Hilbert space average of transition probabilities

Nico Hahn, Thomas Guhr, and Daniel Waltner

Phys. Rev. E 101, 062135 – Published 22 June 2020

Abstract

The typicality approach and the Hilbert space averaging method as its technical manifestation are important concepts of quantum statistical mechanics. Extensively used for expectation values we extend them in this paper to transition probabilities. In this context we also find that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. Furthermore, within our approach we are capable to consider distributions of matrix elements between states that are not orthogonal. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.

Received 21 February 2020
Accepted 26 May 2020

DOI:https://doi.org/10.1103/PhysRevE.101.062135

Physics Subject Headings (PhySH)

Quantum chaos Quantum statistical mechanics Quantum thermodynamics

Random matrix theory

Statistical Physics & ThermodynamicsGeneral Physics

Authors & Affiliations

Nico Hahn^*, Thomas Guhr, and Daniel Waltner

Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany

^*Corresponding author: nico.hahn@uni-due.de

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Issue

Vol. 101, Iss. 6 — June 2020

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Images

Figure 1
Exemplification of the protocol, described in the main text. The density plot on the right is generated by interpolating the data of 100 uniformly sampled vectors after applying $Λ$ and renormalization. The color code represents the likelihood to sample one of these vectors in the corresponding area.
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Figure 2
Blue line: Analytical result for the Hilbert space average (4.8). Red dots: Transition probabilities for 10 individual realizations (5.2). Green diamonds: Their arithmetic mean and the analytical result for the standard deviation according to (5.3) plotted as a tube around the average. The situation is shown for the chaotic chain ( $J = b = π / 4, h = π / 5$ ) with the particle numbers (a) $n = 8$ and (b) $n = 10$ . Remember that the map between $θ$ (lower abscissa) and $|z|$ (upper abscissa) is nonlinear.
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Figure 3
Blue line: Analytical result for the Hilbert space average (5.8). Red dots: Transition probabilities for 100 individual realizations. Green diamonds: Their arithmetic mean and the analytical result for the standard deviation according to (5.9) plotted as a tube around the average. We show the situation for an $n = 8$ -particle spin chain that is (a) chaotic ( $J = b = π / 4, h = π / 5$ ) and (b) interactionless ( $J = h = 0, b = π / 4$ ).
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Figure 4
Histogram of transition probabilities for the Floquet operator $U$ of a kicked Ising chain (a) within the chaotic regime ( $J = b = π / 4, h = π / 5$ ) and (b) that is interactionless ( $J = h = 0, b = π / 4$ ). The inset contains the mean $μ$ , variance $μ_{2}$ , skewness $μ_{3}$ , and kurtosis $μ_{4}$ .
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Figure 5
Blue line: Analytical result for the Hilbert space average (4.12). Red dots: Transition probabilities for 10 individual realizations (5.23). Green diamonds: Their arithmetic mean and the analytical result for the standard deviation according to (5.17) plotted as a tube around the average. The inset shows the purities of the deployed statistical operators, which are determined by the magnetization $m_{z}$ . It is $m_{z} = 0.5$ for (a) and $m_{z} = 7$ for (b).
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Figure 6
Blue line: Analytical result for the Hilbert space average using the approximation (5.20). Red dots: Transition probabilities for 100 individual realizations (5.23). Green diamonds: Their arithmetic mean and the analytical result for the standard deviation according to (5.21) plotted as a tube around the average. The inset shows the purities of the deployed statistical operators $ρ$ and $ρ^{'}$ . They are determined by the magnetizations $m_{z} = 0.5$ , $m_{z}^{'} = - 0.3$ for (a) and $m_{z} = 3$ , $m_{z}^{'} = - 3$ for (b).
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