The Early Universe as a Complex System Driven Far from Equilibrium: Non-Equilibrium Superfluid 4He as an Analogous System

R.V. Duncan, University of New Mexico, and Caltech (University of Missouri, Columbia, effective 8/18/08)

Quantum to Cosmos III

Airlie Center, Warrenton, VA

July 9, 2008

Sponsored by NASA through the Fundamental Physics Discipline within the Microgravity Project Office of NASA, and supported through a no-cost equipment loan from Sandia National Labs. R.V.D. thanks Gordon and Betty Moore for sabbatical support.

Jet Propulsion LaboratoryCalifornia Institute of Technology

CQ (Caltech) and DYNAMX (UNM)

At UNM:Robert DuncanS.T.P. BoydDmitri SergatskovAlex BabkinMary Jayne AdriaansBeverly KlemmeBill MoeurSteve McCreadyRay Nelson Colin Green Qunzhang LiJinyang LiuT.D. McCarsonAlexander Churilov

At Caltech:David Goodstein Richard LeeAndrew Chatto

At JPL:Peter DayDavid ElliottFeng Chuan Liu Talso Chui

Ulf Israelsson

Non-equilibrium Quantum Systems Dense

Sparse

Tc T Q

T = h / prms

prms = (3mkT)½

T = h / (3mkT)½

BEC when T ≈ d

kTc ≈ h2/(3md2)This system may form an SOC state where T = Tc and quantized topological defects are continuously produced.

ei

Properties of Liquid 4He Near its Superfluid Transition

• 4He never freezes under its vapor pressure, due to quantum zero-point motion, and due to weak He – He attractions– Zero-point kinetic energy Eo ~ (m d2)-1

– Eo/kB in 4He ~ 10 K and it never ‘freezes out’

– van der Waals attraction is about 5 K

• Superfluid transition at T under SVP (0.05 bar)• Reduced Temperature: = (T–T) /T

– Zero viscosity – Perfect thermal conductivity– ‘superfluid’ component is effectively at T = 0

‘Two Fluid’ model of the superfluid transition = n + s, j = jn + js (v = nvn + svs) When j = 0, either equilibrium or counterflow

0.2

0.4

0.6

0.8

1.0

s

n

transition

0.5 1.0 1.5 2.0 2.5

Temperature (K)

Long-Range Quantum Order

Below T: ei

Vs = (ħ/m) sm

Hence, x Vs ~ x

• You can not rotate a perfect superfluid, but … • You can rotate a ‘Type II’ superfluid provided

that you create topological defects (quantized vortices)

Quantized Vorticity (QV)

Source: Tilley and Tilley, Figure 6.10 Erkki Thuneberg, http://ltl.tkk.fi/research/theory/vortex.html

( ) ( )S

dlm m

2 o

hn n n

m m

QV / LRQO in many superfluids

• Superconductors (o = h/2e)

• Superfluid 3He (Cooper pairs)

• Bose-Einstein Condensates (2005)

• Neutron Stars

• Cosmology – Topological defects– Symmetry breaking – Event horizon model

Source: Pines, LT12 (1971) p. 7

New Sub-Topic: Superfluids Near Criticality

Correlation length measures range of fluctuationso , where = |T - T|/T with o = 2x10-8 cm and = 0.671. We routinely stabilize ~ 10-9, where the divergence of is limited by gravity

Thermal conductivityDiverges due to fluctuations: = o -x with x ≈ ½

Field Theory: ‘Model F’ of Halperin, Hohenberg, SiggiaSee Hohenberg and Halperin, Rev. Mod. Phys. 49, 435 (1977). Renormalization: Haussmann & Dohm, PRL 67, 3404 (1991).

Pressure dependence of T breaks ‘up-down’ symmetry and stabilizes an interface

He-II

He-I

| = 1.3 K/cm

h

T

(h)He-I

He-II

Non-equilibrium interface stabilizes without gravity! See Weichman et al., Phys. Rev. Lett. 80 4923 (1998)

Now apply a heat flux Q:

Electrical currents in superconductors lower the ‘quench temperature’ in much the same way that a heat flux lowers the temperature where superfluidity abruptly fails in helium.

