State-dependent potentials in a nanofiber-based two-color trap for cold atoms
aa r X i v : . [ phy s i c s . a t o m - ph ] J un State-dependent potentials in a nanofiber-based two-color trap for cold atoms
Fam Le Kien, P. Schneeweiss, and A. Rauschenbeutel
Vienna Center for Quantum Science and Technology, Institute of Atomic and Subatomic Physics,Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria (Dated: September 18, 2018)We analyze the ac Stark shift of a cesium atom interacting with far-off-resonance guided lightfields in the nanofiber-based two-color optical dipole trap realized by Vetsch et al. [Phys. Rev.Lett. , 203603, (2010)]. Particular emphasis is given to the fictitious magnetic field producedby the vector polarizability of the atom in conjunction with the ellipticity of the polarization ofthe trapping fields. Taking into account the ac Stark shift, the atomic hyperfine interaction, anda magnetic interaction, we solve the stationary Schr¨odinger equation at a fixed point in space andfind Zeeman-state-dependent trapping potentials. In analogy to the dynamics in magnetic traps, alocal degeneracy of these state-dependent trapping potentials can cause spin flips and should thus beavoided. We show that this is possible using an external magnetic field. Depending on the directionof this external magnetic field, the resulting trapping configuration can still exhibit state-dependentdisplacement of the potential minima. In this case, we find nonzero Franck-Condon factors betweenmotional states of the potentials for different hyperfine-structure levels and propose the possibilityof microwave cooling in a nanofiber-based two-color trap.
PACS numbers: 42.50.Hz, 37.10.Gh, 32.10.Dk, 32.60.+i, 31.15.ap
I. INTRODUCTION
Optically trapped neutral atoms are among the primecandidates for storing and processing quantum informa-tion [1, 2]. This approach greatly profits from the useof microscopic dipole traps which enable the manipula-tion of individual neutral atoms [3]. Several technicalapproaches have been followed in order to obtain op-tical dipole traps [4] with a microscopic trapping vol-ume. Among these are strongly focused free-space op-tical traps, traps based on evanescent light fields, andplasmonically enhanced optical trapping potentials [5, 6].Tightly confined light fields are known to differ con-siderably from simple light fields which are describedcorrectly in the paraxial approximation. For example,when tightly focusing an initially linearly polarized laserbeam, the latter acquires a longitudinal polarization com-ponent that gives rise to a complex polarization patternin the focal region. Consequently, the laser beam is notcorrectly described by a transversally polarized electro-magnetic wave anymore. Recent experiments with atomsin the motional ground state of a non-paraxial dipoletrap [7, 8] as well as atoms trapped in the evanescentfield of an optical nanofiber [9, 10] contributed to thediscussion of the ground state coherence properties ofatoms in tightly-confining optical potentials [11, 12]. Inthis article, we concentrate on the theoretical descriptionof optical potentials in a nanofiber-based trap. However,our findings have implications for other technical realiza-tions of tightly-confining optical potentials, too.Among the diverse concepts of trapping using opti-cal nanofiber-guided light fields [13–16], only the so-called two-color trap has so far been realized experimen-tally [9, 10] and will be discussed in this work. This trapconfines atoms in the fiber transverse plane by combiningthe attraction and repulsion of a red- and blue-detuned guided light field [13, 14]. For axial confinement, theexperiment of Vetsch et al. [9] used a pair of counter-propagating red-detuned beams and a single, running-wave blue-detuned beam (see Fig. 1). The latter lightfield has an elliptical polarization which leads to a vectorZeeman-state-dependent light shift of the ground stateof the trapped atoms [11]. It has been proposed to re-place the blue-detuned running wave by a rapidly mov-ing blue-detuned standing wave field in order to createa “compensated” trap in which the ellipticity of the sin-gle blue-detuned field and thereby the vector light shifteffect is reduced [11]. This proposal has then been ex-perimentally implemented in [10]. Both versions of thetwo-color trap, the basic scheme and the compensatedscheme have been compared theoretically in [11], in par-ticular concerning the coherence times of ground statespin wave in trapped atomic ensembles.Here, we extend the discussion of state-dependenttrapping potentials and the ground-state coherence ofatoms in a two-color nanofiber-based optical potentialof the type used in Vetsch et al. [9]. Most importantly,we point out deficiencies of the commonly used approachof adiabatic potentials when calculating optical traps. Asan example, we expect Majorana spin flips to play a rolein optical traps that are at the same time steep and ex-hibit a strong gradient of the polarization of the trap-ping laser fields. Inspired from the physics of spin flipsin