Magnetic confinement of neutral atoms based on patterned vortex distributions in superconducting disks and rings

1.8 MB·English

by  B. Zhang

Magnetic confinement of neutral atoms based on patterned vortex distributions in
superconducting disks and rings
B. Zhang,1,2 M. Siercke,1,2 K.S. Chan,1 M. Beian,1,2 M.J. Lim,3 and R. Dumke1,2,∗
1Division of Physics and Applied Physics, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
3Department of Physics, Rowan University, 201 Mullica Hill Road, Glassboro, NJ 08028, USA
(Dated: January 17, 2012)
Weproposeandanalyzeneutralatomtrapsgeneratedbyvorticesimprintedbymagneticfieldpulse
sequences in type-II superconducting disks and rings. We compute the supercurrent distribution
and magnetic field resulting from the vortices in the superconductor. Different patterns of vortices
2 can be written by versatile loading field sequences. We discuss in detail procedures to generate
1 quadrupoletraps,self-sufficienttrapsandringtrapsbasedonsuperconductingdisksandrings. The
0 easeofcreatingthesetrapsandthelowcurrentnoiseinsupercurrentcarryingstructuresmakesour
2 approach attractive for designing atom chip interferometers and probes.

n
PACSnumbers: 37.10.Gh,03.75.Be,74.78.Na
a
J

4
I. Introduction the superconducting structure. Using this numerical ap-
1
proach we design quadrupole traps, self-sufficient traps

and ring traps in the various superconducting geome-
] Recently superconducting atom chips have generated
h tries. We demonstrate that, by simple application of
p a lot of interest due to their attractive properties, such
an additional bias field, a single quadrupole trap may
- as the Meissner effect for type-I superconductors and
m be deformed into a ring. Considering the low current
vortices for type-II superconductors [1–3]. Cold atoms
noise expected from the persistent supercurrents gener-
o have been trapped and manipulated near superconduct-
ating these traps, and the low thermal noise expected
t ingsurfaces[3–5]andBose-Einsteincondensation(BEC)
a fromsuperconductingatomchips,thismethodmaybean
. on a superconducting atom chip has been reported re-
s attractive way to generate low-noise on-chip atom inter-
c cently [6]. Thermal and technical noise in proximity to ferometry. Compared to other low noise traps generated
i superconductingsurfaceshavebeenshownboththeoreti-
s by permanent magnets our system retains one desirable
y callyandexperimentallytobesignificantlyreducedcom-
feature: The ability to control the trap parameters and
h pared to conventional atom chips [7–14]. Superconduct-
geometry from shot to shot.
p ing atom chips have the potential to coherently interface
[ atomic and molecular quantum systems with quantum Thepaperisorganizedasfollows. Sec. IIdescribesthe

solidstatedevices[15–19]. Furthermore,ultracoldatoms equations of motion for the sheet current density J in a
1
v canbeusedtoprobepropertiesofsuperconductors,such type-II superconducting strip, disk and ring in an exter-
0 as the dynamics of vortices in type-II superconductors nal magnetic field. To validate our numerical approach
0 [20, 21]. wecomputethesupercurrentsinasuperconductingstrip,
0 and compare them with analytical results [22]. For the
Previously we reported on the design and realization
3 two-dimensional structures considered in this paper no
1. of several types of magnetic traps involving linear type- explicit analytical solution is known. In Sec. III we de-
II superconducting microstructures [22–24]. These traps
0 signseveraltypesoftrapsforcoldatomsbytailoringthe
are formed by the magnetic field carried by vortices im-
2 magneticfieldpulseswhichinducevortexpatternsinthe
1 printedinthesuperconductortoconfineultracoldatoms. superconducting disk and ring. Sec. IV summarizes our
: Loadingofthevorticesisaccomplishedbytailoredexter-
v results.
nalmagneticpulsesequencesappliedtothesuperconduc-
i
X tor in the mixed state. In a different approach vortex-
r based magnetic traps have been observed after cooling a
a superconductingdiskthroughthecriticaltemperaturein II. Loading vortices
the presence of an external magnetic field [21].
