Light propagation beyond the mean-field theory of standard optics

With ready access to massive computer clusters we may now study light propagation in a dense cold atomic gas by means of basically exact numerical simulations. We report on a direct comparison between traditional optics, that is, electrodynamics of a polarizable medium, and numerical simulations in an elementary problem of light propagating through a slab of matter. The standard optics fails already at quite low atom densities, and the failure becomes dramatic when the average interatomic separation is reduced to around $k^{-1}$, where $k$ is the wave number of resonant light. The difference between the two solutions originates from correlations between the atoms induced by light-mediated dipole-dipole interactions.

We compare propagation of light in a gas of stationary atoms as predicted by the standard electrodynamics of a polarizable medium with an exact numerical solution. We find that the conventional optics fails even at low atom densities, and the failure becomes especially dramatic when the average interatomic separation is reduced to around λ/2π, where λ denotes the resonance wavelength of light. The difference between the two solutions provides an unambiguous signature of the strong correlations between the atoms induced by light-mediated interactions. Laser cooling and trapping together with ready access to massive computer clusters have created the interesting confluence that we now have both the motive and the opportunity to study light propagation in a dense cold gas by means of basically exact numerical simulations. In fact, the small gas samples available in the laboratories [1][2][3][4][5][6] have recently made it possible to simulate atom-light systems numerically [5,[7][8][9][10][11][12][13][14] approaching experimentally realistic conditions [5,15]. Here we report on a direct comparison between traditional optics [16,17] and numerical simulations in an elementary textbook problem of light propagation through a slab of matter. Figure 1 graphically demonstrates that the standard optics and the underlying electrodynamics of a polarizable medium (EDPM) fail qualitatively at high gas densities.
The usual EDPM [16,17] is a mean-field theory (MFT). Atomic ensembles, however, are composed essentially of pointlike dipolar emitters, and the resonant dipole-dipole interactions between the atoms may result in a strongly interacting system [15]. The usual expectation in condensed-matter physics is that a MFT becomes inaccurate when the correlations between the particles become stronger-and here this can be achieved by enhancing the interactions, say, by increasing the density of the sample. The emergence of light-induced correlations between pointlike atoms is the reason for the discrepancy between the traditional optics and our numerical results.
Our follow-up agenda is to set up a precise framework to think about cooperative effects [3,5,[7][8][9][10][11][12][13][14][18][19][20][21][22] in light propagation [23]. In common parlance cooperativity means that each emitter (henceforth, atom) has the radiation from the other atoms fall on it, which in turn alters the field that the radiator emits. Unfortunately, this broad definition would imply that even nearly all of EDPM is about cooperative effects. We will show how the eponymous cooperative Lamb shift [4,[24][25][26][27] arises from elementary optics, and note that in the recent experiments striving to demonstrate cooperative behavior in light scattered from a dense homogeneously broadened (cold) gas [1-3, 5, 6] the standard EDPM has not been ruled out as the cause of the observations.
FIG. 1: Light propagation through a slab of matter and strong light-induced correlations beyond the mean-field theory of traditional optics. Left: Schematic representation of the electric fields at the surfaces of a slab of thickness h and refractive index n. Right: Comparison of the optical response calculated according to standard optics and obtained from an exact numerical solution for stationary strongly correlated point dipoles in the limit of weak excitation. We show the optical thickness D as a function of detuning ∆. The results are for a sample with density ρ = k 3 and thickness h = k −1 . The truncation error in the numerical computations due to the finite area A = 1024 k −2 of the disk-shape sample, about 5%, is irrelevant for a qualitative comparison. The conspicuous difference between the two curves demonstrates the failure of traditional optics.
Accordingly, we propose that a meaningful notion of cooperativity should come with the condition that it reflects strongly correlated behavior of the radiators, i.e., the physics deviates from the usual EDPM. In three dimensions it is surprisingly difficult to solve Maxwell's equations for a continuous polarizable medium even numerically. Ironically, for small atomic samples such as those in Ref. [5] it is easier to simulate the entire strongly interacting theory. However, in our slab example the EDPM solution is on hand for comparison with numerical computations, and we may study the onset of cooperative phenomena quantitatively.
