SLIDE 1
Stony Brook University, SUNY Dmitri V. Averin and Qiang Deng Low-Temperature Lab, Aalto University Jukka P. Pekola and M. Möttönen
Information to energy conversion in an electronic Maxwell’s demon and thermodynamics of measurements.
Publications: PRL 104, 220601 (2010) EPL 96, 6704 (2011) PRB 84, 245448 (2011)
SLIDE 2 Outline
- 1. Statistical distribution of the heat generated in adiabatic
transitions: classical thermal fluctuation-dissipation theorem (FDT).
- 2. Driven single-electron tunneling (SET) transitions as
prototype of reversible information processing.
- 3. Electronic Maxwell’s demons based on the SET pump and
nSQUID array.
- 4. Detector properties in Maxwell’s demon operation and
thermodynamics of quantum measurements.
SLIDE 3
Distribution of heat generated in adiabatic transitions reversible logic operations are adiabatic transitions; potential reversible circuits are based on small, mesoscopic or nano, structures large role of fluctuations Example: driven SET transitions
SLIDE 4 O.-P. Saira et al., PRB (2010); and t.b.p.
. ) ( , ) ( 2 ) ( ) (
2
n E n U n t n E n U n U
C g C
= − = Driven single-electron tunneling (SET) transitions
SLIDE 5 Slow evolution of a system of levels En(t) weakly interacting with an equilibrium reservoir, between the two stationary configurations Heat generated in the reservoir in this evolution
Q T
Q
2
2 =
σ
[ ]
. , p p p p p
n nm m mn n m
Γ = Γ − Γ =∑ & &
( )
. , ) ( ) (
) (
Q S T Q t E t E Q
tot jumps m n
+ ∆ − = − = ∑
:
) ( ) (
= Γ p p
Noise of the generated heat in driven adiabatic evolution
SLIDE 6 . ) ( ) / 1 (
) ( 1 , m m nm m n n
p E E dt T Q & &
−
Γ − =
∫ ∑
Average generated irreversible heat
. ; ; :
1 1 1 1 1 1 1 − − − − − − −
Γ Γ = Γ Γ Γ = Γ Γ Γ Γ = Γ Γ Γ Γ
Γ-1 – “group” inverse:
. 2 ) ( 2 ~
) ( 1 , 2 2
Q T p E E dt Q
m m nm m n n Q
= Γ − = =
−
∫ ∑
& & σ
Noise in generated heat Two-state system
. ) ( ) ( , ) 2 / ( cosh ) 2 / 1 (
2 1 2 10 01 2 2
t E t E T dt
Q
− = Γ + Γ =
−
∫
ε ε ε σ &
Conclusion: irreversibly generated heat vanishes not only
- n average but for each individual transition protocol.
SLIDE 7 Jarzynski equality and statistics of the generated heat
. 1 } / ) ( exp{ = ∆ − − T F Wth
In the limit of adiabatic switching, this relation and thermal FDT imply Gaussian probability density of the heat distribution
. 2 ~2
2
Q T Q
Q
= ≡ σ
In the case of “deterministic” transitions, ∆F=∆U, and
. ) 4 / 1 ( ) (
4 / ) ( 2 / 1
2
Q T Q Q
e Q T Q
− −
= π ρ
{ }
. 1 / exp = − T Q
SLIDE 8 Statistics of heat in driven SET transitions
C Cg ne
Vg
∆U(t)
∫ ∫ ∑
− = + − = ∆ − = ∆ − = ) ( 2 , ) ( ) 1 ) ( 2 ( ) ( t dn n E W W Q U t I t n dt E t U Q
g C th th g C j j β βλ ηβ βσ
Thermodynamics of the transitions
SLIDE 9 Electronic Maxwell’s demon based on SET pump “Information-to-energy conversion”
C C C Cg Cg Vg1 Vg2 C0 V n1 n2
N-n-n1-n2
Alternative approach: G. Schaller et al., PRB (2011).
SLIDE 10 Generated power:
). 3 / ( 3 / eV eV P Γ =
Demon inverts the Landauer principle: bit
a measurement can be used to convert roughly kBT of thermal energy into free energy.
SLIDE 11 Maxwell’s demon based on nSQUID array For M/L≈1 dynamics of individual nSQUIDs reduces to that
- f the differential phase φ describing the circulating
current: . 2 ] , [ , cos cos 2 2 ) ( 4
2 2 2 2
ei Q E L C Q H
J e
= − − Φ + = ϕ ϕ χ ϕ ϕ π
SLIDE 12 Detector properties for demon operation Qualitative similarity to quantum measurements: trade-off between the information acquisition and back-action. Standard quantum detector set-up:
- Error-free and rapid detection:
- No back-action excitations
V I
1
∆
Heisenberg uncertainty relation for detectors:
. ) 4 / (
2
π λ h ≥
V I S
S
Quantitative requirements for Maxwell’s demon:
. , 8 / /
1 2 2 −
<< Γ << τ π τ λ e SI )]. / ln( /[ , / , ) ( / )] ( 1 )[ (
2 2 2 2 2
T e T S S e U f f d d e G
C V V f i f i f i T
ω π γ γ ε ε π γ ε ε ε ε h h << = + ∆ − − − = Γ
∫
SLIDE 13 . ) 2 ( , ) ( 2 ), ( 2 ) (
2 / 1 2 2
L eI C e m x U mv x U H H
C J J z
h h = ≈ + + + = λ λ δ σ
E ,C
J
. . . reciever
r(k) t(k)
U(x)
generator L
. . .
x
qubit (a) (b)
k
D.V. A., K. Rabenstein, V.K. Semenov, PRB 73, 094504 (2006).
The detector employs ballistic propagation of individual fluxons in a JTL. The measured system controls the fluxon scattering potential: JTL-based magnetic flux detector
SLIDE 14
Conclusions Gaussian distribution of the generated heat in the reversible transformations with the width related to average by classical thermal FDT. SET structures can be developed into a prototype of thermodynamically reversible devices and a promising tool for studying basic thermodynamics, e.g., non-equilibrium fluctuation relations, demonstration of the Maxwell’s demon, … ; but at low frequencies. nSQUID arrays would add an advantage of developed support electronics allowing the high-frequency operation both for the development of practical reversible circuits and for fundamental studies of the dynamics of information/entropy in electronic devices, e.g., in thermodynamics of quantum measurements.
