Continuous joint measurement and entanglement of qubits in remote cavities

We present a first-principles theoretical analysis of the entanglement of two superconducting qubits in spatially separated microwave cavities by a sequential (cascaded) probe of the two cavities with a coherent mode, that provides a full characterization of both the continuous measurement induced dynamics and the entanglement generation. We use the SLH formalism to derive the full quantum master equation for the coupled qubits and cavities system, within the rotating wave and dispersive approximations, and conditioned equations for the cavity fields. We then develop effective stochastic master equations for the dynamics of the qubit system in both a polaronic reference frame and a reduced representation within the laboratory frame. We compare simulations with and analyze tradeoffs between these two representations, including the onset of a non-Markovian regime for simulations in the reduced representation. We provide conditions for ensuring persistence of entanglement and show that using shaped pulses enables these conditions to be met at all times under general experimental conditions. The resulting entanglement is shown to be robust with respect to measurement imperfections and loss channels. We also study the effects of qubit driving and relaxation dynamics during a weak measurement, as a prelude to modeling measurement-based feedback control in this cascaded system.

Figure 1

Sequential probe of two spatially separated cavities, each containing a qubit coupled dispersively with strength ${\chi }_i$ to its fundamental cavity mode. The cavities are asymmetric, with the $\kappa$ ports being more transmissive than the $\gamma$ ports. The beam splitter between the cavities models the losses induced by the circulator that enforces one-way field propagation between cavities. We allow for arbitrary coherent-state drives ${\overline{A}}_d\left(t\right)$ and ${\overline{B}}_d\left(t\right)$ into the weakly coupled ports of both cavities. The output field that results after sequential reflection of the probe field from both cavities, $z\left(t\right)$, is measured by a homodyne detector with efficiency ${\eta }_{\mathrm{m}}$ and at a phase $\varphi$ with respect to the probe $\overline{\epsilon }\left(t\right)$.

Figure 2

SLH network decomposition of the apparatus in Fig. 1. The SLH triples $(S,L,H)$ for each block are specified in the main text. (a) shows an SLH block representation superimposed on the experimental apparatus, and (b) shows the SLH block diagram redrawn more conventionally with inputs on the left and outputs on the right. The explicit forms for the resulting $S$ (scattering), $H$ (self-energy), and $L$ (coupling) matrices for the overall model are given in the Appendix. All inputs are in the vacuum state [indicated by dotted lines in (a)] and the single monitored output is $z\left(t\right)$, the field reflected from the output port of cavity 2. All other outputs are not monitored: this is indicated in (a) and (b) by their termination.

Figure 3

Distribution of qubit state populations (200 000 trajectory simulations) as a function of normalized homodyne voltage and measurement pulse width, for simulation parameters corresponding to slightly different qubits. See Sec. 2e for the parameter values. The initial state is given in Eq. (26). The $z$ axis shows the total number of trajectories that result in the state $|00\rangle$ (olive green, far left), $|01\rangle$ (blue, near left), $|10\rangle$ (turquoise, near right), or $|11\rangle$ (yellow, far right).

Figure 4

Bottom panel: Sample trajectory of $|{\rho }_{0110}\left(t\right)|$, the absolute value of an off-diagonal qubit density matrix element simulated using the three dynamical equations derived in Secs. 2b [polaron, Eq. (16)], 2c [laboratory frame compensated polaron, Eqs. (18, 19, 20, 21)], and 2d [reduced equation of motion, Eq. (25)]. We use the ideal setting with identical cavities and lossless transmission. All parameters are described in the main text. Top panel: measurement pulse $\overline{\epsilon }\left(t\right)$ for this trajectory.

Figure 5

Top panel: Coherence between the $\left|01\right.〉$ and $\left|10\right.〉$ states. The blue (lower), purple (middle), and red (upper) lines correspond to ${\kappa }_1=\{1,5,17\}\mathrm{MHz}$, respectively, with $\frac{{\kappa }_2}{2\pi }=\frac{{\kappa }_1}{2\pi }+2.5$ in all cases. Bottom panel: Conditional output field amplitudes (in arbitrary units) when the qubits are in states $\left|01\right.〉$ ($\mathrm{Re}\left\{{\tilde{B}}^{\left(01\right)}\right\}$, solid) and $\left|10\right.〉$ ($\mathrm{Re}\left\{{\tilde{B}}^{\left(10\right)}\right\}$, dashed), for these three values of ${\kappa }_1$, showing suppression of the transients as the cavity decay rate is increased. All other parameters are described in the main text.

Figure 6

Example of a effective drive pulse $A_d\left(t\right)$ and compensation pulses $B_d\left(t\right)$ for two cavities with identical frequencies but different transmittivities, $\frac{{\kappa }_1}{2\pi }=3.9$ MHz and $\frac{{\kappa }_2}{2\pi }=3.5$ MHz. All other parameters are the same as for Fig. 3 (see Sec. 2e). Blue solid line: pulse $A_d\left(t\right)$. Orange short dashed line: compensation pulse $B_d\left(t\right)$ when the adiabatic approximation is used [Eq. (31)]. Red long dashed line: exact solution given by Eq. (37).

Figure 7

Distribution of qubit state populations (over 60 000 trajectory simulations) as a function of normalized homodyne voltage and measurement pulse width. The initial state is given in Eq. (26). Black (low voltage) represents $|00\rangle$ population, blue (high voltage) $|11\rangle$, red (upper, medium voltage) $\frac{1}{\sqrt{2}}\left(\right|01\rangle +|10\rangle )$, and green (lower, medium voltage) represents $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$. The top panel plots relative frequencies of each of the state populations, while the bottom panel plots normalized populations conditioned on the measurement voltage value from the $x$ axis. The bottom plot is essentially a slice of the top panel taken at a measurement pulse duration of $1\mu \mathrm{s}$. For this measurement pulse width, the concurrence achieved is $0.52$. Simulation parameters are specified in Sec. 4.

Figure 8

Maximum concurrence vs measurement efficiency (${\eta }_{\mathrm{m}}$) and photon loss in the channel between the cavities ($1-{\eta }_l$, in dB). Simulation parameters are specified in Sec. 4. Entanglement maximum is calculated by propagating the simulation for different measurement pulse widths and selecting the maximal concurrence achieved for the given parameters averaged over all trajectories.