Abstract
In circuit quantum electrodynamics^{1,2,3,4,5,6,7,8,9,10} (QED), where superconducting artificial atoms are coupled to onchip cavities, the exploration of fundamental quantum physics in the strongcoupling regime has greatly evolved. In this regime, an atom and a cavity can exchange a photon frequently before coherence is lost. Nevertheless, all experiments so far are well described by the renowned Jaynes–Cummings model^{11}. Here, we report on the first experimental realization of a circuit QED system operating in the ultrastrongcoupling limit^{12,13}, where the atom–cavity coupling rate g reaches a considerable fraction of the cavity transition frequency ω_{r}. Furthermore, we present direct evidence for the breakdown of the Jaynes–Cummings model. We reach remarkable normalized coupling rates g/ω_{r} of up to 12% by enhancing the inductive coupling^{14} of a flux qubit to a transmission line resonator. Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of ultrastrong light–matter interaction.
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Main
At microwave frequencies, strong coupling is feasible because of the enormous design flexibility of superconducting circuit QED systems^{1,2}. Here, small cavitymode volumes and large dipole moments of artificial atoms^{15} enable coupling rates g of about^{16,17} 1% of the cavitymode frequency ω_{r}. Nevertheless, as in other systems^{18,19,20,21,22}, the quantum dynamics of these strongly coupled systems follows the Jaynes–Cummings model, which describes the coherent exchange of a single excitation between the atom and the cavity mode. Although the Hamiltonian of a realistic atom–cavity system contains socalled counterrotating terms allowing the simultaneous creation or annihilation of an excitation in both, atom and cavity mode, these terms can be safely neglected for small normalized coupling rates g/ω_{r}. However, when g becomes a significant fraction of ω_{r}, the counterrotating terms are expected to manifest, giving rise to exciting effects in QED. This ultrastrongcoupling regime is difficult to reach in quantumoptical cavity QED (refs 19, 20), but was recently realized in a solidstate semiconductor system^{23,24}. There, quantitative deviations from the Jaynes–Cummings model have been observed, but direct experimental proof of its breakdown by means of an unambiguous feature is still missing.
In this work, we exploit the potential of fluxbased superconducting quantum circuits to reach the ultrastrongcoupling regime^{13,14} and show direct evidence of physics beyond the Jaynes–Cummings model. To this end, we use the large nonlinear inductance of a Josephson junction shared between a flux qubit and a coplanar waveguide resonator. The transmission spectra of the combined system reveal qubit–mode couplings g/ω_{r} of up to 12% and anticrossings that cannot be explained by the Jaynes–Cummings model. Instead, they are caused by the simultaneous creation (annihilation) of two excitations, one in the qubit and one in a resonator mode, while annihilating (creating) only one excitation in a different resonator mode. The size of the anticrossings illustrates the importance of the counterrotating terms for the qubit–cavity dynamics in the ultrastrongcoupling limit.
Images of our quantum circuit and a schematic of the measurement setup are shown in Fig. 1. At a current antinode for the λmode of a niobium superconducting resonator (Fig. 1a–c),a part of the centre conductor is replaced with a narrow aluminium strip interrupted by a largearea Josephson junction (see the Methods section for fabrication details). This junction mediates most of the inductive coupling between a superconducting flux qubit^{25} galvanically connected to the strip. The qubit consists of three nanometrescaled Josephson junctions interrupting a superconducting loop, which is threaded by an external flux bias Φ_{x}. Scanning electron microscope (SEM) images of the qubit loop and the Josephson junctions are shown in Fig. 1d–f. For suitable junction sizes, the qubit potential landscape can be reduced to a doublewell potential, where the two minima correspond to states with clockwise and anticlockwise persistent currents ±I_{p}. At δ Φ_{x}=Φ_{x}−Φ_{0}/2=0, these two states are degenerate and separated by an energy gap Δ. In the qubit eigenbasis, the qubit Hamiltonian reads . Here, is the qubit transition frequency, which can be adjusted by an external flux bias. We note, that for our flux qubit the twolevel approximation is well justified because of its large anharmonicity. The resonator modes are described as harmonic oscillators, , where ω_{n} is the resonance frequency and n is the resonatormode index. The operator () creates (annihilates) a photon in the nth resonator mode. Owing to the inhomogeneous transmission line geometry^{14} (see Fig. 1d), the higher mode frequencies of our resonator are not integer multiples of the fundamental resonance frequency ω_{1}. Throughout this work, we refer to the nth mode as the n λ/2mode. Then, the Hamiltonian of our quantum circuit can be written as
Here, denote Pauli operators, g_{n} is the coupling rate of the qubit to the nth cavity mode and the flux dependence is encoded in sinθ=Δ/ℏω_{q} and cosθ. The operator is conveniently expressed as the sum of the qubit raising () and lowering () operator. Thus, in contrast to the Jaynes–Cummings model, the Hamiltonian in equation (1) explicitly contains counterrotating terms of the form and . Figure 1g shows a schematic of our measurement setup. The quantum circuit is located at the base temperature of 15 mK in a dilution refrigerator. We measure the amplified resonator transmission using a vector network analyser. For qubit spectroscopy measurements, the system is excited with a second microwave tone ω_{s} with power P_{s}, while using the 3λ/2mode at ω_{3}/2π=7.777 GHz for dispersive readout^{9,26}.
