Higher-order nonlinear effects in a Josephson parametric amplifier

Nonlinearity of the current-phase relationship of a Josephson junction is the key resource for a Josephson parametric amplifier (JPA) as well as for a Josephson traveling-wave parametric amplifier, the only devices in which the quantum limit for added noise has so far been approached at microwave frequencies. A standard approach to describe JPA takes into account only the lowest order (cubic) nonlinearity resulting in a Duffing-like oscillator equation of motion or in a Kerr-type nonlinearity term in the Hamiltonian. In this paper we derive the quantum expression for the gain of JPA including all orders of the Josephson junction nonlinearity in the linear response regime. We then analyze gain saturation effect for stronger signals within a semiclassical approach. Our results reveal nonlinear effects of higher orders and their implications for operation of a JPA.

Figure 1

(a) Circuit diagram of the most basic JPA. (b) Description of JPA with the input-output formalism.

Figure 2

Frequency-amplitude $\left(\mathrm{\Omega },r\right)$ stability diagram based on numerical solution of Eq. (13) for a fixed value of the quality factor $Q=30$. Yellow region corresponds to a unique real stable solution while the green region shows parameter range for three real solutions (two stable solutions and one unstable solution). The most right point of the green domain $(0.96988,1.0401)$ is the bifurcation point for the complete sine nonlinearity.

Figure 3

Gain as a function of pump frequency for the fixed pump power $r=0.99$, zero signal detuning $\mathrm{\Delta }=0$, and different quality factors $Q$ of an oscillator. We also choose $f_0=7$ GHz and critical current $I_c=2\mu A$. The color represents a different number of terms $N$ taken into account in Eq. (8). $N=1$ corresponds to the known case of qubic nonlinearity, $N=2,3$ accounts for the next two higher quintic and septic nonlinearities, $N=\infty$ represents the solution for a complete $sin\varphi$ nonlinearity. $\mathrm{\Delta }f$ indicates the frequency difference between the maximum of gain and $\mathrm{\Delta }G$ shows the difference in maximal attainable gain for $N=1$ and one for the complete sine solution $N=\infty$.

Figure 4

Signal gain dependencies on pump frequency for the fixed $Q=10$, zero detuning $\mathrm{\Delta }=0$, natural frequency $f_0=7$ GHz, and critical current $I_c=2\mu A$. The color represents a different number of terms $N$ taken into account in Eq. (8). The pump amplitude $r$ was adjusted to achieve the same values of maximal gain. $\mathrm{\Delta }{f}_n$ with $n=1,2,3$ indicate corresponding frequency shifts of the maximal gain between solutions for $N=\infty -1,N=\infty -2$, and $N=\infty -3$, respectively.

Figure 5

Signal gain saturation due to sine and cubic nonlinearities. Here the natural frequency $f_0=7$ GHz, the critical current $I_c=2\mu A$, and the quality factor $Q=30$ of the JPA are the same as in the linear case depicted in the middle panel of Fig. 3.

Figure 6

Dependence of the dynamic range on the Kerr nonlinearity under constant the natural frequency $f_0=7$ GHz and the quality factor $Q=30$.