Nanoelectromechanics of superconducting weak links
Nanoelectromechanical effects in superconducting weak links are considered. Three different superconducting devices are studied: (i) a single-Cooper-pair transistor, (ii) a transparent SNS junction, and (iii) a single-level quantum dot coupled to superconducting electrodes. The electromechanical c...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1171172017-05-21T03:03:20Z Nanoelectromechanics of superconducting weak links Parafilo, A.V. Krive, I.V. Shekhter, R.I. Jonson, M. Квантовые когерентные эффекты в сверхпроводниках и новые материалы Nanoelectromechanical effects in superconducting weak links are considered. Three different superconducting devices are studied: (i) a single-Cooper-pair transistor, (ii) a transparent SNS junction, and (iii) a single-level quantum dot coupled to superconducting electrodes. The electromechanical coupling is due to electrostatic or magnetomotive forces acting on a movable part of the device. It is demonstrated that depending on the frequency of mechanical vibrations the electromechanical coupling could either suppress or enhance the Josephson current. Nonequilibrium effects associated with cooling of the vibrational subsystem or pumping energy into it at low bias voltages are discussed. 2012 Article Nanoelectromechanics of superconducting weak links / A.V. Parafilo, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2012. — Т. 38, № 4. — С. 348–359. — Бібліогр.: 56 назв. — англ. 0132-6414 PACS: 85.85.+j, 73.23.–b, 74.50.+r, 74.45.+c http://dspace.nbuv.gov.ua/handle/123456789/117117 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы Parafilo, A.V. Krive, I.V. Shekhter, R.I. Jonson, M. Nanoelectromechanics of superconducting weak links Физика низких температур |
| description |
Nanoelectromechanical effects in superconducting weak links are considered. Three different superconducting
devices are studied: (i) a single-Cooper-pair transistor, (ii) a transparent SNS junction, and (iii) a single-level
quantum dot coupled to superconducting electrodes. The electromechanical coupling is due to electrostatic or
magnetomotive forces acting on a movable part of the device. It is demonstrated that depending on the frequency
of mechanical vibrations the electromechanical coupling could either suppress or enhance the Josephson current.
Nonequilibrium effects associated with cooling of the vibrational subsystem or pumping energy into it at low
bias voltages are discussed. |
| format |
Article |
| author |
Parafilo, A.V. Krive, I.V. Shekhter, R.I. Jonson, M. |
| author_facet |
Parafilo, A.V. Krive, I.V. Shekhter, R.I. Jonson, M. |
| author_sort |
Parafilo, A.V. |
| title |
Nanoelectromechanics of superconducting weak links |
| title_short |
Nanoelectromechanics of superconducting weak links |
| title_full |
Nanoelectromechanics of superconducting weak links |
| title_fullStr |
Nanoelectromechanics of superconducting weak links |
| title_full_unstemmed |
Nanoelectromechanics of superconducting weak links |
| title_sort |
nanoelectromechanics of superconducting weak links |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117117 |
| citation_txt |
Nanoelectromechanics of superconducting weak links / A.V. Parafilo, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2012. — Т. 38, № 4. — С. 348–359. — Бібліогр.: 56 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT parafiloav nanoelectromechanicsofsuperconductingweaklinks AT kriveiv nanoelectromechanicsofsuperconductingweaklinks AT shekhterri nanoelectromechanicsofsuperconductingweaklinks AT jonsonm nanoelectromechanicsofsuperconductingweaklinks |
| first_indexed |
2025-07-08T11:40:39Z |
| last_indexed |
2025-07-08T11:40:39Z |
| _version_ |
1837078770702352384 |
| fulltext |
© A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4, pp. 348–359
Nanoelectromechanics of superconducting weak links
(Review Article)
A.V. Parafilo1, I.V. Krive1,2,3, R.I. Shekhter2, and M. Jonson2,4,5
1B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: parafilo_sand@mail.ru
2Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden
3Physical Department, V.N. Karazin National University, Kharkov 61077, Ukraine
4SUPA, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK
5Department of Physics, Division of Quantum Phases and Devices, Konkuk University, Seoul 143-701, Korea
Received December 23, 2011
Nanoelectromechanical effects in superconducting weak links are considered. Three different superconduct-
ing devices are studied: (i) a single-Cooper-pair transistor, (ii) a transparent SNS junction, and (iii) a single-level
quantum dot coupled to superconducting electrodes. The electromechanical coupling is due to electrostatic or
magnetomotive forces acting on a movable part of the device. It is demonstrated that depending on the frequency
of mechanical vibrations the electromechanical coupling could either suppress or enhance the Josephson current.
Nonequilibrium effects associated with cooling of the vibrational subsystem or pumping energy into it at low
bias voltages are discussed.
PACS: 85.85.+j Micro- and nano-electromechanical systems (MEMS/NEMS) and devices;
73.23.–b Electronic transport in mesoscopic systems;
74.50.+r Tunneling phenomena; Josephson effects;
74.45.+c Proximity effects; Andreev reflection; SN and SNS junctions.
Keywords: nanoelectromechanics, single-Cooper-pair box, short SNS junction, quantum dot.
Contents
1. Introduction .......................................................................................................................................... 348
2. Single-cooper-pair box and mechanically mediated Josephson currents .............................................. 349
3. Short vibrating SNS junctions: supercurrent and cooling of the mechanical subsystem ...................... 353
4. Polaronic effects in resonant Josephson current through a vibrating quantum dot ............................... 355
5. Conclusion ........................................................................................................................................... 357
References ................................................................................................................................................ 358
1. Introduction
The Josephson effect [1] is one of the most spectacular
phenomena in quantum physics. Two fundamental quan-
tum phenomena — macroscopic quantum coherence and
electron tunneling — determines the Josephson coupling of
separated superconductors. The theoretical prediction and
experimental observation [2] of the Josephson effect gave
birth to a new science — the superconductivity of weak
links.
In the last decade one has seen a rapid progress in the
formation and development of nanoelectromechanical sys-
tems (NEMS) which can be used as single-molecule mass
sensors, as the basic elements of nanoelectronics (single
electron transistor, relay, etc.) and as effective transducers
and signal processors. Recently, suspended carbon nano-
tube-based NEMS, which have been shown to have ex-
traordinary electrical and mechanical properties (see, e.g.,
[3]), have attracted special interest.
Theory predicts a number of new effects in NEMS that
are due to the interplay of their electronic and vibrational
subsystems. Among the theoretically predicted phenomena
electron shuttling [4,5], phonon assisted single-electron
tunneling [6,7] and the Franck–Condon (“polaronic”) blo-
ckade [8] have already been observed in experiments
[9,10].
Nanoelectromechanics of superconducting weak links
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 349
Single-wall carbon nanotubes have already been used as
weak links in superconducting devices [11] and there is no
doubt that transport properties of superconducting NEMS
will soon be measured. Therefore it is interesting and sig-
nificant to sum up our knowledge of this subject. This is
the aim of the present review.
We have considered three different types of supercon-
ducting NEMS: (i) a vibrating single-Cooper-pair box
coupled to superconducting electrodes, (ii) a transparent
(ballistic) SNS junction with a vibrating normal part (nano-
tube) of the device, and (iii) a vibrating single-level quan-
tum dot coupled by tunneling to superconducting banks.
