Quantum-classical correspondence for a dc-biased cavity resonator–Cooper-pair transistor system111To appear in Fluctuating Nonlinear Oscillators, Edited by Mark Dykman (Oxford University Press).
We investigate the quantum versus classical dynamics of a microwave cavity-coupled-Cooper pair transistor (CPT) system, where an applied dc bias causes the system to self-oscillate via the ac Josephson effect. Varying the dc bias allows the self-oscillation frequency to be tuned. An unusual feature of the system design is that the dc bias does not significantly affect the high quality factor of the cavity mode to which the CPT predominantly couples. The CPT-cavity mode system has a mechanical analogue involving a driven coupled pendulum-oscillator system. The corresponding, nonlinear classical dynamical equations exhibit chaotic, as well as aperiodic motions depending on the initial conditions and the nature and strengths of the damping/noise forces. The quantum master equation exhibits such phenomena as dynamical tunnelling and the generation of nonclassical states from initial classical states. Obviating the need for an external ac-drive line, which typically is harder to noise filter than a dc bias line, the self-oscillating system described here has considerable promise for demonstrating macroscopic quantum dynamical behavior.
The work presented in this chapter has its origins in a seemingly mundane microwave engineering question: is it possible to apply a dc voltage (or current) bias to the center conductor of a superconducting coplanar microwave cavity, without significantly affecting the quality factor of, say, the first and second microwave modes of the cavity? Our original motivation behind this question was to devise a circuit quantum electrodynamics (QED) based scheme wallraffnature04 () that can generate and detect quantum states of a mechanical resonator armournjp08 (); blencowenjp08 (), where the dc bias is required to strongly couple a nanomechanical resonator to a superconducting qubit. However, it turns out that having such a dc bias functionality opens up possibilities for other heretofore difficult-to-realize quantum dynamical investigations, one of which we shall focus on here.
We shall in particular investigate the quantum dynamics of the device shown in Fig. 1, which comprises two Josephson junctions (JJ) in series with a gate electrode, and where the source electrode to one of the JJ’s contacts the center conductor of the microwave cavity, while the drain electrode from the other JJ contacts the ground plane of the microwave cavity. The following section describes how the microwave cavity design allows the application of a dc voltage bias to the center conductor, while maintaining a very large quality factor of the second microwave mode to which the JJ’s strongly couple chenapl11 (). For not too large a bias, the JJ’s operate in the subgap region as a “Cooper pair transistor” (CPT), where the dc bias generates a tunable oscillating supercurrent through the CPT via the ac Josephson effect. The tunneling Cooper pairs will both emit into and absorb photons from the second microwave mode, and it is the resulting coupled CPT-cavity mode quantum dynamics that will be of central interest to us.
Related devices comprising one or more JJ’s embedded in a microwave cavity date back to just a few years following the discovery of the ac Josephson effect josephsonpl62 (), where classical signatures of the resonant microwave modes of the tunnel junctions themselves, interacting with the alternating tunnel currents, were observed and discussed werthamer (); zimmerman (); smith (). Beginning in the ‘90’s, investigations addressed the effect of a structured electromagnetic environment with resonant modes on the current-voltage characteristics of dc voltage biased JJ’s holstprl94 (); ingoldprb94 (); hofheinzprl01 (). And more recently, similar investigations involving double JJ devices were carried out leppakangasprb08 (); pashkinprb11 (). However, the quality factors of the electromagnetic modes in these devices were small, typically less than 10, to be contrasted with quality factors exceeding for the present device design chenapl11 () shown in Fig. 1. As a consequence, emitted microwave photons will now remain in the cavity mode for many Cooper pair tunnel oscillation cycles before leaking out of the cavity mode; it does not make sense to treat the microwave cavity as an electromagnetic environment for the CPT. Instead, the cavity and CPT should be viewed as a strongly-coupled, quantum coherent system.
