Journal of Physics D: Applied Physics J. Phys. D: Appl ...leonid/J_Phys_D_2016.pdf · V Sokolovsky...

1 copy 2016 IOP Publishing Ltd Printed in the UK

1 Introduction

A lattice structure which allows one to trap and manipulate ultracold atoms is a tool for building quantum simulators for the Hubbard models and creating the BosendashEinstein conden-sates These structures are used in experiments in quantum optics quantum information processing atomic and molecular physics (see eg [1ndash5] and the references therein) Most often the optical lattices of atom traps are employed [1 2 6ndash8] Magnetic lattices the periodic arrays of magnetic potential minima created by permanent magnetic microstructures [7 9ndash13] or current-carrying wires [14] and bias fields provide a potentially powerful alternative Magnetic radio-frequency (RF) field was used to create RF dressed state potentials for

neutral atoms causing coherent splitting of a BosendashEinnstein condensate in a magnetic trap [15] A disadvantage of magn-etic traps is that at a distance of 1 μm or less from a metallic surface magnetic fluctuations cause significant loss of trapped atoms [16] Replacing the normal metal by a superconductor increases the atom lifetime in a magnetic trap [17ndash21] Atom traps on superconducting wires have been realized in [16 22ndash30] Additionally the technical noise can be decreased by employing the superconducting chips with a persistent cur-rent or trapped magnetic flux in the mixed state where magn-etic flux penetrates the type-II superconductor in the form of a vortex lattice [20 31ndash36] Superconducting chips were used to build a guided atom interferometer [28] and to measure the quantized magnetic flux in a superconducting ring [37]

Journal of Physics D Applied Physics

Lattices of ultracold atom traps over arrays of nano- and mesoscopic superconducting disks

Vladimir Sokolovsky1 and Leonid Prigozhin2

1 Physics Department Ben-Gurion University of the Negev Beer-Sheva 84105 Israel2 J Blaustein Institutes for Desert Research Ben-Gurion University of the Negev Sede Boqer Campus 84990 Israel

E-mail sokolovvbguacil

Received 16 September 2015 revised 14 February 2016Accepted for publication 2 March 2016Published 24 March 2016

AbstractA lattice of traps for ultracold neutral atoms is a promising tool for experimental investigation in quantum physics and quantum information processing We consider regular arrays of thin film type-II superconducting nanodisks with only one pinned vortex in each of them and also arrays of mesoscopic disks each containing many vortices whose distribution is characterized by the superconducting current density In both cases we show theoretically that the induced magnetic field can create a 3D lattice of magnetic traps for cold atoms without any additional bias field Applying a bias DC field parallel to the superconductor surface one can control the depth and sizes of the traps their heights above the chip surface potential barriers between the traps as well as the structure and dimension of the lattices

In the adiabatic approximation the atom cloud shape is represented by the shape of a closed iso-surface of the magnetic field magnitude chosen in accordance with the atom cloud temperature The computed trap sizes heights and the distances between the neighboring traps are typically from tens to hundreds nanometers for nanodisks and of the order of 1 μm for mesoscopic disks Our calculations show that the depth of magnetic traps on mesoscopic disks is typically between 03 G and 76 G for the nanodisks the depth is about 03 G

Keywords cold atoms superconducting chip magnetic trap lattice

(Some figures may appear in colour only in the online journal)

V Sokolovsky and L Prigozhin

Printed in the UK

165006

JPAPBE

copy 2016 IOP Publishing Ltd

2016

49

J Phys D Appl Phys

JPD

0022-3727

1010880022-37274916165006

Paper

16

Journal of Physics D Applied Physics

IOP

0022-372716165006+12$3300

doi1010880022-37274916165006J Phys D Appl Phys 49 (2016) 165006 (12pp)


2

Theoretically properties of magnetic atom traps over super-conducting chips (both in the Meissner and the mixed states) have been investigated in [38ndash43] in particular splitting and merging traps by varying the bias field [34 35] were simu-lated in [40 44] In these theoretical and experimental works the characteristic sizes of the superconducting chips in the mixed state were much larger than the average vortex separa-tion While the typical sizes of superconducting wires and of the atom cloud were from tens to hundreds of μm a vortex diameter determined by the London penetration depth λ is of the order of 100 nm The magnetic fields were therefore determined by the vortex density and not by the individual vortices (the mesoscopic approximation)

A trap for cold atoms created by the magnetic field of a single vortex pinned in a superconducting nanodisk and an external bias field parallel to the disk surface was theor-etically analyzed in [44] The size of the trap and its height above the superconductor surface are typically tens or hun-dreds of nanometers In the first approximation (without taking into account the response of a superconducting film to the magn etic field applied perpendicularly to the surface) a magnetic lattice for ultracold atoms induced by a nano- engineered vortex array in a thin-film type-II superconductor was analyzed in [45]

In this work we consider lattices of magnetic ultracold atom traps created by the magnetic field of vortices pinned in an array of thin type II superconducting meso- or nanodisks These lattices are created due to the symmetry of the magnetic field induced by the disks and the unique ability of a super-conductor in the mixed state to trap magnetic flux as a vortex set This set can be represented by a system of circular current loops in a disk To illustrate formation of a trap let us consider a pair of current loops (figure 1) By symmetry the x- and y-components of the magnetic field induced by the loop cur-rents cancel each other on the A-B line There is a point on this line where the z-component of this field also vanishes The magnitude of magnetic field increases with the distance form this point This point can be regarded as a center of magnetic trap for ultra-cold neutral atoms in a low field seeking state

Similar arguments can be employed to explain the forma-tion of a magnetic trap lattice above a regular array of super-conducting disks Trap lattices on a square array of disks are considered in section 2 Applying a bias field parallel to the superconducting film surface one can change trap sizes their depth height and number as well as the trap lattice dimension it is also possible to merge several traps into one and split it back into several traps (section 3) We consider also trap lat-tices over a linear array of disks (section 4) and discuss our results in section 5

2 Trap lattice on superconducting disks without bias field

We consider two kinds of thin type-II superconducting disks first the mesoscopic disks in which many vortices are pinned and second the nanodisks pinning a single vortex In the first case the magnetic field is created by currents in an array of disks these currents correspond to the distribution of vor-tices in the superconductors and are computed using the Bean critical state model [43 46] In the second case the magnetic field is induced by single vortices trapped in each disk these currents can be found by solving the modified London equa-tions [44] The disks are prepared from a thin superconducting film deposited on normal material substrate it is assumed that the film thickness a is much less than the disk radius R Hence it is possible to replace the bulk current density by its int egral across the film thickness and use the sheet current density J see [47] We note that an external magnetic field applied parallel to the film surface does not change the sheet current density

