Quantum dynamics in nonequilibrium environments
We present a formalism for studying the behaviour of quantum systems coupled to nonequilibrium environments exhibiting nonGaussian fluctuations. We discuss the role of a qubit as a detector of the statistics of environmental fluctuations, as well as nonMarkovian effects in both weak and strong coupling limits. We also discuss the differences between the influences of classical and quantum environments. As examples of the application of this formalism we study the dephasing and relaxation of a charge qubit coupled to nonequillibrium electron transport through single and double quantum dots.
The standard paradigm of system-environment interactions in quantum mechanics employs an equilibrium environment which is large enough that its fluctuations are Gaussian weiss (). This model is inappropriate, however, if our quantum system couples strongly to a small number of environmental degrees of freedom, which will typically be out of equilibrium and display a full spectrum of fluctuations. The best studied example, both in theorypal02 (); gal05 (); gri05 (); ku05 (); sch06 (); abe08 () and in experimentsim04 (); ith05 (); tia07 (), is the case of two-level fluctuators in the environment of a Josephson qubitmak01 (). If the number of fluctuators coupled to the qubit is small, the decoherence of the qubit shows evidence of nonGaussian environmental fluctuations. Another important class of such environments is provided by mesoscopic transport, in which a quantum system is influenced by the nonequillibrium transport of electrons through some device. Examples of such environments include the partition noise from a quantum point contact ave05 (); ned07 (), and transport through a single-electron transistor (SET) gur08 ().
In this article, we describe a general theory of the influence of nonequillibrium, nonGaussian environments on quantum dynamics. We work within a generalised master equation (GME) framework gur98 (); Brandes04 (), and assume that the environmental degrees of freedom to which our quantum system directly couples may be described by a Markovian GME of the Lindblad form. Under this assumption we derive an effective Liouvillian describing the reduced dynamics of the system alone that can be expressed in terms of environmental correlation functions. We thus obtain an explicit account of the effects on the system of environmental fluctuations of all orders.
Whilst this theory is presented in general terms, we specifically have in mind applications in mesoscopic transport, where we seek to describe the dephasing and relaxation of a quantum system due to the charge fluctuations of a nearby mesoscopic device. In Ref. gur08 () the behaviour of a charge qubit was related to the charge-noise spectrum of SET environment. It is one of the aims of this work to place the results of Ref. gur08 () in a broader context and to generalise not only to arbitrary mesoscopic devices in the Coulomb blockade regime, but also to incorporate the effects of charge-fluctuations of orders beyond Gaussian. For a qubit coupled to the environment via a pure-dephasing coupling, we describe how the long-time behaviour of the qubit is related to the cumulant generating function of the operator through which the system couples to the environment, and show how to calculate this quantity for arbitrary environments.
As illustration of this theory we consider a charge qubit couplied to two mesoscopic environments: i) the SET environment of Ref. gur08 (), which is equivalent to a source of classical telegraph noiseste90 (), and, in certain limits, a model of a single background charge fluctuatorpal02 (); abe08 (), and ii) a double quantum dot (DQD) environment. Whilst transport through a DQD has been extensively studied Brandes04 (); ell02 (); sto96 (); EMAB07 (); kiess07 (), to our knowledge, its role as a decoherence source remains unexplored. Moreover, the DQD environment is an important example because, whereas the SET model can be described in purely classical terms, the inter-dot coherence of the DQD means that transport through it, and hence the fluctuations to which the charge qubit couples, are quantum mechanical in nature. Both these examples exhibit interesting nonMarkovian qubit dynamics, including dramatic visibility oscillations in the strong coupling limitned07 (). The DQD model also exhibits a pronounced quantum Zeno effect Zeno () in this same limit. Comparison of these models highlights the distinctions between quantum and classical fluctuations in determining the dephasing and relaxation of a system coupled to them.
This paper is organised as follows. We first describe the general model considered here and its description in terms of coupled GMEs. We then show how the environment may be traced out to a yield an effective Liouvillian for the system. This Liouvillian is related to environmental correlation functions, used to derive dephasing and relaxation rates for the system in the weak coupling limit. Two special cases are then discussed in which the results are particularly simple: pure dephasing and classical environments with relaxation. We conclude with a study of our two examples and discussions.
