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Dynamic relaxation of a levitated nanoparticlefrom a non-equilibrium steady stateJan Gieseler1, Romain Quidant1,2, Christoph Dellago3* and Lukas Novotny4*

Fluctuation theorems are a generalization of thermodynamics on small scales and provide the tools to characterize thefluctuations of thermodynamic quantities in non-equilibrium nanoscale systems. They are particularly important forunderstanding irreversibility and the second law in fundamental chemical and biological processes that are actively driven,thus operating far from thermal equilibrium. Here, we apply the framework of fluctuation theorems to investigate theimportant case of a system relaxing from a non-equilibrium state towards equilibrium. Using a vacuum-trappednanoparticle, we demonstrate experimentally the validity of a fluctuation theorem for the relative entropy change occurringduring relaxation from a non-equilibrium steady state. The platform established here allows non-equilibrium fluctuationtheorems to be studied experimentally for arbitrary steady states and can be extended to investigate quantum fluctuationtheorems as well as systems that do not obey detailed balance.

One of the tenets of statistical physics is the central limittheorem. It allows systems with many microscopic degreesof freedom to be reduced to only a few macroscopic thermo-

dynamic variables. The central limit theorem states that, indepen-dently of the distribution of microscopic variables, a macroscopicextensive quantity U, such as the total energy of a system withN degrees of freedom, follows a Gaussian distribution with meankUl/N and variance sU

2 /N. Consequently, for large N, the rela-tive fluctuations sU /kUl vanish and the macroscopic quantitybecomes sharp. With the advance of nanotechnology it is now poss-ible to study experimentally systems small enough that the relativefluctuations become comparable to the mean value. This gives riseto new physics where transient fluctuations may run counter tothe expectations of the second law of thermodynamics1.

The statistical properties of the fluctuations of thermodynamicquantities like heat, work and entropy production are describedby exact relations known as fluctuation theorems2–5, which allowus to express the inequalities familiar from macroscopic thermodyn-amics as equalities6,7. Fluctuation relations are particularly impor-tant for understanding fundamental chemical and biologicalprocesses, which occur on the mesoscale where the dynamics aredominated by thermal fluctuations8. For example, they allow us torelate the work along non-equilibrium trajectories to thermodyn-amic free-energy differences9,10. Fluctuation theorems have beentested experimentally on a variety of systems, including pendu-lums11, trapped microspheres1, electric circuits12, electron tunnel-ling13,14, two-level systems15 and single molecules16,17. Most ofthese experiments are described by an overdamped Langevinequation. However, systems in the underdamped regime18, or inquantum systems19 where the concept of a classical trajectoryloses its meaning, are less explored.

Here, we study the thermal relaxation of a highly underdampednanomechanical oscillator from a non-equilibrium steady statetowards equilibrium. Because of the low damping of our system,the dynamics can be precisely controlled, even at the quantumlevel20–22. This high level of control allows us to produce non-thermal steady states and makes nanomechanical oscillators ideal

candidates for investigating non-equilibrium fluctuations for tran-sitions between arbitrary steady states. Although for the initialsteady state, detailed balance is violated, the relaxation dynamicsare described by a microscopically reversible Langevin equationthat satisfies detailed balance23. Under these conditions, a transientfluctuation relation holds7,24 for the relative entropy change character-izing the irreversibility of the relaxation process. Similar relations holdalso for relaxation processes in ageing systems as studied both theor-etically25 and experimentally26–28 in gels and glasses. For the initialnon-equilibrium steady state generated in our experiment we derivean analytical expression for the phase-space distribution, which is

x

Parametric driveΩmod, ε

FeedbackΣ ← Δϕ ← 2Ω0

y

z

ton

toff

+

Figure 1 | Experimental set-up. A nanoparticle is trapped by a tightly

focused laser beam in high vacuum. In a first experiment, the nanoparticle is

initially cooled by parametric feedback. At time t¼ toff, the feedback is

switched off and the nanoparticle trajectory is followed as it relaxes to

equilibrium. After relaxation, the feedback is switched on again and the

experiment is repeated. In a second experiment, the nanoparticle is initially

excited by an external modulation of frequency Vmod in addition to feedback

cooling. Again at a time t¼ toff, both the feedback and the external

modulation are switched off and the nanoparticle is monitored as it relaxes.

1ICFO–Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, 2ICREA–Institucio Catalana de Recerca iEstudis Avancats, 08010 Barcelona, Spain, 3University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria, 4ETH Zurich, PhotonicsLaboratory, 8093 Zurich, Switzerland. *e-mail: [email protected]; [email protected]

ARTICLESPUBLISHED ONLINE: 30 MARCH 2014 | DOI: 10.1038/NNANO.2014.40

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in excellent agreement with the experimental data and directly vali-dates the fluctuation theorem. Our experimental framework can beextended to study transitions between arbitrary steady states and, fur-thermore, lends itself to the experimental investigation of quantumfluctuation theorems29 for nanomechanical oscillators20–22.

The experimental set-up is shown in Fig. 1. We consider a silicananoparticle of radius r ≈ 75 nm and mass m ≈ 3 × 10218 kg that istrapped in vacuum by the gradient force of a focused laser beam.Within the trap, the nanoparticle oscillates in all three spatial direc-tions. To a first approximation, the three motional degrees offreedom are well decoupled. Hence, the time evolution of theparticle position x is described by the one-dimensionalLangevin equation

x + G0x +V20x = 1

mF fluct + Fext

( )(1)

where V0/2p≈ 125 kHz is the particle’s frequency along the directionof interest, G0 is the friction coefficient and Fext is an externally appliedforce. The random nature of the collisions does not only provide deter-ministic damping G0, but also a stochastic force Ffluct, which therma-lizes the energy of the nanoparticle. The fluctuation–dissipation theorem links the damping rate intimately to the strengthof the stochastic force, F fluct(t) =

2mG0 kBT0

√j(t), where T0, kB and

j(t) are the bath temperature, Boltzmann’s constant and white noisecorresponding to kj(t)l¼ 0 and kj(t)j(t′)l¼ d(t 2 t′).

The total energy of the harmonically oscillating nanoparticle isgiven by

E(x, p) = 12

mV20x2 + p2

2m= 1

2mV2

0x(t)2 (2)

where x is the displacement from the trap centre and p is themomentum. The second equality in the above equation followsfrom the slowly varying amplitude approximation,x(t) = x sin(V0t), x ≪ V0x. This approximation is well satisfiedin our experiments because it takes many oscillation periods forthe oscillation amplitude to change appreciably (Fig. 2a, inset).

Applying a time-dependent external force Fext for a sufficientlylong time, the system is initially prepared in a non-equilibriumsteady state with distribution rss(u, a), which, in general, is notknown analytically. Here, u specifies the state of the system and adenotes one or several parameters that determine the initialsteady-state distribution, such as the strength of the external force.At time t¼ toff the external force is switched off and we followthe evolution of the undisturbed system. In this relaxation phase(external force Fext off ) the dynamics satisfies detailed balancewith respect to the equilibrium distribution req/ exp(2b0E(u))at reciprocal temperature b0¼ 1/kBT0. As shown be Evans andSearles24,30 for thermostatted dynamics and by Seifert7 for stochasticdynamics, the time reversibility of the underlying dynamics impliesthe transient fluctuation theorem

0 s

0.1 s

0.9 s

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×10−3

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50

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−150 −75 0 75 150

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10−5

10−3

10−1

ρ(x)

(nm

−1)

ρ(x)

(nm

−1)

ρ(x)

(nm

−1)

p(x)

(nm

–1)

−150

75

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150

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tion

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)

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0

2

4

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8

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1.0a

b

c

dE

/ k BT

0E

/ k BT

0

toff

Tfb

FitExperimental

40

0

x (n

m)

Time (ms)

−400.100.050

0 0.2 0.4 0.6 0.8Time (s)

−15

Figure 2 | Relaxation from a non-equilibrium steady state generated by parametric feedback cooling. The initial non-equilibrium temperature is Tfb.

At time toff, the feedback is switched off and the particle energy relaxes to the equilibrium energy kBT0. a, Time evolution of the average energy evaluated

from 104 individual experiments. The red dashed line is a fit according to equation (10). Inset: The particle oscillates with constant amplitude on short

timescales. b, Four different realizations of the relaxation experiment. Each run yields a different trajectory and the time it takes for the particle to acquire an

energy of kBT0 deviates considerably from the ensemble average shown by the blue curve in a. c, Time evolution of the position distribution, shown as a

density plot. d, Position distributions evaluated at three different times. Distributions correspond to vertical cross-sections in c. Superimposed red curves in

top and bottom panels are theoretical distributions. The initial distribution deviates notably from a thermal equilibrium distribution with the same average

energy (grey dashed line in top panel). The inset in d (top panel) shows a zoom-in of the top region of the distribution r(x) highlighting the deviation from a

thermal distribution.

NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2014.40 ARTICLES

NATURE NANOTECHNOLOGY | VOL 9 | MAY 2014 | www.nature.com/naturenanotechnology 359




p(−DS)/p(DS) = e−DS (3)

holding for the relative entropy change

DS = b0Q + Df (4)

Here, Q is the heat absorbed by the bath at reciprocal temperatureb0. Because no work is done on the system, the heat Q exchangedalong a trajectory of length t starting at u0 and ending at ut equalsthe energy lost by the system, Q¼2[E(ut) 2 E(u0)]. The quantityDf¼ f(ut) 2 f(u0) is the difference of the trajectory-dependententropy f(u)¼2ln rss(u, a) (ref. 31) between the initial andfinal states of the trajectory. Thus, DS is the change in relativeentropy32, or Kullback–Leibler divergence, between the initialsteady-state distribution and the equilibrium distribution observedalong a particular trajectory. Note that the fluctuation theorem (3)holds for any time t at which DS is evaluated and it is not requiredthat the system reaches the equilibrium distribution at time t. Therelative entropy change, which equals the dissipation function intro-duced by Evans and Searles for thermostatted dynamics24,30,33, is thelogarithmic ratio of the probability to observe a particular trajectoryand the probability of the corresponding time-reversedtrajectory7,34,35. As such, DS can be viewed as a measure of theirreversibility occurring during the relaxation process.

From the detailed fluctuation theorem of equation (3), theintegral fluctuation theorem

ke−DSl = 1 (5)

directly follows. Through Jensen’s inequality, the convexity of theexponential function implies the second law-like inequality

kDSl ≥ 0 (6)

such that the average relative entropy change is non-negative. Theaverage relative entropy change is related to the total entropychange of the oscillator and bath together by7

kDSl = DStot + D(rt‖rss) (7)

where D(rt‖rss) is the relative entropy of the statistical state of thesystem at time t with respect to the initial steady-state distribution.Slightly modifying the definition of DS, one can also derive a differ-ent but related integral fluctuation theorem7,31,36, from which thenon-negativity of the total entropy change follows, DStot ≥ 0, pro-viding a direct link to the second law of thermodynamics.However, no detailed fluctuation theorem holds for this case.Analogous fluctuation relations for the total entropy productionhave also been verified for two coupled systems kept in a non-equilibrium steady state by holding each system at a differenttemperature37,38. For further discussion of the fluctuation theoremand the significance of DS, see Supplementary Information, Section 2.

If the initial steady-state distribution is an equilibrium distri-bution, rss(u, a)¼ e2b[E(u)2F(b)], corresponding to a temperatureT¼ 1/kBb and with free energy F(b)¼2kBT ln

du e2bE(u),

the expressions become particularly simple and the fluctuationtheorem for DS acquires a physically very transparent meaning.In this case, f(u)¼ b[E(u) 2 F(b)], such that DS¼ (b0 – b)Qand the fluctuation theorem simplifies to p(2Q)/p(Q)¼exp2(b0 2 b)Q. Note that this particular fluctuation expressionfor the special case of transitions between equilibrium states hasbeen obtained earlier39 and was shown experimentally to holdalso in the case of an ageing bath27. As a consequence of this fluctu-ation relation for the heat, the probability of observing energyflowing from the colder system to the hotter bath is exponentiallysmall compared with the probability of observing energy transferin the other direction. Because Q is an extensive quantity, irreversi-bility for macroscopic systems is a direct consequence of the

ExperimentalTheory

0 2 4 6 80

2

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6

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1

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2

4

5

20 ms50 ms100 ms

150 ms300 ms

−4 −2 2 410−4

10−2ρ fb(

E)

E/kBT0

100

a b

c d

100

10−1

10−2

10−3

0 0.1 0.2 0.3 0.4

ExperimentalFitThermal

0Δ

Δ Δ

∑(Δ

)

p(Δ

)∑(

p(Δ

)t)

Figure 3 | Fluctuation theorem for the relaxation experiment in Fig. 2. a, Energy distribution with feedback on (red circles). The black solid curve is a fit

according to equation (11). Large-amplitude oscillations experience stronger damping and are therefore suppressed relative to an equilibrium distribution

(grey dashed line). b, Probability density p(DS) evaluated for different times after switching off the feedback. c, Function S(DS) evaluated for the

distributions shown in b. d, Function S evaluated for the time-averaged distributions kp(DS)lt. The data are in good agreement with the fluctuation theorem

of equation (3) (black dashed line).

ARTICLES NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2014.40





fluctuation theorem. The integral fluctuation theorem for the rela-tive entropy change further implies that (b0 2 b)kQl ≥ 0, suchthat heat flows from hot to cold on average, in line with thesecond law of thermodynamics.

In the following, we experimentally investigate the fluctuationtheorem (3) for two different initial non-equilibrium steady-state dis-tributions. The first steady state is generated by parametric feedbackcooling (ss¼ fb) and the second one by external modulation (ss¼mod) in addition to feedback cooling. In the case of parametric feed-back cooling we enforce a non-equilibrium state by applying a forceFext¼ Ffb to the oscillating particle through a parametric feedbackscheme (cf. Fig. 1)40. The feedback Ffb = −hmV0x2x adds a colddamping Gfb to the natural damping G0. Here, the parameter hdefines the strength of the feedback. Note that parametric feedbackis different from thermal damping, where an increased damping isaccompanied by an increase in fluctuations. Because parametric feed-back adds an amplitude-dependent damping Gfb/ x2, oscillationswith a large amplitude experience a stronger damping than oscil-lations with a small amplitude. As a consequence, the position distri-bution is non-Gaussian and assumes the form (see SupplementaryInformation, equation 69)

rfb(x,a) =

b0 mV2

0(4 + amV20x2)

8p3

√exp −b0(4+amV2

0x2)2

32a

[ ]erfc

b0/a

√( )× K1/4

b0(4 + amV20x2)2

32a

[ ] (8)

where a¼ h/mG0V0, and erfc and K1/4 are the complementaryerror function and a generalized Bessel function of the secondkind, respectively. In analogy to the thermal equilibrium tempera-ture of the harmonic oscillator, we define an effective temperatureTfb¼ kElfb/kB of the system. Here kElfb denotes the average energywith feedback on. Using distribution (8) to calculate the averageenergy we find the effective temperature

Tfb = T0 2

b0

a

√e−b0/a

p√

erfcb0/a

√( )− 2b0

a

≈

4mG0V0T0

pkBh

√(9)

where the approximation holds for Tfb/T0 ≪ 1.At time t¼ toff , the feedback is switched off and the system

relaxes back to the thermal equilibrium distribution at temperatureT0. The experimental data for this relaxation process are shown inFig. 2c,d. Without the feedback, the collisions with the surroundingmolecules are no longer compensated and the oscillator energyincreases. Exploiting that at low friction the oscillator energychanges slowly, one finds from equations (1) and (2) that the timeevolution of the energy is governed by E = −G0(E − kBT0)+

2EG0 kBT0

√j(t). An average over noise then yields the differential

equation kEl = −G0(kEl − kBT0), which implies that the averageenergy of the oscillator relaxes exponentially to the equilibriumvalue kBT0,

kE(t)l = kBT0 + kB(Tss − T0)e−G0t (10)

HarmonicExperimental

0.1 s

0 s

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k BT0

E /

k BT0

ρ(x)

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−1)

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(nm

−1)

ρ(x)

(nm

−1)

−150 −75 0 75 150Position (nm)

−150 −75 0 75 150Position (nm)

−150 −75 0 75 150Position (nm)

Figure 4 | Relaxation from a non-equilibrium steady state generated by external parametric modulation. The initial effective non-equilibrium temperature is

kBTeff. At time toff, the feedback is switched off and the particle energy relaxes to the equilibrium energy kBT0. a, Time evolution of the average energy evaluated

from repeated individual experiments. The red dashed line is a fit according to equation (10). b, Four different realizations of the relaxation experiment.

Each run yields a different trajectory and the time it takes for the particle to acquire an energy of kBT0 deviates considerably from the ensemble average

shown by the blue curve in a. c, Time evolution of the position distribution shown as a density plot. d, Position distributions evaluated at three different times.