The r- and z-components Br and Bz of the magnetic field of a single disk are a superposition of the fields created by circular current loops in the dimensionless form

( )( )

( )( )

( )int π=

+ ++

minus minusminus +

⎡

⎣⎢

⎤

⎦⎥B r z r

J

r r zK m

r r z

r r zE m d

2

1z

0

1

0

02 2

02 2 2

02 2

(1)

( )( )

( )( )

( )int π=

+ +minus +

+ +minus +

⎡

⎣⎢

⎤

⎦⎥B r z r

J

r

z

r r zK m

r r z

r r zE m d

2r

0

1

0

02 2

202 2

02 2

(2)

where ( )

=+ +

m rr

r r z

4 0

02 2 ( )

intequivπ β

βminusK m

m0

2 d

1 sin2 and

( )int β βequiv minusπ

E m m1 sin d0

2 2 are the complete elliptic

integrals of the first and second kinds respectively Here all dimensions are normalized by R the magnetic field and sheet current density are normalized by micro=b Jn n0 and Jn respec-tively where Jn is differently defined for meso- and nanodisks (see below) and μ0 is the magnetic permeability of vacuum The difference between the meso- and nanodisk cases is in the distribution of sheet current densities

In the mesoscopic case a frozen (trapped) magnetic flux in a thin type-II superconducting disk can be induced by a pulse of orthogonal to the film uniform magnetic field [46] In a single disk only the azimuthal component J of the sheet current density is nonzero Neglecting the lower critical field

Figure 1 A scheme of magnetic trap created by two symmetric circular currents I1 and I2 Magnetic fields H1 and H2 of these currents cancel each other at a point on the AB line

J Phys D Appl Phys 49 (2016) 165006


3

one obtains for the zero-field-cooled disk in a perpendicular increasing magnetic field Ba (see [48])

( )

⎧

⎨⎪

⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟π=

minusminusminus

lt lt

minus lt lt

J r Br

b

b rr b

b r

2

arctan1

0

1 1

a

2

2 2 (3)

where ( )=b B1 cosh 2 a and we set =J Jn c the critical sheet current density assumed independent of the magnetic field If the applied field first increases from zero to Bm and then decreases to Ba (Bm gt 0 Ba lt Bm) the current density is (see [46 48])

( ) ( [ ] )= minus minusJ J r B J r B B 2 2m m a (4)

Magnetization of a mesoscopic disk by a pulse of the magn-etic field perpendicular to its surface 0 rarr Bm rarr 0 results in a changing sign sheet current density in the disk However if Bm ≫ 1 the sheet current density J equals 1 almost in the whole disk Thus for =B 5m we find from (4) that =J 1 for gtr 00135)

For a nanodisk we additionally assume that its thickness is much less than the London penetration depth λ and the single vortex is in the disk center To fix the vortex in this position artificial defects eg a hole or a nano-engineered antidote [15] can be introduced For the GinzburgndashLandau parameter

κ λ ξ= 1 where ξ is the coherence length the core of a vortex is narrow (its radius is close to ξ) and can be neglected The current density should satisfy the modified London equa-tions [49ndash51] and can be represented by a series in powers of ε πλ= Ra 2 the following expression for the sheet current density has been obtained [44]

( ) ( )ε= minus minus minus +J rr

r r1

1565 08 0193 6 (5)

where the normalizing current density is chosen as

=micro λ π

ΦJna

R202

0 Φ = times minus207 10015 Tm2 is the quantum of

magn etic flux If ε 1 it is sufficient to take ( ) =J rr

1If the disks in the array are sufficiently far from each other

ie the distance between the disk centers is larger than R3 (see figure 2) the disk currents are almost independent of each other and can be computed using equations (3)ndash(5) The magn-etic field creating the cold atom traps is the superposition

of fields induced by these currents In the adiabatic approx-imation the atom cloud shape can be represented by the shape of a closed iso-surface of the magnetic field magnitude B chosen in accordance with the atom cloud temperature

Let us consider a regular square disk lattice with the centers ( [ ] [ ] )minus minusm d n d2 1 2 1 in the =z 0 plane Here d2 is the minimal distance between the disk centers and m n = N minus1hellip minus101hellipN For the central part of a large lattice the boundary effects can be neglected and due to symmetry the x- and y-components of the total magnetic field are zero along the parallel to z-axis lines crossing the xy-plane in the points ( )md nd2 2 ( [ ] )minusm d nd2 1 2 and ( [ ] )minusmd n d2 2 1 The points on these lines where the z-component of the total field is zero too are regarded as the trap centers

Results of a simulation of the atom trap lattice (central part) based on a 10 times 10 array of mesoscopic disks with =d 15 (figure 3) show a two-level lattice of traps presented

by closed iso-surfaces of B The trap centers where =B 0 are at the heights =z 213 for the larger traps and =z 151 for the smaller ones the potential barrier between the traps of the different layers (the trap depth) is about 00023 (figure 3(e)) The potential barriers between the traps of the same layer are higher these barriers are about 00028 and 0007 for the larger and smaller traps respectively An iso-surface corresponding to a higher atom temperature (higher B value) forms a con-nected 3D structure (figures 3(b) and (e))

The trap heights and depths depend on the size of the disk lattice Thus the heights of the larger traps are 213 for 10 times 10 mesoscopic disks 267 for 30 times 30 ones and 327 for 100 times 100 disks the x- and y-coordinates of the trap centers remain practically the same This can be explained by the fact that the contribution of a faraway disk to the z-component of the total magnetic field is proportional to r1 3 and such contrib-utions are of the same sign On the other hand the radial comp-onent contributions decay as r1 4 and can cancel each other We note that the trap depth decreases as N1 which can limit the possible number of disks and therefore the trap lattice size

The simulation results obtained for a 10 times 10 nanodisk array with =d 15 (figure 4) are similar to those for the meso-disks (figure 3(a)) However in the chosen dimensionless variables the magnetic field induced by currents in nanodisks is about 16 times stronger than that induced by the currents

Figure 2 The normal to disk component of magnetic field Bz at =z 0 (a) mesoscopic disk for ( ) =J r 1 at ⩽r 1 (b) nanodisk for ( ) =J r r1 In the selected dimensionless units the disk radius equals 1



4

Figure 3 Trap lattice above a 10 times 10 array of mesoscopic disks (central part) =J 1 (a) iso-surfaces =B 0002 the centers of larger traps are at =z 213 of the smaller ones at =z 151 (b) iso-surfaces =B 0003 (c)ndash(e) contour plots of B in the planes =z 213 =z 151 and =y 0 correspondently Here and below the light-colored circles in contour plots (around the magnetic field maximums)

correspond to positions above the disks



5

in mesoscopic disks Hence although the contour plots of nanodisks are similar to the mesoscopic disk plots the levels of B and potential barriers between nanotraps are 16 times higher The trap heights are only slightly higher =z 215 (larger traps) and =z 156 (smaller traps) The differences are caused by different current densities ( )asympJ r 1 and ( ) asympJ r r1 in meso- and nanodisks respectively We note that although in the latter case the current density is very strong near the disk center the region ⩽ εr 1 does not amplify the magn-etic field much because presenting the integrand in (1) as a power series in r0 one can show that the contribution of this region is proportional to ε2

