Cooling A Solid To Its Ground Statetroy/einstein/michigan2012.pdf · 2011. 11. 22. · for a single...

Cooling A Solid To Its Ground State

W. C. Troy

Cooling A Solid To Its Ground State – p.1/41

Acknowledgement

• Stefanos Folias• Chris Horvat• Sashi Marella• Brent Doiron• S. J. Anderson• Richard Field• Stuart Hastings• Anna Vainchtein• Richard Gass

Cooling A Solid To Its Ground State – p.2/41

Contents

1. The Question And Experimental Results.

2. Single Atom Model.

3. Einstein Model For A solid.

4. Debye Model.

5. Dark Matter Detectors, Gravity Wave Detectors.

• W. C. Troy, Quart. Jour. Appl. Math., AMS (2012)

Cooling A Solid To Its Ground State – p.3/41

• R. Feynman - 1982 - proposed that development of aquantum computer is theoretically possible.

The Question. Can a solid be cooled to a temperature

T0 > 0 where all quanta of thermal energy are drained

off, leaving the object in the ground state? If ‘yes,’

quantum effects are expected (superposition of states).

This result may lead to quantum computing devices.

• D. Powell, Moved By Light. Science News, May 7, 2011

Cooling A Solid To Its Ground State – p.4/41

1982- 2006

Laser cooling - techniques based on radiation pressure

were used to remove energy and reduce vibrations inmechanical objects. Two examples:

2006 - vibrations in glass ’doughnuts’ were reduced bycooling to T=11 mK

2006 - wobbles in mirrors were reduced bycooling to T=10 mK

Cooling A Solid To Its Ground State – p.5/41

2006-2009

• Further refinements of laser techniques were developed

to cool sticks, sails, drums and other multi-atom objectsto low temperatures where vibrations are quelled.

• Papers ’flowed in’ as researchers competed to removeevery quantum of energy from an object, leaving it in theground state where quantum effects are expected

• D. Powell, Moved By Light. Science News, May 7, 2011

Cooling A Solid To Its Ground State – p.6/41

2009-2010

• 63 quanta left - Park and Wang, Nature Physics 2009

• 37 quanta left - Schliesser et al, Nature Physics 2009

• 30 quanta left - Groblacher et al, Nature Physics 2009

• 4 quanta left - Rocheleau et al, Nature 2010

Cooling A Solid To Its Ground State – p.7/41

2010 - 2011

• O’Connell et al (Nature, 2010) reduced a quantumdrum (one trillion atoms) to its ground state at

T0 ≈ 20mK

- Science magazine 2010 breakthrough of the year.

• Teufel et al (Nature, 2011) reduced a drum ( 10−13 kg) toits ground state where it stayed for 100 microseconds, muchlonger than the 6 nano second result of O’Connell et al.

Cooling A Solid To Its Ground State – p.8/41

TheoryOur Goal. Answer the question theoretically: can allquanta of thermal energy be drained from a solid?

T0 ≡ Lowest Possible Temperature

Models.

• One Atom Quantum Mech. Based Model: T0 = 0.

• Einstein 1907 Model For A Solid.

(a) Stat. Mech. Partition Function Mds.: T0 = 0.

(b) Stat. Mech. Microcanonical Mds.: T0 > 0.

Today’s goal: show that T0 > 0.

Cooling A Solid To Its Ground State – p.9/41

One Atom Model

Current theory (e.g. Teufel et al, Nature, 2011) - the QMmotivated Bose-Einstein equation

< N >=1

exp( ǫkT

) − 1, 0 < T < ∞,

for a single atom represents the entire solid:

< N >= ave. no. of quanta in 1 particle state with energy ǫ.

T = temperature (K), k = Boltzman’s constant,

ǫ = hν = quantum of energy,

ν = frequency, h= Planck’s constant.

Cooling A Solid To Its Ground State – p.10/41

Statistical Derivation

Partition function method: let Ns denote the number ofphotons that occupy a given energy level Es = hνs.

< Ns > = mean number of photons per degree of freedom

< Ns > =exp(−Nshνs

kT)

∑

∞

Ns=0 exp(−Nshνs

kT)

< Ns > =1

exp(hνs

kT) − 1

, 0 < T < ∞.

Cooling A Solid To Its Ground State – p.11/41

Teufel et al (Nature, 2011) single atom analysis relieson predictions of the Bose-Einstein equation

< N > =1

exp( ǫkT

) − 1, 0 < T < 1.