In this article, we extend our investigation of vortex Consideratype-IIthinsuperconductingstrip,diskand
based microtrap geometries to type-II superconducting ringinamagneticfieldperpendiculartotheirsurface(xy
disksandrings. Tothisendwehavedevelopedanumer- plane) as shown in fig.1. Here d denotes the thickness of
ical approach to calculate the vortex patterns imprinted the superconductors and a denotes the half width of the
after applying various external magnetic field pulses to strip and the (outer) radius of the disk and ring. The
strip is assumed to be infinitely long along y, so that we
canrestrictoursimulationtothexz plane. Similarly,we
canreduceoursimulationtotherzplaneforthediskand
∗Electronicaddress: [email protected] ring due to rotational symmetry. Throughout the paper
2
y z z for the sheet current density J(x,t) of the strip [26]:
Bext z Bext r Bext r (cid:20) dB (cid:90) a ∂J(u,t) (cid:21)
J(x,t)=τ 2πx ext + M(x,u)du , (3)
a x 0.4a a x dt 0 ∂t
d -a a x d d
and J(r,t) for disks and rings [26]:
FIG.1: (Coloronline)Schematicoftheinvestigatedtype-II J(r,t)=τ(cid:20)πrdBext +(cid:90) a ∂J(u,t)Q(r,u)du(cid:21) , (4)
superconductorgeometries. Allgeometriesareassumedtobe dt ∂t
inthethinfilmlimitwithathicknessd. Shownareastripof r0
width 2a (left), a disk with radius a (center) and a ring with where
an outer radius of a and an inner radius of 0.4a (right). The
magneticfieldpulsesareappliedperpendiculartothesurface. M(x,u) = ln(cid:12)(cid:12)x−u(cid:12)(cid:12),
(cid:12)x+u(cid:12)
(cid:32) √ √ (cid:33)
u(cid:90) r/u E( 4ω) K( 4ω)
wewillassumethatthetemperatureofthesuperconduc- Q(r,u) = − 1+ω + 1+ω ωdω,
r 1−ω 1+ω
tors is below the critical value T and that the thickness r0/u
c
d is small compared to the other length scales of the τ = µ0ad . (5)
system. When the external field B exceeds the lower 2πρ(J)
ext
critical field B , but lies below the upper critical value
c1
B , magnetic flux starts to penetrate the superconduc- Here,ρ(J)=ρ (J/J )n−1isthenonlinearresistivitywith
c2 c c
tor in the form of Abrikosov vortices [25]. To facilitate criticalvalueρ ,andn=U /k T whereU isthecharac-
c 0 b 0
our numerical analysis we model these vortices together teristic activation energy of the superconductor [29]. J
c
with the screening current as a macroscopic supercur- isthecriticalsheetcurrentdensity[26,27],definedasthe
rent J(u,t). Our approach closely follows the method maximum current density the superconductor can carry
outlined in [26, 27] which makes use of the continuum withoutatransitionintothenormalstate. Forthescope
approximation where the flux line spacing and the mag- of this paper we have chosen the value of n = 19. This
netic field penetration depth are small compared to all value gives a moderate response of the superconductor
other relevant length scales considered [28]. From the to applied fields. The exact value for n is material de-
distributionofthesupercurrentwethencomputetheto- pendent and does not qualitatively alter the characteris-
talmagneticfieldoutsideofthesuperconductorusingthe ticcurrentdistributionsdiscussedfortheexperimentally
Biot-Savart law. We approximate the current density in applicable values.
the superconductor to be constant over its thickness d,
which is valid in the thin-film approximation where d is
smallerthantheLondonpenetrationdepthofthesuper-
conductor. The sheet current density is then defined as
J(x)=j(x)d, where j(x) is the local current density.
