Brownian motion in superfluid 4 He
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Brownian motion in superfluid He Xiao Li † , ∗ Ran Cheng † , Tongcang Li, and Qian Niu § Department of Physics, The University of Texas at Austin, Texas 78712, USA
We propose to study the Brownian motion of a classical microsphere submerged in superfluid He using therecent laser technology as a direct investigation of the thermal fluctuations of quasiparticles in the quantum fluid.By calculating the temperature dependence of both the friction coefficient and the strength of the random force,we show that the resonant mode of the fluctuational motion can be fully resolved by the present technology.Contrary to the previous work, it is found that the roton contribution is not negligible, and it even becomesdominant when the temperature is above . K. PACS numbers: 05.40.Jc, 67.25.de, 42.55.-f
Since its discovery in 1827, Brownian motion has at-tracted broad interest and become a major thrust of statisticalphysics . Driven by thermal fluctuations among the con-stituent particles, the Brownian motion gave the first directevidence of the particle nature of the background fluid. How-ever, at extremely low temperatures, quantum effects of thefluid particles become important and certain kinds of quantumfluid emerge, where the system can be described by a macro-scopic wave function with long-range phase coherence. Insuch a case, it is the elementary excitations that play the roleof the fluid particles. In thermal equilibrium, those excitationsform a gas of quasiparticles that behaves in a similar way tothe ordinary atomic gas. They are able to create pressure andresistance when colliding at the interface with a solid. How-ever, their unusual dispersion relations are quite different fromthose of ordinary particles. A good example is the superfluid He where two kinds of particle-like excitations, phonons androtons (see Fig. 1), comprise the viscous part of the fluid .A natural question arisen is whether these quasiparticles cancreate Brownian motion. To date, this question has not beenaddressed by any experiment, and the only theoretical studyby Balazs has ignored the roton contribution without justifi-cation. Moreover, Balazs’ prediction was far beyond the ex-perimental capability for two reasons: first, the amplitude ofthe Brownian particle was too small to be monitored at suchlow temperatures; second, the required observation time wastoo long due to the tiny restoring force provided by the quartzfiber. But without a fiber, on the experimental side, no objectcould be suspended in superfluid He for its small density.A recent progress in laser technology, however, removesthese limitations. It not only promotes the position and timeresolutions to a very high level , but also is capable oftrapping a classical microsphere by laser beams of high inten-sity that provides a sufficiently strong restoring force, roughlyabout ∼ times higher than a quartz fiber. In addi-tion, the superfluid medium is transparent to the laser thus thethermal equilibrium of the fluid will not be destroyed by thebeams. This technical development therefore makes it possi-ble to test the Brownian motion experimentally in superfluid He, or potentially, any quantum fluid that does not interactwith the laser. Therefore, a complete study of the Brownianmotion in superfluid He, incorporating contributions fromboth the phonon and the roton, will shed light on both the-oretical and experimental investigations. In this paper, we use the Langevin equation to describe thedynamics of the Brownian particle with the relevant parame-ters taken from recent experiments . The friction coefficientand the strength of the random force are evaluated by the ki-netic theory, and expressed as functions of the temperature.We conclude that the Full-Width-at-Half-Maximum (FWHM)and the mean square of the resonant mode of the fluctuationalmotion are able to be directly measured by the present exper-imental techniques . Interestingly, while the roton contri-bution to the above two quantities is negligible at low temper-atures, it becomes dominant above . K. + R - phonon rotonmaxon ( p ) / k B ( K ) p/ (¯ -1 ) FIG. 1. Quasiparticle spectrum in superfluid He. We have the lin-ear phonon excitations at small momenta and the roton excitationsat