Transition from synchronous to asynchronous superfluid phase slippage in an aperture array

Transcription

1 Transition from synhronous to asynhronous superfluid phase page in an aperture array Y. Sato, E. Hoskinson and R. E. Pakard Department of Physis, University of California, Berkeley CA 94720, USA We have investigated the dynamis of superfluid phase page in an array of apertures. The magnitude of the dissipative phase s shows that they our simultaneously in all the apertures when the temperature is near T λ T 10 mk and subsequently lose their simultaneity as the temperature is lowered. We find that when periodi synhronous phase page ours, the synhroniity exists from the very first phase, and therefore is not due to mode loking of interating osillators. When the system is allowed to relax freely from a given initial energy, the total number of phase s that our and the energy left in the system after the last phase depends reproduibly on the initial energy. We find the energy remaining after the final phase is a periodi funtion of the initial system energy. This dependene diretly reveals the disrete and dissipative nature of the phase s and is a powerful diagnosti for investigation of synhroniity in the array. When the array s synhronously, this periodi energy funtion is a sharp sawtooth. As the temperature is lowered and the degree of synhroniity drops, the peak of this sawtooth beomes rounded, suggesting a broadening of the time interval over whih the array s. The underlying mehanism for the higher temperature synhronous behavior and the following loss of synhroniity at lower temperatures is not yet understood. We disuss the impliations of our measurements and pose several questions that need to be resolved by a theory explaining the synhronous behavior in this quantum system. An understanding of the array phase proess is essential to the optimization of superfluid d- SQUID gyrosopes and interferometers [1]. Superfluid 4 iφ He is desribed by a omplex order parameter ψ e. Phase differenes are proportional to superfluid veloity and vary as d ( Δφ ) dt = Δμ h. Superflow is driven by hemial potential differenes, Δμ = m4 ( ΔP ρ sδt ), where Δ P and ΔT are differenes in pressure and temperature, ρ is the mass density, s is the speifi entropy, and m 4 is the 4 He atomi mass. One of the defining signatures of superfluidity is the absene of flow dissipation below some ritial veloity, v. Whenever the flow through a submiron-size aperture reahes v, dissipation ours in a disrete event wherein the quantum phase differene aross the aperture drops by 2π [ 2].

2 Sine superfluid veloity is proportional to phase gradient, this 2π orresponds to a disrete drop in veloity, v = κ / leff where κ = h/m4 is the quantum of irulation and l eff is the effetive hydrodynami length of the aperture. If one applies a onstant hemial potential differene Δμ aross an aperture, the superfluid veloity inreases linearly to the ritial veloity, followed by an abrupt drop (if the duration of the is short ompared to the aeleration time) and followed again by a linear inrease. The waveform of superfluid veloity v s (t) then resembles a sawtooth in whih the phase events take plae at an average rate equal to the Josephson frequeny f = Δμ/ h. For single apertures, stohasti flutuations in the ritial veloity usually obsure the periodi nature of this proess [3]. Reent work [4] has shown that in superfluid 4 He periodi phase osillations at frequeny f exist in an array of (=4225) apertures. The osillation amplitude near the superfluid transition temperature implies that the phase s our synhronously (i.e. simultaneously) among all the apertures. Josephson osillations an be used as a phase differene sensor in superfluid gyrosopes and interferometers [1]. It is neessary to understand the origin of the synhroniity mehanism in order to optimize the design of suh devies. To investigate the nature of phase s within the array, we have performed three kinds of experiments. In the first, we drive phase osillations by applying a hemial potential differene aross an aperture array and measure the phase osillation amplitude down to T λ T = 160 mk. We find that the amplitude dereases rather dramatially as the temperature is lowered, as ompared to what would be expeted for synhronous behavior. In a seond experiment, we exite transient Josephson osillations lasting from one yle to thousands of yles. We find that the phase size does not hange over many yles of osillation, indiating that when phase s are synhronous, they are synhronous from the very first. In the third experiment, we give the system an initial exitation energy, allow it to deay through the dissipative phase s, then reord the amplitude of the sub-ritial urrent osillation (the so-alled Helmholtz mode) that ours after the last phase. We find that as the temperature dereases, phase s within the array seem to our in a less abrupt manner implying that a phase event is no longer a single simultaneous array-wide event but rather a olletion of unorrelated events loalized to individual apertures. We present these three findings in the first part of this paper and disuss possible interpretations in the seond. Our experimental apparatus is shown in the inset of Fig. 1. Two volumes filled with superfluid 4 He are separated by a diaphragm and an array of (=4225) apertures that are ~ 30 nm in diameter and spaed 3 μm apart in a 50 nm thik silion nitride hip. A thin, flexible, metal-oated diaphragm an be pulled toward an eletrode by the appliation of a voltage between them. A SQUID-based displaement sensor [5] is used

