigner function defines a quasi-probability distribution ME. We are interested in a BMME, obtained by making on the phase space [2]. We derive the time-dependent two approximations [3]. In the first one, the Born ap- equations for the moments of this distribution, show that proximation, we assume that the influence of the particle they have an exact Gaussian solution, and study in de- on the bath is negligible, so that the two systems re- tail its long-time behavior. In particular, we analyze the main uncorrelated (i.e. their joint state is a tensor prod- correlations induced by the environment, which cause a uct) at all times (including the initial one). In the sec- rotation and distortion of the distribution, as well as ond, the Markov approximation, we neglect memory ef- squeezing effects expressed by the widths and the area fects, namely we require that the self-correlations created of the distribution’s effective support. within the environment due to the interaction with the In Sec.III we study a non-linear QBM, corresponding Brownian particle decay over a time scale much shorter to an inhomogeneous environment. In particular we con- than the relaxation time scale of the particle. sider an interaction which is a quadratic function of the Under these hypotheses, one derives from the Hamil- position of the Brownian particle. This model has been tonian Eq. (1) the following ME [1,3, 32], studied in [32], by means of a BMME. We again modify the BMME to obtain an equation in the Lindblad form ∂ρˆ(t) i h ˆ ˆ 2 i and we study its stationary solutions in the phase space = − HS + CxX , ρˆ(t) (3) ∂t ~ (Wigner) representation. For the quadratic QBM, the D D exact stationary state is no longer Gaussian but a Gaus- − x [X,ˆ [X,ˆ ρˆ(t)]] − p [X,ˆ [P,ˆ ρˆ(t)]] sian approximation can be used in certain regimes. How- ~ ~mΩ iC ever, when the damping is strong, the Gaussian ansatz − p [X,ˆ {P,ˆ ρˆ(t)}]. does not converge for large times, showing that it is not ~mΩ a good approximation to a stationary state. To avoid ambiguities, we wish to stress that, throughout the whole paper, we will be working in the Schrödinger picture, where the time-dependence is carried by the state of the system (rather than by the operators), and II. LINEAR QBM average values of operators are calculated as usual as hAit ≡ Tr[ρ(t)A]. A. The Model The effects of the bath on the motion of the Brownian particle are encoded in the spectral density of the bath. In the following, we will focus on the commonly used The QBM model describes the physical behavior of a Lorentz-Drude spectral density: particle interacting with a thermal bosonic bath of harmonic oscillators. In general the potential of the Brown- mγ ω J(ω) = (4) ian particle can be arbitrary, but we will study the har- π 1 + ω2/Λ2 monic case only. The model is described by the Hamil- tonian: which is linear at low frequencies, and decays as 1/ω beyond the cut-off Λ, introduced to regularize the theory. Hˆ = HˆS + HÊ + HÎ , (1) With this choice of the spectral density the coefficients 3 of the BMME at a bath’s temperature T read: B. Lindblad Master Equation

mγΩ Λ2 C = (5) A LME has the form: p 2 Ω2 + Λ2 ∂ρˆ i h ˆ i X ˆ ˆ† 1 ˆ† ˆ Λ = − HS, ρˆ + κij AiρÂj − {Ai Aj, ρˆ} , Cx = − Cp (6) ∂t ~ 2 Ω i,j (10) ~Ω Dx = Cp coth (7) 2kBT where Aˆ are called Lindblad operators and (κ ) is a i ij 2Cp πkBT positive-definite matrix. Dp = + z(T, Λ, Ω) , (8) π ~Λ Following the approach proposed in [38] we will replace the BMME (3), which cannot be brought to a Lindblad with: form, by an equation of the form Eq. (10) with a single Lindblad operator of the form ~Λ i~Ω z(T, Λ, Ω) = ψ − Re ψ (9) Aˆ1 = αXˆ + βP,ˆ with κ11 = 1. (11) 2πkBT 2πkBT Substituting this operator into Eq. (10) we obtain: where ψ(x) is the DiGamma function [32]. In the fol- ∂ρˆ i h ˆ 0 i Γ h ˆ ˆ i lowing, we will refer to T , Λ, and γ as the model’s pa- = − HS, ρˆ − i X, {P , ρ} (12) rameters. The term proportional to C in Eq. (3) leads ∂t ~ ~ x D h h ii D h h ii to a renormalization of the harmonic trapping frequency. XP ˆ ˆ PP ˆ ˆ − 2 X, P, ρˆ − 2 P, P, ρˆ Following the usual approach [1], the latter may be can- ~ 2~ D h h ii celled by including in the Hamiltonian the counter-term − XX X,ˆ X,ˆ ρˆ , 2 2 Vˆc = −CxXˆ . 2~ The evolution defined by Eq. (3) does not preserve pos- with: itivity of the density matrix. As discussed in detail in, ˆ 0 ˆ Γ ˆ ˆ ˆ ˆ e.g., Ref. [31, 32], the lack of positivity leads to viola- H = HS − {X, P } ≡ HS + ∆H (13) S 2 tions of the Heisenberg uncertainty principle away from the Caldeira-Leggett limit discussed below. In particu- and: lar, this prevents the study of the dynamics in the regime 2 2 2 ∗ DXX =~ |α| ,DXP = ~ Re (α β) , (14) of very low temperatures. In fact, these violations in Eq. 2 2 ∗ (3) are driven by the logarithmic divergence at low tem- DPP =~ |β| , Γ = ~Im (α β) . peratures of Dp (which is itself proportional to γ, i.e. to One could obtain the same result employing two Lind- 2 gk). blad operators, proportional to Xˆ and Pˆ respectively. Overcoming this problem is a fundamental step to- Without loss of generality, we may take α to be a posi- wards a correct description of the dynamics of a Brow- tive real number since multiplying Aˆ1 by a phase factor nian particle. In this section we propose a modified ME does not change Eq. (10), and we will restrict ourselves for the linear QBM which has the Lindblad form and, to Imβ > 0, because, as seen from Eq. (14), αIm(β) is consequently, preserves positivity of the density matrix. the damping coefficient Γ, which must be positive. It is well known that the ME (3) cannot be expressed in Eq. (12) differs from Eq. (3) just by two extra terms, the Lindblad form [1,3]. Our equation differs from it by involving DPP and ∆Hˆ . Equating the coefficients of two terms, one of which can be naturally absorbed into the remaining terms with those of the analogous terms the system’s Hamiltonian. appearing in Eq. (3), one finds: Adopting a LME is not the only possible manner to deal with the violations of the Heisenberg uncertainty ~Dp DXX = 2~Dx,DXP = , (15) principle. From a formal point of view, the ME (3) is the mΩ 2 2 result of a perturbative expansion to the second order Cp (~Γ) + DXP Γ = ,DPP = . in the strength of the bath-particle coupling (actually, mΩ DXX expanding to second order requires weaker assumptions than the Born and Markov ones; the resulting equation In the Caldeira-Leggett (CL) limit kBT ~Λ ~Ω, may still take into account some non-Markovian effects these reduce to: which vanish in the limit of large times [1]). In [31] it Γ ≈ γ/2, (16) has been shown that Heisenberg principle violations in D ≈ 2mγk T, the stationary state disappear if one performs a pertur- XX B bative expansion beyond the second order in the coupling kBT DXP ≈ −γ , constant. Obviously, if the exact ME is used, violation Λ γk T of Heisenberg principle cannot occur in any parameter D ≈ B regime. PP 2mΛ2 4

