Quantum and classical nonlinear dynamics in a microwave cavity

Abstract

We consider a quarter wave coplanar microwave cavity terminated to ground via asuperconducting quantum interference device. By modulating the flux through the loop,the cavity frequency is modulated. The flux is varied at twice the cavity frequencyimplementing a parametric driving of the cavity field. The cavity field also exhibitsa large effective nonlinear susceptibility modelled as an effective Kerrnonlinearity, and is also driven by a detuned linear drive. We show that thesemi-classical model corresponding to this system exhibits a fixed point bifurcationat a particular threshold of parametric pumping power. We show the quantum signatureof this bifurcation in the dissipative quantum system. We further linearise about thebelow threshold classical steady state and consider it to act as a bifurcationamplifier, calculating gain and noise spectra for the corresponding small signalregime. Furthermore, we use a phase space technique to analytically solve for theexact quantum steady state. We use this solution to calculate the exact small signalgain of the amplifier.

1 Introduction

Superconducting circuit quantum electrodynamics (circuit QED) [1] is increasingly being used to study systems in the quantumregime. This experimental context sees a superconducting coplanar waveguide act as amicrowave cavity, in contrast to the optical frequency cavities of traditional cavityquantum electrodynamics (cavity QED). The microwave resonator is made from aluminium ona silicon substrate, and Josephson junctions are created by allowing the aluminium tooxidise before adding more aluminium. Such devices are placed in a dilutionrefrigerator, and experiments take place at cryogenic temperatures. Such lowtemperatures, close to the quantum ground state, allow quantum mechanical phenomena tobecome manifest. Recent engineering progress means that fabrication of these devices ispossible [2].

In recent experiments at Chalmers [3], a quarterwave coplanar microwave cavity is terminated to ground via one or more superconductingquantum interference devices (SQUIDs), see Figure 1. Bymodulating the flux through the loop, the cavity frequency can be modulated. If the fluxis varied at twice the cavity frequency this implements a parametric driving of thecavity field. The cavity field also exhibits a large effective nonlinear susceptibilitythat can be modelled as an intensity dependent phase shift [4].

Figure 1

Schematic of the system under consideration in this paper. A schematic ofthe system under consideration. A superconducting microwave cavity of frequency${\omega }_c$has an effective Kerr nonlinearity χ, due primarily to theSQUID-loop connecting it to ground. It is driven both by a linear drive ofamplitude ϵ at a detuning from the cavity of Δ, andalso by a parametric drive at twice the cavity frequency at an amplitudeκ.

This paper is structured as follows. In Section 2 we introduce the nonlinearmicrowave system considered in this paper. We establish a description in the form of aMarkov master equation, one term of which being an effective Hamiltonian we derive. Wealso give the input-output formulation of the microwave system. In Section 3 wepresent a detailed analysis of the fixed point structure of the nonlinear microwavesystem in a semi-classical description, including bifurcation of the fixed points. Weinclude dissipation of the microwave mode. In Section 4 we look at the steady stateof the quantum system. This is done in a phase space representation based on thepositive P-representation, both analytically, and numerically. We look for signatures ofthe semi-classical bifurcations. In Section 5 we analytically compute and plot thesmall signal gain. Then, in Section 6 we linearise the model and extract gain andnoise spectra up to the threshold defined by the semi-classical bifurcation. Finally inSection 7 we summarise our results.

A similar model to that considered here has been given by Wustmann andShumeiko [5]. Their discussion of thesemiclassical steady states and fixed point structure parallels the discussion here butgives a more detailed description of the semiclassical dynamics. They also discuss thequantum noise features of the model using a linearised quantum Langevin approach. Inaddition to a linearised analysis of the gain and signal-to-noise ratio, we give anexact steady state solution for the quantum master equation using the positiveP-representation. The steady state behaviour of the model we describe has beenexperimentally observed by Wilson et al. [6].

2 The dissipative Cassinian oscillator model

2.1 Master equation

We consider a superconducting microwave resonator connected through a superconductingquantum interference device (SQUID) loop to ground. The SQUID loop consists of twoseparated Josephson junctions; a magnetic flux can then be applied to loop to changethe effective resonant frequency of the cavity [3]. The SQUID also induces a significant quartic nonlinearity.Following Wallquist et al. [4] we describe theintegrated cavity + SQUID system in terms of an equivalent circuit composed of acapacitor and a nonlinear inductor. The Hamiltonian for one mode of the cavity fieldmay be written in terms of this effective nonlinear oscillator as [4]

$$\mathcal{H}=E_C{n}^2+E_L{\varphi }^2+\lambda {\varphi }^4,$$

(1)

where $E_C=\frac{{(2e)}^2}{2C}$represents the charging energy of the effective LC oscillator while n is thenumber of elementary charges on the capacitor, $E_L=\frac{{ħ}^2}{2{(2e)}^{2}L}$represents the inductive energy of the effective oscillator, while ϕrepresents the flux though the equivalent inductor and λ represents theinductive nonlinearity. This depends on the inductive energy scale and on the modefunction of the cavity field $\lambda =E_{L}B$,where B is a geometric factor. Further details are given in Wallquist et al.give [4].

The system may be quantised by introducing the bosonic raising and loweringoperators, defined in terms of the canonical variables of the cavity field as,

$$\begin{array}{r}\hat{\varphi }={\left(\frac{E_C}{4E_L}\right)}^{\frac{1}{4}}(\hat{a}+{\hat{a}}^{†}),\\ \hat{n}=-\mathrm{i}{\left(\frac{E_L}{4E_C}\right)}^{\frac{1}{4}}(\hat{a}-{\hat{a}}^{†}),\end{array}$$

(2)

where $\hat{\psi }$ and $\hat{n}$ are the average phase acrossthe junctions and charge on the junctions, respectively and C and Lare the effective lumped capacitance and inductance, respectively, of the equivalentcircuit for the cavity in terms of which the cavity resonant frequency is give by${\omega }_c=\frac{1}{\sqrt{LC}}$.The total Hamiltonian, including the coherent driving and the parametric driving arethen given by

$$\mathcal{H}=ħ{\omega }_c{\hat{a}}^{†}\hat{a}+ħ({ϵ}^{\ast }\hat{a}{\mathrm{e}}^{\mathrm{i}{\omega }_{D}t}+ϵ{\hat{a}}^{†}{\mathrm{e}}^{-\mathrm{i}{\omega }_{D}t})+ħ\chi {{\hat{a}}^{†}}^2{\hat{a}}^2+\frac{ħ}{2}({\kappa }^{\ast }{\hat{a}}^2{\mathrm{e}}^{\mathrm{i}2{\omega }_{D}t}+\kappa {{\hat{a}}^{†}}^2{\mathrm{e}}^{-\mathrm{i}2{\omega }_{D}t}),$$

(3)

where ${\omega }_c$ isthe cavity frequency, $ϵ=|ϵ|{\mathrm{e}}^{\mathrm{i}\upsilon }$represents the coherent driving strength, κ represents the parametricdriving strength, ${\omega }_D$is the coherent driving frequency and we have assumed that the parametric driving isat twice the coherent driving frequency and υ is the phase differencebetween the coherent driving and the parametric driving as we have taken the phase ofthe parametric driving term as zero. In Section 6 we consider coherent homodynedetection of the cavity output. This means there is another phase in this problem;the phase choice for the local oscillator which may not be in phase with either thecoherent or the parametric driving. The term proportional to χrepresents a nonlinear (quartic) phase shift that arises from the nonlinearinductance of the SQUID loop. Quartic non-linearities in oscillators have beendiscussed in [7, 8];parametric terms in the nano-electromechanical context have been discussed in[9–11].

For a realistic device we adopt a dissipative model. We model the microwave cavityresonator as being damped in a zero temperature heat bath. Such a model for the bathis certainly justified as the typical microwave cavity is at mK temperature and thusthe mean excitation photon number is very close to zero [12]. The amplitude decay rate for the microwave cavity isγ. We then describe the dissipative dynamics with the masterequation (with weak damping and the rotating wave approximation for thesystem-environment couplings). In an interaction picture at the coherent drivingfrequency this is

$$\frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}=-\frac{\mathrm{i}}{ħ}[\hat{\mathcal{H}},\hat{\rho }]+\gamma (2\hat{a}\hat{\rho }{\hat{a}}^{†}-{\hat{a}}^{†}\hat{a}\hat{\rho }-\hat{\rho }{\hat{a}}^{†}\hat{a}),$$

(4)

where $\hat{\rho }$ is the density matrix of themicrowave cavity, and $\hat{\mathcal{H}}$ is the Hamiltonian in aninteraction picture in a rotating frame with respect to the linear driving frequency.We have made the rotating wave approximation by ignoring terms with frequency$2{\omega }_D$or above. We thus have

$$\hat{\mathcal{H}}=ħ\Delta {\hat{a}}^{†}\hat{a}+ħ({ϵ}^{\ast }\hat{a}+ϵ{\hat{a}}^{†})+\frac{ħ}{2}({\kappa }^{\ast }{\hat{a}}^2+\kappa {{\hat{a}}^{†}}^2)+\frac{ħ\chi }{2}{{\hat{a}}^{†}}^2{\hat{a}}^2,$$

(5)

where $\Delta ={\omega }_c-{\omega }_D$.In the absence of damping, classical trajectories arising from the parametric andnonlinear portion of this Hamiltonian are the ovals of Cassini, and that system ishence sometimes described as the ‘Cassinian’ oscillator; the quantumversion of that part of the Hamiltonian system has been previously studied byWielinga et al. [13] and more recently byDykman and his collaborators [14].

