Climbing the Jaynes-Cummings Ladder and Observing its Sqrt(n) Nonlinearity in a Cavity QED System
J. M. Fink, M. Goeppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, A. Wallraff
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Climbing the Jaynes-Cummings Ladderand Observing its √ n Nonlinearity in a Cavity QED System
J. M. Fink, M. G¨oppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff Department of Physics, ETH Zurich, CH-8093, Zurich, Switzerland. D´epartement de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1 Canada. (Dated: February 11, 2009)
The already very active field of cavity quantum electrodynamics (QED), traditionallystudied in atomic systems [1, 2, 3], has recently gained additional momentum by the ad-vent of experiments with semiconducting [4, 5, 6, 7, 8] and superconducting [9, 10, 11]systems. In these solid state implementations, novel quantum optics experiments areenabled by the possibility to engineer many of the characteristic parameters at will. Incavity QED, the observation of the vacuum Rabi mode splitting is a hallmark exper-iment aimed at probing the nature of matter-light interaction on the level of a singlequantum. However, this effect can, at least in principle, be explained classically as thenormal mode splitting of two coupled linear oscillators [12]. It has been suggested thatan observation of the scaling of the resonant atom-photon coupling strength in theJaynes-Cummings energy ladder [13] with the square root of photon number n is suffi-cient to prove that the system is quantum mechanical in nature [14]. Here we report adirect spectroscopic observation of this characteristic quantum nonlinearity. Measur-ing the photonic degree of freedom of the coupled system, our measurements provideunambiguous, long sought for spectroscopic evidence for the quantum nature of theresonant atom-field interaction in cavity QED. We explore atom-photon superpositionstates involving up to two photons, using a spectroscopic pump and probe technique.The experiments have been performed in a circuit QED setup [15], in which ultrastrong coupling is realized by the large dipole coupling strength and the long coher-ence time of a superconducting qubit embedded in a high quality on-chip microwavecavity. Circuit QED systems also provide a natural quantum interface between fly-ing qubits (photons) and stationary qubits for applications in quantum informationprocessing and communication (QIPC) [16]. The dynamics of a two level system coupled to a sin-gle mode of an electromagnetic field is described by theJaynes-Cummings Hamiltonianˆ H = ¯ hω ge ˆ σ ee + ¯ hω r ˆ a † ˆ a + ¯ hg ge (ˆ σ † ge ˆ a + ˆ a † ˆ σ ge ) . (1)Here, ω ge is the transition frequency between the ground | g i and excited state | e i of the two level system, ω r is thefrequency of the field and g ge is the coupling strengthbetween the two. ˆ a † and ˆ a are the raising and lower-ing operators acting on the photon number states | n i ofthe field and ˆ σ ij = | i ih j | are the corresponding operatorsacting on the qubit states. When the coherent couplingrate g ge is larger than the rate κ at which photons arelost from the field and larger than the rate γ at which thetwo level system looses its coherence, the strong couplinglimit is realized. On resonance ( ω ge = ω r ) and in the pres-ence of n excitations, the new eigenstates of the coupledsystem are the symmetric ( | g, n i + | e, n − i ) / √ ≡ | n + i and antisymmetric ( | g, n i − | e, n − i ) / √ ≡ | n −i qubit-photon superposition states, see Fig. 1. For n = 1, thesestates are equivalently observed spectroscopically as avacuum Rabi mode splitting [4, 5, 6, 7, 8, 9, 17, 18] or intime resolved measurements as vacuum Rabi oscillations[11, 19, 20, 21] at frequency 2 g ge . The Jaynes-Cummingsmodel predicts a characteristic nonlinear scaling of thisfrequency as √ n g ge with the number of excitations n in the system, see Fig. 1. This quantum effect is in starkcontrast to the normal mode splitting of two classicalcoupled linear oscillators, which is independent of the os-cillator amplitude. Fig. 1:
