Statistical Spectral and Dynamical Properties of Two-Level Systems

Abstract

In a very interesting recent Letter,^ several statistical properties of the energy levels of a single two-level atom interacting with one quantized electromagnetic mode were investigated, and it was found that the nearest-level spacing was highly correlated and regular, in contrast to chaotic dynamical behavior found in a semiclassical version of this model (for references see Ref. 1). Similar regularities in the same system had also been found by Graham and Hohnerbach^ and Graham,^ but for sufficiently strong coupling and suitably chosen initial conditions dynamical properties, e.g., the time-dependent occupation probabilities of the two levels, were found to behave irregularly, i.e., quasiperiodically with a large number of incommensurate frequencies.^'^ In this Comment we wish to examine to what extent the disparity between the regularity of the energy spectrum (cf. Fig. 1 of Ref. 1) and the irregular dynamical behavior found in Refs. 2 and 3 is a consequence of the assumption of a single two-level atom. In the semiclassical version of the model the number of atoms is irrelevant and can be absorbed in rescaled variables and parameters.Concerning the level statistics of the quantum system we find that already for a small number of atoms (iV = 9) and a correspondingly scaled down coupling constant k/VN (X = 0.5) the distribution of relative level spacings Wis) of neighboring levels of equal parity is represented reasonably well (cf. Fig. 1) by the Wigner distribution P(s)^(TTs/2)cxp(-7rs^/4), as one would expect for a quantum system which is chaotic in the classical limit. If we increase the coupling constant X, e.g., to X = 1.5, the level distribution for A^ = 9 further approaches the Wigner distribution, while it retains its regular features for A^= 1.In Fig. 2 we present the power spectrum of the electromagnetic mode gico) for a = 1.33VA^, A = 0.3, and the two values iV = 9, N=l. gio)) is defined as the Fourier transform of the time average of (i;;ol^^(^ + 3(U)) • :'^ : ••;. 1.0-;. •". •. 1.1 Qjj FIG. 2. ^(oj) forA^ = 9 (inset: yv=l).T)a{i)\ilJo}, where |i//o) is chosen as the product of the upper atomic state and a coherent state of amplitude a. For A^= 1 the sharp lines are arranged in two groups corresponding to a broad doublet. The doublet structure corresponds to a well-known regular feature of the dynamics, the periodic "revivals."'^ The distance of the maxima of the doublet is approximately k/a and gives the inverse of the revival time"^ multiplied by 277. The revivals are effectively damped as shown by the width of the doublet. When we increase a the high-frequency component of the doublet decreases in intensity, the two parts of the doublet overlap, the revivals disappear, and a quasiperiodic but effectively irregular dynamics results even for N=l}'^ This happens despite the fact that the level statistics continues to show its regularities. For iV = 9 and much smaller coupling (e.g., X = 0.03) one finds a decaplet of groups of sharp lines. However, for the strong-coupling case corresponding to Fig. 2...

Quantum Irreversibility and Chaos

“…In accordance with the results of [5] and [6], we find that the individual semi-classical trajectories can exhibit chaos for specific values of the system parameters. Using the method of Benettin et al [18], we evaluate the Lyapunov coefficient X which in the case of Fig.…”

“…and similar relations connect the coordinates q and momentum p to the corresponding quantum operators q and p. The operator S£ c{ is identical to the Liouvillian of a classical dipole interacting with a classical oscillator, i.e., this term alone corresponds to the semi-classical set of equations discussed by various authors [5,6]. We refer to the calculations based on the study of the single trajectory solutions of the nonlinear dynamic equations as the semiclassical predictions.…”

Quantum Chaos Triggered by Precursors of a Quantum Phase Transition: The Dicke Model

“…The level statistics for finite N have revealed the existence of quantum chaos in certain isolated regimes of the model [11,12]. On the other hand, the QPT aspect for N → ∞ has been discussed in the context of superradiance [13,14], and recently for exciton condensation [15].…”

The Critical Point Entanglement and Chaos in the Dicke Model

“…For example, the entanglement and the QPT in the rotating-wave approximation (RWA) is reported in [6], which suggests that the entanglement only occurs in the strong coupling limit. The model has also been extensively studied without the RWA, with its quantum-chaotic properties discussed by several other authors [12][13][14][15], without reference to the QPT.…”

“…To date, many aspects of the model have been studied, such as the dynamical properties of quantum entanglement and decoherence [3][4][5][6][7][8], quantum phase transitions (QPT) [2,[9][10][11] and chaotic properties [3,5,9,10,[12][13][14][15]. Furuya et al [3] did the initial work on the entanglement process in a varied version of the model and concluded that for short times, there is a faster increase in decoherence for chaotic initial conditions as compared to regular ones, which have an oscillatory increase.…”