RF dressed atoms beyond the linear Zeeman effect

0.58 MB·English

by  G. Sinuco-León

RF dressed atoms beyond the linear Zeeman effect
G. Sinuco-Le´on and B.M. Garraway
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom
(Dated: September 17, 2012.)
We evaluate the impact that non-linear Zeeman shifts have on resonant radio-frequency dressed
traps in an atom chip configuration. The degeneracy of the resonance between Zeeman levels is
lifted at large intensities of a static field, modifying the spatial dependence of the atomic adiabatic
potential. Inthiscontextwefindeffectswhichareimportantforthenextgenerationofatom-chips
2 with tight trapping.
1
0
PACSnumbers: 37.10.De,37.10.Gh,03.75.Be,32.60.+i
2

p
e I. INTRODUCTION the overall field components are
S
4 BDC =(Gy,Gx,B0). (1)
Theuseofradio-frequencyfields(RF)forthemanipu-
1
lation of ultra-cold atomic samples [1–6] is underpinning HereGisthemagneticfieldgradientnearthequadrupole

] importantdevelopmentsinareassuchasmatter-wavein- centre and B0 is the uniform offset field (see Fig. 1). In
h terferometry [4, 7–11] which are beyond the original use ordertogivequantitativeresults wefocusonthe ground
p of RF fields for evaporative cooling [12]. Nowadays, in state manifold of Rb87 as an example, and evaluate the
m- combination with atom-chip technology [13] or optical dressedpotentialsforthe|F =2,mF =2(cid:105)stateandalso
lattices [14, 15] RF dressing is a well established tech- give some results for the |F = 1,mF = −1(cid:105) state. For
o
nique that allows us routinely to control and engineer simplicity, our analysis is restricted to small amplitudes
t
a atomic quantum states and potential landscapes on mi- of the RF field (BRF), such that beyond rotating wave
s. cron scales [13, 16]. Furthermore, RF dressing plays a approximation (RWA), effects can be ignored [34]. This
c centralroleinseveralproposalsforextendingthescopeof is also helpful to identify clearly the effects due to non-
i
s functions and applications of ultra-cold atomic gases, in- linearityoftheenergyshifts. Toinvestigatetherelevance
y cludingreduceddimensionalityandconnectedgeometries in current experimental situations, we took parameters
h (ring and toroidal traps) [3, 5, 17–22], cooling and prob- from recent experiments, i.e. we took G = 22.67 T/m,
p
[ ing of RF dressed atom traps [6, 23, 24], sub-wavelength B0 =1.0or3.0GandBRF =35.7mG[34]: thisenables
tailoring of potentials [14] and transporting atoms in ustoquantifytheeffectthatthenon-linearZeemanshifts

1 dressed atom traps [25]. have on the shape of dressed potential.
v In the following we first review the theory of the Zee-
2 Experimental realizations of RF dressed magnetic
man effect in non-linear situations (section II) and then
8 traps have worked in a range of static field intensities
applytheresultstoRFdressinginthestrongfieldregime
1 that produce linear Zeeman energy shifts [3, 13, 23, 26–
(section III). We report on the effects on both the vi-
3
28]. Nevertheless, developments in near surface trap-
. brational frequency of RF dressed atom traps and their
9 ping/controlandmicro-fabrication[29]indicatethatpro-
locations, and then the paper concludes in section IV.
0 ductionofstrongtrappingconfigurationsandsub-micron
2
control will be soon on the agenda [30, 31] and a full de-
1
scription of the atomic Zeeman shifts becomes relevant.