magnetic traps for cold atoms, we study the influ-ence of a magnetic offset field for atoms in the two-colornanofiber-based optical trap. Using the concept of light-induced fictitious magnetic fields, we show how an offsetfield allows one to modify the Zeeman-state-dependentpotentials and to reduce spin flip effects. In addition,for a configuration that shows a Zeeman-state-dependentdisplacement of the trap minima, we point out the optionto drive transitions between motional states of the atom nanofiber xyz y -polarized blue-detuned light x -polarized red-detuned light x -polarized red-detuned light atom FIG. 1. Nanofiber atom trapping scheme used in [9]. Thetrapping potential is formed by combining the attraction andrepulsion of red- and blue-detuned guided light beams, re-spectively. A pair of counter-propagating red-detuned beams(thick red arrows) and a single blue-detuned beam (thickblue arrow) are used. The red-detuned light beams are x -polarized (red thin arrows) and the blue-detuned light beamis y -polarized (thin blue arrow). in the trap using a microwave field. We calculate theFranck-Condon factors and suggest microwave cooling ina nanofiber-based optical potential.The paper is organized as follows. In Sec. II we presentthe general formulae to calculate the ac Stark shift of amultilevel atom interacting with a far-off-resonance lightfield of arbitrary polarization and an external magneticoffset field. The concept to describe the influence of thenanofiber surface on the trapping potential is also dis-cussed. In Sec. III we apply the formalism to describe thetwo-color nanofiber-based optical trap realized by Vetsch et al. [9] and present the results of numerical calcula-tions for the adiabatic potentials of a cesium atom in theground state 6 S / or the excited state 6 P / . The defi-ciencies of the common approach of adiabatic potentialsare discussed. In Sec. IV we study the effect of an exter-nal magnetic field on the trap of Vetsch et al. [9] and findcases of non-zero Franck-Condon factors between mo-tional states of different hyperfine-structure levels. Ourconclusions are given in Sec. V. II. FORMALISM
Let us examine the energy shifts of levels of a singlefine-structure state. For this purpose, we will closely fol-low the formalism reviewed in Ref. [17]. In general, due to the degeneracy of atomic levels and the possibility of levelmixing, we must diagonalize the full interaction Hamil-tonian in order to find the energy level shifts [18, 19]. Inthe case of an atom in a two-color nanofiber-based dipoletrap, we have the following interaction Hamiltonian forthe internal state | nJ i of the atom H int = V hfs + V EE + V B . (1)The atomic energy levels can be subject to perturbationsby the Stark interaction V EE , the hyperfine interaction(hfs) interaction V hfs , and the interaction with an exter-nal magnetic field V B . For the description of the two-color trap, we take the hfs interaction from Ref. [20] anddiscuss V EE , V B as well as the interaction of the atomwith the nanofiber surface in the following. A. ac Stark shift of a multilevel atom
Here, review the general theory of the ac Stark shiftof a multilevel atom interacting with a far-off-resonancelight field of arbitrary polarization [17–19, 21, 22].In order to describe the internal states of the multilevelatom, we take a quantization axis z Q , which can be, ingeneral, arbitrary. We use this quantization axis andan associated Cartesian coordinate frame { x Q , y Q , z Q } to specify bare basis states of the atom. Due to the hfsinteraction, the total electronic angular momentum J iscoupled to the nuclear spin I . As a consequence, the pro-jection J z Q of the total electronic angular momentum J onto the quantization axis z Q is not conserved. However,in the absence of the external light field, the projection F z Q of the total angular momentum F = J + I of theatom onto the quantization axis z Q is conserved. We usethe notation | nJF M i ≡ | nJF M i Q for the atomic hfsbasis states, where F is the quantum number for the to-tal angular momentum F of the atom, M is the quantumnumber for the projection F z Q of F onto the quantizationaxis z Q , J is the quantum number for the total angularmomentum J of the electron, and n is the set of the re-maining quantum numbers { nLSI } , with L and S beingthe quantum numbers for the orbital angular momentumand the spin of the electrons, respectively.Consider the interaction of the atom with a classicallight field E = 12 E e − iωt + c . c . = 12 E u e − iωt + c . c ., (2)where ω is the angular frequency and E = E u is thepositive-frequency electric field envelope, with E and u being the field amplitude and the polarization vector,respectively. In general, E is a complex scalar and u is acomplex unit vector. ac Stark interaction operator We assume that the light field is far from resonancewith the atom. In addition, we take J to be a good quan-tum number. This means that we treat only the caseswhere the Stark interaction energy is small compared tothe fine structure splitting and, consequently, the levelmixing