Coherence and entanglement of mechanical oscillators mediated by
coupling to different baths.
We study the non-equilibrium dynamics of two mechanical oscillators with general linear couplings to two uncorrelated thermal baths at temperatures and , respectively. We obtain the complete solution of the Heisenberg-Langevin equations, which reveal a coherent mixing among the normal modes of the oscillators as a consequence of their off-diagonal couplings to the baths. Unique renormalization aspects resulting from this mixing are discussed. Diagonal and off-diagonal (coherence) correlation functions are obtained analytically in the case of strictly Ohmic baths with different couplings in the strong and weak coupling regimes. An asymptotic non-equilibrium stationary state emerges for which we obtain the complete expressions for the correlations and coherence. Remarkably, the coherence survives in the high temperature, classical limit for . This is a consequence of the coherence being determined by the difference of the bath correlation functions. In the case of vanishing detuning between the oscillator normal modes both coupling to one and the same bath, the coherence retains memory of the initial conditions at long time. A perturbative expansion of the early time evolution reveals that the emergence of coherence is a consequence of the entanglement between the normal modes of the oscillators mediated by their couplings to the baths. This suggests the survival of entanglement in the high temperature limit for different spectral densities and temperatures of the baths and is essentially a consequence of the non-equilibrium nature of the asymptotic stationary state. An out of equilibrium setup with small detuning and large produces non-vanishing steady-state coherence and entanglement in the high temperature limit of the baths.
I Introduction, motivation and goals
Progress in quantum information and quantum computing with continuous variables is receiving much attention with platforms that implement quantum optics, cavity quantum electrodynamicsbraun (); girvin () and optomechanicsopto1 (); opto2 (); opto3 (); opto4 (); kipp (). The quantum mechanical degrees of freedom unavoidably couple to other environmental degrees of freedom, or bath, that induce dissipation and decoherence. Therefore, their dynamics are treated as a quantum open systembreuer (); weiss (); gardiner (); ines (); alonso (); marzo (). The motivation for a fundamental study of dissipation and decoherence by environmental degrees of freedom is bolstered by the theoreticalpaavo (); brum (); brunner (); xu () and experimentalbathengi (); ver () potential to engineer the properties of the environmental bath. Quantum brownian motionfeyver (); leggett (); ford1 (); hu1 (); hu2osc (); flehu (); breuer (); weiss () provides a paradigmatic model to study the dynamics of open quantum systems. In this model a quantum mechanical oscillator is linearly coupled to a bath described also by a (large) number of quantum mechanical oscillators; the properties of this bath are determined by its spectral density. This simple yet illuminating model has yielded a deep understanding of the role of an environmental bath on decoherence and dissipation of quantum mechanical degrees of freedom.
This model also provides a theoretical foundation for the emerging field of quantum thermodynamicsquthermo1 (); quthermo2 (); quthermo3 (); paz (); buttner1 (); kosloff () and quantum entanglement induced by coupling to environmental degrees of freedombuttner (); marq (); ronca (); estrada ().
A remarkable experimentgroex () that uses an opto-mechanical resonator probed the spectral properties of an environmental bath coupled to the micro-mechanical oscillators, thus paving the way towards a deeper understanding of the effects of the coupling between quantum mechanical and environmental degrees of freedom in experimentally controlled platforms.
The possibility of engineering the properties of the environmental degrees of freedom including coupling to several different baths, tailoring decoherence and dissipative properties could allow novel cooling techniques of optomechanical systemsxu (). Recent experimental studiesexp1 (); exp2 (); exp3 () have demonstrated the feasibility of coupling various quantum systems to non-equilibrium baths, and of tailoring the environmental degrees of freedom. Recent theoretical analysisbrum () of a V-type molecular system showed the possibility of long-lived coherences despite noise-induced decoherence, and ref.brunner () reports the emergence of steady-state entanglement in the case of finite dimensional coupled systems such as superconducting qubits or quantum dots. Furthermore, coupling to different non-equilibrium environments has been argued to lead to the persistence of entanglement in the high temperature limithiTent (); estrada (); hiTvedral () as well as to the coherent mixing of mechanical excitations in nano-optomechanical structurespainter (). Ref.marq () establishes bounds on the couplings to the baths to maximize entanglement of the quantum degrees of freedom induced by these couplings. A widely used approach to studying decoherence and dissipation relies on the quantum master equation for the reduced density matrix of the system, either in the form of influence functionals obtained in ref.hu1 () or of the Lindblad formbreuer (); weiss (); gardiner () after tracing over the bath degrees of freedom. Recent articlesjoshi (); keeling () study the dynamics of quantum coherence for a system of two oscillators coupled with different couplings to a thermal bath within the framework of the quantum master equation. These studies question the applicability of the Lindblad quantum master equation approach to the non-equilibrium dynamics and suggest important modifications in the case of a system of several oscillators coupled to a bath.
