Dynamical paths and universality in continuous variables open systems
We address the dynamics of quantum correlations in continuous variable open systems and analyze the evolution of bipartite Gaussian states in independent noisy channels. In particular, upon introducing the notion of dynamical path through a suitable parametrization for symmetric states, we focus attention on phenomena that are common to Markovian and non-Markovian Gaussian maps under the assumptions of weak coupling and secular approximation. We found that the dynamical paths in the parameter space are universal, that is they do depend only on the initial state and on the effective temperature of the environment, with non Markovianity that manifests itself in the velocity of running over a given path. This phenomenon allows one to map non-Markovian processes onto Markovian ones and it may reduce the number of parameters needed to study a dynamical process, e.g. it may be exploited to build constants of motions valid for both Markovian and non-Markovian maps. Universality is also observed in the value of Gaussian discord at the separability threshold, which itself is a function of the sole initial conditions in the limit of high temperature. We also prove the existence of excluded regions in the parameter space, i.e. of sets of states which cannot be linked by any Gaussian dynamical map.
As soon as quantum correlations reve ; revd have been recognized as a resource for quantum information processing, it has been realized that decoherence is the main obstacle to overcome in order to effectively implement quantum technologies. Decoherence appears whenever a system interacts with its environment, so that its dynamics is no longer unitary, but rather described by a non unitary completely positive quantum map, irreversibly driving the system towards relaxation and the loss of quantum coherence z1 ; BrePet . The main effect of the interaction with environment is to set up a time scale over which the dynamics of the system is effectively described by a coarse grained Markovian process towards equilibrium. Conversely, for times shorter than , the dynamics is more involved and the correlations with and within the environment play a major role BrePet ; Weiss ; c1 ; smi11 ; nmQJ ; smi13 . In this regime, decoherence may be less detrimental, and the dynamics may even induce re-coherence: this is why a great attention has been devoted to the study of the corresponding non-Markovian maps, e.g. in different continuous variable systems ranging from quantum optics to mechanical oscillators and harmonic lattices man07 ; Paz09 ; rugg ; Cor12 ; And08 . Besides, there are evidences that non-Markovian open quantum systems Bre09 ; m2 ; m3 ; NMBreur can be useful for quantum technologies cry11 ; est11 . As a consequence, much attention is currently devoted to the analysis of system-environment coupling in order to characterize, control, and possibly reduce decoherence in the most effective way d1 ; d2 , e.g. by taking advantage of the back-flow of information from the environment.
As a matter of fact, non-Markovian models are usually more involved than the corresponding Markovian ones, and only few cases can be solved analytically BrePet ; hpz ; for01 . However, these cases are also of great interest, since they display a rich phenomenology which is relevant for practical implementations. This is especially true for continuous variable systems rev12 , where considering a set of quantum oscillators excited in a Gaussian state, and then linearly interacting with their thermal environment, provides an excellent model for a large class of physical systems in order to study non-Markovianity and the decoherence of quantum correlations. Motivated by these considerations, we address in details the dynamics of quantum correlations between two quantum oscillators each interacting with a local thermal environment. We assume a weak coupling between the oscillators and the corresponding environment, as well as the validity of the secular approximation approximation. These are the minimal assumptions to have a model that displays remarkable differences between Markovian and non Markovian dynamics and, at the same time, allows the use of analytic tools to describe results. We also assume the oscillators initially prepared in a symmetric Gaussian state, and study in details the evolution of their quantum correlations as described by their dynamical paths, i.e. the time evolution in a suitably chosen parameter space.
We start by noticing that the set of Gaussian states, i.e. states with a Gaussian Wigner function Oli12 , do not constitute a manifold, nor it is convex, and thus geometrical approaches to their dynamics are not considered particularly appealing. At variance with this belief, we address the study of decoherence by representing dynamical paths in a suitable, overcomplete, parameter space, involving Gaussian entanglement (negativity) sim00 , Gaussian discord GQD10 ; Ade10 and the overall purity of the state. The use of these variables offers a suitable framework to compare non-Markovian maps and their Markovian counterparts, and to show which properties do, and notably do not, distinguish Markovian and non-Markovian processes. In particular, upon describing the dynamics as a path in the three-dimensional space individuated by the above variables, we observe universality: the dynamical paths do not depend on the specific features of the environment spectrum and are determined only by the initial state and the effective temperature of the environment. The non-Markovianity of the system only changes the velocity of running over a given path. This behavior allows one to map non-Markovian processes onto Markovian ones and it may reduce the number of parameters needed to study a dynamical process, e.g. it may be exploited to build constants of motion valid for both Markovian and non-Markovian maps. Universality is also observed in the value of discord at the separability threshold, which moreover is a function of the sole initial conditions in the limit of high temperature. Finally, we find that the geometrical constraints provided by the structure of the parameter space implies the existence of excluded regions, i.e. sets of states which cannot be linked by any Gaussian dynamical map.
