Phase-slip flux qubits

Abstract

In thin superconducting wires, phase-slip by thermal activation near the critical temperature is a well-known effect. It has recently become clear that phase-slip by quantum tunnelling through the energy barrier can also have a significant rate at low temperatures. In this paper it is suggested that quantum phase-slip can be used to realize a superconducting quantum bit without Josephson junctions. A loop containing a nanofabricated very thin wire is biased with an externally applied magnetic flux of half a flux quantum, resulting in two states with opposite circulating current and equal energy. Quantum phase-slip should provide coherent coupling between these two macroscopic states. Numbers are given for a wire of amorphous niobium-silicon that can be fabricated with advanced electron beam lithography. Comment: Submitted to New Journal of Physics, special issue solid state quantum information

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Phase-slip flux qubits

J E Mooij and C J P M Harmans
Kavli Institute of Nanoscience, Delft University of Technology,
2628 CJ Delft, The Netherlands

E-mail: j.e.mooij@tnw.tudelft.nl

Abstract. In thin superconducting wires, phase-slip by thermal activation near
the critical temperature is a well-known effect. It has recently become clear that
phase-slip by quantum tunnelling through the energy barrier can also have a
significant rate at low temperatures. In this paper it is suggested that quantum
phase-slip can be used to realize a superconducting quantum bit without
Josephson junctions. A loop containing a nanofabricated very thin wire is
biased with an externally applied magnetic flux of half a flux quantum,
resulting in two states with opposite circulating current and equal energy.
Quantum phase-slip should provide coherent coupling between these two
macroscopic states. Numbers are given for a wire of amorphous niobium-
silicon that can be fabricated with advanced electron beam lithography.


1. Introduction
The phase difference of the order parameter over a thin, narrow superconducting wire can
change in units of 2π by means of a discrete process that is localized in space and time. The
amplitude of the order parameter becomes smaller over a few coherence lengths, touches zero
in one spot at one instant and then recovers its initial value. After this phase-slip event, the
phase difference has changed by 2π. There is an energy barrier against phase-slip, because the
temporarily reduced order parameter in the phase-slip volume leads to a loss of condensation
energy. The phase-slip process has been studied in detail at temperatures near the critical
temperature, where thermal activation allows crossing over the energy barrier [1]. Phase slip
leads to dissipation for currents below the critical pair-breaking value. For one-dimensional
superconductors it is the equivalent of vortex crossing in two- and three-dimensional systems.
Phase-slip is also possible at zero temperature by means of quantum tunnelling. To observe
this quantum phase-slip, one needs extremely weak superconducting wires. Over the years,
several experiments were performed where significant deviations from thermal activation
were observed at low temperatures [2,3]. It was hard to compare these data with theoretical
predictions, because quantitative calculations were not available. Recently, convincing
experimental evidence for quantum phase-slip at zero temperature was presented by the
Harvard group of Tinkham and collaborators [4,5], who fabricated extremely thin wires by
covering a suspended carbon nanotube with a film of amorphous Mo-Ge. New microscopic
theoretical calculations were performed by Zaikin and collaborators [6,7]. Experimental
results on a series of Mo-Ge wires were compared with this theory and reasonable agreement
was found.

So far, phase-slip has been studied in wires that were connected to an external current source.
Each event is connected with an energy transfer IΦo, where I is the current and Φo= h/2e is the
flux quantum. Induced by thermal activation or by quantum tunnelling, phase slip is
manifested as a stochastic process that leads to dissipation. However, if the concept of
quantum phase-slip is valid, it should also be possible to use the effect as a coherent coupling
between two distinct macroscopic quantum states in a superconducting quantum circuit. We
propose to study the tunnel strength for phase-slip transitions in a closed superconducting
loop that contains a weak wire, while a magnetic flux of half a flux quantum is applied to the
loop. The energy after macroscopic quantum tunnelling is the same but the persistent current

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Phase-slip flux qubits
2
is reversed. Quantum phase-slip should lead to a quantum superposition of the two opposed
current states. One would obtain superconducting quantum bits, more in particular flux qubits,
in which Josephson tunnel junctions are replaced by a weak superconducting wire. Using the
established techniques for experiments on junction-based flux qubits, quantum phase-slip can
be studied under optimally coherent conditions. The fundamental question concerning
superconducting long-range order in an increasingly weak one-dimensional wire can be
addressed without the use of dissipative processes. The application as a phase-slip qubit may
have clear practical advantages over junction-based flux qubits as will be discussed at the end
of this paper.


2. Wires and loops


Figure 1. a. Schematic lay-out of the weak wire in a closed superconducting loop. A SQUID
with two Josephson junctions (crosses) is used to measure the qubit state. b. Energy levels as
a function of applied flux for different fluxoid numbers, according to equation (6) for a loop
with Lk= 4 nH. A phase-slip event changes the fluxoid number n. The arrow indicates the
operating point at f= ½. c. Lowest two energy levels of phase-slip qubit with Ip=0.25 μA and
Γqps/2π= 4 GHz near f= ½..


