From non equilibrium quantum Brownian motion to impurity dynamics in one-dimensional quantum liquids
Impurity motion in one dimensional ultra cold quantum liquids confined in an optical trap has attracted much interest recently. As a step towards its full understanding, we construct a generating functional from which we derive the position non equilibrium correlation function of a quantum Brownian particle with general Gaussian non-factorizing initial conditions. We investigate the slow dynamics of a particle confined in a harmonic potential after a position measurement; the rapid relaxation of a particle trapped in a harmonic potential after a quantum quench realized as a sudden change in the potential parameters; and the evolution of an impurity in contact with a one dimensional bosonic quantum gas. We argue that such an impurity-Luttinger liquid system, which has been recently realized experimentally, admits a simple modeling as quantum Brownian motion in a super Ohmic bath.
Quantum Brownian motion has been the starting point for the understanding of more complex dissipative quantum systems We08 (). Applications to quantum tunnel junctions SchZaik90 (), dissipative two–state systems CalLegg87 () and reaction–rate theory Hanggi90 () are just a few among many. In its simplest form, as proposed in the founding papers FeynVern63 (); Schwinger61 (); CalLegg83 (), the environment induced dissipation is modeled by an ensemble of quantum harmonic oscillators linearly coupled to the particle of interest. So far, in most studies of the dissipative dynamics of a harmonically confined Schwinger61 (); Agarwal71 () or a free Grscin88 (); Pottier00 () quantum particle, the quantity of interest has been the reduced density matrix that is obtained by tracing away the bath degrees of freedom in the density matrix of the coupled system. For generic initial conditions this quantity has been obtained with the help of functional integral methods Grscin88 (); Ingold02 (). An alternative simpler, though in general only approximate, description of the reduced density matrix is given by a master equation. For factorizing initial conditions HuPaz92 () and thermalized initial conditions Grab97 () an exact master equation can be obtained. However, it is also known that there cannot be a master equation – in the form of a partial differential equation local in time – for arbitrary initial conditions Grab97 (). The alternative quantum Langevin approach Ford65 () extensively used in quantum optics Gardiner04 () is not sufficiently powerful either, for quite the same reason: only a few special initial conditions can be successfully treated within this approach and the quantum noise statistics are not tractable in the generic case.
Non equilibrium correlation functions have been recently observed in cold atoms experiments on the dynamics of an impurity atom moving in a one dimensional (1D) quantum liquid Giamarchi11 (); Giamarchi07 (); Palzer09 (). Both the impurity and the quantum liquid are confined in an optical harmonic trap so that the impurity motion resembles the dynamics of a damped quantum harmonic oscillator. In Johnson12 () the authors attempted to describe the impurity dynamics within the Gross-Pitaevskii approach at zero temperature. In this paper we will follow an alternative way by applying quantum Brownian motion theory to the impurity problem. More specifically, we will generalize the path integral formalism in Grscin88 () to the language of generating functionals commonly used in quantum field theory. From the generating functional we will easily deduce the non-equilibrium correlation functions for generic non-factorizing Gaussian initial conditions (for a stochastic description of open quantum systems via a generating functional, see Calzetta02 ()) that cannot be obtained within the density matrix formalism. We will use this technique to treat three problems: the relaxation dynamics of a particle confined in a harmonic potential after a position measurement performed at the “initial time”; the relaxation dynamics of a particle after an abrupt change in the parameters of the confining potential; the motion of an impurity RoschKopp95 (); SchiroZwierlein09 (); Pita04 (); Caux09 (); Palzer09 () in contact with a 1D quantum gas all confined in a harmonic potential. In the latter case, we will explicitly compare our theoretical findings to recent experimental results Giamarchi11 () on ultra cold quantum gases. We will argue that, although the impurity-Luttinger liquid system is described by the Fröhlich polaron Hamiltonian, many aspects of the impurity dynamics can be understood in the framework of quantum Brownian motion.
