Hybrid non-degenerate parametric amplifier for a microwave cavity mode and an NV ensemble

TL;DR

The paper introduces a hybrid non-degenerate parametric amplifier that couples a microwave cavity mode to an NV spin ensemble, driven by a frequency modulation of the spins. By mapping the NV ensemble to a bosonic mode via Holstein-Primakoff and tracking Gaussian-state dynamics through covariance matrices, it analyzes both two-mode and single-mode squeezing under parametric drive, deriving key relations for steady-state gain and added noise. It reports that amplification rates can reach $\sim$dB/$\mu$s and predicts substantial two-mode squeezing, with a protocol to convert to single-mode squeezing, while detailing experimental requirements for room-temperature and cryogenic implementations. The results suggest practical routes to room-temperature, quantum-limited microwave amplification and spin-ensemble-based quantum-state processing, provided suitable dual-resonator cavities and high-$Q$ devices are realized. The work combines analytical modeling, perturbative insights, and numerical simulations to quantify performance and guide experimental design.

Abstract

We introduce an implementation of a non-degenerate parametric amplifier in which the signal and idler modes, respectively, a microwave mode and an ensemble of spins (e.g., nitrogen-vacancy centers in diamond), are operated in their linear regime. This paramp, which amplifies signals in both parts at room and cryogenic temperatures, can be used to generate both the two-mode and single-mode squeezing of either system. It requires merely modulating the frequency of the spin ensemble at the sum of the cavity and spin frequencies (providing the classical pump) with the two systems sufficiently detuned. This effect is remarkable given that modulating a spin ensemble by itself produces neither amplification nor squeezing, unlike modulating an oscillator, and that an off-resonant perturbative analysis would suggest that modulating the spin ensemble merely parametrically drives the cavity mode. With typical cavity parameters including a cavity quality factor~$Q=10^4$, and a 1 GHz modulation amplitude, the microwave signal can be amplified by approximately $18~\mbox{dB}$ in $1.7~\mbox{$μ$s}$, with a resonant bandwidth of about $0.5~\mbox{MHz}$. At $10~\mbox{mK}$ with the same modulation amplitude and a cavity and spin $Q=5\times 10^4$ it generates approximately $5~\mbox{dB}$ of squeezing. We also examine the experimental requirements for implementation.

Figures (5)

• Figure 1: A diagrammatic representation of the system. The diamond crystal containing a high density of NV centers is placed inside a microwave cavity. A pickup loop provides the input signal to the cavity to be amplified/squeezed and receives the cavity output. A microwave horn injects a tone into the cavity that modulates the frequencies of the NV spins. The NV spins are also pumped with a laser (not shown) that cools them close to their ground states. Here $\omega$ and $\Lambda$ are the frequency and amplitude of the microwave tone, $\gamma_{\mathrm{c}}$ is the cavity i/o coupling rate, $\gamma_{\mathrm{l}}$ is the cavity internal loss rate, and $\kappa$ is the effective damping rate of the spins. Not shown is the detuning $\Delta$ between the spins and the cavity and the collective spin/cavity coupling rate $g$.
• Figure 2: Evolution of the eigenvalues of the two-mode covariance matrix under modulation of the frequency of the ensemble of nitrogen-vacancy centers coupled to a cavity mode. The modulation frequency $\omega$ is the sum of the cavity mode frequency and the NV transition frequency. Here we assume ideal conditions in which neither the cavity nor the spins are damped. The covariance matrix has four eigenvalues corresponding to the variances of four quadratures. These eigenvalues are plotted respectively as: orange solid, red dashed, light blue solid, and dark blue dashed lines. Two quadratures are amplified at the same rate, while another two are correspondingly squeezed. The period of the modulation is $\tau = 2\pi/\omega$.
• Figure 3: (a) Amplification rate as a function of the NV modulation amplitude $\Lambda$ under condition \ref{['eq:mod-omega']} for the modulation frequency $\omega$. Blue: the temperature $T=10~\hbox{mK}$ and neither the cavity nor the spins are damped. Orange: $T=10~\hbox{mK}$ with the cavity and spin damped at the rate $200~\hbox{kHz}$. Green: $T=300~\hbox{K}$ with the same damping rates. (b) Steady-state squeezing of joint quadratures of the cavity mode and NV ensemble as a function of the modulation amplitude $\Lambda$ at a temperature of $T=10~\hbox{mK}$. Blue: no damping ($\kappa = \gamma = 0$), purple: $\kappa = \gamma = 5~\hbox{kHz}$, red: $\kappa = \gamma = 50~\hbox{kHz}$, orange: $\kappa = \gamma = 200~\hbox{kHz}$.
• Figure 4: a) Amplification generated by parametric driving of the NV ensemble after 1.7 $\mu$s as a function of temperature for four values of the driving amplitude, $\Lambda$. Above 10 K there is essentially no change in the amplification up to 300 K. b) Two-mode squeezing generated by parametric driving of the NV ensemble after 1.7 $\mu$s as a function of temperature for four values of the driving amplitude, $\Lambda$. No squeezing is generated above 1 K. As above, the parametric drive $\omega = \omega_{\hbox{\scriptsize c}} + \omega_{\hbox{\scriptsize s}}$, where $\omega_{\hbox{\scriptsize c}}$ is the cavity mode frequency and $\omega_{\hbox{\scriptsize s}}$ is the transition frequency of the NV spins.
• Figure 5: The evolution of the squeezing of four quadratures under a simple two-part control protocol that transfers two-mode (joint) squeezing generated by the modulation of the NVs into simultaneous independent single-mode squeezing of the cavity mode and the NV ensemble. The two quadratures that are initially squeezed are plotted in light blue (solid) and dark blue (dashed). The single-mode quadratures two which the squeezing is transferred are plotted in orange (solid) and red (dashed). The horizontal purple dashed line is the standard deviation of a quadrature for a coherent state (the value for no squeezing) and the vertical green dashed line gives the point were the protocol switches from evolution under detuning to evolution under the RWA interaction.