1. The Pennsylvania State University
Einstein’s Legacy in Low Temperature Physics; Superfluids and Supersolids
Moses Chan
The Pennsylvania State University
Supported by National Science Foundation

2. Outline
Quantum mechanics at low temperatures :de-Broglie wave-packets, Bose-Einstein Condensation in vapor and liquid.
Experimental principle for the of observation of Superfluidity: Torsional Pendulum
Observation of superflow in solid helium

3. Temperature Scale (K) 103 Melting of iron 102
10-1
10-2
10-3
10-4
10-5
10-6
Melting of iron
Melting of ice (273K=0oC)
Dilution Fridge
temperature 10mK
Superfluid (2.18K)
Cosmic background(3K)
Evaporation of 4He(4.2K)
Nuclear Demag. 1µK
Superfluid 3He (2mK)
1Kpot
(1.2K)
Still(0.7K)
Mixing Chamber(10mK)
The lowest possible temperature 0 K= oC=-459.7oF
106=1,000,000; 10-6=

4. Superfluidity in liquid 4He was discovered in 1938.
Solid
Superfluid (He II)
Normal
Liquid
(He I)
Superfluid
helium film
can flow up
a wall
Fountain
T=2.176K

5. Persistent current in superfluid
Vanishing viscosity; “The viscosity of He II is at least 1500 times smaller than that of normal helium (He I)”
P. Kapitza, Nature 141, 74 (1938)
Allen & Misener Nature 141,75 (1938)
Liquid helium
Persistent current can be created by r6tating the container while cooling through T. Superfluid will continue to rotate after the container is stopped. Our understanding of superfluidity can be traced to Einstein

6. Superconductivity B I
The phenomenon of superconductivity is analogous to superfluidity. In superconductivity, electric current can flow with no resistance. A similar persistent current of electron pairs can be set up.
MRI uses magnet powered by superconducting current in the persistent mode. In this mode the current and therefore the magnetic field is extremely stable.
Superfluidity and superconductivity are
macroscopic quantum phenomenon

7. Miraculous year in Physics
In 1905, Einstein published 4 monumental papers.
March 1905
Light Quanta; Light wave can behave like
a stream of particles with discrete energy
May 1905
Theory of Brownian Motion; Heat or thermal energy is responsible for the random motions of small particles suspended in a liquid.
June 1905
Theory of Special Relativity
September 1905
Energy equivalent theory, E=mc2
“ A storm broke loose in my mind”

8. Quantum Theory simplified: Thermal de Broglie Wavelength; dB (1924)
* in 3D
* in 1D
Light wave behaves like a particle, conversely a particle, e.g., an atom, electron, elementary particles, and indeed all objects can behave like a wave-packet.
kBT is a measure of “energy of motion”

9. Classical and Quantum pictures of an object (e. g. atom, electron, etc
dB:
de Broglie wave length
kB = 1.38×10-23 Joules-K: Boltzmann constant
T : absolute temperature
h = 6.626×10-34 Joules-s: Planck constant
m: mass of the object of interest

10. Some dB dB * in 1D * in 3D 1) m=70kg (human) at T=300K
dB= 8×10-23cm
= cm
2) m=9.1×10-31kg (electron) at T=300K
dB= 4×10-7cm = 4nm
3) m= 6.69×10-27kg(4He) at T=300K
dB= 5×10-9cm=0.05nm
at T= 2K dB= 6×10-8cm = 0.6nm
at T=0.2K dB= 2×10-7cm = 2nm
4) m=1.42×10-25kg (Rubidium atom) at 1nK
dB= 1×10-3cm = 10µm
dB
* in 3D

11. l dB Collection of identical particles
If the distance between the particles, l,
is much larger than dB then the particles retain their individual identity and their behavior is governed by classical thermodynamics

12. Bose-Einstein Condensation
What if the temperature is reduced so that dB grows to be on the order or even larger than l, the inter-particle spacing?
Einstein, built on the idea of Bose, proposed in 1924 that these identical particles lose their individual identity and begin to behave as one single “giant atom”.
This is known as Bose-Einstein condensation (BEC).
Now we know that this prediction is correct only for bosons* ( with integer spins)
* Named after Satyendra N. Bose of India

13. l dB Collection of identical particles Decreasing temperature
increases dB

14. dB >> l Particles behave coherently like a single “giant atom”
“One for all and all for one”
In the Bose-condensed state particles or atoms do not “run” into each other. Because they act as a single coherent entity they cannot easily lose or gain energy from the surroundings. Hence superfluidity is possible.

