Ensemble Of Quantum Trajectories Research Articles

Effective Phase Space Representation of the Quantum Dynamics of Vibrational Predissociation of the ArBr2(B,ν =16···25) Complex.

We perform trajectory-based simulations of the vibrational predissociation of the ArBr2(B,ν=16···25) van der Waals triatomic complex, constrained to the T-shape geometry. To this aim, we employ a 2-fold mapping of the quantum dynamics into classical-like dynamics in an extended phase space. The effective phase space comprises two distinct sets of degrees of freedom, namely a collection of coupled harmonic oscillators and an ensemble of quantum trajectories. The time evolution of these variables represent bound and unbound motions of the quantum system, respectively. Quantum trajectories are propagated within the interacting trajectory representation. The comparison between the lifetimes of the predissociating complexes computed using the trajectory-based approach and the experimental results available for the target systems indicates that the present method is competitive with wavepacket propagation techniques. The competition between several simultaneous vibrational relaxation pathways was found to have a direct impact on the time scales of vibrational predissociation. Likewise, the analysis of the time evolution of the trajectories reveals the existence of regions in the effective phase space where transitions to vibrational states of higher energy are more likely to occur. The size and location of these regions influence the transient vibrational distributions and therefore the computed lifetimes. Furthermore, the mechanisms of energy redistribution along the dissociation coordinate are analyzed.

Auxiliary open quantum system for the Floquet quantum master equation.

By directly using the probability formulas of quantum trajectories, we construct an auxiliary open quantum system for a periodically driven open quantum system whose dynamics is governed by the Floquet quantum master equation. This auxiliary system can generate a quantum trajectory ensemble that is consistent with the canonical quantum trajectory ensemble. We find that, at a long time limit, though the Lindblad operators are modified, the coherent dynamics of the auxiliary system is the same as that of the original system. A periodically driven two-level quantum system is used to illustrate this construction.

Trajectory-based conservation laws for massive spin-zero relativistic quantum particles in 1 + 1 spacetime

We present novel aspects of a trajectory-based theory of massive spin-zero relativistic quantum particles. In this approach, the quantum trajectory ensemble is the fundamental entity. It satisfies its own action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to conservation laws (via Noether’s theorem). In this paper, we focus on the derivation of the latter. In addition to the usual expected energy and momentum conservation laws, there is also a third law that emerges, associated with the conditions needed to maintain global simultaneity. We also show that the nonrelativistic limits of these conservation laws match those of the earlier, nonrelativistic quantum trajectory theory [J. Chem. Phys. 136, 031102 (2012)].

Photon waiting-time distributions: A keyhole into dissipative quantum chaos.

Open quantum systems can exhibit complex states, for which classification and quantification are still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intracavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I. I. Yusipov et al., Chaos 29, 063130 (2019)], we identify "chaotic" and "regular" regimes there. In particular, we show that chaotic regimes manifest an intermediate power-law asymptotics in the distribution of photon waiting times. This distribution can be retrieved by monitoring photon emission with a single-photon detector so that chaotic and regular states can be discriminated without disturbing the intracavity dynamics.

Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method.

Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work, we consider time-periodically modulated quantum systems that are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a nontrivial computational task. Approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting Hilbert space dimH=N≲300, while the direct long-time numerical integration of the master equation becomes increasingly problematic for N≳400, especially when the coupling to the environment is weak. To go beyond this limit, we use the quantum trajectory method, which unravels the master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of event-driven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long "leaps" forward in time. It is also numerically exact, in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, {η_{1},η_{2},...,η_{n}}, one could propagate a quantum trajectory (with η_{i}'s as norm thresholds) in a numerically exact way. By using a scalable N-particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with N=2000 states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.

Weak measurements, quantum-state collapse, and the Born rule

Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction can be specific to each system-apparatus pair, a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. We analyze the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. This trajectory ensemble distribution can be tested in weak measurement experiments. We also comment on how the required noise may arise in the measuring apparatus.

Schrödinger–Langevin equation with quantum trajectories for photodissociation dynamics

The Schrödinger–Langevin equation is integrated to study the wave packet dynamics of quantum systems subject to frictional effects by propagating an ensemble of quantum trajectories. The equations of motion for the complex action and quantum trajectories are derived from the Schrödinger–Langevin equation. The moving least squares approach is used to evaluate the spatial derivatives of the complex action required for the integration of the equations of motion. Computational results are presented and analyzed for the evolution of a free Gaussian wave packet, a two-dimensional barrier model, and the photodissociation dynamics of NOCl. The absorption spectrum of NOCl obtained from the Schrödinger–Langevin equation displays a redshift when frictional effects increase. This computational result agrees qualitatively with the experimental results in the solution-phase photochemistry of NOCl.

Adsorption of a Hydrogen Atom on a Graphene Flake Examined with Quantum Trajectory/Electronic Structure Dynamics

Adsorption of atomic hydrogen colliding with C37H15, a planar “flake” of graphene, is examined using a hybrid quantum trajectory/electronic structure (QTES) dynamics for the values of the collision energies below 2.0 eV. The entire system of 53 atoms is represented by a quantum trajectory ensemble. The quantum correction on dynamics is included for the adsorbing hydrogen. The electronic structure is computed on-the-fly using the density functional tight binding (DFTB) method. The highest probability of adsorption is obtained for the kinetic energy of incident hydrogen in the range from 0.1 to 0.9 eV. Adsorption mechanism involves transfer of the collision energy to the graphene flake and projection of the carbon atom of the forming CH bond, accompanying sp3 hybridization of orbitals. The QTES–DFTB simulations with and without the quantum correction show that localization of the proton wave function and details of the mixed quantum/classical description of light and heavy nuclei greatly influence the adsor...

