## The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying |

Optics Express, Vol. 19, Issue 7, pp. 6478-6486 (2011)

http://dx.doi.org/10.1364/OE.19.006478

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### Abstract

We use a single ^{133}Cs atom strongly coupled to an optical resonator to induce random binary phase modulation of a near infra-red, ∼ 500pW laser beam, with each modulation edge caused by the dissipation of a single photon (≈ 0.23aJ) by the atom. While our ability to deterministically induce phase edges with an additional optical control beam is limited thus far, theoretical analysis of an analogous, solid-state system indicates that efficient external control should be achievable in demonstrated nanophotonic systems.

© 2011 OSA

1. X. Yang, C. Husko, M. Yu, D.-L. Kwon, and C. W. Wong, “Observation of femtojoule optical bistability in high-Q/V_{m} silicon photonic crystal nanocavities,” Appl. Phys. Lett. **91**, 051113 (2007). [CrossRef]

2. 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 tunneling and blockade,” Nature Physics **4**, 859–863 (2008). [CrossRef]

3. L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photonics **4**, 182–187 (2010). [CrossRef]

4. 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]

*π*phase relative to the driving field. The re-radiated field of such a driven dipole has a phase corresponding to that of the dipole velocity,

*i.e.*, ±

*π/*2 relative to the drive. In the setting of cavity quantum electrodynamics (cQED) with strong coupling [6

6. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science **298**, 1372 (2002). [CrossRef] [PubMed]

8. P. Alsing and H. J. Carmichael, “Spontaneous dressed-state polarization of a coupled atom and cavity mode,” Quantum Opt. **3**, 13–32 (1991). [CrossRef]

9. 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]

*digital*modulation format.

10. 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]

9. 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 is dropped onto a high-finesse Fabry-Perot optical resonator (length 27

*μ*m, field decay rate

*κ/*2

*π*= 9.3MHz) with a standing wave TEM

_{00}mode (waist 18

*μ*m) actively stabilized at a frequency fixed relative to an atomic cycling transition (dipole decay rate

*γ*

_{⊥}

*/*2

*π*=2.6MHz). As individual atoms fall through the spatial profile of the cavity mode, they experience a position-dependent coupling

*g*(

*r*) to the intra-cavity field (maximal value

*g*

_{0}

*/*2

*π*= 56.8MHz at the cavity anti-nodes). A weak optical probe is used to detect atom transits; when an atom reaches a position of near-maximal coupling the probe strength is increased and its frequency brought close to atomic resonance in order to observe binary phase switching [9

9. 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]

*g*(

*r*) on time scales longer than a few

*μ*s. The effects observed here nonetheless typically persist for several tens of

*μ*s, until the atom either completely exits the cavity mode or is pumped into a dark state.

*γ*

_{⊥}

*/*2 switching rates of the most likely models (as opposed to exactly

*γ*

_{⊥}

*/*2, which occurs in the strong-driving limit) is due to the moderate power of the probe. Fig. 2(b) reflects data taken with cavity and probe held 40MHz below atomic resonance. A significant asymmetry in the transition rates is expected and inferred from the likelihood contours. Both figures compare parameter estimates obtained from experimental data to estimates from simulated photocurrent data of comparable aggregate duration and detection efficiency (quantum trajectory simulations of the driven damped Jaynes-Cummings master equation [14

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

*g/κ*is ∼ 1 and the TLS dephasing rate is relatively slow, as can occur in solid-state nanophotonic systems (which are also attractive for integrability and GHz dynamics). A detailed analysis of the efficiency of this and competing strategies would largely distract from the presentation of our initial results, and so will be deferred to a subsequent publication. In Fig. 3 we present simulations of switching-rate control using parameters from a demonstrated quantum dot-photonic crystal cavity system [16

16. A. Faraon, A. Majumder, H. Kim, P. Petroff, and J. Vučković, “Fast electrical control of a quantum dot strongly coupled to a photonic-crystal cavity,” Phys. Rev. Lett. **104**, 047402 (2010). [CrossRef] [PubMed]

*<*5% of the signal power is able to induce switching dynamics an order of magnitude faster than the spontaneous emission background rate. This control beam power corresponds to less than 100aJ per induced phase-switch. Although quite low, this switching energy is not yet at the presumed limit of a single dissipated photon (sub-aJ per edge) exemplified by the spontaneous emission-driven random switching. Control-pulse shaping, coherent feedback and/or reservoir engineering could further reduce the power required for fast deterministic phase modulation.