Motivation and goals:
Experimental advances in cavity electrodynamics and optomechanics along with the possibility of engineering the spectral properties of environmental baths open new avenues for quantum information and quantum computing and provide new platforms for fundamental studies of coherence, dissipation and quantum thermodynamics. Motivated by these developments we study the non-equilibrium dynamics of two coupled “mechanical” oscillators, each one in turn coupled to two different and uncorrelated baths at different temperatures. This model is a generalization of those considered in refs.buttner (); marq (); xu (); estrada (); keeling (); painter () which mostly focused on the influence functionalhu1 (); paz (); estrada () or Lindblad approach to the quantum master equationbreuer (); gardiner (). Instead, we solve exactly the Heisenberg-Langevin equations for general couplings and spectral densities of the baths. Additionally, we focus specifically on the Ohmic case, which allows an analytic treatment, addressing in detail correlations and coherences in the asymptotic stationary regime.
Our main goals are:
To provide a general solution of the Heisenberg-Langevin equations valid for two mechanical oscillators with linear couplings to different environments with general spectral densities. We study the non-equilibrium dynamics as an initial value problem to reveal the approach to a stationary regime in the asymptotic long time limit, allowing one to deal with cases in which correlation functions retain memory of the initial conditions in this limit.
The provide a complete analytic treatment in the case of Ohmic baths. In this exactly solvable case we study the emergence of an asymptotic stationary but non-equilibrium state. We focus on two relevant cases: a strong coupling case that includes vanishing detuning between the normal modes of the mechanical oscillators, and a weak coupling case corresponding to weak damping on the scale of the normal mode frequencies.
To obtain the correlation functions and coherence (off diagonal correlations between the normal modes) at different times. Specifically we focus on the interesting and potentially observable survival of coherence in the classical high temperature limit.
To discuss renormalization aspects that arise from the coupling of individual oscillators to two different baths, which results in “mixing” of the normal modes of the mechanical oscillators, requiring new renormalization counterterms.
To explore the temporal emergence of entanglement and coherence between the normal modes mediated by their couplings to and entanglement with the bath degrees of freedom.
Brief summary of results:
We obtain the general solution of the Heisenberg-Langevin equation for the case of two mechanical oscillators with linear couplings to two different and uncorrelated baths. The different couplings to the baths lead to a “mixing” of the normal modes of the mechanical oscillators. This mixing implies novel renormalization aspects and the necessity for off-diagonal counterterms in the effective Hamiltonian of the mechanical oscillators.
The case of strictly Ohmic bathsbreuer (); weiss () yields an analytic solution for the time evolution of the correlation functions. This solution allows one to obtain correlation functions and coherence (off diagonal correlations of the mechanical normal modes) at different times without the need to invoke the quantum regression theorembreuer (); gardiner (). A stationary non-equilibrium state emerges in the asymptotic long time-limit. We find both in the strong and weak coupling regimes that coherence survives in the high temperature limit if the baths feature different spectral densities and/or temperatures.
We show that this surprising and counterintuitive result is a consequence of the coherence being proportional to the difference of the correlation functions of the baths degrees of freedom.
In the strong coupling case coherence survives in the high temperature limit if the baths feature different temperature, whereas in the weak coupling case coherence survives in this limit if the temperature and/or the couplings are different. Classical equilibrium equipartition follows in both cases when the baths are at the same temperature; in this equilibrium case the coherence is suppressed in the weak coupling case.
When the renormalized normal modes of the mechanical oscillators are degenerate and both couple only to one bath, we find that the asymptotic long time limit retains memory of the initial conditions, possibly suggesting a breakdown of a Markovian approximation.
A perturbative expansion is implemented to learn about the early time evolution of the coherence in the case when both baths are at zero temperature. We find that the emergence of coherence is a manifestation of the entanglement between the normal modes of the mechanical oscillators mediated by their entanglement with the environmental degrees of freedom. This study leads us to conjecture that entanglement may survive in the high temperature limit (of each bath) when the baths are at different temperatures with large relative to the frequencies of the normal modes. We conclude that this result opens the possibility of designing a setup of mechanical oscillators coupled to two baths at different temperatures as a platform that maintains coherence and entanglement in the high temperature limit of both baths.
