Continuous joint measurement and entanglement of qubits in remote cavities

Abstract

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.

Quantum Bayesian approach to circuit QED measurement with moderate bandwidth

“…Such a generalization is useful to describe the process of entanglement of superconducting qubits by measurement [51,52,77,78]. For N qubits the state of the system can be described in the way similar to Eq.…”

Deterministic generation of remote entanglement with active quantum feedback

“…As analyzed in [34], it is possible using a dispersive homodyne readout to implement a half-parity measurement in the superconducting circuit architecture. In reference [4], authors succeeded in applying this measurement to two superconducting transmon qubits separated by over a meter.…”

Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits

“…By reducing the amplitude of the coherent state used to measure the system, we can engineer an entangling measurement with a characteristic measurement time ranging from several hundred nanoseconds to several microseconds: Critically, these time scales are easily resolvable experimentally. The dynamics or the trajectory of the system state can be obtained via the full master equations [46,47], using a two-cavity polaron transformation to account for the cavity degree of freedom, giving the stochastic master equation for the qubit trajectories. Alternatively, in a limit of large cavity decay rate κ ≫ jχj, the qubit evolution can be continuously tracked via the quantum Bayesian approach [2,3], inferring the current states of the system from the measurement readouts and how likely they are to occur.…”

“…Analog feedback control can naturally be applied in the weak continuousmeasurement regimes [42][43][44], and digital feedbackgenerated entanglement has already been demonstrated [45]. Joint measurement is uniquely useful as a means to generate entanglement between remote qubits [12,[46][47][48][49][50], for which no local coupling exists and therefore no unitary means of generating entanglement are available.…”