Without the adjective “vibrating”, all these devices are
familiar systems in the field of superconductivity. The
electromechanical coupling gives them new and unusual
features. In particular, a single-Cooper-pair box could play
the role of a mediator of the Josephson coupling between
remote superconductors. The voltage-driven Andreev
states in a transparent short SNS junction serve as the re-
frigerant responsible for pumping energy from the nano-
tube vibrations to the thermostat of quasiparticle states in
the leads. A single-level vibrating quantum dot coupled to
superconducting leads, depending on the vibration fre-
quency, could either suppress the supercurrent (“hard”
vibrons: 0ω Δ , Δ is the superconducting gap) or even
to enhance the Josephson current if the vibrational subsys-
tem is “soft” 0( 0ω → ) and it is ready to transform to a
new ground state.
As far as we know the present paper is the first review
of the nanoelectromechanics of weak links. We considered
the influence of vibrations on the dc Josephson current at
zero or small bias voltages (adiabatic regime). The review
consist of an introduction, three sections and a conclusion.
In Sec. 2 we discuss the papers on mechanically mediated
Josephson currents. In Sec. 3 the supercurrent and cooling
effects in a vibrating nanotube in a magnetic field are con-
sidered. In Sec. 4 we study the influence of vibrations on
the resonant Josephson current in a superconductor–quan-
tum dot–superconductor tunnel junction.
2. Single-cooper-pair box and mechanically mediated
Josephson currents
The single-Cooper-pair box (SCPB) is a mesoscopic
device in which a small superconducting grain is coupled
bytunneling to a massive superconducting electrode via
Josephson junction and is capacitively connected to a gate
electrode. The Hamiltonian of the Cooper-pair box reads
[12]
2= ( ) cos ,C G JH E n n E− − ϕ (1)
where 2= (2 ) / 2CE e C is the charging energy ( C is the
grain capacity), 2= ( / 2 )J cE e I ( cI is the critical cur-
rent) is the Josephson energy and = / 2G Gn V C e ( GV is
the gate potential). The first term in Eq. (1) represents ki-
netic (electrostatic) energy, the second term is the potential
(Josephson) energy. In a quantum mechanical treatment
the canonically conjugate variables ( )tϕ and ( )n t obey
the commutation relation ˆ ˆ[ , ] = 1nϕ .
The single-Cooper-pair box is realized in the Coulomb
blockade regime [13] (see also [14]),
;C JE EΔ CT E (2)
( 2Δ is the superconducting gap) (and it is assumed that
junction resistance 2
0 /R R h e>> = ), where the single
electron states are energetically unfavorable due to the
parity effect (see, e.g., [15,16]) and the superconducting
properties of the grain are described by a two-level quan-
tum system (qubit) with (2 2)× matrix Hamiltonian (see,
e.g., the review [17])
( )1= .
2SCPB z J xH E− εσ + σ (3)
Here = (1 )C GE nε − , 0 1Gn≤ ≤ and σ are Pauli matric-
es. The eigenenergies of the Hamiltonian (3)
2 21= (1 2 )
2 C G JE E n E± ± − + (4)
are controlled by the gate voltage GV and at special values
of GV , when Coulomb blockade is lifted ( = 1/ 2Gn on
modulus 1), the level splitting = JE EΔ Δ is small and
the levels are well separated from the single electron (hole)
excitations.
The state vector of a SCPB is a coherent superposition
of the states with = 0n and = 1n Cooper pairs on the
grain, | = | 0 |1SCPBψ 〉 α 〉 +β 〉 ( 2 2| | | | = 1α + β ). Notice
that for a closed system any superposition of states with
different electric charges is forbidden by the charge con-
servation law (superselection rules). It means that actually
the SCPB state is entangled with the states of the lead
lead lead| = | 0 | |1 |SCPB ′ψ 〉 α 〉 ψ 〉 +β 〉 ψ 〉 . When the lead is
nonsuperconducting this entanglement results in decohe-
rence of the qubit. For a superconducting lead one has to
distinguish between states with a fixed number of Cooper
pairs | N〉 (strong fluctuations of the superconducting
phase) and states with a fixed superconducting phase | ϕ〉
(strong fluctuations of the number of Cooper pairs). In the
first case lead| = | Nψ 〉 〉 , lead| = | 1N′ψ 〉 − 〉 and the qubit is
characterized by a diagonal density matrix (mixed state)
0 1lead= Tr | | = | 0 0 | |1 1 | .SCPB SCPB SCPB p pρ ψ 〉〈ψ 〉〈 + 〉〈
(5)
The only possibility to create a coherent Josephson hy-
brid | SCPBψ 〉 on the grain is to “entangle” the neutral | 0〉
and the charged |1〉 states with a coherent state of the bulk
superconductor characterized by a fixed superconducting
phase | ϕ〉 . In this case the number of Cooper pairs in the
lead fluctuates strongly ( lead lead| = | = |′ψ 〉 ψ 〉 ϕ〉 ) and the
Josephson coupling of the superconducting grain with the
bulk superconductor creates, in the Coulomb blockade
A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson
350 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
regime defined by Eq. (2), the factorized state
| |SCPBψ 〉 ϕ〉⊗ .
The SCPB Hamiltonian is readily generalized to the
case when the superconducting grain is coupled by tunne-
ling to two superconducting electrodes with fixed phases
/ 2±ϕ . Such a system is called a superconducting single-
electron transistor (SSET) and is described by the Hamil-
tonian
2= ( ) cos ( /2 ) cos ( / 2 ).L R
SSET C G J JH E n n E E− − ϕ −φ − ϕ + φ
(6)
For a symmetric junction = =L R
J J JE E E the potential
energy in Eq. (6) is reduced to 2 cos ( / 2)cos ( )JE− ϕ φ and
the SSET can be described by the qubit Hamiltonian (3)
provided the Josephson energy JE is replaced by
2 cos ( / 2)JE ϕ (ϕ is the superconducting phase differ-
ence).
When the Coulomb blockade is lifted ( = 1/ 2Gn ) the
SSET eigenenergies are = cos( / 2)JE E± ϕ∓ and the cor-
responding eigenstates | = (| 0 |1 ) / 2±〉 〉+ 〉 can (when oc-
cupied) carry the partial supercurrents = (2 / ) / =J e E± ±∂ ∂ϕ
= ( / ) sin ( /2)JeE± ϕ . The average Josephson current
through a SSET takes the form
/
=
2= , = ln 1 e ,
E Tj
j
eJ T
−
±
∂Ω ⎛ ⎞Ω − +⎜ ⎟∂ϕ ⎝ ⎠∑ (7)
where Ω is the grand canonical potential. In equilibrium
at given temperature T and phase difference ϕ one finds
cos ( / 2)
= sin ( / 2) tanh .
2
J JeE E
J
T
ϕ⎡ ⎤ϕ ⎢ ⎥⎣ ⎦
(8)
This formula coincides with the expression for the resonant
Josephson current ([18,19], see also Sec. 4) through a sin-
gle-level quantum dot after the replacement JE →Γ
(Γ Δ , Γ is the level width). Therefore Eq. (8) could be
interpreted as resonant (“macroscopic”) tunneling of Coo-
per pairs through the charge hybrid states | ±〉 on the grain.
Correspondingly the critical current is determined by the
first power of the Josephson energy JE (and not
2R L
J J JE E E∝ ) and the dependence on the phase ϕ is dif-
ferent from the standard Josephson relation for nonreso-
nant electron tunneling sinL R
J JJ E E ϕ∼ .
