U.S. patent application number 10/671651 was filed with the patent office on 2004-05-20 for fluxon injection into annular josephson junctions.
This patent application is currently assigned to D-Wave Systems, Inc.. Invention is credited to Ustinov, Alexey V..
Application Number | 20040095803 10/671651 |
Document ID | / |
Family ID | 26815535 |
Filed Date | 2004-05-20 |
United States Patent
Application |
20040095803 |
Kind Code |
A1 |
Ustinov, Alexey V. |
May 20, 2004 |
Fluxon injection into annular josephson junctions
Abstract
A method and apparatus for inserting fluxons into an annular
Josephson junction is disclosed. Fluxon injection according to the
present invention is based on local current injection into one of
the superconducting electrodes of the junction. By choosing an
appropriate value for the injection current, which depends upon the
spacing between injecting leads among other factors, the residual
fluxon pinning can be reduced to a very small level. Fluxon
injection according to the present invention provides for fully
controlling the trapping of individual fluxons in annular Josephson
junctions and is reversible to a state of zero fluxons without
heating the Josephson above its critical temperature. Fluxon
injection according to the present invention can be used for
preparing the working state of fluxon oscillators, clock
references, radiation detectors, and shaped junctions that may be
used as qubits for quantum computing.
Inventors: |
Ustinov, Alexey V.;
(Effeltrich, DE) |
Correspondence
Address: |
JONES DAY
222 EAST 41ST STREET
NEW YORK
NY
10017
US
|
Assignee: |
D-Wave Systems, Inc.
|
Family ID: |
26815535 |
Appl. No.: |
10/671651 |
Filed: |
September 26, 2003 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
10671651 |
Sep 26, 2003 |
|
|
|
10117696 |
Apr 4, 2002 |
|
|
|
60283477 |
Apr 11, 2001 |
|
|
|
Current U.S.
Class: |
365/162 ;
257/E39.014 |
Current CPC
Class: |
G06N 10/00 20190101;
Y10S 977/933 20130101; H01L 39/223 20130101; B82Y 10/00
20130101 |
Class at
Publication: |
365/162 |
International
Class: |
G11C 011/44 |
Claims
I claim:
1. A fluxon injection system comprising: an annular Josephson
junction; and current injection electrodes electrically connected
to a superconducting electrode of the annular Josephson junction to
form the flux injection system.
2. The system of claim 1, wherein the annular Josephson junction is
a long Josephson junction.
3. The system of claim 1, wherein the annular Josephson junction is
substantially circular in shape.
4. The system of claim 1, wherein the annular Josephson junction is
heart-shaped.
5. The system of claim 1, wherein the annular Josephson junction
has a Lyngby geometry.
6. The system of claim 1, wherein the annular Josephson junction
forms a qubit.
7. The system of claim 1, wherein the annular Josephson junction
comprises Nb/Al--AlO.sub.x/Nb.
8. The system of claim 7, wherein the annular Josephson junction
has a mean diameter of approximately 100 .mu.m and a junction width
of approximately 4 .mu.m.
9. The system of claim 8, wherein spacing between the injection
electrodes is in a range from approximately 10 .mu.m to
approximately 22.mu..
10. The system of claim 1, wherein spacing between the current
injection electrodes is between approximately a value of the
Josephson penetration depth to approximately a width of the current
injection leads.
11. A method of injecting at least one fluxon into an annular
Josephson junction comprising: providing current carrying injection
electrodes electrically connected to a superconducting electrode of
the annular Josephson junction; and, delivering an injection
current through the injection electrodes wherein the injection
current has sufficient magnitude that the magnetic flux generated
in the junction of the Josephson junction by the passage of the
injection current therethrough is sufficient to create at least one
fluxon in the Josephson junction.
12. The method of claim 11, wherein the magnetic flux generated by
the injection current is at least twice the quantum of magnetic
flux.
13. The method of claim 11, wherein the annular Josephson junction
is a long Josephson junction.
14. The method of claim 11, wherein the annular Josephson junction
is substantially circular in shape.
15. The method of claim 11, wherein the annular Josephson junction
is heart-shaped.
16. The method of claim 11, wherein the annular Josephson junction
has a Lyngby geometry.
17. The method of claim 11, wherein the annular Josephson junction
comprises Nb/Al--AlO.sub.x/Nb.
18. The method of claim 17, wherein the annular Josephson junction
has a mean diameter of approximately 100 .mu.m and a junction width
of approximately 4 .mu.m.
19. The method of claim 18, wherein a spacing between the injection
electrodes is in a range from approximately 10 .mu.m to
approximately 22 .mu.m.
20. The method of claim 19, wherein the injection current is
between approximately 3 mA to approximately 10 mA.
21. A fluxon as an article of manufacture wherein the fluxon is
produced according to the process of claim 11.
22. A plurality of fluxons produced according to the process of
claim 11.
23. The method of claim 11, wherein a spacing between the injection
electrodes is in a range from approximately a value of the
Josephson penetration depth to approximately a width of the current
injection leads.
