Information entropic superconducting microcooler
A. O. Niskanen,1, 2 Y. Nakamura,1, 3, 4 and J. P. Pekola5
1CREST-JST, Kawaguchi, Saitama 332-0012,Japan
2VTT Technical Research Centre of Finland, Sensors, PO BOX 1000, 02044 VTT, Finland
3NEC Fundamental Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan
4The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
5Low Temperature Laboratory, Helsinki University of Technology, PO BOX 3500, 02015 TKK, Finland
(Dated: October 25, 2018)
We consider a design for a cyclic microrefrigerator using a superconducting flux qubit. Adiabatic
modulation of the flux combined with thermalization can be used to transfer energy from a lower
temperature normal metal thin film resistor to another one at higher temperature. The frequency
selectivity of photonic heat conduction is achieved by including the hot resistor as part of a high
frequency LC resonator and the cold one as part of a low-frequency oscillator while keeping both
circuits in the underdamped regime. We discuss the performance of the device in an experimentally
realistic setting. This device illustrates the complementarity of information and thermodynamic
entropy as the erasure of the quantum bit directly relates to the cooling of the resistor.
PACS numbers: 74.50.+r,85.80.Fi,03.67.-a
For the purpose of quantum computing, the coher-
ence properties of superconducting quantum bits (qubits)
should be optimized by decoupling them from all noise
sources as well as possible. However, many interesting
experiments can be envisioned also when the decoupling
is far from perfect. One such experiment closely related
to coherence optimization is using a qubit as a spectrom-
eter [1, 2, 3] for the environmental noise by monitoring
the effect of the environment on the quantum two-level
system. Here we focus on the opposite phenomenon, i.e.
the effect of a qubit on the environment. Recently a
superconducting flux qubit [4, 5] with a quite small tun-
neling energy from the point of view of quantum com-
puting was cooled using sideband cooling and a third
level [6] from about 400 mK down to 3 mK. Motivated
by this experiment we consider the possibility of using
a single quantum bit as a cyclic refrigerator for environ-
mental degrees of freedom. The utilized heat conduction
mechanism is photonic which was recently studied also
in experiment [7]. Besides the possible practical uses,
the device is interesting physically as it directly illus-
trates the connection between information entropy and
thermodynamical entropy. For related superconducting
high-frequency cooler concepts see eg. Refs. [8, 9].
Here we study a flux qubit coupled inductively to two
different loops shown in Fig. 1a. In loop j (j = 1, 2) we
have a resistor Rj in series with an inductance Lj and
a capacitance Cj . These form two damped harmonic os-
cillators. The resistors are in general at different tem-
peratures T1 and T2. The coupling of the qubit to these
two admittances Y1 and Y2 is assumed to be sufficiently
large to dominate the relaxation of the qubit. This as-
sumption can be easily validated by e.g. increasing the
mutual inductance. The flux qubit is an otherwise su-
perconducting loop except for three or four Josephson
junctions with suitably picked parameters. In particu-
FIG. 1: (color online) Principle of the flux-qubit cooler.
(a) Layout of the circuit. (b) Energy band diagram. (c)
Schematic of the cooling cycle in the qubit temperature-
entropy plane.
lar one of the junctions is made smaller than others to
form a two-level system. When biased close to half of the
flux quantum Φ0 = h/2e, the qubit can be described (in
persistent current basis) by the Hamiltonian
H/~ = −1
(∆σx + εσz) (1)
where σx and σz are Pauli matrices, ~ε = 2Ip(Φ−Φ0/2)
is the flux-tunable energy bias and Φ is the controllable
flux threading the qubit loop. Away from Φ = Φ0/2 the
eigenstates have the persistent currents ±Ip circulating
in the loop. The tunneling energy ~∆ results in an an-
http://arxiv.org/abs/0704.0845v1
ticrossing at Φ = Φ0/2 and there the energy eigenstates
do not carry average current. The resonant angular fre-
quency of the qubit is ω =
ε2 +∆2.
Consider the ideal cycle shown in Fig. 1b-c where the
bias of the flux qubit is swept slowly (slower than ∆/2π)
between two extreme values ε1 and ε2 corresponding to
two different energy level separations ~ω1 and ~ω2. Let
us further assume that ωj ≈ ωLCj and Qj ≫ 1, where
ωLCj = 1/
LjCj and Qj =
Lj/Cj/Rj. This choice
guarantees that the qubit mainly couples to resistor R1
(R2) at bias point 1 (2). The cooling cycle consists of
steps O, P, Q and R. First in step O the qubit has the
angular frequency ω2 and is allowed to thermalize. Be-
cause of the bandwidth limitations imposed by the reac-
tive elements, the qubit tends to thermalize with resistor
R2 to temperature T2. In the next step P the flux bias is
adiabatically changed to point 1 such that the level popu-
lations do not change but the energy eigenstates do. The
sweep is assumed to be however faster than relaxation.