Tc(Q) = T [1 – (Q/Qo)y], but c =

Theory: Qo = 7 kW/cm2, y = 1/2 = 0.744Onuki, JLTP 55, and Haussman & Dohm, PRL 67, 3404 (1991) Experiment: Qo = 568 W/cm2, y = 0.81± 0.01Duncan, Ahlers, and Steinberg, PRL 60, 1522 (1988) Thermal conductivityx, but now define Q,z = [T(z) – Tc(Q,z)] / T

Experimental Concept

He-SOC

Q + Q

Q

Sidewall Thermometry Probes: Thickness: 50 75 125

Interface

He-II

Science objectives are obtained from the thermal profile data (noise level of < 100 pK/√Hz), while the heat flux is extremely well controlled to Q < 1 pW/cm2

Jet Propulsion LaboratoryCalifornia Institute of Technology

National Aeronautics and Space Administration

DYNAMX Sample Cell wall with Thermal Probes,End Caps, and Structural Supports

1 cm[--------]

‘T5’Critical Thermal Path

Experimental ‘T5’ Apparatus

Open System: ‘Heat from Below’

He-II

He-I

Q

The ‘simple’ critical point (‘lambda’ point) transforms into a complex nonlinear region where the onset of long-range quantum orderis limited by the heat flux Q, and by gravity

HfB: The Nonlinear Region

Theoretical prediction of the nonlinear region [Haussmann and Dohm, PRL 67, 3404 (1991); Z. Phys. B 87, 229 (1992)] with our data [Day et al., PRL 81, 2474 (1998)].

Gravitational Effect: g

As the superfluid transition is approached from above, the diverging correlation length eventually reaches its maximum value g = 0.1 mm, at a distance of 14 nK from T.

HfB: Tc(Q), TNL(Q), T

Tc(Q) and TNL(Q) are expected to extrapolate to T as Q goes tozero in microgravity, but not on Earth, as explained by Haussmann.

0 50 100 150 200

-50

-40

-30

-20

-10

0

10

20

TC(Q)

TNL

(Q)

15 nK = dT/dh * g, but what is T?

[ n

K,

arb

. ori

g.]

Q (nW/cm2)

The Direction of Q is Important :

He-II

He-SOC

He-II

He-I

Q

Q

SOC!

HfA

HfB

HfA: Heat from Above

A: Cell is superfluid (hence isothermal), and slowly warming at about 0.1 nK/s B: SOC state has formed at the bottom and is passing T3 as it advances up

the cell, invading the superfluid phase from belowC: Cell is completely self-organizedD: As heat is added the normalfluid invades the SOC from above

(Suggested by A. Onuki and independently by R. Ferrell in Oregon, 1989)

q

Approach to the SOC State

Simply integrate [(z,Q)] dT/dz = - Q (numerically)

Moeur et al., Phys. Rev. Lett. 78, 2421 (1997)

z (10-3 cm)

T =

T - T

c (z,Q)

Tc(Q) - T

Tsoc(Q)

What is Tsoc(Q)?

T = Q, so soc = Q /

soc soc-x = Q /

soc = [Q/( )]-1/x

T soc = Tsoc(Q,z) – Tc(Q,z)

Tsoc - Tc = T [Q/( )]-1/x

≈ 10-5 W/(cm K), x ≈ 0.48 ≈ 1.27 K / cm

Tsoc - Tc = T [Q / (12.7 pW/cm2)]-2.083

= 2,000 nK (Q = 10 nW/cm2) = 0.14 nK (Q = 1,000 nW/cm2)

Moeur et al., Phys. Rev. Lett. 78, 2421 (1997)

Wave travels only against Q, so ‘Half Sound’Sergatskov et al., JLTP 134, 517 (2004)Weichman and Miller, JLTP 119, 155 (2000)And now Chatto et al., JLTP (2007)

A New Wave on the SOC State

Q

New SOC Anisotropic Wave Propagation

T-wave travels only against Q (bottom to top). See:“New Propagating Mode Near the Superfluid Transition in 4He”, Sergatskov et. al., Physica B 329 – 333, 208 (2003), and“Experiments in 4He Heated From Above, Very Near the Lambda Point”, Sergatskov et. al., J. Low Temp. Phys. 134, 517 (2004), andChatto, Lee, Day, Duncan, and Goodstein, J. Low Temp. Phys. (2007)

The basic physics is simple:

2

dTC Q

dt

Q T

dTC T T

dt

����������������������������

����������������������������

����������������������������

So : u = D, whereD = /C

New Anisotropic Wave SpeedChatto, Lee, Duncan, and Goodstein, JLTP, 2007.

More generally:u = - ∂QSOC/∂T(Chatto et al.)

Similarity in Temperature and Mass SOC

∂T/∂t = (/Cp) 2T + u • T where u = Cp-1

Likewise, for concentration c(z,t) of some agent,

∂c/∂t = Dc 2c + u • c where u = Dc

If Dc is singular along a critical line, then similar SOC waves shouldbe observed.