A single-Cooper-pair box and qubit based on SCPB
were first experimentally realized [20,21]. Later, in the
experiment by Nakamura et al. [22], two additional nor-
mal-metal probe electrodes (a voltage-biased electrode and
a pulse-gate electrode) were attached to the superconduct-
ing island. This allows one to control the quantum states of
the superconductor-based two-level system. In particular
by applying a gate voltage pulse, coherent oscillations be-
tween two charge states (Rabi oscillations) were ob-
served [22].
A movable single-Cooper-pair box can be used for the
transportation of Cooper pairs. In the closed superconduct-
ing circuit shuttling of Cooper pairs results in a mechanical-
ly mediated Josephson current [23]. Even for disconnected
superconducting leads, when initially the superconductors
were in the states with fixed number of Cooper pairs,
Cooper pair shuttling creates long distance phase cohe-
rence [24] and induces a Josephson current through the
system.
What are the requirements one has to fulfill to observe
mechanically assisted Josephson current? It is evident that
phase coherence has to be preserved during the transporta-
tion of the Cooper-pair box between the leads and while
the SCPB interacts with the superconducting bulk elec-
trodes. Phase coherence can be maintained if the few de-
grees of freedom associated with the superconducting qubit
are well separated from the continuum spectrum and there-
fore the characteristic qubit phase coherence time is longer
than the transportation time. Notice that even for a nonmo-
bile SCPB and at low temperatures the “phase breaking
time” is short due to the interaction of the superconducting
qubit with the environment.
As in the case of a stationary SCPB the charging energy
CE should be larger then the Josephson energy JE and
the thermal energy, see Eq. (2). This condition prevents
significant charge fluctuations on the dot. Besides, the
energy quantum 0ω of the mechanical vibrations has to
be much smaller then all other energy scales of the prob-
lem, 0 , ,C JE Eω Δ . This assumption excludes the
creation of quasiparticles and allows one to consider the
mechanical transport of the SCPB as an adiabatic process.
The adiabatic shuttling of Cooper pairs between super-
conducting electrodes can be separated into two stages:
(i) the free motion of the Cooper-pair box, and (ii) the
loading and unloading of charge in the vicinity of leads.
The latter processes are induced by the Josephson coupling
and tunneling of Cooper pairs occur if the Coulomb block-
ade is lifted ( = 0) = ( = 1)E n E n by the appropriate gate
voltage applied near the contacts. The coherent exchange
of a Cooper pair between the grain and the lead creates a
“Josephson hybrid” on the grain:
00 01 11 10| 0 | 0 e |1 ; |1 |1 e | 0 ,
i ij js s s s
− ϕ ϕ
〉 ⇒ 〉 + 〉 〉 ⇒ 〉 + 〉 (9)
where jϕ is the superconducting phase of the left ( = )j L
or right ( =j R ) electrode (we assume that the leads are in
states with a given phase of the superconducting order pa-
rameter). The transition amplitudes ijs are determined by
the Josephson energy JE and the time ct spent by the
grain in contact with the lead
00 11 01 10| | = | | = cos , | | = | | = sin , / .J J J J cs s s s E tθ θ θ
(10)
During the free motion of the SCPB between the leads
the dynamics of the qubit is reduced to the evolution of
the relative phase χ of the = 0n and = 1n states
= ( = 1) ( = 0) = CE n E n Eχ − . The accumulated phases
Nanoelectromechanics of superconducting weak links
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 351
are in general different for left-to-right ( )t+ and right-to-
left ( )t− motion
/ .CE t± ±χ (11)
For a periodic adiabatic motion of the SCPB with fre-
quency 0= / 2f ω π the dc Josephson current was calcu-
lated in Ref. 23. It takes the form
3
22 2
cos sin (cos cos )sin= 2 ,
1 ( cos cos )cos sin
J J
J J
J ef
θ θ Φ Φ + χ
− θ χ − θ Φ
(12)
where = R L + −Φ ϕ −ϕ +χ −χ and = + −χ χ + χ is the total
dynamical phase. The current Eq. (12) is an oscillating
function of the superconducting phase difference
= R Lϕ ϕ −ϕ , which is a direct manifestation of a Joseph-
son coupling between the two remote superconductors.
It is useful to consider Eq. (12) in the limit of weak Jo-
sephson coupling 1Jθ and vanishingly small ( 0t± → )
dynamical phase 0χ → . In this case Eq. (12) is reduced to
the standard Josephson relation = sincJ J Φ , where
/c JJ eE . For weak coupling 1Jθ but a finite dy-
namical phase Jχ θ the mechanically assisted Joseph-
son current is strongly suppressed, 3 2/sinJ JJ ∝ θ χ θ .
However, the direction of the supercurrent will be opposite
to the direction of the “ordinary” Josephson current
( sinJ Φ∼ ) if cos cos < 0.χ + Φ So, for symmetric shuttl-
ing the main qualitative effect of the dynamical phase is
a change of the direction of the supercurrent. In other
words, for a given strength of the Josephson coupling the
direction of the supercurrent is determined by the interplay
of the superconducting (ϕ ) and dynamical (χ ) phases.
It is worth to stress three features, which distinguish a
mechanically assisted supercurrent from the ordinary Jo-
sephson current through weak links. (1) For asymmetrical
phase accumulation ( + −χ ≠ χ ) an anomalous current
( = 0) 0J ϕ ≠ flows through the system. (2) The direction
of the supercurrent at a given superconducting phase dif-
ference depends on the electrostatic phase χ . (3) The su-
percurrent is a nonmonotonic function of the Josephson
coupling strength Jθ . The last feature is connected with
the Rabi oscillations of the population of qubit quantum
states induced by a switching of the Josephson interaction
at the turning points of the shuttle trajectory.
Now we ask the following question: Could mechanical
transportation of Cooper pairs serve as a source for the
creation of phase coherence if the two superconducting
leads were initially in states with a definite number of
Cooper pairs? A positive answer to this question was given
in Ref. 24. The entanglement of the movable Cooper-pair
box with the lead states | jN 〉 ( = ,j L R ) with N extra
Cooper pairs results in the suppression of the relative phase
fluctuations. The manifestation of this phase ordering is the
appearance of an average supercurrent through the junc-
tion. Starting from an initially pure (product) state, the su-
percurrent
= Tr { }, = sin ( )c R LJ J J Jρ φ − φ (13)
(ρ is the density matrix obtained by tracing the density
matrix over the bath variables, jφ is the phase operator)
was shown [24] to stabilize at a fixed value after a large
number of grain rotations. The concrete number of rota-
tions needed for current stabilization depends on the
strength of the Josephson coupling. The calculated average
current [24] oscillates as a function of dynamical phase ±χ
similar to the previously discussed case of a mechanically
assisted Josephson current. Figure 1 illustrates the behavior
of the phase difference distribution as a function of the
number of grain rotations. The graph shows how the sys-
tem evolves (as the number of rotations increases) from a
state of maximum phase fluctuations (i.e., from an initial
pure charge neutral state | = | = 0 | = 0 | = 0 )L Rn N Nψ〉 〉 〉 〉
to an entangled state
, ,0
0,1 ,
| | |N n n N N L RL L R
n N NL R
C n N N+ +
=
δ 〉 〉 〉∑ ∑ (14)
with minimum quantum fluctuations. The corresponding
emergent phase difference depends on the parameters of
the system and the dynamical process. It was also shown
that the current is suppressed with increasing temperatures
due to an increased width of the phase distribution. At low
temperatures the current decays exponentially with a cros-
sover to algebraic decay for high temperatures.
