### 3.1 Classical nonlinear response

Here, we employ the input-output formalism [46, 47] to calculate the nonlinear resonator response discussed qualitatively in the previous section. The derivation presented here is inspired by Ref. [1]. A schematic of the input-output model is shown in Figure 2. The nonlinear resonator is coupled with rate *κ* to a transmission line, through which the pump and signal fields propagate. Based on this model and the Hamiltonian in Eq. (1) we obtain the following equation of motion for the intra-resonator field

\dot{A}=-i{\tilde{\omega}}_{0}A-iK{A}^{\u2020}AA-\frac{\kappa +\gamma}{2}A+\sqrt{\kappa}{A}_{\mathrm{in}}(t)+\sqrt{\gamma}{b}_{\mathrm{in}}(t).

(2)

In addition to the coupling to transmission line modes {A}_{\mathrm{in}}(t) with rate *κ* we account for potential radiation loss mechanisms by introducing the coupling to modes {b}_{\mathrm{in}}(t) with loss rate *γ*, compare Figure 2(a). A boundary condition equivalent to

{A}_{\mathrm{out}}(t)=\sqrt{\kappa}A(t)-{A}_{\mathrm{in}}(t),

(3)

also holds for the loss modes. When operating the device as a parametric amplifier, the input field {A}_{\mathrm{in}} is typically a sum of a strong coherent pump field and an additional weak signal field. Since this signal carries at least the vacuum noise, it is treated as a quantum field. In this formalism this particular situation is accounted for by decomposing each field mode into a sum of a classical part and a quantum part

\begin{array}{c}{A}_{\mathrm{in}}(t)=({a}_{\mathrm{in}}(t)+{\alpha}_{\mathrm{in}}){e}^{-i{\omega}_{p}t},\hfill \\ {A}_{\mathrm{out}}(t)=({a}_{\mathrm{out}}(t)+{\alpha}_{\mathrm{out}}){e}^{-i{\omega}_{p}t},\hfill \\ A(t)=(a(t)+\alpha ){e}^{-i{\omega}_{p}t},\hfill \end{array}

(4)

where *α*, {\alpha}_{\mathrm{in}}, {\alpha}_{\mathrm{out}} represent the classical parts of the field which are associated with the pump, while *a*, {a}_{\mathrm{in}}, {a}_{\mathrm{out}} account for the quantum signal fields. Since all *α*’s are complex numbers the modes *a* satisfy the same bosonic commutation relations as modes *A* do. By multiplying the field modes defined in Eq. (4) with the additional exponential factor {e}^{-i{\omega}_{p}t}, one works in a frame rotating at the pump frequency {\omega}_{p}. The strategy is to first solve the classical response for the pump field *α* exactly and then linearize the equation of motion for the weak quantum field *a* in the presence of the pump. Finally, we derive a scattering relation between input modes {a}_{\mathrm{in}} and reflected modes {a}_{\mathrm{out}}.

The steady state solution for the coherent pump field is determined by

(i({\tilde{\omega}}_{0}-{\omega}_{p})+\frac{\kappa +\gamma}{2})\alpha +iK{\alpha}^{2}{\alpha}^{\ast}=\sqrt{\kappa}{\alpha}_{\mathrm{in}},

(5)

which follows immediately by substituting Eq. (4) into Eq. (2) and collecting only the c-number terms. By multiplying both sides with their complex conjugate we get to the equation

\frac{\kappa}{{(\kappa +\gamma )}^{2}}{|{\alpha}_{\mathrm{in}}|}^{2}=({\left(\frac{{\omega}_{p}-{\tilde{\omega}}_{0}}{\kappa +\gamma}\right)}^{2}+\frac{1}{4}){|\alpha |}^{2}-\frac{2({\omega}_{p}-{\tilde{\omega}}_{0})K}{{(\kappa +\gamma )}^{2}}{|\alpha |}^{4}+{\left(\frac{K}{\kappa +\gamma}\right)}^{2}{|\alpha |}^{6},

(6)

which determines the average number of pump photons {|\alpha |}^{2} in the resonator. Eq. (6) reduces to

1=({\delta}^{2}+\frac{1}{4})n-2\delta \xi {n}^{2}+{\xi}^{2}{n}^{3},

(7)

by defining the scale invariant quantities

\delta \equiv \frac{{\omega}_{p}-{\tilde{\omega}}_{0}}{\kappa +\gamma},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\tilde{\alpha}}_{\mathrm{in}}\equiv \frac{\sqrt{\kappa}{\alpha}_{\mathrm{in}}}{\kappa +\gamma},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\xi \equiv \frac{{|{\tilde{\alpha}}_{\mathrm{in}}|}^{2}K}{\kappa +\gamma},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n\equiv \frac{|{\alpha}^{2}|}{{|{\tilde{\alpha}}_{\mathrm{in}}|}^{2}}.