Following [3], since the quantities represented by P and where the moments of the Wigner function are calculated mΩX have generally the same order of magnitude, one as can argue, as in Eq. (5.56) of [3], that the terms propor- Z ∞ Z ∞ tional to DXP and DPP are negligible in the CL limit, hf(X,P )it = dX dP f(X,P )W (X, P, t). recovering the structure of the usual CL ME. −∞ −∞ The operator ∆Hˆ can be absorbed into the unitary (21) part of the dynamics defined by Eq. (12), so it can be These moments correspond to symmetric ordering of the eliminated by introducing a counter term into the sys- quantum mechanical operators Xˆ and Pˆ [46]. In partic- tem’s Hamiltonian. More generally, we will add to HˆS a ular, note that the time-dependence is solely contained counter term in the Wigner function, in agreement with the fact that we work in the Schrödingerpicture. ˆ ˆ HC = (r − 1)∆H, (17) The solutions for the first moments are: which depends on a parameter r ∈ , leading to the R −Γt 0 modified Hamiltonian: hXit =e X0 cos(βrt) + xr sin(βrt) , (22) −Γt 0 ˆ 0 ˆ ˆ ˆ hP it =e P0 cos(βrt) − pr sin(βrt) , HS =HS − (rΓ/2){X, P } (18) (Pˆ − mrΓXˆ)2 m(Ω2 − r2Γ2)Xˆ 2 where: = + . 2m 2 mΓX0(1 − r) + P0 The effect of r is twofold: it introduces a gauge trans- x0 = (23) r mβ formation which shifts the canonical momentum Pˆ, and r 2 it renormalizes the frequency of the harmonic potential. 0 ΓP0(1 − r) + mX0Ω pr = In the rest of the section we shall study the dynamics βr defined by equation Eq. (12), first for general values of r and then, for the discussion of the stationary state, fo- with: cusing on the case r = 0. We stress that the introduction X ≡ hXi ,P ≡ hP i , (24) of a counter term in the Hamiltonian does not affect the 0 0 0 0 Lindblad character of the LME in Eq. (12), since it just and: enters in its unitary part. p 2 2 2 βr ≡ Ω − Γ (r − 1) . (25)

C. Solution of the LME Similar solutions have been presented in [36, 37, 47]. Eqs. (20) may alternatively be written in terms of the kinetic We are interested in studying the long-time dynamics momentum hP˜it = hP it − mrΓhXit: of the Brownian particle. In particular, we consider its representation in the phase space, employing the Wigner ∂hXit hP˜it function representation [2]. In terms of the Wigner func- = , (26) ˙ ∂t m tion, Eq. (12) becomes W = LW , with ˜ ∂hP it 2 2 ˜ P ∂W ∂W = −m Ω − r(r − 2)Γ hXit − 2ΓhP it, LW = − + mΩ2X (19) ∂t m ∂X ∂P ∂ ∂ or equivalently gathered in the compact form + Γ r (XW ) + (2 − r) (PW ) ∂X ∂P 2 ∂ hXit ∂hXit 2 2 2 2 2 + 2Γ + Ω − r(r − 2)Γ hXi = 0. (27) 1 ∂ W ∂ W ∂ W 2 t + D + D − D . ∂t ∂t 2 XX ∂P 2 PP ∂X2 XP ∂X∂P which, of course, can be derived directly from the equa- Equivalently, one can look at the equations for its mo- tions Eq. (20). For both r = 0 and r = 2 one obtains ments: a damped oscillator with the original frequency of the ∂hXi hP i harmonic trap, Ω. For other values of r the frequency t = t − rΓhXi (20) ∂t m t is renormalized, with the maximal renormalization cor- ∂hP i responding to r = 1. t = −mΩ2hXi − (2 − r)ΓhP i ∂t t t In Eqs. (20) we see that r introduces apparent damp- 2 ing in the position, as already noted in [39]. Because of ∂hX it 2 2hXP it = −2rΓhX it + + DPP this, in the following we will set r = 0. The extra term ∂t m proportional to D , not present in the starting BMME, 2 PP ∂hXP it 2 2 hP it ˙2 = −mΩ hX it − 2ΓhXP it + − DXP appears only in the equation for hX i, without affecting ∂t m the other equations, and in particular those for the first ∂hP 2i t = −2mΩ2hXP i − (4 − 2r)ΓhP 2i + D , moments, so that it may be interpreted as a position dif- ∂t t t XX fusion coefficient. 5

We wish now to focus on the stationary solution of Eq. (19). The latter may be found by means of the following 2.5 Gaussian ansatz:

2 2 1 X P 2ρXP 2.0 0.9 WST = ζ exp 2 2 + 2 + , (28) 2(ρ − 1) σX σP σX σP 1.5 which is normalized to one taking: 0.8

