Proposals for realizing a Josephson diode in Atomtronic circuits

TL;DR

The paper addresses realizing a Josephson diode in tunable atomtronic circuits by breaking inversion symmetry either through asymmetrically placed Josephson junctions in a ring Bose-Einstein condensate or via a biharmonic AC drive, enabling direction-dependent critical currents. It presents a 1D ring-BEC model coupled to an optical cavity, with a dispersive angular optical lattice $V_{\mathrm{opt}}=\hbar U_{0}\cos^{2}(\ell\phi)a^{\dagger}a$ and real-time, non-destructive cavity-optomechanics readout of the Josephson dynamics. Key results include diode efficiencies up to about 15% for position-tunable and up to 91% for drive-tunable configurations, observed via the dc–ac Josephson transition frequency $\omega_{J}=\Delta\mu/\hbar$ in the cavity-output spectrum. The work establishes a highly tunable platform for nonreciprocal Josephson transport in neutral-atom circuits and showcases cavity optomechanical readout as a powerful in situ probe of condensate dynamics.

Abstract

The Josephson diode, a non-reciprocal quantum element analogous to the familiar semiconductor p-n junction diode, has been realized in solid-state systems but remains unexplored in tunable atomtronic circuits. In this work, we propose and numerically demonstrate the realization of the Josephson diode effect in an atomtronic circuit consisting of a ring-shaped Bose-Einstein condensate and with optical barriers serving as Josephson junctions. Our implementation of this macroscopic non-reciprocal quantum phenomenon is based on realizing the required inversion symmetry breaking through asymmetric barrier placement and an asymmetric alternating current (AC) drive, enabling position- and drive-tunable diode effects with efficiencies up to 15% and 91%, respectively. While standard time-of-flight absorption imaging can readily observe these effects, we employ cavity optomechanics for in situ, real-time, and non-destructive measurements of the relevant condensate dynamics. Our results establish a highly tunable platform for nonreciprocal Josephson transport, opening avenues for diode-based neutral-atom technologies in future quantum circuits.

Figures (8)

• Figure 1: (a) Schematic diagram of the proposed experimental setup. A ring BEC is kept at the centre of an optical cavity that is driven by the superposition of two LG beams. The output optical field from the cavity is denoted by $a_{\mathrm{out}}$. (b) Schematic protocol to break the inversion symmetry of the condensate by placing the junction such that it divides the ring into two unequal arms. Here $\Delta\phi$ quantifies deviation from the diametrically opposite positions. (c) Schematic protocol to break the spatio-temporal symmetry by an asymmetric ac drive to the initially symmetric junctions.
• Figure 2: Position tunable diode: [(a)-(b)] Josephson tunneling current during the junction movement period with frequency $f_{\mathrm{bar}} = \pm 0.3$ Hz for (a) initially symmetric barriers ($\Delta \phi = 0$), (b) initially off-centered barriers ($\Delta \phi = 0.43$). [(c)-(d)] Power spectrum of phase quadrature of cavity output field corresponding to the case of (a) and (b), respectively. In the inset, the curves are deliberately displaced horizontally for clarity. [(e)-(f)] Magnitude of peak splitting in the cavity output spectrum versus barrier velocity for (e) $\Delta \phi = 0$ and (f) $\Delta \phi = 0.43$. The vertical dashed lines indicate the critical barrier velocity of the dc-to-ac Josephson transition. (g) Variation of diode efficiency $\eta$ with the amount of initial asymmetry $\Delta \phi$. (h) Variation of critical barrier velocity $f_{\mathrm{c}}$ and $\eta$ vs AQUID rotation rate $\Omega'$. The other parameters used are $N = 3700$, $R_0 = 4$$\mu$m, $V_0 = 17.5\,\mu_0 = 25$ nK , $\sigma = 0.7 \, \zeta = 1.8\,\mu$m, $\zeta = 1/\sqrt{\mu_0/\hbar \omega_\beta}$, $t_{\mathrm{bar}}=0.1$ s, $T = 10$ nK, $P_{\mathrm{in}} = 10 \, \mathrm{fW}$, $\Gamma = 0.0001$, $U_0 = 2\pi \times 212$ Hz, $\Tilde{\Delta} = \Delta_0 - U_0 N/2 = -2 \pi \times 173$ Hz, $\Delta_a = 2\pi \times 4.7$ GHz, $\omega_0 = 2\pi \times10^{15}$ Hz, $\omega_\rho = \omega_z = 2\pi \times 42$ Hz, $\gamma_{0} = 2\pi\times 2$ MHz, and $\Omega' = 0$.
• Figure 3: Applications of atomtronic Josephson diode: [(a)-(d)] Direction dependent modulation detector: Variation of the magnitude of peak splitting ($\Delta \omega$) versus barrier velocity ($f_{\mathrm{bar}}$) for $f_{\mathrm{m}} =40\,\mathrm{Hz},\, \phi_\mathrm{m} = 0.06$, (a) $\Omega' = 0$, $\Delta\phi = 0$, (b) $\Omega' = 0$, $\Delta\phi = 0.43$. [(c)-(d)] Half-wave rectifier: (c) Three cycles of square wave of bias current that corresponds to $f_{\mathrm{bar}} = \pm 0.26$ Hz (lies in between $f_{\mathrm{c}+}$ and $f_{\mathrm{c}-}$). (d) Measured junction voltage ($\Delta \omega$). In the horizontal axes of (c) and (d), $t_{\mathrm{sim}}$ is scaled down to $t_{\mathrm{bar}}/4$ for a clear demonstration of current cycles. Here $t_{\mathrm{bar}} = 0.1$ s, $f_{\mathrm{m}} =0.0\,\mathrm{Hz},\, \phi_\mathrm{m} = 0.0$, $\Omega' = 0$, and the other set of parameters are the same as used in Fig. \ref{['fig:fig2']}.
• Figure 4: Drive tunable diode: temporal evolution of barrier positions for $f_{\mathrm{m}} =10\,\mathrm{Hz},\, \phi_\mathrm{m} = 0.22$, (a) $\theta = \pi/2$, (b) $\theta = 0$, (c) $\theta = -\pi/2$. (d) Variation of the magnitude of peak splitting ($\Delta \omega$) versus barrier velocity ($f_{\mathrm{bar}}$). The other set of parameters used here is the same as used in Fig. \ref{['fig:fig2']} (a).
• Figure SM1: Flowchart of simulation details.