I System-Environment Model
Figure 1 depicts the general situation under discussion here. The environmental degrees of freedom are divided into two sets, labelled E and E’, according to whether they couple to quantum system S directly or not. The Hamiltonian of the system-environment complex is , with the isolated Hamiltonians of our decomposition, and interaction Hamiltonians between system and environment E, and between environmental components, and a dimensionsless coupling constant.
We assume that the EE’ coupling is weak, that reservoir E’ is in equilibrium, and that the Born-Markov approximation is valid for the EE’ coupling. Following a standard master equation derivation we trace out environment E’ and obtain a GME for the SE density matrix:
with system Liouvillian , SE coupling Liouvillian , and Liouvillian given by a Lindblad form obtained by tracing out E’.
In the following we will employ a notation for GMEs in which the elements of the density matrix are arranged into a vector with populations first, followed by coherences jah05 (). In this notation, superoperators are written as matrices, and the GME for the SE density matrix, now a vector, has the form
The GME for the environment reads . Let us denote the eigenvalues of as , assumed distinct, and its right and left eigenvectors as and respectively. These vectors form a biorthogonal set, , but are not adjoint, since is nonHermitian. The stationary state of the environment is given by , the zero-eigenvalue eigenvector of , i.e. . The corresponding left-eigenvector , has elements 1 at locations corresponding to populations and is zero otherwise. Similar definitions hold for the free system Liouvillian and its eigendecomposition; but note that, since contains no damping, its nullspace will be of dimension greater than one.
Finally we assume that the SE interaction has the bilinear form, , where is a dimensionless system operator and an environment operator with dimensions of energy. The corresponding Liouvillian is obtained from , which we write in vector notation as
The superoperators , here represented as matrices, can be obtained by considering matrix elements.
Although we will derive results for arbitrary systems, it is often useful to discuss the case when the system is a qubit. In its diagonal basis, the qubit Hamiltonian is and, in the basis , the corresponding free Liouvillian is
The system part of the SE coupling operator is then a traceless Hermitian matrix with elements where is a unit vector and the vector of Pauli matrices. The relevant operators in Liouville space are
with . We will discuss two particular examples in the following:
Pure dephasing: With , the coupling is diagonal in the same basis as the free evolution of the system. In this case, we have a ‘pure dephasing’ model with both diagonal:
Under this coupling, only the off-diagonal elements of the qubit density matrix evolve in time, and we write with the ‘degree of coherence’ to be determined.
Orthogonal coupling: The other coupling that we will explicitly consider is ‘orthogonal coupling’, with . In this case we have the off-diagonal super-operator matrices
and this coupling will induce relaxation in the system.
Finally, we will also refer to the situation when the environment is classical. In this case, only populations are required to describe the state of environment E, and the GME determining its behaviour is actually a rate equation. It then follows that operators acting on E are diagonal and commute at different times. This in turn implies that the superoperators are identical: . As will be made clear below, this situation represents an environment which experiences no back-action due to its interaction with the system.
Ii Effective system Liouvillian
Having set up our model of coupled master equations, we now proceed to derive a description of the system’s behaviour in terms of environmental quantities. To this end, we derive an effective Liouvillian for the system dynamics. Laplace transform of Eq. (2) gives with SE propagator . This we expand in orders of as
with , the free SE propagator. Assuming that the environment starts in its steady-state , the reduced system propagator is given by , corresponding to a trace over the remaining environmental degrees of freedom. With the expansion of Eq. (8), we have
where we have identified the free system propagator
We can consider the system propagator as arising from an effective system Liouvillian, , which will be nonMarkovian. We write this as a series in :
such that the full system propagator can also be written as
Formal resummation yields
where , the projector out of the environment steady state. Using the eigendecomposition of the free SE Liouvillian, we can write the free SE propagator as
with . The effective Liouvillian may then be written
The effective Liouvillian of Eq. (14) or Eq. (LABEL:LeffALT) is the main formal results of this work; it describes the system dynamics in a compact, self-contained form and includes environmental fluctuations of all orders. In this form it is not particularly instructive, however, since both S and E quantities appear in an intertwined way. In order to see the significance of these results then, we will consider first a weak coupling expansion, and then some special cases where S and E dependencies can be separated.