*g/κ*ratio, which is unfavorable for the control method simulated in Fig. 3, we have attempted to confirm these simulations of the intuitive strategy and demonstrate stimulated switching experimentally using the ‘near-detuned’ system. Figure 4 shows inferred ‘total’ transition rates (the mean rate at which switching in either direction occurs) as a function of the power of the control beam, for three different control beam frequencies. In the strong driving limit the optimum control detuning would be 425MHz, but given our moderate signal power the effective coupling energy (and therefore optimal control detuning) should be somewhat smaller. The statistical significance of the data displayed in Fig. 4 is marginal, but the data do show consistent trends that match expectations: data taken with −375MHz and −425MHz control detuning induce higher transition rates with higher power, with −375MHz consistently higher, while the −525MHz detuning induced essentially no net transition increase over the accessible range of control powers (limited by opto-mechanical effects on the high end). Linear fits to simulations reproduce these general trends of the data and confirm the limited degree of control achievable in our experimental system. We thus infer an apparent control energy requirement of only ∼ 1fJ/edge even in our atomic implementation (with unfavorable parameters for efficient control) of the dressed-state binary phase modulator.

## Appendix

## A. Experimental setup

**103**, 173601 (2009). [CrossRef] [PubMed]

10. 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]

^{133}Cs atoms and a high-finesse Fabry-Perot optical resonator. We attempt to drive only the (6

*S*

_{1}

_{/}_{2},

*F*= 4,

*m*= +4) → (6

_{F}*P*

_{3}

_{/}_{2},

*F*= 5,

*m*= +5) atomic cycling transition at 852nm through frequency- and polarization-selectivity so that the atom may be approximated as a TLS. Optimized mirror paramters and the use of an improved cavity geometry and mirror mounting scheme, as compared to a previous experimental study of single-atom cavity QED in the strong driving regime [9

_{F}**103**, 173601 (2009). [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 radii 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 spacial 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 [17

17. 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 max

*{*

_{r}*g*(

*r*)} ≡

*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*(

*r*)-dependent) cavity transmission amplitude using a relatively weak and near-resonant probe, a free space balanced photodetector, and an actively phase-locked optical local oscillator. Once a near-maximally coupled atom has been detected, the probe power and frequency shift to the desired experimental levels and data acquisition is initiated. Although multiple atom transits per drop may be visible, the atomic ensemble is sufficiently diffuse such that no more than one atom is simultaneously present in the cavity mode and we acquire data from only one transit per ensemble drop. Due to the many sensitive stabilization requirements and slow drifts in the experimental apparatus, data are usually analyzed in groups of 50 atom transits, over which experimental stability can be confidently maintained.

*g*(

*r*)), modulation in the switching variance is typically apparent over timescales greater than a few microseconds; in fact, a significant fraction of the transit segments display near-sinusoidal modulation in the switching variance, corresponding to atomic motion through several standing wave anti-nodes.

## B. The Jaynes-Cummings model and simulation

18. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE **51**, 89 (1963). [CrossRef]

*h̄*= 1): where

*a*is the annihilation operator for the cavity mode,

*σ*= |

*g*〉〈

*e*| is the TLS lowering operator and

^{†}signifies the Hermitian conjugate. Δ is the detuning between the probe and the atomic transition frequencies and Θ is the detuning between the cavity mode resonance and the probe (Θ = 0 in all systems considered here). The third, ‘coupling’ term represents the interaction between the atomic transition and cavity mode and describes the process by which quanta of energy are exchanged (at rate ∝

*g*(

*r*)) between the TLS and mode. The final term represents the coherent driving term, with amplitude

*κ*per intra-cavity photon and the spontaneous emission of an excited TLS at a rate of 2