TheperpendicularfieldcomponentB (x)generatedin We approximate the integration in Eq.3 and 4 by a
z
the specimen plane z =0 by a constant B and by the Riemann Sum. First, we discretise the width of the
ext
sheetcurrentJ(x)alongy forthestripcanbewrittenas strip and disk (or ring) into N elements with the same
[26]: size ∆x = a/N or ∆r = a/N(∆r = a−(r )/N), and
0
denote the middle point of each element by x or r ,
n n
B (x)=B + µ0 (cid:90) aJ(u)(cid:16) 1 − 1 (cid:17)du, (1) n = 1,2...N. Then we rewrite Eq.3 in the form of a
z ext 2π x−u x+u Riemann Sum in matrix form
0
 
Forcylindricallysymmetricstructures,whereJ(r)iscir- j=N
clingclockwise,wecanwritetheperpendicularmagnetic J(xi,t)=τ2πxidBdetxt + (cid:88)M(xi,xj)∂J(∂xtj,t) .
field component with r0 = 0 for disks and r0 = 0.4a for j=1
rings as [26]: (6)
To solve the differential equation, we invert Eq.6 to find
Bz(r)=Bext+ 2µπ0 (cid:90) aJ(u)(cid:16)rE−(ku) − rK+(ku)(cid:17)du, (2) an equation for ∂J(∂xtj,t)
r0
j=N (cid:20) (cid:21)
where E(k) and K(k) are the first and s√econd kind of ∂J(∂xti,t) = (cid:88)M−1(xi,xj) J(xτj,t) −2πxjdBdetxt ,
the complete elliptic integrals with k = 4ru/(r + u)
j=1
and µ is the vacuum permeability [26, 27]. In order to (7)
0
find an expression for the sheet current density J(x) we whereM−1(x ,x )istheinverseofthematrixM(x ,x ).
i j i j
take into account the time-dependence of B and use Now, we can use the Euler method and the initial condi-
ext
Maxwell’s equations, to arrive at the equation of motion tionsJ(x ,t )=0andB (t )=0tocomputethesheet
i 0 ext 0
3
current density J(x ,t ) at t=t by can be found in [30].
i n n
(cid:32)j=N (cid:20) (cid:21)
J(xi,tn)= (cid:88)M−1(xi,xj) J(xτj,t) −2πxjdBdetxt A. Traps generated by a superconducting disk
j=1
(cid:33)
Consider a disk shaped superconductor in the rema-
+J(x ,t ) ×∆t, (8)
i n−1 nent state with a magnetic loading field pulse of B =
ext
0→1.2µ J →0. The second critical field in our calcu-
0 c
where ∆t = tn−tn−1. The same approach can be used lations is approximately 1.8µ0Jc. For symmetry reasons
for Eq.4. From the above Eq.8, we can see that the fi- the radial component of the total external field is always
nal sheet current density J(xi,tn) is only dependent on zero Br(r = 0,z) = 0 above the disk center. The z
the initial conditions and tn, not on dBext/dt. After we component of the total external field Bz(r =0,z) above
obtain J(x ,t ), we compute the B field outside of the the disk center at different heights is plotted in Fig.3.
i n
superconducting film (z >0) using the Biot-Savart law. Bz(r,z) is always zero at the surface center of the disk
In order to validate our numerical model, we consider (r =0,z =0), because the superconductor prevents any
the case of vortices induced in a superconducting strip, magnetic field perpendicular to its surface at this point.
for which an analytical solution exists [22]. Applying a The inset of Fig.4 shows the supercurrent distribution
magnetic field pulse with a magnitude of B perpen- obtained from Eq.4. We see that the supercurrent in the
ext
dicular to the strip surface, the magnetic flux penetrates diskisseparatedintotworegions;onewithcurrentprop-
the strip. We solve Eq.3 numerically and obtain the dis- agating clockwise and one with current in the counter-
tribution of supercurrents in the strip plotted in Fig.2 clockwise direction. Applying a small bias field on the
(solid lines). The small difference in the curves arises order of Bext/10 perpendicular to the disk, allows us to
from the fact that the analytical solution assumes that cancel Bz(r = 0,z = z0) at z0 and create a quadrupole
attheedge,wherethemagneticfluxpenetrates,thesheet trap for low-field seeking atoms. Fig.4 shows such a trap
current density takes the critical value. For the numeri- for Bbias = −0.05µ0Jc and z0 ∼ 1.2a. Increasing the
cal computation the sheet current density is completely bias field will move this trap closer to the disk surface,
determined by Eq.3. however in doing so the bias field will also induce cur-
rents into the disk, ensuring that the field at the disk
surface center remains 0. This puts a J dependent limit
c
1 onthetrapdepthfortrapscreatedveryclosetothedisk
surface.
