Quantum Trajectories Research Articles

Measurement-induced topological entanglement transitions in symmetric random quantum circuits

We study a class of (1+1)D symmetric random quantum circuits with two competing types of measurements in addition to random unitary dynamics. The circuit exhibits a rich phase diagram involving robust symmetry-protected topological (SPT), trivial, and volume law entangled phases, where the transitions are hidden to expectation values of operators and can only be accessed through the entanglement entropy averaged over quantum trajectories. In the absence of unitary dynamics, we find a purely measurement-induced critical point with logarithmic scaling of the entanglement entropy, which we map exactly to two copies of a classical 2D percolation problem. We perform numerical simulations that indicate this transition is a tricritical point that splits into two critical lines in the presence of arbitrarily sparse unitary dynamics with an intervening volume law entangled phase. Our results show how measurements alone are sufficient to induce criticality and logarithmic entanglement scaling, and how arbitrarily sparse unitary dynamics can be sufficient to stabilize volume law entangled phases in the presence of rapid yet competing measurements.

Invariant Measure for Stochastic Schrödinger Equations

Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrödinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a “purification” condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast toward this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.

Euler-Milstein and relevant approaches for high-precision stochastic simulation of quantum trajectories

For quantum diffusive measurements, the system’s dynamics can be described by the Itô stochastic master equation, which works well for an infinitesimal time resolution. However, in practical quantum experiments, one cannot make a time step to be infinitesimal, as it can introduce correlation of noise in time, making the Markovian assumption invalid. On the other hand, increasing a time step can cause errors from non-commuting operations describing the system’s dynamics. We therefore consider implementing the Euler-Milstein and relevant approaches, namely the Itô map, the high-order completely positive map, and the quantum Bayesian, to simulate quajntum trajectory for a quantum system under diffusive continuous measurements. In particular, we numerically simulate trajectories for a qubit measurement in z basis. We show the comparison of individual trajectories and their averaged trajectories among these approaches. We find that the high-order completely positive map approach yields the most accurate averaged quantum trajectory. Furthermore, we also investigate the trace distance from true stochastic quantum trajectories, comparing the four approaches using the numerical simulation. We show that, for a realistic time resolution (as in a superconducting qubit, experiment), the high-order map does give the most accurate estimate of the qubit trajectories.

Nonadiabatic dynamics simulation of photoinduced ring-opening reaction of 2(5H)-thiophenone with internal conversion and intersystem crossing.

In the present work, the quantum trajectory mean-field approach, which is able to overcome the overcoherence problem, was generalized to simulate internal conversion and intersystem crossing processes simultaneously. The photoinduced ring-opening and subsequent rearrangement reactions of isolated 2(5H)-thiophenone were studied based on geometry optimizations on critical structures and nonadiabatic dynamics simulations using this method. Upon 267 nm irradiation, the molecule is initially populated in the 1ππ* state. After a sudden rupture of one C-S bond within 100 fs in this state, the lowest two singlet excited states and the lowest two triplet excited states become quasi-degenerated, and then the intersystem crossing processes between singlet and triplet states accompanied by rearrangement reactions can be observed several times. Compared with our previous nonadiabatic simulations in the absence of intersystem crossing (ChemPhotoChem, 2019, 3, 897-906), some new nonadiabatic relaxation pathways involving triplet states and different ring-opening products were identified. The present work provides new mechanistic insights into the photoinduced ring-opening of thio-substituted heterocyclic molecules and reveals the importance of nonadiabatic dynamics simulation that is able to deal with multiple electronic states with different spin multiplicities.

Entanglement-preserving limit cycles from sequential quantum measurements and feedback

Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local pi-pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e. using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around a Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e. using diffusive quantum trajectories), [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. The method we describe here combines proven techniques in a novel way, complementing existing protocols, and offering a pathway towards generating and protecting entangled states so that they may be used in various applications on demand.

Quantum Trajectory Description of the Time-Independent (Inverse) Fermi Accelerator

We re-examine the (inverse) Fermi accelerator problem by resorting to a quantum trajectory description of the dynamics. Quantum trajectories are generated from the time-independent Schrodinger equation solutions, using a unipolar treatment for the (light) confined particle and a bipolar treatment for the (heavy) movable wall. Analytic results are presented for the exact coupled two-dimensional problem, as well as for the adiabatic and mixed quantum-classical approximations.

Position measurement-induced collapse states and boundary bound diffraction: an idealised experiment

In an earlier work, it was shown that single-slit diffraction of matter waves can be considered as position measurement of particles and that the Fresnel and Fraunhofer diffraction patterns result from the time evolution of the same collapsed quantum wave function. This quantum mechanical treatment of diffraction of particles, based on the standard postulates of quantum mechanics and the postulate of existence of quantum trajectories, leads to the ‘position measurement-induced collapse’ (PMIC) states. In the present work, an idealised experimental set-up to test these PMIC states is proposed. The apparatus consists of a modified Lloyd’s mirror in optics, with two reflectors instead of one. The diffraction patterns for this case predicted by the PMIC formalism are presented. They exhibit quantum fractal structures in space-time called ‘quantum carpets’, first discovered by Berry. The PMIC formalism in this case closely follows the ‘boundary bound diffraction’ analysed in a previous work by Tounli, Alverado and Sanz. In addition to having obtained their results, we have identified that the Fresnel and Fraunhofer patterns are indeed present here. It is anticipated that the verification of fractal and nonfractal features of the wave function at various times and also the predicted revival distances to the screen by this experiment will help to better understand ‘quantum collapse’.

