PHYSICAL REVIEW E 88, 012102 (2013)

Nonequilibrium fluctuation-dissipation inequality and nonequilibrium uncertainty principle

C. H. Fleming and B. L. Hu Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA

Albert Roura Max-Planck-Institut fur¨ Gravitationsphysik (Albert-Einstein-Institut), Am Muhlenberg¨ 1, 14476 Golm, Germany (Received 16 January 2013; revised manuscript received 7 May 2013; published 3 July 2013) The fluctuation-dissipation relation is usually formulated for a system interacting with a heat bath at finite temperature, and often in the context of linear response theory, where only small deviations from the mean are considered. We show that for an open quantum system interacting with a nonequilibrium environment, where temperature is no longer a valid notion, a fluctuation-dissipation inequality exists. Instead of being proportional, quantum fluctuations are bounded below by quantum dissipation, whereas classically the fluctuations vanish at zero temperature. The lower bound of this inequality is exactly satisfied by (zero-temperature) quantum noise and is in accord with the Heisenberg uncertainty principle, in both its microscopic origins and its influence upon systems. Moreover, it is shown that there is a coupling-dependent nonequilibrium fluctuation-dissipation relation that determines the nonequilibrium uncertainty relation of linear systems in the weak-damping limit.

DOI: 10.1103/PhysRevE.88.012102 PACS number(s): 05.30.−d, 03.65.Yz, 05.40.−a

I. INTRODUCTION instantaneous, but where there is nonlocal damping:  The fluctuation-dissipation relation (FDR) is a fundamental t m x¨(t) + 2 dτ γ(t − τ) x˙(τ) − F (x) = ξ(t), (4) relation in nonequilibrium statistical mechanics, dating back to −∞ Einstein’s Nobel Prize-winning work on Brownian motion [1] and Nyquist’s seminal work on electronic conductivity [2]. in which case the classical fluctuation-dissipation relation We distinguish the FDR from the more common fluctuation- generalizes to dissipation theorem (FDT), because, as we will demonstrate, ν˜(ω) = 2kB T γ˜(ω). (5) nonequilibrium environments also permit a FDR. In the nonequilibrium case, there is no universal relation, but we will If the response has some lag, then the thermal noise is colored, show that there is a universal inequality that all such relations and vice versa, whereas above for instantaneous response the satisfy. noise was white. Finally, when we apply this same formalism To give our inequality a physical interpretation, we will to the phase-space characteristics of a quantum oscillator with couch it within the tractable context of a general quantum the same type of Langevin equation [3], the quantum FDR can system under the influence of quantum noise, from a linear be seen to generalize to  environment or where the influence of the environment is hω¯ weak. However, let us first briefly introduce the known results ν˜(ω) = hω¯ coth γ˜(ω). (6) 2k T with the simple example of a one-dimensional Langevin B equation. The equation of motion for a classical damped The qualitative difference between the classical and quantum particle with time-local response coupled to a thermal bath FDR is that at zero temperature there are ground-state fluctu- with temperature T is given by ations proportional toh ¯ . Only at sufficiently high temperature does the quantum FDR asymptote to the classical relation. + − = m x¨(t) 2 γ0x˙(t) F (x) ξ(t) . (1) What we show in this work is that for any nonequilibrium damping noise environment, one has the fluctuation-dissipation inequality (FDI), Deriving this equation from a microscopic model results in a ν˜(ω)  |hω¯ γ˜(ω)|, (7) relation between the damping and noise correlation, or, in other words, for any amount of damping, the correspond- ≡   = − ν(t,τ) ξ(t) ξ(τ) ξ γ0 2kB Tδ(t τ), (2) ing noise must be greater than the corresponding hypothetical ground-state fluctuations. Moreover, the FDI is fundamentally or more simply in the frequency domain, related to the Heisenberg uncertainty principle (HUP). As we will show, if one could have an environment that violated the ν˜(ω) = 2kB Tγ0, (3) FDI, then it would relax the system to a state that violates the  HUP. Furthermore, if one allows the state of the environment where ν(t,τ) = ν(t −τ) andν ˜(ω) = dt e−ıωt ν(t). In short, to violate the HUP, then it may also violate the FDI. the noise is proportional to both the temperature and the Like the HUP, the FDI is mathematical in nature and is amount of damping or resistance. The same can also be said a precise inequality which must be obeyed by all quantum of the diffusion, which is the Einstein-Smoluchowski relation. environments, in all regimes. Whereas, in its most general This model can be easily generalized to a response that is not form, the HUP relates two quantum operators of arbitrary