higher momenta. Rotons with negative and positive slopes arelabeled by R − and R + respectively. The dashed lines representthe analytical expressions to be used to approximate the spectrum: ε ( p ) = c s p and ε ( p ) = ∆ + ( p − p rot ) m , where c s = 239 m/s, m = 1 . × − kg, and p rot / ~ = 1 . ˚A. We consider a microsphere with mass M held in super-fluid He by a laser trap with harmonic angular frequency ω . Quasiparticles excited by thermal fluctuations in the su-perfluid will create random forces on the microsphere, result-ing in the Brownian motion around its equilibrium position.The motion of the microsphere can be well described by theLangevin equation: M ¨ r + γ ˙ r + M ω r = F ( t ) , (1)where r denotes the position of the ball, γ represents the fric-tion to be estimated below, and F ( t ) is the random force thatdrives the Brownian motion. It is apparent that different spa-cial components of r are independent in Eq. (1). Thus inthe following we shall only focus on the z -component of theBrownian motion. As a consequence, we now consider an ef-fective plate with area σ = πr to represent the microsphere,upon which the problem reduces to one dimension.We will solve the Langevin equation under the followingassumption: τ c ≪ π/ω ≪ τ, (2)which says that the observation time τ far exceeds the typicalperiod of free oscillation, and the latter is also much largerthan the time interval τ c between two adjacent collisions fromthe quasiparticles . We decompose the random force into aFourier sum F z ( t ) = P n ( A n cos ω n t + B n sin ω n t ) , where ω n = 2 πn/τ , with n taking integer values, and A n = τ R τ F z ( t ) cos ω k tdt , B n = τ R τ F z ( t ) sin ω n tdt . It is worthmentioning that the Fourier decomposition makes sense onlywhen we regard F z ( t ) as a periodic function of time with pe-riod τ . By expressing the z -component of the displacementas z ( t ) = P n z n ( t ) , Eq. (1) can be solved in the frequencydomain where z n = 12 M A n + B n ( ω − ω n ) + ( γM ) ω n . (3)What subject to direct experimental verifications are theFWHM of the peak and the mean square amplitude of theresonant mode at ω n = ω , which equal to γ/M and h A + B i / γ ω respectively, where h i denotes the ensem-ble average. The latter one can be brought into a more use-ful form if we make the following considerations. Becausethe quasiparticle density is dilute, two adjacent collisions canbe considered uncorrelated, and thus we will adopt the whitenoise assumption that h F z ( t ) F z ( t ) i τ = D ( T ) δ ( t − t ) ,where D ( T ) is a function of the temperature alone. Also,by regarding τ as sufficiently large (According to Ref. 10, ω ∼ π × Hz, thus τ = 1 s meets the requirement),we convert the summation P k to the integral τ π R dω . Thenwe obtain from Eq. (3) the mean square of the resonant mode, h z R i = 1 ω π D ( T ) γ ( T ) . (4)Therefore, what we need to evaluate are γ ( T ) and D ( T ) asfunctions of the temperature. Although the equipartition rela-tion D / γ = k B T / obviates the need to calculate both ofthem, we still do so for strictness.The friction γ originates from the imbalance of the forwardand backward scattering when an object has an instantaneousvelocity with respect to the background fluid. Suppose ∆ σ is a given area on the front side of the effective plate withits normal taken to be the ˆ z -axis. When the plate movesalong z direction with velocity v z , the number of quasipar-ticles within momentum interval [ p , p + d p ] that is able tocollide on ∆ σ during time ∆ t ( τ c ≪ ∆ t ≪ π/ω ) is givenby ∆ σ ∆ td p | v z − u z | N p /h , where u z = ∂ε ( p ) /∂p z is thegroup velocity of the quasiparticle, and N p is the number of particles in each unit volume with vector momentum p . Ifeach collision transfers a momentum δp z to the plate whichwill be specified below, the resulting force from the front isthe average of the total momentum transfer during this timeinterval divided by ∆ t , F f = σh Z d p δp z | v z − u z | e βε − , −∞ < u z < v z , (5)where σ is the total area of the plate. Similarly, force F b frombehind is given by the same expression but the range of u z should be v z < u z < ∞ . Addition of F f and F b gives the netresistant force F r = F f + F b = γv z + O ( v z ) , with the linearterm in v z being the friction. z R- rotonPhonon R+ rotonp Phonon Phononp p p p SuperfulidsSolids
