Kovacs Memory Effect with an Optically Levitated Nanoparticle

Andrei Militaru, Antonio Lasanta, Martin Frimmer, Luis L. Bonilla, Lukas Novotny, and Raúl A. Rica

Phys. Rev. Lett. 127, 130603 – Published 24 September 2021

Abstract

The understanding of the dynamics of nonequilibrium cooling and heating processes at the nanoscale is still an open problem. These processes can follow surprising relaxation paths due to, e.g., memory effects, which significantly alter the expected equilibration routes. The Kovacs effect can take place when a thermalization process is suddenly interrupted by a change of the bath temperature, leading to a nonmonotonic evolution of the energy of the system. Here, we demonstrate that the Kovacs effect can be observed in the thermalization of the center of mass motion of a levitated nanoparticle. The temperature is controlled during the experiment through an external source of white Gaussian noise that mimics an effective thermal bath at a temperature that can be changed faster than any relaxation time of the system. We describe our experiments in terms of the dynamics of a Brownian particle in a harmonic trap without any fitting parameter, suggesting that the Kovacs effect can appear in a large variety of systems.

Received 8 April 2021
Accepted 12 August 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.130603

Physics Subject Headings (PhySH)

Brownian motion Classical statistical mechanics Cooling & trapping Fluctuations & noise Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics

Optical lattices & traps

Statistical Physics & ThermodynamicsGeneral PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Andrei Militaru ¹, Antonio Lasanta ^2,3,4,5, Martin Frimmer¹, Luis L. Bonilla ^6,4,7, Lukas Novotny^1,†, and Raúl A. Rica ^5,8,*

¹Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
²Departamento de Álgebra, Facultad de Educación, Economía y Tecnología de Ceuta, Universidad de Granada, Cortadura del Valle, s/n, 51001 Ceuta, Spain
³Grupo de Teorías de Campos y Física Estadística, Instituto Gregorio Millán, Universidad Carlos III de Madrid, Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Spain
⁴Grupo de Matemática Aplicada a la Física de la Materia Condensada, Instituto Gregorio Millán, Universidad Carlos III de Madrid, Unidad Asociada al Instituto de Ciencias de Materiales de Madrid, CSIC, Spain
⁵Nanoparticles Trapping Laboratory, Universidad de Granada, 18071 Granada, Spain
⁶Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain
⁷Instituto Gregorio Millán, Universidad Carlos III de Madrid, 28911 Leganés, Spain
⁸Universidad de Granada, Department of Applied Physics and Research Unit “Modeling Nature” (MNat), 18071 Granada, Spain

^*rul@ugr.es
^†www.photonics.ethz.ch

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Issue

Vol. 127, Iss. 13 — 24 September 2021

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Images

Figure 1
(a) A 1064 nm wavelength laser is focused through a microscope objective to create an optical trap for a charged silica nanoparticle. We use homodyne detection to record the particle’s position with a quadrant photodiode (QPD). We use a custom programmed FPGA to tune the effective temperature of the particle along the $x$ axis by applying a white Gaussian electrostatic force. The instantaneous temperature $T$ is stored together with the position signal by a data acquisition card (DAQ). (b) Force $F_{v}$ applied on the particle by the FPGA and corresponding temperature $T$ during the Kovacs protocol shown in Fig. 2. $f_{0}$ is a reference force in arbitrary units.
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Figure 2
(a) Temperature of the effective bath during the cooling protocol $T_{H} \to T_{C}$ (dashed light blue line), cooling protocol $T_{H} \to T_{W}$ (dash-dotted silver line), and Kovacs protocol (red solid line). (b) Average potential energy during the protocols described in (a). Discrete symbols (light blue triangles, silver diamonds, and red circles) are used for experimental data, whereas lines are used for the theoretical curves. The horizontal dotted line is the equilibrium value $k_{B} T_{W} / 2$ related to the warm temperature. The vertical dotted line signals the switch time between $T_{C}$ and $T_{W}$ for the Kovacs protocol. The estimation of the error bars is described in [40].
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Figure 3
(a) Measured position distribution $\tilde{ρ} (x, t)$ of the particle during the equilibration $T_{H} \to T_{C}$ , (b) $T_{H} \to T_{W}$ , (c) $T_{H} \to T_{C} \to T_{W}$ (Kovacs protocol). The vertical black line indicates the time $t = 0$ where the effective temperature is first changed. (c) The vertical dotted white line shows the switch time $t = t_{W}$ between $T_{C}$ and $T_{W}$ . The dashed white lines represent the standard deviations inferred from Fig. 2. At all times, the distributions are Gaussian. All the distributions are normalized to the maximum value of (a).
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Figure 4
Theoretical curves of potential $⟨ U ⟩$ , kinetic $⟨ K ⟩$ , and total $⟨ E ⟩ = ⟨ U + K ⟩$ energies during the Kovacs protocol for $⟨ U ⟩$ . At $t = 0$ the effective temperature is switched from $T_{H}$ to $T_{C}$ . At time $t_{W}$ (gray dotted vertical line), the effective temperature is switched to $T_{W}$ . Since the thermal forces affect the velocity instantaneously, we see that the evolution of $⟨ K ⟩$ (and consequently $⟨ E ⟩$ ) presents kink points whenever the effective temperature is changed. In [40], we show that a Kovacs protocol applied to either the kinetic or the total energy would actually generate a standard Kovacs effect, i.e., the hump after $t_{W}$ points upward. Inset: phase space distribution $ρ (x, v, t)$ at $t = t_{W}$ . The distribution is classically squeezed and has a nonzero $⟨ x v ⟩$ correlation. The inset is normalized to its respective maximum and the same color bar as in Fig. 3 is used.
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