*γ*

_{⊥}. Standard quantum trajectory theory [15] based on these processes underlies much of the theoretical analysis and numerical simulation. For example, simulation of a photocurrent given a set of experimental parameters first involves the calculation of a possible trajectory for the internal quantum state vector of the cQED system |

*ψ*(

_{c}*t*)〉 by numerically integrating the stochastic Schrödinger equation [14

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

*dQ*(

*t*) may then be calculated using this state trajectory and calibrated detection efficiency

*η*by where

*g*(

*r*) and a more realistic, multi-level atomic structure, all simulations here utilized the maximal

*g*

_{0}for the static coupling rate and assumed a TLS atomic model.

## C. Hidden Markov model analysis

## Acknowledgments

## References and links

1. | X. Yang, C. Husko, M. Yu, D.-L. Kwon, and C. W. Wong, “Observation of femtojoule optical bistability in high-Q/V |

2. | 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 tunneling and blockade,” Nature Physics |

3. | L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photonics |

4. | 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 |

5. | C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Ch. 6 in |

6. | H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science |

7. | P. Berman, ed., |

8. | P. Alsing and H. J. Carmichael, “Spontaneous dressed-state polarization of a coupled atom and cavity mode,” Quantum Opt. |

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

10. | 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. |

11. | O. Cappé, E. Moulines, and T. Rydén, |

12. | L. R. Welch, “Hidden Markov models and the Baum-Welch algorithm,” IEEE Information Theory Society Newsletter Vol. |

13. | A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory |

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

15. | H. Carmichael, |

16. | A. Faraon, A. Majumder, H. Kim, P. Petroff, and J. Vučković, “Fast electrical control of a quantum dot strongly coupled to a photonic-crystal cavity,” Phys. Rev. Lett. |

17. | 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 |

18. | E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE |

**OCIS Codes**

(020.5580) Atomic and molecular physics : Quantum electrodynamics

(060.5060) Fiber optics and optical communications : Phase modulation

(130.3750) Integrated optics : Optical logic devices

(190.1450) Nonlinear optics : Bistability

(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

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: February 22, 2011

Manuscript Accepted: March 2, 2011

Published: March 22, 2011

**Citation**

Joseph Kerckhoff, Michael A. Armen, Dmitri S. Pavlichin, and Hideo Mabuchi, "The dressed atom as binary phase modulator: towards attojoule/edge optical phase-shift keying," Opt. Express **19**, 6478-6486 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6478

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### References

- X. Yang, C. Husko, M. Yu, D.-L. Kwon, and C. W. Wong, “Observation of femtojoule optical bistability in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. 91, 051113 (2007). [CrossRef]
- 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 tunneling and blockade,” Nature Physics 4, 859–863 (2008). [CrossRef]
- L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photonics 4, 182–187 (2010). [CrossRef]
- 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]
- C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Ch. 6 in Atom-Photon Interactions: Basic Processes and Applications (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
- H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298, 1372 (2002). [CrossRef] [PubMed]
- P. Berman, ed., Cavity Quantum Electrodynamics (San Diego: Academic Press, 1994).
- P. Alsing and H. J. Carmichael, “Spontaneous dressed-state polarization of a coupled atom and cavity mode,” Quantum Opt. 3, 13–32 (1991). [CrossRef]
- 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]
- 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]
- O. Cappé, E. Moulines, and T. Rydén, Inference in Hidden Markov Models (Springer Science+Business Media, New York, 2005).
- L. R. Welch, “Hidden Markov models and the Baum-Welch algorithm,” IEEE Information Theory Society Newsletter Vol. 53, No. 4 (December2003).
- A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory 13260–269 (1967). [CrossRef]
- S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B: Quantum Semiclass. Opt. 1, 424–432 (1999). [CrossRef]
- H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
- A. Faraon, A. Majumder, H. Kim, P. Petroff, and J. Vučković, “Fast electrical control of a quantum dot strongly coupled to a photonic-crystal cavity,” Phys. Rev. Lett. 104, 047402 (2010). [CrossRef] [PubMed]
- 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 3197 (1983). [CrossRef]
- E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89 (1963). [CrossRef]

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