The dimensionless parameter governing cooperative behavior in cold atomic samples turns out to be the combination ρk −3 of the wave number of light k and the density of the sample ρ. In Fig. 1 the value is ρk −3 = 1, but we will demonstrate notable deviations in light scattering between EDPM and an exact solution already at surprisingly low densities and large detunings. Even if not recognized as such, cooperation of the emitters might already affect some of the present experiments with homogeneously broadened resonant emitter systems.
We consider light propagating perpendicularly through a slab of thickness h, see Fig. 1. The light is coming in from vacuum with the refractive index 1. We denote the refractive index of the medium (generally complex) by n, so the wave number inside is K = nk. The incoming light with the reference electric field amplitude at the entrance E 0 gets either reflected or transmitted at the entrance face, with the corresponding amplitude reflection and transmission coefficients being (n − 1)/(n + 1) and 2/(n + 1) [16,17]. Inside the medium we have two amplitudes, E r corresponding to the right-going wave with the propagation factor e iKz , and the left-going amplitude E l . By matching the incoming, reflected and transmitted waves inside the medium at the front face we have Similar matchings can be made at the exit face both inside and outside of the medium, and lead to the relation between the incoming and transmitted amplitudes To complete this exercise in optics we note that, according to the local-field corrections the effective electric field inside the sample is E e = E + P/(3ǫ 0 ) [16,17], the polarization is P = ǫ 0 χE, where χ is the susceptibility, and also P = ραE e , where ρ is the atom density and is the polarizability of an atom. Here ζ = D 2 / , D is the dipole moment matrix element, ∆ = ω − ω 0 denotes the detuning of the light frequency ω from the atomic resonance at ω 0 , and γ is the HWHM linewidth of the transition. We then have where shift of the resonance [28]. The power transmission T = |E T /E 0 | 2 and the optical thickness D = − ln T are then found easily. For comparisons with the standard optics, we solve the quantum field theory of light propagation [29] essentially exactly for stationary atoms in the weak excitation limit [7]. The method is based on classical electrodynamics simulations [7][8][9][10][11][12][13] and stochastic Monte-Carlo sampling of the positions of the atomic dipoles.
In the near-monochromatic case the theory is formulated for the positive-frequency components of electric field, dipole moment, and so on, oscillating at the laser frequency ω. Basically, given an atom at position r i in the electric field E(r), it develops a dipole moment d = αE(r i ), which in turn radiates the dipolar field E D (r) = G(r − r i )d. Here G(r − r ′ ) is the dipole field propagator such that G(r − r ′ )d is the electric field at r from a dipole d at r ′ [16]. Next take a collection of N atoms at positions r i . The field at the position of the i th atom is the sum of the incoming field E 0 and the dipolar fields from the other atoms, But this is a linear set of equations from which one may solve the fields at the positions of the atoms E(r i ).
The electromagnetic fields everywhere readily follow. The transmitted intensity is computed using a method adapted from of Ref. [10]. Finally, to model a gas, one needs to generate sets of random atomic positions compatible with the geometry of the sample, and average the results over these sets [7]. The numerical effort starts getting substantial by the time we have thousands of atoms, so our slabs are actually finite-size circular disks of area A and thickness h. For maximum compatibility with this geometry, we use circularly polarized incoming light. The natural unit of length for this problem is λ = k −1 . For a dense sample with ρ ≃ k 3 , and on resonance, the mean free path of light is a fraction of the wavelength, so our sample thicknesses are generally on the order of k −1 . The convergence to the infinite-slab limit with increasing area of the disk A is quite poor, and to verify it can be expensive in computer time. Here we only offer order-of-magnitude estimates of the truncation error due to the finite A based on a few spot checks. In all cases we discuss the statistical error due to the finite number of samples of atomic positions is at most comparable to, and occasionally orders of magnitude smaller than, the truncation error. Figure 1 dramatically illustrates the breakdown of traditional optics and the corresponding continuousmedium MFT for near-resonant light at the atom density ρ = k 3 . Next we wish to delineate the onset of light-induced correlation effects and the emergence of cooperative phenomenology beyond the MFT. Two limits are relevant where the light-mediated atom-atom interactions are relatively weak. First, the atom-light coupling strength may be reduced by tuning the laser away from resonance. Second, as the short-range dipole-dipole interaction falls off like ∼ 1/r 3 as a function of the interatomic separation r, the corresponding light-mediated interactions between the atoms decrease rapidly with decreasing atom density.
