## Remnants of semiclassical bistability in the few-photon regime of cavity QED |

Optics Express, Vol. 19, Issue 24, pp. 24468-24482 (2011)

http://dx.doi.org/10.1364/OE.19.024468

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### Abstract

Broadband homodyne detection of the light transmitted by a Fabry-Perot cavity containing a strongly-coupled ^{133}Cs atom is used to probe the dynamic optical response in a regime where semiclassical theory predicts bistability but strong quantum corrections should apply. While quantum fluctuations destabilize true equilibrium bistability, our observations confirm the existence of metastable states with finite lifetimes and a hysteretic response is apparent when the optical drive is modulated on comparable timescales. Our experiment elucidates remnant semiclassical behavior in the attojoule (∼ 10 photon) regime of single-atom cavity QED, of potential significance for ultra-low power photonic signal processing.

© 2011 OSA

1. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics **3**, 687–695 (2009). [CrossRef]

2. T. Aoki, A. S. Parkins, D. J. Alton, C. A. Regal, B. Dayan, E. Ostby, K. J. Vahala, and H. J. Kimble, “Efficient Routing of Single Photons by One Atom and a Microtoroidal Cavity,” Phys. Rev. Lett. **102**, 083601 (2009). [CrossRef] [PubMed]

3. L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature **467**, 574–578 (2010). [CrossRef] [PubMed]

4. C. Savage and H. .J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. **24**, 1495–1498 (1988). [CrossRef]

5. G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, “Optical bistability and photon statistics in cavity quantum electrodynamics,” Phys. Rev. Lett. **67**, 1727 (1991). [CrossRef] [PubMed]

6. M. A. Armen and H. Mabuchi, “Low-lying bifurcations in cavity quantum electrodynamics,” Phys. Rev. A **73**, 063801 (2006). [CrossRef]

7. J. Kerckhoff, M. A. Armen, D. S. Pavlichin, and H. Mabuchi, “The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying,” Opt. Express **19**, 6478–6486 (2011). [CrossRef] [PubMed]

8. D. A. B. Miller, “Are optical transistors the logical next step?,” Nature Photonics **4**3–5 (2010). [CrossRef]

6. M. A. Armen and H. Mabuchi, “Low-lying bifurcations in cavity quantum electrodynamics,” Phys. Rev. A **73**, 063801 (2006). [CrossRef]

10. H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett. **98**, 193109 (2011). [CrossRef]

^{133}Cs atom as the nonlinear medium in a Fabry-Perot optical resonator, but we note that the same physics should also be relevant to quantum nonlinear dynamics in strongly coupled nanophotonic systems [11

11. K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature **450**, 862–865 (2007). [CrossRef] [PubMed]

12. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Physics **4**, 859–863 (2008). [CrossRef]

14. A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. **15**, 376 (1969). [CrossRef]

15. S. D. Smith, “Optical bistability, photonic logic, and optical computation,” Appl. Opt. **25**, 1550–1564 (1986). [CrossRef] [PubMed]

4. C. Savage and H. .J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. **24**, 1495–1498 (1988). [CrossRef]

16. S. Ya. Kilin and T. B. Krinitskaya, “Single-atom phase bistability in a fundamental model of quantum optics,” J. Opt. Soc. Am. B **8**, 2289 (1991). [CrossRef]

6. M. A. Armen and H. Mabuchi, “Low-lying bifurcations in cavity quantum electrodynamics,” Phys. Rev. A **73**, 063801 (2006). [CrossRef]

17. C. J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kimble, “Real-time cavity QED with single atoms,” Phys. Rev. Lett. **80**, 4157–4160 (1998). [CrossRef]

18. X. Yang, C. Husko, C. W. Wong, M. Yu, and D. L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. **91**, 051113 (2007). [CrossRef]

19. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nature Photonics **4**, 477–483 (2010). [CrossRef]

8. D. A. B. Miller, “Are optical transistors the logical next step?,” Nature Photonics **4**3–5 (2010). [CrossRef]

21. H. Gang, C. Z. Ning, and H. Haken, “Codimension-two bifurcations in single-mode optical bistable systems,” Phys. Rev. A **41**, 2702 (1990). [CrossRef] [PubMed]

