The message of Hiroshi Ezawa, John Klauder, and Shepp for Euclidean quantum dynamics was contrary to the prevalent method of imposing quantum dynamics on trajectories by killing them; more naturally, quantum dynamics should translate itself into specific dynamical evolutions of the paths. This chapter reviews some of the work that the observation has triggered. In the mid-seventies the prevalent path space approach to quantum dynamics was via Brownian paths that is the paths are killed with probability. Instead of this, Ezawa, Klauder, and Shepp proposed to deform the paths according to the dynamics with obeying a stochastic differential equation, the quantum dynamics is now encoded in the drift term that depends on the potential and more directly on the ground state of the dynamics. Ground states are always smoother than the corresponding potentials, so that the former may survive in limiting cases where the latter fail to exist.
The message of Hiroshi Ezawa, John Klauder, and Shepp for Euclidean quantum dynamics was contrary to the prevalent method of imposing quantum dynamics on trajectories by killing them; more naturally, quantum dynamics should translate itself into specific dynamical evolutions of the paths. This chapter reviews some of the work that the observation has triggered. In the mid-seventies the prevalent path space approach to quantum dynamics was via Brownian paths that is the paths are killed with probability. Instead of this, Ezawa, Klauder, and Shepp proposed to deform the paths according to the dynamics with obeying a stochastic differential equation, the quantum dynamics is now encoded in the drift term that depends on the potential and more directly on the ground state of the dynamics. Ground states are always smoother than the corresponding potentials, so that the former may survive in limiting cases where the latter fail to exist.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
This paper elucidates the dual structure of the Schrödinger dynamics in two correlated stages: (1) We first derive the real-valued Schrödinger equation from scratch without referring to classical mechanics, wave mechanics, nor optics, and thereby attain a concrete and clear interpretation of the Schrödinger (wave) function. Beginning with a factorization of the density distribution function of the particles to two component vectors in configuration space, we impose very simple conditions on them such as translational invariance of space-time and the conservation of flux under a given potential function. A real-valued path-integral is formulated as a Green function for the real-valued Schrödinger equation. (2) We then study a quantum stochastic path dynamics in a manner compatible with the Schrödinger equation. The relation between them is like the Langevin dynamics with the diffusion equation. Each quantum path describes a \textquotedblleft trajectory\textquotedblright\ in configuration space representing, for instance, a singly launched electron in the double-slit experiment that leaves a spot one by one at the measurement board, while accumulated spots give rise to the fringe pattern as predicted by the absolute square of the Schrödinger function. We start from the relationship between the Ito stochastic differential equation, the Feynman-Kac formula, and the associated parabolic partial differential equations, to one of which\ the Schrödinger equation is transformed. The physical significance of the quantum intrinsic stochasticity and the indirect correlation among the quantum paths and so on are discussed. The self-referential nonlinear interrelationship between the Schrödinger functions (regarded as a whole) and the quantum paths (as its parts) is identified as the ultimate mystery in quantum dynamics.
We analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.
Quantum uncertainty emerged from two strikingly different routes to quantum mechanics. Heisenberg's matrix mechanics replaced the classical description of atomic orbits by relations among observable transition quantities and led, in 1927, to the uncertainty relation and its influential measurement-based interpretation. Schrödinger's wave mechanics grew instead from the Bose--Einstein and de Broglie developments and restored a continuous space-time description through matter waves. The two formalisms were soon shown to be closely related, but neither historical route by itself settles the ontological question of whether particles may possess underlying trajectories.
After reviewing these developments, including Einstein's criticism of a simple observability doctrine and Mott's wave-mechanical account of straight cloud-chamber tracks, we distinguish precise quantum uncertainty relations from the stronger claim that quantum mechanics excludes definite microscopic histories. We then compare Feynman paths, Bohmian trajectories, Nelson's stochastic mechanics, and finite-speed persistent Goldstein--Kac dynamics. The Kac process has continuous, piecewise differentiable finite-speed paths; its diffusion limit approaches Wiener kinematics, while analytic continuation of its two-sector dynamics is related to Dirac evolution. These examples show that quantum uncertainty relations constrain observable statistics without, by themselves, uniquely determining an underlying path geometry.
In only 150 pages, not counting appendices, references, or the index, this book is one author's perspective of the massive theoretical and philosophical hurdles in the no-man's-land separating the classical and quantum domains of physics. It ends with him emphasizing his own theoretical contribution to this area. In his own words, he has attempted to answer: 1. 'How can we obtain the quantum dynamics of open systems initially described by the equations of motion of classical physics (quantization process)? 2. 'How can we retrieve classical dynamics from the quantum mechanical equations of motion by means of a classical limiting process (dequantization process)?' However, this monograph seems overly ambitious. Although the publisher's description refers to this book as 'an accessible entrée', we find that this author scrambles too hastily over the peaks of information that are contained in his large collection of 272 references. Introductory motivating discussions are lacking. Profound ideas are glossed over superficially and shoddily. Equations morph. But no new convincing understanding of the physical world results. The author takes the viewpoint that physical systems are always in interaction with their environment and are thus not isolated and, therefore, not Hamiltonian. This impels him to produce a method of quantization of these stochastic systems without the need of a Hamiltonian. He also has interest in obtaining the classical limit of the quantized results. However, this reviewer does not understand why one needs to consider open systems to understand 'quantum-classical correspondence'. The author demonstrates his method using various examples of the Smoluchowski form of the Fokker--Planck equation. He then renders these equations in a Wigner representation, uses what he terms 'an infinitesimality condition', and associates with a constant having the dimensions of an action. He thereby claims to develop master equations, such as the Caldeira--Leggett equation, without use of a Hamiltonian. However, his procedure is seriously flawed. For example, he considers a case of anomalous Brownian motion using a Kolmogorov stochastic equation, and two cases of Brownian motion, the first with a constant force and the second with a linear force. In each of these three cases, his alleged solutions for a function F(x,p,t), which he wishes to interpret as a probability density, are not in fact solutions of their respective evolution equations. The author, in order to interpret these as probability densities, appears to have obtained the correct solutions of these equations, but then has normalized each of them with a time-dependent function, so that their integrals over the whole phase space remains unity for all time. However, an integration of the original equations over phase space demonstrates that an integral of their solutions over the whole phase space must be time-dependent. Despite the poor editing of this monograph, in which errors in equations abound, and in which the dimensionality of quantities is occasionally incorrect, this reviewer's interest in this subject was not extinguished. We are thus thankful for its extensive bibliography.
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(R+)⊗(Cn⊕Cn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B(t),t≥0}, we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.