Applications of the Fokker-Planck equation in circuit quantum electrodynamics

We study exact solutions of the steady-state behavior of several nonlinear open quantum systems which can be applied to the field of circuit quantum electrodynamics. Using Fokker-Planck equations in the generalized $P$ representation, we investigate the analytical solutions of two fundamental models. First, we solve for the steady-state response of a linear cavity that is coupled to an approximate transmon qubit and use this solution to study both the weak and strong driving regimes, using analytical expressions for the moments of both cavity and transmon fields, along with the Husimi $Q$ function for the transmon. Second, we revist exact solutions of a quantum Duffing oscillator, which is driven both coherently and parametrically while also experiencing decoherence by the loss of single photons and pairs of photons. We use this solution to discuss both stabilization of Schrödinger cat states and the generation of squeezed states in parametric amplifiers, in addition to studying the $Q$ functions of the different phases of the quantum system. The field of superconducting circuits, with its strong nonlinearities and couplings, has provided access to parameter regimes in which returning to these exact quantum optics methods can provide valuable insights.

Figure 1

Plot of $|\langle a\rangle |$ as a function of the detuning ${\mathrm{\Delta }}_c$ of the drive from the bare cavity and drive amplitude $\epsilon$ when cavity and qubit are resonant. Other parameters are $g/2\pi =115\mathrm{MHz}, \chi /2\pi =-220\mathrm{MHz}, {\gamma }_c/2\pi =2\mathrm{MHz}$, and ${\gamma }_t/2\pi =0.1\mathrm{MHz}$. We see the characteristic vacuum Rabi splitting, with peaks (A, B) separated by $2g$ at low powers and then demonstrating so-called supersplitting as the power increases. There is extremely low transmitted amplitude at the bare cavity frequency (C). As the power increases, higher order transitions become present in the spectrum (D) and at sufficiently large drive strengths the resonance shifts back to the bare cavity frequency and there is a strong transmission peak (E).

Figure 2

Plot of $|\langle b\rangle |$ as a function of the detuning ${\mathrm{\Delta }}_c$ of the drive from the bare cavity frequency and drive amplitude $\epsilon$ when cavity and qubit are coupled in the strong dispersive regime. Other parameters are ${\mathrm{\Delta }}_{ct}=2.5\mathrm{GHz}, g/2\pi =340\mathrm{MHz}, \chi /2\pi =-220\mathrm{MHz}, {\gamma }_c/2\pi =2\mathrm{MHz}$, and ${\gamma }_t/2\pi =0.1\mathrm{MHz}$. Around the bare transmon frequency, the system behaves like a quantum Duffing oscillator with a dispersively shifted fundamental frequency (A). At higher powers we see peaks corresponding to transitions between higher transmon levels which are separated by $\chi$ (B). Near the bare cavity resonance the transmission peak is dispersively shifted at low power (C), but shifts to the bare cavity frequency at high power (D). This region in shown in greater detail in Fig. 3.

Figure 3

Plots of cavity reflection $R$ and $|\langle a\rangle |$ as a function of the detuning ${\mathrm{\Delta }}_c$ of the drive from the bare cavity frequency and drive amplitude $\epsilon$ when cavity and qubit are coupled in the strong dispersive regime. Other parameters are ${\mathrm{\Delta }}_{ct}/2\pi =2.5\mathrm{GHz}, g/2\pi =340\mathrm{MHz}, \chi /2\pi =-220\mathrm{MHz}, {\gamma }_c/2\pi =2\mathrm{MHz}$ and ${\gamma }_t/2\pi =0.1\mathrm{MHz}$. At lower power, multiple dips in the reflection (less visible peak in the transmission) are visible, corresponding to the cavity frequency changing as a function of transmon state occupation. The reflection dip shifts toward the cavity resonance as the power is increased, approaching it asymptotically at very high power. The same shift of the cavity resonance is seem in the transmission spectrum, in agreement with recent experimental results [34].

Figure 4

(a) Plot of transmon field amplitude $|\langle b\rangle |$ as a function of drive detuning from the bare cavity ${\mathrm{\Delta }}_c$, plotted for various values of the drive amplitude $\epsilon$ for a cavity transmon system with parameters ${\mathrm{\Delta }}_{ct}/2\pi =2.5\mathrm{GHz}, g/2\pi =350\mathrm{MHz}, \chi /2\pi =-220\mathrm{MHz}, {\gamma }_c/2\pi =2\mathrm{MHz}$, and ${\gamma }_t/2\pi =0.1\mathrm{MHz}$. Values of $\epsilon /2\pi$ (from darkest to lightest) are 1, 18, 30, 40, 56, 75, and $100\mathrm{MHz}$. (b) $Q$ function of the transmon field with $\epsilon /2\pi =30\mathrm{MHz}$ and ${\mathrm{\Delta }}_c/2\pi =40\mathrm{MHz}$ and (c) $Q$ function of the transmon field with $\epsilon /2\pi =100\mathrm{MHz}$ and ${\mathrm{\Delta }}_c/2\pi =25\mathrm{MHz}$. These two points are marked with black circles in panel (a). We see that, in addition to the bifurcation of the cavity, the model predicts bistability for the qubit when the system is driven near to the cavity resonance. At higher powers the behavior of the system becomes very similar to that of the standard quantum Duffing oscillator with the characteristic dip due to coherent cancellation of the two steady states. The frequency at which the transmon (and cavity) bifurcation occurs shifts toward the bare cavity frequency as the power is increased, as seen in Fig. 3 for the cavity. A low power, the transmon field response splits into several peaks, corresponding to transmission peaks of the cavity at different transmon occupation numbers, but bistability can still be seen in the transmon $Q$ function.

