Abstract
A fundamental and intrinsic property of any device or natural system is its relaxation time τ_{relax}, which is the time it takes to return to equilibrium after the sudden change of a control parameter^{1}. Reducing τ_{relax} is frequently necessary, and is often obtained by a complex feedback process. To overcome the limitations of such an approach, alternative methods based on suitable driving protocols have been recently demonstrated^{2,3}, for isolated quantum and classical systems^{4,5,6,7,8,9}. Their extension to open systems in contact with a thermostat is a stumbling block for applications. Here, we design a protocol, named Engineered Swift Equilibration (ESE), that shortcuts timeconsuming relaxations, and we apply it to a Brownian particle trapped in an optical potential whose properties can be controlled in time. We implement the process experimentally, showing that it allows the system to reach equilibrium 100 times faster than the natural equilibration rate. We also estimate the increase of the dissipated energy needed to get such a time reduction. The method paves the way for applications in micro and nanodevices, where the reduction of operation time represents as substantial a challenge as miniaturization^{10}.
Main
The concepts of equilibrium and of transformations from an equilibrium state to another, are cornerstones of thermodynamics. A textbook illustration is provided by the expansion of a gas, starting at equilibrium and expanding to reach a new equilibrium in a larger vessel. This operation can be performed either very slowly by a piston, without dissipating energy into the environment, or alternatively quickly, letting the piston freely move to reach the new volume. In the first case, the transformation takes a long (virtually infinite) time to be completed, while the gas is always in a quasiequilibrium state. In the second case instead, the transformation is fast but the gas takes its characteristic relaxation time τ_{relax} to reach the new equilibrium state in the larger volume. This is the time required for the exploration of the new vessel. More generally, once a control parameter is suddenly changed, the accessible phase space changes too^{1,11}; the system adjusts and needs a finite time to reach the final equilibrium distribution. This equilibration process of course plays a key role in outofequilibrium thermodynamics.
An important and relevant question related to optimization theory is whether a targeted statistical equilibrium state can be reached in a chosen time, arbitrarily shorter than τ_{relax}. Such strategies are reminiscent of those worked out in the recent field of shortcut to adiabaticity^{2,3}; they aim at developing protocols, both in quantum and in classical regimes, allowing the system to move as fast as possible from one equilibrium position to a new one, provided that there exists an adiabatic transformation relating the two^{12,13,14}. So far, proofofprinciple experiments have been carried out for isolated systems^{4,5,6,7,8,9} and for photonics circuit design^{15,16,17,18}. Yet, the problem of open classical systems is untouched. Here, we solve this question by putting forward an accelerated equilibration protocol for a system in contact with a thermal bath. Such a protocol shortcuts quasistationarity, according to which a driven open system remains in equilibrium with its environment at all times. This is a key step for a number of applications in nanooscillators^{19}, in the design of nanothermal engines^{20}, or in monitoring mesoscopic chemical or biological processes^{21}, for which thermal fluctuations are paramount and an accelerated equilibration desirable for improved power. We dub the method Engineered Swift Equilibration (ESE).
However, an arbitrary reduction of the time to reach equilibrium will have unavoidable consequences from an energetical point of view^{22}. The question of the corresponding cost is relevant as such, but also for applications, for example in nanodevices^{10}, where the goal is the reduction in size and execution time of a given process. Here, beyond the theoretical derivation of the procedure, we develop an experimental demonstration of ESE, studying the dynamics of a colloidal particle in an optical potential. The energetics of the system will also be analysed in depth, shedding light on the inherent consequences of timescale reduction^{22,23,24,25,26,27}.