24 The method of claim 11, wherein the annular Josephson junction
forms a qubit.
25. A method of creating and destroying fluxons in a Josephson
junction comprising: providing current carrying injection
electrodes electrically connected to a superconducting electrode of
the Josephson junction; and, delivering an injection current
through the injection electrodes wherein the injection current has
sufficient magnitude that the magnetic flux generated in the
junction of the Josephson junction by the passage of the injection
current therethrough is sufficient to create at least one fluxon in
the Josephson junction; and reducing the injection current to zero
thereby returning the Josephson junction to a state without
fluxons.
26. The method of claim 25, wherein the magnetic flux generated by
the injection current is at least twice the quantum of magnetic
flux.
27. The method of claim 25, wherein the annular Josephson junction
is a long Josephson junction.
28. The method of claim 25, wherein the annular Josephson junction
is substantially circular in shape.
29. The method of claim 25, wherein the annular Josephson junction
is heart-shaped.
30. The method of claim 25, wherein the annular Josephson junction
has a Lyngby geometry.
31. The method of claim 25, wherein spacing between the injection
electrodes is in a range from approximately a value of the
Josephson penetration depth to approximately a width of the current
injection leads.
32. The method of claim 25, wherein the annular Josephson junction
forms a qubit.
Description
RELATED APPLICATIONS
[0001] This application derives from Provisional Patent Application
Serial No. 60/283,477, filed Apr. 11, 2001 and incorporated herein
by reference, and claims priority therefrom pursuant to one or more
of 35 U.S.C. .sctn.119, .sctn.120, .sctn.365.
BACKGROUND
[0002] 1. Field of the Invention
[0003] This invention relates generally to Josephson junctions and,
more particularly, to injection of fluxons into annular Josephson
junctions.
[0004] 2. Discussion of Related Art
[0005] Long Josephson junctions are interesting systems from the
perspective of providing a workbench for fundamental investigations
of a variety of superconducting phenomena, as well as having
various applications in cryoelectronics. For an elementary
introduction see SUPERCONDUCTIVITY by Charles P. Poole, Jr.,
Horacio A. Farach and Richard J. Creswick (Academic Press, 1995),
pp. 442-444 and references cited. Long Josephson junctions are also
useful for studying basic properties of solitary waves (solitons).
Solitons of the simplest type are topological kinks and are able to
propagate. A well-known example of such a soliton is the elementary
quantum of magnetic flux .PHI..sub.0 (also called a fluxon, or
Josephson vortex) in a long Josephson junction. See, for example,
A. Barone and G. Patern, PHYSICS AND APPLICATIONS OF THE JOSEPHSON
EFFECT (Wiley, N.Y. 1982); A. V. Ustinov, Physica D 123, 315
(1998). A fluxon in a long Josephson junction can be caused to move
along the junction by the application of a bias current I.sub.B
flowing across the junction. The resulting motion of such a fluxon
gives rise to a dc voltage V.sub.dc across the junction, which is
proportional to the fluxon's mean velocity v. Thus, a measurement
of V.sub.dc as a function of I.sub.B provides a useful way to gain
information about properties of the fluxons, including the number
of fluxons present.
[0006] Our primary concern herein is with long Josephson junctions
which, for economy of language, we refer to simply as "junctions"
understanding thereby that long Josephson junctions are understood.
Explicit descriptions of junctions having other shapes will be
included when necessary for clarity. (00051 An important property
of an annular long Josephson junction results from the quantization
of magnetic flux in a superconducting ring. The annular junction is
a topologically closed system such that the number of initially
trapped fluxons is conserved and new fluxons can be created only in
the form of fluxon-antifluxon pairs. See, e.g., A. Davidson, B.
Dueholm, B. Kryger, and N. F. Pedersen, Phys. Rev. Lett. 55 2059
(1985). Fluxon motion in annular junctions occurs under periodic
boundary conditions and without any reflections from boundaries,
thereby avoiding many mathematical and physical complications that
occur for fluxon motion in other junction shapes. One source of the
interest in investigating annular junctions derives from
fundamental aspects of the Berry phase effect that arises in
annular junctions. (see e.g. F. Gaitan, Phys. Rev. B 63, 104511-1
(2001); and V. Plerou and F. Gaitan, Phys. Rev. B 63, 104512-1
(2001)). Other sources of interest in annular junctions arise from
the phenomena of Cherenkov radiation by solitons that can be
studied therein. (see, for example, E. Goldobin, A. Wallraff, N.
Thyssen, and A. V. Ustinov, Phys. Rev. B 57, 130 (1998); and A.
Wallraff, A. V. Ustinov, V. V. Kuring, J. A. Shereshevsky, and N.
K. Vdovicheva, Phys. Rev. Lett. 84, 151 (2000)). Applications with
a view towards the development of practical devices can also be
investigated with annular junctions.
[0007] Ring-shaped annular junctions have also been proposed as
microwave sources with high stability and very narrow radiation
line width (for example, see U.S. Pat. No. 4,181,902 to A. C.
