Adiabatic elimination of Gaussian subsystems from quantum dynamics under continuous measurement

An ever broader range of physical platforms provides the possibility to study and engineer quantum dynamics under continuous measurements. In many experimental arrangements the system of interest is monitored by means of an ancillary device, whose sole purpose is to transduce the signal from the system to the measurement apparatus. Here we present a method of adiabatic elimination when the transducer consists of an arbitrary number of bosonic modes with Gaussian dynamics while the measured object can be any quantum system. Crucially, our approach can cope with the highly relevant case of finite temperature of the transducer, which is not easily achieved with other methods. We show that this approach provides a significant improvement in the readout of superconducting qubits in circuit QED already for a few thermal excitations and makes it possible to adiabatically eliminate optomechanical transducers relevant for frequency conversion between microwave and optical fields.

Figure 1

(a) Schematic of the considered setup: A quantum system is monitored by coupling it to an ancillary system—the transducer—whose continuously emitted light field is detected. The most simple example of such a setup—a qubit coupled to a cavity mode with monitored output—is shown in (b). The transducer can also be a much more complex device, e.g., an optomechanical converter of microwave to optical photons.

Figure 2

Schematic illustrations of the setups we consider to illustrate the Gaussian adiabatic elimination method. In Secs. 3a and 3c, we analyze dynamics of a qubit coupled to a thermal cavity via dispersive and Jaynes-Cummings interaction, respectively, shown in (a). (b) Setup for entanglement generation by measurement as discussed in Sec. 3b. Here two qubits interact dispersively with the same cavity but not with each other. Starting from a suitable initial state and using the measurement record, it is then possible to postselect an entangled state of the two qubits. In Sec. 3d, we study a qubit coupled to a two-oscillator transducer, where the first oscillator couples to a thermal bath and the second oscillator is used for readout of the qubit state, as shown in (c).

Figure 3

Illustration of determination of the average trace distance. Starting from a single quantum trajectory [expectation values of the Pauli matrices shown in plots (a)–(c)], we calculate the trace distance between the full model, Eq. (37) (dotted blue line), and the result obtained by Gaussian adiabatic elimination, Eq. (39) (solid green line), or using the density operator expansion, Eq. (40) (dot-dashed red line). The resulting trace distances are shown in (d). We further average using 500 quantum trajectories in (e) to obtain an average trace distance. Using time averaging on this result, we further obtain a single figure of merit that determines the quality of the two approaches. For the results shown here with $\overline{n}=2, \chi =0.1\kappa , \varphi =\pi /2$ and initial qubit state $|{\psi }_0\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$, we have the average trace distance $D\approx 0.05$ for Gaussian adiabatic elimination and $D\approx 0.22$ for density operator expansion (cf. Fig. 4).

Figure 4

Average trace distance for Gaussian adiabatic elimination and density operator expansion as compared to the full model. In (a), we plot the trace distance as a function of the measurement phase with green squares showing results for the Gaussian adiabatic elimination and black stars for density operator expansion. The bottom panels show the trace distance versus thermal occupation number (b) and the overall measurement time (c) for two choices of phase—$\varphi =\pi /2$ (green squares for Gaussian adiabatic elimination, black stars for density operator expansion), and $\varphi =0$ (blue circles for Gaussian adiabatic elimination, red crosses for density operator expansion). The parameters used for the simulations are $\chi =0.1\kappa , \overline{n}=2$ [for (a) and (c)], measurement time $T_m=50{\kappa }^{-1}$ [(a) and (b)], and initial qubit state $|{\psi }_0\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. The Fock space of the cavity field for the full model is cut off at $N_{\max }=20$.

Figure 5

Example histograms of the integrated current at the beginning of the readout $[t=5{\kappa }^{-1}$ (a)] and at a later time $[t=100{\kappa }^{-1}$ (b)]. In panel (c), we plot the probability of the two qubits to be in the state $|00\rangle$ (green dot-dashed line), $|11\rangle$ (red dotted line), and $|{\mathrm{\Psi }}_{+}\rangle$ (blue solid line) for an example quantum trajectory. The simulations were run with the parameters $\chi =0.1\kappa , \overline{n}=0$, and 1000 trajectories were used for generating the histograms.

Figure 6

Logarithmic negativity (solid blue line) and success probability (dashed green line) versus the postselection threshold $\nu$ for $\overline{n}=0$ (a) and $\overline{n}=2$ (c). The measurement time is $T_m=100{\kappa }^{-1}$ in (a) and $T_m=250{\kappa }^{-1}$ in (c); moreover, in the insets, we plot the logarithmic negativity and success probability for $T_m=75{\kappa }^{-1}$ (left) and $T_m=150{\kappa }^{-1}$ (right) for both (a) and (c). In addition, in panels (b) and (d), histograms of the integrated currents corresponding to the results in (a) and (c) are shown. We use the coupling $\chi =0.1\kappa$ and average over 1000 quantum trajectories.

Figure 7

(a) Average trace distance for the Gaussian adiabatic elimination (green squares) and density operator expansion (black stars) as a function of the interaction phase $\varphi$. In the bottom panels, we plot the average trace distance versus thermal occupation (b) and measurement time (c) for the choice of phase $\varphi =0$ (blue circles showing Gaussian adiabatic elimination and red crosses for density operator expansion) and $\varphi =\pi /2$ (Gaussian adiabatic elimination shown in green squares, density operator expansion in black stars). The parameters used to run the simulations are $g=0.1\kappa , \overline{n}=2$ [for panels (a) and (c)], measurement time $T_m=50{\kappa }^{-1}$ [for (a) and (b)], and initial qubit state $|{\psi }_0\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. The cavity field for the full model has been cut off at the Fock number $N_{\max }=20$.

Figure 8

(a) Average trace distance as a function of thermal occupation for three regimes: weak coupling ($g=0.2\kappa$, green squares), intermediate coupling ($g=0.5\kappa$, blue circles), and strong coupling ($g=\kappa$, black stars). In (b), we show the average trace distance versus time for $\overline{n}=2, g=\kappa$. Other parameters used in the simulations are $\chi =0.2\kappa , \omega =5\kappa , \gamma =0.1\kappa , T_m=50{\kappa }^{-1}$ for (a), initial qubit state $|{\psi }_0\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$, and we averaged over 100 quantum trajectories.