Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime September 2010

Transcription

1 164-9 Workshop on Nano-Opto-Elctro-Mchanical Systms Approaching th Quantum Rgim 6-1 Sptmbr 1 Nano-Elctro-Mchanics of Suprconducting Wak Links Robrt SHEKHTER Chalmrs Univ. of Tchnology & Univrsity of GothnburgDpt. of Applid Phys., S-4196 Gothnburg SWEDEN

2 * Nano-Elctro-Mchanics of Suprconducting Wak Links Robrt Shkhtr In collaboration with: Lonid Gorlik, Mats Jonson, Gustav Sonn, Milton E. Pna-Aza Univrsity of Gothnburg /Chalmrs Univ. of Tchnology / Hriot-Watt Univrsity Workshop on Nano-Opto-Elctro-Mchanical Systms Approaching th Quantum Rgim, ICTP, Trist, Spt. 6-1, 1

3 Enrgy Transfr in NEM Systms Elctronic subsystm Mchanical subsystm Extrnal powr supply

4 Two Exampls of Controllabl NEM Enrgy Transfr I. Singl Elctronic Shuttl Dvics II. NEM Suprconducting Wak Links I. F. Santandra t al.: Cooling of th vibrations of a dc-biasd nanomchanical rsonator dtrmind by quantum intrfrnc ffcts. II. G.Sonn t al.: Cooling of a suspndd carbon nanotub by an ac Josphson currnt flow.

5 Shuttling in Coulomb Blockad Dvic V > Vc -V/ V/ F V < Vc Thrmally activatd transport F

6 * Nano-Elctro-Mchanics of Suprconducting Wak Links Suprconducting wak link as NEM dvic Suprconductiv pumping of nanovibrations NEM-inducd suprconductiv cooling of mchanical rsonator Conclusions

7 Suspndd Nanowir-Basd Suprconducting Wak Link B

8 Andrv Stats S u L S E 1 Dsin ( / ) D is th normal junction transparncy / F / ( ) v F F

9 Effct of Backscattring E F R 1 R 1 E E () R F E () E () E () Narly ballistic wak link: (cooling of nanovibrations) Josphson wak links: (pumping of nanovibrations)

10 Josphson Wak Links ( D<<1) B Suprconductiv Pumping of Nanovibrations G.Sonn t al. Phys.Rv. B 78, (8)

11 Suprcurrnt-Drivn Nanomchanics Modl: Drivn, dampd nonlinar oscillator G. Sonn t al. PR B 78, (8) Compar: NEM rsonator as part of a SQUID mu u ku HLJ c sin() Driving Lorntz forc ( V / ) ([ HLu ]/ ) V j dc Inducd l.motiv forc u ( t) Buks, Blncow PRB 6 Zhou, Mizl PRL 6 Blncow, Buks PRB 7 Buks t al. EPL 8 Enrgy balanc in stationary rgim dtrmins tim-avragd dc suprcurrnt

12 Giant Magntorsistanc V Altrnating Josphson currnt Altrnating Lorntz forc, F L Mchanical rsonancs F L HLI c sin(vt / 4HLu( t) / ) HLI c sin(vt / ) [4H (I) Forc (I) lads to rsonanc at Forc (II) lads to paramtric rsonanc at L I c / ] u( t)cos(vt / ) (II) V / V / Accumulation and dissipation of a finit amount of nrgy during ach nanowir oscillation priod mans that W V I( t) and thrfor a nonzro avrag (dc) suprcurrnt on rsonanc

13 Giant Magntorsistanc Th onst of th paramtric rsonanc dpnds on magntic fild H. By incrasing H th rsistanc R V / j( t) jumps from R to a finit valu. Rsonanc Paramtric rsonanc Amplitud of wir oscillations small H dc bias voltag largr H dc bias voltag

14 Narly Ballistic Suprconducting NEM Wak Link B Suprconductiv Cooling of Nanovibrations A rfrigrator is a cooling applianc comprising a thrmally insulatd compartmnt and a hat pump to transfr hat from it to th xtrnal nviromnt. (Wikipdia) G.Sonn t al. PRL,14, 68 (1)

15 Andrv Lvls Compltly transparnt junction: normal rflction probability R= Finit small rflction R<<1 rmovs dgnracy at φ=π F J E ( ) J E ( ) E() E() 1 J E R 1 J J J E ( ) cos( / ) E( ) R Dcos ( / ); D 1- R Hˆ cos( / ) ˆ ff z Hˆ ff D cos( / ) ˆ z R ˆ x

16 Quantizd Pumping out th Hat from Nanovibrations ( N 1) 3) ) 3 4 Vt

17 Magnto-motiv NEM Coupling dx x j L J L L ) ˆ( 1 ˆ / / ˆ J J J L S S B u J u F LBJ F u u V BL u V u BLu V u BLu z J ˆ ) / ˆ 1 sin( ˆ ˆ ˆ ˆ ˆ cos ˆ u k M P R BLu D H u x z ff

18 Voltag Biasd NEM Wak Link E( ) ˆ aˆ Ĥff z aˆ ~ B( ) ˆ ( aˆ x aˆ) E( ) R Dcos ( / ) ( ) R sin( / ) R Dcos ( / ) if πn 1 if π( n 1) V xt ( t) V xt t Hˆ V ( t) Undr adiabatic conditions ff E( ( t)) ˆ aˆ z Vxt << R HV (t) ~ a B( ( t)) ˆ ( aˆ x a ˆ)

19 Liouvill- von Numann Equation 1 ˆ () t H ˆ() ( ˆ V t t L ), t (nt,nt T) t i (), Boundary conditions ˆ ( t nt ) ( ), ( ) ˆ in nt in nt Tr( nt ) 1 F nt E(t) E(t) ( n 1)T R i T H ( t) dt ˆ( nt T ) i T H ( t) dt ˆ( nt ) B B T dt ~ B H ( t), ˆ( nt )... int B ( t) a; ~ H ˆ z z H E( ( t)) ˆ aˆ z int ( t) ( ( t)) i t E( t) dt ˆ x i t E( t) dt ( aˆ it aˆ it ) Th main contributions ar givn by th points t = (n+1/)t whr lctromchanical coupling has a maximal valu T E t R R V t

20 Evolution During Singl Priod of Josphson Oscillations P (n) n 1 Final stat Initial stat n P ( n) n Final stat P (n) n 1 Final stat B 1 3 R P ( ) cos n n d 3 P P P B V R V V 1 Undr rsonant conditions R /3 B P ( n) n ; P ( n), P ( n) B V

21 Numrical Estimations 1 B 1T 1 Q k B T 5 1 L nm u V pm V V Final distribution Voltag applid btwn suprconductors gnrats cooling of th nanowir vibrations P(n) Thrmal distribution of th numbr of vibrons n n final.1

22 Suprconducting Nano-Thrmomtr / 3, V B B n j, T k / q B n j n n V Q q q n kt j j q

23 Conclusions 1. Voltag-biasd suprconducting wak link drivs vibrational motion whn an xtrnal magntic fild is switchd on.. Both rsonant pumping and an fficint cooling of nanovibrations can b achivd dpnding on th amount of lctronic backscattring. 3. Th rfrigrating ffct corrsponding to th avrag occupation numbr of vibrations <n>=.1 can b achivd.