Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems

Aurélia Chenu1,2,3,4, Javier Molina-Vilaplana5, and Adolfo del Campo1,2,3,6

1Donostia International Physics Center, E-20018 San Sebastián, Spain
2IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain
3Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, NM 87545, USA
4Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
5Technical University of Cartagena, UPCT, 30202, Cartagena, Spain
6Department of Physics, University of Massachusetts, Boston, MA 02125, USA

Characterizing the work statistics of driven complex quantum systems is generally challenging because of the exponential growth with the system size of the number of transitions involved between different energy levels. We consider the quantum work distribution associated with the driving of chaotic quantum systems described by random matrix Hamiltonians and characterize exactly the work statistics associated with a sudden quench for arbitrary temperature and system size. Knowledge of the work statistics yields the Loschmidt echo dynamics of an entangled state between two copies of the system of interest, the thermofield double state. This echo dynamics is dictated by the spectral form factor. We discuss its relation to frame potentials and its use to assess information scrambling.

Quantum thermodynamics aims at characterizing the laws of thermodynamics in the quantum realm, and thus at understanding the interplay between two fundamental theories of physics. Central to this study is the work performed on a system. At the quantum level, work cannot be measured directly and is rather described by a probability distribution. While the work statistics has been quantified in simple quantum systems, its description is an intrinsically challenging task for complex systems due to the exponential scaling of the problem with the system size. Complex many-body systems constitute a natural choice for the working substance in the design of quantum machines, and their quantitative characterization is highly important.

Our results provide a concrete framework, built on analytic techniques and numerical simulations, to study the thermodynamics of chaotic systems, which constitute a paradigmatic example of complex systems. Remarkably, we recast the problem into a dynamical setup by relating the work statistics to a dynamical quantity known as the Loschmidt echo. Therefore, we show that thermodynamics is related to the time evolution of an entangled state exhibiting nonclassical correlations, and to the emergence of irreversibility in complex quantum systems. In doing so, we not only connect the fields of quantum thermodynamics and quantum evolution, but also tie them to information scrambling: the spreading of information and correlations in many body systems.

Information scrambling was first introduced to address the black hole information paradox from a quantum information perspective. It has later been shown as a useful concept in fields such as condensed-matter and many-body physics to measure quantum chaos. Our research adds quantum thermodynamics to this list. The unexpected and firm connection established here is amenable to experimental tests in a variety of platforms and can help to explore quantum chaos in a new light.

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[1] Ekrem Taha Güldeste and Ceyhun Bulutay, "Loschmidt echo driven by hyperfine and electric-quadrupole interactions in nanoscale nuclear spin baths", Physical Review B 98 8, 085202 (2018).

[2] A. del Campo, J. Molina-Vilaplana, L. F. Santos, and J. Sonner, "Decay of a thermofield-double state in chaotic quantum systems. From random matrices to spin systems", European Physical Journal Special Topics 227 3(2018).

[3] D. M. Kennes, C. Karrasch, and A. J. Millis, "Loschmidt-amplitude wave function spectroscopy and the physics of dynamically driven phase transitions", arXiv:1809.00733 (2018).

[4] Yuanjian Zheng and Dario Poletti, "Irreversible work reduction by disorder in many-body quantum systems", Physical Review E 98 5, 052140 (2018).

[5] Tokiro Numasawa, "Late Time Quantum Chaos of pure states in the SYK model", arXiv:1901.02025 (2019).

[6] Yang-Yang Chen, Gentaro Watanabe, Yi-Cong Yu, Xi-Wen Guan, and Adolfo del Campo, "An Interaction-Driven Many-Particle Quantum Heat Engine: Universal Behavior", arXiv:1812.09327 (2018).

[7] A. Chenu, I. L. Egusquiza, J. Molina-Vilaplana, and A. del Campo, "Quantum work statistics, Loschmidt echo and information scrambling", Scientific Reports 8, 12634 (2018).

[8] Eric G. Arrais, Diego A. Wisniacki, Lucas C. Céleri, Norton G. de Almeida, Augusto J. Roncaglia, and Fabricio Toscano, "Quantum work for sudden quenches in Gaussian random Hamiltonians", Physical Review E 98 1, 012106 (2018).

[9] Qian Wang and Francisco Pérez-Bernal, "Probing excited-state quantum phase transition in a quantum many body system via out-of-time-ordered correlator", arXiv:1812.01920 (2018).

[10] Ekrem Taha Güldeste and Ceyhun Bulutay, "Loschmidt echo driven by hyperfine and electric-quadrupole interactions in nanoscale nuclear spin baths", arXiv:1804.07219 (2018).

[11] Zhenyu Xu, Luis Pedro García-Pintos, Aurélia Chenu, and Adolfo del Campo, "Extreme Decoherence and Quantum Chaos", Physical Review Letters 122 1, 014103 (2019).

[12] Colin Rylands and Natan Andrei, "Loschmidt amplitude and work distribution in quenches of the sine-Gordon model", Physical Review B 99 8, 085133 (2019).

[13] Colin Rylands and Natan Andrei, "Quantum Work of an Optical Lattice", arXiv:1904.07995 (2019).

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