In point 1 the angular frequency is reduced to ω1. Be-
cause the level populations and therefore the Boltzmann
factors do not change the qubit must now be at lower
temperature T̃2 given by T̃2 = T2ω1/ω2 in order to com-
pensate for the change of the qubit splitting. Note that
the quantum mechanical adiabaticity implies also ther-
modynamical adiabaticity: while the energy eigenbasis
changes the level populations and thus also entropy do
not change. In step Q the qubit is allowed to thermalize
to temperature T1 which results in heating of the qubit
and in cooling of resistor 1 if T̃2 < T1. At this point
the ideally pure quantum state of the qubit gets erased
and information stored is lost. The entropy of the qubit
increases, but locally the entropy of resistor 1 decreases
such that one can say that some information is “stored”
in the resistor as it cools but naturally with some loss.
Finally in step R the qubit is adiabatically shifted back
to frequency ω2 which results in heating of the qubit to
the effective temperature T̃1 = T1ω2/ω1 which is assumed
to be higher than T2. The excess energy is dumped to
admittance 2 when the cycle starts again from the be-
ginning. Note that due to the condition T̃2 < T1 resistor
1 can never be cooled below T2ω1/ω2. Since there is no
isothermal stage in the above cycle the present device
is not even in principle a Carnot cooler but rather an
Otto-type device.[10]
The density matrix of the qubit with the resonant an-
gular frequency ω at temperature T (β = (kBT )
−1) is
given by
ρeq(β, ε) =
. (2)
Using this the cooling power and the efficiency of the ideal
cycle in Fig. 1c can be easily calculated. It is given by
the area of the shaded region in the entropy-temperature
plane below points P and Q. In principle one could solve
for the effective temperature of the qubit along the line
between points P and Q as a function of entropy given
by S = −kBTr(ρ ln ρ). Alternatively, we can simply note
that the expectation value of the energy stored in the
qubit in point P is EP = Tr(ρeq(β2ω2/ω1, ε1)H1) while
after relaxation we have EQ = Tr(ρeq(β1, ε1)H1), where
H1 = H(ε1) is the Hamiltonian at point 1. We thus get
for the ideal cooling power
P/f = EQ − EP =
−β1~ω1
e−β1~ω1 + 1
− ~ω1e
−β2~ω2
e−β2~ω2 + 1
≤ ~ω1
where f is the pump frequency. The cooling power
achieves the maximum value of ~ω1f/2 when the ther-
mal population in step O (and P) is small and when
the population in step Q is large, i.e. when β2~ω2 ≫ 1
and β1~ω1 ≪ 1. Naturally a practical device has to be
designed to fulfill the first condition always, in which
case the smallest achievable temperature is on the or-
der of ~ω1/kB below which the cooling power decreases
rapidly. The dynamic range could be made wider by a
tunable ∆ which can be achieved by splitting the small-
est junction into a dc SQUID geometry. Another figure
of merit is the ratio η of the heat removed from resis-
tor 1 divided by the heat added to resistor 2. It can
be obtained as the ratio of the shaded area divided by
the sum of the hatched area and the shaded area, i.e.,
η = (EQ−EP )/(ER−EO) where EO = Tr(ρeq(β2, ε2)H2)
and ER = Tr(ρeq(β1ω1/ω2, ε2)H2). This simplifies neatly
to η = ω1/ω2 < 1 which is in harmony with the second
law of thermodynamics.
