Superfluorescence of a large inverted sphere beyond the linear initiation regime: nonlinear coupling of vector multipole modes, delay-time statistics, and the deferred approximations of the Prasad–Glauber theory

When a large sphere of two-level atoms (radius $R$ with $k_0R \gg 1$, where $k_0$ is the resonant wave number) is fully inverted, its cooperative spontaneous emission — superfluorescence — must be initiated by quantum fluctuations. Prasad and Glauber (1985) solved this initiation problem in the linear regime, in which the population inversion is treated as undepleted: expanding the field in vector spherical harmonics, they showed that only multipole modes of order $l \lesssim k_0R$ amplify, that the initiating polarization fluctuations have Gaussian statistics, and they estimated the mean delay time. They explicitly deferred the rest of the problem: the nonlinear regime in which depletion couples all radiation modes; the fluctuations of the delay time; the effect of finite retardation ($2R/c$); the generalization to non-spherical shapes; and observables requiring vacuum-field (non-normally-ordered) correlations. The later literature supplies partial analogues in other frameworks — delay-time statistics for the sphere in scalar, geometrical-optics treatments (Mostowski and Sobolewska), general three-dimensional delay statistics by mode counting (Kocharovsky and Kocharovsky), and first-principles stochastic nonlinear Maxwell–Bloch modeling in paraxial pencil geometry (Chuchurka et al.) — but the nonlinear regime of the vector-multipole sphere problem and the remaining deferrals are untreated. The problem is to construct and validate that completion, with reduction to the known linear-regime results as a mandatory consistency anchor.

Background

Superfluorescence (SF) is the cooperative form of spontaneous emission from a sample of atoms prepared in a fully inverted state (every atom excited) with no pre-established coherence. Dicke (1954) showed that identical two-level atoms coupled to a common radiation field can decay collectively at a rate growing with the square of their number; the optical realization, in which a macroscopic polarization builds up spontaneously out of quantum fluctuations and then dumps the inversion in a short intense pulse, was observed by Skribanowitz, Herman, MacGillivray, and Feld (1973). The pulse does not start immediately: quantum fluctuations of the atomic polarization and of the electromagnetic vacuum seed a field that is amplified by the inverted medium, and after a stochastic delay time a pulse emerges. The delay time, its statistics, and the pulse shape are the central observables of SF. Early mean-field theories of extended samples (Ernst and Stehle 1968; Rehler and Eberly 1971) treated the nonlinear emission but contained no quantum initiation mechanism, so they could not predict delay times. The quantum theory of the initiation stage in the one-dimensional (pencil-shaped, effectively single-transverse-mode) geometry was developed by Polder, Schuurmans, and Vrehen (1979), Haake, King, Schröder, Haus, and Glauber (1979), and Haake, Haus, King, Schröder, and Glauber (1980, 1981); the fluctuating-source Maxwell–Bloch description and the tipping-angle picture (the effective initial polarization angle that quantum noise prepares) were confirmed experimentally by Vrehen and Schuurmans (1979), and the macroscopic quantum fluctuations were summarized early by Vrehen, Schuurmans, and Polder (1980).