Using two pulses of normal to a single mesoscopic disk magnetic field rarr rarrB0 0m and rarr rarrminusB0 0a the atom trap was realized in an experiment [33] such trap was analyzed theoretically in [42 43] After the second pulse the sheet cur-rent density in a mesoscopic disk acquires the form

( ) ( [ ] ) ( )= minus minus +J J r B J r B B J r B 2 2 2 2m m a a (6)

shown in figure 5 In the nanodisk case any change of the cur-rent density can be caused only by a change of the number of vortices in the disk [52 53] we do not consider this situation here

A lattice of mesodisks with current (6) creates three levels of atom traps presented as above by closed iso-surfaces of magnetic field magnitude this configuration is complicated and we show a scheme of the trap lattice (figure 6) Note that the second pulse not only changes traps formed by the first pulse (their height does not change much but the depth becomes twice smaller) but creates also new traps placed above the disk centers (black ellipsoids in figure 6) The new traps are about 40 times deeper than the traps considered above and are similar to traps created by a single disk (see figure 7)

3 Trap lattice with a bias field

We now consider the traps created by the field of the disk cur-rents supplemented by a parallel to disks bias field Also in this case the results for meso- and nanodisk lattices are quali-tatively similar and the magnitude of magnetic field created by nanodisks is about 16 times stronger Therefore here we present our results only for the same mesoscopic disk lat-tice magnetized by a strong pulse of magnetic field ( )=J 1 Applying a bias field we can change the form and position of atom traps merger of traps is also possible Pairs of traps merge eg for the bias field ( )= minusB 0 0003 0bias (figures 3 and 8) The resulting potential barrier between traps becomes 00028 Further trap merging can be achieved by switching on the x-component of the bias field eg by applying

( )= minus minusB 0003 0003 0bias see figure 9 the depth and height of these traps are respectively 00053 and 165 Turning on and off the bias field we can change the dimension of the trap lattice from 3D to 2D and back

4 A chain of nanodisks

A chain of superconducting disks with or without a bias field can be also used to create magnetic trap lattices Let us con-sider an array of nanodisks pinning a single vortex in their

Figure 4 Trap lattice above a 10 times 10 array of nanodisks (central part) presented by iso-surfaces =B 00032 =J r1 the centers of larger traps are at =z 215 of the smaller ones at =z 156

Figure 5 Sheet current density induced in a mesoscopic disk by two pulses of magnetic field 0 rarr 3 rarr 0 and 0 rarr minus08 rarr 0

Figure 6 Sketch of the magnetic trap lattice created by the currents induced in mesoscopic disks by two magnetic field pulses rarr rarrB0 0m and rarr rarrminusB0 0a Shown are traps with the centers at the points minusm d nd2 1 2 143([ ] ) and ( ] )minusmd n d2 [2 1 143 (red) at ( )md nd2 2 211 (blue) and at ([ ] [ ] )minus minusm d n d2 1 2 1 032 (black) Black circles indicate the superconducting disk positions ( )=d 15



6

centers placed along the y-axis at ( [ ] )minusn d0 2 1 0 Without a bias field a 1D lattice of the traps with the centers at ( )nd0 2 138 is created (figures 10(a) and (b)) the depth of these traps is about 0006 Let ( ( ))=B B x zmax 0x

Mx be the

maximum of the x-component of the magnetic field created by the disk currents in the xndashz plane at y = 0 in our example with =d 2 we found =Bx

M 001 Application of a bias field ( )= minusB B 0 0bbias with gtB Bb x

M destroys these traps and cre-ates new traps in different places (see figures 10(c) and (d) for Bb = 002) the potential barriers between these traps are 0012 We found that these traps are inside a tubular lsquoglobalrsquo trap of the depth Bb (figure 10(d)) similar to the tube-like trap in figures 10(e) and (f) corresponding to a smaller bias field Bb = 0002 As in the case of a trap on a single nanodisk [44] the size and height of these traps and also their distances from the disks increase as the bias field is reduced the trap depth

decreases Further decrease of the bias field causes merger of the traps into a single one (figure 10(e) the blue surface) and appearance of new traps at asympy nd2 and nex 0 (figure 10(e) red closed surfaces) For =B 0006b the potential barrier between traps of these two types is about 0005 Further decrease of the bias field destroys the tube-like trap and transforms the trap lattice into that in figure 10(a)

5 Discussion

Our theoretical study suggests that lattices of magnetic traps for cold atoms can be created above a regular array of supercon-ducting disks These traps can be manipulated by applying a bias field parallel to the disk surfaces Although we considered only the square and linear disk arrays traps can be also created above other forms of disk arrays (rhombic triangular etc)

Figure 7 Contour plots of B in the =y 0 plane after magnetic field pulses 0 rarr 3 rarr 0 and 0 rarr minus08 rarr 0 (a) A single disc (b) one of the central disks in the lattice Here the origin is shifted to the disk center

Figure 8 Trap lattice (a) above a 10 times 10 array of mesodisks (central part) with =J 1 and contour plot (b) of B in the =z 17 plane the bias field ( )= minusB 0 0003 0bias Shown are iso-surfaces =B 00025 The height of the larger traps is 17 of the smaller ones minus147 Orange circles represent disks



7

the disks can be replaced by the circular superconducting wires or thin film rings with an induced persistent cur rent Presented simulation results show that qualitatively the traps above the nano- and mesoscopic disk lattices are similar In the chosen dimensionless variables the depth and gradient are about 16 times larger for the nanodisk based traps

The magnetic field gradient at the trap center and the depth Bdep of the trapping potential are commonly used to charac-terize the confinement of cold atoms in a magnetic trap To ensure atom trapping stability it is desirable that

⩾microB k T10dep B (7)

and also the magnetic field gradient should be strong enough to protect the atoms from the gravity pull Here B is the non-scaled (dimensional) magnetic field micro is the atom magnetic moment kB is the Boltzmann constant T is the atom cloud temperature For the most often employed in experiments 87Rb atoms in the = =F m2 2F state the trap depth at the atom gas temperature 1 μK should be not less than 007 G and the field gradient should be at least 15 G cmminus1 [39] (here F is the total atom spin and mF is its projection on the local field) Let us analyze under which conditions these stability criteria are satisfied for the trap lattices considered in our work

