Dissipative quantum oscillator with two competing heat baths

We study the dissipative dynamics of a harmonic oscillator which couples linearly through its position and its momentum to two independent heat baths at the same temperature. We argue that this model describes a large spin in a ferromagnet. We find that some effects of the two heat baths partially cancel each other. This leads to unexpected features such as underdamped oscillations and long relaxation times in the strong coupling regime. Such a partial frustration of dissipation can be ascribed to the canonically conjugate character of position and momentum. We compare this model to the scenario where a single heat bath couples linearly to both the position and the momentum of the central oscillator. In that case less surprising behaviour occurs for strong coupling. The dynamical evolution of the quantum purity for a single and a double wave packet is also investigated.


Introduction
The dissipative harmonic oscillator has long attracted considerable interest as a prototype of open quantum system. The early work of Magalinskii [1] and Ullersma [2] focusing mainly on an Ohmic environment and the weak coupling limit has been later extended in many different aspects [3]- [14] such as, for instance, to strong coupling [11], to non-Markovian noise [9,12] or to nonlinear coupling [3]. Much activity was boosted by the work of Caldeira and Leggett [15] on the dissipative mechanics of a macroscopic quantum variable. Quantum decoherence has also been addressed in [16]- [18]. A comprehensive review can be found in the textbook by Weiss [19].
In the above context relatively little attention has been paid to effects arising from the coupling to different system variables. The system variable which couples to the heat bath is most often assumed to be the position q. In the following we refer to this model as the q-oscillator. The choice of q as the coupling variable was favoured in [15]. There it was argued that a complex dissipative environment can be modelled by a bath of harmonic oscillators, with the coupling parameters chosen to yield a Langevin equation for the q variable in the semiclassical limit. It was shown later [20] (see also [21]) that, in a superconducting Josephson junction, the oscillator bath model can be derived microscopically from the coupling of the phase variable to the quasiparticle bath, with the phase playing the role of position. At that time the possibility of having a second bath coupled to the momentum p variable was ruled out. However, in a superconducting weak link, the electromagnetic field couples to the relative number variable, which in the above scheme plays the role of momentum [21,22]. If the coupling to the phase is neglected, the semiclassical behaviour of the particle number is governed by an Abraham-Lorentz equation [21,23].

Two independent baths
The general form of a Hamiltonian describing an oscillator coupled to two independent baths is [21] H = ω p 2 (p + δp) 2 + ω q 2 (q + δq) 2 + k ω k a † qk a qk + k ω k a † pk a pk , where [q, p] − = i and all operators are dimensionless andh = 1. The form of the Hamiltonian highlights the symmetry between q and p. The notion of mass is avoided by m = 1/ω p . For the fluctuating pieces, we assume that they are linear in the bath variables and independent of p and q, After two unitary transformations U p = exp(ipδq) and U q = exp(iqδp), one arrives at the Hamiltonian where the short-hand notation a † a = |a| 2 is used. The model is a Caldeira-Leggett-type of Hamiltonian [15]. It describes a harmonic oscillator with momentum p and position q, each variable being coupled to a different oscillator bath. The frequency of the central oscillator is ω 0 = (ω p ω q ) 1/2 . The baths are described by the spectral densities J q (ω) = 2 |λ k | 2 δ(ω − ω k ), J p (ω) = 2 |µ k | 2 δ(ω − ω k ).
Although one could have started directly with equation (3), we prefer to present the Hamiltonian equation (1) as the starting point in order to provide a natural justification for the renormalization terms k |µ k | 2 ω −1 k p 2 and k |λ k | 2 ω −1 k q 2 in equation (3) which otherwise would have to be introduced ad hoc.
For low-lying excitations, and thus for low temperatures, the Hamiltonian (3) becomes equivalent to that of a large spin s in a ferromagnetic environment. This can be seen as follows: the Hamiltonian for a spin in a large magnetic field along the z-direction is where the fluctuating terms model the low-lying bosonic (magnon) excitations of the ferromagnet at the site of the large spin. The fluctuations in the z-direction are neglected since they are quadratic in S x and S y with S x , S y S z . In other words, S z is approximately a constant of motion [29]. For s → ∞, the first term in (5) is a harmonic oscillator S z =h|a| 2 −hs. The action of S i on the eigenstates of this harmonic oscillator is and [S − , S + ] = 2h 2 s. This amounts to keeping only the leading order term in the Holstein-Primakoff transform of S + , S − [30]. The rescaled Hamiltonian H /2s becomes formally identical to equation (3) in the limit s → ∞, with ω 0 = µ B B z /2s and the fluctuating pieces scaling as δB x = √ s λ k (a qk + a † qk ) and δB y = √ s µ k (a pk + a † pk ). Such large spins have been observed in magnetic particles [31].

