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Equilibrium properties and decoherence of an open harmonic oscillator

The equilibrium properties of an open harmonic oscillator are considered in three steps: First the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The second step consists of the introduction of the Heisenberg representation where the dynamical decoherence can be identified without making reference to the state of the system. Finally it is pointed out that the quantum fluctuations generate non-continuous limit for infinitesimal system-environment interactions and at the border of the under- and over-damped oscillator.


I. INTRODUCTION
The systematic construction of the quantization rules for open system is based on the elimination of the environmental degrees of freedom. However this is a quite involved procedure therefore it is important to understand better some simple, generic examples. The simplest open dynamics corresponds to weak interactions within the system and between the system and its environment. The leading approximation to such a dynamics is a harmonic effective theory and the goal of this work is to gain some insight into the relaxed dynamics of an weakly open harmonic oscillator.
The problem of an open harmonic oscillator has attracted lot of attention and impressing advances have already been made. After the initial steps, using the quantum Fokker-Planck [1] and Langevin equations [2][3][4] a simplified master equation was put forward [5,6], to be generalized to a non-local master equation by the elimination of the environment [7][8][9].
The following large number of works, using the operator and the path integral formalism is reviewed in refs. [10] and [11], respectively.
The closed harmonic oscillator has a discrete, equidistant spectrum and the simplest way to explore its stationary states is to use the creation and destruction operators of the elementary excitations. The main goal of the present paper is find the stationary states together with the creation and destruction operators and the extension of the Heisenberg representation. That is realized within the framework of the Closed Time Path (CTP) formalism [12,13] which is well suited to our goal since it offers a natural way to find the reduced system density matrix. The full non-local master equation has already been derived in this formalism for harmonic dynamics [14]. Instead of such a microscopic derivation we have a more phenomenological starting point, the most general harmonic local dynamics in time, defined by an effective Lagrangian. The Fokker-Planck equation, corresponding to this Lagrangian, agrees with the most general master equation for harmonic systems [15,16].
The additional insight, obtained from such a derivation of the master equation, is that there are total time derivative terms in the Lagrangian with kinematic rather than dynamical role on the master equation level. A further bonus of the CTP formalism is a clear and natural way the decoherence can be accessed [19].
The bra and the ket dynamics are equivalent for closed systems, they are complex conjugate of each other. However the bra and ket components are coupled in the non-separable density matrix of a mixed state hence the bra and the ket must be handled independently of each other in the CTP formalism. The harmonic system density matrix, a Gaussian in the two coordinate variables, is reminiscent of the wave function of two harmonic oscillators.
This similarity motivates to work in the Liouville space of operators and construct the creation and destruction operators in the space of density matrices. One finds a relaxed state and a family of excited stationary states in such a manner. The relaxed state turns out to be unique within the Hilbert space of states, defined by the given asymptotic conditions at spatial infinity. The coherent states are constructed as a simple application of this scheme.
It is not obvious how to generalize the Heisenberg representation for open systems with non-unitary time evolution. However the conservation of the total probability proves to be sufficient to find a simple extension of the usual construction, leading to a particularly simple formalism for open harmonic systems where the time dependence is controlled by c-number expressions, containing the classical normal frequencies.
Some of the difficulties in pinning down the environment induced decoherence arises from the lack of simple observables which could be used for its characterization. Another goal of the present work is to find a signature of the environmental induced decoherence which is closely related to observables. The decoherence stands for the the suppression of the different interference terms in our equations. The usual signature, the decay of the density matrix elements with increasing off-diagonality [20][21][22], characterizes the instantaneous decoherence of the actual state and can easiest be seen in the Schrödinger representation. However the environment induced decoherence is independent of the actual system state therefore it should be inferred from the density matrix independent part of the suppression factor of the interference terms in the expectation values. This can be achieved in a natural manner within the Heisenberg representation where the influence of the environment appears in the time evolution of the operators and the decay of their matrix elements with increasing off-diagonality defines the dynamical decoherence.

II. MASTER EQUATION
The master equation for the reduced density matrix of the observed system can be obtained in two steps, by constructing the effective Lagrangian followed by the derivation of the equation of motion, corresponding to it.

