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arXiv:quant-ph/0504135 · Physical Review A

Quantum random walk of the field in an externally driven cavity

G. S. Agarwal, Pradyumna K. Pathak

Published Sep 16, 2005• 28 citations
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Abstract

Using a resonant interaction between atoms and the field in a high-quality cavity, we show how to realize quantum random walks as proposed by Aharonov et al. [Phys. Rev. A 48, 1687 (1993)]. The atoms are driven strongly by a classical field. Under conditions of strong driving we could realize an effective interaction of the form $i{S}^{x}(a\ensuremath{-}{a}^{\ifmmode\dagger\else\textdagger\fi{}})$ in terms of the spin operator associated with the two-level atom and the field operators. This effective interaction generates displacement in the field's wave function depending on the state of the two-level atom. Measurements of the state of the two-level atom would then generate an effective state of the field. Using a homodyne technique, the state of the quantum random walker can be monitored.

DOI
10.1103/physreva.72.033815

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We investigate a novel quantum random walk (QRW) model, possibly useful in quantum algorithm implementation, that achieves a quadratically faster diffusion rate compared to its classical counterpart. We evaluate its asymptotic behavior expressed in the form of a limit probability distribution of a double horn shape. Questions of robustness and control of that limit distribution are addressed by introducing a quantum optical cavity in which a resonant Jaynes-Cummings type of interaction between the quantum walk coin system realized in the form of a two-level atom and a laser field is taking place. Driving the optical cavity by means of the coin-field interaction time and the initial quantum coin state, we determine two types of modification of the asymptotic behavior of the QRW. In the first one the limit distribution is robustly reproduced up to a scaling, while in the second one the quantum features of the walk, exemplified by enhanced diffusion rate, are washed out and Gaussian asymptotics prevail. Verification of these findings in an experimental setup that involves two quantum optical cavities that implement the driven QRW and its quantum to classical transition is discussed.
We investigate a novel quantum random walk (QRW) model, of possible use in quantum algorithm implementation, that achieves a quadratically faster diffusion rate compared to its classical counterpart. We evaluate its asymptotic behaviour expressed in the form of a limit probability distribution of a double-horn shape. Questions of robustness and control of that limit distribution are addressed by introducing a quantum optical cavity in which a resonant Jaynes–Cummings type of interaction between the quantum walk coin system realized in the form of a two-level atom and a laser field is taking place. Driving the optical cavity by means of the coin–field interaction time and the initial quantum coin state, we determine two types of modification of the asymptotic behaviour of the QRW. In the first one the limit distribution is robustly reproduced up to a scaling, while in the second one the quantum features of the walk, exemplified by an enhanced diffusion rate, are washed out and Gaussian asymptotics prevail. Verification of these findings in an experimental set-up that involves two quantum optical cavities that implement the driven QRW and its quantum to classical transition is discussed.
Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality cavity is studied. We consider the strongly coupled atom-field system as a quantum-classical hybrid with dynamically coupled quantum and classical degrees of freedom. It is shown that at given initial conditions the regime of atomic motion is governed mainly by the detuning of the atom-field resonance /spl delta/. We show that, even in the absence of any other interaction with environment, the interaction of a purely quantum atom-field system with the external atomic degree of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics (QED). It manifest itself in the classical degree of freedom as a random walking of an atom inside the cavity and in the quantum degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependence of variance of entanglement on /spl delta/, correlates strongly with the dependence of maximum Lyapunov exponent on /spl delta/. This result provides a quantum-classical correspondence in a closed physical system.
We propose a scheme for measuring the mean value of quadrature moments of one- and two-mode cavity fields by considering the resonant atom–cavity interaction in the presence of strong classical driving fields. With the aid of strong classical fields, it is easy to realize the quadrature operators for cavity fields, which are correlated to the internal state of the two-level atom. It is shown that the mean value of quadrature moments of cavity fields is determined by the time derivative of the population of the internal state at the initial interaction time. The scheme can be directly used to test quantum entanglement of two-mode cavity fields.
We propose the implementation of a strongly driven one-atom laser, based on the off-resonant interaction of a three-level atom in $\ensuremath{\Lambda}$ configuration with a single cavity mode and three laser fields. We show that the system can be described equivalently by a two-level atom resonantly coupled to the cavity and driven by a strong effective coherent field. The effective dynamics can be solved exactly, including a thermal field bath, allowing an analytical description of field statistics and entanglement properties. We also show the possible generation of quantum superposition (Schr\"odinger cat) states for the whole atom-field system and for the field alone after atomic measurement. We propose a way to monitor the system decoherence by measuring atomic populations. Finally, we confirm the validity of our model through numerical solutions.