The dynamical framework is the Maxwell–Bloch system: the coupled semiclassical equations for the resonant field envelope, the atomic polarization, and the population inversion. In the linear initiation regime the inversion is treated as undepleted (the number of atoms that have emitted, $N_{\mathrm{em}}(t) \ll N/2$), so the field–polarization equations are linear and driven by quantum noise; this stage is also called amplified spontaneous emission (ASE). Prasad and Glauber (1985) treated this initiation problem for the geometry that is intrinsically three-dimensional: $N$ two-level atoms uniformly distributed in a sphere of radius $R$ with $k_0R \gg 1$, a sample whose linear dimension greatly exceeds the wavelength. They expanded the field and polarization in vector spherical harmonics — multipole modes of electric and magnetic type and order $(l, m)$ — and solved the linearized equations. Their main results: (i) only 'amplifying modes' with multipole order $l \lesssim k_0R$ undergo significant gain; (ii) the multipole-coefficient operators of the initial atomic polarization (their operators $c$ and $d$) have Gaussian statistics, being sums of many independent single-atom contributions, and the two opposite-circular-polarization components are mutually uncorrelated; (iii) the average delay time $\langle t_D\rangle$, estimated as the time at which $N/2$ photons have been emitted (their Eq. (80)), has — because the gain parameter is proportional to $N$ — the characteristic one-dimensional form $(1/N)\ln N$; (iv) only a fraction of order $1/(k_0R)$ of the atoms participate in the cooperative radiation in a single pulse, the emission in any single realization being confined close to one dominant radial direction, with far-field intensity correlations restricted to the diffraction-limited angular range of order $\lambda_0/R$ ($\lambda_0 = 2\pi/k_0$ being the resonant wavelength).

The same paper explicitly defers five things. First, the nonlinear regime: the analysis 'is, of course, confined to the linear regime'; one effect of the nonlinearity is 'to couple all of the radiation field modes to one another,' which 'may be expected to alter significantly the result for later times.' Second, the fluctuations of the delay time about its average, which 'can be calculated in the present model by following the procedure of Ref. 13' (Haake et al. 1980) — a procedure indicated but not carried out. Third, retardation: the propagation kernels are simplified by neglecting the finite transit time $2R/c$ of light across the sphere, justified only when the time scale of variation of the kernels greatly exceeds $2R/c$. Fourth, the generalization to non-spherical samples, offered as the conjecture that 'many of the results may be carried over from the spherical case simply by substituting $L$ for $2R$,' where $L$ is the maximum linear dimension. Fifth, vacuum-field contributions: the theory computes normally ordered observables (those measured by photon absorption), and the vacuum field operators are dropped, so any observable sensitive to vacuum fluctuations — a non-normally-ordered field correlation function — lies outside the treatment.

The later literature bears on pieces of this but leaves the core open. Mostowski and Sobolewska (1983) developed a three-dimensional quantum theory of SF initiation from a sphere using the scalar wave equation and a Debye (ray) expansion, finding the same initiation physics as the one-dimensional theory with the sample length replaced by the sphere diameter; with the same machinery they derived delay-time statistics for large Fresnel numbers (the Fresnel number $F = S/(\lambda_0 B)$, with $S$ the cross-sectional area and $B$ the sample length, counts the transverse diffraction modes of an extended sample) and a three-dimensional initiation theory for pencil-like samples (1984). Kocharovsky and Kocharovsky (1985) obtained delay-time statistics for general three-dimensional samples by counting the amplifying 'hot' modes (of order $(k_0R)^2$ for a sphere), as presented in detail in the review of Kocharovsky, Zheleznyakov, Kocharovskaya, and Kocharovsky (2017). These supply delay-time distributions for the sphere, but in scalar/geometrical-optics or mode-counting frameworks, not within the vector-multipole Gaussian model, and the correspondence between the frameworks for the sphere is not established in the cited literature. On the nonlinear side, Watson, Gibbs, Mattar, Cormier, Claude, McCall, and Feld (1983) simulated SF in three dimensions including transverse effects and quantum fluctuations, and Chuchurka, Benediktovitch, Krušič, Halavanau, and Rohringer (2024) derived first-principles stochastic three-dimensional Maxwell–Bloch equations with fluctuation–dissipation-consistent noise sources covering spontaneous emission, ASE, SF, and saturation — but both treatments are for paraxial (narrow-angle, pencil-like) samples, not for the non-paraxial sphere with its competing vector multipole modes. Shape effects in the linear regime were analyzed in adjacent settings (Hazak 1990 for a refractively guided cylinder; Friedberg and Manassah 2009, 2010 and Manassah 2012 for cooperative decay rates and frequency shifts in spherical and other geometries) without testing the $L \leftrightarrow 2R$ conjecture. No later treatment of the retardation corrections or of non-normally-ordered observables for the sphere exists in the cited literature, and the authors' own later 'polarium' paper (Prasad and Glauber 2000) addresses a different problem (coherent radiation by a resonant medium in the near-cooperative regime).