3 to monitor the position of the diaphragm that serves as a mirophone to determine the magnitude of the phase osillation. In our first type of measurement, we apply a DC step voltage between the diaphragm and the eletrode at t = 0. This pulls the flexible diaphragm toward the eletrode reating a pressure head (and therefore a hemial potential differene) aross the array. If the initial pull is large enough, the flow veloity inside the apertures reahes v and the fluid undergoes 2 π phase s at the Josephson frequeny. These dissipative events ontinue until there is no energy left to drive the fluid up to the ritial veloity. The phase osillation ends, and the system begins to osillate about Δμ = 0 at a different frequeny the Helmholtz frequeny. The restoring fore of the diaphragm, the inertia of the fluid moving in the apertures, and the heat apaity of the fluid in the inner volume determine the frequeny of this resonant mode [6]. Fig. 1(b) shows a typial diaphragm displaement x(t) during one of these relaxation transients. The disontinuities in fluid veloity due to phase events show up as sudden slope hanges in x(t). These an be seen in the first half of Fig. 1(). To determine whether or not phase s are ourring synhronously throughout the array, we measure the peak-to-peak amplitude of the phase urrent osillations,, and ompare this number to the expeted magnitude if all apertures are loked I together, I. This expeted magnitude is determined by diretly measuring the urrent phase relation [ 7] I ( φ ) for the array during periods of sub-ritial flow (i.e. the Helmholtz osillation) where the flow is synhronous aross the array. We are onerned with the strong oupling regime T λ T 10 mk, where I ( φ ) is linear. The expeted magnitude of the urrent osillation for synhronous 2π phase s is then, di( φ) I = 2π 1. d φ If the phase s are synhronous, I = I and if the array loses synhroniity, I < I. We determine mass urrents through the array by monitoring the diaphragm position x() t. The urrent through the array is given by, I = ρax& 2. where ρ is the total fluid density, and A is the diaphragm area.

4 Figure 1: (a) Experimental apparatus. A: Fixed eletrode. B: Soft diaphragm. C: Heater. D: Aperture array. Above the eletrode is a SQUID-based transduer (not shown) whih monitors the position of the diaphragm. (b) Typial diaphragm transient response. This data was taken at T λ T 9 mk. The pressure aross the array is diretly proportional to the displaement of the diaphragm from equilibrium. The initial steep rise after the pressure step at t = 0 is a linear relaxation during whih the fluid is exhibiting phase s at frequeny f. The lightly damped Helmholtz osillation begins when the system reahes Δμ =0 near t = 0.24 se. The slowly urvature in the mean of the Helmholtz osillation (between t = 0.24 se and t = 10 se) reflets hanges in pressure head in response to a relaxing thermo-mehanial temperature differential, suh that the mean Δμ remains zero. The dotted irle shows when the phase osillation ends and the Helmholtz mode begins. The lose-up is shown in (). When a hemial potential differential exits aross the array, the diaphragm 1 exhibits osillations at the Josephson period, f. If the amplitude of suh diaphragm osillations is xd, the magnitude of the mass urrent osillations at frequeny f is given by, 2πf ρaxd I = 3. γ where γ is the Fourier oeffiient of the first harmoni of the displaement sensor signal. We assume here that the urrent exhibits a sawtooth waveform, a ase where γ = 2/ π.

5 Figure 2: Measured phase urrent osillation amplitude I (for f < 300 Hz) and the expeted value for a fully synhronous ase I. The lines are a guide to the eye. To determine x d, we reord the signal x ( t) preeding the Helmholtz mode and ompute the Fourier transform of the diaphragm osillations. By analyzing the spetral ontent in small time intervals, we extrat the frequeny and the amplitude of the phase osillations as a funtion of time throughout the transient. The amplitude varies with frequeny due to ell resonanes but levels off at lower frequenies (typially below 300Hz). We use this limiting value for x d. This Fourier analysis beomes more diffiult at lower temperatures beause the duration of the phase flow beomes shorter due to inreasing ritial veloity. To extend the duration of phase flow, we use a heater installed inside of the inner ell. First, we apply a step voltage to the heater whih reates a temperature differential Δ T aross the array and starts the phase osillation. We then ontinuously inrease the heater power during the transient to ounterat ooling due to net superfluid flow through the array (the thermo-mehanial effet). In this way, we slow the rate at whih hemial potential goes to zero. The extended transient allows us to apply the Fourier analysis desribed above and find the amplitude of osillations at lower temperatures. One we obtain the amplitude of the diaphragm osillations x d, we use Eq. 3 to ompute I. Figure 2 shows the variation of I with temperature. For omparison, we also plot I defined by Eq. 1 using data derived from Ref.[ 7]. As seen in the figure, at the highest temperatures where phase s appear, ( T λ T 9 mk), we find I = I, whih implies that phase s are ourring synhronously among all the apertures. However as the temperature dereases, the