1 1.0 ζ ≡ , |ρ| ≤ 1, (29) 0.7 p 2 2πσX σP 1 − ρ θ 0.5 0.6 with: hXP i p 2 p 2 2 4 6 8 10 12 14 16 σX = hX i, σP = hP i, ρ = − , (30) σX σP and, in the remainder of this Section, the variances are computed using the time-independent Gaussian Ansatz Figure 1. Plot of the angle θ/π at γ/Ω = 0.8. This an- in Eq. (28)[48]. Inserting the Gaussian ansatz in Eq. gle is represented in the ellipse at the bottom of the picture. (28) into Eq. (19) we ﬁnd: Here, the orange-solid (green-dashed) line represents the mi- nor (major) axis of the Wigner function, i.e., that related to D − 4mΓD + m2(4Γ2 + Ω2)D δl (δL). The axes X and P are those of the phase space. σ2 = XX XP PP (31) X 4m2ΓΩ2 2 2 2 DXX + m Ω DPP σP = 4Γ 2.5 0.8 σP σX ρ = mDPP /2 2.0 0.6 We introduce the adimensional variables: 1.5

r 2 r 2 1.0 2mΩσX 2σP 0.4 δx = , δp = (32) ~ mΩ~ 0.5

0.2 With this parametrization, the Heisenberg inequality 2 4 6 8 10 12 14 16 σX σP ≥ ~/2 reads δxδp ≥ 1. The Lindbladian character of Eq. (19) guarantees that the second moments will satisfy the Heisenberg relation at all times. We furthermore note that the term with Figure 2. Eccentricity of the Wigner function introduced in coefficient D , i.e. the extra term induced by the Lind- Eq. (28), at γ/Ω = 0.8. The red dashed line represents the val- PP ues of T and Λ yielding δ2 = 1, and we have genuine squeezing blad form of the ME, leads to a correlation between the l below it. two canonical variables. Geometrically, this correlation can be interpreted as a rotation of the stationary solution in the phase space, see We now aim to quantify such a rotation, calculating the the black sketches in Fig.1. In the CL limit, the term angle θ between the major axis of the Wigner function with the coefficient D is negligible, and the solution PP (i.e. the eigenvector corresponding to δ ), and the X-axis is an ellipse with its axes parallel to the canonical ones, L of the phase space. In Fig.1 we present the behavior of θ reproducing the well-known results. as function of T and Λ, at fixed γ. At high Λ the major To analyze the properties of the stationary state in axis aligns approximately with the P -axis of the phase the phase space, we consider the variances of the major space (θ = π/2), while at low Λ, it is close to the X-axis and minor axes of the Wigner function. These axes are (θ = π), in agreement with the behavior of the BMME defined as the eigenvectors of the covariance matrix: discussed in [32], where hXP i was identically zero. On δ2 −ρδ δ the other hand, at low temperatures the Wigner function cov(X,P ) = x x p (33) −ρδ δ δ2 associated to the stationary solution of the LME may be x p p significantly rotated with respect to the axes of the phase space. The smaller and larger eigenvalues of this matrix, δl and δL, are given respectively by: In [32] it has been shown that, going to low temperature, the position of the Brownian particle governed by q 2 1 2 2 2 22 2 2 2 the BMME experiences genuine squeezing along x in the δl,L = δx + δp ∓ δx − δp + 4δxδpρ (34) 2 Wigner function representation, i.e. δx < 1. Similar 6

1.6 1.4 1.00 γ/Ω=0.25 γ/Ω=0.25 1.2 γ/Ω=0.5 0.95 γ/Ω=0.5 γ/Ω=0.75 γ/Ω=0.75 1.0 0.90 γ/Ω=1 γ/Ω=1 0.8 0.85 0.6 6 8 10 12 14 16 18 20 10 12 14 16 18 20

2 Figure 3. Minimum value of δl over all temperatures, as Figure 5. Minimum value of the cooling parameter χ over all a function of the cut-oﬀ frequency, at several values of the temperatures, as a function of the cut-oﬀ frequency, at several damping constant. values of the damping constant.

2.5 2.0 effective heating if the effective phase space area is larger 2.0 1.8 than the one occupied by a quantum Gibbs-Boltzmann 1.5 1.6 distribution at the same temperature. We thus define the system to be cooled if1: 1.0 1.4

0.5 1.2 δlδL χ = < 1, (36) coth ~Ω 2 4 6 8 10 12 14 16 1.0 2kB T

and heated otherwise. The degree of heating/cooling χ Figure 4. Cooling parameter χ introduced in Eq. (36), plot- is shown in Fig.4. In Fig.5 we present the minimal ted for γ/Ω = 0.8. The system exhibits cooling to the right value achieved by χ as the temperature is varied. We of the solid line, and heating to its left. For comparison, the note that to obtain small values of χ one needs to choose dashed line represents the cooling/heating boundary obtained large values of both Λ and γ. with the BMME (3), which is independent of γ. There is a difference between the configuration of the cooling areas arising in the Lindblad dynamics studied here, and the ones produced by the BMME (3) studied in [32]. In the latter, the cooling/heating boundary squeezing effects are pointed out in [18], by studying the coincides with the line defined by δ = δ , and this condi- numerical solution of the exact ME. In the case of the x p tion does not depend on γ, while in the present Lindblad LME, it was checked numerically that δ introduced in x model, the location of the boundary varies with γ. How- Eq. (31) is always bigger than one. However, the mi- ever, the boundary calculated within the LME converges nor axis of the ellipse describing the Wigner function to the BMME one in the γ → 0 limit. Moreover, the can display genuine squeezing. To quantify the degree LME discussed here displays heating at very low tem- of squeezing of the Wigner function, Fig.2 shows the peratures. values of eccentricity defined as In Figs.3 and5 we have not extended the range of val- p 2 ues of the damping constant beyond γ = 1. In fact, the η = 1 − (δl/δL) , (35) expressions for the coefficients of the equation Eq. (12) computed for different values of temperature T and UV- have been obtained by comparing it with the equation cutoff Λ. The eccentricity is largest at low temperatures. Eq. (3). The latter is perturbative to second order in the In particular, below the red dashed line, we find an area strength of the coupling between the Brownian particle where δl < 1, corresponding to genuine squeezing along and the environment. The square of the coupling con- the minor axis of the Wigner Function, while in the CL stant is proportional to the damping coefficient, so the limit the eccentricity η approaches zero, and we obtain validity of the perturbative expansion fails for γ large. In a Wigner function with circular symmetry. In Fig.3 we particular, in the case of QBM this perturbative expan- 2 present the minimal value of δl obtained by choosing the sion holds for γ . Ω[1, 23]. appropriate (low) temperature. This picture highlights the range of values of Λ and γ where genuine squeezing occurs. We find that the eccentricity is an increas- ing function of the damping constant, i.e. squeezing be- 1 For the Gibbs-Boltzmann distribution we have 2 2 2 comes more pronounced as γ grows. In particular, at hX iGB hP iGB ∼ coth (~Ω/2kB T ). So the denominator least γ/Ω > 0.5 is needed to obtain δl < 1. of Eq. (36) provides an information regarding the area of the We may say that the Brownian particle experiences an Gibbs-Boltzmann distribution. 7

