Session C48: Multimode Circuit QED and Strong Coupling

Multimode cavity QED 1: State preparation and readout

Quantum information processing requires the creation of scalable architectures with long lived, highly coherent, readily addressable quantum states. A promising architecture consists of Fock states of photons in coupled superconducting microwave cavity arrays, with state preparation and readout achieved by coupling to superconducting qubits via circuit QED. We describe experiments on such multimode circuit QED devices consisting of 1D chains of 10-20 tunnel coupled 2D high-Q microwave resonators coupled to a single, flux-tunable transmon qubit. We use parametric sideband transitions [1], implemented via flux modulation of the transmon, to realize arbitrary states of the photonic qubits. We also discuss ongoing efforts to engineer multimode architectures comprised of coupled 3D microwave cavities in which photon lifetimes can exceed 10 ms [2]. [1] J. D. Strand et al, Physical Review B 87.22 (2013) [2] M. Reagor et al, Applied Physics Letters 102, 192604 (2013)   

Multimode cavity QED 2: Parameter dependence and limitations through theoretical modeling

Superconducting circuits are well-established as promising building blocks for future quantum information processing devices. While in recent years gate and readout fidelities have improved significantly, superconducting qubits can still benefit greatly from added intrinsic robustness and improved error resilience. In this talk, we present results for qubits based on the modes of a 1d resonator array, where qubit manipulation and readout are achieved by interaction with a parametrically driven superconducting transmon. Through theoretical modeling, we provide insight into mode addressability as well as crosstalk, and their dependence on the system's size in various parameter regimes.   

Multimode cavity QED 3: Universal quantum gates

A promising architecture for scalable quantum computation consists of photonic qubits in multimode superconducting cavities, coupled to superconducting qubits. In this talk, we describe schemes to implement pairwise universal gate operations between the photonic qubits. We use parametric sideband interactions [1] mediated by a superconducting qubit to realize arbitrary single photonic qubit gates, as well as the C-phase gate between arbitrary pairs of photonic qubits, thereby realizing universal gate operations. We also describe schemes to realize beam splitter and phase shifter elements in this multimode architecture, allowing for circuit QED realizations of linear optical quantum computation schemes. [1] ~J. D. Strand et al, Physical Review B 87.22 (2013)   

Multimode cavity QED 4: Quantum state tomography

One of the challenges of large scale quantum information processing is the ability to perform quantum state tomography of massively entangled states. We implement multiplexed tomography of quantum states of multimode cavity arrays comprising several photonic qubits. Quantum state tomography is performed via sequential parametric transitions [1] with a single, flux-tunable transmon qubit, in conjunction with multimode photonic gates and transmon readout. We describe schemes to prepare and characterize W states of several modes of the multimode cavity, and our progress towards extending such schemes to multiphoton entangled states. The ability to create and measure arbitrary quantum states, in conjunction with the large coherence time of microwave cavities, makes multimode cavity QED a promising architecture for scalable quantum computation and bosonic quantum simulation. [1] J. D. Strand et al, Physical Review B 87.22 (2013)   

Multimode Strong Coupling in Circuit QED

We present experimental and theoretical studies in the multimode strong coupling (MMSC) regime of cavity quantum electrodynamics (QED). In MMSC, a single atom is simultaneously coupled to a large, but discrete, number of cavity harmonics, with atom-mode coupling strengths comparable to the free spectral range (FSR). This regime is readily accessible in circuit QED, by strongly coupling a transmon qubit to a low fundamental frequency microwave cavity. We present some key results from our original experiment (PRX 5, 021035, 2015), in which a transmon qubit, resonant with the 75th harmonic of a 90 MHz cavity, reached qubit-mode coupling strengths exceeding 30MHz. When this system is coherently driven, we observed complex multimode fluorescence, with the notable formation of ultra-narrow linewidths. To better understand these unique features of multimode resonance fluorescence we developed a quantum formalism, which attributes the spectral linewidth narrowing to the correlated spontaneous emission of doubly dressed states. Finally we will share preliminary experimental results from our continuing study of MMSC, this time from a system where qubit-mode coupling strengths approach and even exceed the FSR.   

Multiphoton Quantum Rabi Oscillations in Ultrastrong Cavity QED

When an atom is strongly coupled to a cavity, the two systems can exchange a \emph{single} photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED has been reached experimentally where the interaction between light and artificial atoms (qubits) becomes ultrastrong, \emph{i.e.}, its strength is comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit-QED systems can undergo \emph{two}- and \emph{three}-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits.   