Our experimental system consists of a microsphere immersed in water^{28} (see Methods). The particle is trapped by an optical harmonic potential U(x, t) = κ(t)x^{2}/2, where x is the particle position and κ(t) is the stiffness of the potential, which can be controlled by the power of the trapping laser^{20}. The system is affected by thermal fluctuations; its dynamics is overdamped and described by a Langevin equation. Our Brownian particle has a relaxation time defined as τ_{relax} = γ/κ, where γ is the fluid viscous coefficient. At equilibrium, the probability density function (pdf) ρ(x) of x is Gaussian , with variance σ_{x}^{2} = k_{B}T/κ, as prescribed by the equipartition theorem. Here, k_{B} is the Boltzmann constant and T the bath temperature. In this system, we consider the compression process sketched in Fig. 1, in which the stiffness is changed from an initial value to a larger one. The evolution of the system during the relaxation towards the new equilibrium state is monitored through the position pdf ρ(x, t), which is Gaussian at all times (see Methods and Supplementary Information). Thus, the distribution ρ(x, t) is fully characterized by the time evolution of its mean and its standard deviation σ_{x}(t). The main question is now that of finding, provided it exists, a suitable time evolution of the stiffness κ(t) (our control parameter), for which the equilibration process is much faster than τ_{relax}. This question can be affirmatively answered using a particular solution of the Fokker–Planck equation (see Methods and Supplementary Information). We emphasize that the ESE idea is not restricted to manipulating Gaussian states, and that nonharmonic potentials U can be considered, along the lines presented in the Supplementary Information.
In this Letter, two methods are compared. On the one hand, at a given instant t_{i} = 0, we suddenly change κ from the initial value κ_{i} to the final value κ_{f}. During this protocol, referred to as STEP, the particle mean position does not change, whereas the spread σ_{x} equilibrates to the new value in about 3 relaxation times τ_{relax} = γ/κ_{f}. On the other hand, following the ESE procedure, κ(t) is modulated in such a way that σ_{x} is fully equilibrated at t_{f} ≪ τ_{relax}. The protocol which meets our requirements is given by equation (8) (in Methods). In the experiment, we select κ_{i} = 0.5 pN μm^{−1} and κ_{f} = 1.0 pN μm^{−1} in such a way that τ_{relax} ≃ 15 ms. Furthermore, to have a welldefined separation between timescales, we choose t_{f} = 0.5 ms, which is roughly 100 times smaller than the thermalization time in the STEP protocol. Both protocols are shown in Fig. 2a, where we can appreciate the rather complex time dependence of the ESE control procedure. This is a necessity to allow for a quick evolution to the new equilibrium state. The faster the evolution (smaller t_{f}), the stiffer the transient confinement must be (the maximum stiffness reached in Fig. 2a is 37 κ_{i}). To study the evolution of ρ(x, t) for the two protocols, we perform the following cycle. First, the particle is kept at κ_{i} for 50 ms to ensure proper equilibration. Then, at t = 0 ms we apply the protocol (either STEP or ESE) and x(t) is measured for 10 ms in the case of ESE and 100 ms for STEP. Finally, the stiffness is set again to κ_{i} and this cycle is repeated N times. The evolution of σ_{x}(t) for t > 0 is obtained by performing an ensemble average over N = 2 × 10^{4} cycles.
The results are shown in Fig. 2b, where σ_{x}(t) is plotted as a function of time for the two protocols, from one equilibrium configuration to the other. It appears that the engineered system reaches the target spread precisely at t_{f}, and subsequently does not evolve. On the other hand, the STEP equilibration occurs after a time close to 3τ_{relax}. Figure 2c, d represents the complete STEP and ESE dynamics of ρ(x). The Gaussian feature is confirmed experimentally during ESE, even far from equilibrium, as the kurtosis is Kurt(x) = (3.00 ± 0.01). The results of Fig. 2 clearly show the efficiency of ESE, driving the system into equilibrium in a time which is 100 times shorter than the nominal equilibration time 3τ_{relax}.