Low Temperature Regime the bath, but is quadratic in the position of the Brownian particle: We consider here in detail the stationary state in the low temperature regime kBT < ~Ω. Such a study was impossible in [32] because solutions violated the Heisen- X gk Hˆ = xˆ Xˆ 2. (38) berg principle there. Here, the Lindblad form of the ME I R k in Eq. (12) ensures the positivity of the density matrix k at all times, so no violations of the Heisenberg principle occur. Here R is a characteristic length related to the motion of In the discussion above, we noticed that the time- p dependent equations of motion of the LME admit as the Brownian particle and we set it to be R = ~/mΩ. an exact solution a Gaussian with non-zero correla- The interaction term in Eq. (38) describes an interac- tions between the two canonical variables X and P . In tion of the particle with an inhomogeneous environment, the stationary state, in particular, one finds hXP i = giving rise to position-dependent damping and diffusion. −mDPP /2 6= 0. This is a novelty in comparison with the stationary solution of the BMME in Eq. (3), which A concrete example where we may encounter this kind shows no correlations between X and P . In the range of of nonlinearity is the model of an impurity in a Bose- Λ explored in Fig.1, the correlation between X and P Einstein condensate. In [44, 45] it has been shown that the dynamics of such a system can be described by the becomes noticeable for kBT . 0.5~Ω. So, an important feature of the stationary solution of our LME at low tem- FröhlichHamiltonian. In an inhomogeneous gas, i.e. perature is that its major axis is rotated with respect to a gas with a spatially dependent density profile, this those of the phase space. Hamiltonian differs from the QBM one due to the non- In Fig.2 we analyze the eccentricity of the stationary linear dependence of the interaction term on the posi- state. We point out that as the temperature decreases, tion of the impurity. When we consider, for instance, the distribution becomes increasingly more squeezed. In a Thomas-Fermi density profile, i.e. a density profile particular, at low temperature we find a region displaying varying quadratically with the position, the interaction genuine squeezing of the probability distribution in the Hamiltonian is an even function of the position. Here, the direction of l. coupling in Eq. (38) provides the first-order correction to In Fig.4 we also note the presence of a cooling area in the zero-order term in the expansion of the interaction the low temperature regime. Nevertheless, in the zero- between the impurity and the bath. In short, QBM with temperature limit the stationary state shows again heat- a quadratic coupling is not just a mathematical exercise, ing. but opens modeling possibilities in new contexts. In Ap- The zero-temperature limit of the Lindblad model de- pendix F of Ref. [32] it has been shown in detail that the serves special attention, as the two limits T → 0 and Hamiltonian of an impurity in a BEC can be expressed γ → 0 do not commute. Taking first the zero-coupling in the form of that of QBM with a generic coupling. and then the zero-temperature limit, one simply finds The dynamics induced by the interaction term in Eq. δx = δp (in agreement with the general result for a free (38) has already been discussed in detail in [32]. There, harmonic oscillator), but no further information on their the ME for the Brownian particle has been derived, in the specific value. If instead one takes first T → 0 and then BM approximations, for a Lorentz-Drude spectral den- γ → 0, one finds δx = δp and the additional condition: sity. Nevertheless, this ME is not in a Lindblad form, nor 2 is exact. Accordingly, the stationary solution is not de- 5 [log(Λ/Ω)] fined for some values of the model’s parameters because δxδp = δlδL = + > 1, (37) 4 π2 of violations of the Heisenberg uncertainty principle at indicating that for the Lindblad model the Heisenberg low temperatures. inequality is not saturated in the limit when the particle In this Section, we aim to find a LME as similar as becomes free. This is in contrast with the behavior of possible to that derived in [32]. Just like in the case of the non-Lindblad BMME (3), for which, in this limit, we linear QBM, we expect it to differ from the BMME by have δxδp = 1. Summarizing, the effect of DPP is to in- some extra terms. To achieve this goal we consider a troduce extra heating at low temperatures and couplings, single Lindblad operator: manifested by a small constant, and a weak logarithmic dependence on the UV cut-off Λ. III. QUADRATIC QBM 2 2 Aˆ1 = µXˆ + ν{X,ˆ Pˆ} + Pˆ (39) A. The Hamiltonian and the Lindblad ME

In this section we consider the quadratic QBM, whose where µ, ν and are nonzero complex numbers. Substi- coupling is still linear in the positions of the oscillators of tuting it into Eq. (10) we obtain: 8

∂ρˆ i h i Dµ h 2 h 2 ii Dν h h ii D h 2 h 2 ii = − HˆS + ∆Hˆ2, ρˆ − Xˆ , Xˆ , ρˆ − {X,ˆ Pˆ}, {X,ˆ Pˆ}, ρˆ − Pˆ , Pˆ , ρˆ (40) ∂t ~ 2~2 2~2 2~2 D h h ii D h h ii D h h ii − µν Xˆ 2, {X,ˆ Pˆ}, ρˆ − µ Xˆ 2, Pˆ2, ρˆ − ν Pˆ2, {X,ˆ Pˆ}, ρˆ ~2 ~2 ~2 C h i C h i C h i − i µν Xˆ 2, {{X,ˆ Pˆ}, ρˆ} − i µ Xˆ 2, {Pˆ2, ρˆ} − i ν Pˆ2, {{X,ˆ Pˆ}, ρˆ} , ~ ~ ~

where: and Dµ = 2mΩDxx. The remaining coefficients are then uniquely determined as: D D C µ ≡ |µ2|, µν ≡ Re(µ∗ν), µν ≡ Im(µ∗ν), (41) ~2 ~2 ~ and similarly for the other combinations of indices. We 1 2 could have obtained the same result by means of three Dν = Dµν Dµ + ~ Cµν Cµ , (44) Lindblad operators (rather than a single one), each pro- Dµ portional to one of the terms appearing on the right-hand 1 Cν = [Cµν Dµ − Dµν Cµ] , side of Eq. (39). Dµ Similarly to Sec.II, there is a term which appears in 1 h 2 2i the unitary part of the ME: D = Dµ + (~Cµ) , Dµ 2 2 1 h i ∆Hˆ2 = 2Dµν Xˆ − 2Dν Pˆ + 2Dµ{X,ˆ Pˆ} (42) 2 2 Dν = Dµν + (~Cµν ) . 1 1 Dµ − C {{X,ˆ Pˆ}, Xˆ 2} − C {Pˆ2, Xˆ 2} 2 µν 2 µ 1 + C {{X,ˆ Pˆ}, Pˆ2}. 2 ν It is easy to check that in the CL limit kBT ~Λ ~Ω, We eliminate it by introducing appropriate counter terms the coefficients of all extra terms vanish, and Eq. (40) re- in the Hamiltonian. covers the structure of the BMME introduced in Ref. [32]. The ME in Eq. (40) is in a Lindblad form. Proceeding as in Sec.II, equating the coefficients on the right hand B. Stationary State of the Quadratic QBM side of Eq. (40) to the corresponding ones in the BMME for quadratic QBM derived in [32], we obtain: We turn now to the study of the stationary state of the Dpp Brownian particle in the case of quadratic coupling. To Dµ = ,Dµν = Dxp, (43) mΩ this end we express the LME in Eq. (40) in terms of the Cpp Cxp Wigner function W , and obtain an equation of the form Cµ = ,Cµν = , ~mΩ ~ W˙ = LW , with:

