Spontaneous dressed-state polarization in the strong driving regime of cavity QED
aa r X i v : . [ qu a n t - ph ] J u l Spontaneous dressed-state polarization in the strong driving regime of cavity QED
Michael A. Armen,
1, 2
Anthony E. Miller, and Hideo Mabuchi Edward L. Ginzton Laboratory, Stanford University, Stanford CA 94305, USA Physical Measurement and Control 266-33, California Institute of Technology, Pasadena CA 91125, USA (Dated: August 31, 2018)We utilize high-bandwidth phase quadrature homodyne measurement of the light transmittedthrough a Fabry-Perot cavity, driven strongly and on resonance, to detect excess phase noise inducedby a single intracavity atom. We analyze the correlation properties and driving-strength dependenceof the atom-induced phase noise to establish that it corresponds to the long-predicted phenomenonof spontaneous dressed-state polarization. Our experiment thus provides a demonstration of cavityquantum electrodynamics in the strong driving regime, in which one atom interacts strongly with amany-photon cavity field to produce novel quantum stochastic behavior.
PACS numbers: 42.50.Pq,42.50.Lc,42.65.Pc,42.79.Ta
Current research in single-atom optical cavity quan-tum electrodynamics (cavity QED) [1] largely empha-sizes the input-output properties of strongly coupled sys-tems [2], from normal-mode splitting [3] to photon block-ade [4, 5]. While theory has predicted a wide rangeof quantum nonlinear-optical phenomena in the strongdriving regime [6], experiments have with few excep-tions [7, 8, 9, 10] focused on relatively weak drivingconditions with average intracavity photon number . ≫ Cs atoms is droppedover a high-finesse Fabry-Perot cavity (cavity length l ≈ µ m, field decay rate κ/ π ≈ S / , F = 4) → (6 P / , F = 5) transition (dipole de-cay rate γ ⊥ / π ≈ . ≈ . µ m), the strength g of the coherentatom-cavity coupling is a function of the atomic position(with maximum value ≈
16 MHz in our setup). We selecttransit events in which optical pumping and the atomictrajectory lead to maximal values of g by initializing thecavity probe in a detuned configuration that producesa real-time photocurrent directly related to g . Whena set threshold is reached during a single-atom transit,the probe frequency and power are quickly shifted to de-sired values for data acquisition. Using this triggeringscheme we obtain phase-quadrature homodyne data inwhich near-maximal atom-cavity coupling strength is ap-parently maintained for up to ∼ µ s, limited by opticalpumping into the dark (6 S / , F = 3) hyperfine state.Fig. 1a shows a representative example of the single-shot photocurrents thus obtained. A distinct transitionin the signal can be seen at time t ∗ ∼ µ s. The pho- FIG. 1: (a) Black solid trace is a photocurrent segmentrecorded with input power such that N ≈
20, filtered to abandwidth of 4 MHz. Blue dashed horizontal lines indicatethe standard deviation of the optical shot noise. The atom isoptically pumped to a dark state at time t ∼ µ s, resultingin an abrupt disappearance of the atom-induced excess phasenoise. Units are referred to the phase quadrature amplitude ofthe intra-cavity field. (b) Histogram of filtered photocurrentsegments (0.1–8 MHz bandpass) from multiple atom transits;see text for explanation of the theoretical curve. tocurrent variance for t > t ∗ corresponds to optical shotnoise while for t < t ∗ the variance is clearly larger, indi-cating a significant effect of the atom on light transmittedthrough the cavity. We interpret the change as an opti-cal pumping event in which the atom is transferred tothe dark hyperfine ground state, and have verified thatsuch events can be suppressed by adding an intracavityrepumping beam. Although the signal-to-noise ratio inour measurements is limited, a histogram of photocurrentsegments from multiple atom transits (Fig. 1b) reveals aflat-topped distribution supporting the theoretical expec-tation of random-telegraph (rather than Gaussian) statis-tics of the atom-induced phase noise. The smooth curveis a theoretical prediction produced by fitting the sum oftwo Gaussian functions (constrained to have width corre-sponding to optical shot noise) to histograms generatedvia quantum trajectory simulations of our cavity QEDmodel [18, 19]. In the limit of low sampling noise ourexperimental histogram would be expected more clearlyto display such a bimodal distribution, although somedeviations from ideal theory would presumably emerge FIG. 2: Driving-strength dependence of the splitting (in unitsof phase quadrature amplitude of the intra-cavity field) be-tween the centroids of bi-Gaussian fits to photocurrent his-tograms as in Fig. 1b. The experimental data (points witherror bars) are directly compared with theoretical predictionsbased on quantum trajectory simulations (solid curves) andthe cavity QED Master Equation (see text). Blue points andcurves are computed with data and simulations filtered to10 MHz; red points and curves reflect filtering at 2 MHz,which results in a decrease in apparent switching amplitudesince 2 MHz is well below the purported switching frequency.Black horizontal dotted lines indicate the splitting predictedby steady-state solution of the Master Equation for large N . because of residual atomic motion and optical pumpingamong Zeeman sub-states.In Fig. 2 we summarize a comparison of experimentaland theoretical photocurrent histograms, such as the onedepicted in Fig. 1b, across a range of probe powers. Theprobe power is conventionally parametrized by the meanintracavity photon number N that would be produced inan empty cavity; note that for N & ∼ N −
