PHYSICAL REVIEW A 91, 063826 (2015) Quantum spectra of Raman photon pairs from a mesoscopic particle C. H. Raymond Ooi, 1, 2 W. M. Edmund Loh, 1 and C. H. Kam 2 1 Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, 639798 Singapore (Received 13 February 2015; published 19 June 2015) Quantum Langevin formalism with noise operators is used to provide quantum descriptions of photon pairs (the Stokes and anti-Stokes fields) emitted by a mesoscopic spherical particle composed of quantum particles in a double Raman configuration. The spectra of the fields obtained are sensitive to the dimension of the microsphere and can be controlled by pump and control laser fields. Spectral peaks due to quantum coherence are Stark shifted by the laser fields experiencing autofocusing inside the spherical particle, causing broadening of peaks as the size of the microsphere increases. The antinormal-order spectrum is found to be identical to the normal-order spectrum. The anti-Stokes spectrum is identical to the Stokes spectrum when the linear dispersion is neglected. Frequency-dependent dielectric functions of the Stokes and anti-Stokes spectra corresponding to the linear dispersions of the particle yield narrow morphology-dependent resonance gain peaks at certain frequencies of the Stokes and anti-Stokes spectra that depend not only on the particle size but also on the angle of observation. DOI: 10.1103/PhysRevA.91.063826 PACS number(s): 42.50.Dv, 42.50.Gy, 42.50.Lc, 42.25.Fx I. INTRODUCTION Quantum properties of photon pairs in the double Raman scheme [1] have been studied in various systems such as single atoms [2], two atoms with dipole-dipole interaction [3], an array of atoms [4], a single-atom two-photon laser [5], and a one-dimensional amplifier [6]. These systems are relevant for generation of nonclassical photons in quantum information [7]. Recent progress in nanotechnology has stretched the applicability of quantum entanglement to nanophotonics [8]. The studies of quantum effects now extend to nanoparticles as well as microparticles [9]. Coherent Raman scattering of light by microparticles composed of atoms with quantum coherence has been studied using semiclassical theory [10] in the interest of enhancing backscattered signal. Also, optical bistability in a similar system has been investigated [11]. In particular, light scattered from spherical microparticles has prompted many theoretical studies that encompass areas such as stimulated emission processes [12], electronic Raman scattering [13], Raman coupling coefficients [14], and second-harmonic generation [15]. In this work we use quantum Langevin formalism [16] to describe the interactions of particles with pump and control laser fields inside a small spherical particle with arbitrary dimension (Fig. 1). This formalism correctly expresses the scattered Stokes and anti-Stokes electric fields as quantum operators in terms of the noise operators [17], which enables us to compute the quantum-mechanical expressions for field-field correlation functions in a transparent manner. In particular, we obtain the spectra [18] for both normal and antinormal-order expressions of the Stokes and anti-Stokes electric fields. The normal-order spectrum, being the Fourier transform of the first-order correlation function G (1) (r,t ), plays an essential role in the description of the experimentally observed quantities such as photoelectron statistics. The differences between the normal and antinormal-order correlations can be understood as follows. Normal-order correlation functions are utilized more frequently than the antinormal-order ones in the photodetection theory due to the ubiquity of the photon detection experiment based on the photoelectric effect [19]. However, there exists another possible photodetection method using the quantum counter introduced by Mandel [20], which can only be described by the antinormal-order correlation functions. Such a photon- counting device functions by stimulated emission rather than by absorption of photons and would be useful when the average number of photons is not too small. Thus, it is the creation operator instead of the annihilation operator that plays the central role. A comparison between the two distinct correlations is interesting in terms of the photodetected spectrum. II. QUANTUM LANGEVIN FORMALISM FOR COHERENCES The quantum Langevin formalism for quantum particles with the double Raman scheme in four levels a–d gives 16 coupled equations. If the Stokes ˆ E s and anti-Stokes fields ˆ E a are weak the populations and the coherences ˆ σ dc and ˆ σ ab can be approximated as complex numbers. This enables us to solve just the following four coupled equations We have the closed coupled equations for the slowly varying atomic envelope operators of the coherences ˆ p ac = ˆ σ ac e −iν a t , ˆ p ad = ˆ σ ad e −iν cs t ,ˆ p bc = ˆ σ bc e −iν ac t , and ˆ p bd = ˆ σ bd e iν s t , where ν ij = ν i − ν j and ν i (i ∈ p,s,c,a) are the carrier frequen- cies of the pump, Stokes, control, and anti-Stokes fields, respectively, which satisfy ν p + ν c = ν s + ν a for the closed (parametric four photons) transitions d dt ˆ p ac =−T ac ˆ p ac − i g ∗ a · ˜ E † a (ˆ p cc − ˆ p aa ) − i ( ∗ c ˆ p bc −  ∗ p ˆ p ad ) + e −iν a t ˆ F ac , (1) d dt ˆ p ad =−T ad ˆ p ad + i (ˆ p ab g s · ˜ E s − g ∗ a · ˜ E † a ˆ p cd ) + i ( p ˆ p ac −  ∗ c ˆ p bd ) + e −iν cs t ˆ F ad , (2) 1050-2947/2015/91(6)/063826(12) 063826-1 ©2015 American Physical Society