Inductively guided circuits for ultracold dressed atoms

2.1 MB·

by  German Sinuco-Leon

Inductively guided circuits for ultracold dressed atoms
German Sinuco-Le´on,1 Kathryn Burrows,1 Aidan S. Arnold,2 and Barry M. Garraway1
1Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom
2Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
(Dated: January 28, 2014)
We propose a flexible and robust scheme to create closed quasi-one dimensional guides for ultra-
cold atoms through the dressing of hyperfine sub-levels of the atomic ground state. The dressing
fieldisspatiallymodulatedbyinductiveeffectsoveramicro-engineeredconductingloop,freeingthe
trapping region from leading wires in its proximity. We show that arrays of connected ring traps
can also be created by carefully designing the shape of the conducting loop. We report on charac-
teristics of the trap and mechanisms that limit the range of parameters available for experimental
implementation,includingnon-adiabaticlossesandheatdissipationbyinducedcurrents. Weoutline
conditions to select appropriate parameters for operation of the trap with atom-chip technology.
PACSnumbers: 37.10.Gh,67.85.-d
4
1
Techniques for trapping and manipulating ultra-cold isfreefrompotentialsymmetrybreakingcurrentcarrying
0
2 atomic matter on the micron scale have dramatically de- wires in the vicinity of the trapping volume [16, 21–24].
veloped during the last two decades. In general terms, In addition, the system can be designed to create mul-
n
such fine control is possible thanks to precise tempo- tiply connected atomic circuits, e.g. arrays of connected
a
J ral and spatial resolution of electric and magnetic fields, ringtraps,havinginmindapplicationsthatbenefitfrom
7 demonstratedinseveralexperimentalconfigurationsthat matter-wave interferometry as in [5].
2 benefit from advances in micro-fabrication, electronic For illustrative purposes, we present calculations for
control, and laser technology [1]. These technical de- thehyperfinelevelstructureof87Rb,denotedby|F,m (cid:105),
F
] velopmentshaveleadtoimpressiveexperimentaldemon- andshowninFig. 1(a). Nevertheless,ourconclusionsare
h
p strations of macroscopic quantum phenomena, such as straightforwardly extended to other atomic species with
- matterwaveinterferometry[2]andpersistentmatterflux similar energy level structure.
m [3, 4], and are at the heart of promising developments of
to tfoercmhnaotlioognictaelchanpoplloicgayti[o6n]sanindmqueatrnotluomgys[i5m],ulqautaonrstu[7m, 8in].- (a)Ω Ω Ω (b) BAC  BDC
a -1 0 +1 ΔE
. Ring traps (and toroidal traps) are of particular inter- 1
s
c est because of the possibility they offer to study physical F=2 z
i phenomena in a non-trivial geometry with true periodic y
s
y boundary conditions, and to create atomic analogues of x
I
h solid state electronic devices (e.g. [5]). Trapping of cold
Ω
p gases in such geometries has been demonstrated with a -1
1 [ voaprtiiectaylofifeeldxspe[3ri,m4e,n9t–a1l1t]ecohrnmiqaugens,erteicqufiierlidngdcisotnrtirboultoiovnesr ΔE-1 Ω-1 Ω0 Δ F=1 (c) BAC Bind Bt1otal zz
ΔE 1 + =
v [12–15]. In addition, there are several proposals for ring Zeeman x
6 traps that rely solely on the field produced by current m= -2 -1 0 1 2 2a
F
9 carrying conductors, being suitable to be implemented
7 with atom-chip technology (e.g. [14, 16]), in which feed- FIG. 1. (a) Ground state energy level structure of 87Rb.
6 ing wires can break desirable symmetries. Such an ef- Arrows indicate magnetic dipole couplings between pairs of
.
