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arXiv:2609.17576

Non-homogeneous Partial Differential Wave Equation for the Description of Correlated Pair States in the Microworld

Isayeva Elmira, Alireza Khalili Golmankhaneh

Published Jul 15, 2026• 0 citations
arXivHTMLPDF

Abstract

In this paper, a new non-homogeneous linear second-order partial differential equation (PDE) is proposed and investigated, designed to describe the dynamics of correlated pair objects of the microworld (analogues of Cooper pairs or biphotons). The fundamental complex solution of the proposed equation initially contains a spatial background and a doubled phase, which allows for a natural localization of the pair condensate in space. In this work, the continuity and energy balance equations are derived, the discrete spectrum in potential fields is calculated, and specific effects of tunneling and nonlinear interaction of the pair conglomerate with electromagnetic radiation are described.

DOI
10.13140/rg.2.2.15209.86889

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We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, Φ\PhiΦ. For a scattering length with periodic microstructure of subscale ϵ\epsilonϵ, we formally describe via classical homogenization the lowest many-body bound state in terms of Φ\PhiΦ and K up to second order in ϵ\epsilonϵ. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.
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We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.
We provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-$\mathrm{\ensuremath{\Lambda}}$ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. We find that a complex-valued nonlinear coupling coefficient can lead to noises even without linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.