The CPT-cavity mode device has a mechanical analogue involving a driven coupled pendulum-oscillator system (Sec. III.2). The corresponding, nonlinear classical dynamical equations exhibit chaotic, as well as aperiodic motions depending on the initial conditions and the nature and strengths of the damping/noise forces. Thus, the device in principle allows the experimental investigation of the quantum dynamics of a system for which the corresponding classical dynamics is chaotic. There is a long tradition of using Josephson junction devices for investigating macroscopic quantum dynamics in systems with corresponding nonlinear classical equations srivastavapr87 (); leggettcp09 (). The Sussex group carried out some of the first, pioneering work in the ‘80’s prancehpa83 (), which was followed by the demonstration of quantum tunneling by the Clarke group at Berkeley clarkescience88 (), and which culminated in demonstrations over a decade later of superposition states by the Lukens friedmannature00 () and Mooij chiorescuscience03 () groups at Stonybrook and Delft, respectively. Subsequent, related developments have largely focussed on the realization of superconducting quantum bits for quantum computing applications neeleynature10 (); dicarlonature10 (), although JJ devices still occasionally are used for exploring macroscopic quantum dynamics and the transition to classical dynamics fedorovprl11 ().
A large body of theoretical work concerning the quantum-classical correspondence for driven systems has focused on the Duffing and other anharmonic oscillators dykmanzetf88 (); zurekprl94 (); brunjpa96 (); kohlerpre97 (); habibprl98 (); bhattacharyaprl00 (); monteolivapre01 (); habibprl02 (); peanoprb04 (); everittpre05 (); marthalerpra06 (); dykmanpre07 (); greenbaumpre07 (); serbanprl07 (); katznjp08 (); versopra10 (); ketzmerickpre10 (), as well as on various rigid rotor models todaptps89 (); casatiptps89 (); foxpra90 (); foxpra91 (); grahampra91 (); grahamprl91 (); foxpre94 (); latkapra94 (); mirbachprl95 (); gorincp97 (); mouchetpre01 (); mouchetpre06 (); reichl04 (); haake10 (); many insights have been gained by investigating dynamical properties using a quantum phase space (i.e., Wigner or Husimi function takahashiptps89 (); leepre93 ()) description, and by examining the Floquet states and associated quasienergy spectra. However, relevant experimental results have been few steckscience01 (); chaudhurynature09 (); vijayrsi09 (). One of the key difficulties is that most experimental realisations require an external ac signal to drive the system, which can be one of the most significant sources of noise, preventing the system from displaying manifest quantum dynamical behavior. In contrast, the CPT-resonator system in Fig. 1 generates its own ac drive, i.e., it self-oscillates. As a consequence of the ac-Josephson effect, only a dc voltage bias is required, and by varying , the drive frequency can be tuned. Since it is considerably easier to noise filter a dc bias line than an ac-drive line, the device described here has considerable promise for exhibiting macroscopic quantum dynamical behavior.
The outline of this chapter is as follows. In Sec. II, we give a description of the CPT-cavity system. The classical system equations are derived in Sec. III, and solutions to these equations are discussed in Sec. IV. The corresponding quantum master equation is derived and a quantum phase space representation of the system state is described in Sec. V. Solutions to the quantum master equation are discussed in Sec. VI and in Sec. VII we investigate the classical limit of the quantum master equation. We conclude in Sec. VIII.
Much of the analysis will in fact deal with a simplified system comprising the driven ‘pendulum’ part of the device. An analysis of the full CPT-resonator mode system dynamics, including results from experiment, will be published elsewhere.
Ii The cavity-Cooper pair transistor device
To introduce a dc bias into a high- microwave cavity, we begin with a standard coplanar-waveguide-based resonator that is one wavelength long at the operating frequency, illustrated schematically in Fig. 1(a). As is usually the case, the ends of the cavity are terminated by small capacitors (on the order of a few fF) that even at a typical operating frequency of 5 GHz have a large impedance. To a first approximation, then, we can treat these terminations as open circuits, so that the cavity voltage is a local maximum at the cavity end, and the cavity current a local minimum. At a distance from each end of the cavity, the situation is reversed: the cavity voltage is minimal and the current maximal, so that the points are low impedance points.
At the low impedance points we then introduce dc bias lines consisting of sections of waveguide terminated with an inductance . These lines are chosen to have a length , so that the impedance they present to the main cavity line at the point is the same as their terminating impedance . For even a small inductance of a few nH, this impedance can be substantial at the operating frequency of the cavity.