22. H. Gang, C. Z. Ning, and H. Haken, “Distribution of subcritical Hopf bifurcations and regular and chaotic attractors in optical bistable systems,” Phys. Rev. A **41**, 3975 (1990). [CrossRef] [PubMed]

**73**, 063801 (2006). [CrossRef]

*a priori*as they ignore correlations between the atom and intra-cavity photons. However, the MBEs do retain some relevance as a projection of the quantum master equation onto the sub-manifold of semiclassical atom-field states [23

23. H. Mabuchi, “Derivation of Maxwell-Bloch-type equations by projection of quantum models,” Phys. Rev. A **78**, 015801 (2008). [CrossRef]

**73**, 063801 (2006). [CrossRef]

*E*in the range of [1.5,2.3]

*κ*. Using identical parameters, the steady-state solution of the quantum master equation can be determined for each value of the drive strength. Contours of the corresponding intracavity field Wigner functions (tracing over the atomic states) are displayed for several drive strengths in Fig. 1(b). A double-peaked structure emerges along the amplitude quadrature in a range of

*E*/

*κ*similar to the bistable region of the MBEs. The bimodal steady states correspond to an incoherent mixture of two states: a weakly excited atom and low-amplitude field state, and a fully saturated atom and a high-amplitude field state. Thus, the two peaks of the distribution may be qualitatively associated with the low- and high-amplitude branches of the semiclassical absorptive bistability curve, but in the quantum model neither is truly stable [4

4. C. Savage and H. .J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. **24**, 1495–1498 (1988). [CrossRef]

7. J. Kerckhoff, M. A. Armen, D. S. Pavlichin, and H. Mabuchi, “The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying,” Opt. Express **19**, 6478–6486 (2011). [CrossRef] [PubMed]

24. M. A. Armen, A. E. Miller, and H. Mabuchi, “Spontaneous dressed-state polarization in the strong driving regime of cavity QED,” Phys. Rev. Lett. **103**173601 (2009). [CrossRef] [PubMed]

^{133}Cs atoms are dropped into a Fabry-Perot optical resonator (length 27

*μ*m, 10cm radius of curvature mirrors, field decay rate

*κ*/2

*π*= 9.3MHz) supporting a circularly-polarized, 852nm TEM

_{00}mode actively frequency-stabilized relative to the (6

*S*

_{1/2},

*F*= 4,

*m*= +4) → (6

_{F}*P*

_{3/2},

*F*= 5,

*m*= +5) atomic cycling transition (with dipole decay rate

_{F}*γ*

_{⊥}/2

*π*= 2.6MHz). As an atom falls through the cavity mode it experiences a position-dependent coupling rate

*g*(maximum

*g*

_{0}/2

*π*=56.8MHz at the cavity anti-nodes) that can be monitored via the transmission of a weak and detuned circularly-polarized optical probe. Once a strongly coupled atom is detected, the probe power and detuning are adjusted for optical homodyne detection and 200MS/s data acquisition (first arrow at 2

*μ*s on the time axis in Fig. 1(c). Fig. 1(c) depicts a representative signal (all data has been post-filtered at 20MHz bandwidth for clarity); with the drive amplitude held at a fixed value of

*E*= 2.6

*κ*, the amplitude quadrature of the transmitted field fluctuates with a large variance until 14

*μ*s (red arrow) when the atom is abruptly lost and and the measured transmission settles to the shot noise-variance signal with intermediate mean amplitude expected for our cavity when empty (perturbed only by slight ∼1–10kHz mechanical instabilities). See Appendix A for more explanation of the apparatus.

*E*} = {−1.1,.7,2.6}

*κ*, wide/bimodal and narrow/normal distributions are apparent in the amplitude- and phase-quadrature distributions, respectively (Fig. 3(d)). However, when the atom-drive detuning is increased to Δ = 3

*κ*for the same drive amplitude, the low-amplitude transmitted field dominates (Fig. 3(e)) as the drive threshold for ‘bistability’ increases with |Δ|. The bimodal amplitude distribution reemerges when the drive amplitude is increased to

*E*= 3.7

*κ*in Fig. 3(f).

