Criticality in the quantum kicked rotor with a smooth potential

Rina Dutta and Pragya Shukla

Phys. Rev. E 78, 031115 – Published 10 September 2008

Abstract

We investigate the possibility of an Anderson-type transition in the quantum kicked rotor with a smooth potential due to dynamical localization of the wave functions. Our results show the typical characteristics of a critical behavior, i.e., multifractal eigenfunctions and a scale-invariant level statistics at a critical kicking strength which classically corresponds to a mixed regime. This indicates the existence of a localization to delocalization transition in the quantum kicked rotor. Our study also reveals the possibility of other types of transition in the quantum kicked rotor, with a kicking strength well within the strongly chaotic regime. These transitions, driven by the breaking of exact symmetries, e.g., time reversal and parity, are similar to weak-localization transitions in disordered metals.

Received 14 May 2008

DOI:https://doi.org/10.1103/PhysRevE.78.031115

Authors & Affiliations

Rina Dutta and Pragya Shukla

Department of Physics, Indian Institute of Technology, Kharagpur, India

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Vol. 78, Iss. 3 — September 2008

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Figure 1
(Color online) Spectral measures of QKR1. (a) nearest-neighbor spacing distribution $P (S)$ of the eigenvalues for various sizes $N$ , with inset showing behavior on the linear scale; (b) number variance $Σ^{2} (r)$ for various sizes. The convergence of the curves for different sizes indicates scale invariance of the statistics. The behavior is critical, being different from that of the two end points, namely, Poisson and CUE statistics, even in the infinite-size limit.Reuse & Permissions
Figure 2
(Color online) Spectral measures of QKR2. (a) Distribution $P (S)$ of the nearest-neighbor eigenvalue spacings $S$ for various sizes, with inset showing behavior on the linear scale; (b) number variance $Σ^{2} (r)$ for various sizes. Again, the statistics being intermediate between COE and CUE and convergence of the curves for different sizes indicate its critical behavior.Reuse & Permissions
Figure 3
(Color online) Multifractality of QKR1. (a) Fractal dimension $τ_{q}$ along with the fit $y (q) = (1 + c) q - 1 - c q^{2}$ with $c = 0.06$ (good only for $q ⩽ 3$ ). A fit for the large- $q$ regime suggests the behavior $τ_{q} = q - 1 + 0.02 q^{2}$ . (b) Multifractal spectrum $f (α)$ for various sizes along with the parabolic fit $f (α) = d - {(α - α_{0})}^{2} ∕ 4 (α_{0} - d)$ with $α_{0} = 1.09$ and $d = 1$ . (c) Distribution $P_{u} (u^{'})$ with $u^{'} = (\ln u - ⟨ \ln u ⟩) ∕ ⟨ \ln^{2} u ⟩$ of the local intensity of an eigenfunction for QKR1. The solid line represents the function $f (u^{'})$ given by Eq. (13) with $a = 5.2$ , $b = 1.2$ , and $c = 0.78$ [corresponding to an approximate log-normal behavior of $P_{u} (u)$ ], a good approximation in the tail region as expected. The inset shows the behavior on a linear scale.Reuse & Permissions
Figure 4
(Color online) Multifractality of QKR2. (a) Fractal dimension $τ_{q}$ along with the fit $y (q) = (1 + c) q - 1 - c q^{2}$ with $c = 0.075$ (good only for $q < 4$ ). (b) Mulifractal spectrum $f (α)$ for various sizes along with a parabolic fit, of the same form as in Fig. 3b with $α_{0} = 1.045$ and $d = 1$ . (c) Distribution $P_{u} (u^{'})$ of the local intensity of an eigenfunction for QKR2, with $u^{'}$ the same as in Fig. 3c. The solid line represents the function $f (u^{'})$ with $a = 5.3$ , $b = 0.95$ , and $c = 0.73$ [corresponding to a log-normal behavior of $P_{u} (u)$ ], which fits well in the tail region of $P_{u} (u^{'})$ . The inset shows the behavior on a linear scale.Reuse & Permissions
Figure 5
(Color online) Comparison of spectral statistics of the Anderson ensemble with QKR3 and QKR4. (a) $P (S)$ , with inset showing the linear behavior; (b) $Σ^{2} (r)$ . Note that the statistics for QKR3 and QKR4 turn out to be close to that suggested by the relation $Λ_{AE} = Λ_{KR, T}$ (giving $χ = 0.95$ for QKR3 and $ϕ = 0.84 π^{2}$ for QKR4). (c) Phase-space behavior of the classical kicked rotor at $K = 4.5$ [see Eq. (2)].Reuse & Permissions
Figure 6
(Color online) Comparison of the multifractality of the eigenfunctions of QKR3 with the Anderson ensemble. (a) Local intensity distribution $P_{u} (u^{'})$ of an eigenfunction with $u^{'}$ the same as in Figs. 3, 4. Also shown is the function $f (u^{'})$ with $a = 15.9$ , $b = 7.7$ , $c = 0.99$ , a good fit in the tail region of $P_{u} (u^{'})$ . The inset shows the behavior on a linear scale. (b) Multifractal spectrum $f (α)$ ; note that the QKR3 analog of the Anderson ensemble here is the same as the one in Fig. 5. Again the fit has the same parabolic form as in Figs. 3, 4, with $α_{0} = 1.1$ and $d = 1$ .Reuse & Permissions

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