Abstract
On the level of single atoms and photons, the coupling between atoms and the electromagnetic field is typically very weak. By using a cavity to confine the field, the strength of this interaction can be increased by many orders of magnitude, to a point where it dominates over any dissipative process. This strongcoupling regime of cavity quantum electrodynamics^{1,2} has been reached for real atoms in optical cavities^{3}, and for artificial atoms in circuit quantum electrodynamics^{4} and quantum dot systems^{5,6}. A signature of strong coupling is the splitting of the cavity transmission peak into a pair of resolvable peaks when a single resonant atom is placed inside the cavity, an effect known as vacuum Rabi splitting. The circuit quantum electrodynamics architecture is ideally suited for going beyond this linearresponse effect. Here, we show that increasing the drive power results in two unique nonlinear features in the transmitted heterodyne signal: the supersplitting of each vacuum Rabi peak into a doublet and the appearance of extra peaks with the characteristic spacing of the Jaynes–Cummings ladder. These findings constitute direct evidence for the coupling between the quantized microwave field and the anharmonic spectrum of a superconducting qubit acting as an artificial atom.
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Main
Circuit quantum electrodynamics (QED) realizes the coupling between a superconducting qubit and the microwave field inside an onchip transmissionline resonator^{4,7}. The drastic reduction in mode volume for such a quasionedimensional system^{8,9}, the recent improvement in coherence times^{10} and the absence of atomic motion render circuit QED an ideal system for studying the strongcoupling limit. In the customary linearresponse measurement of the transmitted microwave radiation, the vacuum Rabi splitting manifests itself as an avoided crossing between the qubit and the resonator.
It may be argued that the observation of the splitting is not yet a clear sign for quantum behaviour^{11,12}, because avoided crossings are also ubiquitous in classical physics. However, quantum mechanics gives rise to a distinct anharmonicity of these splittings when initializing the resonator in a higher photon Fock state: when the resonator mode is occupied by n photons, the splitting is enhanced by a factor as compared with the vacuum Rabi situation. This anharmonicity has been observed with single atoms in both microwave^{13} and optical cavities^{14,15}. In circuit QED, this characteristic trait has been observed in timedomain measurements^{16}. In a system similar to ours, the position of the n=2 peaks was recently demonstrated in a twotone pump–probe measurement^{17}.
Here, we present a theoretical analysis and experimental investigation of the vacuum Rabi resonance in a circuit QED system, where we study the anharmonicity up to n=5 by exploring the power dependence of the heterodyne transmission (see the Methods section). This type of detection is particularly simple, because it is a continuous measurement involving only a single driving frequency. We find that, surprisingly, each vacuum Rabi peak develops into a new doublet (supersplitting), which can be simply explained by the saturation of a twolevel system comprising both photon and qubit degrees of freedom. For even stronger driving, a series of extra peaks due to multiphoton transitions^{18,19} up the Jaynes–Cummings ladder appears, but the complete shape of the spectrum closely matches the predictions from a master equation treatment including only relaxation, dephasing and the known multilevel spectrum of the superconducting qubit. The quantitative agreement confirms the validity of the extended Jaynes–Cummings Hamiltonian for our system, rendering circuit QED an excellent tool for probing the fundamental interaction of matter and light.
We consider a system consisting of a superconducting transmon qubit^{10,20} with its multiple quantum levels coupled to a single mode of a transmissionline resonator. Our arguments, however, will be general and should apply to any other realization of cavity QED with sufficiently large coupling and long enough atom coherence times. Taking into account the relatively small anharmonicity of the transmon qubit, the system is modelled in terms of an extended Jaynes–Cummings Hamiltonian including a coherent drive at frequency ω_{d}/2π,
Here, the operator a (a^{†}) annihilates (creates) a photon in the resonator mode with frequency ω_{r}/2π. The energies corresponding to the transmon eigenstates are ℏω_{j} (j=0,1,…), and they are determined by the Cooper pair box Hamiltonian^{21} for given Josephson energy E_{J} and charging energy E_{C}. In the transmon regime considered here, E_{J}/E_{C}≫1, the levels form a weakly anharmonic ladder with anharmonicity α=ω_{12}−ω_{01}∼−E_{C}/ℏ (where ω_{i j}=ω_{j}−ω_{i} is the transition frequency between levels i and j). The transmon–resonator coupling strengths are , where g_{0}/2π is the vacuum Rabi frequency. We have neglected nonnearestneighbour terms, , because they are small in this regime^{20}. We are interested in the case where the drive, the qubit and the cavity are all tuned close to the same frequency. As the drive is small, ξ≪ω_{d}, we have taken a rotatingwave approximation and dropped rapidly oscillating terms. The resulting energy level diagram for vanishing drive, ξ=0, is schematically shown in Fig. 1.
