Experiments on distribution of
energy in non-equilibrium electron
tunneling
Jukka Pekola, Low Temperature Laboratory (OVLL),
Aalto University, Helsinki
1. Energy relaxation, heating and cooling in nanoelectronic
circuits at low temperatures
2. Statistics of dissipated work in single-electron tunneling
in classical regime: charge counting detection
3. Quantum regime? Work in a two-level system
4. Calorimetric detection, distribution of heat
A thermal model for an electronic
conductor
The energy distribution of electrons in a
small metal conductor
Equilibrium –Thermometer measures the temperature of the ”bath”
Quasi-equilibrium –Thermometer measures the temperature of the electron system which
can be different from that of the ”bath”
Non-equilibrium –There is no well defined temperature measured by the ”thermometer”
Illustration: diffusive
normal metal wire
H. Pothier et al. 1997
Electron-electron and electronphonon relaxation
e-e relaxation drives
the system towards
quasi-equilibrium
e-p relaxation drives
the system towards
equilibrium
Electromagnetic transfer of heat
(photons)
Electron
system
Gn
Electrical
environment
Lattice
Schmidt et al., PRL 93, 045901 (2004)
Meschke et al., Nature 444, 187 (2006)
Ojanen et al., PRB 76, 073414 (2007), PRL 100, 155902 (2008)
D. Segal, PRL 100, 105901 (2008)
L. Pascal et al., PRB 83, 125113 (2011)
Heat transported between two resistors
R1
T1
R2
T2
Coupling constant:
Linearized expression
for small temperature
difference
DT = T1 – T2:
Radiative contribution to net heat flow between
electrons of 1 and 2:
Demonstration of photonic heat
conduction
x
Gen
x
Thermal
R2
R1
model
F
x
Tunable
x
impedance
n
170
T0 = 167mK
165
160
Te1 (mK)
LJ
CJ
matching using
DC-SQUIDs
155
157mK
125
118mK
120
115
105mK
110
105
100
10 µm
75mK
60mK
95
90
F (a.u.)
M. Meschke, W. Guichard and J.P., Nature 444, 187 (2006)
Experiment on the quantum of thermal
conductance
SAMPLE A in a loop (”matched”)
[SAMPLE B without loop (”not
matched”)]
A. Timofeev et al., PRL 2009
Results in the two
sample geometries
Heat transported by
residual quasiparticles
at T > 0.3 K and by
photons (in the loop
sample) at T < 0.3 K
Quasiparticles
Energy current in a tunnel junction
Energy current (from conductor 1)
For a NIN junction
The Joule power is divided equally between 1 and 2 in this
case.
NIS junction as a refrigerator
Cooling power of a
NIS junction:
e2RT
Optimum cooling power is
reached at V  D/e:
Optimum cooling power of a NIS junction at TS ,TN<< TC
Efficiency (coefficient of performance) of a NIS
junction refrigerator:
Experimental status
Nahum, Eiles, Martinis 1994 Demonstration of NIS cooling
Leivo, Pekola, Averin 1996, Kuzmin 2003, Rajauria et al. 2007 Cooling electrons 300 mK
-> 100 mK by SINIS
Manninen et al. 1999 Cooling by SIS’IS see also Chi and Clarke 1979 and Blamire et al.
1991, Tirelli et al. 2008
Manninen et al. 1997, Luukanen et al. 2000 Lattice refrigeration by SINIS
Savin et al. 2001 S – Schottky – Semic – Schottky – S cooling
Clark et al. 2005, Miller et al. 2008 x-ray detector refrigerated by SINIS
Prance et al. 2009 Electronic refrigeration of a 2DEG
Kafanov et al. 2009 RF-refrigeration
Quaranta et al 2011 Cooling from 1 K to 0.4 K
Refrigeration of a ”bulk” object
Nguyen et al 2012 NIS cooling at 1 nW power
For reviews, see Rev. Mod. Phys. 78, 217 (2006);
Reports on Progress in Physics 75, 046501 (2012).
