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Quantum Electronics and Laser Science Conference

Weak values, quantum trajectories, and the Stony-Brook cavity QED experiment

Howard M. Wiseman

Published Jun 25, 2003• 0 citations
Publisher

Abstract

Summary form only given. I apply quantum trajectory theory to the recent Stony-Brook cavity QED experiment demonstrating waveparticle duality. I show that the left half of the fractional correlation function he measured in that experiment can be recast as a dynamical weak value. The system consists of a damped cavity, very weakly driven, through which a beam of resonant atoms pass. The weak measurement is a homodyne measurement, the initial system state is the steady state, and the ensemble is post-selected on the detection of a photon from the cavity.

DOI
10.1109/qels.2002.1031013

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•Jan 1, 2003•0 citations
Summary form only given. I apply quantum trajectory theory to the recent Stony-Brook cavity QED experiment demonstrating waveparticle duality. I show that the left half of the fractional correlation function he measured in that experiment can be recast as a dynamical weak value. The system consists of a damped cavity, very weakly driven, through which a beam of resonant atoms pass. The weak measurement is a homodyne measurement, the initial system state is the steady state, and the ensemble is post-selected on the detection of a photon from the cavity.
Weak values as introduced by Aharonov, Albert, and Vaidman (AAV) are ensemble-average values for the results of weak measurements. They are interesting when the ensemble is preselected on a particular initial state and postselected on a particular final measurement result. It is shown that weak values arise naturally in quantum optics, as weak measurements occur whenever an open system is monitored (as by a photodetector). The quantum-trajectory theory is used to derive a generalization of AAV's formula to include (a) mixed initial conditions, (b) nonunitary evolution, (c) a generalized (nonprojective) final measurement, and (d) a non-back-action-evading weak measurement. This theory is applied to the recent cavity-QED experiment demonstrating wave particle duality [G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, Phys. Rev. Lett. 85, 3149 (2000)]. It is shown that the ``fractional-order'' correlation function measured in that experiment can be recast as a weak value in a form as simple as that introduced by AAV.
We use the quantum trajectory (stochastic wave function) approach to simulate the dynamics of a cavity QED system. The system consists of one atom interacting on resonance with one mode of a coherently driven optical cavity. We simulate a variety of different measurements on this system. The simulation of direct photoelectric detection produces trajectories that illustrate the collapse of the wave function in delayed photon coincidence measurements. It also allows us to calculate waiting-time distributions, pho­toelectron counting distributions, and the mean transmitted intensity as a function of the excitation frequency and field strength. A simulation with frequency filtering of the stochastic wave fuction gives results for optical spectra. From a simulation of homodyne detection we obtain trajectories showing the fluctuations in the quadrature phase amplitudes of the intracavity field and the atomic dipole. These simulations are used to calculate spectra of squeezing. We show that when the cavity is excited near one of the "vacuum" Rabi resonances, the composite atom-cavity system behaves like a two-state system. Under these conditions the optical spectrum shows Mollow sidebands caused by a dynamic Stark shift of the Jaynes-Cummings eigenstates. Thus, in this measurement we are observing the "dressing" of the (one-quantum) dressed states.
Ab initio quantum trajectory simulations of a cavity QED system comprising an atomic beam traversing a coherently driven standing-wave cavity are carried out. The intensity correlation function in transmission is computed and compared with the experimental measurements of Rempe et al. [Phys. Rev. Lett. 67, 1727 (1991)] and Foster et al. [Phys. Rev. A 61, 053821 (2000)]. It is shown that atomic beam density fluctuations induced by the motion of the atoms can account for the reported disagreement of the experimental results with theory (by an overall scale factor of 2 to 4). Moderate misalignments of the atomic beam produce large intracavity photon number fluctuations which significantly degrade the quantum correlations. One parameter fits to the experimental data are made in the weak-field limit with the adjustable parameter being the atomic beam tilt. Departures of the experimental conditions from the weak-field limit are discussed.
Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state as it heads towards one of the eigenstates of the measured operator. This evolution can be understood as a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories, consisting of noisy fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. The rate of evolution is specific to each system-apparatus pair, and the Born rule constraint requires the magnitudes of the noise and the attraction to be precisely related. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum theory.
We investigate the quantum fluctuations of a single atom in a weakly driven cavity, with an intracavity optical lattice. The weak driving field is on resonance with the atoms and the cavity, and is the second-harmonic of the lattice beam. In this special case we can find eigenstates of the Hamiltonian in terms of Mathieu functions.We present analytic results for the second order intensity correlation function g(2)(τ)g^{(2)}(τ)g(2)(τ) and the intensity-field correlation function hθ(τ)h_θ(τ)hθ​(τ), for both transmitted and fluorescent light for weak driving fields. We find that the coupling of the center of mass motion to the intracavity field mode can be deleterious to nonclassical effects in photon statistics; less so for the intensity-field correlations, and compare the use of trapped atoms in a cavity to atomic beams.
We study the time evolution of the quantum field inside a cavity coupled to a beam of two-level atoms of temperature T, given that each atom, after having crossed the cavity, interacts with a classical field and finally with a detector measuring its state. It is shown numerically that, if the atom-quantum field coupling is weak and % is large, for any given realization of the measurements, any initial state of the cavity field localizes after some time into a squeezed state. The center a of the squeezed state moves randomly on a line in the complex plane, but the squeezing parameters r and Φ show very little fluctuation. Their mean values r and Φ are independent of the realization, the initial state, and the atom-field coupling constant. The time evolution of r and Φ is computed in the limit of weak atom-field coupling and large. It is found that r increases with T; i.e., the squeezing is enhanced by increasing the temperature of the atomic beam. The limitation on the degree of squeezing that can be reached is discussed.