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arXiv:1512.02307 · Nature Communications

Mapping quantum state dynamics in spontaneous emission

Mahdi Naghiloo, Neda Foroozani, D. Tan, Arian Jadbabaie, Kater W. Murch

Published May 11, 2016• 42 citations
arXivHTMLPDF

Abstract

The evolution of a quantum state undergoing radiative decay depends on how its emission is detected. If the emission is detected in the form of energy quanta, the evolution is characterized by a quantum jump to a lower energy state. In contrast, detection of the wave nature of the emitted radiation leads to different dynamics. Here, we investigate the diffusive dynamics of a superconducting artificial atom under continuous homodyne detection of its spontaneous emission. Using quantum state tomography, we characterize the correlation between the detected homodyne signal and the emitter's state, and map out the conditional back-action of homodyne measurement. By tracking the diffusive quantum trajectories of the state as it decays, we characterize selective stochastic excitation induced by the choice of measurement basis. Our results demonstrate dramatic differences from the quantum jump evolution associated with photodetection and highlight how continuous field detection can be harnessed to control quantum evolution.

DOI
10.1038/ncomms11527

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The evolution of a quantum state undergoing radiative decay depends on how its emission is detected. If the emission is detected in the form of energy quanta, the evolution is characterized by a quantum jump to a lower energy state. In contrast, detection of the wave nature of the emitted radiation leads to different dynamics. Here, we investigate the diffusive dynamics of a superconducting artificial atom under continuous homodyne detection of its spontaneous emission. Using quantum state tomography, we characterize the correlation between the detected homodyne signal and the emitter's state, and map out the conditional back-action of homodyne measurement. By tracking the diffusive quantum trajectories of the state as it decays, we characterize selective stochastic excitation induced by the choice of measurement basis. Our results demonstrate dramatic differences from the quantum jump evolution associated with photodetection and highlight how continuous field detection can be harnessed to control quantum evolution.
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.
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We use homodyne detection to monitor the radiative decay of a superconducting qubit. According to the classical theory of conditional probabilities, the excited-state population differs from an exponential decay law if it is conditioned upon a later projective qubit measurement. Quantum trajectory theory accounts for the expectation values of general observables, and we use experimental data to show how a homodyne detection signal is conditioned upon both the initial state and the finally projected state of a decaying qubit. We observe, in particular, how anomalous weak values occur in continuous weak measurement for certain pre- and postselected states. Subject to homodyne detection, the density matrix evolves in a stochastic manner, but it is restricted to a specific surface in the Bloch sphere. We show that a similar restriction applies to the information associated with the postselection, and thus bounds the predictions of the theory.
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 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.
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.