1539-3755/2013/88(1)/012102(10)012102-1 ©2013 American Physical Society C. H. FLEMING, B. L. HU, AND ALBERT ROURA PHYSICAL REVIEW E 88, 012102 (2013) systems, the FDI relates the two-time noise and dissipation (or dissipation kernels will always obey our inequality. However, equivalently, susceptibility) kernels of arbitrary environments. the physical interpretation of these kernels is less obvious Though it is exact, we will mostly discuss its context and in the case of strong coupling to nonlinear environments physical implications in one of two regimes, both with station- with significant higher-order cumulants for their processes, ary correlations: (1) the regime of weak coupling to a general such as fermionic environments. For example, for a thermal environment and (2) the regime of nonperturbative coupling environment consisting of two-level systems the noise kernel is to a linear environment. Moreover, for linear systems we temperature independent, whereas the dissipation kernel has additionally show that the nonequilibrium FDR (which must explicit temperature dependence, as shown in Appendix A. satisfy the FDI) determines the nonequilibrium uncertainty This nonintuitive behavior is opposite to that of typical relation (which must satisfy the HUP) for weak coupling. bosonic environments, such as photons and phonons. Leaving In the following section we present the necessary back- this consideration aside, let us briefly discuss next how the ground material of quantum open systems, wherein we correlation function arises in the most common formalisms formally categorize quantum noise in terms of its time for open-system analysis. dependence, dissipation, and microscopic origin. Readers In the influence functional formalism [8] for the quantum familiar with this may skip to our results in the later sections Brownian model with continuous system couplings Ln(x; t) and refer back as needed. In Sec. III we derive the FDI from a (in the Lagrangian) and a linear environment [9–11]the microscopic model and contrast it to the usual thermal FDR. correlation function appears as the kernel in the exponent of On the other hand, in Sec. IV we work from the other end a Gaussian influence functional, called the influence kernel ζ and motivate the FDI phenomenologically, but less generally. in Refs. [12,13]. The reduced density matrix ρ for the open This result also produces the (weak-coupling) nonequilibrium system evolves according to the double path integral uncertainty relation for quantum Brownian motion, which can x|ρ(t)|y   be contrasted to the finite-temperature uncertainty relation x y + ı S[x(t)]− ı S[y(t)]− [x(t),y(t)] [4–7]. = Dx(t) Dy(t) e h¯ h¯ ,

x0 y0 x0,y0 II. NOISE AND BACK-REACTION (12) A. Open systems and noise with the initial-value average given by  + Consider the closed system environment Hamiltonian ··· ≡  |ρ |  ··· x0,y0 dx0 dy0 x0 (0) y0 , (13) environment renormalization = + + + for factorized initial states ρ(0) ⊗ ρ (0). The Gaussian influ- HC H HE HI HR , (8) E  ence phase is given by system interaction   1 t t with the interaction Hamiltonian expanded as a sum of tensor =    [x(t),y(t)] 2 dτ dτ n(τ) νnm(τ,τ ) m(τ ) products (in the Schrodinger¨ picture): 2¯h 0 0 n,m   t τ = ⊗ ı    HI(t) Ln(t) ln(t), (9) + dτ dτ (τ) μ (τ,τ ) (τ ), h¯ n nm m n n,m 0 0 where Ln(t) and ln(t) are system and environment operators (14) respectively, possibly with some intrinsic time dependence. in terms of the difference and sum coordinates The environment coupling operators ln will typically be collective observables of the environment, with dependence n(t) = Ln(x; t) − Ln(y; t), (15) upon very many modes. For linear environments or weak coupling to general environments, the central ingredient of n(t) = Ln(x; t) + Ln(y; t). (16) any open-systems analysis is the second-order (multivariate) The Gaussian influence is exact for a linear environment correlation function of the environment (E): and perturbative for general environments. The conventional =  αnm(t,τ) ln(t) lm(τ) E, (10) influence functional here requires the system only to be coupled with continuous variables, though the formalism can where ln(t) represents the time-evolving ln(t) in the interaction be extended to fermionic systems with Grassman numbers. The (Dirac) picture. The correlation function can always be written noise kernel ν appears in the influence kernel as the correlation in terms of two real kernels, respectively symmetric and of an ordinary real stochastic source, whereas the dissipation antisymmetric, and traditionally referred to as the noise kernel kernel μ alone would produce a purely homogeneous (though and dissipation kernel: not positivity preserving in general) evolution. α = ν + μ These same roles can also be inferred from the Heisenberg  (t,τ) (t,τ) ıh¯  (t,τ) . (11) equations of motion for the system operators after integrating complex noise noise dissipation the environment dynamics, producing the so-called quantum Note that despite this choice of name the dissipation kernel Langevin equation [14]. In Appendix B we generalize the can also give rise to nondissipative effective forces. quantum Langevin equation to nonlinear systems, although The correlation function α(t,τ) exists for any quantum restricted to Gaussian influence functionals and, hence, pertur- system and environment, and its corresponding noise and bative for nonlinear environments. The Heisenberg equations

012102-2 NONEQUILIBRIUM FLUCTUATION-DISSIPATION ... PHYSICAL REVIEW E 88, 012102 (2013) of motion for an open system operator S(t), where the for all vector functions f(t) indexed by the noise. Hermiticity environment has been integrated out, can be expressed as is relatively straightforward to see, whereas the positivity in