(8)

*δ* is the detuning between pump and resonator frequency in units of the total resonator linewidth, {\tilde{\alpha}}_{\mathrm{in}} is the dimensionless drive amplitude, and *ξ* is the product of drive power and nonlinearity, also expressed in dimensionless units. Finally, *n* is the mean number of pump photons in the resonator relative to the incident pump power. As an important consequence, we notice from Eq. (8) that only the product of drive power and nonlinearity determines the dynamics but not each quantity itself. Therefore, a small nonlinearity can at least in principle be compensated by increasing the drive power. Properties such as the gain-bandwidth product are therefore independent of the strength of the nonlinearity as long as the pump power is much larger than the power of amplified fluctuations. Furthermore, the solutions of Eq. (7) for negative *ξ* values are identical to those for positive *ξ* up to a sign change in *δ*. Since *ξ* is negative for the Josephson parametric amplifier, we focus on this particular case.

Equation (7) is a cubic equation in *n* and can therefore be solved analytically. We do not present the lengthy solutions here explicitly, but assume in the following that we have an explicit analytical expression for *n* in terms of *δ* and *ξ*. In Figure 2(b) we plot *n* for various parameters *ξ* as a function of *δ*. At the critical value {\xi}_{\mathrm{crit}}=-1/\sqrt{27} the derivative \partial n/\partial \delta diverges and thus the response of the parametric amplifier becomes extremely sensitive to small changes. For even stronger effective drive powers \xi /{\xi}_{\mathrm{crit}}>1 the cubic Eq. (7) has three real solutions. The solutions for the high and low photon numbers are stable, while the intermediate one is unstable. The system bifurcates in this regime as mentioned earlier. The critical detuning below which the system becomes bistable is {\delta}_{\mathrm{crit}}=-\sqrt{3}/2. The critical point ({\xi}_{\mathrm{crit}},{\delta}_{\mathrm{crit}}) is the one at which both \partial \delta /\partial n and {\partial}^{2}\delta /{\partial}^{2}n vanish. In scale invariant units the maximal value of *n* is 4, which is reached at the detuning \delta =4\xi.

Experimentally, the system parameters are characterized by measuring the complex reflection coefficient \mathrm{\Gamma}\equiv {\alpha}_{\mathrm{out}}/{\alpha}_{\mathrm{in}}. Based on the input-output relation {\alpha}_{\mathrm{out}}=\sqrt{\kappa}\alpha -{\alpha}_{\mathrm{in}} and Eq. (5) we evaluate this reflection coefficient as

\mathrm{\Gamma}=\frac{\kappa}{\kappa +\gamma}\frac{1}{\frac{1}{2}-i\delta +i\xi n}-1.

(9)

In Figure 2(c) we plot the absolute value of the reflection coefficient at \xi ={\xi}_{\mathrm{crit}} for various loss rates *γ*. For vanishing losses \gamma =0 all the incident drive power is reflected from the device and |\mathrm{\Gamma}|=1. Note that also in this case the resonance is clearly visible in the phase of the reflected signal (not shown here). When the loss rate *γ* becomes similar to the external coupling rate *κ* part of the radiation is dissipated into the loss modes. In the case of critical coupling \gamma =\kappa all the coherent power is transmitted into the loss modes at resonance. This is equivalent to the case of a symmetrically coupled \lambda /2 resonator, for which the transmission coefficient is one at resonance [48].

### 3.2 Linearized response for weak (quantum) signal fields

Under the assumption that the photon flux associated with the signal \u3008{a}_{\mathrm{in}}^{\u2020}{a}_{\mathrm{in}}\u3009 is much smaller than the photon flux of the pump field {|{\alpha}_{\mathrm{in}}|}^{2}, we can drop terms such as K{a}^{\u2020}a\alpha, because they are small compared to the leading terms K{a}^{\u2020}{\alpha}^{2} and Ka{|\alpha |}^{2}. By neglecting these terms we obtain a linearized equation of motion for *a* in the presence of the pump field. In order to preserve the validity of this approximation even for larger input signals, the amplitude *α* of the pump field needs to be increased. Experimentally, this can be achieved by reducing the strength of the nonlinearity *K* as discussed in Section 3.3 in more detail. Substituting Eq. (4) into Eq. (2) and keeping only terms which are linear in *a* one finds

\dot{a}(t)=i({\omega}_{p}-{\tilde{\omega}}_{0}-2K{|\alpha |}^{2}+i\frac{\kappa +\gamma}{2})a(t)-iK{\alpha}^{2}{a}^{\u2020}(t)+\sqrt{\kappa}{a}_{\mathrm{in}}(t)+\sqrt{\gamma}{b}_{\mathrm{in}}(t).

