Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers
David Grass, Julian Fesel, Sebastian G. Hofer, Nikolai Kiesel, Markus Aspelmeyer
OOptical trapping and control of nanoparticles inside evacuated hollow core photoniccrystal fibers
David Grass, ∗ Julian Fesel, Sebastian G. Hofer, Nikolai Kiesel, and Markus Aspelmeyer † Vienna Center for Quantum Science and Technology (VCQ),Faculty of Physics, University of Vienna, A-1090 Vienna, Austria (Dated: April 1, 2016)We demonstrate an optical conveyor belt for levitated nano-particles over several centimetersinside both air-filled and evacuated hollow-core photonic crystal fibers (HCPCF). Detection of thetransmitted light field allows three-dimensional read-out of the particle center-of-mass motion. Anadditional laser enables axial radiation pressure based feedback cooling over the full fiber length.We show that the particle dynamics is a sensitive local probe for characterizing the optical intensityprofile inside the fiber as well as the pressure distribution along the fiber axis. In contrast to previousindirect measurement methods we find a linear pressure dependence inside the HCPCF extendingover three orders of magnitude from 0.2 mbar to 100 mbar. A targeted application is the controlleddelivery of nano-particles from ambient pressure into medium vacuum.
Optically levitated nano-particles are a new paradigmin the development of high-quality mechanical resonators[1]. This approach has been proposed as a route forultra-sensitive force measurements [2], studies of stochas-tic out-of-equilibrium physics in the underdamped regime[3, 4], and for room-temperature quantum optomechanics[5–7]. Early experiments by Ashkin have demonstratedoptical levitation of micrometer-scale dielectric objects inhigh vacuum [8]. Recent efforts focus on sub-micron par-ticles and have already demonstrated mechanical qualityfactors up to 10 (see ref. [9]), thereby surpassing thoseof state-of-the-art clamped nano-mechanical devices [10].Further examples include the realization of zepto-Newtonforce sensing [11], a test of fluctuation theorems [12], andoptical feedback- and cavity-cooling [13–16].Hollow-core photonic crystal fibers [17, 18] (HCPCF)add a particularly intriguing instrument to the tool-box of levitated optomechanics. Free-space optical trapstypically offer only small volumes of high light inten-sity, and in this respect limited capabilities for opticalmicro-manipulation. By contrast, HCPCF allow a tighttransversal confinement of optical fields over the full dis-tance of the fiber length, which can extend over severalmeters. Recently, precise control over micrometer sizedparticles in HCPCFs has been established in several ex-periments [19–21] and first sensing capabilities have beendemonstrated [22]. Here we present new methods foroptical micro-manipulation of particles inside HCPCFand extend the application regime to sub-micron sizes.Specifically, we demonstrate an optical conveyor belt fornano-particles inside a HCPCF, in which particles areoptically trapped in a standing wave field and can betransported and precisely positioned along the fiber. Weshow three-dimensional (3d) read-out of the center-of-mass (COM) motion together with feedback control inaxial direction. We demonstrate that a particle can be ∗ [email protected] † [email protected] delivered into vacuum (0.2 mbar) with a pressure differ-ence of three orders of magnitude between the fiber ends.We use the read-out and control capabilities to investi-gate the pressure distribution inside the HCPCF.To establish the optical conveyor belt two counter-propagating laser fields with equal polarization andwavelength λ tr excite the eigenmodes of a HCPCF,which are approximated with linear polarized modes[23] (LP ij ).The resulting standing wave, mainly formedby the fundamental mode (LP ), creates an opticaltrap for Rayleigh particles with trapping potential U = − αI ( r, z ) / (2 ε c ) (particle polarizability α = 4 πε a ( ε − / ( ε + 2), particle radius a (cid:28) λ tr , dielectric constant ε ,vacuum permeability ε , speed of light c , intensity dis-tribution I ( r, z ), r = (cid:112) x + y and z denote radial andaxial direction inside the HCPCF, respectively). For asufficiently deep potential the particle is trapped closeto an intensity maximum through the gradient force [24] (cid:126)F grad = −∇ U . By introducing a frequency detuning ∆ ν between the counterpropagating lasers the standing wavepattern moves along the fiber axis carrying the trappedparticle at a velocity v z = ∆ νλ tr /
