Nelson's stochastic quantum mechanics provides an ideal arena to test how the Born rule is established from an initial probability distribution that is not identical to the square modulus of the wave function. Here, we investigate numerically this problem for three relevant cases: a double-slit interference setup, a harmonic oscillator, and a quantum particle in a uniform gravitational field. For all cases, Nelson's stochastic trajectories are initially localized at a definite position, thereby violating the Born rule. For the double slit and harmonic oscillator, typical quantum phenomena, such as interferences, always occur well after the establishment of the Born rule. In contrast, for the case of quantum particles free-falling in the gravity field of the Earth, an interference pattern is observed \emph{before} the completion of the quantum relaxation. This finding may pave the way to experiments able to discriminate standard quantum mechanics, where the Born rule is always satisfied, from Nelson's theory, for which an early subquantum dynamics may be present before full quantum relaxation has occurred. Although the mechanism through which a quantum particle might violate the Born rule remains unknown to date, we speculate that this may occur during fundamental processes, such as beta decay or particle-antiparticle pair production.
Recently, the properties of bouncing oil droplets, also known as "walkers", have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They constitute indeed a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to develop a phenomenological theory for the behavior of walkers in terms of de Broglie-Bohm and Nelson dynamics. We study in particular how modifications of the de Broglie pilot-wave theory, à la Nelson, affect the …
We propose a generalization of quantum mechanical equations in the hydrodynamic form by introducing, into the Lagrangian density, terms taking into account the diffusion velocity at zero and finite temperatures and the diffusion pressure energy of the warm vacuum. Based on this, for the model of one-dimensional hydrodynamics, we construct a system of equations that are analogous to the Euler equations but with the inclusion of quantum and thermal effects. They are a generalization of equations of the Nelson stochastic mechanics. The numerical analysis of the behavior of solutions of this syste…
We illustrate through explicit numerical calculations how the Born-rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie-Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie-Bohm ontology, the box contains an objectively-existing 'pilot wave' which guides the electron trajectory, and this is represented mathematically by a Schroedinger wave func…
In this work we derive Born's rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to a environement made of "qubits" (i.e., Bohmian pointers) we show that entanglement together with deterministic chaos lead to a fast relaxation from any statistitical distribution $ρ(x)$ (of finding a particle at point $x$) to the Born probability law $|Ψ(x)|^2$. Our model is discussed in the context of Boltzmann's kinetic theory and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.
We generalize the standard Born-Oppenheimer approximation to the case of open quantum systems. We define the zeroth order Born-Oppenheimer approximation of an open quantum system as the regime in which its effective Hamiltonian can be diagonalized with fixed slowly changing variables. We then establish validity and invalidity conditions for this approximation for two kinds of dissipations--the spin relaxation and the dissipation of center-of-mass motion. As an example, the Born-Oppenheimer approximation of a two-level open system is analyzed.
Typicality has always been in the minds of the founding fathers of probability theory when probabilistic reasoning is applied to the real world. However, the role of typicality is not always appreciated. An example is the paper "Foundations of statistical mechanics and the status of Born's rule in de Broglie-Bohm pilot-wave theory" by Antony Valentini, where he presents typicality and relaxation to equilibrium as distinct approaches to the proof of Born's rule, while typicality is in fact an overriding necessity. Moreover the "typicality approach" to Born's rule of "the Bohmian mechanics schoo…
Consideration is given to the behaviour of de Broglie trajectories that are separated from the bulk of the Born distribution with a view to describing the quantum relaxation properties of more `extreme' forms of quantum nonequilibrium. For the 2-dimensional isotropic harmonic oscillator, through the construction of what is termed the `drift field', a description is given of a general mechanism that causes the relaxation of `extreme' quantum nonequilibrium. Quantum states are found which do not feature this mechanism, so that relaxation may be severely delayed or possibly may not take place at …