(10)

Since Eq. (10) is linear, we can solve it by decomposing all modes into their Fourier components

a(t)\equiv \frac{\kappa +\gamma}{\sqrt{2\pi}}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\mathrm{d}\mathrm{\Delta}{e}^{-i\mathrm{\Delta}(\kappa +\gamma )t}{a}_{\mathrm{\Delta}}

(11)

and equivalently for {a}_{\mathrm{in},\mathrm{\Delta}} and {b}_{\mathrm{in},\mathrm{\Delta}}. Note that the detuning Δ between signal frequencies and the pump frequency, is expressed here in units of the linewidth \kappa +\gamma. Substituting the Fourier decompositions into Eq. (10) and comparing the coefficients of different harmonics, results in

0=(i(\delta -2\xi n+\mathrm{\Delta})-\frac{1}{2}){a}_{\mathrm{\Delta}}-i\xi n{e}^{2i\varphi}{a}_{-\mathrm{\Delta}}^{\u2020}+{\tilde{c}}_{\mathrm{in},\mathrm{\Delta}},

(12)

where {\tilde{c}}_{\mathrm{in},\mathrm{\Delta}}\equiv (\sqrt{\kappa}{a}_{\mathrm{in},\mathrm{\Delta}}+\sqrt{\gamma}{b}_{\mathrm{in},\mathrm{\Delta}})/(\kappa +\gamma ) is the sum of all field modes incident on the resonator. Furthermore, in Eq. (12) *ϕ* is the phase of the intra-resonator pump field, defined by \alpha =|\alpha |{e}^{i\varphi}. The fact that Eq. (12) couples modes {a}_{\mathrm{\Delta}} and {a}_{-\mathrm{\Delta}}^{\u2020} can be interpreted as a wave mixing process. In order to express {a}_{\mathrm{\Delta}} in terms of the input fields {c}_{\mathrm{in},\mathrm{\Delta}}, Eq. (12) is rewritten as a matrix equation

\left(\begin{array}{c}{\tilde{c}}_{\mathrm{in},\mathrm{\Delta}}\\ {\tilde{c}}_{\mathrm{in},-\mathrm{\Delta}}^{\u2020}\end{array}\right)=\left(\begin{array}{cc}i(-\delta +2\xi n-\mathrm{\Delta})+\frac{1}{2}& i\xi n{e}^{i2\varphi}\\ -i\xi n{e}^{-i2\varphi}& i(\delta -2\xi n-\mathrm{\Delta})+\frac{1}{2}\end{array}\right)\left(\begin{array}{c}{a}_{\mathrm{\Delta}}\\ {a}_{-\mathrm{\Delta}}^{\u2020}\end{array}\right).

(13)

By inverting the matrix on the right hand side, the quantum part of the intra-resonator field {a}_{\mathrm{\Delta}} is expressed in terms of the incoming field {\tilde{c}}_{\mathrm{in},\mathrm{\Delta}}

{a}_{\mathrm{\Delta}}=\frac{i(\delta -2\xi n-\mathrm{\Delta})+\frac{1}{2}}{(i\mathrm{\Delta}-{\lambda}_{-})(i\mathrm{\Delta}-{\lambda}_{+})}{\tilde{c}}_{\mathrm{in},\mathrm{\Delta}}+\frac{-i\xi n{e}^{2i\varphi}}{(i\mathrm{\Delta}-{\lambda}_{-})(i\mathrm{\Delta}-{\lambda}_{+})}{\tilde{c}}_{\mathrm{in},-\mathrm{\Delta}}^{\u2020}

(14)

with {\lambda}_{\pm}=\frac{1}{2}\pm \sqrt{{(\xi n)}^{2}-{(\delta -2\xi n)}^{2}}. Using Eq. (3), the final transformation between input and output modes is

{a}_{\mathrm{out},\mathrm{\Delta}}={g}_{S,\mathrm{\Delta}}{a}_{\mathrm{in},\mathrm{\Delta}}+{g}_{I,\mathrm{\Delta}}{a}_{\mathrm{in},-\mathrm{\Delta}}^{\u2020}+\sqrt{\frac{\gamma}{\kappa}}({g}_{S,\mathrm{\Delta}}+1){b}_{\mathrm{in},\mathrm{\Delta}}+\sqrt{\frac{\gamma}{\kappa}}{g}_{I,\mathrm{\Delta}}{b}_{\mathrm{in},-\mathrm{\Delta}}^{\u2020}

