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International Topical Meeting on Microwave Photonics

Quantum Master Equation for a Lossy Josephson Traveling-Wave Parametric Amplifier

Yongjie Yuan, Michael Haider, Christian Jirauschek

Published Oct 1, 2022• 3 citations
Publisher

Abstract

In this paper, we present a theoretical framework for investigating losses and thermal fluctuations in a Josephson traveling-wave parametric amplifier (JTWPA). Our model is based on a discrete-mode Hamiltonian which includes a four-wave-mixing process and system-reservoir interactions. From this Hamiltonian, we derive a quantum master equation that describes the lossy Josephson-junction-embedded transmission line. The resulting equation of motion for the reduced density operator is applied for evaluating the average photon number in the signal mode. We present an analytic solution for the case of a dispersionless transmission line, where the phase-modulation effects can be neglected, and evaluate the photon number spectrum of a JTWPA structure with parameters from the literature. The quantum master equation is especially advantageous for a direct treatment of the expectation values of the photon number, in comparison to other strategies.

DOI
10.1109/mwp54208.2022.9997682

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In this paper, we present a theoretical framework for investigating losses and thermal fluctuations in a Josephson traveling-wave parametric amplifier (JTWPA). Our model is based on a discrete-mode Hamiltonian which includes a four-wave-mixing process and system-reservoir interactions. From this Hamiltonian, we derive a quantum master equation that describes the lossy Josephson-junction-embedded transmission line. The resulting equation of motion for the reduced density operator is applied for evaluating the average photon number in the signal mode. We present an analytic solution for the case of a dispersionless transmission line, where the phase-modulation effects can be neglected, and evaluate the photon number spectrum of a JTWPA structure with parameters from the literature. The quantum master equation is especially advantageous for a direct treatment of the expectation values of the photon number, in comparison to other strategies.
In this paper, we present a density operator description for a dissipative-dispersive Josephson traveling-wave parametric amplifier. A quantum master equation has been derived for the description of dissipation and thermal fluctuations that arise from lossy dielectric substrates. The formalism is especially effective in the dispersive case. Based on the equation of motion of the system density operator, we investigate the amplification performance by predicting the signal gain and the total photon number, considering amplifier phase-mismatch present in real devices. In addition to the gain investigation, we perform a two-mode squeezing analysis for such structures.
In this paper, we are constructing a circuit quantum electrodynamic model describing dissipative effects in a Josephson traveling-wave parametric amplifier (JTWPA). Losses and noise are treated using the quantum Langevin method. The total Hamiltonian of our model describes the coupling of the modes in a parametric four-wave-mixing scheme, as well as the interaction with the environment. Based on the Heisenberg equations of motion, we obtain the temporal evolution of the signal photon number and analyze the signal gain of the JTWPA.
Josephson traveling-wave parametric amplifiers (JTWPAs) achieve high parametric gain through four-wave-mixing with near quantum-limited noise performance and extremely low energy consumption. This makes them suitable for the detection of ultra-low-power microwave signals in the lowest temperature stage of a dilution refrigerator. Distributed circuit representations of JTWPAs have been studied in the framework of classical theory. A nonlinear wave equation describing the flux along a nonlinear transmission line was derived in [1]. Dispersion engineering was introduced in [2] to increase the amplifier’s gain and bandwidth by means of resonant phase-matching (RPM). In the classical treatment [1], dielectric loss is considered for the formulation of the wave equation, however, a closed-form solution was only given when dissipation was neglected. Our work focuses on the effects of noise and dissipation in the nonlinear transmission line. Our model differs from [1],[2], where we derive the telegrapher’s equations taking into account the loss and noise contributions to construct the current-flux-relation. In the super-conducting state, the dissipation due to the imperfect electric insulation in the substrate needs to be taken into account. It