Eq. (25) can be proven by inserting Eq. (10) and manipulating h¯ S˙(t) =+ı[H,S(t)] + ı {l (t),[L (t),S(t)]}, (17) n n this expression into the form n   t t t ≡ ξ − † † ln(t) n(t) 2 dτ μnm(t,τ) Lm(τ), (18) dτ1 dτ2 f (τ1) α(τ1,τ2) f(τ2) =Q QE, (26) m 0 0 0 where H is the free system Hamiltonian and ξ (t)isan in terms of the operator n  operator-valued stochastic process with two-time correlations t = Q dτ1 fn(τ1) ln(τ1), (27) ξ (t) ξ (τ)ξ = αnm(t,τ) = νnm(t,τ) + ıhμ¯ nm(t,τ). (19) n m n 0 In the classical limit, the dissipation kernel vanishes in the and so positivity must follow, as the environment’s density ma- commutator expectation value for the operator noise process, trix is positive definite. Positivity and the noise decomposition but not in the Langevin equation’s memory kernel. (11) are the key properties from which the FDI arises. Finally, using the notation of Ref. [15], the second-order (in Stationary correlations are defined by their invariance under the interaction) master equation [16–18] of the reduced density time translations, matrix ρ can be represented in terms of the noise correlation as α = α − ı (t,τ) (t τ), (28) ρ˙ =− [H,ρ] + L {ρ}, (20) h¯ 2 and can produce asymptotically stationary (time-independent) with the second-order contribution given by the operation master equations. Such correlations are produced when the environment is in an initially stationary state and its coupling 1 L {ρ}≡ [L ,ρ (A  L )† − (A  L ) ρ], (21) operators in the Schrodinger¨ picture are constant in time: 2 2 n nm m nm m h¯ nm ρ (0) = pE(εi )|εi εi |, (29) A  E where the operators and product define the second-order i operators  yielding the correlation function t  ≡ {G } + − (Anm Lm)(t) dτ αnm(t,τ) 0(t,τ) Lm(τ) , (22) α (t,τ) = p (ε )ε |l |ε ε |l |ε  e ıεij (t τ), (30) 0 nm E i i n j i m j ij given the free system propagator G (t,τ):ρ(τ) → ρ(t). For 0 ∗ example, for a time-independent system Hamiltonian H we where z = z denotes the complex conjugate, εij ≡ εi −εj , have the free system propagator |εi  denotes the energy basis of the environment, and pE(εi ) + ı − − ı − are its stationary probabilities at the initial time. The associated G ρ = h¯ H(t τ) ρ h¯ H(t τ) 0(t,τ) e e . (23) characteristic function can be obtained quite directly from the This formalism also makes no assumptions as to the structure mode sum  of the system, e.g., continuous or discrete; however, the master +∞ 1 +ıωt equation is strictly perturbative, even for linear environments. α(t) = dωe α˜ (ω), (31) 2π −∞ Therefore the system Hamiltonian must contain sufficiently large transition energies to justify a perturbative expansion in yielding the Fourier transform the interaction, and the environment correlations cannot be ∝  | | −  | | −  excessively long ranged, or else the expansion can be secular α˜ nm(ω) 2π pE(εi ) εi ln εi ω εi lm εi ω , (32) in time. i One context in which the influence functional, Langevin where the underscored proportionality here is strictly in equation, and master equation all work together seamlessly is reference to the continuum limit of the reservoir which relates in the quantum Brownian motion of linear systems [14,19,20]. environmental mode sums to integrals given the infinitesimal In addition to a a quantum Langevin equation for noncommut- strength of individual environmental mode couplings. This can ing operators, linearity makes it possible in that case to have a be more rigorously defined through the use of a finite spectral Langevin equation for real classical stochastic processes from density function in place of the infinitesimal environment which general quantum correlation functions and the master couplings. equation can be exactly derived [19,20]. Also of note are quasistationary correlations of the form

α(t,τ) = αS(t −τ) + δα(t +τ), (33) B. The correlation function where α (t −τ) denotes a stationary correlation function, From its microscopic definition, Eq. (10), the environmental S or, more specifically, the stationary projection of α(t,τ), correlation function is Hermitian in the sense of while δα(t +τ) is an additional nonstationary contribution. α(t,τ) = α†(τ,t) (24) Such correlations will result from (time-independent) linear coupling to an environment with nonstationary initial state, and also positive definite in the sense of   such as a squeezed thermal state. In these cases the stationary t t † projection of the correlation function does correspond to the dτ1 dτ2 f (τ1) α(τ1,τ2) f(τ2)  0, (25) 0 0 stationary projection of the initial state of the environment.

