Quantum Trajectories Research Articles

Adiabatic and Nonadiabatic Dynamics with Interacting Quantum Trajectories.

We present a quantum dynamics method based on the propagation of interacting quantum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys.2010,370, 4-14] and Schiff and Poirier [J. Chem. Phys.2012,136, 031102] on quantum dynamics without wavefunctions. It consists of determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quantum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

Probing non-Markovian quantum dynamics with data-driven analysis: Beyond “black-box” machine-learning models

A precise understanding of the influence of a quantum system's environment on its dynamics, which is at the heart of the theory of open quantum systems, is crucial for further progress in the development of controllable large-scale quantum systems. However, existing approaches to account for complex system-environment interaction in the presence of memory effects are either based on heuristic and oversimplified principles or give rise to computational difficulties. In practice, one can leverage on available experimental data and replace first-principles simulations with a data-driven analysis that is often much simpler. Inspired by recent advances in data analysis and machine learning, we propose a data-driven approach to the analysis of the non-Markovian dynamics of open quantum systems. Our method allows, on the one hand, capturing the most important characteristics of open quantum systems such as the effective dimension of the environment and the spectrum of the joint system-environment quantum dynamics, and, on the other hand, reconstructing a predictive model of non-Markovian quantum dynamics, and denoising the measured quantum trajectories. We demonstrate the performance of the proposed approach with various models of open quantum systems, including a qubit coupled with a finite environment, a spin-boson model, and the damped Jaynes-Cummings model.

Quantum fluctuations and semiclassicality in an inflaton-driven evolution

A semiclassical description of quantum systems is applied to probe the dynamics of the cosmological model of an inflationary universe with quadratic inflaton potential, described in a quantum framework of geometrodynamics. The systematic analysis, focusing in particular on the inflationary and post-inflationary epochs, revealed several surprising and counterintuitive features: (i) during inflation the universe rapidly spreads out in volume which leads to significant relative variance by the end of inflation; (ii) despite that, the quantum evolution can still be described to high accuracy by semiclassical methods; (iii) moreover, in the post-inflationary epoch, as the order of included quantum corrections increases, the quantum trajectory approaches the classical one and the description involving second-order corrections only is actually the least accurate there. The consequence of the latter is that the effects of the quantum variances are washed out by the higher-order quantum corrections.

Zeno crossovers in the entanglement speed of spin chains with noisy impurities

We use a noisy signal with finite correlation time to drive a spin (dissipative impurity) in the quantum XY spin chain and calculate the dynamics of entanglement entropy (EE) of a bipartition of spins, for a stochastic quantum trajectory. We compute the noise averaged EE of a bipartition of spins and observe that its speed of spreading decreases at strong dissipation, as a result of the Zeno effect. We recover the Zeno crossover and show that noise averaged EE can be used as a proxy for the heating and Zeno regimes. Upon increasing the correlation time of the noise, the location of the Zeno crossover shifts at stronger dissipation, extending the heating regime.

Diverging current fluctuations in critical Kerr resonators

The parametrically pumped Kerr model describes a driven-dissipative nonlinear cavity, whose nonequilibrium phase diagram features both continuous and discontinuous quantum phase transitions. We consider the consequences of these critical phenomena for the fluctuations of the photocurrent obtained via continuous weak measurements on the cavity. Considering both direct photodetection and homodyne detection schemes, we find that the current fluctuations diverge exponentially at the discontinuous phase transition. However, we find strikingly different current fluctuations for these two detection schemes near the continuous transition, a behaviour which is explained by the complementary information revealed by measurements in different bases. To obtain these results, we develop formulas to efficiently compute the diffusion coefficient -- which characterises the long-time current fluctuations -- directly from the quantum master equation, thus connecting the formalisms of full counting statistics and stochastic quantum trajectories. Our findings highlight the rich features of current fluctuations near nonequilibrium phase transitions in quantum-optical systems.