We now turn our attention to the energy required to achieve such a large time reduction. Developments in the field of stochastic thermodynamics^{29} endow work W and heat Q with a clear mesoscopic meaning, from which a resolution better than k_{B}T can be achieved experimentally (see Methods for an explicit definition). In Fig. 3, the complete energetics of our system is shown for the ESE and STEP protocols. The evolution of the mean cumulative work 〈W(t)〉 reveals the physical behaviour of the system undergoing ESE. In the first part of the protocol (t < 0.2 ms), confinement is increased, which provides positive work to the system. In the subsequent evolution (0.2 < t < 0.5 ms), work is delivered from the system to the environment through the decrease of the stiffness. In striking contrast to an adiabatic transformation, the value of heat increases monotonically, as the system dissipates heat all over the protocol. In the inset of Fig. 3, 〈Q〉 and 〈W〉 are shown for the STEP process. Notice how the work exerted on the system is almost instantaneous, whereas heat is delivered over a wide interval of time, up to complete equilibration. As expected, there is a price to pay for ESE. A significant amount of work is required to speed up the evolution and beat the natural timescale of our system^{22}. It can be shown that the cost 〈W(t_{f})〉 behaves like τ_{relax}/t_{f} for t_{f} → 0. More precisely, this amounts to a time–energy uncertainty relation: t_{f} 〈W(t_{f})〉 ∼ 0.106 (2τ_{relax}) k_{B}T. If instead, one proceeds in a quasistatic fashion (t_{f} ≫ τ_{relax}), the cost reduces to the free energy difference, k_{B}Tlog(κ_{f}/κ_{i})/2, which is 0.35 k_{B}T when κ_{f} = 2κ_{i}. For the ESE experiments shown, we have 〈W(t_{f})〉 ≃ 7.0 k_{B}T, about twenty times larger.
Our results show the feasibility and expediency of accelerated protocols for equilibrating confined Brownian objects. The ESE path allows a gain of two orders of magnitude in the thermalization time, as compared to an abrupt change of control parameter (STEP process). The associated energetic cost has been assessed. Finally, although an overdamped problem has been solved here, the generalization of the ESE protocol to nonisothermal regimes for underdamped systems can in principle be worked out theoretically. Its application to AFM tips, vacuum optical traps, or to transitions between nonequilibrium steady states, constitutes a timely experimental challenge in this emerging field.
Methods
Experimental setup.
Silica microspheres of radius 1 μm were diluted in milliQ water to a final concentration of a few spheres per millilitre. The microspheres were inserted into a fluid chamber, which can be displaced in three dimensions by a piezoelectric device (Nanomax TS MAX313/M). The trap is realized using a nearinfrared laser beam (Lumics, λ = 980 nm with maximum power 500 mW) expanded and injected through an oilimmersed objective (Leica, 63 × NA 1.40) into the fluid chamber. The trapping laser power, which determines the trap stiffness, is modulated by an external voltage V_{κ} by means of a Thorlabs ITC 510 laser diode controller with a switching frequency of 200 kHz. V_{κ} is generated by a National Instrument card (NI PXIe6663) managed by a custommade Labview program. The detection of the particle position is performed using an additional HeNe laser beam (λ = 633 nm), which is expanded and collimated by a telescope and passed through the trapping objective. The forwardscattered detection beam is collected by a condenser (Leica, NA 0.53), and its back focalplane field distribution projected onto a custom position sensitive detector (PSD from First Sensor with a bandpass of 257 kHz) whose signal is acquired at a sampling rate of 20 kHz with a NI PXIe4492 acquisition board.
Energetics measurement.
From the experimental observables, the stiffness κ and the particle position x, it is possible to infer the energetic evolution of our system within the stochastic energetics framework^{29}. The notion of work W is related to the energy exchange stemming from the modification of a given control parameter—here the trap stiffness. Alternatively, heat Q pertains to the energy exchanged with the environment, either by dissipation or by Brownian fluctuations. The work W(t) and dissipated heat Q(t) are expressed as W(t) = ∫ _{0}^{t}(∂U/∂κ)°(dκ/dt′)dt′, Q(t) = −∫ _{0}^{t}(∂U/∂x)°(dx/dt′)dt′, where ° denotes the Stratonovich integral and U is the potential energy. Under this definition, the first law reads as ΔU(t) = W(t) − Q(t), where W(t), Q(t) and ΔU(t) are fluctuating quantities. Because T is fixed, both ESE and STEP processes share the same value 〈ΔU(t_{f})〉 = 0 between the initial and the final state. As a consequence, we have 〈W(t_{f})〉 = 〈Q(t_{f})〉.
ESE protocol for the harmonic potential.