The semi-classical dynamics and the exact quantum steady state can be found using thecomplex P-function of Drummond and Gardiner [15]. In this approach the density operator in terms of the offdiagonal projectors onto oscillator coherent states

$$\hat{\rho }(t)=\int \mathrm{d}\alpha \mathrm{d}\beta P(\alpha ,\beta ,t)\frac{|\alpha 〉〈{\beta }^{\ast }|}{〈{\beta }^{\ast }|\alpha 〉}.$$

(6)

This function determines the normally ordered moments by

$$〈{{\hat{a}}^{†}}^m{\hat{a}}^n〉=\int \mathrm{d}\alpha \mathrm{d}\beta P(\alpha ,\beta ,t){\alpha }^n{\beta }^m.$$

(7)

It may seem surprising at first sight to notice that the Positive P-function hassupport in a phase space with twice as many canonical variables as the correspondingclassical problem. There is a direct physical interpretation of the extra variablesbased on a measurement model in which there are twice as many readout channels forthe canonical phase space variable [16]. Thisis required if the distributions are to give normally ordered moments directly viaintegration. In [17] a direct implementationusing circuit QED of these additional channels is demonstrated and connection is madeto the stationary normal ordered moments.

The master equation can then be converted into a Fokker-Planck like equation for theP-function,

$$\begin{array}{rl}\frac{\partial P(\alpha ,\beta )}{\partial t}=& \{{\partial }_{\alpha }[(\gamma +\mathrm{i}\Delta )\alpha +\mathrm{i}ϵ+\mathrm{i}\chi \beta {\alpha }^2+\mathrm{i}\kappa \beta ]+{\partial }_{\beta }[(\gamma -\mathrm{i}\Delta )\beta -\mathrm{i}{ϵ}^{\ast }-\mathrm{i}\chi \alpha {\beta }^2-\mathrm{i}{\kappa }^{\ast }\alpha ]\\ +{\partial }_{\alpha \alpha }^2[-\mathrm{i}\frac{1}{2}(\kappa +\chi {\alpha }^2)]+{\partial }_{\beta \beta }^2\left[\mathrm{i}\frac{1}{2}({\kappa }^{\ast }+\chi {\beta }^2)\right]\},\end{array}$$

(8)

where ${\partial }_{\alpha }=\frac{\partial }{\partial \alpha }$and ${\partial }_{\alpha \alpha }^2=\frac{\partial 2^{}}{\partial \alpha \partial \alpha }$etc. The corresponding stochastic differential equations are

$$\begin{array}{r}\mathrm{d}\alpha =-(\gamma +\mathrm{i}\Delta )\alpha \mathrm{d}t-\mathrm{i}ϵ\mathrm{d}t-\mathrm{i}(\chi {\alpha }^2+\kappa )\beta \mathrm{d}t+{[-\mathrm{i}(\kappa +\chi {\alpha }^2)]}^{\frac{1}{2}}\mathrm{d}{z}_1,\\ \mathrm{d}\beta =-(\gamma -\mathrm{i}\Delta )\beta \mathrm{d}t+\mathrm{i}{ϵ}^{\ast }\mathrm{d}t+\mathrm{i}(\chi {\beta }^2+{\kappa }^{\ast })\alpha \mathrm{d}t+{\left[\mathrm{i}({\kappa }^{\ast }+\chi {\beta }^2)\right]}^{\frac{1}{2}}\mathrm{d}{z}_2.\end{array}$$

(9)

The semi-classical equations are obtained by setting $\beta ={\alpha }^{\ast }$in the drift term and ignoring the diffusion term, and are thus

$$\frac{\mathrm{d}\alpha }{\mathrm{d}t}=-(\gamma +\mathrm{i}\Delta +\mathrm{i}\chi |\alpha {|}^2)\alpha -\mathrm{i}ϵ-\mathrm{i}\kappa {\alpha }^{\ast },$$

(10)

from which it is apparent that the nonlinearity appears as a nonlinear detuning.This ensures that the instability in the $\chi =0$model when $\kappa =\gamma$does not arise.

3 Semi-classical fixed point structure

3.1 No coherent driving, $ϵ=0$

We first consider the case of no driving field $ϵ=0$.The semi-classical equations of motion (10) are then given by

$$\frac{\mathrm{d}\alpha }{\mathrm{d}t}=-(\gamma +\mathrm{i}\Delta )\alpha -\mathrm{i}\chi \alpha |\alpha {|}^2-\mathrm{i}\kappa {\alpha }^{\ast }.$$

(11)

The semi-classical steady states, ${\alpha }_0=\sqrt{n_0}{\mathrm{e}}^{\mathrm{i}{\theta }_0}$,are given by ${\overline{n}}_0=0$,and

$$\begin{array}{r}{\overline{n}}_0=-{\Delta }^{\prime }\pm \sqrt{{\left({\kappa }^{\prime }\right)}^2-1},\\ sin(2{\theta }_0)=-\frac{1}{{\kappa }^{\prime }},\end{array}$$

(12)

where we have defined the scaled variables

$$\begin{array}{r}{\overline{n}}_0=\frac{\chi }{\gamma }{n}_0,\\ {\kappa }^{\prime }=\frac{\kappa }{\gamma },\\ {\Delta }^{\prime }=\frac{\Delta }{\gamma }.\end{array}$$

(13)

In order to determine the stability of the fixed points, we linearise the equationsof motion around the fixed points. Thus we have the semi-classical linearisedequation of motion for $\delta \alpha =\alpha -{\alpha }_0$and $\delta {\alpha }^{\ast }={\alpha }^{\ast }-{\alpha }_0^{\ast }$

$$\left[\begin{array}{c}\frac{\mathrm{d}(\delta \alpha )}{\mathrm{d}t}\\ \frac{\mathrm{d}(\delta {\alpha }^{\ast })}{\mathrm{d}t}\end{array}\right]\approx {\mathbf{M}}_{\alpha {\alpha }^{\ast }}\left[\begin{array}{c}\delta \alpha \\ \delta {\alpha }^{\ast }\end{array}\right],$$

(14)

where

$${\mathbf{M}}_{\alpha {\alpha }^{\ast }}=\left[\begin{array}{cc}-\gamma -\mathrm{i}\Delta -\mathrm{i}2\chi |{\alpha }_0{|}^2& -\mathrm{i}\chi {\alpha }_0^2-\mathrm{i}\kappa \\ \mathrm{i}\chi {({\alpha }_0^{\ast })}^2+\mathrm{i}\kappa & -\gamma +\mathrm{i}\Delta +\mathrm{i}2\chi |{\alpha }_0{|}^2\end{array}\right].$$

(15)

In the limit of no parametric pumping ($\kappa =0$),this Jacobian matches the result obtained by Babourina-Brooks et al. in[8]. Stability of the fixed pointrequires all the eigenvalues of the Jacobian to have a real part less than or equalto zero [18]. A real part of exactly zeroindicates marginal stability in that parameter direction, where the fixed point isneither attractive nor repulsive. Real parts strictly less than zero are attractingfixed points which draw in nearby regions in phase space. In general, stability maydepend on more coupling parameter combinations than those which define the fixedpoints.

The origin is a fixed point for all parameter values. Indeed, the origin is the onlyfixed point for $\kappa <\gamma$,the ‘below threshold’ regime. This fixed point is stable for${\kappa }^2<{\gamma }^2+{\Delta }^2$.Four additional fixed points occur as antipodal pairs for $\kappa >\gamma$;the ‘above threshold’ regime. The first additional pair of fixed points,which we will call the ‘stable pair’, and are given by

$$({\overline{n}}_0,2{\theta }_0)=(-{\Delta }^{\prime }+\sqrt{{\left({\kappa }^{\prime }\right)}^2-1},-\pi +arccsc{\kappa }^{\prime }),$$

(16)

exists for $\Delta <\sqrt{{(\kappa )}^2-{\gamma }^2}$,and is always stable. The second additional pair of fixed points, which we will callthe ‘unstable pair’

$$({\overline{n}}_0,2{\theta }_0)=(-{\Delta }^{\prime }-\sqrt{{\left({\kappa }^{\prime }\right)}^2-1},-arccsc{\kappa }^{\prime }),$$

(17)

exists for $\Delta <-\sqrt{{(ka)}^2-{\gamma }^2}$,and is always unstable. We note that the unstable pair of fixed points can only existfor negative detuning, and that whenever the unstable pair of fixed points exists,the first pair of fixed points exists also. Also, we note that for$ϵ=0$,$\hat{a}\to -\hat{a}$ is a symmetry of the system,and thus pairs of antipodal fixed points are the expected semi-classical result. Weplot the radial and angular components of the semi-classical fixed points inFigure 2. The colour in these plots shows thestability.

Figure 2

Radial and angular components of the semi-classical fixedpoints. (a) Radial ${\overline{n}}_0=\frac{\chi }{\gamma }{n}_0$and (b) angular ${\theta }_0$components of the semi-classical fixed points. The existence and components ofthe semi-classical fixed points are functions of the two non-dimensional ratiosof the parametric pumping magnitude κ, detuning Δ,and dissipation rate γ of the system: ${\kappa }^{\prime }=\frac{\kappa }{\gamma }$and ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$.Note then that all non-zero values in (a) represent not just a single fixedpoint, but a pair of antipodal fixed points. The fixed point at origin is notplotted in (b) for the obvious reason that its angular component is undefined.The colours of the plot indicate the stability: green indicates stable fixedpoints and checkered red indicates unstable fixed points. Visible in thisdiagram is a clear semi-classical threshold where the origin becomes unstable,and above which the stable semi-classical fixed points separate.