For more quantitative analysis we have to consider the
details of the relaxation rates due to the baths. The
Golden Rule transition rates due to resistor j are given
↓,↑ =
|〈0|dH/dΦ|1〉|2M2j S
I (±ωj)
M2j S
I(±ωj) (4)
where the positive sign corresponds to relaxation. The
total thermalization rate is Γ
↑ + Γ
↓. Here the
unsymmetrized noise spectrum is given by
I(ω) =
e−iωt〈δIj(0)δIj(t)〉dt
2~ωReYj(ω)
1− exp(−βj~ω)
. (5)
where ReYj(ω) = R
j /[1+Q
− ωLCj
)2] is the real
part of admittance of circuit j. The total relaxation rate
is thus
2(Ip∆Mj)
2 coth
1 +Q2j
− ωLCj
) . (6)
To model the behavior of the device we utilize the Bloch
master equation (see e.g Ref. [11]) given in our case by
~̇M = − ~B× ~M−Γ1th( ~M‖− ~MT1)−Γ2th( ~M‖− ~MT2)−Γ2 ~M⊥,
where ~M = Tr(~σρ) is the “magnetization” of the qubit,
and ~B = ∆~x + ε~z is the fictitious magnetic field. Note
however that the z-component of ~B and ~M do correspond
to real magnetic field and magnetization, respectively. In
Eq. (7) ~M‖ and ~M⊥ are the components of the magnetiza-
tion parallel and perpendicular to ~B, respectively. These
are explicitly
~M‖ =
(∆Mx + εMz)(∆~x + ε~z) (8)
~M⊥ =
ε2Mx−∆εMz
~x+My~y +
2Mz−∆εMx
~z. (9)
Here ~MT stands for the ε-dependent equilibrium mag-
netization of a qubit at temperature T given explicitly
~MT =
and Γ2 = (Γ
th + Γ
th)/2 + Γϕ is the dephasing rate. The
possibility of pure dephasing at the rate Γϕ has been
included. In the simulation we neglect pure dephasing
due to the intentionally large dominating thermalization
rate. Equation (7) describes relaxation towards instan-
taneous equilibrium with two competing rates due to
two different thermal baths. Equations of this type are
usually used in the stationary case, but for driving fre-
quencies slower than ∆/~ it should be also valid. As
is obvious from Eq. (7), the qubit actually tends to re-
lax towards an effective ε-dependent equilibrium mag-
netization (Γ1th
~MT1 + Γ
~MT2)/(Γ
th + Γ
th) at the rate
To illustrate the practical potential of the device we
show in Fig. 2 the simulated cooling power with si-
nusoidal driving of ǫ(t) compared to the ideal case
along with the actual loop in the entropy temperature
plane. The heat flow Pj from resistor j to the qubit
is simply obtained by integrating the product of the
thermalization rate and the energy deficit, i.e., Pj =
∫ 1/f
[Tr(ρeq(βj , ǫ(t))H)− Tr(ρ(t)H)] . The den-
sity matrix ρ(t) = 1
~M(t) · ~σ is solved numerically using
the Bloch equation (system is followed over a few periods
until it has converged to the limit cycle). We see that the
actual simulated behavior does not significantly deviate
at low f from the ideal behavior and that cooling pow-
ers on the order of fW can be achieved with reasonable
sample parameters. The oscillatory behavior at high f is
interpreted as Landau-Zener interference [12, 13].
However, the cooling power has to be compared with
realistic heat loads to evaluate the utility of the flux qubit
cooler. On one hand, resistor 1 is subject to heat load
0  ln 2
(a) (c)
(b) (d)
0 0.5 1 1.5
f (GHz)
0  ln 2
0 0.5 1 1.5
f (GHz)
FIG. 2: (color online) Example of the simulated cooling power
with ω1/2π = ∆/2π = 5 GHz (ǫ = 0 GHz), ω2/2π = 20.62
GHz (ǫ = 20 GHz), Q1 = Q2 = 10, ωj = ωLCj and
2(Ip∆Mj)
2/(Rj~ωj) = 20 × 10
9s−1. This can be achieved
e.g. with Ip = 200 nA, M1 = 29 pH, M2 = 59 pH and
R1 = R2 = 1 Ω. The driving is sinusoidal. (a) The solid
line illustrates the path in the T − S plane for the ideal cy-
cle described in the text while the dashed (dotted) line is a
result of simulation for f = 0.05 GHz (f = 1 GHz) with
T1 = T2 = 0.3 × ~ω2/kB ≈ 300 mK. (b) Simulated cooling
power vs. f for the same temperatures as in (a) is shown
with the dashed line while the solid line is the ideal result of
Eq. 3 (c-d) Same as (a-b) but with T1 = 0.5× T2 ≈ 150 mK.