Distributions correspond to vertical cross-sections in c. Superimposed red curves in top and bottom panels are theoretical distributions. The initial distribution

features a sharply peaked double-lobe distribution, characteristic for a harmonic oscillator at constant energy. As the system evolves, the two peaks smear

out and merge into a single Gaussian distribution.






where Tss denotes an arbitrary initial steady-state temperature, forexample Tfb.

To verify this equation, we repeated the relaxation experiment104 times. Each time, the same initial distribution rfb(u0, a) wasestablished by parametric feedback and, after switching off the feed-back, the system was followed as it evolved from u0 to ut within timet. Along each 1 s trajectory we sampled the particle position at arate of 625 kHz and, from integration over 64 successive positionmeasurements, we obtained the energy at a rate of 9.8 kHz. InFig. 2a we show the average over the individual time traces togetherwith a fit to equation (10). Equilibrium is reached after a time oforder t0¼ 1/G0¼ 0.17 s. According to equation (10) and the datashown in Fig. 2, the average energy of the particle increases mono-tonically. However, due to the small size of the particle, the fluctu-ating part,

2EG0 kBT0

√j(t), is comparable to the deterministic part

2G0(E 2 kBT0), so an individual trajectory can be quite differentfrom the ensemble average of equation (10). Figure 2b shows fourrealizations of the relaxation experiment. Each particle trajectoryx(t) results from switching off the feedback at initial time t¼ toff.

The 104 trajectories allow us to evaluate the distributionspfb(DS) = kd[DS − DS(ut)]lfb for different times t. Here, the sub-script ‘fb’ denotes the average over the initial distributions obtainedunder the action of feedback. For this initial non-equilibrium steadystate, the energy distribution is calculated analytically as (seeSupplementary Information, equation 64)

rfb(E,a) =ab0

p

√exp −b0/a

( )erfc

b0/a

√( ) exp −b0 E + a

4E2

[ ]( )(11)

This distribution has the form of a Boltzmann–Gibbs distributionfor the generalized energy Eþ aE2/4, where the term aE2/4arises from the feedback and strongly penalizes high energystates. It is consistent with the phonon number distribution of anoptomechanical system with a quadratic coupling term41.

Inserting the above distribution into equation (4) we find that, forthe relaxation from rfb , the relative entropy change is given byDS = b0a E2

t − E20

( )/4. In this case, the integral fluctuation

theorem implies that kDE2l ≥ 0; that is, the average of the squaredenergy does not decrease during the relaxation process. Figure 3ashows the measured steady-state distribution of the energy to bein excellent agreement with the prediction of equation (11). Forsmall energies, the measured distribution features a small dipcaused by measurement noise. For comparison, we also show thecorresponding equilibrium distribution with the same averageenergy (grey dashed line). It is evident that it deviates strikinglyfrom the true distribution rfb(E, a). In Fig. 3b we plot the distri-butions rfb(DS) for different times t. They become increasinglyasymmetric for long times, with higher probabilities for positiveDS and lower probabilities for negative DS. To test fluctuationtheorem (3) for our measurements we define

S(DS) = lnp(DS)

p(−DS)

[ ]= DS (12)

where S(DS) is predicted to be time-independent. Using thedistributions for DS shown in Fig. 3b, we compute S(DS) andshow the resulting data in Fig. 3c. Because the fluctuationtheorem (3) is time-independent, we evaluate the time-average foreach r(DS) in Fig. 3c and render it in the plot shown in Fig. 2d.The averaging improves the statistics and leads to excellent agree-ment with the fluctuation theorem for DS. The offset for smallDS results from measurement noise.

The experimental scheme introduced here allows us to studynon-equilibrium processes for arbitrary initial states and for arbi-trary transitions between states. To demonstrate that the fluctuationtheorem holds for arbitrary non-equilibrium initial states, we applyan external harmonic drive signal in addition to the parametricfeedback as illustrated in Fig. 1. The harmonic drive generates aforce Fmod¼ emV0

2 cos(Vmodt)x acting on the nanoparticle, with

ExperimentalTheory

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150 ms300 ms

0 1 2 3 4 50

1

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2.0 2.5 3.0 3.5 4.0

100

a b

c d

10−1

10−2

10−3

10−1

10−2

10−3

ρ fb(

E)

E/kBT0 Δ

ΔΔp(

Δ )

∑(Δ

)

∑(p(

Δ)

t)

Figure 5 | Fluctuation theorem for the relaxation experiment of Fig. 4. a, The energy distribution with external modulation on (red circles) differs

significantly from an equilibrium distribution with identical average energy (grey dashed line). b, Probability density p(DS) evaluated for different times after

switching off the modulation. c, Function S(DS) evaluated for the distributions shown in b. d, Function S evaluated for the time-averaged distributions

kp(DS)lt. The data are in good agreement with the fluctuation theorem of equation (3) (black dashed line).






modulation frequency Vmod/2p¼ 249 kHz and modulation depthe ¼ 0.03. Modulation at Vmod brings the particle into oscillationat frequency 124.5 kHz and amplitude x. The resulting steady-state position distribution rmod(x) deviates strongly from an equili-brium Gaussian distribution and resembles the characteristicdouble-lobe function

rmod(x) =p−1

x2 − x2√ (13)

of a harmonic oscillator with constant energy. As in the previousexperiment, at t¼ toff the modulation and the feedback are switchedoff, and the nanoparticle dynamics is measured during relaxation.Figure 4 shows the relaxation of the particle’s average energy andthe evolution of the position distribution.

Due to the additional driving, the average initial energy is largerthan the thermal energy kBT0. After the driving is switched off, theaverage energy relaxes exponentially to the equilibrium valueaccording to equation (10). As in the previous experiment, individ-ual realizations of the switching experiment differ significantly fromthe average (Fig. 4b). As the system relaxes, the two lobes of theinitial position distribution broaden until they merge into a singleGaussian peak corresponding to temperature T0.

In the case of parametric modulation, the form of the initialenergy distribution rmod(E) is not known analytically and thereforeneeds to be determined experimentally. Using the measured initialdistribution together with the energies E0 and Et evaluated attimes 0 and t, respectively, we calculate DS¼ b0Qþ Df.Figure 5a shows the initial energy distribution rmod(E), which hasa narrow spread around a non-zero value and therefore differssignificantly from a thermal distribution with identical effectivetemperature (grey dashed line). The measured distributions of DSevaluated at different times after switching off the modulation areshown in Fig. 5b. As before, we use the distributions p(DS) toevaluate S(DS) and plot it in Fig. 5c. To reduce the variance wetime-average the distributions p(DS) and plot the correspondingS function in Fig. 5d. As in the previous experiment, we find excel-lent agreement with theory (black dashed line), providing solidexperimental validation for the fluctuation theorem (3) beingvalid for initial steady states that are out of equilibrium.

In conclusion, we have experimentally demonstrated the validityof a fluctuation theorem for the relaxation from a non-equilbriumstate towards equilibrium. The theorem holds for the relativeentropy change DS, which is related (but not identical) to thetotal entropy production. Using a levitated nanoparticle in highvacuum we have verified the fluctuation theorem for differentinitial non-equilibrium states, demonstrating that this theoreticalframework can be used to understand fluctuations in nanoscalesystems. Our experimental approach allows us to measure thedynamics of a nanoparticle during relaxation from an arbitraryinitial state and to study its statistical properties. We succeeded inderiving an analytic expression for the non-equilibrium steadystate under the action of a feedback force and demonstrated excel-lent agreement with experimental data. The presented experimentalframework naturally extends to the study of transitions betweenarbitrary steady states and to quantum fluctuation theorems,similar to recent proposals for trapped ions19,29. We envision thatour approach of using highly controllable nanomechanical oscil-lators will open up experimental and theoretical studies of fluctu-ation theorems in complex settings, which arise, for instance,from the interplay of thermal fluctuations and nonlinearities42

where detailed balance does not hold43,44. Furthermore, it servesas an experimental simulator platform in analogy to quantumsimulators based on ultracold gases, superconducting circuits ortrapped ions45.

Received 5 November 2013; accepted 6 February 2014;published online 30 March 2014

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AcknowledgementsThis research was supported by ETH Zurich, ERC-QMES (no. 338763), ERC-Plasmolight(no. 259196), Fundacio Privada CELLEX and the Austrian Science Fund (FWF) within theSFB ViCoM (grant F41). The authors acknowledge support from the ESF NetworkExploring the Physics of Small Devices.

Author contributionsL.N. and J.G. conceived and designed the experiments. J.G. performed the experiments.J.G., C.D. and L.N. analysed the data. C.D. developed the theoretical framework. R.Q.contributed materials/analysis tools. J.G., C.D. and L.N. co-wrote the paper.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to L.N. and C.D.