General results
Elimination of the bath variables yields the Heisenberg equations of motion for q and ṗ The response kernel is defined as and the force operator F q (t) = λ k a qk exp(−iω k t) + H.c., with F p (t) defined accordingly. In Fourier space, equation (6) reads whereJ n (ω) is the symmetrized Riemann transform [32] The oscillation modes are given by the zeros of the function where χ(ω) is the generalized susceptibility. For the two baths at the same temperature, the symmetrized correlation functions for the position C (+) qq (t) ≡ 1 2 [q(t), q(0)] + and the momentum C (+) pp (t) ≡ 1 2 [p(t), p(0)] + are obtained from χ(ω) as follows The corresponding expression for C (+) pp (t) is obtained by interchanging p ↔ q in equation (12). The antisymmetrized correlation functions for position and momentum C (−) nn (t) ≡ [n(t), n(0)] − (with n = q, p) are obtained from equation (12) in the standard way [19], i.e. by substituting sin(ωt) for cos(ωt) coth(βω/2).
We wish to emphasize that, within the dissipation model studied here, the equations of motion (6) are symmetric in q and p for arbitrary coupling strength. Thus the situation is different from that of a q-oscillator when described within the rotating-wave approximation, which is known to yield a model symmetric in q and p but which holds only in the weak coupling limit [33].

