Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect

We present a theoretical study of a superconducting charge qubit dispersively coupled to a transmission line resonator. Starting from a master equation description of this coupled system and using a polaron transformation, we obtain an exact effective master equation for the qubit. We then use quantum trajectory theory to investigate the measurement of the qubit by continuous homodyne measurement of the resonator out field. Using the same polaron transformation, a stochastic master equation for the conditional state of the qubit is obtained. From this result, various definitions of the measurement time are studied. Furthermore, we find that in the limit of strong homodyne measurement, typical quantum trajectories for the qubit exhibit a crossover from diffusive to jumplike behavior. Finally, in the presence of Rabi drive on the qubit, the qubit dynamics is shown to exhibit quantum Zeno behavior.

Figure 1

(Color online) Schematic layout and lumped element version of the circuit QED implementation. A superconducting charge qubit (green) is fabricated inside a superconducting 1D transmission line resonator (blue). Here, we have taken the input [left-hand side (LHS)] and output [right-hand side (RHS)] capacitance of the resonator to be unequal. With the output capacitance much larger than the input one, a larger fraction of the photons in the resonator will escape on the output side. In this way, the signal to be measured is enhanced.

Figure 2

(Color online) In-phase $I=\mathrm{Re}[⟨a⟩]=⟨a+a^{†}⟩∕2$ and quadrature $Q=\mathrm{Im}[⟨a⟩]=⟨i{a}^{†}-ia⟩∕2$ component of the $Q$-function solution for the resonator state ${\mathrm{Tr}}_{\mathrm{qubit}}\left[{\varrho }_s\right]$. Here the parameters are $\kappa ∕2\pi =10\mathrm{MHz}$, $\chi ∕2\pi =5\mathrm{MHz}$, $ϵ∕2\pi =20\mathrm{MHz}$, and ${\Delta }_r=0$. The measurement drive is turned on at $t=50\mathrm{ns}$ and the initial state is $\mid 0⟩\otimes (\mid e⟩+\mid g⟩)∕\sqrt{2}$. For illustration purposes, ${\gamma }_1$ was taken to be zero. (a) $t=0$, (b) $t=75\mathrm{ns}$, (c) Steady-state solution.

Figure 3

(Color online) (a) Numerical solution of the three components of the Bloch vector as obtained from the full master equation: $x$ (red dashed line), $y$ (green dashed-dotted line), and $z$ (solid blue line). For this simulation, a measurement drive tuned at the resonator frequency $({\Delta }_r=0)$ and with envelope ${\epsilon }_{\mathrm{d}}\left(t\right)=A\tanh \left[\right(t-t_{\mathrm{on}})∕\sigma ]$ is turned on at $t_{\mathrm{on}}=20\mathrm{ns}$ [green dashed-dotted line in panel (b)]. The turn-on time is indicated with the vertical dashed line. (b) Purity $p$ (solid blue line) and the trace distance $d_{\mathrm{Tr}}$ (red dashed line) with respect to the effective model derived here. The latter shows the excellent agreement between the model and full numerical results. For these results, we have taken $A∕2\pi =5\mathrm{MHz}$, $\sigma =5\mathrm{ns}$, $T_1=7\mathrm{\mu }\mathrm{s}$, and $T_2=500\mathrm{ns}$. The other parameters are the same as in Fig. 2.

Figure 4

(Color online) ${\Gamma }_{\mathrm{d}}\left(t\right)$ (solid blue line) and ${\Gamma }_{\mathrm{m}}\left(t\right)∕2$ (dashed red line) for a measurement drive turned on at the time $t=50\mathrm{ns}$. The drive frequency is tuned to the bare resonator frequency $({\Delta }_r=0)$ and the amplitude is ${\epsilon }_{\mathrm{m}}∕2\pi =\sqrt{5}$. The rest of the parameters are the same as in Fig. 2. The inset shows the ratio between the total dephasing and the total measurement-induced dephasing. The vertical black dashed line is the turn on time for the measurement drive. The horizontal dashed line is the steady-state dephasing rate.

Figure 5

(Color online) (a) Numerical solution of the three components of the Block vector as obtained from the full homodyne SME: $x$, red dashed line; $y$, green dashed-dotted line; $z$, solid blue line. The measurement frequency and envelope is the same as in Fig. 3. We have assumed perfect efficiency $\eta =1$ and taken the LO phase to be $\varphi ={\theta }_{\beta }=\pi$, corresponding to maximal information gain at ${\Delta }_r=0$. (b) Conditional purity $p_J$ (full blue line) and measurement pulse amplitude (dotted-dashed green line). Also shown is one minus the trace distance (dashed red line), as calculated from the full homodyne SME and the effective SME. (c) One minus the trace distance minimized over the simulation time, $300\mathrm{ns}$, for a range of measurement amplitude and two values of $T_1$: $7\mathrm{\mu }\mathrm{s}$ (full red dots), $100\mathrm{ns}$ (empty blue squares). Here, the trace distance is averaged over 70 trajectories.

Figure 6

(Color online) The signal-to-noise ratio (SNR) at ${\Delta }_r=0$ and for a fixed number of photons in the resonator $n=2$. The efficiency was taken to be ideal $\eta =1$ and the relation rate ${\gamma }_1∕2\pi =1\mathrm{MHz}$. The three insets show the phase space representation of the resonator state for $\kappa =0.01\chi$, $\kappa =2\chi$, and $\kappa =5\chi$, respectively. The optimal SNR occurs at $\kappa =2\chi$.