P1: Thermal energy is present at every positive T > 0.

P2: T → 0 as < N > → 0. i.e. T0 = 0.

P2: Quantum effects become important when < N > < 1.

< N > < 1 when T <ǫ

k ln(2).

Cooling A Solid To Its Ground State – p.12/41

Comparison With Experiment

O’Connell et al experiment: ground state achieved when

< N > = .07, ν = 6GHz and T = 20mK

The one atom model

< N > =1

exp( ǫkT

) − 1

predicts

Ground State At T = 106mK

Cooling A Solid To Its Ground State – p.13/41

Einstein Theory For A Solid

A1. Each atom is a 3D quantum oscillator, which

is attached to a preferred position by a spring.

A2. q > 0 quanta of energy have been added to the solid.

A3. Each quantum has energy ǫ = νh.

Einstein Goal (1907): develop a formula for specific heat

Cv = 1n

∂U∂T

in terms of T and the quantum ǫ = hν.

Cooling A Solid To Its Ground State – p.14/41

Petit-Dulong Law of Specific Heat

• P. L. Dulong (1785 - 1838) - French physicist and chemist.

• A. T. Petit (1791-1820) - French physicist.

• 1819 - Petit-Dulong Law of specific heat:

Cv =1

n

∂U

∂T= 5.94

(

cal

gm K

)

.

• A. T. Petit and P. L. Dulong, "Recherches sur quelquespoints importants de la Théorie de la Chaleur," Annales deChimie et de Physique 10 (1819)

Cooling A Solid To Its Ground State – p.15/41

Specific Heat:Cv =1n

∂U∂T

• Dulong and Petit (1819): for all solids,

Cv = 5.94

(

cal

gm K

)

.

• Weber (1875), Kopp (1904), Dewar (1904):

0 < Cv << 5.94

for many solids at low T.

Cooling A Solid To Its Ground State – p.16/41

Einstein Formula (1907):

Cv = 5.94( ǫ

kT

)2 exp( ǫkT

)(

exp( ǫkT

) − 1)2 , 0 < T < ∞.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6C

v

Figure 1. Cv vs. scaled temperature for diamond.

Cooling A Solid To Its Ground State – p.17/41

θ = temperature where Cv = 12 (5.94) ≈ 2.97

Lewis and Randall: ‘Thermodynamics And The FreeEnergy Of Chemical Substances‘ (1923)

Cooling A Solid To Its Ground State – p.18/41

Behavior of Cv asT → 0+

Cv = 5.94( ǫ

kT

)2 exp( ǫkT

)(

exp( ǫkT

) − 1)2 , 0 < T < ∞.

It is easily verified that

limT→0+

Cv = 0. (1)

Can T → 0 as the number of quanta decreases to zero?To answer this study the ’ derivation of Cv.

Cooling A Solid To Its Ground State – p.19/41

ODE Derivation of Cv

Solve Layton’s BVP

dU

dT=

1

kN ′T 2

(

U2 −(

N ′ǫ

2

)2)

,

U(0+) =N ′ǫ

2, U(∞) = ∞.

Theorem Uniqueness of solutions implies that

U =N ′ǫ

eǫ

kT − 1+

N ′ǫ

2,

1

nU ′(T ) = Einstein Specific Heat Function.

Cooling A Solid To Its Ground State – p.20/41

Micro. Canonical Derivation

N = no. of atoms. N ′ = 3N = no. of degrees of freedom.

W = no. of ways to distribute q quanta of energyover N ′ degrees of freedom:

W =(q + N ′ − 1)!

q!(N ′ − 1)!.

Entropy:

S = k ln(W ) = k ln

(

(q + N ′ − 1)!

q!(N ′ − 1)!

)

.

Simplification: de Moivre - Stirling approximation to n!

Cooling A Solid To Its Ground State – p.21/41

James Stirling 1692-1770

• Scottish mathematician.

• 1715 - Oxford - expelled from Baliol college for

correspondence supporting the 1708 Scottish

"Gathering of the Brig o’ Turk" uprising against the Stuarts.

• 1725 - Venice - feared assassination for discovering

trade secrets of Venice glassmkaers. Newton helped

him return to England.

Cooling A Solid To Its Ground State – p.22/41

Abraham de Moivre 1667-1754

• French mathematician.

• 1687 - left France under Hugenout persecution.

