Theory of bistable correlated nonlinear oscillators in circuit quantum electrodynamics

We will describe our theoretical and experimental work towards understanding the joint activated dynamics exhibited by two superconducting quantum oscillators. The open quantum systems approach can describe a cavity mode oscillator which is strongly coupled to a superconducting qubit in the strongly coherently driven dispersive regime. We will start by introducing previous work ranging from the quantum optical stochastic methods, exact solutions in phase space and methods of non-equilibrium field theory. Next we will explain the challenge of extending the application of these methods to two oscillators and to the non-semi-classical regime. We will survey our early work on this problem which started by earlier theoretical work on the use of strongly driving pulses for quantum state detection [1,2,3]. Recent collaboration with experiments have encouraged us to apply a range of methods and we uncovered some surprising dynamical behaviour of the including a joint bistability of the generalised Jaynes-Cummings model [4,5], meta-stable dark states [6] and critical slowing down of the system [7] which are properties of the bistability in two degrees of freedom.

[1] E. Ginossar and Lev S Bishop and D. I Schuster and S. M Girvin, Protocol for high fidelity readout in the photon blockade regime of circuit QED,Phys. Rev. A 82,022335 (2010)

[2] Lev S. Bishop, Eran Ginossar, and S. M. Girvin, Response of the Strongly Driven Jaynes-Cummings Oscillator, Phys. Rev. Lett. 10 , 100505 (2010)

[3] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, High-Fidelity Readout in Circuit Quantum Electrodynamics Using the Jaynes-Cummings Nonlinearity, Phys. Rev. Lett. 173601 (2010)

[4] Th. K. Mavrogordatos, G. Tancredi, M. Elliott, M. J. Peterer, A. Patterson, J. Rahamim,2 P. J. Leek, E. Ginossar, and M. H. Szymanska, Simultaneous Bistability of a Qubit and Resonator in Circuit Quantum Electrodynamics, Phys. Rev. Lett. 040402 (2017)

[5] Matthew Elliott and Eran Ginossar, Applications of the Fokker-Planck equation in circuit quantum electrodynamics,  Phys. Rev. A 94, 043840 (2016)

[6] Th. K. Mavrogordatos, F. Barratt, U. Asari, P. Szafulski, E. Ginossar, and M. H. Szymańska ,Rare quantum metastable states in the strongly dispersive Jaynes-Cummings oscillator, Phys. Rev. A 97, 033828 (2018)

[7] P. Brooks, G. Tancredi et al., in preparation.