The paper is structured as follows. In Section II we introduce the interaction model and briefly review its solution for Gaussian states. We also introduce symmetric Gaussian states and the quantities used to quantify their quantum correlations, i.e. Gaussian entanglement and Gaussian discord. The dynamics of the system is then described in details in both the Markovian and the non Markovian regimes, illustrating universality of the dynamical paths. Section III closes the paper with some concluding discussions and remarks.
Ii Kinematics and dynamics
Here we consider the dynamical decoherence of two oscillators of frequency , each coupled to its own bosonic environment made of modes at frequencies . The baths are separated and of equal structure (see Paz09 ; Cor12 for the interaction with a common bath). The system-bath interaction Hamiltonian is given by (we use natural units)
where is the coupling, the complex modulate the dispersion over the bath’s modes, and , and denote the canonical operators of the systems’ and baths’ modes respectively, i.e. , and . In the weak coupling limit and performing the secular approximation we can write a non-Markovian time-local master equation for the dynamical evolution of the density operator describing the quantum state of the two oscillators in the interaction picture hpz
where and denote commutator and anticommutator between the operators and . Upon defining the spectrum of environment as
the diffusion (heating) and dissipation (damping) coefficients are given by SabLin :
respectively, with . At high temperatures the damping coefficient is negligible and the diffusion is dominant, while at lower temperatures they have the same order of magnitude.
It is worth noticing that the assumptions of weak coupling and secular approximation are the minimal ones to have a model that displays differences between Markovian and non Markovian dynamics. At the same time the dynamical equations remain simple enough to allow the use of analytic tools to describe results.
where , , is the displacement operator and , . The solution of this equation may be written as follows
where is the characteristic function at time and the corresponding quantity at . The quantity represents an effective time-dependent damping factor, given by
The matrices and are given by
Gaussian states have Gaussian characteristic function and they are fully characterized by the mean values of the canonical operators and and by the covariance matrix (CM) , that writes
with . Since the Gaussian character of an input state is preserved by the master equation (1), and we are considering Gaussian states, we need to address the sole evolution of the first two moments. In addition, we can focus attention to the evolution of the CM only, being the quantum correlations independent of the first moments.
In particular, we assume that the initial state is a two-mode Gaussian state with zero amplitude, i.e. , , and with covariance matrix . According to Eq. (3) the evolved state at time is still a Gaussian state with zero mean values, and with covariance matrix given by rugg ; RWA ; ale04
Upon retaining only the terms consistent with the secular approximation we arrive at the expression
is a time-dependent effective diffusion factor.
The non-Markovian features are embodied in the time dependence of the coefficients and , which describes diffusion and damping respectively. For times both coefficients are strongly influenced by the whole spectrum of the environment rugg . On the other hand, for times the coefficients achieve their Markovian limiting values. In particular we have
and the solution (8) rewrites as
where is the CM of the stationary state, i.e. a thermal equilibrium state at temperature and, in turn, a population of thermal photons.
ii.1 Symmetric Gaussian states
A bipartite Gaussian state is symmetric if its CM can be recast (via local operations) in a form depending on two real parameters and , that is
the ’s being Pauli matrices. Note that uncertainty relations impose a constraint which reads sim87 . The evolution under the master equation (1) preserves the symmetry [see Eq. (8)], therefore the evolved CM at time may be still written as , where
with and .
A symmetric CM of the form (9) corresponds to prepare the two oscillators in a squeezed thermal state (STS), i.e a state with density operator of the form
where denotes a single mode thermal state with photons and is the two-mode squeezing operator. For the state reduces to the so-called two-mode squeezed vacuum state or twin-beam state, i.e. the maximally entangled state of two oscillators at fixed energy.
The parameters of the CM are connected to the physical parameters as follows
Furthermore, by introducing the (equal) population (mean photon number) of the two subsystems
the diagonal elements may be written as , while the coefficients describe the correlations among them. It is worth noting that any two-mode entangled Gaussian state can be converted into a symmetric one by local operations and classical communication sym1 ; sym2 . Therefore, our results about the dynamics of quantum correlations actually holds for more general initial states than the symmetric ones, including any initially entangled state.