More precisely, the paper is organized as follows. In Sec. II we present the model and we review the main results obtained in Grscin88 (). In Sec. III we employ path integral methods to derive the generating functional of out of equilibrium correlations. Our results cover both factorized and non-factorized Gaussian initial conditions as well as the effects of an initial position measurement performed on the particle. In Sec. IV we study the equilibration processes after an initial position measurement and after a quench in the harmonic potential and we derive the equilibration times for low and high bath temperatures. In Sec. V we apply our formalism to impurity motion in a quantum gas described by the Luttinger theory. To keep the discussion simple, we choose to use a simplified modeling of the experiment in which we neglect polaronic effects Devreese09 (); Tempere09 () as well as the possible renormalization of the external potential Giamarchi11 (). These subtle effects will be analyzed elsewhere. The Luttinger liquid is found to behave as an exotic quantum bath of harmonic oscillators with a highly non Ohmic spectral density and non-linearly coupled to the particle. This is shown to lead to the curious behavior that the oscillator frequency can increase upon increasing the coupling constant between the “bath” and the impurity in strong contrast to the behavior of an Ohmic damped oscillator. We further calculate the non equilibrium equal time correlation function (the variance of the position) and we compare our theoretical results to experimental evidence. In the last section, Sec. VI, we conclude and we present further possible applications of our work.
Ii The model
We study the evolution of a particle of mass evolving in a (possibly time-dependent) potential where is the position operator. The Brownian motion stems from its interaction with a quantum heat bath which is usually modeled by an infinite set of harmonic oscillators linearly coupled to the position operator . The full system is then described by the Hamiltonian , with
is the momentum operator of the particle. The last term in Eq. (1) introduces a time-dependent source , a c-number, that couples linearly to the particle’s position . and are the position and momentum operators of the –th harmonic oscillator, with mass and frequency and , respectively. is the coupling strength between the particle and the -th oscillator’s position. The last term in Eq. (3) compensates for the bath-induced renormalization of the potential. Indeed, the sum of Eqs. (2) and (3) can be rewritten as
which shows the absence of any drift force induced by the bath and ensures that corresponds to the physical potential right from the start. The model Hamiltonian Eqs. (2)-(3) has been widely used in the literature as a generic model for the dissipative dynamics of a quantum particle CalLegg83 (); Grscin88 (); We08 ().
In the Heisenberg representation the time evolution of all possible observables is governed by
with the time-ordering operator. By introducing the density matrix of the initial state the -time average of a set of operators, is
where we took the product of the s to be time ordered (with ) so that we can more easily make the connection between Eq. (II) and its path integral representation. We assumed that is normalized to one. Note that for a generic initial matrix this, as well as any other, correlation function is not necessarily stationary, i.e., it may depend on the times explicitly.
In all cases the model has to be supplemented by information on the initial condition of the coupled system. These are incorporated in the initial density matrix . Equilibrium dynamics can be studied by choosing to be the Boltzmann weight, that is
where is the full coupled Hamiltonian and the normalization constant has been ignored. This truly equilibrium density matrix has to be distinguished from , a case in which each component of the “universe” (the whole particle–bath system) is in equilibrium on its own at the same temperature. This subtle point is often overlooked in the literature.
Non equilibrium dynamics can be studied whenever the initial density matrix is not of the form in Eq. (7). The simplest choice is an initial product state for which the initial density matrix factorizes into two contributions and which solely depend on particle and bath variables, respectively:
Brownian motion Hanggi05 (); We08 () as well as the dynamics of more complex macroscopic systems Cugliandolo06 (); Culo98 (); Culo99 (); Cugrlolosa (); Kech (); Kechye (); Bipa (); Arbicu10 (); Arbicu10b () with a factorized initial density matrix have been studied in a variety of physical situations. However, in many cases it is not appropriate to assume Eq. (8) since one has no command over the bath and it is impossible to “switch it on and off” at will. In addition, with recent developments in cold atom experiments, new classes of initial conditions become of relevance.
The first one covers all situations in which the particle is in equilibrium in a potential and either it is released or the potential is suddenly modified at . In this case Eq. (7) holds with replaced by describing the initial state.