Before doing so, let us note that an expression similar to Eq. (9) can be written down for the reduced environmental propagator, . The initial state of our system, which we take to be a qubit here, is arbitrary and can be written in the form
which defines the vectors , and coefficients and . The only conjugate state we shall need here is . With the system starting in this state, the environmental propagator can be written as
With this expression we can calculate the effects of back-action of the system on the environment. We will not follow this calculation further, except to note what happens for classical environments. In this case, we have and thus . The effective environment propagator thus contains terms like and, as is easy to verify, . All terms beyond the first in Eq. (LABEL:OMeffE) start in just this fashion, and therefore, for classical environments we have , and there is no back-action. It is possible to construct models with a classical environment that do experience back-action with, for example, the rates of the free environmental Liouvillian depending on the state of the qubit mak01 (); gur08 (); oxt06 () Such models, however, are outside the class discussed here in which all back-action effects arise from the quantum-mechanical nature of the system-environment coupling.
Iii Weak coupling: dephasing and relaxation rates
We now consider the situation where the SE coupling is small and describe the weak coupling expansion of Eq. (LABEL:LeffALT). In this case, we can consider the partial Liouvillians of Eq. (13) as successive approximations to the full Liouvillian. At first order, we have , with the steady-state expectation value of operator .
At second order we have
From the form of , we see that this expression depends on environmental quantities like , which can be evaluated straightforwardly for any particular model. Moreover, they can be related to correlation functions of operator via the quantum regression theorem (QRT) lax68 (); carBOOK (). Let us define the second-order correlation function
with . Here the time-dependence of the operators is given by the evolution of the full environmental Hamiltonian . Using the QRT to express this correlation function in terms of quantities acting on E alone, we obtain . Similarly, by recalling that super-operator is equivalent to operator acting from the right, we obtain
Putting these results together, we obtain our final form for the second-order effective Liouvillian
This Liouvillian determines the system behaviour for all times in the weakly coupled limit for arbitrary system and environment. Its form is simply that of a matrix in system-space, the elements of which contain environmental correlation functions evaluated at various frequencies.
The long-time behaviour of a qubit can be described by a pair of rates, and describing dephasing and relaxation respectively. These rates are determined from , or as is the case here, its second-order approximation. We first diagonalise , such that the effective system propagator may be written . We then find the poles of each , with the th eigenvalue of . In the long time limit, only the pole lying rightmost in the complex plane contributes. For a qubit then, assumes the asymptotic form
with constants, some frequency of coherent oscillation and and , the aforementioned rates.
We now discuss Eq. (22) and the corresponding rates for two illustrative couplings.
Pure dephasing: Let us consider a qubit and set , since we can always incorporate it into a redefinition of . The effective Liouvillian of Eq. (22) is diagonal, has zeroes at first and second diagonal elements, and has the third element
and . The off-diagonal elements of the qubit density matrix there evolve as with . The rightmost-lying pole of is
correct to second order in . The dephasing rate is therefore
with the full, symmetrised correlation operator
with denoting the anticommutator.
Orthogonal coupling: We now consider the orthogonal coupling and set for simplicity’s sake. The non-zero poles of the second-order effective Liouvillian with system operators of Eq. (7) are
The corresponding relaxation and dephasing rates are
correct to second order in . The stationary state of the qubit is found to be
as a measure of the purity of a qubit density matrix. For a completely mixed state and pure state . For a pure dephasing model, the final purity depends on the initial state — the generic final state is , where is given by the initial conditions. The purity of this state is , which is unity if we start in a pure localised state, , and zero if we start in the superposition . For the stationary state of the orthogonal coupling model, Eq. (30), the purity is
with , the commutator. In words: the purity is determined by the ratio of the Fourier transforms of the commutator and the anticommutator of fluctuation-operator at different times. For a classical environment, the commutator is zero, and the purity is zero. Nonzero values of the purity are an indicator of the existence of back-action of the system on the environmentgur08 (), and in the current class of models, the back-action is always quantum.
These results illustrate the generality of the connexion between these rates and the second-order correlation functions given in Ref. gur08 (). In principle, we can extend the above analysis to arbitrary order in , making the connexion to environmental correlations functions with the QRT. At third order, for example, we have the effective Liouvillian
With third-order correlation functions defined as in appendix A, this Liouvillian can be written as
Iv Pure Dephasing
The Liouvillian of Eq. (14) contains a mixture of system and environment operators in the inverse and in general this means that this inverse can not be carried out explicitly. Useful results can be obtained by expansion, as above, but the results can becomes unwieldy. In certain cases, however, we can effect a separation of S and E components, expressing the matrix elements of in terms of environmental quantities. In this section and the next, we consider two cases: pure dephasing and orthogonal coupling to a classical environment.
In the pure dephasing model, the effective Liouvillian is diagonal, and only has non-zero elements at positions 3 and 4. The degree of coherence is given by the third diagonal element of . In an interaction picture for the system (obtained by shifting ), we obtain
where we have used the shorthand . This quantity contains information on environmental fluctuations of all orders, as may be seen by expansion. However, expansion is not necessary because, for a finite-dimensional environment E at least, it may be evaluated directly in closed form through matrix inversion. Evaluation for an infinite-dimensional environment may still be possible using phase-space methods carBOOK ().
Equation Eq. (36) also suggests the point-of-view of the pure-dephasing qubit acting as a detector of the environmental fluctuations. As we now show, , the inverse Laplace transform of , is related to the generating function for the zero-frequency cumulants of operator . The inverse Laplace transform of is obtained from the roots of fli08 (). In the long time limit, we only need the pole lying rightmost in the complex plane, which we denote , such that we can approximate with some constant. Considering as infinitesimally small, we can write as an expansion about : and solve order-by-order. For example, the first three terms are
with the shorthand . Application of the QRT, shows that these quantities are equal to the zero-frequency limits of the Keldysh-ordered correlation functions: , is equal to the zero-frequency limit of from Eq. (27), and is the zero-frequency limit of defined in appendix A, and so on. Therefore, within the approximations made here, may be identified as the cumulant generating function for zero-frequency, Keldysh-ordered correlation functions of the operator, which we denote , . The coupling parameter is identified with the ‘counting field’ such that
This result mirrors that of full counting statistics naz03 (); FCS () in which the dephasing of a probe qubit is related to the cumulants of the number of electrons passed through the device, viz. the zero-frequency correlation functions of current fluctuations . In this context, the relationship between Keldysh-ordered CGF and dephasing was to be expected. However, by proceeding in the above manner, we have obtained explicit expression for the full CGF, as the inverse Laplace transform of with as in Eq. (36), as well as for the lowest cumulants from Eq. (37). Obtaining these expressions from standard Keldysh approach would be non-trivial. The simplicity arises here from the introduction of the two superoperators , which correspond to the two branches of the Keldysh contour.
We have talked considerably about the long-time limit of the behaviour of the qubit. However, the short-time behaviour is also of interest, particularly with an eye to applications in quantum information processing where we definitely want to avoid the heavy dephasing of the long-time limit. In such circumstances, only the weak-coupling limit is of interest and thus we restrict our short-time discussion to the second-order approximation. We have as before, but from truncating Eq. (36), we approximate
with first-order contribution removed by considering the appropriate rotating frame. As shown in appendix B, we can explicitly perform the Laplace transform of the arising from Eq. (39) and, correct to order and leading term in , the coherence decays as
The leading term is proportional to , which is to be compared with the linear dependence that would arise from a simple Markovian decay . Note also that it is the quantity , and not , that determines the time-scale of this initial behaviour.
V Orthogonal coupling and classical environment
We can also effect the separation between system and environment quantities if environment is classical. For simplicity, we consider just the orthogonal coupling of Eq. (7). To proceed, we define the system operator
and use it to transform the effective system Liouvillian of Eq. (14). With , we find
The stationary state in the original basis is , the fully incoherent superposition of the two qubit states. This holds true for any non-pure-dephasing coupling to any classical environment without back-action.
Vi Single Electron Transistor
As a first example, we consider a charge qubit coupled to a SET environment as discussed in Ref. gur08 (). The environment consists of a single level coupled to electron reservoirs that can either be occupied or empty. The infinite bias limit is assumed such that transport is unidirectional, with electrons entering the level with rate and leaving with rate . The only pertinent density matrix elements are the two populations of an empty or occupied level. In this empty/full basis, the Liouvillian of single level reads
The stationary state of the environment is , with , and has a single non-zero eigenvalue: . The charge qubit is taken to couple to the charge in the SET, and we take the coupling operator to be with , the number operator for the single-level, and setting the energy scale. The two coupling superoperators are
which are observed to be equal since we have a classical environment.
With the qubit coupled to this environment in a pure dephasing configuration, the effective Liouvillian term of Eq. (36) evaluates
The simplicity of this model means that the inverse Laplace transform of can be performed analytically. With for simplicity, we have