The cooling threshold at 0.14 GHz in (d) is caused by finite
Q-factor.
from the phonons of the substrate on which the device
rests. On the other hand, resistor 2 should be coupled
well enough to phonon bath such that the unavoidable
work done on it does not raise T2 excessively. The heat
flow between the electron system of resistor j and the
phonon system is given by Pel−ph = ΣV (T
j −T 5ph) where
Vj is the volume of resistor j and Σ is typically on the
order of 109 Wm−3K−5. Thus resistor 1 needs to have a
sufficiently small volume while resistor 2 should be large
enough physically in order to serve as a heat sink. In ad-
dition the photonic heat conduction between the resistors
due to temperature gradient may in principle contribute
also. Following an analysis similar to Ref. [14], the heat
flow from admittance Y2(ω) to Y1(ω) can be written as
ReY1(ω)ReY2(ω)(n2(ω)− n1(ω))
where nj(ω) = [exp(βj~ω − 1)]−1 are the boson occu-
pation factors and M is the mutual inductance between
the loops. For detuned high-Q resonators the photonic
heat conduction turns out to be quite negligible. For
instance for the values of Fig. 3 with M = 5 pH and
R1 = R2 = 1 Ω we get only Pγ = 2 × 10−18 W even
0 0.5 1 1.5
f (GHz)
FIG. 3: (color online) Equilibrium temperature as a function
of pump frequency for three different phonon bath temper-
atures. The temperature of resistor 1 (volume 10−21m3) is
shown with dashed line while the temperature of resistor 2
(volume 10−18m3) is shown with solid line. The bath tem-
peratures Tph ≈ T2 from top to bottom are 0.3 × ~ω2/kB,
0.2 × ~ω2/kB and 0.1 × ~ω2/kB. Otherwise the parameters
are like in Fig. 2.
if T1 = 0 K and T2 = 300 mK. Figure 2 illustrates the
calculated equilibrium temperature versus operation fre-
quency obtained numerically by finding the balance be-
tween the dominating phononic heat conduction and the
integrated cooling power. We see that almost a factor of
2 reduction of T1 is possible with realistic parameters.
In practice the drop of T1 can be measured e.g. using
an additional SINIS thermometer, in which resistor 1 will
serve as the normal metal N. Its reading is sensitive to
the electronic temperature of N only, and self-heating can
be made very small. The resistors should be made out of
thin film normal metal such as copper or gold with typ-
ically sub 1 Ω square resistance. Volume can be picked
freely. To get the resonant frequencies and quality factor
as above we need L1 = 320 pH, C1 = 3.2 pF, L2 = 80
pH and C2 = 0.8 pF which are also realistic. For the
inductor one may use either Josephson or the kinetic in-
ductance of superconducting wire while the capacitance
values are similar to those in typical flux qubits [2]. To
satisfy the conditions of the above numerical example we
need quite large mutual inductances which however can
be easily achieved using e.g. kinetic inductance [15]. The
strong driving requires also rather large inductance be-
tween the microwave line and the qubit, which should not
result in uncontrolled relaxation. For instance, Mmw=5
pH coupling to the control line is acceptable as it would
result in at most 3 × 107 s−1 relaxation rate assuming a
50 Ω environment at 0.3 K. This choice will not degrade
the performance of the device significantly since driving
is much faster. Yet sufficiently strong driving can be
achieved with a modest 3 µA ac current. Fabrication
process will require most likely three lithography steps.
In conclusion, we have described a method of using a
superconducting flux qubit driven strongly at microwave
frequency to cool an external metal resistor. Here we con-
sidered LC resonators to achieve the required frequency
selectivity but a coplanar wave-guide resonator or a me-
chanical oscillator could be used in principle, too. We
demonstrated by a numerical example that it is possible
to observe the associated temperature decrease experi-
mentally. This effect is directly related to the loss of
information and thus to the increase of entropy of the
quantum bit.
J.P.P thanks NanoSciERA project ”NanoFridge” of
EU for financial support.
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mamoto, and J. S. Tsai, Phys. Rev. Lett. 93, 267007
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J.S. Tsai, Phys. Rev. Lett. 97, 167001 (2006).
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van der Wal, S. Lloyd, Science 285, 1036 (1999).
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E. Mooij, Science 299, 1869 (2003).
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Berggren, L. S. Levitov, and T. P. Orlando, Science 314,
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[8] J. Hauss, A. Fedorov, C. Hutter, A. Shnirman, and G.
Schön, cond-mat/0701041.
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Lett. 98, 037201 (2007).
[10] H.T. Quan, Y.X. Liu, C. P. Sun, and Franco Nori,
quant-ph/0611275.
