Vacuum Rabi oscillation induced by virtual photons in the ultrastrong coupling regime C. K. Law Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region, People’s Republic of China Wepresentaninteractionschemethatexhibitsadynamicalconsequenceofvirtualphotonscarried by a vacuum-field dressed two-level atom in the ultrastrong coupling regime. We show that, with theaidofanexternaldrivingfield,virtualphotonsprovideatransitionmatrixelementthatenables theatom toevolvecoherentlyandreversiblytoanauxiliary levelaccompanied bytheemission ofa real photon. The process corresponds to a type of vacuum Rabi oscillation, and we show that the effective vacuum Rabi frequency is proportional to the amplitude of a single virtual photon in the groundstate. Therefore theinteraction schemecouldserveasaprobeofgroundstatestructuresin theultrastrong coupling regime. 3 1 PACSnumbers: 42.50.Pq,42.50.Ct,42.50.Lc 0 2 n A single-mode electromagnetic field interacting with a f a two-level atom has been a fundamental model in quan- J ω tumopticscapturingthephysicsofresonantlight-matter p 2 interaction. In particular, the Jaynes-Cummings (JC) model [1, 2], which describes the regime where the inter- e ] h action energy ~λ is much smaller than the energy scale ω p of an atom ~ωA and a photon ~ωc, has tremendous ap- c g - plications in cavity QED [3, 4] and trapped ion systems t n [5]. Recently, there has been considerable researchinter- FIG.1: (Coloronline)InteractionschemeofaΞ−typethree- a u est in the ultrastrong coupling regime where λ becomes level atom in a cavity. The atomic states |gi and |ei and a q comparable to ωc and ωA. Such a regime has been ex- cavityfieldmodeoffrequencyωcformaquantumRabimodel [ plored by experiments in various related systems with described by HR, and an external classical field of frequency artificial atoms and cavity photon resonators, including ωp drives thetransition between |ei and |fi. 2 superconducting qubit in coplanar waveguide [6] or LC v resonator [7], microcavities embedding doped quantum 4 wells [8, 9], and two-dimensional electron gas coupled oretical aspects of three-level artificial atoms in circuit 6 8 to metamaterial resonators [10]. In addition, theoreti- QEDwasdiscussedin[17],andΞ−typesuperconducting 5 cal investigations have also found novel phenomena in atoms have been demonstrated in experiments [18–20]. . the ultrastrong coupling regime, such as the asymmetry Recently Carusotto et al. have studied the dynamics of 2 1 ofvacuumRabi-splitting[11],photonblockade[12],non- a related system in a different driving configuration[21]. 2 classical states generation [13], superradiance transition The Hamiltonian of our system is given by (~=1), 1 [14], and collapse and revivals dynamics [15]. v: Akeyfeatureinthe ultrastrongcouplingregimeisthe H =HR+ωf|fihf|+Ωcosωpt(|fihe|+|eihf|) (1) i significantnumberofvirtualphotonsexistingaroundthe X where H is the Hamiltonian of the Rabi model [22], vacuum-field dressed atom. These virtual photons are R r generatedby counter-rotatingterms in the Hamiltonian, ω a and they can have direct physical consequences. For ex- HR = 0(|eihe|−|gihg|)+ωca†a+λ(a+a†)(|gihe|+|eihg|). 