The bifurcating behaviour of the semi-classical steady state is depicted in the‘phase digram’ of Figure 3. There are twoqualitatively different transitions that can take place in the semi-classical systemas it moves from being ‘below threshold’ to ‘abovethreshold’. First, for positive detuning ($\Delta >0$),the threshold parametric pumping ${\kappa }^{\prime }=1$is effectively increased by the detuning to ${\kappa }^{\prime }=\sqrt{1+{\Delta }^{\prime 2}}$(the solid green/checkered red boundary in Figure 3. Atthis threshold, the semi-classical system undergoes a supercritical pitchforkbifurcation where the stable origin goes unstable, and the stable pair of fixedpoints emerges from the origin and grows in separation with increasing parametricpumping (the checkered red region in the upper half-plane in Figure 3). Alternatively, for all values of negative detuning($\Delta <0$), atthe threshold parametric pumping ${\kappa }^{\prime }=1$two saddle-node bifurcations produce both the stable and unstable fixed point pairs(the solid green/striped blue boundary in Figure 3). Theorigin remains stable, and the two newly created pairs of fixed points then exist forparametric pumping above threshold (${\kappa }^{\prime }>1$)until pumping reaches the even higher value ${\kappa }^{\prime }=\sqrt{1+{\Delta }^{\prime 2}}$.Between these two values (the striped blue region in Figure 3) with increasing parametric pumping, the stable pair increases inseparation and the unstable pair moves to the origin. At the higher parametricpumping ${\kappa }^{\prime }=\sqrt{1+{\Delta }^{\prime 2}}$(the striped blue/checkered red boundary in Figure 3) theunstable pair annihilates in a subcritical pitchfork bifurcation at the origin, andthe origin becomes unstable for all higher parametric pumping ${\kappa }^{\prime }>\sqrt{1+{\Delta }^{\prime 2}}$.The stable pair of fixed points continues to grow in separation with furtherincreased parametric pumping (the checkered red region in the lower half-plane inFigure 3).

Figure 3

‘Phase diagram’ of the semi-classical system. The‘phase diagram’ of the semi-classical system. The existence andcomponents of the semi-classical fixed points are functions of the twonon-dimensional ratios of the parametric pumping magnitude κ,detuning Δ, and dissipation rate γ of the system:${\kappa }^{\prime }=\frac{\kappa }{\gamma }$and ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$.We also show the parameter regimes chosen for numerically computing the quantumsteady state. There are three different classes of fixed points: the origin isa fixed point for all parameter values (stable in the green and striped blueregions, and unstable in the checkered red region); a stable pair of antipodalfixed points exists for ‘above threshold’ parametric pumping (thestriped blue and checkered red regions); and an unstable pair exists for smallvalues of ‘above threshold’ parametric pumping if the detuningΔ is negative (the striped blue region only). Thesemi-classical steady states at the specific various black circles and crossesare depicted in Figures 4 and 5 respectively. These are for comparison with the quantum steadystates discussed in Section 4 and depicted in Figures 4 and 5.

We now illustrate the steady state behaviour of the semi-classical microwave systemin different ‘phases’. We choose a point in parameter space from eachregion of the semi-classical ‘phase diagram’ of Figure 3. These choices are marked with the black circles in that figure.These steady states are shown in Figure 4. Some other nearidentical steady states, corresponding to the black crosses in Figure 3 are depicted in Figure 5. These areplotted for comparison with the quantum version of the system in Section 4.

Figure 4

Semi-classical steady states of the microwave system for different regionsof the phase diagram. Semi-classical steady states of the microwavesystem for different regions of the ‘phase diagram’ inFigure 3. These nine plots show thesemi-classical fixed points that correspond to the parameters of the nine blackcircles in Figure 3. Specifically, from top-left tobottom-right, moving form left to right, then top to bottom, these parametervalues are: $({\kappa }^{\prime },{\Delta }^{\prime })=(0,10),(5,10),(0,0),(5,0),(0,-10),\text{ and }(5,-10)$. The coloursindicate stability; the green circle markers are stable fixed points, and thered cross markers are unstable fixed points. Clearly visible are thequalitative differences in each region separated by the semi-classicalbifurcations. These semi-classical fixed points can be compared with thequantum steady state Wigner functions for the same set of parameters inFigure 6.

Figure 5

Semi-classical steady states of the microwave system for different regionsof the phase diagram. Semi-classical steady states of the microwavesystem for different regions of the ‘phase diagram’ inFigure 3. These eight plots show thesemi-classical fixed points that correspond to the parameters of the eightblack crosses in Figure 3. Specifically theseparameter values, fro left to right, are: $({\kappa }^{\prime },{\Delta }^{\prime })=(3,-10),(3.25,-10),(3.5,-10),\text{ and }(3.75,-10)$. The coloursindicate stability; the green circle markers are stable fixed points, and thered cross markers are unstable fixed points. All fixed points in the sameregions of parameter space are very similar qualitatively. These semi-classicalfixed points can be compared with the quantum steady state Wigner functions forthe same set of parameters in Figure 7.

As well as the bifurcation structure, an important observation to make at this pointis that the existence and bifurcations of the fixed points depend upon only threeparameters: the magnitude of the parametric pumping rate κ; thedetuning Δ; and the dissipation rate γ. Andspecifically, only the two non-dimensional ratios of them, here we chose${\kappa }^{\prime }=\frac{\kappa }{\gamma }$and ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$.Thus, the below threshold to above threshold transition of the parametric oscillatoris independent of the size of the induced Kerr nonlinearity χ. However,the separation of the semi-classical fixed points $\sqrt{{\overline{n}}_0}=\sqrt{\frac{\chi }{\gamma }{n}_0}$,and thus the degree and visibility of the above-threshold oscillations, depends onthe scaling parameter $\frac{\chi }{\gamma }$.Thus, to see the semi-classical fixed points move significantly away from the origin,and thus to observe significant above-threshold behaviour we require a significantlylarge nonlinearity χ as well as parametric pumping κ.

It will also prove useful to consider the non-dissipative limit($\gamma \to 0$) ofthe semi-classical equations on resonance ($\Delta \to 0$).The fixed points are no longer stable zero dimensional attractors but ratherelliptical fixed points corresponding to stable small oscillations in thecorresponding Hamiltonian model. One easily sees that the fixed points occur at

$${\alpha }_0^2=-\frac{\kappa }{\chi }.$$

(18)

3.2 Including coherent driving, $ϵ\ne 0$

Similar to our definitions ${\alpha }_0=\sqrt{{\overline{n}}_0}{\mathrm{e}}^{\mathrm{i}{\theta }_0}$and ${\overline{n}}_0$,we introduce the scaled Cartesian coordinates ${\overline{x}}_0$and ${\overline{y}}_0$such that ${\alpha }_0=x_0+\mathrm{i}{y}_0$and

$$\begin{array}{r}{\overline{x}}_0=\sqrt{\frac{\chi }{\gamma }}{x}_0,\\ {\overline{y}}_0=\sqrt{\frac{\chi }{\gamma }}{y}_0.\end{array}$$

(19)

We also define the scaled linear driving

$${\overline{ϵ}}^{\prime }=\sqrt{\frac{\chi }{\gamma }}\frac{ϵ}{\gamma }.$$

(20)

In terms of these, the fixed points of the semi-classical equation of motion (10)satisfy the quintic equation

$$\begin{array}{rl}0=& 4{\kappa }^{\prime 2}{\overline{y}}_0^5-4\mathrm{\Im }\left\{{\overline{ϵ}}^{\prime }\right\}{\kappa }^{\prime }{\overline{y}}_0^4+[4({\kappa }^{\prime }+{\Delta }^{\prime }{\kappa }^{\prime 2}-{\kappa }^{\prime 3})+|{\overline{ϵ}}^{\prime }{|}^2]{\overline{y}}_0^3\\ +2[\mathrm{\Re }\left\{{\overline{ϵ}}^{\prime }\right\}({\Delta }^{\prime }-2{\kappa }^{\prime })+\mathrm{\Im }\left\{{\overline{ϵ}}^{\prime }\right\}(1+2{\kappa }^{\prime }({\Delta }^{\prime }-2{\kappa }^{\prime }))]{\overline{y}}_0^2\\ +[1+{\Delta }^{\prime 2}-{\kappa }^{\prime 2}-3\mathrm{\Re }\left\{{\overline{ϵ}}^{\prime }\right\}\mathrm{\Im }\left\{{\overline{ϵ}}^{\prime }\right\}+\mathrm{\Im }{\left\{{\overline{ϵ}}^{\prime }\right\}}^2({\Delta }^{\prime }-5{\kappa }^{\prime })]{\overline{y}}_0\\ +[\mathrm{\Re }\left\{{\overline{ϵ}}^{\prime }\right\}+\mathrm{\Im }\left\{{\overline{ϵ}}^{\prime }\right\}({\Delta }^{\prime }+{\kappa }^{\prime })+\mathrm{\Im }{\left\{{\overline{ϵ}}^{\prime }\right\}}^3].\end{array}$$

(21)

We notice that were there to be only a small input signal and not a large lineardrive with a small input signal on top ($ϵ={\overline{ϵ}}^{\prime }=0$),then the quintic factorises into the quadratic

$$0={\overline{y}}_0(4{\kappa }^{\prime 2}{\left({\overline{y}}_0^2\right)}^2+4({\kappa }^{\prime }+{\Delta }^{\prime }{\kappa }^{\prime 2}-{\kappa }^{\prime 3}){\overline{y}}_0^2+1+{\Delta }^{\prime 2}-{\kappa }^{\prime 2}).$$

(22)

This of course defines the tractable analytic fixed points given inSection 3.1.