of atomic states with different quantum numbers J can be neglected. Then, the operator for the ac Starkinteraction in the second-order perturbation theory withrespect to the light field amplitude E is given by [17–19] V EE = − |E| (cid:26) α snJ − iα vnJ [ u ∗ × u ] · J J + α TnJ u ∗ · J )( u · J ) + ( u · J )( u ∗ · J )] − J J (2 J − (cid:27) . (3)The quantities α snJ , α vnJ , and α TnJ are the conventionaldynamical scalar, vector, and tensor polarizabilities, re-spectively, of the atom in the fine-structure level nJ .They are given in [17–19]. Note that the tensor polariz-ability vanishes for J = 1 / u ∗ × u ] vanishes, making thecontribution of the vector polarizability to the ac Starkshift to be zero.In the hfs basis {| nJF M i} , the hfs interaction opera-tor V hfs is diagonal. However, the ac Stark interaction V EE is, in general, not diagonal in both indices F and M of the hfs basis. Therefore, in order to find the dressedeigenstates and eigenvalues of the atom, one has to diag-onalize the total Hamiltonian (1). Fictitious magnetic fields
As the idea of fictitious magnetic fields will be of greatimportance to understand the nanofiber-based two-colortrap, we will discuss this concept in detail. It is clearfrom Eq. (3) that the second term on the right-handside of this equation, which is responsible for the vectorlight shift, can be written in a form that is similar to theoperator for the interaction between a static magneticfield and an atom [see Eq. (5)]. This means that theeffect of the vector polarizability on the Stark shift isequivalent to that of a magnetic field with the inductionvector [23–34] B fict = α vnJ µ B g nJ J i [ E ∗ × E ] . (4)Here, µ B is the Bohr magneton and g nJ is the Land´efactor for the fine-structure level nJ .The direction of the light-induced fictitious magneticfield B fict is determined by the vector i [ E ∗ × E ], which isa real vector. Note that B fict is independent of F , that is, B fict is the same for all hfs levels nJF of a fine-structurelevel nJ . If the light field is linearly polarized, we have i [ E ∗ × E ] = 0 and hence B fict = 0.In general, the vector Stark shift operator can be ex-pressed in terms of the operator J as V EE vec = µ B g nJ ( J · B fict ). In the case of the ground state nS / of alkali-metal atoms, since the hfs splitting of is very large com- pared to the light shift, the vector Stark shift opera-tor can be given in terms of the operator F as V EE vec = µ B g nJF ( F · B fict ). Here, g nJF is the Land´e factor forthe hfs level nJF . For the hfs levels F = I ± / nS / , we have g nJF | F = I ± / = ± g nJ / (2 I + 1). Hence, when the direction of the ficti-tious magnetic field B fict is taken as the quantizationaxis z , the vector Stark shifts of the sublevels M ofthe hfs levels F = I ± / V EE vec | F = I ± / M = ± µ B g nJ M B fict / (2 I + 1). These shiftsare integer multiples of the quantity µ B g nJ B fict / (2 I + 1).It is clear that the sublevels M and − M of the hfs levels F = I + 1 / F = I − /
2, respectively, of the groundstate have the same vector Stark shift. In contrast, thesublevels with the same number M of two different hfslevels F = I ± / B fict is invariant under space reflec-tion. Both the real and fictitious magnetic fields changesign under time reversal. Both fields shift the magneticsublevels of the atom in the same manner, expressed bythe scalar product of the magnetic dipole of the atomand the magnetic induction vector of the field [23–34].Consequently, in calculations for level shifts, the ficti-tious magnetic field B fict can be vector-added to a realstatic magnetic field B if the latter is present in the sys-tem. Precession and echos of the atomic spin in an fic-titious magnetic field have been demonstrated [25, 29],and an optical Stern-Gerlach experiment has been per-formed [27]. Localized changes of the spin of an opticallytrapped ensemble as well as RF/MW-induced evapora-tive cooling of atoms in magnetic traps with contributionof fictitious magnetic fields are possible [25, 32, 34].There are limitations to the concept of fictitious mag-netic fields [23]. As the ratio α vnJ /g nJ J is a quantitythat depends on the considered atomic level nJ , so dothe magnitude and sign of the fictitious magnetic field.This can be a restriction when the fictitious magneticfield is used as a quantization magnetic field for, e.g.,optical pumping. Furthermore, for real magnetic fields,there can be no local maximum of the magnetic field infree-space [28]. This restriction does not apply to fic-titious magnetic fields: A circularly-polarized focussedGaussian beam produces a fictitious magnetic field witha local maximum at the intensity maximum of the beam,which should allow one to build a “magnetic” trap forhigh-field-seeking paramagnetic atoms. In this context,the much faster switching time for optically induced ficti-tious magnetic fields as compared to real magnetic fieldscan turn out to be a technical advantage. nanofiber xyz z B θ B ϕ B ζ x B y B FIG. 2. Orientation of the external magnetic field with re-spect to the nanofiber.
