A 2010 cavity-QED experiment (Norris, Orozco, Barberis-Blostein, Carmichael, Phys. Rev. Lett. 105, 123602) observed spontaneously generated coherence (SGC) directly, as long-lived ground-state quantum beats in the second-order correlation function of photons spontaneously emitted by a continuously driven cold 85Rb atomic beam inside a two-mode optical cavity, and acknowledged a set of open issues: the deferred quantitative study of drive-induced and atom-number-dependent light shifts, exact model-to-data parameter extraction, and extension to quantum feedback based on the spontaneously generated coherence. Later work by the same group resolved part of this agenda: the anomalous drive-induced light shift was quantified (Phys. Rev. A 85, 021804(R) (2012)), a detailed parameter-space model of the beats was published (Phys. Rev. A 86, 053816 (2012)), and a first, deliberately simple feedback protocol (drive gating with dark evolution) was implemented (New J. Phys. 15, 013017 (2013)). Two questions in this program remain open: closing the quantitative model-to-data gap — independently calibrated parameters and a full accounting of the observed decoherence — and realizing feedback control of the SGC beyond drive gating, up to the conditional-measurement and external-field protocols with probabilistic error correction proposed in New J. Phys. 12, 023002 (2010).

Background

Quantum beats are oscillations in the intensity of radiation emitted by atoms, produced by interference between indistinguishable emission pathways. In Type-I (V-shaped) systems a superposition of excited states decays to a common lower level and beats appear in the mean intensity, lasting only an excited-state lifetime; in Type-II (Lambda-shaped) systems, where decay proceeds to distinct final ground states, no beat appears in the mean intensity because the orthogonal ground states record which path was taken. Spontaneously generated coherence (SGC) is coherence between non-degenerate ground states created by spontaneous emission itself, possible when the emitted photon carries no information about which ground state was reached; before 2010 it had been discussed mainly theoretically, with only indirect experimental evidence.

Norris, Orozco, Barberis-Blostein, and Carmichael (Phys. Rev. Lett. 105, 123602 (2010)) observed SGC directly. A dilute beam of cold 85Rb atoms (speed about 22 m/s) traverses a 2.2 mm Fabry-Perot optical cavity — a resonator confining the light to two degenerate, orthogonally polarized modes, H and V — operated in the intermediate regime of cavity QED (the study of atoms coupled to quantized cavity fields), with single-atom cooperativity C1=g2/γκ≈0.12C_1=g^2/\gamma\kappa\approx 0.12, where gg is the single-photon atom-field coupling, κ\kappa the cavity field decay rate, and γ\gamma the atomic dipole decay rate. The V mode is weakly and continuously driven on the F=3,m=0→F′=4,m′=0F=3, m=0 \to F'=4, m'=0 transition; a weak magnetic field (about 5 G) sets the quantization axis. A photon spontaneously emitted into the undriven H mode carries an undetermined helicity, so its detection projects the atom onto the superposition (∣g−1⟩+∣g+1⟩)/2(|g_{-1}\rangle+|g_{+1}\rangle)/\sqrt{2} of Zeeman ground states — a quantum eraser, since the detection erases the which-path information the photon could have carried. The superposition then Larmor-precesses, i.e. precesses about the magnetic field at the frequency set by the ground-state Zeeman splitting, and when the driven mode re-excites the atom, emission of a second H-mode photon closes two indistinguishable time orderings. The result is a long-lived oscillation, at twice the Larmor frequency, in the second-order intensity correlation function g(2)(τ)g^{(2)}(\tau) — the normalized rate of photon-pair coincidences at delay τ\tau — visible only in the fluctuations, never in the mean intensity. Additional beats arise from pairs of atoms (interference of emission time order) and from interference with drive light mixed into the H mode by mirror birefringence (a homodyne beat at the Larmor frequency itself).

The theoretical treatment uses the quantum-trajectory (Monte Carlo wavefunction) method — a stochastic unraveling of the open-system master equation in which the atomic ensemble evolves conditioned on simulated photon detections — with two approximations: (i) the driven V mode is treated as a classical field, with its absorption by the atoms retained, and (ii) reabsorption of H-mode photons is neglected. Both are justified only at moderate-to-weak dipole coupling and weak drive. Quantitative comparison with data is limited by parameters that are hard to determine independently: the fidelity of optical pumping into m=0m=0, the atomic-beam tilt, and small birefringence and background amplitudes. The beat frequency also depends on the drive: the observed anomalous light shift — an intensity-dependent shift of the Larmor frequency, with sign opposite to the naive AC Stark expectation, reaching about 1% per intracavity photon — arises stochastically from phase accumulated in discrete quantum jumps (spontaneous emissions that map the ground-state coherence among Zeeman sublevels), the same mechanism that causes Rayleigh-scattering decoherence of ground-state superpositions analyzed by Uys et al. (Phys. Rev. Lett. 105, 200401 (2010)).

The source paper closes by proposing a new class of quantum feedback in this system: measurement-based control in which the photon-detection record conditions real-time actuation. Barberis-Blostein, Norris, Orozco, and Carmichael (New J. Phys. 12, 023002 (2010)) worked out the proposal — conditional measurements combined with external fields to shield and manipulate the ground-state superposition and to reverse detected errors (probabilistic quantum error correction, in the sense of Koashi-Ueda measurement reversal) — and showed it should be robust under realistic experimental conditions. The apparatus itself is described in Norris, Cahoon, and Orozco (Phys. Rev. A 80, 043830 (2009)). Subsequent same-group work delivered part of this agenda: the anomalous light shift was quantified (Phys. Rev. A 85, 021804(R) (2012)), a detailed parameter-space model of the beats was published (Phys. Rev. A 86, 053816 (2012)), and a first, deliberately simple feedback protocol — switching the drive off after the first detection, letting the coherence evolve in the dark, and switching it back on — was implemented (New J. Phys. 15, 013017 (2013)), preventing the light-shift-induced frequency shift and more than doubling the beat amplitude.