A. Effective Lagrangian
The effective theory for the reduced density matrix is the easiest to derive in the CTP path integral formalism. Let us suppose that the observed system and its environment together make up a closed dynamics, defined by the action S[x, y] in the path integral expressions where x and y denote the system and the environment coordinates, respectively. The full density matrix, given in terms of the time evolution operator U(t, t i ), can obtained by integrating over the trajectory pairs,x = (x + , x − ),ỹ = (y + , y − ), where x ± (t) = x ± , y ± (t) = y ± and x ± (0) = x i,± , y ± (0) = y i,± . The reduced system density matrix, can be written in the form where the trajectoryỹ is closed, y + (t) = y − (t) and the convolution with the initial density matrix, assumed to be factorisable in the system and its environment, is suppressed. By , which characterizes the system dynamics, where the influence functional is given by The unitarity of the time evolution of the observed system plus its environment preserves of the total probability, Tr[ρ] = 1, keeps ρ Hermitean and implies the CTP symmetry, Another important consequence of the relation UU † = 1 1 is that the final time can be sent to infinity and the expectation value of are obtained by the help of the generator functional, where the integration is over the CTP trajectory pairs, spanning −∞ < t < ∞. It proves to be advantageous to use the coordinates x = (x + + x − )/2 and x d = x + − x − , representing the average physical coordinate and the quantum fluctuations, respectively.
We are interested in local harmonic effective dynamics where the influence functional arises from a local influence Lagrangian. The most general quadratic Lagrangian consists of The CTP symmetry excludesẋ 2 , xẋ and x 2 . The coefficient of the time derivatives are given in the mid-point prescription [23].
The total time derivatives lead to boundary contributions to the action [17] and influence the expectation values in a time-independent manner. There are two total time derivative The effect of the first term in the exponent is a gauge transformation which can be taken into account by changing the momentum operator, p → p + αx, in the observables. We set α = 0 below for the sake of simplicity. The β-dependent term is not a gauge transformation and is retained, it introduces a static decoherence (β < 0) or recoherence (β > 0) in the coordinate basis. Our Lagrangian therefore is chosen to be The equation of motion for x, obtained by varying x d , indicates that mν is Newton's friction constant. The parameters d 0 and d 2 control the decoherence in the coordinate basis and have to be non-negative to render the path integral convergent. The environment parameters are quadratic in the system-environment coupling strength, denoted by g below, [23].

B. Equation of motion
The equation of motion for the reduced density matrix can be derived by making an infinitesimal time step, t → t + ∆t, in the path integral (5). The resulting "Fokker-Planck equation", contains the the generator of the time evolution, written in terms of dimensionless variables It is useful to employ the bra-ket notation in the Liouville space of operators [24,25] by considering the density matrix as a "wave function", ρ(x, and using the scalar product To render the transition between the Liouville space operators and the usual operators, acting on the Hilbert space of pure states, we place a hat above the latter. The trace is associated to a particular bra, Tr[Â] = Tr|A , with Tr|x, x d = δ(x d ). While the operators in the transition amplitudes of the closed dynamics are constructed by the help of the canonical pair, (x,p), the scalar product in the space of operators needs two canonical and the left and the right hand sides of the arrow display the Liouville and the Hilbert space expressions, respectively. Similar relations, needed below are The simplified expression for the momentum p ± → p d ↔p can be used for normalizable states.
We need four different bases in the subspace of canonical operators, span by x ± and p ± .
The basis X = (x, p, x d , p d ) is useful in dealing with the solution of the master equation.
The operators handle the elementary excitations of the closed dynamics and their representation in the The dagger is used for the operators, acting on the Hilbert space of pure states. Two further other bases are introduced later.
Though the master equation is linear there is an essential difference in the use of the linear algebra in the space of pure states and density matrices. Namely, the expectation values are linear in the density matrix, Tr[Aρ] = A † |ρ , hence they do not contain the interference terms [18], The basis where the density matrix is diagonal can be called decohered basis because the additive terms of the density matrix are completely decohered.

III. NON-STATIONARY BASIS STATES
The general relaxed stationary solutions of the master equation with Ω = 0 can be specified by prescribing ρ(x, 0), ρ(0, x d ) and ∇ρ(0, x d ) for x d ≥ 0. We seek a solution in the linear space with a Gaussian asymptotic as |x ± | → ∞, in particularly a simple Gaussian, It is easy to see that the choice yields the desired solution. The the positivity of the second moment of the canonical operators, x ± and p ± , It is advantageous to introduce the basis because b and b d annihilate the relaxed state, bρ 0 = b d ρ 0 = 0. By the help of the creation operators one can build orthogonal vectors, which are Hermitian owing to the Hermiticity of the relaxed state, The linear space of the density matrices of the open oscillator is span by the vectors (17).

