Session X26: Superconducting Qubits: Coherent Phases in Superconducting Resonators

Galvanic coupling of two superconducting microwave resonators

Thermal entanglement forms when lowering the temperature sufficiently and can be detected via suitable correlation measurements on, e.g., a system of two degenerate coupled microwave resonators. In an actual experiment, this requires a coupling strength of 10-20\% of the transition frequency. For superconducting flux quantum circuits, such large couplings are usually obtained using the kinetic inductance: the circuits have to be galvanically connected and form a single entity. Nevertheless, the physics can often be described by two separate systems with an enhanced coupling constant. In contrast, we show that the mode structure of two coupled superconducting microwave resonators changes discontinuously when connecting them galvanically instead of placing them very close to each other and explain this phenomenon with a toy model.   

Fourth order dispersive regime of circuit QED with a transmon and a CJBA

In most of the recent circuit QED experiments [1,2], a transmon qubit [3] is fabricated inside a transmission-line resonator. While the transmon qubit has the advantage of providing both better coupling and longer dephasing time than the Cooper-pair box, its anharmonicity is also reduced. Higher levels must therefore be taken into account. We study a transmon coupled to a cavity bifurcation amplifier (CJBA) in the dispersive regime, where the qubit-resonator detuning is larger than the coupling strength. Building on previous work [4], we use perturbation theory to fourth order and obtain analytical solutions for the system. We compare these to numerical simulations and experimental data and analyze how the non-linearity of the CJBA and higher transmon levels affect the physics of the system. We finally show how the many-level structure of this system can be used to help for the measurement. [1] DiCarlo et al, Nature, 2009, 460, 240, [2] Mallet et al, Nat. Phys, 2009, 5, 791, [3] Koch et al, PRA, 2007, 76, 042319, [4] Boissonneault et al, PRA, 2008, 77, 060305(R)   

RF SQUID-Mediated Coherent Tunable Coupling Between a Superconducting Phase Qubit and Lumped Element Resonator

We demonstrate coherent tunable coupling between a superconducting phase qubit and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase qubit and resonator. We verify the coupling strength modulation by observing modulation in the size of the avoided crossing in the phase qubit's spectroscopy as well as coherently by observing modulation in the vacuum Rabi oscillation frequency at zero detuning. The measured spectroscopic splittings and vacuum Rabi oscillations are in good agreement with theoretical predictions.   

Coupled Harmonics in Nonlinear Coplanar Waveguide Resonators

The coplanar waveguide resonator has become a ubiquitous design element in superconducting quantum information experiments. By necessity, many of these resonators are loaded with nonlinear elements and, as such, incorporate a mechanism for coupling power between harmonic modes of the system. In this talk, we will present a specific example in which a dc SQUID terminates a quarter wave resonator and how the mode coupling via this nonlinear element can both enhance and degrade the performance of this device as a parametric amplifier and frequency converter.   

Selection rules for multiphoton excitations in a qubit-resonator system

The theoretical analysis of a qubit-resonator system reveals selection rules at certain symmetry points of the qubit. In the case of a superconducting flux qubit, this symmetry of the double-well potential can be broken in a controlled way by changing the external magnetic field. Only then, odd and even multiphoton processes can coexist at the same flux bias. We illustrate this phenomenon with spectroscopy measurements of a superconducting flux qubit strongly coupled to an on-chip coplanar waveguide resonator. Our studies extend up to three-photon driving and clearly show the transition from strict selection rules to a regime of coexistent multiphoton excitations. We acknowledge support from SFB631, NIM, UPV/EHU Grant GIU07/40, and European project EuroSQIP.   

Two-resonator Circuit Quantum Electrodynamics: A superconducting Quantum Switch

We introduce a systematic formalism for two-resonator circuit QED, where two on-chip microwave resonators are simultaneously coupled to one superconducting qubit. Within this framework, we demonstrate that the qubit can function as a quantum switch between the two resonators. We also show that our quantum switch represents a flexible architecture for the manipulation and generation of nonclassical microwave field states as well as the creation of controlled multipartite entanglement in circuit QED. In addition, we propose a realistic implementation of two-resonator circuit QED.   