Competing financial interestsThe authors declare no competing financial interests.





http://www.nature.com/reprints



SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2014.40

NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology 1

Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state

Supplementary Information for

Dynamic Relaxation of a Levitated Nanoparticle from a

Non-Equilibrium Steady State

Jan Gieseler1, Romain Quidant1,2, Christoph Dellago3 and Lukas Novotny4

1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860

Castelldefels (Barcelona), Spain

2ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

3 University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria

4 ETH Zürich, Photonics Laboratory, 8093 Zürich, Switzerland

Abstract

The supplementary information provides the theory of parametric feedback cooling

and a discussion of the fluctuation theorem, which is demonstrated experimentally in

the main text.

Contents

1 Theory of parametric feedback cooling S2

1.1 Stochastic differential equation for the energy . . . . . . . . . . . . . . . S3

1.2 Energy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S7

1.3 Effective temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S9

1.4 Relaxation of average energy . . . . . . . . . . . . . . . . . . . . . . . . S11

1.5 Phase space distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . S12

2 Fluctuation theorem for ∆S S16

2.1 Relative entropy change ∆S . . . . . . . . . . . . . . . . . . . . . . . . . S16

2.2 Detailed fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . . . S19

2.3 Integral fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . . . . S20

2.4 Relaxation from an initial equilibrium state . . . . . . . . . . . . . . . . S21

S1



2.5 Relaxation from a steady state generated by parametric feedback . . . . S22

2.6 Distributions of ∆S for t → ∞ . . . . . . . . . . . . . . . . . . . . . . . S22

1 Theory of parametric feedback cooling

The trapped particle experiences both the trap force generated by the laser as well as a

viscous force (Stokes force) due to the random impact of gas molecules. For small oscil-

lation amplitudes, the trapping potential is harmonic and the three spatial dimensions

are decoupled. Each direction can be characterised by a frequency Ω0, which is defined

by the particle mass m and the trap stiffness k as Ω0 =√

k/m. The equation of motion

for the particle’s motion along q = x, y, z is therefore

q(t) + Γ0 q(t) + Ω20 q(t) =

1

m[Ffluct(t) + Ffb(t)] , (1)

where Γ0 is the friction coefficient and Ffluct is a random Langevin force that satisfies the

fluctuation dissipation theorem 〈Ffluct(t)Ffluct(t′)〉 = 2mΓ0 kBT0 δ(t− t′). In the above

equation, Fopt(t) = Ω0ηq2p is a time-varying, non-conservative optical force introduced

by parametric feedback of strength η and p = mq is the momentum.

Starting from Equ. (1), we now derive a stochastic differential equation for the energy

E(q, p) =1

2mΩ2

0q2 +

p2

2m(2)

in the limit of a highly underdamped system (Q = Ω0/Γ0 ≫ 1). As shown below, in

the low friction limit, the stochastic equation of motion for the energy (or rather, for

the square root of the energy) can be written in a form which resembles over-damped

Brownian motion in energy space. As a result, we can consider the energy as the only

relevant variable1. As a side product, we also obtain the energy and the position distri-

butions in the non-equilibrium steady state generated by the application of the feedback

loop. In fact, in energy space the dynamics of the system with feedback can be viewed

as an equilibrium dynamics occurring in a system with an additional force term.

1note that this works only in the low friction limit

S2


1.1 Stochastic differential equation for the energy

For the developments below it is more convenient to write (1) as a stochastic differential

equation (SDE),

dq =p

mdt, (3a)

dp = (−mΩ20q − Γ0p− Ω0ηq

2p)dt+√

2mΓ0kBT0dW, (3b)

where W (t) is a Wiener process with

〈W (t)〉 = 0, (4a)

〈W (t)W (t′)〉 = t′ − t. (4b)

Note that in particular 〈W 2(t)〉 = t for any time t ≥ 0. Accordingly, for a short

(infinitesimal) time interval dt we have

〈dW 〉 = 0, (5)

〈(dW)2〉 = dt. (6)

The time derivative of the Wiener process, ξ(t) = dW (t)/dt, is white noise and it is

related to the random force by Ffluct(t) =√2mΓ0kBT0ξ(t).

We determine the energy change dE that occurs during the short time interval dt

during which position and momentum change by dq and dp as specified by the equations

of motion (3a) and (3b). To lowest order, the energy change is given by

dE =

(

∂E

∂q

)

dq +

(

∂E

∂p

)

dp+1

2

(

∂2E

∂p2

)

(dp)2. (7)

Note that this equation differs from the usual chain rule because we have to keep the

term proportional to (dp)2. The reason is that according to Equ. (3b), dp depends on

dW which is of order√dt. Hence, if we want to keep all terms at last up to order dt,

we cannot neglect the second order term in the above equation because (dp)2 is of order

dt. In contrast, we can safely neglect the terms proprtional to (dq)2 and dqdp, because

they are of order dt2 and (dt)3/2, respectively.

Computing the derivatives of the energy with respect to q and p we obtain

dE = mΩ20qdq +

p

mdp+

1

2m(dp)2. (8)

S3


Inserting dq and dp from Eqs. (3a) and (3b) and neglecting all terms of order (dt)3/2

or higher yields

dE = −m(Γ0 +Ω0ηq2)( p

m

)2dt+

p

m

√

2mΓ0kBT0dW + Γ0kBT0dW2. (9)

To avoid the multiplicative noise of Equ. (9) we consider the variable ǫ =√E

instead of the energy E. The change dǫ due to the changes dq and dp occurring during

an infinitesimal time interval dt is given by

dǫ =

(

∂ǫ

∂q

)

dq +

(

∂ǫ

∂p

)

dp+1

2

(

∂2ǫ

∂p2

)

(dp)2 (10)

as all other terms are of order (dt)3/2 or higher. Evaluation of the partial derivatives

yields

dǫ = mΩ20

q

2ǫdq +

1

2ǫ

p

mdp+

1

2

(

1

2mǫ− 1

4ǫ3p2

m2

)

(dp)2. (11)

Using the equations of motion (3a) and (3b) and exploiting that (dp)2 = 2mΓ0kBT0(dW )2

up to order dt we obtain

dǫ = −Γ0 +Ω0ηq2

2ǫ

p2

mdt+

√2mΓ0kBT0

2ǫ

p

mdW +

Γ0kBT0

2ǫ

(

1− p2

2mǫ2

)

(dW )2. (12)

We now integrate this equation over an oscillation period τ = 2π/Ω0 to obtain the

change ∆ǫ =∫ τ0 dǫ over one oscillation period,

∆ǫ = −Γ0

2

∫ τ

0

p2

mǫdt− Ω0η

2

∫ τ

0

q2p2

mǫdt+

√

2mΓ0kBT0

∫ τ

0

p

2mǫdW (13)

+Γ0kBT0

∫ τ

0

1

2ǫ

(

1− p2

2mǫ2

)

(dW )2. (14)

To compute the integrals on the right hand side of the above equation, we assume

that in the low-friction limit the energy E, and hence also ǫ remains essentially constant

over one oscillation period. We also assume that the feedback mechanism changes the

energy of the system slowly and that the motion of the system during one oscillation

period is practically not affected by the feedback either. In the low friction regime, where

the coupling to the bath is weak, a small feedback strength (i.e., a small η) is sufficient

for considerable cooling. Accordingly, during one oscillation period the position q and

the momentum p are assumed to evolve freely:

q(t) = ǫ

√

2

mΩ20

sinΩ0t, (15)

p(t) = mq(t) = ǫ√2m cos Ω0t, (16)

S4


where we have selected the phase of the oscillation such that the position q = 0 at time

0. Hence, the first two integrals of (14) are given by

∫ τ

0

p2

mǫdt = 2ǫ

∫ τ

0cos2Ω0t dt = ǫτ (17)

and∫ τ

0

q2p2

mǫdt =

4ǫ3

mΩ20

∫ τ

0sin2Ω0t cos

2 Ω0t dt =ǫ3τ

2Ω20m

. (18)

Insertion of these results and of the harmonic expressions for q(t) and p(t) from above

into Equ. (14) gives

∆ǫ = −Γ0ǫ

2τ − ηǫ3

4mΩ0τ +

√

Γ0kBT0∆R1 +Γ0kBT0

2ǫ∆R2. (19)

where ∆R1 and ∆R2 are given by

∆R1 =

∫ τ

0cos Ω0t dW (20)

and

∆R2 =

∫ τ

0sin2 Ω0t (dW )2. (21)

Since W (t) is a Wiener process, ∆R1 and ∆R2 are random numbers. Next we will

determine the statistical properties of ∆R1 and ∆R2.

