We introduce jumptime unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jumptime-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jumptime evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jumptime-averaged states may decay into the dark states and the jumptime evolution may eventually terminate. Jumptime-averaged quantum states and the associated jumptime evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jumptime evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. The latter emerge in the unraveling of Markovian quantum master equations and/or in continuous quantum measurements. Ensemble-averaging quantum trajectories at the occurrence of quantum jumps, i.e., the jump times, gives rise to a discrete, deterministic evolution which is highly sensitive to the presence of dark states. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hami…
An important and well established area of quantum optics is the theory of Markovian stochastic Schrödinger equations (or by another name quantum trajectory theory). Recently stochastic Schrödinger equations have been developed for non-Markovian systems. In this paper we extend the current known stochastic Schrödinger equations for non-Markovian systems to include the position unraveling. We also discuss and illustrate that this stochastic Schrödinger equation can have an interpretation under both the orthodox and the de Broglie-Bohm hidden variable interpretation of quantum mechanics. We concl…
We derive a family of Gaussian non-Markovian stochastic Schrödinger equations for the dynamics of open quantum systems. The different unravelings correspond to different choices of squeezed coherent states, reflecting different measurement schemes on the environment. Consequently, we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the Gaussian process. By construction, the reduced dynamics of the open system does not depend on the squeezing parameters. They determine the non-hermitian Gaussian correla…
The development and use of Monte Carlo algorithms plays a visible role in the study of non-Markovian quantum dynamics due to the provided insight and powerful numerical methods for solving the system dynamics. In the Markovian case, the connections between the various types of methods are fairly well-understood while for non-Markovian case there has so far been only a few studies. We focus here on two jumplike unravelings of non-Markovian dynamics, the non-Markovian quantum jump (NMQJ) method and the property state method by Gambetta, Askerud, and Wiseman (GAW). The results for simple quantum …
Non-Markovian evolution of an open quantum system can be `unraveled' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schrödinger equation, as introduced by Diósi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jump-like unravelings. To illustrate the jump-like behavior we consider a simple system: A classically driven (at Rabi frequency $Ω$) two-level atom coupled linearly to a three mode optical bath, with a cent…
Do stochastic Schrödinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system {\em on average} obeys a master equation, the answer is yes. Markovian stochastic Schrödinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic \sch equation introduced by Strunz, Di\' …
By using a condition of average trace preservation we derive a general class of non-Markovian Gaussian diffusive unravelings [L. Diosi and L. Ferialdi, Phys. Rev. Lett. \textbf{113}, 200403 (2014)], here valid for arbitrary non-Hermitian system operators and noise correlations. The conditions under which the generalized stochastic Schrodinger equation has the same symmetry properties (invariance under unitary changes of operator base) than a microscopic system-bath Hamiltonian dynamics are determined. While the standard quantum diffusion model (standard noise correlations) always share the sam…
Non-Markovian stochastic Schrödinger equations (NMSSE) are important tools in quantum mechanics, from the theory of open systems to foundations. Yet, in general, they are but formal objects: their solution can be computed numerically only in some specific cases or perturbatively. This article is focused on the NMSSE themselves rather than on the open-system evolution they unravel and aims at making them less abstract. Namely, we propose to write the stochastic realizations of linear NMSSE as averages over the solutions of an auxiliary equation with an additional random field. Our method yields…
The state matrix $ρ$ for an open quantum system with Markovian evolution obeys a master equation. The master equation evolution can be unraveled into stochastic nonlinear trajectories for a pure state $P$, such that on average $P$ reproduces $ρ$. Here we give for the first time a complete parameterization of all diffusive unravelings (in which $P$ evolves continuously but non-differentiably in time). We give an explicit measurement theory interpretation for these quantum trajectories, in terms of monitoring the system's environment. We also introduce new classes of diffusive unravelings that a…
In the context of a charge qubit under continuous monitoring by a single electron transistor, we propose an unraveling of the generalized quantum Markovian master equation into an ensemble of individual quantum trajectories for stochastic point process. A suboptimal feedback algorism is implemented into individual quantum trajectories to protect a desired pure state. Coherent oscillations of the charge qubit could be maintained in principle for an arbitrarily long time in case of sufficient feedback strength. The effectiveness of the feedback control is also reflected in the detector's noise s…