The interaction of the cavity–transmon system with its environment enables control and measurement by means of microwave drive and detection^{4,7}, but also leads to decoherence. The appropriate framework to capture all of these effects is a description in terms of the reduced density matrix ρ for the cavity and transmon. The Markovian master equation that governs the evolution of ρ is given by
where is the usual Lindblad damping superoperator defined by . The three damping terms on the righthand side model the loss of cavity photons at rate κ, the intrinsic relaxation of transmon excitations at rate γ_{1}, and the pure dephasing of transmon state superpositions at rate γ_{ϕ} and relative strengths α_{j}, β_{j} (see the Methods section).
The quantity of interest for detection of the vacuum Rabi splitting is the transmitted amplitude A(ω_{d}) of a microwave signal at frequency ω_{d}/2π. Here, we use a heterodyne measurement of the field quadratures I=V_{0}〈a+a^{†}〉 and Q=V_{0}〈i a^{†}−i a〉, where V_{0} is a voltage related to the gain of the experimental amplification chain. We can express the steadystate transmission amplitude as
Our calculations thus require only the steadystate solution ρ_{s} of the master equation (1). We explore the transmission intensity A^{2} as a function of the drive strength ξ, and compare our results to experimental data obtained for a transmon qubit in resonance with a ω_{r}/2π=6.92 GHz coplanar waveguide (CPW) resonator with coupling strength g_{0}/π=347 MHz (see the Methods section for experimental details). The intensity A^{2} is conveniently expressed in units of A_{1}^{2}=4V_{0}^{2}, the intensity resulting from a coherent state with a mean cavity occupancy of one photon.
In the linearresponse regime, the vacuum Rabi peaks have the characteristic Lorentzian line shape. Their separation and width are given by 2g_{0} and (γ_{1}+2γ_{ϕ}+κ)/2, respectively. For our specific sample, the splitting is observed to exceed 260 linewidths, shown in Fig. 2a. Strikingly, when increasing the drive power beyond the linearresponse regime, the shape of the transmission curve changes drastically as shown in Fig. 2b–f. Each vacuum Rabi peak develops a central dip and eventually supersplits into a doublet of peaks. We find excellent agreement between our prediction based on equations (1) and (2) and the measured intensity.
The physics of this supersplitting can be elucidated within a reduced twolevel model. In the case of the left vacuum Rabi peak, this model (represented in Fig. 1b) takes into account only the Jaynes–Cummings ground state and the antisymmetric superposition of transmon and photon excitation. (The right vacuum Rabi peak may be modelled by instead taking the symmetric superposition.) These two states, ↓〉=0,0〉 and , form an effective twolevel system composed of both transmon and cavity degrees of freedom. Within the effective twolevel subspace, the photon operators are mapped^{22} to Pauli operators , , so that the microwave tone acts as a drive on the effective twolevel system, that is,
a scenario that Carmichael and coworkers^{12,22} have referred to as ‘dressing of dressed states’. The Hamiltonian H_{eff} refers to the frame rotating at the drive frequency, Δ=ω_{01}−g_{0}−ω_{d} is the detuning between the drive and the vacuum Rabi peak and is the effective drive strength. With the notable exception of the recent work by Schuster et al. ^{14}, previous investigations were primarily concerned with effects on photon correlations and fluorescence, as observed in photoncounting measurements^{12,15}. According to the operator mapping, photon counting can be related to the measurement of 〈Σ_{z}〉, whereas detection of the heterodyne amplitude A corresponds to 〈Σ_{−}〉. As a result, heterodyne detection fundamentally differs from photon counting and the vacuum Rabi supersplitting is a characteristic only of heterodyne detection.
