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physica status solidi (b)

On Parametric Amplification in Josephson Transmission Line

Eugene Kogan

Published Mar 4, 2023• 2 citations
Publisher

Abstract

The series‐connected lossy Josephson transmission line (JTL), constructed basically from Josephson junctions, capacitors, and resistors, is considered. The continuum approximation equation describing the JTL is derived. From the latter, equations describing pump, signal, and idler interaction are derived, and the thresholds for the parametric amplification are calculated.

DOI
10.1002/pssb.202300005

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Parametric Interaction in Josephson Junction Circuits and Transmission Lines

Hamid Reza Mohebbi

This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). \nThe unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. \nMoving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. \nA novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. \nA rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. \nKnowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.

•May 20, 2011•3 citations

Comment on "Parametric amplification in Josephson junction embedded transmission lines"

Saptarshi Chaudhuri, Jiansong Gao

Recently Yaakobi and co-workers [Phys. Rev. B 87, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the wrong expression of the gain factor and invalidates their conclusions on the gain and bandwidth. In this comment, we present the correct expression for the parametric gain. We show that for a transmission line with weak dispersion or positive dispersion ($Δk>0 $), as is the case discussed by Yaakobi et al, while quadratic (power) gain can occur around the pump frequency, exponential gain is impossible. Furthermore, for a transmission line with proper intrinsic or engineered dispersion, exponential gain occurs at frequencies where the phase matching condition is met, while around the pump frequency the gain is still quadratic.

arXiv:1308.2951•Aug 13, 2013•1 citation

Parametric amplification in Josephson junction embedded transmission lines

Oded Yaakobi, L. Frièdland, Chris Macklin +1 more

An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.

Physical Review B•Apr 1, 2013•118 citations

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An analysis is presented describing four-photon parametric amplification in an unbiased Josephson junction. Central to the theory is the model of the Josephson effect as a nonlinear inductance. Linear, small signal analysis is applied to the « two fluid » model of the Josephson junction. The gain, gain-bandwidth product, high frequency limit and effective noise temperature is calculated for a cavity reflection amplifier. The analysis is extended to multiple (series-connected) junctions and sub-harmonic pumping.

Revue de Physique Appliquée•Jan 1, 1974•13 citations

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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 ${\tilde f_n}$ 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.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 21, 2021•0 citations

Mesoscopic Hamiltonian for Josephson traveling-wave parametric amplifiers

Thomas H. A. van der Reep

We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [O'Brien et al., Phys. Rev. Lett. 113, 157001 (2014)] is briefly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantization techniques in terms of discrete-mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.

arXiv:1812.05907•Physical Review A•Jun 28, 2019•26 citations
The series‐connected lossy Josephson transmission line (JTL), constructed basically from Josephson junctions, capacitors, and resistors, is considered. The continuum approximation equation describing the JTL is derived. From the latter, equations describing pump, signal, and idler interaction are derived, and the thresholds for the parametric amplification are calculated.
This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). \nThe unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. \nMoving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. \nA novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. \nA rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. \nKnowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.
Recently Yaakobi and co-workers [Phys. Rev. B 87, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the wrong expression of the gain factor and invalidates their conclusions on the gain and bandwidth. In this comment, we present the correct expression for the parametric gain. We show that for a transmission line with weak dispersion or positive dispersion ($Δk>0 $), as is the case discussed by Yaakobi et al, while quadratic (power) gain can occur around the pump frequency, exponential gain is impossible. Furthermore, for a transmission line with proper intrinsic or engineered dispersion, exponential gain occurs at frequencies where the phase matching condition is met, while around the pump frequency the gain is still quadratic.
An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.
An analysis is presented describing four-photon parametric amplification in an unbiased Josephson junction. Central to the theory is the model of the Josephson effect as a nonlinear inductance. Linear, small signal analysis is applied to the « two fluid » model of the Josephson junction. The gain, gain-bandwidth product, high frequency limit and effective noise temperature is calculated for a cavity reflection amplifier. The analysis is extended to multiple (series-connected) junctions and sub-harmonic pumping.
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.
We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [O'Brien et al., Phys. Rev. Lett. 113, 157001 (2014)] is briefly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantization techniques in terms of discrete-mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.