Ultrastrong coupling in a flux qubit-transmission line system

Recent advances in circuit QED have enabled the study of light-matter interactions in new regimes of coupling strength. Experiments based on flux qubits coupled to resonators observed indications of the so-called ultrastrong coupling regime, where the coupling strength is comparable to the qubit energy splitting. We have realized an experiment where a flux qubit is coupled to an open transmission line with an adjustable coupling strength, which can be tuned into the ultrastrong coupling regime. When the coupling strength is low, the qubit behaves like an isolated dipole scatterer, reflecting over 97\% of the incident coherent probe. At larger coupling strengths, the qubit linewidth exceeds its energy splitting, indicating that the system operates deeply in the ultrastrong coupling regime. We find that qualitative features of the qubit response evolve with the coupling strength in ways unexpected based on scattering calculations within the rotating-wave approximation. Some features of the evolution can be understood in the broader context of the spin-boson model.   

Ultra-strong coupling in a transmon circuit architecture

New unexplored phenomena are predicted in cQED for the ultra-strong coupling (USC) regime and beyond. Here, we explore two strategies to increase the coupling between a transmon qubit and a microwave resonator. In the first approach, we increase the impedance of the resonator, enhancing it’s voltage zero-point fluctuations, and measure a vacuum Rabi splitting of 916 MHz. In a second approach, we create a transmon qubit by making a superconducting island suspended above the center conductor of the resonator and which is shorted to ground by two Josephson junctions. Doing so, we maximize the dipole moment of the qubit and observe a vacuum Rabi splitting of 1.2 GHz with a qubit linewidth of 1 MHz. This first transmon qubit in the USC regime improves the coherence time by a factor of 100 compared to other systems in the USC limit. Finally we predict that by combining both approaches, a coupling of $\sim 3.6$\ GHz is possible, reaching close to the deep strong coupling limit.   

Deep strong coupling in a circuit QED system (1) - Introduction -

Recently, light-matter interaction at the single-photon level has been demonstrated in superconducting circuits (circuit-QED). The interaction energy between a superconducting artificial atom and an excitation quantum of a harmonic oscillator in the microwave region has been shown to be very large, at least a few thousand times that of the atom-photon interaction obtained using Rydberg atoms [1]. It is also intriguing that, depending on the circuit design, the relevant physical parameters of this system can be controlled at will. In particular, an interaction energy as large as the transition energy of a superconducting artificial atom or a harmonic oscillator is possible, where totally new states, such as a spontaneously generated Schrödinger-cat-like correlated ground state of light and matter, have been predicted [2,3]. In this talk, I will introduce the motivation and the significance of the research, methods to achieve such a strong interaction, and a brief overview of the obtained results. [1] J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda, K. Semba, and H. Takayanagi, Phys. Rev. Lett. 96, 127006 (2006). [2] S. Ashhab and Franco Nori, Phys. Rev. A 81, 042311 (2010). [3] S. Ashhab Phys. Rev. A 87, 013826 (2013).   

Deep strong coupling in a circuit QED system (2) - experiment -

Among a variety of cavity/circuit-QED systems, the superconducting flux qubit is a promising candidate for increasing the coupling strength further because of its huge magnetic moment. Using a flux qubit, $g / \omega_r = 0.12$ ($g$: coupling strength, $\omega_r$: bare resonator frequency) has been reported [1]. However, $g / \omega_r$ is still lower than 1. Here, instead of the widely used coplanar waveguide (CPW) resonators, we use a lumped-element resonator consisting of an inductor (L) and a capacitor (C). While CPW resonators are distributed-element circuits and are therefore restricted by impedance matching constraints, one can freely choose the ratio, $L/C$, of a lumped-element resonator. This allows us to design a much smaller inductance and to make the zero-point current fluctuation much larger. Using a flux qubit and a lumped-element resonator, we have achieved $g / \omega_r$ comparable to or larger than 1, which is the deep strong coupling regime, where a variety of interesting physics is expected [2]. In this presentation, the sample design and spectroscopy data will be shown. [1] T. Niemczyk et al., nature physics 6, 772 (2010). [2] S. Ashhab and F. Nori, PRA 81, 042311 (2010).   