The Schrödinger equation with friction from the quantum trajectory perspective

Similarity of equations of motion for the classical and quantum trajectories is used to introduce a friction term dependent on the wavefunction phase into the time-dependent Schrödinger equation. The term describes irreversible energy loss by the quantum system. The force of friction is proportional to the velocity of a quantum trajectory. The resulting Schrödinger equation is nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of reactions proceeding with energy transfer from the system to the environment. Examples of dynamics are given for single and symmetric and asymmetric double well potentials.

Efficient quantum trajectory representation of wavefunctions evolving in imaginary time

The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential--computed approximately to be practical in high-dimensional systems--influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(-S(x, t)/ħ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.

Three-component superposition states of light in a dissipative medium

The process of the simultaneous absorption of three photons in a medium possessing weak one-photon absorption is analyzed. The mode experiencing absorption is perturbed externally via a three-photon parametric process. It is shown that a steady-state, three-component superposition state of light can exist in this system in the range where the amplitude of the state is small (small perturbations of the system). The latter condition is due to the fact that, in this range of interactions, the field spends significantly more time in one of three types of superposition states of light that form an ensemble of quantum trajectories of the system than in the other two.

Calculation of reaction rate constants using approximate evolution of quantum trajectories in imaginary and real time

Reaction rate constants can be directly obtained from evolution of the flux operator eigenvectors under the Boltzmann and Hamiltonian operators. This is achieved by evolving the quantum trajectory ensemble, representing a wavefunction, in imaginary time seamlessly switching to the real-time dynamics. Quantum–mechanical effects are incorporated through the quantum potential dependent on the trajectory momenta or on the derivatives of the wavefunction amplitude. For practicality the quantum potential and wavefunction nodes are described using linear basis, which is exact for Gaussian wavefunctions. For the Eckart barrier approximate rate constants show significant improvement over the parabolic barrier rate constants.

Extending Bell’s beables to encompass dissipation, decoherence, and the quantum-to-classical transition through quantum trajectories

In this paper, employing the It\^o stochastic Schr\odinger equation, we extend Bell's beable interpretation of quantum mechanics to encompass dissipation, decoherence, and the quantum-to-classical transition through quantum trajectories. For a particular choice of the source of stochasticity, the one leading to a dissipative Lindblad-type correction to the Hamiltonian dynamics, we find that the diffusive terms in Nelsons stochastic trajectories are naturally incorporated into Bohm's causal dynamics, yielding a unified Bohm-Nelson theory. In particular, by analyzing the interference between quantum trajectories, we clearly identify the decoherence time, as estimated from the quantum formalism. We also observe the quantum-to-classical transition in the convergence of the infinite ensemble of quantum trajectories to their classical counterparts. Finally, we show that our extended beables circumvent the problems in Bohm's causal dynamics regarding stationary states in quantum mechanics.

Superposition states of light in a three-photon absorbing dissipative medium

We consider the process of simultaneous absorption of three photons in a medium in the presence of weak one-photon absorption. We show that in such a system stationary three-component superposition states of light may be formed in the range of small values of state amplitude (weak perturbation). This circumstance is associated with the fact that in this range of interaction the field spends more time in one of the three types of superposition states of light (constituting an ensemble of quantum trajectories of the system) than in two other types. We also show that in the range of large values of state amplitude it is possible to obtain three types of non-stationary superposition states of light. By using numerical simulation of quantum trajectories of the system we study the dynamics of the quantum entropy of the field. We calculate the Wigner functions of the field states. We also obtain analytical results for the density matrix of the steady state of the system.

Quantum Dissipative Chaos in the Statistics of Excitation Numbers

A quantum manifestation of chaotic classical dynamics is found in the framework of oscillatory number statistics for the model of a nonlinear dissipative oscillator. It is shown that the probability distributions and variances of oscillatory number states are strongly transformed in the order-to-chaos transition. A nonclassical, sub-Poissonian statistics of oscillatory excitation numbers is established for chaotic dissipative dynamics in the framework of the Fano factor and Wigner functions. It is proposed to use these results in experimental studies of the quantum dissipative chaos.

Chaos in a double driven dissipative nonlinear oscillator.

We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the framework of the statistical ensemble of quantum trajectories in a quantum state diffusion approach. The quantum dynamical manifestations of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement, are studied by numerical simulation of the ensemble averaged Wigner function and von Neumann entropy.

Quantum trajectories generalized

By definition, an ensemble of quantum trajectories reproduces the density matrix of an open system after ensemble averaging. We present a general class of stochastic Schrodinger equations for quantum trajectories which unifies and extends existing methods. Various quantum jump and state-diffusion methods are recognized as opposite limiting cases. The inherent freedom of choice can be used to significantly reduce the calculation time in specific cases. Examples of this are methods with adaptive noise, for which a specific observable becomes noise free to first order. Linear stochastic equations can also be constructed, which allow analytical solutions. As an illustration, the decoherence of Schrodinger cat states of a cavity mode is discussed.