Consider a long superconducting wire with length L and normal state resistance Rn at
temperature zero. The transverse dimensions are much smaller than the coherence length ξ=
(ξoℓ)½, where ξo is the BCS coherence length and ℓ the electronic mean free path (ℓ<< ξo). The
kinetic inductance Lk quantifies the superconducting current response to a gauge-invariant
phase difference γ, according to
1
2
k
L
π
The increase of energy due to current and phase difference is
2
2
11
2 2
k
L
π
⎝⎠
Within a Ginzburg-Landau-like description, with the density of superconducting electrons
derived from BCS/Gorkov theory for dirty superconductors at T=0, one finds
ћ
n
k
Bc
where Τc is the critical temperature and a is a numerical constant (an estimate is a ≈ 0.14).
The maximum (pair-breaking) supercurrent in the wire can be estimated in a similar
approach:
k T
Ib
eRξ

o
s
I
γ
Φ
=

(1)

o
E
γ
γ
Φ
=
⎛
⎜
⎞
⎟

(2)

R
L ak T
=
(3)

B c
o
=
(4)

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Phase-slip flux qubits
3
Here Rξ is the resistance of a length of wire equal to ξ, so that Rξ/Rn=ξ/L. The prefactor b is a
constant of order unity, an estimated value is b≈1.3.

Now bend the wire to form a closed superconducting loop. A wider section can be included
with negligibly low kinetic induction (figure 1a). An external source applies a magnetic flux
Φext=fΦo through the loop. The induced current in the loop I generates a flux I Lg where Lg is
the geometric inductance of the loop. Single-valuedness of the order parameter requires that
L IL
fnf
L
Φ
with n an integer. The value of n can, after cool-down, only be changed by a phase-slip event.
In practice, the kinetic inductance of the wires that exhibit quantum phase-slip is of order nH,
while the geometric inductance is of order 10 pH. In calculating γ, the flux generated by the
current can be ignored. As a consequence we can write

(
2
)
fn
γπ=−

Φ
=−


22222
gsg
ok
n
γππππγπ=−−=−−

(5)

(
2
o
2
1
2
k
)
Efn
L
(6)
Figure 1b gives an example of the levels for various values of n when Lk= 4 nH. In the
immediate neighbourhood of f=½ where the lines for n=0 and n=1 cross, defining δf= f-½, one
finds
2
11
24
k
L
⎠
The next higher energy levels for f≈ ½ are at a large distance, about Φo
The ‘classical’ circulating persistent currents are I1,2=±Ip with Ip=Φo/2Lk. For Lk equal to 4 nH,
Ip is 0.25 μA. This is similar to values of persistent currents in Josephson junction flux qubits.

If the quantum phase slip rate is Γqps, the wire flux qubit has the following Hamiltonian
(viewed as a pseudospin in the basis where the persistent currents create magnetization in the
z-direction):

opzqpsx
HfI
δσσ=Φ+ Γ
ћ


σz and σx are Pauli spin operators. The qubit energy splitting is
(
ћ
op qps
dEfI
δ = ±Φ+Γ
In figure 1c the qubit levels near f= ½ are plotted for Ip= 0.25 μA and a phase-slip rate
Γqps/2π= 4 GHz.

3. Phase-slip rates
To estimate the rate of quantum phase slip in the wire at low temperature we follow the
discussion by Lau et al. [5], based on a heuristic extension to low temperatures of the well-
established theory for thermally activated phase-slip near the critical temperature [1]. The
barrier height for a phase-slip event at zero temperature in their description is
R
E k T
Rξ
where Rq=h/4e2=6.5 kΩ. Counterintuitively, in this expression a longer coherence length leads
to a lower barrier, as Rξ= Rn’ξ. This is due to the fact that the condensation energy density
scales with ξo
Rn’ and Tc are easily determined, but the coherence length is not well known in the relevant
materials.

The attempt frequency is

1,2
o
E
δ
Φ
=
⎛⎜⎝
± f⎞⎟ (7)
2/Lk higher.
(8)

)()
22 (9)

0.83
q
BB c
=
(10)
-2 and all other factors combine to give the elegant form above. For a given strip

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Phase-slip flux qubits
4

1
0.57
0.83
GL
q
L
ξ
R
Rξ
τ
Ω =
(11)
with τGL=πħ/8kBT
of the BCS gap). The phase-slip rate amounts to
R
k T
L
c
R
ξ
ξ
c and d are unknown constants of order unity that express the uncertainties in the derivation.
Lau et al. could fit the resistance due to driven phase-slip in a large set of experimental wires,
with known values of their resistance per unit length, quite accurately to equation (12). From
their fit, one would induce that d=1 within the uncertainty of knowing the precise value of ξ.