It is illustrative to consider first the far-off resonance case. Coherently scattered light off the atoms could be determined by measuring the reflection coefficient R or the transmission coefficient T = 1 − R = e −D . For the MFT of Eqs.
(2) and (4), an expansion in 1/∆ gives The factor in front is the absorption coefficient of independent atoms far-off resonance when the cross section for photon scattering ≃ 6πγ 2 /(k 2 ∆ 2 ) is asymptotically small. Maybe surprisingly, even in the MFT and far-off resonance, the sample does not have to behave like a collection of independent radiators. Here we see remnants of the etalon effect due to the reflections of light at the faces of the slab ∝ |∆ LL | ∝ ρ. To demonstrate beyond-MFT effects in light scattering one cannot compare with independent radiators, but must expressly exclude the optics of the sample as the cause of the observations. We are in a position to do so in our numerical experiments.
We fix the thickness of the slab arbitrarily at h = π/k. For various values of density ρ we find from the MFT exactly, numerically, the corresponding positive detuning ∆(ρ) for which the reflection coefficient is R M = 0.01. The slab should thus be optically thin. To corroborate cooperative light scattering, we compute the reflection coefficients R C from our beyond-MFT classical electrodynamics simulations for the same densities ρ and detunings ∆(ρ). We choose the numbers of the atoms, obviously integers, so that for a given density the area of the disk is as close to A = 2000 k −2 as possible, whereupon the truncation error in the reflection coefficient R C is at most a few per cent. In Fig. 2 we plot the ratio R C /R M , the cooperative enhancement of light scattering, for a number of sample densities. It exceeds two already for a sample as dilute as ρk −3 = 0.1, and is more than ten at ρk −3 = 1. While the log-lin plot emphasizes the onset of cooperativity, it obscures the fact that for these parameters the cooperative enhancement is nearly linear in density.
According to the prevalent view [5,6,30] coopera-tive effects should suppress light scattering. However, in the far-off resonant case we see enhanced scattering. Our example clearly shows that there are two distinct dimensionless density parameters relevant to light propagation, ρk −3 that governs cooperative phenomena, and on-resonance optical thickness, here ∝ ρhk −2 , that is a MFT concept. This distinction is not always made in the literature on cooperative effects. The preceding example illustrated how cooperative phenomena emerge in the weakly interacting limit that is achieved in a far-off resonance atomic ensemble, but light-mediated interactions can be weak even close to the atomic resonance if the density is sufficiently low. We now turn to low atom density to study the onset of beyond-MFT correlations.
We expand the optical depth derived from the analytical expressions (2) and (4) in density ρ, and choose the parameters K 0 and K 1 independent of ρ in the expression D = ρ/(K 0 + K 1 ρ) in such a way that to second order in ρ we have the same expansions. The quantity D will then present a usual resonance lineshape as a function of the detuning ∆, with the resonance shifted from ∆ = 0 by s: This shows the LL shift, plus etalon effects. The etalon contribution equals the "cooperative Lamb shift" [24], so the cooperative Lamb shift is not cooperative according to our definition. The formula (7) was recently tested experimentally, albeit in an inhomogeneously broadened (hot) gas [4]. For a comparison with numerical results we pick two densities ρ = 0.01 k 3 and ρ = 0.005 k 3 , compute the optical thickness as a function of detuning, and fit the results to a Lorentzian with a variable width and shift s. In Fig. 3 we show the shifts as a function of the sample thickness in units of the LL shift for each density. Similar results were mentioned in Ref. [15], although without giving details. Here the area of the disk is as close to 4,000 k −2 as possible, whereupon the truncation errors in the data points are at most on the order of one per cent. We also display the prediction from Eq. (7), omitting the leading LL shift that would shift the curve down by one unit.