Since we assumed a great number of vortices Nv are pinned in each mesoscopic disk the disk cannot be arbitrary small This number can be estimated as

( ) ( )int intπ π micro

=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1

For =J 1 we found numerically using (1) that the integral on the right is 018 Hence to pin Nv vortices the disk radius should not be less than

microasymp

ΦR

N

Jv 0

0 c (8)

The critical sheet current density Jc of the superconducting film depends on the superconductor material substrate temper ature fabrication technology etc see eg [54ndash56] In atom trap experiments the niobium (Nb) [22 23 25ndash27 29] magnesium diboride (MgB2) [20 31] and high-temper-ature YBCO [28 33 35] superconducting films have been employed The critical sheet current density (see table 1) varies in the range from 11 times 104 A mminus1 to 30 times 104 A mminus1 For Nv = 200 and Jc = 11 times 104 A mminus1 (YBCO thin film with thickness of 300 nm at 77 K) the minimal radius of a mesoscopic disk is about 6 μm For other materials orand a lower chip temperature the disk may be smaller Thus for Jc = 30 times 104 A mminus1 the minimal radius is estimated as 1 μm It is easy to see that the depth of a trap does not depend on the disk radius and is proportional to micro J0 c the gradient of magn-etic field is scaled as micro J R0 c

The characteristic dimensionless trap depth in figures 3 4 7ndash9 can be taken as 0002 this means that depending on the superconductor material and chip temperature (table 1) the depth can vary from 03 G to 76 G According to stability criterion (7) this is sufficient to trap atoms at temperature of 4 μK (for the trap depth 03 G) and up to 100 μK (for 76 G) Furthermore the magnetic field gradient in these traps is approximately 0002 times 140 G6 μm = 470 G cmminus1 or more which is much higher than the required 15 G cmminus1 The field gradient requirement limits the disk radius from above

The radius of a nanodisk should be of the order of the London penetration depth λ This depth for the type-II super-conductors depends on superconductor film material produc-tion technology temperature etc For example for Nb3Sn λ is estimated as 65 nm for MgB2 it is about 110 nm [61] and about 200 nm for YBCO films [57] at the superconductor operation temperature Hence the typical radius of a nanodisk should be of the order of 100 nm Modern technology allows produc-tion of superconducting thin film structures with character-istic size of ~25 nm [62] In the nanodisk case the magnetic

Figure 9 Trap lattice (a) above a 10 times 10 array of mesodisks (central part) with =J 1 and contour plot (b) of B in the plane at trap height =z 165 the bias field ( )= minus minusB 0003 0003 0bias Shown are iso-surfaces =B 00025 The orange circles represent disks



8

field is normalized by micro πλ= ΦJ a R2n0 02 and contrary to the

case of mesodisks the magnetic fields and the trap depth are inversely proportional to the disk radius For R = λ = 100 nm and the film thickness a = 03 R the value of micro Jn0 is estimated as 100 G then the trap depth (see figure 3) is of the order

of micro J0003 n0 =03 G and the magnetic field gradient is about R03 = 3104 G cmminus1 which meets the stability requirements

for the potential barrierThe maximal dimensionless bias magnetic field in the

examples above is 002 For nanodisks this corresponds to 2 G

Figure 10 Trap lattice above a linear chain of 10 nanodisks (central part) =J r1 Top no bias field (a) the iso-surfaces =B 0005 (b) contour plot of B at the trap height =z 138 middle the bias field =B 002b (c) the iso-surfaces =B 001 the trap centers are ( )nd175 2 128 (d) contour plot of B at =z 128 bottom the bias field =B 0002b (e) the iso-surfaces =B 0004 (red) and =B 00009 (blue) (f) contour plot of B in the =y 0 plane



9

In the mesoscopic disk case the maximal bias field micro J002 0 c is also much less than the lower critical field (see table 1) Hence application of such a field parallel to the disk surface cannot lead to appearance of new vortices or influence the cur-rent density

Although we showed assuming the adiabatic approx-imation that the trap lattices above a superconducting disk array can be realized the atom cloud lifetime is limited by such factors as the Majoran instability and Johnson thermal magnetic noise The detailed analysis of these processes is beyond the scope of our work we will use the published results and estimates to evaluate these harmful influences

First in the center of all traps considered above the magn-etic field is zero This negatively influences on the lifetime of atoms in a trap because of the spontaneous spin flips (the Majorana instability) occurring if B ~ 0 [63] The Majorana instability of trapped atoms has been studied for nanotraps [44 45] and for mesoscopic traps (see eg [63ndash65]) To decrease the instability in the case of a trap on wires carrying a transport current it is possible to apply an additional DC field parallel to the wires [14 64] Applying such a field to the trap lattices considered in our work changes the form height and depth of the traps and can vary the trap lattice dimension but cannot increase the magnetic field in the trap center The atom lifetime in a trap upon the mesoscopic disks can be estimated using the experimental data as follows At the cloud temper-ature 120 μK the lifetime of ~10 s was achieved in a quadru-pole trap on the mesoscopic superconducting disk [33] The characteristic trap size in this work was about 015 mm Since the lifetime of a cloud in a quadrupole trap is proportional to the squared trap size [63] for the 6 μm traps on mesodisks considered in our work the time can be estimated as 10 ms

Assuming quantum adiabatic approximation and applying Fermirsquos rule the average lifetime of 87Rb atoms (in the ground state and subjected to thermal escape and Majorana spin flips) was estimated to be in the range 005ndash10 ms for a 100 nm nanotrap [44] The semiclassical estimate presented in the same work yields a similar range 005ndash35 ms

Applying a radio-frequency field can significantly increase the atom lifetime [45 63] This method allows one to achieve a 20 time longer lifetime of a micrometer-size cloud [63] The results [45] also indicate that the radio-frequency field can increase the atom lifetime in a nano-trap up to 015 s Similar

results can be expected for the trap lattices considered in our work

The heights of nano-traps are of the order of 100 nm at such trap distances from the surface of a conventional con-ductor the CasimirndashPolder force and the Johnson thermal magnetic noise exceed all other harmful influences on atom cloud and dominantly limit its lifetime (as long as technical noise is kept to a minimum) [39 64] Replacement of usual conductors by superconductors significantly decreases this noise and according to the theoretical estimates [17 19] the lifetime of atoms trapped near a superconducting layer in the Meissner state can be at least six orders of magnitude longer Analysis [18] suggests that in this case even at the trap height of 1 μm above a superconducting layer the cloud lifetime is limited mainly by environmental noises and may reach 5000 s while the lifetime of an atom cloud at such a distance from a normal metal current-carrying layer would not exceed 01 s Other advantages of superconductors are zero heat genera-tion and the ability to create magnetic fields due to trapped magnetic flux or a persistent current the latter enables one to eliminate the current supply fluctuations and increases the lifetime An estimated lifetime of 10 min in a magnetic trap 300 μm above an atom chip based on a niobium strip covered by a gold layer was reported in [16] Yet the experimental data [20 21] for superconducting chips with the trap height of 30 μm show an enhancement of the lifetime of only one order of magnitude indicating that additional noises reduce the life-time One may expect the atom cloud lifetime in nanotraps to be limited by other mechanisms rather than the thermal magn-etic noise