The classical mechanical motion of a simple Cooper-
pair box can be modelled by letting the Josephson energy
and the charging energies be time dependent. Therefore the
properties of a voltage biased Josephson junction subject to
an ac gate potential ( ( )CE t ) could be useful for a better
0.8
0.6
0.4
0.2
0
0
0.5
1.0
1.5
2.0
10
1
10
2
10
3
10
4
Number of rotations
��
�/
Fig. 1. Probability density for the difference ΔΦ between the
phases of the superconducting condensates of two bulk supercon-
ductors in contact via a movable superconducting grain. The
grain periodically moves from one bulk superconductor to the
other and the graph shows how the system evolves from a state
with maximum to a state with minimum fluctuations as the num-
ber of rotation periods increases.
A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson
352 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
understanding of superconducting nanoelectromechanics.
In Ref. 25 the microwave dynamics and transport proper-
ties of a voltage-biased single Cooper pair transistor were
considered.
A bias voltage V makes the phase difference change
with time, 0( ) = Jt tϕ ω +ϕ ( = 2 /J eVω and we set
0 =ϕ −π for definiteness in what follows). This time de-
pendence generates a periodic variation of the relative po-
sition of the qubit levels E± , Eq. (4), as well as a periodic
change in the partial currents J± . The total Josephson cur-
rent depends on the relative “population” pδ of the levels
which for a finite bias ( 0Jω ≠ ) cross each other at times
= 2n Jt nω π . Consequently pδ is controlled now only by
relaxation processes. Weak relaxation Jν ω results in
equalization of the level population and hence the Joseph-
son current J p∝ δ ∝ ν . In the limit 0ν → not only the dc
but also the ac current through the considered weak link
vanishes. A time dependent gate voltage ( )Gn t ∝
cos ( )t∝ ω +χ may significantly affect the current by in-
ducing resonant interlevel transitions and thereby changing
their populations.
This situation was considered in [25] where the interlevel
transitions at resonance condition 2 sin ( /2)J J rE tω= ω
were used for current stimulation. Landau–Zener interlevel
transitions occur, and the level populations are changed,
during the short time interval rtδ near the times ( )n
rt . Be-
tween these “scattering events” the system is in “free mo-
tion” and the level populations are “frozen”. The two dif-
ferent frequencies = /2JΩ ω and ω , which determine the
ac properties of the superconducting qubit, make the dy-
namics of this two-level system highly nontrivial. If the
ratio /ω Ω is an irrational number the dynamics is quasi-
periodic and the dc Josephson current is still zero even
under microwave irradiation. When / = /N p qω Ω +
(where N and <p q are integers) a finite dc supercurrent
flows through the system even in the limit of weak dissipa-
tion 0ν → .
To obtain an analytical result one has to evaluate the dc
Josephson current, which in our case is given by the ex-
pression
0
1= Tr { ( ) ( )},
Tq
q
J dt t J t
T
ρ∫ ( ) = cos ,J
z
eE
J t tσ Ω (15)
where 0= = 2 /qT qT q π Ω , ( )tρ is a density matrix obey-
ing the Bloch equation
0= [ ( ), ] ( )i H t i
t
∂ρ
ρ − ν ρ−ρ
∂
(16)
and ( )H t is the Hamiltonian of the two-level system,
0( ) = sin ( ) cos ( ).z J xH t E t tσ Ω +σ ε ω + χ (17)
Here /ε ω is the rate of the microwave-induced inter-
level transitions and 0 ( )tρ is the quasistatic density matrix
of the unperturbed ( = 0ε ) Hamiltonian (17), and ν is the
relaxation rate.
The problem can be analytically solved in the weak dis-
sipation limit. The microwave-induced dc current takes the
form [25]
2
0
2 2 2
0
tan sin (2 )
= arccos ,
2 1 ( )cos
q
q
J
qeJ
E q
⎛ ⎞ τ θ χω ω
⎜ ⎟
π − τ χ⎝ ⎠
(18)
where ( ) = 2 ( ) 1 ( ) cosw wτ ε ε − ε θ is the probability of in-
terlevel transition ( ( )w ε is the probability amplitude of
the standard Landau–Zener scattering matrix [26]) and
/20 0
0 0
0
= sin ( /4 ) ,
T t
J
t
E
dt t T t
−
θ Ω −ω −∫
(19)
1
0 = arcsin .
2 J
t
E
− ω
Ω
This current plotted as a function of / = / eVω Ω ω de-
monstrates many sharp features at rational values of
/ eVω . The smooth dependence = ( / )J f ω Ω obtained
by an interpolation procedure is shown in Fig. 2. The sharp
peaks in the –I V characteristics of the microwave irra-
diated single Cooper pair transistor at “fractional” values
of the bias voltage, = / ( / )eV N p qω + , are a signature
of quantum interference effects caused by the resonant
interaction of the SSET with the microwave field. These
peaks are finite when N →∞ and do not vanish in the
limit of weak microwave irradiation 1τ . The calculated
–I V characteristics differ qualitatively from the Shapiro
effect (manifested as voltage steps in the –I V characteris-
tics of a voltage-biased Josephson junction in an ac field,
see, e.g., [25]) in classical Josephson junctions.
Fig. 2. Microwave-induced current J in units of 2
0 = /I eω π ,
plotted as a function the microwave frequency ω normalized to
= / 2JΩ ω . The result was obtained with 0 = / 11χ π ,
/ 2 = 0.001JEω , = 0.5w and max = 7q .
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
J
I/
0
10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5
���
Nanoelectromechanics of superconducting weak links
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 353
3. Short vibrating SNS junctions: supercurrent and
cooling of the mechanical subsystem
In this section we consider the nanoelectromechanical
properties of a suspended single-wall metallic carbon na-
notube coupled to superconducting electrodes. Depending
on the coupling strength this system can be modelled as a
vibrating quantum dot (weak coupling) or SNS junction
(strong coupling). The interaction of electronic and me-
chanical degrees of freedom in this device can be mediated
either by electrical charges or currents. The electrical for-
ces are most pronounced in the weak coupling (tunneling)
regime when the number of particles on the dot is a well-
defined quantity. This case will be analyzed in detail in the
next section. Here we consider electromechanical effects
induced by the interaction of a supercurrent with nanotube
vibrations in a magnetic field.
We will model the S/SWNT/S junction as a short SNS
junction (the length L of the suspended nanotube is as-
sumed to be much shorter than the superconducting cohe-
rence length, 0 /FL vξ Δ ). It is well known (see,
e.g., [19,27,28]) that in a one-dimensional (single channel)
short SNS junction the spectrum of Andreev bound states
is reduced to two states with energies ( ) = ( ),AE E± ϕ ± ϕ
2 1/2( ) = [1 sin ( /2)]AE Dϕ Δ − ϕ , where 0 1D≤ ≤ is the
junction transparency. When occupied these levels carry
supercurrents in opposite directions. The equilibrium Jo-
sephson current at temperature T reads
2 2
sin= tanh .
