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Advances in Continuous and Discrete Models

On the statistical background of quantum mechanics: generalities and a concrete example

Martine Le Berre, Yves Pomeau A

Published May 19, 2023• 0 citations
Publisher

Abstract

Abstract We revisit our description of randomness in quantum processes that began in collaboration of Jean Ginibre. The calculations were performed on a worked example: the fluorescence of a single two-level atom pumped by a resonant laser field. This pump laser is described classically (by a function, not an operator). Our aim is first to built a Kolmogorov-type equation (K-equation) for the atomic state, so that the two parameters θ, φ that define this density matrix are random functions of time, therefore the atomic density matrix is a random density matrix. Such an approach, initiated for gas kinetics, was not yet applied to quantum phenomena, whereas it is especially tailored to very quick events well separated (in time) like the quantum jumps observed in spontaneous emission of photons by an atom. Here, we try to clarify the basis of our statistical approach leading to the K-equation below, and we present the main results deduced from it. We explain finally that our approach can be interpreted in terms of Everett’s theory of many-worlds, because at every emission a new history begins for the atom, with two nonoverlapping wave functions.

DOI
10.1186/s13662-023-03776-2

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Abstract We revisit our description of randomness in quantum processes that began in collaboration of Jean Ginibre. The calculations were performed on a worked example: the fluorescence of a single two-level atom pumped by a resonant laser field. This pump laser is described classically (by a function, not an operator). Our aim is first to built a Kolmogorov-type equation (K-equation) for the atomic state, so that the two parameters θ, φ that define this density matrix are random functions of time, therefore the atomic density matrix is a random density matrix. Such an approach, initiated for gas kinetics, was not yet applied to quantum phenomena, whereas it is especially tailored to very quick events well separated (in time) like the quantum jumps observed in spontaneous emission of photons by an atom. Here, we try to clarify the basis of our statistical approach leading to the K-equation below, and we present the main results deduced from it. We explain finally that our approach can be interpreted in terms of Everett’s theory of many-worlds, because at every emission a new history begins for the atom, with two nonoverlapping wave functions.
The randomness of some irreversible quantum phenomena is a central question because irreversible phenomena break quantum coherence and thus yield an irreversible loss of information. The case of quantum jumps observed in the fluorescence of a single two-level atom illuminated by a quasi-resonant laser beam is a worked example where statistical interpretations of quantum mechanics still meet some difficulties because the basic equations are fully deterministic and unitary. In such a problem with two different time scales, the atom makes coherent optical Rabi oscillations between the two states, interrupted by random emissions (quasi-instantaneous) of photons where coherence is lost. To describe this system, we already proposed a novel approach, which is completed here. It amounts to putting a probability on the density matrix of the atom and deducing a general "kinetic Kolmogorov-like" equation for the evolution of the probability. In the simple case considered here, the probability only depends on a single variable θ describing the state of the atom, and p(θ,t) yields the statistical properties of the atom under the joint effects of coherent pumping and random emission of photons. We emphasize that p(θ,t) allows the description of all possible histories of the atom, as in Everett's many-worlds interpretation of quantum mechanics. This yields solvable equations in the two-level atom case.
A major question in our understanding of the fabric of the world is where the randomness of some quantum phenomena comes from and how to represent it in a rational theory. The statistical interpretation of quantum mechanics made its way progressively since the early days of the theory. We summarize the main historical steps and then we outline how the randomness gains to be depicted by using tools adapted to Markov processes. We consider a model system corresponding to experimental situations, namely a single two-level atom submitted to a monochromatic light triggering transitions from the ground to the excited state. After a short summary of present quantum approaches, we explain how a general "kinetic-like" Kolmogorov equation yields the statistical properties of the fluorescent light radiated by the atom which makes at once Rabi oscillations between the two states, and random quantum jumps with photo-emission. As an exemple we give the probability distribution of the time intervals between two successive emitted photons by using the Kolmogorov equation.
Among theories of the physical world, quantum mechanics remains a topic of lively discussions on its so-called interpretation. For some it remains an open question to understand how deterministic equations of this theory, as established long ago, may combine with a fundamental uncertainty. We consider the process of spontaneous emission by an atom interacting with infinite number of degrees of freedom of the electromagnetic field. There is uncertainty in the evolution of the photo-emission process which was characterized as Markovian by using the equations of quantum mechanics when the decay of the atom is due to the coupling with the vacuum field. The Markovian property leads us naturally to describe spontaneous emission by using the classical Kolmogorov equation for the probability evolution of a parameter defining the state of the atom. We explain that Everett’s many-worlds interpretation weld together our description, and appears therefore as a consequence of the equations of quantum mechanics, that solves in this case the riddle of deterministic equations describing random events.
Abstract We discuss the statistics of emission of photons by a single atom or ion illuminated by a laser beam at the frequency of quasi-resonance between two energy levels, a situation that corresponds to real experiments. We extend this to the case of two laser beams resonant with the energy differences between two excited levels and the ground state (three level atom in V-configuration). We use a novel approach to this type of problem by considering a Kolmogorov equation for the probability distribution of the atomic state, which takes into account first the deterministic evolution of this state under the effect of the incoming laser beam and second the random emission of photons during the spontaneous decay of the excited state(s) to the ground state. This approach yields solvable equations in the two level atom case. For the three level atom case we set the problem and clearly define its frame. The results obtained are valid in both opposite limits of rare and frequent spontaneous decay, compared to the period of the optical Rabi oscillations due to the interaction between resonant excitation and atomic levels. Our analysis gives access to various statistical properties of the fluorescence light, including one showing that its fluctuations in time are not invariants under time reversal. This result makes evident the fundamentally irreversible character of quantum measurements, represented here by the emission of photons of fluorescence.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.