Becoming prominent for small atom-surface distances is the CasimirndashPolder force FCP which decreases the magnetic bar-rier and allows atoms to tunnel to the surface as was already observed in [66] In our case a superconducting disk array is deposited on a dielectric substrate and to estimate the CasimirndashPolder force we consider two limiting cases atoms above a superconductor and atoms above a dielectric substrate In both cases the distance between the trap center and surface zt is in the range from 013 μm to 022 μm (see figures 3 4 7ndash10) In this range the CasimirndashPolder potential can be approximated as

micro micro

π= minus

sim

Uz32 t

ss

203

(9)

Table 1 Parameters of superconducting films [22 23 25ndash27 29 57ndash60]

Superconductor Tc (K) Ts (K) Bc1 (G) Bc2 (kG)Film thickness (nm)

Critical sheet current density Jc (A mminus1) micro J0 c (G)

Nb ~95 4ndash6 gt1200 gt18 400ndash900 (16ndash36) times 104 200ndash450MgB2 ~40 4 gt250 gt100 1600 16 times 105 2000YBCO ~90 77 80ndash300 gt100 300 11 times 104 140

600ndash800 (12ndash21) times 104 150ndash260YBCO doped by Ag ~90 77 80ndash300 gt100 1000 3 times 104 380

30 times 104 380010 200ndash1000 gt1000

Note Tc and Ts are the critical and operation temperatures of a superconductor Bc1 and Bc2 are the lower and upper magnetic critical fields at the temperature Ts The first critical field of YBCO superconductors strongly depends on the field direction parallel or perpendicular to the c-axis Superconducting materials exhibit the Meissner effect below the lower critical field and pass into the normal state if the field is larger than the upper critical field



10

for a superconductor [67] and

= minussim

U C zd tCP4 (10)

for a dielectric surface [66 68] (here micros is the projection of the atom magnetic moment micro on the superconductor surface which can be estimated as micro micro= FB where microB is the Bohr magneton) Our estimation using (9) showed that the CasimirndashPolder force is much less than the magnetic force in the trap

asympsim micro

FmB

R

dep

tr where Rtr is the characteristic trap size (lsquo~rsquo means

the dimensional units) From the experiments with a 300 μm thick silicon substrate with a 1 μm thick Si3N4 layer the coef-ficient CCP was estimated as 82 10minus56 Jm4 at ⩾z 05t μm [66] a close value 11 10minus56 Jm4 was theoretically predicted for sapphire [68] Let us estimate the ratio of the CasimirndashPolder force from a dielectric substrate to the magnetic force in the trap lattice presented in figure 4 asymp asympR z04 17ttr and asymp asympR z1 23ttr for the smaller and larger traps corre-spondently =B 00035dep and =F 2 for both traps (to take the superconductor layer thickness into account we slightly increased the trap height above the dielectric surface) Our calculation showed that this ratio is approximately 014 and 0078 for the smaller and larger traps respectively Hence the CasimirndashPolder force is significantly weaker than the magn-etic forces and the CasimirndashPolder interaction decreases the potential barrier by about 10 It is worth to note that these values are upper estimates proximity of the superconducting disks can decrease the CasimirndashPolder interaction and it is also predicted that at low distances from a dielectric surface the force can be several times less than the estimate (10) (see figure 3 in [68]) In the experiments with normal-metal-based magnetic traps for cold atoms above a dielectric surface the CasimirndashPolder force limits the atom-surface distance to ~1 μm see eg [66] In our smaller traps (figure 4) the magnetic field gradient is about sdot minus85 10 G cm4 1 at asympz 170 nmt while for the data presented in [66] this gradient can be estimated as lt minus100 G cm 1 at the atom-surface distance of ~1 μm and decreases closer to the surface The larger magnetic field gra-dient in nanotraps allows one to overcome the CasimirndashPolder force closer to the surface

The estimation above was done for the central part of the trap lattice Limited size of the disk array causes a non-uniformity of magnetic traps in the lattice Let us compare the characteris-tics of the most different traps created by the 10 times 10 nanodisk array a central trap and a corner one Our calculation predicts that the depth of a smaller corner trap is ~15 times higher and its radius is twice smaller than those of the central trap However the corner trap height is 14 ie by about 03 smaller Hence for the corner trap the ratio of the CasimirndashPolder and magnetic forces is approximately the same as for the central trap

We expect that the lifetime of atoms in both meso- and nanotraps is mainly determined by the Majorana instability and does not exceed 10 ms applying a radio-frequency field can significantly increase the atom lifetime up to the order of 01 s

At a low temperature less than asymp 200 nK for 87Rb in the

= =F m2 2F state [69] the BosendashEinstein condensate is

created and three-body recombination plays a crucial role in

atom loss (see eg [69 70] and the references therein) The rate of the atom loss is mainly determined by the squared atomic density and in an experiment with the macro-scopic magnetic trap [69] atomic density in the condensate decreased at asymp 75 nK from times minus2 10 cm14 3 to times minus7 10 cm13 3 in 16 s Therefore the characteristic lifetime for a macroscopic trap lattice can be estimated as of the order of 10 s To estimate the time for a nanotrap lattice the tunneling and surface prox-imity effects should be also taken into account According to experimental results [71] the lifetime in optical lattices can be several times smaller than in magnetic traps Using this result and taking into account that the optical trap sizes are ~500 nm ie of the same order as the size of considered nanotraps we expect the BosendashEinstein condensate lifetime of few seconds in the nanotrap lattices obviously these questions need fur-ther investigation

Development of the atom loading procedure is a nontrivial problem which is out of the scope of this work we sup-pose however that some of the existing techniques can be used also in the case of trap lattices considered above The loading procedures have been developed for magnetic traps (without any bias field) on a single superconducting disk and a single square in [33] and [34] respectively The character-istic chip size in these works was 1 mm Since a 10 times 10 array of mesodisks with the radius of 10 μm occupies only about 04 mm times 04 mm square the same technique can possibly be employed to load atoms into the lattices of traps

Application of a bias field to a trap lattice above a linear disk chain leads to appearance of a long trap (figure 10(e)) similar to traps created by a long wire current and a bias field [64] Hence the atoms can be first loaded into this long trap using the loading technique [64] Then changing the bias field it should be possible to split this long trap into a linear lattice of traps (see figure 10)

