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arXiv:1308.2951

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

Saptarshi Chaudhuri, Jiansong Gao

Published Aug 13, 2013• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1308.2951

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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>0Δk>0 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.
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.
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.
We extend the theory for a Josephson-junction traveling-wave parametric amplifier (TWPA) operating in the three-wave-mixing regime and we propose a scheme for achieving high gain. The continuous three-mode model [P.K. Tien, J. Appl. Phys. 29, 1347 (1958)] is, on one hand, extended to describe a discrete chain of Josephson junctions at high frequencies close to the spectral cutoff where there is no up-conversion. On the other hand, we also develop a continuous multimode theory for the small frequency limit where the frequency dispersion is close to linear. We find that in both cases the gain is significantly reduced compared to the prediction by the continuous three-mode model as the result of increasingly strong dispersion at the high frequencies and generation of up-converted modes at the small frequencies. The developed theory is in quantitative agreement with simulations of the full solution of the dynamical equations. To recover the high gain, we propose to engineer a chain with dispersive features to form a two-band frequency spectrum and to place the pump frequency within the upper band close to the spectral cutoff. We prove that there exists a sweet spot, where the signal and the pump are phase matched, while the up-conversion is inhibited. This results in a high gain, which grows exponentially with the length of the TWPA.
We develop a concept of traveling-wave Josephson parametric amplifier based on a ladder-type superconducting transmission line containing serial array of one-junctions SQUIDs. The circuit operates in a three-wave mixing mode and exploits both quadratic and cubic (Kerr) nonlinearities in the current-phase relation of the SQUIDs controlled by external magnetic flux. These two types of nonlinearities allow efficient three-wave mixing and perfect phase matching, respectively. The transmission line exhibits a power gain with exponential dependence on its length (number of cells NNN) in the widest frequency range, 0≲fs,fi≲fp0 \lesssim f_s, f_i \lesssim f_p0≲fs​,fi​≲fp​, where fsf_sfs​, fif_ifi​ and fpf_pfp​ are the signal, idler and pump frequencies, respectively, obeying relation fs+fi=fpf_s + f_i = f_pfs​+fi​=fp​. Estimation of the amplifier characteristics with typical experimental parameters, number of cells NNN = 700, length of 28 mm and pump frequency of 12.5 GHz yields a flat (-3 dB) gain of about 20 dB in the bandwidth from 3.35 GHz to 9.15 GHz and a saturation power of -90 dBm.
The recently proposed Josephson traveling-wave parametric amplifier (JTWPA) based on a ladder transmission line consisting of radio-frequency superconducting quantum interference devices and exploiting three-wave mixing has great potential in achieving both a gain of 20 dB and a flat bandwidth of at least 4 GHz. To realize this concept in practical amplifiers, we model the advanced JTWPA circuit with periodic modulation of the circuit parameters (engineered dispersion loadings), which allow the basic mixing process, i.e., ωs=ωp−ωi, where ωs, ωp, and ωi are the signal, the pump, and the idler frequencies, respectively, and efficiently suppress propagation of unwanted higher tones, including ω2p=2ωp, ωp+s=ωp+ωs, ωp+i=ωp+ωi, etc. The engineered dispersion loadings allow achieving a sufficiently wide 3 dB-bandwidth from 3 to 9 GHz combined with a reasonably small ripple (±2 dB) in the gain-vs-frequency dependence.