## A. Experimental apparatus

^{133}Cs atoms and a high finesse Fabry-Perot optical resonator [24

24. M. A. Armen, A. E. Miller, and H. Mabuchi, “Spontaneous dressed-state polarization in the strong driving regime of cavity QED,” Phys. Rev. Lett. **103**173601 (2009). [CrossRef] [PubMed]

7. J. Kerckhoff, M. A. Armen, D. S. Pavlichin, and H. Mabuchi, “The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying,” Opt. Express **19**, 6478–6486 (2011). [CrossRef] [PubMed]

28. H. Mabuchi, Q. A. Turchette, M. S. Chapman, and H. .J. Kimble, “Real-time detection of individual atoms falling through a high-finesse optical cavity,” Opt. Lett. **21**, 1393–1395 (1996). [CrossRef] [PubMed]

*S*

_{1/2},

*F*= 4,

*m*= +4) → (6

_{F}*P*

_{3/2},

*F*= 5,

*m*= +5) atomic cycling transition at 852nm in order to approximate the atom as a two-level system. Although the cavity used in the experiment was constructed to optimize signatures of spontaneous dressed-state polarization at high drive amplitudes [7

_{F}**19**, 6478–6486 (2011). [CrossRef] [PubMed]

^{−9}Torr) chamber and placed on a multi-stage vibration-isolation stack, the Fabry-Perot optical resonator is formed by two high-reflectivity (8ppm transmission, 2ppm loss), 10cm radius of curvature dielectric mirrors with roughly 27

*μ*m of separation, yielding a 300,000-finesse optical resonator for the standing wave, TEM

_{00}, 18

*μ*m-waist transverse spatial mode with a field decay rate of

*κ*= 2

*π*× 9.3MHz. We took particular care to mount the mirrors in a rotationally-symmetric manner to minimize stress-induced birefringence in the mirror coatings, allowing for full polarization-selectivity of the atomic transitions. The cavity length is tuned and actively stabilized by two shear-mode piezoelectric plates underlying the two mirror mounts. The precise cavity length and resonance frequency is continually stabilized by the Pound-Drever-Hall (PDH) [29

29. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B **31**97 (1983). [CrossRef]

^{6}atoms is formed roughly 1cm above the cavity mode in the UHV chamber. After cooling, the ensemble trap is switched off, allowing the cold atoms to fall under gravity towards the cavity mode and by the time they reach the cavity mode their free-fall velocity tends to dominate any residual thermal motion. Due to the strong coupling between the targeted atomic transition and the cavity mode (with calculated maximum value

*g*

_{0}= 2

*π*× 56.8MHz at the cavity anti-nodes, using the dipole strength of the atomic transition and cavity mode volume), individual atom transits are detected by monitoring the (

*g*-dependent) cavity transmission amplitude using a relatively weak and near-resonant probe [28

28. H. Mabuchi, Q. A. Turchette, M. S. Chapman, and H. .J. Kimble, “Real-time detection of individual atoms falling through a high-finesse optical cavity,” Opt. Lett. **21**, 1393–1395 (1996). [CrossRef] [PubMed]

*C*≡

*g*

^{2}/2

*κγ*

_{⊥}≫ 1), the atomic ensemble is sufficiently diffuse such that no more than one strongly-coupled atom is simultaneously present in the cavity mode and we acquire data from only one transit per ensemble drop. It is in principle possible that a weakly coupled “haze” of background atoms are also coupled to the mode, but there is now a considerable literature (starting with [28

28. H. Mabuchi, Q. A. Turchette, M. S. Chapman, and H. .J. Kimble, “Real-time detection of individual atoms falling through a high-finesse optical cavity,” Opt. Lett. **21**, 1393–1395 (1996). [CrossRef] [PubMed]