∂ P L = − X + mΩ2∂ X + 2D ∂2 X2 + 2D (∂ P − ∂ X)2 + 2D ∂2 P 2 (45) m P µ P ν P X X 2 2 + 4Dµν (∂P XP − ∂P ∂X X + ∂P X) − 4Dµ(∂X X − 1)∂P P −4Dν P ∂X (∂P P − ∂X X) 2 h i + 8C ∂ PX2 + ~ ∂2 (∂ X − 1) + C 4∂ XP 2 − 2∂ ∂2 X + 2 2∂ ∂ µν P 4 P X µ P ~ P X ~ P X 2 − 2Cν P ∂X 4XP + ~ ∂P ∂X .

We now ﬁnd the stationary solution of the above equa- higher-order moments by their Wick expressions in terms tion. In this case the Gaussian ansatz in Eq. (28) may of second moments (which would be exact in a Gaussian at best provide an approximate solution, in contrast with case), obtaining the following closed, nonlinear system of the case of the linear QBM, since the system of equations equations in the variables δx, δp and ρ: for the second moments is not closed. We approximate 9

1 ∂δ2 x = 4m ΩC [1 + δ2δ2(1 + 2ρ2)] + 2m2Ω2D δ2 + 4D δ2 − Ωδ δ ρ (46) 2 ∂t ~ ν x p p ν x x p 1 ∂δ2 2D 4 C p = µ δ2 − ~ C + 6 C δ δ3ρ + Ωδ δ ρ + 4δ2 D − D − ~ µν 1 + 2ρ2 δ2 , (47) 2 ∂t m2Ω2 x mΩ µν ~ µ x p x p p ν µ mΩ x and:

1 ∂(δ δ ρ) 12 − x p = 4 C + Ωδ2 − 8mΩD δ2 + ~ C δ δ3ρ 2 ∂t ~ µ p ν p mΩ µν p x D + 8D − 12m ΩC δ2 δ δ ρ − Ω + 8 µν + 2 1 + 2ρ2 C δ2 δ2. (48) µ ~ ν p x p mΩ ~ µ p x

This system of equations could admit more than one 3.0 stationary solution, so we have to study the proper one. 2.5 0.8 We choose the solution that coincides with that obtained 2.0 with the non-Lindblad dynamics in the CL limit, since in 0.6 this limit the coeﬃcients of the extra terms of the LME 1.5 in Eq. (40) vanish. In [32] the stationary state in the case 0.4 of the non-Lindblad dynamics has been studied in detail, 1.0 and the variances have been calculated analytically. 0.2 0.5

Similarly to the linear QBM studied in the previous 2 4 6 8 10 12 14 16 section, we characterize the stationary state in terms of the variances of the Wigner function, and deﬁne the eccentricity, the cooling parameter, and the angle between Figure 6. Eccentricity η of the Wigner function at γ/Ω = 0.1, the major axis and the X axis of the phase space as be- for quadratic coupling. fore. These quantities are shown in Figs.6,7, and8, as functions of Λ and T , when γ/Ω = 0.1. In Fig.6 we point out that the eccentricity tends to zero in the CL 3.0 limit, while it increases away from it. This behavior is similar to that found for the linear QBM. We found that 2.5 4.0 for γ/Ω ≤ 0.1 the Brownian particle experiences neither 3.5 2.0 cooling nor genuine squeezing. 3.0 1.5 In contrast to the linear case, we do not ﬁnd a notice- 2.5 able rotation at low temperature in the quadratic one. 1.0 2.0

We would expect to observe this at larger values of γ, 1.5 as in the case of linear coupling. However, for larger 0.5 values of the damping constant the many stationary so- 2 4 6 8 10 12 14 16 lutions of the system of Eqs. (46-48) cross, and therefore it is not straightforward to determine the stationary solution of (45) that coincides with the one obtained in the Figure 7. Cooling parameter χ for quadratic coupling, at CL limit. Moreover, for larger values of γ the Gaussian γ/Ω = 0.1. ansatz given in Eq. (28) may fail to approximate any stationary states. To show this point, in Fig.9 we plotted 2 the time dependence of δx for several values of γ, at ﬁxed values of T and Λ. Above a certain value of γ, the po- We conclude this Section pointing out that, although sition variance does not converge to a stationary value. in Eqs. (46-48) we performed the Gaussian approxima- This suggests that in these cases the Gaussian solution tion at the level of the equations for the moments, it is of Eq. (45) is not stationary. Fig.9 is plotted for the possible to obtain exactly the same result applying the 2 2 initial conditions δx = δp = 1, corresponding to the case approximation directly on the original LME in Eq. (40), when the harmonic oscillator is in its ground state. The or on that LME expressed in terms of the Wigner func- choice of the initial conditions is not crucial, as we ob- tion, Eq. (45). In AppendixB we show, by a very general serve a very similar behavior with quite diﬀerent initial analytical demonstration, that the Gaussian approxima- conditions. tion applied to the original LME yields again a ME of 10

3.0 3.0

2.5 0 2.5

20 2.0 -0.1 2.0

15 1.5 -0.2 1.5

1.0 -0.3 1.0 10

0.5 -0.4 0.5 5

2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16

Figure 8. Angle θ/π between the major axis of the Wigner Figure 10. Plot of the product δxδp at γ/Ω = 0.1, for function, and the X axis of the phase space at γ/Ω = 0.1, for quadratic coupling. This quantity is always larger than 1, in quadratic coupling. accord with the HUP.

