Transition from phase slips to the Josephson effect in a superfluid 4 He weak link

Transcription

1 1 Tranition from phae lip to the Joephon effet in a uperfluid 4 He weak link E. Hokinon 1, Y. Sato 1,. Hahn 2 and R. E. Pakard 1 1 Department of Phyi, Univerity of California, Berkeley, CA, 94720, USA. 2 Jet Propulion Laboratory, California ntitute of Tehnology, Paadena, CA 91109, USA The rih dynami of flow between two weakly oupled maroopi quantum reervoir ha led to a range of important tehnologie. Pratial development ha o far been limited to uperonduting ytem, for whih the bai building blok i the o-alled uperonduting Joephon weak link 1. With the reent obervation of quantum oillation 2 in uperfluid 4 He near 2K, we an now enviion analogou pratial uperfluid helium devie. The harateriti funtion whih determine the dynami of uh ytem i the urrent-phae relation (ϕ ), whih give the relationhip between the uperfluid urrent flowing through a weak link and the quantum phae differene ϕ aro it. Here we report the meaurement of the urrent-phae relation of a uperfluid 4 He weak link formed by an array of nanoaperture eparating two reervoir of uperfluid 4 He. A we vary the oupling trength between the two reervoir, we oberve a tranition from a trongly oupled regime in whih (ϕ ) i linear and flow i limited by 2π phae lip, to a weak oupling regime where (ϕ ) beome the inuoidal ignature of a Joephon weak link. The dynami of flow between two weakly oupled maroopi quantum reervoir an be highly ounterintuitive. n both uperondutor and uperfluid,

2 2 urrent will oillate through a ontrition (weak link) between two reervoir in repone to a tati driving fore whih, in a laial ytem, would imply yield flow in one diretion. n uperondutor, uh juntion have given rie to a range of tehnologie. Although promiing analogou devie 345 baed on weak link have been demontrated in uperfluid 3 He, pratial development will be hampered by the diffiulty of working at the very low temperature (<10-3 K) required. Quantum oillation were reently oberved in uperfluid 4 He at a temperature 2000 time higher. To undertand the fundamental nature of thee oillation, and to make progre toward devie development, it i neeary to know the relationhip between urrent and phae differene aro the juntion, (ϕ ). The meaurement of (ϕ ) reported here reveal a tranition between two ditint quantum regime, and open the door to the development of uperfluid 4 He interferene devie analogou to the d- SQUD, whih will be highly enitive to rotation. When uperfluid 4 He, well below it tranition temperature T λ = 2.17 K, i fored through a ontrition, it will aelerate until it reahe a ritial veloity, v, at whih a quantized vortex i nuleated. Thi i hown hematially in Fig. 1. The vortex move aro the path of the fluid, dereaing the quantum phae differene between the reervoir by 2π and dereaing the fluid veloity 6 by a quantized amount v. Thi phae lip proe repeat, uh that the flow through the ontrition follow a awtooth waveform. The ritial veloity dereae toward zero a T i inreaed toward T λ, but v i motly independent of T. When v < v, the flow atually revere diretion whenever a phae lip our. f thi ituation were to ontinue a T T λ and v drop below v /2, upon phae lipping the flow would end up with a veloity greater than v in the oppoite diretion. Thi ould not be an energy onerving proe. At about the ame temperature that thi would our, the healing length of the uperfluid, ξ d = 0.34(1 T Tλ ) nm, beome omparable to the diameter of the ontrition, d. Superfluidity i then uppreed in the onfined

3 3 geometry of the ontrition, whih now at a a barrier between the two reervoir of uperfluid, analogou to a Dayem bridge in uperondutor 7. n thi limit, where the wave funtion on either ide of the barrier partially overlap, the dynami of flow through the aperture i expeted to be deribed by the Joephon effet equation, whih predit inuoidal, rather than awtooth, oillation. Previouly, flow feature of a hydrodynami reonator were found to be onitent with uh inuoidal behavior 8. The ytem an be brought from one limit to the other by varying the trength of the oupling through the aperture. Thi oupling trength depend on the ratio ξ d and we ontrol ξ 4 by varying T. 4 Figure 1. Shemati of flow through an aperture and orreponding veloity evolution. Superfluid with veloity v aelerate in repone to a driving fore up to a ritial veloity v at whih a ingly quantized vortex i nuleated, roe the flow path, and aue a drop in v. Repeated vortex nuleation event give rie to a awtooth waveform. The ritial veloity v drop a T T λ. At ome T (lower urve, part b) the uperfluid flow will atually revere diretion.