The second class concerns all situations in which the position of the free particle is measured at . This procedure projects the initial density matrix onto the quantum states of the measurement outcome. We focus on the case where no quantum quench is performed in addition to the position measurement so that . If the position is exactly determined at the initial density matrix is
the projection operator onto the state . If, instead, we take the measured position of the particle to be Gaussian distributed around the projection operator takes the form
where measures the uncertainty of the particle’s position at . Once again we neglected the irrelevant normalization factor.
A third important class of initial conditions are the factorized density operators, see Eq. (8), in which the initial state of the system is a pure state. Since any state can be expanded in terms of displaced Gaussians (or coherent states) it suffices to consider initial states of the form
to cover the whole class of initially factorized pure states.
In this article we derive a generating functional that allows us to obtain the -time correlators for these types of initial conditions. We are mainly interested in the evolution and averages of the particle’s position observables for which with some function depending on the position of the particle. Note that due to the coupling of the bath to the particle’s position the momentum dynamics follow from the Heisenberg equation . Therefore, by focusing on the particle position operator we simultaneously describe the dynamics of the particle’s momentum. While in Grscin88 () the authors derived an explicit expression for the equilibrium correlation functions the generating functional will allow us to go beyond the equilibrium case.
Iii The generating functional
In this section we derive the generating functional of all (non-equilibrium) correlation functions. This goes beyond the analysis in Grscin88 () where explicit expressions for equilibrium correlation functions were given. More precisely, we derive a functional of two time-dependent sources such that the two-time correlation is given by
and similarly for higher order correlations.
We obtain the path integral formulation of the generating functional by making use of the coherent states of the bath variables which are defined in App. A. The ensuing functional integration includes paths over particle and bath variables. Since we are not interested in the degrees of freedom of the bath, we average over all bath variables to find a “reduced action” that only depends on the particle variables. In the special cases discussed below (e.g., harmonic potential) the remaining path-integrals can also be performed and the functional can be fully determined. In this section we sketch all steps in the derivation. Further technical details are reported in App. A. The reader who is not interested in these technical details can jump directly to Eq. (54) where its rather lengthy final expression is given.
iii.1 The density matrix
In terms of the product states between the particle and the bath eigenstates the matrix elements of the time evolution operator read
and of its Hermitian conjugate
is the anti-chronological time ordering operator. The elements of the time-dependent density matrix, , are given by
where the matrix elements of the initial density matrix have been denoted by
and we used the short-hand notation
The path integral representations of and are
where we made clear with the superscripts and which paths belong to and , respectively. The functional integration measures are defined in App. A.
iii.2 Reduced density matrix for a system initially coupled to an equilibrium bath
The time-dependent density matrix in Eq. (16) still contains information about the degrees of freedom of the bath which we are not interested in. Therefore, we average (trace) over all the bath variables to find a reduced density matrix that depends only on the particle variables and the external sources.
We are interested in a system that is initially coupled to an equilibrium bath. Therefore,
where all initial Hamiltonians are labeled with a subscript . At this point it is not necessary to make explicit since this term involves only particle variables that are not affected by the trace over the bath variables. The matrix element of the initial density operator Eq. (17) in Eq. (16) can be represented by an imaginary time path integral
where the initial action is in general different from reflecting the fact that . The reduced density matrix can now be recast as
where is the “influence functional” that depends only on the particle variables, as also do and . The path integral runs over all paths with
which is the reason for the name “closed-time path integral”. It is convenient to introduce the linear combinations
Expected values evaluated at different times are now expressed in terms of a path-integral over , and with an effective action ,
where is given by
We introduced the initial mass of the particle and the initial potential that are in general different from the “bulk” mass and potential . This allows for quenches in these parameters. Note that the case in which the initial state is a pure state [e.g. Eq. (11)] can be easily recovered by setting and or, equivalently, by noting that the path shrinks identically to zero (since there is no initial Hamiltonian for this simple type of initial condition).