The proposed atom trap lattices possess several advantages in comparison with optic lattices [1 2 6ndash8] trap lattices based on RF dressed state potentials [15] magnetic lattices created by permanent magnetic microstructures [7 9ndash13] or current-carrying wires [14] The main advantages are the possibility to create trap lattices without any external field and transport current reduced technical noise and absence of the conductor heating achievable trap height of the order of 100 nm allows one to study the atom-surface interactions at distances which are very difficult to achieve using the usual traps employment of superconductors decreases the Johnson noise and increases the atom lifetime In addition merging and splitting the atom traps as well as changing the trap lattice dimension can be relatively easy realized by varying the bias field Finally replacing a superconducting chip by another one enables one to vary the trap lattice configuration keeping the same set-up and loading procedure

Summarizing 3D lattices of cold atom traps can be cre-ated without any bias field using an array of superconducting meso- or nanodisks Varying the bias field one can control the characteristics of traps merge several traps into one trap and then split it again into several traps and even change the trap lattice dimension The trap sizes heights and the dis-tances between the neighboring traps are typically hundreds



11

nanometers for nanodisks and of the order of 1 μm for mes-oscopic disks Such lattices can be used for experimental investigation of coherence and decoherence of atom clouds tunneling of cold atoms including atoms in the BosendashEinstein condensate state

Acknowledgments

The authors appreciate helpful comments by R Folman

References

[1] Bloch I Dalibard J and Zwerger W 2008 Many-body physics with ultracold gases Rev Mod Phys 80 885

[2] Bloch I 2005 Ultracold quantum gases in optical lattices Nat Phys 1 23

[3] Elliott T J Kozlowski W Caballero-Benitez S F and Mekhov I B 2015 Multipartite entangled spatial modes of ultracold atoms generated and controlled by quantum measurement Phys Rev Lett 114 113604

[4] Goldman N Beugnon J and Gerbier F 2013 Identifying topological edge states in 2D optical lattices using light scattering Eur Phys J Spec Top 217 135

[5] Scarola V W and Das Sarma S 2006 Cold-atom optical lattices as quantum analog simulators for aperiodic 1D localization without disorder Phys Rev A 73 041609

[6] Miranda M Inoue R Okuyama Y Nakamoto A and Kozuma M 2015 Site-resolved imaging of ytterbium atoms in a 2D optical lattice Phys Rev A 91 063414

[7] Leung V Y F Tauschinsky A van Druten N J and Spreeuw R J C 2011 Microtrap arrays on magnetic film atom chips for quantum information science Quantum Inf Process 10 955

[8] Fallani L and Kastberg A 2015 Cold atoms a field enabled by light Europhys Lett 110 53001

[9] Whitlock S Gerritsma R Fernholz T and Spreeuw R J C 2009 2D array of microtraps with atomic shift register on a chip New J Phys 11 023021

[10] West A D Weatherill K J Hayward T J Fry P W Schrefl T Gibbs M R J Adams C S Allwood D A and Hughes I G 2012 Realization of the manipulation of ultracold atoms with a reconfigurable nanomagnetic system of domain walls Nano Lett 12 4065

[11] Herrera I et al 2015 Sub-micron period lattice structures of magnetic microtraps for ultracold atoms on an atom chip J Phys D Appl Phys 48 115002

[12] Leung V Y F et al 2014 Magnetic-film atom chip with 10 μm period lattices of microtraps for quantum information science with Rydberg atoms Rev Sci Instrum 85 053102

[13] Singh M Volk M Akulshin A Sidorov A McLean R and Hannaford P 2008 1D lattice of permanent magnetic microtraps for ultracold atoms on an atom chip J Phys B At Mol Opt Phys 41 065301

[14] Grabowski A and Pfau T 2003 A lattice of magneto-optical and magnetic traps for cold atoms Eur Phys J D 22 347

[15] Hofferberth S Lesanovsky I Fischer B Verdu J and Schmiedmayer J 2006 Radiofrequency-dressed-state potentials for neutral atoms Nat Phys 2 710

[16] Emmert A Lupaşcu A Nogues G Brune M Raimond J M and Haroche S 2009 Measurement of the trapping lifetime close to a cold metallic surface on a cryogenic atom-chip Eur Phys J D 51 173

[17] Skagerstam B-S K Hohenester U Eiguren A and Rekdal P K 2006 Spin decoherence in superconducting atom chips Phys Rev Lett 97 070401

[18] Hohenester U Eiguren A Scheel S and Hinds E A 2007 Spin-flip lifetimes in superconducting atom chips BardeenndashCooperndashSchrieffer versus Eliashberg theory Phys Rev A 76 033618

[19] Skagerstam B-S K and Rekdal P K 2007 Photon emission near superconducting bodies Phys Rev A 76 052901

[20] Hufnagel C Mukai T and Shimizu F 2009 Stability of a superconductive atom chip with persistent current Phys Rev A 79 053641

[21] Kasch B Hattermann H Cano D Judd T E Scheel S Zimmermann C Kleiner R Koelle D and Fortaacutegh J 2010 Cold atoms near superconductors atomic spin coherence beyond the Johnson noise limit New J Phys 12 065024

[22] Nirrengarten T Qarry A Roux C Emmert A Nogues G Brune M Raimond J M and Haroche S 2006 Realization of a superconducting atom chip Phys Rev Lett 97 200405

[23] Minniberger S et al 2014 Magnetic conveyor belt transport of ultracold atoms to a superconducting atom chip Appl Phys B 116 1017

[24] Cano D Hattermann H Kasch B Zimmermann C Kleiner R Koelle D and Fortaacutegh J 2011 Experimental system for research on ultracold atomic gases near superconducting microstructures Eur Phys J D 63 17

[25] Wang S X Ge Y Labaziewicz J Dauler E Berggren K and Chuang I L 2010 Superconducting microfabricated ion traps Appl Phys Lett 97 244102

[26] Emmert A Lupaşcu A Brune M Raimond J M Haroche S and Nogues G 2009 Microtraps for neutral atoms using superconducting structures in the critical state Phys Rev A 80 061604

[27] Roux C Emmert A Lupascu A Nirrengarten T Nogues G Brune M Raimond J M and Haroche S 2008 BosendashEinstein condensation on a superconducting atom chip Europhys Lett 81 56004

[28] Muumlller T Wu X Mohan A Eyvazov A Wu Y and Dumke R 2008 Towards a guided atom interferometer based on a superconducting atom chip New J Phys 10 073006

[29] Bernon S et al 2013 Manipulation and coherence of ultra-cold atoms on a superconducting atom chip Nat Commun 4 2380