2 21 ( / 2)sin
AEe DJ
TD
Δ ϕ ⎛ ⎞
⎜ ⎟
⎝ ⎠− ϕ
(20)
The supercurrent produced by the continuum spectrum
(scattering states) is zero in the considered limit
0/ 0L ξ → . Since the continuum spectrum does not con-
tribute to the current the Hamiltonian of a short SNS junc-
tion can in many cases be represented by the Hamiltonian
of a two-level system (Andreev qubit [29,30])
= cos( / 2) sin( / 2),A z xH Rσ Δ ϕ + σ Δ ϕ (21)
where = 1R D− is the reflection coefficient. The super-
current operator is defined as = (2 / )( / )A zJ e E∂ ∂ϕ σ and
in equilibrium it results in the average Josephson current
Eq. (20). The qubit Hamiltonian Eq. (21) is readily diago-
nalized by the unitary transformation
†= = ( ) ,A A A zH U H U E ϕ σ = exp ( ),yU i− θσ
(22)
tan 2 = tan( / 2).Rθ ϕ
In the basis of Andreev levels the current operator takes
the form
2= ( cos 2 sin 2 ).A
z x
EeJ
∂
σ θ−σ θ
∂ϕ
(23)
Although the Hamiltonian (21) is formally valid for all
values of the reflection coefficient in the interval 0 1R≤ ≤
it is “in practice” used as a qubit Hamiltonian only for
transparent junctions with 1R . In this limit the energy
gap (at ϕ ≈ π ) between Andreev states is small,
= 2gE Rδ Δ Δ , and the energy levels are well sepa-
rated from the continuum states, which introduce dissipa-
tion in the dynamics of the Andreev qubit.
The electromechanical coupling in our system is phe-
nomenologically introduced by applying a magnetic field
H perpendicular to the direction of the current. It is as-
sumed that the magnetic field acts only on the normal part
of the SNS junction. Then the nanotube is deflected due to
the Laplace force = (1/ )F c LJH . The corresponding in-
teraction term int =H F y represents the coupling of elec-
trical and mechanical degrees of freedom. Vibrations of the
nanotube are modelled by a harmonic potential and the
total Hamiltonian reads
† †
0= 2 ( ) ,A
A z
dE
H H b b b b
d
+ α + σ + ω
ϕH (24)
where † ( )b b is the creation (destruction) operator of a
vibrational mode with frequency 0ω , 0 = /hc eΦ is the
flux quantum, 0= 2 /α πΦ ΦH H is the dimensionless
strength of electron–vibron interaction ( 0= LlΦH H ,
0 0= / 2l mω is the amplitude of zero-point vibrations).
Notice that the electron–vibron coupling results in an
effective electron–electron interaction and hence the con-
cept of Andreev qubit Eq. (21) (derived for noninteracting
electrons) is not valid in the general case. Besides, the inte-
raction term is L-independent while the level energies in
Eq. (21) were obtained in the limit 0L → . Therefore one
can justify Eq. (24) only in perturbation theory with re-
spect to the small parameter αH and we neglect the influ-
ence of the magnetic field on the level energies. Then the
current J can be expressed in terms of the unperturbed
energies of the Andreev levels. When additionally the SNS
junction is transparent ( 1R ) the derivative of the level
energies with respect to ϕ can be taken to be a constant,
2 ( = 0) / = sin( / 2)AE Rδ δϕ −Δ ϕ Δ at ϕ ≈ π . This mod-
el is the starting point of the considerations in Refs. 31, 32,
where a new mechanism for cooling the vibrational sub-
system was proposed.
The physical idea underlying the superconductivity-in-
duced electromechanical cooling mechanism is rather sim-
ple. When a bias voltage is applied over the SNS junction
the voltage-driven Andreev states play the role of a media-
tor responsible for pumping energy from the nanomechani-
cal vibrations to the quasiparticle states in the leads. The ac
Josephson dynamics of a short SNS junction induced by a
weak dc driving voltage 2= ( ) / = 4ceV eV E R≤ δ Δ Δ is de-
scribed by an adiabatic evolution of the Andreev states. At
the start of each cooling cycle the energy separation of the
Andreev levels, ( ) = 2 ( )AE Eδ ϕ ϕ , initially shrinks in such
a way as to bring them into thermal contact (at )ϕ π
with the vibrational subsystem ( =gEδ ω ) and resonant
energy exchange between electronic and vibronic degrees
A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson
354 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
of freedom occurs. Being thermally populated at the mo-
ment of their separation from the continuum spectrum, the
Andreev states are over-cooled at ϕ π if the thermal
relaxation is not sufficiently fast to follow the level dis-
placement. This work is done by the bias voltage. In the
vicinity of the resonance point = / 2rt eVπ interlevel
transitions with absorbtion and emission of vibronic excita-
tions take place. It is physically evident (and justified by
calculations) that scattering from the lower electronic
branch (–) into the upper (+) branch with the absorption of
a vibron after passing through the resonance is more prob-
able than scattering that involves vibron emission. Analo-
gously, inelastic transitions from the upper to the lower
branch occur mostly with the emission of a vibron. This
means that transitions between the over-cooled states in-
duced by the electromechanical coupling will result in the
absorption of energy from the vibronic subsystem. At the
second stage (ϕ π ) of the adiabatic evolution the ab-
sorbed energy is transferred into electronic quasiparticles
when the Andreev states merge with the continuum spec-
trum ( 2ϕ π ). The process of cooling is continued through
the formation of new thermally populated Andreev states
and their time evolution in the next cooling cycle and so on.
A calculation of the transition rates induced by the time
independent weak ( 1αH ) coupling term of the Hamil-
tonian (24) in the Andreev level basis eff =H
3 1= [ ( )]AE tϕ τ +α ΔτH ( iτ are the Pauli matrices) yields a
simple expression for the probability of inelastic scattering
| , | , 1n n− 〉 → + − 〉 with the absorption of a vibron ( n is the
number of vibrons) [32],
2 2/3
0( ) , = ( / )( / ) 1.cp n n V V+ π Γ Γ α Δ ωH (25)
As the opposite process | , | , 1n n+ 〉 → − + 〉 is forbidden if
initially only the lower branch is populated (T Δ ), the
mechanical subsystem would thus approach the vibrational
ground state. Figure 3 illustrates how the probability np for
the mechanical subsystem to have n excited quanta depends
on n after 1N cooling cycles. Initially np is thermally
distributed 0 0= exp ( / ) [1 exp ( / )]np n T T− ω − − ω and
after many periods ( 310N ∼ ) the vibrational subsystem if
effectively cooled down to a small final average vibron
population 0.1n〈 〉 ∼ .
In the end of this section we briefly discuss magnetic
field-induced superconducting pumping of nanomechani-
cal vibrations in a nanotube-based Josephson junction [33].