Although the idea is general (as discussed in Supplementary Information), we first start by a presentation applying the method to harmonic confinement. The dynamics of the system is then ruled by the Langevin equation
where a dot denotes time derivative and x is for the position of the Brownian particle. The friction coefficient γ = 6πνR is here constant, ν being the kinetic viscosity coefficient and R the radius of the bead. The diffusion constant then reads as D = k_{B}T/γ. The stiffness κ has an explicit dependence on time and ξ(t) is a white Gaussian noise with autocorrelation 〈ξ(t)ξ(t + t′)〉 = 2δ(t′). Equation (1) is overdamped (there is no acceleration term in ), which is fully justified for colloidal objects^{30}. The Langevin description equation (1) can be recast into the following Fokker–Planck equation for the probability density^{31}:
At initial and final times (t_{i} and t_{f}), ρ(x, t) is Gaussian, as required by equilibrium. A remarkable feature of the ESE (nonequilibrium) solution is that, for intermediate times, ρ(x, t) remains Gaussian,
We demand that
Combining equation (2) with equation (3), we obtain
Requiring that the equality holds for any position x, the equation is simplified into:
This relation was obtained in refs 23,24 by studying the evolution of the variance σ_{x}^{2}. However, unlike in these works, we supplement our description with the constraints , as a fingerprint of equilibrium for both t < 0 and t > t_{f}.
Next, the strategy goes as follows. We choose the time evolution of α, complying with the above boundary conditions. To this end, a simple polynomial dependence of degree 3 is sufficient. Other more complicated choices are also possible. Introducing the rescaled time s = t/t_{f}, we have
where Δκ = κ_{f} − κ_{i}. Finally, equation (6) has been satisfied, from which we infer the appropriate evolution κ(t) that is then implemented in the experiment:
The analysis, restricted here to the onedimensional problem, can be easily recast in three dimensions. It is also straightforward to generalize the idea to account for a timedependent temperature T(t), which can be realized experimentally^{20}. In this latter situation, the key relation equation (6) is unaffected, and therefore indicates how κ should be chosen, for prescribed α(t) and T(t). This highlights the robustness of the ESE protocol.
The mean work exchanged in the course of the transformation takes a simple form in our context:
According to our ansatz (3), 〈x^{2}〉 = 1/(2α(t)), and using the relation (6), equation (9) can be written in the following form^{23,24}:
where τ_{relax} = γ/κ_{f} and η is a numerical factor given by
Notice that equation (10) coincides with expressions derived in previous works^{32,33} using linear response theory.
For our parameters, we find η ≃ 0.106, as indicated in the main text. Interestingly, expression (10) gives the free energy difference value in the limit t_{f} ≫ τ_{relax}, 0.5k_{B}Tlog(κ_{f}/κ_{i}), which appears as the minimal mean work. In the opposite limit, we have a time–energy relation: t_{f}〈W〉 = k_{B}Tτ_{relax}(κ_{f}/κ_{i})η. We emphasize that the scaling in 1/t_{f} when t_{f} → 0 is ansatz independent, although the specific value of the η parameter depends on the ansatz. It can be shown that the lowest η value for all admissible protocols is , which gives here. Thus, our protocol, although suboptimal in terms of mean work, nevertheless has an η value close to the best achievable.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
References
Boltzmann, L. Lectures on Gas Theory (Univ. California Press, 1964).
Torrontegui, E. et al. Shortcuts to adiabaticity. Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
Deffner, S., Jarzynski, C. & del Campo, A. Classical and quantum shortcuts to adiabaticity for scaleinvariant driving. Phys. Rev. X 4, 021013 (2014).
Couvert, A., Kawalec, T., Reinaudi, G. & GuéryOdelin, D. Optimal transport of ultracold atoms in the nonadiabatic regime. Europhys. Lett. 83, 13001 (2008).
Schaff, J.F., Song, X.L., Vignolo, P. & Labeyrie, G. Fast optimal transition between two equilibrium states. Phys. Rev. A 82, 033430 (2010).
Schaff, J.F., Song, X.L., Capuzzi, P., Vignolo, P. & Labeyrie, G. Shortcut to adiabaticity for an interacting Bose–Einstein condensate. Europhys. Lett. 93, 23001 (2011).
Bason, M. G. et al. Highfidelity quantum driving. Nature Phys. 8, 147–152 (2012).
Bowler, R. et al. Coherent diabatic ion transport and separation in a multizone trap array. Phys. Rev. Lett. 109, 080502 (2012).