numerics
0.5 analytics
]
Jc 0.3
of
nits 0 J]c
u 0
J [−0.5 fo stin µ 0.2
u
[
−10 0.2 0.4 0.6 0.8 1 Bz0.1
x [units of a]
FIG.2: (Coloronline)Distributionofthesheetcurrentden-
sity across the strip for B = 0 → µ J → 0. Solid line: 0 0.5 1 1.5
ext 0 c
sheet current density obtained by Eq.3; Dashed line: sheet z[units of a]
current density our former results [22].
FIG. 3: (Color online) Vertical component B (r,z) of the
z
magnetic field carried by vortices loaded by B = 0 →
ext
1.2µ J →0 vs z above the disk center r=0.
0 c
III. Magnetic traps
The superconductor has a memory of the history of
In this section, we apply Eq.4 to compute the sheet the external fields. Therefore it is possible to design a
current density induced in the superconducting disk and pulse sequence in which the role of the bias field is as-
ring by various loading fields. From the distribution of sumed by another induced supercurrent carried by the
the sheet current density we then compute the magnetic disk. Here we use two loading pulses with amplitudes
field carried by the vortices by means of the Biot-Savart of 0.8µ J and −0.4µ J respectively to write a vortex
0 c 0 c
law. An intuitive picture of the vortex loading process pattern in the disk, which results in a self-sufficient trap
4
lowed by a second pulse −0.6µ J . Due to the nonlinear
0 c
1.4 response of the material, this pulse sequence does not
result in a self-sufficient trap, and instead a quadrupole
trap trap can be formed far away from the chip surface by
1.2
applying an additional bias field B . By increasing
bias
]
a the value of B this trap is brought closer to the chip
of 1 surface wherebiiatseventually deforms into a ring trap.
s 0.5
unit0.8 ]of Jc 0 (cid:16164) The perpendicular bias field lifts the field zero on the
z [ nits
u−0.5 (a) 1.4
0.6 J [
−1
0 0.2 0.4 0.6 0.8 1 1.2
r [units of a]
0−.41.4 −1 −0.6 0 0.6 1 1.4 a]
r [units of a] of 1
s
nit 0.8
u
FIG. 4: (Color online) Equipotential lines showing a z [ trap
quadrupole trap above the disk center formed by the vortex 0.6
field together with a perpendicular field B = −0.05µ J .
bias 0 c
The vortex loading field pulse has an amplitude of 1.2µ J . 0.4
0 c −1.4 −1 −0.6 0 0.6 1 1.4
The field variation per contour line is 5×10−3µ0Jc. Inset: r [units of a]
Distribution of the sheet current density along the radial di-
rection resulting from the loading field pulse. (b) 1.4
1.2
above the disk center, as shown in Fig.5. Choosing typi- a]
cal experimental values for the critical current density of of 1
s
YBCO at liquid Nitrogen temperatures jc = 1MA/cm2 unit 0.8
we calculate a self-sufficient trap depth of 35µK for a z [ trap
disk with a diameter of 1mm. This trap has the feature
0.6
oftransformingintoaringtrapbyapplyinganadditional
perpendicular bias field B =0.01µ J . 0.4
bias 0 c −1.4 −1 −0.6 0 0.6 1 1.4
r [units of a]
(c) 1.4
1.4
trap
1.2
1.2
a]
s of a]1.0 units of 0.81 trap
nit z [
u0.8
z [ 0.6
0.6 0−.41.4 −1 −0.6 0 0.6 1 1.4
r [units of a]
0.4
−1.4 −1 −0.6 0 0.6 1 1.4 FIG. 6: (Color online) Equipotential lines for an atom
r [units of a]
with nonzero magnetic moment. The loading field pulses are
B = 0 → 1.2µ J → −0.6µ J → 0. (a) With an ad-
ext 0 c 0 c
FIG. 5: (Color online) A self-sufficient trap above the disk ditional bias field B = 0.03µ J , two traps are obtained
bias 0 c
centerformedbythevortexfieldcreatedbytwoloadingfield above the disk center. The field variation per contour line is
pulses with amplitudes 0.8µ0Jc and −0.4µ0Jc respectively. 1.5×10−3µ0Jc. (b) Bbias =0.035µ0Jc, the two traps merge.