1 fect can be mitigated by employing inductive coupling hyperfine sub-levels, corresponding to linear (solid lines) and
0 [17], which has been demonstrated in millimetre sized circular(dashedlines)polarizationsofthemagneticfield. (b)
4 Sketchofanatom-chipconfigurationtocreateaninductively
ring traps [15] and proposed for microscopic ring traps
1 coupled guide for ultra-cold atoms. It shows the magnetic
based on generalizing the radio-frequency dressing ap-
: field configuration (arrows), a closed conductor (orange) and
v proach [18] to an inductive system [19].
the generated trapping region (green). (c) Side view of the
i
X In this contribution we show that highly configurable magnetic field distribution in the neighbourhood of the con-
one-dimensionalmicroscopicguidesforultra-coldatomic ductor: theuniformexternalfieldcombineswiththeinduced
r
a matter result from the response of an inductive loop to field and produces a total field with a quadrupole-like distri-
AC magnetic fields tuned near the atomic ground state bution.
hyperfine splittingofalkaliatoms. Thistrappingscheme
isidealforatomiccoherentmanipulationduetothenegli- Asketchofthephysicalset-upisshowninFig. 1(b). It
giblespontaneousemissionassociatedwithhyperfinelev- comprises a micro-engineered conducting loop (metallic
elsoftheatomicgroundstate[20]. Inaddition, thispro- or superconducting), a static magnetic field, B zˆ, and
DC
posaldoesnotrequiresophisticatedopticalcontrolandit a homogeneous AC magnetic field, B cos(ωt)zˆ, both
AC
2
transverse to the plane of the loop. In response to the increases arbitrarily by reducing the detuning with re-
electro-motive force induced by the time variation of the specttopairsoftransitions,resultinginthedivergentbe-
magnetic flux across the area enclosed by the loop, an haviourinFig. 2(a)-(b)(verticaldashedlines)atinteger
electric current circulates withinit. The induced current multiples of ∆=|g µ B |≈0.7MHz for B =1G.
F B DC DC
produces, in its turn, an inhomogeneous magnetic field This trapping scheme provides confinement of atoms
of the form B (r)cos(ωt+δ), that modifies the total in two hyperfine states in overlapping regions. In
ind
ACmagneticfield. Forsufficientlylargefrequenciesthat our example of Fig. 2, detuning in the range ∆ ∈
0
the inductive reactance of the loop dominates its Ohmic [−0.5,0.5]MHzproduceenergy-shiftlandscapesforstates
resistance, the external and induced fields are almost |F =2,m =1(cid:105) and |F =1,m =−1(cid:105) with approxi-
F F
in anti-phase. Thus, the resulting field has an approx- mately equal curvatures for both states. Even better,
imately quadrupole distribution, schematically shown in these two states experience exactly the same potential
Fig. 1(c),whosecentreislocatedclosetotheconducting landscape for a driving field resonant to the hyperfine
loop at the position where the amplitude of induced and splitting, ∆ = 0. Note that the static magnetic field
0
externalfieldssatisfyB =B cos(δ)[17],whereδ+π makesthisresonantdrivingtobeblue(red)detunedwith
ind AC
istherelativephasebetweenexternalandinducedfields. respect to coupling of states with m = −1 (m = 1),
F F
By tuning the driving frequency ω near the atomic as schematically shown by the solid arrows in Fig. 1(a).
ground state hyperfine transition, the AC magnetic field The detuning of the driving field also provides control
couples hyperfine Zeeman split sub-levels as depicted in overtheshapeofthetrappingcross-section,asseeninthe
Fig. 1(a), leading to state-dependent potential energy potential landscapes in Fig. 2(c-f). This is because the
landscapes for the atomic centre-of-mass motion [25]. relative weights of the terms in Eq. (1) can be adjusted
The energy shifts are conveniently described in terms of by changing the offset field and the driving frequency
thefi√eldcomponentsinsphericalu√nitvectorsuˆ−1 =(xˆ− that determine ∆mF.
