Quantum Trajectory Method Research Articles

Exact steady states of the impurity-doped XXZ spin chain coupled to dissipators

We give an exact matrix product steady state and matrix product forms of local observables for the bulk impurity-doped XXZ spin model coupled to dissipators at both ends, whose dynamics is described by the Lindblad quantum master equation. We find that local magnetization is induced at the impurity site when the spin current flows, which is contrary to the usual situation where the current suppresses magnetization due to heating. It is proved that this current-induced magnetization survives in the thermodynamic limit, and the spin current does not depend on the impurity strength. We discuss the role of bulk integrability by comparing the results with those of non-integrable models solved numerically by the quantum trajectory method.

Synchronization of a superconducting qubit to an optical field mediated by a mechanical resonator

We study the synchronization of a superconducting qubit to an external optical field via a mechanical resonator in a hybrid electrooptomechanical system. The quantum trajectory method is employed to investigate synchronization. The bistability in one of the qubit polarization vectors, where the qubit rotates about the polarization vector, is observed for a single quantum trajectory run. The rotation in one of the stable states is synced with the external optical drive. When the number of trajectories is significantly increased, the qubit no longer displays bistability. However, synchronization with less quantum fluctuations is still observed. The scheme could be used to prepare and monitor the state of the microwave qubit to a long-lived optical photon through synchronization, which may find applications in long-distance quantum communication. Also, this hybrid system can be used to study quantum synchronization.

Recollision of excited electron in below-threshold nonsequential double ionization

Recollision is the most important post-tunneling process in strong-field physics, but so far has been restricted to interaction between the first ionized electron and the residual ion in nonsequential double ionization. Here we identify the role of recollision of the second ionized electron in the below-threshold nonsequential double ionization process by introducing a Coulomb-corrected quantum-trajectories method. We will reproduce the experimentally observed cross-shaped and anti-correlated patterns in correlated two-electron momentum distributions, and the transition between them. Both the cross-shaped and anti-correlated patterns are attributed to recolliding trajectories of the second electron. The effect of recollision of the second electron is significantly enhanced by the stronger Coulomb potential of the higher valence residual ion, and is further strengthened by the recapture process of the second electron. Our work paves a potential way to image ultrafast dynamics of atoms and molecules in intense laser field.

Measuring time-dependent induced quantum coherences via two-dimensional coherence spectroscopy

We propose a two-dimensional spectroscopic protocol for measuring the time-dependent coherences between the stationary states of a system induced by a time-dependent system-bath interaction. We also investigate the role of temporally-correlated noise on coherence dephasing. This protocol enables dynamical information about the system and its coupling to the environment to be determined. Our results are based on the quantum-trajectory method, and are obtained from both approximate, analytical and exact, numerical solutions of the time-dependent Schroedinger equation. As an example, we show how this protocol can be used to investigate exciton dynameds in conjugated polymers induced by the coupling of their torsional modes with the environment.

Squeezed-vacuum-induced optical tristability in an atom-cavity system

In this paper we investigate the model of $N$ two-level atoms coupled to a squeezed vacuum inside a coherently driven cavity. By a linear stability analysis in the semiclassical description, we find that the optical tristability emerges in our system owing to the presence of the squeezed vacuum. Moreover, the transition among monostable, bistable, and tristable states can be realized by manipulating the squeezing phase or the squeezing strength of the squeezed vacuum. We also show the signatures of quantum tristability by using the $Q$ function and quantum trajectory methods for the situation of a single atom and low cavity photon number. Our paper offers a way to realize optical tristability in a two-level atomic system.

Photon statistics of the light transmitted and reflected by a two-dimensional atomic array

This work proposes to investigate the photon statistics of the light transmitted and reflected by a two-dimensional array of interacting atoms. The reflected beam is characterized by photon antibunching. On the other hand, in the transmitted beam the indistinguishability between the driving laser photons and the photons re-emitted by the atoms results in photon bunching. The overlap between the driving and scattered fields is enhanced by the cooperative optical response of the atomic array. In the examples used in this paper, up to 25\% of the transmitted photons are grouped in pairs. The simulations are carried out using the stochastic method of quantum trajectories.

Fundamental Limit of Phase Coherence in Two-Component Bose-Einstein Condensates.

We experimentally and theoretically study phase coherence in two-component Bose-Einstein condensates of ^{87}Rb atoms on an atom chip. Using Ramsey interferometry we determine the temporal decay of coherence between the |F=1,m_{F}=-1⟩ and |F=2,m_{F}=+1⟩ hyperfine ground states. We observe that the coherence is limited by random collisional phase shifts due to the stochastic nature of atom loss. The mechanism is confirmed quantitatively by a quantum trajectory method based on a master equation which takes into account collisional interactions, atom number fluctuations, and losses in the system. This decoherence process can be slowed down by reducing the density of the condensate. Our findings are relevant for experiments on quantum metrology and many-particle entanglement with Bose-Einstein condensates and the development of chip-based atomic clocks.

Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes

Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.

QTM: Computational package using MPI protocol for Quantum Trajectories Method.

The Quantum Trajectories Method (QTM) is one of the frequently used methods for studying open quantum systems. The main idea of this method is the evolution of wave functions which describe the system (as functions of time). Then, so-called quantum jumps are applied at a randomly selected point in time. The obtained system state is called as a trajectory. After averaging many single trajectories, we obtain the approximation of the behavior of a quantum system. This fact also allows us to use parallel computation methods. In the article, we discuss the QTM package which is supported by the MPI technology. Using MPI allowed utilizing the parallel computing for calculating the trajectories and averaging them—as the effect of these actions, the time taken by calculations is shorter. In spite of using the C++ programming language, the presented solution is easy to utilize and does not need any advanced programming techniques. At the same time, it offers a higher performance than other packages realizing the QTM. It is especially important in the case of harder computational tasks, and the use of MPI allows improving the performance of particular problems which can be solved in the field of open quantum systems.

Effects of imperfect pulses on dynamical decoupling using quantum trajectory method**Project supported by the National Basic Research Program of China (Grant No. 2016YFA0301903), the National Natural Science Foundation of China (Grant Nos. 11174370, 11304387, 61632021, 11305262, and 61205108), and the Research Plan Project of the National University of Defense Technology (Grant No. ZK16-03-04).

The dynamical decoupling (DD) method is widely adopted to preserve coherence in different quantum systems. In the case of ideal pulses, its effects on the suppression of noise can be analytically described by the mathematical form of filter function. However, in practical experiments, the unavoidable pulse errors limit the efficiency of DD. In this paper, we study the effects of imperfect pulses on DD efficiency based on quantum trajectories. By directly generating a pseudo noise sequence correlated in time, we can explore the performance of DD with different pulse errors in the typical noise environment. It shows that, for the typical 1/f noise environment, the phase error of operational pulses severely affects the performance of noise suppression, while the detuning and intensity errors have less influence. Also, we get the thresholds of these errors for efficient DD under the given experimental conditions. Our method can be widely applied to guide practical DD experimental implementation.

Quantum trajectories for time-dependent adiabatic master equations

We develop a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension $N^2$, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor $N^2$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for $8$-qubit quantum annealing examples. We also apply the quantum trajectories method to a $16$-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.

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.

Conditional counting statistics of electrons tunneling through quantum dot systems measured by a quantum point contact

We theoretically study the conditional counting statistics of electron transport through a system consisting of a single quantum dot (SQD) or coherently coupled double quantum dots (DQD's) monitored by a nearby quantum point contact (QPC) using the generating functional approach with the maximum eigenvalue of the evolution equation matrix method, the quantum trajectory theory method (Monte Carlo method), and an efficient method we develop. The conditional current cumulants that are significantly different from their unconditional counterparts can provide additional information and insight into the electron transport properties of mesoscopic nanostructure systems. The efficient method we develop for calculating the conditional counting statistics is numerically stable, and is capable of calculating the conditional counting statistics for a more complex system than the maximum eigenvalue method and for a wider range of parameters than the quantum trajectory method. We apply our method to investigate how the QPC shot noise affects the conditional counting statistics of the SQD system, going beyond the treatment and parameter regime studied in the literature. We also investigate the case when the interdot coherent coupling is comparable to the dephasing rate caused by the back action of the QPC in the DQD system, in which there is considerable discrepancy in the calculated conditional current cumulants between the population rate (master-) equation approach of sequential tunneling and the full quantum master-equation approach of coherent tunneling.

Cascaded Kerr photon-blockade sources and applications in quantum key distribution

To raise the repetition rate, a single-photon source based on Kerr quantum blockade in a cascaded quantum system is studied. Using the quantum trajectory method, we calculate and simulate the photon number distributions out of a two-cavity system. A high quality single-photon source can be achieved through optimizing parameters. The designed photon source is further applied to the decoy state quantum key distribution (QKD). With and without statistical fluctuation, the key rate can be both raised drastically.

Simulating generic spin-boson models with matrix product states

The global coupling of few-level quantum systems ("spins") to a discrete set of bosonic modes is a key ingredient for many applications in quantum science, including large-scale entanglement generation, quantum simulation of the dynamics of long-range interacting spin models, and hybrid platforms for force and spin sensing. We present a general numerical framework for treating the out-of-equilibrium dynamics of such models based on matrix product states. Our approach applies for generic spin-boson systems: it treats any spatial and operator dependence of the two-body spin-boson coupling and places no restrictions on relative energy scales. We show that the full counting statistics of collective spin measurements and infidelity of quantum simulation due to spin-boson entanglement, both of which are difficult to obtain by other techniques, are readily calculable in our approach. We benchmark our method using a recently developed exact solution for a particular spin-boson coupling relevant to trapped ion quantum simulators. Finally, we show how decoherence can be incorporated within our framework using the method of quantum trajectories, and study the dynamics of an open-system spin-boson model with spatially non-uniform spin-boson coupling relevant for trapped atomic ion crystals in the presence of molecular ion impurities.