B. Interaction with a static magnetic field
The Hamiltonian for the interaction between the mag-netic field and the atom is [20] V B = µ B g nJ ( J · B ) . (5)It can be shown that the matrix elements of the operator V B in the basis {| F M i} are given by the expression V BF MF ′ M ′ = µ B g nJ ( − J + I − M p J ( J + 1)(2 J + 1) × p (2 F + 1)(2 F ′ + 1) (cid:26) F F ′ J I J (cid:27) × X q =0 , ± ( − q B q (cid:18) F F ′ M q − M ′ (cid:19) . (6)Here, B − = ( B x Q − iB y Q ) / √ B = B z Q , and B = − ( B x Q + iB y Q ) / √ B = { B x Q , B y Q , B z Q } .We note that Eq. (6) is valid for an arbitrary quantiza-tion axis z Q .When F is a good quantum number, the interactionoperator (5) can be replaced by the operator V B = µ B g nJF ( F · B ) . (7)In the absence of the light field, the energies of theZeeman sublevels are ¯ hω nJF M = ¯ hω nJF + µ B g nJF BM .Here, ¯ hω nJF is the energy of the hfs level nJF in theabsence of the magnetic field and M = − F, . . . , F is themagnetic quantum number. This integer number is aneigenvalue corresponding to the eigenstate | F M i B of theprojection F z B of F onto the z B axis.In general, the quantization axis z Q may be differ-ent from the magnetic field axis z B and, consequently, | F M i ≡ |
F M i Q may be different from | F M i B . In order to find the level energy shifts, we must add the magneticinteraction operator V B to the combined hfs-plus-Starkinteraction operator and then diagonalize the resultingoperator (1).When we use the direction z B of the applied magneticfield as the quantization axis for the internal states ofthe atom (see Fig. 2), we need to describe the nanofiber-guided light field E in a Cartesian coordinate system { x B , y B , z B } . Let θ B be the angle between the direc-tion z B of the magnetic field and the fiber axis z . As-sume that the plane ( z, z B ) intersects with the fiber cross-section plane ( x, y ) at a line ζ . Let ϕ B be the angle be-tween ζ and x . We choose the axes x B and y B such that x B is in the plane ( z B , z ) and y B is in the plane ( x, y ).Then, the transformation for an arbitrary vector E fromthe coordinate system { x, y, z } to the coordinate system { x B , y B , z B } is given by the equations E x B = ( E x cos ϕ B + E y sin ϕ B ) cos θ B − E z sin θ B , E y B = −E x sin ϕ B + E y cos ϕ B , E z B = ( E x cos ϕ B + E y sin ϕ B ) sin θ B + E z cos θ B . (8) C. Atom-surface interaction
We approximate the surface-induced potential by thevan der Waals potential U surf = − C / ( r − a ) , where C isthe van der Waals coefficient for the interaction betweenan atom and a dielectric surface. This approximation isjustified by the fact that, in the close vicinity of the fibersurface, the geometry of the surface is not essential and,consequently, the atom sees the fiber surface as a flatsurface [14]. Meanwhile, in the region of large distances(comparable to or much larger than the fiber radius),the surface-induced potential is small and falls off fasterthan the optical potential [14]. In the case consideredhere, the surface is silica and the atom is cesium. Fornumerical calculations, we take C (6 S / ) = 2 π ¯ h × . µ m [35] and C (6 P / ) = 2 C (6 S / ) [36, 37]. III. NANOFIBER-BASED TWO-COLOR TRAP
In the experiment of Vetsch et al. [9], atomic cesiumwas trapped in a nanofiber-based two-color optical po-tential [13, 14]. Two light fields, a red-detuned standingwave field and a blue-detuned running wave field, arelaunched into the fiber and are guided in the fundamen-tal modes (see Fig. 1). The parameters of the experimentare the fiber radius a = 250 nm, the wavelengths of thetrapping fields λ = 1064 nm (red-detuned with respectto the cesium D lines) and λ = 780 nm (blue-detunedwith respect to the cesium D lines), and the respectivepowers P = 2 × . P = 25 mW. P o l a r i za b iliti e s ( a . u . ) Wavelength of light (nm)
FIG. 3. Polarizabilities of the ground state 6 S / (red color)and the excited state 6 P / (blue color) of atomic cesium inthe region of wavelengths from 775 nm to 785 nm. The scalar,vector, and tensor components α snJ , α vnJ , and α TnJ are shownby the solid, dashed, and dotted curves, respectively. P o l a r i za b iliti e s ( a . u . ) Wavelength of light (nm)
FIG. 4. Same as Fig. 3 but in the region of wavelengths from1060 nm to 1070 nm.