*S*

_{1/2},

*F*= 4) → (6

*P*

_{3/2}

*, F*= 5), 852nm hyperfine transition (so that the diode laser is on atomic resonance). The purpose of this ∼ 80,000 finesse, 10kHz-linewidth transfer cavity is two-fold: provide a stable, Cs-locked frequency reference for all the lasers and act as a cleaning cavity for the diode laser, producing a more narrow linewidth 852nm laser source in transmission than is easily achievable in the laser lock. The 852nm beam transmitted by the transfer cavity seeds a high power slave laser, which sources the light used in both the science cavity drive and optical local oscillator. The ‘drive’ arm of the slave output is frequency shifted to the red of the atomic transition again before entering an electro-optic modulator (EOM 1), which adds the optical carrier sideband near atomic resonance that serves as the science cavity’s dynamic drive (the carrier and other sidebands are sufficiently detuned from the cQED system and have no measurable effect). The science cavity is actively stabilized by a second, titanium-sapphire (Ti:S) laser at roughly 826nm and frequency locked relative to the 852nm diode laser and the atomic transition via a frequency lock of the Ti:S laser to the transfer cavity: an optical sideband exactly two free spectral ranges to the red of the desired science cavity resonance near the atomic resonance is defined by a second EOM (EOM 2) and used to PDH resonance lock the science cavity in transmission. The power of the locking laser was kept at <100nW so that the AC Stark shift it induced in the atoms could be neglected in our analysis. As implied by the main article, the effects of the uncertainties in system parameters (e.g. laser noise, cavity resonance frequency instabilities, AC Stark shifts of the atom) are dominated by the effects of the variations in the atom-cavity coupling rate.

*μ*s transit through the mode, the drive is initially tuned several MHz from the desired frequency for the experiment and its amplitude is set below the level of atomic saturation, while the several mW optical LO is tuned to the experimental drive frequency. Both heterodyne quadratures of the transmitted atom-probe are detected via their interference with the optical LO. The heterodyne phase quadrature is used to stabilize the relative phase of the free-space LO against slow drifts in the signal and LO path lengths via a 1kHz bandwidth optical phase lock loop. The amplitude quadrature is monitored by a Schmitt trigger that fires when the measured field amplitude drops below a threshold indicating a strongly coupled atom in the cavity. This trigger shifts the drive amplitude, frequency and LO-relative phase to the desired experimental configuration and initiates data acquisition of the homodyne photocurrent at 200MS/s. Although collected at full bandwidth, the data presented in the main article has been additionally filtered for clarity using a 20MHz two-pole low pass, filtering out as much of the high frequency shotnoise as possible while still preserving the visibility of the fast, quantum fluctuations.

## B. cQED modeling and photocurrent predictions

*h̄*= 1) where

*a*is the cavity mode annihilation operator,

*σ*is the atomic lowering operator and

^{†}denotes the Hermitian conjugate. From left to right, the RHS terms correspond to the atom-drive detuning, the mode-drive detuning, the atom-mode coupling, and the external drive, respectively. The complete quantum model of this system comes from extending these Hamiltonian dynamics to include processes associated with the dissipation of photons through the cavity mirrors at mean rate 2

*κ*per intra-cavity photon and excited atomic state spontaneous emission at mean rate 2

*γ*

_{⊥}, as modeled by

*a*and

*σ*operator-coupling to external quantum fields [30, 31]. Only one of these fields, the transmitted mode, is monitored by our homodyne detection setup. For many applications, the entire model may be effectively represented by a

*master equation*that describes the unconditional evolution of any mode and/or atom operator

*O*(in the Heisenberg picture) [32, 31] For example, the (one dimensional) null space of the analogous, Schrödinger picture ‘Liouvillian,’ ℒ

*, corresponds to the steady state density matrix utilized in Figs. 1(b) and 3 in the main article. This model may be trivially extended to describe the interaction of any number of two level atoms with the mode (or even with multiple modes).*

_{S}**73**, 063801 (2006). [CrossRef]

*e.g.*〈

*aσ*

^{†}〉 ≈ 〈

*a*〉〈

*σ*

^{†}〉 (see also [23

23. H. Mabuchi, “Derivation of Maxwell-Bloch-type equations by projection of quantum models,” Phys. Rev. A **78**, 015801 (2008). [CrossRef]

*a*〉 and other expectations. Steady state solutions for 〈

*a*〉 (proportional to the mean field transmitted by a Fabry-Perot cavity) and their dynamical stability may be found. Properly scaled, these mean-field solutions depend on the atom(s)-mode coupling only through a dimensionless parameter known as the ‘cooperativity,’

*C*=

*Ng*

^{2}/2

*κγ*

_{⊥}, where

*N*is the number of coupled atoms and

*g*the rate of coupling to a single atom [6

**73**, 063801 (2006). [CrossRef]

*C*≫ 1 systems, however, the correlations between discrete excitations in the mode (photons) and the atom(s) that are ignored in the mean-field model may play a significant role in the overall dynamics when

*N*∼ 1, as demonstrated in the main article, with our experimental

*C*≲67,

*N*=1 system.