Ohmic coupling
First we focus on the important special case that both spectral densities are Ohmic, α q = α p = 1. Then the real parts ofJ q (ω) andJ p (ω) vanish to lowest order in −1 . The susceptibility has poles at the roots of the quadratic polynomial [5] which are where κ ≡ ω p γ q + ω q γ p 2ω 0 (1 + γ q γ p ) 1/2 .
The solutions of equation (13) are either purely imaginary or a pair of complex conjugates depending on whether κ is greater or smaller than 1. Thus, is the commonly accepted criterion to distinguish between underdamped and overdamped oscillations. The underdamped region lies in a stripe of width = 4η(1 + η 4 ) −1/2 , with η ≡ (ω q /ω p ) 1/2 , limited by the graphs of the functions In figure 1, the stripes of underdamped oscillations, marked in the (γ q , γ p ) plane are plotted for three different values of η. For large η = mω 0 , corresponding to a large mass of the central oscillator, the stripe of underdamped oscillations becomes smaller. In the limit η → ∞, the introduction of an infinitesimal coupling to the momentum can induce a transition from overdamped to underdamped oscillations. However, the range of values of γ p allowing for underdamped oscillations also becomes increasingly small as ∝ η −1 .
The behaviour of the system as a function of γ p for fixed γ q is also interesting. We set η = 1. The inverse damping time τ −1 is a monotonously increasing function of γ p for γ q < 2 and a monotonously decreasing function for γ q > 2. Therefore an additional bath coupling to  (16)). Right panel: regions of underdamped oscillations with criterions B (equation (19)) and C (equation (21)), for η = 1.
p has opposite effects on the damping time τ depending on whether the original q-oscillator starts in the underdamped (γ q < 2) or in the overdamped (γ q > 2) regime. In the first case, the additional bath always reduces the damping time, leading to infinitely strong damping in the limit γ p → ∞. However, in the regime γ q > 2, we can drive the system from the overdamped into the underdamped regime by increasing the coupling strength γ p . What is more, we can take γ p = γ q = γ, which corresponds to a completely symmetric Hamiltonian in p and q. At this point, the system is always in the underdamped regime and for large γ the oscillator frequency is close to its maximum. We find for the inverse damping time Therefore we are led to the paradoxical situation that the higher the friction coefficient γ the larger τ becomes. In particular, for γ → ∞ one gets ω 0 τ → ∞. This time dilatation is in itself a remarkable effect, since it contrasts with the behaviour of the q-oscillator, for which τ → 0 as γ q → ∞ (see equation (14)).
In the general, non-symmetric case, we note that for a given finite γ q the damping time τ cannot be made to acquire arbitrary values by tuning γ p . Rather, it is bounded between 2γ q /ω 0 and 2/γ q ω 0 , which is higher depending on γ q ≷ 1.
However, from these striking observations, one should not conclude that in the limit of infinite γ the central oscillator recovers the dynamics of a free oscillator. It rather leads to a dilatation of all timescales. For instance, the renormalized oscillator frequency ζ vanishes even faster than the inverse damping time, since ζ = ω 0 /(1 + γ 2 ). An analysis of the correlator in equation (12) shows that for t τ the particle behaves as a free ballistic (though very slow) particle, C (+) qq (t) ∝ t/τ. Instead of using criterionA, we may focus on D q (ω) ≡ ImC (−) qq (ω)/ω.As a spectral function, ImC (−) qq (ω) is independent of temperature. Of special interest is the slope of D q (ω) near ω = 0. Since lim ω→∞ D q (ω) = 0, the condition for the existence for D q (ω) displaying a maximum can be written as which may be viewed as an indicator of underdamped oscillations or, equivalently, coherent transitions. It is often employed for the spin-boson model [24,26]. For Ohmic damping, one finds and the critical curve for γ p is thus given by the relation γ crit p = γ q (γ 2 q /η 2 − 2)/η 2 , so that criterion B is satisfied for γ p > γ crit p . Both being based on exact expressions, we observe that criterion B differs substantially from criterion A, see figure 1. Surprisingly, even for the q-oscillator there is a region √ 2 < γ q < 2 where criterion A and criterion B are different. In figure 2, D q (ω) is plotted for different coupling strengths γ q , γ p . For large couplings γ p , we observe that, although we are (according to criterion B) in the underdamped region, the maxima of the curves are not very pronounced. This is related to the fact that the renormalized oscillator frequency ζ vanishes for large γ p faster than the inverse damping time τ −1 .
The foregoing discussion suggests the introduction of a third, more restrictive criterion for underdamped oscillations, namely [27], According to this condition, the region of underdamped oscillations is convex, i.e. by increasing the coupling strength, no transition from overdamped to underdamped oscillations can be realized. Table 1 summarizes the different criteria for coherent, underdamped dynamics as they are applied to three prototypical cases.
For high temperatures (k B T ω 0 ) the integral for the coordinate autocorrelation function equation (12) can easily be solved yielding the classical solutions of the Heisenberg equations of motion. For κ < 1, they read For κ > 1, equation (22) describes the corresponding non-oscillating solution. Analytical expressions for C (+) qq (t) are also available for T = 0. In this case, the integral in equation (12) can be expressed in terms of exponential integrals [19]. We do not show the result here and limit ourselves to note that for long times only the lowest order term, i.e. the term linear in ω, contributes in the numerator of the right-hand side of equation (12). Thus the long time behaviour is the same as for the q-oscillator, namely, we have For the classical solution equation (22) criterion A (equation (16)) is the only natural criterion to differentiate between underdamped and overdamped oscillations, since it distinguishes solutions with infinitely many zeros from solutions with no zeros at all. For zero temperature, the number of zeros of C (+) qq is always finite and criterion A is less informative. The zero temperature mean squares q 2 , p 2 can be calculated exactly. They are given by The corresponding expressions for p 2 are obtained by interchanging p ↔ q in equations (24) and (15). For small γ q , γ p we have f(κ) = 1 − 2κ/π + O(κ 2 ) and the position mean square becomes and, correspondingly, For γ q = 0, γ p = 0, we recover the results for the q-oscillator, with the characteristic logarithmic dependency of p 2 on the cutoff [19]. In the general case, the Heisenberg uncertainty product diverges as p 2 q 2 ∝ ln p ln q . Thus the reduced density matrix at equilibrium becomes approximately the identity, its off-diagonal elements being essentially zero in both the position and the momentum representation.