As ∆R1 is the result of a (weighted) sum of Gaussian random numbers, it will be a

Gaussian random number, too. The mean of ∆R1 is given by

〈∆R1〉 = 〈∫ τ

0cos Ω0t dW 〉 =

∫ τ

0cos Ω0t〈dW 〉 = 0, (22)

where the angular brackets imply an average over all noise histories. The variance of

∆R1 is given by

〈(∆R1)2〉 = 〈

∫ τ

0cos Ω0t dW

∫ τ

0cos Ω0t

′ dW ′〉

=

∫ τ

0

∫ τ

0cos Ω0t cos Ω0t

′〈dWdW ′〉

=

∫ τ

0cos2 Ω0t dt =

τ

2, (23)

where we have exploited that 〈dWdW ′〉 = δ(t′ − t)dt. Hence, the random variable ∆R1

can be written as

∆R1 =

√

1

2W (τ), (24)

S5


where W (τ) is a Wiener process at τ , i.e., a Gaussian random variable with variance τ .

In a similar way, we can show that the mean of ∆R2 is given by

〈∆R2〉 = 〈∫ τ

0sin2Ω0t (dW )2〉

=

∫ τ

0sin2Ω0t〈(dW )2〉 (25)

=

∫ τ

0sin2Ω0tdt =

τ

2,

because of 〈(dW )2〉 = dt. For the second moment of ∆R2 we obtain

〈(∆R2)2〉 = 〈

∫ τ

0sin2 Ω0t (dW )2

∫ τ

0sinΩ0t

′ (dW ′)2〉

=

∫ τ

0

∫ τ

0sin2 Ω0t sin

2Ω0t′〈(dW )2(dW ′)2〉

=

∫ τ

0

∫ τ

0sin2 Ω0t sin

2Ω0t′dtdt′

=

(∫ τ

0sin2 Ω0tdt

)2

=τ2

4, (26)

where we have used that (dW )2 and (dW ′)2 are uncorrelated and that 〈(dW )2〉 = dt.

Thus, the variance of ∆R2 vanishes,

〈(∆R2)2〉 − 〈∆R2〉2 =

τ2

4− τ2

4= 0. (27)

This result implies that the random variable ∆R2 is sharp such that it can be replaced

by its average, ∆R2 = τ/2.

Putting everything together we obtain

∆ǫ =

(

−Γ0ǫ

2− ηǫ3

4mΩ0+

Γ0kBT0

4ǫ

)

τ +

√

Γ0kBT0

2W (τ). (28)

Since the oscillation period τ is assumed to be short compared to the dissipation time

scale 1/Γ0, we can finally write the stochastic differential equation for the variable ǫ,

dǫ =

(

−Γ0ǫ

2− ηǫ3

4mΩ0+

Γ0kBT0

4ǫ

)

dt+

√

Γ0kBT0

2dW. (29)

This equation, in which ǫ is the only variable, is the main result of this section. It

implies that the relaxation process can be understood as a Brownian motion of ǫ (or,

equivalently, of the energy) under the influence of an external “force”.

S6


Similarly, we can derive the corresponding stochastic differential equation for the

energy E = ǫ2:

dE =

(

−Γ0E − ηE2

2mΩ0+ Γ0kBT0

)

dt+√

2EΓ0kBT0dW. (30)

Note that, in contrast to ǫ, the energy is subject to multiplicative noise.

1.2 Energy distribution

Equation (29) derived in the previous section resembles the Langevin equation of a

variable ǫ evolving at temperature kBT0 under the influence of an external force f(ǫ) at

high friction ν:

dǫ =1

νf(ǫ)dt+

√

2kBT0

νdW. (31)

The isomorphism is established by setting ν = 4/Γ0 and

f(ǫ) = −2ǫ− ηǫ3

mΩ0Γ0+

kBT0

ǫ. (32)

Interestingly, a low friction Γ0, which determines the magnitude of the frictional force

acting on the particle, corresponds to a high friction ν for the time evolution of ǫ and,

thus, of the energy E. The Langevin equation (31) is known to sample the Boltzmann-

Gibbs distribution

ρ(ǫ) ∝ exp −β0U(ǫ) , (33)

where β0 = 1/kBT0 is the reciprocal temperature and U(ǫ) is the potential correspond-

ing to the force f(ǫ) = −dU(ǫ)/dǫ. In our case, integration of the force f(ǫ) of Equ.

(32) yields the potential

U(ǫ) = ǫ2 +α

4ǫ4 − kBT0 ln ǫ, (34)

where we have introduced

α =η

mΩ0Γ0(35)

to simplify the notation. Hence, Equ. (29) generates the distribution

ρ(ǫ, α) ∝ ǫ exp

−β0

(

ǫ2 +α

4ǫ4)

, (36)

which can be viewed as the equilibrium distribution of the potential U(ǫ). In the above

equation, we have included the feedback strength α explicitly as a parameter for ρ(ǫ, α)

in order to indicate that this distribution is valid also for the non-equilibrium steady

S7


0 1 2 3 4 5E

-12

-10

-8

-6

-4

-2

0

log

P(E

)η=0.000η=0.001η=0.010η=0.100η=1.000

Figure S1: Logarithm lnP (E) of the energy distribution shifted by a constant such that

lnP (0) = 0. The symbols are simulation results obtained for η = 0.000, 0.001, 0.010, 0.100

and 1.000 and the solid lines are the corresponding predictions of Equ. (37). The simulations

were carried out in phase for kBT0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm

of Sivak, Chodera and Crooks [1] to integrate the Langevin equation of motion for a total of

2× 109 time steps of length ∆t = 0.006 for each value of η.

state generated by the feedback mechanism. The effect of the feedback is, however,

reduced to a particular term in this potential. As can be easily seen by a change of

variables from ǫ to E = ǫ2, the distribution of Equ. (36) corresponds to the energy

distribution

ρ(E,α) =1

Zαexp

−β0E − β0α

4E2

, (37)

where the normalisation factor Zα =∫

dEρ(E,α) is given by

Zα =

√

π

αβ0eβ0/αerfc

(√

β0α

)

. (38)

Thus, the energy distribution has the form of the Boltzmann-Gibbs distribution for the

“energy” H = E + αE2/4, where E is the energy of the system and αE2/4 can be

viewed as a feedback energy. Some energy distributions obtained for different feedback

strengths η ranging from 0 to 1 are shown in Fig. S1.

S8


1.3 Effective temperature

The average energy is obtained by integration over the energy distribution of Equ. (37),

〈E〉 = Z−1α

∫

dEE exp

−β0E − β0α

4E2

. (39)

Evaluation of the integral yields

〈E〉 = 2

β0

√

β0α

e−β0/α

√π erfc

(

√

β0

α

) − β0α

, (40)

Thus, the effective temperature Teff obtained by applying the feedback mechanism is:

kBTeff = kBT0

2√αkBT0

e−1/αkBT0

√π erfc

(

1√αkBT0

) − 2

αkBT0

. (41)

For small values of α this expression turns into

kBTeff = kBT0 (1− αkBT0) . (42)

Hence, for weak feedback, the relative effective temperature Teff/T0 decreases linearly

as function of α with slope kBT0. The effective temperature Teff is shown in Fig. S2 as

a function of the feedback strength α.

For large values of α, the asymptotic behaviour of the effective temperature is given

by

kBTeff ≈√

4kBT0

πα=

√

4kBT0mΩΓ0

πη. (43)

Hence, at low friction the effective temperature decreases as√Γ0 and is inversely pro-

portional to√η.

An alternative but approximate expression for the effective temperature of the oscil-

lator with feedback can be derived from Equ. (53) for the time evolution of the average

energy. In the steady state, the time derivative of the average energy must vanish such

that

−Γ0〈E〉 − η

2mΩ0〈E2〉+ Γ0kBT0 = 0, (44)

which implies1

2α〈E2〉+ 〈E〉 − kBT0 = 0. (45)

S9


This equation cannot be simply solved for 〈E〉, because in general the second moment of

the energy, 〈E2〉, cannot be expressed in terms of the average energy 〈E〉. In equilibrium,

however, the energy is distributed exponentially and therefore the relation 〈E2〉 = 2〈E〉2

holds. Using this relation in the steady state away from equilibrium, where it is valid

only approximately, we obtain an equation for the average energy 〈E〉,

α〈E〉2 + 〈E〉 − kBT0 = 0. (46)

Solving this quadratic equation yields

〈E〉 = −1 +√

1 + 4α/β02α

, (47)

which, because of 〈E〉 = kBTeff , is equivalent to

kBTeff =

√1 + 4αkBT0 − 1

2α. (48)

For small values of α, the effective temperature in this approximation turns into

kBTeff = kBT0 (1− αkBT0) , (49)

which is identical to the result of Equ. (42) obtained from the exact energy distribution.