33. A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. **2**423–441 (1990). [CrossRef]

33. A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. **2**423–441 (1990). [CrossRef]

*β*is some complex-valued scalar function of time,

_{s}***denotes complex conjugation,

*B*is the QSDE annihilation process of the measured, transmitted field, 𝒯 is the time-ordering operator, and the expectation is taken over both the system and external field degrees of freedom.

_{out,s}*β*

_{θ}_{+}

_{π}_{/2}-quadrature produces the characteristic functional where

*k*is an aribtrary real-valued scalar function of time and

_{s}*dY*/

_{θ,s}*dt*≡

*I*is the homodyne photocurrent operator of quadrature

_{s}*θ*. This functional may be used to calculate moments of instantaneous photocurrent measurements [33

33. A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. **2**423–441 (1990). [CrossRef]

*t*is calculated to be where

*t*<

*t*′ <

*T*) Note that the first term on the RHS of Eq. (9) may be identified as the shotnoise contribution to the photocurrent correlation function, while the second is the contribution from the system. As the detector-filtered photocurrent operator is

*f*(

_{t}*s*) is some filter function imposed by the detection at time

*t*(e.g. a low-pass filter initiated at time

*t*), the above two equations may in principle be used to calculate the mean and variance of filtered photocurrent measurements. In practice, though, we again invoke a small integration time approximation, justified by our high bandwidth detection. If we approximate the state

*ρ*as static over the effective integration time (as far as detection is concerned), we may take

_{t}*ρ*in the RHS of Eqs. (8) and (9) as independent of time over the effective integration interval and approximate ℒ(

_{t}*t*′ –

*t*) ≈ 0, greatly simplifying these calculations. These approximations again allow us to model the detector filter function by a simple time-averaging filter of width

*τ*= 2/(

*πf*), where

_{c}*f*= 20MHz is the cut-off frequency of the two-pole low pass filter that sets the band-width of the data presented in the main article. Thus the mean and variance of the photocurrent measurements expected from our detector when the system is in the state

_{c}*ρ*may be calculated from These methods and approximations were used to calculate the theory curves in Fig. 5 of the main article. Integrating the Schrodinger-picture master equation for AM drive (initiating system and drive in the ground state) returns an expected ensemble system state

_{t}*ρ*at each point in a drive cycle. These states were then used to calculate the expected ensemble mean and standard deviation of the photocurrent measurements using Eqs. (10), given the bandwidth of the data. Relating the master equation’s quantum ensemble, single cycle predictions to the several cycle, single shot data in Fig. 5 is justified by an assumption of the system’s ergodicity (see below).

_{t}*ψ*(

_{c}*t*)〉 by numerically integrating the stochastic Schrodinger equation [26

26. S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B: Quantum Semiclass. Opt. **1**, 424–432 (1999). [CrossRef]

*denotes expectation with respect to |*

_{c}*ψ*(

_{c}*t*)〉 and the state vector is forcibly re-normalized after each recursive update. The simulated photo-increment

*dY*may then be obtained using this state trajectory and calibrated detection efficiency

_{t}*η*by where

*i.e.*reproducing the amplitude quadrature component of Fig. 3(d) in the main article using simulated data). Finally, Fig. 7(d) compares the sample mean and sample standard deviation of simulated amplitude quadrature measurements during 50 cycles of a 1MHz AM external drive, for comparison to Fig. 5(d) in the main article. Both Figs. 7(c) and (d) utilize several times more aggregate data than the experimental data presented in the main article in order to articulate the convergence of predictions from quantum trajectories to those of master equation simulations. As we expect both types of systems to be stationary ergodic processes, we have empirically confirmed that the main discrepancies between quantum trajectory and steady state master equation based simulations observed in Fig. 7 arise from the marginal appropriateness of the

*τ*ℒ → 0 approximation when modeling 20MHz bandwidth signals derived from our system.

