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arXiv:2105.06786 · Physical Review A

Beyond lowest order mean-field theory for light interacting with atom arrays

F. Robicheaux, Deepak A. Suresh

Published Aug 4, 2021• 47 citations
arXivHTMLPDF

Abstract

Results from higher order mean-field calculations of light interacting with atom arrays are presented for calculations of one- and two-time expectation values. The atoms are approximated as two levels and are fixed in space. Calculations were performed for mean-field approximations that include the expectation value of one operator (mean field), two operators (mean field 2), and three operators (mean field 3). For the one-time expectation values, we examined three different situations to understand the convergence with increasing order of mean field and some limitations of higher order mean-field approximations. As a representation of a two-time expectation value, we calculated the ${g}^{(2)}(\ensuremath{\tau})$ for a line of atoms illuminated by a perpendicular plane wave at several emission angles and two different intensities. For many cases, the mean field 2 will be sufficiently accurate to quantitatively predict the response of the atoms as measured by one-time expectation values. However, the mean-field-3 approximation will often be needed for two-time expectation values.

DOI
10.1103/physreva.104.023702

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Results from higher order mean-field calculations of light interacting with atom arrays are presented for calculations of one- and two-time expectation values. The atoms are approximated as two levels and are fixed in space. Calculations were performed for mean-field approximations that include the expectation value of one operator (mean field), two operators (mean field 2), and three operators (mean field 3). For the one-time expectation values, we examined three different situations to understand the convergence with increasing order of mean field and some limitations of higher order mean-field approximations. As a representation of a two-time expectation value, we calculated the g(2)(\ensuremathτ){g}^{(2)}(\ensuremath{\tau})g(2)(\ensuremathτ) for a line of atoms illuminated by a perpendicular plane wave at several emission angles and two different intensities. For many cases, the mean field 2 will be sufficiently accurate to quantitatively predict the response of the atoms as measured by one-time expectation values. However, the mean-field-3 approximation will often be needed for two-time expectation values.
We theoretically and computationally investigate the behavior of infinite atom arrays when illuminated by nearly resonant light. We use higher-order mean-field equations to investigate the coherent reflection and transmission and incoherent scattering of photons from a single array and from a pair of arrays as a function of detuning for different values of the Rabi frequency. For the single array case, we show how increasing the light intensity changes the probabilities for these different processes. For example, the incoherent scattering probability initially increases with light intensity before decreasing at higher values. For a pair of parallel arrays at near resonant separation, the effects from increasing light intensity can become apparent with incredibly low-intensity light. In addition, we derive the higher-order mean-field equations for these infinite arrays, giving a representation that can be evaluated with a finite number of equations.
The second-order intensity correlation function for the light transmitted from a driven optical cavity containing identical two-level atoms is calculated in the weak-field limit by using the equations for expectation values of operators. Atomic nonradiative dephasing is taken into account in addition to cavity-field damping and atomic radiative decay. The correlation function including the atomic nonradiative dephasing rate is numerically calculated. An analytic expression for the correlation function is also derived in the case of purely radiative atomic decay.
The configuration space, i.e. the Hilbert space, of compound quantum systems grows exponentially with the number of its subsystems: its dimensionality is given by the product of the dimensions of its constituents. Therefore a full quantum treatment is rarely possible analytically and can be carried out numerically for fairly small systems only. Fortunately, in order to obtain interesting physics, approximations often very well suffice. One of these approximations is given by the cumulants expansion, where expectation values of products of operators are approximated by products of expectation values of said operators, neglecting higher-order correlations. The lowest order of this approximation is widely known as the mean field approximation and used routinely throughout quantum physics. Despite its ubiquitous presence, a general criterion for applicability and convergence properties of higher order cumulants expansions remains to be found. In this paper, we discuss two problems in quantum electrodynamics and quantum information, namely the collective radiative dissipation of a dipole-dipole interacting chain of atoms and the factorization of a bi-prime by annealing in an adiabatic quantum simulator. In the first case we find smooth, convergence behavior, where the approximation performs increasingly better with higher orders, while in the latter going beyond mean field turns out useless and, even for small system sizes, we are puzzled by numerically challenging and partly non-physical solutions.
Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially. In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom. Dynamical variables for a two-level atom We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.
We apply the dynamical mean field theory (DMFT) approximation to the real, scalar ${\ensuremath{\varphi}}^{4}$ quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$. We also map out the phase diagram in the most relevant case of four dimensions. In two and three dimensions, DMFT incorrectly predicts a first-order phase transition for all bare quartic couplings, which is problematic, because the second-order nature of the phase transition of lattice ${\ensuremath{\varphi}}^{4}$ theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behavior away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first- to second-order transition in the local limit of DMFT which is computationally much cheaper. We also discuss technical issues related to the convergence of the nonlinear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a ${\ensuremath{\varphi}}^{4}$ interaction.