Other spectral densities
For general spectral densities, α q and α p are arbitrary positive real numbers. While we always have we find First, we briefly recall the behaviour of q 2 and p 2 as functions of α q and γ q in the case of the q-oscillator. For q 2 , it may be summarized by the formula [19] A measurement of the position, i.e. a diagonalization of the density matrix in the position basis takes place only for α q < 2. The behaviour of p 2 is opposed to that of q 2 in such a way that the product q 2 p 2 is always 1/4, as required by the Heisenberg relation. For p 2 , we may write In the forthcoming discussion, we focus on q 2 . We have a contribution from J q (ω), which reduces q 2 , and another contribution from J p (ω) enhancing q 2 (see equation (25)). From the results for the q-oscillator, we expect that the J p (ω) contribution should always dominate for α p < 2 and the J q (ω) contribution should be negligible for α q 2. The opposite regime α q < 2 and α p > 2 is more interesting (the case α q = 1 and α p = 3 has been studied in [21]). There we have a sum of two terms of order 1 in −1 with opposite sign. In the following, we focus on that case.
We assume the susceptibility χ(ω) to be an algebraic function. This entails rational exponents α n = β n /m with integer β n and m. The integrand in equation (12) has r ≡ max(β q + β p , 2m) poles on the m Riemann sheets which split into two classes. For α q , α p > 1, there are 2m poles close to the oscillator frequency ±ω 0 on each sheet. In addition there are r ≡ max(β q + β p − 2m, 0) poles of the order of i . If either α q or α p becomes equal to or smaller than one, the 2m poles detach from ±ω 0 moving into the complex plane. The equilibrium mean square can be written as a sum of the contributions from the two types of poles.
For the second term, we find (γ q , γ p small) For a sketch of the derivation of equation (32), see appendix A. Therefore, the high energy poles yield, to first order in γ p , a finite contribution to the position mean square. This contribution is positive for superohmic coupling to the momentum but vanishes for Ohmic and subohmic coupling. It has a singularity for α p = 1 which marks the transition to the logarithmic dependence on the cutoff frequency, see equation (24). We note that the right-hand side of equation (32) does not depend on the oscillator frequency ω 0 , i.e. it does not depend on the properties of the system itself. The contribution from the poles close to ω 0 , C ω 0 , can in principle be calculated in a similar way as C for arbitrary exponents α q , α p ; however, the calculations become increasingly messy. A general treatment is also hampered by the wide variety of casuistic behaviour, with different regimes defined by the conditions α n ≷ 1, α n ≷ 2 and α q + α p ≷ 2, α q + α p ≷ 4. We focus on the specific case α q 1 and α p > 2, which is the most relevant, since most baths occurring in nature are either Ohmic (Markovian approximation, electron gas [34]) or superohmic (photon or acoustic phonon baths). In that case, the low energy poles are essentially determined by the lower spectral exponent (here, α q ). This reflects the general property of dissipative quantum systems that an environment becomes increasingly efficient with decreasing spectral exponent α n . To leading order in γ q , we find If we compare C ω 0 with equation (32), we observe that the two contributions have opposite sign.
In particular, the contributions of C ω 0 and C cancel each other to first order in γ q and γ p provided that In figure 3, q 2 is plotted against the coupling strengths γ p and γ q for α q = 1 and α p = 3. q 2 is monotonously increasing as a function of γ p . For γ p = 0, it is a monotonously decreasing function of γ q . For the parameters chosen in figure 3, equation (34) holds on the diagonal γ q = γ p . There q 2 remains close to its unperturbed value 1/2 if γ q , γ p 1.  (28)).
Finally, on the right-hand side of figure 3, the position mean square q 2 is plotted as a function of γ q for different spectral exponents α p . For small γ q , the enhancement of q 2 due to the coupling to the momentum is larger for smaller values of the spectral exponent α p . This is the expected behaviour. This behaviour, however, is inverted when γ q increases. Then the relative enhancement of q 2 as compared with the q-oscillator is bigger for higher spectral exponent α p . A crossing occurs always for some value of the coupling constant γ q , although sometimes this can be very large.

Time evolution
To study the nonequilibrium properties and in particular the loss of coherence for an initially pure state, the calculation of C (+) qq (t) is not sufficient. Instead, one should perform a full nonequilibrium calculation with inclusion of specific initial conditions by means of a Laplace transformation. The position-momentum symmetry of the oscillator Hamiltonian suggests the use of the reduced Wigner function W(q, p, t) instead of the reduced density matrix. It was shown in [35] for the qoscillator that the time evolution of W(q, p, t) can only for certain initial conditions be described by an exact master equation. Here, we consider decoupled initial conditions , which fall into this class. In this case, we are able to derive not only an exact Master equation for the reduced Wigner function but a solution in terms of a two-fold convolution integral.
Generally W(q, p, t) can be expressed as where the bracket denotes the average over initial conditions [11] · · · 0 ≡ dq dp k,k da qk da pk da * qk da * pk (· · ·)W S (q , p , {a q , a p , a * q , a * p }), and q(t), p(t) are the classical solutions of the equations of motion equation (6). The Wigner function of the thermalized baths is given by the system itself being in an arbitrary pure state characterized by W S . In this case, one can express equation (35) as a two-fold convolution integral where W S is the Fourier transform of W S in both arguments. We have introduced the auxiliary functions and the temperature-dependent quantities The functions X q0 , X qp and Y qp are defined accordingly by interchanging p and q. For a better understanding of the forthcoming discussion in section 4, we notice that the functions defined in equation (39) are the time-dependent coefficients of the solutions q(t), p(t) of the initial value problem equation (6). For instance This necessarily requires φ 0 (0) = 1 and φ n (0) = 0. Finally, we note that W fulfils a Fokker-Planck type of equation [19], where ∇ ≡ (∂/∂q, ∂/∂p), with g = (g q , g p ) the phase-space drift term, and with (the 2 × 2 matrix) B(t) the state-independent phase-space diffusion term, B qp (t) is obtained from B pq (t) by exchanging q and p. To derive the coefficients (44) and (45), we have employed the ansatz equation (42) for the Fokker-Planck operator.Acting with yet undefined functions f n ,B nm (n, m = p, q) on the right-hand side of equation (42), and comparing with its time derivative, yields conditions for the functions f n , B nm . An exact Fokker-Planck equation in the form of equation (42) for the q-oscillator has been first derived in [11] and later by a different method in [13,16].