After restricting the master equation (1) to the twolevel subspace, the system evolution can be expressed in terms of simple Bloch equations for the three components of the reduced density matrix ,
(An intuitive approach ignoring dissipation and avoiding the Bloch equations is discussed in Supplementary Information, Discussion.) Here, T′_{1} and T′_{2} are the effective relaxation and dephasing times, which are related to γ_{1}, γ_{ϕ} and κ via T′_{1}^{−1}=(γ_{1}+κ)/2 and T′_{2}^{−1}=(γ_{1}+2γ_{ϕ}+κ)/4. The usual steadystate solution of the Bloch equations for x and y gives the heterodyne amplitude
This expression correctly describes the crossover from the linear response at small driving strength, Ω≪(T′_{1}T′_{2})^{−1/2}, producing a Lorentzian of width 2T′_{2}^{−1}, to the doublet structure observed for strong driving. Specifically, as the drive power is increased, the response saturates and the peak broadens, until at Ω=(T′_{1}T′_{2})^{−1/2} the peak undergoes supersplitting with peak–peak separation . The fact that we use heterodyne detection is indeed crucial for the supersplitting. It is easy to verify within the twolevel approximation that photon counting always results in a Lorentzian. For photon counting, probing beyond the linearresponse regime merely results in power broadening; specifically, the width of the Lorentzian is given by (see Supplementary Information, Fig. S4). That there is a difference between photon counting and heterodyne detection is a characteristic of a single qubit. For a manyqubit system, both types of measurement would typically give the same result, and this manyqubit nonlinear response would be rather different to the singlequbit case, developing first as a frequency pulling and eventually yielding hysteresis^{23}.
In Fig. 2c–f, the analytical expression (4) is plotted for comparison with the full numerical results and the experimental data. We find good agreement for low to moderate drive power, confirming that the supersplitting can be attributed to driving the vacuum Rabi transition into saturation while measuring the transmission with the heterodyne technique. For higher drive power, a left–right asymmetry appears in the true transmission spectrum, which is not reproduced by equation (4), and which is partly due to the influence of levels beyond the twolevel approximation.
Higher levels of the extended Jaynes–Cummings Hamiltonian become increasingly important when the drive power is raised further. Figure 3 shows the emergence of extra peaks in the transmission spectrum. Each of the peaks can be uniquely identified with a multiphoton transition from the ground state to an excited Jaynes–Cummings state. For simplicity, we consider the situation where the anharmonicity α and the coupling strength g_{0} are sufficiently different that mixing between higher transmon levels and the regular Jaynes–Cummings states is minimal for the lowexcitation subspaces. Accordingly, the experiments are carried out using a different transmon qubit in the same sample, with a smaller coupling of g_{0}/π=94.4 MHz (see the Methods section). In this case, the nphoton transitions to the nexcitation subspace occur at frequencies E_{n±}/n h=(ω_{r}±n^{−1/2}g_{0})/2π, and thus reveal the anharmonicity of the Jaynes–Cummings ladder. The features associated with unsaturated nphoton transitions have a width set by the characteristic decay rates from the n,±〉 states, [(2n−1)κ+γ_{1}]/2≃1 MHz. As the drive increases, each nphoton transition begins to saturate in turn and develop extra structure analogous to the supersplitting of the vacuum Rabi peaks. The detailed comparison between experimental data and numerical simulation in Fig. 3c–e shows superb agreement down to the narrowest features observed. (Supplementary Information, Fig. S5 shows the quadratures of the simulated spectrum.)
The possibility of multiphoton transitions at sufficiently large drive powers also affects the shape of the vacuum Rabi splitting when tuning the qubit frequency ω_{01} through resonance with the cavity, shown in Fig. 4. Instead of the simple avoided crossing commonly observed at low drive powers^{4}, the presence of multiphoton transitions leads to a fanlike structure where individual branches can again be identified onetoone with the possible transitions in the Jaynes–Cummings ladder. In the experimental data of Fig. 4a, processes up to the fivephoton transition are clearly visible. Detailed agreement with the theory verifies that the more general situation of nonzero detuning between the qubit and the resonator is correctly described by our model.
In summary, we have shown the supersplitting and the emergence of multiphoton transitions in the nonlinear response of the vacuum Rabi resonance, presenting both theoretical predictions and their experimental verification. This enables the observation of the characteristic anharmonicity in the Jaynes–Cummings ladder in a heterodyne transmission measurement, and is direct evidence for the strong coupling between the superconducting qubit and the quantized microwave field. The high precision of agreement between predictions and experiment demonstrates the robustness of the Jaynes–Cummings physics in a circuit QED system. This opens up pathways for further investigations in quantum optics and quantum information, such as generating numbersqueezed states or using multilevel quantum logic.
Methods
Theory.
The modelling of the transmon follows ref. 20. Although some expressions given in the main text are asymptotic results valid only for E_{J}/E_{C}≫1, our calculations are based on a full diagonalization of the Cooper pair box Hamiltonian.