• 1697 - generalized Newton’s Binomial Theorem to the

Multinomial Theorem, elected to the Royal Society.

• 1712 - appointed to a commission (along with Halley,

Bonet, Aston and Taylor) to review the claims of Newton

and Leibniz as to who discovered Calculus.

Cooling A Solid To Its Ground State – p.23/41

Stirling’s Formula

• 1733 - de Moivre, "Miscellanes Analytica,"

N ! ∼ [constant]√

N

(

N

e

)N

as N → ∞.

• 1733 - Stirling, "Methodus Differentials,"

N ! ∼√

2πN

(

N

e

)N

as N → ∞.

• Statistical Mechanics and Physics textbooks:

ln(N !) = N ln(N) − N, N >> 1.

Cooling A Solid To Its Ground State – p.24/41

Derivation of Cv

Apply ln(M !) = M ln(M) − M when M >> 1.

S = k(

ln((q + N ′ − 1)!) − ln(q!) − ln((N ′ − 1)!))

becomes

S = k(

(q + N ′ − 1) ln(q + N ′ − 1) − q ln(q) − (N ′ − 1) ln(N ′ − 1))

Therefore,

dS

dq= k ln

(

1 +N ′ − 1

q

)

.

Cooling A Solid To Its Ground State – p.25/41

Internal Energy:

U = qǫ + N ′ǫ

2and

dU

dq= ǫ.

Temperature:

1

T=

∂S

∂U=

dSdq

dUdq

=k

ǫln

(

1 +N ′ − 1

q

)

.

Conclusion I:

T0 = limq→0+

T = 0.

Cooling A Solid To Its Ground State – p.26/41

Invert Temperature Equation:

q =N ′ − 1

eǫ

kT − 1, N ′ = 3nNA.

U = qǫ + N ′ǫ

2=

(N ′ − 1)ǫ

eǫ

kT − 1+ N ′

ǫ

2.

Specific Heat: Cv = 1n

∂U∂T

.

Cv = 5.94( ǫ

kT

)2 exp( ǫkT

)(

exp( ǫkT

) − 1)2 , 0 < T < ∞.

Conclusion II:

Cv exists for all T > 0 and limT→0+

Cv = 0.

Cooling A Solid To Its Ground State – p.27/41

Widely quoted:

limT→0+

Cv = 0. (2)

Property (2) is mathematically questionable becauseits derivation is based on

ln(q!) = q ln(q) − q,

which loses accuracy as q → 0+.

Cooling A Solid To Its Ground State – p.28/41

The Error In Stirling’s Approximation At Low q:

ln(q!) = q ln(q) − q

When q=10 Relative Error = 13 Percent.

When q=2 Relative Error = 188 Percent.

When q=2 Stirling’s Approximation Gives

.69 = −.61

When q=1 Stirling’s Approximation Gives

0 = −1

Cooling A Solid To Its Ground State – p.29/41

New Derivation

Replace Stirling ’s approximation

ln(M !) = M ln(M) − M

with the exact formula

ln(M !) = ln(Γ(M + 1)),

where the Gamma function Γ(z) is defined by

Γ(z) =

∫

∞

0tz−1e−tdt, Re(z) > 0.

Basic Property: M ! = Γ(M +1) when M is a positive integer.

Cooling A Solid To Its Ground State – p.30/41

Entropy S = k ln(

(q+N ′−1)!

q!(N ′−1)!

)

becomes

S = k(

ln(Γ(q + N ′)) − ln(Γ(q + 1)) − ln(Γ(N ′)))

.

U = qǫ + N ′ǫ

2.

1

T=

dSdq

dUdq

1

T=

k

ǫ

(

Γ′(q + N ′)

Γ(q + N ′)− Γ′(q + 1)

Γ(q + 1)

)

∀q ≥ 0.

Cooling A Solid To Its Ground State – p.31/41

Temperature:

T =ǫ

k

(

Γ′(q + N ′)

Γ(q + N ′)− Γ′(q + 1)

Γ(q + 1)

)

−1

> 0 ∀q ≥ 0.

Specific heat:

CV =1

n

dU

dT=

1

n

dUdq

dTdq

=

−k

n

(

Γ′(q + N ′)

Γ(q + N ′)− Γ′(q + 1)

Γ(q + 1)

)2 [d

dq

(

Γ′(q + N ′)

Γ(q + N ′)− Γ′(q + 1)

Γ(q + 1)

)]

−1

> 0 ∀q ≥ 0.