Figure 7

(Color online) (a) Typical trajectories for the conditional variance (blue solid line) with ${\Gamma }_{\mathrm{d}}∕2\pi =5\mathrm{MHz}$ and $\eta =1$. Average conditional variance (red dashed line) found using Eq. (5.10) for the same parameters. Note that the calculated ensemble average over 1000 typical trajectories falls directly over the average conditional variance, and is therefore not explicitly shown. The vertical dash line shows the time at which the conditional variance crosses $1∕2$, $t_m=13.5\mathrm{ns}$. The green dashed-dotted line is the ensemble average conditional variance for $T_1=159\mathrm{ns}$ $({\gamma }_1∕2\pi =1\mathrm{MHz})$ and the rest of the parameters identical as above. For this value of ${\gamma }_1$, the SNR is 10. (b) Average fidelity of the measurement at estimating an unknown but pure initial state for ${\Gamma }_{\mathrm{d}}∕2\pi =5\mathrm{MHz}$, $\eta =1$, and ${\gamma }_1∕2\pi =0$ (red dashed line) or ${\gamma }_1∕2\pi =1\mathrm{MHz}$ (blue dots). The green solid line is a fit using Eq. (5.18) with fit parameters $A_{\mathrm{fit}}=0.846$ (horizontal dashed line) and ${\Gamma }_{\mathrm{fit}}∕2\pi =14.6\mathrm{MHz}$. The two vertical lines correspond to the times where the fidelity crosses $\mathrm{erf}(1∕\sqrt{2})$. For ${\gamma }_1∕2\pi =0$ this yields $t_m=15.9\mathrm{ns}$, while for ${\gamma }_1∕2\pi =1$ we find $t_m=18.5\mathrm{ns}$.

Figure 8

(Color online) Measurement time (full blue line) as extracted from the requirement that the fidelity reaches $\mathrm{erf}(1∕\sqrt{2})$ as a function of the SNR $(2{\Gamma }_{\mathrm{d}}∕{\gamma }_1)$. This is plotted for ${\gamma }_1∕2\pi =1\mathrm{MHz}$. The vertical dashed line corresponds to $\mathrm{SNR}=3.16$. This defines the minimal SNR for which we can define a measurement time. This is also presented in the inset which shows the long time fidelity as a function SNR, with the vertical dashed line being the SNR where the long time fidelity crosses $\mathrm{erf}(1∕\sqrt{2})$. The red dashed line in the main panel corresponds $t_m=1∕{\Gamma }_{\mathrm{ci}}$, the measurement time for ${\gamma }_1=0$. At low SNR, it always underestimate the measurement time.

Figure 9

(Color online) Time evolution of the $z$ component of the conditional state for a qubit initially in the excited state and $\chi ∕2\pi =5\mathrm{MHz}$, $\kappa ∕2\pi =10\mathrm{MHz}$, ${\Delta }_r=0$, $\eta =1$, and $T_1=7\mathrm{\mu }\mathrm{s}$. (a) Measurement drive strength of ${\epsilon }_{\mathrm{d}}∕2\pi =0\mathrm{MHz}$ (red dashed line) and $0.2\mathrm{MHz}$ (solid blue line). (b) ${\epsilon }_{\mathrm{d}}∕2\pi =0.9\mathrm{MHz}$ (solid blue line) and $20\mathrm{MHz}$ (dashed red line).

Figure 10

(Color online) Integrated signal $S\left(t\right)$ as a function of time. (a) Measurement drive strength ${\epsilon }_{\mathrm{d}}∕2\pi =0.9\mathrm{MHz}$ and integration steps $\delta t=48\mathrm{ns}$. (b) ${\epsilon }_{\mathrm{d}}∕2\pi =20\mathrm{MHz}$ and $\delta t=32\mathrm{ns}$. The vertical dashed line indicates the time where the conditional state crosses 0 in Fig. 9. The two horizontal dashed lines are the expected results corresponding to $z=\pm 1$ constant.

Figure 11

(Color online) Time evolution of the conditional state for the full homodyne SME for a qubit initially in the ground state: $x$, red dashed line; $y$, green dashed-dotted line; $z$, solid blue line. The system parameters are $\chi ∕2\pi =5\mathrm{MHz}$, $\kappa ∕2\pi =10\mathrm{MHz}$, ${\Delta }_r=0$, $\eta =1$, ${\gamma }_1={\gamma }_{\varphi }=0$, and ${\Omega }_{\mathrm{R}}∕2\pi =2.5\mathrm{MHz}$. (a) Measurement drive amplitude of ${\epsilon }_{\mathrm{d}}∕2\pi =0.9\mathrm{MHz}$. (b) ${\epsilon }_{\mathrm{d}}∕2\pi =20\mathrm{MHz}$. (c) Integrated signal with $\delta t=32\mathrm{ns}$ and for a measurement amplitude of ${\epsilon }_{\mathrm{d}}∕2\pi =20\mathrm{MHz}$.

Figure 12

(Color online) Average jump time as a function of the measurement drive strength ${\epsilon }_d$. The jump time is defined as the time for the system state to go from $z=\pm 0.95$ to $z=\mp 0.95$. The red dashed line corresponds to the Rabi period $\pi ∕{\Omega }_{\mathrm{R}}$ of $200\mathrm{ns}$ for a Rabi drive of ${\Omega }_{\mathrm{R}}∕2\pi =2.5\mathrm{MHz}$. The blue solid line corresponds to the expected mean jump time expected from the Zeno effect, ${\gamma }_{\mathrm{jump}}={\Omega }_R^2∕2({\gamma }_2+{\Gamma }_{\mathrm{d}})$. This agrees very well with the numerically simulations of Eq. (7.1) (green filled diamonds) where the mean jump time was extracted from a total simulation time of $16\mathrm{ms}$. The red open diamonds are the mean jump time as extracted from the full numerical homodyne SME for a total simulation time of about $15\mathrm{ms}$.