We present a formulation for the emergence of classical dynamics in a quantum world using a path integral approach that incorporates continuous measurements. Our approach complements decoherence and coarse-grained quantum-to-classical transition frameworks. The path-integral formulation provides the joint statistics of a sequence of measurements, with each Feynman path picking up an additional random phase. Its magnitude is proportional to the measurement strength, and we give conditions under which the dominant contribution to the probability amplitude comes from trajectories near classical paths. Information proliferates across the environment, a key feature of quantum Darwinism, via plane-wave probe scattering. Extending to repeated measurements, we show that in the continuous limit each system trajectory picks up an additional phase due to momentum kicks from the probes--the origin of the back-action force. We provide conditions under which measurements yield enough ``which-path'' information while keeping the wave packet localized. This allows the quantum-to-classical transition to be described from individual measurement records, complementing the ensemble description from density matrices. We further show that the same scattering that decoheres a trajectory heats it, tying decoherence and measurement back-action together. This bounds how redundantly an individual classical trajectory can be recorded before back-action randomises it into Brownian motion. For a trapped particle, the ceiling is fixed by the resolution measured in units of the zero-point motion. It is not restrictive for macroscopic systems; it collapses to a single record where the semiclassical description of a trajectory fails, delimiting the regime in which objective classical trajectories exist. The deterministic-to-Brownian crossover is accessible in levitated optomechanics.
We present a formulation for the emergence of classical dynamics in a quantum world using a path integral approach that incorporates continuous measurements. Our approach complements decoherence and coarse-grained quantum-to-classical transition frameworks. The path-integral formulation provides the joint statistics of a sequence of measurements, with each Feynman path picking up an additional random phase. Its magnitude is proportional to the measurement strength, and we give conditions under which the dominant contribution to the probability amplitude comes from trajectories near classical paths. Information proliferates across the environment, a key feature of quantum Darwinism, via plane-wave probe scattering. Extending to repeated measurements, we show that in the continuous limit each system trajectory picks up an additional phase due to momentum kicks from the probes--the origin of the back-action force. We provide conditions under which measurements yield enough ``which-path'' information while keeping the wave packet localized. This allows the quantum-to-classical transition to be described from individual measurement records, complementing the ensemble description from density matrices. We further show that the same scattering that decoheres a trajectory heats it, tying decoherence and measurement back-action together. This bounds how redundantly an individual classical trajectory can be recorded before back-action randomises it into Brownian motion. For a trapped particle, the ceiling is fixed by the resolution measured in units of the zero-point motion. It is not restrictive for macroscopic systems; it collapses to a single record where the semiclassical description of a trajectory fails, delimiting the regime in which objective classical trajectories exist. The deterministic-to-Brownian crossover is accessible in levitated optomechanics.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
This thesis considers quantum systems which interact with their surrounding environment. Specifically we consider the underlying stochastic dynamics which arise from the back-action of measurements performed in the environment. These quantum trajectories can be tracked by continuously monitoring the environment, as is now possible in modern experiments. We consider phenomena understood at the average level of the system state density matrix, and explore these at this more information rich level. This is especially relevant for experimental setups and implementation of quantum technologies. Firstly, we consider how systems exhibiting metastability behave at the level of stochastic quantum trajectories. In doing so we make comparisons to classical theories of metastability, where trajectories exhibit fast relaxation into distinct phases and slow transitions between them. We also illustrate when this phenomenology does not hold, with instead the emergence of a slow manifold in the form of a decoherence free subspace. We also highlight the useful role that quantum reset processes can play in the analysis of stochastic quantum trajectories. The dynamics of the stochastic quantum trajectories depends on the choice of environmental measurements. There are many choices which leave the average dynamics unchanged, but give different statistical ensembles of quantum trajectories. This gauge freedom of the average dynamics is well known. We derive the analogous result at the level of quantum trajectory ensembles; giving the measurement protocols which leave the generator of the stochastic trajectory dynamics unchanged. We also extend this to include joint trajectories of conditional system states and their associated measurement records, either full records or coarse-grained. Using these results on gauge freedoms of quantum trajectories, we consider the conditions under which the trajectory dynamics exhibit a weak unitary symmetry. We then consider under what conditions the joint state of the system and environment exhibit symmetric dynamics, including under various levels of dephasing. We find that the joint dynamics always features a separable symmetry directly related to that of the quantum master equation, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.
Quantum chaos, a phenomenon that began to be studied in the last century, still does not have a rigorous understanding. By virtue of the correspondence principle, the properties of the system that lead to chaotic dynamics at the classical level must also be present in the underlying quantum system. In the classical case, the exponential divergence of nearby trajectories in time is described in terms of the Lyapunov exponent. However, in the quantum case, a similar description of chaos is, strictly speaking, impossible due to the absence of trajectories. There are different approaches to remedy this situation, but the universal criterion of quantum chaos is absent. We propose a quantum chaos definition in a manner similar to the classical one using decoherent histories as a quantum analog of trajectories. For this purpose, we consider the model of an open quantum kicked top interacting with the environment, which is a bosonic bath, and illustrate this idea. Here, the environment plays the role of a trajectory-recording device. For the kicked-top model at the classical level, depending on the kick strength, crossover occurs between the integrable and chaotic regimes. We show that for such a model, the production of entropy of decoherent histories is radically different in integrable and chaotic regimes. Thus, the entropy of an ensemble of quantum trajectories can be used as a signature of quantum chaos.
Abstract: Quantum tunneling is conventionally described through semiclassical WKB methods and imaginary-time formulations of the Feynman Path Integral, which successfully reproduce transmission probabilities but leave the local kinematic structure of the barrier transition largely implicit. In this work, we introduce a finite-rate temporal integration framework in which localized quantum states evolve through successive stochastic re-integrations over a characteristic interval . Within this picture, a macroscopic potential barrier acts as an open fluctuating electromagnetic environment rather than a rigid static boundary. As a localized state partially penetrates the barrier, transient geometric interlocking suppresses forward integration and prolongs the effective interaction window. The under-barrier configuration is modeled not as a single coherent leap across the forbidden region, but as a metastable sequence of stochastic local re-integrations continuously driven by environmental fluctuations. Transmission through thick barriers therefore emerges as a rare cumulative process requiring repeated successful geometric alignments across many finite integration intervals, naturally producing exponential suppression with barrier thickness. To formalize this mechanism, the tunneling process is represented as a coupled stochastic escape problem governed by bipartite Langevin dynamics and an associated Fokker–Planck description. The transition probability acquires a bilinear structure arising from simultaneous compatibility conditions between particle-side and environment-side fluctuations. In the self-consistent limit, this naturally produces the quadratic probability weighting characteristic of the Born measure, while remaining compatible with normalized Hilbert-space amplitudes. Within this framework, the Euclidean-time contribution appearing in standard tunneling theory admits a local kinematic interpretation as the extended stochastic integration interval required for compatible transition geometries to accumulate across the barrier. The formalism further predicts enhanced sensitivity of tunneling dynamics to high-frequency external probing, potentially relevant to attoclock tunneling-delay experiments and strong-field manifestations of the Quantum Zeno effect.
We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions---which occur in the large size limit---through the metastable behavior of many-body systems of moderate sizes accessible to numerics.