2, analogous to stand-ing wave conveyor belt techniques in free space [25, 26].For small displacements, the potential U can be ap-proximated as a 3d harmonic oscillator potential. Theequation of motion for the trapped particle’s COM mo-tion along the fiber axis is¨ z + Γ p ˙ z + Ω z = F therm m (1)with m the mass and Ω the mechanical frequency of theparticle. Collisions with gas molecules result in a pressuredependent Stokes friction [13, 14, 27] force F = m Γ p ˙ z and in Brownian force noise F therm . The latter is a zero-mean stochastic process (cid:104) F therm (cid:105) = 0 satisfying the corre-lation function (cid:104) F therm ( t ) F therm ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) )2 m Γ p k B T for the case of a Markovian heat bath (Boltzmann’s con-stant: k B , environment temperature: T = 293 K).In our experiment (figure 1a), two vacuum chambersare connected through a 15 cm long HCPCF (HC-1060,NKT Photonics). Each chamber has an independent a r X i v : . [ phy s i c s . op ti c s ] M a r read-out ν tr ν tr + Δ vacHCPCF a) b) BS vacx(t) z(t)y(t) t [s] d [ n m ] d [ n m ] c)xy z FIG. 1. a) Schematic drawing of the experimental setup:A hollow core photonic crystal fiber (HCPCF) connects twovacuum chambers (vac) whose pressures can be individuallycontrolled. Two equally polarized, counterpropagating laserswith frequencies ν tr and ν tr + ∆ are focused into the HCPCF.For ∆ = 0 they form a standing wave optical trap for silicanano-particles. A beam splitter (BS) is used to split off 10%of the clockwise propagating light for 3-dimensional detectionof the particle motion (read-out). Particles from a nebulizersource (not shown) are trapped in front of the HCPCF; theycan be transported along the fiber by detuning the clockwisepropagating laser ∆ (cid:54) = 0. b) Image of a 200 nm radius sil-ica particle trapped inside the HCPCF (indicated with whitedashed lines). c) 3d trajectory of a trapped nano-particle (z:along fiber axis, x,y: radial directions) pressure control and one chamber is connected to a nebu-lizer that supplies airborne silica nano-particles from anisopropanol solution (Appendix A). The counterpropa-gating fields for trapping and transport are derived froma single-frequency Nd:YAG laser ( λ tr = 1064 nm). Eachof the beams is individually frequency shifted by anacousto-optical modulator (AOM), which allows controlover their relative detuning ∆ ν . The AOMs are drivenby a dual-frequency source where both channels are syn-thesized from the same master oscillator allowing a rela-tive detuning ∆ ν between the outputs in discrete steps of10 kHz. This allows to realize step-sizes for the particlemotion of approx. 12 µ m. Fine-grained positioning onthe sub-micron scale (around 0 . µ m) is achieved using amanual position stage to change the optical path lengthdifference by ∆ l , thereby displacing the standing waveby ∆ l/ P/P I/I -50 0 50 100 150120140160180200 z [µm] Ω / � [ k H z ] H z x [ µ m ] a)b) c)d) z [µm] FIG. 2. Image of a 200 nm radius silica particle (white arrow)trapped inside a) and outside b) the HCPCF (indicated by thewhite dashed line). Note that additional scattering occurs atthe fiber cleave. c) Simulated intensity distribution inside theHCPCF due to interference between the fundamental LP and a higher order LP mode. Different trapping positions(white dots) also exhibit different axial frequencies. d) Axialmechanical frequency of a trapped particle inside ( z >
0) andin front of the fiber ( z < µ m in front of the HCPCFat a pressure of p = 0 . The geometry of the optical trap has a periodicity λ sw = 2 π/ (2 β ) ≈ . µ m that is determined by thewave-vector β of the fundamental mode. A weak ex-citation of the higher-order LP mode with a differentwave vector β = β − ∆ β results in a modulation of theradial intensity profile of the optical trap. Specifically,due to interference between the eigenmodes, the radialposition of the intensity maxima oscillate along the z-axiswith a beat-note modulation of η mod = 2 π/ ∆ β . This isillustrated in figure 2c, where the white dotted line marksthe intensity maximum, along which particles would betrapped[28]. Note that the same interference effect givesalso rise to a modulation of the axial intensity profilewith a periodicity of η mod / . ± . µ m. This corresponds to a core radius of r co = (4 . ± . µ m of the fiber, which is in agreementwith the manufacturer specification r co = (5 ± µ m.Furthermore, the amplitude of the modulation allowsto estimate the power ratio between the higher ordermode (LP ) and the fundamental mode (LP ) withP / P = (0 . ± . z <
