Unified linear response theory of quantum electronic circuits

TL;DR

The paper addresses the challenge of modeling the high-frequency electrical response of multi-level quantum systems in the presence of relaxation and dephasing. It introduces a Lindblad perturbation theory that yields a universal small-signal circuit model, mapping each pair of levels to an LC resonator and splitting the admittance into Hamiltonian, Sisyphus, and Hermes contributions: $Y(ω)=2 i ω (α e)^2 χ^*(ω)$. The Hermes term captures decoherence-induced effects and, in the fast-decoherence limit, recovers the semiclassical quantum-capacitance picture, while the Hamiltonian term encodes resonant Rabi-like dynamics. The approach is demonstrated on a double quantum dot charge qubit and a Majorana qubit, showing continuous transitions between adiabatic, resonant, coherent, and incoherent regimes and enabling frequency-tuned readout strategies. Overall, the framework enables realistic circuit-level modeling of quantum devices and supports design of hybrid quantum-classical circuits and enhanced measurement protocols.

Abstract

Modelling the electrical response of multi-level quantum systems at finite frequency has been typically performed in the context of two incomplete paradigms: (i) input-output theory, which is valid at any frequency but neglects dynamic losses, and (ii) semiclassical theory, which captures well dynamic dissipation effects but is only accurate at low frequencies. Here, we develop a unifying theory, valid for arbitrary frequencies, that captures both the quantum behaviour and the non-unitary effects introduced by relaxation and dephasing. The theory allows a multi-level system to be described by a universal small-signal equivalent circuit model, a resonant RLC circuit, whose topology only depends on the number of energy levels. We apply our model to a double quantum-dot charge qubit and a Majorana qubit, showing the capability to continuously describe the systems from adiabatic to resonant and from coherent to incoherent, suggesting new and realistic experiments for improved quantum state readout. Our model will facilitate the design of hybrid quantum-classical circuits and the simulation of qubit control and quantum state readout.

Figures (4)

• Figure 1: Universal small-signal model of a quantum device embedded in a classical circuit. Circuit elements are color-coded according to the description they originate from: input-output (blue) or semiclassical (red). Moreover, in this work we introduce a third branch (black), arising from a consistent quantum description of (detuning-dependent) dynamical decoherence. Notably, the input-output and semiclassical models overlap only in the description of the quantum capacitance (upper left).
• Figure 2: Coherent charge qubit. Capacitance (a-c), conductance(d-f) and absolute value of the admittance (g-i) of the DQD charge qubit for increasing frequency at $k_{\rm B} T = 2.5 \Delta$ and $\Gamma_0 = \Delta_0/10$. The line cuts are taken in the adiabatic ($\omega_b = 0.2 \Delta$) and resonant ($\omega_t = 4 \Delta$) regimes. Panels b-c and e-f show the contribution to real and imaginary parts of the admittance from the Hamiltonian (cyan), Sisyphus (orange) and Hermes (pink) components, while panels h-i show a comparison between our model and the semiclassical model Esterli_Otxoa_Gonzalez-Zalba_2019 and input-output theory kohler_dispersive_2018. We additionally carry out a brute-force integration of the LME (blue dots), showing perfect agreement with our analytical model. Color bars are linear for the $\pm 5$% of the values on either side of zero and logarithmic outside.
• Figure 3: Effect of dephasing. Capacitance (a-c), conductance(d-f) and absolute value of the admittance (g-i) of the DQD charge qubit in the resonant regime ($\omega = 4 \Delta$) for increasing dephasing rate $\Gamma_\phi$ at $k_{\rm B} T = 2.5 \Delta$, and $\Gamma_0 = \Delta_0/10$. The line cuts are taken in the low-dephasing ($\Gamma_\phi = 0.02 \Delta$) and high-dephasing ($\Gamma_\phi = 7.5 \Delta$) regimes. Panels b-c and e-f show the contribution to real and imaginary part of the admittance from the Hamiltoinan (cyan), Sisyphus (orange) and Hermes (pink) components, while panels h-i show a comparison between our model and the semiclassical model Esterli_Otxoa_Gonzalez-Zalba_2019 and input-output theory kohler_dispersive_2018. We additionally carry out a brute-force integration of the LME (blue dots), showing perfect agreement with our analytical model. Scale bars are linear within $\pm 5$% and logarithmic outside.
• Figure 4: Majorana qubit. (a-b) Absolute value of the admittance of the auxiliary QD in case of the even (a) and odd (b) joint Majorana parity. (c) Absolute value of the admittance difference between the two parity states, which directly translates to the visibility of the readout of the two states. (d-e) $|Y_{\textnormal{even}} - Y_{\textnormal{odd}}|$ in the resonant (d) and adiabatic (e) regime, showing the benefit of performing readout at $\Delta_{\textnormal{odd}} < \omega < \Delta_{\textnormal{even}}$ to suppress the admittance of the odd state at $\varepsilon = 0$, which instead has a strong response at $\varepsilon_{\textnormal{odd}} = \sqrt{\omega - \Delta_{\textnormal{odd}}}$ where it is resonant with the excitation. Color bars are linear below $10^{-2}$ and logarithmic above.