We first present measurements allowing the extraction of the coupling constants of the qubit to the first three resonator modes. The spectroscopy data in Fig. 2a show the dressed qubit transition frequency^{1,26} with the expected hyperbolic flux dependence and a minimum at δ Φ_{x}=0. Furthermore, the two lowest resonator modes (ω_{1} and ω_{2}) are visible. In principle, a fit to the Hamiltonian in equation (1) would yield all system parameters. However, our measurement resolution does not allow us to reliably determine the system parameters, in particular the undressed qubit energy gap Δ and the coupling constants g_{n} in this situation. Instead, we extract them from a cavity transmission spectrum with negligible photon population. For that purpose, we first measure the powerdependent a.c.Zeeman shift of the qubit transition frequency at δ Φ_{x}=0. The data are shown in the inset of Fig. 2a. The average photon number can be estimated using the relation (refs 6, 8), where κ_{3}/2π≈3.7 MHz is the fullwidth at halfmaximum of the cavity resonance and P_{rf} is the probe power referred to the input of the resonator. Figure 2b shows a colourcoded transmission spectrum for the 3λ/2mode as a function of δ Φ_{x}. The data are recorded at an input power P_{rf}≈−140 dBm (green data point in Fig. 2a, inset) corresponding to .
We observe a spectrum with a large number of anticrossings resulting from the multimode structure of our cavity system. To extract the individual coupling constants g_{n}, we compute the lowest nine transition frequencies of the Hamiltonian given in equation (1) incorporating the first three resonator modes. Fitting the results to the spectrum of the 3λ/2mode shows excellent agreement with the measured data as shown in Fig. 2c. We note that the spectrum for the λmode shown in Fig. 2d can be well described without additional fitting using the parameters extracted from the 3λ/2mode. For the qubit, we obtain 2I_{p}=630 nA and Δ/h=2.25 GHz. The latter deviates significantly from the dressed qubit transition frequency at δ Φ_{x}=0 (see Fig. 2a, inset) because of the strong qubit–cavity interaction. Most importantly, we find coupling rates of g_{1}/2π=314 MHz, g_{2}/2π=636 MHz and g_{3}/2π=568 MHz. The values for g_{n} correspond to normalized coupling rates g_{n}/ω_{n} of remarkable 11.2%, 11.8% and 7.3%, respectively. These large coupling rates allow us to enter the ultrastrongcoupling regime and, as we will show below, lead to significant deviations from the Jaynes–Cummings physics.
In the following, we analyse the features in our data that constitute unambiguous evidence for the breakdown of the rotatingwave approximation inherent to the Jaynes–Cummings model. In Fig. 3, we compare the energylevel spectrum of the Hamiltonian in equation (1) with that of a threemode Jaynes–Cummings model. We note that, depending on δ Φ_{x}, there are regions where our data can be well described by the Jaynes–Cummings model, and regions where there are significant deviations (see Fig. 3a). For our analysis we use the notation , where q={g,e} denote the qubit ground or excited state, respectively, and N_{n}〉={0〉,1〉,2〉,…} represents the Fock state with photon occupation N in the nth resonator mode. At the outermost anticrossings (Fig. 3b), where ω_{3}≈ω_{q}, the eigenstates ψ_{±}〉 of the coupled system are in good approximation symmetric and antisymmetric superpositions of e,0,0,0〉 and g,0,0,1〉. This exchange of a single excitation between the qubit and the resonator is a characteristic of the Jaynes–Cummings model. On the contrary, the origin of the anticrossing shown in Fig. 3c is of a different nature: the dominant contributions to the eigenstates ψ_{±}〉 are approximate symmetric and antisymmetric superpositions of the degenerate states ϕ_{1}=e,1,0,0〉 and ϕ_{2}=g,0,0,1〉. The transition from ϕ_{1} to ϕ_{2} can be understood as the annihilation of two excitations, one in the λ/2mode and one in the qubit, while, simultaneously, creating only one excitation in the 3λ/2mode. Such a process can result only from counterrotating terms as they are present in the Hamiltonian (1), but not within the Jaynes–Cummings approximation. Here, only eigenstates with an equal number of excitations can be coupled. Although counterrotating terms in principle exist in any real circuit QED system, their effects become prominent only in the ultrastrongcoupling limit with large normalized couplings g_{n}/ω_{n} as realized in our system. Hence, the observed anticrossing shown in Fig. 3c is a direct experimental manifestation of physics beyond the rotatingwave approximation in the Jaynes–Cummings model. As shown in Fig. 3d, the latter would imply a crossing of the involved energy levels, which is not observed. A similar argument applies to the innermost anticrossings of the 3λ/2mode (see Fig. 3a), although the involved eigenstates have a more complicated character, and to the innermost anticrossings of the λmode shown in Fig. 2d.
