1. Transmission of emitting tunneling atoms in cavities

Spontaneous emission in atomic tunneling has been virtually unexplored before our recent work¹. Since tunneling is a distinct manifestation of wavelike properties, it is important to raise the basic questions: can spontaneous decay of internal excitations in tunneling atoms be viewed as a decoherence process that is analogous to its counterpart in diffracted atoms? and if so, how would such decoherence manifest itself?

We have put forward a theory of spontaneous emission from a two-level atom as it tunnels through a square potential barrier¹. Our theory demonstrates that the emission process is describable as loss of coherence between interfering classical trajectories in space-time, which constitute the atom tunneling motion. The emitted photon at each frequency is correlated to particular atomic classical trajectories, in a way which makes them measurably distinguishable. This distinguishability destroys their interference², as does “which-way” (“Welcher-Weg”) information, which is obtainable from spontaneous emission in diffracted atoms^3,4.

The ensuing analysis rests on two observations: (i) The overall duration of the decay process is much longer than the inverse transition frequency ${\omega }_{\mathit{\text{eg}}}^{-1}$ (see below). This allows us to resort to the rotating wave approximation (RWA), which is used in the Wigner-Weisskopf (WW) treatment of spontaneous emission^5,6. (ii) Nearly all of the cavity-enhanced spontaneous emission is funneled into the continuum of nearly resonant modes with wave-vectors q ≈ (ω/c)ẑ, which are aligned with the cavity axis z, perpendicular to the atomic incidence axis x. This allows us to use the dipole approximation, since q∙x ≈ 0, and neglect off-axis photon recoil effects on the atomic wavepacket. Hence, the RWA interaction Hamiltonian of the atom with the cavity-mode continuum becomes effectively one-dimensional, H_int = -ζ(x)∫dωρ(ω)[g_ωa_ω |e〉〉g| +h.c.]. Here ζ(x) = 1 for 0 ≤ x ≤ L and 0 elsewhere, i.e., the interaction is confined to the cavity, whose x-axis extent coincides with that of the barrier; ρ(ω) is a Lorentzian mode-density distribution associated with the cavity-mode linewidth η ⁷; g_ω is the coupling of the atom to the cavity mode at ω and a_ω is the corresponding annihilation operator. The transition frequency ω_eg is shifted (renormalized) by the difference between the AC Stark shifts of |e〉 and |g〉, ${\Delta }_{\mathit{AC}}=\frac{1}{4}\left(\frac{{\Omega }_e^2}{{\delta }_e}-\frac{{\Omega }_g^2}{{\delta }_g}\right).$

In order to analyze the entanglement of emitted photon states with the trans-lational degrees of freedom of the tunneling atom, we have developed a theoretical approach which combines the WW treatment^5,6, resulting in exponential decay of the excited state, with the Feynman path-integral method, which yields a coherent sum over the atomic classical trajectories contributing to tunneling⁸.

The above analysis yields the probability for an atom incident as a nearly monochromatic wave-packet to be transmitted in the excited state

where σ(E_k ,V) is the transmission amplitude for a structureless particle of kinetic energy E_k through a square potential barrier of height V and length L,

k = √2mE_k /ħ and p = √2m(E_k - V)/ħ being the corresponding wavevectors outside and inside the barrier, respectively. The effect of spontaneous emission is to shift the effective potential V by - iħΓ.

Plots of Eq. (1) reveal the overall diminishing of $P_e^{\mathit{\text{tr}}}$ with γ in both the tunneling (below-barrier) and allowed (above-barrier) regimes of E_k . The corresponding probability $P_g^{\mathit{\text{tr}}}$ of the transmitted ground state wave-packet is an incoherent sum (integral) of partial wave-packet transmission probabilities P_ω associated with photon emission at ω

where F(ω) = ρ(ω)|g_ω |²|/(Δ² + γ ²) and σ_ω (E_k , V) is a complicated function of E_k , V and ω. The most salient effect of spontaneous emission is seen to be (Fig. 1a) the huge enhancement of $P_g^{\mathit{\text{tr}}}$ as a function of γ for atoms initially in the deep tunneling regime p_L = √2m(V - E_k )L/ħ > 1.