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edited by R. Fazio, V. F. Gantmakher, and Y. Imry
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http://arxiv.org/abs/cond-mat/0701041
http://arxiv.org/abs/quant-ph/0611275
http://arxiv.org/abs/cond-mat/0309049

  We consider a design for a cyclic microrefrigerator using a superconducting
flux qubit. Adiabatic modulation of the flux combined with thermalization can
be used to transfer energy from a lower temperature normal metal thin film
resistor to another one at higher temperature. The frequency selectivity of
photonic heat conduction is achieved by including the hot resistor as part of a
high frequency LC resonator and the cold one as part of a low-frequency
oscillator while keeping both circuits in the underdamped regime. We discuss
the performance of the device in an experimentally realistic setting. This
device illustrates the complementarity of information and thermodynamic entropy
as the erasure of the quantum bit directly relates to the cooling of the
resistor.

Information entropic superconducting microcooler
A. O. Niskanen,1, 2 Y. Nakamura,1, 3, 4 and J. P. Pekola5
1CREST-JST, Kawaguchi, Saitama 332-0012,Japan
2VTT Technical Research Centre of Finland, Sensors, PO BOX 1000, 02044 VTT, Finland
3NEC Fundamental Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan
4The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
5Low Temperature Laboratory, Helsinki University of Technology, PO BOX 3500, 02015 TKK, Finland
(Dated: October 25, 2018)
We consider a design for a cyclic microrefrigerator using a superconducting flux qubit. Adiabatic
modulation of the flux combined with thermalization can be used to transfer energy from a lower
temperature normal metal thin film resistor to another one at higher temperature. The frequency
selectivity of photonic heat conduction is achieved by including the hot resistor as part of a high
frequency LC resonator and the cold one as part of a low-frequency oscillator while keeping both
circuits in the underdamped regime. We discuss the performance of the device in an experimentally
realistic setting. This device illustrates the complementarity of information and thermodynamic
entropy as the erasure of the quantum bit directly relates to the cooling of the resistor.
PACS numbers: 74.50.+r,85.80.Fi,03.67.-a
For the purpose of quantum computing, the coher-
ence properties of superconducting quantum bits (qubits)
should be optimized by decoupling them from all noise
sources as well as possible. However, many interesting
experiments can be envisioned also when the decoupling
is far from perfect. One such experiment closely related
to coherence optimization is using a qubit as a spectrom-
eter [1, 2, 3] for the environmental noise by monitoring
the effect of the environment on the quantum two-level
system. Here we focus on the opposite phenomenon, i.e.
the effect of a qubit on the environment. Recently a
superconducting flux qubit [4, 5] with a quite small tun-
neling energy from the point of view of quantum com-
puting was cooled using sideband cooling and a third
level [6] from about 400 mK down to 3 mK. Motivated
by this experiment we consider the possibility of using
a single quantum bit as a cyclic refrigerator for environ-
mental degrees of freedom. The utilized heat conduction
mechanism is photonic which was recently studied also
in experiment [7]. Besides the possible practical uses,
the device is interesting physically as it directly illus-
trates the connection between information entropy and
thermodynamical entropy. For related superconducting
high-frequency cooler concepts see eg. Refs. [8, 9].
Here we study a flux qubit coupled inductively to two
different loops shown in Fig. 1a. In loop j (j = 1, 2) we
have a resistor Rj in series with an inductance Lj and
a capacitance Cj . These form two damped harmonic os-
cillators. The resistors are in general at different tem-
peratures T1 and T2. The coupling of the qubit to these
two admittances Y1 and Y2 is assumed to be sufficiently
large to dominate the relaxation of the qubit. This as-
sumption can be easily validated by e.g. increasing the
mutual inductance. The flux qubit is an otherwise su-
perconducting loop except for three or four Josephson
junctions with suitably picked parameters. In particu-
FIG. 1: (color online) Principle of the flux-qubit cooler.
(a) Layout of the circuit. (b) Energy band diagram. (c)
Schematic of the cooling cycle in the qubit temperature-
entropy plane.
lar one of the junctions is made smaller than others to
form a two-level system. When biased close to half of the
flux quantum Φ0 = h/2e, the qubit can be described (in
persistent current basis) by the Hamiltonian
H/~ = −1
(∆σx + εσz) (1)
where σx and σz are Pauli matrices, ~ε = 2Ip(Φ−Φ0/2)
is the flux-tunable energy bias and Φ is the controllable
flux threading the qubit loop. Away from Φ = Φ0/2 the
eigenstates have the persistent currents ±Ip circulating
in the loop. The tunneling energy ~∆ results in an an-
http://arxiv.org/abs/0704.0845v1
ticrossing at Φ = Φ0/2 and there the energy eigenstates
do not carry average current. The resonant angular fre-
quency of the qubit is ω =
ε2 +∆2.