Walther, A. et al. Controlling fast transport of cold trapped ions. Phys. Rev. Lett. 109, 080501 (2012).
Peercy, P. S. The drive to miniaturization. Nature 406, 1023–1026 (2000).
Maxwell, J. C. On the dynamical theory of gases. Phil. Trans. R. Soc. Lond. 157, 49–88 (1867).
Chen, X. et al. Fast optimal frictionless atom cooling in harmonic traps: shortcut to adiabaticity. Phys. Rev. Lett. 104, 063002 (2010).
GuéryOdelin, D., Muga, J., RuizMontero, M. & Trizac, E. Nonequilibrium solutions of the Boltzmann equation under the action of an external force. Phys. Rev. Lett. 112, 180602 (2014).
Papoular, D. & Stringari, S. Shortcut to adiabaticity for an anisotropic gas containing quantum defects. Phys. Rev. Lett. 115, 025302 (2015).
Tseng, S.Y. & Chen, X. Engineering of fast mode conversion in multimode waveguides. Opt. Lett. 37, 5118–5120 (2012).
Tseng, S.Y. Robust coupledwaveguide devices using shortcuts to adiabaticity. Opt. Lett. 39, 6600–6603 (2014).
Ho, C.P. & Tseng, S.Y. Optimization of adiabaticity in coupledwaveguide devices using shortcuts to adiabaticity. Opt. Lett. 40, 4831–4834 (2015).
Stefanatos, D. Design of a photonic lattice using shortcuts to adiabaticity. Phys. Rev. A 90, 023811 (2014).
Kaka, S. et al. Mutual phaselocking of microwave spin torque nanooscillators. Nature 437, 389–392 (2005).
Martínez, I. A. et al. Brownian Carnot engine. Nature Phys. 12, 67–70 (2016).
Collin, D. et al. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437, 231–234 (2005).
Cui, Y.Y., Chen, X. & Muga, J. G. Transient particle energies in shortcuts to adiabatic expansions of harmonic traps. J. Phys. Chem. A http://dx.doi.org/10.1021/acs.jpca.5b06090 (2015).
Schmiedl, T. & Seifert, U. Optimal finitetime processes in stochastic thermodynamics. Phys. Rev. Lett. 98, 108301 (2007).
Schmiedl, T. & Seifert, U. Efficiency at maximum power: an analytically solvable model for stochastic heat engines. Europhys. Lett. 81, 20003 (2008).
Aurell, E., Gawedzki, K., MejiaMonasterio, C., Mohayaee, R. & MuratoreGinanneschi, P. Refined second law of thermodynamics for fast random processes. J. Stat. Phys. 147, 487–505 (2012).
Acconcia, T. V., Bonança, M. V. S. & Deffner, S. Shortcuts to adiabaticity from linear response theory. Phys. Rev. E 92, 042148 (2015).
Zheng, Y., Campbell, S., De Chiara, G. & Poletti, D. Cost of transitionless driving and work output. Preprint at http://arXiv.org/abs/1509.01882 (2016).
Neuman, K. C. & Block, S. M. Optical trapping. Rev. Sci. Instrum. 75, 2787–2809 (2004).
Sekimoto, K. Stochastic Energetics Vol. 799 (Springer, 2010).
Barrat, J.L. & Hansen, J.P. Basic Concepts for Simple and Complex Liquids (Cambridge Univ. Press, 2003).
Risken, H. The FokkerPlanck Equation (Springer, 1984).
Sivak, D. A. & Crooks, G. E. Thermodynamic metrics and optimal paths. Phys. Rev. Lett. 108, 190602 (2012).
Bonança, M. & Deffner, S. Optimal driving of isothermal processes close to equilibrium. J. Chem. Phys. 140, 244119 (2014).
Acknowledgements
We would like to thank B. Derrida for useful discussions. I.A.M., A.P. and S.C. acknowledge financial support from the European Research Council Grant OUTEFLUCOP.
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Martínez, I., Petrosyan, A., GuéryOdelin, D. et al. Engineered swift equilibration of a Brownian particle. Nature Phys 12, 843–846 (2016). https://doi.org/10.1038/nphys3758
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DOI: https://doi.org/10.1038/nphys3758
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