: II. ZEEMAN SHIFT THEORY
v
OneimpressiveapplicationofRFdressingofmagnetic
i
X traps is the miniaturized matter-wave interferometer for In the limit of slow atomic motion, the magnetic mo-
r coherent spatial splitting and subsequent stable interfer- ment of the atoms keeps its orientation relative to a spa-
a
ence of matter waves on an atom chip [4, 11, 32]. In tially varying magnetic field, B . The dynamics can
DC
theinterferometer,thepotentiallandscapetypicallycom- be described by the Hamiltonian [36, 37]
prises a double well in a transverse plane, accompanied
H =AI·J +µ |B |(g J −g(cid:48)I ), (2)
by longitudinal weak trapping [4, 32–34]. It has been B DC J z I z
proposed that the double well potential can be helpful
where A is a measure of the hyperfine splitting, J and I
tostudyentanglementandsqueezingphenomena,andto
are the spin and nuclear angular momentum operators,
study phase coherence dynamics and many-body quan-
and the electronic and nuclear g-factors are g and g(cid:48),
tum physics [32, 35]. Here we establish the relevance of J I
respectively. The Bohr magneton is denoted by µ , as
non-linear Zeeman shifts on the production of a double B
usual.
well potential in a typical atom chip configuration.
Asiswellknown, theoperatorF =J +I commutes
z z z
For the purposes of this work, the potential landscape withtheHamiltonian(2)andthusm isagoodquantum
F
isproducedbyresonantlydressingastaticmagneticfield number (along with I and J). Since we consider Rb87
comprisedofaquadrupolefieldwithanoffsetfield. Then with I =3/2 and J =1/2, the interaction mixes pairs of
2
These states have energies
A αA
EZ,± = I± (g −2g(cid:48)I) . (4)
mF=±(I+1/2) 2 2 J I
In the weak field regime, corresponding to α (cid:28) 1, the
Zeeman shifts become linear in |B | and all these en-
DC
ergy levels [Eq. (3) and Eq. (4)] are approximated by
(see e.g. Ref. [37])
A A
EZ,± ≈ − + (I+1/2)+m µ g |B |, . (5)
mF 4 2 F B F DC
Here the signs in front of the square root contribution in
Eq. (3) have been absorbed by the definition of the g-
factor associated with the total angular momentum F =
I+J. BecauseJ =1/2thereareonlytwopossiblevalues
F =I+J =2 and F =|I−J|=1:
F(F +1)−I(I+1)+J(J +1)
g = g
F J 2F(F +1)
F(F +1)+I(I+1)−J(J +1)
−g(cid:48) . (6)
I 2F(F +1)
Figure 2(a) shows the atomic energy levels, correspond-
(c) _m_2w_2(x-xw)2 ing to the upper hyperfine manifold, at different posi-
0.4
tions in the field distribution Eq. (1). In the case of
Hz) RF dressed atom traps with Rb87 atoms in the trapping
M states m =1,2 follow such energy curves adiabatically
(E/h 0.2 and becoFme trapped at positions of minimum field am-
h plitude [16].
0 x
-2 -1 0 1 w 2 III. DOUBLE WELL DRESSED POTENTIAL
x (µm)
Figure 1: (a) Schematic of a magnetic field quadrupole dis- We consider a magnetic trap with static field distri-
tribution[Eq. (1)]andtheRFcouplingfield. Thehorizontal bution (1), dressed by a uniform and linearly polarized
double-headed arrow (red) indicates the polarization of the oscillating magnetic field (RF) [1, 4]. The coupling be-
coupling RF field. The offset field B points out of the page. tween atomic states is dominated by the component of
0
(b) Contours of a typical RF dressed double well potential. theRFfieldorthogonaltoB [5],vanishingatpositions
DC
(c)Potentialenergyalongy=0(thex-axis)inpanel(b). Pa- where they are parallel (e.g. at positions of coordinates
rameters characterizing the double well are defined as: well
(x = 0,y)). A typical resulting double well potential is
positionx ,inter-wellbarrierheighthandwellfrequencyw.
w showninFig. 1(b). Tocharacterizethepotentialenergy,
we evaluate the dressed energy along the x axis, where a
double well appears as a consequence of the dressing.
states|m −1/2,1/2(cid:105)and|m +1/2,−1/2(cid:105)inam ,m
F F I J
We define the z-axis parallel to direction of the static
basis: |m ,m (cid:105). These two states have the same value
I J
field, take the RF field polarized along the x-axis and
of m by construction. In this case, because J = 1/2,
F
applythestandardrotatingwaveapproximation(RWA).