Single-photon switch: Controllable scattering of photons inside a one-dimensional resonator waveguide

We analyze the coherent transport of a single photon, which propagates in a one-dimensional coupled-resonator waveguide and is scattered by a controllable two-level system located inside one of the resonators of this waveguide. Our approach, which uses discrete coordinates, unifies low and high energy effective theories for single-photon scattering. We show that the controllable two-level system can behave as a quantum switch for the coherent transport of a single photon. This study may inspire new electro-optical single-photon quantum devices. We also suggest an experimental setup based on superconducting transmission line resonators and qubits. References: L. Zhou, Z.R. Gong, Y.X. Liu, C.P. Sun, F. Nori, Controllable scattering of photons inside a one-dimensional resonator waveguide, Phys. Rev. Lett. 101, 100501 (2008). L. Zhou, H. Dong, Y.X. Liu, C.P. Sun, F. Nori, Quantum super-cavity with atomic mirrors, Phys. Rev. A 78, 063827 (2008).   

Entanglement and Shot Noise in a two-qubit c-QED

It has been proved in the last few years that circuit quantum electrodynamics (c-QED) is a promising architecture for quantum information processing and quantum computation realizations [1]. We theoretically study a c-QED system consisting of two charge-qubits (defined in two Cooper-Pair boxes) coupled to a common bosonic mode (photon, phonon) inside a transmission line cavity. In particular, we analyze a transport setup in which each qubit is attached to a drain electrode such that the Josephson Quasiparticle cycle induces an electrical current flow across them [2]. By means of a Quantum Master Equation approach, shot noise cross-correlations of the electrical current are studied and their remarkable behavior is related with the degree of entanglement induced between the qubits by the coupling with the boson mode.\\[4pt] [1] L. DiCarlo et al., Nature 460, 240 (2009); J. Mayer et al., Nature 449, 443 (2007), Y. Bishop et al., New J.Phys. 11, 073040 (2009), Hutchinson et al., Can. J. Phys. 87, 225 (2009). \\[0pt] [2] For example O. Astafiev, et al., Nature 449, 588 (2007).   

One-step multi-qubit GHZ state generation in a circuit QED system

Based on a circuit QED setup of superconducting flux qubits or charge qubits coupled to a transmission line resonator, we propose a scheme to generate multi-qubit GHZ states. The states are prepared by applying a single dc pulse to the superconducting qubits. Our scheme is based on the non-dispersive dynamic evolution of the system, hence the operation time is short and independent of the number of qubits. Using realistic parameters, we estimate that the GHZ state can be produced within the coherence time of the multi-qubit system. Our scheme is independent of the initial state of the transmission line resonator and works in the presence of higher harmonic modes. Our analysis also shows that the scheme is robust to various operation errors and environmental noise.   

Arbitrary Control of Entanglement between Two Superconducting Resonators

We consider the problem of synthesizing an arbitrary entangled state of two superconducting resonators. We show that this problem can be solved efficiently for entangled superpositions of photon number (Fock) states, utilizing a coherent interaction of each resonator with a single artificial atom. We theoretically analyze this approach for synthesizing high NOON states using existing technology.   

Deterministic generation of entangled photon pairs in coupled superconducting resonators

Entangled photon pairs play an important role in quantum information. In this work, we propose a scheme for the on-demand generation of entangled photon pairs in a circuit of four superconducting transmission-line resonators connected in a ring geometry. Each resonator in this circuit couples with a superconducting qubit. Strong coupling between superconducting resonators and superconducting qubits has been demonstrated in recent experiments. This coupling can generate a tunable Kerr-like nonlinearity in the resonators. Combining this nonlinearity with the photon tunneling between adjacent resonators, the system can be described as a Bose-Hubbard-like model. We will show that entangled photon pairs, i.e. EPR pairs, can be prepared by microwave pulses with high fidelity when the nonlinearity dominates over the photon tunneling; and can be transferred to flying modes when the photon tunneling dominates over the nonlinearity. We also studied the effect of resonator damping on this scheme.   