Indeed, the representation in terms of the coefficients and does not fully illustrate the correlation properties of a state. In particular, it does not allow to analyze the relations between different kinds of quantum correlations, as entanglement or discord, in a dynamical context, and to compare their robustness against dissipation and noise. To this aim we introduce a different (overcomplete) parametrization involving the overall purity of the state
its Gaussian entanglement expressed in terms of the minimum symplectic eigenvalue
(the state is separable iff ) and the Gaussian quantum discord, which for symmetric Gaussian states may be written as GQD10
The parameter space individuated by , , and is overcomplete and
the third parameter is a function of the other two a1 at any
time. In the following, we will describe the dynamics of the system by paths
in the three-dimensional space according to the
Definition (Dynamical path):
a dynamical path for a symmetric Gaussian state is a line
in the three-dimensional space individuated by the
overall purity of the state , its least symplectic eigenvalue
, and its Gaussian discord .
Dynamical paths lay on the surface individuated by the constraint (II.1) and in the region satisfying the uncertainty relations. In terms of the parameters these constraints correspond to
A dynamical path describes the evolution of a symmetric Gaussian state in a noisy Gaussian channel with no explicit dependence on time. This allows one to compare non-Markovian maps and their Markovian counterparts, and to show which properties do, and do not, distinguish Markovian and non- Markovian processes. At the same time, it allows us to reveal the relationships among the different kinds of quantum correlations in a dynamical context. In other words, each dynamical path actually describes an equivalence class of dynamical time-dependent trajectories (including both Markovian and non-Markovian ones), characterized by a specific dependence of the Gaussian discord on the other two parameters.
ii.2 Markovian dynamics
The Markovian master equation depends on the (effective) environment temperature and on the damping , nonetheless the Markovian dynamical paths depend exclusively on the (effective) temperature of the environment. The damping only affects the speed of running over a dynamical path, but not its shape, and the rate determines in a unique way the rate . In the left panel of Fig. 1 we show few Markovian paths for different values of the temperature, assuming the two oscillators initially prepared in a two-mode squeezed vacuum (TWB) , i.e a pure maximally entangled state of the two oscillators. As it is apparent from the plot, two limiting paths emerge at low and high temperatures. The transition from one regime to the other occurs continuously by raising the temperature, and we see that the high temperatures limit is already achieved for temperatures corresponding to
Two other phenomena are revealed by this representation: (i) the value of the discord at the separability threshold () depends only on the initial squeezing and approaches a universal curve in the high temperature limit; (ii) for a given initial state there are STS that cannot be reached during any Markovian decoherence process, despite the fact that they have reduced entanglement and purity compared to the initial state.
ii.3 Non Markovian dynamics
As mentioned in the introduction, non-Markovian dynamics may display remarkable differences from their Markovian counterpart during the initial transient when . Entanglement oscillations may occur, and the separability threshold may be delayed or accelerated depending on the spectrum of the environment. A question thus arises on whether these differences also affect significantly the dynamical path in the space of parameters. As we will see, the answer is negative, and universality occurs. The results about the dynamics that we are going to discuss are independent of the particular choice of environment spectrum, and this is a crucial point of our analysis. However, in order to show some numerical solutions, we employ few examples corresponding to white noise and to both Ohmic and super-Ohmic spectral densities with cut-off . More specifically, we are going to consider Ohmic spectrum
which leads to non-Markovian features when out of resonance, i.e. when , SuperOhmic spectrum
and white noise spectrum .
Let us start by analyzing the high temperature regime, where over a timescale we can neglect the damping (it becomes relevant over times , which is definitely in the Markovian regime). Short time non-Markovian dynamics is thus due to the behavior of the heating function and, in turn, is very sensitive to the details of the environment spectrum . In this limit non-Markovian effects can be seen during the whole decoherence process, with entanglement oscillation across the separability threshold man07 . The dynamics is driven by the approximate dynamical equation
The minimum symplectic eigenvalue is thus given by
The condition imposes a constraint to the dynamical paths, which is the same independently on whether the dynamic of is Markovian or displays oscillations, as long as and . In other words, the paths are the same of the Markovian case, and the possible oscillations of only influences the speed of running over the dynamical path. In the right panel of Fig. 1 we show the dynamical paths for different values of the initial squeezing .
ii.4 Discord at the separability threshold
The condition also implies that the Gaussian discord may be written as
i.e. it depends on the temperature and on the initial squeezing isa12 only through the minimum symplectic eigenvalues. At the separability threshold, i.e. for such that , we have
i.e. the discord at separability is a universal function of the initial squeezing. In Fig.2 we show the Gaussian discord at separability as a function the initial squeezing. The solid black line correspond to the above high temperature approximation , whereas the colored symbols correspond to the full non-Markovian solutions for , obtained taking into account the damping and different environment spectra. As it is apparent from the plot, there is an excellent agreement among the two solutions, independently on the environment spectrum. We also notice that saturates to a limiting value
as far as the initial squeezing increases. The initial squeezing needed for achieve the saturation regime increases with temperature. As it may be seen from the plot, for high temperatures, i.e. for , it is about .