the diagonalization of Hamiltonian (2) reduces to 2×2
Then, the Hamiltonian (2) in a rotating frame becomes
matrix blocks and some uncoupled terms. This leads to
a Breit-Rabi formula for the hyperfine energy spectrum
H = AI·J +µ |B |(g J −g(cid:48)I )±(cid:126)w (J +I )
of an alkali atom in a magnetic field [36]: B DC J z I z RF z z
µ |B |
+ B RF (g J −g(cid:48)I ). (7)
EZ,± = ± A(cid:113)(I+1/2)2+2m α(g +g(cid:48))+α2(g +g )2 2 J x I x
mF 2 F J I J I
A(1+4αg(cid:48)m ) BecauseJ =1/2thesign±ischosenaccordingtowhich
− I F (3)hyperfine manifold we are interested in: F = I +J or
4
F =|I−J|. Thisisbecauseeachpolarizationcomponent
where α = µ |B |/A. However, there are two uncou- of the dressing field couples magnetic sublevels within
B DC
pledstatesinthem ,m basis: |I,1/2(cid:105)and|−I,−1/2(cid:105). a subspace with a given total angular momentum (see
I J
3
0.01
0.4 (a) (b)
m =2
F
0
0.2 m =1
F
E
z) /h
GH 0 mF=0 -0.01 (M
(h (c) 0.03Hz
E/ )
-0.2 mF=-1
0
m =-2
F
-0.4 -0.03
0 5 10 15 20 25 30 -0.1 -0.05 0 0.05 0.1
x (µm) x-x (µm)
w
Figure 2: (a) Zeeman shifts for the upper manifold (F = 2) of the ground state of Rb87, in a field distribution as Eq. (1).
Arrowsindicatelocationsofresonanceatlow(left)andhigh(right)fieldintensities. Panels(b)and(c)showdressedpotentials
(solid) and bare states (dashed) for resonance located at (b) x =2 µm, with w /2π=2.31 MHz and (c) x =30 µm with
w RF w
w /2π = 14.4 MHz. Notice how, at large fields, several crossings of bare states appear. In all cases the field gradient is
RF
G=22.67 T/m, the RF amplitude is B =35.7 mG, and the offset field is B =3 G.
RF 0
Appendix A). The diagonalization of Eq. (7) produces well separation is 2x in Fig. 3.)
0
thedressedstateenergiesdisplayedinFig. 3(solidcurve)
Withafixedstaticfieldconfiguration,thedistancebe-
for the magnetic field configuration of this paper.
tween the wells is controlled by the RF frequency w .
RF
In the weak field regime and for RF frequencies much
At large distances, the non-linear character of Zeeman
smaller than the hyperfine splitting (w (cid:28) A/(cid:126)), the
RF shifts impacts on the dressed potential, and Eq. (8)
dressed energies for states of the upper hyperfine mani-
is no longer valid. In particular, the well frequency is
fold are [1]
significantly modified, as seen for m = 2 in Fig. 4.
F
For well separations of a few microns, the correction
(cid:115)
(cid:18) (cid:126)w (cid:19)2 (cid:18)|B |(cid:19)2 (∆x = x −x0) is about 5% (or 10%) for offset fields
E+ ≈ m µ g |B |+ RF + RF w w w
mF F B F DC µ g 2 of 1 G (or 3 G), respectively. The relative effect on the
B F
frequency increases approximately linearly with the well
A A
− + (I+1/2), (8) separation, and can become a significant fraction of w
4 2
for separations of tens of microns. The weakening of the
well(orequivalently,thereductionofw),canbequalita-
which produces, for the field distribution of Eq. (1) and
tively understood as a consequence of the appearance of
for weak-field seeker states [16], the dressed potentials
multiple resonant positions due to non-linearity of Zee-
shown in Fig. 3 (dashed line). The double well is con-
man shifts [see Fig. 2(b)]: as the distance between bare
veniently described by three parameters: the well min-
state crossings increases, the dressed potential softens in
imum position (x ), the height of the inter-well barrier
w
comparisonwiththepotentialfromasingleresonantpo-
(h) and the well frequency w which is defined through
sition. This becomes one of the main effects of the non-
a harmonic approximation centred around the potential
linear Zeeman effect on dressed RF potentials.
