Nonideal quantum measurement effects on the switching-current distribution of Josephson junctions

The quantum character of Josephson junctions is ordinarily revealed through the analysis of the switching currents, i.e., the current at which a finite voltage appears: A sharp rise of the voltage signals the passage (tunnel) from a trapped state (the zero voltage solution) to a running state (the finite voltage solution). In this context, we investigate the probability distribution of the Josephson-junction switching current taking into account the effect of the bias sweeping rate and introducing a simple nonideal quantum measurement scheme. The measurements are modeled as repeated voltage samplings at discrete time intervals, that is, with repeated projections of the time-dependent quantum solutions on the static or the running states, to retrieve the probability distribution of the switching currents. The distribution appears to be immune to the quantum Zeno effect, and it is close to, but distinguishable from, the Wentzel-Kramers-Brillouin approximation. For energy barriers comparable to the quantum fundamental energy state and in the fast bias current ramp rate the difference is neat, and remains sizable in the asymptotic slow rate limit. This behavior is a consequence of the quantum character of the system that confirms the presence of a backreaction of quantum measurements on the outcome of mesoscopic Josephson junctions.

Figure 1

Sketch of the measurement process at the time $t=nT/N$. The red (thin, light gray) solid curve sketches, in arbitrary units, the washboard potential, Eq. (3). The vertical line denoted by ${\phi }^{*}$ corresponds to the separation between the trapped state (region I) and the running state (region II). The dashed black curve (also reported in the inset), except for a normalization constant, is the square modulus of the wave function, just after the measurement, $|{\psi }_2{(\phi ,t)|}^2$; see Eq. (6). The solid curve represents its evolution, as per Eq. (2), after ${10}^3$ normalized units: $|{\psi }_2(\phi ,t+{10}^3){|}^2$. The presence of a finite probability in region II behind the energy barrier limited by ${\phi }^{*}$ corresponds to the possibility of escapes, i.e., to tunnel towards nonzero voltage solutions. The nonzero current of probability at the boundary demonstrates that the perfect matched layer [32] gives the expected decaying towards infinity.

Figure 2

(a) Radiated probability absorbed by the radiative boundary condition: red (light gray) dashed curve, initial condition, the fundamental resonance without measurement; black solid line, initial condition, the fundamental resonance with measurement as per Eq. (11). (b) The probability that the wave function is found in the integration domain as a function of the time for different values of the bias current $\gamma$, from $\gamma =0.45$ to 0.6 with step $\gamma =0.05$. The normalized maximum energy barrier is $V_0=4$.

Figure 3

Decay rate of the solution after a quantum measurement (dots-solid line) compared to the WKB rate, Eq. (13) as a function of the bias current. The dimensionless energy barrier reads $V_0=4$. We underline that both axes are on a log scale.

Figure 4

Effect of $V_0$ value on relaxation time. The curves show the behavior of the absorbed norm at infinity (radiative condition) computed starting from a quantum measured fundamental resonance having the same WKB decay rate. The reference rate is displayed in the dashed line (nonmeasured fundamental state) for $V_0=4$ and $\gamma =0.4$

Figure 5

The probability distribution ${\mathcal{P}}_{{\gamma }_{\text{sw}}}\left(n/N\right)$ of the switching currents for repeated measurements. The ramp time is $T=800$; during this time the number of measurements changes from $N=50$ to 3200. The normalized maximum energy barrier is $V_0=4$.

Figure 6

Behavior of the maximum probability of switch as a function of the current ramp time. For each ramp time we report the effect of the number of measurements as per Eqs. (2) and (12). The dashed line represents the WKB approximation given by Eq. (13). The normalized maximum energy barrier is $V_0=4$.

Figure 7

Cumulative distribution of quantum escape times in the long $T/N$ regime. The black circles represent the WKB switching-current cumulative distribution corresponding to the PDF reported in Eq. (14). Gray squares represent numerical simulations. The normalized maximum energy barrier is $V_0=4$, the number of measurement reads $N=100$, and the ramp time is $T={10}^4$.

Figure 8

The probability distribution ${\mathcal{P}}_{{\gamma }_{\text{sw}}}\left(n/N\right)$ of the switching currents for repeated measurements (triangles, squares, and diamonds) and the WKB approximation (circles) for $T={10}^4$. The number of measurements is $N=100$ while the ramp time changes from $T=3200$ to $10000$. The normalized maximum energy barrier is $V_0=4$.

Figure 9

The prefactor relevant for Eq. (A4) taking into account relaxation time correction to the decaying of measured fundamental resonance. The barrier height reads $V_0=4$.