2 ample, by modulating the atom-field coupling strength (2) virtual photons can be released as a form of quantum Hereω isthe(bare)transitionfrequencybetween|eiand 0 vacuum radiation [16]. In this paper we address a dif- |gi,andω −ω /2isthetransitionfrequencybetween|fi f 0 ferent effect of the vacuum-field dressedatom, namely, a and |ei. The parameter λ denotes the atom-cavity cou- kind of vacuum Rabi oscillations that would not occur if pling strength, and the classical driving field has a fre- virtual photons are absent. quencyω andaninteractionstrengthΩ. InwritingH , p R Specifically, we investigate the quantum dynamics of we have kept counter-rotating terms because λ is com- a driven quantum Rabi model. The configuration of our parable to ω in the ultrastrong coupling regime. Note c system is shown in Fig. 1 in which a Ξ-type three-level that the coupling between the cavity mode and the level atomisconfinedinasingle-modecavity. Theatomiclev- |fi is assumed to be weak and so that it is not included els |gi and |ei are coupled to a cavity field of frequency in the Hamiltonian. ω . These two atomic levels and the cavity field mode Initially the system is prepared in the ground state of c constitute a Rabi model. In addition, there is an ex- H ,whichisthelowest-energystateofthesysteminthe R ternal classical field driving the transition between |ei absenceofthe drivingfield. Ourtaskistodeterminethe and the third atomic level |fi. We note that some the- dynamics after the driving field is turnedon. To analyze 2 the problem, we apply a unitary transformation to sim- whereΩ′ =η1/4Ωisarenormalizeddrivingfieldstrength, plify the Hamiltonian. It is known that for low energy and the last term indicates a new coupling between |gi states ofthe Rabi model, H canbe transformedto into and |fi through the cavity field mode. R a form of Jaynes-Cummings Hamiltonian approximately Afurthersimplificationcanbemadebyexploitingres- by a unitary operatore−S [23]. Here the operatorS and onancewhenω istunedtoacertainresonancefrequency p its parameters are defined by: defined by the undriven system. In this paper we con- sider the resonance at λξ S = (|gihe|+|eihg|)(a†−a), (3) ω λ2ξ ω′ c ω =ω +ω − (ξ−2)− 0 , (11) ξ = ωc , (4) p f c (cid:20) ωc 2 (cid:21) ω +ηω c 0 which corresponds to the transition between |g,0i to 2λ2ξ2 η =exp(− ). (5) |f,1i, since the square bracket term is the approxi- ωc2 mate ground state energy of HR by the transformation Then it can be shown that H′ = eSH e−S is approxi- method. By the condition (11), |g,0i and |f,1i are res- R R onantly coupled, but |f,1i and |e,1i is far away from mately given by [23–26] resonance (the corresponding detuning is of order ω ). c ω′ Therefore if Ω′ is not too strong, the system is confined H′ ≈ 0(|eihe|−|gihg|)+ω a†a+λ′(a|eihg|+a†|gihe|) R 2 c tothe tworesonantlycoupledstates,i.e., alloff-resonant λ2ξ transitions may be ignored. In this way H′ in the inter- + (ξ−2)(|eihe|+|gihg|) action picture is reduced to ω c ≡ HJC (6) H′ ≈ − λξ Ω′(|g,0ihf,1|+|f,1ihg,0|). (12) I 2ω whereH describesaJCmodelinwhichtheatomicfre- c JC quency and cavity-atom interaction strength are renor- Eq. (12) indicates that the systemwould execute a form malized as ω0′ =ηω0 and λ′ =2ηω0ξλ/ωc, respectively. ofvacuumRabioscillations,inwhich|g,0ibehavesasan Note that HJC in Eq. (6) is an approximationto HR′ , excited atom in the vacuum field, and |f,1i behaves as andthedifferenceHR′ −HJC describesmulti-photonpro- an ground atom with a single photon. In cavity QED, cesses that correspond to higher order corrections [23– such oscillations lead to vacuum Rabi splitting [27–29]. 