A microwave photon approaching either dc biasing “T” junction will therefore see a short circuit (the low impedance of the main line) in parallel with a large impedance (the dc bias line) and to first order the cavity photons will be unaffected by the presence of the dc bias lines. The second (full wave) resonance of the cavity should still enjoy a very large of up to several thousand in the presence of a dc bias. By placing a CPT at the center of the cavity (the black dot in Fig. 1(a)), where there is an antinode in the cavity voltage, it should be possible to strongly couple the CPT to the cavity and use an applied dc bias to produce self oscillations of the CPT/cavity system via the ac Josephson effect.
Electron and optical micrographs of a device based on these ideas are shown in Fig. 2. The cavity itself is fabricated out of a Nb film on an undoped Si substrate, as shown in Fig. 2(a). Input and output lines on the left and right are coupled to the main line by small capacitors. The dc bias lines extend toward the top of the image; each is terminated by a small on-chip spiral inductor. Cavities based on this design have been shown to posses a large of several thousand for the full wave mode at a temperature of 4 K even when a dc bias voltage or current is applied to the central conductor of the cavity chenapl11 ().
At the center of the cavity, a narrow wire to be used as a gate line for the CPT is brought through the ground plane of the waveguide, as shown in Fig. 2(b). Two thin Ti/Au contact pads are added to the central conductor and ground plane of the cavity to the right of the entry point for the gate wire. These contact pads, which are driven superconducting by the proximity effect, allow for good metal-to-metal contact between the CPT and cavity. Finally, the CPT and its gate are added to the structure using standard electron beam lithography and shadow evaporation techniques, as in Fig. 2(c).
Iii Classical model of device
iii.1 Closed system equations
The effective lumped element model description of the dc voltage biased microwave cavity-coupled Cooper pair transistor (CPT) device is illustrated in Fig. 3. It is supposed that, for the considered bias range, the CPT couples predominantly to a particular mode of the cavity. We neglect for the time being the cavity and CPT sources of dissipation, modeled by the parallel network admittance and series network impedance, respectively, focusing first on writing down the closed system equations of motion. For a typical device, the cavity effective capacitance is a few pF, while the Josephson junction (JJ) capacitance is at least a few hundred aF, and the gate bias capacitance is about . Furthermore, the effective bias line inductance is a few nH and the cavity effective inductance is a few tenths of nH. Thus, the typical size hierarchies are and . We shall make use of these to simplify by approximation the equations of motion. Using Kirchhoff’s Laws and the constitutive relations for the various lumped circuit elements, it is straightforward to obtain the equations of motion. In terms of the phase differences across the two JJs, the equations are
where is the flux quantum, is the JJ critical current, is an integration constant, and we have assumed . Transforming to ‘center-of-mass’ (CoM) and relative phase coordinates , Eqs. (2) and (4) become
where we have assumed .
The Hamiltonian is
where is minus the number of excess Cooper pairs on the island, is the polarization charge induced by the applied gate voltage bias in units of Cooper pair charge, is the approximate CPT charging energy (neglecting ), i.e., the electrostatic energy cost for putting one additional Cooper pair on the CPT island, and is the Josephson energy of a single JJ.
It is convenient to work instead in terms of the shifted CoM coordinate: , where the driving frequency is
Performing this canonical transformation with the appropriate generating function, we obtain the following transformed Hamiltonian:
where we have dropped the tilde on the shifted coordinate and have set .
The key observation to make about Hamiltonian (13) is the presence of the time-dependent drive, which originates from the ac Josephson effect, and can be controlled via the externally applied bias [Eq. (11)]; the nonlinear system self-oscillates. In contrast to most other driven nonlinear system investigations, no externally applied ac drive is required, thus eliminating one of the main sources of noise that hinders the demonstration of macroscopic quantum dynamics.
iii.2 Mechanical analogue model
In order to gain insights into the cavity mode-CPT dynamics, as well as motivate other parameter choices, it is useful to consider a mechanical analogue. Hamiltonian (13) can be reexpressed in the following form:
where , , , and . From Eq. (14), we see that the cavity mode-CPT system is equivalent to a system consisting of two coupled rotors with moments of inertia and angular momentum . Neglecting the rotor coupling, the ‘+’ rotor behaves as a torsional oscillator with frequency . For small angular displacements and with the drive turned off (i.e., ), the ‘-’ rotor behaves as a pendulum. For small displacements, the pendulum oscillates approximately harmonically with frequency . With the drive turned on (i.e., ), the pendulum rotor’s ‘gravitational acceleration’ is sinusoidally modulated at frequency , periodically switching sign as a result. The gravitational acceleration is also modulated by the torsional oscillator coordinate. The ratio of the rotors’ moments of inertia is
where is the cavity impedance and is the von Klitzing constant. The frequency ratio for small angle, undriven oscillations is
For typical capacitance values, CPT charging and Josephson energies of a few Kelvins (in units of ), and for a cavity mode frequency (), we see that the ratios in Eqs. (15) and (16) are large: the moment of inertia ratio is of order and the frequency ratio of order . Thus, the mechanical analogue corresponds to a fast pendulum with a small moment of inertia that is coupled to a slow torsional oscillator with a large moment of inertia.
A measure of the zeropoint fluctuations in the pendulum angular coordinate is
For typical CPT parameter values, we have , i.e., the zeropoint uncertainty is comparable to the size of the coordinate space ( radians). Thus, we don’t expect the driven quantum pendulum dynamics to resemble much the dynamics of the driven classical pendulum, which can be chaotic. Recovering the classical pendulum limit requires a smaller charging energy than Josephson energy, for example a ‘transmon’-like CPT kochpra07 (). The classical limit will be discussed in detail in Sec. VII.
How do the mechanical analogue moments of inertia compare in magnitude to those of actual mechanical systems? The hydrogen molecule has a rotational moment of inertia horizfp27 (). For (), we have . Thus, the typical CPT pendulum equivalent moment of inertia is an order of magnitude larger than that of the hydrogen molecule. For a transmon-like CPT, the moment of inertia is about two orders of magnitude larger than that of the hydrogen molecule. From Eq. (15), we see that the cavity mode torsional oscillator equivalent moment of inertia is about times larger than that of the hydrogen molecule.
iii.3 Open system equations
The cavity-CPT device is subject to several sources of dissipation and noise. Two significant electromagnetic environment sources arise from the capacitive couplings between cavity and input/output microwave lines and the capacitive coupling between the CPT island and gate voltage bias line. Referring to Fig. 3 , we model the cavity noise/dissipation by an infinite parallel network of ‘bath’ oscillators, and the gate voltage noise/dissipation by an infinite series network of ‘bath’ oscillators yurke83 (); devoret95 (); burkard04 (). The actual dissipative mechanisms can be modeled by such infinite oscillator networks by making appropriate choices for the oscillator frequency distribution spectra. In the following, we will analyze the two noise/dissipation sources independently, beginning first with the cavity noise source.
Extending Hamiltonian (13) to include the infinite parallel network of oscillators, we obtain:
where is the phase coordinate across the network capacitance . Integrating Hamilton’s equations of motion for the network oscillator coordinate , we obtain:
where , the network oscillator “masses” are and the system-network oscillator couplings are . Following the approach of Ref. cortes85 (), we integrate (21) by parts and substitute into the equations for and to obtain the following Langevin equation:
where we have assumed that the couplings are small and we have neglected frequency renormalization terms and where
is the damping kernel and
is the noise force. Assuming the network oscillator initial coordinates , are randomly distributed according to the Maxwell-Boltzmann thermal distribution at temperature , we find for the force-force correlation function:
With being the “mass” of the coordinate, we see that (25) obeys the usual fluctuation-dissipation relation. With the Markovian approximation , Eq. (22) describes a dissipative cavity mode where the admittance in Fig. 3 is simply replaced by a resistance .
