Voltage-Current curves for small Josephson junction arrays
Voltage-Current curves for small Josephson junction arrays
B. Douçot1 and L.B. Ioffe2
Laboratoire de Physique Théorique et Hautes Énergies, CNRS UMR 7589,
Universités Paris 6 et 7, 4, place Jussieu, 75252 Paris Cedex 05 France
Center for Materials Theory, Department of Physics and Astronomy,
Rutgers University 136 Frelinghuysen Rd, Piscataway NJ 08854 USA
We compute the current voltage characteristic of a chain of identical Josephson circuits charac-
terized by a large ratio of Josephson to charging energy that are envisioned as the implementation
of topologically protected qubits. We show that in the limit of small coupling to the environment
it exhibits a non-monotonous behavior with a maximum voltage followed by a parametrically large
region where V ∝ 1/I . We argue that its experimental measurement provides a direct probe of the
amplitude of the quantum transitions in constituting Josephson circuits and thus allows their full
characterization.
I. INTRODUCTION
In the past years, the dramatic experimental progress
in the design and fabrication of quantum two level sys-
tems in various superconducting circuits1 has raised a
hope that such solid state devices could eventually serve
as basic logical units in a quantum computer (qubits).
However, a very serious obstacle on this path is the ubiq-
uitous decoherence, which in practice limits the typical
life-time of quantum superpositions of two distinct log-
ical states of a qubit to microseconds. This is far from
being sufficient to satisfy the requirements for implement-
ing quantum algorithms and providing systematic error
correction.2
This has motivated us to propose some alternative
ways to design Solid-State qubits, that would be much
less sensitive to decoherence than those presently avail-
able. These protected qubits are finite size Josephson
junction arrays in which interactions induce a degener-
ate ground-state space characterized by the remarkable
property that all the local operators induced by couplings
to the environment act in the same way as the identity
operator. These models fall in two classes. The first
class is directly inspired by Kitaev’s program of topo-
logical quantum computation,3 and amounts to simulat-
ing lattice gauge theories with small finite gauge groups
by a large Josephson junction lattice.4,5,6 The second
class is composed of smaller arrays with sufficiently large
and non-Abelian symmetry groups allowing for a persis-
tent ground-state degeneracy even in the presence of a
noisy environment.7,8 All these systems share the prop-
erty that in the classical limit for the local superconduct-
ing phase variables (i.e. when the Josephson coupling is
much larger than the charging energy), the ground-state
is highly degenerate. The residual quantum processes
within this low energy subspace lift the classical degen-
eracy in favor of macroscopic coherent superpositions of
classical ground-states. The simplest example of such
system is based on chains of rhombi (Fig. 1) frustrated
by magnetic field flux Φ = Φ0/2 that ensures that in the
classical limit each rhombus has two degenerate states.8
Practically, it is important to be able to test these ar-
rays and optimize their parameters in relatively simple
experiments. In particular one needs the means to verify
the degeneracy of the classical ground states, the pres-
ence of the quantum coherent processes between them
and measure their amplitude. Another important pa-
rameter is the effective superconducting stiffness of the
fluctuating rhombi chain. The classical degeneracy and
chain stiffness can be probed by the experiments dis-
cussed in9; they are currently being performed10. The
idea is that a chain of rhombi threaded individually by
half a superconducting flux quantum, the non-dissipative
current is carried by charge 4e objects,11,12 so that the
basic flux quantum for a large closed chain of rhombi
becomes h/(4e) instead of h/(2e) which can be directly
observed by measuring the critical current of the loop
made from such chain and a large Josephson junction.
The main goal of the present paper is to discuss a prac-
tical way to probe directly the quantum coherence associ-
ated with these tunneling processes between macroscop-
ically distinct classical ground-states. In principle, it is
relatively simple to implement, since it amounts to mea-
suring the average dc voltage generated across a finite
Josephson junction array in the presence of a small cur-
rent bias (i. e. this bias current has to be smaller than
the critical current of the global system). The physi-
cal mechanism leading to this small dissipation is very
interesting by itself; it was orinally discussed in a sem-
inal paper by Likharev and Zorin16 in the context of a
single Josephson junction. Consider one element (single
junction or a rhombus) of the chain, and denote by φ
the phase difference across this element. When it is dis-
connected from the outside world, its wave-function Ψ
is 2πζ-periodic in φ where ζ = 1 for a single junction
and ζ = 1/2 for a rhombus. This reflects the quanti-
zation of the charge on the island between the elements
which can change by integer multiples of 2e/ζ. If φ is
totally classical, the element’s energy is not sensitive to
the choice of a quasi-periodic boundary condition of the
form Ψ(φ+ 2πζ) = exp(i2πζq)Ψ(φ), where q represents
the charge difference induced across the rhombus. In the
presence of coherent quantum tunneling processes for φ,
the energy of the element ǫ(q) will acquire q-dependence,
with a bandwidth directly related to the basic tunnel-
http://arxiv.org/abs/0704.0900v1
ing amplitude. Whereas q is constrained to be integer
for an isolated system, it is promoted to a genuine con-
tinuous degree of freedom when the array is coupled to
leads and therefore to a macroscopic dissipative environ-
ment. So, as emphasized by Likharev and Zorin16, the
situation becomes perfectly analogous to the Bloch the-
ory of a quantum particle in a one-dimensional periodic
potential, where the phase φ plays the role of the posi-
tion, and q of the Bloch momentum. A finite bias cur-
rent tilts the periodic potential for the phase variable, so
that in the absence of dissipation, the dynamics of the
phase exhibits Bloch oscillations, very similar to those
which have been predicted17 and observed18,19 for elec-
trons in semi-conductor super-lattices. If the driving cur-
rent is not too large, it is legitimate to neglect inter-band
transitions induced by the driving field, and one obtains
the usual spectrum of equally spaced localized levels of-
ten called a Wannier-Stark ladder. In the presence of
dissipation, these Wannier-Stark levels acquire a finite
life-time, and therefore the time-evolution of the phase
variable is characterized by a slow and uniform drift su-
perimposed on the faster Bloch oscillations. This drift is
translated into a finite dc voltage by the Josephson re-
lation 2eV = ~(dφ/dt). This voltage decreases with cur-
rent until one reaches the current bias high enough to
induce the interband transition. At this point the phase
starts to slide down fast and the junction switches into
a normal state. In the context of Josephson junctions
these effects were first observed in the experiments on
Josephson contacts with large charging energy20,21,22,23
and more recently24,25 in the semiclassical (phase) regime
of interest to us here. Bloch oscillations in the quantron-
ium circuit driven by a time-dependent gate voltage have
also been recently observed.26
This picture holds as long as the dissipation affecting
the phase dynamics is not too strong, so that the radia-
tive width of the Wannier-Stark levels is smaller than the
nearest-level spacing (corresponding to phase translation
by 2πζ) that is proportional to the bias current. This
provides a lower bound for the bias current which has
to be compatible with the upper bound coming from the
condition of no inter-band transitions. As we shall see,
this requires a large real part of the external impedance
Zω ≫ RQ as seen by the element at the frequency of the
Bloch oscillation, where the quantum resistance scale is
RQ = h/(4e
2). This condition is the most stringent in or-
der to access experimentally the phenomenon described
here. Note that this physical requirement is not lim-
ited to this particular experimental situation, because
any circuit exploiting the quantum coherence of phase
variables, for instance for quantum information process-
ing, has to be imbedded in an environment with a very
large impedance in order to limit the additional quantum
fluctuations of the phase induced by the bath. The intrin-
sic dissipation of Josephson elements will of course add to
the dissipation produced by external circuitry, but we ex-
pect that in the quantum regime (i.e. with sizable phase
fluctuations) considered here, this additional impedance
will be of order of RQ at the superconducting transition
temperature, and will grow exponentially below. Thus,
the success of the proposed measurements is also a test of
the quality of the environment for the circuits intended
to serve as protected qubits.