Scott). Annular junctions with trapped fluxons have also been
suggested as radiation detectors in which they have an advantage of
a stable operation point at a finite voltage. (See for example, C.
Nappi and R. Christiano, Appl. Phys. Lett. 70, 1320 (1997); M. P.
Lisitskii et al., Nucl. Instr. and Methods in Phys. Research A 444,
476 (2000)). More recently, annular junctions of special shapes
have been proposed for the creation, storage and manipulation of
quantum bits ("qubits") in the form of fluxons (see A. Wallraff, Y.
Koval, M. Levitchev, M. V. Fistul, and A. V. Ustinov, J. Low Temp.
Phys. 118, 543 (2000)); and fluxon ratchets (E. Goldobin, A. Sterk,
and D. Koelle, Phys. Rev. E 63, 031111 (2001), and Carapella, Phys
Rev. B 63, 054515 (2001)). Fluxons in Josephson transmission lines,
which are discrete analogs of long Josephson junctions, have been
proposed as on-chip clocks by V. Kaplunenko, V. Borzenets, N.
Dubash, and T. Van Duzer, Appl. Phys. Lett. 71, pp 128-130 (1997),
Y. Zhang and D. Gupta, Supercond. Sci. Technol., 12, pp 769-772
(1999), D. Gupta and Y. Zhang, App. Phys. Let. 76, pp. 3819-3821
(2000), and U.S. Pat. No. 6,331,805, "On-Chip long Josephson
Junction (LJJ) Clock Technology", to Gupta et al.
[0008] A significant problem in utilizing fluxon states in annular
junctions is preparation of the initial state of the system
containing a single or a predetermined number of fluxons. For
example, in order to realize a state having a single fluxon, a
single magnetic flux quantum has to be trapped in the junction,
i.e., between its superconducting electrodes. The only reliable and
reproducible technique for trapping fluxons in an annular junction
that has been previously known and used requires rather exotic and
complicated apparatus, namely a low temperature scanning electron
(or laser) microscope. See e.g. A. V. Ustinov, T. Doderer, B.
Mayer, R. P. Huebener and V. A. Oboznov, Europhys Lett. 19, 63
(1992). Other known methods for trapping magnetic flux in an
annular junction can be used while cooling the sample below the
critical temperature of its superconducting electrode(s). These
other methods are based on either sending a current through an
additional specially designed coil placed on top of the annular
junction (see I. V. Vemik, V. A. Oboznov and A. V. Ustinov, Phys.
Lett.A 168, 319 (1992)), or applying a small bias current directly
through the junction (see A. V. Ustinov, Pis'ma Zh. Eksp. Teor.
Fiz. 64, 178 (1996) [Sov. Phys. JETP Lett. 64, 191 (1996)]; and I.
V. Vemik, S. Keil, N. Thyssen, T. Doderer, A. V. Ustinov, H.
Kohlstedt, and R. P. Huebener, J. Appl. Phys. 81, 1335 (1997)).
Unfortunately, the latter techniques are not sufficiently
reproducible and require heating of a junction to high temperature.
Moreover, fluxons trapped in such ways often suffer from parasitic
pinning due to Abrikosov vortices which become trapped in
superconductive electrodes. Thus, there is a need for a system to
inject a single fluxon, or a known number of fluxons, in a
controlled manner into an annular Josephson junction. The present
invention is directed to providing such a system.
SUMMARY
[0009] The present invention relates to a fluxon injection system
including injection electrodes separated by a distance D in contact
with one terminal of an annular Josephson junction. Fluxons are
trapped on the annular Josephson junction when an injection current
of sufficient magnitude is injected through the injection
electrodes.
[0010] Application of an injection current causes current to flow
from one of the injection electrodes into a superconducting
electrode and across the Josephson barrier. The current is
collected by another injection electrode on the same
superconducting electrode so that the total current across the
Josephson barrier remains zero. A magnetic flux is thus created in
the region between the injection electrodes. As the magnitude of
the magnetic flux created by the injection current increases and
becomes larger than the elementary quantum of magnetic flux,
.PHI..sub.0, it may become energetically favorable for a
compensating negative flux to be created. If the induced flux
exceeds .PHI..sub.0, the remaining positive flux on the annular
Josephson junction can exist in solitary form and become a fluxon.
The induced flux on the annular Josephson junction is removed when
the injection current is removed. In this case, the compensating
negative flux annihilates the remaining solitary positive flux and
the junction is then free of fluxons. Thus, control of the
properties of the system, including the current flow between the
injection electrodes and the electrode spacing, results in
controlled insertion of fluxons into the annular junction. Such
controlled fluxon initial states on annular Josephson junctions can
be used in connection with clock references, radiation detectors,
and fluxon oscillators, among other applications. Shaped junctions
can advantageously be employed along with the injection systems of
the present invention for initialization of qubits for quantum
computing, among other applications.
[0011] These and other embodiments are further described below with
respect to the following figures.
BRIEF DESCRIPTION OF THE FIGURES
[0012] FIG. 1(a) shows a plan view of an annular Josephson junction
with a trapped fluxon (dimensions are not to scale).