[30] Cano D Kasch B Hattermann H Kleiner R Zimmermann C Koelle D and Fortaacutegh J 2008 Meissner effect in superconducting microtraps Phys Rev Lett 101 183006

[31] Mukai T Hufnagel C Kasper A Meno T Tsukada A Semba K and Shimizu F 2007 Persistent supercurrent atom chip Phys Rev Lett 98 260407

[32] Imai H Inaba K Tanji-Suzuki H Yamashita M and Mukai T 2014 BosendashEinstein condensate on a persistent-supercurrent atom chip Appl Phys B 116 821

[33] Shimizu F Hufnagel C and Mukai T 2009 Stable neutral atom trap with a thin superconducting disc Phys Rev Lett 103 253002

[34] Siercke M Chan K S Zhang B Beian M Lim M J and Dumke R 2012 Reconfigurable self-sufficient traps for ultracold atoms based on a superconducting square Phys Rev A 85 041403

[35] Muumlller T Zhang B Fermani R Chan K S Lim M J and Dumke R 2010 Programmable trap geometries with superconducting atom chips Phys Rev A 81 053624

[36] Muumlller T Zhang B Fermani R Chan K S Wang Z W Zhang C B Lim M J and Dumke R 2010 Trapping of ultra-cold atoms with the magnetic field of vortices in a thin-film superconducting micro-structure New J Phys 12 043016

[37] Weiss P et al 2015 Sensitivity of ultracold atoms to quantized flux in a superconducting ring Phys Rev Lett 114 113003

[38] Cano D Kasch B Hattermann H Koelle D Kleiner R Zimmermann C and Fortaacutegh J 2008 Impact of the Meissner effect on magnetic microtraps for neutral atoms near superconducting thin films Phys Rev A 77 063408



12

[39] Dikovsky V Sokolovsky V Zhang B Henkel C and Folman R 2009 Superconducting atom chips advantages and challenges Eur Phys J D 51 247

[40] Zhang B Fermani R Muumlller T Lim M J and Dumke R 2010 Design of magnetic traps for neutral atoms with vortices in type-II superconducting microstructures Phys Rev A 81 063408

[41] Sokolovsky V Prigozhin L and Dikovsky V 2010 Meissner transport current in flat films of arbitrary shape and a magnetic trap for cold atoms Supercond Sci Technol 23 065003

[42] Sokolovsky V Prigozhin L and Barrett J W 2014 3D modeling of magnetic atom traps on type-II superconductor chips Supercond Sci Technol 27 124004

[43] Zhang B Siercke M Chan K S Beian M Lim M J and Dumke R 2012 Magnetic confinement of neutral atoms based on patterned vortex distributions in superconducting disks and rings Phys Rev A 85 013404

[44] Sokolovsky V Rohrlich D and Horovitz B 2014 Trapping neutral atoms in the field of a vortex pinned by a superconducting nanodisk Phys Rev A 89 053422

[45] Romero-Isart O Navau C Sanchez A Zoller P and Cirac J I 2013 Superconducting vortex lattices for ultracold atoms Phys Rev Lett 111 145304

[46] Mawatari Y Sawa A and Obara H 1996 Critical state of YBa2Cu3Oy disc in perpendicular fields Physica C Supercond 258 121

[47] Brandt E H and Indenbom M 1993 Type-II-superconductor strip with current in a perpendicular magnetic field Phys Rev B 48 12893

[48] Mikheenko P N and Kuzovlev Y E 1993 Inductance measurements of HTSC films with high critical currents Physica C Supercond 204 229

[49] Fetter A L 1980 Flux penetration in a thin superconducting disk Phys Rev B 22 1200

[50] Clem J 1975 Simple model for the vortex core in a type II superconductor J Low Temp Phys 18 427

[51] Carneiro G and Brandt E H 2000 Vortex lines in films fields and interactions Phys Rev B 61 6370

[52] Buzdin A I and Brison J P 1994 Vortex structures in small superconducting disks Phys Lett A 196 267

[53] Kanda A Baelus B J Peeters F M Kadowaki K and Ootuka Y 2004 Experimental evidence for giant vortex states in a mesoscopic superconducting disk Phys Rev Lett 93 257002

[54] Parinov I A 2012 Microstructure and Properties of High-Temperature Superconductors 2nd edn (Berlin Springer)

[55] Pan A V Pysarenko S V Wexler D Rubanov S and Dou S X 2007 Multilayering and Ag-doping for properties and performance enhancement in YBa2Cu3O7 films IEEE Trans Appl Supercond 17 3585

[56] Moon S H Yun J H Lee H N Kye J I Kim H G Chung W and Oh B 2001 High critical current densities in

superconducting MgB2 thin films Appl Phys Lett 79 2429

[57] Karasik V R and Shebalin I Y 1970 Superconducting properties of pure niobium Sov PhysmdashJETP 30 1068

[58] Buzea C and Yamashita T 2001 Review of the superconducting properties of MgB2 Supercond Sci Technol 14 R115

[59] Liang R Dosanjh P Bonn D A Hardy W N and Berlinsky A J 1994 Lower critical fields in an ellipsoid-shaped YBa2Cu3O695 single crystal Phys Rev B 50 4212

[60] Lamura G Aurino M Andreone A and Villeacutegier J-C 2009 First critical field measurements of superconducting films by third harmonic analysis J Appl Phys 106 053903

[61] Jin B B Klein N Kang W N Kim H-J Choi E-M Lee S-I Dahm T and Maki K 2002 Energy gap penetration depth and surface resistance of MgB2 thin films determined by microwave resonator measurements Phys Rev B 66 104521

[62] Sochnikov I Shaulov A Yeshurun Y Logvenov G and Bozovic I 2010 Large oscillations of the magnetoresistance in nanopatterned high-temperature superconducting films Nat Nano 5 516

[63] Petrich W Anderson M H Ensher J R and Cornell E A 1995 Stable tightly confining magnetic trap for evaporative cooling of neutral atoms Phys Rev Lett 74 3352

[64] Folman R Kruumlger P Schmiedmayer J Denschlag J and Henkel C 2002 Advances in Atomic Molecular and Optical Physics vol 48 ed B Benjamin and W Herbert (New York Academic) p 263

[65] Brink D M and Sukumar C V 2006 Majorana spin-flip transitions in a magnetic trap Phys Rev A 74 035401

[66] Lin Y-J Teper I Chin C and Vuletić V 2004 Impact of the CasimirndashPolder potential and Johnson noise on BosendashEinstein condensate stability near surfaces Phys Rev Lett 92 050404

[67] Haakh H Intravaia F Henkel C Spagnolo S Passante R Power B and Sols F 2009 Temperature dependence of the magnetic CasimirndashPolder interaction Phys Rev A 80 062905