1. We display the best-fit values of thetwo Gaussian centroids obtained in fits to data and sim-ulations; the blue points and theoretical curves are forthe maximal signal bandwidth of 10 MHz while the redpoints and curves are computed with signals and simula-tions filtered to 2 MHz. The horizontal lines indicate thesplitting predicted by steady-state solution of the cavityQED Master Equation in the asymptotic region of large N . Our data closely match the predicted sharp onsetof atom-induced phase fluctuations for lower values of N and also asymptote correctly for high N . That the split-ting becomes independent of N for high N is a distinctivefeature of spontaneous dressed-state polarization [12].In Fig. 3 we display the autocorrelation functions ofexperimental photocurrent segments for four character-istic sets of probe parameters, together with theoreti-cal predictions. Data points and theoretical curves aredisplayed for N = (4 , ,
56) with the atom, cavity and τ [ µ s] Y ( t + τ ) Y ( t ) N=4N=20N=20, ∆ =40MHzN=56 FIG. 3: Autocorrelation functions of the photocurrent Y ( t )(units as in Figs. 1, 2) obtained under a range of driving pa-rameters. Experimental autocorrelations are computed afterac-filtering the photocurrents (20 kHz cutoff) to suppress ar-tifacts caused by atomic motion and optical pumping. Pointsare experimental data and curves are theoretical predictionsbased on the cavity QED Master Equation for four differentparameter sets (see legend and text). probe frequencies all coincident, and for N = 20 withthe cavity and probe frequencies coincident but 40 MHzbelow the atomic resonance frequency. The vertical lineindicates the predicted average period (2 /γ ⊥ ) of sponta-neous phase-switching events. In the resonant cases, therapid growth with N of predominantly short-timescalephotocurrent fluctuations agrees with theory and rein-forces our identification of the atom-induced phase noisewith switching caused by spontaneous emission. The in-creased correlation timescale for the detuned case followsfrom a tendency of the atom-cavity system to favor one ofthe sub-ladders discussed by Alsing and Carmichael [12].We should note that theory predicts identical autocorre-lation functions for atom-cavity detuning of either posi-tive or negative sign, but in the experiment setting ∆ < g : κ : γ ⊥ realized in our experiment. We have beenlimited in this regard by the properties of the mirrors wehad available at the time the experiment was assembled(significant improvements should be possible with com-mercial mirror technology [19]), and by a pronouncedbirefringence of the assembled cavity that forced us touse linear rather than circular probe polarization [20].Furthermore our maximum digitization rate in record- FIG. 4: (a) Theoretical reconstruction (red) produced via pos-terior decoding [21] of a two-state switching trajectory from asegment of the experimental homodyne photocurrent (black,10 MHz bandwidth, N ≈ t ∼ µ sthe reconstruction algorithm correctly identifies the end ofthe atom-induced fluctuations and infers a ‘dark’ signal statewith zero mean and Gaussian shot-noise fluctuations only. (b)Quantum trajectory simulation of the expected switching be-havior in a cavity QED system (conditional expectation valueof the phase quadrature amplitude of the intra -cavity field)using parameter values of the current experiment. (c) Simu-lated phase-quadrature homodyne photocurrent correspond-ing to (b), including shot-noise and finite bandwidth as in theexperimental data. Note that the duration of the simulationsin (b) and (c) are ≈ µ s. ing the photocurrent was 2 . × samples per second,whereas the switching rate induced by atomic sponta-neous emission should be γ ⊥ / ≈ ∼ ≈
12 aJ, representing an operating regimefor optical switching that lies significantly below foresee-able improvements in existing technology but stays abovethe single-photon level where propagation losses and sig-nal regeneration would seem to be dominant engineer-ing concerns. Ultra-low energy nonlinear optical effects should be achievable with nanophotonic implementationsof cavity QED [28, 29], providing a potential path towardlarge-scale integration. Even with our modest values of g and κ , the atom-induced phase shift of light transmittedthrough the cavity is ± .