For large values of α, on the other hand, the approximate effective temperature of Equ.

(48) becomes

kBTeff =

√

kBT0

α=

√

kBT0mΩ0Γ0

η. (50)

In this approximation the effective temperature at low friction (or strong feedback)

has the same scaling with Γ0 and η as the one predicted by Equ. (43). However

this approximate expression underestimates the exact low-friction effective temperature

given in Equ. (43) by a factor of√

π/4 ≈ 1.1284. As can be seen in Fig. S2, Equ. (48)

correctly reproduces the effective temperature for weak feedback, but underestimates

the effective temperature by about 10% for stronger feedback.

It is interesting to note that Equ. (45) implies that

kBTeff = kBT0 −α

2〈E2〉, (51)

which is an exact result holding for the non-equilbrium steady state generated by the

feedback. For the dissipation rate, this implies

P = Γ0(kBT0 − kBTeff). (52)

S10


0 10 20 30 40α

0

0.2

0.4

0.6

0.8

1

Tef

f/T

0 10 20 30 40α

0.6

0.7

0.8

0.9

1

1.1

Tef

f/Tth

Figure S2: Effective temperature Teff as a function the feedback parameter α = η/mΩ0Γ0.

The symbols are results of simulations and the solid lines are the effective temperatures

predicted by Equ. (41) (red) and Equ. (48) (blue), respectively. The simulations were

carried out for kBT0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm of Sivak,

Chodera and Crooks [1] to integrate the Langevin equation of motion for a total of 108 time

steps of length ∆t = 0.006 for each value of α. Inset: same effective temperatures normalized

by the effective temperature Tth predicted by Equ. (41).

1.4 Relaxation of average energy

Carrying out an average over noise realizations in Equ. (30) and exploiting that 〈dW 〉 =0, the time derivative of the ensemble average of the energy is given by

d〈E〉dt

= −Γ0〈E〉 − η

2mΩ0〈E2〉+ Γ0kBT0. (53)

Note that in general this average is over a non-equilibrium distribution.

Without feedback (A = 0) the average energy evolves according to

d〈E〉dt

= −Γ0〈E〉+ Γ0kBT0. (54)

This differential equation can be easily solved yielding an exponential relaxation of the

average energy with such that the average energy approaches its equilibrium value of

kBT0 exponentially,

〈E(t)〉 = kBT0 + [〈E(0)〉 − kBT0]e−Γ0t, (55)

S11


where 〈E(0)〉 is the average energy at time 0. In principle, Equ. (53) can be used to

compute the time evolution of the average energy also in the presence of the feedback.

In this case, however, one needs to be able to compute the second moment of the energy

at each instant during the relaxation process.

In the non-equilibrium steady state obtained with feedback on, the energy change

vanishes on the average. Hence, the energy flow from the oscillator to the heat bath

is exactly compensated by the energy extracted from or added to the oscillator by the

feedback mechanism,

Γ0 (〈E〉 − kBT0) = − η

2mΩ0〈E2〉 (56)

as already expressed in Equ. (44). Here, the left hand side is the average energy flow

(energy per unit time) from the oscillator to the heat bath and the right hand side is

the energy change due to the feedback. Using the parameter α = η /mΩ0Γ0 , Equ. (56)

turns intoα

2〈E2〉 = kBT0 − 〈E〉. (57)

This equation, relating the first and second moments of the energy distribution, holds

in the non-equilibrium steady state generated by the feedback mechanism.

We also introduce the average dissipation rate

P =αΓ0

2〈E2〉, (58)

which is the average energy per unit time transferred from the heat bath to the oscil-

lator. Note that for positive η, the system is colder than the heat bath and, therefore,

on average the energy flows from the heat bath to the system. This is an interesting

difference to most other non-equilibrium steady states, in which energy flows from the

system to the heat bath compensating for the energy dissipated by an external pertur-

bation.

1.5 Phase space distribution

Based on the energy distribution ρ(E, η) of Equ. (37) we derive the phase space distri-

bution ρ(q, p, η) of the non-equilibrium steady state generated by the application of the

feedback. We start by writing the joint probability distribution function ρ(q,E, η) to

S12


observe the pair (q,E) as

ρ(q,E, η) = ρ(q|E, η)ρ(E, η), (59)

where ρ(q|E, η) is the conditional probability to observe the position q for a given energy

E at feedback strength η. Assuming that the motion of the oscillator is essentially

undisturbed during an oscillation period as is the case in the low friction limit, the

distribution of positions is given by

ρ(q|E, η) =

Ω0

π√

2E/m−Ω2

0q2

if q2 ≤ 2EmΩ2

0

0 else(60)

simply because the probability to find the system at q is inversely proportional to the

magnitude of the velocity, |p|/m =√

2E/m− Ω20q

2 the system has at q. This condi-

tional probability distribution diverges at the turning points q0 = ±√

2E/mΩ20 and it

vanishes for |q| > q0. Multiplying the conditional distribution ρ(q|E, η) with the energy

distribution from Equ. (37) one obtaines the desired joint distribution ρ(q,E, η).

Next, we change variables from (q,E) to (q, p). The respective distributions are

related by

ρ(q, p, η) =1

2ρ(q,E, η)

∣

∣

∣

∣

∂(q,E)

∂(q, p)

∣

∣

∣

∣

. (61)

The Jacobian of the transformation is given by

∣

∣

∣

∣

∂(q,E)

∂(q, p)

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

∂q∂q

∂q∂p

∂E∂q

∂E∂p

∣

∣

∣

∣

∣

∣

=|p|m

. (62)

In Equ. (61) we have exploited that the distribution ρ(q, p, η) is symmetric in p, and the

factor 1/2 arises because p and −p correspond to the same energy E = kq2/2 + p2/2m.

Thus, the phase space density ρ(q, p, η) becomes

ρ(q, p, η) =1

2ρ[E(q, p), η]

Ω0

π√

2E/m− Ω20q

2

|p|m

=Ω0

2πρ[E(q, p), η], (63)

where we have used that |p|/m =√

2E/m− Ω20q

2. Note that the second case of Equ.

(60) does not need to be taken into account, because for given q and p, the condition

q2 ≤ 2E/mΩ20 is always obeyed. Using the energy distribution from Equ. (37) we finally

obtain

ρ(q, p, η) =Ω0

2πZexp

−β0E(q, p)− β0η

4mΓ0Ω0E(q, p)2

. (64)

S13


Hence, the motion of the oscillator with feedback samples the equilibrium distribution

of a system with energy

H(q, p) = E(q, p) +η

4mΓ0Ω0E(q, p)2. (65)

In the phase space distribution of Equ. (64) the term depending on the squared energy

E(q, p)2 causes correlations between q and p that are absent in equilibrium with feedback

off.

The average of the effective energy H(q, p), which contains the energy E(q, p) plus

a "feedback" energy αE(q, p)2/4 is given by

〈H〉 = 〈E〉 + α

4〈E2〉. (66)

Using Equ. (57) and noting that 〈E〉 = kBTeff we obtain

〈H〉 = kBT0 + kBTeff

2(67)

So the average effective energy H(q, p) is the arithmetic average of the energy of two

harmonic oscillators, one at temperature T0 and the other one at temperature Teff .

From the phase space density ρ(q, p, η) one can get the distribution ρ(q, η) of the

positions by integration over the momenta:

ρ(q, p, η) =

∫ ∞

−∞dp ρ(q, p, η). (68)

Carrying out the integral yields

ρ(q, η) =

√

β0mΩ20(4 + αmΩ2

0q2)

8π3

×exp

[

−β0(4+αmΩ2

0q2)2

32α

]

erfc

(

√

β0

α

) K 1

4

[

β0(4 + αmΩ20q

2)2

32α

]

, (69)

where K1/4 is a generalised Bessel function of the second kind. This expression is

compared with simulations in Fig. S3.