Figure 5

Classical phase diagram of the parametrically driven Duffing oscillator in the $(\mathrm{\Delta },{\epsilon }_2)$ plane based on number of fixed points. The plane is dived into three regions by the lines ${\epsilon }_2^2-{\mathrm{\Delta }}^2={\gamma }_1^2$ and ${\epsilon }_2={\gamma }_1$, corresponding to the well-known threshold of the parametric oscillator. The system has up to five fixed points, of which up to three can be stable. The classical diagram is unaffected by the value of $U$, which only modifies the amplitude and therefore separation of the solutions. Figures 6 and 7 show the evolution of the steady state as the phase space is traversed parallel to the arrow, but for larger negative detunings.

Figure 6

Analytical $Q$-function plots for a parametrically driven Duffing oscillator with ${\gamma }_1/2\pi =1\mathrm{MHz}, U=5{\gamma }_1$, and $\mathrm{\Delta }=-12{\gamma }_1$ driven at four different drive strengths: (a) ${\epsilon }_2=2{\gamma }_1$, (b) ${\epsilon }_2=4.25{\gamma }_1$, (c) ${\epsilon }_2=4.75{\gamma }_1$, and (d) ${\epsilon }_2/2\pi =6.25\mathrm{MHz}$. Over this range of drives, the resonator state crosses two classical phase boundaries and we see the emergence of three stable points, followed by a return to two. The addition of a significant $U$ makes the threshold at ${\epsilon }_2={\gamma }_1$ appear much later than when $U=0$, as it is harder to add photons to the resonator, while the second boundary seems to appear much earlier than predicted, as there is extremely low probability of being in the $\alpha =0$ state in much of the phase.

Figure 7

Analytical $Q$-function plots for a parametrically driven Duffing oscillator with ${\gamma }_1/2\pi =1\mathrm{MHz}, U=5{\gamma }_1$, and $\mathrm{\Delta }=-8{\gamma }_1$ driven at four different drive strengths: (a) ${\epsilon }_2=2.75{\gamma }_1$, (b) ${\epsilon }_2=3.5{\gamma }_1$, (c) ${\epsilon }_2=3.75{\gamma }_1$, and (d) ${\epsilon }_2=5{\gamma }_1$. At a smaller detuning than in Fig. 6, the effect of a large $U$ is to prevent all of the stable states from being resolved. While the system crosses two classical phase boundaries, we see only a single transition, with the single fixed point becoming bistable at sufficiently high driving powers.

Figure 8

Minimum quadrature uncertainty for a nonideal degenerate parametric amplifier as a function of parametric drive ${\epsilon }_2$ for different values of the nonlinearity $U$ (given in MHz). Dissipation occurs at a rate ${\gamma }_1=1\mathrm{MHz}$. (a) For very small nonlinearities, the system behaves like an ideal degenerate parametric amplifier with a sharp threshold at ${\epsilon }_2={\gamma }_1$, where the minimum quadrature uncertainty goes to $1/2$ of that of the vacuum, corresponding to an uncertainty $\mathrm{\Delta }{X}_{min}=0.25$. As $U$ is increased, this threshold is moved initially slightly lower, and then to higher powers, while the maximum squeezing that can be achieved is reduced. Past the minimum there is a period where the steady state has bifurcated, but the uncertainty in some direction is still less than that of the vacuum. (b) For $U\gg {\gamma }_1, \mathrm{\Delta }{X}_{min}$ has a fixed minimum value at around 0.36 (dashed line) and the position of the minimum is at approximately $U/3$. For very large drives the state is made up of two well-separated coherent states, each with the same uncertainty as the vacuum.

Figure 9

$Q$ functions of nonlinear oscillator driven by a two-photon process and with different ratios of single-photon to two-photon loss rates. The system has $U/2\pi =0.1\mathrm{MHz}, {\gamma }_1/2\pi =1\mathrm{MHz}$, and the drive ${\epsilon }_2$ adjusted so that the average number of photons in the mode is 2.2. The other parameters are (a) ${\gamma }_1/{\gamma }_2=20, {\epsilon }_2/2\pi =1.15\mathrm{MHz}$; (b) ${\gamma }_1/{\gamma }_2=2, {\epsilon }_2/2\pi =2.28\mathrm{MHz}$; (c) ${\gamma }_1/{\gamma }_2=1, {\epsilon }_2/2\pi =3.4\mathrm{MHz}$; and (d) ${\gamma }_1/{\gamma }_2=.1, {\epsilon }_2/2\pi =23.5\mathrm{MHz}$. When single-photon loss is the dominant loss mechanism, the resonator nonlinearity causes distortions of the stable coherent states, reducing the fidelity of any stored state. Once ${\gamma }_2$ becomes comparable to ${\gamma }_1$, however, the distortions are reduced, with only a small bridge between the two states. Once the two-photon loss is much faster than the single-photon loss, the distortions are eliminated completely.

Figure 10

Plots of the number of excitations in the transmon field $\langle b^{†}b\rangle$ as a function of the detuning of the drive from the bare cavity ${\mathrm{\Delta }}_c$ and drive amplitude $\epsilon$ in (a) the resonant regime and (b) the dispersive regime. The system parameters are the same as in Figs. 1 and 2. Contours mark boundaries with one, three, and five excitations. There are fewer than three excitations over most of the parameter space, suggesting that the Duffing approximation of the transmon holds in these regions. Only the very bright peaks where the cavity frequency has almost returned to the bare cavity frequency do we see the number of excitations increase above five, suggesting that further terms in the cosine potential are required to describe these regions.