Level diagram of a resonant ( ν r = ν ge ) cavityQED system. The uncoupled qubit states | g i , | e i and | f i from left to right and the photon states | i , | i , ..., | n i frombottom to top are shown. The dipole coupled dressed statesare shown in blue and a shift due to the | f, i level is indi-cated in red. Pump ν g , − , ν g , and probe ν − , − , ν , transition frequencies are indicated accordingly. Since the first measurements of the vacuum Rabi modesplitting with, on average, a single intra-cavity atom [17]it remains a major goal to clearly observe this character-istic √ n nonlinearity spectroscopically to prove the quan-tum nature of the interaction between the two-level sys-tem and the radiation field [12, 14, 22]. In time domainmeasurements of vacuum Rabi oscillations, evidence forthis √ n scaling has been found with circular Rydbergatoms [19] and superconducting flux qubits [11] interact-ing with weak coherent fields. Related experiments havebeen performed with one and two-photon Fock states[20, 21]. We now observe this nonlinearity directly us-ing a scheme similar to the one suggested in Ref. [22] bypumping the system selectively into the first doublet | ±i and probing transitions to the second doublet | ±i . Thistechnique realizes efficient excitation into higher doubletsat small intra cavity photon numbers avoiding unwanteda.c. Stark shifts occurring in high drive and elevated tem-perature experiments.In a different regime, when the qubit is detuned by anamount | ∆ | = | ω ge − ω r | ≫ g ge from the cavity, photonnumber states and their distribution have recently beenobserved using dispersive quantum non-demolition mea-surements in both circuit QED [23] and Rydberg atomexperiments [24].In our experiments, in the resonant regime a super-conducting qubit playing the role of an artificial atomis strongly coupled to photons contained in a coplanarwaveguide resonator in an architecture known as circuitQED [9, 15]. We use a transmon [25, 26], which is acharge-insensitive superconducting qubit design derivedfrom the Cooper pair box (CPB) [27], as the artificialatom. Its transition frequency is given by ω ge / π ≃ p E C E J (Φ) with the single electron charging energy E C ≈ . E J (Φ) = E J,max | cos ( π Φ / Φ ) | and E J,max ≈ . ν r ≈ .
94 GHz and decay rate κ/ π ≈ . g ge . A simplified electrical circuit diagram ofthe setup is shown in Fig. 2b.The system is prepared in its ground state | g, i bycooling it to temperatures below 20 mK in a dilution re-frigerator. We then probe the energies of the lowest dou-blet | ±i measuring the cavity transmission spectrum T and varying the detuning between the qubit transitionfrequency ν ge and the cavity frequency ν r by applying amagnetic flux Φ, see Fig. 3a. The measurement is per-formed with a weak probe of power P ≈ −
137 dBm ap-plied to the input port of the resonator populating itwith a mean photon number of ¯ n ≈ . P is calibrated in a dispersive a.c. Stark shift measure- ment [28]. At half integers of a flux quantum Φ , thequbit energy level separation ν ge approaches zero. Atthis point the bare resonator spectrum peaked at the fre-quency ν r is observed, see Fig. 3b. We use the measuredmaximum transmission amplitude to normalize the am-plitudes in all subsequent measurements. At all otherdetunings | ∆ | ≫ g ge the qubit dispersively shifts [25] thecavity frequency ν r by χ ≃ − g ge E C / (∆(∆ − E C )).Measuring cavity transmission T as a function of fluxbias Φ in the anti-crossing region yields transmissionmaxima at frequencies corresponding to transitions tothe first doublet | ±i in the Jaynes-Cummings ladder asshown in Fig. 3c. On resonance (∆ = 0), we extracta coupling strength of g ge / π = 154 MHz, see Fig. 3d,where the linewidth of the individual vacuum Rabi splitlines is given by δ ν ≈ . g ge /π of over 100 linewidths δ ν , clearly demonstrating that the strong coupling limitis realized [9, 29]. Solid white lines in Figs. 3 (and 4)are numerically calculated dressed state frequencies withthe qubit and resonator parameters as stated above, be-ing in excellent agreement with the data. For the cal-culation, the qubit Hamiltonian is solved exactly in thecharge basis. The qubit states | g i and | e i and the flux Fig. 2:
Sample and experimental setup. a , Optical im-ages of the superconducting coplanar waveguide resonator(top) with the transmon type superconducting qubit embed-ded at the position indicated. On the bottom, the qubitwith dimensions 300 × µ m close to the center conductoris shown. b , Simplified circuit diagram of the experimentalsetup, similar to the one used in Ref. [9]. The qubit is capaci-tively coupled to the resonator through C g and the resonator,represented by a parallel LC circuit, is coupled to input andoutput transmission lines via the capacitors C in and C out . Us-ing ultra low noise amplifiers and a down-conversion mixer,the transmitted microwave signal is detected and digitized. dependent coupling