An analogous calculation yields the non-equilibrium momentum distribution ρ(p, η) =∫∞−∞ dq ρ(q, p, η),

ρ(p, η) =

√

β0(4m+ αp2)

8π3m2

×exp

[

−β0(4m+αp2)2

32αm2

]

erfc

(

√

β0

α

) K 1

4

[

β0(4m+ αp2)2

32αm2

]

. (70)

S14


-4 -2 0 2 4q

-12

-10

-8

-6

-4

-2

0

2

ln P

(q)

η=0.000η=0.001η=0.010η=0.100η=1.000

Figure S3: Logarithm lnP (q) of the distribution of position q for different feedback strengths.

The symbols are simulation results obtained for η = 0.000, 0.001, 0.010, 0.100 and 1.000 and

the solid lines are the corresponding predictions of Equ. (69). The simulations were carried

out for kBT0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm of Sivak, Chodera

and Crooks [1] to integrate the Langevin equation of motion for a total of 2× 109 time steps

of length ∆t = 0.006 for each value of η.

S15


2 Fluctuation theorem for ∆SIn this section we define the relative entropy ∆S and discuss the fluctuation theorem

holding for this quantity [10, 11, 12, 3].

2.1 Relative entropy change ∆SWe consider a system with energy E(u), including both potential and kinetic energy,

with u specifying the state of the system. The system is in contact with a heat bath at

reciprocal temperature β0 = 1/kBT0. If no external perturbation acts on the system, u

is distributed according to the equilibrium distribution

ρeq(u) =1

Z(β0)e−β0E(u), (71)

where Z(β0) =∫

due−β0E(u) is the partition function related to the free energy by

F (β0) = −kBT lnZ(β0). If left undisturbed, the system evolves according to a dynamics

which is microscopically reversible, i.e., it obeys the detailed balance condition for the

equilibrium distribution,

e−β0E(u)p(u → v, t) = e−β0E(v)p(v∗ → u∗, t). (72)

Here, p(u → v, t) is the probability to move from state u at time 0 to state v at time

t and the star denotes a state with inverted momenta. The dynamics generated by the

Langevin equation (with feedback off) obeys this condition.

The system is initially prepared in a steady state with distribution ρss(u, α), for

instance by letting a feedback mechanism act on it. Here, α denotes one or several pa-

rameters such as the strength of the feedback mechanism, which determine the steady

state distribution. In general, ρss(u, α) is not known analytically. At time t = 0 the

feedback is switched off and the system relaxes back to equilibrium. During the relax-

ation process, the system exchanges energy (heat) with the heat bath. Since no work

is done on the system during the relaxation, the total heat Q exchanged with the heat

bath by the system evolving from u0 at time 0 to ut at time t is given by

Q = −[E(ut)− E(u0)]. (73)

Note that the heat is defined in such a way that a positive Q corresponds to energy

absorbed by the bath and lost by the system. For a given steady state distribution

S16


ρss(u, α), we also define the quantity

φ(u) = − ln ρss(u, α) (74)

as well as its difference between the initial and final state,

∆φ = φ(ut)− φ(u0). (75)

Based on this definition, we introduce the relative entropy change

∆S = β0Q+∆φ, (76)

which depends only on the state of the system at times 0 and t.

We call ∆S the relative entropy change for the following reason. Defining

S = lnρeq(u)

ρss(u)(77)

we can write

∆S = S(ut)− S(u0), (78)

such that ∆S is the change in the quantity S accumulated along the trajectory evolving

from u0 to ut. The average of S(u) over the equilibrium distribution is the relative

entropy [2] between the equilibrium distribution ρeq(u) and the steady state distribution

ρss(u),

D(ρeq‖ρss) =∫

du ρeq(u) lnρeq(u)

ρss(u). (79)

This quantity, also known as Kullback-Leibler divergence, is a non-symmetric measure

of the difference between two distributions and it vanishes if the two distributions are

identical. Since the ensemble average of S(u) is a relative entropy, we view S(u) as the

relative entropy associated with state u. While strictly speaking the relative entropy is

a property of the entire distributions, entropies assigned to individual configurations or

trajectories have been considered before and shown to be useful [3]. Accordingly, ∆Scan be viewed as the relative entropy change of the trajectory connecting u0 with ut.

In Equ. (76), the first term on the right hand side is the entropy change of the reservoir

at reciprocal temperature β0 and the second term is the entropy change with respect

to the initial steady state distribution. Note that in contrast to the definition of the

generalized work Y of Hatano and Sasa [4, 5], in the definition of ∆S the initial steady

state distribution (rather than the time-evolved distribution) is evaluated both at the

beginning u0 and the end ut of the trajectory.

S17


It is also interesting to note that the relative entropy change is ∆S is equal to the

logarithmic ratio of the probability to observe a particular trajectory and the probability

to observe the time reversed trajectory [3, 8, 9]. To be more explicit, consider the

probability P [u(t)] of observing a particular trajectory u(t) of length t evolving from

u0 to ut and the probability P [u∗(t)] of the time-reversed trajectory u∗(t). In the time

reversed, or conjugate trajectory, the same microscopic states are visited, but in reversed

order and with inverted momenta, ut′ = u∗t−t′ , where the star indicates momentum

inversion [3, 9, 13]. For dynamics that is microscopically reversible (i.e., detailed balance

holds) and assuming that both the forward and the reversed trajectories are started from

the same steady state distribution ρss(u), the ratio of the probabilities to observe a pair

of conjugate trajectories is then given by [3, 9, 14]

P [u(t)]

P [u∗(t)]= eβQ+∆S = e∆S . (80)

Hence, the relative entropy of the distributions of the forward and the reversed trajec-

tories, obtained as average over the logarithmic probability ratio, reads

D(P [u(t)]‖P [u∗(t)]) =∫

du(t) P [u(t)] lnP [u(t)]

P [u∗(t)]= 〈∆S〉, (81)

where the integration extends over all trajectories u(t). Therefore, the ensemble average

〈∆S〉 of the relative entropy change is the relative entropy of the forward and reversed

ensembles of trajectoris and, as such, provides a measure for the irreversibility (time-

asymmetry) of the relaxation process. As a consequence of Equ. (81), in an equilibrium

system the relative entropy change vanishes, 〈∆S〉eq = 0, because in equilibrium a

particular forward trajectory and its time-reversed trajectory have the same probability

to be observed.

For deterministic thermostatted dynamics ∆S equals the dissipation function intro-

duced by Evans and Searles [10, 11, 12]. It is also worth noting that the relative entropy

change and the total entropy change are related by [3]

∆S = ∆Stot − lnρss(ut)

ρt(ut), (82)

where ρt(u) is the statistical state of the system at time t. The relation between relative

entropy change and total entropy change is discussed further below.

S18


2.2 Detailed fluctuation theorem

As shown by Evans and Searles [10, 11, 12] for thermostatted dynamics and by Seifert

for stochastic dynamics [3], the following transient fluctuation theorem holds for time-

independent driving and microscopically reversible dynamics

Pt(−∆S)Pt(∆S) = e−∆S . (83)

Here, Pt(∆S) is the distribution of ∆S observed at an arbitrary time t over many repe-

titions of the relaxation experiment. In particular, this fluctuation theorem is valid for a

system relaxing to equilibrium from a non-equilibrium steady state as considered here.

Since it is instructive and emphasises the significance of the microscopic reversibility

of the underlying dynamics, we provide a short derivation of the detailed fluctuation

theorem (83) in the following. The derivation is based on two conditions: (1) the initial

steady state distribution and the equilibrium distribution are symmetric with respect to

momentum reversal, (2) the dynamics is microscopically reversible, i.e., detailed balance

holds. Since the energy is quadratic in the momentum p, the first condition is always

obeyed if the distribution is a function of the energy only, as it is the case for parametric

feedback cooling. The latter condition is fulfilled, for instance, for a system evolving

according to a Langevin equation.

The fluctuation theorem follows most easily by considering the probability P [u(t)] of

observing a particular trajectory u(t) and the probability P [u∗(t)] of the time-reversed

trajectory u∗(t). As mentioned above, for dynamics that is microscopically reversible,

this ratio is given by P [u(t)]/P [u∗(t)] = exp(∆S) [3, 9]. The distribution of ∆S at time

t can be expressed in terms of the probability P [u(t)],

Pt(∆S) =∫

du(t) P [u(t)] δ(∆S[u(t)]−∆S) , (84)

where δ(·) is the Dirac δ-function. Transforming integration variables from u(t) to u∗(t)

and taking advantage of the symmetry of Q and ∆φ with respect to momentum reversal,

one finds

Pt(∆S) =∫

du∗(t) P [u∗(t)]e−∆S[u∗(t)] δ(−∆S[u∗(t)]−∆S) = e∆SPt(−∆S), (85)

which holds for any time t > 0. Thus, the transient fluctuation theorem of Equ. (83) is a

direct consequence of the microscopic reversibility of the dynamics during the relaxation

process.