iyˆ)/ 2,uˆ =zˆ,uˆ =−(xˆ+iyˆ)/ 2, and corresponding
0 +1
Rabi frequencies Ωi = µBgJBi(cid:104)F(cid:48),m(cid:48)F|Jˆi|F,mF(cid:105) with ∆0/h (MHz)
i = −1,0,1 and g the Land´e factor of the electronic -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
J
angular momentum J. After the rotating-wave approx- z) 2 (a)
H
imation and utilizing second order perturbation theory, k
nearthequadrupolecentretheenergyshiftsaregivenby ν (x
0
[20] z) 2 (b)
H
∆E (r)=±1(cid:18)|Ω0(r)|2 ν (kz
mF 4 ∆ 0
|Ω (mr)F|2 |Ω (r)|2 (cid:19) 24 (c) (d) µK 30
+ ∆mF −−1∆Zeeman + ∆mF ++1∆Zeeman (1) µz (m) 0 25
with ∆ =µ g B and the detuning 20
Zeeman B F DC
-24
24 15
∆ =2A+µ B m (g −g )−(cid:126)ω. (2) (e) (f)
mF B DC F F F−1 m) 10
wherethezerofieldhyperfinesplittingofthegroundstate µz ( 0 5
is 2A, and g the hyperfine Land´e factor [26].
F -24 0
To give an explicit example of the potential landscape 0 25 50 75 100 0 25 50 75 100
emerging from Eq. (1) we consider a circular loop of x (µm) x (µm)
gold with radius a = 100µm and diameter s = 10µm,
FIG. 2. (a,b) Trap frequencies corresponding to states
corresponding to approximate resistance R≈0.26Ω and
|F =2,m =1(cid:105) (solid) and |F =1,m =−1(cid:105) (dashed) of
inductance L ≈ 0.33nH [27]. In this case, the total 87Rb, asFfunction of the AC detuningF, with B = 2G,
AC
fielddistributionproducesacirculartrappingregionwith
B =1G, along the (a) x and (b) z directions. Lower pan-
DC
typical landscapes as shown in Figs. 2(c)-(f), for states els (c-f): Trapping potentials for ∆ = −1.1MHz (left col-
0
|F =2,mF =1(cid:105) and |F =1,mF =−1(cid:105) of 87Rb, applied umn)and∆0 =0.5MHz(rightcolumn),forthestates(c),(d)
fields of B =1G and B =2G. |F =2,m =1(cid:105)and(e),(f)|F =1,m =−1(cid:105). Gravitational
DC AC F F
The resulting quadrupole AC field distribution pro- attraction is included.
ducesharmonicconfinement,sincethelineardependence
ofthefieldamplitudewiththedistancetothequadrupole So far we have focused on the trapping produced by a
centre translates into a quadratic variation of the energy circular conductor. However, our scheme offers the pos-
shift in Eq. (1). The tightness of the trap, quantified by sibility of creating complex atomic guides shaped by the
the spatial curvature of the ∆E (r) along the xˆ and conducting loop. We illustrate this by considering a de-
mF
zˆdirections in units of frequency, is shown in Fig. 2(a)- manding case where we impose a severe ‘pinch’ in the
(b) as function of the detuning of the driving field (see shape of the conducting loop, as depicted in Fig. 3, cre-
Eq. (2)). According to Eq. (1), the trapping tightness ating a double loop with a variety of junction geome-
3
tries. The field distribution corresponding to this case ble with the loop length [15, 28]. In such a case, the
can be understood as follows: away from the pinch cen- induced current distributes unevenly across the conduc-
tre,thefielddistributionissimilartothequadrupolefield torandproducesamagneticfieldthatdiffersignificantly
inFig.1(c),whileinitsneighbourhoodthetotalfieldre- from the one produced by a single filament, having di-
sults from combining two quadrupole-like distributions rect impact on the quality of the trapping potential [29].
associated with conducting segments at each side of the An illustration of these effects is shown in Fig. 4, where
constriction. In particular, when the induced field bal- weconsidercircularloopswithsquareandcircularcross-
ancestheappliedoneatthecentreofthepinch,thefield sectionsmadeoftwodifferentconductingmaterialscom-
distribution acquires a hexapolar character. The geom- monly used in atom-chip experiments: gold (Au) and
etry of the resulting potential landscape is sensitive to superconducting niobium (Nb) [28].
the shape of the conductor, while its energy scale is de-
termined by the amplitude and detuning of the applied 16
10-1
fields. This is illustrated in Fig. 3(b)-(c), where field dis-
dttrhuirbceiuentgdioisffniesgrnaeinnfidtcaceonnntelsrytgrydiciltffaieonrndesnscitzaepjusendscihfftiaeovrniengbgeeboeymn≈eotbr1tieaµsimn.e,Cdporfonor-- 0 Hz) 12 Power (5 mW) Current (X 0.5A)
0
stoidmeroarteiocnomofptlehxisgceaosmeectarinesboefstthreaicgohntdfourcwtaorrd,lwyhaicphplciaend ν (X 1z 8 10-2 1 s (µm) 10
be used to create more involved atomic guides.