In Sec.II we dealt with QBM with a linear coupling. In this case, the stationary state can be represented ex- 20 γ/Ω=0.35 γ/Ω=0.2 actly by a Gaussian Wigner function. We put particu- γ/Ω=0.1 lar emphasis on the analysis of its properties in the low 15 temperature regime, where its properties are most interesting. 10 At low temperature we found that the Brownian particle exhibits genuine squeezing of the probability distri- 5 bution. An important feature of the stationary state in this regime is its rotation in the phase space, a direct consequence of the extra terms introduced to obtain a 5 10 15 20 25 30 Lindblad form for the equation. Another important effect experienced by the stationary state can be quantified Figure 9. Time dependence of δ2 for several values of γ, at x by the degree of cooling, expressing the ratio of the area Λ/Ω = 16 and kB T/~Ω = 4. The thin solid lines represent 2 of the effective support of the Wigner function to that the stationary value of δx in the state, namely the stationary solution of Eqs. (46-48) for such a quantity. of the Gibbs-Boltzmann distribution at the same temperature. A concrete physical system where cooling and squeezing can be encountered is suggested in [49]. In Sec.III we performed the same analysis for QBM the Lindblad form, guaranteeing therefore that the ap- with a coupling which is quadratic in the coordinates proximated solutions will preserve the HUP at all times. of the test particle. Importantly, we found that there We provide further numerical evidence of this fact in Fig. exists a critical value of the damping constant over which 10, where we plot the product of the two uncertainties δx the Gaussian ansatz fails to approximate any stationary and δp resulting by Eqs. (46-48), on which the Gaussian solutions. approximation has been carried out. As may be noticed Our procedure of adding extra terms to the BMME in the figure, the approximated equations do not produce derived in [32], so that the resulting equation is in a Lind- any violation of the HUP. blad form, is just one of the ways to obtain a Markovian dissipative LME. Other approaches have been presented, e.g., in Refs. [50, 51]. Moreover, for Gaussian dynamics IV. CONCLUSIONS AND OUTLOOK an exact (non-Markovian) closed master equation with time dependent coefficients can be derived [52–54]. In We studied a modification of the QBM model, focusing a forthcoming work we plan to derive a LME describing on the description of the stationary state of the Brownian QBM with a general class of couplings, and study its var- particle in the phase space, using the Wigner function ious limiting behaviors, in particular the small mass limit representation. To perform this analysis we considered a of the Brownian particle. ME of the Lindblad form, which ensures the positivity of The method which we used to treat the LME in this the density matrix at all times. In this way we got rid manuscript is not the only suitable one. Another possible of the Heisenberg principle violations discussed in [32], manner to solve this kind of equations, and in particular which prohibited the study of the dynamics in the low to characterize the stationary solution has been presented temperature regime. in Ref. [55]. The core of this procedure is turning LMEs 11 into partial first-order differential equations for a phase- ACKNOWLEDGMENTS space distribution (PSD) which generalizes well-known ones such as the Wigner function. The main point lies This work has been funded by a scholarship from in removing the evolution generated by the free Hamil- the Programa Mástersd’Excel-léncia of the Fundació tonian by including it in the interaction representation. Catalunya-La Pedrera, ERC Advanced Grant OSYRIS, Accordingly, the time dependence of the PSD originates EU IP SIQS, EU PRO QUIC, EU STREP EQuaM solely from the interaction term. Although the interac- (FP7/2007-2013, No. 323714), Fundació Cellex, the tion picture adopted in [55] could be used in the context Spanish MINECO (SEVERO OCHOA GRANT SEV- we are treating, its usefulness is not necessarily guaran- 2015-0522 and FOQUS FIS2013-46768), the Generalitat teed. In fact, the interaction picture represents a suitable de Catalunya (SGR 874). P.M. is funded by a “Ramón tool when the free part of the Hamiltonian describes a dy- y Cajal” fellowship. S.H.L. and J.W. were partially sup- namics much faster than that induced by the interaction ported by the NSF grant MS 131271. They are grateful term. In general this is not the case for the Brownian to L. Torner and ICFO for hospitality in the Summer of motion of a trapped particle, where the time scales re- 2015. lated to both processes can approach the same order of magnitude. On the other hand, employing this method looks like a very interesting task, which maybe can allow Appendix A: Heisenberg Uncertaintly Principle for us to go beyond the Gaussian approximation underlying density operators Sec. (III). This task, however, lies outside of the focus of the current paper. We thus reserve it for future works. The purpose of this Appendix is to present a self- There are other methods to correct the Heisenberg contained derivation of the Heisenberg uncertainty prin- principle violations highlighted in [32]. The BMME for ciple for density operators. We start from the pure state the quadratic QBM derived in [32] is based on a second case. Consider an arbitrary state |ψi and observables Aˆ order perturbative ME in the bath-particle coupling con- and Bˆ. Denoting by hAî the mean of the observable Aˆ stant. Going to higher orders permits one, in principle, to in the state |ψi, get rid of the violations of the Heisenberg principle. This task can be pursued by means of the time-convolutionless hAî = hψ|Aˆ|ψi, (A1) method presented in [1]. An advantage of this approach is that the resulting ME incorporates non-Markovian ef- for the variance of Aˆ we have fects. Nevertheless, since it arises from a perturbative 2 expansion, it does not allow to investigate the strong cou- 2 ˆ ˆ σA = ψ A − hAi ψ , (A2) pling regime γ > Ω, where cooling and squeezing effects are expected to be stronger. ˆ The ideas presented here can be used to investigate the and similarly for B. For future reference, let us also note physical behavior of an impurity in a BEC by open quan- that for any real number a, tum systems techniques. In this framework the impurity 2 ˆ 2 plays the role of the Brownian particle, while the set of ψ A − a ψ ≥ σA. (A3) the BEC Bogoliubov modes represents the environment. The linear QBM provides a useful tool to study the dy- The claim we want to prove is namics of the impurity in a uniform medium, while the QBM with a generic coupling may be used to investigate * ˆ ˆ +2 2 2 1 [A, B] impurities immersed in an inhomogeneous background, σAσB ≥ , (A4) such as the one provided by an harmonic trap. 4 i In conclusion, in AppendixB we proved that the Gaus- where the right-hand side contains the mean value of the sian approximation preserves the Lindblad form of a ME, observable [A,ˆ Bˆ]/i in the state |ψi. Introducing the vec- and so it does not yield any HUP violation, regardless of tors whether it is performed on the equations for the moments E E or directly on the LME. We developed this demonstra- |fi = (Aˆ − hAî)ψ and |gi = (Bˆ − hBî)ψ (A5) tion starting from a LME related to a Lindblad operator which is just quadratic in the creation and annihilation we have operators, because it is enough to cover the situation analysed in Sec.III. In general, one could extend the 2 2 2 σAσB = hf|fi hg|gi ≥ |hf|gi| (A6) proof to LMEs associated to Lindblad operators containing nth powers of creation and annihilation operators. applying the Cauchy-Schwarz inequality. The right-hand This, as far as we know, has never been shown and consti- side of the last inequality can be rewritten as tutes an interesting motivation for future projects. Also, 2 2 a generalization of this proof to LMEs for fermionic sys- hf|gi + hg|fi hf|gi − hg|fi |hf|gi|2 = + (A7) tems [56] is apparently possible and interesting. 2 2i 12 with both terms on the right-hand side nonnegative. In the case most important for us, when Aˆ = Xˆ is the Rewriting the second term as the square of the mean position operator and Bˆ = Pˆ is the momentum opera- of the observable [A,ˆ Bˆ]/2i, and leaving the first term tor, the commutator of Aˆ and Bˆ is a multiple of identity, h i [X,ˆ Pˆ] out (keeping it would lead to a stronger inequality, called X,ˆ Pˆ = i Iˆ. The mean value of in any state Robertson-Schrödingerinequality), we obtain the desired ~ i bound is thus equal to ~ and in particular, for the density op- eratorsρ ˆt, solving a Lindblad equation we obtain at all * +2 times the standard form of the Heisenberg Uncertainty 1 [A,ˆ Bˆ] σ2 σ2 ≥ (A8) Principle, A B 4 i 2 2 2 ~ P σX σP ≥ . (A15) in the pure state case. Now, if ρ = j pj |φji hφj| is an 4 arbitrary density operator, with pj non-negative coefficients summing up to 1, the mean of Aˆ in the state ρ equals Appendix B: Gaussian Approximation hAî(ρ) = Tr ρÂˆ . (A9) The purpose of this Appendix is to prove that the Gaussian approximation performed on the LME for Quadratic QBM preserves its Lindblad form. The For the variance of Aˆ in the state ρ we have demonstration we are about to present considers a 2 2 Gaussian approximation carried out directly on the ME, (ρ) ˆ ˆ (ρ) σA = Tr ρˆ A − hAi , (A10) while in SectionIIIB it has been done on the equations for the moments. As we will show, the two procedures are completely equivalent. and similarly for Bˆ. We thus have