FIG. 2. Reflections and transmissions of quasiparticles on the in-terface separating the effective plate and the superfluid He. Thisdiagram only show the case when a phonon with momentum p (in) in-cident on the interface with four outgoing channels: elastic reflectionto a phonon p with possibility R , to a R − roton p with possibil-ity R , and to a R + roton p with possibility R ; inelastic collisionthat creates a phonon p into the solid with possibility R . Arrowson each line represents the group velocity of that quasiparticle. Noteespecially that the group velocity for R − is in the opposite directionof its momentum. Specific computation of γ along this line needs detailed in-formation on δp z . Let us now focus on an individual collisionthat respects the momentum and energy conservations, p z + M v z = ( p z − δp z ) + M v ′ z ,ε ( p ) + 12 M v z = ε ( p p − p z δp z + ( δp z ) ) + 12 M v ′ z , and we regard the mass of the Brownian particle M as suffi-ciently large ( M = 2 . × − kg experimentally) so thateach process is a hard wall collision. Problem arises when theenergy of the incoming quasiparticle ε ( p ) lies in the region (∆ roton , ∆ maxon ) , for which the above equations admit three in-equivalent solutions of δp z . That is to say, for example, fora phonon carrying definite momentum and energy incident onthe plate, it has three different outgoing channels labeled by i = 1 , , representing the phonon, R − and R + rotons re-spectively. We may also add the possibility of inelastic col-lision that creates phonons in the Brownian particle labeledas i = 4 . This multi-channel collision process is illus-trated in Fig. 2. The momentum transfer δp z then dependson the transition probability R ij connecting the i -th and the j -th channels. However, it is remarkable that the final expres-sion of γ turns out to be independent of R ij as if there wereno inter-channel transitions. We omit the detailed argument ofthis result, as it shares similar logic with the problem of quasi-particle pressure in superfluid He , where all inter-channeltransitions mutually cancel. After some manipulations, boththe phonon and the roton contributions can be brought into thesame form as γ ( T ) = 4 πσh Z ∞ d p p e βε ( p ) − . (6)Insertion of the dispersion relations ε ( p ) = c s p and ε ( p ) =∆ + ( p − p rot ) m (See Fig. 1) yields the contributions from thephonon and the roton respectively, γ ph ( T ) = σπ ~ c ( k B T ) , (7) γ rot ( T ) = σp rot ~ π r m e − ∆ kBT p k B T (cid:18) mk B Tp rot (cid:19) , (8)and the total friction coefficient is γ ( T ) = γ ph ( T ) + γ rot ( T ) .For a Brownian particle with M = 2 . × − kg , theFWHM γ ( T ) /M is depicted in Fig. 3. The crossing point of γ ph ( T ) and γ rot ( T ) is at 0.76 K. While the roton contributionis negligible at low temperatures, it becomes dominate above0.76 K. We see from the figure that the typical FWHM rangesfrom 10 Hz to 1 kHz, and that the departure from pure phononcontribution above 0.76 K is of the order of 1 kHz. These arewell within the capability of current experiments where theresolution of frequency is down to 1 Hz. All Phonon Roton F W H M o f t he M S D z k ( k H z ) Temperature (K)
FIG. 3. The temperature dependence of the FWHM of the meansquare displacement spectra. The thick black line denotes the totalwidth, while the dashed and dash-dotted lines are contributions fromthe phonon and the roton respectively. The effective area of the plateis taken to be π/ µ m . Having fully evaluated the friction coefficient γ ( T ) , nowwe turn to the more involved quantity D ( T ) which comes from the fluctuation of the random force exerted on theBrownian particle. Mathematically, the fluctuation is em-bodied in the statistical deviation of the quasiparticles dis-tribution n ( r , p ) in the six-dimensional phase space whichsatisfies N p = R d r n ( r , p ) . Neglecting the influence on n ( r , p ) of the scattering among quasiparticles, we againassume that n ( r , p ) on different phase points are inde-pendent so that h n ( r , p ) n ( r ′ , p ′ ) i ∼ δ ( r − r ′ ) δ ( p − p ′ ) , i.e., only contributions from the same phase point arekept. Then with the help of the Bose relation h ( N p − ¯ N p ) i = ¯ N p + ¯ N p , we know after some