The field variation per contour line is 9.5×10−4µ0Jc. The field variation per contour line is 1.54×10−3µ0Jc. (c)
B = 0.06µ J , a ring trap is formed. The field variation
bias 0 c
Whileitispossibletosplittheself-sufficienttrapintoa per contour line is 1.2×10−3µ0Jc.
ring, it does not necessarily imply a simple loading pro-
cedure with a high transfer efficiency from an external disk surface and lowers the quadrupole trap, producing
trap via the self-sufficient trap. A more straightforward two traps above the disk center, see Fig.6(a). Increasing
wayofproducingaringtrapfromanexternalquadrupole the bias field brings the two traps closer together, until
trap is to use a first magnetic field pulse of 1.2µ J fol- they merge at B = 0.03µ J , see Fig.6(b). Increas-
0 c bias 0 c
5
ing B further will turn the trap into a ring trap, as
bias (a) 0.8
showninFig.6(c). ContinuingtoincreaseB increases
bias
theradiusoftheringtrapuntilthetrapisloweredtothe 0.7 trap
surface. a]
of 0.6 0.4
B. Traps generated by a superconducting ring z [units 00..45 J [units of J]c−−−00000....2642
−0.8
0.3
0 0.2 0.4 0.6 0.8 1
r [units of a]
0.2
In this section we consider an annular super- −0.75 −0.5 −0.25 0 0.25 0.5 0.75
r [units of a]
conductor. The inset of Fig.8(a) shows the su-
percurrents carried by the vortices in the rema- (b) 0.7
nent state, when we apply a field pulse with a
0.6
magnitude of 0.8µ0Jc to a superconducting ring. nits of a]0.5 trap
z [u0.4
0.3
0.03
0.2
0.1
J]c −0.75 −0.5 −0.25r [unit0s of a]0.25 0.5 0.75
0
µof 0.02
s FIG. 8: (Color online) (a) A quadrupole trap formed by
nit a loading field pulse Bext = 0 → 0.8µ0Jc → 0 and a bias
u
[z field Bbias = −0.028µ0Jc. The field variation per contour
B 0.01 line is 2×10−3µ J . Inset: Distribution of the sheet cur-
0 c
rentdensityalongtheradialdirectionofthesuperconducting
ring. (b) A ring trap formed when increasing the bias field
0 0.5 1 1.5
to B =−0.06µ J . The field variation per contour line is
z [units of a] bias 0 c
5×10−3µ J .