4
(a)
5 10 15 20
s (µm)
FIG. 4. Main panel: Trap frequency (in 100Hz) as a func-
tionofconductorthickness(s),correspondingtocircularand
squarecross-sectionsofsuperconductingNb(solidanddashed
lines), and gold (short-dashed and dot-dashed lines). Inset:
Peakvaluesofpowerdissipated(infactorsof5mW,solidand
short-dashed) and total current (in factors of 0.5A, dashed
anddot-dashedlines)ingoldloopsofcircular(solid,dashed)
(b)
andsquarecrosssections(short-dashedanddot-dashed). Pa-
m) rameters as in Fig. 2 with ∆0 =0.
z (μ
In the case of a normal conductor, the combination of
y (μm) x (μm) x (μm) x (μm) smallskindepthathighfrequencywitharadiallydepen-
dent magnetic flux, pushes the induced current towards
(c)
the outer edge of the conductor, spreading the current
along the conductor surface. Adding the Meissner effect
according to the London description of superconductors
[30],thecurrentisconfinedevenmoredramaticallyinthe
case of superconducting loops. As a consequence of dis-
tributingthecurrentoverawidearea,thegradientofthe
magnetic field is reduced in comparison to the single fil-
ament case. In terms of the atomic potential landscape,
FIG. 3. A figure-of-eight guide for atoms in the state
this translates to modifying the trapping position (i.e.
|F =2,m =1(cid:105) of 87Rb, produced by a loop with a central
F
the centre of the quadrupole field distribution) and re-
symmetric constriction (orange dashed line in (a)). The con-
ductor shape is defined by circles of radius 70µm centred at ducingitstightness(herequantifiedthroughthetrapfre-
x = ±100µm and a pair of parabolas that cuts the circle quency along the x direction, νx). Our numerical results
with matching first derivative. (a) Magnetic field landscape indicate that both position and trap frequency, although
in the loop plane, z = 0, for the applied fields BDC = 1G, dependent on the conducting material and cross-section
BAC =2G. (b) Iso-energy surface at 0.5µK corresponding to shape, do not vary strongly with these parameters. In
central gaps of 35.2 µm (left), 33.9µm (centre) and 32.9µm both cases, the most relevant parameter is the thickness
(right). (c) Potential energy landscape and field distribution
of the conductor, favouring the use of thin conductors to
in the plane x = 0, corresponding to surface plots directly
produce strong trapping potentials.
above, in panel (b). In (b) and (c) ∆ =0.35MHz.
0 The design of atom-chip configurations with current
carrying elements is limited by several technical issues
Modelling the loop as a single current filament is in- that restrict the range of experimentally accessible pa-
sufficient to describe the potential landscape associated rameters [1]. In the present case, for example, the goal
with conductors whose cross-section radius is compara- ofobtainingthetightestpossibletrap,e.g. withsmallde-
4
tuningorlargedrivingfields,shouldbebalancedagainst trapping geometry produced by a circular loop of induc-
an increase in heating and atom-loss rates. In what fol- tance L and radius a, the rate of transitions between
lows, we briefly consider these two problems. pairs of dressed states is approximately [33]:
Ohmic loses due to the induced current must be re-
stricted to avoid thermal destruction of the conductive 1 (cid:114) 2 (cid:18) 2L (cid:19)3(cid:18)(cid:126)u(cid:19)2 |Ω |3
loop, or undesirable alteration of the trapping track due Γ ≈ 0 (4)
|1(cid:105)→|2(cid:105) 2π m µ a2 4 (∆ )9/2
to thermal deformation of the conductor. For typical 0 mF
experimental parameters, such as those in Fig 2, the av-
erage current densities (see inset Fig. 4) are significantly Under typical experimental conditions, e.g. an atom
lower than the maximal tolerable values demonstrated moving with speed u ≈ 10 mm/s (corresponding to a
in experiments with normal and superconducting mate- temperatureof1µK),andforthetrapconfigurationpre-
rials operating under DC and high frequency conditions sented in Fig. 2, Eq. (4) predicts non-adiabatic transi-
(≈ 106 A/cm2) [31, 32], suggesting that the heat gen- tionswitharateof∼10−5 s−1,allowingenoughtimefor
erated in our proposed trapping setup can be efficiently manipulation of the trapped atoms.