manipulationsthe total fluctuation of the momentum transfer during ∆ t is: h G z i ∆ t = σ ∆ t R d p /h | u z | ( δp z ) ( ¯ N p + ¯ N p ) . The corre-sponding fluctuation of the random force should be h F z i ∆ t = h G z i ∆ t / (∆ t ) and the quantity of central interest is D = h F z i ∆ t ∆ t , which gets rid of the ∆ t dependence, D ( T ) = σh Z d p (cid:12)(cid:12)(cid:12)(cid:12) ∂ε ( p ) ∂p z (cid:12)(cid:12)(cid:12)(cid:12) ( δp z ) e βε ( p ) ( e βε ( p ) − . (9)Again, by inserting the dispersion relations, a straightforwardcalculation leads us to: D ph ( T ) = σπ ~ c ( k B T ) , (10) D rot ( T ) = σp rot √ m ~ π e − ∆ kBT ( k B T ) (cid:18) mk B Tp rot (cid:19) , (11)and D ( T ) = D ph ( T ) + D rot ( T ) . We mention in passing thatthe equipartition relation D / γ = k B T / does hold sepa-rately for phonon and roton excitations.Equipped with the friction coefficient and the fluctuation ofrandom force, we are able to evaluate the temperature depen-dence of the resonant mode, which serves as another quantityfor direct experimental test. In view of the equipartition rela-tion, the resonant mode in Eq. (4) is h z R i = 2 k B T /ω πγ ph ( T ) + γ rot ( T ) . (12)This is plotted in Fig. 4, where . K is again identified as theturning temperature. The higher the temperature, the lowerthe amplitude of the resonant mode is. Fortunately, the lowest p h z R i seen from the plot is about − ∼ − nm/ √ Hz, farbeyond the experimental resolution . × − nm/ √ Hz . Atlow temperatures, however, the resonant mode is dominatedby the phonon excitation and diverges as T − . This seem-ingly counterintuitive result is resolved when we consider theassumption made in Eq. (2), which has been discussed inRef. 9. As the temperature goes down, τ c sharply increasesand the above assumption becomes invalid, where a new the-ory is required. Fortunately, for a typical angular frequency ω ∼ π × Hz, Eq. (2) holds until the temperature islowered to − K.Before conclusion, two further comments are in order.First, the motion of the Brownian particle will transfer ki-netic energy to the background fluid. As a consequence themass M in Eq. (1) should be the effective mass instead of the All Phonon Roton z R ( t ) ( n m ) / H z Temperature (K)
FIG. 4. The temperature dependence of the square amplitude of theresonant mode. The thick line represents the result of Eq. (12). Incomparison, we also plotted the result in existing literatures by thedashed line, where the roton contribution is absent. The dash-dotline depicts the case if the roton contributes alone. Parameters relat-ing to experiment are the same as those in Fig. 2, and the resonantfrequency ω is taken to be π × Hz. bare mass m . Below T = : M = m + 12 ρ s × πr , where ρ s is the superfluid densityand r is the radius of the microsphere. For the microspherewith a diameter of 3 µ m and mass of m = 2 . × − kg, the correction term is roughly about 5%.Moreover, the white noise assumption on the random forceas well as the delta correlated distribution n ( r , p ) implies theindependence of different quasiparticle modes. This is quitereasonable when the density of quasiparticles is dilute, as therelative strength of quasiparticle scattering is proportional toits square. At roughly T = , and wewould expect a negligible effects of quasiparticle scattering.In conclusion, we have studied the Brownian motion ofa classical microsphere driven by thermally excited quasi-particles in superfluid He. Contrary to previous work, weclaim the importance of contributions from both the phononand the roton excitations, and find the turning temperature oftheir relative importance at 0.76 K. More importantly, the twopredictions we give on the FWHM and the resonant mode areable to be tested in current experiments. Generalization toother types of quantum fluids are left for future inquires.The authors are grateful to Prof. M. Raizen for help-ful discussions. This work is supported by NSF (GrantNo. DMR0906025), DOE (Grant No. DE-FG03-02ER45958,Division of Materials Science and Engineering), and theWelch foundation (Grant No. F-1255). † These authors contribute equally to this work. § On leave from the University of Texas at Austin. ∗ [email protected] A. Einstein, Ann. Phys. , 549 (1905). A. Einstein,
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