Duality between operator ordering factor and massless scalar field

In order to investigate the role of quantum effects in the evolution of the universe, one can either use the Wheeler–DeWitt equation (WDWE) that contains an operator ordering factor, or add an item called massless scalar field to WDWE. In this paper, we study the relationship between operator ordering factor and massless scalar field, by applying de Broglie–Bohm quantum trajectory approach to WDWE. In theory, the evolution of the universe is determined by action, i.e., the phase part of the wavefunction of the universe. For the case of operator ordering factor and the case of massless scalar field, the functions that determine the phase part of the wavefunction of the universe satisfy the same differential equation, both in the minisuperspace model and in the Kantowski–Sachs model. This shows the equivalence of using operator ordering factor or massless scalar field to study evolution of the universe. Since there is no accelerating solution of WDWE with operator ordering factor for a grownup universe in the minisuperspace model, the equivalence of the operator ordering factor and the massless scalar field rules out the possibility of a massless scalar field as the candidate for dark energy, if the current universe is indeed homogeneous and isotropic.

Heavy quarkonium dynamics at next-to-leading order in the binding energy over temperature

Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological application, we numerically solve the resulting NLO Lindblad equation using the quantum trajectories algorithm. To achieve this, we map the solution of the three-dimensional Lindblad equation to the solution of an ensemble of one-dimensional Schrödinger evolutions with Monte-Carlo sampled quantum jumps. Averaging over the Monte-Carlo sampled quantum jumps, we obtain the solution to the NLO Lindblad equation without truncation in the angular momentum quantum number of the states considered. We also consider the evolution of the system using only the complex effective Hamiltonian without stochastic jumps and find that this provides a reliable approximation for the ground state survival probability at LO and NLO. Finally, we make comparisons with our prior leading-order pNRQCD results and experimental data available from the ATLAS, ALICE, and CMS collaborations.

Key for a Hidden Quantum State.

Quantum trajectories are crucial to understanding the evolution of open systems. We consider an open cavity mode undergoing up and down multistate quantum jumps due to the emission and absorption of photons. We prove that among all subtrajectories, starting simultaneously from different photon number states, only one survives a long single-run evolution. A random Fock state terminating the subtrajectory becomes known for the ergodic case via the key-the processed record of the input and output photocounts, and the trajectory duration. Based on this result, we propose a robust protocol to infer the Fock state, a valuable resource for quantum applications.

Collisional-model quantum trajectories for entangled-qubit environments

We study the dynamics of quantum systems interacting with a stream of entangled-qubit pairs in the weak coupling limit. For a large class of two-qubit bath states, we present a detailed framework describing conditional dynamical maps for the system, known as quantum trajectories, that arise when the qubits are measured. Depending on the measurement basis, these quantum trajectories can be jump-type or diffusive-type, and can successively transfer entanglement from the bath qubits other quantum systems. They also exhibit features not present in standard quantum optical trajectories due to the fact that collisional models are not confined by the standard white-noise limit for optical-mode baths. As an example of this formalism, we consider the case of two remote two-level systems, where jump-type quantum trajectories herald the birth and death of entanglement.

Monitoring Fast Superconducting Qubit Dynamics Using a Neural Network

Weak measurements of a superconducting qubit produce noisy voltage signals that are weakly correlated with the qubit state. To recover individual quantum trajectories from these noisy signals, traditional methods require slow qubit dynamics and substantial prior information in the form of calibration experiments. Monitoring rapid qubit dynamics, e.g. during quantum gates, requires more complicated methods with increased demand for prior information. Here, we experimentally demonstrate an alternative method for accurately tracking rapidly driven superconducting qubit trajectories that uses a Long-Short Term Memory (LSTM) artificial neural network with minimal prior information. Despite few training assumptions, the LSTM produces trajectories that include qubit-readout resonator correlations due to a finite detection bandwidth. In addition to revealing rotated measurement eigenstates and a reduced measurement rate in agreement with theory for a fixed drive, the trained LSTM also correctly reconstructs evolution for an unknown drive with rapid modulation. Our work enables new applications of weak measurements with faster or initially unknown qubit dynamics, such as the diagnosis of coherent errors in quantum gates.

Quantum trajectory theory and simulations of nonlinear spectra and multiphoton effects in waveguide-QED systems with a time-delayed coherent feedback

We study the nonlinear spectra and multi-photon correlation functions for the waveguide output of a two-level system (including realistic dissipation channels) with a time-delayed coherent feedback. We compute these observables by extending a recent quantum trajectory discretized-waveguide (QTDW) approach which exploits quantum trajectory simulations and a collisional model for the waveguide to tractably simulate the dynamics. Following a description of the general technique, we show how to calculate the first and second order quantum correlation functions, in the presence of a coherent pumping field. With a short delay time, we show how feedback can be used to filter out the central peak of the Mollow triplet or switch the output between bunched and anti-bunched photons by proper choice of round trip phase. We further show how the loop length and round trip phase effects the zero-time second order quantum correlation function, an indicator of bunching or anti-bunching. New resonances introduced through the feedback loop are also shown through their appearance in the incoherent output spectrum from the waveguide. We explain these results in the context of the waiting time distributions of the system output and individual trajectories, uniquely stochastic observables that are easily accessible with the QTDW model.