The interplay of elastic and superconducting properties of
S/suspended nanotube/S junction is provided by a magnet-
ic field H applied perpendicular to the nanotube. Then the
nanotube vibrations ( , )u x t are influenced by the Laplace
force = (1/ ) ( )LF c J LϕH acting on a current-carrying
tube of length L ( ( ) = sincJ Iϕ ϕ is the Josephson cur-
rent). The dynamics of the superconducting phase differ-
ence ϕ , controlled by the Josephson relation = (2 / )e Vϕ
(V is the bias voltage), is affected by the magnetic field
due to an electromotive force ( ( / ) ( , ))V V c dxu x t→ − ∫H
experienced by the wire moving in the static magnetic
field. The set of nonlinear dynamical equations for the am-
plitude ( )a t of vibrations ( ( , ) = ( ) ( )u x t u x a t , where ( )u x
is the profile of the nanotube bending mode) and the phase
( )tϕ in dimensionless variables reads [33]
= sin , = .Y Y Y є V Y+ γ + ϕ ϕ − (26)
Here ( ) = (4 / ) ( )Y t eL a tH , 0= 2 /V eV ω ( 0ω is the
frequency of the bending mode), 2 2 2
0= 8 /cє eL I mωH
and γ is the dimensionless damping coefficient which is
assumed to be small. The dimensionless time t in Eq. (26)
is measured in units of 1
0
−ω . The dc Josephson current
through the system is 2= ( / ) ( )dcj V a tγ 〈 〉 , where ...〈 〉 de-
notes time-averaged quantity. Numerical simulations of
Eq. (26) performed in [33] when both dimensionless para-
meters are small ( , 1є γ ) revealed distinct resonance
peaks in the vibration amplitude at integer values of bias
voltage. For small vibration amplitudes there is a resem-
blance between the considered resonances in the Josephson
junction coupled to elastic vibrations and the Fiske effect
(see, e.g., [34]) in Josephson junctions coupled to an elec-
tromagnetic resonator. In particular = 1V corresponds to a
direct resonance and = 2V represents a parametric reson-
ance. However, in the nonlinear regime, which holds when
the driving force is large >є γ , the resonances in the con-
sidered system are significantly different from those of the
Fiske effect. It was shown [33] that for realistic experimen-
tal parameters the system can be driven into a multistable
regime by varying the strength of magnetic field. The ac
Josephson current on resonance initially grows with in-
creasing magnetic field, but then falls off as 21/H as the
vibration amplitude is saturated. The predicted in [33] mul-
Fig. 3. (Color online). Evolution of the distribution of the me-
chanical modes, np , as a function of the quantum state n for
different number of periods N ( = / 20 nsVT eVπ ∼ ). Initially,
np is thermally distributed, 0exp ( / )np n T∝ − ω , with
0= 5T ω . Here 6
0 = 10−ω eV, 0= 10Δ ω , 0 = 20l pm,
= 100L nm, = 1H T. The inset shows the probability amplitude
for the system to scatter out of the initial Andreev state as a func-
tion of n for the same parameters.
0 5 10 15 20
0.05
0.10
0.15
0.20
0.25
0.30
n
N = 0
N = 1·10
2
N = 5·10
2
N = 2·10
2
N = 3·10
2
0 5 10 15 20 25
0.05
0.1
n
25
|
(
)|
�
2
n
2
p
n
1
1
2
2
3
3
44
55
Nanoelectromechanics of superconducting weak links
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 355
tistability of nanotube vibrations could result in hysteresis-
like behavior of dc Josephson current as a function of bias
voltage.
4. Polaronic effects in resonant Josephson current
through a vibrating quantum dot
In this section we consider the influence of a strong
electron–vibron interaction on the Josephson current. Re-
cently experiments with suspended carbon nanotubes re-
vealed a remarkably large electron–vibron coupling in
normal electron transport through nanotube-based quantum
dots [10,35–37]. In the cited transport experiments the vi-
brational effects were observed in the regime of Coulomb
blockade and the electromechanical coupling was induced
by the interaction of an extra charge on the vibrating tube
with the gate potential. Carbon nanotube-based junctions
have already been used in tunneling superconducting de-
vices [11]. Therefore one could expect strong electrome-
chanical effects in superconducting transport in SNS junc-
tions with suspended nanotubes as well.
Here we consider the simplest model of a vibrating
quantum dot (QD) coupled to superconducting electrodes
via tunneling junctions. The dot is modelled by a single
spin-degenerate ( = ,σ ↑ ↓ ) level interacting with a single
vibronic mode ( 0ω )
† † †
0 0
= ,
ˆ= ( ) ,QD iH d d Un n n b b b bσ σ ↑ ↓
σ ↑ ↓
ε + + ε + + ω∑ (27)
where †( )d dσ σ is the destruction (creation) operator for an
electron on the dot with spin projection = ,σ ↑ ↓ and ener-
gy 0ε measured from the Fermi level, †ˆ =n d dσ σ σ ,
ˆ ˆ ˆ=n n n↑ ↓+ , †( )b b is the vibronic destruction (creation)
operator, U is the energy of electron–electron interaction
and iε is the energy of electron–vibron interaction. Using
this model is a standard approach to studying vibrational
effects in single-molecule transistors (see, e.g., the reviews
[38,39]). In Ref. 40 this Hamiltonian was used for studying
the effects of electron–vibron interactions on the Joseph-
son current through superconductor–QD–superconductor
(S/QD/S) junction (see, e.g., [41]). The left ( = )j L and
right ( = )j R superconducting electrodes are described by
the standard BCS Hamiltonian
† † †
,
, = ,
= h.c.j k j k j jk j kj kj
k k
H c c c cσσ ↑ − ↓
σ ↑ ↓
⎛ ⎞
ε − Δ +⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑ (28)
( = | | e
i j
j
ϕ
Δ Δ is the superconducting order parameter),
and the QD–S coupling is described by the tunneling Ha-
miltonian
†
, = ,
= h.c.tj kj k j
k
H t c dσσ
σ ↑ ↓
+∑ (29)
The standard trick used in treating the Hamiltonian (27)
(see, e.g., [42]) is to eliminate the electron–vibron interac-
tion by the unitary transformation ˆ ˆ= exp ( )U i pnλ
†ˆ( = ( )/ 2p i b b− is the dimensionless momentum opera-
tor, 0= 2 /iλ − ε ω is the dimensionless electron–vibron
interaction strength). The transformation results in a (pola-
ronic) shift of the dot level, 2
0 0=pε ε − λ ω , and the
Coulomb interaction energy, 2
0= 2pU U − λ ω , in the
QD Hamiltonian. The electron–vibron interaction reap-
pears in the transformed tunneling Hamiltonian via the
replacement ei p
kj kjt t λ⇒ in Eq. (29). The average current
= =L RJ J J− is represented as the thermal average of the
tunneling Hamiltonian
†= ( / ) [ , ] = 2( / ) Im ,jj kj k j
k
J i e H N e t d c∗
σ σ
σ
∑ (30)
where H is the total Hamiltonian, jN is the number op-
erator for electrons on the left or right electrode and the
average ...〈 〉 is taken with the total Hamiltonian. In pertur-
bation theory with respect to the tunneling Hamiltonian the
averages of fermionic (electrons) and bosonic (vibrons)
operators factorize and can be evaluated analytically in
limiting cases (see below). The critical Josephson current
( ) = sincJ Iϕ ϕ (ϕ is the superconducting phase differ-
ence = R Lϕ ϕ −ϕ ) reads [40]
2
1 2 3 1 22
0 0 0
= ( )L R
c
e
I d d d
β β βΓ Γ Δ
− τ τ τ τ − τ ×
π ∫ ∫ ∫ H
3 1 2 3 1 2 3( ) ( , , ) ( , , ),× τ τ τ τ τ τ τH F B (31)
where =1/Tβ , 0 0( ) = (| |) (( | |) | |)K Kτ τΔ − β− τ ΔH 0( ( )K x is
a modified Bessel function), 2= 2 | | ( )j k kj kjtΓ π δ ε − εΣ
is the partial level width, which is energy independent in
the wide band approximation. The fermion and vibron cor-
relation functions are
† †
1 2 3 1 2 3( , , ) = { ( ) ( ) ( ) (0)} ,T d d d dτ ↓ ↑↓ ↑
τ τ τ 〈 τ τ τ 〉F
(32)
( )( ) ( ) 31 21 2 3( , , ) = {e e e e } ,i pi p i p i pT λ τ− λ τ − λ τ λ
ττ τ τ 〈 〉B
where now the averages are taken with the transformed QD
Hamiltonian (which is a quadratic in the vibron operators).
