Real-time Path Integrals Research Articles

Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

Picard–Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem.

Nonperturbative stochastic method for driven spin-boson model

We introduce and apply a numerically exact method for investigating the real-time dissipative dynamics of quantum impurities embedded in a macroscopic environment beyond the weak-coupling limit. We focus on the spin-boson Hamiltonian that describes a two-level system interacting with a bosonic bath of harmonic oscillators. This model is archetypal for investigating dissipation in quantum systems, and tunable experimental realizations exist in mesoscopic and cold-atom systems. It finds abundant applications in physics ranging from the study of decoherence in quantum computing and quantum optics to extended dynamical mean-field theory. Starting from the real-time Feynman-Vernon path integral, we derive an exact stochastic Schr\"odinger equation that allows us to compute the full spin density matrix and spin-spin correlation functions beyond weak coupling. We greatly extend our earlier work [P. P. Orth, A. Imambekov, and K. Le Hur, Phys. Rev. A 82, 032118 (2010)] by fleshing out the core concepts of the method and by presenting a number of interesting applications. Methodologically, we present an analogy between the dissipative dynamics of a quantum spin and that of a classical spin in a random magnetic field. This analogy is used to recover the well-known noninteracting-blip approximation in the weak-coupling limit. We explain in detail how to compute spin-spin autocorrelation functions. As interesting applications of our method, we explore the non-Markovian effects of the initial spin-bath preparation on the dynamics of the coherence ${\ensuremath{\sigma}}^{x}(t)$ and of ${\ensuremath{\sigma}}^{z}(t)$ under a Landau-Zener sweep of the bias field. We also compute to a high precision the asymptotic long-time dynamics of ${\ensuremath{\sigma}}^{z}(t)$ without bias and demonstrate the wide applicability of our approach by calculating the spin dynamics at nonzero bias and different temperatures.

Non-equilibrium quantum dynamics of the magnetic Anderson model

We study the non-equilibrium dynamics of a spinful single-orbital quantum dot with an incorporated quantum mechanical spin-1/2 magnetic impurity. Due to the spin degeneracy, double occupancy is allowed, and Coulomb interaction together with the exchange coupling of the magnetic impurity influence the dynamics. By extending the iterative summation of real-time path integrals (ISPI) to this coupled system, we monitor the time-dependent non-equilibrium current and the impurity spin polarization to determine features of the time-dependent non-equilibrium dynamics. We particulary focus on the deep quantum regime, where all time and energy scales are of the same order of magnitude and no small parameter is available. We observe a significant influence of the non-equilibrium decay of the impurity spin polarization both in the presence and in the absence of Coulomb interaction. The exponential relaxation is faster for larger bias voltages, electron–impurity interactions and temperatures. We show that the exact relaxation rate deviates from the corresponding perturbative result. In addition, we study in detail the impurity's back action on the charge current and find a reduction of the stationary current for increasing coupling to the impurity. Moreover, our approach allows us to systematically distinguish mean-field Coulomb and impurity effects from the influence of quantum fluctuations and flip-flop scattering, respectively. In fact, we find a local maximum of the current for a finite Coulomb interaction due to the presence of the impurity.

Laser driven dynamics of a quantum dot coupled to phonons: Dependence of the reappearance of Rabi rotations on the pulse length and shape

AbstractWe study the dynamics of a semiconductor quantum dot coupled to phonons and laser light in the limit of strong electronic confinement. Using real time path integrals our numerical results are nonperturbative with respect to both, the carrier‐light and the carrier‐phonon coupling. As it is known the damping of Rabi rotations is a non‐monotonic function of the pulse area.Here, we analyze how pulse parameters affect the resulting decay‐reappearance of Rabi rotations, which arises from the resonant nature of carrier‐phonon coupling. For longer pulses, the maximal reduction of the amplitude of Rabi rotations is shifted to higher pulse areas. In addition, for Gaussian pulses the reappearance rate is significantly reduced compared to rectangular profiles. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Iterative summation of path integrals for nonequilibrium molecular quantum transport