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As for the nonstationary contributions, due to their highly induced forces. A canonical example of this is nonequilibrium oscillatory behavior in the late-time limit they typically lose electrodynamics, wherein these corrections contain both the effect asymptotically. Therefore, quasistationary correlations mass renormalization of the electron as well as magnetostatic can produce an asymptotically stationary master equation with forces between electrons [22]. equivalent asymptotics as generated by their corresponding stationary correlation. 1. The energy-damping property To demonstrate that a symmetric damping kernel can be C. The damping kernel given a physical interpretation we will consider the time Here we will use the Langevin equation (B24) to determine evolution of the effective system Hamiltonian. First, we will the role of the dissipation kernel and to motivate an appropriate assume the following dynamical evolution of the system- method of renormalization. The correlation function α(t,τ)is coupling operators: positive definite, and therefore the noise kernel ν(t,τ)mustalso be positive definite, which follows from Eq. (25) by taking a h¯ L˙ m(t) =+ı [Heff(t),Lm] . (38) real f(t). However, the dissipation kernel μ(t,τ) is not positive definite, but it is related to the damping kernel γ (t,τ), which This relation is exact if all of the system coupling operators Ln is given by commute, otherwise it is only perturbative. Now we substitute Eq. (37) into Eq. (36), apply the above relation, and integrate ∂ μ(t,τ) =− γ (t,τ), (34) to obtain the following open-system energy as a function of ∂τ time when neglecting the transient slip: and can be positive definite, negative definite, or indefinite, as will be discussed in Sec. II C2. For nonstationary noise, Heff(t) = Heff(0) − Hγ (t) + Hξ (t), (39) Eq. (34) is an incomplete definition, and constructing a sym-   t t metric damping kernel will require additional considerations. H (t) ≡+ dτ dτ γ (τ,τ){L˙ (τ ),L˙ (τ)}, Assuming that relation (34) is sufficient, which is the case γ nm n m nm 0 0 for stationary correlations, the nonlocal term present in the (40) Langevin equation can be represented:   t t ≡− 1{ξ ˙ } = ˙ Hξ (t) n(t),Ln(t) . (41) dτ μnm(t,τ) Lm(τ) dτ γnm(t,τ) Lm(τ) 2 0   0   n dissipation damping This relation reveals that a positive-definite damping kernel − + γnm(t,t) Lm(t) γnm(t,0)Lm(0) . will only decrease the system energy, whereas a negative- renormalization slip definite damping kernel will only increase the system energy. The criteria for each condition will be more thoroughly covered (35) in Sec. II C2. This expression for Hγ (t) also contrasts nonlocal In the damping-kernel representation we now have three terms: damping to local damping. Evaluation with a delta-correlated nonlocal damping, renormalization forces, and transient “slip” damping kernel yields damping which is strictly dissipative forces. The nonlocal damping and renormalization forces will at every instant of time whereas nonlocal damping is only be more thoroughly considered in the following subsections. guaranteed to have an accumulated dissipative effect since the The transient slip is a pathology associated with factorized initial time. initial conditions and can be avoided with the consideration of a properly correlated initial state [21]. 2. Classification of stationary damping Given a symmetric damping kernel, which is the case for stationary correlations, our quantum Langevin equation can Two-time correlation functions can always be decomposed then be expressed as into a real noise kernel and dissipation kernel as in Eq. (11). The Hermiticity stated in Eq. (24) leads to the relations ˙ =+ + ı {ξ } h¯ S(t) ı[Heff(t),S(t)] n(t),[Ln(t),S(t)] 2 1 T n ν t,τ = α t,τ + α τ,t ,  ( ) [ ( ) ( )] (42) t 2 + ı dτ γ (t,τ){L˙ (τ),[L (t),S(t)]}, (36) 1 nm m n μ(t,τ) = [α(t,τ) − αT(τ,t)]. (43) nm 0 2ıh¯ when discarding the transient slip and where the effective For stationary correlations α(t −τ), which lead to a symmetric Hamiltonian is given by damping kernel, one can introduce the Fourier transform α = +∞ −ıωτ α = − ˜ (ω) −∞ dτ e (τ). The noise and damping kernels Heff(t) H(t) Ln(t) γnm(t,t) Lm(t). (37) are then Hermitian in both noise index and frequency argu- nm ment: We denote this term as the “effective Hamiltonian” and not the “renormalized Hamiltonian,” because the correction γ˜ (ω) = γ˜ †(ω) = γ˜ ∗(−ω), (44) may contain both divergences which require renormalization and terms which describe completely new environmentally ν˜(ω) = ν˜ †(ω) = ν˜ ∗(−ω). (45)

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The first equality in Eqs. (44)–(45) follows from the Fourier dissipation kernel is given by their difference. The essential transform of Eq. (24), whereas the second equality follows point is that if there is any damping, or amplification, there will from the fact that ν(τ,t) = νT(t,τ) and γ (τ,t) = γ T(t,τ). be quantum noise, and Eq. (50) determines its lower bound. Moreover, by Bochner’s theorem both α˜ (ω) and ν˜(ω)are This is quite a departure from classical physics where noise can positive-definite for all frequencies; see Appendix C.For be made to vanish in the zero-temperature limit, although the stationary correlations, the damping kernel is uniquely defined lower bound of this inequality is satisfied by zero-temperature by relation (34). Again the damping kernel γ˜ (ω) may be quantum noise since α˜ (|ω|) = 0 in that case. positive definite, negative definite, or indefinite. For the case of a single collective system coupling, coupled As proven in Sec. II C1, environments with positive-definite to one or more environments, it is sufficient to define a damping kernels are damping or resistive environments, while fluctuation-dissipation relation those with negative-definite damping kernels are amplifying. ν˜(ω) = κ˜(ω)˜γ (ω), (52) If the system coupling operators Ln are position operators, the damping terms correspond to forces linear in momentum. ν˜(ω) κ˜(ω) ≡ , (53) Stationary correlations are the easiest to dissect and the most γ˜(ω) well behaved. Their dissipation and damping kernels are related by withκ ˜(ω) being the fluctuation-dissipation kernel [10,11] which relates fluctuations to dissipation. For multivariate noise μ = γ ˜ (ε) ıε ˜ (ε), (46) one might use the symmetrized product and from the definition of the dissipation kernel in Eq. (43) and ν˜(ω) = 1 [κ˜(ω) γ˜ (ω) + γ˜ (ω) κ˜(ω)], (54) the double Hermiticity in Eq. (44)–(45), the damping kernel 2 κ γ will be exactly positive or negative definite, respectively, if we which would ensure ˜(ω) to be positive definite if ˜ (ω)is,in have a strict inequality between positive and negative energy accord with this being a (continuous) Lyapunov equation [23]. argumented environmental correlations: We will use this particular definition for quantum Brownian motion in the next section. Inequality (50) can now be α −| | α∗ +| | ˜ ( ω ) > ˜ ( ω ) (Damping), (47) restated as ∗ κ  | | α˜ (−|ω|) < α˜ (+|ω|) (Amplifying), (48) ˜(ω) hω¯ , (55) κ with inequality in the sense of Appendix C.FromEq.(32), one for damping environments. Typically ˜(ω) will contain depen- can show that damping environments result when the initial dence upon the precise nature of the environment couplings ln. stationary probability of the environment pE(ε) is a mono- tonically decreasing function of the environment energy. Am- B. Equilibrium fluctuation-dissipation relation plifying environments result from monotonically increasing Let us consider a time-independent system-environment functions or population inversion. The most common example interaction and environment Hamiltonian as well as initial of each being positive and negative temperature reservoirs. stationary probabilities of the environment given by pE(ε). If Furthermore, in the perturbative master-equation formalism, the FDR is to be independent of precisely how the system the above equations imply that damping environments induce and environment are coupled, then one can work out from the a higher probability of transitions to lower energy states, microscopic theory (32) that the FDR kernel must be a scalar while amplifying environments induce a higher probability quantity, directly related to the initial state of the environment of transitions to higher energy states. by way of κ˜(ω) p (ε − ω) + p (ε) III. NONEQUILIBRIUM RELATIONS = E E , (56) hω¯ pE(ε − ω) − pE(ε) A. Nonequilibrium fluctuation-dissipation relation for all ε. To prove this one first applies relation (32) to and inequality definitions (42)–(43), and notes that if the dissipation and From the definitions of the multivariate noise kernel ν (42), noise are related in a manner independent of the coupling dissipation kernel μ (43), and damping kernel γ (46), one can then the two kernels must be related term by term in a sum prove the fluctuation-dissipation inequality: over couplings. ν  ±ıh¯ μ, (49) But such an equality between the FDR kernel and mode probabilities implies the functional relation with inequality in the sense of Appendix C. In the case of  κ˜(ω) + stationary noise it takes the more useful form hω¯ 1 pE(ε − ω) = pE(ε), (57) κ˜(ω) − 1 ν˜(ω)  ±hω¯ γ˜ (ω), (50) hω¯ in the Fourier domain where the ω would denote energy-level where the ω translations can factor out. This factorization transitions of the system; When the noise is univariate, the property is unique to exponential functions; therefore, only the ∝ −βε inequality can also be written thermal distribution pE(ε) e can produce a fluctuation- dissipation relation which is generally coupling independent. ν˜(ω)  |hω¯ γ˜(ω)|. (51) We then have that  To prove the general inequality, one simply notes that the noise hω¯ κ˜T (ω) ≡ hω¯ coth , (58) kernel is the sum of two positive-definite kernels, whereas the 2kB T