Completely positive quantum trajectories with applications to quantum state smoothing

Here, we are concerned with comparing estimation schemes for the quantum state under continuous measurement (quantum trajectories), namely quantum state filtering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)], quantum state smoothing. Unfortunately, the cumulative errors in the most typical simulations of quantum trajectories with a total time of simulation $T$ can reach orders of $T \Delta t$. Moreover, these errors may correspond to deviations from valid quantum evolution as described by a completely positive map. Here we introduce a higher-order method that reduces the cumulative errors in the complete positivity of the evolution to of order $T\Delta t^2$, whether for linear (unnormalised) or nonlinear (normalised) quantum trajectories. Our method also guarantees that the discrepancy in the average evolution between different detection methods (different `unravellings', such as quantum jumps or quantum diffusion) is similarly small. This equivalence is essential for comparing quantum state filtering to quantum state smoothing, as the latter assumes that all irreversible evolution is unravelled, although the estimator only has direct knowledge of some records. In particular, here we compare, for the first time, the average difference between filtering and smoothing conditioned on an event of which the estimator lacks direct knowledge: a photon detection within a certain time window. We find that the smoothed state is actually {\em less pure}, both before and after the time of the jump. Similarly, the fidelity of the smoothed state with the `true' (maximal knowledge) state is also lower than that of the filtered state before the jump. However, after the jump, the fidelity of the smoothed state is higher.

Numerical evaluation of two-time correlation functions in open quantum systems with matrix product state methods: a comparison

We compare the efficiency of different matrix product state (MPS) based methods for the calculation of two-time correlation functions in open quantum systems. The methods are the purification approach[1] and two approaches[2,3] based on the Monte-Carlo wave function (MCWF) sampling of stochastic quantum trajectories using MPS techniques. We consider a XXZ spin chain either exposed to dephasing noise or to a dissipative local spin flip. We find that the preference for one of the approaches in terms of numerical efficiency depends strongly on the specific form of dissipation.

Dynamical hysteresis properties of the driven-dissipative Bose-Hubbard model with a Gutzwiller Monte Carlo approach

We study the dynamical properties of a driven-dissipative Bose-Hubbard model in the strongly interacting regime through a quantum trajectory approach with a cluster-Gutzwiller Ansatz for the wave function. This allows us to take classical and quantum correlations into account. By studying the dynamical hysteresis surface that arises by sweeping through the coherent driving strength we show that the phase diagram for this system is in qualitative correspondence with the Gutzwiller mean-field result. However, quantitative differences are present and the inclusion of classical and quantum correlations causes a significant shift of the critical parameters. Additionally, we show that approximation techniques relying on a unimodal distribution such as the mean field and $1/z$ expansion drastically underestimate the particle number fluctuations. Finally, we show that a proposed mapping of the driven-dissipative many-body Bose-Hubbard model onto a single driven-dissipative Kerr model is not accurate for parameters in the hysteresis regime.

Wavepacket phase-space quantum Monte Carlo method

We develop a novel numerical method for solving the Wigner equation which does not rely upon evaluation of the Wigner kernel and utilizes quantum trajectories. This method is a generalization of the effective potential approach widely used in semiconductor TCAD tools while also solving the full Wigner equation. The method’s accuracy is demonstrated with several standard examples.

Quantum trajectories for the variational description of closed systems: A case study with Gaussian states.

In this paper, it is proposed to improve the quality of a variational description of a closed quantum system by adding fictitious dissipation that reduces the entanglement. The proposal is implemented for a small Bose-Hubbard chain in the mean field regime, which shows chaotic behavior and associated fast growth of quantum fluctuations. For appropriately chosen dissipation, good agreement with the truncated Wigner approximation is found. This shows that, when chaos leads to the breakdown of a variational method, adding fictitious dissipation can be a cure.