In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This procedure was based on a Koopman-von Neumann approach where classical equations are reformulated into a quantumlike form. In this article, we develop a different derivation of quantum equations, based on purely classical stochastic arguments, taking some elements from the Bohm-Fenyes-Nelson approach. This study starts from a remark noticed by different authors…
For sufficiently low reservoir temperatures, it is known that open quantum systems subject to decoherent interactions with the reservoir relax towards their ground state in the weak coupling limit. Within the framework of quantum master equations, this is formalized by the Born-Markov-secular (BMS) approximation, where one obtains the system Gibbs state with the reservoir temperature as a stationary state. When the solution to some problem is encoded in the (isolated) ground state of a system Hamiltonian, decoherence can therefore be exploited for computation. The computational complexity is t…
We have recently developed a new understanding of probability in quantum gravity. In this paper we provide an overview of this new approach and its implications. Adopting the de Broglie-Bohm pilot-wave formulation of quantum physics, we argue that there is no Born rule at the fundamental level of quantum gravity with a non-normalisable Wheeler-DeWitt wave functional $Ψ$. Instead the universe is in a perpetual state of quantum nonequilibrium with a probability density $P\neq\left\vert Ψ\right\vert ^{2}$. Dynamical relaxation to the Born rule can occur only after the early universe has emerged i…
We discuss the main mechanisms generating chaotic behavior of the quantum trajectories in the de Broglie - Bohm picture of quantum mechanics, in systems of two and three degrees of freedom. In the 2D case, chaos is generated via multiple scatterings of the trajectories with one or more `nodal point - X-point complexes'. In the 3D case, these complexes form foliations along `nodal lines' accompanied by `X-lines'. We also identify cases of integrable or partially integrable quantum trajectories. The role of chaos is important in interpreting the dynamical origin of the `quantum relaxation' effec…
We show that cosmological quantum relaxation predicts an anisotropic primordial power spectrum with a specific dependence on wavenumber k. We explore some of the consequences for precision measurements of the cosmic microwave background (CMB). Quantum relaxation is a feature of the de Broglie-Bohm pilot-wave formulation of quantum theory, which allows the existence of more general physical states that violate the Born probability rule. Recent work has shown that relaxation to the Born rule is suppressed for long-wavelength field modes on expanding space, resulting in a large-scale power defici…
Dynamics of two interacting dipolar two-level systems in a multi-mode electromagnetic cavity: sudden death and revival of the entanglement within the Born-Markov approximation
École Normale Supérieure Paris-Saclay（巴黎萨克雷高等师范学院）；The University of Tokyo（东京大学）
In the context of the de Broglie-Bohm pilot wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium - thus violating the Born rule - usually relax over time to the expected $|ψ|^2$ distribution on a coarse-grained level. We analyze the relaxation of nonequilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing in the influence of different parameters such as the number of modes, the coarse-graining length…
Numerical simulations indicate that the Born rule does not need to be postulated in the de Broglie-Bohm pilot-wave theory, but arises dynamically (relaxation to quantum equilibrium). These simulations were done for a particle in a two-dimensional box whose wave-function obeys the non-relativistic Schroedinger equation and is therefore scalar. The chaotic nature of the de Broglie-Bohm trajectories, thanks to the nodes of the wave-function which act as vortices, is crucial for a fast relaxation to quantum equilibrium. For spinors, we typically do not expect any node. However, in the case of the …
In the context of de Broglie-Bohm pilot-wave theory, violations of the Born rule are allowed and can be considered as describing nonequilibrium distributions. We study the effects of interactions on quantum relaxation towards equilibrium for a system of one-dimensional coupled harmonic oscillators. We show by numerical simulations that interactions can delay or even prevent complete relaxation for some initial states. We also discuss how this effect might be relevant for cosmological scenarios and how nonequilibrium could be detected in some models.