Quantum trajectory framework for general time-local master equations

Master equations are one of the main avenues to study open quantum systems. When the master equation is of the Lindblad–Gorini–Kossakowski–Sudarshan form, its solution can be “unraveled in quantum trajectories” i.e., represented as an average over the realizations of a Markov process in the Hilbert space of the system. Quantum trajectories of this type are both an element of quantum measurement theory as well as a numerical tool for systems in large Hilbert spaces. We prove that general time-local and trace-preserving master equations also admit an unraveling in terms of a Markov process in the Hilbert space of the system. The crucial ingredient is to weigh averages by a probability pseudo-measure which we call the “influence martingale”. The influence martingale satisfies a 1d stochastic differential equation enslaved to the ones governing the quantum trajectories. We thus extend the existing theory without increasing the computational complexity.

Enhanced entanglement negativity in boundary-driven monitored fermionic chains

We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying nonequilibrium states. We focus on a prototypical one-dimensional model of boundary-driven noninteracting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, which otherwise decays to zero in the absence of measurements. This result is in contrast with the case of unitary evolution where monitoring suppresses entanglement production. For small values of $\ensuremath{\gamma}$, the stationary-state negativity shows a logarithmic scaling with system size, transitioning to an area-law scaling as $\ensuremath{\gamma}$ is increased beyond a critical value. Similar critical behavior is found in the mutual information, while the late-time purity shows no apparent signature of a transition, being $O(1)$ for all values of $\ensuremath{\gamma}$. Our work unveils the double role of weak monitoring in current-driven open quantum systems, simultaneously damping transport and enhancing entanglement.

Bottomonium suppression and elliptic flow in heavy-ion collisions

In this proceedings contribution I review recent progress concerning the suppression of bottomonium production in the quark-gluon plasma. Making use of open quantum system methods applied to potential non-relativistic quantum chromodynamics one can show that the dynamics of heavy-quarkonium bound states satisfying the scale hierarchy 1/a0 » πT ∼ mD » E obey a Lindblad equation whose solution provides the quantum evolution of the heavy-quarkonium reduced density matrix. To solve the resulting Lindblad equation we use a quantum trajectories algorithm which allows one to include all possible angular momentum states of the quark-antiquark probe in a scalable manner. We solve the Lindblad equation using a tuned 3+1D dissipative hydrodynamics code for the background temperature evolution. We then consider a large number of Monte-Carlo sampled bottomonium trajectories embedded in this background. This allows us to extract the centrality- and pT-dependence of the nuclear suppression factor RAA[Υ] and elliptic flow v2[Υ]. We find good agreement between our model predictions and available √ sNN = 5.02 TeV = 5.02 TeV Pb-Pb collision experimental data from the ALICE, ATLAS, and CMS collaborations.

Monitored open fermion dynamics: Exploring the interplay of measurement, decoherence, and free Hamiltonian evolution

The interplay of unitary evolution and local measurements in many-body systems gives rise to a stochastic state evolution and to measurement-induced phase transitions in the pure state entanglement. In realistic settings, however, this dynamics may be spoiled by decoherence, e.g., dephasing, due to coupling to an environment or measurement imperfections. We investigate the impact of dephasing and the inevitable evolution into a non-Gaussian, mixed state, on the dynamics of monitored fermions. We approach it from three complementary perspectives: (i) the exact solution of the conditional master equation for small systems, (ii) quantum trajectory simulations of Gaussian states for large systems, and (iii) a renormalization group analysis of a bosonic replica field theory. For weak dephasing, constant monitoring preserves a weakly mixed state, which displays a robust measurement-induced phase transition between a critical and a pinned phase, as in the decoherence-free case. At strong dephasing, we observe the emergence of a new scale describing an effective temperature, which is accompanied with an increased mixedness of the fermion density matrix. Remarkably, observables such as density-density correlation functions or the subsystem parity still display scale invariant behavior even in this strongly mixed phase. We interpret this as a signature of gapless, classical diffusion, which is stabilized by the balanced interplay of Hamiltonian dynamics, measurements, and decoherence.