B c the Ginzburg-Landau time at zero temperature (1/τGL=1.4Δ(0)/ħ in terms

1.5exp 0.3
ћ
q
Bc
qps
q
R
d
R
ξ
Γ=−
⎛
⎜
⎝
⎞
⎟
⎠
(12)

Figure 2. Quantum phase-slip rate as a function of resistance per unit length for four values of
ξ, according to equation (12). Assumptions: L= 50 nm, Tc= 1.2 K, c=d=1.


In figure 2 we plot Γqps/2π versus Rn’ as calculated from (12) for various values of ξ. Desired
values of Γqps for qubits are in the range of 1-10 GHz. The figure makes clear that Rξ=Rn’ξ has
to be around 0.3 kΩ, and also illustrates how sensitively the quantum tunnelling depends on
the precise value of Rξ. For practical purposes as a qubit one must control Rξ, and therefore
the line width, to within a few percent.

4. Practical wire qubits
Clearly superconducting wires with very high resistance are needed. Can such wires in
circuits be fabricated with established clean room techniques? In the following we analyze
realistic values for a wire of amorphous Nb-Si. These wires can be fabricated from very thin
amorphous films that are homogeneous to below a nanometer scale. The material is similar to
the Mo-Ge of [8] and [4]. In our group, we studied a Nb-Si wire of 130 nm width that clearly
exhibited deviations from thermal activation in its low temperature resistance [3]. Electron
beam lithography at this time allows fabrication of wires with width down to 10 nm, starting
from ultrathin films. Co-sputtered films of Nb-Si with composition around Nb50%-Si50%
showed a resistivity of about 2.0 μΩm. The critical temperature changed with thickness and
with composition [9]. In the following we choose as an example a Nb-Si film with

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Phase-slip flux qubits
5
composition 42% Nb-58%Si and a nominal thickness of 2.5 nm, to be the starting material.
The film showed a sharp superconducting transition at 1.2 K, and metallic temperature
dependence above Tc. The sheet resistance was 1 kΩ/square. A 10 nm wide wire of this
material will have a resistance Rn’= 0.1 kΩ/nm. The electronic mean free path is estimated to
be ℓ= 0.4 nm. The BCS coherence length is not known, but given the low critical temperature
it is likely to exceed the value for Nb which is 20 nm. This would lead to values of ξ that
could range from 3 to 8 nm. For that range of ξ and for Rn’= 0.1 kΩ, equation (12) gives rates
Γqps/2π ranging from 4 to 60 GHz. One cannot draw firm conclusions, but it seems there is a
good chance that with practical wires one can achieve a quantum phase-slip rate in the GHz
range.

We now discuss the practical design of a phase-slip flux qubit that is based on a wire of this
Nb-Si material. The phase-slip rate depends exponentially on the resistance in a length ξ of
the wire; the length of the wire has only linear influence. Given the desired phase-slip rate and
the corresponding desired resistance per unit length near 0.1 kΩ/nm, one can use the length of
the wire to optimize the energy and the persistent current level. The example discussed above
with L= 50 nm leads to Lk= 4 nH and Ip= 0.25 μA, a convenient level for SQUID
measurement.

The phase-slip qubit potentially has two strong advantages. The first is connected with 1/f-
type noise. In the existing superconducting qubits, based on Josephson junctions, charge noise
and critical current noise are very significant sources of decoherence, usually limiting
performance [12,13]. The phase-slip qubit is highly insensitive to charge noise, as it has no
islands between tunnel junctions. The equivalent of critical current noise would be
fluctuations of the energy barrier or the loop inductance. Phase-slip is an exponential process
as is electron tunnelling through oxide barriers. However, the barrier is determined by the
geometry and superconducting properties on the scale of the coherence length. Ionization or
microscopic motion of a single atom will not influence the tunnel barrier by a significant
amount. The loop inductance is even more macroscopic. The qubit will be sensitive to flux
noise in the same way as junction flux qubits. Trapping and detrapping of vortices in wider
parts of the qubit loop should be kept in mind, but is unlikely with the smooth homogeneous
films of Nb-Si.

The other advantage of the phase-slip qubit is the large distance of more than 500 GHz
between the qubit energy levels and the next higher states of the loop. In superconducting
phase and charge/phase qubits, the next higher levels are close on the scale of the qubit
splitting. Strong excitation may lead to leakage of quantum information. In junction flux
qubits, the distance to the next higher level is a few qubit splittings away. In the phase-slip
qubit, extremely fast excitation can be allowed.

5. Conclusions
Quantum phase-slip in superconducting wires that can be fabricated with advanced electron
beam lithographic techniques can have a rate of the order of several GHz. Superconducting
flux quantum bits can be developed where Josephson junctions have been replaced by a
superconducting quantum wire. The two states of the wire qubit will be very well separated
from the higher levels. The wire qubit will likely have much lower 1/f noise than the presently
used superconducting qubits.

Acknowledgements. We thank M. Tinkham, G. Schön and A. Zaikin for very helpful
discussions. This research is supported by FOM and by EU through SQUBIT2.

References
[1] Tinkham M 1996 Introduction to superconductivity, McGraw Hill, New York 2nd edition
p.288