Quantum measurements and the associated state changes are properly described in the language of instruments. We investigate the properties of a time continuous family of instruments associated with the recently introduced family of general Gaussian non-Markovian stochastic Schrödinger equations. In this Letter we find that when the covariance matrix for the Gaussian noise satisfies a particular $δ$-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator. This class contains, for example the spin-boson and quantum Brownian motion…
We solve two long standing problems for stochastic descriptions of open quantum system dynamics. First, we find the classical stochastic processes corresponding to non-Markovian quantum state diffusion and non-Markovian quantum jumps in projective Hilbert space. Second, we show that the diffusive limit of non-Markovian quantum jumps can be taken on the projective Hilbert space in such a way that it coincides with non-Markovian quantum state diffusion. However, the very same limit taken on the Hilbert space leads to a completely new diffusive unraveling, which we call non-Markovian quantum diff…
We present a Heisenberg operator based formulation of coherent quantum feedback and Pyragas control. This model is easy to implement and allows for an efficient and fast calculation of the dynamics of feedback-driven observables as the number of contributing correlations grows in systems with a fixed number of excitations only linearly in time. Furthermore, our model unravels the quantum kinetics of entanglement growth in the system by explicitly calculating non-Markovian multi-time correlations, e.g., how the emission of a photon is correlated with an absorption process in the past. Therefore…
A new method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs which are not of Lindblad form. So it opens new large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local…
Thirty years ago, the present author discussed pure state unraveling (stochastic quantum trajectories) of Markovian open system dynamics. The fact that he considered all positive dynamics, not restricted for the Lindblad-Gorini-Kossakowski-Sudarshan complete-positive subclass, has remained unnoticed so far. Here we point out the importance of the transition rate operator and the merit of invariant (representation-independent) approach, with reference to invariant condition of positive dynamics and extension of quantum state diffusion beyond complete-positive dynamics. A new result, description…
A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space ${\mathcal{H}}$, the extended state space is the triple Hilbert space ${\mathcal{H}}\otimes{\mathbb C}^3$ which is obtained by combining the open system with a three state system. This embedding is used to derive an unraveling fo…
Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common ``quantum trajectory'' techniques corresponding to continuous measurement schemes, which solve the master equation by unraveling its evolution into stochastic trajectories in Hilbert space, correspond closely to particular sets of decoherent (or consistent) histories. This is illustrated by a simple model of photon counting. An equivalence is shown for these models between standard quantum jumps and the orthogonal jumps of Diósi, which have already been shown to correspond to dec…
In natural light harvesting systems, the sequential quantum events of photon absorption by specialized biological antenna complexes, charge separation, exciton formation and energy transfer to localized reaction centers culminates in the conversion of solar to chemical energy. A notable feature in these processes is the exceptionally high efficiencies ($>$ 95\%) at which excitation is transferred from the illuminated protein complex site to the reaction centers. The high speeds of excitation propagation within a system of interwoven biomolecular network structures, is yet to be replicated in a…
Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common ``quantum jumps'' techniques, which solve the master equation by unraveling its evolution into stochastic trajectories in Hilbert space, correspond closely to a particular set of decoherent histories. This is illustrated by a simple model of a photon counting experiment.
We extend quantum trajectory theory to encompass the evolution of a large class of open quantum systems interacting with an environment at {arbitrary coupling strength}. Specifically, we prove that general time-local quantum master equations admit an unraveling described by ordinary jump-stochastic differential equations. The sufficient condition is to weigh the state vector Monte Carlo 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 traject…
Lindbladian spectral statistics beyond no-jump Hamiltonians: roles of recycling and Liouville-space structure
Singapore University of Technology and Design（新加坡科技设计大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；School of Physics, Beihang University（北京航空航天大学物理学院）；EPD Pillar, Singapore University of Technology and Design（新加坡科技设计大学工程产品与设计支柱）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654（MajuLab，CNRS-UNS-NUS-NTU国际联合研究单位，UMI 3654）