In the absence of electron–vibron ( = 0λ ) and Coulomb
( = 0U ) interactions an evaluation of the integrals in
Eq. (31) for 0| | Tε Δ and 0ε Δ results in the sim-
ple expression (see, e.g., [43])
2
0
0
= tanh .
2 2c
eI
T
εΓ ⎛ ⎞
⎜ ⎟ε ⎝ ⎠
(33)
The perturbative result (33) does not describe the resonant
transport 0 0ε → , 0T → . For noninteracting electrons a
nonperturbative (in Γ ) analysis of Eq. (31) predicts a satu-
ration of the critical resonant ( 0 = 0ε ) current /cI eΓ
at 0T → . The resonant supercurrent through a single-level
noninteracting QD can be calculated by using the spectrum
of Andreev levels in a short SINIS (“I” stands for insulat-
ing barrier) with strong barriers at the NS boundaries (see,
e.g., [18,19,41]). In our notation the spectrum of bound
A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson
356 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
states is 2 2 2
0( ) = ( / 2)cosAE ϕ ± ε + Γ ϕ and the corres-
ponding Josephson current reads
2 2 22
0
2 2 2
0
/ 2sin cos( ) = tanh .
2 2/ 2cos
eJ
T
⎛ ⎞ε +Γ ϕΓ ϕ ⎜ ⎟ϕ
⎜ ⎟ε +Γ ϕ ⎝ ⎠
(34)
In the perturbative region ( 0Γ → ) Eq. (34) reproduces the
critical current Eq. (33). For resonant transport 0( = 0)ε
the Josephson current = ( / 2 )(sin / | cos( / 2) |)rJ eΓ ϕ ϕ is
strongly enhanced.
A finite Coulomb interaction, 0U ≠ , tends to suppress
the Josephson current by splitting the dot level. If U Γ
the conditions for resonant tunneling can not be satisfied
and the critical current 2
cI Γ∼ . In the limit of strong Cou-
lomb interaction U →∞ (physically U Δ ) the critical
current can be evaluated analytically. Here we consider the
most interesting case, where 0,Γ ε Δ . Then for tempera-
tures T Δ and for 0| | >ε Γ the critical current takes the
form (up to a numerical factor of order one)
2
0 .cI
T
εΓ
Δ
(35)
We see that the supercurrent direction depends on the
sign of 0ε and for 0 < 0ε the considered superconducting
weak link acts as a “ π ”-junction [44]. The appearance in
Eq. (35) of an additional small factor 0| | / 1ε Δ in com-
parison with the analogous noninteracting expression,
Eq. (33), is explained by virtual depairing of Cooper pairs
in transition through a single spin-polarized electronic
level. At 0T → , 0 0ε → the critical current reads
2
0( / )( / ) sgn ( )cI e Γ Δ ε∼ . The current is strongly sup-
pressed (by “depairing” factor / 1Γ Δ ) in comparison
with the resonant critical current /eΓ∼ .
The electron–vibron interaction introduces an extra
energy scale, the vibron energy quantum 0ω , to the prob-
lem. It is clear that one could expect maximum effect of
zero-point fluctuations of dc Josephson current in the limit
when superconducting transport affects only the ground
state of the vibrational subsystem. In the case of strong
electron–electron interaction | |pU Δ the considered re-
gime is realized when 0ω Δ . For a weak effective
interaction | | 0pU → the corresponding inequality reads
0 0max { , }ω ε Γ . The bosonic correlation function
1 2 3( , , )τ τ τB can be expressed as exponential of the sum of
two-point correlation functions 2ˆ ˆ ˆ ˆ ˆ( ) = ( )p p p p p〈〈 τ 〉〉 〈 τ 〉 − 〈 〉
which are readily evaluated for equilibrated vibrons. At
low temperatures 0T → this correlation function in the
considered high-frequency limit does not depend on τ and
2exp ( 2 )− λB . This current suppression is known as the
Franck–Condon (polaronic) blockade of low-temperature
and low-voltage electron transport [8,38]. The additional
factor 2 in the exponent, compared to the normal transport
result, accounts for the correlated tunneling of two elec-
trons. In other words the Josephson current through a vi-
brating QD is strongly suppressed at low temperatures due
to a polaronic narrowing of the level width,
2= exp ( )λΓ ⇒ Γ Γ −λ . Contrary to the normal-transport
case, where the considered current suppression is absent
for resonant tunneling (when the conductance ceases to
depend on λΓ ) the Josephson current is suppressed by
zero-point fluctuations of the vibrating QD even for reson-
ance conditions.
This result is confirmed by a direct calculation [45] of
the resonant Josephson current through a single-level vi-
brating quantum dot. In particular the approach used in the
cited paper allows one to evaluate the resonant current for
an asymmetric S–QD–S junction ( L RΓ ≠ Γ ), which will
be important for us when considering the adiabatic regime
of vibrations (see below).
Since in the superconducting leads the quasiparticles
have a gap Δ in their excitation spectrum, the bulk fer-
mions can be integrated out, which leads to an effective
Hamiltonian for the dot degrees of freedom. For
0 0, , ,T bΓ ε ω Δ and = 0U the effective Hamiltonian
reads [45]
† †
eff 0 0
= ,
1= [ ( )]
2iH b b n b bσ
σ ↑ ↓
⎛ ⎞ε − ε + − + ω +⎜ ⎟
⎝ ⎠
∑
† ( cos / 2 sin / 2) ,t x yd d+ Γ σ ϕ + γσ ϕ (36)
where †† = ( , )d d d↓↑
, =t L RΓ Γ +Γ is the total level width
and = ( ) / ( )L R L Rγ Γ −Γ Γ +Γ is the asymmetry parame-
ter. It is clear that for superconducting transport in the con-
sidered regime Δ →∞ only two fermion states on the dot
are relevant: unoccupied | 0〉 and double occupied |↑↓〉
fermion level (in Eq. (36) the total energy was shifted so
that 0 0=E −ε , 0=E↑↓ ε ). In this basis (represented by jτ
Pauli matrices) Hamiltonian Eq. (36) after rotation
/2 /23 3eff eff= e ei iH H− τ χ τ χ , = arctan [ tan ( / 2)]χ γ ϕ takes
the form of the Hamiltonian for a two-level system (qubit)
interacting with harmonic oscillator
† †
eff 0 3 0= [ ( )]iH b b b b− ε − ε + τ + ω +
22 2
1 / 2 / 2.cos sint+ Γ τ ϕ + γ ϕ (37)
We analyze this model in the limit of a strongly asym-
metric junction 1γ → ± . The opposite case of a symmetric
junction ( 0, = =L Rγ → Γ Γ Γ ) results, as expected, in
a resonant supercurrent with a renormalized level width
2
= ( / ) e sin ( / 2) sgn [cos ( / 2)]rJ e −λΓ ϕ ϕ . For an asym-
metric junction the Josephson current at resonance reads
22
22 2
(1 ) e sin
= .