15. Bose-Einstein Condensation in the vapor phase
1) Introduce Rb vapor into a vacuum space
2) Cool the Rb atoms by colliding them with appropriate laser beam and other clever techniques so that their dB is larger than the separation of the atoms.
3) First accomplished by Carl Wieman and Eric Cornell in 1995 on Rb atoms

16. Bose-Einstein Condensation in the vapor phase
1) Introduce Rb vapor into a vacuum space
2) Cool the Rb atoms by colliding them with appropriate laser beam and other clever techniques so that their dB is larger than the separation of the atoms.
3) First accomplished by Carl Wieman and Eric Cornell in 1995 on Rb atoms
Wieman
Cornell
Nobel Prize in 2001
Ketterle

17. BEC of Rubidium gases JILA BEC group(1995) 400nK 200nK 50nK
This is a famous picture showing BEC of Rubidium gases taken by University of Colorado
The contour map shows velocity profile of Rubidium gases in the magnetic trap with decreasing temperature.
At 50 nK, most of Rb gases condensed in to the zero momentum state as predicted by Bose & Einstein.
400nK nK nK

18. Fritz London is the first person to recognize that superfluidity in liquid 4He is a BEC phenomenon.
Solid
Superfluid (He II)
Normal
liquid
(He I)
Liquid helium
Persistent current is possible in the superfluid phase.
At 2K, dB of 4He = 0.6nm , separation of 4He atoms l = 0.3nm

19. Principle for the observation of liquid helium behaving as a “Macroscopic Atom” : torsional pendulum
rigid support point
Period of Oscillation
When I decreases,
Resonant period decreases
I : Rotational Inertia (proportional to mass)
K: Spring constant of torsion rod (Stiffness)

20. Torsional Pendulum Period of Oscillation
rigid support point
Period of Oscillation
I : Rotational Inertia (proportional to mass)
K: Spring constant of torsion rod (Stiffness)
What if we have a set of infinitely smooth ball bearings?

21. Torsional Pendulum Period of Oscillation
rigid support point
Period of Oscillation
Period also decreases because a fraction of I remains stationary and decouples from the oscillation
I : Rotational Inertia (proportional to mass)
K: Spring constant of torsion rod (Stiffness)

22. Torsional oscillator ideal for detection of superfluidity
Resolution
Resonant period (o) ~ 1 ms
stability in  is 0.1ns
/o = 5×10-7
Mass sensitivity ~10-7g
Be-Cu
Torsion Rod
Torsion Bob
containing helium f
Drive
Amp
Detection f0
Q=f0/f ~ 2×106

23. Measurement of superfluidity
liquid helium
Above 2.176K, liquid helium behaves as a normal fluid. It will oscillate with the disk if d is smaller than the viscous penetration depth (). ω R d
( ~ 3µm,
if the oscillating frequency
is 2*1000 rad/s )
d <  of normal fluid
In the normal fluid phase,
I total = I cylinder+ Ihelium
 : viscosity

24. T< T
Superfluid fraction stays still when the container is being oscillated at a velocity less than a certain critical value. ω R d
I total = I cylinder + I n(T);
I total = I cylinder; at T=0, ρn=0
Non-Classical Rotational Inertia
d <  of normal fluid

25. Empty torsional cell
I total = I torsion cell
*=1,723,000ns

26. Helium is introduced into the cell
I total = I torsion cell + I normal helium
Period shifted by 2155ns
due to mass loading of
normal liquid helium
*=1,723,000ns

27. Expected background if there is no superfluid transition

28. ω A certain fraction of the liquid, known as
superfluid fraction decouples from the oscillation
of the torsional cell and does not contribute to
the rotational inertia
Superfluid
Decoupling = ω R
*=1,723,000ns

29. Superfluid fraction =ρs+ρn ; two fluid model of Tisza and Landau =
total mass loading
At T=0K
100% superfluid

30. Does Bose-Einstein Condensation also occur in a solid?
1) In principle it is possible, however “conventional wisdom” said it is unlikely to happen or immeasurably small.
2) If it is going to occur, the most likely candidate is solid 4He, the most quantum mechanical solid.

31. Search for the supersolid phase in solid 4He.
Be-Cu Torsion Rod
OD=2.2mm
I D=0.4mm
Filling line
Detection
Drive
Mg disk
Solid helium in annulus channel
Torsion cell
with helium in annulus
Al shell
Channel OD=10mm
Width=0.63mm
Eunseong Kim

32. Torsional Oscillator
Torsion rod
Torsion cell
3.5 cm
Drive
Detection

33. Solid 4He at 51 bars The supersolid fraction is on the order of 1.3%
Amplitude of oscillation is 0.7Å
A decrease in the resonant period, similar to that found in superfluid liquid helium, appears below 0.25K
0= 1,096,465ns at 0 bar
1,099,477ns at 51 bars
(total mass loading=3012ns
due to filling with helium)
The supersolid fraction
is on the order of 1.3%
Nature, 425, 227 (2004); Solid helium in porous glass
Science 305, 1941 (2004); Bulk solid

34. Different Speed of Oscillation
4µm/s is equivalent to oscillation amplitude of 0.7Å

35. Supersolid fraction
The supersolid fraction at T=0K is on the order of 1.3%

36. Control experiment I : Solid 3He?
Nature 427,225(2004)

37. Torsion cell with blocked annulus
Control experiment II
With a barrier in the annulus, there should be NO simple superflow and the measured superfluid decoupling should be vastly reduced
Torsion cell with blocked annulus
Mg barrier
Mg disk
Filling line
Solid helium in annulus channel
Al shell
Mg barrier
Mg Disk
Al shell
Solid helium
Channel OD=15mm
Width=1.5mm

39. Normal solid
Normal liquid
Supersolid
superfluid
Phase Diagram

40. Pressure dependence
As a function of pressure the supersolid fraction shows a maximum near 55bars. The supersolid fraction extrapolates to zero near 170 bars.