2 2 Theorem For a quadratic Lindblad operator: (ρ) X ˆ ˆ (ρ) σ = pj φj A − hAi φj A ˆ 2 ˜ † 2 † ˜ † j L =α ãˆ + β(â ) +γ ãˆ aˆ + δaˆ + ãˆ +η ˜ (B1) 2 X (φj ) ≥ pj σA (A11) the self-consistent Gaussian approximation preserves the j Lindblad form (and thus the positivity ofρ ˆ and HUP). The annihilation and creation operators are repre- 2 † ˜ ˜ (φj ) sented respectively bya ˆ anda ˆ , whileα, ˜ β, γ,˜ δ, ,˜ η˜ where σ denotes the variance of Aˆ in the state A are complex parameters. It is immediate to prove that |φji, and in the last step we used inequality Eq. (A3). the Lindblad operator introduced in Eq. (39) can be Similarly, expressed in the form showed in Eq. (B1). Note that it is possible to assume haî = 0, since it just shifts the 2 2 (ρ) X (φj ) parameters. σB ≥ pj σB (A12) j Lemma 1 The parameterη ˜ in Eq. (B1) can be shifted By the (discrete version of) the Cauchy-Schwarz inequal- arbitrarily. ity (it is crucial that pj ≥ 0 here!) we obtain Proof: the core of the proof lies in the fact that any ad-  2 ditive constant in the definition of the Lindblad operator 2 2 (ρ) (ρ) X (φj ) (φj ) can be compensated by a re-definition of the Hamilto- σA σB ≥  pjσA σB  (A13) j nian, namely: ∂ρˆ i which, using the pure-state version of the uncertainty = − [H,ˆ ρˆ] + DL+∆˜η(ˆρ) (B2) principle, is bounded from below by ∂t ~ i ˆ ˆ 2 = − [H + ∆H∆˜η, ρˆ] + DL(ˆρ),  h i 2  h i (ρ) ~ * A,ˆ Bˆ + * A,ˆ Bˆ + 1 X 1  pj φj φj  =   where: 4 i 4  i  j D (ˆρ) = LˆρˆLˆ† − Lˆ†Lˆρ/ˆ 2 − ρˆLˆ†L/ˆ 2, (B3) (A14) L which is the desired mixed-state version of the inequality. is the Lindblad dissipator, and: In the last application of the Cauchy-Schwarz inequality, it is crucial that we are dealing with a density operator, i ∆Hˆ = − [(∆˜η)Lˆ† − (∆˜η)∗Lˆ], (B4) so that the eigenvalues pj are nonnegative. ∆˜η 2 13 with ∆˜η ∈ C. The non-trivial terms are: Of course changing of Hamiltonian is allowed, since ˆ† ˆ ˆ ˆ† ˆ† ˆ ˆ ˆ† it just modifies the time dependence ofa ˆ anda ˆ† in the hd2d1id2ρˆd1 + hd1d1id2ρˆd2 (B13) interaction picture. ˆ† ˆ ˆ ˆ† ˆ† ˆ ˆ ˆ† +hd2d2id1ρˆd1 + hd1d2id1ρˆd2 ρˆ Lemma 2 It is possible to perform the factorization: − hdˆ†dˆ idˆ†dˆ + hdˆ†dˆ idˆ†dˆ , 2 1 1 2 2 2 1 1 2 Lˆ = dˆ dˆ (B5) † † † † ρˆ 1 2 − hdˆ dˆ idˆ dˆ + hdˆ dˆ idˆ dˆ , 1 1 2 2 1 2 2 1 2 with: The resulting ME has a dissipator of the form: ˆ ˜ ˜ † ˜ d1 = Aaˆ + Baˆ + C (B6) X ˜ ˆ ˆ† 1 ˆ† ˆ ˆ † DL(ˆρ) = Γij diρˆdj − {djdi, ρˆ} , (B14) d2 =a ˆ + Dãˆ + E˜ 2 i,j=1,2