A. Clark et al., Appl. Phys. Lett. 86, 173508 (2005).
Early experiments
M. Leivo et al., 1996
Quantum dot cooler
Edwards, 1993 (proposal)
Prance et al, 2009
Cooling 280 mK -> 187 mK
demonstrated
Dissipation in electron tunneling
m1
E
DU
m2
Dissipation generated by tunneling
in a biased junction
= (m1-E)+(E-m2) = m1-m2 = DU
Generated heat Q = DU due to relaxation (first distributed to
the electron system, then typically to the lattice by electronphonon scattering)
Steady-state fluctuation theorems
Evans et al., 1993
Bochkov and Kuzovlev, 1977
Experiment on a double quantum dot circuit: B. Kung et al., PRX 2, 011001 (2012);
D. Golubev et al., PRB 84, 075323 (2011); Y. Utsumi et al., PRB 86, 075420 (2012).
Dissipation in driven single-electron
transitions
C
n
Cg
1
Vg
1
n
ng
Single-electron box
Buttiker 1987
Lafarge 1991
0
0
0
time
t
0
time
t
ENERGY
0.4
0.2
The total dissipated heat in a ramp:
n=1
n=0
0.0
-0.5
0.0
0.5
1.0
1.5
ng
D. Averin and J. P., EPL 96, 67004 (2011).
Distribution of dissipated energy
Take a normal-metal SEB
with a linear gate ramp
1.0

1
ng
0
0
time
t
0.5
0.0
-5
0
Q
5
10
n = 0.1, 1, 10 (black, blue, red)
Jarzynski equality
Systems driven by control parameter(s),
starting in equilibrium
FB
= dissipated work
Jarzynski 1997
FA
Clausius inequality and FDT for near-equilibrium drive follow from
JE:
Work done by the gate
J. P. and O.-P. Saira, JLTP 169, 70 (2012)
In general:
For a SEB box:
for the gate sweep 0 -> 1
This is to be compared to:
Comparison of work and heat
Quite generally:
Experiment on a single-electron box
O.-P. Saira, Y. Yoon, T. Tanttu, M. Möttönen, D. Averin, J.P., arXiv:1206.7049,
PRL to appear
Detector
current
Gate drive
P(Q)
TIME (s)
Q/EC
Measurements of the heat distributions at
various frequencies and temperatures
sQ /EC
<Q>/EC
symbols: experiment;
full lines: theory;
dashed lines:
Integral fluctuation relation
U. Seifert, PRL 95, 040602 (2005)
In single-electron transitions with overheated island:
However,
should be valid also for unequal temperatures.
J. P., A. Kutvonen, and T. Ala-Nissila, arXiv:1205.3951
Experiments with un-equal
temperatures
TH
TN
TS
T0
Coupling to two different baths
J. Koski et al, unpublished.
TN=139 mK = Tbath, TS = 180 mK
Experimental test of the integral
fluctuation theorem
SEB with an overheated island
Distribution of dissipation in a
quantum system?
Work and FRs in quantum systems have been discussed intensively over the
past decades:
J. Kurchan, cond-mat/0007360.
H. Tasaki, cond-mat/0009244.
S. Yukawa, J. Phys. Soc. Jpn. 69, 2367 (2000).
S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003).
V. Chernyak and S. Mukamel, Phys. Rev. Lett. 93, 048302 (2004).
A. E. Allahverdyan and Th.M. Nieuwenhuizen, Phys. Rev. E 71, 066102 (2005).
A. Engel and R. Nolte, EPL 79, 10003 (2007).
P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102(R) (2007); see also Erratum Rev.
Mod. Phys. 83, 1653 (2011).
M. Esposito, U. Harbola and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009).
M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009).
P. Talkner , M. Campisi, and P. Hänggi, J. Stat. Mech. P02025 (2009).
M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. 83, 711 (2011); 83, 1653 (2011) (E).
Isolated quantum system
Results have been obtained for systems that are isolated from
the environment during the driving.