The superscripts in the path integral in Eq. (23) remind us of the constraint that the paths are subject to. One has
Note that due to the periodic boundary conditions of the trace .
iii.3 Generic Gaussian initial conditions
It is very easy to include the change of induced by the initial position measurement in Eq. (9). By using the explicit Gaussian form of the projector [see Eq. (10)] the dependence on the initial measurement can be simply incorporated in by an additional term of the form
In the limit of strong uncertainty the effect of the initial measurement is blurred.
In order to recover the case where the initial state of the system is pure and decouples from the environment [which corresponds to the factorized initial density matrix with given by Eq. (11)] the action in Eq. (III.2) has to be supplemented by
with the notation
In the following expressions we will write only the terms that depend on or since the ones depending on and contribute only to an overall constant.
iii.4 The sources
The source term appears as in [see Eq. (14)] and as in [see Eq. (15)]. For convenience, we distinguished the function existing on the positive running branch of the closed time contour, which we still call , from the one existing on the negative running branch of the same contour, which we call . This implies that the potentials in Eq. (III.2) are given by and .
After the transformation of variables in Eq. (20) we obtain two external time-dependent sources and which couple linearly to the variables and , respectively. All correlation functions can be computed from the generating functional as derivatives of with respect to or evaluated at . A physical force is represented by , that is by and . Therefore, the linear response of the mean value Eq. (23) to an external force can be obtained for .
The generating functional, that is to say, the trace over the reduced density matrix in the presence of the external sources reads
where the path integral is subject to the constraints in Eqs. (25). The overall normalization factor depends on , and all parameters in the model but not on the fields. We can now write
and all other correlation functions can be obtained in a similar way by noting that and . At this point it has become obvious why two sources are needed in order to obtain all non-equilibrium correlation functions.
iii.5 The harmonic case
To go further we restrict ourselves to the study of a quantum Brownian particle in a harmonic potential for which
The choice of a quadratic potential renders the problem analytically solvable. The generating functional can be calculated by simply evaluating the action on its minimal action path (over the initial condition branch and the time-dependent branches) as Gaussian fluctuations yield only pre-factors that are independent of the sources and can be determined at the end of the calculation from the normalization of the density matrix. Note that, although both initial and bulk potentials are harmonic, they are not necessarily the same thus allowing for the study of quantum quenches.
iii.6 Integration over the initial condition
with the fixed end-points and . As the path is part of the whole closed-time path it implicitly depends on the fixed end-time as well. In Grscin88 () one can find a detailed analysis of this equation of motion which uses a Fourier expansion of the path on the interval . By using the results found therein we obtain
We introduced the complex “force”
with the functions and
The constants and are given by
with , , [for the definition of see Eq. (42) below] and
where is the spectral density of the bath. The two-time function reads
the real part of the kernel . The time-dependent bath kernel is given by [see Eq. (144)]
The functions and as well as the kernel are not to be confused with the correlation functions and the linear response function that will be denoted by and , respectively.
iii.7 Real time minimal action paths with external sources
The equations of motion for and read
The action evaluated along the minimal action paths can be determined by inserting the solutions to Eqs. (III.7) and (44) into Eq. (34). However, the authors of Grscin88 () noted a simplification of the calculation which can be generalized to our case where the source is also present (in Grscin88 () no external source for was used). The idea is the following. After a partial integration in the second line of Eq. (34) the action takes the form
On the other hand, one can split the force Eq. (35) into its real and imaginary parts . Then, the minimal action path splits into , where satisfies the boundary conditions . The trick is to show now that one can simply focus on the real part of the minimal action path in order to obtain the complete stationary phase action. Indeed, if we evaluate the action Eq. (34) only along the minimal and we obtain
where we used the fact that satisfies the real part of Eq. (III.7). We now want to show that Eqs. (III.7) and (45) are indeed equal. With the help of the imaginary part of Eq. (III.7) and the equation of motion (44) we can easily prove by integration by parts that
and by using this identity in Eq. (III.7) we recover Eq. (45). Therefore, the right-hand-side (rhs) of Eq. (45) and the rhs of Eq. (III.7) coincide. It is sufficient to evaluate the action Eq. (34) along the real component that satisfies a much simpler equation than .