Unfortunately, solving a quintic equation analytically in terms of radicals can leadto unhelpful expressions, and is not even always possible. We can of coursenumerically solve for the fixed points for certain parameter values, but we leavenon-perturbative exploration of the steady states of the $ϵ\ne 0$system for a later study. Instead, we will ultimately expand the Positive P functionas a power series in ϵ in Section 5.

4 Quantum steady state

In the previous section we described the fixed point bifurcations of the semi-classicalsystem. Here, we investigate whether there is a signature of those semi-classicalbifurcations present in the full quantum system. This can be done exactly using thepositive P function, or numerically by computing the quantum steady state densityoperator in a truncated number basis and then constructing a phase space quasiprobability density (e.g. a Q function) in different regions of the semi-classical‘phase diagram’ of Figure 3. As we will show, bychanging the coupling parameters so as to be on different sides of a semi-classicalbifurcation, there is a corresponding qualitative change in the quantum steady state.This kind of correspondence principle has proven to be the case for other dissipativenonlinear quantum systems [19–21].

4.1 Steady state via the positive P function

The steady state solution of (8) can be found as the potential conditions aresatisfied [22]. The steady state solution canbe written as

$$P_s(\alpha ,\beta )=\mathcal{N}{e}^{-V(\alpha ,\beta )},$$

(23)

where the potential function is given by

$$\begin{array}{rcl}V(\alpha ,\beta )& =& -2\alpha \beta -\lambda ln(\chi {\alpha }^2+\kappa )-{\lambda }^{\ast }ln(\chi {\beta }^2+\kappa )\\ -\frac{2ϵ}{\sqrt{\chi \kappa }}arctan\frac{\alpha }{A_0}-\frac{2{ϵ}^{\ast }}{\sqrt{\chi \kappa }}arctan\frac{\beta }{A_0^{\ast }},\end{array}$$

(24)

where

$$\lambda =-1+\frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi },$$

(25)

and $A_0=\sqrt{-{\alpha }_0^2}=\sqrt{\frac{\kappa }{\chi }}$determines the semi-classical fixed points of the corresponding Hamiltonian(non-dissipative) model, see (18). This may be written in an alternate form by notingthat $arctanx=\mathrm{i}\frac{1}{2}ln\frac{1-\mathrm{i}x}{1+\mathrm{i}x}$,

$$P_s(\alpha ,\beta )=\mathcal{N}{\left(\frac{{\alpha }_0-\alpha }{{\alpha }_0+\alpha }\right)}^{\mu }{\left(\frac{{\alpha }_0^{\ast }-\beta }{{\alpha }_0^{\ast }+\beta }\right)}^{{\mu }^{\ast }}{(\chi {\alpha }^2+\kappa )}^{\lambda }{(\chi {\beta }^2+\kappa )}^{{\lambda }^{\ast }}{\mathrm{e}}^{2\alpha \beta },$$

(26)

where $\mu =\mathrm{i}\frac{ϵ}{\sqrt{\chi \kappa }}$.

Before we can compare this distribution to the phase space structure of thesemi-classical fixed points we must face the unusual feature that the Positive Pfunction has support on a phase space with twice as many dimensions as thecorresponding classical problem. The semi-classical subspace corresponds to$\beta ={\alpha }^{\ast }$.If it were not for the noise terms in the stochastic differential equations, (9), wecould start on this subspace and never leave it. The noise however will drive thedynamics off the semi-classical subspace. Despite this we can find a very closecorrespondence between the semi-classical fixed points and the form of the steadystate Positive P function.

We first discuss the correspondence for the case of no coherent driving,$ϵ=0$. Thepeaks of the steady state positive P function will be located at the minimum of thecorresponding potential function, that is to say, the solutions of,${\partial }_{\alpha }V={\partial }_{\beta }V=0$.This gives

$$\begin{array}{r}\beta =-\frac{\lambda \alpha }{{\alpha }^2-{\alpha }_0^2},\\ \alpha =-\frac{{\lambda }^{\ast }\beta }{{\beta }^2-{\alpha }_0^{\ast 2}},\end{array}$$

(27)

where we have used (18). A little algebra shows that these are equivalent to

$$({\beta }^2-{\alpha }_0^{\ast 2})({\alpha }^2-{\alpha }_0^2)=|\lambda {|}^2,$$

(28)

$$\frac{1-{(\frac{{\alpha }_0}{\alpha })}^2}{1-{(\frac{{\alpha }_0^{\ast }}{\beta })}^2}=\frac{\lambda }{{\lambda }^{\ast }}.$$

(29)

There are two classes of solutions: $\beta ={\alpha }^{\ast }$and $\beta =-{\alpha }^{\ast }$.We will refer to the first of these as the semi-classical subspace and the second asthe nonclassical.

We first consider the semi-classical subspace. With $\beta ={\alpha }^{\ast }$,the first equation in (27) should be compared with the semi-classical steady statefrom (11), which may be written as

$${\alpha }^{\ast }=-\frac{(\frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi })\alpha }{{\alpha }^2+\frac{\kappa }{\chi }}.$$

(30)

In the limit of small quantum noise, $\chi \to 0$,$\kappa \to 0$,such that $\frac{\kappa }{\chi }=\mathrm{constant}$ we find that

$$\lambda \approx \frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi },$$

(31)

and, in the semi-classical subspace, the P-function is peaked on the semi-classicalsteady states.

In the model of Wolinsky and Carmichael [23]the nonlinear detuning χ becomes complex, thus describing nonlineardamping, and the dynamics of the positive P-function takes a very similar form tothat considered here. In particular the additional fixed points of the non classicaldimension are also present. As they describe, the non classical subspace allows thenoise to drive a stochastic process that corresponds to the nonclassical features ofthe steady state solution. In the case of strong nonlinearity they show that thesteady state positive P-function on the non classical subspace is localised on thenon classical fixed points and that these peaks reflect the fact that the steadystate is close to a superposition of two coherent states localised on the classicalfixed points.

The explicit solutions to (27) are not straightforward; they are

$$\begin{array}{r}(\alpha ,\beta )=(0,0),\\ ({\pm }_1\frac{1}{{\alpha }_0^{\ast }}\sqrt{|{\alpha }_0{|}^4+\mathrm{i}\lambda \mathrm{\Im }\{\lambda \}{\pm }_2\lambda \sqrt{|{\alpha }_0{|}^4-\mathrm{\Im }{\{\lambda \}}^2}},\\ {\mp }_2{\pm }_1\frac{1}{{\alpha }_0}\sqrt{|{\alpha }_0{|}^4-\mathrm{i}{\lambda }^{\ast }\mathrm{\Im }\{\lambda \}{\pm }_2{\lambda }^{\ast }\sqrt{|{\alpha }_0{|}^4-\mathrm{\Im }{\{\lambda \}}^2}}).\end{array}$$

(32)

These are very close, though not exactly coincident, with the semi-classical fixedpoints derived in Section 3.

4.2 Numerical steady state

To perform the numerical computation of the quantum steady state we use the QuantumOptics MATLAB toolbox [24].To do this we approximate the infinite basis of the microwave cavity oscillator; wechoose to do this by truncating in the Fock (number) basis. This means that we mustchoose couplings such that the bifurcation takes place sufficiently close to theorigin to be accurately approximated by the truncation. This is roughly because acoherent state of amplitude α has a mean occupation number of$|\alpha {|}^2$. Given thequantum steady state typically (as we shall see direct evidence of in this section)has support centred on the semi-classical steady state, fixed points far from theorigin (high $|\alpha |$) will produce high occupations and thusinaccurate results if we truncate in the Fock (number) basis.

We choose a point in parameter space from each region of the semi-classical‘phase diagram’ of Figure 3. These choices aremarked with the black circle markers in that figure. Semi-classically, thecorresponding steady states were shown in Figure 4. We nowlook at the quantum steady state through the Wigner function of the steady statedensity matrix. The Wigner function is defined as $W(x,y)=\frac{1}{\pi ħ}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}z〈x-z|\hat{\rho }|x+z〉{\mathrm{e}}^{\mathrm{i}\frac{2yz}{ħ}}$;for more on the Wigner function see [15, 25]. These Wigner functions are shown in Figure 6. There are clear signature of the semi-classical bifurcations.The quantum steady state has support centred on the stable semi-classical fixedpoints, something which has been previously observed in [19–21].

Figure 6

Steady state Wigner functions of the quantum microwave system. Densityplots of steady state Wigner functions of the quantum microwave system forvarious parameter regimes. The Wigner function $W(x,y)$ is plotted wherex and y are two quadratures of the microwave field. Thesenine plots show the quantum steady states that correspond to the parameters ofthe nine black circles in Figure 3. Specifically,from top-left to bottom-right, moving form left to right, then top to bottom,these parameter values are: $({\kappa }^{\prime },{\Delta }^{\prime })=(0,10),(5,10),(0,0),(5,0),(0,-10),\text{ and }(5,-10)$. The otherparameters are set to unity $\chi =\gamma =1$for the purpose of having a Wigner density well inside the number basistruncation. The quantum steady state shows clear signs of the semi-classicalbifurcations it undergoes. Particular comparison can be made to thesemi-classical steady states of Figure 4. Thequantum steady state has support centred on the semi-classical stable fixedpoints.