The description of the transmon–cavity system in terms of a master equation requires a model for the relaxation and dephasing of higher transmon levels. As detailed studies of the microscopic origin of the dominant relaxation and dephasing channels are still outstanding, we have chosen plausible superoperators for our master equation (1). Assuming that relaxation of higher transmon levels may arise owing to a coupling of external degrees of freedom to the charge on the superconducting island, we take the relative strengths of relaxation to be related to the coupling parameters as α_{j}=g_{j}/g_{0}. Dephasing of higher levels is likely to be due to charge noise. Denoting the charge dispersion^{20} of level j by ε_{j}=ω_{j}(n_{g}=0)−ω_{j}(n_{g}=1/2), we therefore take the relative dephasing strengths to be β_{j}=2ε_{j}/(ε_{1}−ε_{0}). (The normalization of α_{j} and β_{j} is chosen to allow the usual interpretation of γ_{1} and γ_{ϕ} as relaxation and dephasing rates for the first two levels of the transmon spectrum.) In fact, the pure dephasing rate is sufficiently small for our qubits that we set γ_{ϕ}=0. A comparison of extra simulations with auxiliary experimental results at increased temperatures enables us to place an approximate upper bound of 0.003 on the number of thermal photons in the cavity.
For the steadystate solution of equation (1), the Hilbert space is truncated to a subspace with maximum number of excitations N, using the projector . In our simulations, we keep up to N=7 excitations. To reach agreement with the experimentally measured signal for the strongest drive powers, it is necessary to account for a small amount (∼−58 dB) of leakage of the drive past the cavity. In addition, there is a small bias introduced by measuring the intensity as the square of the I and Q quadratures, each of which is subject to noise. Accordingly, the quantity that corresponds to the experimental signal is A^{2}=2V_{0} tr (a ρ_{s})+b ξ^{2}+2σ_{n}^{2}, where b describes the leakage of the drive bypassing the cavity and σ_{n} is the measurement noise in each of the I and Q channels.
Fits are obtained by minimizing the mean squared deviation between experiment and calculation over the full power range, with fit parameters being b and the two scaling factors describing the attenuation and amplification for input and output signals. (In our plots, the output scaling factor is applied to the experimental data, converting them to units of A_{1}^{2}.) To obtain optimal agreement, we also make small adjustments to the system parameters γ_{1}, κ, g_{0}, ω_{r}, ω_{01} and E_{C}. These parameters are confined by separate experiments to narrow ranges, and all values used in fits are consistent within the experimental uncertainties. Once obtained, the same set of parameters was used in generating Figs 3 and 4, and Supplementary Information, Movies.
Experiment.
Measurements are carried out in a dilution refrigerator at 15 mK. The sample consists of two superconducting transmon qubits^{10,20}, coupled to an onchip CPW cavity. Fabrication of the sample followed the description given in ref. 10. The CPW resonator has a halfwavelength resonant frequency of ω_{r}/2π=6.92 GHz and a photon decay rate of κ/2π=300 kHz. Transmission measurements are carried out using a heterodyne detection scheme. The transmitted radiofrequency voltage signal through the cavity is mixed down to a 1 MHz carrier signal, and then digitally mixed down to d.c. to obtain the transmitted voltage amplitude as a function of frequency. The vacuum Rabi coupling strengths for the two qubits are obtained as g_{0}/π=347 MHz (qubit 1) and g_{0}/π=94.4 MHz (qubit 2). Timedomain measurements of the qubits show that they are Purcelllimited and completely homogeneously broadened at their flux sweet spots^{24}. The coherence times are T_{1}=1.7 μs and T_{2}=0.7 μs (qubit 1, away from the flux sweet spot) and T_{1}=1.4 μs and T_{2}=2.8 μs (qubit 2, at flux sweet spot). The charging energies of the two qubits are measured to be E_{C}/h=400 MHz and E_{C}/h=340 MHz. (See Supplementary Information, Methods, for further details.)
Change history
15 December 2008
In the pdf version of this Letter originally published online, a production error caused the character '&' to appear in place of 'V_{0}'. This has now been corrected.
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Acknowledgements
This work has been supported by Yale University via a Quantum Information and Mesoscopic Physics Fellowship (A.A.H., J.K.), the LPS/NSAARO grant W911NF0510365, NSF grants DMR0653377, DMR0603369 and PHY0653073 and by the Academy of Finland. We thank J. Gambetta, A. Blais and A. Wallraff for helpful discussions, and L. Frunzio and B. Johnson for fabrication of the sample.
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J.M.C. led the experimental effort. L.S.B. and J.K. carried out the calculations and did most of the writing. A.A.H. gave technical support and conceptual advice. E.T. contributed to the early theory. M.H.D., S.M.G. and R.J.S. provided support and supervised the project.
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Bishop, L., Chow, J., Koch, J. et al. Nonlinear response of the vacuum Rabi resonance. Nature Phys 5, 105–109 (2009). https://doi.org/10.1038/nphys1154
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DOI: https://doi.org/10.1038/nphys1154
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