26]. Since|g,0iisthegroundstateofHJC,e−S|g,0iisan Note that the effective vacuum Rabi frequency here is approximated ground state of HR in the original frame. λξΩ′/ωc, which is significant in the ultrastrong coupling The accuracy of such an approximation has been tested regime where λ is comparable to ω . c in Ref. [23]. Specifically, if λ is comparable but smaller It is useful to go back to the original frame in which than ωc, the ground state energy of HJC has a good the Rabi oscillations occur between the states e−S|f,1i agreement with that of HR obtained by exact numerical and e−S|g,0i. Since e−S|f,1i = |f,1i, an initial ground calculations over a range of parameters. For example in state will evolve to |f,1i after half of a Rabi period. If the case ωc = ω0 = 2λ, the approximated ground state weswitchofftheexternalfieldatthismoment,thesingle energy obtained by HJC has the percentage error about photondescribedby|f,1iwillbefreetoescapethecavity 0.65%. because the atom in the state |fi does not couple to Now we perform the transformation for our system the cavity field when Ω = 0, i.e., the photon cannot be Hamiltonian H, which becomes, reabsorbed by the atom. In this way, a π pulse of the H′ = eSHe−S drivingfieldcangeneratearealphotondeterministically while the atom is excited to the |fi state. ≈ H +ω |fihf| JC f Togainabetterinsightofthephysicalprocesswithout +Ωcosω t(eS|eihf|+|fihe|e−S). (7) relyingonthe approximationmade inEqs. (6) and(10), p weexpressthe HamiltonianbytheeigenbasisofH . Let SinceeS|ei=cosh[λξ(a†−a)]|ei+sinh[λξ(a†−a)]|gi,we R ωc ωc |ψnibeaneigenvectorofHR withtheeigenvalueλn,i.e., expandthehyperbolicsineandcosineoperatorfunctions H |ψ i=λ |ψ i (the ground state is denoted by |ψ i), R n n n 0 in normal order up to first order in λξ/ω , 0 and consider the expansion |e,ni = c |ψ i with m nm m λξ the coefficients cnm =hψm|e,ni. TherPefore cosh (a†−a) ≈ η1/4, (8) (cid:20)ω (cid:21) c |fihe|= |f,nihe,n|= c∗ |f,nihψ |. (13) nm m sinh λξ(a†−a) ≈ η1/4λξ(a†−a) (9) Xn Xnm (cid:20)ω (cid:21) ω c c Inthisway,theHamiltonian(1)intheinteractionpicture Therefore the transformed Hamiltonian becomes, becomes, H′ ≈ HJC +ωf|fihf|+Ω′cosωpt(|fihe|+|eihf|) H =Ωcosω t ei(ωf+nωc−λm)tc∗ |f,nihψ |+h.c. I p nm m +λξΩ′cosω t(|gihf|−|fihg|)(a†−a) (10) Xnm ω p (14) c 3 λ/ω c 1.0 0 0.2 0.4 0.6 0.8 1.0 0.8 −0.1 −0.2 0.6 P c10 −0.3 1f 0.4 −0.4 0.2 −0.5 0 20 40 60 80 100 FIG. 2: (Color online) Probability amplitude of |e,1i in the ω t c ground state of HR as a function of the coupling strength λ for theω0 =ωc case. The solid red line corresponds to exact FIG. 3: (Color online) Probability of |f,1i as a function numerical values, and the dashed blue line is obtained from theapproximatedgroundstatee−S|g,0iaccordingtoEq. (6). of time for Ω = 0.2ωc(blue long dashed), 0.4ωc(green short dashed) and 0.8ωc (red solid). The parameters used are: λ = 0.5ωc, ωc = ω0 = ωf/3, ωp = ωf +ωc −λ0, and the numerical ground state energy λ0 =−0.633ωc. The figure is At the resonant frequency ω = ω +ω −λ , |ψ i and p f c 0 0 essentially thesame if ωp in Eq. (11) is used. |f,1iareresonantlycoupled. Ifwekeeponlytheresonant terms, then we have Ωc∗ oscillations are less perfect in the sense that the maxi- HI ≈ 210|f,1ihψ0|+h.c. (15) mum P1f ≈ 0.9 is smaller than one. Such a behavior is understood because the off-resonance transitions ne- Comparing with H′ in Eq. (12) and noting that |ψ i ≈ glected in Eq. (12) or (15) would generate energy shifts I 0 e−S|g,0i, H describes the same type of resonant inter- which in turn could bring the driven system out of reso- I action as HI′. However, we emphasize that HI in Eq. nance. Asaresult,theamplitudeofoscillationsinP1f is (15)isa moreaccurateinteractionHamiltonianthanH′ reduced. Since these energy shifts are generally propor- I because H is deriveddirectly fromthe eigenbasisof H tionalto Ω2, as long as Ω is small comparedwith detun- I R without making use of the approximation in Eq. (6). In ings associated with off-resonance transitions, it would this sense, the resonant condition (11) can be improved be safe to use Eq. (15), and this is demonstrated in Fig. by replacing the square bracket term by λ0. 