minimum [see Fig. 1(c)]. To distinguish these proper-
ties being evaluated according to Eq. (8) or Eq. (7), Figure 5 shows similar results for what is the F = 1
we denote the former quantities (i.e. from the weak field manifold in the weak field regime. However, in this case
expression) with a superscript 0. weseethatthetrapistightened bythenon-linearZeeman
The potential well’s position and frequency are de- effect. The result is understood from the fact that g ,
F
termined by the location of the resonant coupling, as Eq. (6),changessignforF =1. Asaresultthesequence
suggested in Fig. 2(b)-(c), and by the intensity of the of levels seen for F = 2 in Fig. 2(c) is reversed, which
RF field. Figure 3 shows typical dressed potentials in a enables us to have tighter trapping as seen (inverted)
double-well regime (where we show only one of the two at the bottom of the manifold of Fig. 2(c). Since the
wells). WecompareboththedressedeigenvalueofHamil- trappingonRb87F =2involvespotentialstwiceassteep
tonian(7),andEq. (8)fordifferentRFfrequenciesw . as those on Rb87 F = 1 we would not expect the trap
RF
A rather significant change in the well shape is seen for to become tighter in F = 1 overall. (The factor two
quitemodestwellseparationsforthisfieldgradient. (The in steepness is because the two manifolds have the same
4
4
3 (a)
2
1
0
) 4
z
H
3 (b)
M
h ( 2
E/ 1
0
4
3 (c)
2
1
0
-0.5 -0.25 0 0.25 0.5
x - x (µm)
0
Figure 3: Comparison of non-linear and linear dressed potentials. There are two potential wells at ±x and we show here
0
just the well located at x=+x with m =2: Red (solid) eigenvalue of Hamiltonian (7). Blue (dashed) line is the adiabatic
0 F
potential (8). The dressing frequencies and well positions are (c) 2.6 MHz, x =10 µm (b) 5.2 MHz, 30 µm and (a) 8.2 MHz,
0
50 µm. The quadrupole field gradient is G = 22.67 T/m [34] and the bias field is B = 3.0 G as in Fig. 2. In each of (a-c),
0
w has been adjusted to produce a minimum at x0 =1.0 µm using the weak field expansion Eq. (10).
RF w
√
magnitude of g , which means a factor of two in the of the well frequency scaled by 2 (dash and dash-dot
F
potentials because for F = 2 we can use m = 2, while line in Fig. 4, respectively).
F
for F = 1 we can only use m = −1.) However, Figs. Figure6presentsacomparisonofthenon-lineareffects
F
4 and 5 show that, in absolute terms, F = 1 trap does onthewellposition(∆x =x −x0)andinter-wellbar-
w w w
become tighter than the F = 2 trap for quite modest rier height (∆h = h−h0). Again, these are calculated
values of the RF frequency. In essence, the weakening from the diagonalization of the Hamiltonian (7) and by
of the F = 2 trap by the non-linear Zeeman effect, and comparing results to the weak field approximation, Eq.
correspondingtighteningofF =1,becomessufficientfor (8). We see that the shift in the position of a potential
the trap frequency in F =1 to become higher. wellcanbelargeinboththelowRFfrequencylimitand
The potential well’s frequency for the dressed state the high RF frequency limit (see Fig. 6(a), inset). The
|F = 1,m = −1(cid:105) can be evaluated analytically using shiftathighRFfrequencyseemsreasonableasthehigher
F
the solution of a three level system (see Appendix B), frequencies usually result in RF resonance in regions of
and approximating the well’s position by taking the po- strongermagneticfieldswherethenon-linearZeemanef-
sitions of the resonant couplings (large dots in Fig. 2(c)) fectisexpectedtoplayarole. Noteherethatthevertical
and averaging them. The frequency obtained following asymptotes in Fig. 6 correspond to the threshold were
this procedure coincides very closely with the numerical theRFfrequencyisjustsufficienttoexcitetransitionsin
results shown in Fig. 5. the static field configuration, i.e. (cid:126)wRF →µBgF|B0|.