41. Is the supersolid phase unique with 4He?
Apparently not!
Preliminary torsional oscillator data of Tony Clark and Xi Lin indicate similar supersolid-like decoupling in solid H2.
de Boer parameter
3He  3.09
4He  2.68
H2  1.73
HD  1.41
D2  1.22
More quantum
mechanical

42. Period increased by 2900ns when the torsional cell is filled with H2
Torsional oscillator
Q ~ 600,000
o = 980,486ns
 ~ 0.1ns
H2 - o = 2900ns

44. Supersolid fraction in hydrogen is
~1ns/2900ns = 0.035%

45. No decoupling seen in solid HD within 0.05ns
Torsional Oscillator:
Q ~ 1.3 million
o = 560,395ns
 ~ 0.05ns
HD - o = 4014ns

46. Supersolid fraction 4He at 51 bars 1.3% 4He at 136 bar 0.5%

47. What is the precise mechanism for the supersolid phase and why the supersolid fraction is so small? We do not know yet.

48. At high temperature the de Broglie matter waves of the helium atoms in the solid are disconnected, then it is not possible for superflow. U ρ
ρ=Ψ*Ψ

49. At sufficiently low temperature, T<0
At sufficiently low temperature, T<0.2K, the de Broglie waves of the atoms overlap each other slightly allowing the appearance of superflow ρ U
overlapping~ ρS

50. Bose-Einstein Condensation and Superfluidity were seen in vapor and liquid, and now also in solid

51. Thanks to Philip W. Anderson, Jayanth R. Banavar, John Beamish,
John M. Goodkind,
T. L. Ho,
Jainendra K. Jain,
Anthony J. Leggett,
John D. Reppy ,
Wayne Saslow ,
David S. Weiss,

52. Thanks to Walter Hardy for the beautiful torsional pendulum demo.
Phillip W. Anderson,
Jayanth R. Banavar,
Milton W. Cole,
John M. Goodkind,
T. L. Ho,
Jainendra K. Jain,
Anthony J. Leggett,
Jay D. Maynard,
John D. Reppy ,
Wayne Saslow ,
David S. Weiss,
Walter Hardy for the
beautiful torsional
pendulum demo.
Eunseong
Tony Xi

54. Control experiment II : Solid 3He?
Nature 427,225(2004)

55. Solid 4He at 51 bars Science 305, 1941(2004)
4µm/s corresponds to amplitude of oscillation of 0.7Å
NCRI appears below 0.25K
Strong |v|max dependence
(above 14µm/s)
Amplitude minimum, Tp
0= 1,096,465ns at 0 bar
1,099,477ns at 51 bars
(total mass loading=3012ns
due to filling with helium)
Science 305, 1941(2004)

56. Non-Classical Rotational Inertia Fraction
|v|max
ρS/ρ
NCRIF
Total mass loading
=3012ns at 51 bars

57. Velocity dependence of NCRIF
at low temperature

58. With a block in the annulus, irrotational flow of the supersolid fraction contributes about 1% (Erich Mueller) of the barrier-free decoupling.
Δ~1.5ns
-*[ns]
Irrotational flow pattern
in a blocked annular channel
(viewed in the rotating frame)
A. L. Fetter, JLTP(1974)

59. With a block in the annulus, irrotational flow of the supersolid fraction contributes about 1% (Erich Mueller) of the barrier-free decoupling.
Δ~1.5ns
-*[ns]
Irrotational flow pattern
in a blocked annular channel
(viewed in the rotating frame)
A. L. Fetter, JLTP(1974)

60. Similar reduction in superfluid response is seen in liquid helium at 19 bars in the blocked cell
measured superfluid decoupling in the blocked cell
Δ(T=0)≈ 93ns.
While the expected decoupling in unblocked cell is 5270ns.
Hence the ratio is 1.7% similar to that for solid.
[ns]
Conclusion :
superflow in solid as in superfluid is irrotational.

61. Pressure dependence of supersolid fraction

62. ρs near Tλ Greywall and Ahlers, Lipa and Chui
3D-XY model??
Greywall and Ahlers, Lipa and Chui

63. Temperature Scale(K) 103 Melting of iron 102 Melting of ice (273K=0oC)
1Kpot
(1.2K)
Still(0.7K)
Mixing Chamber(10mK)
Temperature Scale(K)
103
102 10 1
10-1
10-2
10-3
10-4
10-5
10-6
Melting of iron
Melting of ice (273K=0oC)
Dilution Fridge
temperature 10mK
Superfluid (2.18K)
Cosmic background(3K)
Evaporation of 4He(4.2K)
Nuclear Demag. 1µK
Superfluid 3He (2mK)