Problem Statement

Consider $N$ identical two-level atoms with transition wave number $k_0$ (resonant wavelength $\lambda_0 = 2\pi/k_0$), uniformly and randomly distributed in a sphere of radius $R$ with $k_0R \gg 1$, all prepared in the excited state at $t = 0$, radiating into free space with no cavity, no external seed field, and no dephasing or inhomogeneous broadening (the idealized setting of Prasad and Glauber 1985). Construct the theory of the complete superfluorescent emission of this system — from quantum initiation through the nonlinear saturation regime — and resolve the approximations that the linear theory explicitly left open. Concretely, the completion consists of the following five open points, each an independent part of the research target:

(1) Nonlinear regime. The dynamics of the field, polarization, and inversion past the linear stage, in which depletion of the inversion couples the amplifying multipole modes to one another and to the remaining modes, terminating in the saturated pulse(s): the total emitted energy and its temporal profile, the number and ordering by growth rate of the participating modes, the far-field angular and polarization distributions, and the intensity correlations across the diffraction-limited angular range of order $\lambda_0/R$.

(2) Delay-time statistics. The probability distribution of the delay time $t_D$ (the time at which $N/2$ photons have been emitted), including its dependence on $N$ and $k_0R$.

(3) Retardation. A quantitative delimitation of the approximation that neglects the light-transit time $2R/c$ across the sphere: the condition, in terms of the physical parameters, under which the instantaneous-kernel treatment is valid, and the leading corrections when it is not.

(4) Shape generalization. A determination of for which non-spherical sample shapes the results carry over by replacing the sphere diameter $2R$ with the maximum linear dimension $L$ (the Prasad–Glauber conjecture), and for which shapes they do not, together with the mode structure that replaces the vector spherical harmonics.

(5) Vacuum-sensitive observables. The extension to observables that are not normally ordered — correlation functions to which the vacuum field contributes — at least in the linear regime.

Consistency requirement (mandatory): any solution, analytic or numerical, must reduce in the early-time linear regime to the results of Prasad and Glauber (1985): the amplifying-mode spectrum (multipole orders $l \lesssim k_0R$), the Gaussian statistics and mutual uncorrelation of the two circular-polarization components of the initiating fluctuations, and the mean delay time of their Eq. (80) with its characteristic $(1/N)\ln N$ dependence on atom number.