In the case of F = 2, we find that, similarly, the To understand the shifts in position of the potential
well frequency corresponding to the dressed state |F = wells we look carefully at the issue of how we define the
2,m =2(cid:105)canbeestimatedbyconsideringcouplingsbe- resonancelocationandthepotentialwelllocation. Shifts
F
tweenthebarelevels|F =2,mF =−1(cid:105),|F =2,mF =0(cid:105) of the well positions, ∆xw = (xw −x0w), can be under-
and |F = 2,m = 1(cid:105). This is shown in Fig. 4 (short- stood by investigating where resonant RF coupling oc-
dash) and indicFates that at high RF frequencies the cur- curs, i.e, positions where (cid:126)wRF matches the energy sep-
vature of the dressed energy is dominated by the cross- aration between levels with |∆mF|=1. That is,
ings of just these three levels (see Fig. 2(c)). In the
EZ,+−EZ,+ =(cid:126)w (9)
intermediate regime of dressing frequency, the coupling mF mF±1 RF
of all levels contribute significantly to the dressed state
which,intheweakfieldregime,reducestoauniquevalue
and the three level approximation is not valid. However,
of w given by
RF
at low RF dressing frequencies, we observe that the lin-
ear regime result coincides with the three level estimate µ g |B |−(cid:126)w =0. (10)
B F DC RF
5
24 24
(a) (b)
18 18
) )
z z
H H
k k
( (
) 12 ) 12
π π
2 2
( (
w/ w/
6 6
1 3 5 7 9 3 5 7 9
w /(2π) (MHz) w /(2π) (MHz)
RF RF
Figure 4: Well frequency as a function of RF frequency, obtained from the eigenvalue of Hamiltonian (7) (solid) and the
linear approximation to the eigenvalues (8) (dashed), for (a) B =1.0 G and (b) B =3.0. Other parameters are as in Fig. 2
0 0 √
(includingF =2,m =2). Thewellfrequencycorrespondingtoathreelevelsystemanditsvaluescaledby 2areshownby
F
the short-dash and dot-dash lines, respectively.
24 24
(a) (b)
18 18
) )
z z
H H
k k
) ( 12 ) ( 12
π π
2 2
( (
w/ 6 w/ 6
0 0
1 3 5 7 9 3 5 7 9
w /(2π) (MHz) w /(2π) (MHz)
RF RF
Figure5: WellfrequencyasafunctionofRFfrequencyinthecaseF =1,m =−1. Resultsareobtainedfromtheeigenvalue
F
of Hamiltonian (7) (solid) and the linear approximation to the eigenvalues (8) (dashed), for (a) B =1.0 G and (b) B =3.0.
0 0
Other parameters are as in Fig. 2.
In the regime of weak magnetic field, multiple reso- IV. CONCLUSIONS
nances occur at the same position [see Fig. 2(b)], since
thespacingbetweenadjacentsub-levelsisdegenerate. In
Our calculations show that for a resonant RF dressed
contrast, for large fields the degeneracy of transition fre-
double well potential, the well frequency is a sensitive
quenciesisdestroyedandtheresonancelocationsbecome
parameter to the non-linearity of Zeeman shifts. This
separatedinspace,asshowninFig. 2(c). Asinthetreat-
is relevant for investigations of tunnelling processes in
mentofthewellfrequenciesabove,asimpleandgoodes-
double well potentials, BEC interferometry, and phase
timateforthelocationofthepotentialwellminimumx
w
coherencedynamics,sincethesephenomenaaresensitive
canbefoundbytakinganaverageofthepositionswhere
tothepotentialshape[32,35]. Thesemodificationsofthe
Eq. (9) is satisfied for different (m ,m ±1) pairs [dots
F F
well frequency can be large enough to be tested exper-
in Fig. 2(c)]. This quantity, x¯ , has been evaluated us-
w
imentally via interferometry or atom cloud oscillations
ing both the procedure of Appendix B (Eq. B7), and
[3, 4].