S19


2.3 Integral fluctuation theorem

From the detailed fluctuation relation of Equ. (83) one easily obtains an integral fluc-

tuation theorem by integration over the probability density P (∆S),

〈e−∆S〉 =∫

d∆S P (∆S)e−∆S =

∫

d∆S P (−∆S) = 1, (86)

where the last step involves a variable change from ∆S to −∆S.

Applying Jensen’s inequality, i.e., 〈ex〉 ≥ e〈x〉, to the integral fluctuation theorem

one obtains

1 = 〈e−∆S〉 ≥ e−〈∆S〉, (87)

which is equivalent to

〈∆S〉 ≥ 0. (88)

Thus, the average change in relative entropy is non-positive. Using the definition of ∆S,

the average of the relative entropy change can be written as

〈∆S〉 = β0〈Q〉+∆I +D(ρt‖ρss), (89)

where β0 is the reciprocal temperature of the bath. The first term on the right hand

side of the above equation is the change of thermodynamic entropy of the bath,

∆Sbath = β0〈Q〉. (90)

The second term, ∆I, is the change in Shannon entropy (or information entropy) I be-

tween the initial statistical state characterised by the distribution ρss and the statistical

state with distribution ρt at time time after the relaxation has started,

∆I = I[ρt]− I[ρss], (91)

where the Shannon entropy I of a distribution ρ(u) is defined as

I[ρ] = −∫

du ρ(u) ln ρ(u). (92)

If one identify the Shannon entropy with the thermodynamic entropy, then ∆I is nothing

else than the entropy change of the system,

∆Ssystem = ∆I. (93)

S20


Finally, the last term in Equ. (103) is the relative entropy of the distribution at time t

with respect to the steady state distribution at time 0,

D(ρt‖ρss) =∫

du ρt(u) lnρt(u)

ρss(u). (94)

Putting things together, one obtains

〈∆S〉 = ∆Sbath +∆Ssystem +D(ρt‖ρss) = ∆Stotal +D(ρt‖ρss), (95)

where ∆Stotal = ∆Ssystem + ∆Sbath is the total entropy change of system and bath

together. The inequality that follows from the integral fluctuation theorem implies the

second law-like inequality

∆Stotal +D(ρt‖ρss) ≥ 0. (96)

An integral fluctuation theorem can be derived [3, 6] also for the quantity

R = β0Q− lnρt(ut)

ρss(u0), (97)

which, in contrast to ∆S, depends also on the time propagated distribution ρt(ut).

From the fluctuation theorem for R it follows that the average total entropy change is

non-negative, 〈Stot〉 ≥ 0, providing a microscopic statement of the second law.

2.4 Relaxation from an initial equilibrium state

If the initial steady state ρss(u, α) is an equilibrium distribution e−βE(u)/Z(β) corre-

sponding to the temperature T = 1/kBβ differing from the temperature T0 of the heat

bath, the expressions become particularly simple. In this case,

φ(u) = − lne−βE(u)

Z(β)= βE(u) − βF (β) (98)

such that

∆φ = β [E(ut)− E(u0)] = −βQ. (99)

Hence, the relative entropy production is given by

∆S = β0Q− βQ = (β0 − β)Q. (100)

The fluctuation theorem for ∆S then becomes a fluctuation theorem for the heat Q

exchanged with the reservoir during the relaxation,

Pt(−Q)

Pt(Q)= e−(β0−β)Q, (101)

S21


as shown earlier by Jarzynski [7]. The integral fluctuation theorem then turns into

〈e−(β0−β)Q〉 = 1, (102)

Due to the convexity of the exponential function, this result implies that

(β0 − β)〈Q〉 ≥ 0. (103)

Now, if the system is initially colder than then bath, i.e., β > β0, then (β0 − β) < 0

and the above inequality implies that 〈Q〉 ≤ 0, i.e., the system absorbs energy from the

bath. In other words, heat flows from hot to cold as expected from the second law of

thermodynamics.

2.5 Relaxation from a steady state generated by parametric

feedback

If the initial steady state ρss(u, α) is due to parametric feedback cooling, the total

effective “energy” is given by

H(u, α) = E(u) +α

4E2(u). (104)

Then, ∆φ = β0∆H and

∆S = β0α

4

[

E2(ut)− E2(u0)]

. (105)

In this case, the inequality following from the integral fluctuation theorem implies that

〈∆E2〉 ≥ 0. (106)

Thus, the average of the squared energy does not decrease during the relaxation pro-

cess. Experimental results obtained for the relaxation from steady states generated by

parametric feedback are presented and discussed in the main paper.

2.6 Distributions of ∆S for t → ∞We now consider the distribution Pt(∆S) of the quantity ∆S = (β0α/4)(E

2t − E2

0).

Since ∆S is completely determined by E0 and Et, the distribution of ∆S can be written

as

Pt(∆S) =∫∫

dE0dEt P0(E0)P (Et|E0)δ(∆S −∆S(Et, E0)), (107)

S22


where P (Et|E0) is the conditional probability that the energy is Et at time t provided

it was E0 at time 0. In general, P (Et|E0) is unknown and to determine it one would

have to know the Green’s function of the SDE for the energy. But what can be done

easily is to compute Pt(∆S) for long times, i.e., in the limit t → ∞. In this case the

final energy, Et, is statistically independent from the initial energy, E0, such that

P (Et|E0) = P∞(Et), (108)

where P∞(E) is the asymptotic distribution of the energy reached in the long time

limit. For the relaxation process after turning off the feedback, P∞(E) is the equilibrium

distribution of the energy for temperature T0. In this limit, the distribution is given by

P∞(∆S) =∫∫

dE0dEt P0(E0)P∞(Et)δ(∆S −∆S(Et, E0)). (109)

To solve the integral, we transform variables from E to M ,

M =αβ04

E2. (110)

Then, ∆S is given by

∆S = (Mt −M0). (111)

The distributions of E and M are related by

P (M) = P (E)

∣

∣

∣

∣

dM

dE

∣

∣

∣

∣

−1

= P (E(M))/√

αβ0M. (112)

The distributions of M with and without feedback are then given by

P0(M) = C0 exp

(

−√

4β0α

√M −M

)

/√

αβ0M (113)

and

P∞(M) = C∞ exp

(

−√

4β0α

√M

)

/√

αβ0M. (114)

Using the new variable M , the long time distribution of ∆S can be written as

P∞(∆S) =∫∫

dM0dMt P0(M0)P∞(Mt)δ[∆S − (Mt −M0)]. (115)

Integration over M0 yields

P∞(∆S) =∫

dM P0(M −∆S)P∞(M). (116)

S23


-40 -20 0 20 40∆S

10-30

10-25

10-20

10-15

10-10

10-5

100

P(∆S

)

η=0.001η=0.010η=0.100η=1.000

Figure S4: Logarithm lnP (∆S) of the long time distribution ∆S for different feedback

strengths. The distributions were obtained by numerical integration of Equ. (116) using the

the distributions of Eqs. (113) and (114). The feedback strengths were η = 0.001, 0.010, 0.100

and 1.000 and for the other parameters were used kBT0 = 1, m = 1, k = 1, and Γ0 = 0.01.

The corresponding values of α were α = 0.1, 1, 10 and 100. For η = 0, the distribution is a

delta function centered at ∆S = 0. The dashed line indicates the distribution for η → ∞(see Equ. (117)).

Note that we have defined the distributions of M such that they vanish for negative

M and the integration extends from −∞ to +∞. The integral of the above equation

cannot be calculated analytically, but we can determine the distribution P∞(∆S) by

numerical integration to arbitrary precision using the distributions of Eqs. (113) and

(114). Some distributions of ∆S obtained in this way for various values of the feedback

strength η are shown in Fig. S4 and are in excellent agreement with the experimental

data presented in the main text..

In the limit of large η, the long time distribution of ∆S becomes:

P∞(∆S) = Ce∆S/2K0(|∆S|/2), (117)

where K0 is a modified Bessel function of the second kind. This limiting distribution is

shown in Fig. S4 as a dashed line. This distribution manifestly satisfies the fluctuation

theorem.

S24


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[2] D. A. Sivak and G. E. Crooks, Near-Equilibrium Measurement of Non-Equilibrium

free energy, Phys. Rev. Lett. 108, 150601 (2012).

[3] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular ma-

chines, Rep. Prog. Phys. 75, 126001 (2012).

[4] T. Hatano and S.-I. Sasa, Steady-State Thermodynamics of Langevin Systems.

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