Consider the ideal cycle shown in Fig. 1b-c where the
bias of the flux qubit is swept slowly (slower than ∆/2π)
between two extreme values ε1 and ε2 corresponding to
two different energy level separations ~ω1 and ~ω2. Let
us further assume that ωj ≈ ωLCj and Qj ≫ 1, where
ωLCj = 1/
LjCj and Qj =
Lj/Cj/Rj. This choice
guarantees that the qubit mainly couples to resistor R1
(R2) at bias point 1 (2). The cooling cycle consists of
steps O, P, Q and R. First in step O the qubit has the
angular frequency ω2 and is allowed to thermalize. Be-
cause of the bandwidth limitations imposed by the reac-
tive elements, the qubit tends to thermalize with resistor
R2 to temperature T2. In the next step P the flux bias is
adiabatically changed to point 1 such that the level popu-
lations do not change but the energy eigenstates do. The
sweep is assumed to be however faster than relaxation.
In point 1 the angular frequency is reduced to ω1. Be-
cause the level populations and therefore the Boltzmann
factors do not change the qubit must now be at lower
temperature T̃2 given by T̃2 = T2ω1/ω2 in order to com-
pensate for the change of the qubit splitting. Note that
the quantum mechanical adiabaticity implies also ther-
modynamical adiabaticity: while the energy eigenbasis
changes the level populations and thus also entropy do
not change. In step Q the qubit is allowed to thermalize
to temperature T1 which results in heating of the qubit
and in cooling of resistor 1 if T̃2 < T1. At this point
the ideally pure quantum state of the qubit gets erased
and information stored is lost. The entropy of the qubit
increases, but locally the entropy of resistor 1 decreases
such that one can say that some information is “stored”
in the resistor as it cools but naturally with some loss.
Finally in step R the qubit is adiabatically shifted back
to frequency ω2 which results in heating of the qubit to
the effective temperature T̃1 = T1ω2/ω1 which is assumed
to be higher than T2. The excess energy is dumped to
admittance 2 when the cycle starts again from the be-
ginning. Note that due to the condition T̃2 < T1 resistor
1 can never be cooled below T2ω1/ω2. Since there is no
isothermal stage in the above cycle the present device
is not even in principle a Carnot cooler but rather an
Otto-type device.[10]
The density matrix of the qubit with the resonant an-
gular frequency ω at temperature T (β = (kBT )
−1) is
given by
ρeq(β, ε) =
. (2)
Using this the cooling power and the efficiency of the ideal
cycle in Fig. 1c can be easily calculated. It is given by
the area of the shaded region in the entropy-temperature
plane below points P and Q. In principle one could solve
for the effective temperature of the qubit along the line
between points P and Q as a function of entropy given
by S = −kBTr(ρ ln ρ). Alternatively, we can simply note
that the expectation value of the energy stored in the
qubit in point P is EP = Tr(ρeq(β2ω2/ω1, ε1)H1) while
after relaxation we have EQ = Tr(ρeq(β1, ε1)H1), where
H1 = H(ε1) is the Hamiltonian at point 1. We thus get
for the ideal cooling power
P/f = EQ − EP =
−β1~ω1
e−β1~ω1 + 1
− ~ω1e
−β2~ω2
e−β2~ω2 + 1
≤ ~ω1
where f is the pump frequency. The cooling power
achieves the maximum value of ~ω1f/2 when the ther-
mal population in step O (and P) is small and when
the population in step Q is large, i.e. when β2~ω2 ≫ 1
and β1~ω1 ≪ 1. Naturally a practical device has to be
designed to fulfill the first condition always, in which
case the smallest achievable temperature is on the or-
der of ~ω1/kB below which the cooling power decreases
rapidly. The dynamic range could be made wider by a
tunable ∆ which can be achieved by splitting the small-
est junction into a dc SQUID geometry. Another figure
of merit is the ratio η of the heat removed from resis-
tor 1 divided by the heat added to resistor 2. It can
be obtained as the ratio of the shaded area divided by
the sum of the hatched area and the shaded area, i.e.,
η = (EQ−EP )/(ER−EO) where EO = Tr(ρeq(β2, ε2)H2)
and ER = Tr(ρeq(β1ω1/ω2, ε2)H2). This simplifies neatly
to η = ω1/ω2 < 1 which is in harmony with the second
law of thermodynamics.