[0013] FIG. 1(b) illustrates fluxon insertion according to some
embodiments of the present invention (dimensions are not to
scale).
[0014] FIG. 2 shows a schematic view (dimensions are not to scale)
of an experimentally studied annular Josephson junction having
local current injection leads according to some embodiments of the
present invention.
[0015] FIGS. 3(a) and 3(b) show experimentally measured
current-voltage characteristics of annular junction at injection
currents I.sub.L=3.37 mA and I.sub.L=6.68 mA, respectively.
[0016] FIG. 4 shows the measured dependence of critical current of
the annular junction I.sub.c on injection current I.sub.L for some
embodiments of the present invention.
[0017] FIG. 5 shows the measured dependence of the critical current
of the annular junction I.sub.c on the applied magnetic field H
generated by a current I.sub.H delivered to a coil for I.sub.H=0 mA
(solid line, a) and for I.sub.H=3.51 mA (line with dots, b).
[0018] FIGS. 6a and 6b show numerically calculated dependencies of
the normalized critical current of the annular junction
.gamma..sub.c on the injection current amplitude F for normalized
spacing between injection electrodes, d=2 and d=1, respectively,
along with insets showing the injection current profile.
[0019] FIGS. 7a and 7b show numerically calculated current voltage
characteristics of annular junction normalized injection electrode
spacing d=2 and d=1, respectively, between the injectors and
various injection currents.
[0020] FIG. 8 shows the spatial-temporal evolution of the
instantaneous normalized voltage in the simulated annular system
for .epsilon.=8, d=1, and .gamma.=0.4.
[0021] FIG. 9a shows a flux injection system according to the
present invention utilizing a heart-shaped junction as for qubit
initialization, among other purposes.
[0022] FIG. 9b shows a flux injection system according to the
present invention utilizing a compact, symmetric and uniformly
biased annular junction as for a soliton oscillator or clock, among
other purposes.
[0023] FIGS. 10(a) and 10(b) depict experimental current-voltage
plots for two different values of I.sub.L. 10(a): I.sub.L.=5.27 mA.
10(b): I.sub.L.=10.06 mA.
[0024] FIG. 11 depicts the effect of local injection current on the
critical current.
[0025] FIG. 12 depicts the dependence of the critical current on
the applied magnetic field (applied by means of a coil current
I.sub.H) for two different values of local current I.sub.L.
DETAILED DESCRIPTION OF THE INVENTION
[0026] The present invention relates to fluxon injection systems.
Further, the fluxon injection systems according to some embodiments
of the present invention are demonstrated both experimentally and
numerically. Experimental demonstrations of the fluxon injection
systems according to some embodiments of the present invention show
injection and removal of a desired number of fluxons into or out of
an annular Josephson junction. Further, a theoretical model of the
fluxon injection systems is described. Numerical simulations based
on the proposed model describing the fluxon injection systems show
good agreement with the experimental data and provide further
incite into the fluxon injection process and fluxon interaction
with small pinning potential remaining in the injection region.
Several embodiments of the fluxon injection systems according to
the present invention are described.
[0027] An annular Josephson junction with a trapped fluxon is shown
schematically in FIG. 1(a) comprising superconducting electrodes 1a
and 1b, Josephson tunneling barrier, 2 and fluxon, 3. The depiction
of FIG. 1(a) corresponds to the special case in which the fluxon,
3, is trapped between the electrodes 1a and 1b when the junction is
in the process of cooling down through the critical temperature
T.sub.c of the superconducting electrodes. Usually, however, when
cooling a Josephson junction through the superconducting critical
temperature of the electrodes, no such fluxon becomes trapped in
the junction. Fluxon injection must be accomplished during cooling
through T.sub.c since it is usually not possible to inject a fluxon
into the junction when the junction is in its superconducting
state, that is, below T.sub.c.
[0028] The present invention relates to a fluxon injection system
including local injection of current I.sub.L into the
superconducting electrodes of the junction by means of injection
leads attached thereto, schematically shown as 4a and 4b in FIG.
1(b). Unlike the special case depicted in FIG. 1(a), it is assumed
for the schematic depiction of FIG. 1(b) that there is no magnetic
flux trapped in the junction in the absence of injection current,
that is when I.sub.L=0. When the current I.sub.L is turned on,
current flows from the injection lead 4a into the superconducting
electrode 1a as 5, and also flows across the Josephson barrier, 2.