Circuit QED with superconducting flux qubits controlled by a bias current

We propose a circuit QED scheme for the superconducting flux qubits. It is shown that the bias-current-control of the flux qubit can be performed for the qubit loop with asymmetry. Hence the three-Josephson-junction flux qubit rather than the rf-SQUID qubit or the dc-SQUID loop can be controlled by a bias current as in the usual superconducting phase qubit. The qubit operation and the qubit state detection are performed by bias current, instead of the external magnetic field used for the flux qubit. Our current-biased flux qubit has the advantages of the optimal point operation of the flux qubit and the scalability and fast operation of the phase qubit as well. By coupling the flux qubit with the oscillating current modes in a superconducting cavity, the two-qubit gate between flux qubits can be achieved.   

Quasi-Lumped Element Resonator Coupled to a Cooper-Pair Box

We have coupled a ``quasi-lumped element'' microwave resonator ($f_{0}$ = 5.433 GHz), made of superconducting Al on sapphire, to an Al/AlO$_{\mbox{x}}$/Al Cooper-pair box (CPB) qubit. In zero magnetic field, the CPB is far detuned from the resonator and we measure a 50 kHz decrease in $f_{0}$ with the qubit in the ground state and near the degeneracy point of the CPB. By exciting the CPB from the ground state using a second microwave tone and monitoring the transmission through our resonator, we have determined that our CPB has a charging energy $E_{C}/h$ = 12.5 GHz and a maximum $E_{J}/h$ = 9 GHz. By changing the external magnetic field, we can also decrease the effective $E_ {J}$ of the CPB. This decreases the detuning between the CPB and resonator and increases the frequency shift of the resonator. From modeling our data, we extract a coupling strength of $g/2\pi$ = 11 MHz [1,2]. Single and two tone spectroscopy of this system will be presented as well as preliminary measurements of $T_{1}$ and $T_{2}^*$. [1] Alexandre Blais \emph{et al}., Phys. Rev. A \textbf{69}, 062320 (2004). [2] A. Wallraff \emph{et al}., Nature \textbf{431}, 162 (2004).   

Cavity QED at the quantum-classical boundary

The quantum limit of cavity QED is characterized by a well resolved vacuum Rabi mode splitting spectrum. If the number of excitations $n$ in the resonantly coupled matter-light system is increased from one, the nonlinear $\sqrt{n}$ scaling of the dressed eigenstates is observed [1]. At very large photon numbers the transmission spectrum turns into a single Lorentzian line as expected from the correspondence principle. This classical limit emerges when the occupancy of the low energy dressed states is increased until the quantum nonlinearity of the available transitions becomes small compared to dephasing and relaxation rates $[2]$. We explore this quantum-classical crossover in a circuit QED system where we vary the thermal occupation of the resonator by 5 orders of magnitude using a quasi-thermal noise source. From vacuum Rabi spectra measured in linear response and from time resolved vacuum Rabi oscillation measurements we consistently extract cavity field temperatures between 100 mK and 10 K using a master equation model. The presented experimental approach is useful to determine the thermal occupation of a quantum system and offers the possibility to study entanglement and decoherence at elevated temperatures. $[1]$ J.~M.~Fink et al.~\textit{Nature} \textbf{454}, 315 (2008). $[2]$ I.~Rau, et al.~\textit{Phys. Rev. B} \textbf{70}, 054521 (2004).   

Multiple mode circuit QED with engineered photon lifetimes

We present the realization of a cavity quantum electrodynamics setup in which photons of strongly different lifetimes are engineered in different harmonic modes of the same cavity. We achieve this in a superconducting transmission line resonator with superconducting qubits strongly coupled to the different resonator modes. We use a short lifetime externally coupled mode for dispersive readout of the qubits, while simultaneously performing experiments with maximized photon coherence in an adjacent long lifetime mode of the multiple quality factor resonator. We demonstrate sideband based generation of n photon Fock states and the scaling of the sideband Rabi frequency with $\sqrt{n}$ using schemes that may be extended to realize sideband based two-qubit logic gates.