For a fixed density and toward zero sample thickness, it is obviously the area density that counts, and it tends to zero. Hence the shift vanishes in the limit h → 0 [15]. Starting from about h = k −1 an oscillatory dependence on sample thickness is found, as predicted by Eq. (7). Nonetheless, there is an approximately constant difference between the numerical results and the prediction, which without the scaling by |∆ LL | would amount to a difference proportional to sample density [4]. This difference is, by our definition, a cooperative effect. In fact, at low sample densities one expects a shift of a resonance proportional to ρk −3 from dimensional analysis, but dimensional analysis alone does not provide the multiplicative constant. The LL shift amounts to a specific prediction for the constant. The usual local-field corrections and the ensuing Lorentz-Lorenz and Clausius-Mossotti formulas also depend on this particular value of the constant. Curiously, none of our simulations here or in Ref. [15] agree with the LL shift. The enduring success of the textbook local-field corrections remains a puzzle to us.
The EDPM is a MFT in that it treats the surroundings of each radiator in the average sense, as if the other atoms made a continuous polarizable medium. In the present case of discrete atoms the MFT may be implemented by factoring the two-point correlation function of polarization and ground-state atom density as the product of polarization and density, as in P 2 (r 1 ; r 2 ) = P(r 1 )ρ(r 2 ) [15]. This approximation amounts to neglecting position-dependent correlations between the atoms: In the presence of the dipole-dipole interactions the dipole moment of an atom obviously depends on exactly where the nearby atoms are. Our simulations treat all of such correlations exactly.
Nonetheless, the simulations are classical, and one should wonder if quantum mechanics makes a difference. We have shown that if photon recoil may be ignored and the light intensity is sufficiently low, for a J = 0 → J ′ = 1 transition the results from our classical-electrodynamics simulations agree with a full quantum analysis of the light-matter system [7,29]. Quantum mechanical calculations that allow only one photon at a time [3,22,30,31] come down to the same observation. An obvious corollary is that there is nothing inherently quantum mechanical about cooperative response; or, putting this the other way round, a system of pointlike classical radiators may perfectly well exhibit cooperative phenomena. This is why we may find a stark difference from the traditional classical optics even in classical simulations.
Subwavelength thickness inhomogeneously broadened (hot) atomic samples bounded by glass walls [4,32] and ellipsoid-shape small homogeneously broadened (cold) trapped atom samples [3- 6,30] have been used to a good effect in studies of light-matter interactions, but the combination of wall-less confinement and homogeneous broadening is yet to be achieved experimentally for a slab. At the moment our analysis constitutes a thought experiment, but we surmise that the lessons are generic.
The traditional electrodynamics of a polarizable medium is a MFT, and may fail dramatically for a cold atomic sample by the time the density reaches ρ ∼ k 3 . We propose a clarification in the nomenclature in that genuine cooperative response in light propagation should expressly deviate from EDPM. This point is not trivial; to establish cooperativity one would have to account for the optics of the medium. While studying quantitatively the onset of cooperative effects, we have found notable deviations from EDPM already at surprisingly low atom densities and for off-resonant ensembles. These could serve as unambiguous experimental signatures for the emergence of light-induced correlations beyond the MFT of traditional optics.
There is more to light propagation in a dense cold gas than we have learned from the venerable textbooks on electrodynamics. The new phenomenology is currently becoming accessible to dedicated experiments, and might even occur inadvertently, say, where cold and dense gases are used. Although we focus on low-temperature atomic vapors, the general features of cooperative phenomena we have discussed here should be relevant to other dense resonant emitter systems that are being realized in a growing number of experiments, such as with trapped ions [33], Rydberg atoms [34][35][36][37][38], and nanofabricated arrays of circuit elements [39][40][41].
We acknowledge support from NSF, Grant Nos. PHY-0967644 and PHY-1401151, and EPSRC. Most of the computations were done on Open Science Grid, VO Gluex, and on the University of Southampton Iridis 4 High Performance Computing facility.