Levitated Nanoparticles for Microscopic Thermodynamics-A Review

Simulation of optomechanical interaction of levitated nanoparticle with photonic crystal micro cavity

We propose and analyze theoretically a promising design of an optical trap for vacuum levitation of nanoparticles based on a one-dimensional (1D) silicon photonic crystal cavity (PhC). The considered cavity has a quadratically modulated width of the silicon wave guiding structure, leading to a calculated cavity quality factor of 8 × 105. An effective mode volume of approximately 0.16 μm3 having the optical field strongly confined outside the silicon structure enables optical confinement on nanoparticle in all three dimensions. The optical forces and particle-cavity optomechanical coupling are comprehensively analyzed for two sizes of silica nanoparticles (100 nm and 150 nm in diameter) and various mode detunings. The value of trapping stiffnesses in the microcavity is predicted to be 5 order of magnitudes higher than that reached for optimized optical tweezers, moreover the linear single photon coupling rate can reach MHz level which is 6 order magnitude larger than previously reported values for common bulk cavities. The theoretical results support optimistic prospects towards a compact chip for optical levitation in vacuum and cooling of translational mechanical degrees of motion for the silica nanoparticle of a diameter of 100 nm.

Adiabatic computing for optimal thermodynamic efficiency of information processing

Landauer's principle makes a strong connection between information theory and thermodynamics by stating that erasing a one-bit memory at temperature [Formula: see text] requires an average energy larger than [Formula: see text], with [Formula: see text] Boltzmann's constant. This tiny limit has been saturated in model experiments using quasistatic processes. For faster operations, an overhead proportional to the processing speed and to the memory damping appears. In this article, we show that underdamped systems are a winning strategy to reduce this extra energetic cost. We prove both experimentally and theoretically that, in the limit of vanishing dissipation mechanisms in the memory, the physical system is thermally insulated from its environment during fast erasures, i.e., fast protocols are adiabatic as no heat is exchanged with the bath. Using a fast optimal erasure protocol, we also show that these adiabatic processes produce a maximum adiabatic temperature [Formula: see text], and that Landauer's bound for fast erasures in underdamped systems becomes the adiabatic bound: [Formula: see text].

A Review of Single-Cell Microrobots: Classification, Driving Methods and Applications

Single-cell microrobots are new microartificial devices that use a combination of single cells and artificial devices, with the advantages of small size, easy degradation and ease of manufacture. With externally driven strategies such as light fields, sound fields and magnetic fields, microrobots are able to carry out precise micromanipulations and movements in complex microenvironments. Therefore, single-cell microrobots have received more and more attention and have been greatly developed in recent years. In this paper, we review the main classifications, control methods and recent advances in the field of single-cell microrobot applications. First, different types of robots, such as cell-based microrobots, bacteria-based microrobots, algae-based microrobots, etc., and their design strategies and fabrication processes are discussed separately. Next, three types of external field-driven technologies, optical, acoustic and magnetic, are presented and operations realized in vivo and in vitro by applying these three technologies are described. Subsequently, the results achieved by these robots in the fields of precise delivery, minimally invasive therapy are analyzed. Finally, a short summary is given and current challenges and future work on microbial-based robotics are discussed.

Shortcuts to Equilibrium with a Levitated Particle in the Underdamped Regime

We report on speeding-up equilibrium recovery in the previously unexplored general case of the underdamped regime using an optically levitated particle. We accelerate the convergence toward equilibrium by an order of magnitude compared to the natural relaxation time. We then discuss the efficiency of the studied protocols, especially for a multidimensional system. These results pave the way for optimizing realistic nanomachines with application to sensing and developing efficient nanoheat engines.

Stroboscopic thermally-driven mechanical motion

Unstable nonlinear systems can produce a large displacement driven by a small thermal initial noise. Such inherently nonlinear phenomena are stimulating in stochastic physics, thermodynamics, and in the future even in quantum physics. In one-dimensional mechanical instabilities, recently made available in optical levitation, the rapidly increasing noise accompanying the unstable motion reduces a displacement signal already in its detection. It limits the signal-to-noise ratio for upcoming experiments, thus constraining the observation of such essential nonlinear phenomena and their further exploitation. An extension to a two-dimensional unstable dynamics helps to separate the desired displacement from the noisy nonlinear driver to two independent variables. It overcomes the limitation upon observability, thus enabling further exploitation. However, the nonlinear driver remains unstable and rapidly gets noisy. It calls for a challenging high-order potential to confine the driver dynamics and rectify the noise. Instead, we propose and analyse a feasible stroboscopically-cooled driver that provides the desired detectable motion with sufficiently high signal-to-noise ratio. Fast and deep cooling, together with a rapid change of the driver stiffness, are required to reach it. However, they have recently become available in levitating optomechanics. Therefore, our analysis finally opens the road to experimental investigation of thermally-driven motion in nonlinear systems, its thermodynamical analysis, and future quantum extensions.