Moving on now to modelling the gate voltage noise, we insert an infinite series network between the gate voltage source and the gate capacitance . Hamiltonian (13) is then modified approximately as follows:
where is the phase coordinate across the network capacitance, and we assume is small compared to the other capacitances. In the following, we neglect the coupling between the infinite series network and the coordinates, since this results simply in adding to the dissipation due to the cavity mode loss considered above. Integrating Hamilton’s equations of motion for the network oscillator coordinate , we obtain:
where the network oscillator frequencies and masses are the same as for the cavity mode, while the system-network oscillator couplings are now . Substituting Eq. (30) into Hamilton’s equation for , integrating by parts and neglecting renormalization and shift terms, we obtain the following equation:
Assuming the network oscillator initial coordinates , are randomly distributed according to the Maxwell-Boltzmann thermal distribution at temperature , we find for the correlation relation:
Now, we have:
where is the fluctuating voltage across the unloaded series network. But in the Markovian approximation, the voltage noise across a resistance is
where is the effective resistance characterizing the loss associated with the gate voltage noise. Substituting Eq. (41) into the damping term of Eq. (34), we obtain for the , coordinate equations in the presence of gate voltage noise and associated damping within the Markov approximation:
Now that we have analyzed both the cavity noise and gate voltage noise, we finally write down in dimensionless form the classical Markovian Langevin equations for the cavity-CPT system in the presence of both noise sources. In first order form, the equations of motion are:
where the dimensionless conversions are and , with . The dimensionless drive force amplitude and frequency are and , respectively [with the tildes subsequently dropped in Eq. (47) and below]. The cavity mode quality factor is in terms of the cavity mode resistance , while denotes the gate voltage resistance. The associated dimensionless cavity and gate bias noise “forces” satisfy the respective correlation relations
where we distinguish the cavity mode environment and gate voltage effective noise temperatures, since they are not necessarily the same in experiment.
Iv Classical dynamics
The set of Langevin equations (47) provides a full description of the classical stochastic dynamics of the system. Numerical integration of these equations averaged over many different realizations of the noise allows one to obtain probability distributions for all of the system variables. Ultimately these distributions could then be compared with appropriately chosen quasiprobability distributions for the corresponding quantum degrees of freedom. However, this approach is rather demanding from a computational point of view, especially for the quantum dynamics. We will restrict ourselves to outlining the behavior of the simpler system consisting of the driven Cooper-pair transistor alone. In effect this corresponds to the limit of small , and .
Looking at Eq. (47), it is clear that for a strongly damped and weakly driven cavity, the variable will remain small and hence to a good approximation it will be possible to drop the dependence of the equations so that the latter become entirely decoupled from the evolution of the cavity variables. In this limit we are left with just the pair of equations,
We start by solving Eq. (51) in the limit where . In this regime the equations are simple classical equations of motion for and . Nevertheless, they reveal a complex dynamical behavior which has already been investigated in different contexts (see e.g. mouchetpre06 ()). Depending on the initial conditions and the choice of parameters, the system typically has a mixed phase space in which the behavior is either chaotic or quasiperiodic. The phase space is visualized in a stroboscopic plot in which a point is plotted after each period of the drive, examples of which are shown in Fig. 4. In the limit the system is integrable with natural frequencies , hence for very small values of the phase space is perturbed around resonances reichl04 () which occur at (see Fig. 4a); as is increased the resonances get larger and a chaotic sea forms when they overlap. Islands of stability (where the orbits remain quasiperiodic) are found near even when (see Fig. 4b).
We can explore the sensitivity of the system to dissipation (as opposed to noise) by setting and changing the value of . We find that even rather low levels of dissipation can have a significant effect on the the long time behavior. For example, for the parameters used in Fig. 4b with , the phase space appears to contain only two attractive fixed points (one associated with each of the resonances). However, the chaotic sea is present as a transient, albeit one which can be rather long-lived: for certain initial conditions it only disappears after periods of the drive.
