Nonequilibrium field theory for dynamics starting from arbitrary athermal initial conditions

Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate “$n-\mathrm{particle}$ Green's functions” that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.

Figure 1

(a) Keldysh contours showing forward and backward propagation in time. In our formalism, the matrix element of the initial density matrix ${\widehat{\rho }}_0$ is written as the derivative of $exp\left[\mathbf{i}\delta S\right(u\left)\right]$, where $\delta S\left(u\right)$ is an added quadratic term in the action which couples to the initial bilinear source $\widehat{u}$. The set of derivatives $\mathcal{L}({\partial }_u,{\rho }_0)$, to be taken, is completely dictated by initial ${\widehat{\rho }}_0$. (b) The Kadanoff-Baym contour with an extension along the imaginary time axis, from $t=0$ to $t=-\mathbf{i}\beta$, in the Konstantinov-Perel formalism. In the KP formalism, ${\widehat{\rho }}_0=exp(-\beta {\widehat{H}}_0)$, with some many-body operator ${\widehat{H}}_0$, is written as a path integral along the imaginary axis.

Figure 2

Evolution of a linear chain of bosons, starting from a Fock state. Each site $l$ is connected to a bath with temperature $T=g$ and chemical potential ${\mu }_l$, where ${\mu }_l={\mu }_1+\nu (l-1)$ with ${\mu }_1=-4.05g$ and $\nu =0.75g$. Here, $g$ is the tunneling amplitude in the linear chain. (a) The initial Fock state in a $N=9$ site system where the $l\mathrm{th}$ site is occupied by $l$ particles. Each circle represents a particle. (b) The same Fock state with the origin shifted to the central site and local densities defined in terms of their deviations from the occupation of the central site, i.e., 5. The filled red circles indicate positive deviations while empty red circles indicate negative deviations. In terms of the deviations, the initial profile is antisymmetric under reflection about the central site. (c) Color plot of density $n_l\left(t\right)$ and (d) current $I_l\left(t\right)$ in the system as a function of site (link) number and time in underdamped regime with system bath coupling $\epsilon =0.35g$. The density profile executes a seesaw motion keeping the density of the central site almost constant at short times. The current shows a maximum at the center at short times. At long times, system settles to a density profile decreasing from left to right, governed by the chemical potential gradient in the baths. We use $g=1$ to set the unit of time, $t$ and $l$ is measured in units of lattice spacing.

Figure 3

Dynamics of a fermionic OQS starting from arbitrary initial condition. A 20 site linear chain of spinless fermions with nearest-neighbor tunneling amplitude $g$ is coupled at each site to the baths at temperature $T_l=g$ and the chemical potential ${\mu }_l=-4.05g$ through a coupling $\epsilon =0.2g$. (a) The initial state $\left|\right\{n\}\rangle$, with left half occupied, the right half empty, and a domain wall at the center. (b) Another initial state $\left|\right\{m\}\rangle$, obtained from moving the rightmost particle in $\left|\right\{n\}\rangle$, one site to its right. Color plot of (c) for density and (d) for current as a function of site (link) number and time for a system starting with $\left|\right\{n\}\rangle \langle \{n\left\}\right|$. The diamonds are defined by ballistic motion of domain walls and their reflection from the edges. [(e)–(h)] Current as a function of time for off-diagonal ${\widehat{\rho }}_{0S}=a\left|\left\{n\right\}\right.〉\left.〈\left\{n\right\}\right|+\left(1-a\right)\left|\left\{m\right\}\right.〉\left.〈\left\{m\right\}\right|+[\mathbf{i}b\left|\left\{n\right\}\right.〉\left.〈\left\{m\right\}\right|+\text{H.c}.]$. (e) Current at the central link ($l=10$) for $b=0$ and $a=1/4,1/2,3/4$. (f) Same as (e) for a link far from the center ($l=5$). The initial oscillations at the central link are more pronounced for smaller $a$. (g) Current at the central link ($l=10$) and (h) current at the link far from the center $l=5$ for ${\widehat{\rho }}_{0S}$ with $a=1/2$ and $b=0.0,0.2,0.5$. The initial current on the central link is controlled by the value of $b$. We use $g=1$ to set the unit of time, $t$ and $l$ is measured in units of lattice spacing.