We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave fun…
The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the \emph{exact} solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the \emph{exact} c…
The de Broglie-Bohm pilot-wave formulation of quantum theory allows the existence of physical states that violate the Born probability rule. Recent work has shown that in pilot-wave field theory on expanding space relaxation to the Born rule is suppressed for long-wavelength field modes, resulting in a large-scale power deficit ξ(k) which for a radiation-dominated expansion is found to have an approximate inverse-tangent dependence on k (assuming that the width of the initial distribution is smaller than the width of the initial Born-rule distribution and that the initial quantum states are ev…
The spreading of a particle along a chain, and its relaxation, are central themes in statistical and quantum mechanics. One wonders what are the consequences of the interplay between coherent and stochastic transitions. This fundamental puzzle has not been addressed in the literature, though closely related themes were in the focus of the Physics literature throughout the last century, highlighting quantum versions of Brownian motion. Most recently this question has surfaced again in the context of photo-synthesis. Here we consider both an infinite tight-binding chain and a finite ring within …
We study the intrinsic optical bistability displayed by a small sample of $V$-type three-level atoms induced by the near dipole-dipole interaction. The use of the coherent state properties in the limit of the generalized second-order Born approximation for BBGKY-hierarchy of equations for the reduced density operators allows one to derive the operator describing the near dipole-dipole interaction (local field correction) with interference terms involving no additional assumptions. The dynamics of populations of the excited states and the total spontaneous intensity are analysed as functions of…
We argue that in quantum gravity there is no Born rule. The quantum-gravity regime, described by a non-normalisable Wheeler-DeWitt wave functional $Ψ$, must be in quantum nonequilibrium with a probability distribution $P\neq\left\vert Ψ\right\vert ^{2}$ (initially and always). A Born rule can emerge only in the semiclassical regime of quantum systems on a classical spacetime background, with normalisable Schrödinger wave functions $ψ$. Conditioning on the underlying quantum-gravitational ensemble yields a nonequilibrium distribution $ρ\neq\left\vert ψ\right\vert ^{2}$ at the beginning of the s…
We show using a realistic Hamiltonian-type model that definite outcomes of quantum measurements may emerge from quantum evolution of pure states, i.e quantum dynamics provides a deterministic collapse of the wavefunction in a quantum measurement process. The relaxation of the wavefunction into a pointer state with classical properties is driven by the interaction with an environment. The destruction of superpositions, i.e. choosing a preferred attraction basin and thereby a preferred pointer state, is caused by a tiny nonlinearity in the macroscopic measurement apparatus. In more details, we n…
We consider Bohm's second-order dynamics for arbitrary initial conditions in phase space. In principle Bohm's dynamics allows for 'extended' nonequilibrium, with initial momenta not equal to the gradient of phase of the wave function (as well as initial positions whose distribution departs from the Born rule). We show that extended nonequilibrium does not relax in general and is in fact unstable. This is in sharp contrast with de Broglie's first-order dynamics, for which non-standard momenta are not allowed and which shows an efficient relaxation to the Born rule for positions. On this basis w…
We study incomplete relaxation to quantum equilibrium at long wavelengths, during a pre-inflationary phase, as a possible explanation for the reported large-scale anomalies in the cosmic microwave background (CMB). Our scenario makes use of the de Broglie-Bohm pilot-wave formulation of quantum theory, in which the Born probability rule has a dynamical origin. The large-scale power deficit could arise from incomplete relaxation for the amplitudes of the primordial perturbations. We show, by numerical simulations for a spectator scalar field, that if the pre-inflationary era is radiation dominat…
Branching stochastic mechanics: Clustering and connected correlations within a branching-process representation of the Schrödinger equation
Université Paris-Saclay, CEA Institut de Recherche sur les Lois Fondamentales de l’Univers（巴黎萨克雷大学，法国原子能与替代能源委员会基础规律研究学院）；Université Paris-Saclay, Ecole Normale Supérieure Paris-Saclay（巴黎萨克雷大学，巴黎萨克雷高等师范学院）
Tensor cores, along with tensor processing units, represent a new form of hardware acceleration specifically designed for deep neural network calculations in artificial intelligence applications. Tensor cores provide extraordinary computational speed and energy efficiency, but with the caveat that they were designed for tensor contractions (matrix-matrix multiplications) using only low-precision floating point operations. In spite of this, we demonstrate how tensor cores can be applied with high efficiency to the challenging and numerically sensitive problem of quantum-based Born-Oppenheimer m…
We propose a universal quantum computing scheme in which the orthogonal qubit states $|0>$ and $|1>$ are identical in their single-particle spin and charge properties. Each qubit is contained in a single quantum dot and gate operations are induced all-electrically by changes in the confinement potential. Within the computational space, these qubits are robust against environmental influences that couple to the system through single-particle channels. Due to the identical spin and charge properties of the $|0>$, $|1>$ states, the lowest-order relaxation and decoherence rates $1/T_1$ and $1/T_2$…