Purity
A convenient quantity to measure the degree of global decoherence is the purity P(t), defined as the average of the density matrix itself [17], which is basis independent. Of special interest is the equilibrium value P β ≡ lim t→∞ P(t) which measures the efficiency of the environment in destroying quantum coherence. Here, we implicitly assume ergodic behaviour which, as we shall see, applies in the presence of Ohmic baths. For a harmonic oscillator in thermal equilibrium, the reduced Wigner function is [19] W β (q, p) = 1 2π q 2 which leads to

Coherence decay for Ohmic damping
Although the equilibrium decoherence (as measured by the product q 2 β p 2 β ) is enhanced by the additional noise term, one may wonder whether for low temperatures the decoherence time becomes larger than for the damped q-oscillator. To answer this question exhaustively, one would have to calculate the time evolution of the purity for an arbitrarily pure initial condition.
We consider two cases. Firstly, we choose a coherent (Gaussian) state as initial state. This case should present the greatest robustness against decoherence [17,36]. Secondly, we choose a superposition of two Gaussian wave packets. Then a new aspect of decoherence comes into play, namely, the fast vanishing of the relative coherence between the two Gaussian wave packets. We distinguish the two manifestations of decoherence by introducing a new quantity, which we call relative purity.

Decoherence for a Gaussian wave packet as initial state
We consider here the case where the system starts in a coherent state at t = 0. The Wigner function for a Gaussian wave packet is where p 0 = √ 2η Im (α) and q 0 = √ 2/η Re (α) are defined in terms of the complex eigenvalue α (amplitude) of the coherent state (recall η = ω q /ω p ). Since W S (q, p, 0) is Gaussian, the convolution integral of equation (38) can be performed. The integrals become particularly simple when both baths are equal (γ p = γ q = γ and η = 1), corresponding to the completely symmetric case. Then we have φ p = φ q ≡ φ 1 and, by the same token, X pq = X qp ≡ X 1 in equation (39) and also X p0 = X q0 ≡ X 0 in equation (40). The crossed terms in equation (38) vanish and the Wigner function becomes a product of two Gaussians for all times W(q, p, t) = 1 4π For the purity, one obtains the simple expression The remaining task is to calculate the quantities φ 0 , φ 1 , X 0 , X 1 from their definition in equations (39) and (40). Using the naïve form ( → ∞) of the spectral function of an Ohmic bath as given in section 2, we would find φ 0 (0) = 1/(1 + γ p γ q ) = 1 leading to inconsistencies (see the discussion after equation (40)). The reason for this lies in an initial slippage caused by the somewhat unphysical character of the decoupled initial condition [37]- [40]. Technically it stems from the fact that for J(ω) ∝ ω the integrals in equations (39) do not converge at t = 0 [11]. This problem is overcome by regularizing the Ohmic spectral functions, i.e. by reintroducing a finite cutoff . The explicit calculation with a Drude regularized spectral function is sketched in appendix A. Here, we state only the main result: at T = 0, the purity decays on two timescales, given by −1 and τ in (18), for 0 t −1 , with P −1 β = 2 q 2 β and the oscillator frequency = ω 0 /(1 + γ 2 ). Coherence is reduced immediately after the start of the coupling. Although afterwards it decreases more slowly, on a timescale τ, for larger couplings the curves of P(t) for different values of γ never cross, i.e., P(t) is a monotonously decreasing function of γ for all t.
An initial slip similar to that discussed in this section also occurs for the q-oscillator [11]. However, in that case its effect on purity is much less severe. Specifically, out of the set of functions φ q (t), φ p (t) and φ 0 (t) defined in equations (39), only φ q (t) is affected by the initial slip. After a time −1 it becomes φ q (t) ∼ γ p = 0 [11]. However, we note that φ q appears in the temperature-dependent terms X qp , X pq , Y qp , Y pq of equation (40) only in combination with the spectral density of the momentum coupling J p (ω), which vanishes for the q-oscillator by definition. We thus reach the important conclusion that the purity evolution of the q-oscillator is insensitive to the initial slip stemming from the use of decoupled initial conditions: at t ∼ −1 , the purity is still approximately unity, decreasing afterwards at a rate ∝ γ q .
Isar et al [41] studied in detail the decay of purity for the q-oscillator. In particular they found constraints that must be satisfied by the bath if the purity is to remain constant and close to unity during the whole time evolution of the oscillator. Such a high purity is not possible in the present case due to the initial slip (see equation (53)).
Another possible preparation of the initial state is in a constrained equilibrium. The expectation values q(0) = q 0 and p(0) = p 0 are held fixed with the bath equilibrated around those values. Then the Wigner function is [11] W(q, p, t) = 1 while the purity is given by its equilibrium value equation (49). A more detailed discussion of the purity decay of the q-oscillator, together with the general formulae, is given in appendix C.