For lower temperature the approximation is no longer valid and the Gaussian discord at separability is given by (Markovian expression)
In Fig. 2 we show as a function of for different values of (dashed gray lines). We also report the values obtained from the full non-Markovian solutions for different environment spectra and not so low temperature, i.e. . As it is apparent from the plot the two solutions are in excellent agreement and this may be understood as follows. At low temperatures the damping and the heating function become of the same order of magnitude and thus the separability threshold does depend on the environment spectrum. On the other hand, separability is always achieved in the Markovian regime, and thus is a universal quantity. The plot confirms that this argument holds also if the temperature is not so low, i.e. for . For times , there is a competition between and and in principle, one would not expect a universal behavior. However, low temperature and weak coupling make the effect of damping and heating very weak, with appreciable perturbation of the initial state only after a long time. In other words, any dynamical effect of the interaction is taking place in the Markovian regime, thus re-gaining universality and independence on the environment spectrum. This also means that the dynamical paths in the left panel of Fig. 1 legitimately describe non-Markovian dynamical trajectories at low temperatures.
ii.5 Universality of constants of motion
Any path-dependent property may be checked analytically using the set of Markovian equations and then extended to the non-Markovian regime, where an analytic approach would be unfeasible. In particular, let us introduce the rescaled time , and recall that in the Markovian regime we have
where pedices refer to initial/stationary state. Then, any constant of motion, e.g. , with built using the Markovian dynamical equation is a constant of motion also in the non-Markovian regime, independently on the environment spectrum, and with potential application for the development of general channel engineering strategies. The temperature dependence disappears in the high-temperatures limit.
Iii Discussion and conclusions
We have addressed the dynamics of quantum correlations in continuous variable open systems and analyzed the evolution of bipartite Gaussian states in independent noisy channels. We have assumed weak coupling between the system and the environment, as well as the secular approximation. These are the minimal assumptions to have a model that displays remarkable differences between Markovian and non-Markovian dynamics and, at the same time, allows the use of analytic tools to describe results.
In describing the noisy evolution of two-mode symmetric Gaussian states we introduced the concept of dynamical paths, i.e. lines in the three-dimensional space individuated involving Gaussian entanglement, Gaussian discord and the overall purity of the state. Dynamical paths describe the evolution of symmetric Gaussian states with no explicit dependence on time. This has been proven suitable to address the decoherence effects of both Markovian and non-Markovian Gaussian maps, and to reveal which properties do, and do not, distinguish Markovian and non-Markovian processes. At the same time, dynamical paths allow us to reveal the relationships among the different kinds of quantum correlations in a dynamical context. Each dynamical path actually describes an equivalence class of dynamical time-dependent trajectories (including both Markovian and non-Markovian ones), characterized by a specific dependence of the Gaussian discord on the other two parameters.
Upon describing the dynamics as a path in the three-dimensional space individuated by the above variables, we have observed universality: The dynamical paths do not depend on the specific features of the environment spectrum and are determined only by the initial state and the effective temperature of the environment. Non Markovianity manifests itself in the velocity of running over a given path. This phenomenon allows one to map non-Markovian processes onto Markovian ones and it may reduce the number of parameters needed to study a dynamical process, e.g. it may be exploited to build constants of motions valid for both Markovian and non-Markovian maps.
Universality is also observed for the value of discord at the separability threshold, which moreover depends on the sole initial squeezing in the high temperature limit. We also found that the geometrical constraints provided by the structure of the parameter space implies the existence of excluded regions, i.e. sets of Gaussian states which cannot be linked by any Gaussian dynamical map, despite the fact that they have reduced entanglement and purity compared to the initial one.
Our results have been obtained for Gaussian states and are not directly transferable to the non Gaussian sector of the Hilbert space. Indeed, there are no necessary and sufficient criteria to individuate and quantify non Gaussian entanglement, and there are no analytic formulas to evaluate non Gaussian quantum discord. The interplay between Gaussian and non Gaussian quantum correlations has been discussed in the recent years nGL ; ng1 ; ng2 ; ng3 , but a complete understanding has not yet been achieved.
Finally, we emphasize once again that the universality of dynamical paths does not depend on the environment spectrum, i.e. it is a consequence of the sole assumptions of weak coupling and the linear interaction between system and environment. It may therefore be conjectured that universality represents a more general feature, characterizing any open quantum system admitting a Markovian limit.
Acknowledgements.This work has been supported by MIUR (FIRB LiCHIS-RBFR10YQ3H), EPSRC (EP/J016349/1), the Finnish Cultural Foundation (Science Workshop on Entanglement), the Emil Aaltonen foundation (Non-Markovian Quantum Information) and SUPA. SO, and MGAP thanks Ruggero Vasile for useful discussions.
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