by using the numerical solution of Eq. (9) itself. These
results agree with the full solution of Eq. (7) and are We have also seen that in taking account of the non-
presented as one curve in Fig. 6. linear Zeeman effect the sign of the g-factor is an im-
portant consideration in the tightness of the resonantly
RF dressed atom trap. In the case of Rb87 we found
that for modest RF frequencies the F = 1 dressed trap
could be tighter than the F = 2 dressed trap, which is
quite against expectation in the linear regime. By using
a model 3-level system, we calculate the well’s frequency
6
10-1 Appendix A: RF dressing in the strong field regime
(a) 90
An alkali atom interacting with a static magnetic field
m)
(nw50 plus an orthogonal RF dressing field is described by the
0/xww 10-2 ∆x Hamiltonian:
x
∆ H = AI·J +µ |B |(g J −g(cid:48)I )
10 B DC J z I z
1 3 5 7 9
wRF/2π (MHz) +µB|BRF|(gJJx−gI(cid:48)Ix)cos(wRFt) (A1)
where the first term arises from the hyperfine interac-
10-3 tion. Assuming that the hyperfine coupling is stronger
1 3 5 7
−h wRF/µB gF B0 than the interaction with the static field, the state space
can be split into a direct sum of spaces with two angular
(b)
momentum operators F↓ (which has spin I−J) and F↑
0
(which has spin I +J). Left and right circular polar-
ization components of a linearly polarized dressing field
Hz)-0.5 couple magnetic levels within only one of the hyperfine
M subspaces. Thus, the relevant component in each case is
H ( selected by an observer rotating in an appropriate sense.
∆
With this in mind, the unitary transformation between
-1
lab and rotating frames can be defined by [38]:
U =exp(−iw t(F↑−F↓)), (A2)
RF z z
1 3 5 7 9
wRF/(2π) (MHz) where F↑ and F↓ work in subspaces of F as described
z z
above. Then, in the rotating frame, the time evolution
Figure 6: (a) Main panel: Relative displacement of well po- follows the Schr¨odinger Eq. i(cid:126)∂t|ψ(cid:105)=H(cid:48)|ψ(cid:105) with
sitions (∆x /x0 =(x −x0)/x0) versus (cid:126)w /µ B , eval-
w w w w w RF B 0
uatedviaEq. (7). Onthisscale, ∆x /x0 areapproximately H(cid:48) =U†HU −i(cid:126)U†U˙. (A3)
w w
independent of B . Inset: Well position displacement for off-
0
setfieldB =1G(dashed)andB =3G(short-dashed)(b) The hyperfine term and the interaction with the static
0 0
Modification of the double well barrier height (∆h=h−h0) field are invariant under the unitary transformation Eq.
for B0 =1 G (solid) and B0 =3 G (dashed) respectively, as (A2). The interaction with the dressing field transforms
function of the RF frequency. In all cases BRF = 35.7 mG into:
and G = 22.67 T/m [34]. Dashed vertical lines indicate the
limiting frequencies at µBgFB0/h. H(cid:48) = µBBRF,x(cos2w t xˆ∓sin2w t yˆ)·(g J −g(cid:48)I)
RF 2 RF RF J I
µ B
+ B RF,x(g J −g(cid:48)I ). (A4)
for F = 1. In case of the F = 2 manifold, this model 2 J x I x
works well in a regime of non-linear Zeeman shifts.