Decoherence of two Gaussian wave packets
Formula (38) also allows us to investigate more complicated initial conditions such as, for example, the superposition of two Gaussian wave packets. This case has been studied for a single bath by Caldeira and Leggett [42]. It is an interesting case study because it displays two different aspects of decoherence. On the one hand, there is the decoherence which either wave packet would experience alone. This part is essentially described by P(t), which we will call Gaussian purity (see equations (52) and (53)). On the other hand, there is the decoherence due to the spatial separation of the two packets. This second contribution is expected to become increasingly important when the distance a between the two packets becomes large. As initial wavefunction we choose which translates into an initial Wigner function Here, c ≡ √ 2[1 + exp(−a 2 /8σ 2 )] 1/2 is a normalization constant. We assume the most symmetric case and set σ = 1/ √ 2. Plugging (56) into (38), we find that the purity can be expressed in terms of the Gaussian purity as The function φ(t) is given by where φ 0 and φ 1 are defined in (39) and computed in (B.2) for the symmetric case. φ(t) evolves from φ(0) = 1 to lim t→∞ φ(t) = 0. We define the relative purity as the ratio As expected, P rel (t) → 1 as a → 0 and P rel (t) → 1/2 as a → ∞.
Interestingly, the structure of (57) is such that, as time passes and φ(t) P(t) evolves from 1 to 0, the ratio P rel (t) starts at unity, as corresponds to a pure state, then decreases and finally, at long times, goes back to unity. When a is large P rel (t) decays rapidly to 1/2, on a timescale ∼1/4a 2 γ. There it stays for a time which increases with distance as ∼γ −1 ln a. Afterwards it returns to one. The ratio 1/2 can be rightly interpreted as resulting from the incoherent mixture of the two wave packets. Thus it comes as a relative surprise that P rel (t) becomes unity again at long times, as if coherence among the two wave packets were eventually recovered. The physical explanation lies in the ergodic character of the long time evolution, with both wave packets evolving towards the equilibrium configuration (see equation (24) and the subsequent discussion).