Problem Statement

The problem is the surviving open core of the issues acknowledged in Norris et al. (Phys. Rev. Lett. 105, 123602 (2010)), after later same-group work delivered the deferred light-shift study and a first simple feedback protocol. For the driven two-mode cavity-QED system in which spontaneously generated ground-state coherence is observed as quantum beats in g(2)(τ)g^{(2)}(\tau), two linked questions remain open. They share one objective — establishing the spontaneously generated coherence as a quantitatively understood and actively controllable resource — and each feeds the other: a validated quantitative model is prerequisite to feedback design, and feedback performance is a stringent test of the model. Both questions are mandatory and are judged on their own deliverables. In (2), the required minimum is a feedback protocol that uses the conditional detection record and demonstrably exceeds drive gating, characterized by its coherence lifetime and phase-control accuracy; the external-field and probabilistic-error-reversal program of New J. Phys. 12, 023002 (2010) is the target beyond that minimum, with error-correction fidelity required only of protocols that reverse detected errors.

(1) Quantitative model-to-data closure. Establish a quantitatively validated description of the ground-state quantum beats in which the parameters presently fitted rather than measured (optical-pumping fidelity, atomic-beam tilt and velocity distribution, birefringence and background amplitudes) are determined independently, and the observed decoherence of the beat — including the excess noise left unaccounted for in the 2012 light-shift analysis — is fully explained. The relevant data are the parameter values and measured g(2)(τ)g^{(2)}(\tau) curves reported in the cited works; closure is to be argued against those published quantities rather than against raw apparatus records.

(2) Feedback control of the spontaneously generated coherence. Implement and characterize quantum feedback protocols based on the spontaneously generated ground-state coherence beyond the demonstrated drive-gating scheme: protocols that use the conditional detection record together with external fields to preserve the coherence, control its phase, and manipulate it, up to the reversal of detected errors (probabilistic quantum error correction) proposed in New J. Phys. 12, 023002 (2010); and characterize the achieved coherence lifetime, phase-control accuracy, and error-correction fidelity.

References

  1. D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, "Observation of Ground-State Quantum Beats in Atomic Spontaneous Emission", Phys. Rev. Lett. 105, 123602 (2010). DOI: 10.1103/PhysRevLett.105.123602, https://doi.org/10.1103/PhysRevLett.105.123602 (arXiv:1008.3920, https://arxiv.org/abs/1008.3920)
  2. P. Barberis-Blostein, D. G. Norris, L. A. Orozco, and H. J. Carmichael, "From quantum feedback to probabilistic error correction: manipulation of quantum beats in cavity QED", New J. Phys. 12, 023002 (2010). DOI: 10.1088/1367-2630/12/2/023002, https://doi.org/10.1088/1367-2630/12/2/023002 (arXiv:1001.0952, https://arxiv.org/abs/1001.0952)
  3. D. G. Norris, A. D. Cimmarusti, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, "Anomalous light shift through quantum jumps in quasiresonant Rayleigh scattering", Phys. Rev. A 85, 021804(R) (2012). DOI: 10.1103/PhysRevA.85.021804, https://doi.org/10.1103/PhysRevA.85.021804 (arXiv:1202.3159, https://arxiv.org/abs/1202.3159)
  4. D. G. Norris, A. D. Cimmarusti, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, "Spontaneous creation and persistence of ground-state coherence in a resonantly driven intracavity atomic ensemble", Phys. Rev. A 86, 053816 (2012). DOI: 10.1103/PhysRevA.86.053816, https://doi.org/10.1103/PhysRevA.86.053816 (arXiv:1211.6364, https://arxiv.org/abs/1211.6364)
  5. A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. Carmichael, "Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts", New J. Phys. 15, 013017 (2013). DOI: 10.1088/1367-2630/15/1/013017, https://doi.org/10.1088/1367-2630/15/1/013017 (arXiv:1301.3000, https://arxiv.org/abs/1301.3000)
  6. A. D. Cimmarusti, B. D. Patterson, C. A. Schroeder, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, "Feedback in a cavity QED system for control of quantum beats", EPJ Web of Conferences 57, 03005 (2013). DOI: 10.1051/epjconf/20135703005, https://doi.org/10.1051/epjconf/20135703005
  7. A. D. Cimmarusti, "Control protocols for manipulation of ground-state quantum beats in a cavity QED system", PhD thesis, University of Maryland, College Park (2014). DOI: 10.13016/M2CS34, http://hdl.handle.net/1903/15699
  8. H. Uys, M. J. Biercuk, A. P. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, and J. J. Bollinger, "Decoherence due to Elastic Rayleigh Scattering", Phys. Rev. Lett. 105, 200401 (2010). DOI: 10.1103/PhysRevLett.105.200401, https://doi.org/10.1103/PhysRevLett.105.200401
  9. D. G. Norris, E. J. Cahoon, and L. A. Orozco, "Atom detection in a two-mode optical cavity with intermediate coupling: Autocorrelation studies", Phys. Rev. A 80, 043830 (2009). DOI: 10.1103/PhysRevA.80.043830, https://doi.org/10.1103/PhysRevA.80.043830