3 for Ω up to 0.4ωc. TheroleofvirtualphotonsisnowexplicitlyseeninEq. Finally,it is worthnoting thatthe Hamiltonianin Eq. (15)throughtheeffectivevacuumRabifrequencyΩ|c |. (14)hashigherresonancesatω =ω +nω −λ forodd 10 p f c 0 Thisisbecausec ispreciselythe probabilityamplitude positive integers n. The requirement of an odd n is be- 10 of a single virtual photon state in |ψ i. In other words, cause|ψ ihasadefiniteparityinwhichtheatomicstate 0 0 we may interpret that the interaction described in Eq. |ei and odd photon numbers are connected. In the case (15) is induced or mediated by a virtual photon. In Fig. n = 3, the driving field at the corresponding ω would p 2, we plot c (solid line) as a function of λ/ω for the resonantly excite the atom to |fi with the emission of 10 c caseω =ω ,andthefigureshowsthatthemagnitudeof three real photons. The effective Hamiltonian would be c 0 c is appreciable in the ultrastrongcoupling regime. As ofthe sameformof(15),butwith|f,1iandc∗ replaced 10 10 a comparison, we also plot the approximate amplitude by|f,3iandc∗ , i.e., the effective Rabifrequencyis pro- 30 c ≈−η1/4ξλ/ω (dashed line) obtained from e−S|g,0i. portional to |c |. Such a three-photon resonance was 10 c 30 FortheparametersusedinFig. 2,weseethattheapprox- also observed in our numerical calculations. imation agrees well with the exact numerical calculation To conclude, we have shown that virtual photons in up to λ/ω <0.6. the ultrastrong coupling regime can play a key role in c We have tested our prediction of the virtual-photon- quantumdynamicsbyprovidingthetransitionmatrixel- induced Rabi oscillations by solving numerically the ementsthatallowthesystemtoaccessrelevantquantum Schr¨odinger equation defined by the Hamiltonian (1) statesofinterest. Inourscheme,thesystemcanexhibita withtheinitialstate|ψ i. InFig. 3weplottheexactnu- formofvacuumRabioscillationswhichcanbeconsidered 0 mericalprobabilityP ofthesysteminthestate|f,1ias asasignatureofvirtualphotons. Sinceourmainfocusin 1f a function of time. The parameter λ=ω /2 used in the this paper is on the interaction induced by virtual pho- c figureisservedasanexampleofultrastrongcoupling. We tons, decoherence effects have not been included in the seetheRabicyclesaspredictedbytheHamiltonians(12) discussion. However, as long as the decoherence times or (15) for relatively weak driving fields with Ω≤0.4ω . is sufficiently short, coherent dynamics predicted by the c AtastrongerdrivingfieldwithΩ=0.8ω (redsolidline), Hamiltonian(12)or (15)wouldbe justified. Specifically, c and there is a high frequency pattern due to counter ro- given a vacuum Rabi period T ≈ 2πω /λξΩ′, the cav- c tating terms of the classical driving field, and the Rabi ity field damping rate γ and atomic decay rate γ , the c A 4 condition γ T ≪ 1 (j = c,A) ensures that the system the corresponding virtual photon amplitude, our scheme j can execute a Rabi cycle without being affected by the can be used to probe the ground state structure of the damping, and this is achievable in the ultrastrong cou- quantum Rabi model. plingregimewithmoderatesmallγ’s. Fortheparameters used in Fig. 3, for example, γ <10−2ω would be suffi- j c cient. Weemphasizethatafiniteinteractiontimewithin Acknowledgments T isofpracticalimportance,sincetheinteraction(12)or (15)isswitchableviathedrivingfield. Thisfeaturecould be a tool for performing quantum operations on qubits The author thanks Dr. H. T. Ng for