Multi-photon resonances in Josephson junction-cavity circuits

Abstract

We explore the dissipative dynamics of nonlinearly driven oscillator systems tuned to resonances where multiple excitations are generated. Such systems are readily realised in circuit QED systems combining Josephson junctions with a microwave cavity and a drive achieved either through flux or voltage bias. For resonances involving three or more photons the system undergoes a sequence of two closely spaced dynamical transitions (the first one discontinuous and the second continuous) as the driving is increased leading to steady states that form complex periodic structures in phase space. In the vicinity of the transitions the system displays interesting bistable behaviour: we find that coherent effects can lead to surprising oscillations in the weight of the different dynamical states in the steady state of the system with increasing drive. We show that the dynamics is well-described by a simple effective rate model with transitions between states localised at different points in the phase space crystal. The oscillations in the weights of the dynamical states is reflected in corresponding oscillations in a time-scale that describes transitions between the states.

Josephson photonics with simultaneous resonances

“…The idea is to assume that each mode is in a coherent state, ρ α = N l=1 |α l α l |, described by a complex amplitude α l . Substituting this into the master equation ( 5) leads to a set of equations of motion for the amplitudes, the fixed points of which provide a valuable framework for understanding the dynamics of the system 9,18,32,33 . This approach can be thought of as providing an essentially classical description of the dynamics as (quantum) fluctuations in the amplitudes are neglected 34 .…”

“…When applied to systems with a unique resonance, the resulting fixed point amplitudes have been shown to provide an increasingly accurate way of predicting properties like the average occupation numbers of the modes as the strength of the quantum fluctuations (measured by ∆ l ) are reduced 14 . However, one key limitation is that no information is provided about the way in which the density operator spreads between two or more coexisting stable fixed points 18 .…”

“…Furthermore, the energy transferred by tunneling Cooper pairs into microwave modes can be tracked by monitoring either the resulting dc current or the microwaves leaking out of the circuit 1 . Recent experimental, [1][2][3][4][5][6] and theoretical work [7][8][9][10][11][12][13][14][15][16][17][18] has explored a wide range of ways in which JJ-cavity systems can be used to generate non-classical microwave states.…”

“…Such resonances can be selected by simply tuning the voltage and are modelled theoretically using a rotating wave approximation (RWA) which leads to a convenient time-independent Hamiltonian for the system 9,10,12,19 . The simplest resonances involve a single mode and can be exploited to provide a single photon source 4,5 , although higher order resonances in which two or more photons are generated within a particular mode also occur 9,18,20 .…”

Emission of photon multiplets by a dc-biased superconducting circuit

“…The impact of quantum fluctuations for large α masks all sharp transitions predicted by semiclassics. Besides blurring all transitions, quantum fluctuations may also allow dynamical transitions between solutions and crucially affect certain observables [39]. J ≡ 1 and a parametric transition at lower EJ (see Ref.…”