However, in two of the Wigner functions of Figure 6 (thosecorresponding to the striped blue region of Figure 3)there are three semi-classical stable fixed points, yet only two main regions ofquantum steady state density. To investigate this further, we consider the quantumsteady states corresponding to small parameter changes in this region. In particular,we look at the quantum steady states corresponding to the parameter space choicesmarked with black crosses in Figure 3. Semi-classically,the corresponding steady states were shown in Figure 5.The corresponding Wigner functions are shown in Figure 7.Interestingly, there is a gradual transition from quantum steady state supportcentred on the semi-classical stable fixed point at the origin, to support centred onthe separated stable pair. This transition does not correspond to any semi-classicalbifurcation, and at this stage is a quantum feature we cannot explain or predictsemi-classically. We mention it here to suggest one direction for futureinvestigation of this system.

Figure 7

Steady state Wigner functions of the quantum microwave system. Densityplots of steady state Wigner functions of the quantum microwave system forsmall parameter changes in the blue parameter region of Figure 2. The Wigner function $W(x,y)$ is plotted wherex and y are two quadratures of the microwave field. Theseeight plots show the quantum steady states that correspond to the parameters ofthe eight black crosses in Figure 3. Specificallyfrom left to right these parameter values are: $({\kappa }^{\prime },{\Delta }^{\prime })=(3,-10),(3.25,-10),(3.5,-10),\text{ and }(3.75,-10)$. The otherparameters are set to unity $\chi =\gamma =1$for the purpose of having a Wigner density well inside the number basistruncation. Comparison should be made with the semi-classical steady states ofFigure 5. The quantum steady state shows supportthat shifts from being centred on the semi-classical stable fixed point at theorigin, to being centred on the separated stable pair. This transition does notcorrespond to any semi-classical bifurcation. While the semi-classical steadystates of Figure 5 are quite insensitive to smallparameter shifts in regions bounded by semi-classical bifurcations, thecorresponding quantum steady states have a marked qualitative change.

5 The small signal gain

The positive P-function directly determines the normally ordered steady state moments ofthe intracavity field. We now need to chose the contour of integration so that thenormalization constant is fixed. To this end we define the integrals

$$A_{mn}={\mathcal{N}}^{-1}\int \mathrm{d}\alpha \mathrm{d}\beta {\alpha }^n{\beta }^m{P}_s(\alpha ,\beta ),$$

(33)

and express the normally ordered moments as

$$〈{{\hat{a}}^{†}}^m{\hat{a}}^n〉=\frac{A_{mn}}{A_{00}}.$$

(34)

If we wish to regard this system as an amplifier, we need to calculate the mean cavityfield amplitude $〈\hat{a}〉$ as a function of ϵ for thecase that $ϵ\ll \kappa$.With this in mind we expand the solution in a Taylor series in ϵ

$$\begin{array}{rcl}{P}_s(\alpha ,\beta )& =& P_s^{(0)}(\alpha ,\beta )-2\mu \sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{\alpha }{{\alpha }_0}\right)}^{2k+1}{P}_s^{(0)}(\alpha ,\beta )\\ -2{\mu }^{\ast }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{\beta }{{\alpha }_0^{\ast }}\right)}^{2k+1}{P}_s^{(0)}(\alpha ,\beta ),\end{array}$$

(35)

where $P_s^{(0)}(\alpha ,\beta )$ is the exact steady statesolution for $ϵ=0$. Then

$$\begin{array}{rcl}{A}_{mn}& =& A_{mn}^{(0)}-2\mu \sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{\alpha }{{\alpha }_0}\right)}^{2k+1}{A}_{m,n+2k+1}^{(0)}\\ -2{\mu }^{\ast }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{\beta }{{\alpha }_0^{\ast }}\right)}^{2k+1}{A}_{m+2k+1,n}^{(0)}.\end{array}$$

(36)

In this form we can see that the normalisation for $P_s(\alpha ,\beta )$ is the same as that for$P_s^{(0)}(\alpha ,\beta )$ as the integrals$A_{0,2k+1}^{(0)}$ vanish.

If we now substitute (26) (with $ϵ=0\Rightarrow \mu =0$)into (33), and use the Beta function identity

$$(1-{\mathrm{e}}^{\mathrm{i}2\pi \alpha })(1-{\mathrm{e}}^{\mathrm{i}2\pi \beta })\mathrm{B}(\alpha ,\beta )={\int }_C{t}^{\alpha -1}{(1-t)}^{\beta -1}\mathrm{d}t,$$

(37)

then we obtain the moments for zero coherent driving

$$\begin{array}{rcl}{A}_{mn}^{(0)}& =& \frac{{\kappa }^{\lambda +{\lambda }^{\ast }+1}(1-{\mathrm{e}}^{\mathrm{i}2\pi (\lambda +1)})(1-{\mathrm{e}}^{\mathrm{i}2\pi ({\lambda }^{\ast }+1)})}{-4\chi }{(-\frac{\kappa }{\chi })}^{(n+m)/2}\\ \times \sum _{l=0}^{\mathrm{\infty }}\{\frac{1}{l!}{\left(\frac{2\kappa }{\chi }\right)}^l[1+{(-1)}^{l+n}][1+{(-1)}^{l+m}]\\ \times \mathrm{B}(\lambda +1,\frac{l+n+1}{2})\mathrm{B}({\lambda }^{\ast }+1,\frac{l+m+1}{2})\}.\end{array}$$

(38)

Since we will always be interested in ratios of these, we can omit the leadingconstant; this then exactly matches the expression found by Kryuchkyan and Kheruntsyan[26]

$$\begin{array}{rl}{A}_{mn}^{(0)}=& {(-\frac{\kappa }{\chi })}^{(n+m)/2}\sum _{l=0}^{\mathrm{\infty }}\{\frac{1}{l!}{\left(\frac{2\kappa }{\chi }\right)}^l[1+{(-1)}^{l+n}][1+{(-1)}^{l+m}]\\ \times \mathrm{B}(\lambda +1,\frac{l+n+1}{2})\mathrm{B}({\lambda }^{\ast }+1,\frac{l+m+1}{2})\}.\end{array}$$

(39)

We first consider the steady state mean intra-cavity photon number with no coherentsignal,

$${〈{\hat{a}}^{†}\hat{a}〉}^{(0)}=\frac{A_{11}^{(0)}}{A_{00}^{(0)}}.$$

(40)

In Figure 8 we plot this as a function of the parametricdriving strength. Note that we do not see a bistable curve as in Figure 2. The reason for this is that the quantum steady state gives a longtime average which averages over all possible switching events between the twosemi-classical steady states in the bistable region. The quantum steady state is adouble peaked distribution in the complex P representation with each peak localised nearone or the other semi-classical fixed points in the bistable region.

Figure 8

The steady state mean photon number in the cavity for no linear driving.The steady state mean photon number in the cavity for no coherent driving$ϵ=0$as a function of the parametric pump magnitude κ and the detuningΔ. The corresponding semi-classical fixed points plotted inFigure 2 showed bi-stability for negative detuning$\Delta <0$which does not occur in the quantum steady state. Time units are chosen so that$\gamma =1$and $\chi =0.25$.

We can now write the moments of the intra-cavity field as,

$$\begin{array}{rl}〈{{\hat{a}}^{†}}^m{\hat{a}}^n〉=& {〈{{\hat{a}}^{†}}^m{\hat{a}}^n〉}^{(0)}-2\mu \sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{1}{{\alpha }_0}\right)}^{2k+1}{〈{{\hat{a}}^{†}}^m{\hat{a}}^{n+2k+1}〉}^{(0)}\\ -2{\mu }^{\ast }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{1}{{\alpha }_0^{\ast }}\right)}^{2k+1}{〈{{\hat{a}}^{†}}^{m+2k+1}{\hat{a}}^n〉}^{(0)},\end{array}$$

(41)

where ${〈\cdots 〉}_0$denotes the steady state average for $ϵ=0$. Inparticular, the average amplitude in the cavity at steady state is

$$\begin{array}{rcl}〈\hat{a}〉& =& {〈\hat{a}〉}^{(0)}-2\mu \sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{1}{{\alpha }_0}\right)}^{2k+1}{〈{\hat{a}}^{2k+2}〉}^{(0)}\\ -2{\mu }^{\ast }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2k+1}{\left(\frac{1}{{\alpha }_0^{\ast }}\right)}^{2k+1}{〈{{\hat{a}}^{†}}^{2k+1}\hat{a}〉}^{(0)},\end{array}$$

(42)

where we have used ${〈\hat{a}〉}_0=0$.Explicitly, this average amplitude is

$$\begin{array}{rcl}〈\hat{a}〉& =& -2\frac{{\alpha }_0}{N}\sum _{k=0}^{\mathrm{\infty }}\sum _{r=0}^{\mathrm{\infty }}\frac{1}{2k+1}\frac{2^{2r}|{\alpha }_0{|}^{4r}}{(2r)!}\mathrm{\Gamma }(\frac{1}{2}+r)\mathrm{\Gamma }(\frac{3}{2}+r+k)\\ \times [\frac{\mu }{\mathrm{\Gamma }(\frac{3}{2}+r+{\lambda }^{\ast })\mathrm{\Gamma }(\frac{5}{2}+r+k+\lambda )}+\frac{{\mu }^{\ast }|{\alpha }_0{|}^2}{\mathrm{\Gamma }(\frac{5}{2}+r+\lambda )\mathrm{\Gamma }(\frac{5}{2}+r+k+{\lambda }^{\ast })}],\end{array}$$