transferred to the supportive structures of the device. Feeding the external field into the conducting loop
Also, although our numerical results for heating power presents another potential challenge. However, in the
favours using thick conductors, this should be balanced case of 87Rb, and atoms with similar mass, the driving
againstthehighertrappingfrequencyandbetterthermal fieldshouldhaveafrequencyintheGHzrange,forwhich
coupling achievable with thin wires, which can support the near-surface field of a microwave co-planar cavity
large current densities and are also convenient for fabri- couldbesuitable[20]. Forthecaseof6Liandlightatoms,
cation [1]. the driving frequency falls in the MHz range, where ad-
We estimate non-adiabatic atom losses in our trap- ditional techniques can easily be employed [1].
ping setup by considering an atom moving at speed u
In summary, we have shown that complex one-
in the plane defined by the conducting loop. After the
dimensional guides for ultra-cold matter can be defined
rotating-wave approximation, the atom-field interaction
by inductive effects over metallic and superconducting
is described by the two-level Hamiltonian [6]:
loops. For operation, theloop should befedwith amag-
netic field that oscillates near resonance to the hyperfine
∆ Ω
H = mF σ + 0(cos(ϕ)σ +sin(ϕ)σ ) (3) splitting of the atomic ground state, which induces an
2 z 2 x y
electric current on the conductor without the need of
where σ with i = x,y,z are Pauli matrices, and the leading wires that might introduce undesired features in
i
spatially-dependentphaseϕ,andRabifrequencyΩ ,are thepotentiallandscape. Ournumericalinvestigationsin-
0
defined by the combination of the applied and induced dicatethatexperimentalrealizationofthistypeoftrapis
fields. Atom-loss processes are modelled as transitions realisticwithcurrenttechnology,predictingtrappingfre-
betweentheposition-dependenteigenvectorsofHamilto- quenciesvaryingfromafewhundredHztoafewkHz. In-
nian Eq. (3), denoted by {|1(cid:105),|2(cid:105)} in the present treat- terestingly,ourschemecanproduceoverlappingtrapping
ment [33]. Such dressed states consist of linear combi- regionsfortwodifferenthyperfinestates,whichmightbe
nations for hyperfine states with the same projection of of interest for atomic species where a low magnetic field
angular momentum m , that depend on the amplitude Feshbachresonanceisavailable,suchasin6Li,aswellas
F
of the magnetic field. For example, at the centre of the complex quasi-one dimensional circuits for cold matter.
quadrupole field distribution, where the field is null, the We acknowledge fruitful comments and input from
dressed states |1(cid:105),|2(cid:105) coincide with the hyperfine states Brage Gording, David Lucas, Michael K¨ohl and Peter
|F,m (cid:105),|F −1,m (cid:105), while very far from the zero they Kru¨ger. This work was supported by EPSRC grant
F F
are an equal superposition of these two states. In the EP/I010394/1.
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F
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6
SUPPLEMENTARY INFORMATION
∇×∇×A=−iσωA (5)
INDUCED CURRENTS IN CONDUCTORS WITH
FINITE CROSS-SECTION
where σ is the ring conductivity [27].