Before examining the full behavior of Eq. (51) with dissipation and noise, it is also worth considering the effect of averaging over an ensemble of initial conditions. In order to make a comparison with the quantum dynamics we need to consider how an initial distribution of values evolves. Because of the chaotic behavior of the system the effects of considering a range of initial coordinates can be very dramatic even after a relatively short period of time. Starting from a Gaussian distribution of initial states centered on a point in the chaotic sea, leads to a set of trajectories that spreads out rapidly over the chaotic sea as can be seen in Fig. 5a. The islands within the chaotic sea stand out (the handful of points that lie within the islands come from initial points that didn’t fall within the chaotic sea). Clearly averaging over a range of initial conditions has a dramatic effect on the dynamics of the averages of the system, this is particularly clear for the quantity which very rapidly becomes a periodic oscillation with period as shown in Fig. 5b.
Examples of the probability distribution for the classical, noisy, evolution of the system are shown in Fig. 6. The numerical interaction is carried out using a generalization of the Heun method used for deterministic differential equations breuer (). In this case an average is carried out both over realizations of the noise and the initial conditions which are chosen from a Gaussian distribution with variances centered on a given point in phase space. When noise is added to the system the trajectories eventually diffuse between the chaotic sea and the quasiperiodic orbits so that the difference in the probability distribution over the island and chaotic sea regions gets washed out over time. In Fig. 6 the remnants of the island can be seen at , but by they have disappeared completely.
V Quantum model of device
v.1 Quantum master equation
The Poisson bracket relations for the classical canonical coordinates are
(where recall ). Applying the correspondence principle, the quantum commutation relations are
However, the phase coordinates are not periodic functions of their associated system configuration spaces; the representations of the commutation relations (53) give the unbounded eigenvalue spectrum for the corresponding phase operators. While this is not a problem for the ‘torsional’ oscillator because of the strong harmonic confining potential, which limits the accessible region of configuration space, the ‘pendulum’ typically explores the whole of its unit circle () configuration space. A suitable pendulum configuration space function is with Poisson bracket relation:
The corresponding commutation relation is then
Introduce raising and lowering operators for the torsional (CoM) coordinate:
where recall , the ‘moment of inertia’ is , and we have dropped the hats on the operators for notational convenience. The CoM phase coordinate oscillator zero-point uncertainty is
where recall is the cavity impedance and is the von Klitzing constant, so that . The Hamiltonian operator corresponding to (13) is
The parameter appearing in the Hamiltonian operator is a purely quantum signature of the nontrivial topology of the corresponding classical pendulum’s configuration space . An interesting question concerns the particular value for that Nature chooses and why kastruppra06 (); kowalski (); bahr (). However, it is likely not possible to measure in experiment, since from (LABEL:quantumhamiltonianeq) it is clear that the effect of a nonzero value is indistinguishable from that due to the presence of an excess charge on the CPT island. From now on, we shall set .
We now derive the open system quantum master equations within the self-consistent Born approximation (SCBA) following the approach reviewed in Ref. paz01 (). In the following, we analyze the two noise/dissipation sources independently, beginning first with the cavity mode environment. We write the Hamiltonian (20) as where the Hamiltonian describes the cavity mode-CPT system [Eq. (LABEL:quantumhamiltonianeq)], the Hamiltonian describes the infinite parallel network environment, and the interaction part is
where . Defining , we obtain for the master equation within the SCBA:
where the operators and are in the interaction picture and the expectation values are performed assuming the environment (infinite parallel network) is in a thermal state. We have
where is the thermal occupation number of the environment at frequency and where the spectral function is
with and .
In Sec. VII, we compare the quantum versus classical dynamics and establish conditions under which the former is well approximated by the latter–the so-called classical limit. A necessary condition to be in the classical limit is that the environment temperature must be sufficiently large such that we can make the approximation . This requires . The environment correlation function then becomes
where is the classical damping kernel (23). If, furthermore, the spectral function upper cut-off satisfies , then we can make the Markovian approximation and
where is the damping rate. Eq. (71) is just the standard Born-Markov master equation for a quantum Brownian particle in the high temperature limit paz01 (), where the second term on the right hand side describes damping and the third term on the right hand side describes diffusion.
While Eq. (71) is appropriate for investigating the classical limit, under the cryogenic conditions of an actual experiment and for say an cavity mode, we expect that , so that a low temperature limit is more appropriate. Using Eq. (57) to express the master equation in terms of raising and lowering operators, making the rotating wave approximation and the replacement