In many physical realizations Zω has a significant fre-
quency dependence and the condition Zω ≫ RQ is sat-
isfied only in a finite frequency range ωmax > ω > ωmin.
This situation is realized, for example, when the Joseph-
son element is decoupled from the environment by a
long chain of larger Josephson junctions (Section V).
In this case the superconducting phase fluctuations are
suppressed at low frequencies implying that a phase co-
herence and thus Josephson current reappears at these
scales. The magnitude of the critical current is however
strongly suppressed by the fluctuations at high frequen-
cies. This behavior is reminiscent of the reappearance
of the Josephson coupling induced by the dissipative en-
vironment observed in27. At higher energy scales fluc-
tuations become relevant, the phase exhibits Bloch os-
cillations resulting in the insulating behavior described
above. Thus, in this setup one expects a large hierarchy
of scales: at very low currents one observes a very small
Josephson current, at larger currents an almost insulat-
ing behavior and finally a switching into the normal state
at largest currents.
In the case of a chain of identical elements, the total
dc voltage is additive, but Bloch oscillation of different
elements might happen either in phase or in antiphase.
In the former case the ac voltages add increasing the
dissipation in the external circuitry; while in the latter
case the dissipation is low and the individual elements
get more decoupled from the environment. As we show in
Section III a small intrinsic dissipation of the individual
elements is crucial to ensure the antiphase scenario.
This paper is organized as follows. In section II, we
present a semi-classical treatment of the voltage versus
bias current curves for a single Josephson element. We
show that this gives an accurate way to measure the effec-
tive dispersion relation ǫ(q) of this element, which fully
characterizes its quantum transition amplitude. Further,
we show that application of the ac voltage provides a
direct probe of the periodicity (2π versus π) of each ele-
ment. In Section III we consider the chain of these ele-
ments and show that under realistic assumptions about
the dynamics of individual elements, it provides much
more efficient decoupling from the environment. Sec-
tion IV focusses on the dispersion relation expected in
a practically important case of a fully frustrated rhom-
bus which is the building block for the protected arrays
considered before.5,8 In this case, the band structure has
been determined by numerical diagonalizations of the
quantum Hamiltonian. An important result of this anal-
ysis is that even in the presence of relatively large quan-
tum fluctuations, the effective band structure is always
well approximated by a simple cosine expression. Finally,
in section V we discuss the conditions for the experimen-
tal implementation of this measurement procedure and
the full V (I) characteristics expected in realistic setup.
After a Conclusion section, an Appendix presents a full
quantum mechanical derivation of the dc voltage when
the bias current is small enough so that inter-band tran-
sitions can be neglected, and large enough so that the
level decay rate can simply be estimated from Fermi’s
golden rule.
II. SEMI-CLASSICAL EQUATIONS FOR A
SINGLE JOSEPHSON ELEMENT
Let us consider the system depicted on Fig. (1). In
the absence of the current source, the energy of the one
dimensional chain of N Josephson elements is a 2πζ pe-
riodic function of the phase difference φ across the chain.
The current source is destroying this periodicity by in-
troducing the additional term −~(I/2e)φ in the system’s
Hamiltonian. Because φ is equal to the sum of phase
differences across all the individual elements, it seems
that the voltage generated by the chain is N times the
voltage of a chain reduced to a single element. This is,
however, not the case: the individual elements are cou-
pled by the common load, and furthermore, as we show
in the next section, their collective behavior is sensitive
to the details of the single element dynamics. In this sec-
tion, we consider the case of a single Josephson element
(N = 1), rederive the results of Likharev and Zorin16 for
single Josephson contact and generalize them for more
complicated structures such as rhombus and give ana-
lytic equations convenient for data comparison.
The dynamics of a single Josephson contact is analo-
gous to the motion of a quantum particle (with a charge
e) in a one-dimensional periodic potential (with period
a) in the presence of a static and uniform force F , the
phase-difference φ playing the role of the spatial coordi-
nate x of the particle.16 In the limit of a weak external
force, it is natural to start by computing the band struc-
ture ǫn(k) for k in the first Brillouin zone [−π/a, π/a],
n being the band label. A first natural approximation is
to neglect interband transitions induced by the driving
field. This is possible provided the Wannier-Stark energy
gap ∆B = Fa is smaller than the typical band gap ∆ in
zero external field. As long as ∆B is also smaller than
the typical bandwidth W , the stationary states of the
Schrödinger equation spread over many (roughlyW/∆B)
periods, so we may ignore the discretization (i.e. one
quantum state per energy band per spacial period) im-
posed by the projection onto a given band. We may
therefore construct wave-packets whose spacial extension
∆x satisfies a≪ ∆x≪ aW/∆B, and the center of such a
wave-packet evolves according to the semi-classical equa-
tions:
dǫn(k)
F (2)
In the presence of dissipation, the second equation is
modified according to:
F − m
where m∗ is the effective mass of the particle in the n-th
band and τ is the momentum relaxation time introduced
by the dissipation.
