Shift a laser beam back and forth to exchange heat and work in thermodynamics

Although the equivalence of heat and work has been unveiled since Joule’s ingenious experiment in 1845, they rarely originate from the same source in experiments. In this study, we theoretically and experimentally demonstrated how to use a high-precision optical feedback trap to combine the generation of virtual temperature and potential to simultaneously manipulate the heat and work of a small system. This idea was applied to a microscopic Stirling engine consisting of a Brownian particle under a time-varying confining potential and temperature. The experimental results justified the position and the velocity equipartition theorem, confirmed several theoretically predicted energetics, and revealed the engine efficiency as well as its trade-off relation with the output power. The small theory–experiment discrepancy and high flexibility of the swift change of the particle condition highlight the advantage of this optical technique and prove it to be an efficient way for exploring heat and work-related issues in the modern thermodynamics for small systems.

Here we collect some basic calculations for the particle dynamics in the optical feedback trap in the main text. More general treatments are referred to the literatures of stochastic processes. More discussions about numerical error control can be found in Ref.

A. Langevin equation for two noises
Let us consider the Langevin equation in Sec. II of the main text γẋ + kx = ξ f v + ξ f and its alternative expressionẋ where the thermal noise ξ f and the external noise ξ f v are Gaussian and white, with Following a standard derivation, the particle dynamics starting at x(0) = 0 is Its correlation is (without loss of generality, we assume t ≤ t ) has a unit of "force 2 ·time". Throughout this calculation, we keep ·1 and /1 to remind the correct dimension. For t = t, Two special cases of Eq. (S3): 1. For a free particle (k → 0) at t < ∞ , Here, the noise strength, A + A v , has been expressed as a "diffusion constant" D through the relation For ξ f (t) = A = 0 , assuming an analogous relation x 2 (t) = k B Tv k , one can define As a result, the correlation functions Eq. (S2) can be reexpressed as where g(1) is a Gaussian white noise of zero mean and unite variance.

B. A difference equation of the Langevin equation
Equation (S1) has a difference equation (See Sekimoto for more general cases) with two random variables ξ f v and ξ f to be determined and a short time-span ∆. The particle dynamics following this equation is where Multiplying these equations by proper Ω n , one obtains After summing up all these equations, we get the particle position after N steps of duration ∆ at time t = N ∆: where Ξ denotes the sum of all terms in the parenthesis [. . .]. For x 0 = 0, where Z ≡ Ω 2(N −1) + Ω 2(N −2) + . . . + Ω 2 + 1. (S14) Two special cases: 1. For a free particle ( k = 0 ) and t < ∞, which gives 2. For a confined particle ( k > 0 ) and t → ∞ , Eq. (S14) implies where normally Ω = 1 − ∆ k γ ≈ 0.9 . . . < 1 , because ∆ k γ needs to be sufficiently small.

C. Consistency between differential and difference equations
Since γ 2 from Eq. (S15) must describe the same particle dynamics as x 2 (t) In analogy to the arguments for Eq. (S5) and (S6), and subsequently respectively, as in the main text.

D. The dynamics for the center of the optical tweezers
Given ∆ = t u the acquisition time in the main text, the differential equation Eq. (S1), with ξ f v ≡ 2γk B Tv 1 g (1) and ξ f ≡ 2γk B T 1 g (1) in Eq. (S9), has a difference equation Eq. (S10),