Realistic-imperfection analysis of the two-atom quantum eraser and of the two-photon correlation imaging protocols of Glauber, Kleber, Patnaik, Scully and Walther (2005)

The 2005 study by Glauber, Kleber, Patnaik, Scully and Walther (J. Phys. B 38, S521) derived idealized two-level-atom protocols for Young-type and Hanbury Brown-Twiss-type interference, two quantum-eraser schemes that sort joint photon counts into fringe and antifringe subensembles (one of them using a micromaser cavity read out by an idealized probe detector), and a two-photon correlation imaging scheme whose Stokes-anti-Stokes coincidence fringes have an effective halved wavelength of order $\lambda/2$ and were promised to improve imaging resolution. These protocols were derived under idealizations, and how they behave under realistic imperfections - finite detector efficiency, nonideal cavity coupling, finite atomic lifetimes, noise, imperfect excitation timing, partial temporal overlap and partial distinguishability of the emitted photons - was not quantified there; the quantitative assessment of the achievable resolution improvement (source brightness, background counts, detector timing resolution) was deferred to a companion work then in preparation, which was never published. The feasibility and realistic performance of these two-atom protocols therefore remains an open question: no work is known to quantify the eraser sorting fidelity, the visibility budget, or the end-to-end resolution assessment for these specific protocols.

Background

Photon-correlation optics runs from the Hanbury Brown-Twiss (HBT) experiment - the 1954 stellar intensity interferometer, which correlates photon counts at two separated detectors and thereby revealed photon bunching - through Glauber's quantum theory of optical coherence to present-day quantum imaging. In that theory the field statistics are characterized by correlation functions: the first-order correlation $G^{(1)}$ determines amplitude (Young-type) interference in the mean intensity registered by a single detector, and the second-order correlation $G^{(2)}$ determines the joint detection rate at two detectors, i.e. the photon-photon correlations. The source paper (Glauber, Kleber, Patnaik, Scully and Walther, J. Phys. B: At. Mol. Opt. Phys. 38, S521-S534 (2005)) analyzes a deliberately minimal model: two identical two-level atoms A and B - atoms with a ground state $|b\rangle$ and an excited state $|a\rangle$ separated by a transition of frequency $\omega$ and natural linewidth $\Gamma$ - held a distance $d \gg \lambda$ apart, where $\lambda$ is the transition wavelength. Comparing the semiclassical theory of radiation (the field treated as classical and radiated by dipoles proportional to atomic expectation values) with the full quantum theory of the radiation field, the paper shows that weak-pulse excitation (pulse duration $\delta\tau \ll \Gamma^{-1}$ and small excitation probability) and single-photon excitation, which prepares the entangled state $\tfrac{1}{\sqrt{2}}(|a_A b_B\rangle + |b_A a_B\rangle)$, both yield $G^{(1)}$ interference fringes that the semiclassical theory misses (the dipole expectation value vanishes for the entangled state), whereas when both atoms are fully excited the emitted one-photon wavepackets $|\gamma_A\rangle$ and $|\gamma_B\rangle$ are orthogonal ($\langle\gamma_A|\gamma_B\rangle = 0$, so each photon carries which-way - literally which-path, i.e. which atom emitted - information), $G^{(1)}$ loses its fringes, but the two indistinguishable two-photon detection paths in $G^{(2)}$ still interfere and produce fringes in the coincidence rate.

The paper then constructs two quantum erasers - schemes that discard which-way information by measuring a correlated degree of freedom and thereby restore interference. One eraser conditions only on detector placement; the other, the atom-photon eraser, passes the atoms through a micromaser: a microwave cavity of side length on the order of the transition wavelength in which single atoms interact strongly with one highly excited cavity mode. Assuming the atom-cavity coupling to be much stronger than the atoms' coupling to the external vacuum, the collective Dicke states (the symmetric and antisymmetric basis states of the two-atom system, e.g. the antisymmetric $|0,0\rangle = \tfrac{1}{\sqrt{2}}(|b_A a_B\rangle - |a_A b_B\rangle)$, which has zero electric dipole moment and so does not radiate) decay selectively inside the cavity, and an idealized cavity probe detector $\mathcal{D}$ reading out the cavity after the passage lets one sort the joint photon counts into complementary fringe and antifringe subensembles - the eraser signature. Finally, for two atoms that each emit a Raman Stokes photon $\gamma$ and, later, an anti-Stokes photon $\phi$ (the two sequentially emitted photons of a Raman process on a three-level atom), the factorized two-photon correlation $G^{(2)}_{\gamma\phi}$ carries fringes of spatial period set by $k_\gamma + k_\phi \simeq 2k$, where $k$ is the optical wavenumber - an effective halved wavelength $\lambda/2$. Since classical imaging resolution is bounded by the Rayleigh criterion (angular resolution $1.22\lambda/d_r$ for an aperture $d_r$), coincidence fringes of period $\lambda/2$ promise resolution beyond the classical diffraction limit, the goal of the quantum-lithography program initiated by Boto et al. (2000) and advanced experimentally by D'Angelo, Chekhova and Shih (2001).