(43)

where

$$N=\sum _{s=0}^{\mathrm{\infty }}\frac{2^{2s}|{\alpha }_0{|}^{4s}}{(2s)!}\frac{|\mathrm{\Gamma }(\frac{1}{2}+s){|}^2}{|\mathrm{\Gamma }(\frac{3}{2}+s+\lambda ){|}^2}$$

(44)

we recall that ${\alpha }_0^2=-\frac{\kappa }{\chi }$,$\lambda =-1+\frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi }$,and $\mu =\mathrm{i}\frac{ϵ}{\sqrt{\chi \kappa }}=\mathrm{i}\frac{1}{\sqrt{\frac{\kappa }{\chi }}}\frac{ϵ}{|\chi |}$. Writing$ϵ=|ϵ|{\mathrm{e}}^{\mathrm{i}\upsilon }$,we can obtain the magnitude of the cavity field at steady state $|〈\hat{a}〉|$,

$$|〈\hat{a}〉|=G(\frac{\kappa }{\chi },\frac{\Delta }{\chi },\frac{\gamma }{\chi },\upsilon )\left|\frac{ϵ}{\chi }\right|,$$

(45)

where the gain $G=G(\frac{\kappa }{\chi },\frac{\Delta }{\chi },\frac{\gamma }{\chi },\upsilon )\ge 0$ is

$$\begin{array}{rcl}G& =& \frac{2}{R}|\sum _{k=0}^{\mathrm{\infty }}\sum _{r=0}^{\mathrm{\infty }}\frac{1}{2k+1}\frac{2^{2r}}{(2r)!}{\left(\frac{\kappa }{\chi }\right)}^{2r}\mathrm{\Gamma }(\frac{1}{2}+r)\mathrm{\Gamma }(\frac{3}{2}+r+k)\\ \times [\frac{1}{S}-\frac{(\frac{\kappa }{\chi }){\mathrm{e}}^{-2\mathrm{i}\upsilon }}{S^{\ast }}]|,\end{array}$$

(46)

where

$$\begin{array}{r}R=\sum _{s=0}^{\mathrm{\infty }}\frac{2^{2s}}{(2s)!}{\left(\frac{\kappa }{\chi }\right)}^{2s}\frac{|\mathrm{\Gamma }(\frac{1}{2}+s){|}^2}{|\mathrm{\Gamma }(\frac{1}{2}+s+\frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi }){|}^2},\\ S=\mathrm{\Gamma }(\frac{1}{2}+r+\frac{\Delta }{\chi }+\mathrm{i}\frac{\gamma }{\chi })\mathrm{\Gamma }(\frac{3}{2}+r+k+\frac{\Delta }{\chi }-\mathrm{i}\frac{\gamma }{\chi }).\end{array}$$

In Figure 9 we plot the maximum gain $G_{\max }=max\{G\}$ for a given parametric pump strengthκ and detuning Δ. We have plotted the maximum gain bychoosing the optimal signal phase υ at each set of parameters. Comparingthis to Figure 8 for the case when there is no coherentdriving, we see that the gain is a maximum around the critical parametric drivingstrength in the bi-stable, negatively detuned region.

Figure 9

The maximum gain versus the pump magnitude and detuning. The maximum gain$G_{\max }=max\{G\}$ at the optimal signal phaseυ, versus the parametric pump magnitude κ and detuningΔ, with time units chosen so that $\gamma =1$and $\chi =0.25$.The two plots show different camera perspectives of the same plotted data. Theplot is made from summing 300 terms of the appropriate hypergeometric series. Thesummation is not normalised, and thus the gain values are only correct up to ascale; however, the shape of the plot is indicative.

6 Linearised quantum system

6.1 Input-output formalism

We consider the microwave cavity with the input-output formulation of quantum optics,as originally described by Collett and Gardiner in [27]. To do this, we model the superconducting microwave resonatoras a single-sided cavity as depicted in Figure 10.

Figure 10

The superconducting microwave resonator modelled as a single-sided cavityfor use as an input-output formulation. The superconducting microwaveresonator modelled as a single-sided cavity for use as an input-outputformulation. The incoming field mode operator is ${\hat{a}}_i(t)$ and the outgoing field mode operator is${\hat{a}}_o(t)$. Loss from the microwave cavityoccurs at a rate γ. Note here that γ is thecoefficient of the amplitude decay, the coefficient for the photon number lossis 2γ. The Hamiltonian dynamics of the cavity mode$\hat{a}$ are governed by theInteraction picture Hamiltonian ℋ.

The input and output fields are treated explicitly with their mode annihilationoperators ${\hat{a}}_i$and ${\hat{a}}_o$respectively. With this formulation, the quantum stochastic differential equation weobtain for the microwave resonator field mode operators $\hat{a}$ and ${\hat{a}}^{†}$are

$$\begin{array}{r}\frac{\mathrm{d}\hat{a}}{\mathrm{d}t}=-\frac{\mathrm{i}}{ħ}[\hat{a},\mathcal{H}]-\gamma \hat{a}+\sqrt{2\gamma }{\hat{a}}_i(t),\\ \frac{\mathrm{d}{\hat{a}}^{†}}{\mathrm{d}t}=-\frac{\mathrm{i}}{ħ}[{\hat{a}}^{†},\mathcal{H}]-\gamma {\hat{a}}^{†}+\sqrt{2\gamma }{{\hat{a}}_i}^{†}(t),\end{array}$$

(47)

where the input field is effectively white noise, uncorrelated in time,

$$[{\hat{a}}_i(t),{{\hat{a}}_i}^{†}\left(t^{\prime }\right)]=\delta (t-t^{\prime })\hat{I}.$$

(48)

The probability per unit time to detect a photon in the input field is$2\gamma 〈{{\hat{a}}_i}^{†}(t){\hat{a}}_i(t)〉$. Finally, therelationship between the input, output, and cavity fields is given by

$${\hat{a}}_o(t)=\sqrt{2\gamma }\hat{a}(t)+{\mathrm{e}}^{\mathrm{i}\xi }{\hat{a}}_i(t),$$

(49)

where the phase of the second term, the reflected input, may vary with the system.For an almost perfectly reflecting mirror of an optical cavity we have$\xi =\pi$and ${\mathrm{e}}^{\mathrm{i}\xi }=-1$,here we choose this phase as an appropriate approximation.

We also look at the various fields in the frequency domain, by defining thefrequency-domain operators as the time-domain operators’ Fourier transforms,

$$\begin{array}{r}\hat{\tilde{a}}(\omega )={\mathcal{F}}_{t\to \omega }\{\hat{a}(t)\},\\ {\hat{\tilde{a}}}_i(\omega )={\mathcal{F}}_{t\to \omega }\{{\hat{a}}_i(t)\},\\ {\hat{\tilde{a}}}_o(\omega )={\mathcal{F}}_{t\to \omega }\{{\hat{a}}_o(t)\},\end{array}$$

(50)

where we have used the Fourier Transform convention $\tilde{f}(\omega )={\mathcal{F}}_{t\to \omega }\{f(t)\}=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}xf(t){\mathrm{e}}^{-\mathrm{i}\omega t}$.In the frequency domain, the input field is also uncorrelated in frequency,

$$[{\hat{\tilde{a}}}_i(\omega ),{\hat{\tilde{a}}}_i^{†}\left({\omega }^{\prime }\right)]=\delta (\omega -{\omega }^{\prime })\hat{I},$$

(51)

and the relationship between the input, output, and cavity fields is then given by

$${\hat{\tilde{a}}}_o(\omega )=\sqrt{2\gamma }\hat{\tilde{a}}(\omega )-{\hat{\tilde{a}}}_i(\omega ).$$

(52)

6.2 Gain spectra

Recall our quantum equations of motion for the microwave cavity field (47). Welinearise the system about a semi-classical fixed point ${\alpha }_0$as we did semi-classically in (14). This gives us the linearised equation of motionfor the fluctuation

$$\left[\begin{array}{c}\frac{\mathrm{d}(\widehat{\delta a}(t))}{\mathrm{d}t}\\ \frac{\mathrm{d}({\widehat{\delta a}}^{†}(t))}{\mathrm{d}t}\end{array}\right]={\mathbf{M}}_{\alpha {\alpha }^{\ast }}\left[\begin{array}{c}\widehat{\delta a}(t)\\ {\widehat{\delta a}}^{†}(t)\end{array}\right]+\sqrt{2\gamma }\left[\begin{array}{c}{\hat{a}}_i(t)\\ {{\hat{a}}_i}^{†}(t)\end{array}\right],$$

(53)

where

$$\widehat{\delta a}(t)=\hat{a}(t)-{\alpha }_0.$$

(54)

In the frequency domain the linearised equation of motion (53) becomes

$$\mathrm{i}\omega \left[\begin{array}{c}\widehat{\tilde{\delta a}}(\omega )\\ {\widehat{\tilde{\delta a}}}^{†}(-\omega )\end{array}\right]={\mathbf{M}}_{\alpha {\alpha }^{\ast }}\left[\begin{array}{c}\widehat{\tilde{\delta a}}(\omega )\\ {\widehat{\tilde{\delta a}}}^{†}(-\omega )\end{array}\right]+\sqrt{2\gamma }\left[\begin{array}{c}{\hat{\tilde{a}}}_i(\omega )\\ {\hat{\tilde{a}}}_i^{†}(-\omega )\end{array}\right].$$

(55)

We rewrite this to obtain an expression for the microwave cavity field fluctuationin terms of the input radiation,

$$\left[\begin{array}{c}\widehat{\tilde{\delta a}}(\omega )\\ {\widehat{\tilde{\delta a}}}^{†}(-\omega )\end{array}\right]=\sqrt{2\gamma }{(\mathrm{i}\omega \mathbf{I}-{\mathbf{M}}_{\alpha {\alpha }^{\ast }})}^{-1}\left[\begin{array}{c}{\hat{\tilde{a}}}_i(\omega )\\ {\hat{\tilde{a}}}_i^{†}(-\omega )\end{array}\right].$$

