Stony Brook University, SUNY Dmitri V. Averin and Qiang Deng

Low-Temperature Lab, Aalto UniversityJukka 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)

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 andnSQUID array.

4. Detector properties in Maxwell’s demon operation and thermodynamics of quantum measurements.

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

O.-P. Saira et al., PRB (2010); and t.b.p.

.)(

,)(2)()(2

0

0

nEnU

ntnEnUnU

C

gC

=

−=

Driven single-electron tunneling (SET) transitions

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

QTQ 22 =σ

[ ] ., pppppn

nmmmnnm Γ=Γ−Γ=∑ &&

( ) .,)()( )( QSTQtEtEQ tot

jumpsmn +∆−=−= ∑

0: )0()0( =Γ pp - local equilibrium

Noise of the generated heat in driven adiabatic evolution

.)()/1( )0(1

,mmnm

mnn pEEdtTQ && −Γ−= ∫ ∑

Average generated irreversible heat

.;;: 1111111 −−−−−−− ΓΓ=ΓΓΓ=ΓΓΓΓ=ΓΓΓΓΓ-1 – “group” inverse:

.2)(2~ )0(1

,

22 QTpEEdtQ mmnmmn

nQ =Γ−== −∫ ∑ &&σ

Noise in generated heat

Two-state system

.)()(,)2/(cosh)2/1( 212

1001

22 tEtETdtQ −=

Γ+Γ= −∫ εεεσ

&

Conclusion: irreversibly generated heat vanishes not only on average but for each individual transition protocol.

Jarzynski equality and statistics of the generated heat

.1}/)(exp{ =∆−− TFWth

In the limit of adiabatic switching, this relation and thermal FDT

imply Gaussian probability density of the heat distribution

.2~22 QTQQ =≡σ

In the case of “deterministic” transitions, ∆F=∆U, and

.)4/1()( 4/)(2/1 2 QTQQeQTQ −−= πρ

{ } .1/exp =− TQ

Statistics of heat in driven SET transitions

C Cgne

Vg

∆U(t)

∫

∫∑

−=+−=∆

−=∆−=

)(2,

)()1)(2()(

tdnnEWWQU

tItndtEtUQ

gCthth

gCj

j

β

βλ

ηβ

βσ

Thermodynamics of the transitions

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 N-n-n1-n2

Alternative approach: G. Schaller et al., PRB (2011).

Generated power:).3/(3/ eVeVP Γ=

Demon inverts the Landauer principle: bit of information gained in a measurement can be used to convert roughly kBT of thermal energy into free energy.

Maxwell’s demon based on nSQUID array

For M/L≈1 dynamics of individual nSQUIDs reduces to that of the differential phase φ describing the circulating current:

.2],[

,coscos22

)(4

2

2

20

2

eiQ

ELC

QH Je

=

−−Φ

+=

ϕ

ϕχϕϕπ

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

0 1

∆ Heisenberg uncertainty relation for detectors:

.)4/( 2πλh≥VI SS

Quantitative requirements for Maxwell’s demon:

.,8// 122 −<<Γ<< τπτλ eSI

)]./ln(/[

,/,)(

/)](1)[(

2

2222

TeTS

SeU

ffddeG

CV

Vfi

fifiT

ω

πγγεε

πγεεεε

h

h

<<

=+∆−−

−=Γ ∫

.)2

(,)(2),(2

)( 2/122

0 LeIC

emxUmvxUHH

CJ

Jz

hh=≈+++= λ

λδσ

E ,CJ . . . 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

ConclusionsGaussian 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.