0) with different colors for different trappingbeam powers and fits to Gaussian beam envelopes (solidlines). The waist of the mode is a free fit parameter andis determined as w = (2 . ± . µ m. This value issmaller than the waist of the fundamental fiber mode(3 . µ m) which we attribute to a tighter focus of thebeam that is coupled into the HCPCF. All shown datawere obtained at a pressure of p = 0 . µ min front of the fiber. At lower pressures the particle is lostfrom the trap, both inside and in front of the HCPCF.This is a common phenomenon [11, 15, 16, 29–31] thatis poorly understood so far and likely related to residualnoise in the trapping potential. For the specific purposeof interfacing the particle with another optical field, likea cavity mode [32], the distance from the fiber is an im-portant benchmark to avoid scattering or shadowing ef-fects. At higher pressures ( p ≈ µ m in front of theHCPCF (Appendix C).Next, we determine the damping rate Γ p of the particleCOM motion. Depending on the pressure regime underinvestigation we either perform energy relaxation in thetime domain or linewidth measurements in the frequencydomain (Appendix D). While the latter method is lim-ited by power drifts in the trap laser (and hence driftsin the trap frequency) to a regime where the linewidthΓ p is above 4 kHz, the first method requires feedbackcooling and therefore works well for underdamped mo-tion (i.e., low pressures). We realize feedback coolingwith an additional laser ( λ fb = 1064 nm) that is coupledinto the HCPCF (figure 3a). To avoid interference effectswith the trapping laser it is well separated in frequency( ν fb (cid:54) = ν tr ) and orthogonally polarized. Modulating thefeedback laser power proportional to the velocity of theoscillator around a constant offset generates a feedbackforce F fb = m Γ fb ˙ z on the particle at a new equilibriumposition (Appendix E). Such a force modifies the frictionterm to m (Γ p + Γ fb ) ˙ z and hence, can be used for coolingof the COM motion, as already demonstrated in [33, 34].When applied in 3d, feedback cooling is known to allow ν fb ν tr AOMa) b)PBS p=0.7mbar F fb =Γ fb ż Q -1 T [ K ] -6 -5 -4 t [s] E / E c) FIG. 3. Feedback cooling and rethermalization of the nano-particle COM motion inside a HCPCF: a) A laser for feedbackcooling ( ν fb ) is superimposed on a polarizing beam splitter(PBS) with the trapping laser. Its intensity is modulated withan acusto-optical modulator (AOM) to cool the COM motionof a trapped particle. b) Time dependence of the potentialenergy (red curve) of the nano-particle COM motion whenfeedback cooling is switched on (left half) and off (right half).Fitting an exponential dependence to the transient processes(black curves) determines the values for damping and coolingrates Γ p and Γ fb . The two peaks (within the grey shadedareas) are caused by the switching processes. c) Relationbetween pressure dependent mechanical quality and effectivemode temperature for constant feedback strength. trapping in the high vacuum regime [13, 14].Without any feedback control the COM motion of theparticle is in thermal equilibrium and has the mean po-tential energy E = 1 / k B T = 1 / m Ω (cid:104) z (cid:105) . When thefeedback is switched on, energy is extracted from theCOM motion decaying as E ( t ) = E c + ( E − E c ) e − Γ fb t to a lower value E c < E (figure 3c). After switching offthe feedback cooling, the system relaxes back to thermalequilibrium. During this transient process the time evo-lution can be approximated by an exponential [35] (Ap-pendix F) yielding E ( t ) = E − ( E − E c ) e − Γ p t . For eachdata point we calculate the potential energy as ensem-ble average over approximately 2000 switching processes.Figure 3b shows an explicit example for the energy curveat p = 0 . p , Γ fb , E and E c are obtained by fitting the exponential models.The corresponding effective temperature attained dur-ing the feedback process is T eff = T E c /E = T Γ p / (Γ p +Γ fb ) + T ro where T ro is a residual offset caused by noisein the position detection [36]. In our current imple-mentation, we find T ro = (4 . ± .
28) K, constrainedmainly by the shot-noise limited read-out and hence bythe signal-to-noise ratio. The minimal effective temper-ature is T eff = (5 . ± .
12) K. Further improvementof the signal-to-noise ratio should enable feedback cool-ing to much lower temperatures [37, 38], and eventually,in combination with optimal filtering, into the quantumground state of motion [39–41].The ability to measure the local damping rate of theparticle motion enables us to use it as a nano-scale pres- relaxationspectral Γ / π [ H z ] p [mbar] × × p [ m b a r] -6 -5 -4 -3 -2 -1 z [m] FIG. 4. Pressure distribution measured along the HCPCF.A particle is trapped in front of the HCPCF at a pressure of p ≈ . p = 10 mbar (grey shaded area right).The particle is moved in steps towards the second chamber.At each position the damping Γ p is measured via the energyrelaxation method (red data points) or via a spectral evalua-tion (blue data points). The black curve is a linear fit to thedata. Inset: Pressure calibration. The particle is trapped infront of the HCPC where local pressure and particle linewidthare independently measured by a vacuum gauge and opticalread-out, respectively. The relation between damping Γ p andpressure p is measured (points) and fitted (solid line). sure sensor. It turns out that directly determining thepressure gradient inside a HCPCF that interconnectstwo reservoirs at different pressure, is a non-trivial task.It is related to the more general question of how pres-sure is distributed over a range that covers different flowregimes [42]. In our case, the HCPCF establishes a 15cm long channel of 10 µ m diameter between two differ-ent pressure reservoirs: one at p = 0 . Kn ≈ p = 10 mbar ( Kn ≈ . Kn = λ free /r co is the Knudsen number, which character-izes the flow regime by comparing the mean free path ofair molecules λ free with the channel radius r co . To relatethe mechanical damping Γ p of the particle COM mo-tion to the local pressure p , we calibrate a nano-particletrapped at the edge of the HCPCF inside one vacuumchamber with the reading of a standard vacuum gaugethat is connected to this chamber. The calibration datais shown in the inset of figure 4. For the actual measure-ment of the pressure distribution we launch the particleapproximately 15 µ m in front of the HCPCF at the low-pressure end ( Q ≈ ±
10) and shuttle it towardsthe second chamber, i.e., the high-pressure end. Thered data points shown in figure 4 correspond to energyrelaxation measurements and the blue data points to aspectral measurement of the oscillator linewidth, as de-scribed above. The overlap region shows the consistencyof the two measurement methods. The solid black lineis a linear fit to the data excluding those points wherethe relaxation method (light red) and those where thespectral evaluation (light blue) breaks down.The data is consistent with a linear pressure distribu-tion. For the simple case of molecular flow (
Kn >
Kn < . . µ m for a pressure gra-dient of approximately 10 mbar/m between the fiberends. In contrast to previous indirect predictions we finda linear pressure dependence in the molecular flow regimethat even extends into the slip flow regime. The evalua-tion is based on spectral analysis for high pressures andequilibration from a feedback-cooled steady state for lowpressures. The excellent level of control achieved in ourexperiment shows that HCPCFs provide a versatile toolfor optical micro-manipulation of nano-particles.As a first relevant application of this method we en-vision delivery of nano-particles into a high-vacuum en-vironment, well controlled in arrival time, position andeffective temperature. This may enable further applica-tions for force sensing, cavity optomechanics or matter-wave interferometry [44, 45] ACKNOWLEDGMENTS
We would like to thank U. Delic, O. A. Schmidt, T. G.Euser and P. Russel for stimulating discussions. We ac-knowledge funding from the European Commission viathe Collaborative Project TherMiQ (Grant Agreement618074) and the ITN cQOM, from the European Re-search Council (ERC CoG QLev4G) and from the Aus-trian Science Fund FWF under projects F40 (SFB FO-QUS). D.G. is supported by the FWF under projectW1210 (CoQuS).
Appendix A: Particle preparation and loading
The nano-particles used in this experiment are SiO -F-0.4 from microParticles GmbH with a diameter d =387 nm. They come in a water solution with a 10% massconcentration. The particles are diluted with Isopropanolto a mass concentration of 10 − . An Omron Air U22asthma spray [46] is used in a nitrogen environment tonebulize the Isopropanol-particle solution. A cloud ofairborne nano-particles from the nebulzer is sucked intothe vacuum chamber. When a particle is trapped in frontof the HCPCF the optical conveyor belt is switched onand particles are transported into the HCPCF. Appendix B: Read-out of the particle motion
The readout of the particle COM motion inside theHCPCF is based on interference between scattered lightfrom the particle and the fundamental LP trappingmode. We treat the nano-particle as dipole scatterer E dp which is preeminently excited by the fundamental LP trapping mode. Here, the axial readout ( z ) relies on scat-tering into the LP mode, and the transversal readout( x, y ) on scattering into the LP mode, respectively. X X SknifeZ Z SLOa) b)
FIG. 5. a) Light that interacted with the nano-particle ( S ),sampled from the BS, is detected with photodiode Z anda local oscillator ( LO ) is detected with photodiode Z . Thedifference between both photocurrents is proportional to theaxial particle displacement. b) The signal ( S ) is focused on aknife edge such that 50% of S is reflected and send on pho-todiode X and 50% is transmitted on photodiode X . Theknife position is locked to minimize | X − X | and the differ-ence photocurrent between both diode is proportional to theradial displacement of the nano-particle. Let us assume the particle is axially displaced by δz .The mode overlap between scattered light from the parti-cle and the fundamental mode is η ( δz ) = (cid:104) LP , E dp (cid:105) = | η | e iβ δz + iφ with | η | , φ independent of δz . Thesuperposition of both fields arriving at detector Z is I Z1 = ε c (cid:12)(cid:12) LP + η LP (cid:12)(cid:12) ≈ I − I | η | ( β − k ) δz (B1)with I the intensity in the trapping mode and for φ − π/ , | η | (cid:28)
1. The first term of equation B1is constant and the second is a signal proportional to theaxial displacement δz of the nano-particle. The detec-tor Z integrates the intensity given in equation B1. Asshown in figure 5a, a fraction of the trapping laser (LO)that does not interact with the nano-particle with the in-tensity distribution I is integrated by a second detectorZ such that the difference signal between both detectorsresults in the signal S Z = S Z − S Z (B2)= 2 P | η | ( β − k ) δz (B3)proportional to the particle displacement δz .The radial read-out relies on interference between scat-tered light from the particle into the higher order LP mode and the trapping mode. A particle moving radi-ally (without loss of generality we assume motion alongthe x axis, the treatment in the orthogonal direction y isanalogous) excites the antisymmetric LP mode witha position dependent phase and amplitude η ( δx ) = (cid:104) LP , E dp (cid:105) = E ( a + ib ) δx . The interference betweenthe excited mode by the particle and the fundamentalmode is I X = ε c | LP + η LP | (B4)= I + | η | ε + ε cE (cid:60){ η ∗ e i ∆ βz } (B5) ≈ I + ε ε ε cE | η | sin(∆ βz + φ ) δz (B6)with | η | = √ a + b and φ = arctan( b/a ). It is im-portant to note that the first term I = ε c/ E ε ( x, y )is symmetric with respect to the x-axis and the secondterm ∝ ε ( x, y ) ε ( x, y ) is anti-symmetric with respectto the x-axis. Here ε ( x, y ) and ε ( x, y ) are the radialfield distributions of the LP and LP mode. Secondlythe term proportional to the displacement δx also de-pends on the axial particle position z . Therefore, thesensitivity of the radial read-out is modulated with thewavelength of 2 π/ ∆ β ≈ µ m.As shown in figure 5b the intensity distribution I X issplit on a knife-edge such that half of the mode is reflectedto detector X and half of the mode is transmitted to de-tector X . The difference signal between both detectorsis S X = S X1 − S X2 = (cid:90) − x dA I X − x (cid:90) dA I X (B7) ∝ P | η | sin(∆ βz + φ ) δz (B8)with x the extend of the detector. Altogether, the sym-metric part of I X vanishes and the anti-symmetric partstays due to the integration and results in a signal di-rect proportional to the radial particle motion δx . Theread-out along the y -direction has the same underlyingprinciple and, accordingly, a second knife-edge that isrotated by 90 ◦ .In order to monitor all three directions simultaneously,the light sampled at the beam splitter (BS, figure 1a,main text) is split into three parts for each spatial di-rection such that all can be monitored simultaneously,indicated with S in figure 5. Appendix C: Trapping in front of HCPCF at highpressure