(56)

Using this expression, together with our input-output expression (52), allows us toobtain an expression for the output fluctuation of the microwave cavity in terms ofthe input,

$$\left[\begin{array}{c}{\widehat{\stackrel{˘}{a}}}_o(\omega )\\ {\widehat{\stackrel{˘}{a}}}_o^{†}(-\omega )\end{array}\right]=\mathbf{G}(\omega )\left[\begin{array}{c}{\hat{\tilde{a}}}_i(\omega )\\ {\hat{\tilde{a}}}_i^{†}(-\omega )\end{array}\right],$$

(57)

where the gain matrix $\mathbf{G}(\omega )$ is

$$\mathbf{G}(\omega )=\left[\begin{array}{cc}{G}_{11}(\omega )& G_{12}(\omega )\\ G_{21}(\omega )& G_{22}(\omega )\end{array}\right]=2\gamma {(\mathrm{i}\omega \mathbf{I}-{\mathbf{M}}_{\alpha {\alpha }^{\ast }})}^{-1}-\mathbf{I}.$$

(58)

Note that ${\widehat{\stackrel{˘}{a}}}_o(\omega )$ is the output fluctuation in the frequencydomain. If we re-introduced the coherent term we would obtain the full outputamplitude in the frequency domain, ${\hat{\tilde{a}}}_o(\omega )={\widehat{\stackrel{˘}{a}}}_o(\omega )+\sqrt{4\pi \gamma }{\alpha }_0\delta (\omega )\hat{I}$.

Now, we can rewrite our Jacobian ${\mathbf{M}}_{\alpha {\alpha }^{\ast }}$from (15) in terms of the parameters we defined in (13) for our semi-classical steadystates,

$${\mathbf{M}}_{\alpha {\alpha }^{\ast }}=\gamma \left[\begin{array}{cc}-1-\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)& -\mathrm{i}{\overline{n}}_0{\mathrm{e}}^{\mathrm{i}2{\theta }_0}-\mathrm{i}{\kappa }^{\prime }\\ \mathrm{i}{\overline{n}}_0{\mathrm{e}}^{-\mathrm{i}2{\theta }_0}+\mathrm{i}{\kappa }^{\prime }& -1+\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)\end{array}\right].$$

(59)

We now introduce two other useful parameters, ${\mathrm{\Lambda }}_0\in \mathbb{R}$, and${\omega }^{\prime }\in \mathbb{R}$,

$$\begin{array}{r}{\mathrm{\Lambda }}_0={\left({\kappa }^{\prime }\right)}^2-{\Delta }^{\prime 2}+{\overline{n}}_0(-4{\Delta }^{\prime }+2{\kappa }^{\prime }cos(2{\theta }_0)-3{\overline{n}}_0),\\ {\omega }^{\prime }=\frac{\omega }{\gamma }.\end{array}$$

(60)

In terms of these parameters, the gain matrix $\mathbf{G}(\omega )$ is

$$\begin{array}{rl}\mathbf{G}(\omega )=& \frac{1}{{\mathrm{\Lambda }}_0+{({\omega }^{\prime }-\mathrm{i})}^2}\\ \times \left[\begin{array}{cc}-1-{\mathrm{\Lambda }}_0-{\omega }^{\prime 2}+2\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)& 2\mathrm{i}{\overline{n}}_0{\mathrm{e}}^{\mathrm{i}2{\theta }_0}+2\mathrm{i}\kappa \\ -2\mathrm{i}{\overline{n}}_0{\mathrm{e}}^{-2\mathrm{i}{\theta }_0}-2\mathrm{i}\kappa & -1-{\mathrm{\Lambda }}_0-{\omega }^{\prime 2}-2\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)\end{array}\right].\end{array}$$

(61)

To calculate the gain measured at an arbitrary phase, we first define the quadratureoperator in the frequency domain ${\hat{X}}_{\varphi }(\omega )={\widehat{\stackrel{˘}{a}}}_o(\omega ){\mathrm{e}}^{\mathrm{i}\varphi }+{\widehat{\stackrel{˘}{a}}}_o^{†}(-\omega ){\mathrm{e}}^{-\mathrm{i}\varphi }$,which can be written in terms of our gain matrix as

$${\hat{X}}_{\varphi }(\omega )=\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}\varphi }& {\mathrm{e}}^{-\mathrm{i}\varphi }\end{array}\right]\mathbf{G}(\omega )\left[\begin{array}{c}{\hat{\tilde{a}}}_i(\omega )\\ {\hat{\tilde{a}}}_i^{†}(-\omega )\end{array}\right].$$

(62)

This expression reduces to

$${\hat{X}}_{\varphi }(\omega )=g_{\varphi }(\omega )({\hat{\tilde{a}}}_i(\omega ){\mathrm{e}}^{\mathrm{i}(\varphi +\zeta (\omega ))}+{\hat{\tilde{a}}}_i^{†}(-\omega ){\mathrm{e}}^{-\mathrm{i}(\varphi +\zeta (\omega ))}),$$

(63)

where our signal gain $g_{\varphi }(\omega )$ at phase ϕ is

$$g_{\varphi }(\omega )=\frac{|-1-{\mathrm{\Lambda }}_0-{\omega }^{\prime 2}-2\mathrm{i}{\mathrm{e}}^{-2\mathrm{i}\varphi }({\kappa }^{\prime }+{\overline{n}}_0{\mathrm{e}}^{-2\mathrm{i}{\theta }_0})+2\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)|}{{\mathrm{\Lambda }}_0+{({\omega }^{\prime }-\mathrm{i})}^2},$$

(64)

and the frequency-dependent phase shift $\zeta (\omega )$ is

$$\zeta (\omega )=arg(-1-{\mathrm{\Lambda }}_0-{\omega }^{\prime 2}-2\mathrm{i}{\mathrm{e}}^{-2\mathrm{i}\varphi }({\kappa }^{\prime }+{\overline{n}}_0{\mathrm{e}}^{-2\mathrm{i}{\theta }_0})+2\mathrm{i}({\Delta }^{\prime }+2{\overline{n}}_0)).$$

(65)

Note that while our signal gain $g_{\varphi }(\omega )$ is complex for non-zero frequency, it is realfor the DC frequency in this frame.

If we consider our analytically solved case of no linear driving bias($ϵ={\overline{ϵ}}^{\prime }=0$),and linearise about the ‘below threshold’ stable fixed point at theorigin, then ${\mathrm{\Lambda }}_0={\kappa }^{\prime 2}-{\Delta }^{\prime 2}<1$and our gain $|g_{\varphi }(\omega )|$ at aphase ϕ can be plotted against the scaled parametric pumping${\kappa }^{\prime }$and detuning ${\Delta }^{\prime }$.We plot the maximum gain $|g_{{\varphi }_{\max }}(\omega )|$ for eachvalue of parametric pumping and detuning (optimising ϕ to find themaximum gain at DC for each pair of these parameters) in Figure 11.

Figure 11

The maximum and minimum gain spectra. (a) The maximum gain spectra$|g_{{\varphi }_{\max }}(\omega )|$,and (b) the minimum gain spectra $|g_{{\varphi }_{\min }}(\omega )|$,for linearisation about the fixed point at the origin for$ϵ=0$.The gain is dependent upon four variables: the scaled parametric pumping${\kappa }^{\prime }=\frac{\kappa }{\gamma }$;the scaled detuning ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$;and the scaled probed frequency ${\omega }^{\prime }=\frac{\omega }{\gamma }$and phase ϕ. Here, we plot the spectra against only theparametric pumping and the probed frequency. The different coloured sheets showdifferent detunings: the red (innermost) surface shows the gain for no detuning${\Delta }^{\prime }=0$;the green (central) surface shows the gain for $|{\Delta }^{\prime }|=1$; and theblue (outermost) surface shows the gain for $|{\Delta }^{\prime }|=2$. Where thesurface is not plotted (other than where it is truncated around thesingularities at $({\kappa }^{\prime 2})=1+{\Delta }^{\prime 2}$)it is because the origin does not exist as a stable semi-classical fixed pointto be linearised about for those parameter values. We choose the optimal phaseϕ to produce the (a) maximum gain, and (b) minimum gain, at DCfor each pumping power and detuning.

