Atomic Josephson Parametric Amplifier

Abstract

We study the dynamics of a driven atomic Josephson junction that we propose as a parametric amplifier. By periodically modulating the position of the barrier, we induce a small current across the junction, serving as our input signal. The pump field is implemented by modulating the barrier height at twice the Josephson plasma frequency. The resulting dynamics exhibit parametric amplification of the signal through nonlinear mixing between the signal and pump fields, which is encoded in a specific microscopic pattern of density waves and phase excitations that can be addressed within the experimental cold atoms capabilities. This work paves the way for tunable amplifiers in atomtronic circuits, with potential applications in several fields including precision measurements and quantum information processing. At the same time, our analysis provides the microscopic explanation of the general notion of parametric amplification occurring in nonlinear coherent devices.

Figures (8)

• Figure 1: Atomic Josephson parametric amplifier (JPA). (a) Simulation of a Josephson junction, which consists of two 2D clouds separated by a Gaussian tunnel barrier of height $V_0$ and width $w$ (indicated by two dotted vertical lines). We use $V_0/\mu=1.5$ and $w/\xi=1.1$, where $\mu$ is the mean-field energy and $\xi$ is the healing length. (b) Sketch of the JPA protocol. We periodically modulate the barrier position using $x(t) = x_0 + x_1 \sin(2\pi f_\mathrm{s} t)$, where $x_1$ is the amplitude and $f_\mathrm{s}$ is the frequency. For the pump we modulate the barrier height, i.e., $V(t)= V_0\bigl(1+ A_\mathrm{d} \cos(2\pi f_\mathrm{d} t) \bigr)$, where $A_\mathrm{d}$ is the amplitude and $f_\mathrm{d}$ is the frequency. (c) Sketch of the operation of a JPA. We modulate the barrier height near twice the plasma frequency $f_0$ of the JJ. The signal mode at $f_\mathrm{d}/2 + \delta f$ is amplified and an idler mode at frequency $f_\mathrm{d}/2 - \delta f$ is created, with $\delta f$ being the tuning parameter.
• Figure 2: JPA dynamics. Time evolution of the imbalance $z(t)$ demonstrating the amplification for the signal at frequency $f_\mathrm{s}=30\, \mathrm{Hz}$ and two different sets of signal and pump amplitudes: (a) $v_1/v_c=0.21$ and $A_\mathrm{d}=0.9$. (b) $v_1/v_c=0.42$ and $A_\mathrm{d}=0.5$. $v_c$ is the critical velocity above which the junction is resistive. The results at $A_\mathrm{d}=0$ represent the system without the pump. $T_\mathrm{s} =1/f_\mathrm{s}$ is the signal oscillation period. (c, d) Power spectrum of the time evolution in (a, b).
• Figure 3: Nonlinear mixing in the power spectrum $S_\omega=|z(\omega)|^2$ for signal frequencies between $f_\mathrm{s}=28$ and $52\, \mathrm{Hz}$ at $\tilde{I} _1 \equiv I_1/I_c=0.42$. For each column the spectrum is normalized by its maximum value $S_{\omega, \mathrm{m}}$. (a) Without pump the spectrum mainly displays the central peak at $f_\mathrm{s}=f$. (b, c) In the presence of pump there is another peak at $f_\mathrm{d}- f_\mathrm{s}$ (indicated by white dashed line), located at the frequency of the signal mirrored at the plasma frequency $f_0$. The case of $f_\mathrm{s}=f_0$, where signal and idler are at the same frequency, is a special case known as the degenerate case. (d) The result of the RSJ circuit model for $\tilde{I} _1=0.42$ and $A_\mathrm{d}=0.8$; see text.
• Figure 4: JPA characteristics. (a) Dimensionless gain $G_\mathrm{s}=S_\omega(A_\mathrm{d})/S_\omega(0) - 1$ as a function of $A_\mathrm{d}$ is shown for signal amplitudes between $\tilde{I} _1=0.12$ and $0.42$. The results of the RSJ model are indicated by the dashed lines. (b) Signal-to-noise ratio (SNR) corresponding to the results shown in (a). (c, d) Simulation and RSJ model results at temperature $T/T_0=0.3$, using the same relative signal frequency with respect to the Josephson plasma frequency as in panels (a, b); see text.
• Figure 5: Dephasing dynamics. (a) Average vortex number $N_v$ as a function of $A_\mathrm{d}$ for the parameters used in Fig. \ref{['Fig:gain']}(a). (b, c) Time evolution of the junction phase $\phi(t)$ for two different samples at $\tilde{I} _1=0.42$ with (continuous line) and without pump (dashed line). (d) Imbalance $z(t)$, averaged over many samples, corresponding to the parameters in (b, c).