Proof: comparing Eqs. (B5) and (B1), one obtains: ˜ ˆ† ˆ 0 0 where Γij = hdj0 di0 i, where 1 = 2 and 2 = 1. This matrix is evidently positive definite, as follows from the ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ A =α, ˜ AD + B =γ, ˜ AD + CE =η ˜ (B7) Schwartz inequality, so that the dissipator is again of B˜D˜ = β,˜ A˜E˜ + C˜ = δ,˜ B˜E˜ + C˜D˜ = δ,˜ Lindblad form. Note that the generalization to many oscillators, many so that Lindblad operators is straightforward. Note also that the non-generic case is simple to treat. It requires, however, D˜ = β/˜ B,˜ α˜β/˜ B˜ + B˜ =γ ˜ (B8) a direct calculation. The quartic Lindblad term in this case is treated as above, while the quadratic one does not provide in general two solutions B˜1 and B˜2 for B˜, and need to be touched, since it already describes a Gaussian quantum process. The third order term on the other α˜E˜ + C˜ = δ,˜ B˜E˜ + (β/˜ B˜)C˜ =η. ˜ (B9) hand partially vanishes and partially gives contributions to the Hamiltonian in the Gaussian approximation. If we can solve these linear equations for E˜ and C˜, we The remaining question is whether the approximation may plug the solution intoα ˜D˜ + C˜E˜ =η ˜, and adjust η that we perform on the level of the ME is the same as adequately (which we can do according to Lemma 1). the Gaussian de-correlation we performed according to It is easy to check that the two equations for E˜ and the Wick’s theorem prescription at the level of the equa- C˜ cannot be solved if B˜1 = B˜2 = 0, which impliesγ ˜ = tions for the moments in Sec.IIIB. To illustrate this, we 0 andα ˜β˜ = 0, i.e. the non-generic case Lˆ =α ãˆ2 + consider an arbitrary operator Oˆ and we derive the dy- δãˆ + ãˆ† +η ˜, and the related one withα ˜ = 0 , β˜ 6= namical equations for its average value starting by the 0. The caseα ˜ = β˜ = 0 is trivial, as it corresponds to ME induced by the superoperator in Eq. (B10). linear Lindblad operator: for such a case, the Gaussian The dynamical equation for the average value of an approximation is not needed, since there exists an exact operator Oˆ presents the following form: solution of Gaussian form. ∂hOî 1 Now we prove the Theorem in the generic case: u (1) (2) (3) = hO + hO − hO + hO (B15) Proof of the Theorem: We look to the Lindblad dissipator ∂t 2 related to the factorized Lindblad operator in Eq. (B5): in which: h i 1 u i ˆ ˆ D (ˆρ) = dˆ dˆ ρˆdˆ†dˆ† − {dˆ†dˆ†dˆ dˆ , ρˆ} (B10) hO = − Tr O H, ρˆ (B16) L 1 2 2 1 2 2 1 1 2 ~ (1) ˆ ˆ ˆ † ˆ† ˆ† ˆ† ˆ ˆ ˆ h = Tr(Od1d2ρdˆ 2d1) = hd2d1Od1d2i In the Gaussian approximation, one replaces pairs of op- O (2) ˆ ˆ† ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ erators by their mean values. “Anomalous” terms gener- hO = Tr(Od2d1d1d2ρˆ) = hOd2d1d1d2i ate contributions that may be reabsorbed in the Hamil- h(3) = Tr(Oˆρˆdˆ†dˆ†dˆ dˆ ) = hdˆ†dˆ†dˆ dˆ Oî. tonian, such as O 2 1 1 2 2 1 1 2 Performing the Gaussian approximation at the level of 1 1 hdˆ dˆ i ρˆdˆ†dˆ† − {dˆ†dˆ†, ρˆ} = − hdˆ dˆ i[dˆ†dˆ†, ρˆ] the equation for the moments means to carry out such 1 2 2 1 2 2 1 2 1 2 2 1 an approximation on the average values in Eqs. (B16), (B11) (1) ˆ ˆ ˆ † ˆ† ˆ† ˆ† ˆ ˆ ˆ hO = Tr(Od1d2ρdˆ 2d1) = hd2d1Od1d2i (B17) and: ˆ† ˆ† ˆ ˆ ˆ † ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ ˆ ' hd2d1ihOd1d2i + hd2d1Oihd1d2i − hd2d1ihOihd1d2i 1 1 + hdˆ†dˆ ihdˆ†Oˆdˆ i + hdˆ†Oˆdˆ ihdˆ†dˆ i − hdˆ†dˆ ihOîhdˆ†dˆ i hdˆ†dˆ†i dˆ dˆ ρˆ − {dˆ dˆ , ρˆ} = hdˆ†dˆ†i[dˆ dˆ , ρˆ]. (B12) 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 1 1 2 ˆ† ˆ ˆ† ˆ ˆ ˆ† ˆ ˆ ˆ† ˆ ˆ† ˆ ˆ ˆ† ˆ +hd2d2ihd1Od1i + hd2Od2ihd1d1i − hd2d2ihOihd1d1i, 14

(2) ˆ ˆ† ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ hO = Tr(Od2d1d1d2ρˆ) = hOd2d1d1d2i (B18) B13), obtained by performing the Gaussian approximation on the ME related to a dissipator in Eq. (B10). ' hdˆ†dˆ†ihOˆdˆ dˆ i + hOˆdˆ†dˆ†ihdˆ dˆ i − hdˆ†dˆ†ihOîhdˆ dˆ i 2 1 1 2 2 1 1 2 2 1 1 2 This proves that performing the Gaussian approximation ˆ† ˆ ˆ ˆ† ˆ ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ ˆ ˆ† ˆ + hd2d1ihOd1d2i + hOd2d1ihd1d2i − hd2d1ihOihd1d2i at the level of the ME is equivalent to doing it at the ˆ† ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ † ˆ ˆ† ˆ level of the equations for the moments of an observable. +hd d2ihOˆd d1i + hOˆd d2ihd d1i − hd d2ihOîhd d1i, 2 1 2 1 2 1 Note that the equations resulting by this approximation will always admit a Gaussian solution, although it is not guaranteed that the latter is stationary. h(3) = Tr(Oˆρˆdˆ†dˆ†dˆ dˆ ) = hdˆ†dˆ†dˆ dˆ Oî (B19) O 2 1 1 2 2 1 1 2 The demonstration we developed holds for Lindblad ˆ† ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ ˆ ' hd2d1ihd1d2Oi + hd2d1Oihd1d2i − hd2d1ihOihd1d2i operators which are quadratic in the creation and anni- † † † † † † hilation operators. This case covers the situation studied + hdˆ dˆ ihdˆ dˆ Oî + hdˆ dˆ Oîhdˆ dˆ i − hdˆ dˆ ihOîhdˆ dˆ i 2 1 1 2 2 1 1 2 2 1 1 2 in Sec. (III), but it is not the most general one. In fact, ˆ† ˆ ˆ† ˆ ˆ ˆ† ˆ ˆ ˆ† ˆ † ˆ ˆ ˆ† ˆ +hd2d2ihd1d1Oi + hd2d2Oihd1d1i − hd2d2ihOihd1d1i. one could consider also LMEs with Lindblad operators containing higher powers of creation and annihilation op- It is now tedious but easy to check that replacing the erators. Extending the proof we presented to this general expressions in Eq. (B17-B19) in Eq. (B15) we get the dy- case is an interesting perspective that we reserve for fu- namical equations generated by the terms in Eqs. (B11- ture works.

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