Single bath with linear coupling to both position and momentum
It is instructive to compare equation (3) with the Hamiltonian of a harmonic oscillator interacting with a single heat bath in the most general form of linear coupling, with complex parameters λ k , µ k . This model is different from that discussed in the previous sections in that here time reversal invariance is broken. By this we mean the following. In the models described by equation (3), the bath modes might describe for example a magnetic field coupled to the momentum of a charged particle, which clearly would break time reversal invariance. However in that case, such a symmetry breaking is somewhat fictitious since, due to the linear nature of the coupling and to the modelling of the bath as a set of harmonic oscillators, one can always find a unitary transformation which restores time reversal invariance, i.e. which renders all parameters in the Hamiltonian real quantities. It is easy to see that, for general (complex) λ k and µ k , such a unitary transformation cannot be found for equation (60).
By an analysis similar to that of section 2, one finds general expressions for the symmetrized equilibrium correlation functions The same expression applies for C (+) pp (t) with the indexes p, q interchanged everywhere. As before, we define a generalized 'susceptibility' for the system which now reads Equations (61) and (62) involves four different spectral functions.J q (ω) andJ p (ω) are defined as in equations (4) and (10). Now we introduce the spectral functions which reflect the mixing of the bath modes. The 'hat' symbol denotes the transformation In equation (64), the real part is an antisymmetric function and the imaginary part is symmetric. This is exactly reverse to the 'tilde' (Riemann) transform, defined in equation (10). Before we calculate equation (61) for a specific case, we analyse some generic features. First we notice that, by settingĴ − (ω) andJ + (ω) to zero, we recover the autocorrelation function for two independent baths (see equation (12)). On the other hand, the two spectral densities J + (ω) and J − (ω) are not independent from J p (ω) and J q (ω); rather, they satisfy Actually this relation has already been used to simplify the integrand of equation (61). We note that it holds for the spectral densities themselves but in general not for their Riemann transforms J + (ω) andĴ − (ω). Apart from the above constraint one can, at least in principle, freely choose three of the four spectral functions. If, in addition, we make the physically reasonable assumption that the 'mixing angle' in equation (63) is frequency independent (θ k = θ), condition (65) fixes J + (ω) and J − (ω) completely.
Therefore, in the important case where both J q (ω) and J p (ω) have the same spectral exponent (α q = α p = α), J + and J − also obey the same power law. Using equation (65), we observe that the term (ImJ q ImJ p ) implicit in equation (62) drops out. We recall that this interference term has been responsible for the non trivial 'phase-space' diagram figure 1. What is more, only ReJ + appears in equations (61) and (62). That means, according to equation (28), that for α q + α p < 4 the symmetric spectral function J + does not contribute at all. The relation between the 'hat' and the 'tilde' transformation iŝ from which all properties off can easily be deduced. For example from equations (27), (28) and (67), it follows that ReĴ − (ω) = 0, only for α q + α p < 2.
It is illustrative to look at the 'generalized' susceptibility equation (62) for the case α q + α p < 2, where all real parts of the spectral functions vanish. We obtain This is exactly the susceptibility of a q-oscillator coupled to a single bath with an additive spectral function We conclude that, when only one bath is involved, the particular structure of the coupling of the bath to the position or momentum variable can always be modelled by a q-oscillator with an appropriately chosen spectral function. In this context, the combination ωJ − (ω) can be considered as stemming from an effective additional noise source that couples to position. Note however, that this is not the same as saying that a single-bath coupling to q and p can always be unitarily transformed to a coupling only to q (q-oscillator), since this is not true in general. For one thing, the noise properties, which are not uniquely given by χ(ω), would be different.
We also wish to emphasize that a double-bath dissipative oscillator cannot be described in terms of an oscillator coupled to an effective single bath. In particular, it can never be modelled by a susceptibility like (69). This is the reason why the physics explored in section 2 is so different from that which could be found in any possible single-bath scenario.
We do not repeat here the analysis of section 2 for arbitrary spectral functions but focus instead on the case of Ohmic coupling (α q = α p = 1) in order to compare with the results obtained for two independent baths. As ReJ n (ω) = ReJ + (ω) = 0 vanish (see subsection 2.3), we obtain The form of the susceptibility varies depending on whether the mixing angle is a multiple of π or not. We discuss the two cases separately.

Mixing angle θ = 0, π
For θ = 0 or θ = π, one can bring, by an appropriate redefinition of a k , a † k , the interaction part of the Hamiltonian equation (60) into the form so that the bath couples to the main oscillator either through the position of their oscillators or through their momentum, but not through both. From equation (66), it can be seen that the antisymmetric spectral function vanishes identically. Apart from the susceptibility, the integrand of equation (70) becomes equal to that of equation (12). Thus all results of subsection 2.2, in particular equations (22) and (24), carry formally over. The only but crucial difference lies in the poles of the generalized susceptibility χ(ω). The set of zeros can be written as with κ defined as According to criterion A, equation (16), the regions of overdamped and underdamped oscillations in the (γ q , γ p ) plane are separated by a straight line with negative slope given as before by the condition κ = 1 (see figure 4). Therefore, a crossover from overdamped to underdamped oscillations by increasing one of the coupling constants is now impossible. We also note that, in the symmetric case (γ q = γ p = γ), the relaxation time is the same as for the q-oscillator (τ −1 = ω 0 γ/2). Increasing γ q or γ p leads inevitably to an enhancement of dissipation. This is in blatant contrast to the results obtained previously for the double-bath model and one of the most remarkable results of this study. It best illustrates the importance of the specific structure of the bath and justifies a posteriori the detailed study of a model with two independent baths.