For more quantitative analysis we have to consider the
details of the relaxation rates due to the baths. The
Golden Rule transition rates due to resistor j are given
↓,↑ =
|〈0|dH/dΦ|1〉|2M2j S
I (±ωj)
M2j S
I(±ωj) (4)
where the positive sign corresponds to relaxation. The
total thermalization rate is Γ
↑ + Γ
↓. Here the
unsymmetrized noise spectrum is given by
I(ω) =
e−iωt〈δIj(0)δIj(t)〉dt
2~ωReYj(ω)
1− exp(−βj~ω)
. (5)
where ReYj(ω) = R
j /[1+Q
− ωLCj
)2] is the real
part of admittance of circuit j. The total relaxation rate
is thus
2(Ip∆Mj)
2 coth
1 +Q2j
− ωLCj
) . (6)
To model the behavior of the device we utilize the Bloch
master equation (see e.g Ref. [11]) given in our case by
~̇M = − ~B× ~M−Γ1th( ~M‖− ~MT1)−Γ2th( ~M‖− ~MT2)−Γ2 ~M⊥,
where ~M = Tr(~σρ) is the “magnetization” of the qubit,
and ~B = ∆~x + ε~z is the fictitious magnetic field. Note
however that the z-component of ~B and ~M do correspond
to real magnetic field and magnetization, respectively. In
Eq. (7) ~M‖ and ~M⊥ are the components of the magnetiza-
tion parallel and perpendicular to ~B, respectively. These
are explicitly
~M‖ =
(∆Mx + εMz)(∆~x + ε~z) (8)
~M⊥ =
ε2Mx−∆εMz
~x+My~y +
2Mz−∆εMx
~z. (9)
Here ~MT stands for the ε-dependent equilibrium mag-
netization of a qubit at temperature T given explicitly
~MT =
and Γ2 = (Γ
th + Γ
th)/2 + Γϕ is the dephasing rate. The
possibility of pure dephasing at the rate Γϕ has been
included. In the simulation we neglect pure dephasing
due to the intentionally large dominating thermalization
rate. Equation (7) describes relaxation towards instan-
taneous equilibrium with two competing rates due to
two different thermal baths. Equations of this type are
usually used in the stationary case, but for driving fre-
quencies slower than ∆/~ it should be also valid. As
is obvious from Eq. (7), the qubit actually tends to re-
lax towards an effective ε-dependent equilibrium mag-
netization (Γ1th
~MT1 + Γ
~MT2)/(Γ
th + Γ
th) at the rate
To illustrate the practical potential of the device we
show in Fig. 2 the simulated cooling power with si-
nusoidal driving of ǫ(t) compared to the ideal case
along with the actual loop in the entropy temperature
plane. The heat flow Pj from resistor j to the qubit
is simply obtained by integrating the product of the
thermalization rate and the energy deficit, i.e., Pj =
∫ 1/f
[Tr(ρeq(βj , ǫ(t))H)− Tr(ρ(t)H)] . The den-
sity matrix ρ(t) = 1
~M(t) · ~σ is solved numerically using
the Bloch equation (system is followed over a few periods
until it has converged to the limit cycle). We see that the
actual simulated behavior does not significantly deviate
at low f from the ideal behavior and that cooling pow-
ers on the order of fW can be achieved with reasonable
sample parameters. The oscillatory behavior at high f is
interpreted as Landau-Zener interference [12, 13].
However, the cooling power has to be compared with
realistic heat loads to evaluate the utility of the flux qubit
cooler. On one hand, resistor 1 is subject to heat load
0  ln 2
(a) (c)
(b) (d)
0 0.5 1 1.5
f (GHz)
0  ln 2
0 0.5 1 1.5
f (GHz)
FIG. 2: (color online) Example of the simulated cooling power
with ω1/2π = ∆/2π = 5 GHz (ǫ = 0 GHz), ω2/2π = 20.62
GHz (ǫ = 20 GHz), Q1 = Q2 = 10, ωj = ωLCj and
2(Ip∆Mj)
2/(Rj~ωj) = 20 × 10
9s−1. This can be achieved
e.g. with Ip = 200 nA, M1 = 29 pH, M2 = 59 pH and
R1 = R2 = 1 Ω. The driving is sinusoidal. (a) The solid
line illustrates the path in the T − S plane for the ideal cy-
cle described in the text while the dashed (dotted) line is a
result of simulation for f = 0.05 GHz (f = 1 GHz) with
T1 = T2 = 0.3 × ~ω2/kB ≈ 300 mK. (b) Simulated cooling
power vs. f for the same temperatures as in (a) is shown
with the dashed line while the solid line is the ideal result of
Eq. 3 (c-d) Same as (a-b) but with T1 = 0.5× T2 ≈ 150 mK.