Neglecting the rapidly rotating field (rotating with an-
Finally, we note that near the limiting frequency set
gular velocity of 2w ), the total Hamiltonian becomes
bythestaticoffsetfield(seeFig. 6),thedoublewellsep- RF
the expression given in Eq. (7).
aration and frequency of the dressed potential are quite
sensitive to the dressing frequency w (see also Figs. 4
RF
and5). Thisregimeisrelevantfordoublewellswithsub-
Appendix B: Frequency and position of strong-field
micronseparations,asituationlikelytooccurinnearsur-
potential wells
facetrappingconfigurations[30]. Ourworkshowsthatto
achievestableconfigurationswithsmallwellseparations,
Consider the dressed energy as function of the static
goodcontroloftheRFfrequencyisneeded. However,the
magnetic field, as given, e.g. in Eq. (8), and produce a
mainresultofthisworkisthatthevibrationalfrequency
Taylor expansion around the minimum occurring at the
can be sensitive to the chosen RF frequency because of
magnetic field |B |,
the non-linear nature of Zeeman shifts. DC,min
β
E(|B |)=E + (|B |−|B |)2+... (B1)
DC 0 2 DC DC,min
Acknowledgments with β,
We gratefully acknowledge funding from the EPSRC β = d2E(|BDC|)(cid:12)(cid:12)(cid:12) . (B2)
(grant EP/I010394/1). d|BDC|2 (cid:12)|BDC,min|
7
The well’s frequency is parametrized by a quadratic de- with
pendence of the dressed energy with the distance to the
position of minimum 1 (cid:18) g −g (cid:19)2
G = ∓ J I . (B8)
mw2 F I+1/2 2(I+1/2)
E(x)=E + (x−x )2 (B3)
0 2 w
In Eqs. (B7)-(B8) the upper and lower signs are chosen
where
according to F =I+J and F =|I−J| respectively.
|B |=|B (x )|. (B4) Analytic expressions for the dressed energy can only
DC,min DC w
be obtained in simple cases. For our example of Rb87
In the case of a quadrupole magnetic field of gradi- in its ground state, after ignoring couplings between the
ent G plus offset field, where the field is |B (x)| = F = I +J and F = |I −J|, the dressed energies of the
DC
(cid:112)(Gx)2+B2,thewell’sfrequencyintermsofexpansion F =1 manifold are:
0
Eq. (B1) is given by:
√
C 2 C2−3D (cid:18)θ+φ (cid:19)
(cid:114) (cid:115) E (|B |) = − + cos i
w = µ2BβG 1− B0 . (B5) i DC 3 3 3
mA |B | (cid:18) (cid:19)
DC,min 1 g
+A − + F (I+0.5) (B9)
4 2|g |
Then in the regime of linear Zeeman shifts, and after a F
with φ =0,2π,4π corresponding to i=1,2,3, and
i
Taylor expansion to second order of Eq. (8) we obtain
|B | = (cid:126)wRF C = −(E1Z,−+E0Z,−+E1Z,−)
DC,min gFµB D = EZ,−EZ,−+EZ,−EZ,−+EZ,−EZ,−
1 −1 1 0 −1 0
8Ag
β = F . (B6) −2d2+(cid:126)w (EZ,−−EZ,−−(cid:126)w )
µ |B | RF 1 −1 RF
B RF
E = d2(EZ,−+EZ,−)−EZ,−EZ,−EZ,−
1 −1 1 0 −1
Forthistobevalid,theminimumshouldoccuratamag-
−(cid:126)w EZ,−(EZ,−−EZ,−−(cid:126)w )
netic field such that µB|BDC,min|(cid:28)2A. RF 0 1 −1 RF
For more intense fields, where the multi-level crossing R = (9CD−27E−2C3)/54
degeneracy is lifted, but levels can still grouped in man- Q = (3D−C2)/9
ifolds F = I + J and F = |I − J|, |B | can be
DC,min (cid:112)
obtained as an average of the crossing points between θ = arccos(R/ −Q3), (B10)
consecutive m levels. The field at which such crossings
F
occur can be obtained analytically by solving Eq. (9), whered= µBBRFgF(cid:104)m ±1|J +I |m (cid:105)andtheZeeman
expanding the Zeeman shifted energies (3) up to second 2 F x x F
shifted levels are given by Eq. (3).
order in µ B/A. This procedure gives us
B
(cid:113)
A mF(cid:88)=F−1gF − gF2 ∓4(1−2mF)GF(cid:126)wARF
|B |=
DC,min 2FµB 2(1−2mF)GF References
mF=−F
(B7)
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