Ii The model and the general solution of the equations of motion.
We consider two interacting oscillators of equal unit mass coupled to two different baths in equilibrium at different temperatures, specifically
The Hamiltonian describes two independent baths of harmonic oscillators to be taken in thermal equilibrium at different temperatures respectively. The system-bath coupling is taken to be
where is an arbitrary mixing angle and
This system-bath Hamiltonian generalizes the cases studied in refs.keeling (); marq (), and as will be seen below, it describes mixing and coherence among the mechanical oscillators similar to the coherent mixing of excitations in nano-optomechanical structures described in ref.painter () and the case of two-mode coupling in multimode cavity quantum optomechanicsmeystre (). In ref.keeling () the case was considered with only one bath; after phase redefinitions the system bath coupling was taken as . Writing and absorbing into a redefinition of the couplings in the system-bath Hamiltonian (II.4,II.5), the model studied in ref.keeling () is equivalent to the generalized model described above with and .
It is convenient to introduce the vector
and the frequency matrix
to write the potential term in as ; diagonalizing this quadratic form we obtain the normal modes of the coupled oscillators. Introducing
the matrix is written as
which yields a straightforward diagonalization by a unitary transformation, namely
are the normal mode frequencies.
The coordinates and momenta of the normal modes (), are related to the original ones () by
The system Hamiltonian (II.2) becomes diagonal in the normal mode basis, the coordinates and momenta describing independent, uncoupled harmonic oscillators of frequencies respectively.
while in the normal mode basis () it becomes
In the normal mode basis the total Hamiltonian is
where is given by (II.16) in the normal mode basis.
The main conclusion from this analysis is that the form of the total Hamiltonian (II.17) is general even when the mechanical oscillators are coupled.
The equations of motion in the normal mode basis are
Treating the dynamics as an initial condition problem will allow us to examine the approach to a stationary state and correlations in this state in what follows. We proceed to solve the equations of motion for the bath variables and insert the solution into the equations of motion for the system coordinates, namely
are the “free-field” operator solutions of the homogeneous equation in terms of independent annihilation () and creation () bath operators. The independent baths are assumed in equilibrium at temperatures respectively, with statistical averages
The self-energy kernels are given by
Here the self-energies for each bath are given by
in terms of the spectral densities of the baths given by
and the noise terms
We note that the spectral densities are odd functions of .
where correspond to statistical averages over the bath variables and are given by (II.33,II.34), respectively. This relationship between the noise correlation functions and the self-energies in (II.32) is a manifestation of the fluctuation-dissipation relation independently for each bath.
We note that as a consequence of the couplings to the baths, the equations of motion (II.27,II.28) are off diagonal in the normal mode basis; in other words the coupling to the baths induces a “mixing” between the normal modes. This mixing is at the heart of the bath-induced coherencekeeling (), namely off diagonal correlations between the normal modes and, as will be shown in a later section, of the entanglement between the normal modes.
with . From equation (II.32) it follows that
We note that the couplings to the bath induce off diagonal entries in the matrix equations of motion; as discussed below these off-diagonal, bath-induced terms will lead to coherence between the different normal modes.
Furthermore, the analysis presented above shows that the coupling to the baths induces renormalization of the frequencies (Lamb-shifts) along with the couplings between the oscillators. The resulting effective Hamiltonian can again be brought to the normal mode general form (II.17) by a rotation as discussed above.
Anticipating this renormalization of the normal mode frequencies and mode coupling induced by the interactions with the bath, and following the arguments leading to the general form (II.17) we write the system Hamiltonian in the renormalized normal mode basis corresponding to the renormalized frequencies by introducing a matrix of counterterms and write the total Hamiltonian as
where are the renormalized normal mode frequencies and
is a counterterm frequency matrix that will be required to cancel the contributions from the real part of the self-energy corrections (Lamb-shifts) that diverge in the limit of large bandwidths of the baths.
Where in eqn. (II.43) is given by (II.16) in terms of the renormalized angle . Since the self-energies associated with the normal modes are given by (II.29-II.31) and the counterterms are chosen to cancel the divergent contributions from these, we write following eqns. (II.29-II.31)
where will be chosen to cancel the divergent contribution from . This aspect will be discussed in detail below for the case of Ohmic baths (see section III) but similar considerations should apply for any spectral density of a bath with a bandwidth large compared to the (renormalized) frequencies in the system.
