Experimental Verification of Demon-Involved Fluctuation Theorems

L.-L. Yan, J.-T. Bu, Qian Zeng, K. Zhang, K.-F. Cui, F. Zhou, S.-L. Su, L. Chen, Jin Wang, Gang Chen, and M. Feng

Phys. Rev. Lett. 133, 090402 – Published 29 August 2024

Abstract

The limit of energy saving in the control of small systems has recently attracted much interest due to the concept refinement of the Maxwell demon. Inspired by a newly proposed set of fluctuation theorems, we report the first experimental verification of these equalities and inequalities in an ultracold ${}^{40}{Ca}^{+}$ ion system, confirming the intrinsic nonequilibrium in the system due to involvement of the demon. Based on elaborately designed demon-involved control protocols, such as the Szilard engine protocol, we provide experimentally quantitative evidence of the dissipative information and observe tighter bounds of both the extracted work and the demon’s efficacy than the limits predicted by the Sagawa-Ueda theorem. Our results substantiate a close connection between the physical nature of information and nonequilibrium processes at the microscale, which help to further understand the thermodynamic characteristics of information and the optimal design of nanoscale and smaller systems.

Received 16 March 2024
Accepted 29 July 2024

DOI:https://doi.org/10.1103/PhysRevLett.133.090402

Physics Subject Headings (PhySH)

Fluctuation theorems Irreversible processes Quantum simulation Quantum thermodynamics

Trapped ions

Atom & ion cooling Atom & ion trapping & guiding Jaynes-Cummings model Master equation Two-level models

Statistical Physics & ThermodynamicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

L.-L. Yan ^1,2,*, J.-T. Bu ^3,4,*, Qian Zeng^5,*, K. Zhang ^6,7, K.-F. Cui¹, F. Zhou^3,†, S.-L. Su ^1,2, L. Chen ^3,8, Jin Wang^9,10,‡, Gang Chen^1,2, and M. Feng ^3,8,§

¹Key Laboratory of Materials Physics, Ministry of Education, School of Physics and Laboratory of Zhongyuan Light, Zhengzhou University, Zhengzhou 450001, China
²Institute of Quantum Materials and Physics, Henan Academy of Sciences, Zhengzhou 450046, China
³State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy of Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
⁴School of Physics, University of the Chinese Academy of Sciences, Beijing 100049, China
⁵State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun 130022, China
⁶School of Physics, Northwest University, Xi’an 710127, China
⁷Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China
⁸Research Center for Quantum Precision Measurement, Guangzhou Institute of Industry Co. Ltd., Guangzhou 511458, China
⁹Center for Theoretical Interdisciplinary Sciences, Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
¹⁰Departments of Physics and Astronomy, State University of New York, Stony Brook, New York 11794-3400, USA

^*These authors contributed equally to this work.
^†Contact author: [email protected]
^‡Contact author: [email protected]
^§Contact author: [email protected]

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Issue

Vol. 133, Iss. 9 — 30 August 2024

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Images

Figure 1
Schematic for verifying the demon-involved fluctuation theorems. (a) The Szilard engine feedback control protocol, where the demon (denoted by $y$ ) makes a noisy measurement on the system (black circle) at the initial time, where the short-lived state $| a ⟩$ (green dashed line) is introduced to simulate measurement noise [as explained in (b) and its caption]. The demon can successfully extract work of $E$ from the system only if it makes a correct measurement when the system is in the higher energy level (left). In all other cases, including a wrong measurement or a lower energy level (right), the demon extracts no work. After the work extraction, the demon couples the system to the environmental heat bath with temperature $T$ to recharge the system for the next cycle. (b) Experimental scheme based on a trapped ${}^{40}{Ca}^{+}$ ion, where the system is a qubit encoded by two energy levels of electronic states, i.e., the ground state $| ↓ ⟩ = | 4 S_{1 / 2}, m_{J} = 1 / 2 ⟩$ and the metastable state $| ↑ ⟩ = | 3 D_{5 / 2}, m_{J} = + 3 / 2 ⟩$ with $m_{J}$ being the magnetic quantum number. An equilibrium Boltzmann distribution and thermalization process of the qubit system is generated by combining the irradiation of a 729-nm laser pulse (Rabi frequency $Ω_{1}$ ) with a dephasing process of the system. A short-lived state $| a ⟩ = | 3 P_{3 / 2}, m_{J} = + 3 / 2 ⟩$ and an 854-nm laser pulse with Rabi frequency $Ω_{2}$ are used for mimicking the noisy measurement by controlling the unlocking and duration of the laser. The demon adds a phonon into the vibrational mode through a red-sideband transition carried by the 729-nm laser irradiation, depending on the measurement result “1,” i.e., in the upper state $| ↑ ⟩$ , where the auxiliary vibrational mode works for storing the energy due to the work output from the system. (c) The probability flow in the Szilard engine controlled by the demon, where the abbreviation $p_{x m | n} ≔ p (x_{0} = x, a_{0} = m | y = n)$ with $x = ↓, ↑$ , $m = 0, 1, 2, \dots$ , and $n = 0$ , 1; for example, $p_{↑ 0 | 1}$ means that the system is initially in the spin state $| ↑ ⟩$ and the vibrational state $| 0 ⟩$ , and the measurement result is “1” [40]. (d) The measurement error under different pulse lengths of the 854-nm laser in the generation of the mixed state $ρ = p_{↓} | ↓ ⟩ ⟨ ↓ | + p_{↑} | ↑ ⟩ ⟨ ↑ |$ with $p_{↓} = (1 + \cos θ_{c}) / 2$ and $p_{↑} = 1 - p_{↓}$ . (e),(f) The joint and conditional probabilities for $θ_{c} = π / 3$ after the measurement.
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Figure 2
The fluctuation theorems (a1)–(c1) and ensemble average (a2)–(c2) of the dissipative information $σ_{I}$ , the conditional entropy production $σ_{X | Y}$ , and the unconditional entropy production $σ_{X}$ for different initial mixed states, prepared by laser irradiation with durations $θ_{c} = π / 2$ , $π / 3$ , and $π / 6$ , respectively. Their entropy production difference $Δ σ ≔ ⟨ σ_{X | Y} ⟩ - ⟨ σ_{X} ⟩ - ⟨ σ_{I} ⟩$ ; see the inset in (c2). The dot data and curves (also in the following figures) represent the experimental observation and analytical calculation, respectively.
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Figure 3
The thermodynamic second law inequalities of work (a) for the conventional and tighter form and (b) the conventional and tighter low bound, under $θ_{c} = π / 3$ . (c) The coarse-grained version of SUT for different mixed states. (d) The output work $W_{out}$ and stored work $W_{ext}$ in the vibrational mode after the control process.
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Figure 4
The output work efficacy $η_{out}$ (red data), actual efficacy $η_{ext}$ (blue data), and the best efficacy $η_{\max}$ (black data) of the demon in the whole control process.
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