In order to gain more insight into the above general results, we shall henceforth assume that the cavity linewidth η and E_k satisfy the following inequalities

The spectrum of spontaneous emission is then limited to |Δ| ≪ E_k and becomes Lorentzian in this range, F(ω) ≈ ℒ_γ(Δ), since the spectral variation of ρ(ω) and |g_ω |² is slow, ρ(ω)|g_ω |² ≈ 2πγ, in accordance with the WW approximation. The equation for σ_ω can now be simplified to

It is seen from Eqs. (3) and (5) that the dramatic enhancement effects in the tunneling regime are due to the first term in (5), corresponding to atoms that have decayed to the ground state shortly after entering the barrier and are subsequently transmitted through the barrier as unexcited atoms with kinetic energy E_k - ħΔ, which can be above the barrier if Δ < 0. By contrast, the second term in (5) corresponds to atoms that have decayed shortly before exiting the barrier after having effectively been transmitted as excited atoms with the initial kinetic energy E_k , whence this term is exponentially small in the tunneling regime. The use of Eq. (5) in Eq. (3) therefore leads to the enhancement of P_ω (Fig. 1a) and $P_g^{\mathit{\text{tr}}}$ due to the possibility to gain kinetic energy from the broad vacuum field reservoir by emitting a photon detuned below the resonance ħω_eg . In the deep tunneling regime, assuming that γ ≪ (V - E_k ), Eqs. (3)-(5) allow us to roughly estimate that the atoms have probability of order

to jump over the barrier into the allowed energy regime by emitting a photon with Δ < E_k - V < 0 (Fig. 1).

Under the assumptions leading to Eq. (5), along with ħΔ ≪ E_k , we can obtain a simplified expression for the total transmission probability

where $\widehat{\sigma }$(t, V), the Fourier transform of σ(E, V), is the impulse response (to a temporal δ-function) for transmission of a structureless particle. We thus obtain the following important result: the total transmission probability $P_{\mathit{\text{tot}}}^{\mathit{\text{tr}}}$ coincides, in the limit of narrow spontaneous linewidth γ [Eq. (4)], with the transmission probability of a partially incoherent wavepacket of a structureless particle with coherence time γ ^-1 (See Ref. 9).

 

Figure 1: (a) The energy spectrum of transmitted ground state atoms: Solid curve - transmission probability P_ω [Eq. (3)] (in units of ħ/V) for E_k /V = 0.8, L = 2.5λ_DB(E_k /V = 1), γ = 0.05V/ħ, ω_eg = 100V/ħ as a function of kinetic energy following emission. Dashed curve - spontaneous lineshape. Inset: Idem, on a small scale. Dotted curve - cavity lineshape. (b) Schematic description of the experiment

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The following conclusions can be inferred from the above analysis: (a) The probability distribution of the transmitted atoms is approximately Lorentzian for final kinetic energies E_k - ħΔ above the barrier, whereas their counterparts below the barrier only contribute an exponentially small tail to this distribution. (b) The fact that fast atoms emerging from the barrier are almost always unexcited means that the barrier acts as a “filter” that transmits almost only atoms that have already decayed.

These results open a new vista into the transition from quantum dynamics to classicality via decoherence by focusing on the effects of excitation decay on atomic tunneling. In the limit of negligible decay γ → 0, which is realizable by detuning the cavity off resonance with ω_eg , the excited atomic wavepacket with E_k < V exhibits tunneling, which is a result of interference between many classical trajectories, and is characterized by exponentially low transmission $P_e^{\mathit{\text{tr}}}$ [Eq. (1)]. When γ is appreciable, the wavepacket is dominated by the portion that has decohered by decay into the field-mode continuum and has thereby lost its tunneling properties: its energy spread becomes classical (statistical), giving rise to a Lorentzian tail into the above-barrier energy range, thereby allowing for enhancement of the transmission [Eqs. (3),(7)]. The effects of this decoherence on barrier traversal times will be discussed elsewhere.

The results predicted here can be experimentally realized by a variety of cold atoms. In accord with Eq. (4), the lifetime of the |e〉 → |g〉 transition should preferably be long, above 10^-6 sec. A confocal cavity whose finesse is ~ 10⁵ and subtends a solid angle of ~ 0.1 steradians can enhance spontaneous emission rate γ by a factor of ~ 30. The cavity linewidth η should be much larger than γ, i.e., preferably above 10MHz. Correspondingly, the potential energy V and the kinetic energy E_k must be above 0.1GHz, which requires the laser Rabi frequency Ω_e(g) and detuning δ _e,(g) to be well within the GHz range. This implies that the transition frequency ω_eg can lie anywhere between the GHz and the optical ranges.

2. Atomic reflection and localization at cavity interfaces

We have recently considered an excited atomic wavepacket or an atomic beam propagating from a region where spontaneous emission is negligible (x < 0) to a region where spontaneous emission is strongly enhanced (x > 0), due to the high density of the electromagnetic field modes. The wavefunction of the total system (atom plus field) can be written in the following general form in the rotating wave approximation

where the ket-vector |e, {0}) denotes the atom in the excited state with no photons in the field, whereas |g, {q}) corresponds to the ground state of the atom with a photon emitted at a mode q, and $\tilde{\psi }$ _e(q) are the corresponding amplitudes. One obtains coupled Schrödinger equations for the envelopes of these states given an atom with initial energy E and transition frequency ω ₀, ψ_e (r) and ψ _q(r) by assuming $\overline{\psi }$ _e,q(r,t) = ψ _e,q(r)e ^{-i(E+ħω₀/2)t} Far from the interaction region the solution describes propagation of the atomic wavepacket. The total energy of the incident excited atom E + ħ ω ₀ is then equal to the kinetic energy of the ground state atom plus the emitted photon energy ħω _q.