IV. STATIONARY STATES
The operator set B is useful to generate a basis for the density matrix however the basis states follow a rather involved time-dependence. The stationary states can be obtained from ρ 0 by the help of the ladder operator set, A. These operators are defined by the commutation and perform the shift, Ω → Ω + λ, when acting on a stationary state. The density matrix of a harmonic system, being the product of bra and ket components, involves the normal frequencies of the classical dynamics forward and backward in time. Hence, the spectrum of (19), where ω µ = 1 − ν 2 /4 and τ, τ ′ = ±1, is on the unit circle on the complex frequency plan for an under-damped oscillator, ν < 2, and on the real axis in the over-damped case, ν > 2.
The ladder operators, A = (a + , a − ,ā + ,ā − ), corresponding to the frequency shift λ +± and These operators satisfy the commutation relations and the equation a τ ρ 0 = 0 suggesting to interpret a ± andā ± as annihilation and creation operators of the elementary excitations, respectively.
We write the linear transformations, connecting the operator bases X, A, B and C in the form These transformations have unit determinant, det T = 1, we record here only few matrix elements, T p,a ± = 0, and used below.
The action of the ladder operators on the density matrix can be seen clearer by presenting the action of the ladder operators in the Hilbert space of pure states,ā ± ρ ↔ [â ± ,ρ] and The ladder operators, written in the Hilbert space, are linear superpositions of the creation and annihilation operators of the closed dynamics,â ± andâ † ± , defined by the normal frequency λ ∓,− .
The generator of the time evolution assumes a simple form, in terms of the ladder operators with the eigenstates The Hermitian conjugate of the density matrix (29) is ρ †(a) m,n = ρ giving the overlap with the basis set (17). These density matrices are eigenfunctions of the annihilation oper- provide a resolution of the identity, and become Hermitean for real u and v.
Another family of coherent states, is defined by the help of the ladder operators. They satisfy the eigenvalue conditions a ± ρw = w ± ρw, the equation Tr[ρw] = 1 and are Hermitean for w * ± = w ∓ . The matrix elements of the linear transformation between the bases A and B, given by eqs. (26), can be used to establish a relation between the families (33) and (37). For this end we introduce the vectors U = (u, v), W = (w + , w − ), and the matrix yielding The insertion of this expression of ρ u,v into (36) gives The conservation of the total probability leads to the extension of eq.
obeying the equation of motion in particular L S = L H . In the case of the harmonic oscillator the generator (28) yields a H± = e tλ −,∓ a ± , a H± = e tλ +,±ā ± .
According to the basis transformation A → X, given by (25), the ladder operators can be considered as those linear superpositions of the canonical operators x ± and p ± , building up U and U † in eq. (3), which follow an oscillatory time dependence with a normal frequency in the Heisenberg representation.
As simple an application of eqs. (46) the expectation value of the canonical operators, x and p, in the coherent state of the ladder operators, The Hermitean density matrices, w ± = w ± iw d /2 with real w and w d , give In particular, the expressions (15) for the second moments follows in a simple manner and one can see that the disconnected components to the expectation value in a non-relaxed state receives a time-dependent damping factor e −µt . Hence the energy expression of the closed oscillator approaches the asymptotic value with the dissipative relaxation time scale, ν.
For instance the expectation value of the projector |ψ ψ|, represented by the operator

