1 Introduction

Well-known textbooks have noted for many years that the group velocity v_g in anomalously dispersive media can exceed the speed of light c in vacuum. They assure us that there is no violation of special relativity because, when v_g >c, a pulse becomes so distorted in shape that the whole concept of group velocity breaks down. This argument is misleading in two important respects: (a) undistorted pulses with superluminal group velocities have been observed [1]–[4] and (b) group velocity does not in general represent velocity of information transmission, and so its value is irrelevant to special relativity.

The model considered here is conceptually attractive for several reasons: (1) It is completely quantum-mechanical; (2) the total electric field operator due to the source atom and all the atoms of the medium is obtained self-consistently and exactly; (3) it is shown that the radiation from the source atom can register a “click” at an ideal detector sooner, on average, than it could if there were no medium between the source atom and the detector, i.e., there is an observable superluminal effect; (4) we can prove a theorem analogous to the classical one that the frontal velocity of a sharp wave front cannot exceed c.

2 The Model

The source atom has transition frequency and electric dipole moment ω₀ and d, respectively, and is located at a point (x=0) outside the dielectric. There are N_T identical atoms making up the dielectric. They have transition frequency ω_d and transition dipole moment µ, respectively, and are located at points x_j. All the atoms are coupled to the quantized electromagnetic field in the electric dipole approximation. The Hamiltonian is

where the σ’s are two-state Pauli operators in the standard notation and the a _kλ are the field annihilation operators for plane-wave modes with wave vectors k (|k|=ω_k ) and polarization indices λ. The coupling constant C _kλ=µ(2πω_k /ħV)^1/2, where V is a quantization volume.

It will be convenient for our purposes to simplify the model by restricting the field modes to plane waves propagating along a single (z) axis and with a single polarization, so that k, λ→k. (In reality, of course, the field from the source atom will have a dipole radiation pattern. This can be dealt with easily enough, but it only complicates the results without affecting our conclusions.) Then, using the formal solution of the Heisenberg equation of motion for the field operators a_k (t), we obtain

for the electric field operator at any point z. Now the sum over field modes

where A is the cross-sectional area of our quantization volume V=AL, which we have allowed to become infinite by taking L→∞. Thus

for z>0, where θ(t) is the unit step function and E₀ (z,t) is the source-free electric field operator.

It is convenient to deal with Fourier-transformed operators defined by writing

Then equation (5) implies

where the atoms of the dielectric are assumed to be uniformly distributed with density N and to occupy the half-space z>z ₀.

From the Hamiltonian (1) it follows from the Heisenberg equations of motion and the commutation relations for the two-state operators that the σ_xj ’s satisfy

where in the last step we have replaced the operator σ_zj by -1 under the assumption that the atoms making up the dielectric remain with high probability in their ground states. 2β is the rate of spontaneous emission of the dielectric atoms, which in the present model undergo no other relaxation. Equation (10) implies

for the induced dipole moment at frequency ω, where α(ω)=(2µ ² ω_d /ħ)/[${\omega }_d^2$ -ω ²-2iβω] is the polarizability. Then

where we have written F(z, ω) as f(ω)e^{iωz /c}. To solve this integral equation we write

and determine g(ω) and n(ω) by substitution. The result of the algebra is

for z>z₀ , where n(ω) is the complex refractive index [n ²(ω)=1+4πNα(ω)].

Combining these results, we obtain the following expression for the electric field operator at (z,t) inside the medium:

where E_o (z,t) is the source-free (vacuum) field inside the medium and for simplicity we take z_o =0 for the position of the initially excited atom.

It may be worthwhile to note that, aside from the approximation that the host atoms remain with probability one in their ground states, the expression (15) is exact for the case of a dilute dielectric medium. We began with the Hamiltonian (1) for the field and the atoms in vacuum, and obtained (a) the relation between the refractive index and the polarizability, and (b) the electric field operator (15), by solving exactly the self-consistent integral equation (12).

Now the probability that the radiation from the source atom at z=0 will register a count at an ideal broadband detector at z>0 at time t involves an expectation value over the initial state |ψ〉 in which all the atoms of the dielectric are in their ground states, the field is in the vacuum state, and the state of the source atom is arbitrary. The photon counting rate is proportional to [5]

In terms of the two-state lowering and raising operators σ and σ ^†, respectively

Aside from much more slowly varying contributions, σ(t) and σ ^†(t) vary in time pre­dominantly as $e^{-i{\omega }_{o}t}$ and $e^{i{\omega }_{o}t}$, respectively. Therefore only the second term on the right-hand side of (17) will contribute to (16) over the time scales of interest (i.e., “energy-conserving” times, long compared with 1/ω_o ). The rate at which radiation emitted by the source atom will register counts at the detector at z is therefore proportional in this approximation to

3 Superluminal Photon Detection

Suppose the source atom is excited by a time-dependent mechanism, e.g., by a resonant laser pulse E(t)=ℜ[F_o (t) exp(-iω_ot)]. If the Rabi frequency is small compared with the radiative decay rate, and the pulse duration is long compared with the radiative lifetime, then σ ^†(t ^′)σ(t ^″) ∝ F_o (t ^′)F_o (t ^″)exp[iω_o (t ^′-t ^″)] and

where a(t) is the probability amplitude that the source atom is in the upper state at time t, and we have assumed that the source atom is initially in the lower state [σ(-∞)|ψ〉=0]. If F_o (t)=C exp(-t ²/2τ ²),

Except for the factor ω/[n(ω)+1], the integral here is the same as that studied in detail, in the classical context, by Garrett and McCumber [6]. [See their equation (8).] We will make a much cruder approximation than these authors in order to bring out the main feature we wish to emphasize: the detector can register a count sooner on average than it could if there were no dielectric medium between it and the source atom.