constant g ge are then incorporated inthe Jaynes-Cummings Hamiltonian Eq. (1). Its numericdiagonalization yields the dressed states of the coupledsystem without any fit parameters.In our pump and probe scheme we first determine theexact energies of the first doublet | ±i at a given fluxΦ spectroscopically. We then apply a pump tone at thefixed frequency ν g , − or ν g , to populate the respec-tive first doublet state | ±i . A probe tone of the samepower is then scanned over the frequency range of thesplitting. This procedure is repeated for different fluxcontrolled detunings. The transmission at the pump andprobe frequencies is spectrally resolved in a heterodynedetection scheme.Populating the symmetric state | i , we observe anadditional transmission peak at a probe tone frequencythat varies with flux, as shown in Fig. 4a. This peak cor-responds to the transition between the symmetric dou-blet states | i and | i at frequency ν , . Similarly,in Fig. 4c where the antisymmetric state | −i is popu-lated we measure a transmission peak that correspondsto the transition between the two antisymmetric doublet Fig. 3:
Vacuum Rabi mode splitting with a single pho-ton. a , Measured resonator transmission spectra versus ex-ternal flux Φ. Blue indicates low and red high transmission T . The solid white line shows dressed state energies as ob-tained numerically and the dashed lines indicate the bare res-onator frequency ν r as well as the qubit transition frequency ν ge . b , Resonator transmission T at Φ / Φ = 1 / a , with a Lorentzian line fit inred. c , Resonator transmission T versus Φ close to degen-eracy. d , Vacuum Rabi mode splitting at degeneracy withLorentzian line fit in red. states | −i and | −i at frequency ν − , − . The trans-mission spectra displayed in Figs. 4b and d recorded atthe values of flux indicated by arrows in Figs. 4a andc show that the distinct transitions between the differ-ent doublets are very well resolved with separations oftens of linewidths. Transitions between symmetric andantisymmetric doublet states are not observed in this ex-periment, because the flux-dependent transition matrixelements squared are on average smaller by a factor of 10and 100 for transitions | i → | −i and | −i → | i ,respectively, than the corresponding matrix elements be-tween states of the same symmetry.The energies of the first doublet | ±i , split by g ge /π on resonance, are in excellent agreement with the dressedstates theory (solid red lines) over the full range of fluxΦ controlled detunings, see Fig. 5. The absolute energiesof the second doublet states | ±i are obtained by addingthe extracted probe tone frequencies ν − , − and ν , to the applied pump frequencies ν g , − or ν g , , see bluedots in Fig. 5. For the second doublet, we observe twopeaks split by 1 . g ge /π on resonance, a value very closeto the expected √ ∼ .
41. This small frequency shiftcan easily be understood, without any fit parameters, bytaking into account a third qubit level | f, i which is at Fig. 4:
Vacuum Rabi mode splitting with two photons.a , Cavity transmission T as in Fig. 3 with an additional pumptone applied to the resonator input at frequency ν g , pop-ulating the | i state. b , Spectrum at ∆ = 0, indicatedby arrows in a . c , Transmission T with a pump tone ap-plied at ν g , − populating the | −i state. d , Spectrum atΦ / Φ ≈ .
606 as indicated by arrows in c . Fig. 5:
Experimental dressed state energy levels.
Mea-sured dressed state energies (blue dots) reconstructed by sum-ming pump and probe frequencies, compared to the calculateduncoupled cavity and qubit levels (dashed lines), the calcu-lated dressed state energies in the qubit two-level approxima-tion (dotted) and to the corresponding calculation includingthe third qubit level (solid red lines). frequency ν gf ≃ ν ge − E C for the transmon type qubit[25], just below the second doublet states | ±i . In orderto find the energies of the dressed states in the presenceof this additional level we diagonalize the Hamiltonianˆ H = ˆ H + ˆ H , where ˆ H = ¯ hω gf ˆ σ ff + ¯ hg ef (ˆ σ † ef ˆ a + ˆ a † ˆ σ ef )and g ef / π ≈
210 MHz (obtained from exact diagonaliza-tion) denotes the coupling of the | e i to | f i transition tothe cavity. The presence of the | f, i level is observedto shift the antisymmetric state | −i , being closer in fre-quency to the | f, i state, more than the symmetric state | i , see Figs. 1 and 5, leading to the small difference ofthe observed splitting from √
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