Quantum Trajectories Research Articles

Improving quantum parameter estimation by monitoring quantum trajectories

Quantum-enhanced parameter estimation has widespread applications in many fields. An important issue is to protect the estimation precision against the noise-induced decoherence. Here we develop a general theoretical framework for improving the precision for estimating an arbitrary parameter by monitoring the noise-induced quantum trajectorie (MQT) and establish its connections to the purification-based approach to quantum parameter estimation. MQT can be achieved in two ways: (i) Any quantum trajectories can be monitored by directly monitoring the environment, which is experimentally challenging for realistic noises; (ii) Certain quantum trajectories can also be monitored by frequently measuring the quantum probe alone via ancilla-assisted encoding and error detection. This establishes an interesting connection between MQT and the full quantum error correction protocol. Application of MQT to estimate the level splitting and decoherence rate of a spin-1/2 under typical decoherence channels demonstrate that it can avoid the long-time exponential loss of the estimation precision and, in special cases, recover the Heisenberg scaling.

Theory for frequent measurements of spontaneous emissions in a non-Markovian environment: Beyond the Lorentzian spectrum

The measurement-result-conditioned evolution of a system (e.g., an atom) with spontaneous emissions of photons is described by the quantum trajectory (QT) theory. In this work we generalize the associated QT theory from an infinitely wide bandwidth Markovian environment to the finite bandwidth non-Markovian environment. In particular, we generalize the treatment for an arbitrary spectrum, which is not restricted by the specific Lorentzian case. We rigorously prove the general existence of a perfect scaling behavior jointly defined by the bandwidth of the environment and the time interval between successive photon detections. For a couple of examples, we obtain analytic results to facilitate the QT simulations based on the Monte-Carlo algorithm. For the case where the analytical result is not available, a numerical scheme is proposed for practical simulations.

Selection of electron quantum trajectories in the macroscopic high-order harmonics generated by near-infrared lasers

We theoretically investigate the selection of electron quantum trajectories in the macroscopic high harmonic generation (HHG) with few-cycle, 1200-nm near-infrared lasers by changing gas pressure. The selected continuous HHG spectra are able to support the isolated attosecond pulses (IAPs) in the soft x rays. The time-frequency analysis of high harmonics indicates that the long-trajectory electron emissions are dominant at lower pressure and the short-trajectory emissions are suffered at higher and optimal pressure. The quantitative analysis of phase mismatch uncovers the physical origin of dynamic changes in the harmonics fields after propagation in the gas medium. Furthermore, we check the dependency of IAP on the carrier-envelope phase (CEP) of laser pulses and find that IAP can still be obtained even if the CEP changes.

Position Measurement-Induced Collapse: A Unified Quantum Description of Fraunhofer and Fresnel Diffractions

Position measurement-induced collapse states are shown to provide a unified quantum description of diffraction of particles passing through a single slit. These states, which we here call `quantum location states', are represented by the conventional rectangular wave function at the initial moment of position measurement. We expand this state in terms of the position eigenstates, which in turn can be represented as a linear combination of energy eigenfunctions of the problem, using the closure property. The time-evolution of the location states in the case of free particles is shown to have position probability density patterns closely resembling diffraction patterns in the Fresnel region for small times and the same in Fraunhofer region for large times. Using the quantum trajectory representations in the de Broglie-Bohm, modified de Broglie-Bohm and Floyd-Faraggi-Matone formalisms, we show that Fresnel and Fraunhofer diffractions can be described using a single expression. We also discuss how to obtain the probability density of location states for the case of particles moving in a general potential, detected at some arbitrary point. In the case of the harmonic oscillator potential, we find that they have oscillatory properties similar to that of coherent states.

The role of quantum coherence in non-equilibrium entropy production

Thermodynamic irreversibility is well characterized by the entropy production arising from non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences. This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes. We discuss how these features emerge both in Lindblad-Davies differential maps and finite maps subject to the constraints of thermal operations. We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environment and the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state. Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurements leads to fundamental limitations in the description of quantum trajectories and fluctuation theorems.

Quantum trajectories for a system interacting with environment in N-photon state

We derive stochastic master equation for a quantum system interacting with an environment prepared in a continuous-mode N-photon state. To determine the conditional evolution of the quantum system depending on continuous in time measurements of the output field the model of repeated interactions and measurements is applied. The environment is defined as an infinite chain of harmonic oscillators which do not interact between themselves and they are prepared initially in an entangled state being a discrete analogue of a continuous-mode N-photon state. We provide not only the quantum trajectories but also the analytical formulae for the whole statistics of the output photons and the solution to the master equation. The solution in the continuous case is represented by a simple diagrammatic technique with very transparent ‘Feynmann rules’. This technique considerably simplifies the structure of the solution and enables one to find physical interpretation for the solution in terms of a few elementary processes.