The main contribution to the integral over t ^′ in equation (19) comes from frequencies ω near ω_o . Write n(ω)=n_R (ω)+in_I (ω), where n_R (ω) and n_I (ω) are real. Then

and the double integral is approximately

where ${\upsilon }_g=c⁄{[d\left(\omega n_R\right)⁄d\omega ]}_{\omega ={\omega }_o}$ is the group velocity and it is assumed that group velocity dispersion is negligible. Then, from (19),

where P(t)=|a(t)|² is the probability at time t that the source atom is excited.

The group velocity can exceed c near the resonance frequency of the absorbing medium [6]. The result (23) then shows that the peak probability that a single photon is counted at z can occur sooner than it could if there were no medium between the source atom and the detector, albeit the probability that the photon is counted is reduced by the factor exp(-αz), where α=2ω_on_I (ω_o )/c is the absorption coefficient. Note also that the argument t-z/v_g does not have to be positive, nor does t-z/c have to be positive in the limiting case where the medium of propagation is the vacuum: because of our assumption that the pumping mechanism for exciting the atom is on at all times, there is a finite probability at all times that a photon count will be recorded at the detector. However, if a peak excitation probability for the atom occurs at time T, say, then the photon counting rate corresponding to this peak will itself peak at time T+z/v_g when the photon propagates in the medium, compared with the later time T+z/c at which it would peak were it propagating in vacuum.

4 Quantum Counterpart of the Classical Proof of Einstein Causality

Consider now the case where the source atom is suddenly put in its excited state at t=0, having been in its lower state prior to that time. Then, since the lowering operator σ(t ^″) acting on the lower state gives 0 for all times t ^″ earlier than t=0, we can replace (18) by

For our purposes we need only observe a general property of the last integral. Namely since n(ω ^')+1 only has poles in the lower half of the complex ω ^′-plane, and n(ω ^′)→1 as ω ^′→∞, the integral over ω ^′ has to vanish unless t>t ^″+z/c. Consequently, since the integration over t ^″ starts at t ^″=0, P(z,t) vanishes unless t>z/c. In other words, a suddenly excited atom cannot cause a photon to be counted at z before the time it takes for light to propagate in vacuum from the atom to the detector.

This is the analogue of the classical result that a sharp wave front cannot propagate faster than c. In our quantum-mechanical model, however, we cannot create a sharp “front” of photon probability because the radiative lifetime of the excited state is finite and the emitted radiation must have a finite spectral width. And of course the emitted “pulse” in our model represents a probability distribution for finding a single photon. Nevertheless the limiting case of an atom excited by a delta-function pulse, and with a very small radiative lifetime, provides a quantum analogue of the idealized sharp classical wave front. Our result is then analogous to the classical Brillouin-Sommerfeld proof of Einstein causality, i.e., the proof that the “frontal velocity” cannot exceed c [7].

5 Summary and Remarks

We have treated the propagation of a field in an absorbing medium from a microscopic viewpoint, assuming a single atom as the source of the field. Starting from the electric-dipole Hamiltonian for two-state atoms interacting with the field, and assuming all the atoms making up the medium remain with high probability in their lower states, we obtained an essentially exact expression for the electric field operator in the medium. We then considered two examples for the case where the group velocity v_g >c. In the first example the source atom has an excitation probability that has a smooth dependence on time, as would be the case if the atom were excited by a laser pulse with a smooth temporal profile. Then the probability at time t of registering a count at a detector at z>0 varies as P(t-z/v_g ) rather than P(t-z/c), where P is the upper-state occupation probability of the source atom. In other words, the photon counting probability is delayed by a time (z/v_g ) shorter than the delay time (z/c) associated with propagation in vacuum. In the second example the source atom is sharply excited at a time t=0, in the sense that P(t)=θ(t), where θ is the unit step function, and we proved that the probability to produce a photon count at the detector is exactly zero

before the time z/c. This is a quantum counterpart of the proof by Sommerfeld and Brillouin that the “frontal velocity” of an idealized sharp classical wave front cannot exceed c.

Regarding the model employed here, we note that the assumption that the atoms of the dielectric remain with high probability in their ground states renders our treatment of propagation very similar to a purely classical one. Our treatment of the source atom, or course, is completely quantum-mechanical. It might also be noted that the spontaneous emission rate of the source atom is modified by the presence of the dielectric, and can in general be either greater than or smaller than the free-space spontaneous emission rate. This does not affect our conclusions.