We formulate and apply a nonperturbative numerical approach to the nonequilibrium current, $I(V)$, through a voltage-biased molecular conductor. We focus on a single electronic level coupled to an unequilibrated vibration mode (Anderson-Holstein model), which can be mapped to an effective three-state problem. Performing an iterative summation of real-time path integral (ISPI) expressions, we accurately reproduce known analytical results in three different limits. We then study the crossover regime between those limits and show that the Franck-Condon blockade persists in the quantum-coherent low-temperature limit, with a nonequilibrium smearing of step features in the $IV$ curve.

Long-time dynamics and stationary nonequilibrium of an optically driven strongly confined quantum dot coupled to phonons

We study the long-time dynamics and the stationary nonequilibrium state of an optically driven quantum dot coupled to acoustic phonons using numerically exact real-time path integrals and a fourth-order correlation expansion. By exploring wide ranges of temperatures, carrier-phonon and carrier-light coupling strengths, we characterize the stationary nonequilibrium state and compare the exact solution to known, approximatively derived results. It is found that analytical calculations tend to overestimate the influence of the carrier-phonon coupling, particularly at low temperatures and in the weak-coupling regime. The possibility of controlling the dot occupation in the stationary nonequilibrium by varying the laser detuning is discussed. A comparison between the numerical methods identifies the range of validity of the correlation expansion, which in the long-time limit is found to be surprisingly wide.

Real-time path integrals for quantum dots: Quantum dissipative dynamics with superohmic environment coupling

Real-time path integrals for quantum dots: Quantum dissipative dynamics with superohmic environment coupling

Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering

Several path integral representations for the $T$-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.

Time-dependent transport through a correlated quantum dot with magnetic impurity

We investigate electronic- and spin transport through a single level quantum dot with magnetic impurity in a symmetric forward bias setup. On the quantum dot, electrons either interact with each other due to Coulomb interaction or with the spin 1/2 magnetic impurity. For certain configurations, the tunnel coupling to the leads induces an exponential relaxation of the impurity spin, which has been prepared in a polarized state initially. Furthermore, we study the influence of the nonequilibrium transport current on the relaxation dynamics. We obtain the respective numerical result by means of the iterative summation of path integral (ISPI) scheme. Within this approach, observables of interest are calculated from a functional derivative with respect to appropriate source terms in the Keldysh partition function. The real-time path integral extends over all possible paths (i) of the impurity spin and (ii) of the Ising like fluctuating spin fields we have to introduce in order to decouple the quartic interaction term of the Anderson model. The ISPI scheme allows us to sum up all paths including the time non-local self energies of the leads.

Comparative study of theoretical methods for non-equilibrium quantum transport

We present a detailed comparison of three different methods designed to tackle non-equilibrium quantum transport, namely the functional renormalization group (fRG), the time-dependent density matrix renormalization group (tDMRG) and the iterative summation of real-time path integrals (ISPI). For the non-equilibrium single-impurity Anderson model (including a Zeeman term at the impurity site), we demonstrate that the three methods are in quantitative agreement over a wide range of parameters at the particle–hole symmetric point as well as in the mixed-valence regime. We further compare these techniques with two quantum Monte Carlo approaches and the time-dependent numerical renormalization group method.

The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies

In this paper we focus on a new computational procedure, which permits an efficient calculation within the classical auxiliary field methodology. As has been previously reported, the method suffers from a sign problem, typically encountered in methodologies based on a field-theoretical approach. To ameliorate its statistical convergence, the efforts have so far exclusively been concentrated on the development of efficient analytical integral transformation techniques, such as the method of Gaussian equivalent representation of Efimov et al. In the present work we reformulate the classical auxiliary field methodology according to the concepts of the stationary phase Monte Carlo method of Doll et al., a numerical strategy originally developed for the simulation with real-time path integrals. The procedure, which is here employed for the first time for auxiliary field computation, utilizes an importance sampling strategy, to identify the regions of configuration space that contribute most strongly to the functional integral averages. Its efficiency is here compared to the method of Gaussian equivalent representation.