Dynamics of Information Erasure and Extension of Landauer's Bound to Fast Processes

Using a double-well potential as a physical memory, we study with experiments and numerical simulations the energy exchanges during erasure processes, and model quantitatively the cost of fast operation. Within the stochastic thermodynamics framework we find the origins of the overhead to Landauer's bound required for fast operations: in the overdamped regime this term mainly comes from the dissipation, while in the underdamped regime it stems from the heating of the memory. Indeed, the system is thermalized with its environment at all times during quasistatic protocols, but for fast ones, the inefficient heat transfer to the thermostat is delayed with respect to the work influx, resulting in a transient temperature rise. The warming, quantitatively described by a comprehensive statistical physics description of the erasure process, is noticeable on both the kinetic and potential energy: they no longer comply with equipartition. The mean work and heat to erase the information therefore increase accordingly. They are both bounded by an effective Landauer's limit k_{B}T_{eff}ln2, where T_{eff} is a weighted average of the actual temperature of the memory during the process.

Nonequilibrium Control of Thermal and Mechanical Changes in a Levitated System

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small nonequilibrium systems. While work and heat are equally important forms of energy exchange, fluctuation relations have not been experimentally assessed for the generic situation of simultaneous mechanical and thermal changes. Thermal driving is indeed generally slow and more difficult to realize than mechanical driving. Here, we use feedback cooling techniques to implement fast and controlled temperature variations of an underdamped levitated microparticle that are 1 order of magnitude faster than the equilibration time. Combining mechanical and thermal control, we verify the validity of a fluctuation theorem that accounts for both contributions, well beyond the range of linear response theory. Our results allow the investigation of general far-from-equilibrium processes in microscopic systems that involve fast mechanical and thermal changes at the same time.

Detailed fluctuation theorem bound for apparent violations of the second law

The second law of thermodynamics is a statement about the statistics of the entropy production, 〈Σ〉≥0. For small systems, it is known that the entropy production is a random variable and negative values (Σ<0) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average 〈Σ〉, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak-coupling approximation, a levitated nanoparticle, and a classical particle in a box.

Direct and Clean Loading of Nanoparticles into Optical Traps at Millibar Pressures

Nanoparticles levitated by optical fields under vacuum conditions have applications in quantum science, the study of nanothermodynamics and precision sensing. The existing techniques for loading optical traps require ambient conditions and often involve dispersion in liquids, which can contaminate delicate optics and lead to enhanced optical absorption and heating. Here, we present a clean, dry and generic mechanism for directly loading optical traps at pressures down to 1 mbar, exploiting Laser Induced Acoustic Desorption and allowing for the rapid and efficient trapping of nanoparticles.

Levitodynamics: Levitation and control of microscopic objects in vacuum

[Figure: see text].

Kovacs Memory Effect with an Optically Levitated Nanoparticle

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.

Information bound for entropy production from the detailed fluctuation theorem

Fluctuation theorems impose fundamental bounds in the statistics of the entropy production with the second law of thermodynamics being the most famous. Using information theory, we quantify the information of entropy production and find an upper tight bound as a function of its mean from the strong detailed fluctuation theorem. The bound is given in terms of a maximal distribution, a member of the exponential family with nonlinear argument. We show that the entropy produced by heat transfer using a bosonic mode at weak coupling reproduces the maximal distribution in a limiting case. The upper bound is extended to the continuous domain and verified for the heat transfer using a levitated nanoparticle. Finally, we show that a composition of qubit swap engines satisfies a particular case of the maximal distribution regardless of its size.

Negative entropy production rates in Drude-Sommerfeld metals

It is commonly accepted that in typical situations the rate of entropy production is non-negative. We show that this assertion is not entirely correct, not even in the linear regime, if a time-dependent, external perturbation is not compensated by a rapid enough decay of the response function. This is demonstrated for three variants of the Drude model to describe electrical conduction in noble metals, namely the classical free electron gas, the Drude-Sommerfeld model, and the extended Drude-Sommerfeld model. The analysis is concluded with a discussion of potential experimental verifications and ramifications of negative entropy production rates.

Robust Optical-Levitation-Based Metrology of Nanoparticle's Position and Mass

Light has shown an incredible capability in precision measurement based on optomechanic interaction in high vacuum by isolating environment noises. However, there are still obstructions, such as displacement and mass estimation error, highly hampering the improvement of absolute accuracy at the nanoscale. Here, we present a nonlinearity based metrology to precisely measure the position and mass of a nanoparticle with optical levitation under 10^{-5}  mbar. By precisely controlling the oscillation amplitude of the levitated nanoparticle at the nonlinear regime for high accuracy calibration, we realized a feasible sub-picometer-level position measurement with an uncertainty of 1.0% without the prior information of mass, which can be further applied to weigh the femtogram-level mass with an uncertainty of 2.2%. It will also pave the way to construct a fine-calibrated optomechanic platform in high vacuum for high sensitivity and accuracy measurement in force and acceleration at the nanoscale and the study in quantum superposition at the mesoscopic scale.