The coupled equations for ψ_e and ψ _q yield a complicated integro-differential wave equation for ψ_e (r _e), with Γ(r, ŕ) acting as a non-local complex potential whose shape and strength are determined by the confined mode eigenfunctions ε_q(r). If the linewidth of the spatially confined modes ħη_c is much larger than the atomic energy E, the recoil energy E_rec ≡ ħ ² ${\omega }_0^2$/2mc ² and the spontaneous linewidth in the confined reservoir, ħγ_c , then the correlation length of the interaction of the emitted photon with the atom is much shorter than the spontaneous decay length and the deBroglie wavelength λ_DB. Such an atom effectively moves in a local complex potential

where μ is the atomic dipole matrix element, ε _q(r) are the field mode amplitudes and Δ_c is the detuning of the atomic transition frequency ω ₀ from the center of the spectral line of the reservoir.

In order to concentrate on the atomic motion along the axis of incidence x and avoid diffraction effects caused by the local potential in the directions perpendicular to x, we consider a multimode confocal cavity where the many degenerate modes contributing to Γ(r render it approximately uniform in the directions perpendicular to x. We assume that the transition frequency ω ₀ is resonant with the Lorentzian center of the degenerate modes. Then the real part of Γ(x) is much less than the imaginary part γ_c (x) = Im{Γ(x)}. We then obtain

For a step-like interaction profile γ_c (x) = γ_c Θ(x), where Θ(x) is the Heaviside step function, the probability to detect an excited atom decreases as eik_γx, where k_γ = √2m(E + iħγ_c )/ħ, so that only the fraction |r|² of excited atoms remains at large negative x (to the left of the interface). This reflection increases with the spontaneous emission rate γ_c . The atomic interaction with the confined vacuum reservoir for ħ_γ > E is thus analogous to the skin effect of light reflection from metals. If the energy of the incident atom is comparable to E_rec , the width Δx of the interface should satisfy Δx ≈ λ_BD(E) ~ λ_opt. A realistic description of the atomic entry into a confocal cavity shows a much lower reflection probability, even for subrecoil energies. However, when the real part of Γ(x) contributes too, for ω ₀ well off the center of the Lorentzian spectrum (large Δ_c), the cavity can be strongly reflective. This spectral dependence of the reflectivity on the detuning is characteristic of the atomic skin effect.

The spatial variation of the q-mode amplitude in Eq. (8) can be estimated for a strong decay ħγ_c ≪ E and incidence energy well above the recoil limit. Then ψ _q ∞ e ^±ik_qx, where ħk _q = √2m(E - ħΔ_q) and Δ_q = ω _q - ω _q. Whenever E > ħΔ_q, k _q becomes imaginary and ψ _q(r) is exponentially localized at the interface between free space and the confined-field region. A solution with imaginary $k_{\mathrm{q}}^x$ represents a transient atomic wavepacket which disappears after the incident atomic wavepacket decays or leaves the interface, and is accompanied by a transient bound photon, which eventually disappears with it, after the time ~ ħ/ΔE, the inverse of the energy bandwidth ΔE of the incident atom. If such a photon is detected, then a localized atomic state is formed. The subsequent evolution of the atomic wavepacket is governed by the free-space Schrodinger equations with the localized atomic distribution serving as the initial condition.

In Fig. 2 and Refs. (12–16) we show a movie of a cold excited atom incident on an open cavity This movie assumes a step-function profile of γ(x) and reproduces the qualitative features of the atomic skin effect and localization at the interface.

To conclude, we have found that excited-atom reflection from the interface between two spatial regions with different spontaneous emission rates is appreciable for cold atoms and enhanced coupling to the mode continuum, when the effective width of the interface is smaller than the atomic deBroglie wavelength. This reflection is analogous to the optical skin effect of metal surfaces. Transient localized atomic state appear at the interface while an excited two-level atom is crossing it, due to detection of spontaneously emitted “bound photons” at “forbidden” energies, having short lifetime and range of propagation. The regime considered here is essentially different from Refs. 10 and Ref. 11, where the correlation time of the atom with the emitted photon is large, thereby responsible for the oscillation of the atomic population.

 

Figure 2: Numerical simulation of an initially excited atomic wavepacket approaching a sharp interface between a region of enhanced spontaneous emission and free space, showing excited-state reflection (orange) and different ground-state components: (a) Total ground-state envelope (b) Δ_q = 0 (no change of kinetic energy) (c) Δ_q > 0 (slowing down) (d) Δ_q < 0 (acceleration) (e) the incident and reflected wavepackets at the moment of incidence on the interface and a bound component observable by “forbidden” (near-field) photon detection. [Media 1] [Media 2] [Media 3] [Media 4] [Media 5]

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