We consider protocols to generate quantum entanglement between two remote qubits, through joint time-continuous detection of their spontaneous emission. We demonstrate that schemes based on homodyne detection, leading to diffusive quantum trajectories, lead to identical average entanglement yield as comparable photodetection strategies; this is despite substantial differences in the two-qubit state dynamics between these schemes, which we explore in detail. The ability to use different measurements to achieve the same ends may be of practical significance; the less-well-known diffusive scheme appears far more feasible on superconducting qubit platforms in the near term.
We consider protocols to generate quantum entanglement between two remote qubits, through joint time-continuous detection of their spontaneous emission. We demonstrate that schemes based on homodyne detection, leading to diffusive quantum trajectories, lead to identical average entanglement yield as comparable photodetection strategies; this is despite substantial differences in the two-qubit state dynamics between these schemes, which we explore in detail. The ability to use different measurements to achieve the same ends may be of practical significance; the less-well-known diffusive scheme appears far more feasible on superconducting qubit platforms in the near term.
We discuss recent developments in measurement protocols that generate quantum entanglement between two remote qubits, focusing on the theory of joint continuous detection of their spontaneous emission. We consider a device geometry similar to that used in well-known Bell state measurements, which we analyze using a conceptually transparent model of stochastic quantum trajectories; we use this to review photodetection, the most straightforward case, and then generalize to the diffusive trajectories from homodyne and heterodyne detection as well. Such quadrature measurement schemes are a realistic two-qubit extension of existing circuit QED experiments, which obtain quantum trajectories by homodyning or heterodyning a superconducting qubit’s spontaneous emission, or an adaptation of existing optical measurement schemes to obtain jump trajectories from emitters. We mention key results, presented from within a single theoretical framework, and draw connections to concepts in the wider literature on entanglement generation by measurement (such as path information erasure and entanglement swapping). The photon which-path information acquisition, and therefore the two-qubit entanglement yield, is tunable under the homodyne detection scheme we discuss, at best generating equivalent average entanglement dynamics as in the comparable photodetection case. In addition to deriving this known equivalence, we extend past analyses in our characterization of the measurement dynamics: we include derivations of bounds on the fastest possible evolution toward a Bell state under joint homodyne measurement dynamics and characterize the maximal entanglement yield possible using inefficient (lossy) measurements.
Continuous weak measurements of quantum systems are of great relevance in quantum foundations and applications. They can be achieved by probing the quantum system of interest by repeatedly measuring an auxiliary system weakly coupled to it. Here we study two qubits coupled in different points to a common one-dimensional electromagnetic field and simultaneously monitored in the right- and left-propagating output channels by homodyne detection. Using a collision-model description, we derive an analytical Stochastic Master Equation (SME) governing the resulting diffusive quantum trajectories, including the interference between the two measurement channels. For nonlinear functions of the quantum state, such as entropies, averages over quantum trajectories generally differ from the corresponding quantities evaluated on the unconditional state. Through this mechanism, we show that continuous monitoring generates entanglement, absent in the unconditional dynamics, and enhances quantum magic in the qubit pair during the decay. Both resources can be tuned through the optical phase accumulated between the qubits and the phases of homodyne local oscillators. Our results establish continuous homodyne monitoring of multiple emitters as a tunable mechanism for generating quantum resources.
Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local $\ensuremath{\pi}$ pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e., using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e., using diffusive quantum trajectories) [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. Our method provides a pathway towards generating and protecting entangled states, complementing others in the literature with similar aims, such that two-qubit entanglement may be used in various applications on demand.
We review the continuous monitoring of a qubit through its spontaneous emission, at an introductory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring and illustrate its equivalence to the stochastic master equation formalism throughout. Special emphasis is given to homodyne (phase-sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results.
Solid-state quantum emitters, such as quantum dots, color centers, rare-earth dopants, and organic molecules, offer qubit systems that integrate well with chip-scale photonic and electronic devices. To fully harness their potential for quantum applications requires the generation of entanglement between two remote qubits with high fidelity and efficiency. In this article, we compare the performance of three common photon-mediated entanglement schemes under realistic noise for solid-state quantum emitters, including optical dephasing and spectral diffusion. We identify the optimal scheme across different noise regimes and calculate the measurement parameters needed to achieve the highest entanglement fidelity at a given rate. Additionally, we explore the effects of temporal and spectral filtering in enhancing entanglement fidelity. Our findings provide practical guidelines for selecting optimal entanglement schemes and outline the measurement strategies for achieving better entanglement fidelity.
In this paper we consider a scheme for achieving long-time sustainable Bell states of two distant qubits mediated by a one-dimensional waveguide whose outputs are subjected to time-continuous photon counting or homodyne detection. In both of the detection cases, it is shown that different Bell states can be obtained, for different initial states, in the long-time regime. In particular, we find that, in the case of photon counting, a cyclic jump among Bell states can be formed once the first photon is registered. While in the homodyne-detection case we further reveal that any steady Bell state can be achieved independent of detection efficiency, with a probability of 50% . The underlying physical reason for this is also analyzed. Our scheme is advantageous over previous studies in which only transient or intermittent Bell states can be generated. The long-time Bell states of distant qubits can be used for constructing quantum networks.