This current I.sub.L is collected by lead 4b, such that the total
current across the Josephson barrier 2 remains equal to zero. The
current I.sub.L generates a local magnetic flux .PHI..sub.L which,
without loss of generality, can be taken to satisfy
.PHI..sub.L>0 in the region between the injection leads 4a and
4b. As the injection current I.sub.L is increased, .PHI..sub.L also
increases. As .PHI..sub.L becomes larger than .PHI..sub.0, it may
become energetically favorable to have (L compensated by a negative
magnetic flux -.PHI..sub.0 (with a magnetic field component
directed outside the ring, see FIG. 1(b)). Note that the energy
contained in a magnetic field increases as the square of magnetic
field. Thus, two separated flux quanta .PHI..sub.0+.PHI..sub.0 will
have lower energy (.about.2.PHI..sub.0.sup.2) than a single
2.PHI..sub.0 fluxon (.about.4.PHI..sub.0.sup.2). Since the total
magnetic flux in the junction barrier has to remain zero because of
the quantization of magnetic flux, there also has to appear a
positive magnetic fluxon .PHI..sub.0 somewhere else in the
junction. As soon as the current I.sub.L gets large enough such
that .PHI..sub.L>.PHI..sub.- 0, the induced magnetic flux
+.PHI..sub.0 may exist in the long junction in the solitary form,
i.e. as a free fluxon. Moreover, if .PHI..sub.0>2.PHI..sub.0,
two free fluxons should appear, and so on.
[0029] One anticipates that the existing N free fluxons will
interact with the remaining magnetic flux
(.PHI..sub.L-N.PHI..sub.0) near the injection leads. An important
feature of a fluxon injection system according to some embodiments
of the present invention is that the interaction of free fluxons
with the remaining magnetic field (which leads to pinning of free
fluxons) can be minimized by appropriate choices for the distance D
between the injection leads and for the injection current value
I.sub.L.
[0030] Experiments have been performed with Nb/Al--AlO.sub.x/Nb
Josephson annular junction with the mean diameter 2R=100 .mu.m and
the width W=4 .mu.m. (.mu.m=10.sup.-6 meter), both as depicted in
FIG. 1. The so-called Lyngby geometry furnished with additional
local current injection leads 4a and 4b as shown in FIG. 2 has been
used in these experiments. The Lyngby geometry is often chosen for
experiments as it provides rather uniform bias current distribution
over the junction (I.sub.B in FIG. 2) and, at the same time, it has
proven to be suitable for fluxon trapping at T.sub.c.
[0031] Fluxon insertion using local current injection pursuant to
the present invention is not limited to the Lyngby geometry of FIG.
2, but can be done for essentially any annular junction geometry,
as discussed further below.
[0032] The junction used for the experimental examples presented
herein has a critical current density of about 1.1 kA/cm.sup.2
which corresponds to a Josephson length .lambda..sub.J.apprxeq.11
.mu.m and a plasma frequency .omega..sub.p/(2.pi.).apprxeq.135 GHz.
This implies the ratio 2.pi.R/.lambda..sub.J.ident.l.apprxeq.28 of
the junction's length to the fluxon's size and the junction width
W/.lambda..sub.J<1, i.e., the junction can be regarded as long
and quasi-one-dimensional. The local current I.sub.L was applied
via 2 .mu.m-wide leads spaced by a distance D=22 .mu.m. A magnetic
field H was applied in the plane of the tunnel barrier (shown as 6
in FIG. 2) using a coil with a conversion ratio about 0.35 Oe/mA.
The measurements were done at the temperature of 4.2 K.
[0033] Measurements were performed in the junction state with no
fluxons trapped in the junction barrier during cooling down. The
critical current I.sub.c at I.sub.L=0 and H=0 was about 7.8 nmA.
Increasing I.sub.L, causes the critical current to decrease and
fluxon steps appear in the current-voltage (I.sub.B-V.sub.dc)
characteristics (i.e. bias current vs dc voltage). FIGS. 3a and 3b
show two examples of I.sub.B-V.sub.dc curves obtained at two
different values of the injection current I.sub.L, which was kept
on and constant during the course of every measurement. At
I.sub.L=3.37 mA (FIG. 3(a)), there is a clear single fluxon step
with an asymptotic voltage of about 64 .mu.V. This I.sub.B-V.sub.dc
curve indicates the fluxon depinning current I.sub.db to be rather
small, about 13% of the junction's critical current I.sub.c
measured at I.sub.L=0. From the top of the step, the junction
switches to superconducting gap voltage.
[0034] FIG. 3(b) shows the I.sub.B-V.sub.dc curve obtained at
injection current I.sub.L=6.68 mA. Here we find the double-fluxon
step with an asymptotic voltage of about 128 .mu.V. Again, from the
top of this step the junction switches to the gap voltage. We note
also the remaining depinning current and the single-fluxon step
have substantially reduced current amplitude.
[0035] The effect of the local current injection I.sub.L on the
critical current of the studied annular junction is presented in
FIG. 4. Since this dependence was found to be not completely
symmetric, all four quadrants of the (I.sub.c, I.sub.L) plane are
presented. The similarity of the observed dependence of I.sub.c on
I.sub.L of FIG. 4 to the conventional Fraunhofer-like pattern of
the critical current of a small Josephson junction on magnetic
field is striking. It can be noted that the single and double
fluxon I.sub.B-V.sub.dc curves observed in FIG. 3 correspond
approximately to the first and second minima of the Fraunhofer
pattern, respectively.