## Acknowledgments

## References and links

1. | J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics |

2. | T. Aoki, A. S. Parkins, D. J. Alton, C. A. Regal, B. Dayan, E. Ostby, K. J. Vahala, and H. J. Kimble, “Efficient Routing of Single Photons by One Atom and a Microtoroidal Cavity,” Phys. Rev. Lett. |

3. | L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature |

4. | C. Savage and H. .J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. |

5. | G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble, “Optical bistability and photon statistics in cavity quantum electrodynamics,” Phys. Rev. Lett. |

6. | M. A. Armen and H. Mabuchi, “Low-lying bifurcations in cavity quantum electrodynamics,” Phys. Rev. A |

7. | J. Kerckhoff, M. A. Armen, D. S. Pavlichin, and H. Mabuchi, “The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying,” Opt. Express |

8. | D. A. B. Miller, “Are optical transistors the logical next step?,” Nature Photonics |

9. | S. H. Strogatz, |

10. | H. Mabuchi, “Coherent-feedback control strategy to suppress spontaneous switching in ultralow power optical bistability,” Appl. Phys. Lett. |

11. | K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature |

12. | A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nature Physics |

13. | L. A. Lugiato, in |

14. | A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. |

15. | S. D. Smith, “Optical bistability, photonic logic, and optical computation,” Appl. Opt. |

16. | S. Ya. Kilin and T. B. Krinitskaya, “Single-atom phase bistability in a fundamental model of quantum optics,” J. Opt. Soc. Am. B |

17. | C. J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kimble, “Real-time cavity QED with single atoms,” Phys. Rev. Lett. |

18. | X. Yang, C. Husko, C. W. Wong, M. Yu, and D. L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. |

19. | K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nature Photonics |

20. | H. J. Carmichael in |

21. | H. Gang, C. Z. Ning, and H. Haken, “Codimension-two bifurcations in single-mode optical bistable systems,” Phys. Rev. A |

22. | H. Gang, C. Z. Ning, and H. Haken, “Distribution of subcritical Hopf bifurcations and regular and chaotic attractors in optical bistable systems,” Phys. Rev. A |

23. | H. Mabuchi, “Derivation of Maxwell-Bloch-type equations by projection of quantum models,” Phys. Rev. A |

24. | M. A. Armen, A. E. Miller, and H. Mabuchi, “Spontaneous dressed-state polarization in the strong driving regime of cavity QED,” Phys. Rev. Lett. |

25. | A. C. Doherty, A. S. Parkins, S. M. Tan, and D. F. Walls, “Motion of a two-level atom in an optical cavity,” Phys. Rev. A |

26. | S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B: Quantum Semiclass. Opt. |

27. | J. Kerckhoff, H. I. Nurdin, D. S. Pavlichin, and H. Mabuchi, “Designing quantum memories with embedded control: Photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett. |

28. | H. Mabuchi, Q. A. Turchette, M. S. Chapman, and H. .J. Kimble, “Real-time detection of individual atoms falling through a high-finesse optical cavity,” Opt. Lett. |

29. | R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B |

30. | P. Berman., Ed., |

31. | H. J. Carmichael, |

32. | C. W. Gardiner and P. Zoller, |

33. | A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(020.5580) Atomic and molecular physics : Quantum electrodynamics

(130.3750) Integrated optics : Optical logic devices

(190.1450) Nonlinear optics : Bistability

(190.3100) Nonlinear optics : Instabilities and chaos

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(020.1335) Atomic and molecular physics : Atom optics

(130.4815) Integrated optics : Optical switching devices

**ToC Category:**

Atomic and Molecular Physics

**Citation**

Joseph Kerckhoff, Michael A. Armen, and Hideo Mabuchi, "Remnants of semiclassical bistability in the few-photon regime of cavity QED," Opt. Express **19**, 24468-24482 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24468

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### References

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