[68] Antezza M Pitaevskii L P and Stringari S 2004 Effect of the CasimirndashPolder force on the collective oscillations of a trapped BosendashEinstein condensate Phys Rev A 70 053619

[69] Soumlding J Gueacutery-Odelin D Desbiolles P Chevy F Inamori H and Dalibard J 1999 Three-body decay of a rubidium BosendashEinstein condensate Appl Phys B 69 257

[70] Harter A Krukow A Deisz M Drews B Tiemann E and Denschlag J H 2013 Population distribution of product states following three-body recombination in an ultracold atomic gas Nat Phys 9 512

[71] Laburthe-Tolra B OrsquoHara K M Huckans J H Phillips W D Rolston S L and Porto J V 2004 Observation of reduced three-body recombination in a correlated 1D degenerate bose gas Phys Rev Lett 92 190401



2

Theoretically properties of magnetic atom traps over super-conducting chips (both in the Meissner and the mixed states) have been investigated in [38ndash43] in particular splitting and merging traps by varying the bias field [34 35] were simu-lated in [40 44] In these theoretical and experimental works the characteristic sizes of the superconducting chips in the mixed state were much larger than the average vortex separa-tion While the typical sizes of superconducting wires and of the atom cloud were from tens to hundreds of μm a vortex diameter determined by the London penetration depth λ is of the order of 100 nm The magnetic fields were therefore determined by the vortex density and not by the individual vortices (the mesoscopic approximation)

A trap for cold atoms created by the magnetic field of a single vortex pinned in a superconducting nanodisk and an external bias field parallel to the disk surface was theor-etically analyzed in [44] The size of the trap and its height above the superconductor surface are typically tens or hun-dreds of nanometers In the first approximation (without taking into account the response of a superconducting film to the magn etic field applied perpendicularly to the surface) a magnetic lattice for ultracold atoms induced by a nano- engineered vortex array in a thin-film type-II superconductor was analyzed in [45]

In this work we consider lattices of magnetic ultracold atom traps created by the magnetic field of vortices pinned in an array of thin type II superconducting meso- or nanodisks These lattices are created due to the symmetry of the magnetic field induced by the disks and the unique ability of a super-conductor in the mixed state to trap magnetic flux as a vortex set This set can be represented by a system of circular current loops in a disk To illustrate formation of a trap let us consider a pair of current loops (figure 1) By symmetry the x- and y-components of the magnetic field induced by the loop cur-rents cancel each other on the A-B line There is a point on this line where the z-component of this field also vanishes The magnitude of magnetic field increases with the distance form this point This point can be regarded as a center of magnetic trap for ultra-cold neutral atoms in a low field seeking state

Similar arguments can be employed to explain the forma-tion of a magnetic trap lattice above a regular array of super-conducting disks Trap lattices on a square array of disks are considered in section 2 Applying a bias field parallel to the superconducting film surface one can change trap sizes their depth height and number as well as the trap lattice dimension it is also possible to merge several traps into one and split it back into several traps (section 3) We consider also trap lat-tices over a linear array of disks (section 4) and discuss our results in section 5

2 Trap lattice on superconducting disks without bias field

We consider two kinds of thin type-II superconducting disks first the mesoscopic disks in which many vortices are pinned and second the nanodisks pinning a single vortex In the first case the magnetic field is created by currents in an array of disks these currents correspond to the distribution of vor-tices in the superconductors and are computed using the Bean critical state model [43 46] In the second case the magnetic field is induced by single vortices trapped in each disk these currents can be found by solving the modified London equa-tions [44] The disks are prepared from a thin superconducting film deposited on normal material substrate it is assumed that the film thickness a is much less than the disk radius R Hence it is possible to replace the bulk current density by its int egral across the film thickness and use the sheet current density J see [47] We note that an external magnetic field applied parallel to the film surface does not change the sheet current density

The r- and z-components Br and Bz of the magnetic field of a single disk are a superposition of the fields created by circular current loops in the dimensionless form

( )( )

( )( )

( )int π=

+ ++

minus minusminus +

⎡

⎣⎢

⎤

⎦⎥B r z r

J

r r zK m

r r z

r r zE m d

2

1z

0

1

0

02 2

02 2 2

02 2

(1)

( )( )

( )( )

( )int π=

+ +minus +

+ +minus +

⎡

⎣⎢

⎤

⎦⎥B r z r

J

r

z

r r zK m

r r z

r r zE m d

2r

0

1

0

02 2

202 2

02 2

(2)

where ( )

=+ +

m rr

r r z

4 0

02 2 ( )

intequivπ β

βminusK m

m0

2 d

1 sin2 and

( )int β βequiv minusπ

E m m1 sin d0

2 2 are the complete elliptic

integrals of the first and second kinds respectively Here all dimensions are normalized by R the magnetic field and sheet current density are normalized by micro=b Jn n0 and Jn respec-tively where Jn is differently defined for meso- and nanodisks (see below) and μ0 is the magnetic permeability of vacuum The difference between the meso- and nanodisk cases is in the distribution of sheet current densities

In the mesoscopic case a frozen (trapped) magnetic flux in a thin type-II superconducting disk can be induced by a pulse of orthogonal to the film uniform magnetic field [46] In a single disk only the azimuthal component J of the sheet current density is nonzero Neglecting the lower critical field

Figure 1 A scheme of magnetic trap created by two symmetric circular currents I1 and I2 Magnetic fields H1 and H2 of these currents cancel each other at a point on the AB line



3


( )

⎧

⎨⎪

⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟π=

minusminusminus

lt lt

minus lt lt

J r Br

b

b rr b

b r

2

arctan1

0

1 1

a

2

2 2 (3)







r r1

1565 08 0193 6 (5)


=micro λ π

ΦJna

R202














4





5
















6


Mx be the





5 Discussion






7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































3


( )

⎧

⎨⎪

⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟π=

minusminusminus

lt lt

minus lt lt

J r Br

b

b rr b

b r

2

arctan1

0

1 1

a

2

2 2 (3)







r r1

1565 08 0193 6 (5)


=micro λ π

ΦJna

R202














4





5
















6


Mx be the





5 Discussion






7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































4





5
















6


Mx be the





5 Discussion






7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































5
















6


Mx be the





5 Discussion






7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































6


Mx be the





5 Discussion






7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































7







=Φ

=Φ

N B r r rR J

B r r r2

0 d2

0 d v

R

z z0 0

20 c

0 0

1


microasymp

ΦR

N

Jv 0

0 c (8)







8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































8









9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































9









micro micro

π= minus

sim

Uz32 t

ss

203

(9)






30 times 104 380010 200ndash1000 gt1000




10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































10


= minussim

U C zd tCP4 (10)


asympsim micro

FmB

R

dep















11


Acknowledgments


References









































12





































11


Acknowledgments


References









































12





































12




































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