Will spinoidal decompositions create singular mass diffusivities?

Intermittency in ion channels?

Iodine waves near the basement membrane of the thyroid gland?

What is going on microscopically?

How does helium develop the huge thermal conductance (> 10 W/(cmK)) necessary to self-organize at high Q? – Weichman and Miller, JLTP 119, 155 (2000):

• Q-induced vortex sheet shedding• Vortex production rate increases as Q decreases

– New experiments to measure vorticity creation– Experiments that sense second sound noise

How does He-II do this?

Synchronous phase slips – each slip creates a sheet of quantized vortices! See: Weichman and Miller, JLTP 119, 155 (2000):

Recent Results (Chatto et al.)

• Csoc has been measured and agrees with microgravity CP for 250 nK ≤ T – T ≤ 650 nK – Csoc agrees with static heat capacity from

Lambda Point Experiment on Earth orbit– NO OBSERVED GRAVITATIONAL

ROUNDING!

• Reason suggested for why Csoc diverges at T and soc diverges at Tc(Q)

Recent results on the Self-Organized Critical StateChatto, Lee, Duncan, and Goodstein, JLTP (2007)

Predicted by

Haussm

ann

‘Unrounded Earth-based data in 3 cm tall cell!High-Q

data

R. Haussmann, Phys. Rev. B 60, 12349 (1999).

Future Work: Fly Us!See Barmatz, Hahn, Lipa, and Duncan, “Critical Phenomena Measurements In Microgravity: Past, Present, and Future”, Reviews of Modern Physics 79, 1 (2007)

Next Point:

This ‘CMP’ measurement technologyhas opened the door to a new class of experimental possibilities that may be useful in observational cosmology …

Fundamental Noise Sources

Heat fluctuations in the linkone independent measurement per time constant = RC(noise)2 / C(TQ)2 = 4RkBT2

so TQ √R and TQ TSee: Day, Hahn, & Chui, JLTP 107, 359 (1997)

Thermally induced electrical current fluctuationsmutual inductance creates flux noise()2 T N2 r4 / L M = / s, s ≈ 1 so M √T

SQUID noise(SQ)21/2 ≈ 4 √Hz with shorted inputexternal circuit creates about three times this noise levelso ≈ 12 √Hz and SQ ≈ 12 pK/√Hz

T, C Tbath

R

r

N

L

Heat Fluctuation Noise Across the Link

R = 40 K/W

(TQ)2 = 4RkBT2

so TQ = 0.10 nK/√Hz

3 dB point at 10 Hz,suggesting ≈ 50 ms

(collaboration with Peter Day)

HRT Time Constant

Method:

• Controlled cell temperature with T1

• Pulsed a heater located on T2

• Cell in superfluid state

• Contact area of only 0.05 cm2

• Rise time ~ 20 ms

• Decay time = 48 ms

Collaboration with Peter Day

New Ultra-Stable Platform

The helium sample is contained within the PdMn thermometric element. Power dissipation is precisely controlled with the rf-biased Josephson junction array (JJA). Stabilized to T ~ 10-11 K.

1

2

Green, Sergatskov, and Duncan, J. Low Temp. Phys. 138, 871 (2005)

Reduce the Heat Fluctuation Noise

Reduce R from 40 to 0.25 K/W

Now TQ ≈ 7 pK/√Hz

Minimize TM with a gap toreduce mutual inductance to the SQUID loop

is PdMn thickness = 0.76 mm

r4 4 r 3 r3/(43) ≈ 18

Noise: Thermally Driven Current Fluctuations

Thermal current fluctuations: = 38 /(Hz K)1/2 √SQUID circuit noise: SQ = 12.5 /√Hz

RF-biased Josephson Junctions for Heater Control

Vn = n (h/2e) fh/2e = o = 2.07 V/GHzf = 94.100000000 GHzRel = 1,015 Pn = Vn

2 / Rel = 37.3 n2 pW

Standoff vs. Josephson Quantum Number n

Rel = 1,015

Rso = 4,456 K/W

Tcool

T

Rso

n = 7

n = 10

n = 0

A New ‘Fixed-Point’ Standard

T = T – 125 K

Future Work: Radiometric Comparisons

Three independent BB references

Inner shield maintained at T± 50 pK

Each reference counted up from T

to 2.7 K using Josephson heater control

Compare to each other and eventuallyto the CMB with 0.05 nK resolution

New control theory has been developed

(suggested in part by Phil Lubin, UCSB)