FIG. 1: The experimental setup discussed in this paper: a
chain of identical building blocks represented by shaded rect-
angle that are biased by the external current source charac-
terized by the impedance Z(ω). The internal structure of the
block that is considered in more detail in the following sec-
tions is either a rhombus (4 junction loop) frustrated by half
flux quantum, or a single Josephson junction but the the re-
sults of the section II can be applied to any circuit of this
form provided that the junctions in the elementary building
blocks are in the phase regime, i.e. EJ ≫ EC .
In the context of a Josephson circuit, we have to diago-
nalize the Hamiltonian describing the array as a function
of the pseudo-charge q associated with the 2πζ periodic
phase variable φ. The quantity q controls the periodic
boundary condition imposed on φ, namely the system’s
wave-function is multiplied by exp(i2πq) when φ is in-
creased by 2πζ. From this phase-factor, we see that
the corresponding Brillouin zone for q is the interval
[−1/2, 1/2]. For a simple Josephson contact (ζ = 1), the
fixed value of q means that the total number of Cooper
pairs on the site carrying the phase φ is equal to q plus an
arbitrary integer. For a doubly periodic element, such as
rhombus (ζ = 1/2), charge is counted in the units of 4e.
To simplify the notations we assume usual 2π periodicity
(ζ = 1) in this and the following Sections and restore
the ζ-factors in Sections IV, V. From the band structure
ǫn(q), we may write the semi-classical equations of mo-
tion in the presence of the bias current I and the outer
impedance Z as:
dǫn(q)
where we used the Josephson relation for the voltage drop
V across the Josephson element as V = (~/2e)(dφ/dt)
and defined ZQ = ~/(4e
This semi-classical model exhibits two different
regimes. Let us denote by ωmax the maximum value of
the “group velocity” |dǫn(q)/(~dq)|. If the driving cur-
rent is small (I < Ic = 2eωmaxZQ/Z), it is easy to see
that after a short transient, the system reaches a station-
ary state where q is constant and:
that is: V = ZI. Thus, at I < Ic the current flows en-
tirely through the external impedance, i.e. the Joseph-
son elements become effectively insulating due to quan-
tum phase fluctuations. Indeed, a Bloch state writ-
ten in the phase reprentation corresponds to a fixed
value of the pseudo-charge q and non-zero dc voltage
(1/2e)(dǫn/dq). Note that the measurement of the maxi-
mal value Vc of the voltage on this linear branch directly
probes the spectrum of an individual Josephson block,
because Vc = ~ωmax/2e
At stronger driving (I > Ic), it is no longer possible
to find a stationary solution for q. The system enters
therefore a regime of Bloch oscillations. In the absence of
dissipation (Z/ZQ → ∞), the motion is periodic in time
for both φ and q. A small but finite dissipation preserves
the periodicity in q, but induces an average drift in φ or
equivalently a finite dc voltage. To see this, we first note
that the above equations of motion imply:
Since the right-hand side is a periodic function of q with
period 1, q(t) is periodic with the period T (I) given by:
T (I) =
∫ 1/2
f(q)dq (8)
f(q) =
On the other hand, the instantaneous dissipated power
reads:
ǫn(q)−
= −~ZQ
)2 (10)
Because q(t) is periodic, averaging this expression over
one period gives:
〉 = 2e
)2〉 (11)
or, equivalently:
〈V 〉 = ~
)2〉 (12)
Using the equations of motion, we get more explicitely:
〈V 〉 = 1
) ∫ 1/2
−1/2(
)2f(q)dq
∫ 1/2
−1/2 f(q)dq
Here we emphasized by the subscript that Zω might have
some frequency dependence. As we show in Appendix,
the dissipation actually occurs at the frequency of Bloch
oscillations that becomes ωB = 2πI/2e in the limit of
large currents. In the limit of large currents, I ≫ Ic,
(that can be achieved for large impedances) we may ap-
proximate f(q) by a constant, so the voltage is given by
the simpler expression:
〈V (I ≫ Ic)〉 =
4e2ZωI
∫ 1/2
)2dq (14)
On the other hand, when I approaches Ic from above,
Bloch oscillations become very slow and f(q) is strongly
peaked in the vicinity of the maximum of the group ve-
locity. Since this velocity is in general a smooth function
of q, we get in this limit for the maximal dc voltage:
2ω2max
4e2Z0Ic
= Z0Ic (15)
0 1 2 3 4 5
0 1 2 3 4 5
FIG. 2: Typical I − V curve of a single Josephson element
measured by a circuits shown in Fig. 1
In the simplest case of a purely harmonic dispersion,
ǫ(q) = 2w cos 2πq, the maximal voltage Vc = 4πw/(2e).
If one can further neglect the frequency dependency of
Z, the V (I) can be computed analytically:
〈V 〉 = ZI I < Ic (16)
〈V 〉 = ZIc
I2 − I2c
I > Ic (17)
We show this dependence in Fig. 2. This expression (16),
(17) is related to the known result for Z ≪ ZQ13,14 by
the duality15 transformation:
V → I,
I → V,
Z → 1
The semi-classical approximation is valid when the os-
cillation amplitude of the superconducting phase is much
larger than 2π, which allows the formation of the semi-
classical wave-packets. When I is much larger than Ic,
this oscillation amplitude is equal to 2eW/~I, whereW is
the total band-width of ǫn(q). This condition also ensures
that the work done by the current source when the phase
increases by 2π is much smaller than the band-width. In
order to observe the region of negative differential resis-
tance, corresponding to the regime of Bloch oscillations,
we require therefore that:
2π~Ic
≪W ≃ 2eVc
, (18)
where the last equality becomes exact in the case of a
purely harmonic dispersion. This translates into:
Z ≫ RQ. (19)
For large currents one can compute dc voltage di-
rectly by using the golden rule (without semiclassics);
we present the results in Appendix A. The result is con-
sistent with the large I limit of Eq. (17), 〈V 〉 = V 2c /2ZI.