(15a)

\stackrel{\gamma /\kappa \to 0}{=}{g}_{S,\mathrm{\Delta}}{a}_{\mathrm{in},\mathrm{\Delta}}+{g}_{I,\mathrm{\Delta}}{a}_{\mathrm{in},-\mathrm{\Delta}}^{\u2020},

(15b)

with

{g}_{S,\mathrm{\Delta}}=-1+\frac{\kappa}{\kappa +\gamma}\frac{i(\delta -2\xi n-\mathrm{\Delta})+\frac{1}{2}}{(i\mathrm{\Delta}-{\lambda}_{-})(i\mathrm{\Delta}-{\lambda}_{+})}

(16)

and

{g}_{I,\mathrm{\Delta}}=\frac{\kappa}{\kappa +\gamma}\frac{-i\xi n{e}^{2i\varphi}}{(i\mathrm{\Delta}-{\lambda}_{-})(i\mathrm{\Delta}-{\lambda}_{+})}.

(17)

Eq. (15b) is the central result of this calculation. The output field at detuning Δ from the pump frequency is a sum of the input fields at frequencies Δ and −Δ multiplied with the signal gain factor {g}_{S,\mathrm{\Delta}} and the idler gain factor {g}_{I,\mathrm{\Delta}}, respectively. The additional noise contributions introduced via the loss modes {b}_{\mathrm{in},\mathrm{\Delta}} vanish in the limit \gamma /\kappa \to 0. In the ideal case \gamma =0, the coefficients {g}_{S,\mathrm{\Delta}} and {g}_{I,\mathrm{\Delta}} satisfy the relation

{G}_{\mathrm{\Delta}}\equiv {|{g}_{S,\mathrm{\Delta}}|}^{2}={|{g}_{I,\mathrm{\Delta}}|}^{2}+1

(18)

and Eq. (15b) is identical to a two-mode squeezing transformation [19, 49] with gain {G}_{\mathrm{\Delta}}. The two-mode squeezing transformation describes a linear amplifier in its minimal form (compare Ref. [50]), of which we discuss characteristic properties in the following section.

### 3.3 Gain, bandwidth, noise and dynamic range

For simplicity we consider the case of no losses \gamma =0, for which the parametric amplifier response is described by Eq. (15b). An incoming signal at detuning Δ is thus amplified by the power gain {G}_{\mathrm{\Delta}}={|{g}_{S,\mathrm{\Delta}}|}^{2} and mixed with the frequency components at the opposite detuning from the pump. Characteristic properties of the parametric amplifier, such as the maximal gain and the bandwidth, are thus encoded in the quantity {g}_{S,\mathrm{\Delta}} as a function of pump-resonator detuning *δ*, effective drive strength *ξ* and detuning between signal and pump Δ.

In Figure 3(a) we plot the gain {G}_{0} for zero signal detuning \mathrm{\Delta}=0 as a function of *δ* and *ξ*. We find that the maximal gain increases with increasing drive strength *ξ* while the optimal value for *δ* at which this gain is reached, shifts approximately linearly with increasing *ξ*. The optimal values for *δ* are indicated as a dashed white line in Figure 3(a). Mathematically, the gain diverges when *ξ* approaches the critical value {\xi}_{\mathrm{crit}}. In practice, the gain is limited to finite values due to the breakdown of the stiff pump approximation (see discussion below).

By changing the pump parameters *ξ* and *δ* we can adjust the gain {G}_{0} to a desirable value, which is typically about 20 dB. Note that the gain can take values smaller than one, in the presence of finite internal losses \gamma >0. Once the pump parameters are fixed we characterize the bandwidth of the amplifier by analyzing the gain as a function of the signal detuning Δ. In Figure 3(b) we plot the gain as a function of Δ for the indicated values of \xi /{\xi}_{\mathrm{crit}} and the corresponding optimal pump detunings *δ* (compare dashed white line in (a)). The gain curves are well approximated by Lorentzian lines as indicated by the dashed black lines in Figure 3(b). When the gain is increased, the band of amplification becomes narrower. This is quantitatively expressed by the gain-bandwidth relation \sqrt{{G}_{0}}B\approx 1, where *B* is the detuning Δ for which the gain reaches half of its maximal value. This gain-bandwidth relation follows from the Lorentzian approximation of the gain curves shown as black dashed lines in Figure 3(b)) and holds for gain values above a few dB. Remember that Δ is defined in units of the resonator linewidth \kappa +\gamma, which means that the amplifier bandwidth equals approximately the resonator linewidth divided by the square root of the gain.