We have presented measurements on a superconducting circuit QED system in the ultrastrongcoupling regime. Our transmission spectra are in excellent agreement with theoretical predictions and show clear evidence for physics beyond the Jaynes–Cummings model. This system can act as an onchip prototype for unveiling the physics of ultrastrong light–matter interaction. Future explorations may include squeezing, causality effects in quantum field theory^{27}, the generation of bound states of qubits and photons^{28}, in situ switching of distinct physical regimes^{29} and ultrafast quantum operations in circuit QED for quantum information protocols.
Methods
Fabrication details.
The coplanar waveguide resonator is fabricated using optical lithography and reactive ion etching. We use a thermally oxidized (50 nm) silicon substrate with a 100nmthick niobium film, deposited by d.c.magnetron sputtering before patterning. The coplanar waveguide centre conductor is 20 μm wide and separated from the lateral ground planes by a gap of 12 μm, resulting in a characteristic impedance of approximately 50 Ω. The resonator with a length of 23 mm is defined by two interdigital coupling capacitors (see Fig. 1b) with a numerically calculated capacitance of about 10 fF. The centre conductor is interrupted by a gap of 80 μm at a maximum of the current distribution for the λmode (see Fig. 1c). At this point, the amplitude of the standing current wave for the λ/2 and 3λ/2mode is smaller by a factor of . The aluminium strip connected to the flux qubit and the large Josephson junction is fabricated by electronbeam lithography and Al/AlO_{x}/Al shadow evaporation techniques. For the bottom and top aluminium layer, we use a thickness of 50 and 80 nm, respectively. For the in situ oxidation of the bottom layer, pure oxygen ({p}_{{\text{O}}_{\text{2}}}=2×10^{−4} mbar; 22 min) is used. Two of the qubit Josephson junctions have an area A≈250×140 nm^{2} whereas the other junction is smaller by a factor of α≈0.7. The critical current density of the Josephson junctions is j_{c}≈1.3 kA cm^{−2}, leading to a Josephson energy of E_{J}≈224 GHz for a junction with area A. The area of the qubit loop is roughly 20×9 μm^{2} and the large Josephson junction mediating the ultrastrong qubit–cavity coupling has an area of about 7A.
Theoretical description of the coupling.
The qubit–cavity coupling is determined by the local inductance M=L_{J}+L. Here, L_{J} is the Josephson inductance of the coupling junction and L is the inductance of the shared edge between the centre conductor and the qubit. Although L_{J}>L dominates M, it has negligible influence on the vacuum current in the resonator because the total resonator inductance L_{r}≫L_{J},L. Consequently, the coupling strengths can be written as ℏg_{n}=M I_{n}I_{p}. This result can also be obtained analytically from a more thorough theoretical treatment^{14}. From the experimental values of g_{n}, we obtain L_{J}≈60 pH in agreement with our junction parameters.
References
Blais, A., Huang, RS., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Chiorescu, I. et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature 431, 159–162 (2004).
Johansson, J. et al. Vacuum Rabi oscillations in a macroscopic superconducting qubit LC oscillator system. Phys. Rev. Lett. 96, 127006 (2006).
Schuster, D. I. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515–518 (2007).
Astafiev, O. et al. Single artificialatom lasing. Nature 449, 588–590 (2007).
Deppe, F. et al. Twophoton probe of the Jaynes–Cummings model and controlled symmetry breaking in circuit QED. Nature Phys. 4, 686–691 (2008).
Fink, J. et al. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. Nature 454, 315–318 (2008).
Abdumalikov, A., Astafiev, O., Nakamura, Y., Pashkin, Y. & Tsai, J. Vacuum Rabi splitting due to strong coupling of a flux qubit and a coplanarwaveguide resonator. Phys. Rev. B 78, 180502 (2008).
