Quantum Brownian Motion at Strong Dissipation Probed by Superconducting Tunnel Junctions

We have investigated the phase dynamics of a superconducting tunnel junction at ultralow temperatures in the presence of high damping, where the interaction with environmental degrees of freedom represents the leading energy scale. In this regime, theory predicts the dynamics to follow a generalization of the classical Smoluchowski description, the quantum Smoluchowski equation, thus, exhibiting overdamped quantum Brownian motion characteristics. For this purpose, we have performed current-biased measurements on the small-capacitance Josephson junction of a scanning tunneling microscope placed in a low impedance environment at milli-Kelvin temperatures. We can describe our experimental findings with high accuracy by using a quantum phase diffusion model based on the quantum Smoluchowski equation. In this way we experimentally demonstrate that overdamped quantum systems follow quasiclassical dynamics with significant quantum effects as the leading corrections.

Figure 1

(a) Parameter diagram for the dynamics of overdamped diffusion $\eta ={\gamma }^2/{\omega }_0^2>1$: The regime of OQPD (QSM) emerges at lower temperatures from the CSM when the dimensionless friction $\eta$ sufficiently exceeds the dimensionless inverse temperature $\mathrm{\Theta }=\gamma \hslash \beta /(2\pi )\gg 1$. (b) The phase dynamics corresponds to the dissipative quantum dynamics of a particle in a washboard potential (grey solid), which appears as MQT for weak friction, i.e., the tunneling of the phase wave function through the potential wall (red wave package), and as OQPD in the QSM domain, for which quantum fluctuations effectively reduce the barrier height (blue solid). Classical phase diffusion at $\mathrm{\Theta }\ll 1$ is illustrated as the thermally activated escape of a particle (green dot) over the potential barrier.

Figure 2

(a) Circuit diagram with the current source $I_B$, the source impedance $R_B$, the load-line resistor and shunt capacitor $R_0$ and $C_0$, respectively. The junction and its direct environment thermalized at $T=15\mathrm{mK}$ are highlighted by the blue box. $I_0$ and $C_J$ denote the junction element and capacitor, respectively, and $Z$ the environmental impedance. Right: Schematic representation of the STM tip on top of the reconstructed $V(100)$ surface [22]. (b) IVC measured at $G_N=0.004G_0$. (c) Simulated real part $\Re [Z(\nu )]$ of the environmental impedance [28]. (d) In-gap region of an IVC from a current-biased measurement at $G_N=0.074G_0$.

Figure 3

(a) Experimental (red dots) and calculated (dashed lines) IVCs at indicated values of $G_N/G_0$. The deviation between $I_S$ and $I_{J,\max }^{\mathrm{QSM}}$ is highlighted via the grey bars. (b) Top: Calculated $I_0$ (open circles) and linear fit (dashed line) as a function of $G_N/G_0$. Bottom: Experimental, $I_S$ (open triangles), and calculated switching current, $I_{J,\max }^{\mathrm{QSM}}$ (open squares), as well as a quadratic fit (dashed line) as a function of $G_N/G_0$. Error bars are contained within the symbols. (c) Experimental (Exp, open circles) and calculated from MQT (Calc, plus signs) zero bias resistance as well as a square fit to the experimental data (Fit, dashed line) as a function of $E_C/E_J$.

Figure 4

(a) Relative deviation between theoretical and experimental current maxima $\mathrm{\Delta }{I}_{\mathrm{QSM}}$ and ratio $\eta /\mathrm{\Theta }$ as a function of $G_N/G_0$. The QSM regime boundary is marked as a dashed line at $\eta /\mathrm{\Theta }=1$. (b) Comparison between experimental (dots) and calculated IVC (dashed line) from the purely classical CIZ model at $G_N=0.051G_0$.