Including this counterterm matrix in , the matrix (II.41) for the equations of motion in Laplace variable now reads
The solution for is obtained by inverting this matrix. It is convenient to introduce the quantities
in terms of which one finds
It is now straightforward to obtain
where is the identity matrix and
The matrix is traceless with determinant , hence its eigenvalues are . Therefore are projectors onto the eigenvectors of with eigenvalues respectively.
The solution of the Heisenberg-Langevin equations can now be obtained by inverse Laplace transform, with
The Bromwich contour is parallel to the imaginary axis in the complex plane to the right of all the singularities of . Stability requires that the singularities have ; therefore the Bromwich contour corresponds to with and
Therefore the solution for the time evolution of the normal mode Heisenberg operators is given by
In order to highlight the separate contributions from initial conditions and noise terms, we write the solution (II.61) as
Correlation functions of Heisenberg operators require two different averages:
Average over the initial conditions denoted by correspond to averaging in the initial state in terms of the averages of the Heisenberg operators at the initial time . We will assume that the initial state is uncorrelated for the normal modes with leading to . These assumptions on the initial state of the system can be relaxed with the corresponding expectation values and initial state correlations changing accordingly.
Averages over the noise terms (see eqn. II.35) in terms of , with the thermal noise averages given by eqn. (II.37). We will denote averages over the noise as . The bath averages (II.37) also imply that .
Because the theory is Gaussian, only the one and two point correlation functions must be obtained; higher order correlation functions are obtained from Wick’s theorem.
The results above describe how to extract the real time dynamics of relaxation and coherence in the general case of two mechanical oscillators coupled to each other as in (II.2) and to respective thermal baths as in (II.4). For general spectral densities, the analysis of the time evolution, correlation functions and coherences will likely involve a numerical study. However, we can make analytic progress in the case of Ohmic baths described in section (III) below.
Iii Ohmic Baths
We consider a Drude model for an Ohmic bath with
from which we find
Alternatively, we also consider the case of sharp cutoffs for the respective spectral densities
where are the cutoffs or bandwidths of the respective baths; in this case we find for the complex self-energy
In the limit both Ohmic spectral densities yield a linearly () divergent real part and the same imaginary part. We are primarily interested in the low energy, long time dynamics which, after renormalization, should be insensitive to the high frequency degrees of freedom of the bath. Therefore in the following we will consider the cut-off spectral density (III.3) for which the self-energies simplify, namely
where it is implicit that . Following ref.weiss () we refer to this as the strict Ohmic case.
We choose the counterterms to cancel the real part of the self-energies, namely
yielding in the normal mode basis
It is convenient to define
where is the detuning between the normal modes. For the renormalized normal modes are degenerate, we emphasize that this is different from the non-interacting case where the (un-renormalized) normal mode frequencies are given by eqn. (II.13) which become degenerate for .
It now remains to input these expressions into the Green’s function to obtain the real time evolution; see below. Because there are several scales in the problem we focus on two limits of particular interest.
iii.1 Strong Coupling.
We refer to the case when the term in (III.10) can be neglected as the strong coupling regime because the contributions from the couplings to the bath, which determine the off-diagonal terms in the “mixing matrix” in (II.56) have the same magnitude as the diagonal terms. We study separately the cases and in the strong coupling regime because the case of vanishing detuning, , is particularly relevant and describes a similar case in ref.keeling (); marq ().
Vanishing detuning corresponds to degenerate renormalized normal modes for the mechanical oscillators. We note that because of renormalization effects from the interactions with the bath, the conditions of degeneracy of the renormalized normal modes are different from that for degeneracy of the “bare” (unrenormalized) normal mode frequencies. From the relations given by equations (II.45-II.47) and (III.6) one can find the relation between the bare frequencies and the spectrum of the baths that leads to the degeneracy of the renormalized normal modes.
from which we identify
which, upon using eqn. (II.58), yields
The form of eqn. (III.16) indicates that performing the unitary transformation
the solution in this new basis is given by
The interpretation of this result is clear: For vanishing detuning the normal modes are degenerate, therefore one can make a unitary transformation (rotation) that diagonalizes the coupling to the independent baths in , eqn. (II.16). Equations (III.15) and (III.16) clearly show that the Green’s function is diagonal in the bath basis. The normal mode coordinates evolve as linear combinations of the modes, which evolve independently in time with simple complex frequencies.
We are now in position to obtain the correlation functions and coherences for the Ohmic case with zero detuning.
Assuming that the system is in the ground state for the independent normal modes at , and using it follows that