[0036] FIG. 5 presents experimental data on the critical current
dependence on the applied magnetic field, H generated by a current
I.sub.H delivered to a coil, both with an injection current
(I.sub.L=3.51 mA) and with no injection current (I.sub.L=0). One
notes that the zero-injection pattern shown in FIG. 5 (curve (a))
is slightly asymmetric but, in general, looks rather as expected
for a long annular junction with no trapped fluxons. In contrast,
the I.sub.L=3.51 mA curve (b) has a pronounced minimum around zero
field. Such behavior looks very similar to that of an annular
junction with one trapped fluxon, as studied in detail by I. V.
Vernik, S. Keil, N. Thyssen, T. Doderer, A. V. Ustinov, H.
Kohlstedt, and R. P. Huebener, J. Appl. Phys. 81, 1335 (1997), and
A. V. Ustinov, B. A. Malomed, and N. Thyssen, Phys. Lett. A 233,
239 (1997). Indeed, this is consistent with the single-fluxon
I.sub.B-V.sub.dc curve observed in FIG. 3(a) which corresponds to a
similar range of the injection current I.sub.L For some ranges at
low values of the field H (between coil current values between
approximately 2 mA and approximately 3 mA for both polarities),
linear increase of I.sub.c with H is observed.
[0037] A long quasi-one-dimensional Josephson junction is described
by the perturbed sine-Gordon equation for the superconducting phase
difference .phi. across the junction (see A. Barone and G. Patern,
PHYSICS AND APPLICATIONS OF THE JOSEPHSON EFFECT (Wiley, N.Y.
1982)).
.phi..sub.xx-.phi..sub.tt=sin .phi.-.alpha..phi..sub.t+.gamma.+f(x)
Eq. (1)
[0038] As used in Eq. 1, x is the spatial coordinate along the
junction and t is time, measured, respectively, in units of the
Josephson length .lambda..sub.J and inverse plasma frequency
.omega..sub.p.sup.-1, and where subscripts refer to partial
derivatives with respect to the indicated variable. The coefficient
.alpha. accounts for the damping due to quasi-particle tunneling
across the junction, while .gamma. and f(x) are the
spatially-uniform and spatially-varying bias current densities,
both normalized to the junction's critical current density
j.sub.c.
[0039] In the case of an annular junction with no fluxon trapped,
solutions of Eq. (1) are subject to periodic boundary conditions,
.phi..sub.x(x+l)=.phi..sub.x(x) and .phi.(x+l)=.phi.(x), where
l=2.pi.R/.lambda..sub.J is the normalized circumference of the
junction.
[0040] In order to model the local current injection described
above, the spatially-varying bias current term can be taken in the
form
f(x)=.epsilon.[.delta.(x.sub.0)-.delta.(x.sub.0+d)], Eq. (2)
[0041] where d=D/.lambda..sub.J is the normalized spacing between
the injection leads, .epsilon. is the injected current amplitude
and .delta.(x) is the Dirac delta function. The net current
.intg.f(x)dx over the junction is zero.
[0042] A related model has been studied almost two decades ago by
Aslamazov and Gurovich (see L. G. Aslamazov and E. V. Gurovich,
Pis'ma Zh. Eksp. Teor. Fiz. 40, 22 (1984) [Sov. Phys. JETP Lett.
40, 746 (1984)). They considered interaction of fluxons with an
Abrikosov vortex that is trapped in one of the junction electrodes,
with its normal core parallel to the tunnel barrier. The Abrikosov
vortex was modeled by
f(x)=.epsilon..delta..sub.x(x.sub.0), Eq. (3)
[0043] where .delta..sub.x(x.sub.0) is the spatial derivative of
the .delta. function at x=x.sub.0. Later, the influence of the
Abrikosov vortex was considered, in that the approach used in the
references L. G. Aslamazov and E. V. Gurovich, Pis'ma Zh. Eksp.
Teor. Fiz. 40, 22 (1984) [Sov. Phys. JETP Lett. 40, 746 (1984)] and
M. V. Fistul and G. F. Giuliani, Phys. Rev. B 58, 9343 (1998)
assumes the condition .epsilon.<<1. In the case described by
Eq. (2), the coefficient e may be, in general, arbitrarily
large.
[0044] Recently, the same problem of an Abrikosov vortex trapped
near a long Josephson junction was studied in the theoretical paper
M. V. Fistul and G. F. Giuliani, Phys. Rev. B 58, 9343 (1998). The
Abrikosov vortex configuration considered by these authors is
substantially the same as that of Aslamazov and Gurovich discussed
above (L. G. Aslamazov and E. V. Gurovich, Pis'ma Zh. Eksp. Teor.
Fiz. 40, 22 (1984) [Sov. Phys. JETP Lett. 40, 746 (1984)). The main
result obtained in M. V. Fistul and G. F. Giuliani, Phys. Rev. B
58, 9343 (1998) is qualitatively similar to the findings obtained
here in that the locally induced magnetic flux generates a stable
state of two fluxons with opposite polarity, with one of them
(antifluxon) pinned by the Abrikosov vortex and another (fluxon)
moving freely in the rest of the junction. For an Abrikosov vortex
the magnitude of the induced flux cannot exceed .PHI..sub.0.
