Linear response theory of Josephson junction arrays in a microwave cavity

Recent experiments on Josephson junction arrays (JJAs) in microwave cavities have opened up a new avenue for investigating the properties of these devices while minimizing the amount of external noise coming from the measurement apparatus itself. These experiments have already shown promise for probing many-body quantum effects in JJAs. In this work, we develop a general theoretical description of such experiments by deriving a quantum phase model for planar JJAs containing quantized vortices. The dynamical susceptibility of this model is calculated for some simple circuits, and signatures of the injection of additional vortices are identified. The effects of decoherence are considered via a Lindblad master equation.

Figure 1

Cartoon of a Josephson junction array inside a microwave cavity. Such setups have been used experimentally to probe quantum behavior of Josephson junction devices via spectroscopy measurements.

Figure 2

Illustration of the mixed representation, which allows us to incorporate Josephson junctions into a loop-charge approach, and the Born-Oppenheimer approximation which allows us to replace the capacitively-shunted JJ with an effective QPS element. The branch with the Josephson junction (colored blue) is initially considered within the node-flux representation, while all other branches are incorporated via the loop-charge representation. The fictitious loop charge $Q_{ij}^{\prime }$ will be reduced to an algebraic constraint. When fast-moving degrees of freedom are integrated out, the Josephson junction is approximated by an effective quantum phase slip element.

Figure 3

(a) A two-loop circuit with hard boundaries, so that vortices cannot enter or exit the circuit. (b) The dynamical response $|{\chi }_n\left(\omega \right)|$ of the circuit depicted in (a). The response consists only of sharp peaks at the resonance frequencies given in Eq. (24). The susceptibility has been normalized at each value of $f$ to make peaks equally visible across the whole spectrum, so that the color axis is arbitrary and does not represent the actual peak height.

Figure 4

(a) A two-loop circuit with with junction boundaries, such that particle number is no longer conserved. (b) The dynamical response $|{\chi }_n\left(\omega \right)|$ of the two-loop circuit depicted in (a). Since particle number is no longer conserved, the zero-particle and one-particle ground states are adiabatically connected. Energy levels curve as they approach the crossover, and additional resonant frequencies appear when compared with the response in Fig. 3.

Figure 5

Energy gaps for a two-site circuit with closed boundary (gray) and open boundary (dashed green). Vertical blue lines illustrate estimates for width of crossover region given by Eqs. (26) and (27). The color of the thick green line indicates the magnitude of the matrix element $|\langle {\psi }_m|\widehat{n}|{\psi }_n\rangle |$, which gives the magnitude of the linear response in accordance with Eq. (20).

Figure 6

Energy gaps of a two-site system as the junctions on the boundary are turned on with an external frustration of $f=0.24$, in the center of the crossover region described by Eqs. (26) and (27). Thick green lines indicate the matrix element, as in Fig. 5. Black dotted lines give the results of second-order perturbation theory, given in the Appendix. It can be seen that some levels acquire a finite matrix element as tunneling through the boundary increases. This occurs as the total number of particles is no longer conserved, and eigenstates of the Hamiltonian consist of superpositions of states of different particle number.

Figure 7

The dynamical response ${\chi }_n\left(\omega \right)$ for a two-site system with closed (top) or open (bottom) edges with dephasing, calculated using Eq. (29), with dynamics given by the Lindblad equation (30). Compared with the (pure) ground state calculations in Figs. 3 and 4, respectively, additional lines appear in the spectrum, corresponding to energy gaps relative to other states appearing in the steady state mixture. Some of these additional states are listed in Fig. 8.

Figure 8

The linear dynamical response ${\chi }_n\left(\omega \right)$ for $f=0$ as the charge dephasing rate ${\mathrm{\Gamma }}_1$ is adjusted while all other dephasing rates are fixed at 0. The peaks $A, B_1, B_2$, and $C$ labeled in (a) and (b) correspond to transitions between states listed in (c) (using the notation introduced in Appendix pp1). At zero dephasing, the only peak present is $C$, corresponding to the transition between the ground and first excited state (see Fig. 5). Dephasing drives the system from the ground state into a mixed state, so that other transitions can contribute. As the dephasing rate is increased, the peaks broaden until, at strong dephasing, important features are washed out completely.

Figure 9

(a) Circuit diagrams for a $2\times 3$ loop flux-based circuit. The relevant degrees of freedom for this circuit are vortices in the loops, which may tunnel across the QPS elements on the branches. In the top circuit, total vortex number is conserved, whereas in the bottom circuit vortices may enter and exit the array by tunneling across the outer edges. (b),(d) The dynamical response $|{\chi }_n\left(\omega \right)|$ for the $2\times 3$ grids shown in Fig. 9 with hard (top) and junction (bottom) boundaries. (c) The energy gaps about the ground state energy plotted as a function of external flux (dotted lines). Color of thick, solid lines corresponds to the amplitude squared of the matrix element for the vortex number operator between that state and the ground state, $|\langle {\psi }_n|{\widehat{\mathrm{\Phi }}}_1|{\psi }_0{\rangle |}^2$, c.f. Eq. (20).

Figure 10

Energy gaps above ground $E_i-E_0$ as a function of the boundary tunneling $t_{\text{edge}}$. The darkness of the solid green lines indicates the value of the matrix element $|\langle {\psi }_i|\widehat{n}|{\psi }_0{\rangle |}^2$. These lines are not visible where the matrix element vanishes.

Figure 11

Active duals of the circuits in Figs. 3 and 4, respectively, obtained via a duality transformation. These circuits exhibit the same dynamics as their duals but with different variables.