6.3 Squeezing spectra

Having derived the gain matrix (61) relating the input to the output, we are now alsoin a position to investigate the squeezing spectrum of the microwave system. Recallour quadrature operator in the frequency domain ${\hat{X}}_{\varphi }(\omega )={\widehat{\stackrel{˘}{a}}}_o(\omega ){\mathrm{e}}^{\mathrm{i}\varphi }+{\widehat{\stackrel{˘}{a}}}_o^{†}(-\omega ){\mathrm{e}}^{-\mathrm{i}\varphi }$written in terms of our gain matrix in (62). Our squeezing spectrum is the varianceof this quadrature operator. We thus define this squeezing spectrum$S_{\varphi }(\omega )$, again in the frequency domain, to be

$$S_{\varphi }(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}〈{\hat{X}}_{\varphi }(\omega ),{\hat{X}}_{\varphi }\left({\omega }^{\prime }\right)〉\mathrm{d}{\omega }^{\prime },$$

(66)

where the notation for the covariance bracket is $〈\hat{A},\hat{B}〉=〈\hat{A}\hat{B}〉-〈\hat{A}〉〈\hat{B}〉$. We then use the linearity of botharguments of the covariance bracket to express the variance of the quadratureoperator as

$$\begin{array}{rl}{S}_{\varphi }(\omega )=& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}\varphi }& {\mathrm{e}}^{-\mathrm{i}\varphi }\end{array}\right]\mathbf{G}(\omega )\left[\begin{array}{cc}〈{\hat{\tilde{a}}}_i(\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉& 〈{\hat{\tilde{a}}}_i(\omega ),{\hat{\tilde{a}}}_i^{†}(-{\omega }^{\prime })〉\\ 〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉& 〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i^{†}(-{\omega }^{\prime })〉\end{array}\right]\\ \times \mathbf{G}{\left({\omega }^{\prime }\right)}^T\left[\begin{array}{c}{\mathrm{e}}^{\mathrm{i}\varphi }\\ {\mathrm{e}}^{-\mathrm{i}\varphi }\end{array}\right]\mathrm{d}{\omega }^{\prime }.\end{array}$$

(67)

Then, using the commutation relation of (51) we can rewrite $S_{\varphi }(\omega )$ in terms of normally-ordered variances of theinput field as

$$\begin{array}{rl}{S}_{\varphi }(\omega )=& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}\varphi }& {\mathrm{e}}^{-\mathrm{i}\varphi }\end{array}\right]\mathbf{G}(\omega )\left[\begin{array}{cc}〈{\hat{\tilde{a}}}_i(\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉& \begin{array}{c}〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉\\ +\delta (\omega +{\omega }^{\prime })\end{array}\\ 〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉& 〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i^{†}(-{\omega }^{\prime })〉\end{array}\right]\\ \times \mathbf{G}{\left({\omega }^{\prime }\right)}^T\left[\begin{array}{c}{\mathrm{e}}^{\mathrm{i}\varphi }\\ {\mathrm{e}}^{-\mathrm{i}\varphi }\end{array}\right]\mathrm{d}{\omega }^{\prime }.\end{array}$$

(68)

To proceed we now use the statistics of the input field. A coherent input field haszero normally-ordered variances ($〈{\hat{\tilde{a}}}_i(\omega ),{\hat{\tilde{a}}}_i({\omega }^{\prime })〉=〈{\hat{\tilde{a}}}_i^{†}(-{\omega }^{\prime }),{\hat{\tilde{a}}}_i(\omega )〉=〈{\hat{\tilde{a}}}_i^{†}(-\omega ),{\hat{\tilde{a}}}_i^{†}(-{\omega }^{\prime })〉=0$).Thus, the only non-zero term in central matrix is the delta function term arisingfrom the commutation relations. Using this, we can compute the integral of the matrixexpression, and our squeezing spectrum reduces to

$$\begin{array}{rl}{S}_{\varphi }(\omega )=& G_{11}(\omega )G_{12}(-\omega ){\mathrm{e}}^{\mathrm{i}2\varphi }+G_{11}(\omega )G_{22}(-\omega )\\ +G_{21}(\omega )G_{12}(-\omega )+G_{21}(\omega )G_{22}(-\omega ){\mathrm{e}}^{-\mathrm{i}2\varphi },\end{array}$$

(69)

or explicitly,

$$S_{\varphi }(\omega )=\frac{|-1-{\mathrm{\Lambda }}_0-{\omega }^{\prime 2}-\mathrm{i}2{\mathrm{e}}^{-\mathrm{i}2\varphi }({\kappa }^{\prime }+{\overline{n}}_0{\mathrm{e}}^{-\mathrm{i}2{\theta }_0})+\mathrm{i}2({\Delta }^{\prime }+2{\overline{n}}_0){|}^2}{|{\mathrm{\Lambda }}_0+{({\omega }^{\prime }-\mathrm{i})}^2{|}^2}.$$

(70)

If we consider our analytically solved case of no linear driving bias($ϵ={\overline{ϵ}}^{\prime }=0$),and linearise about the ‘below threshold’ stable fixed point at theorigin, then ${\mathrm{\Lambda }}_0={\kappa }^{\prime 2}-{\Delta }^{\prime 2}<1$and our squeezing $S_{\varphi }(\omega )$ at a phase ϕ can be plottedagainst the scaled parametric pumping ${\kappa }^{\prime }$and detuning ${\Delta }^{\prime }$.We plot the squeezing spectrum for each value of parametric pumping and detuning(setting the phase ϕ to be that which gives the minimum noise searchingover all frequencies for each pair of these parameters) in Figure 12.

Figure 12

The minimum possible noise spectra. The minimum possible noise spectra$S_{{\varphi }_{\min }}(\omega )$ for linearisation about the fixedpoint at the origin for $ϵ=0$.The noise spectrum is dependent on four parameters: the scaled parametricpumping ${\kappa }^{\prime }=\frac{\kappa }{\gamma }$;the scaled detuning ${\Delta }^{\prime }=\frac{\Delta }{\gamma }$;and the scaled probed frequency ${\omega }^{\prime }=\frac{\omega }{\gamma }$and phase ϕ. Here we plot the noise spectrum against only theparametric pumping and the probed frequency. The different coloured sheets showdifferent detunings: the red (innermost) surface shows the noise for nodetuning ${\Delta }^{\prime }=0$;the green (central) surface shows the noise for $|{\Delta }^{\prime }|=1$; and theblue (outermost) surface shows the noise for $|{\Delta }^{\prime }|=2$. Where thesurface is not plotted it is because the origin does not exist as a stablesemi-classical fixed point to be linearised about for those parameter values.We choose the optimal phase ϕ to produce the minimum noise$S_{\varphi }(\omega )$ searching over all frequencies foreach pumping power and detuning.

6.4 Signal to noise ratio

For operation of the microwave system as a bifurcation amplifier, the parameterswhich result in maximum gain may not result in minimum noise. Instead, rather thanoptimising for maximum gain or minimum noise individually, the quantity which we wishto maximise is the signal to noise ratio. However, we see from our expressions forthe gain (64) and noise (70), that $S_{\varphi }(\omega )=|g_{\varphi }(\omega ){|}^2$, and that our signalto noise ratio is thus unity,

$${\mathrm{SNR}}_{\varphi }(\omega )=\frac{|g_{\varphi }(\omega )|}{\sqrt{S_{\varphi }(\omega )}}=1.$$

(71)

For the linearised system, this equality holds for all values of all parameters(parametric pumping, detuning, and cavity dissipation), all probed frequencies andphases, and regardless of which semi-classical fixed point we choose to lineariseabout.

Physically the means that our system is acting as a parametric amplifier. Thequadrature of maximum gain is the same as the quadrature of maximum noise, andvice-versa for the minimum gain and noise. We can thus use this microwave system toamplify a signal to a measurable level without affecting its signal to noiseratio.

7 Conclusion

In this paper we detailed the quantum and semi-classical structure of a superconductingmicrowave resonator connected through a SQUID loop to ground. In particular we observedthat the semi-classical model contains a bifurcation structure, and that the remains ofthis structure are still visible in the full quantum mechanical steady state.Furthermore, we showed it can be used as a bifurcation amplifier. We did this analysisby: linearising about the semi-classical steady state below the ‘threshold’of the amplifier; by truncating the oscillator basis in the Fock basis and numericallycomputing the quantum phase space at steady state; and also by computing the exactquantum steady state by using an analytical phase space technique.

First, we showed that the corresponding semi-classical model has its fixed pointsdetermined by a quintic polynomial. We showed that for the small linear signal regime$ϵ=0$, thatthis quintic factors and is analytically solvable. This semi-classical system thenundergoes a bifurcation of its semi-classical steady state with increased parametricpumping power. This bifurcation gives a threshold for the amplifier and occurs when theparametric pumping power equals the cavity decay, with adjustment for a detuned drive,$|\kappa {|}^2={\gamma }^2+{\Delta }^2$.The sign of the detuning specifies the form of the bifurcations. For a positive detuning$\Delta \ge 0$,the origin undergoes a supercritical pitchfork bifurcation at the threshold. Fornegative detuning $\Delta <0$, theorigin instead loses its stability at $|\kappa {|}^2={\gamma }^2+{\Delta }^2$in a subcritical pitchfork bifurcation with two intermediate pairs of fixed pointscreated in saddle-node bifurcations when the parametric pumping power reaches$|\kappa {|}^2={\gamma }^2$.The numerically calculated quantum steady states were shown to have clear signatures ofthese semi-classical steady state bifurcations. Specifically, the Wigner functionrepresentation of the quantum phase space was seen to have support on the semi-classicalfixed points.

In addition to numerically computing the quantum phase space at steady state bytruncating the oscillator basis, we also calculated the exact quantum steady state. Thiswas done following the work of Kryuchkyan and Kheruntsyan [26] by using the Positive P representation. The method tookadvantage of the fact that the potential conditions were satisfied. The exact quantumphase space density at steady state was seen to be peaked in the vicinity of thecorresponding semi-classical fixed points.

We showed that the quantum device functioned as a bifurcation amplifier until threshold.We calculated the small signal gain of the amplifier using the exact quantum steadystate. We also approximated this by linearising the steady state about thesemi-classical below-threshold fixed point using the input-output formalism of Collettand Gardiner [27]. With this procedure we alsocalculated noise spectra, and we showed that the signal to noise ratio at allfrequencies and phases was equal to unity. We thus showed that the quarter-wavemicrowave resonator considered can be made to act as a parametric amplifier. This devicecan take a signal from a nano-electromechanical system and amplify it to a measurablelevel without affecting its signal to noise ratio.