Rigorous real-time Feynman path integral for vector potentials

we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the $L^2$ transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities.

A rigorous real-time Feynman path integral and propagator

We will derive a rigorous real time propagator for the Non-relativistic Quantum Mechanic $L^2$ transition probability amplitude and for the Non-relativistic wave function. The propagator will be explicitly given in terms of the time evolution operator. The derivation will be for all self-adjoint nonvector potential Hamiltonians. For systems with potential that carries at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real time, time sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in Nonstandard Analysis Formulation. Finally, we will compute the propagator for the harmonic oscillator using the Nonstandard Analysis Feynman path integral formuluation; we will compute the propagator without using any knowledge of classical properties of the harmonic oscillator.

Corrections to the semiclassical approximation by numerical evaluation of real time path integrals in complex space

A numerical method for evaluating real time path integrals by analytical continuation of the coordinate space variables into the complex plane is presented. With this method, a Monte Carlo gaussian sampling function around the classical paths is obtained. The function to be sampled is smooth near the stationary path and good Monte Carlo convergence is expected.

Effects of periodic driving on asymmetric two-level systems coupled to dissipative environments

We study the dynamics of initially localized states in biased dissipative TLSs evolving under a time periodic force simulating monochromatic light. Accurate numerical calculations are presented using an iterative procedure to evaluate realtime path integrals. In the high-temperature and moderate to high friction regime, the system exhibits exponential decay. In this regime the decay constant approaches a plateau value that depends only on the intensity of the driving field. Simple analytical calculations using a Landau-Zener approach predict correctly the value of the decay rate. Bias does not affect significantly the plateau value of the decay constant provided that the asymmetry is small relative to the amplitude of the driving field. In the low-friction regime, phase relations are important and there is nonmonotonic variation of the delocalization rate. Weak friction broadens the window for extending the lifetime of localized states. At very strong friction the decay rate diminishes monotonically.

Convergence and path cancellation in quantum Monte Carlo real time path integration

We find that poor resolution of the tails of the distribution of integrands obtained inhibits convergence in Monte Carlo calculation of real time path integrals. We show that many methods previously tried to improve convergence neither resolve nor diminish the tails effectively. We find that large contributions to the integrand come from paths that have a large variance from the zero-path and/or that have few zero crossings. The results of crude dampings based on where such paths are poorly sampled suggest that exploring cancellations of paths characterized by path variance and zero crossings may be effective in improving convergence.

On the mechanism of the primary charge separation in bacterial photosynthesis

We present a detailed analysis of the mechanism of the primary charge separation process in bacterial photosynthesis using real-time path integrals. Direct computer simulations as well as an approximate analytical theory have been employed to map out the dynamics of the charge separation process. Two distinct parameter regions, one characteristic of sequential transfer and the other characteristic of superexchange, have been found to yield charge separation dynamics in agreement with experiments. Nonadiabatic theory provides accurate rate estimates for low-lying and very high-lying bacteriochlorophyll state energies, but it breaks down in between these two regimes.

Numerical path integral techniques for long time dynamics of quantum dissipative systems

Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.

Recursive evaluation of the real-time path integral for dissipative systems. The spin-boson model

The tunneling dynamics of an electronic two-state system coupled to a bath of harmonic oscillators at zero temperature is treated using a real-time path-integral approach. In the present work the previously developed recursive class-average approach is extended to include the Feynman-Vernon influence functional and to evaluate electronic population probabilities. The efficiency and accuracy of the method is demonstrated by application to two model problems taken from recent literature.

Comment on "Monte Carlo evaluation of real-time Feynman path integrals for quantal many-body dynamics: distributed approximating functions and Gaussian sampling"

Comment on "Monte Carlo evaluation of real-time Feynman path integrals for quantal many-body dynamics: distributed approximating functions and Gaussian sampling"