When operating the JPA, we also have to understand its behavior in terms of added noise. In the ideal case with zero loss rate (\gamma =0), the input-output relation of the parametric amplifier in Eq. (15b) has the minimal form of a scattering mode amplifier [19]. The amplification process reaches the vacuum limit as long as the input modes are cooled into the vacuum. In practice, however, the device may have finite loss *γ* which increases the effectively added noise by a factor of (\kappa +\gamma )/\kappa. This is due to the additional amplified noise, which originates from the modes {b}_{\mathrm{in},\mathrm{\Delta}} and contributes to the output field {a}_{\mathrm{out},\mathrm{\Delta}} (compare Eq. (15a)). Another potential source of noise is related to the stability of the resonance frequency of the parametric amplifier. Magnetic flux noise in the SQUID loop may lead to a fluctuating resonance frequency and thus a fluctuating effective gain.

In the derivation made in the previous sections we have assumed that the solution of the classical drive field is unaffected by the presence of additional signal and quantum fluctuations at the input. This is known as the stiff pump approximation [8], which assumes that the pump power at the output is equal to the pump power at the input of the JPA. This is of course an approximation, since the pump field provides the energy which is necessary for amplifying the input signal. The stiff pump approximation is valid as long as the pump power is significantly larger than the total output power of all amplified (quantum) signal and vacuum fields. In order to quantitatively analyze the pump depletion due to the presence of amplified fields we add the terms 2iK\u3008{a}^{\u2020}a\u3009\alpha and iK\u3008{a}^{2}\u3009{\alpha}^{\ast} to the left hand side of Eq. (5) and solve Eq. (5) and Eq. (10) self-consistently. This mean-field approach is similar to the one used in Ref. [8]. For our calculation we model the incoming signal field, which is to be amplified, as white noise with average photon number {n}_{\mathrm{th}} per unit time and bandwidth. Based on this model we find that the gain decreases when the signal strength exceeds a certain value, see Figure 4(a). The number of input photons {n}_{\mathrm{th}} at which this happens becomes smaller with decreasing ratio \kappa /|K|. This is expected because the pump power close to the bifurcation point is proportional to \kappa /|K| and provides the energy required for amplification. For small values \kappa /|K| the gain is reduced even for {n}_{\mathrm{th}}=0 due to the amplification of vacuum fluctuations (see blue data points in Figure 4(a)). As a measure of the dynamic range we specify the 1 dB compression point of the JPA, *i.e.* the value {n}_{1\phantom{\rule{0.2em}{0ex}}\mathrm{dB}} of input photons {n}_{\mathrm{th}} at which the JPA gain {G}_{0} decreases by 1 dB compared to the stiff pump approximated gain value. As shown in Figure 4(b), the 1 dB compression point increases proportionally to \kappa /|K| in the limit of {n}_{1\mathrm{dB}}\gg 1. The presence of constant vacuum fluctuations leads to a gain compression by more than 1 dB even for {n}_{\mathrm{th}}=0 when \kappa /|K| becomes smaller. We can qualitatively explain this behavior by comparing the pump power {P}_{p}=\u0127{\omega}_{p}{|{\alpha}_{\mathrm{in}}|}^{2} with the power of amplified signal and vacuum fields

{P}_{\mathrm{out}}\stackrel{\gamma =0}{=}\u0127{\omega}_{p}\kappa (2{n}_{\mathrm{th}}+1)\int \frac{\mathrm{d}\mathrm{\Delta}}{2\pi}({G}_{\mathrm{\Delta}}-1).

(19)

Making use of the gain-bandwidth relation we find the following scaling of the ratio between the two powers

\frac{{P}_{\mathrm{out}}}{{P}_{p}}\propto (2{n}_{\mathrm{th}}+1)\frac{|K|}{\kappa}\sqrt{{G}_{0}}.

(20)

The results shown in Figure 4(b) together with Eq. (20) indicate that the validity of the stiff pump approximation is essentially determined by the ratio {P}_{\mathrm{out}}/{P}_{p}. The calculations furthermore show that the dynamic range can be increased by reducing the ratio |K|/\kappa of the JPA, which seems to be the case also for flux driven parametric amplifiers [40]. In Section 5 we discuss how to achieve small nonlinearities by making use of multiple SQUIDs connected in series.