Deep in the classical regime (EJ ≫ EC), the bandwidth
and the generated voltage become exponentially small.
In this regime the bandwidth is much smaller than the
energy gaps, so these formulas are applicable (asuming
(19) is satisfied) until the splitting between Wannier-
Stark levels becomes equal to the first energy gap given
by the Josephson plasma frequency, i.e. for I < eωJ/π.
Upon a further increase of the driving current in this
regime the generating voltage experiences resonant in-
crease for each splitting that is equal to the energy gap:
Ik = e(Ek−E0)/π. Physically, at these currents the phase
slips are rare events that lead to the excitation of the
higher levels at a new phase value that are followed by
their fast relaxation. At very large energies, the band-
width of these levels becomes larger than their decay rate
due to relaxation, (RQ/Z)EC . At these driving currents,
the system starts to generate large voltage and switches
to a normal state. At a very large EJ this happens at
the driving currents very close to the Josephson criti-
cal current 2eEJ , but in a numerically wide regime of
100 & EJ/EC & 10 the generated voltage at low curents
is exponentially small but switching to the normal state
occurs at significantly smaller currents than 2eEJ .
In the intermediate regime where EJ and EC are com-
parable, we expect a band-width comparable to energy
gaps so that the range of application of the quantum
derivation is not much larger than the one for the semi-
classical approach.
Negative differential resistance associated to Bloch os-
cillations has been predicted long ago,28 and observed
experimentally29 in the context of semi-conductor su-
perlattices. For Josephson junctions in the cross-over
regime (EJ/EC ≃ 1), a negative differential resistance
has been observed in a very high impedance environ-
ment,24 in good agreement with earlier theoretical pre-
dictions.30 More recently, the I − V curve of the type
shown on Fig 2 have been reported on a junction with
a ratio EJ/EC = 4.5
25. These experiments show good
agreement with a calculation which takes into account
the noise due to residual thermal fluctuations in the
resistor.31
Although the above results allows the extraction of the
band structure of an individual Josephson block from the
measurement of dc I−V curves, the interpretation of ac-
tual data may be complicated by frequency dependence
of the external impedance Zω. Additional information
independent on Zω can be obtained from measuring the
dc V (I) characteristics in the circuit driven by an ad-
ditional ac current. In this situation, the semi-classical
equations of motion become:
dǫn(q)
I + I ′ cos(ωt)
A small ac driving amplitude I ′ strongly affects the V (I)
curve only in the vicinity of resonances where nωB(IR) =
mω, with m and n integers. The largest deviation occurs
for m = n = 1. Furthermore, for I ′ ≪ I the terms
with m > 1 are parametrically small in I ′/I while for
I ≫ Ic the terms with n > 1 are parametrically small
in Ic/I. Experimental determination of the resonance
current, IR, would allow a direct measurement of the
Bloch oscillation frequency and thus the periodicity of the
phase potential (see next Section). Observation of these
mode locking properties have in fact provided the first
experimental evidence of Bloch oscillations in a single
Josephson junction.20,21
We now calculate the shape of V (I) curve in the vicin-
ity ofm = n = 1 point when both I ′ ≪ I and I ≫ Ic. We
denote by φ0(t) and q0(t) the time-dependent solutions
of the equations at I = IR in the absence of ac driving
current. We shall look for solutions which remain close
to φ0(t) and q0(t) at all times and expand them in small
deviations φ1 = φ − φ0, q1 = q − q0. We can always
assume that q1 has no Fourier component at zero fre-
quency because such component can be eliminated by a
time translation applied to q0. The equations for φ1, q1
become
ǫ′′n(q0)q1 (22)
I − IR + I ′ cos(ωt)
Because the main component of
d2ǫn(q0)
oscillates with
frequency ω and q1 has no dc component, the average
value of the voltage
is due to the part of q1 that os-
cillates with the same frequency, q1ω = I
′/(2eω) sin(ωt).
Because q0 = ω(t−t0)+χ(ω(t−t0)) where χ(t) is a small
periodic function, the first equation implies that
> = 〈
ǫ′′n(ω(t− t0)) sin(ωt)〉
ǫ′′n(q) cos(2πq)dq
The deviation q1 remains small only if the constant parts
cancel each other in the right hand side of the equation
(23). This implies
I − IR
ǫn(q) cos(2πq)dq (24)
We conclude that in the near vicinity of the reso-
nances the increase of the current does not lead to addi-
tional current through the Josephson circuit, so the re-
lation between current and voltage becomes linear again
δV = ZδI. In other words, the Josephson circuit be-
comes insulating with respect to current increments. The
width of this region (in voltage) is directly related to the
first moment of the energy spectrum of the Josephson
block providing one with the direct experimental probe
of this quantity. In particular, a Josephson element such
as rhombus in a magnetic flux somewhat different from
Φ0/2 displays a phase periodicity 2π but a very strong
deviations from a simple cos 2πq spectrum that will man-
ifest themselves in first moment of the spectrum. Note
finally, that the discussion above assumes that the ex-
ternal impedance Zω has no resonances in the important
frequency range. The presence of such resonances will
modify significantly the observed V (I) curves because it
would provide an efficient mechanism for the dissipation
of Bloch (or Josephson) oscillations at this frequency.
III. CHAIN OF JOSEPHSON ELEMENTS
We shall first consider the simplest example of a two-
element chain, because it captures the essential physics.
This chain is characterized by two phase differences (φ1
and φ2) and two pseudo-charges (q1 and q2). The equa-
tions of motion for the pseudo-charges (5) implies that
the charge difference q1− q2 is constant, because the cur-
rents flowing through these elements are equal, and thus
the right-hand sides of the evolution equations (5) are
identical. Because of this conservation law, even the long-
term physical properties depend on the initial conditions.
