Relative phase and Josephson dynamics between weakly coupled Richardson models

We consider two weakly coupled Richardson models to study the formation of a relative phase and the Josephson dynamics between two mesoscopic attractively interacting fermionic systems. Our results apply to superconducting properties of coupled ultrasmall metallic grains and to Cooper-pairing superfluidity in neutral systems with a finite number of fermions. We discuss how a definite relative phase between the two systems emerges and how it can be conveniently extracted from the many-body wave function, finding that a definite relative phase difference emerges even for very small numbers of pairs ($\sim$10). The Josephson dynamics and the current-phase characteristics are then investigated, showing that the critical current has a maximum at the BCS-BEC crossover. For the considered initial conditions a two-state model gives a good description of the dynamics and of the current-phase characteristics.

Figure 1

Chemical potential $\mu$ per level versus $g/N$, as computed from Eqs. (11) and (12) for $M=N/2-1$. Here and in the captions of the following figures the pairing coefficient $g$ and the energies are expressed in units of $d$.

Figure 2

Second-order processes associated to the Hamiltonian (23).

Figure 3

Particle-hole states.

Figure 4

Energy spectrum vs $\lambda$ for $N=8$ and $M_T=8$, with $g=0.1$ (left), $g=1.2$ (center), $g=6.2$ (right); for simplicity, thereafter also $\lambda$ is expressed in units of $d$.

Figure 5

(Left) Standard deviation ${\sigma }_{\chi }$ vs $g$ associated with the level structure, given by ($N_{\text{bin}}=100$, $\lambda =0.05$, $N=8$, $M_T=6$). (Right) Energy difference between the first excited state and the ground state as a function of $g$ for the same values of the parameters.

Figure 6

Phase differences $\delta {\varphi }_w\left(t\right)$ (solid blue line) and $\delta {\varphi }_z\left(t\right)$ (dotted red line) vs $t$ for the coupled systems with $N=8$ levels each, total number of pairs $M_T=8$, pairing strength $g=0.6$, tunneling parameter $\lambda =0.1$, $D=2$ (corresponding to $M_0=5$), and initial imbalance $z(t=0)=0.25$. Time here and in the following figures is in units of $\hslash /d$.

Figure 7

Phase difference means $\delta {\varphi }_{w,z}\left(t\right)$ (dashed blue lines) and standard deviations ${\sigma }_{w,z}$ (red solid lines), as determined from the correlation functions $w$ (left) and $z$ (right), for two coupled grains with the same parameters as in the previous figure.

Figure 8

Same quantities as in Fig. 7 with $g=0.2$ (other parameters unchanged).

Figure 9

$C_w^{\delta \varphi }$ (lower dashed line), $S_w^{\delta \varphi }$ (upper dashed line), $C_z^{\delta \varphi }$ (lower solid line), and $S_z^{\delta \varphi }$ (upper solid line) vs $g$. Parameters are as in Figs. 6, 7, 8: $N=8$, $M_T=8$, $\lambda =0.1$, $D=2$ (moreover, $t_{\text{max}}=1000$).

Figure 10

(Left) Phase difference, as a function of time, for $N=8$, $M_T=6$, $\delta M_0=2$, $g=5$, $\lambda =0.05$. (Right) Averaging over short times $t_{\text{av}}$ (here, $t_{\text{av}}=6$) to remove fluctuations.

Figure 11

Dynamical phase-portrait $z$–$\delta \varphi$ for different values of the parameter $\xi$ in the BCS regime, with $N=8$, $M_T=7$, $D=1$, $g=0.57$, $\lambda =0.05$. The chosen values of $\xi$ are $\xi =0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$, corresponding, respectively to $\delta M(t=0)=0.98,0.92,0.83,0.72,0.60,0.47,0.34,0.22,0.10$.

Figure 12

Phase portrait for different initial phases in the BEC regime [$N=8$, $M_T=7$, $D=1$, $\delta M(t=0)=1$, $g=9.7$, $\lambda =0.05$].

Figure 13

(Left) Fit for the $I$–$\delta \varphi$ characteristics with $g=2.3$, $\lambda =0.05$, $D=1$, $\delta M(t=0)=1$, $N=8$, $M_T=7$; blue circles are numerical points obtained from the quantum dynamics, and the red line is the fit according Eq. (41). (Right) Critical current fit; the blue circles are numerical results, while the red line is Eq. (42) with $c\simeq 0.27$.

Figure 14

Number-phase diagram for $N=8$, $D=2$, $\delta M(t=0)=2$, $M_T=6$, $\lambda =0.05$, and $g=0.4$.

Figure 15

(Left) Current-phase characteristics for $g=3.8$, $\lambda =0.05$, $D=2$, $\delta M(t=0)=2$, $N=8$, $M_T=7$; blue circles are numerical points obtained from the quantum dynamics, and the red line is the fit according Eq. (43). (Right) Critical current vs $g$; the blue circles are the numerical results, while the red line is Eq. (42).