012102-5 C. H. FLEMING, B. L. HU, AND ALBERT ROURA PHYSICAL REVIEW E 88, 012102 (2013) for the thermal distribution. One should be careful to note that Wigner function representation [27], HUP violating states of the thermal FDR is not special because it exists, nor because of the environment can be constructed which violate the quantum its simple form, but because of its complete independence of FDI. Such is the case for the classical vacuum, which has the details of the system-environment interaction. “Observable vanishing fluctuations yet finite damping. Now we shall show dependence” for nonequilibrium correlations was also noticed that FDI violating noise can also relax the system into a HUP in Ref. [24]. In a more general context, the thermal FDR is violating state. also special because it ensures a relaxation to detailed balance Let us consider weakly influencing a system of oscillators in a coupling-invariant manner. In fact, these properties can be at resonance, all with mass m and frequency ω, via position- shown to be equivalent [15]. position coupling to some phenomenological set of noise As a concrete example of an elegant yet nonequilibrium processes with resistive correlation α˜ (ω). We do not assume FDR, the late-time dominating stationary correlations for the system-environment couplings to be identical, nor do linear coupling to a squeezed thermal reservoir [25,26] will we neglect the presence of cross-correlations among the noise produce the FDR kernel processes. Furthermore, the system may relax in general  to stationary states which do not correspond to thermal hω¯ r = κ˜T (ω) cosh[2 r(ω)]¯hω coth , (59) equilibrium. A simple example would be a situation where 2kB T there is a constant heat flow through the system between two where r(ω) is the squeezing parameter, which may be allowed thermal baths at different temperatures. to vary with the energy scale. One can easily see that this FDR From the results of Ref. [20,28] and the second-order master also obeys inequality (50) as it must. equation coefficients (21), the damping kernel γ˜ (ω) will play the role of the dissipation coefficients and the noise kernel ν˜(ω) will play the role of the normal diffusion coefficients C. Fluctuation-dissipation equality in the Fokker-Planck or master equation. Integrals over the For stationary environments, we can more easily determine two kernels will then provide the system renormalization when equality is achieved between the fluctuations and and antidiffusion coefficients, respectively. Given sufficient dissipation in the sense of dissipation and bandwidth-limited correlations, the system ν˜(ω) =±hω¯ γ˜ (ω). (60) will relax into a Gaussian state which satisfies the Lyapunov equation Working backwards through the FDI proof, this implies    1 2 2 ± α + ∓ α − T = ν˜(ω) = σpp γ˜ (ω) + γ˜ (ω) σ pp , (64) (1 1) ˜ (ω) (1 1) ˜ ( ω) 0, (61) 2 m m   which can only occur for all ω if either the environment is for the momentum covariance, which has elements p p , and damping with α˜ (+|ω|) = 0 or amplifying with α˜ (−|ω|) = 0. i j In the perturbative master-equation formalism, these equalities 1 σxx = σpp , (65) correspond to there being no environmentally induced transi- (mω)2 tions to higher energies and lower energies, respectively. From σ xp = 0, (66) relation (32), we then have the constraint upon the environment couplings for the remaining covariances in the phase-space (Wigner func- tion) representation [27], and to lowest order in the system-  | | − = ε ln ε ω 0 (Damping), (62) environment interaction. Comparing Eq. (64) to Eq. (54),we can express our covariances as ε|ln|ε + ω=0 (Amplifying), (63) 1 σ = κ˜(ω), (67) which must hold for all environment energies ε for which xx 2mω2 there is any population pE(ε). Assuming that the environment m σ = κ˜(ω), (68) coupling l is not sparse at particular energies, then the pp 2 damping environment must have all of its population in in terms of the FDR kernel κ˜(ω). So far our FDR kernel remains the lowest-energy state, T =+0orβ =+∞, while the phenomenological and not microscopically derived. However, amplifying environment must have all of its population in the it must be positive definite for this to describe a physical state. highest-energy state, T =−0orβ =−∞. If the environment Since κ˜(ω) is positive definite, we can transform to the basis coupling l is sparse, then it must only be the case that the in which it diagonalizes. If κ˜(ω) is a scalar quantity, then this population of states lies on a boundary of the energy spectrum is simply the normal basis of the free system. For each mode accessible by l. in this basis we have the phase-space covariance   xp  IV. NONEQUILIBRIUM UNCERTAINTY PRINCIPLE σ xx σ 1 κ˜ (ω)0 σ = n n = 2mω2 n n px pp m , (69) In the context of second-order perturbation theory, quantum σn σn 0 2 κ˜n(ω) noise is effectively Gaussian in the influence functional, which must then satisfy the generalized Heisenberg uncer- and Gaussian noise is equivalent to that arising from linear tainty relation due to Schrodinger¨ [29,30]: coupling to a bath of harmonic oscillators. Therefore, any violations of Eq. (50) must correspond to an environment 1 h¯ 2  x2 p 2− { x, p}  , (70) of oscillators in a nonquantum state. In the phase-space or 2 4