Therefore the free energy of the above fluxon-antifluxon state is
higher than that of the conventional (fluxon-free) state. In the
examples presented herein, local flux is induced by the external
current I.sub.L and, thus, may exceed .PHI..sub.0. The
lowest-energy state then becomes the dissociated fluxon-antifluxon
state.
[0045] In order to model the process and correctly interpret the
obtained experimental results on fluxon injection, numerical
simulations were performed by solving the partial-differential
equation of Eq. (1). In the simulations, each .delta.-function in
Eq. (2) was approximated by the more smooth function 1 ( x - x 0 )
[ 1 - tanh 2 2 ( x - x 0 ) ] , Eq . ( 4 )
[0046] such that .eta..xi.=.epsilon.. The numerically injected bias
is spread over a distance of about .xi..lambda..sub.J. Through this
paper, results obtained with .xi.=0.5 are presented, which
approximate the actual experimental case. It has been checked,
however, that taking .xi.=1 produces rather similar results,
indicating that the results are not sensitively dependent on the
precise value of .xi.. In the numerically calculated
current-voltage relationships, characteristics are shown in
normalized limits as .gamma.(.upsilon.), were .upsilon. is the
normalized average fluxon velocity. With this normalization
.upsilon.=1 corresponds to the asymptotic voltage of the
single-fluxon step. In order to save the computation time, the
simulations are performed with the reduced normalized junction
length l=10 and the dissipation coefficient .alpha.=0.1.
[0047] Numerical results, in general, turn out to be very similar
to the above presented experimental data. Two different values for
the spacing d between the current injecting points, d=2 and d=1,
have been chosen for simulation.
[0048] FIG. 6 presents the calculated dependence of the critical
current .gamma..sub.c on the injection current amplitude .epsilon..
As has been already seen in experiment, this dependence resembles
the conventional Fraunhofer pattern of the critical current of a
small Josephson junction in a magnetic field. The actual length of
the corresponding small junction is associated with the distance d
between the injecting points. As may be expected, the overlap
between the lobes is larger for larger d, see, e.g. FIG. 6(a) and
(b). Due to this overlap, the minimum value of the critical (fluxon
depinning) current between the lobes is decreasing with d.
[0049] The calculated curves for normalized bias current .gamma. vs
normalized voltage v for various injection currents (indicated on
the plots) are shown in FIG. 7. Both single fluxon and double
fluxon steps can be clearly recognized. Altogether, these curves
look very similar to those of FIG. 3, numerical data show that the
steps on curves of FIG. 7 account for free moving fluxons under the
action of the uniformly distributed bias current.
[0050] It can be seen in FIG. 7 that there is a residual pinning of
fluxon(s) due to the effect of the local bias injectors. According
to FIG. 6, this pinning is the smallest at the local injection
current values which lie between the lobes of the
.gamma..sub.c(.epsilon.) curve. Thus, for a given injector spacing
d the residual fluxon pinning can be minimized by choosing an
appropriate value for the injection current .epsilon.. The smallest
pinning can be achieved when the lobes of the
.gamma..sub.c(.epsilon.) curve join at .gamma..sub.c=0, that is for
the cases in which d<1. Thus, the optimum spacing D between the
injectors is about .lambda..sub.J or less (but advantageously
larger than the width of the injection leads, 7, in order to avoid
edge effects).
[0051] This section of numerical results is concluded by FIG. 8. It
shows a two dimensional gray scale plot of the spatial and temporal
evolution of the instantaneous normalized voltage .phi..sub.t(x, t)
in the annular junction for .epsilon.=8, d=1 and .gamma.=0.4. The
moving fluxon is recognized as a solitary wave packet moving with a
nearly steady velocity across the junction. One can see that there
is just a very tiny disturbance of the fluxon motion arising in the
region of local current injection. Obviously, this remaining
pinning may be important only at low fluxon velocities, where the
fluxon's kinetic energy is of comparable scale with the pinning
potential.
[0052] Additional experimental results have been obtained for a
reduced spacing between injectors of D=10 .mu.m for comparison with
the numerical results presented above. Other experimental
conditions for the annular Jospehson junction related to the
composition, geometry, temperature, among others are the same as
given above in connection with the previous experimental results.
D=10.mu. is approximately the same as the numerical results
presented in FIGS. 6b, 7b and 8. (The experimental results
presented herein have d=D/.lambda..sub.J.apprxeq.1.1).
[0053] Measurements were commenced in the state having no trapped
fluxons in the junction barrier during cooling. The critical
current, I.sub.c, at I.sub.L=0 and H=0 is about 10.7 mA. FIG. 10(a)
and 10(b) give current-voltage plots for two different values of
I.sub.L. As above, the injection current is kept on and at a
constant value during the measurements presented in FIG. 10. FIG.