4 4 A hemati of our experimental ell, deribed in more detail elewhere 9, i hown in Fig 2a. A ylindrial inner reervoir of diameter 8mm and height 0.6mm i bounded on the top by an 8µm thik flexible Kapton diaphragm on whih a 400nm thik layer of uperonduting lead ha been evaporated. An array of 4225 aperture paed on a 3µm quare lattie in a 50nm thik ilion nitride membrane i mounted in a rigid aluminum plate forming the wall and bottom of the inner reervoir. Flow meaurement both above and below T λ indiate that the aperture are d = 38 ± 9 nm in diameter. Preure an be indued aro the array by appliation of an eletrotati fore between the diaphragm and a nearby eletrode, thereby pulling up on the diaphragm. Above the eletrode i a uperonduting oil (not hown) in whih a peritent eletrial urrent flow, produing a magneti field whih i modified by the uperonduting plane of the diaphragm. Motion of the diaphragm, indiating fluid flow through the array, indue hange in the peritent urrent flowing in the oil, whih are deteted with a SQUD. The output of the SQUD i proportional to the 15 diplaement of the diaphragm x(t). We an reolve a diplaement of 2 10 m in one eond. The inner reervoir it in a ealed an filled from room temperature with 4 He through a apillary whih, in order to deouple the an and inner reervoir from environmental flutuation, i then bloked loe to the an with a ryogeni valve. The an i immered in a pumped bath of liquid helium whih i temperature tabilized to within ± 50 nk uing a ditributed reitive heater and high reolution thermometer 10 in a feedbak loop.

5 5 Figure 2. Shemati of the experimental ell and two typial flow tranient. (a) Shemati. (b), () Diaphragm poition x(t) a a funtion of time within two flow tranient. Thee plot eah how both lower amplitude Joephon frequeny oillation, for whih µ > 0, and larger amplitude Helmholtz oillation, for whih µ = 0. n (b), T T = 7.4 mk, λ 4 d = 0. 4 ξ and (ϕ ) i motly linear. The Joephon frequeny oillation our by the phae lip mehanim here the uperfluid veloity, proportional to dx dt, follow a awtooth waveform. n (), T λ T = 0.8 mk, ξ d = 1. 8 and (ϕ ) i motly inuoidal. Near t = 0.6 e where the 4 Joephon frequeny oillation deay into Helmholtz oillation, ϕ approahe a loal maximum of jut le than π. The urrent dx dt low here, refleting the inuoidal nature of (ϕ ). Superfluid 4 He with uperfluid denity ρ i deribed by a maroopi quantum iφ wave funtion ψ = ρ. n unretrited pae, the flow veloity i proportional to the e gradient of the phae: v = m ) φ. Here i Plank ontant h divided by 2 π ( 4 and m 4 i the 4 He atomi ma. The uperfluid urrent through our array i a funtion of the phae differene φ between the two reervoir. n general thi phae differene evolve aording to the Joephon-Anderon phae evolution equation,

6 6 d φ dt = µ, where µ = m4 ( P ρ T ) i the hemial potential differene aro the array. Here ρ i the fluid total ma denity, i the entropy per unit ma, and P and T are the preure and temperature differene aro the array. f ( φ) i 2π periodi, a ontant µ give rie to oillation at the Joephon frequeny, f J = µ h. Thi an our in either the trong oupling phae lip regime or the weak oupling Joephon regime. Our goal here are to determine the detailed time evolution of thee oillation and the underlying urrent-phae relation ( φ) a we hange the oupling trength by varying the temperature. Hereafter we ue ϕ to denote φ. preure P aro the array. The urve how the diplaement x(t) of the diaphragm in the poitive diretion (poitive lope). n panel b, where T λ -T = 7.4 mk, ξ 4 d = 0. 4 ξ 4 n Fig 2, panel b and are etion of two flow tranient exited by a tep in the a fluid i driven though the aperture under the influene of a time dependent hemial potential gradient. The lope of thee urve, dx / dt, i proportional to the total ma urrent through the aperture array, ( t ) = ρadx dt, where A i the diaphragm area. For both flow tranient, the preure tep i uh that a onventional fluid would be driven and the oupling i relatively trong. The lear lope diontinuitie are the ignature of phae lip that our in the firt half of the plot whenever the ontinuouly aelerating flow reahe a maximum urrent. By ontrat in panel, where T λ -T = 0.8 mk, d = 1. 8 and the ytem i in the Joephon weak link regime. The harp phae lip diontinuitie have been moothed out into inuoidal Joephon oillation. The µ indued by the initial preure tep relaxe to equilibrium throughout eah tranient. When µ reahe zero the Joephon frequeny oillation eae and lower frequeny reonant Helmholtz or pendulum-mode oillation begin, with urrent amplitude le than panel b and.. Thee are the larger diplaement oillation in the eond halve of