012102-6 NONEQUILIBRIUM FLUCTUATION-DISSIPATION ... PHYSICAL REVIEW E 88, 012102 (2013) or in terms of the phase-space covariance determined by the commutator of the field operators and not by the environment’s state (see, for instance, Ref. [31]). Now h¯ 2 det(σ)  , (71) let us say that we can solve the zero temperature case (or 4 the result is already available), but we cannot solve the more and, therefore, it must be the case that realistic nonequilibrium case, either because the realistic case is too difficult or because we lack a sufficient understanding of κ˜ (ω)  hω,¯ (72) n the nonequilibrium state of the combined environment. If we for all ω>0. But this is equivalent to our previous statement, solve the zero-temperature case, and in our derivation we track all instances of the noise kernel and its resulting diffusion coef- κ˜(ω)  hω,¯ (73) ficients, then at the end of the calculation we can apply the FDI in terms of positive definiteness as ω is a scalar quantity. So not to obtain bounds on the true solution. As a trivial example, one only do FDI violating correlations arise from HUP violating can apply the FDI to our resulting Eq. (74). However, in a more environment states, they can also produce HUP violating complicated system, and particularly if the full-time evolution system states via dissipation and diffusion (and decoherence). is considered, results of such an analysis can be very nontrivial. Furthermore we can say that in the weak-damping limit, the scalar FDR kernelκ ˜(ω) precisely determines the (asymptotic) ACKNOWLEDGMENTS nonequilibrium uncertainty product   This work is supported in part by NSF grants PHY-0426696, 1 κ˜(ω) 2 det(σ) = , (74) PHY-0801368, DARPA grant DARPAHR0011-09-1-0008 and 2 ω the Laboratory for Physical Sciences. for a single system mode of frequency ω. Larger FDR kernels naturally produce larger uncertainty, and minimal FDR kernels APPENDIX A: KERNELS FOR TWO-LEVEL (zero temperature) produce minimal uncertainty. Nonpertur- ENVIRONMENTS bative results require evaluation of the exact expressions A model with an environment consisting of a set of found in Refs. [20,28] for a single-system oscillator and two-level systems provides a useful illustration of how very multiple-system oscillators, respectively. nonlinear environments can give rise to nonintuitive noise and dissipation kernels, which nevertheless respect fluctuation- V. DISCUSSION dissipation relations. We consider the time-independent In this paper we have derived a fluctuation-dissipation environment Hamiltonian ω inequality (FDI) for an open quantum system interacting H = k σ (k), (A1) E 2 z with a nonequilibrium environment from the microscopi- k cally derived environment correlation function and recovered the well-known fluctuation-dissipation relation (FDR) for a where σ (k) denotes a Pauli spin matrix of the kth particle. For thermal environment. The FDI is a very general statement the environment coupling operator, we take the most general contrasting quantum noise to classical noise, and is satisfied time-independent noncommuting operator even for nonequilibrium fluctuations. Simply put, quantum = · σ (k) fluctuations are bounded below by quantum dissipation, ln gkn , (A2) whereas classically the fluctuations can be made to vanish. k The lower bound of this inequality is exactly satisfied by with the vector of coefficients and spin matrices given by zero-temperature noise and is in accord with the Heisenberg   ≡ x y uncertainty principle (HUP). FDI violating correlations arise gkn gkn,gkn,0 , (A3) from HUP violating states of the environment and can relax σ ≡ σ σ σ the open system into HUP violating states. Therefore, the ( x , y , z). (A4) FDI can be viewed as an open-system corollary to the HUP Assuming a factorized initial state for the environment, with both from microscopic and phenomenological considerations. the state of each spin initially given by Analogously, the nonequilibrium FDR also determines the   1 − δ(ω ) q(ω ) nonequilibrium uncertainty product, most directly in the limit ρ(k) = k k E ∗ , (A5) of weak damping [see Eq. (74)], and the corresponding FDI q(ωk) 1 + δ(ωk) implies the HUP. As an example of a practical application of the FDI, we note it is then straightforward to calculate the noise and damping kernels to be that there is a large class of situations where applying the FDI to the zero-temperature solution of an open system can produce νnm(t) = (g kn · gkm) cos(ωkt), (A6) general bounds on all related nonequilibrium solutions. For k instance, consider a possibly multipartite quantum optical = · δ(ωk) and/or mechanical system with linear photonic and phononic γnm(t) (gkn gkm) cos(ωkt) , (A7) hω¯ k environments, as would be the case with a system linearly k which satisfies the nonequilibrium FDR coupled to the environment field operators and where the system moves very slowly relative to the speeds of sound hω¯ ν˜(ω) = γ˜ (ω), (A8) and light. In the case we consider, the damping kernels are δ(ω)