10(a) clearly shows a single fluxon step with an asymptotic voltage
value about 64 .mu.V. FIG. 10(a) indicates that the fluxon
depinning current is small, about 6% of the junction's critical
current measured for I.sub.L=0 (I.sub.dp.apprxeq.0.64 mA). This
value is approximately a factor of 2 smaller than that measured
above in which D was approximately a factor of 2 larger.
[0054] FIG. 10(b) depicts the current-voltage plot for an injection
current I.sub.L=10.06 mA, or roughly twice the injection current of
FIG. 10(a). The double-fluxon step is apparent with an asymptotic
voltage of about 128 .mu.V.
[0055] We observe in FIG. 10 that the current amplitude of the
fluxon steps is increased by almost 50% in comparison with the
previous measurements for D=22 .mu.m. At the same time, the
remaining fluxon depinning current is reduced. These observations
taken together indicate higher quality and uniformity of the fluxon
states achieved in the annular Josephson junction.
[0056] FIG. 11 depicts the effect of the local injection current on
the critical current. The similarity to the conventional Fraunhofer
patters noted above is not as pronounced in FIG. 11.
[0057] FIG. 12 depicts the dependence of critical current on the
applied magnetic field (as applied by means of a coil current,
I.sub.H). The pattern for I.sub.L=0 is more symmetric than that
observed in connection with FIG. 5 above and is the type of
functional dependence typical of a long, uniform annular Josephson
junction with no trapped fluxons. The curve in FIG. 12 for
I.sub.L=5.75 mA shows a pronounced minimum around zero applied
magnetic field, a typical behavior for one trapped fluxon. For
small values of H, a nearly linear increase of I.sub.c with H is
observed.
[0058] In summary, local current injected by an injection system
according to the present invention modulates the critical current
of the junction in a manner which is similar to the Fraunhofer
pattern. Locally produced magnetic fields act on the junction
region between the injection points, which reacts by creating a
magnetic field pattern of a small Josephson junction. By choosing
the spacing D between the injecting current leads smaller than
.lambda..sub.J, one can tune the injection current I.sub.L such
that the residual fluxon pinning is reduced to a very small
level.
[0059] FIGS. 9a and 9b illustrate additional examples of annular
junctions with a fluxon injection system according to the present
invention. A heart-shaped junction shown in FIG. 9(a) can be used
as vortex qubit (see A. Wallraff, Y. Koval, M. Levitchev, M. V.
Fistul and A. V. Ustinov, J. Low Temp. Phys. 118, 543 (2000)) when
the fluxon quantum state is a superposition of the two lowest
energy states in the (upper) lobes of the heart. In this case the
junction uniformity is required only for the upper junction region
shown in FIG. 9(a), where the fluxon resides. The bottom part of
the heart can be comfortably used for preparing the initial state
of the qubit, including application of a fluxon injection
system.
[0060] Applications other than qubits often require a compact,
highly symmetric and uniformly biased annular junction with a
fluxon trapped in its barrier. A possible layout of such a soliton
oscillator or clock is shown in FIG. 9(b). The bias current is to
be injected via uniformly distributed thin film resistors (not
shown) attached to the superconducting film surrounding the
junction. The bias current then flows uniformly through the barrier
and is collected via a contact to the ground plane in the center of
the ring. It is known that it is nearly impossible to trap a fluxon
in such a symmetric structure when cooling down through T.sub.c.
Therefore, local current injection pursuant to various embodiments
of the present invention can be used in this structure as well as
in any other, see FIG. 9(b). The same principle can be also applied
to the design of radiation and/or particle detectors based on
annular junctions (C. Nappi and R. Christiano, Appl. Phys. Lett.
70, 1320 (1997) and M. P. Lisitskii et. al. Nucl. Instr. and
Methods in Phys. Research A 444, 476 (2000)). Moreover, the working
junction area can be substantially increased as local fluxon
injection pursuant to various embodiments of the present invention
are expected also to work in larger two dimensional annular
junctions, as are typically required for detectors.
[0061] The fluxon injection procedures described herein offer the
advantage of reversibility not present in conventional fluxon
injection techniques. Reducing the injection current to zero causes
the fluxon(s) present in the junction to disappear, returning the
junction to the fluxon-free state without the necessity of heating
the junction above its critical temperature. The junction remains
in the fluxon-free state until further injection current is applied
sufficient to re-create fluxons. Thus, practical and convenient
resetting of the junction to the state of zero fluxons is
accomplished.
[0062] In conclusion, fluxon injection systems have been described
herein and demonstrated both experimentally and numerically. Such
fluxon injection systems allow for trapping, retaining and removing
any desired number of fluxons to, within or from the junction.
Fluxon injection systems according to the present invention
substantially simplify the use of annular junctions as oscillators,
radiation detectors, vortex qubits among other applications.
[0063] Having described the invention in detail, those skilled in
the art will appreciate that, given the present disclosure,
modifications may be made to the invention without departing from
the spirit of the inventive concept described herein. Therefore, it
is not intended that the scope of the invention be limited to the
specific embodiments illustrated and described.
* * * * *