References

1. S. Prasad and R. J. Glauber, Initiation of superfluorescence in a large sphere, Physical Review A 31, 1583–1597 (1985). https://doi.org/10.1103/PhysRevA.31.1583
2. S. Prasad and R. J. Glauber, Polarium model: Coherent radiation by a resonant medium, Physical Review A 61, 063814 (2000). https://doi.org/10.1103/PhysRevA.61.063814
3. J. Mostowski and B. Sobolewska, Initiation of superfluorescence from a sphere, Physical Review A 28, 2943–2952 (1983). https://doi.org/10.1103/PhysRevA.28.2943
4. J. Mostowski and B. Sobolewska, Delay-time statistics in superfluorescence for large Fresnel numbers, Physical Review A 28, 2573–2575 (1983). https://doi.org/10.1103/PhysRevA.28.2573
5. J. Mostowski and B. Sobolewska, Three-dimensional theory of initiation of superfluorescence, Physical Review A 30, 1392–1400 (1984). https://doi.org/10.1103/PhysRevA.30.1392
6. V. V. Kocharovsky and Vl. V. Kocharovsky, Superradiance statistics for three-dimensional samples, Optics Communications 53, 345–348 (1985). https://doi.org/10.1016/0030-4018(85)90275-5
7. V. V. Kocharovsky, V. V. Zheleznyakov, E. R. Kocharovskaya, and Vl. V. Kocharovsky, Superradiance: the principles of generation and implementation in lasers, Physics–Uspekhi 60, 345–384 (2017). https://doi.org/10.3367/ufne.2017.03.038098
8. S. Chuchurka, A. Benediktovitch, Š. Krušič, A. Halavanau, and N. Rohringer, Stochastic modeling of x-ray superfluorescence, Physical Review A 109, 033725 (2024); preprint arXiv:2303.00853. https://doi.org/10.1103/PhysRevA.109.033725
9. G. Hazak, Amplification of spontaneous emission and coherence in three-dimensional systems, Physical Review A 42, 4169–4176 (1990). https://doi.org/10.1103/PhysRevA.42.4169
10. F. Haake, H. King, G. Schröder, J. Haus, and R. Glauber, Fluctuations in superfluorescence, Physical Review A 20, 2047–2063 (1979). https://doi.org/10.1103/PhysRevA.20.2047
11. F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, Delay-time statistics and inhomogeneous line broadening in superfluorescence, Physical Review Letters 45, 558–561 (1980). https://doi.org/10.1103/PhysRevLett.45.558
12. F. Haake, J. W. Haus, H. King, G. Schröder, and R. Glauber, Delay-time statistics of superfluorescent pulses, Physical Review A 23, 1322–1333 (1981). https://doi.org/10.1103/PhysRevA.23.1322
13. D. Polder, M. F. H. Schuurmans, and Q. H. F. Vrehen, Superfluorescence: Quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source, Physical Review A 19, 1192–1203 (1979). https://doi.org/10.1103/PhysRevA.19.1192
14. Q. H. F. Vrehen and M. F. H. Schuurmans, Direct measurement of the effective initial tipping angle in superfluorescence, Physical Review Letters 42, 224–227 (1979). https://doi.org/10.1103/PhysRevLett.42.224
15. Q. H. F. Vrehen, M. F. H. Schuurmans, and D. Polder, Superfluorescence: macroscopic quantum fluctuations in the time domain, Nature 285, 70–71 (1980). https://doi.org/10.1038/285070a0
16. E. A. Watson, H. M. Gibbs, F. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, Quantum fluctuations and transverse effects in superfluorescence, Physical Review A 27, 1427–1434 (1983). https://doi.org/10.1103/PhysRevA.27.1427
17. R. Friedberg and J. T. Manassah, Eigenmode expansion of the polarization for a spherical sample of two-level atoms, Physics Letters A 373, 4416–4425 (2009). https://doi.org/10.1016/j.physleta.2009.10.009
18. R. Friedberg and J. T. Manassah, Analytic expressions for the initial Cooperative Decay Rate and Cooperative Lamb Shift for a spherical sample of two-level atoms, Physics Letters A 374, 1648–1659 (2010). https://doi.org/10.1016/j.physleta.2010.02.012
19. J. T. Manassah, Cooperative radiation from atoms in different geometries: decay rate and frequency shift, Advances in Optics and Photonics 4, 108 (2012). https://doi.org/10.1364/AOP.4.000108
20. R. H. Dicke, Coherence in spontaneous radiation processes, Physical Review 93, 99–110 (1954). https://doi.org/10.1103/PhysRev.93.99
21. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, Observation of Dicke superradiance in optically pumped HF gas, Physical Review Letters 30, 309–312 (1973). https://doi.org/10.1103/PhysRevLett.30.309
22. V. Ernst and P. Stehle, Emission of radiation from a system of many excited atoms, Physical Review 176, 1456–1479 (1968). https://doi.org/10.1103/PhysRev.176.1456
23. N. E. Rehler and J. H. Eberly, Superradiance, Physical Review A 3, 1735–1751 (1971). https://doi.org/10.1103/PhysRevA.3.1735