7 7 The method we ue to determine (ϕ ) i oneptually imilar to that ued 11 by Marhenkov et al for 3 He. The uperfluid urrent a a funtion of time (t) i determined from tranient data uh a that hown in Fig. 2, with a mall orretion due to a mall normal flow omponent (t). The phae differene aro the aperture array n t () i determined by integrating the phae evolution equation: = 1 ϕ t ϕ( 0) µ(τ )dτ, where the phae offet i determined by the fat that ϕ = 0 when = 0. Elimination between (t) and ϕ () t of the ommon variable of time then yield the urrent-phae relation (ϕ ). 0 ntegration of the phae evolution equation require knowledge of both P(t) and T (t). P(t) i diretly determined by the diaphragm diplaement: P = kx A, where k i a meaured pring ontant. An abolute alibration of P i provided by the Joephon frequeny relation for T = 0 : P = ρhf J m4 (with k and A thi in turn provide the alibration for x ). A temperature differene T (t) i reated whenever uperfluid flow into or out of the inner ell (the thermo-mehanial effet) and i alulated uing the meaured urrent and a imple heat flow equation 9. The urrent-phae funtion for everal temperature are hown in Fig. 3. A mooth tranformation our from the low temperature trong oupling regime where (ϕ) i linear with limiting value, into the weak oupling regime, within a few mk of T λ, where (ϕ ) morph into a ine funtion. For T T > 5 mk, (ϕ ) i motly linear and the ytem i in the phae lip regime. Under the influene of a ontant µ, ϕ will inreae linearly until it reahe a ritial value ϕ (the maximum value of ϕ for λ eah plot) then lip bak diontinuouly by 2π. drop from ϕ ) to ( ϕ 2π ). ( Beaue ϕ i le than 2π at thee temperature, ϕ and revere diretion when a phae lip our. A the temperature i inreaed, going from top to bottom in Fig. 3, ϕ dereae. Around the temperature at whih ϕ reahe π, (ϕ ) morph into a inuoid, whih i the ignature of an ideal Joephon weak link. Eah of the urve in

8 8 Fig. 3 i obtained by averaging the (ϕ ) data from between 5 and 70 tranient uh a thoe in Fig. 2. Figure 3. Evolution of (ϕ ) with temperature near T λ. Eah urve ha been normalized by it maximum value and hifted vertially. The orreponding T λ -T i indiated in mk to the right of eah urve. An intriguing quetion that remain i why, in the temperature regime invetigated here, the array appear to at like a ingle aperture, or a ingle weak link. The amplitude of the oillation in the phae lip regime, but loe to the tranition to weak oupling behavior, indiate that all the aperture are ating together 2. One imple argument for phae oherene aro the array i that phae gradient parallel to the wall ontaining the aperture orrepond to lateral urrent, whih are not energetially favorable. t ha been uggeted that wherea thermal flutuation an be trong in a ingle aperture, they may be uppreed in an array 12. t i not at all lear in the trong oupling limit how the aperture interat and give rie to the ynhronou generation of

9 9 phae lip. We are working on extending thee meaurement to lower temperature, and there i preliminary evidene that the array beome le ynhronou a T drop. We find that the meaured (ϕ ) i well deribed by an empirial model oniting of a purely linear kineti indutane in erie with an ideal (purely inuoidal) weak link. For the latter, θ ) = in( ). For the linear indutane, ( 1 θ1 ( θ 2 ) = θ 2 m L. Here θ 1 i the phae aro the ideal weak link, θ 2 i the phae 4 aro the linear indutane L, and ϕ = θ 1 + θ 2. The model an be haraterized in term of and the ratio of two indutane, α = L. Here i the kineti L J L J indutane of the ideal weak link evaluated at L J = 1 ( m )( d dθ ) = m 4 4 θ = θ = 0, 1. The overall urrent-phae relation an be written parametrially: = in( θ1), ϕ = θ 1 + α in( θ 1 ). An analogou model ha been applied to uperonduting Joephon juntion 13. t ha been found to be inappliable to 3 He weak link 14. The model parameter an be determined from the meaured (ϕ ) in a very imple way: i the maximum of (ϕ ), whih our at ϕ = ϕ m, and α = ϕ m π 2 i the deviation of the peak poition from π 2. n the limit α 0, the linear indutane i negligible and ( ϕ) = in( ϕ). n the limit α >> 1, (ϕ ) i linear exept near ϕ m. For α 1, there exit a ritial phae ϕ at whih d dϕ =. The model i multiple valued in thi ae, and a phae lip our when the ytem fall off the liff at ϕ onto an adjaent branh of (ϕ ). When α >> 1, ϕ ϕm and the ize of the phae lip i = 2 π α. The tranition between the diontinuou phae lip regime and the ontinuou weak oupling regime our when α = 1, ϕ m = π 2 + 1, ϕ = π, and = 0. The meaured parameter and α (independent of any model) are plotted veru temperature in Fig. 4 a and b. The model predition for (ϕ ), uing only the meaured and α value, i plotted in Fig. 4 for four different temperature, along with the atual meaured (ϕ ). The agreement i triking, and how that, within the