discussions. This formed by the atom or the field, as well as for determin- workispartiallysupportedbyagrantfromtheResearch istic single-photon generation [30–32]. In addition, since Grants Council of Hong Kong, Special Administrative the effective vacuum Rabi frequency is proportional to Region of China (Project No. CUHK401812). [1] E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 [18] Mika A. Sillanp¨a¨a, Jian Li, Katarina Cicak, Fabio Al- (1963). tomare, Jae I. Park, Raymond W. Simmonds, G. S. [2] Bruce W. Shore and Peter L. Knight, J. Mod. Opt. 40, Paraoanu, and Pertti J. Hakonen,Phys. Rev.Lett.103, 1195 (1993). 193601 (2009). [3] H.J. Kimble, Phys.Scr., T76, 127 (1998). [19] A. A. Abdumalikov, Jr., O. Astafiev, A. M. Zagoskin, [4] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Yu.A.Pashkin,Y.Nakamura,andJ.S.Tsai,Phys.Rev. Phys.73, 565 (2001). Lett. 104, 193601 (2010). [5] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, [20] R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, C. Lang, and D. J. Wineland, Phys. Rev.Lett. 76, 1796 (1996). L. Steffen, M. Boissonneault, A. Blais, and A. Wallraff, [6] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Phys. Rev.Lett. 105, 223601 (2010). Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. [21] I. Carusotto, S. De Liberato, D. Gerace, and C. Ciuti, Hu¨mmer, E. Solano, A. Marx, and R. Gross, Nat. Phys. Phys. Rev.A 85, 023805 (2012). 6, 772 (2010). [22] I. I. Rabi, Phys.Rev.49, 324 (1936); 51, 652 (1937). [7] P.Forn-D´ıaz,J.Lisenfeld,D.Marcos,J.J.Garc´ıa-Ripoll, [23] C.J.GanandH.Zheng,Eur.Phys.J.D59,473(2010). E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Phys. [24] Zhiguo Lu¨ and Hang Zheng, Phys. Rev. B 75, 054302 Rev.Lett. 105, 237001 (2010). (2007). [8] G. Gu¨nter, A.A.Anappara,J. Hees,A. Sell, G. Biasiol, [25] Xiufeng Cao, J. Q. You, H. Zheng, A. G. Kofman, and L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Franco Nori, Phys.Rev.A 82, 022119 (2010). Leitenstorfer, and R. Huber,Nature458, 178(2009). [26] Hai-BinLiu,Jun-HongAn,ChongChen,Qing-JunTong, [9] Y.Todorov,A.M.Andrews,R.Colombelli, S.DeLiber- Hong-Gang Luo, and C. H. Oh,arXiv:1208.4295. ato,C.Ciuti,P.Klang,G.Strasser,andC.Sirtori,Phys. [27] J. J. Sanchez-Mondragon, N.B. Narozhny, and J. H. Rev.Lett. 105, 196402 (2010). Eberly, Phys. Rev. Lett. 51, 550 (1983); G. S. Agarwal, [10] G. Scalari, C. Maissen, D. Turcinkova, D. Hagenmu¨ller, Phys. Rev.Lett. 53, 1732 (1984). S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. [28] R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Wegscheider, M. Beck, and J. Faist, Science 16, 1323 Rev. Lett.68, 1132 (1992). (2012). [29] A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. [11] X.Cao,J.Q.You,H.Zheng,andF.Nori,New.J.Phys. McKeever,andH.J.Kimble,Phys.Rev.Lett.93,233603 13, 073002 (2011). (2004). [12] A. Ridolfo, M. Leib, S. Savasta, and M. J. Hartmann, [30] C. K. Law and H. J. Kimble, J. Mod. Opt. 44, 2067 Phys.Rev.Lett. 109, 193602 (2012). (1997). [13] S.AshhabandF.Nori,Phys.Rev.A81,042311 (2010). [31] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. [14] S.Ashhab,Phys. Rev.A (to be published). Buck,A.Kuzmich,and H.J. Kimble,Science303, 1992 [15] J. Casanova, G. Romero, I. Lizuain, J. J. Garc´ıa-Ripoll, (2004). and E. Solano, Phys. Rev.Lett.105, 263603 (2010). [32] B. Darqui´e, M. P. A. Jones, J. Dingjan, J. Beugnon, S. [16] S. De Liberato, D. Gerace, I. Carusotto, and C. Ciuti, Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Phys.Rev.A 80, 053810 (2009). Grangier, Science 309, 454 (2005). [17] J. Q. You and Franco Nori, Nature474, 589 (2011).