We use the open-source software package FEMM [35]
In this section we provide more information about the to solve Eq. (5) for rings of gold with a range of cross-
induced current within conducting rings in the setup section sizes, under the action of a magnetic field oscil-
schematically shown in Fig. 1(b) of the main text. We lating at a frequency ω = 6.7GHz. Figure 6 shows the
focus on oscillating magnetic fields with an associated current distribution for R = 2.5µm and R = 7µm. In
wavelength(λ=c/ω)muchlargerthanthedimensionof rings of size comparable to the skin-depth at high fre-
the ring, and apply a quasi-static approximation to the quencies, the current distributes across the hole area of
Maxwell equations for the electromagnetic field [27, 30]. the cross-section. In the case of large rings, the current
In Sec. A we present results for the current distribu- concentratesalongtheconductorsurfaceleavingthecon-
tion in metallic rings taking parameters corresponding ductorcentrefreefromcurrentflow. Thisconfinementof
to gold. In Sec. B we detail a procedure to evaluate the current impacts the power dissipated by the electric
the current distribution in rings described by the Lon- flow, as shown in Fig. 4 of the main text.
don theory of superconductivity, with parameters corre-
sponding to superconducting Niobium, adapting results π MA/cm120-3
from references [28] and [34].
(a) (b)
We evaluate the current distribution using the coordi-
nate systems in Figs. 5(a)-(b). Exploiting the circular
symmetry of the ring cross-section, the current density θπ/2 10-7
is evaluated at points defined by the polar-coordinate
system with origin at its centre, as shown in Fig. 5(a).
We express the Maxwell equations coupled to a consti-
tutive relation between the fields and the current in the
0 10-11
ring (Ohm and London equations for metallic and su- 0 1 2 0 3 6
r (µm) r (µm)
perconducting materials, respectively) in the cylindrical
coordinate system with origin at the centre of the ring,
FIG.6. Currentdensitydistributionacrossthecross-section
as defined in Fig. 5(b).
of conducting loops of gold with thickness (a) 2.5µm and (b)
7µm. Inbothcases,theringradiusisa=100µm,B =2G
AC
(a) (b) z and ω=2π×6.7GHz. The vertical dashed line indicates the
z a position of the skin-depth.
q r
r i
r
FEMM also provides us results of the magnetic field
j
s r s r distribution (not shown) which is then used to evaluate
the trapping frequencies displayed in Fig. 4 of the main
text.
FIG. 5. (a) The current density in circular rings are eval-
uated at points defined by the polar coordinate system with
B. Superconducting rings.
originatthecentreoftheconductorcross-section. (b)Cross-
section of the coordinate system defined to evaluate the cur-
rent distribution in conducting rings. For this work, we con- We consider superconducting rings of uniform cross-
siderconductorswithrotationalsymmetryaroundthez axis. section, described by the London theory [30], where the
In both panels, the circular region represents the conductor supercurrent and the potential vector are related by:
cross-section.
e2n
J(r)=− sA(r) (6)
m
where m and e are the electron mass and charge, re-
A. Metallic rings
spectively, andn isthedensityofsuperconductingelec-
s
trons. Using this expression implies neglecting non-local
The time-variation of magnetic flux across a metallic effects on the current distribution as well as restricting
conductor induces an electric current whose distribution the frequency of the oscillating field to values smaller
depends on the properties and geometry of the ring as than the superconducting gap (typically of the order of
well as the frequency of the field. For a harmonic vari- a few ∼100GHz) [30].
ation of the magnetic field with frequency ω, the quasi- In the presence of an external field and a given cur-
static Maxwell equation for the vector potential is: rentdistribution, underquasi-staticconditions, thetotal
7
vector potential is: whereI =J ∆A,isthecurrentflowinginthej-thloop,
j j
and:
µ (cid:90) J(r(cid:48))
A(r)=AAC(r)+ 4π0 dV(cid:48)|r−r(cid:48)| (7) (cid:73) (cid:73)
V L = Q(r ,r )d(cid:96) ·d(cid:96) (14)
i,j i j i j
where the integral is over the volume of the current-
carrying conductors. A is the vector potential asso- isthemutualinductancebetweenthei-thandj-thloops,
AC
ciate with the applied field, which, in the case of a uni- which for i(cid:54)=j becomes:
formmagneticfieldalongthez axisisA =φˆρB /2,
AC AC
imposingtheCoulombgaugecondition∇·A =0[30].
