Proposal for a 3-Wave Mixing Element with Quantum Paraelectric Materials

Abstract

At cryogenic temperatures and microwave frequencies, the perovskite crystals strontium titanate (STO) and potassium tantalate (KTO) have large, tunable permittivity arising from a quantum paraelectric phase. As such, these materials hold promise as a platform to realize compact, variable capacitance elements for use in quantum devices. From modulating this capacitance, we propose the development of a parametric mixing element: a quantum paraelectric nonlinear dielectric amplifier (PANDA). We calculate that a PANDA made from a nanofabricated parallel plate capacitor and realistic design constraints can demonstrate a three-wave mixing strength of order MHz, in comparison to an effective Kerr strength of sub-Hz. This suggests excellent performance as a three-wave mixing element, with high compression power in analogy to superconducting parametric amplifiers based on kinetic inductance. Beyond parametric amplifiers, we predict that compact, tunable capacitors based on STO, KTO, and related materials can enable a wide class of cryogenic quantum circuits including novel filters, switches, circulators, and qubits.

Figures (6)

• Figure 1: (a) The perovskite crystals strontium titanate (STO) and potassium tantalate (KTO) have a cage-like structure where a central Ti or Ta atom lies within a cell formed by O and Sr or K atoms, respectively. At cryogenic temperatures these materials have high permittivity and low loss. Permittivity is tunable via an applied electric field $\Vec{E}$, which displaces the central atom and polarizes the crystal. (b) We consider a parallel plate capacitor, with plate area $A$ and separation $d$, and with the dielectric made from a thin film of STO or KTO. A voltage $v$ across the plates induces an electric field $E=v/d$, which changes $\varepsilon(v)$ and $C(v)=\varepsilon_0 \varepsilon_r(v) A / d$. Capacitance is plotted choosing $A=(4 \,\mathrm{\mu m})^2$ and $d=200 \,\mathrm{nm}$, and the dielectric properties given in Appendix \ref{['sec:dielectric_properties']}.
• Figure 2: (a) A resonator is formed by combining an inductor ($L$) and capacitor ($C$). We model that a DC bias voltage $v_0$ can be applied across the capacitor plates, changing the capacitance as described in Fig. \ref{['fig:resonator']}. (b) This nonlinear resonant circuit is analogous to an oscillating mass on a spring whose spring constant stiffens when stretched. We are interested in its dynamics, Eq. \ref{['eq:Hamiltonian_nonzero_bias']}, given both a static restoring force set by $v_0$, and, a parametric driving force set by $v_\mathrm{ac}$. (c) The resonant frequency $\omega_0/2\pi$ choosing $L=0.5~\,\mathrm{nH}$, (d) the three-wave mixing (3WM) strength $\xi/2\pi$ choosing $v_\mathrm{ac} = 1\,\mathrm{mV}$ (AC bias charge $q_\mathrm{ac}=v_\mathrm{ac}C(v)$), (e) and the effective Kerr strength $K_\mathrm{eff}/2\pi$, all as functions of $v_0$.
• Figure 3: (a) Input-output model for a parametric amplifier made from an STO or KTO capacitor. Modulation of the capacitance around $\omega_p \approx 2\omega_0$ yields degenerate parametric amplification. (b,c) Reflection $R(\omega)$ of a STO and KTO device, respectively, plotted for different values of $\xi$ normalized to resonator loss $\kappa/2$. The model (Eq. \ref{['eq:gain_3WM']}) uses the parameters as given in Fig. \ref{['fig:nonlinearity']} and an external quality factor of $Q_\mathrm{ext}=100$.
• Figure 4: At cryogenic temperature the perovskites STO and KTO have multiple low-energy states associated with positive or negative displacement of the central atom geyer:2005fujishita:2016. This leads to a quantum paraelectric phase in which the quantum fluctuations (horizontal white arrows) are significant compared to the energy barrier between these two potential wells. Permittivity is proportional to the inverse curvature of the free energy compared to relative displacement. An electric field, $E$, tilts the potential, which constraints the magnitude of fluctuations and reduces permittivity. STO and KTO differ in the depth of the double well potential and its susceptibility to an applied field esswein:2022. (b) Relative permittivity $\varepsilon_r=\varepsilon/\varepsilon_0$ and (c) loss tangent $\mathrm{tan}(\delta)$, modeled vs. $E$. Calculations use Eq. \ref{['eq:epsilon_vs_bias']} which is based on the model and material parameters in Ref. vendik:1999.
• Figure 5: (a) The three-wave mixing (3WM) term $\xi/2\pi$ and (b) the effective Kerr term $K_\mathrm{eff}/2\pi$, both as functions of capacitor plate separation $d$, which is swept while keeping the ratio $A/d=(4~\,\mathrm{\mu m})^2/(200~\,\mathrm{nm})=8\times10^{-5}~\,\mathrm{m}$ fixed. The 3WM term is reported at its maximum value, occuring at $v_0\neq0$ and taking $v_\mathrm{ac} = 1 \,\mathrm{mV}$. The effective Kerr term is also reported at its maximum value, occuring $v_0=0$. Both nonlinearities decreases as the capacitor plates become further apart.