Thermodynamics of continuous non-Markovian feedback control

Feedback control mechanisms are ubiquitous in science and technology, and play an essential role in regulating physical, biological and engineering systems. The standard second law of thermodynamics does not hold in the presence of measurement and feedback. Most studies so far have extended the second law for discrete, Markovian feedback protocols; however, non-Markovian feedback is omnipresent in processes where the control signal is applied with a non-negligible delay. Here, we experimentally investigate the thermodynamics of continuous, time-delayed feedback control using the motion of an optically levitated, underdamped microparticle. We test the validity of a generalized second law which bounds the energy extracted from the system, and study the breakdown of feedback cooling for very large time delays.

Work distribution in thermal processes

We find the moment generating function (mgf) of the nonequilibrium work for open systems undergoing a thermal process, i.e., when the stochastic dynamics maps thermal states into time-dependent thermal states. The mgf is given in terms of a temperaturelike scalar satisfying a first-order ordinary differential equation. We apply the result to some paradigmatic situations: a levitated nanoparticle in a breathing optical trap, a Brownian particle in a box with a moving piston, and a two-state system driven by an external field, where the work mgfs are obtained for different timescales and compared with Monte Carlo simulations.

Optomechanics with levitated particles

Optomechanics is concerned with the use of light to control mechanical objects. As a field, it has been hugely successful in the production of precise and novel sensors, the development of low-dissipation nanomechanical devices, and the manipulation of quantum signals. Micro- and nano-particles levitated in optical fields act as nanoscale oscillators, making them excellent low-dissipation optomechanical objects, with minimal thermal contact to the environment when operating in vacuum. Levitated optomechanics is seen as the most promising route for studying high-mass quantum physics, with the promise of creating macroscopically separated superposition states at masses of 10⁶ amu and above. Optical feedback, both using active monitoring or the passive interaction with an optical cavity, can be used to cool the centre-of-mass of levitated nanoparticles well below 1 mK, paving the way to operation in the quantum regime. In addition, trapped mesoscopic particles are the paradigmatic system for studying nanoscale stochastic processes, and have already demonstrated their utility in state-of-the-art force sensing.

Resolved-Sideband Cooling of a Levitated Nanoparticle in the Presence of Laser Phase Noise

We investigate the influence of laser phase noise heating on resolved sideband cooling in the context of cooling the center-of-mass motion of a levitated nanoparticle in a high-finesse cavity. Although phase noise heating is not a fundamental physical constraint, the regime where it becomes the main limitation in Levitodynamics has so far been unexplored and hence embodies from this point forward the main obstacle in reaching the motional ground state of levitated mesoscopic objects with resolved sideband cooling. We reach minimal center-of-mass temperatures comparable to T_{min}=10  mK at a pressure of p=3×10^{-7}  mbar, solely limited by phase noise. Finally we present possible strategies towards motional ground state cooling in the presence of phase noise.

Stochastic thermodynamics of nonharmonic oscillators in high vacuum

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time-dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time-dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jump which correspond to adiabatic processes linked by a smooth exponential time-dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nanoparticles, the levitated particle in a harmonic trap, and the free particle in a box, as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

Cavity-Based 3D Cooling of a Levitated Nanoparticle via Coherent Scattering

We experimentally realize cavity cooling of all three translational degrees of motion of a levitated nanoparticle in vacuum. The particle is trapped by a cavity-independent optical tweezer and coherently scatters tweezer light into the blue detuned cavity mode. For vacuum pressures around 10^{-5}  mbar, minimal temperatures along the cavity axis in the millikelvin regime are observed. Simultaneously, the center-of-mass (c.m.) motion along the other two spatial directions is cooled to minimal temperatures of a few hundred millikelvin. Measuring temperatures and damping rates as the pressure is varied, we find that the cooling efficiencies depend on the particle position within the intracavity standing wave. This data and the behavior of the c.m. temperatures as functions of cavity detuning and tweezer power are consistent with a theoretical analysis of the experiment. Experimental limits and opportunities of our approach are outlined.

Quantum heat distribution in thermal relaxation processes

We analyze the heat exchange distribution of open quantum systems undergoing a thermal relaxation process with a time-dependent effective temperature. We show that such processes arise, for example, if the dynamics maximizes the entropy production. Using a two-point measurement scheme, we find an expression for the heat moment generating function that depends solely on the system's partition function and on the thermalization function (i.e., the law of cooling) describing the effective temperature. Applications include the relaxation of free bosonic and fermionic modes, for which closed-form expressions for the time-dependent heat distribution function are derived. Multiple free modes with arbitrary dispersion relations are also briefly discussed. In the semiclassical limit our formula agrees with previous results of the literature for the heat distribution of an optically trapped nanoscopic particle far from equilibrium.