Cooling A Solid To Its Ground State – p.32/41

Lowest Temperature:

T0 = limq→0+

T =νh

k

(

Γ′(N ′)

Γ(N ′)+ γ

)

−1

> 0.

γ = Euler’s constant. T0 is large when ν is large!

Lowest Specific heat:

limq→0+

Cv =

−k

n

(

Γ′(N ′)

Γ(N ′)− Γ′(1)

Γ(1)

)2[

d

dq

(

Γ′(q + N ′)

Γ(q + N ′)− Γ′(q + 1)

Γ(q + 1)

∣

∣

∣

∣

q=0

)]

−1

> 0.

Cooling A Solid To Its Ground State – p.33/41

Derivative Free Method

Goal: Derive temperature formula without differentaition.

S = k ln

(

(q + N ′ − 1)!

q!(N ′ − 1)!

)

.

U = qνh + N ′νh

2.

1

T=

∆S

∆U=

S(q + 1) − S(q)

U(q + 1) − U(q)=

k

νhln

(

q + N ′

q + 1

)

.

T 0 = limq→0

T =νh

k ln(N ′)> T0 =

νh

k

(

Γ′(N ′)

Γ(N ′)+ γ

)

−1

> 0.

Cooling A Solid To Its Ground State – p.34/41

Comparison With Experiment

O’Connell et al (Nature, 2010): ν = 6 × 109Hz, one trillion atoms.

Ground state at N = .07, T0 = 20mK

One atom model:

Ground state at N = .07, T0 = 106mK

New Temperature Formula:

Ground state at q = 0, T0 = 9.8mK

Cooling A Solid To Its Ground State – p.35/41

Diamond

T0 = limq→0+

T = 23.205 (K) and limq→0+

Cv = .63 × 10−20

250 500 750 1000 12500

1

2

3

4

5

6C

v Diamond

T(Kelvin)

New Cv

23.1 23.205 23.3

0.3

0.63

0.9

x 10−20

Cv

Blowup

Einstein Cv

New Cv

C0

T0 T

Cooling A Solid To Its Ground State – p.36/41

Diamond

T0 = limq→0+

T = 23.205 (K) and limq→0+

Cv = .63 × 10−20

250 500 750 1000 12500

1

2

3

4

5

6C

v Diamond

T(Kelvin)

New Cv

23.1 23.205 23.3

0.3

0.63

0.9

x 10−20

Cv

Blowup

Einstein Cv

New Cv

C0

T0 T

Cooling A Solid To Its Ground State – p.37/41

Answer to the original question is Yes!

Question. Can a solid be cooled to a temperature T0 > 0

where all quanta are drained off, leaving the object in

its ground state?

For the Einstein solid we have proved that

limq→0+

T = T0 > 0 and limq→0+

U = N ′ǫ

2= Ground State.

Therefore, as the quanta are drained off, the solid does

cool to temperature T0 > 0 where its internal energy

reaches the ground state.

Cooling A Solid To Its Ground State – p.38/41

Debye Formula

Einstein Specific Heat (1907):

Cv = 5.94( ǫ

kT

)2 exp( ǫkT

)(

exp( ǫkT

) − 1)2 , 0 < T < ∞.

Debye Specific Heat (1913):

Cv = 9Nk

(

T

TD

)3 ∫ TD

T

0

x4ex

(ex − 1)2dx, 0 < T < ∞.

Cooling A Solid To Its Ground State – p.39/41

Dark Matter Detectors

Dark Matter - 23 per cent of the universe.

Properties - very cold, does not absorb or emit light.

Detector - germainium is cooled to a temperature where allthermal energy has been removed.

Experiment - Gran Sasso Lab. - check to see if a tiny amount ofheat or recoil energy is created when dark matter interactswith an atom of the detector.

• L. Randall, Discover Magazine, November, 2011

Cooling A Solid To Its Ground State – p.40/41

Gravity Wave Detectors

Goal - detect ripples in space-time thoughtto be created by colliding black holes.

LIGO - $350 Million NSF Project - large mirrors weighing 10kghave been cooled to 234 quanta.

AURIGA - Italy - aluminum bars weighing more thanone ton have been cooled to 4000 quanta.

• D. Powell, Science News, May 7, 2011

Cooling A Solid To Its Ground State – p.41/41