Suppression of macroscopic quantum tunneling in a large Josephson junction coupled to a resonator

We employ a functional integral technique to calculate the macroscopic quantum tunneling rate at zero temperature of a current-biased Josephson junction weakly coupled to a resonator. We allow for the effects of environmental dissipation on both the junction and the resonator via baths of harmonic oscillators, and we consider both cases of weak and strong junction damping. The resonator can be, in principle, any quantum harmonic oscillator that couples to the junction bilinearly in either the coordinate or the velocity. The Josephson phase difference $\varphi$ is the tunneling variable in this system, and the low-lying bound states in the junction’s potential energy well are weakly metastable due to tunneling through a finite-width energy barrier. There has been some interest in using such biased junctions as qubits with the resonator providing multiqubit coupling. The main result of this work is that coupling to the resonator has a suppressive effect on the junction’s tunneling: the stronger the coupling strength between the junction and resonator, the greater the reduction of the tunneling rate. Including damping to the junction also suppresses tunneling, a well-known result, but damping of the resonator actually reduces the magnitude of suppression; i.e., damping the resonator partially counteracts the suppression provided directly by the junction-resonator coupling. Details of the junction-resonator coupling yield interesting variations on this theme that may be useful for qubit design. For example, the coupling $U_{\mathit{int}}$ between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\omega }_R$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically $U_{\mathit{int}}\propto {\left({\omega }_R\right)}^n$, where $n=1∕2$. We find that for any value of $n$ in the range $0<n<1$ the junction’s tunneling rate exhibits a nonmonotonic dependence on ${\omega }_R$, and for $n=1∕2$, the tunneling rate is maximally suppressed for ${\omega }_R∕{\omega }_J\approx 1$, where ${\omega }_J$ is the bias current-dependent plasma frequency of the junction.

Figure 1

(Color online) The tilted washboard potential energy of a current-biased Josephson junction, $U_J∕E_J$, where $E_J=\hslash I_c∕2e$ is the Josephson energy for a JJ of critical current $I_c$ (solid line). Also shown is the quadratic-plus-cubic approximation (dashed line). Both curves were constructed for a bias current of $I_B∕I_C=0.25$. The vertical arrow represents the height of the potential energy barrier of the quadratic-plus-cubic well. Inset: circuit diagram of an underdamped JJ with critical current $I_c$ and effective capacitance $C$. The JJ is biased by a dc current source $I_B$ and connected to a resonator, labeled as $R$.

Figure 2

(Color online) Tunneling rate for weak JJ-resonator coupling and weak damping as a function of the scaled frequency $\Omega ={\omega }_R∕{\omega }_J$ defined in Eq. (27). The tunneling rate is scaled by its value ${\Gamma }_0$ in the absence of JJ-resonator coupling and dissipation. For simplicity, we choose $C=1$ in Eq. (32) and ${\alpha }_J={\alpha }_R=1∕8$. In all plots, the increasingly longer-dashed lines correspond to $\overline{g}=0,0.5,0.75,1$, respectively. At a fixed value of $\Omega$, the tunneling rate is increasingly suppressed with larger coupling strength. (a) The exponent in Eq. (33) has a value of $n=-1$, for which ${\Gamma }_1∕{\Gamma }_0\to 0$ as $\Omega \to 0$ for finite $\overline{g}$. (b) $n=0$. As a function of frequency $\Omega$, maximal suppression occurs for $\Omega \to 0$, where ${\Gamma }_1∕{\Gamma }_0$ approaches a nonzero limit determined by $\overline{g}$, ${\alpha }_J$, and ${\alpha }_R$. (c) $n=1∕2$. ${\Gamma }_1∕{\Gamma }_0$ exhibits nonmonotonic dependence on $\Omega$ for finite $\overline{g}$, with maximum tunneling suppression occurring for $\Omega \approx 1$. For large $\Omega$ and for $n=-1$, 0, and 0.5 the scaled tunneling rate approaches a constant value determined by the amount of JJ damping, ${\alpha }_J$. (d) $n=1$. ${\Gamma }_1∕{\Gamma }_0$ exhibits monotonic suppression as $\Omega$ is increased but levels off to a constant value independent of $\Omega$ that is determined by $\overline{g}$ and ${\alpha }_J$. (e) $n=3∕2$. ${\Gamma }_1∕{\Gamma }_0$ is exponentially suppressed for large $\Omega$ and finite $\overline{g}$ and asymptotically approaches zero as $\Omega$ is increased. Note that the vertical scales on the plots are different.

Figure 3

(Color online) $\mathcal{I}(\Omega ,{\alpha }_R)$ defined in Eq. (31) as a function of the scaled frequency $\Omega ={\omega }_R∕{\omega }_J$ defined in Eq. (27). The increasingly longer-dashed lines correspond to ${\alpha }_R=0$, 0.125, and 0.25, respectively. The function tends to a finite value dependent on ${\alpha }_R$ in the low-frequency limit, and it asymptotically approaches zero for high frequencies.