Similar problems have already been discussed in the con-
text of a chain of Josephson junctions driven by a current
larger than the critical current.32,33,34,35 This unphysical
behavior disappears if we take into account the dissipa-
tion associated with individual elements. Physically, it
might be due to stray charges, two-level systems, quasi-
particles, phonon emission, etc.36,37 A convenient model
for this dissipation is to consider an additional resistor
in parallel with each junction. For the sake of simplicity,
we assume that each element has a low energy band with
a simple cosine form. This physics is summarized by the
equations:
φ̇j = 4πw sin 2πqj (25)
q̇j =
I − 1
φ̇i −
Eliminating the phases gives:
(q̇1 +Ω1 sin 2πq1) = ν −
(sin 2πq1 + sin 2πq2)
(q̇2 +Ω2 sin 2πq2) = ν −
(sin 2πq1 + sin 2πq2)
where
(2e)2Ri
(2e)2Z
Here we allowed for different effective resistances asso-
ciated with each element because this has an important
effect on their dynamics. Indeed the difference between
the currents flowing through the resistors changes the
charge accumulated at the middle island and therefore
violates the conservation law mentioned before. Using
the notations δΩ = (Ω2 − Ω1)/2 and q± = (q2 ± q1)/2,
we have:
˙q− +Ωsin 2πq− cos 2πq++
+δΩcos 2πq− sin 2πq+ = 0 (27)
˙q+ + (ν0 +Ω) sin 2πq+ cos 2πq−−
−δΩ sin 2πq− cos 2πq+ = ν (28)
Significant quantum fluctuations imply that internal re-
sistance of the element R ∼ ZQ for individual elements
at T . TC ; at lower temperature it grows exponen-
tially. Thus, in a realistic case R ≫ Z which implies that
Ωi ≪ ν. In the insulating regime the equations (27-28)
have stable stationary solution (ν0 +Ω) sin 2πq+ = ν,
q− = 0. This solution exists for (ν0 +Ω) < ν , i.e. if
the voltage drop across both junctions does not exceed
Vc = 8πw/(2e). The conducting regime occurs when
ν > (ν0 +Ω); to simplify the analytic calculations we
assume that ν ≫ ν0. This allows to solve the equations
(27-28) by iterations in all non-linear terms. In the ab-
sence of non-linearity q+ = νt , q− = const; the first
iteration gives periodic corrections ∝ cos 2πνt. Averag-
ing the result of the second order iteration over the period
we get
˙〈q−〉 = −
ν0 cos
2 2πq− + 2Ω
The second term in the right hand side of this equation
is much smaller than the first if Ω ≪ ν0. In its absence
the dynamics of q− has fixed points at cos 2πq− = 0. At
these fixed points the periodic potentials generated by in-
dividual elements cancel each other and the dissipation in
external circuitry (which is proportional to cos2(2πq−))
is strictly zero. In a general case the equation (29) has
solution
cos2(2πq−) =
1 + ν0+2Ω
2Ω(ν0 + 2Ω)t
that corresponds to the short bursts of dissipation in
external circuitry that occur with low frequency νb =
2Ω(ν0 + 2Ω). The average value of cos
2(2πq−)
< cos2(2πq−) >=
ν0+2Ω
ν0 + 2Ω
is small implying that the effective dissipation introduced
by the external circuitry is strongly suppressed because
the pseudocharge oscillations on different elements al-
most cancel each other. The effective impedance of the
load seen by individual junction is strongly increased:
Zeff =
ν0 + 2Ω
Z (30)
Similar to a single element case discussed in the previous
Section, an additional dissipation in the external circuit
implies dc current across the Josephson chain
V = Vc
I ≫ Ic = Vc/Zeff
We conclude that a chain of Josephson elements has a
current-voltage characteristics similar to the one of the
single element with one important difference: the effec-
tive impedance of the external circuitry is strongly en-
hanced by the antiphase locking of the individual Joseph-
son elements. In particular, it means that the condition
Z ≫ RQ is much easier to satisfy for the chain of the
elements than for a single element. The analytical equa-
tions derived here describe the chain of two elements but
it seems likely that similar suppression of the dissipation
should occur in longer chains.
To substantiate this claim, lets us generalize the av-
eraging method which led to Eq. (29) for N = 2. The
coupled equations of motion read:
q̇j +Ωj sin 2πqj = ν −
sin 2πqk (31)
To second order in Ωj and ν0, the averaged equations of
motion are:
〈q̇j〉 = −
cos(2π(qk − ql))
− ν0Ωj
cos(2π(qj − qk)) (32)
This set of coupled equations is similar to the Kuramoto
model for coupled rotors38 defined as:
q̇j = ωj −
sin(2π(qj − qk) + α) (33)
The equation of motion (33) exhibits synchronisation of
a finite fraction of the rotors only for K > Kc(α).
39,40
The last term in Eq. (32) is equivalent to the interac-
tion term of Kuramoto model with α = π/2. The ad-
ditional (third) term in the model (32) is the same for
all oscillators, it is thus not correlated with individual
qj and thus can not directly lead to their synchroni-
sation. Remarkably, it turns out that for model (33)
Kc(α = π/2) = 0
39,40, suggesting that in our case, syn-
chronization never occurs on a macroscopic scale. Note
that the coupling K arising from Eq. (32) is not only j-
dependent, but it is also proportional to N . This could
present a problem in the infinite N limit, but should not
present a problem in a finite system. It is striking to see
that α = π/2 is the value for which synchronization is
the most difficult.
IV. ENERGY BANDS FOR A FULLY
FRUSTRATED JOSEPHSON RHOMBUS
In order to apply general results of the previous sec-
tion to the physical chains made of Josephson junctions
or more complicated Josephson circuits we need to com-
pute the spectrum of these systems as a function of the
pseudocharge q conjugated to the phase across these ele-
ments. In all cases the superconducting phase in Joseph-
son devices fluctuates weakly near some classical value
φ0 where the Josephson energy has a minimum in the
limit EJ/EC ≫ 1. In the vicinity of the minimum, the
phase Hamiltonian is H = −4EC d
E′′(φ0)(φ−φ0)2,
so a higher energy state of the individual element (at a
fixed q) can be approximated by one of the oscillator
En = (n +
)ωJ where the Josephson plasma frequency
8E′′(φ0)EC ≈
8EJEC . The Josephson en-
ergy is periodic in the phase with the period 2π but the
amplitude of the transitions between these minima is ex-
ponentially small:
w = a~ωJ(EJ/EC)
1/4 exp(−c
EJ/EC)
where a, c ∼ 1. In this limit one can neglect the contribu-
tion of the excited states (separated by a large gap ωJ )
to the lower band, so the low energy spectrum acquires
a simple form ǫ(q) = 2w cos 2πq. The numerical coeffi-
cients c, a in the formulae for the transition amplitude
depend on the element construction. For a single junc-
tion as = 8 2
π , cs =
8 while for the rhombus
ar ≈ 4.0 , cr ≈ 1.61. In case of the rhombus in mag-
netic field with flux Φ0/2 the Hamiltonian is periodic in
phase with period π provided that the rhombus is sym-
metric along its horizontal axis: indeed in this case the
combination of the time reversal symmetry and reflec-
tion ensures that the Josephson energy has a minimum
for φ0 = ±π/2. Thus, in this case the period in q dou-
bles and the low energy band becomes ǫ(q) = 2w cosπq.