Deep strong coupling in a circuit QED system (3) - data and analysis -

We have experimentally achieved deep-strong coupling between a superconducting flux qubit and a superconducting LC circuit, where the coupling energy, $\hbar g$, exceeds both the transition energy of the flux qubit, $\hbar\omega_{\rm q}$, and the resonant energy of the LC circuit, $\hbar\omega_{\rm r}$. At the optimal flux bias of the flux qubit, the qubit-resonator system is described by the Rabi model, which is one of the simplest quantum models of atom-cavity systems. The Hamiltonian of the Rabi model can be written as $\mathcal{H}_{\rm Rabi}$ = $-\frac{\hbar}{2}\omega_{\rm q}\sigma_z$ + $\hbar\omega_{\rm r}(a^{\dagger}a + \frac{1}{2})$ + $\hbar g \sigma_x (a + a^{\dagger})$, where $\sigma_{x(z)}$ is a Pauli matrix and $a(a^{\dagger})$ is an annihilation (creation) operator. In this presentation, we will show the spectroscopy data of qubit-resonator systems in the deep-strong-coupling regime. Transition frequencies calculated from $\mathcal{H}_{\rm Rabi}$ fit the measured data well. We have also observed that $\hbar\omega_{\rm q}$ is largely suppressed due to the Lamb shift caused by the deep-strong coupling to the resonator. In this regime, the ground state is predicted to be an entangled state of the qubit's persistent-current states and the resonator's coherent states.   

Quantum electrodynamics near a photonic band-gap

Quantum electrodynamics predicts the localization of light around an atom in photonic band-gap (PBG) medium or photonic crystal.~Here we report the first experimental realization of the strong coupling between a single artificial atom and an one dimensional PBG medium using superconducting circuits. In the photonic transport measurement, we observe an anomalous Lamb shift and a large band-edge avoided crossing when the artificial atom frequency is tuned across the band-edge. The persistent peak within the band-gap indicates the single photon bound state. Furthermore, we study the resonance fluorescence of this bound state, again demonstrating the breakdown of the Born-Markov approximation near the band-edge. This novel architecture can be directly generalized to study many-body quantum electrodynamics and to construct more complicated spin chain models.~   

Experimental study of a 72-site Jaynes-Cummings lattice in the nonlinear dispersive regime

The building blocks of circuit-QED provide useful tools for the study of nonequilibrium and highly nonlinear behavior. In particular, the inherent dissipation in circuit-QED systems naturally gives rise to crossovers between different steady-states and dynamical phase transitions in even as few as two-site lattices. We explore the steady-state behavior of a 72-site Jaynes-Cummings lattice in the dispersive regime, highlighting the sharp transition in fluorescence at critical drive powers as well as strong nonlinear wave-mixing phenomena. ~   

The driven-dissipative Jaynes-Cummings lattice in the nonlinear dispersive regime

Experiments studying circuit-QED lattices have great potential for advancing our understanding of nonequilibrium many-body phenomena, including dissipative and dynamical phase transitions. One particular model realizable in this architecture is the driven-dissipative Jaynes-Cummings lattice. Motivated by the experimental measurements in the Houck lab, we theoretically investigate the dispersive regime including sub-leading nonlinear contributions from Kerr terms, employing a semi-classical approximation and numerics based on the quantum master equation. We explore the features of the experimentally detected crossover which is observed for an increase of the driving strength beyond a certain threshold.   

Spin-boson model with an engineered reservoir in circuit QED

A superconducting qubit coupled to an open transmission line represents an implementation of the spin-boson model with an engineered environment. Using a flux qubit with a large mutual inductance to the transmission line, we confirm in a resonance fluorescence experiment that the spectral function $J(\omega)$ of this environment is Ohmic over a frequency range of several gigahertz. Furthermore, partial reflectors implemented into the transmission line modify the spectral function of the transmission line. For weak enough reflectors, we find that the resulting broad peak can be interpreted in terms of an enhanced spontaneous emission rate. Our work [M. Haeberlein \emph{et al.}, arXiv:1506.09114 (2015)] lays the ground for future quantum simulations of other, more involved, impurity models with superconducting circuits.   

Strongly Correlated Photons at Full Transmission

We show how to make strongly correlated photons in a fully transmitted pulse. The system consists of three-level qubits (3LS) coupled to a one-dimensional waveguide. Our two-photon scattering approach naturally connects photon correlations with inelastic scattering. We find that the total inelastically scattered flux is much larger than in the case of two-level systems, making 3LS better candidates for experimental study of non-classical light. Strikingly, there is a further substantial increase in inelastic flux upon adding either more 3LS or a mirror. Typically, resonant probe photons at electromagnetically induced transparency are not correlated --- the correlations occur off resonance and so involve backscattering. Remarkably, we show that for three qubits, the qubit frequencies and the pump beam can be engineered such that correlated photons are fully transmitted, thereby greatly improving the efficiency of generating photon correlation.