Figure 4

(Color online) Scaled tunneling rate for weak resonator coupling and weak damping as a function of the scaled frequency $\Omega ={\omega }_R∕{\omega }_J$ defined in Eq. (27). We choose $C=1$, $\overline{g}=1$, and ${\alpha }_J={\alpha }_R=1∕8$. The increasingly longer-dashed lines correspond to values of the exponent $n$ [see Eq. (33)] of 0.1, 0.25, 0.5, and 0.75, respectively. The nonmonotonic behavior becomes more pronounced as $n\to 0^{+}$.

Figure 5

Contour plots of the scaled tunneling rate ${\Gamma }_1∕{\Gamma }_0$ for weak resonator coupling and weak damping as a function of the scaled frequency $\Omega ={\omega }_R∕{\omega }_J$ defined in Eq. (27) and the coupling strength $\overline{g}$ defined in Eq. (28). We choose ${\alpha }_J={\alpha }_R=1∕8$ and $C=1$. (a) $n=0.5$. The largest plotted contour corresponds to ${\Gamma }_1∕{\Gamma }_0=0.78$ while the smallest corresponds to a value of 0.7 for the same quantity. The intermediate contours represent successive increments of 0.01 in the tunneling rate. The region of greatest suppression occurs for $\Omega \approx 1$, and the amount of suppression—i.e., the depth of the “well” in (a)—increases with increasing resonator coupling strength $\overline{g}$. (b) $n=-1$. The largest plotted contour corresponds to ${\Gamma }_1∕{\Gamma }_0=0.78$ while the smallest corresponds to a value of 0.1 for the same quantity. The intermediate contours represent values in the tunneling rate between 0.1 and 0.78. The tunneling rate is monotonically suppressed toward zero as $\Omega \to 0$.

Figure 6

(Color online) Scaled tunneling rate as a function of (a) junction damping or (b) resonator damping. For each graph $n=0.5$, $C=1$, and $\Omega =1$. The increasingly longer dashed lines correspond to $\overline{g}=0$, 0.5, 0.75, and 1, respectively. (a) ${\alpha }_R=0.125$ is fixed, and ${\alpha }_J$ varies. (b) ${\alpha }_J=0.125$ is fixed, and ${\alpha }_R$ varies.

Figure 7

(Color online) Scaled tunneling rate ${\Gamma }_2∕{\Gamma }_{0,2}$ for an overdamped JJ as a function of the scaled frequency $\Omega ={\omega }_R∕{\omega }_J$ defined in Eq. (27). ${\Gamma }_{0,2}$ represents the MQT rate of an overdamped JJ in the absence of junction-resonator coupling [see Eq. (50)]. We choose ${\alpha }_R=0.25$, ${\alpha }_J=10$, and $C=1$ for all graphs. The increasingly longer dashed lines correspond to $\overline{g}=0$, 0.5, 0.75, and 1, respectively. (a) Frequency exponent $n=-1$, (b) $n=0$, (c) $n=0.5$, (d) $n=1$, and (e) $n=1.5$.

Figure 8

(Color online) Scaled tunneling rate ${\Gamma }_2∕{\Gamma }_{0,2}$ for an overdamped JJ as a function of the junction damping, ${\alpha }_J$, where ${\Gamma }_{0,2}$ represents the MQT rate of an overdamped JJ in the absence of junction-resonator coupling [see Eq. (50)]. We choose $n=0.5$, $\Omega =1$, and $C=1$. (a) The increasingly longer-dashed lines correspond to $\overline{g}=0$, 0.5, 0.75, and 1, respectively. Resonator damping was fixed at ${\alpha }_R=0.25$. (b) The increasingly longer-dashed lines correspond to ${\alpha }_R=0$, 0.1, 0.25, and 0.5, respectively. The resonator-junction coupling strength was fixed at $\overline{g}=0.5$. Weakly nonmonotonic dependence of the tunneling rate on ${\alpha }_J$ is evident for ${\alpha }_R=0.5$

Figure 9

(Color online) Scaled tunneling rate ${\Gamma }_1∕{\Gamma }_{0,1}$ for an underdamped JJ weakly coupled to a resonator as a function of the scaled dc bias current $i_B=I_B∕I_c$. We choose $n=0.5$ and ${\Omega }_0=1$, where ${\Omega }_0={\omega }_R∕{\omega }_{J0}$ and ${\omega }_{J0}$ is the junction plasma frequency for zero bias current. The ratio of the Josephson coupling and charging energies, $E_J∕E_c=10000$, which is typical for a phase qubit. The increasingly longer-dashed lines correspond to ${\overline{g}}_0=0$, 0.25, 0.5, and 1, respectively. (a) ${\alpha }_{J0}={\alpha }_{R0}=0$ and (b) ${\alpha }_{J0}={\alpha }_{R0}=0.1$.