The maximal voltage generated by the chain of N such
elements at I = Ic = (8πζew/~)(ZQ/Z) is
Vc = N
The voltage generated at larger currents depend on the
collective behaviour of the elements in the chain. For a
single element it is simply
〈V (I)〉 = (2πζw)
I2 − I2c
For more than one element the total volage is sufficiently
reduced due to the antiphase correlations. Generally, one
expects that
〈V (I)〉 = N (2πζw)
I2 − I2c
, (35)
where Zeffω is the effective impedance of the environment
affecting each Josephson element which is generally much
larger than its ’bare’ impedance Zω. For two elements the
exact solution (see previous Section) gives Zeffω ≈
that shows the increase of the effective impedance by a
large factor
R/Z. We expect that a similar enhance-
ment factor appears for all N & 2. Finallly, For I < Ic,
the system is ohmic with:
〈V (I)〉 = Z0I (36)
As discussed in Section II, application of a small addi-
tional ac voltage produces features on the current-voltage
characteristics for the currents that produce Bloch oscil-
lation with frequencies commensurate with the frequency
of the applied ac field ωB = 2πζI/2e = (m/n)ω. At these
currents the system becomes insulating with respect to
current increments, the largest such feature appears at
m = n = 1 that allows a direct measurement of the
Josephson element periodicity.
For smaller EJ/EC ∼ 1 the quasiclassical formulas
for the transition amplitudes do not work and one has
to perform the numerical diagonalization of the quan-
tum system in order to find its actual spectrum. As
EJ/EC → 1 the higher energy band approaches the
low energy band and the dispersion of the latter de-
viates from the simple cosine form shown in Figure 3.
These deviations, lead to higher harmonics in the dis-
persion: ǫ(q) = 2w cos 2πζq + 2w′ cos 4πζq and change
the equations (34,35). Our numerical diagonalization of
a single rhombus shows, however, that even at relatively
small EJ/EC ∼ 1 the second harmonics w′ does not ex-
ceed 0.15w, so its additional contribution to the voltage
current characteristic (∝ w′2) can always be neglected.
Thus, in the whole range of EJ/EC > 1 the voltage cur-
rent characteristic is given by Eqs. (34,35) where the ef-
fective value of transition amplitude t can be found from
the band width W = E1 − E0 = 4w plotted in Fig. 3.
For comparison we show the variation of the lower band
width for a single junction in Fig. 4
0 2.5 5 7.5 10 12.5 15
EJ/EC
(E1-E0)/EC
(E2-E0)/EC
0 0.1 0.2 0.3 0.4
E2/EC
E1/EC
E0/EC
EJ/EC=4
FIG. 3: Spectrum of a single rhombus biased by magnetic flux
Φ = Φ0/2. The upper pane shows the bands of the rhombus
characterized by Josephson eneergy EJ/EC = 4 as a function
of bias charge, q. The two lower levels are fitted by the first
two harmonics (dashed line), the coefficient w′ of the second
harmonics is w′ = 0.1w. One observes period doubling of the
first two states that reflects the symmetries of the rhombus
frustrated by a half flux quantum. The second excited level is
doubly degenerate that makes its period doubling difficult to
observe. Physically, these states correspond to an excitation
localized on the upper or lower arms of the rhombus. The
lower pane shows the dependence of the gaps for q = 0 as a
function of EJ/EC . Because higher order harmonics are very
small for all EJ/EC > 1, the gap E1 − E0 coincides with 4w
where w is the tunneling amplitude between the two classical
ground states.
V. PHYSICAL IMPLEMENTATIONS
Generally, the effects described in the previous sections
can be observed if the environment does not affect much
the quantum fluctuations of individual elements and the
resulting quasiclassical equations of motion. These physi-
cal requirements translate into different conditions on the
impedance of the environment at different frequencies.
We begin with the quantum dynamics of the individual
elements. The effect of the leads impedance on it can
be taken into account by adding the appropriate current
term to the phase equation of motion before projecting
on a low energy band and requiring that their effect on
the phase dynamics is small at the relevant frequencies.
For instance, for a single junction
= E′J (φ) +
0 1 2 3 4
EJ/EC
FIG. 4: Band width W = 4w of a single Josephson junction
The characteristic frequency of the quantum fluctuations
responsibe for the tunneling of a single element is Joseph-
son plasma frequency, ωJ =
8EJEc, so the first condi-
tion implies that
|Z(ωJ)| ≫
Ec/EJZQ (37)
For a typical ωJ/2π ∼ 10GHz, the impedance of a sim-
ple superconducting lead of the length ∼ 1cm is smaller
than ZQ and the condition (37) is not satisfied. The
situation is changed if the Josephson elements are decou-
pled from the leads by a large resistance or by a chain
of M ≫ 1 large junctions with
ẼJ/Ẽc ≫ 1 that has
no quantum tunneling transitions of their own (the am-
plitude of such transitions is ∝ exp(−
8ẼJ/Ẽc ). As-
suming that elements of this chain have no direct ca-
pacitive coupling to the ground (M2C0 ≪ C), the chain
has an impedance Z =
8Ẽc/ẼJMZQ at the relevant
frequencies, so a realistic chain with M ∼ 50 junctions
8ẼJ/Ẽc ∼ 10 provides the contribution to the
impedance needed to satisfy (37).