0 c
FIG. 7: (Color online) B (r = 0,z) carried by the vortices
z
atdifferentz abovetheringcenter. Theloadingfieldpulseis
B =0→0.8µ J →0. 0.8
ext 0 c
trap
0.7
Similar to the superconducting disk, B (r = 0,z)
r ]
is always zero by symmetry. We plot the z compo- a0.6

f
nent of the magnetic field B (r = 0,z) generated by o
z
these supercurrents at different height z in Fig.7. If ts0.5
an additional perpendicular field is applied to cancel ni
u
B (r = 0,z = z ), a quadrupole trap can be formed [
z 0 z 0.4
at (0,z ). When B = −0.028µ J a quadrupole
0 bias 0 c
trap is formed at z ∼ 0.7a, as shown in Fig.8(a). For
0 0.3
small bias fields, increasing the magnitude of B
bias
reduces the trap distance to the chip surface. When
0.2
|B | ∼ 0.06µ J , the quadrupole trap transforms −1 −0.5 0 0.5 1
bias 0 c
r [units of a]
into a ring trap, as shown in Fig.8(b). Further in-
creasing the bias field will enlarge the radius of the
ring trap until the trap is lowered to the surface. FIG.9: (Coloronline)Aringtrapformedbyaloadingfield
Alternatively, a ring trap may be formed by apply- sequence Bext =0→µ0Jc →−0.5µ0Jc →0 and a bias field
ing two field pulses and a bias field as indicated in Bbias = 0.005µ0Jc. The double pulse sequence increases the
confinement of the atoms in the ring compared to the single
Fig.9. The first pulse has a magnitude of µ J , the
0 c
pulse loading sequence. The field variation per contour line
second has a magnitude of −0.5µ J and the bias field
0 c is 1.8×10−3µ J .
B = 0.005µ J . The ring trap of Fig.9 has a larger 0 c
bias 0 c
radius and distance to the surface as well as tighter
confinement than the one shown in Fig.8(b).
6
IV. Conclusion advantageofalowtraplossrateduetothereducedspin
flips of the atoms close to the chip surface. The poten-
We have investigated and designed a variety of con- tialcontributiontothespinflipratebyvortexmotionin
fining geometries for ultra cold neutral atoms based on the superconductor has yet to be experimentally investi-
type-II superconducting disk and ring structures. These gated.
geometriesincludequadrupoletraps,aself-sufficienttrap An additional control over the trap geometries, com-
and ring traps. We have shown that a quadrupole trap plementary to the magnetic field pulses, can be obtained
above a superconducting disk and ring can be formed bydressingtheatomicstateswitharadio-frequencyfield
by applying a single magnetic field loading pulse and [32, 33]. This can help to lift the magnetic field zero in-
adding an external bias field. Using a double pulse se- trinsicallypresentinquadrupoletraps,thusreducingthe
quenceweareabletoproduceaself-sufficienttrapabove Majorana spin flip rate. The low field noise in the pro-
the disk center. We have developed a procedure for the posedringtrapsmaymakethemanattractivecandidate
formation of ring traps based on superconducting disk fortheconstructionofaSagnac-typeinterferometer[34].
and ring structures. Furthermore we have shown an effi- Apersistentflowofultracoldatoms,evenananalogofthe
cient loading procedure for these ring traps. Apart from superconducting quantum interference device (SQUID),
methodsoutlinedinthispaperbasedonsuperconductors may be realized in these ring traps [35, 36]. Taking
in the mixed state, it is possible to produce quadrupole into account all the intrinsic advantages discussed high-
trapswitharing-shapedsuperconductorintheMeissner lights the potential of magnetic traps for neutral atoms
state using flux trapped inside the ring and a bias field. basedonpatternedvortexdistributionsinsuperconduct-
By varying the trapped flux and bias field magnitudes ing disks and rings.
ring traps can be produced. However, the limitations
on the imprinted current distribution make it impossible
to generate versatile magnetic field patterns [31]. The
Acknowledgments
key point of using type-II superconducting structures is
that a single microstructured element is capable of gen-
erating tailored complex trap geometries for a variety of WeacknowledgefinancialsupportfromNanyangTech-
experimental implementations. This allows us to control nological University (grant no. WBS M58110036), A-
thegeometryofconfiningpotentialsandtheirparameters Star (grant no. SERC 072 101 0035 and WBS R-144-
without having to fabricate a new atom chip. In addi- 000-189-305) and the Centre for Quantum Technologies,
tiontothisflexibility, superconductorshavetheintrinsic Singapore. WethankS.A.Cheongforfruitfuldiscussion.
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