Unconventional quantum correlations of light emitted by a single atom in free space

D. Goncalves, M. W. Mitchell, and D. E. Chang

Phys. Rev. A 104, 013724 – Published 29 July 2021

Abstract

We present an approach to engineer the photon correlations emerging from the interference between an input field and the field scattered by a single atom in free space. Nominally, the inefficient atom-light coupling causes the quantum correlations to be dominated by the input field alone. To overcome this issue, we propose the use of separate pump and probe beams, where the former increases the atomic emission to be comparable to the probe. Examining the second-order correlation function $g^{(2)} (τ)$ of the total field in the probe direction, we find that the addition of the pump formally plays the same role as increasing the coupling efficiency, even though the physical atom-light coupling efficiency remains unchanged. We show that one can tune the correlation function $g^{(2)} (0)$ from zero (perfect antibunching) to infinite (extreme bunching) by a proper choice of pump amplitude. We further elucidate the origin of these correlations in terms of the transient atomic state following the detection of a photon, and show that these correlations can be observed under realistic conditions.

Received 28 April 2020
Revised 2 October 2020
Accepted 12 July 2021

DOI:https://doi.org/10.1103/PhysRevA.104.013724

Physics Subject Headings (PhySH)

Light-matter interaction Photon statistics Quantum optics

Atoms

Quantum correlations, foundations & formalism Two-level models

Atomic, Molecular & Optical

Authors & Affiliations

D. Goncalves ^1,*, M. W. Mitchell ^1,2, and D. E. Chang ^1,2

¹ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain

^*daniel.goncalves@icfo.es

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Vol. 104, Iss. 1 — July 2021

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Images

Figure 1
Conceptual scheme of the proposed technique. Two coherent, continuous beams (pump and probe) illuminate resonantly a single atom in a “Maltese cross” configuration. Along the transmitted direction of the probe beam, the total field consists of the coherent sum of the incident probe field (small blue arrow) and the quantum field rescattered by the atom, which contains contributions from both the probe and pump beams (small and large yellow arrows, respectively). A sufficiently large pump beam allows the rescattered field to be comparable to the incident probe in intensity, despite a low collection efficiency $η$ of emitted photons in the transmitted direction. This enables strong quantum correlations to emerge in the collected field.
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Figure 2
Transmission spectra of a weak, coherent probe beam as a function of the laser detuning $Δ$ from the atomic resonance, in units of the free-space decay rate $Γ_{0}$ . (a) Transmission spectrum without the pump. We take a realistic coupling efficiency of $η = 0.05$ , consistent with typical experimental values. (b) Same as (a) but now with the additional pump beam, with pump-probe relative phase $ϕ = 0$ . The pump amplitude is tuned to obtain different effective coupling efficiencies $Λ = {η, 0.25, 0.5, 1.1}$ (colors from light to dark blue). We note a total extinction of the resonant transmission for $Λ = 0.5$ and transmission higher than 100% for $Λ > 1$ .
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Figure 3
Second-order correlation function $g^{(2)} (τ)$ of the transmitted field for a weak, coherent probe beam, as a function of time delay $τ$ scaled by the free-space decay rate $Γ_{0}$ . In (a) the pump amplitude is set to zero and we take an experimentally realistic atom-light coupling efficiency of $η = 0.05$ . Due to the low coupling efficiency, the transmitted field mostly consists of the incident coherent field, and thus $g^{(2)} (τ) \approx 1$ at all times $τ$ . In (b) and (c), a nonzero pump yields an effectively enhanced coupling efficiency of $Λ = 0.3$ and $Λ = 0.4$ (respectively), giving rise to nontrivial photon correlations. In (d) we consider the case of $Λ = 10$ , where a pump beam much stronger than the probe causes the total field correlations to be dominated by the atomic scattered field, which is trivially antibunched. Light blue regions correspond to notably antibunched correlations ( $g^{(2)} < 0.5$ ), while light yellow is used to indicate large photon bunching ( $g^{(2)} > 10$ ).
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Figure 4
Second-order correlation function $g^{(2)} (0)$ as a function of the coupling efficiency $η$ , the pump-probe relative phase $ϕ$ , and the amplitude ratio $| Ω_{pump} / Ω_{probe} |$ . In (a) we fix $η = 0.05$ and plot $g^{(2)} (0)$ as a function of $| Ω_{pump} / Ω_{probe} |$ and $ϕ$ . In (b) we show again the values of $g^{(2)} (0)$ but now fixing $ϕ = 0$ and exploring different $η$ . Additionally, we show the beam waist $w_{0}$ associated to the values of $η$ assuming a spatial Gaussian detection mode within the paraxial approximation. As extreme values of bunching are possible, we represent all the values $g^{(2)} (0) \geq 10$ by the same color.
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Figure 5
(a) Schematic representation of an atom displaced by an amount $r_{d} = (Δ y, Δ z)$ from the origin, which also constitutes the common focal point of the probe and pump beams propagating along $y$ and $z$ , respectively. (b),(c) Transmission $T (r_{d})$ (b) and second-order correlation function $g^{(2)} (0; r_{d})$ (c) for different atomic displacements $r_{d} = (Δ y, Δ z)$ . The amplitude, beam waists, and phase of the pump and probe fields are chosen such that $η (0) = 0.05$ and $Λ (0) = 1 / 2$ at the focal point. Central regions of (b) and (c) are also shown magnified.
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Figure 6
(a) Energy levels of the maximum angular momentum ground- and excited-state manifolds of ${}^{87}{Rb}$ , with $F_{g} = 2$ and $F_{e} = 3$ , respectively. Each manifold has $2 F_{g (e)} + 1$ hyper-fine levels labeled by $m_{g (e)} \in [- F_{g (e)}, F_{g (e)}]$ . The dashed lines indicate the allowed decay paths of the excited states, while the blue lines represent the transitions driven by a strong pump field with $σ_{+}$ polarization. For clarity, the transitions of the $x$ -polarized probe field are not indicated here. In the steady state, the strong pump nearly perfectly restricts the population within the cycling transition involving states $| F_{e} = 3, m_{e} = 3 〉$ and $| F_{g} = 2, m_{g} = 2 〉$ (golden line). (b) Illustration of the most relevant states. The strong pump restricts most of the population to the cycling transition $| F_{e} = 3, m_{e} = 3 〉 \to | F_{g} = 2, m_{g} = 2 〉$ , as stated earlier. The weak probe (red) has polarization $x = (σ_{-} - σ_{+}) / \sqrt{2}$ , while the pump field (blue) is $σ_{+}$ polarized. Under the conditions relevant to our scheme, the amount of population driven out of the two-level subspace, to $| F_{e} = 3, m_{e} = 1 〉$ , is negligible. (c) Transmission $T$ (blue) and second-order correlation function $g^{(2)} (0)$ (green) for a multilevel atom as a function of the pump-probe amplitude ratio. The solid lines are obtained from Eqs. (8) and (10) after substituting the coupling parameter $\bar{η}$ and taking into account the suggested pump and probe polarizations. The dashed lines are obtained with a numerical simulation considering all the hyperfine levels in (a). In the simulations, we take $\bar{η} = 0.025$ .
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