Similar decoupling from the leads of the individual el-
ements can be achieved by a sufficiently long chain of
similar Josephson elements, e.g. rhombi. Consider a
long (N ≫ 1) chain of similar elements connected to
the leads characterized by a large but finite capacitance
Cg ≫ C. For a short chain the tunneling of a single
element changes the phase on the leads resulting in a
huge action of the tunneling process. However, in a long
chain of junctions, a tunneling of individual rhombus may
be compensated by a simultaneous change of the phases
δφ/N of the remaining rhombi, and subsequent relax-
ation of δφ from its initial value π towards the equilib-
rium value which is zero. For N ≫ 1, this later process
can be treated within the Gaussian approximation, with
the Lagrangian (in imaginary time):
where Eg = e
2/(2Cg). So the total action involved in the
relaxation is: S = π
If this action S is less
than unity this relaxation has strictly no effect on the
tunneling amplitude of the individual rhombus.
We now turn to the constraints imposed by the qua-
siclassical equations of motion. The solution of these
equations shows oscillation at the Bloch frequency that
is ωB = 2πζI/(2e) for large currents and approaches
zero near the Ic. Thus, for a single Josephson ele-
ment the quasiclassical equations of motion are valid if
Re(RQ/Z(ωB)) ≪ 1 . A realistic energy band for a
Josephson element, W ∼ 0.3K and Z/ZQ ∼ 100 cor-
respond to Bloch frequency ωB/2π ∼ 0.1GHz . In this
frequency range a typical lead gives a capacitive contri-
bution to the dynamics. The condition that it does not
affect significantly the equations of motion implies that
the lead capacitance C . 10fF . As discussed in Section
(III) the individual elements in a short chain oscillate in
antiphase decreasing the effective coupling to the leads
by a factor
R/Z where R is the intrinsic resistance
of the contact. This factor can easily reach 102 at suf-
ficiently low temperatures making much less restrictive
the condition on the lead capacitance.
Large but finite impedance of the environment
Re(RQ/Z(ωB)) . 1 modifes the observed current-voltage
characteristics qualitatively, specially in the limit of very
small driving current. When I vanishes, and with infi-
nite external impedance, the wave function of the phase
variable is completely extended, with the form of a Bloch
state, and the pseudo-charge q is a good quantum num-
ber. As discussed at the end of Sec. II, the system be-
haves as a capacitor. But when the external impedance
is finite, charge fluctuations appear, which in the dual
description means that quantum phase fluctuations are
no longer unbounded. To be specific, consider a realistic
example of N rhombi chain (or two ordinary junctions)
attached to the leads with Z(ω) = Z0 in a broad but fi-
nite frequency interval ωmin < ω < ωmax and decreases as
Z(ω) = Z0(ωmax/ω) for ω > ωmax, Z(ω) = Z0(ω/ωmin)
for ω < ωmin. Such Z(ω) is realized in a long chain
of M Josephson junctions between islands with a finite
capactive coupling to the ground C0: ωmax = ωJ and
ωmin = (
C/C0/M)ωJ . The effective action describ-
ing the phase dynamics across the chain has contribu-
tions from the tunneling of individual rhombi and from
impedance of the chain
Ltot =
8π2ζ2Nw
Here the first term describes the effect of the tunneling of
the Josephson element between its quasiclassical minima
which we approximate by a single tunneling amplitude
w resulting in a spectrum ǫ(q) = −2w cos 2πζq that in a
Gaussian approximation becomes ǫ(q) = 4π2ζ2wq2. This
approximation is justified by the fact that, as we show
below, the main effect of the phase fluctuations comes
from the broad frequency range where the action is dom-
inated by the second term while the first serves only as
a cutoff of the logarithmical divergence. Its precise form
is therefore largely irrelevant.
This action leads to a large but finite phase fluctuations
8π2ζ2Nw
min(ωmax, ω
where ω′max = 8π
2ζ2Nw(ZQ/Z0). These fluctuations
are only logarithmically large, so they result in a finite
renormalization of the Josephson energy of the rhombi
chain and the corresponding critical current. In the ab-
sence of such renormalization the Josephson energy of a
finite chain of elements can be approximated by the lead-
ing harmonics E(φ) = −E0 cos(φ/ζ) with E0 ∼ EJ for
N ∼ 1 and EJ & Ec. Renormalization by fluctuations
replaces E0 by
ER = exp(−
)E0 =
min(ωmax, ω
]− Z0
In the limit of ωmin → 0 or Z0 → ∞ the phase fluc-
tuations renormalize Josephson energy to zero. But for
realistic parameters this suppression of Josephson energy
is finite which thus results in a small but non-zero value of
the critical current. In this situation the current-voltage
characterictics sketched in Fig. 1 is modified for very
small values of currents and voltages: instead of insulat-
ing regime at very low currents and voltages one should
observe a very small supercurrent (ER/2e) followed by a
small voltage step as shown in Fig. 2 by a dashed line.
As is clear from the above discussion the value of the
resulting critical current is controlled by the phase fluc-
tuations at low ω ≪ ωmax; these frequencies are much
smaller than the typical internal frequencies of a chain of
Josephson elements which can be thus lumped together
into an effective object characterized by the bare Joseph-
son energy E(φ) and transition amplitude between its
minima w. We thus expect the same qualitative behav-
ior for a small chain of Josephson elements as for a single
element at low currents.
VI. CONCLUSION
The main results of the present work are the expres-
sions (34), (35) for the I-V curves of a chain of N identi-
cal basic Josephson circuits. They are derived within the
assumption that the Josephson coupling is much larger
than the charging energy, but in fact, the numerical cal-
culations show that they remain very accurate even if
EJ ≈ EC . These equations predict a maximum dc volt-
age when I = Ic and V (I) ∝ 1/I for I ≫ Ic. The anoma-
lous V versus I dependence exhibited by these equations
is a signature of underdamped quantum phase dynamics.
It occurs only if the impedance of the external circuitry
is sufficiently large both at the frequency of Bloch oscil-
lations and at the Josephson frequency of the individual
elements. The precise conditions are given in Section V.
Observation of this dependence and the measurement of
the maximal voltage would provide the proof of the quan-
tum dynamics and the measurements of the tunneling
amplitude which is the most important characteristics of
these systems. It would also provide a crucial test of the
quality of decoupling to the environment.