6 amplitude of urrent osillation starts to rapidly derease (relative to I ) showing a loss of synhroniity among apertures. This is the entral finding of this experiment. either the mehanism for the initial synhronizations nor the reason for its subsequent loss is yet understood. However, systems of interating nonlinear osillators often exhibit synhronization after multiple yles [8]. If suh nonlinear mode loking is present in the array, one would expet the size of first phase to be smaller than that of the n th where n >> 1. Our seond type of experiment is direted toward determining if there is a hange in overall size between the first and nth phase osillation. Equation 2 shows that when a dissipative phase ours, the sudden urrent drop in the aperture array is refleted by a sudden hange in the slope of the diaphragm position urve x(t). By adusting the voltage step applied to the diaphragm we vary the length of the phase osillation train from as little as one to as many as several thousands of s. We then ompare the abrupt slope hanges, shown in Fig. 1(), at the th first and the n. The hange in the slope, Δx&(t) is determined as follows. The fluid aeleration is proportional to the hemial potential differene Δμ aross the array. If Δμ is onstant in the viinity of a, the urrent inreases linearly in time and the displaement of the diaphragm follows a parabola. We fit two parabolas at the usp in the diaphragm position x() t (one before the phase and another right after) and find the hange in the slope Δx&(t). The result is plotted in Fig. 3. We find that the phase size does not hange over many yles. This result shows that when the osillations are synhronous, they are synhronous from the very first. We onlude then that the synhronization is not due to a nonlinear mode loking proess. Our third experiment sheds additional light on the nature of olletive phase page in the array. We apply a small step voltage, V, between the diaphragm and the eletrode to reate hemial potential differentials whih are suffiiently small to keep the fluid veloity inside the apertures sub-ritial. In the subsequent flow transient, the hemial potential reahes zero without induing any phase s and the diaphragm osillates at the Helmholtz frequeny with an initial amplitude x h. In the absene of phase page, the initial energy in the Helmholtz osillation, E h, should be proportional to E 0, whih is the energy that we put into the system by the appliation of a voltage step. As we inrease the initial kik on the diaphragm and plot E h versus E 0, we expet a line with onstant slope until E 0 is large enough to aelerate the fluid up to v, triggering a phase. At that point, energy is dissipated. If phase s our simultaneously in all the apertures, E h should then drop disontinuously due to the abrupt extration of energy. After suh an event, as we inrease inrease linearly again until the proess repeats. Eo further, E should h

7 Figure 3. Diaphragm veloity hange at the 1 st and the n th where n is on the order of The temperature dependene omes from the inreasing superfluid density as the temperature dereases. 2 Sine the equilibrium diaphragm displaement is proportional to V, the energy 4 that we put into the system, E 0, sales as V. The initial energy in the Helmholtz osillation, E h, is proportional to the square of the initial Helmholtz diaphragm osillation amplitude, x h. Thus a plot of x h versus V (whih orresponds to E h versus E 0 ) should be a sawtooth if the phase page ours abruptly and simultaneously throughout the array. If the phase page proess is distributed in time, as individual apertures independently of others, the sawtooth would be rounded. 2 4 Figure 4 shows our measurements of x h vs. V at various temperatures. As the 2 4 temperature is lowered below T λ, the shape of x h vs V evolves from a sharp sawtooth indiative of an abrupt olletive phase event to a smoother urve that implies a ontinuous phase slide proess. This suggests that some apertures are experiening a phase before the others, allowing the array to dissipate energy in a more ontinuous manner. Figure 4 also illustrates the striking rossover from a dissipative phase regime to the non-dissipative Josphson regime. The ritial veloity v (or Helmholtz amplitude) at whih a ours inreases as the temperature dereases. At T λ T 15 mk, v v, and a single array phase event removes almost all the energy in the fluid and leaves none for the Helmholtz mode. Therefore, the Helmholtz osillation

8 amplitude goes to 0 every time a phase ours. As one gets loser to T λ, v beomes smaller than v, and a phase event auses a reversal in the flow diretion. Phase s are no longer fully dissipative the system retrieves some of the energy involved in the reversal of flow. At T λ T 5 mk, where v 2v, dissipation due to the osillations, whih are still present as Josephson osillations instead of phase s, eases. One an view this to be the omplete transition into a weakly oupled Josephson regime. In the weakly oupled regime the dominant dissipation ours through thermal ondution and normal flow Josephson osillations ease and Helmholtz osillations begin when there is no longer enough energy (the flat limiting value in the 5 mk data) to reah the ritial urrent and drive Josephson osillations. This alternate form of dissipation, although small ompared to the phase s, explains why (in the phase regime) the period of the x 2 h versus V 4 urves inreases with V 4 : for larger initial energy, the system takes longer to reah the Helmholtz mode and more energy is dissipated through thermal ondution and normal flow. 2 4 Figure 4: Measured versus V. Temperatures shown are x h T λ T in mk.

