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arXiv:1705.04287 · Physical Review A

Homodyne monitoring of postselected decay

D. Tan, Neda Foroozani, Mahdi Naghiloo, Alexander Holm Kiilerich, Klaus Mølmer, Kater W. Murch

Published Aug 3, 2017• 25 citations
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Abstract

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.

DOI
10.1103/physreva.96.022104

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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.
In quantum mechanics, the measurement outcome of an observable in a quantum system is intrinsically random, yielding a probability distribution. The state of the quantum system can be described by a density matrix (t), which depends on the information accumulated until time t, and represents our knowledge about the system. The density matrix (t) gives probabilities for the outcomes of measurements at time t. Further probing of the quantum system allows us to refine our prediction in hindsight. In this thesis, we experimentally examine a quantum smoothing theory in a superconducting qubit by introducing an auxiliary matrix E(t) which is conditioned on information obtained from time t to a final time T. With the complete information before and after time t, the pair of matrices [(t), E(t)] can be used to make smoothed predictions for the measurement outcome at time t. We apply the quantum smoothing theory in the case of continuous weak measurement unveiling the retrodicted quantum trajectories and weak values. In the case of strong projective measurement, while the density matrix (t) with only diagonal elements in a given basis |n> may be treated as a classical mixture, we demonstrate a failure of this classical mixture description in determining the smoothed probabilities for the measurement outcome at time t with both diagonal (t) and diagonal E(t). We study the correlations between quantum states and weak measurement signals and examine aspects of the time symmetry of continuous quantum measurement. We also extend our study of quantum smoothing theory to the case of resonance fluorescence of a superconducting qubit with homodyne measurement and observe some interesting effects such as the modification of the excited state probabilities, weak values, and evolution of the predicted and retrodicted trajectories.
The evolution of two-level quantum systems (qubits) in the presence of continuous-time weak measurements can be described by stochastic differential equations, which is dominated by the Hamiltonian and decoherence channels. In more concrete terms, the conventional density matrix evolves according to the stochastic master equation in the Schrödinger picture. Particularly, for a monitored qubit, the deterministic submanifolds concerning the dynamical evolution of the density matrix has been previously obtained. As a complement, We have recently developed deterministic submanifolds for the backward-evolving quantum state for a monitored qubit. In this paper, in order to better understand the behavior of a monitored qubit, by using the Bloch sphere representation and the properties of Pauli matrices, we provide a closed-form expression for probability distribution over the deterministic submanifolds of the backward evolving dynamics in different scenarios. We further analyze the backward-evolving trajectories of the quantum state by taking into account the analytic solutions and the corresponding submanifolds under heterodyne/homodyne detection associated with non-demolition or fluorescence measurements.
A qubit can relax by fluorescence, which prompts the release of a photon into its electromagnetic environment.By counting the emitted photons, discrete quantum jumps of the qubit state can be observed.The succession of states occupied by the qubit in a single experiment, its quantum trajectory, depends in fact on the kind of detector.How are the quantum trajectories modified if one measures continuously the amplitude of the fluorescence field instead?Using a superconducting parametric amplifier, we have performed heterodyne detection of the fluorescence of a superconducting qubit.For each realization of the measurement record, we can reconstruct a different quantum trajectory for the qubit.The observed evolution obeys quantum state diffusion, which is characteristic of quantum measurements subject to zero point fluctuations.Independent projective measurements of the qubit at various times provide a quantitative validation of the reconstructed trajectories.By exploring the statistics of quantum trajectories, we demonstrate that the qubit states span a deterministic surface in the Bloch sphere at each time in the evolution.Additionally, we show that when monitoring fluorescence, coherent superpositions are generated during the decay from excited to ground state.Counterintuitively, measuring light emitted during relaxation can give rise to trajectories with increased excitation probability.
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.
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.
In a Bayesian analysis, the likelihood that specific candidate parameters govern the evolution of a quantum system are conditioned on the outcome of measurements which, in turn, cause measurement backaction on the state of the system [Tsang, Phys. Rev. Lett. 108, 170502 (2012)]. Specializing to the distinction of two candidate hypotheses, we study the achievements of continuous monitoring of the radiation emitted by a quantum system followed by an optimal projective measurement on its conditioned final state. Our study of the radiative decay of a driven two-level system shows an intricate interplay between the maximum information available from photon counting and homodyne detection and the final projective measurement on the emitter. We compare the results with theory predicting a lower bound for the probability to assign a wrong hypothesis by any combined measurement on the system and its radiative environment.