Classical analysis of capacitively coupled superconducting qubits

Abstract

An electrical circuit consisting of two capacitively coupled inductive loops, each interrupted by a Josephson junction, is analyzed through the classical resistively and capacitively shunted junction ͑RCSJ͒ model. The same circuit has recently been studied experimentally and the results were used to demonstrate quantummechanical entanglement in the system by observing the correlated states of the two inductive loops after initial microwave perturbations. Our classical analysis shows that the observed phenomenon exists entirely within the classical RCSJ model, and we provide a detailed intuitive description of the transient dynamics responsible for the observations.

Tomography and Entanglement in Coupled Josephson Junction Qubits

“…The externally applied fluxes through loops 1 and 2 are represented by È 1x and È 2x . As discussed in [17], in dimensionless form the governing equations of this system turn out to be:…”

“…As in [17], parameter values were set at ¼ 5 Â 10 À5 , L ¼ 2:841, g ¼ 0:9954, I c ¼ 1:1 A, C ¼ 1:3 pF, and ! À1 J ¼ 0:02 ns.…”

“…Our previous work [17] outlined the dynamical modes exhibited by this system. One of these modes is consistent with the experimental observations and can be obtained in the linear limit of oscillation amplitude.…”

“…Following Ref. [17], we therefore take M b ¼ 0, % 0, and for small amplitude oscillations of ' 1 and ' 2 , we write ' a ¼ ' 0 þ c a , where ' 0 is a constant and jc a j ( 1. Similarly we assume j' b j ( 1.…”

“…This expression directly gives the experimentally [16] and theoretically [17] observed slow envelope modulation of the two coupled oscillators, and calculating the modulation frequency from the experimentally provided system parameters yields I ¼ 2! d % 0:0014, which is in very good agreement with observations [16,17].…”