A. Dynamic polarizability of cesium
For calculating the optical potentials of the two-colortrap with the formalism presented in II, the polarizabil-ity of cesium for the trapping wavelength is required. Weplot in Figs. 3 and 4 the polarizabilities α snJ (solid lines), α vnJ (dashed lines), and α TnJ (dotted lines) of the groundstate 6 S / (red color) and the excited state 6 P / (bluecolor) in the regions around the wavelengths 780 nm and1064 nm of the trapping light fields. The calculationis based on the formalism and the data set provided inRef. [17]. We observe from Figs. 3 and 4 that the mag-nitude of the vector and the tensor polarizabilities α vnJ α TnJ can be, in general, substantial. This means that theac Stark shift can be different for each Zeeman state,depending on the polarization of the light field.
B. Guided modes of optical nanofibers
When the radius of the nanofiber is well below thewavelengths of the guided light fields, the nanofiber sup-ports only the hybrid fundamental modes HE corre-sponding to each given wavelength [38]. The light fieldin such a mode is strongly guided. It penetrates into theoutside of the nanofiber in the form of an evanescent wavecarrying a significant fraction of the optical power [39]. Inorder to describe guided light fields, we use Cartesian co-ordinates { x, y, z } and associated cylindrical coordinates { r, ϕ, z } , with z being the fiber axis.Suppose, the nanofiber is a silica cylinder of radius a and refractive index n and is surrounded by an infinitevacuum of refractive index n = 1. For a fundamentalguided mode HE of a light field of frequency ω (free-space wavelength λ = 2 πc/ω and free-space wave number k = ω/c ), the propagation constant β is determined bythe fiber eigenvalue equation [38]. The cylindrical com-ponents of the unnormalized mode-profile vector function e ( r ) of the electric part of the fundamental guided modeare given, for r > a , by [38] e r = i [(1 − s ) K ( qr ) + (1 + s ) K ( qr )] ,e ϕ = − [(1 − s ) K ( qr ) − (1 + s ) K ( qr )] ,e z = 2 qβ K ( qr ) . (9)Here the parameter s is defined as s = (1 /h a +1 /q a ) / [ J ′ ( ha ) /haJ ( ha ) + K ′ ( qa ) /qaK ( qa )]. The pa-rameters h = ( n k − β ) / and q = ( β − n k ) / characterize the fields inside and outside the fiber, re-spectively. The notations J n and K n stand for theBessel functions of the first kind and the modified Besselfunctions of the second kind, respectively. We notethat the axial component e z is significant in the case ofnanofibers [39]. This makes guided modes of nanofibersvery different from plane-wave modes of free-space andfrom guided modes of conventional, weakly guiding opti-cal fibers [38, 39].In the experiment by Vetsch et al. [9], an optical po-tential with a trapping minimum sufficiently far from thefiber surface was produced by two guided light fields E and E in the fundamental modes with red- and blue-detuned frequencies ω and ω , respectively. In the ex-periment, the field E is a pair of counter-propagating x -polarized beams, while the field E is a single y -polarizedbeam (see Fig. 1). The combined field of the pair ofcounter-propagating x -polarized red-detuned beams is E = A { [ˆ x ( e r cos ϕ − ie ϕ sin ϕ ) + ˆ y ( e r + ie ϕ ) × sin ϕ cos ϕ ] cos βz + i ˆ z e z cos ϕ sin βz } . (10)In deriving the above equation we have assumed, withoutloss of generality, that the point z = 0 corresponds to anantinode of the transverse component of the field. Thepolarization of E is linear at every point in space.Meanwhile, the single y -polarized blue-detuned field is E = A [ˆ x ( e r + ie ϕ ) sin ϕ cos ϕ + ˆ y ( e r sin ϕ − ie ϕ cos ϕ ) + ˆ z e z sin ϕ ] e iβz . (11)The above equations for the trapping light fields inconjunction with the atomic polarizability given in III Aare used to calculate the optical potential of the two-colortrap. C. Adiabatic trapping potentials for a cesium atom
