Quantum and classical nonlinear dynamics in a microwave cavity
© Meaney et al.; licensee Springer on behalf of EPJ. 2014
Received: 29 January 2014
Accepted: 24 April 2014
Published: 10 June 2014
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.
Superconducting circuit quantum electrodynamics (circuit QED)  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 .
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 . 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. .
2 The dissipative Cassinian oscillator model
2.1 Master equation
where represents the charging energy of the effective LC oscillator while n is thenumber of elementary charges on the capacitor, 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 ,where B is a geometric factor. Further details are given in Wallquist et al.give .
where isthe cavity frequency, represents the coherent driving strength, κ represents the parametricdriving strength, 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].
where .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.  and more recently byDykman and his collaborators .
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 . Thisis required if the distributions are to give normally ordered moments directly viaintegration. In  a direct implementationusing circuit QED of these additional channels is demonstrated and connection is madeto the stationary normal ordered moments.
from which it is apparent that the nonlinearity appears as a nonlinear detuning.This ensures that the instability in the model when does not arise.
3 Semi-classical fixed point structure
3.1 No coherent driving,
In the limit of no parametric pumping (),this Jacobian matches the result obtained by Babourina-Brooks et al. in. Stability of the fixed pointrequires all the eigenvalues of the Jacobian to have a real part less than or equalto zero . 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.
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 choseand .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 ,and thus the degree and visibility of the above-threshold oscillations, depends onthe scaling parameter .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.
3.2 Including coherent driving,
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 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
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.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.
There are two classes of solutions: and .We will refer to the first of these as the semi-classical subspace and the second asthe nonclassical.
and, in the semi-classical subspace, the P-function is peaked on the semi-classicalsteady states.
In the model of Wolinsky and Carmichael 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.
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 .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. 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 ) will produce high occupations and thusinaccurate results if we truncate in the Fock (number) basis.
5 The small signal gain
In this form we can see that the normalisation for is the same as that for as the integrals vanish.
6 Linearised quantum system
6.1 Input-output formalism
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 haveand ,here we choose this phase as an appropriate approximation.
6.2 Gain spectra
Note that is the output fluctuation in the frequencydomain. If we re-introduced the coherent term we would obtain the full outputamplitude in the frequency domain, .
Note that while our signal gain is complex for non-zero frequency, it is realfor the DC frequency in this frame.
6.3 Squeezing spectra
6.4 Signal to noise ratio
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.
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, 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,.The sign of the detuning specifies the form of the bifurcations. For a positive detuning,the origin undergoes a supercritical pitchfork bifurcation at the threshold. Fornegative detuning , theorigin instead loses its stability at in a subcritical pitchfork bifurcation with two intermediate pairs of fixed pointscreated in saddle-node bifurcations when the parametric pumping power reaches.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  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 . 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.
This work was supported by the Australian Research Council grants FF0776191 andCE110001014.
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