These constructions, however, rest on idealizations: the atom-photon eraser assumes a wavelength-scale cavity, dominant cavity coupling and perfectly efficient probe detection, with no analysis of how finite detector efficiency, nonideal cavity coupling, finite atomic lifetimes and noise degrade the subensemble sorting and the recovered fringes; the interference formulas assume well-separated atoms, controlled excitation timing ($\delta\tau \ll \Gamma^{-1}$), causally ordered non-overlapping wavepackets and perfectly indistinguishable polarization and mode degrees of freedom; and the promised resolution improvement is not quantified against realistic source brightness, background counts and detector timing resolution, the quantitative assessment being deferred to a companion work then in preparation. The unresolved question is precisely this feasibility analysis for the paper's own two-atom protocols.

Problem Statement

Determine, by quantitative theory and/or experiment, how the two-atom protocols of Glauber, Kleber, Patnaik, Scully and Walther (2005) perform under realistic imperfections, covering the three idealization clusters of the paper's protocols:

(i) The atom-photon quantum eraser, which assumes a micromaser cavity of side length of order the transition wavelength, atom-cavity coupling much stronger than the atoms' coupling to the external vacuum, and an ideal cavity probe detector $\mathcal{D}$ that registers counts perfectly: within a single model in which the probe detector has finite efficiency, the cavity coupling is finite, the atomic excited states have finite lifetimes, and noise (cavity loss, dephasing, dark counts) is present all together, compute the fidelity with which joint detection events are sorted into the fringe and antifringe subensembles (the sorting fidelity), and the visibility of the recovered fringes, as functions of each of these imperfection parameters (the others held at stated values); and identify the parameter regimes in which erasure still restores interference, taking 'restored' to mean that the recovered-fringe visibility, relative to its ideal (imperfection-free) value, lies above a threshold that the answer states explicitly.

(ii) The two-atom interference fringes derived under weak-pulse excitation ($\delta\tau \ll \Gamma^{-1}$) and single-photon excitation with well-separated, causally ordered wavepackets: quantify the degradation of the $G^{(1)}$ and $G^{(2)}$ fringe visibility under imperfect excitation timing, partial temporal overlap of the emitted-photon wavepackets, and partial distinguishability of the photons' polarization or mode degrees of freedom.

(iii) The two-photon correlation imaging claim, in which Stokes-anti-Stokes photon pairs from two atoms produce $G^{(2)}_{\gamma\phi}$ fringes of effective wavelength $\lambda/2$: provide the complete quantitative assessment of the achievable resolution improvement, including realistic pair-source brightness, background counts, and detector timing resolution - i.e. whether and by how much the promised sub-Rayleigh imaging benefit survives realistic operating conditions.

The three clusters address distinct protocols of the paper - the micromaser atom-photon eraser (i), the two-atom interference fringes (ii), and the Raman-pair correlation imaging scheme (iii) - and together concern one objective: establishing the experimental feasibility and realistic performance of the paper's proposals. The target for each cluster is quantitative results, predicted or measured, as functions of the relevant imperfection parameters, for the experimental parameter regime assumed (micromaser, atomic species and lifetimes, detector efficiencies and timing): closed-form or numerical sorting fidelities - the sorting fidelity of cluster (i) being the probability, under a stated assignment rule, that a joint detection event is assigned to its true fringe or antifringe subensemble - fringe visibilities for clusters (i) and (ii), and a resolution-improvement factor for cluster (iii) together with the uncertainty propagated into that factor from the assumed ranges of detector efficiency, brightness, background counts and detector timing.

References

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