Solving quantum trajectories for systems with linear Heisenberg-picture dynamics and Gaussian measurement noise

We study solutions to the quantum trajectory evolution of $N$-mode open quantum systems possessing a time-independent Hamiltonian, linear Heisenberg-picture dynamics, and Gaussian measurement noise. In terms of the mode annihilation and creation operators, a system will have linear Heisenberg-picture dynamics under two conditions. First, the Hamiltonian must be quadratic. Second, the Lindblad operators describing the coupling to the environment (including those corresponding to the measurement) must be linear. In cases where we can solve the $2N$-degree polynomials that arise in our calculations, we provide an analytical solution for initial states that are arbitrary (i.e. they are not required to have Gaussian Wigner functions). The solution takes the form of an evolution operator, with the measurement-result dependence captured in $2N$ stochastic integrals over these classical random signals. The solutions also allow the POVM, which generates the probabilities of obtaining measurement outcomes, to be determined. To illustrate our results, we solve some single-mode example systems, with the POVMs being of practical relevance to the inference of an initial state, via quantum state tomography. Our key tool is the representation of mixed states of quantum mechanical oscillators as state vectors rather than state matrices (albeit in a larger Hilbert space). Together with methods from Lie algebra, this allows a more straightforward manipulation of the exponential operators comprising the system evolution than is possible in the original Hilbert space.

Revisiting Bohmian Quantum Mechanics with Quantum Trajectories

Revisiting Bohmian Quantum Mechanics with Quantum Trajectories

Noise and charge discreteness as ultimate limit for the THz operation of ultra-small electronic devices

To manufacture faster electron devices, the industry has entered into the nanoscale dimensions and Terahertz (THz) working frequencies. The discrete nature of the few electrons present simultaneously in the active region of ultra-small devices generate unavoidable fluctuations of the current at THz frequencies. The consequences of this noise remain unnoticed in the scientific community because its accurate understanding requires dealing with consecutive multi-time quantum measurements. Here, a modeling of the quantum measurement of the current at THz frequencies is introduced in terms of quantum (Bohmian) trajectories. With this new understanding, we develop an analytic model for THz noise as a function of the electron transit time and the sampling integration time, which finally determine the maximum device working frequency for digital applications. The model is confirmed by either semi-classical or full- quantum time-dependent Monte Carlo simulations. All these results show that intrinsic THz noise increases unlimitedly when the volume of the active region decreases. All attempts to minimize the low signal-to-noise ratio of these ultra-small devices to get effective THz working frequencies are incompatible with the basic elements of the scaling strategy. One can develop THz electron devices, but they cannot have ultra-small dimensions. Or, one can fabricate ultra-small electron devices, but they cannot be used for THz working frequencies.

Entanglement statistics in Markovian open quantum systems: A matter of mutation and selection.

Controlling dynamical fluctuations in open quantum systems is essential both for our comprehension of quantum nonequilibrium behavior and for its possible application in near-term quantum technologies. However, understanding these fluctuations is extremely challenging due, to a large extent, to a lack of efficient important sampling methods for quantum systems. Here, we devise a unified framework-based on population-dynamics methods-for the evaluation of the full probability distribution of generic time-integrated observables in Markovian quantum jump processes. These include quantities carrying information about genuine quantum features, such as quantum superposition or entanglement, not accessible with existing numerical techniques. The algorithm we propose provides dynamical free-energy and entropy functionals which, akin to their equilibrium counterpart, permit one to unveil intriguing phase-transition behavior in quantum trajectories. We discuss some applications and further disclose coexistence and hysteresis, between a highly entangled phase and a low entangled one, in large fluctuations of a strongly interacting few-body system.

Probing spectral and temporal resolved dynamics in high-order harmonic generation of atom driven by multicycle near-infrared laser pulses

We present an ab initio investigation of the time-frequency characteristics of high-order harmonic generation (HHG) calculated by solving accurately and efficiently the time-dependent Schrödinger equation by means of the time-dependent generalized pseudospectral method. We extend a new synchrosqueezing transform (SST) to obtain the time-frequency spectra of the HHG and the dynamical phase associated with the dipole-emission time profile. The SST time-frequency analysis allows us to explore in depth the dynamics and contributions of the quantum trajectories in HHG. Particularly it uncovers the subtle details of the spectral and temporal fine structures of the HHG and provides novel insights regarding the dynamical origin of the HHG in below-threshold harmonic regimes.

Quantum Zeno effect appears in stages

In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for a continuous partial measurement. We show that the onset of the Zeno regime is marked by a $\textit{cascade of transitions}$ in the system dynamics as the measurement strength is increased. Some of these transitions are only apparent in the collective behavior of individual quantum trajectories and are invisible to the average dynamics. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These newly predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.

Entanglement transition in the projective transverse field Ising model

Discrete quantum trajectories of systems under random unitary gates and projective measurements have been shown to feature transitions in the entanglement scaling that are not encoded in the density matrix. In this paper, we study the projective transverse field Ising model, a stochastic model with two noncommuting projective measurements and no unitary dynamics. We numerically demonstrate that their competition drives an entanglement transition between two distinct steady states that both exhibit area law entanglement, and introduce a classical but nonlocal model that captures the entanglement dynamics completely. Exploiting a map to bond percolation, we argue that the critical system in one dimension is described by a conformal field theory, and derive the universal scaling of the entanglement entropy and the critical exponent for the scaling of the mutual information of two spins exactly. We conclude with an interpretation of the entanglement transition in the context of quantum error correction.

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.