Effects of Spatial Nonlocality versus Nonlocal Causality for Bound Electrons in External Fields.

Using numerically exact solution of the time-dependent Schrödinger equation together with time-dependent quantum Monte Carlo (TDQMC) calculations, here we compare the effects of spatial nonlocality versus nonlocal causality for the ground state and for real-time evolution of two entangled electrons in parabolic potential in one spatial dimension. It was found that the spatial entanglement quantified by the linear quantum entropy is predicted with good accuracy using the spatial nonlocality, parameterized naturally within the TDQMC approach. At the same time, the nonlocal causality predicted by the exact solution leads to only small oscillations in the quantum trajectories which belong to the idler electron as the driven electron is subjected to a strong high frequency electric field, without interaction between the electrons.

Observing a topological transition in weak-measurement-induced geometric phases

Measurement plays a quintessential role in the control of quantum systems. Beyond initialization and readout which pertain to projective measurements, weak measurements in particular, through their back-action on the system, may enable various levels of coherent control. The latter ranges from observing quantum trajectories to state dragging and steering. Furthermore, just like the adiabatic evolution of quantum states that is known to induce the Berry phase, sequential weak measurements may lead to path-dependent geometric phases. Here we measure the geometric phases induced by sequences of weak measurements and demonstrate a topological transition in the geometric phase controlled by measurement strength. This connection between weak measurement induced quantum dynamics and topological transitions reveals subtle topological features in measurement-based manipulation of quantum systems. Our protocol could be implemented for classes of operations (e.g. braiding) which are topological in nature. Furthermore, our results open new horizons for measurement-enabled quantum control of many-body topological states.

The Entanglement-Assisted Communication Capacity Over Quantum Trajectories

The unique and often-weird properties of quantum mechanics allow an information carrier to propagate through multiple trajectories of quantum channels simultaneously. This ultimately leads us to quantum trajectories with an indefinite causal order of quantum channels. It has been shown that indefinite causal order enables the violation of bottleneck capacity, which bounds the amount of the transferable classical and quantum information through a classical trajectory with a well-defined causal order of quantum channels. In this treatise, we investigate this beneficial property in the realm of both entanglement-assisted classical and quantum communications. To this aim, we derive closed-form capacity expressions of entanglement-assisted classical and quantum communication for arbitrary quantum Pauli channels over classical and quantum trajectories. We show that by exploiting the indefinite causal order of quantum channels, we obtain capacity gains over classical trajectory as well as the violation of bottleneck capacity for various practical scenarios. Furthermore, we determine the operating region where entanglement-assisted communication over quantum trajectory obtains capacity gain against classical trajectory and where the entanglement-assisted communication over quantum trajectory violates the bottleneck capacity.

Quantum thermodynamics under continuous monitoring: A general framework

The thermodynamics of quantum systems driven out of equilibrium has attracted increasing attention in the last decade, in connection with quantum information and statistical physics, and with a focus on non-classical signatures. While a first approach can deal with average thermodynamics quantities over ensembles, in order to establish the impact of quantum and environmental fluctuations during the evolution, a continuous quantum measurement of the open system is required. Here, we provide an introduction to the general theoretical framework to establish and interpret the thermodynamics for quantum systems whose nonequilibrium evolution is continuously monitored. We review the formalism of quantum trajectories and its consistent application to the thermodynamic scenario, where primary quantities such as work, heat, and entropy production can be defined at the stochastic level. The connection to irreversibility and fluctuation theorems is also discussed together with some recent developments, and we provide some simple examples to illustrate the general theoretical framework.

Continuous measurement of a qudit using dispersively coupled radiation

We analyze the continuous monitoring of a qudit coupled to a cavity using both phase-preserving and phase-sensitive amplification. The quantum trajectories of the system are described by a stochastic master equation, for which we derive the appropriate Lindblad operators. The measurement back-action causes spiraling in the state coordinates during collapse, which increases as the system levels become less distinguishable. We discuss two examples: a two-level system and an $N$-dimensional system and meter with rotational symmetry in the quadrature space. We also provide a comparison of the effects of phase-preserving and phase-sensitive detection on the master equation and show that the average behavior is the same in both cases, but individual trajectories collapse at different rates depending on the measurement axis in the quadrature plane.