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Cavity QED: a quantum trajectory point of view

Dominique Spehner, Miguel Orszag

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

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.

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Temperature-enhanced squeezing in cavity QED

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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 found that, if the coupling between the atoms and the quantum field is weak and is not too small, for any given realization of the measurements, an arbitrary initial state of the field localizes after some time into squeezed states. The centre α of the squeezed state moves randomly in time in the complex plane, but the squeezing amplitude r and phase ϕ show very small fluctuations. Their mean values and are independent of the random results of the measurements, of the initial state and of the atom–field coupling constant λ. The time evolution of r and ϕ is determined analytically by deriving and solving the quantum state diffusion equation describing the field dynamics in the limit of small λ, keeping finite. It is shown that increases with T , i.e., the squeezing is enhanced by increasing the temperature of the atomic beam.

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Squeezing with coupled pendulums: two-level atoms in a cavity

Mark G. Raizen, Luis A. Orozco, Min Xiao +2 more

A theoretical and experimental investigation of the generation of squeezed states of light has been carried out for a system of two-level atoms inside a high-finesse cavity. The regime that we consider is such that the coupling of atoms to the cavity mode produces a splitting in the normal mode structure of the atom-field system (the so-called vacuum-field Rabi splitting). We have employed this splitting to generate squeezing with observed reductions in photocurrent noise of 30 % below the shot-noise level. A degree of squeezing of ~50% is inferred for the field state in the absence of propagation and detection losses. Squeezing has been observed in two different cavity geometries with quite different coupling coefficients. In each case, the frequency of best squeezing corresponds to that of the normal mode splitting g N (g= single atom coupling coefficent and N = number of atoms) in good agreement with our theoretical predictions. Regions of parameter space with very large degrees of squeezing (>20×) are identified. Preliminary extensions of the theory into a transient regime with (cavity lifetime) ≪ (interaction time) ≪ (atomic lifetime) are described.

Annual Meeting Optical Society of America•Jan 1, 1987•0 citations
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.
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 found that, if the coupling between the atoms and the quantum field is weak and is not too small, for any given realization of the measurements, an arbitrary initial state of the field localizes after some time into squeezed states. The centre α of the squeezed state moves randomly in time in the complex plane, but the squeezing amplitude r and phase ϕ show very small fluctuations. Their mean values and are independent of the random results of the measurements, of the initial state and of the atom–field coupling constant λ. The time evolution of r and ϕ is determined analytically by deriving and solving the quantum state diffusion equation describing the field dynamics in the limit of small λ, keeping finite. It is shown that increases with T , i.e., the squeezing is enhanced by increasing the temperature of the atomic beam.
We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field ℰ and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state |ψ(t)〉 of the cavity field. For weak atom-field coupling, the evolution of |ψ(t)〉 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field ℰ. For ℰ=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector |ψ(t)〉〈ψ(t)| over all results of the measurements, is independent of ℰ and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large ℰ, an arbitrary initial field state |ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states |α,re2iφ〉, moving in the α-complex plane but with almost constant squeezing parameters r and φ. The values of r and φ are determined analytically. On the other hand, for ℰ=0, the dynamics transforms the initial state into Fock states |n〉 with fluctuating numbers of photons n, as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.
Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, ${\ensuremath{\alpha}}^{2},$ initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing $\ensuremath{\alpha},$ even though the field more closely resembles a classical field for which no spin squeezing could be achieved.
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.
We present the conditional time evolution of the electromagnetic field produced by a cavity QED system in the strongly coupled regime. We obtain the conditional evolution through an intensity–amplitude correlation function that measures the time evolution of the field after the detection of a photon. A connection exists between this correlation function and the spectrum of squeezing, which permits the study of squeezed states in the time domain. We calculate the spectrum of squeezing from the master equation for the reduced density matrix using both the quantum-regression theorem and quantum trajectories. Our calculations not only show that spontaneous emission degrades the squeezing signal, they also point to the dynamical processes that cause this degradation.
A theoretical and experimental investigation of the generation of squeezed states of light has been carried out for a system of two-level atoms inside a high-finesse cavity. The regime that we consider is such that the coupling of atoms to the cavity mode produces a splitting in the normal mode structure of the atom-field system (the so-called vacuum-field Rabi splitting). We have employed this splitting to generate squeezing with observed reductions in photocurrent noise of 30 % below the shot-noise level. A degree of squeezing of ~50% is inferred for the field state in the absence of propagation and detection losses. Squeezing has been observed in two different cavity geometries with quite different coupling coefficients. In each case, the frequency of best squeezing corresponds to that of the normal mode splitting g N (g= single atom coupling coefficent and N = number of atoms) in good agreement with our theoretical predictions. Regions of parameter space with very large degrees of squeezing (>20×) are identified. Preliminary extensions of the theory into a transient regime with (cavity lifetime) ≪ (interaction time) ≪ (atomic lifetime) are described.