Hofheinz, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546–549 (2009).
Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89–109 (1963).
Ciuti, C. & Carusotto, I. Input–output theory of cavities in the ultrastrong coupling regime: The case of timeindependent cavity parameters. Phys. Rev. A 74, 033811 (2006).
Devoret, M., Girvin, S. & Schoelkopf, R. CircuitQED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be? Ann. Phys. 16, 767–779 (2007).
Bourassa, J. et al. Ultrastrong coupling regime of cavity QED with phasebiased flux qubits. Phys. Rev. A 80, 032109 (2009).
Niemczyk, T. et al. Fabrication technology of and symmetry breaking in superconducting quantum circuits. Supercond. Sci. Technol. 22, 034009 (2009).
Schoelkopf, R. J. & Girvin, S. M. Wiring up quantum systems. Nature 451, 664–669 (2008).
Bishop, L. et al. Nonlinear response of the vacuum Rabi resonance. Nature Phys. 5, 105–109 (2008).
Thompson, R. J., Rempe, G. & Kimble, H. J. Observation of normalmode splitting for an atom in an optical cavity. Phys. Rev. Lett. 68, 1132–1135 (1992).
Walther, H., Varcoe, B. T. H., Englert, BG. & Becker, T. Cavity quantum electrodynamics. Rep. Prog. Phys. 69, 1325–1382 (2006).
Haroche, S. & Raimond, JM. Exploring the Quantum (Oxford Univ. Press, 2006).
Reithmaier, J. P. et al. Strong coupling in a single quantum dot semiconductor microcavity system. Nature 432, 197–200 (2004).
Gröblacher, S., Hammerer, K., Vanner, M. R. & Aspelmeyer, M. Observation of strong coupling between a micromechanical resonator and an optical cavity field. Nature 460, 724–727 (2009).
Günter, G. et al. Subcycle switchon of ultrastrong light–matter interaction. Nature 458, 178–181 (2009).
Anappara, A. et al. Signatures of the ultrastrong light–matter coupling regime. Phys. Rev. B 79, 201303 (2009).
Mooij, J. E. et al. Josephson persistentcurrent qubit. Science 285, 1036–1039 (1999).
Schuster, D. I. et al. Ac stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field. Phys. Rev. Lett. 94, 123602 (2005).
Sabin, C., GarciaRipoll, J. J., Solano, E. & Leon, J. Dynamics of entanglement via propagating microwave photons. Phys. Rev. B 81, 184501 (2010).
Hines, A. P., Dawson, C. M., McKenzie, R. H. & Milburn, G. J. Entanglement and bifurcations in JahnTeller models. Phys. Rev. A 70, 022303 (2004).
Peropadre, B., FornDiaz, P., Solano, E. & GarciaRipoll, J. J. Switchable ultrastrong coupling in circuit QED. Phys. Rev. Lett. 105, 023601 (2010).
Zueco, D., Reuther, G. M., Kohler, S. & Hanggi, P. Qubitoscillator dynamics in the dispersive regime: Analytical theory beyond the rotatingwave approximation. Phys. Rev. A 80, 033846 (2009).
Acknowledgements
We thank G. M. Reuther for discussions and T. Brenninger, C. Probst and K. Uhlig for technical support. We acknowledge financial support by the Deutsche Forschungsgemeinschaft through SFB 631 and the German Excellence Initiative through NIM. E.S. acknowledges financial support from UPV/EHU Grant GIU07/40, Ministerio de Ciencia e Innovación FIS200912773C0201, Basque Government Grant IT47210, European Projects EuroSQIP and SOLID. D.Z. acknowledges financial support from FIS200801240 and FIS200913364C020 (MICINN).
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T.N. fabricated the sample, conducted the experiment and analysed the data presented in this work. F.D. provided important contributions regarding the interpretation of the results. T.N. and F.D. cowrote the manuscript. J.J.GR. provided the basic idea and the techniques for the numerical analysis of the data. E.S. and J.J.GR. supervised the interpretation of the data. D.Z. and T.H. contributed to the understanding of the results and developed an analytical model of our system. H.H. contributed to the numerical analysis and helped with the experiment. E.P.M. contributed strongly to the experimental setup. M.J.S. and F.H. contributed to discussions and helped edit the manuscript. A.M. and R.G. supervised the experimental part of the work.
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Niemczyk, T., Deppe, F., Huebl, H. et al. Circuit quantum electrodynamics in the ultrastrongcoupling regime. Nature Phys 6, 772–776 (2010). https://doi.org/10.1038/nphys1730
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DOI: https://doi.org/10.1038/nphys1730
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