Specifically:
1. System is in thermal equilibrium initially
2. During the drive, system obeys unitary evolution (no coupling
to environment)
3. At the end, the state of the system is measured
Then:
Kurchan 2000
Talkner et al. 2007
Example: two-level system – unitary
evolution (G << dq/dt)
PLZ = probability of
Landau-Zener transition
W/EC
Average work in a driven open
quantum system
With the help of the power operator
:
<Work> = <Change of internal energy> + <Dissipated heat>
Difficult to proceed beyond the average work in general
P. Solinas et al., arXiv:1206.5699
Energy conservation, calorimetry
1. System initially in thermal equilibrium
2. Drive the system (it can interact with the environment during
the application of work)
3. Wait ”infinitely” long time (after applying the work) to reach
equilibrium again
SOURCE OF
WORK
W
DE
ENVIRONMENT
Then Work = Heat absorbed by the environment, i.e. if we can
measure energy absorbed from t = 0 to t >> t (in each
realization), we know the whole distribution of work performed
(Consider for simplicity a periodic system with drive over one period)
E g , Ee
A basic quantum two-level system: Cooper
pair box
-0.5
EJ
0.0
q
Ec
0.5
In the basis of adiabatic eigenstates:
(in the charge basis)
Measurement of work distribution
of a two-level system (CPB)
Calorimetric measurement:
Measure temperature of the
resistor after relaxation.
”Typical parameters”:
-
DTR ~ 10 mK over 0.1 - 1 ms
time
TR
TIME
Technicalities
Can be determined via a Joule heating measurement
Unwanted dissipation in the biasing resistor:
Tunnel junction calorimetry
Meschke et al., Nature 2006; Timofeev et al., PRL 2009
Calorimetric detection – results in the
LZ model
Numerical results by
simulations (MCWF):
EC /k = 1 K, EJ /EC = 0.1, Cg /CS
= 0.1, T = 0.2 K, t = 10-9 s,
R/RQ = (0,) 0.01, 0.1, 1.0, 3.0
Analytical result:
J.P. et al., unpublished
Results in the strong dissipation model
P(E) calculation for Josephson current
Summary
Discussed energy relaxation, average heat current and
electron cooling
Distribution of dissipation measured in a singleelectron box via charge counting statistics
Work and dissipation in a quantum system: two-level
system analyzed and a calorimetric measurement
proposed for a superconducting ”qubit”
Collaboration: Olli-Pentti Saira, Matthias Meschke,
Youngsoo Yoon, Ville Maisi, Andrey Timofeev,
Alexander Savin, Mikko Möttönen, Paolo Solinas, Jonne
Koski
Dmitri Averin (SUNY)
JE in the presence of LZ transitions
For a trajectory where detailed balance holds:
For a trajectory which is possibly interrupted by a LZ transition:
By a straightforward analysis one then finds:
Relaxation after driving
Internal energy
Heat
Dissipation during the gate ramp
various e
various T
t
Solid lines: solution of the full master equation
Dashed lines:
t
Problem of non-commuting
hamiltonians
M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. 83, 1653 (2011),
Erratum.
”
”
Different expressions of work
cf. C. Jarzynski, C. R. Phys. 8, 495 (2007)
Integral fluctuation relation
U. Seifert, PRL 95, 040602 (2005)
G. Bochkov and Yu. Kuzovlev,
Physica A 106, 443 (1981)
In single-electron transitions with overheated island:
Inserting
we find that
is valid also for unequal temperatures.
Quantum ”FDT”
Unitary evolution of a two-level
system during the drive
(Gt << 1)
in classical regime at finite T
Experiments with un-equal
temperatures
TH
TN
TS
T0
Coupling to two different baths
TN=139 mK = Tbath, TS = 180 mK
Preliminary experiments with un-equal
temperatures
P(Q)
TH
TN
TS
T0
Coupling to two different baths
Q/EC
Calibrations
Maxwell’s demon
Negative heat
Possible to extract heat
from the bath
0.4

P(Q<0)
0.5
0.0
-3 -2 -1 0
1
Q
2
3
4
0.3
0.2
0.1
0.0
1
Provides means to make Maxwell’s demon using SETs
n
10
Maxwell’s demon in an SET trap
n
D. Averin, M. Mottonen, and J. P., PRB 84, 245448
(2011)
Related work on quantum dots: G. Schaller et al., PRB
84, 085418 (2011)
”watch and move”
S. Toyabe et al., Nature Physics 2010
Demon strategy
Adiabatic ”informationless” pumping: W = eV per cycle
Ideal demon: W = 0
n
Energy costs for the
transitions:
Rate of return (0,1)->(0,0)
determined by the energy
”cost” –eV/3. If G(-eV/3) << t-1,
the demon is ”successful”.
Here t-1 is the bandwidth of the
detector. This is easy to satisfy
using NIS junctions.
Power of the ideal demon: