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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Localization of an atom by homodyne measurement

A. M. Herkommer, H. J. Carmichael, Wolfgang P. Schleich

Published Feb 1, 1996• 47 citations
Publisher

Abstract

We study a continuous homodyne measurement of the field transmitted from an optical cavity. In particular, we investigate the back-action of this measurement onto an atom whose centre-of-mass motion is entangled with the cavity field. Using the method of quantum trajectories we calculate analytically and numerically the time evolution of the entangled quantum state, and demonstrate the localization of the atom relative to the nodes of the cavity field. We compare the quantum trajectory formalism of the continuous homodyne measurement to a projection onto quadrature eigenstates of the field and show that in the long-time limit both descriptions are identical.

DOI
10.1088/1355-5111/8/1/014

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•Jan 1, 2003•0 citations
We study a continuous homodyne measurement of the field transmitted from an optical cavity. In particular, we investigate the back-action of this measurement onto an atom whose centre-of-mass motion is entangled with the cavity field. Using the method of quantum trajectories we calculate analytically and numerically the time evolution of the entangled quantum state, and demonstrate the localization of the atom relative to the nodes of the cavity field. We compare the quantum trajectory formalism of the continuous homodyne measurement to a projection onto quadrature eigenstates of the field and show that in the long-time limit both descriptions are identical.
The quantum effects on a cavity mode of the electromagnetic field caused by measuring one of its quadrature components is analyzed. We consider three measurement schemes: an intracavity quantum-nondemolition coupling to another mode, simple homodyne detection, and balanced homodyne detection. It is shown that, for suitable initial conditions, the first scheme has an effect which approaches that of a projective collapse of the state vector for long measurement times. However, the two homodyne schemes (which are shown to be equivalent for large local-oscillator amplitudes) do not approximate a projective measurement in any limit. In particular, it is shown that homodyne measurement cannot produce a squeezed state from a classical initial state. All three schemes are analyzed in terms of ``quantum trajectories'' which link measurement theory with stochastic quantum-jump processes.
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.
We are interested in dynamics of quantum many-body systems under continuous observation, and its physical realizations involving cold atoms in lattices. In the present work we focus on continuous measurement of atomic currents in lattice models, including the Hubbard model. We describe a Cavity QED setup, where measurement of a homodyne current provides a faithful representation of the atomic current as a function of time. We employ the quantum optical description in terms of a diffusive stochastic Schr\"odinger equation to follow the time evolution of the atomic system conditional to observing a given homodyne current trajectory, thus accounting for the competition between the Hamiltonian evolution and measurement back action. As an illustration, we discuss minimal models of atomic dynamics and continuous current measurement on rings with synthetic gauge fields, involving both real space and synthetic dimension lattices (represented by internal atomic states). Finally, by ``not reading'' the current measurements the time evolution of the atomic system is governed by a master equation, where---depending on the microscopic details of our CQED setups---we effectively engineer a current coupling of our system to a quantum reservoir. This provides interesting scenarios of dissipative dynamics generating ``dark'' pure quantum many-body states.
We present a detailed theoretical description of a recently proposed atomic scanning microscope in a cavity QED setup [D. Yang et al., Phys. Rev. Lett. 120, 133601 (2018)]. The microscope continuously observes atomic densities with optical subwavelength resolution in a nondestructive way. The superresolution is achieved by engineering an internal atomic dark state with a sharp spatial variation of population of a ground level dispersively coupled to the cavity field. Thus, the atomic position encoded in the internal state is revealed as a phase shift of the light reflected from the cavity in a homodyne experiment. Our theoretical description of the microscope operation is based on the stochastic master equation describing the conditional time evolution of the atomic system under continuous observation as a competition between dynamics induced by the Hamiltonian of the system, decoherence effects due to atomic spontaneous decay, and the measurement backaction. Within our approach we relate the observed homodyne current with a local atomic density and discuss the emergence of a quantum nondemolition measurement regime allowing continuous observation of spatial densities of quantum motional eigenstates without measurement backaction in a single experimental run.
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.