Quantum-Limited Amplification and Entanglement in Coupled Nonlinear Resonators

We demonstrate a coupled cavity realization of a Bose-Hubbard dimer to achieve quantum-limited amplification and to generate frequency entangled microwave fields with squeezing parameters well below $-12\mathrm{dB}$. In contrast to previous implementations of parametric amplifiers, our dimer can be operated both as a degenerate and as a nondegenerate amplifier. The large measured gain-bandwidth product of more than 250 MHz for the nondegenerate operation and the saturation at input photon numbers as high as 2000 per $\mu s$ are both expected to be improvable even further, while maintaining wide frequency tunability of about 2 GHz. Featuring flexible control over all relevant system parameters, the presented Bose-Hubbard dimer based on lumped element circuits has significant potential as an elementary cell in nonlinear cavity arrays for quantum simulations.

Figure 1

(a) Optical frequency representation of the Bose-Hubbard dimer, illustrated as two cavities each with on-site interaction strength $U$ and coupled with hopping rate $J$. The left cavity emits into a transmission line at rate $\kappa$. (b) Mode structure of the dimer and drive induced redshifts. See text for details. (c) Calculated phase diagram of the dimer driven with a coherent input field ${\alpha }_{\text{in}}$ at detuning $\delta$ from the bare resonance frequency ${\omega }_0$ for $J=0.7\kappa$ and $U<0$. The red line indicates drive configurations with a vanishing field amplitude in the left cavity. The dashed blue lines indicate redshifted frequencies, ${\tilde{\omega }}_{-}$ and ${\tilde{\omega }}_{+}$. (d) Driving the system at frequency ${\tilde{\omega }}_{+}$ (gray vertical arrow) results in degenerate parametric amplification, with signal gain $G_s$ (red) and idler gain $G_i$ (blue) both occupying the symmetric mode. (e) Signal and idler gain become nondegenerate when driving the system in between ${\tilde{\omega }}_{+}$ and ${\tilde{\omega }}_{-}$.

Figure 2

(a) False-color micrograph of the sample. The interdigitated finger structures form the capacitors of two coupled oscillators. An effective nonlinear inductance is realized as an array of SQUIDs in each resonator, also shown enlarged. (b) Simplified circuit diagram of the experimental setup. The circuit is driven with a pump field at frequency ${\omega }_p={\omega }_0+\delta$ through a $-20\mathrm{dB}$ directional coupler, of which the second port is used to interferometrically suppress the pump field reflected from the sample by more than $-60\mathrm{dB}$. Input and output signal fields are separated using a circulator. (c) Argument $\mathrm{Arg}[\mathrm{\Gamma }]$ of the measured (blue dots) and fitted (red line) reflection coefficient $\mathrm{\Gamma }$ vs probe frequency. (d) Measured $\mathrm{Arg}[\mathrm{\Gamma }]$ vs external magnetic flux which is controlled by a voltage applied to a coil biasing filter. The white dashed line indicates the data trace shown in (c).

Figure 3

(a) Measured signal (red) and idler gain (blue) vs frequency for two pump powers $P_p\approx \{-75.9,-75.6\}\mathrm{dBm}$ fitted to a Lorentzian (black lines) for degenerate operation. (b),(c) Measured idler $G_I$ and signal gain $G_S$ for a nondegenerate operation together with Lorentzian fits (black lines) for drive powers $P_p\approx \{-75.8,-75.4,-75.,-74.6\}\mathrm{dBm}$.

Figure 4

(a) Two-mode squeezing spectrum $S_{+-}^{\varphi }(\mathrm{\Delta })$ for local oscillator phases $\varphi =\pi \{8,48,69,79,89\}/180$ with a global fit to the theoretical model. (b) Value of $S_{+-}^{\varphi }(\mathrm{\Delta })$ at the sideband frequency $\mathrm{\Delta }/2\pi =157.5\mathrm{MHz}$ indicated by the vertical dashed line in (a) vs local oscillator phase $\varphi$. The inset shows the value of squeezing at $\varphi =\pi /2$ vs gain in comparison to the ideal theoretical value (red line). (c) Real and imaginary part of the measured cumulants $⟨⟨(a_{+}^{†}{)}^n(a_{+}{)}^m(a_{-}^{†}{)}^k(a_{-}{)}^l⟩⟩$ for indicated orders ($n$, $m$, $k$, $l$) up to $n+m+k+l\le 4$.