As mentioned in the main text the largest distancebetween HCPCF end and the position a particle can bestably trapped in front of the HCPCF is an importantbenchmark of our experiment. The data shown in themain text was taken at the lowest pressure at which wecould keep a particle trapped. At higher pressures ( p ≈ µ m from the HCPCF tip (Figure 6). -100 0 100 2000100200300 z [µm] Ω / � [ k H z ] FIG. 6. The mechanical frequency of a trapped nano-particleat p ≈ Appendix D: Linewidth determination in thefrequency domain
The spectrum of damped, thermally driven harmonicoscillator is [47] S x ( ω ) = 2 k B T Γ p πm − ω ) + (Γ p + Γ fb ) ω . (D1)Equation D1 has a peak at the mechanical frequencyΩ with a linewidth Γ = Γ p + Γ fb . Both values can beobtained from fitting. To resolve the linewidth Γ fromthe spectrum a measurement time of at least t (cid:38) / Γis necessary. However,in our experiment the mechanicalfrequency fluctuates due to drifts in the trapping laser,leading to inhomogeneous broadening of the spectrum.This effect becomes significant for a linewidth Γ (cid:46) p ≈ Appendix E: Feedback control and center of massmotion temperature
The scattering force [24] of the feedback laser is F scatt = σ scatt c I ( r, z ) with σ scatt = πk a ( ε − / ( ε + 2) the scattering cross section, k = 2 π/λ the wave vector, ε the dielectric constant of the particle, c the speed of lightand I ( r, z ) the intensity distribution of the electromag-netic field.The scattering force pushes a trapped particle awayfrom an intensity maximum of the standing wave. Fora constant power P fb = P a trapped particle finds anew equilibrium position z where F scatt ( z ) = F grad ( z ).Modulation of the feedback laser around P either pushesthe particle further away from z or the gradient forcepulls the particle back towards intensity maximum ofthe standing wave. By modulating the laser power wecan therefore effectively apply a force in both directionsfor a particle trapped at the new equilibrium position.Modulating the feedback laser proportional to the parti-cle velocity results in a force F fb = m Γ fb ˙ z . The read-out of the axial COM motion is used forfeedback control. The voltage signal from the detec-tor is highpass filtered (corner frequency ν hp ≈
50 kHz)and connected to a feedback circuit. It consist of threeparts, a phase shifter, a variable amplifier and a con-trolled switch. The phase shifter delays the signal by t = π/ (2Ω) effectively leading to a velocity feedback inthe high-Q limit. The variable amplifier is used to changethe strength of the feedback signal. The control switchis used to enable or disable the feedback control, whichis used for the energy relaxation measurements.With appropriate settings of delay and gain the feed-back laser applies a feedback force F fb = m Γ fb ˙ z whereΓ fb is proportional to the gain. The equation of motionbecomes to ¨ z + (Γ p + Γ fb ) ˙ z + Ω z = F therm m . (E1)An effective temperature T COM of the COM motion [36]can be defined T COM = T Γ p Γ p + Γ fb (E2)and for Γ p (cid:28) Γ fb the expression simplifies to T COM ≈ T Γ p / Γ fb . Appendix F: Transient behaviour of the potentialenergy
The equation of motion (equation 1, main text) de-scribing the levitated nano-particle can be rewritten asa system of differential equations of first order intro-ducing the velocity v = ˙ x , a normalized state vector q = ( x/x th , v/v th ) (cid:62) with x th = (cid:112) k B T / ( m Ω ) the RMSthermal amplitude, v th = x th / Ω, n = (0 , (cid:112) Γ p ) ζ and ζ (cid:112) Γ p = F therm / ( m Ω x th ) to˙ q = A q + n ζ, (F1)with A a constant matrix A = (cid:18) − Ω − Γ p (cid:19) . (F2)The time dependence of the covariance matrix Σ( t ) = (cid:104) qq (cid:62) (cid:105) is given by˙Σ = A Σ + Σ A (cid:62) + nn (cid:62) . (F3)The first element of the covariance matrix contains thepotential energy and is plotted in figure 7 for an ini-tial state E = 0 . E (lowest temperature achieved, seemain text) relaxing back to thermal equilibrium E . Theenergy relaxation from a cold state back to thermal equi-librium or from thermal equilibrium to a cold state in themain text is approximated with an exponential function.The analytical solution has an additional modulation, de-picted in the inset in figure 7, which is not resolved byour experimental data and not necessary to infer the re-laxation constant Γ p or Γ fb . -7 -6 -5 -4 t [s] E / E -3 -2 -5 -5 FIG. 7. Thermalization of the potential energy from an initialstate with E = 0 . E to thermal equilibrium Appendix G: Pressure distribution in the molecularflow regime