The cooling threshold at 0.14 GHz in (d) is caused by finite
Q-factor.
from the phonons of the substrate on which the device
rests. On the other hand, resistor 2 should be coupled
well enough to phonon bath such that the unavoidable
work done on it does not raise T2 excessively. The heat
flow between the electron system of resistor j and the
phonon system is given by Pel−ph = ΣV (T
j −T 5ph) where
Vj is the volume of resistor j and Σ is typically on the
order of 109 Wm−3K−5. Thus resistor 1 needs to have a
sufficiently small volume while resistor 2 should be large
enough physically in order to serve as a heat sink. In ad-
dition the photonic heat conduction between the resistors
due to temperature gradient may in principle contribute
also. Following an analysis similar to Ref. [14], the heat
flow from admittance Y2(ω) to Y1(ω) can be written as
ReY1(ω)ReY2(ω)(n2(ω)− n1(ω))
where nj(ω) = [exp(βj~ω − 1)]−1 are the boson occu-
pation factors and M is the mutual inductance between
the loops. For detuned high-Q resonators the photonic
heat conduction turns out to be quite negligible. For
instance for the values of Fig. 3 with M = 5 pH and
R1 = R2 = 1 Ω we get only Pγ = 2 × 10−18 W even
0 0.5 1 1.5
f (GHz)
FIG. 3: (color online) Equilibrium temperature as a function
of pump frequency for three different phonon bath temper-
atures. The temperature of resistor 1 (volume 10−21m3) is
shown with dashed line while the temperature of resistor 2
(volume 10−18m3) is shown with solid line. The bath tem-
peratures Tph ≈ T2 from top to bottom are 0.3 × ~ω2/kB,
0.2 × ~ω2/kB and 0.1 × ~ω2/kB. Otherwise the parameters
are like in Fig. 2.
if T1 = 0 K and T2 = 300 mK. Figure 2 illustrates the
calculated equilibrium temperature versus operation fre-
quency obtained numerically by finding the balance be-
tween the dominating phononic heat conduction and the
integrated cooling power. We see that almost a factor of
2 reduction of T1 is possible with realistic parameters.
In practice the drop of T1 can be measured e.g. using
an additional SINIS thermometer, in which resistor 1 will
serve as the normal metal N. Its reading is sensitive to
the electronic temperature of N only, and self-heating can
be made very small. The resistors should be made out of
thin film normal metal such as copper or gold with typ-
ically sub 1 Ω square resistance. Volume can be picked
freely. To get the resonant frequencies and quality factor
as above we need L1 = 320 pH, C1 = 3.2 pF, L2 = 80
pH and C2 = 0.8 pF which are also realistic. For the
inductor one may use either Josephson or the kinetic in-
ductance of superconducting wire while the capacitance
values are similar to those in typical flux qubits [2]. To
satisfy the conditions of the above numerical example we
need quite large mutual inductances which however can
be easily achieved using e.g. kinetic inductance [15]. The
strong driving requires also rather large inductance be-
tween the microwave line and the qubit, which should not
result in uncontrolled relaxation. For instance, Mmw=5
pH coupling to the control line is acceptable as it would
result in at most 3 × 107 s−1 relaxation rate assuming a
50 Ω environment at 0.3 K. This choice will not degrade
the performance of the device significantly since driving
is much faster. Yet sufficiently strong driving can be
achieved with a modest 3 µA ac current. Fabrication
process will require most likely three lithography steps.
In conclusion, we have described a method of using a
superconducting flux qubit driven strongly at microwave
frequency to cool an external metal resistor. Here we con-
sidered LC resonators to achieve the required frequency
selectivity but a coplanar wave-guide resonator or a me-
chanical oscillator could be used in principle, too. We
demonstrated by a numerical example that it is possible
to observe the associated temperature decrease experi-
mentally. This effect is directly related to the loss of
information and thus to the increase of entropy of the
quantum bit.
J.P.P thanks NanoSciERA project ”NanoFridge” of
EU for financial support.
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