As a deeply quantum mechanical system, the chain
of Josephson devices is very sensitive to an additional
ac driving. It exhibits resonances when the driving
frequency is commensurate with the frequency ωB =
2πζI/2e of the Bloch oscillations. This would provide
additional ways to characterize the quantum dynamics
of these circuits and confirm the period doubling of the
rhombi frustrated by exactly half flux quantum.
Acknowledgments
LI is thankful to LPTMS Orsay, and LPTHE Jussieu
for their hospitality through a financial support from
CNRS while BD has enjoyed the hospitality of the
Physics Department at Rutgers University. This work
was made possible by support from NSF DMR-0210575,
ECS-0608842 and ARO W911NF-06-1-0208.
APPENDIX A: QUANTUM-MECHANICAL
CALCULATION OF THE DC VOLTAGE
In the large current regime where I ≥ Imax, the energy
drop ∆B = hI/2e induced by the driving current when
φ increases by 2π becomes comparable to or larger than
the bandwidth W . In this regime, the semi-classical ap-
proach is no longer reliable. But as long as ∆B remains
small compared to the typical gap ∆ between nearby
bands, we may still construct the system wave functions
in the presence of the driving field from Wannier orbitals
belonging to a single band. In such quantum-mechanical
approach, dissipation is described as the result of cou-
pling the single degree of freedom (φ, q) to a continuum
of oscillator modes (qα, pα). The corresponding Hamil-
tonian has the form:
Hn = ǫn
(q2α+p
α) (A1)
where we have chosen the following commutation rela-
tions:
[φ, q] = i, [qα, pβ ] = iδαβ (A2)
and all other commutators between these operators van-
ish. The form of (A1) is plausible on the physical ground
because when the superconducting islands are coupled
to macroscopic leads, the charge q undergoes quantum
fluctuations, so that it has to be replaced by a “dressed
charge” q −
α gαqα in the effective Hamiltonian. A
more explicit justification is that the corresponding semi-
classical equations of motion have the same form as
Eqs. (4), (5) which simply mean that the effective cur-
rent going through the superconducting circuit is the bias
current minus the current going through the external im-
pedence. The semi-classical equations deduced from (A1)
read:
gαqα)
= ωαpα
= −ωαqα +
gαqα)
It is then natural to introduce q′ = q −
α gαqα, so that:
(q′) (A3)
To show that (A4) has the same form as (5), we notice
that the driving term for the bath oscillators is directly
proportional to dφ/dt. Specifically:
+ ω2αqα = ωαgα
Going to Fourier space, we see that after averaging over
initial conditions for the bath oscillators, Eq. (A4) takes
the form:
− iωq̃′(ω) = I
2πδ(ω)− i
ω2 − ω2α
−iωφ̃(ω)
This is exactly the frequency space version of Eq. (5),
where as usual, the dissipation is related to the spectral
density of the bath by:
ω2 − ω2α
Here we emphasize that Z is typically frequency depen-
dent, in which case the term (ZQ/Z)(dφ/dt) in Eq. (5)
becomes a convolution with ZQ/Z replaced by a non-local
kernel in time.
Now we turn to the solution of the quantum prob-
lem (A1) in the large driving current regime. Let us first
consider the Josephson array without dissipation. It is
straightforward to express its eigenstates in the q repre-
sentation41 because the Schrödinger equation reads then:
Eψ(q) = ǫn(q)ψ(q)− i
(q) (A8)
so that:
ψ(q) = ψ(0) exp
−i 2e
(ǫn(q
′)− E)dq′
The energy spectrum is determined via the boundary
condition ψ(q + 1) = ψ(q) so that:
∫ 1/2
′)dq′ +∆WSν, ν integer (A10)
This yields a Wannier-Stark ladder of spacially localized
states, with a constant level spacing equal to ∆B. Note
that increasing ν by one unit multiplies the wave-function
ψ(q) by exp(i2πq). In the phase representation, this is
equivalent to a translation by −2π. Of course, in the
absence of dissipation, these levels have infinite life-time,
and therefore, no dc voltage is generated.
Let us now consider the limit of a weak coupling to
the dissipative bath. This means that the decay rate Γ
of the Wannier-Stark levels is much smaller than the level
spacing ∆B. Assuming that transitions take place mostly
between two adjacent levels, we get an average voltage:
〈V 〉 = ~
〉 = hΓ
(A11)
The rate Γ is estimated via Fermi’s golden rule which we
prefer to use in the correlation function formulation :
Γν→ν′ =
|ν′〉|2C̃AA((ν − ν′)ωB) (A12)
where ωB = ∆B/~ = 2πI/(2e). In this expression,
C̃AA is the Fourier transform of the correlation function,
CAA(t − t′) = 〈A(t)A(t′)〉 of the Heisenberg operators
A(t) =
α gαqα(t), taken in the equilibrium state of the
dissipative bath.
We evaluate now the matrix element of the velocity
operator dǫn/dq between Wannier-Stark states:
〈ν|dǫn
|ν′〉 =
∫ 1/2
(q) exp(i2π(ν′ − ν)q) dq (A13)
As we have seen, in most physically interesting situations,
we can approximate the periodic function ǫ(q) by a single
harmonic 2w cos(2πq). In this case:
|ν′〉 = 2πwδ|ν′−ν|,1 (A14)
In the zero temperature limit, the bath correlation func-
tion is:
C̃AA(ω) =
πg2αδ(ω−ωα) =
θ(ω) (A15)
where θ(ω) is the Heaviside step function. Putting all
these elements together gives:
〈V (I ≫ Ic)〉 =
(2πw)2
2e2ZωI
(A16)
where Zω denotes the external impedance taken for
ω = ωB, and this result is in perfect agreement with the
large current limit of the semi-classical treatment, shown
as Eq. (35).
When the band structure is replaced by 2w cos(2πζq)
as in the case of a rhombus (for which ζ = 1/2), we have
〈V 〉 = ζ hΓ
(A17)
The frequency of Bloch oscillations becomes ωB(ζ) =
ζωB(ζ = 1), and the matrix element is multiplied by
ζ, so that the voltage is multiplied by ζ2. Again, this is
compatible with the semi-classical result (35).
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http://arxiv.org/abs/cond-mat/0411174
http://arxiv.org/abs/cond-mat/0605061
