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)
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• 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
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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
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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
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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
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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.
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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.
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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.
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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 < ∞.
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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).
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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.
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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)
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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.
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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.
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θ = temperature where Cv = 12 (5.94) ≈ 2.97
Lewis and Randall: ‘Thermodynamics And The FreeEnergy Of Chemical Substances‘ (1923)
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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.
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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.
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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!
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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.
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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.
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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
)
.
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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.
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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.
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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+.
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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
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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.
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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.
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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.
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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
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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
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