15 rad for input power corre-sponding to N = 20. Each atomic spontaneous emissionevent that switches the phase of the transmitted light dis-sipates a mere 0 .
23 aJ of energy, while the light transmit-ted through the cavity during a typical interval betweenswitching events carries an optical signal energy ≈ . N = 20. Such a 1:10 ratio between the switching en-ergy and the energy of the controlled signal in nonlinear-optical phase modulation would be highly desirable forthe implementation of cascadable photonic logic devices,and is not generally achieved in schemes based on single-photon saturation effects in cavity QED [30, 31]. While itis certainly not clear whether the specific phenomenon ofspontaneous dressed-state polarization can be exploitedin the design of practicable switching devices, we hopethat our demonstration will serve to draw some atten-tion up from the bottom of the Jaynes-Cummings laddertowards the strong-driving regime of cavity QED.This research has been supported by the NSF underPHY-0354964, by the ONR under N00014-05-1-0420 andby the ARO under W911NF-09-1-0045. AEM acknowl-edges the support of a Hertz Fellowship. We thankDavid Miller for enlightening conversations and DmitriPavlichin for crucial technical assistance in the prepara-tion of Fig. 4a. [1] H. Mabuchi and A. C. Doherty, Science , 1372 (2002).[2] H. J. Kimble, Phys. Scr. T76 , 127 (1998).[3] R. J. Thompson, G. Rempe and H. J. Kimble, Phys. Rev.Lett. , 1132 (1992).[4] K. M. Birnbaum et al. , Nature , 87 (2005).[5] B. Dayan et al. , Science , 1062 (2008).[6] M. A. Armen and H. Mabuchi, Phys. Rev. A , 063801(2006).[7] J. A. Sauer et al. , Phys. Rev. A , 213601 (2007).[9] I. Schuster et al. , Nature Phys. , 382 (2008).[10] M. Brune et al. , Phys. Rev. Lett. , 1800 (1996).[11] S. Ya. Kilin and T. B. Krinitskaya, J. Opt. Soc. Am. B , 2289 (1991).[12] P. Alsing and H. J. Carmichael, Quantum Opt. , 13(1991).[13] J. E. Reiner, H. M. Wiseman and H. Mabuchi, Phys.Rev. A , 042106 (2003).[14] E. T. Jaynes and F. W. Cummings, Proc. IEEE , 89(1963).[15] A. Boca et al. , Phys. Rev. Lett. , 233603 (2004).[16] H. Mabuchi, Q. A. Turchette, M. S. Chapman andH. J. Kimble, Opt. Lett. , 1393 (1996).[17] H. Mabuchi, J. Ye and H. J. Kimble, Appl. Phys. B ,1095 (1999).[18] H. J. Carmichael, An Open Systems Approach to Quan-tum Optics: Lectures Presented at the Universite Libre De Bruxelles October 28 to November 4, 1991 (LectureNotes in Physics New Series M) (Springer, Berlin, 1993).[19] H. Mabuchi and H. M. Wiseman, Phys. Rev. Lett. ,4620 (1998).[20] K. M. Birnbaum, Cavity QED with Mul-tilevel Atoms (California Institute ofTechnology Ph.D. Dissertation, 2005);http://resolver.caltech.edu/CaltechETD:etd-05272005-103306.[21] R. Durbin, S. Eddy, A. Krogh and G. Mitchison,
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