Superfluid helium weak links:
physics and applications
Richard Packard
University of California,
Berkeley
Collaborators
3He
Alex Loshak
Sergei Pereversev
Scott Backhaus
Alosha Marchenkov
Ray Simmonds
Seamus Davis
4He
Emile Hoskinson
Yuki Sato
Supported by NSF and NASA
What is a superfluid weak link?
Josephson Equations
Generic technique for observing the physics
3He
Josephson oscillations (the quantum whistle)
The current-phase relation
The 3He superfluid gyroscope
4He
Quantized phase slips
Search for phase slip sound
Josephson oscillations again!
The current phase relation
The coherence question
Prospects for a practical gyroscope
A superfluid weak link:
L
R
R   R eiR
L   L eiL
  R  L
I  I c sin(  )
  
if   const ., I  I c sin 
t ,
  
superconductor
  2eV
superfluids
d  

dt


fj 
h
  m (P   sT )
Criterion for weak link barrier dimensions
Josephson Junction:
Superconducting weak link:

Superfluid weak link:

2m3  Pdc  m4  Pdc

 or

fj 
 sT 
h   
h  


f j 3  183 Hz / mPa or f j 4  68.7 Hz / mPa
fj 
2eV
 483THz / Volt
h
Size of barrier must be on the order of or less than
the coherence length 
For superfluid 3He, nm
For superfluid 4He, nm
60nm
3 
1/ 2
1  T / Tc 
For 3He cool to below 1mK
0.3nm
4 
0.67
1  T / T 
For 4He cool to below 2K
Use 3He to exploit longer healing length
3
4225 holes in a 50 nm
thick silicon nitride
membrane
60nm
Able to detect ~ 10-15 m
Determination of the current-phase relation
12
10
x [picometers]
8
6
0
1
2
3
4
tim e [s]
5
4
2
0
-2
-4
0.6
0.8
1.2
1.6
2.0
tim e [s]
2m
d

  3 P (t )
 2m3  t
dt

  x d
   (t )   (0)   



0

P  x(t )

Eliminate common variable time and plot ...
I (t )  A
dx
dt
Two Distinct Current-Phase Relations.
Experimental Data.
60
State L
20
State H
10
-12
I [kg/s] (x10 )
-12
I [kg/s] (x10 )
40
0
-10
20
0
-20
-40
-20
-60
0


2
0


2
Superfluid dc-SQUID: a gyroscope
It
I2 = Ic sin( )


I1 = Icsin( )
m4
 
vs

It
I  I c sin(  )
*
 
   d  2 n
It
 
 2  A 
*

I c  2 I c cos

h
2
m
3 

The critical current is
modulated by the rotation flux
Superfluid 3He gyroscope
Whistle amplitude vs. reorientation wrt
the Earth’s axis
prediction
 
 2  A 
*

I c  2 I c cos

 h 2m3 
What about 4He?
Variation of coupling strength
via the healing length, 4(T)
I (  )
d
4 
0.3nm
1  T T 0.67
Strong coupling: 4 < d
Weak coupling: 4 > d
1.
A superfluid “strong link”
aperture
leff
4He
Vs
Vs
vortex

 drives superfluid 4He through an aperture
2. Vs increases till it reaches Vc
3. At Vc, a vortex is nucleated and the flow velocity drops by a fixed amount Vslip
Vslip 

leff
h m4

leff
2 phase slip
2 Phase Slip
aperture
4He
Vs

1.

drives superfluid 4He through an aperture
2. Vs increases till it reaches Vc
3. At Vc, a vortex is nucleated and the flow velocity drops
by a fixed amount Vslip
V
slip


leff
h m4

leff
leff
Phase Slip Oscillations

What happens if we keep applying
aperture
4He
1. V increases till it reaches Vc
2. Phase slip event takes place and
V drops by Vslip
3. V increases again due to 
to repeat the same proc
Vs

vc
?
Phase slips occur at Josephson Frequen
Vslip
t

fj 
h
Generic apparatus
3m
T+T, P+P
4225 holes in a 50 nm thick
silicon nitride membrane
T, P
60nm
Demonstration of Josephson frequency
relation when  = m4P/
Chemical potential difference
  m4 P   sT 
Josephson frequency:
 m4 P
fj 

h
h
T+T, P+P
T, P
T = 0 at
beginning
of transient
E. Hoskinson, R. E. Packard, T. M. Haard, Nature 433, 376 (2005)
Thermally driven quantum oscillations
  m4 P   sT 
f j   h
T+T, P+P
T, P
E. Hoskinson, R. E. Packard, PRL 94, 155303 (2005).
How do you get from a sawtooth to a
sinusoid?
?
I   s v s    
I  I c sin 
1
When Vc  Vslip ,   d
2
Determining the current-phase relation
I (t )  x (t )



P
(t )      sT 
 

 (t )  
1


(
t
)
dt

given I (t ) and  (t )
eliminate t to get I ( )
T+T, P+P
T, P
movie
Are the phase slip oscillations synchronicity
or quantum phase rigidity?
• A Fourier transform of the quantum whistle shows
very narrow spectral width.
• Does the periodicity evolve over time or is it present
from the first instant?
The end of an impulse transient
The magnitude of the slips implies that
all 4225 apertures slip together.
Phase slip osc.
Vslip  Vc
T+T, P+P
T, P
The slip removes all of the energy
Movie
2Vc  Vslip  Vc
Slips remove some but not all of the energy
Movie
Equality of 1st and Nth slip
Equality of 1st and Nth slip
• Slip size is the same for the 1st slip and
the Nth. Maybe synchronization plays
no role.
• Synchronization cannot be ruled if the
interactions are strong.
More Questions
• Is phase slippage a global process
wherein a single vortex filament passes
over the complete array?
• Is phase slippage a cooperative process
wherein each aperture slips but they all
are locked together? N vortex events
• Does phase slippage in one aperture
trigger an “avalanche” slip in all N?
Slip size is not 2N as T decreases
2N
One more mystery
• Why does the slip size decrease as T is
lowered?
Next step is to build a superfluid
4He dc squid gyroscope
Future possibilities:
A superfluid gyroscope operating near 2K
cooled by a mechanical cryocooler
Useful for geodesy, seismology and
navigation
Summary
3He: Arrays of nanometer size apertures behave as ideal Josephson weak
links. Quantum coherent dynamics
Josephson oscillations, Shapiro steps, plasma mode.
Discovery of novel dissipation, novel I() relation
Proof-of-principle of superfluid dc-SQUID gyroscope
4He:
Aperture arrays behave quantum coherently near T
The current-phase relation has been mapped from the “strongly
coupled” linear regime to the “weakly coupled” Josephson I()
regime.
Near T all apertures phase slip coherently. Dynamics
unknown.
At lower temperatures the phase slip sound amplitude
decreases but the quantum whistle remains well defined. Why??