AC µ ρ ρ (cid:90) 2π cosu
Superconductingringswithhomogeneouscross-section L = 0 i j du
haveacurrentdistributionindependentoftheazimuthal i,j 4π 0 (ρ2i +ρ2j +(zi−zj)2−2ρiρjcosu)1/2
angle φ, and flowing tangentially to the perimeter of the (15)
conductor, i.e., along the direction φˆ. This symmetry This last integral is evaluated following [34].
argument and Eq. (6) allow us to simplify Eq. (7) to: For the self-inductance Li,i we follow [28]:
ρBACφˆ =(cid:90) dV(cid:48)φˆ(cid:48)J(ρ(cid:48),z(cid:48))(cid:26) m δ(r−r(cid:48))+ µ0 1 (cid:27) L =µ ρ (cid:20)log(cid:18)8ρi(cid:19)− 7(cid:21)+µ λ22πρi (16)
2 e2ns 4π|r−r(cid:48)| i,i 0 i R 4 0 ∆Ai
(8)
where we have used an elementary property of the Dirac which includes the kinetic inductance term with λ2 =
delta distribution [34]. m .
Itisconvenienttoseparatetheintegraloverthevolume µ0nInse2this work we consider superconducting ring of size
oftheconductorintoanintegralovertheconductorcross- a = 100µm, and circular cross-section in the range s ∈
section and one over its circumference (see Fig. 5(b)): [1,20]µm. For Niobium, the London penetration depth
(cid:90) (cid:90)(cid:90) (cid:73) λ≈100nm [28].
dV(cid:48)φˆ(cid:48) = dρ(cid:48)dz(cid:48)× d(cid:96)(cid:48) (9) Figure7presentsthecurrentdistributioninringswith
Ring s=2.5µm and s=7µm, for an applied field B =1G.
AC
In comparison to the case of metallic conductors shown
where d(cid:96)(cid:48) =ρ(cid:48)dφ(cid:48)φˆ(cid:48). Thus Eq. (8) becomes:
in Fig. 6, the current distribution concentrates more
ρB (cid:90) (cid:73) strongly near the surface of the conductor. Neverthe-
ACφˆ = dρ(cid:48)dz(cid:48)J(ρ(cid:48),z(cid:48)) Q(r,r(cid:48))d(cid:96)(cid:48) (10) less, the impact on the trapping properties of the setup
2
Ring in Fig. 1 of the main text is similar in both cases, as
with Q(r,r(cid:48)) defined as: shown in Fig. 4 also of the main text.
Q(r,r(cid:48))= m δ(r−r(cid:48))+ µ0 1 (11) π MA/cm120-3
e2n 4π|r−r(cid:48)|
s (a) (b)
Equation (8) can be recast in terms of magnetic flux
across the loop C defined by {r =(ρ,φ,z)|φ∈[0,2π)},
using the relation Φ =(cid:72) A·d(cid:96): θπ/2 10-7
C C
(cid:90) (cid:73) (cid:73)
πρ2B = dρ(cid:48)dz(cid:48)J(ρ(cid:48),z(cid:48)) Q(r,r(cid:48))d(cid:96)(cid:48)·d(cid:96)
AC 0 10-11
C Ring 0 1 2 0 3 6
(12) r (µm) r (µm)
Thisequationimpliesthatthemagneticfluxacrossthe
loopC,createdbythecurrentdistribution,compensates FIG.7. Currentdensitydistributionacrossthecross-section
exactly the magnetic flux imposed by the external field. ofsuperconductingloopsofNiobiumwiththickness(a)2.5µm
This corresponds to the well known Meissner effect in and (b) 7µm. In both cases, the ring radius is a = 100µm,
superconductors, and implies that the induced current BAC =2G. The vertical dashed line indicates the position of
adjust instantaneously in order to null the total flux of the skin-depth.
magnetic field across any loop defined within the super-
conducting ring.
ToobtainasolutionofEq. (12),wediscretizethecon-
ductor cross-section in elements of area ∆A centred at
i
positions r , as schematically shown in Fig. 5(b). Then,
i
we obtain the equation:
(cid:88)
πρ2B = L I (13)
i AC i,j j
j