VII. DECOHERENCE
The decoherence denotes the suppression of the interference terms in a given basis with increasing off-diagonality. One can separate two different decoherence phenomena, depending on the choice of the interference terms in question. The simplest possibility is to follow the decrease of the density matrix elements with increasing off-diagonality [20][21][22]. Such an instantaneous decoherence is a property of the actual state and can be best identified in the Schrödinger representation as the decay of O * S (x, x d )ρ(x, x d , t), in the expectation value of an observable O with x d due to the second factor.
The dynamical feature of the environment induced decoherence, the gradual build up of the suppression in time, originates from the environment rather than the system. Hence, one would like to locate decoherence independently of the reduced density matrix which refers only to the system. That goal can be reached by monitoring the interference term of an expectation value in the Heisenberg representation, where the time dependence of the first factor, the dynamical aspect, is independent of the state. The decay [19], accumulates the decohering effects of the environment in the observation of O during the time evolution.
We start with the instantaneous decoherence in the coordinate representation, the decay of the off-diagonal matrix elements of the density matrix with the off-diagonality, x d . Since the normalization of the density matrix ρ(x, x d ) is handled by the x-dependence the x ddependence always defines a decoherence length scale of the state which may depend on x. In the case of harmonic systems one finds a translation invariant decoherence length, ℓ dec,ρ = 1/R for the relaxed state, (14). Since the density matrix is exponentially decreasing even in the pure ground state the further increase of compared to 1/2 is an indication of the environment induced decoherence. i.e. the environment may be either re-or de-cohering.
Another measure of the suppression in the off-diagonality in the coordinate representation is the absolute magnitude of the expectation value of the shift operator in the coordinate space, using the coefficients (25). When its expectation value is calculated in a coherent state (37) then the operators a ± andā ± can be replaced by w ± and 0, respectively, defines the decoherence length by the gradual orthogonalization of the bra and the ket components of the state during a shift. A Hermitean coherent state with w ± = w ± iw d /2, supports a time-independent decoherence length. However the decoherence is more involved in the state, ρ nd = (ρ w + ρ † w )/2, constructed by the help of a non-Hermitean coherent state with w ± = iw, where and z = |z|e −iφ . The density matrix, as the function of the off-diagonality, displays an oscillatory maximum which diminishes with the dissipation rate.
The dynamical decoherence is found as the function of the initial off-diagonality. To gain some intuitive idea about its origin we note that the Euler-Lagrange equations of the Lagrangian (9).  (14) contain the ratio of the environmental parameters, κ = ν/(d 0 + d 2 ), rendering the limit g → 0 discontinuous, Q 2 → 2κ, R 2 → 1/2κ while Q 2 = 1/R 2 = 2 for g = 0.
The other degeneracy is in the space of the closed ladder operators C, within the subspaces C ± , span by (c ∓ , c ± ), performing the shift Ω → Ω ± i. However an infinitesimal systemenvironment interaction prefers a unique basis since the ladder operators, are singled out by the discontinuous limit of ρ 0 as g → 0. The expectation value of the energy expression of the closed harmonic oscillator, c.f. (15), may be above or below the ground state energy of its the closed counterpart.
The localisation and decoherence lengths scales are related by ℓ dec,ρ = ℓ dec,sh = 1/ℓ loc = √ 2κ. However the practical importance of these characteristic scales is limited by the slow approach to the relaxed state, the relaxation time being O(g −2 ). The classical dynamics, expressed by the first moments of the canonical operators, is regular at g = 0 and the singularity arises from the quantum fluctuations in the relaxed state.
The relaxed state of the translation invariant dynamics ω = 0, the quantum Brownian motion is a Gibbs operator, corresponding to an O( ) temperature [26] and it is natural to expect a similar result for the harmonic oscillator, too. Let us compare the Gibbs operator of a harmonic oscillator of mass m T , frequency ω and temperature T , where the dimensional coordinate is xℓ T with ℓ T = /m T ω and β = ω/k B T . The comparison with (13)- (14) relates the length scales, and gives the equation for the temperature. In the case of infinitesimal interactions the two length scales are identical, ℓ 2 T = ℓ 2 , and the relaxed states within the interval 0 < κ < 1 supports the temperature There are non-thermal relaxed states states for 1 < κ < κ 0 ias long as the upper limit, set by the positivity condition of the master equation, allows.
The high temperature relaxed states are always available for small κ however the pure ground state of the closed oscillator with κ = 1 can be approached by infinitesimally weak environment interaction only if κ 0 ≥ 1. As κ is increased from 0 the temperature and the decoherence length drop and the localisation length increases. The pure ground state of the closed dynamics is reached at κ = 1 and the relaxed state becomes more localized and recohered and ceases to be a thermal state when κ > 1.

B. Border between under-and over-damped oscillator
The normal frequencies λ ±,+ and λ ±,− become degenerate at ν = 2, the transition between the under-and the over-damped oscillators. Such a degeneracy takes place in the classical equation of motion, as well, whose solution is The initial conditions x(0) = x i , andẋ(0) = p i /m yield where the singular factor 1/ω ν comes from the application of the relation x p − x m = (p i /m + x i ν/2)/iω n in eliminating x p and x m . We have the approximate regular solution, The arguments, presented in this work raise several questions. The most pressing is the nature of the true relaxation when the initial state is far from the relaxed state. It is known that the excitations above the relaxed state die out with the dissipative time scale and the relaxation of the Gaussian states is similar process [26]. It remains to see how a general, non-Gaussian initial state approaches the relaxed state.
Another question, raised by this work concerns the conditions, to be imposed on a realistic master equation. Namely the conservation of the total probability is not sufficient to guarantee the stability of the dynamics owing to the emergence of traceless components of the density matrix with exponentially growing weight. They are not present in the space of states, constructed from the relaxed Gaussian state by the help of the ladder operators but can be generated if there is a relaxation in the Gaussian state. How to recognize the existence of such states and to make sure that they are excluded?
Yet another question is raised by the family of relaxed states. The translation invariant open dynamics of a free particle with infinitesimal system-environment relaxes to thermal equilibrium states. However the harmonic oscillator can relax to non-thermal states, as well.
In general, the relaxed state can be more or less localized and more or less decohered than the pure ground state. It would be interesting to identify the dynamical mechanism which stabilizes these unusual states.
Finally, the present results can be the staring point to develop the Heisenberg and the interaction representations in open quantum field theories. However to be useful, these construction must cover non-local interactions, too.