is represented by a parallel conductance and an additional noise source describing temporal fluctuations [3]. The flux along the transmission line is expanded into three discrete modes, representing pump, signal, and idler contributions. The equations of motion of the signal and idler modes result in\begin{equation*}{\partial _z}{\tilde A_{\text{s}}} = - \left( {{\Gamma _{\text{s}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{s}}} + {\text{i}}{\kappa _{\text{s}}}\tilde A_{\text{i}}^{\ast} + {\tilde f_{\text{s}}}\left( {i_{\text{s}}^{{\text{noise}}}} \right),\quad {\partial _z}{\tilde A_{\text{i}}} = - \left( {{\Gamma _{\text{i}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{i}}} + {\text{i}}{\kappa _{\text{i}}}\tilde A_{\text{s}}^{\ast} + {\tilde f_{\text{i}}}\left( {i_{\text{i}}^{{\text{noise}}}} \right),\tag{1}\end{equation*}where Ãnare the mode amplitudes in a co-rotating frame, ∆kTis the total phase mismatch, κndescribe the coupling due to four-wave-mixing, Γnare the damping factors and f~n{\tilde f_n}f~​n​ represent thermal fluctuations on the transmission line, with n ∈ {s,i}. Pump-depletion was neglected when deriving the equations above. We define the signal power gain Gsas the ratio of output to input power. Assuming no initial idler contribution at the input of the device, we obtain\begin{equation*}{G_{\text{S}}} = {\left| {\cosh (gz) + \frac{{ - {\Gamma _{\text{s}}} + {\Gamma _{\text{i}}} - 2{\text{i}}\Delta {k_{\text{T}}}}}{{4g}}\sinh (gz)} \right|^2}{{\text{e}}^{ - \frac{1}{2}\left( {{\Gamma _{\text{s}}} + {\Gamma _{\text{i}}}} \right)z}},\tag{2}\end{equation*}where g is the gain factor, which depends on the coupling strength, the pump amplitude and the total phase mismatch. The gain spectrum of the JTWPA setup from [4] was calculated according to our model with good agreement to the results in the literature. The gain spectrum of the JTWPA is plotted in Fig. 1 (a) without RPM and in Fig. 1 (b) with RPM, both for the case with and without dissipation.
In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum nature of the device. A quantum mechanical treatment of the JTWPA was only given in a few recent papers, where [1] derives a Hamiltonian for a chain of unit cells without resonant phase-matching (RPM), and [2] solves the dynamics of the system with and without RPM, also partly considering noise and squeezed state generation. However, our approach differs from [2] in that we give an explicit solution to the resulting nonlinear wave equation for narrow-band signals, starting from discrete chain Hamiltonians with and without RPM. We focus on describing the dynamics of the device in a noiseless quantum setting and present solutions for the governing nonlinear wave equation in Fig. 1. The starting point of this work is the amplifier design as given in [3]. The mathematical description relies on the Hamiltonian of a discrete chain of unit cells, which was derived in [1], and which we extended by a RPM circuit [2]. As the size Δl of a single chain element is relatively small compared to the wavelength, the chain of discrete Josephson elements can be considered in terms of a continuous transmission line.
We present a quantum-mechanical model for a four-wave-mixing Josephson traveling-wave parametric amplifier including substrate losses and associated thermal fluctuations. Under the assumption of a strong undepleted classical pump tone, we derive an analytic solution for the bosonic annihilation operator of the weak signal photon field using temporal equations of motion in a reference timeframe, including chromatic dispersion. From this result, we can predict the asymmetric gain spectrum of a Josephson traveling-wave parametric amplifier due to nonzero substrate losses. We also predict the equivalent added input noise including quantum fluctuations as well as thermal noise contributions. Our results are in excellent agreement with recently published experimental data.
In this contribution, we investigate different types of Josephson junction arrays that are embedded in coplanar microwave transmission lines, acting as traveling-wave parametric amplifiers. For the realization of traveling-wave parametric amplifiers, different Josephson junction array geometries have been proposed in the literature. Here, we investigate the amplifier’s Josephson nonlinearity along with the gain performance, following a general method to derive the nonlinear wave-mixing coefficients from the current-phase relation of different junction array topologies. These nonlinear coefficients are then incorporated into a corresponding Hamiltonian, which is used in order to solve the Heisenberg equations of motion of the photon number operator, determining the gain of the amplifier.