10 10 temperature range we have invetigated, the entire (ϕ ) an be aurately reprodued by thi model at a given temperature from the meaured and α alone. Although the model i empirial, it lend inight into how the evolution of (ϕ ) an be viewed a the tranition from a multiple-valued hytereti funtion to one that i ingle-valued. Figure 4. Evolution of uperfluid 4 He weak link parameter with temperature and an empirial model whih an generate (ϕ) from thee parameter. (a) Meaured maximum urrent. (b) Meaured deviation of (ϕ ) peak poition from π 2. The inet how an expanded view of the data within 4 mk of T λ. () Normalized (ϕ ), meaured (olid point), and generated uing model (dotted line). The experiment deribed here reveal the evolution of the funtion (ϕ ) haraterizing the union of two uperfluid 4 He reervoir. Thi evolution how a tranition between two important and ditint quantum phenomena: phae lip, aoiated with the generation of ingly quantized vortie, and the Joephon effet, aoiated with the weak oupling of two quantum ytem through a potential barrier. We find that a imple two parameter model aurately deribe the entire temperature regime under tudy. The in(ϕ ) behavior revealed at the higher temperature will lead to the development of a uperfluid 4 He interferometer, an analog of the uperonduting

11 11 d-squd. Suh a devie, operating near 2K, a regime aeible by mehanial ryoooler, will lead to pratial devie ueful in inertial navigation, geodey and bai phyi. Correpondene and requet for material hould be addreed to R.E.P. (rpakard@berkeley.edu). Aknowledgement Dr. T. Haard made important ontribution to the ontrution and deign of the apparatu. M. Abreau alo helped with the ontrution of the apparatu. The aperture array were fabriated by A. Lohak. We thank D. H. Lee and H. Fu for helpful diuion. Thi work wa upported in part by the NSF (grant DMR ) and NASA. 1 Van Duzer, T., Turner, C. W., Superondutive Devie and Ciruit 2 nd edn (Prentie-Hall, Upper Saddle River, 1999). 2 Hokinon, E., Pakard, R. E., Haard, T. M. Quantum whitling in uperfluid helium-4. Nature 433, 376 (2005). 3 Mukharky, Y., Avenel, O., Varoquaux, E. Rotation meaurement with a uperfluid 3 He gyrometer. Phyia B, , (2000) 4 Avenel, O., Mukharky, Y., Varoquaux, E. Superfluid Gyrometer. J. Low Temp. Phy. 135, (2004) 5 Simmond, R. W., Marhenkov, A., Hokinon, E., Davi, J. C., Pakard, R. E. Quantum interferene of uperfluid 3 He. Nature 412, (2001). 6 Avenel, O., Bernard, M., Burkhart, S., Varoquaux, E. Autopy of uperflow ollape and multiple phae lip. Phyia B, 210, (1995). 7 Anderon, P. W., Dayem, A. H. Behavior of Thin-Film Superonduting Bridge in a Mirowave Field. Phy. Rev. Lett. 13, (1964). 8 Sukhatme, K., Mukharky, Y., Chui, T., Pearon, D. Obervation of the ideal Joephon effet in uperfluid 4 He. Nature 411, (2001). 9 Hokinon, E., Pakard, R. E. Thermally Driven Joephon Oillation in Superfluid 4 He. Phy. Rev. Lett. 94, (2005). 10 Welander, P. B., Hahn,., Miniature high-reolution thermometer for low-temperature appliation. Rev. Si. ntrum. 72, (2001). 11 Marhenkov, A., Simmond, R. W., Bakhau, S., Lohak, A., Davi, J. C., Pakard, R. E. Bi-tate Superfluid 3 He Weak Link and the Stability of Joephon π State. Phy. Rev. Lett. 83, (1999). 12 Chui, T., Holme, W., Penanen, K., Flutuation of the Phae Differene aro an Array of Joephon Juntion in Superfluid 4 He near the Lambda Tranition. Phy. Rev. Lett. 90, (2003).

12 12 13 Deaver, B. S., Piere, J. M. Relaxation oillator model for uperonduting bridge. Phy. Lett. 38A, (1972). 14 Davi, J. C., Pakard, R. E. Superfluid 3 He Joephon weak link. Rev. Mod. Phy. 74, (2002).