2 ( / 2) ( / 2)cos sin
teJ
−λ− γ Γ ϕ
ϕ + γ ϕ
(38)
We see that the maximum Josephson current flows in
symmetric junctions and that the supercurrent in a strongly
asymmetric ( ( ) ( )R L L RΓ Γ , i.e. 1γ ≈ ± ) junction is de-
termined (as it should be) by the smallest transparency of
Nanoelectromechanics of superconducting weak links
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 357
the barriers, 2
asym = (2 / )( / ) exp ( )sinL R tJ e Γ Γ Γ −λ ϕ .
In the considered limit of “hard” vibrons ( 0 0,ω Γ ε )
the vibration-induced suppression of current 2( exp ( ))−λ∼
does not depend on the asymmetry parameter γ .
It is worth to mention here that a novel type of Andreev
bound state spectroscopy based on a dispersive measure-
ment of polariton states on a quantum dot strongly coupled
to a bosonic subsystem (QED cavity) was suggested in
Ref. 46.
In the end of this section we briefly consider the oppo-
site (adiabatic) limit of “soft” vibrons 0 0ω → . In this
case the elastic energy associated with the vibrons is small
and the vibronic subsystem can be easily excited and trans-
formed to a new ground state ( 0x〈 〉 ≠ ) of an interacting
fermion–boson system. In the adiabatic limit the fast fer-
mionic degrees of freedom can be integrated out, resulting
in an effective nonquadratic potential effU for the vibrons.
When calculating the Josephson current one can distin-
guish two cases: (i) the level widths jΓ do not depend on
coordinate, and (ii) a shift of the oscillator-center-of-mass
strongly affects the tunneling rates /jΓ .
The last case can be realized for instance in a supercon-
ducting variant of the C60-based molecular transistor [9].
Notice that for normal transport the assumption (ii) could
result in electron shuttling [47]. For a dc Josephson effect
energy pumping in the vibrational subsystem is impossible
and, instead of shuttling, one could expect a static shift of
the center-of-mass of the vibrating QD if the dot displace-
ment will increase the supercurrent. As it was shown above
the maximum resonant supercurrent flows in a symmetric
( =L RΓ Γ ) junction. Therefore, if initially the QD posi-
tion in the gap between the superconducting electrodes
corresponds to an asymmetric junction (0) (0)
( ) ( )L R R LΓ Γ the
S–QD–S junction will nevetheless act as perfectly symme-
tric junction ( = ) = ( = )L m R mx x x xΓ Γ due to a shift of
the oscillator mx x→ for some values of phase difference
and the energy of resonant level.
A more subtle quantum effect is the appearance of a
new (shifted) quantum state of the vibrational subsystem
due to quantum fluctuations of the fermion vacuum. In
general, fermion loops (polarization “bubble” diagrams)
contribute negatively to the ground-state energy. It means
that for a sufficiently strong electron–vibron interaction the
classical ground state of vibrons, = 0,x becomes unstable
and a new minimum of the effective vibronic potential
appears.
In Ref. 48 it was shown by numerical calculations that
in the limit 0ω Δ Γ the effective potential eff ( )U x
for vibrons takes the form of an asymmetric double-well
potential in a certain region of phase-difference space ϕ
(the effective electron–vibron coupling depends on ϕ and
becomes strong, see Ref. 48). The frequency of vibrons
0′ω in the new (shifted) ground state is smaller then 0ω
and hence the effective dimensionless electron–vibron
coupling 3/2
0
−λ ω∼ is increased >′λ λ . Correspondingly,
the Josephson current is decreased. Naively, one would
expect the appearance of sharp features in the phase de-
pendence of the current at critical values of ϕ when the
vibronic system is shifted to a new ground state. Numerical
calculations performed in [48] for the case Γ Δ , when
continuum states strongly affect the current, revealed only
cusps in the = ( )J J ϕ dependence, which, however, could
be significant for the noise properties of S–QD–S junc-
tions.
Notice that in the regime of almost transparent junc-
tions (Γ Δ , 1λ ) the electron–vibron interaction can
be taken into account by a vibron-induced renormalization
of the junction transparency in n SNINS junction [49].
Scattering of tunneling electrons on the zero-point fluctua-
tions results in an effective transmission probability
2 2
eff 0= 1 / 8( / )T −λ ω Γ of the SNINS junction [19,49].
5. Conclusion
It is useful to compare vibrational effects in normal
metal and superconducting transport through a quantum
dot. If the bare tunneling matrix elements are coordinate-
independent quantities, the electron–vibron interaction
tends to suppress the electrical current (both normal and
superconducting) by “dressing” the tunneling electrons
with vibron excitations on the dot.
For normal electron transport the vibron-induced sup-
pression is most pronounced in the regime of sequential
electron tunneling (T Γ ) where the peak conductance
(at 0 ( ) = 0gVε ) scales as / TλΓ at low temperatures
0T ω with a renormalized (suppressed) tunneling
width
2
= e / ( )L R L R
−λ
λΓ Γ Γ Γ +Γ . In superconducting
transport we found an identical vibron-induced suppression
of the resonant Josephson current in the limit of “hard”
vibrons 0ω Δ . The Franck–Condon blockade (FCB)
of normal transport is manifested as an enhancement of the
satellite peaks and in the anomalous (nonmonotonic) tem-
perature dependence of the conductance at 0T ω [50].
For the Josephson current a partial lifting of the FCB could
be expected for “soft” vibrons 0Γ ω Δ in the tem-
perature region 0 Tω Δ . So far this interesting
problem has not been considered in the literature.
Another experimentally observed nanoelectromechani-
cal effect in normal electron transport through quantum
dots is electron shuttling. This phenomenon occurs at finite
bias voltage (in ideal situation at 0>eV ω ) when both the
electron–vibron interaction and a dependence of the tunne-
ling matrix elements on coordinate are taken into account
(see, e.g., the reviews [51,52]). Electron shuttling is a
strongly nonequilibrium process when energy from the
electrons (provided by the battery) is pumping into the
vibrational subsystem. For equilibrium superconducting
transport ( = 0V ), instead of electron shuttling one could
expect the transition of “soft” vibrons 0 0ω → to a new
ground state. The problem of Cooper pair shuttling through
A.V. Parafilo, I.V. Krive, R.I. Shekhter, and M. Jonson
358 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
a vibrating single-level quantum dot at finite bias voltages
is still an open question.
Superconductivity introduces two special features to
nanoelectromechanics, namely, coherence and the addi-
tional low-energy scale Δ . In our review we considered
coherent effects mostly associated with the electron trans-
port near the Fermi level. The peculiarities of vibrational
effects when continuum spectrum is involved in supercon-
ducting transport (although they are partly studied in the
literature, see, e.g., Refs. 48, 49), were not in the center of
our considerations. Notice also, that among a number of
papers on superconducting nanoelectromechanics where
the vibrational subsystem is modelled by external time-
dependent field (see, e.g., [53–56]) we reviewed only the
first publications.
Despite the fact that our review is brief and we could
not comment on all published papers in the field, we hope
that the first retrospective view on the already solved prob-
lems in nanoelectromechanics of weak links will induce
further interest both in theoreticians and experimentalists
to this new area of physics.
Acknowledgments
We thank L. Gorelik, S. Kulinich and V. Shumeiko for
valuable discussions. Financial support from the National
Academy of Science of Ukraine (grant No. 4/10-H ”Quan-
tum phenomena in nanosystems and nanomaterials at low
temperatures”), the Swedish VR, and the Korean WCU
program funded by MEST/NFR (R31-2008-000-10057-0)
is gratefully acknowledged. I.V.K. acknowledges the hos-
pitality of the Department of Physics at the University of
Gothenburg.
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