Does the measurement take place when nobody observes it?

We consider non‐selective continuous measurements of a particle tunneling to a reservoir of finite band‐width (Λ). The particle is continuously monitored by frequent projective measurements (“quantum trajectory”), separated by a time‐interval τ. A simple analytical expression for the decay rate has been obtained. For Markovian reservoirs (), no effect of the measurements is found. Otherwise for a finite Λ, the decay rate always depends on the measurement time τ. This result is compared with alternative calculations, with no intermediate measurements, but when the measurement device is included in the Schrödinger evolution. We found that the detector affects the system by the decoherence rate (), related to the detector's signal. Although both treatments are different, the final results become very close for . This τ corresponds to the minimal time for which the detector's signal can be distinguished by an “observer”. This indicates a fundamental role of information in quantum motion and can be used for the extension of the quantum trajectory method for non‐Markovian environments.

On-demand single-photon sources via quantum blockade and applications in decoy-state quantum key distribution.

Quantum blockades as a nonlinear quantum optical process have been well studied in recent years. Using the quantum trajectory method, we calculate and discuss the output photon number distributions of a single-photon blockade process in a Kerr nonlinear dissipative resonator, revealing that the probability of the single-photon state can be optimized. Then we show through numerical simulations that such a quasi-single-photon source can drastically raise the key rate in the decoy-state quantum key distribution.

Calculation of Bohmian quantum trajectories for STEM.

We present in this work the calculation of Bohmian quantum trajectories representing the wave function propagation in a crystal for a focused electron probe in a scanning transmission electron microscope (STEM). The wave function and quantum trajectories are obtained from the calculation of time-dependent Schrödinger equation by fast Fourier transformation multislice algorithm. In our work, the Bohmian quantum trajectories of a scanning probe penetrating a Cu crystal are studied as an example of this calculation scheme. The results help us to better understand the electron diffraction process in a microscopic imaging from a trajectory-based point of view. This Bohmian quantum trajectory method can be used to extend the application of classical Monte Carlo method from the study of electron interaction with amorphous solid to crystalline structure.

Quantum, Statistical, and Quasiclassical Trajectory Studies For the Ne + HeH(+) → NeH(+) + He Reaction on the Ground Electronic State.

Real wave packet, statistical quantum, and quasiclassical trajectory methods were employed to study the dynamics of Ne + HeH(+)(v0,j0) → He + NeH(+) reaction on an ab initio potential energy surface [J. Phys. Chem. A 2013, 117, 13070-13078]. Quantum and statistical quantum calculations were performed within the centrifugal sudden (CS) approximation as well as including the Coriolis coupling (CC). Dense oscillatory structures of the quantum reaction probabilities and fair agreement between quantum and statistical cross sections suggest a complex forming mechanism for the reaction. No significant differences between cross sections obtained within the CS and CC approaches are observed. Quasiclassical trajectory results give an excellent average description of the quantum CC results. At low collision energies, there is a substantial decrease in reactivity for the reaction upon rovibrational excitation. Initial state selected rate constants for the title reaction are calculated between 20 and 1000 K, and the calculated value at 300 K agrees quite well with the available experimental result. Reaction cross sections and rate constants are also compared with those calculated via the Langevin capture model for exothermic reactions.

Cluster Luttinger liquids and emergent supersymmetric conformal critical points in the one-dimensional soft-shoulder Hubbard model

We investigate the quantum phases of hard-core bosonic atoms in an extended Hubbard model where particles interact via soft-shoulder potentials in one dimension. Using a combination of field-theoretical methods and strong-coupling perturbation theory, we demonstrate that the low-energy phase can be a conformal cluster Luttinger liquid (CLL) phase with central charge $c=1$, where the microscopic degrees of freedom correspond to mesoscopic ensembles of particles. Using numerical density-matrix renormalization-group methods, we demonstrate that the CLL phase [first predicted in M. Mattioli et al., Phys. Rev. Lett. 111, 165302 (2013)] is separated from a conventional Tomonaga-Luttinger liquid by an exotic critical point with central charge $c=\frac{3}{2}$. The latter is expression of an emergent conformal supersymmetry, which is not present in the original Hamiltonian. We discuss the observability of the CLL phase in realistic experimental settings with weakly dressed Rydberg atoms confined to optical lattices. Using quantum Monte Carlo simulations, we show that the typical features of CLLs are stable up to comparatively high temperatures. Using exact diagonalizations and quantum trajectory methods, we provide a protocol for adiabatic state preparation as well as quantitative estimates on the effects of particle losses.