Holographic interferences in photoelectron spectra: different approaches

We perform a theoretical study of the holographic structures in photoelectron spectra for ionization of hydrogen atoms induced by short laser pulses. To elucidate the nature of the holographic structures present in the momentum distributions of photoelectrons, we use several quantum approximations, such as the strong field and Coulomb–Volkov approximations up to second order, as well as semiclassical Monte Carlo simulations. In a single-cycle pulse, we eliminate the intracycle interference from the spectra isolating the holographic structure formed in the photoionization process. We probe the different approaches and analyze the role of electron–core interaction numerically by solving the time dependent Schrodinger equation. We show that the two-step semiclassical model of Shvetsov-Shilovski et al. [Phys. Rev. A 94, 013415 (2016)] fully considers the effect of the Coulomb potential on the electron dynamics and semiclassical phase reproducing the holographic structure in full quantum calculations. Contrarily, perturbative quantum (strong field and Coulomb–Volkov) and semiclassical (quantum trajectory Monte Carlo) methods account only partially for some of the characteristics of the holographic interference pattern.

Open quantum walks

Open quantum walks (OQWs) are a class of quantum walks, which are purely driven by the interaction with the dissipative environment. In this paper, we review theoretical advances on the foundations of discrete time OQWs, continuous time OQWs and a scaling limit of OQWs called open quantum Brownian motion. The main focus of the review is on the results and developments of discrete time OQW, covering general formalism, quantum trajectories for OQWs, central limit theorems, the microscopic derivation as well as possible generalisations and applications of OQWs.

Time-reversed quantum trajectory analysis of micromaser correlation properties and fluctuation relations

The micromaser is examined with the aim of understanding certain of its properties based on a time-reversed quantum trajectory analysis. The background theory of master equations derived from a repeated interaction model perspective is briefly reviewed and extended by taking into account the more general renewal process description of the sequence of interactions of the system with incoming ancilla, and results compared with other recent (and not so recent) approaches that use this generalisation. The results are then specialised to the micromaser, and a quantum trajectory unravelling of the micromaser dynamics is formulated that enables time-reversed quantum trajectories, defined according to the Crooks approach, to, first, be shown to arise naturally in the analysis of micromaser and atomic beam correlations, and second used in the formulation of a fluctuation relation for the probabilities of trajectories and their time-reversed counterparts.

Surface Hopping without Momentum Jumps: A Quantum-Trajectory-Based Approach to Nonadiabatic Dynamics.

We describe a new method for simulating nonadiabatic dynamics using stochastic trajectories. The method, which we call quantum trajectory surface hopping (QTSH), is a variant of the popular fewest-switches surface-hopping (FSSH) approach, but with important differences. We briefly review and significantly extend our recently described consensus surface-hopping (CSH) formalism, which captures quantum effects such as coherence and decoherence via a collective representation of the quantum dynamics at the ensemble level. Using well-controlled further approximations, we derive an independent trajectory limit of CSH that recovers the FSSH stochastic algorithm but rejects the ad hoc momentum rescaling of FSSH in favor of quantum forces that couple classical and quantum degrees of freedom and lead to nonclassical trajectory dynamics. The approach is well-defined in both the diabatic and adiabatic representations. In the adiabatic representation, the classical dynamics are modified by a quantum-state-dependent vector potential, introducing geometric phase effects into the dynamics of multidimensional systems. Unlike FSSH, our method obeys energy conservation without any artificial momentum rescaling, eliminating undesirable features of the former such as forbidden hops and breakdown of the internal consistency of quantum and ensemble-based state probabilities. Corrections emerge naturally in the formalism that allow approximate incorporation of decoherence without the computational expense of the full CSH approach. The method is tested on several model systems. QTSH provides a surface-hopping methodology that has a rigorous foundation and broader applicability than FSSH while retaining the low computational cost of an independent trajectory framework.

Treating branch cuts in quantum trajectory models for photoelectron holography

Most implementations of Coulomb-distorted strong-field approaches that contain features such as tunneling and quantum interference use real trajectories in continuum propagation, while a fully consistent approach must use complex trajectories throughout. A key difficulty in the latter case are branch cuts that appear due to the specific form of the Coulomb potential. We present a method for treating branch cuts in quantum-trajectory models, which is subsequently applied to photoelectron holography. Our method is not numerically intensive, as it does not require finding the location of all branching points and branch cuts prior to its implementation, and is applicable to Coulomb-free and Coulomb-distorted trajectories. We show that the presence of branch cuts leads to discontinuities and caustics in the holographic fringes in above-threshold ionization (ATI) photoelectron angular distributions (PAD). These artefacts are removed applying our method, provided they appear far enough from the polarization axis. A comparison with the full solution of the time-dependent Schr\"odinger equation is also performed, and a discussion of the applicability range of the present approach is provided.

Phase diagram of the dissipative quantum Ising model on a square lattice

The competition between interactions and dissipative processes in a quantum many-body system can drive phase transitions of different order. Exploiting a combination of cluster methods and quantum trajectories, we show how the systematic inclusion of (classical and quantum) nonlocal correlations at increasing distances is crucial to determine the structure of the phase diagram, as well as the nature of the transitions in strongly interacting spin systems. In practice, we focus on the paradigmatic dissipative quantum Ising model: in contrast to the non-dissipative case, its phase diagram is still a matter of debate in the literature. When dissipation acts along the interaction direction, we predict important quantitative modifications of the position of the first-order transition boundary. In the case of incoherent relaxation in the field direction, our approach confirms the presence of a second-order transition, while does not support the possible existence of multicritical points. Potentially, these results can be tested in up-to date quantum simulators of Rydberg atoms.