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−   + ρ(k) so that the driven solutions of the environment operators are and, since 1 δ(ω) 1 by the positivity of E ,this relation satisfies the FDI. For a thermal state we have given by   t z (t) = (t) z (0) + dτ (t,τ) ε g (τ) L (τ), = hω¯ k k k k kn n δ(ω) tanh , (A9) 0 n 2kB T (B11) which also satisfies the FDT. However, notice that the noise in terms of the free environment transition matrix , which kernel has no relation to the temperature or even to the k satisfies the relations initial state of the environment, other than the fact that the environment spins are taken to be initially uncorrelated. ˙ k(t) = Fk(t) k(t), (B12) Instead, the damping kernel decreases with temperature to satisfy the FDT. k(0) = 1, (B13)

(t,τ) ≡ (t) (τ)−1. (B14) APPENDIX B: THE QUANTUM LANGEVIN EQUATION k k k The driven solutions can then be written as The quantum Langevin equation [14] is well understood  t in the context of bilinear position-position couplings between = ξ + ln(t) n(t) 2 dτ μnm(t,τ) Lm(τ), (B15) system and environment. Here we would like to extend the 0 simpler theory by considering quantum Langevin equations m ξ = T that correspond to the same class of Gaussian influences which n(t) gkn(t) k(t) zk(0), (B16) appear in the second-order master-equation and influence- k 1 functional formalisms. We begin with the same kind of μ (t,τ) = g (t)T (t,τ) ε g (τ), (B17) nm 2 kn k km Hamiltonian for our system and environment: k where μ is so far only proven to be a memory kernel in the H = H + H + H , (B1) driven solution. We further wish to prove that, when assuming C I E ξ the environment to be initially in a Gaussian state, n(t)isan HI(t) = Ln(t) ⊗ ln(t). (B2) operator-valued Gaussian stochastic process, with two-time n correlation function The Heisenberg equations of motion for any system operator ξ (t) ξ (τ)ξ = αnm(t,τ) = νnm(t,τ) + ıhμ¯ nm(t,τ), S(t), as driven by the environment, are therefore given by n m (B18) h¯ S˙(t) =+ı[H(t),S(t)] + ı [ln(t) ⊗ Ln(t),S(t)]. (B3) in terms of the noise kernel ν and dissipation kernel μ,given n by

To generate a Gaussian influence, we specify the environment = 1{ξ ξ } νnm(t,τ) n(t), m(τ) , (B19) to be linear in its driven dynamics 2 ξ 1  = −1 π 2 + 2 1 HE(t) mk (t) k(t) ck(t) qk(t) , (B4) = ξ ξ 2 μnm(t,τ) [ n(t), m(τ)] . (B20) k   2ıh¯ ξ = q + π π ln(t) gkn(t) qk(t) gkn(t) k(t) . (B5) To prove this, we must relate the memory kernel to the k commutator. From Eq. (B16), the commutator is given by To approximate nonlinear environments perturbatively, one ξ ξ = T ε would match the influence kernels at the end of the calculation. [ n(t), m(τ)] ıh¯ gkn(t) k(t) k(τ) gkm(τ). (B21) Let us define the “phase-space” vectors k Note that for any two-dimensional matrix A zk ≡ (qk,π k), (B6) ε A ε   −1 = ≡ q π A , (B22) gkn(t) gkn(t),gkn(t) . (B7) det A ε2 = = The Heisenberg equations of motion for the environment with 1. It therefore suffices to prove that det k(t) 1. = = operators, as driven by the system, can then be expressed First, note that det k(0) det 1 1. Then consider its conveniently as dynamics: d d d log det k (t) Tr log k (t) z˙ (t) = F (t) z (t) + ε g (t) L (t), (B8) det k(t) = e = e k k k kn n dt dt dt n    d −1 Tr log k (t) 0 m (t) = Tr log k(t) e F (t) ≡ k , (B9) dt k − ck(t)0 −1   = Tr[ ˙ k(t) k(t) ] det k(t) 01 ε ≡ , (B10) = Tr[F (t)] det (t) = 0. (B23) −10 k k 012102-8 NONEQUILIBRIUM FLUCTUATION-DISSIPATION ... PHYSICAL REVIEW E 88, 012102 (2013)