Mixing angle θ = 0, π
We briefly consider the case of non-vanishing mixing angle. The effects should be largest for θ = π/2, corresponding to an interaction Hamiltonian This form of coupling is called amplitude coupling in [43]. Now J − (ω) is no longer zero but we have which is exactly the spectral function of a superohmic bath with exponent α = 2. The logarithm in ReĴ − inhibits further analytic treatment of equation (70) even in the high temperature limit; thus we limit ourselves to a qualitative analysis. For simplicity, we omit the terms containing logarithms in the generalized susceptibility equation (62). The contributions to C (+) qq stemming from those terms should decay on a timescale ∼ −1 . The remaining terms define a susceptibility which is identical to the susceptibility of a q-oscillator coupled linearly to one bath with spectral density It is well known [19] that, in the case of a polynomial spectral function, the term with the lowest exponent dominates (see also the discussion in subsection 2.3). Therefore we may conclude that the crossover diagram figure 4 remains unchanged by a mixing angle θ = 0.

Conclusions
We have discussed the behaviour of a quantum Brownian particle in a harmonic potential subject to two independent noise sources, one of which couples to its position and the other one to its momentum. In the symmetric case where both baths are Ohmic and their coupling strength is the same, we find underdamped oscillations of the central oscillator for all coupling strengths. This indicates that the two baths partially cancel each other. The effect is due to the mutually conjugate character of position and momentum. It was first noted in [26] for the (cylindrically) symmetric spin-boson model with spin s = 1 2 , i.e. in the deep quantum regime. 'Quantum frustration' of the spin can pictorially be described as the result of having two observers attempting to measure simultaneously, with equal efficiency, two non-commuting components of the spin. Because of the uncertainty principle, both of them fail to measure anything. Here, we have investigated the analogous effect for a quantum oscillator coupled to two independent baths. We have found a moderate form of cancellation which we have labelled 'quasiclassical frustration' because our dissipative oscillator may describe a large spin impurity coupled to the magnon bath in a ferromagnetic medium. The most notable features of quasiclassical frustration become manifest in the strong coupling limit, where underdamped dynamics survives and all timescales diverge. It remains to be investigated whether the occurrence of frustration in the classical regime is a general property or an artifact of the harmonic oscillator.
We have compared the double-bath model with the case where a single bath couples linearly to the position and momentum of the oscillator. In the latter case, the 'phase-space' diagram is simpler in that transitions from overdamped to underdamped oscillations can never occur. Comparison of the two models indicates that bath correlations can qualitatively change the behaviour of a dissipative system.
A point of caution is needed in the interpretation of our results. We have definitely ruled out the at first sight the enticing but on closer inspection unphysical conjecture that the effects of the two observers cancel each other completely and the particle is not affected by the environment. Indeed we have seen in the specific example of a decoupled initial state that, in destroying quantum coherence, two baths are always more efficient than one. This is true for both the overdamped and the underdamped regime. This demonstrates that, at least for the harmonic oscillator, underdamped dynamics is by no means a reliable signature of high global coherence, which here we identify with purity (see section 4). Our work provides further evidence that decoherence and dissipation are not necessarily correlated. By dissipation we mean here the net transfer of energy from the central oscillator to the thermal baths. It is characterized by the classical equation of motion or, equivalently, by the properties of the spectral function. We have seen that, depending on the situation, an increase of dissipation can be accompanied by either a reduction or an increase of decoherence. Vice versa, a source of decoherence may or may not lead to dissipation. This is the case in the so-called pure dephasing (not considered here), where the interaction part of the Hamiltonian commutes with the system Hamiltonian and no dissipation occurs at all, see i.e. [43].
The fact that we have focused on a quantum oscillator coupled linearly to oscillator baths has allowed us to investigate analytically the equilibrium and dynamical properties. The question of frustration is also raised in other dissipative quantum systems, such as a small spin coupled to a boson [26] or a fermion [34] bath, which are not amenable to an exact analytical treatment. The extension of the present study to less tractable physical scenarios provides a theoretical challenge.

Appendix C. Purity decay of the q-oscillator and general formulae
Here, we state the general formulae for the purity decay of a coherent (Gaussian) initial state. To calculate the purity decay for two Ohmic baths, both with decoupled initial conditions but with different coupling strengths, we proceed as in subsection 4.