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.

Non-reversal Open Quantum Walks

A model of non-reversal quantum walk is introduced. The process is introduced in 1D and 2D using the formalism of open quantum walks (OQWs). In such a walk, a particle cannot go back to previously visited sites but it can jump on the same site or move to a new site. Examples of some non-reversal quantum trajectories and distributions are given and their difference with normal OQWs is discussed. Finally, a linear relationship is formulated between the radius of the spread and the number of steps of the walk that might have interesting applications.

Catching and reversing quantum jumps and thermodynamics of quantum trajectories

A recent experiment by Minev et. al [arXiv:1803.00545] demonstrated that in a dissipative (artificial) 3-level atom with strongly intermittent dynamics it is possible to "catch and reverse" a quantum jump "mid-flight": by the conditional application of a unitary perturbation after a fixed time with no jumps, the system was prevented from getting shelved in the dark state, thus removing the intermittency from the dynamics. Here we offer an interpretation of this phenomenon in terms of the dynamical large deviation formalism for open quantum dynamics. In this approach, intermittency is seen as the first-order coexistence of active and inactive dynamical phases. Dark periods are thus like space-time bubbles of the inactive phase in the active one. Here we consider a controlled dynamics via the (single - as in the experiment - or multiple) application of a unitary control pulse during no-jump periods. By considering the large deviation statistics of the emissions, we show that appropriate choice of the control allows to stabilise a desired dynamical phase and remove the intermittency. In the thermodynamic analogy, the effect of the control is to prick bubbles thus preventing the fluctuations that manifest phase coexistence. We discuss similar controlled dynamics in broader settings.

Probing time-resolved emission in laser-driven electron-multirescattering in a high-order harmonic generation

ABSTRACTWe present an ab initio study of quantitative extraction of the time–frequency spectra of multiple rescattering processes of the electrons driven by the mid-infrared laser field in a high-order harmonic generation (HHG). The HHG is calculated by solving the three-dimensional time-dependent Schrödinger equation by means of the time-dependent generalized pseudospectral method. We extend a synchrosqueezed transform (SST) technique to extract the individual contributions of multiple rescattering processes in HHG. Combining with an extended semiclassical analysis and the SST time–frequency spectrum, the role of quantum trajectories in multiple rescattering processes in HHG is clarified. We find that the SST allows us to distinguish the individual contribution of multiple rescattering processes in HHG and show the details of the spectral and temporal fine structures of the HHG, which provides an important tool for a deep understanding to the dynamics of multiple rescattering processes in HHG.

Quantum Zeno effect and the many-body entanglement transition

We introduce and explore a one-dimensional "hybrid" quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a "quantum Zeno phase" where strong measurements suppress the entanglement growth to saturate in an area-law. Extensive numerical simulations of the quantum trajectories of the many-particle wavefunctions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable "weak measurement phase" that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a novel continuous quantum dynamical phase transition between the "weak measurement phase" and the "quantum Zeno phase", driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this novel quantum entanglement transition are accessed.

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.

An exponential quantum projection filter for open quantum systems

An approximate exponential quantum projection filtering scheme is developed for a class of open quantum systems described by Hudson–Parthasarathy quantum stochastic differential equations, aiming to reduce the computational burden associated with online calculation of the quantum filter. By using a differential geometric approach, the quantum trajectory is constrained in a finite-dimensional differentiable manifold consisting of an unnormalized exponential family of quantum density operators, and an exponential quantum projection filter is then formulated as a number of stochastic differential equations satisfied by the finite-dimensional coordinate system of this manifold. A convenient design of the differentiable manifold is also presented through reduction of the local approximation errors, which yields a simplification of the quantum projection filter equations. It is shown that the computational cost can be significantly reduced by using the quantum projection filter instead of the quantum filter. It is also shown that when the quantum projection filtering approach is applied to a class of open quantum systems that asymptotically converge to a pure state, the input-to-state stability of the corresponding exponential quantum projection filter can be established. Simulation results from an atomic ensemble system example are provided to illustrate the performance of the projection filtering scheme. It is expected that the proposed approach can be used in developing more efficient quantum control methods.

Spatially resolved common-path high-order harmonic interferometry.

Spatially resolved interference is observed between high-order harmonics generated in two longitudinally separated gas targets. High-contrast modulations in the intensity of each harmonic order up to the cutoff are observed on-axis in the far field of the source as the separation between the gas targets is increased. For low-order harmonics, additional off-axis modulations are observed, which are attributed to the interference between the contributions from the long quantum trajectories from each gas target. The inherent synchronization of this setup offers the prospect for high-stability metrology of quantum states with ultrafast temporal resolutions.