9 We have onsidered possible mehanisms for the observed derease in phase amplitude as exhibited in Fig 2. Disrete phase page in superfluid 4 He is usually assoiated with the passage of quantized vorties that are stohastially nuleated near the aperture surfae [2]. The intrinsi flutuations ause the ritial veloity to be spread out over a range Δv. This finite distribution width an ause the phase osillation to lose its well-defined periodiity [3]. The ritial veloity width Δv is a funtion of temperature, and the relevant quantity in determining the temporal oherene of phase osillations in a given aperture is Δv v /. If Δv / v > 1, the periodiity at f J is lost. Previous work [3, 9, 10] suggests that this ratio Δv / v inreases with dereasing temperature near the superfluid transition temperature. The observed deline in the osillation amplitude therefore ould be a manifestation of loss of periodiity in any individual aperture. Another possible mehanism for the loss of synhroniity at lower temperatures may involve variations in the surfae mirostruture among the array apertures. With the fluid flowing fastest near asperities, the ritial veloity for an aperture must be affeted by the surfae inhomogeneities. Sine the superfluid healing length ξ is a funtion of temperature, how muh of these nano-sale inhomogeneities the fluid atually sees should depend on temperature as well. The healing length is given by 0.3nm ξ(t) = (4) 0.67 ( 1 T ) Tλ and it dereases from 10 nm to 1.5 nm as the temperature is lowered from T λ T 10 mk to T λ T 160 mk. If the surfae variations are on the order of a few na nometers, this ould very well provide a ritial veloity distribution whose width inreases with dereasing temperature while allowing the individual apertures to maintain well-defined periodi osillations. Several overarhing questions remain. Is it possible for apertures to at independently in the presene of a marosopi wavefuntion? Cirulation around every loop drawn through the apertures must be quantized while minimizing the energy assoiated with the phase gradient aross the array. It is not lear how this ondition is satisfied when phase s are ourring in random positions within the array. What are the dynamis of vorties near the transition temperature when the energy removed in a single phase beomes omparable to the flow energy itself? What is even meant by a vortex when the vortex ore ~ ξ ( T ) is omparable to the size of the apertures? Perhaps then phase s our by ollapse of the wave funtion rather than by vortex dynamis [ 11]. The superfluid order parameter may already be so weakened that at v the fluid in the aperture beomes momentarily normal before superfluidity is restored to a state in whih the phase differene aross the array has dropped by 2 π. This might lead to synhroniity if the wave funtion is so weak in all of the apertures

10 that an exitation that auses the wave funtion to ollapse in one aperture perturbs the other apertures enough to ause them all to ollapse. The experiments desribed above show that near T λ phase page ours olletively in all the apertures in an array and the related osillations at the Josephson frequeny are not due to nonlinear synhronization. The observed deline in phase osillation amplitude and the rounding of the sawtooth in the x 2 h versus V 4 plot both indiate that array phase page loses its olletive nature as the temperature is lowered. The results reported herein raise fundamental questions about the phase page proess and the dynamis of a weakened superfluid onfined in a multiply onneted region. We aknowledge stimulating onversations with Prof. Dung Hai Lee and Henry Fu. We thank Aditya Joshi for his suggestions to the manusript. This work was supported in part by the SF grant DMR and ASA. 1 E. Hoskinson, Y. Sato, and R. E. Pakard, arxiv:ond-mat/ O. Avenel and E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985). 3 S. Bakhaus and R. E. Pakard, Phys. Rev. Lett. 81, 1893 (1998). 4 E. Hoskinson, R. E. Pakard, and T. M. Haard, ature, 433, 376 (2005). arxiv:ond-mat/ H. J. Paik, J. Appl. Phys. 47, 1168 (1976). 6 E. Hoskinson, PhD thesis, University of California, Berkeley, USA (2005). 7 E. Hoskinson, Y. Sato, I. Hahn, and R. E. Pakard, ature Physis, 2, 23 (2006). arxiv:ond-mat/ R. E. Mirollow and S. H. Strogatz, SIAM J. Appl. Math. 50, 1645 (1990). 9 C. A. Lindensmith, J. A. Flaten, and Jr. W. Zimmermann, In S. Danis, V. Gregor, and K. Zaveta, editors, Proeedings of st the 21 International Conferene on Low Temperature Physis, Prague, page 131 (1996). 10 E. Varoquaux, O. Avenel, Y. Mukharsky, and P. Hakonen, In C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, editors, Quantized Vortex Dynamis and Superfluid Turbulene, volume 571, page 36. Springer, Berlin (2001). 11 W. J. Skopol, M. R. Beasley, M. Tinkham, JLTP, 16, 145 (1974)