It was erroneously reported in Ref. [15] that the relation (B20) that we have two sets of indices, time t and noise n, did not always hold for parametric bath oscillators. is indicative of a convenient block-matrix structure being Substituting (B15) back into (B3) results in a quantum employed. Langevin equation for the open system. However, due to the Given a notion of positivity for a matrix, there is a natural fact that l(t) commutes with system operators at all times notion of inequality that can exist between matrices. If a matrix whereas the operator noise ξ(t) and dissipation separately do K is positive definite in the above sense, then we can express not, there is no unique method of expressing the quantum this more succinctly as Langevin equation in a manifestly Hermitian manner. All +K > 0, (C3) representations generate equivalent solutions, but some are more robust with respect to approximations than others. In −K < 0, (C4) particular, non-Hermitian representations are rather patholog- ical as they only preserve Hermiticity after noise averaging. where 0 is the matrix with zero in every entry. Therefore if The second-order adjoint master equation [15] would appear K−J is a positive-definite matrix, then we can say to suggest the following representation: K > J, ı (C5) h¯ S˙(t) =+ı[H(t),S(t)] + {l (t),[L (t),S(t)]}, (B24) 2 n n and if K ± J is a positive-definite matrix, then we can say  n t K > ±J, = ξ + ln(t) n(t) 2 dτ μnm(t,τ) Lm(τ). (B25) (C6) † † m 0 F KF > |F JF|, Although we have derived this Langevin equation under for all F and whereby the notation ± we mean to imply the assumption of a linear environment, we may apply it the inequality for both + and − cases. Note that, unlike self-consistently for any Gaussian influence. The quantum scalar quantities, kernels and matrices cannot always be Langevin equation here is nonperturbative. However, it only ordered in this way. The relationship between matrices can approximates non-Gaussian environments in a perturbative be indetermined. manner. A famous theorem for stationary kernels, where κ(t,τ) = Finally note that in the corresponding classical Langevin κ(t −τ), is due to Bochner and states that if κ is positive equation, ξ (t) would be real Gaussian noise with two-time n definite, then it follows that the Fourier transform is non- correlation ν , and the dissipation kernel would only appear nm negative:κ ˜(ω)  0 for all ω. For stationary matrix kernels as given in Eq. (B25). That the dissipation kernel appears in κ(t −τ), Bochner’s theorem naturally implies that the Fourier the operator-noise average is a quantum feature. transform is positive definite: κ˜(ω)  0 for all ω. The basic idea behind the theorem in this case can be understood APPENDIX C: INEQUALITY OF OPERATORS by realizing that for stationary kernels the time integration The positivity of a matrix kernel κ(t,τ) in the sense of domain in Eq. (C1) can be extended to (−∞,∞). Taking then   t t into account that the Fourier transform of a convolution is † dτ1 dτ2 f (τ1) κ(τ1,τ2) f(τ2)  0, (C1) simply the product of the Fourier transforms, together with 0 0 Parseval’s theorem for Fourier transforms, Eq. (C1) becomes for all vector functions f(t), is the natural extension of the equivalent to positivity of a matrix K in the sense of  ∞ † F† KF  0, (C2) dωf˜ (ω) κ˜(ω) f˜(ω)  0, (C7) −∞ for all vectors F. The time arguments of a kernel can be viewed as the (continuous) indices of a matrix. The fact for arbitrary f˜(ω), which implies κ˜(ω)  0 for all ω.

[1] A. Einstein, Ann. Phys. 322, 549 (1905). [12] A. P. Raval, B. L. Hu, and J. Anglin, Phys. Rev. D 53, 7003 [2] H. Nyquist, Phys. Rev. 32, 110 (1928). (1996). [3] C. H. Fleming, Ph.D. thesis, University of Maryland College [13] A. P. Raval, B. L. Hu, and D. Koks, Phys.Rev.D55, 4795 Park, 2011. (1997). [4] B. L. Hu and Y. Zhang, Mod. Phys. Lett. A 8, 3575 (1993). [14] G. W. Ford, J. T. Lewis, and R. F. O’Connell, Phys.Rev.A37, [5] B. L. Hu and Y. Zhang, Int. J. Mod. Phys. A 10, 4537 (1995). 4419 (1988). [6] A. Anderson and J. J. Halliwell, Phys.Rev.D48, 2753 (1993). [15] C. H. Fleming and B. L. Hu, Ann. Phys. 327, 1238 (2012). [7] C. Anastopoulos and J. J. Halliwell, Phys. Rev. D 51, 6870 [16] N. Kampen and I. Oppenheim, J. Stat. Phys. 87, 1325 (1995). (1997). [8] R. P. Feynman and F. L. Vernon, Ann. Phys. 24, 118 (1963). [17] H. P. Breuer, A. Ma, and F. Petruccione, in Quantum Computing [9] A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983). and Quantum Bits in Mesoscopic Systems,editedbyA.J. [10] B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992). Leggett, B. Ruggiero, and P. Silvestrini (Kluwer, Dordrecht, [11] B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 47, 1576 (1993). 2003).

012102-9 C. H. FLEMING, B. L. HU, AND ALBERT ROURA PHYSICAL REVIEW E 88, 012102 (2013)

[18] W. T. Strunz and T. Yu, Phys. Rev. A 69, 052115 [24] K. Martens, E. Bertin, and M. Droz, Phys.Rev.Lett.103, 260602 (2004). (2009). [19] E. Calzetta, A. Roura, and E. Verdaguer, Physica A 319, 188 [25] B. L. Hu and A. Matacz, Phys.Rev.D49, 6612 (1994). (2003). [26] D. Koks, A. Matacz, and B. L. Hu, Phys. Rev. D 55, 5917 (1997). [20] C. H. Fleming, A. Roura, and B. L. Hu, Ann. Phys. 326, 1207 [27] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, (2011). Phys. Rep. 106, 121 (1984). [21] C. H. Fleming, A. Roura, and B. L. Hu, Phys. Rev. E 84, 021106 [28] C. H. Fleming, A. Roura, and B. L. Hu, arXiv:1106.5752 [quant- (2011). ph] (2011). [22] C. H. Fleming, P. R. Johnson, and B. L. Hu, J. Phys. A 45, [29] H. P. Robertson, Phys. Rev. 46, 794 (1934). 255002 (2012). [30] D. A. Trifonov, Eur.Phys.J.B29, 349 (2002). [23] R. Bhatia, Positive Definite Matrices (Princeton University [31] C. H. Fleming, N. I. Cummings, C. Anastopoulos, and B. L. Hu, Press, Princeton, 2007). J. Phys. A 45, 065301 (2012).

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