In this section, we present the results of numerical cal-culations for the adiabatic potentials of the ground state6 S / and the excited state 6 P / of a cesium atom inthe two-color nanofiber-based trap realized in the exper-iment [9, 40].The total atomic trap potential U consists of the opti-cal potential U opt and the surface-induced potential U surf (see II C), that is, U = U opt + U surf . (12)In the work by Vetsch et al. [9], there was no exter-nal magnetic offset field present and we therefore neglectsuch an interaction for the moment. The optical potential U opt is produced by the light shifts of the atomic energylevels and obtained by diagonalizing the total interactionHamiltonian (see II) H int = V hfs + V EER + V EEB (13)at each point in space. Here, V EER and V EEB are theoperators for the Stark interaction caused by the red-and blue-detuned light fields, respectively. In view of thelarge mutual detuning, the interference between the red-and blue-detuned light fields has been neglected in Eq.(13).We plot in Figs. 5, 6, and 7 the radial, azimuthal, andaxial dependences of the potentials of the sublevels ofthe hfs levels F = 3 and F = 4 of the ground state 6 S / and of the hfs level F = 4 of the excited state 6 P / ofa cesium atom in the nanofiber-based trap. In order toavoid overcrowding, we do not show the potentials of theother 6 P / hfs levels. The dashed black and solid redcurves of the figures, corresponding to the potentials forthe sublevels of the hfs levels F = 3 and F = 4 of theground state 6 S / , respectively, show clearly that thereare trap minima at the positions with the coordinates r − a = 224 nm, ϕ = 0 , ± π , and z = 0. Since the red-detuned field is a standing wave along the fiber axis, thereare two arrays of trapping minima [9]. The trap depthis about 0 .
43 mK, 1 .
75 mK, and 0 .
87 mK in the radial,azimuthal, and axial directions, respectively. The bluecurves, corresponding to the potentials for the sublevelsof the hfs level F = 4 of the excited state 6 P / , showthat the excited states are not trapped.The intersection of the dashed black and solid redcurves in Fig. 6(a) occurs at ϕ = 0 , ± π , that is, on the Atom-to-surface distance r-a (nm) P F =4 T r a p po t e n ti a l ( m K ) S F =3 and 4
FIG. 5. Radial dependence of the potentials of the ground andexcited states of a cesium atom in the nanofiber-based trap.The sublevels of the excited-state hfs level 6 P / F = 4 areshown as solid blue curves, and the sublevels of the ground-state hfs levels 6 S / F = 3 and 6 S / F = 4 are shownas dashed black and solid red curves, respectively. The az-imuthal and axial coordinates of the atom are ϕ = 0 and z = 0, respectively. For parameters of the trap, see the text. P F =4 S F =3 and 4 - π /2 π /2- π π π π ππ Azimuthal position ϕ T r a p po t e n ti a l ( m K ) S F =3 and 4 (a)(b)
FIG. 6. (a) Azimuthal dependence of the potentials of theground and excited states of a cesium atom in the nanofiber-based trap. The sublevels of the excited-state hfs level6 P / F = 4 are shown as solid blue curves, and the sublevelsof the ground-state hfs levels 6 S / F = 3 and 6 S / F = 4 areshown as dashed black and solid red curves, respectively. Theradial distance and axial position of the atom are r − a = 224nm and z = 0, respectively. For parameters of the trap,see the text. (b) Expanded view of the potentials of theground-state hfs levels in Fig. 6(a) around the trap mini-mum at ϕ = 0. The sublevels of the ground-state hfs levels6 S / F = 3 and 6 S / F = 4 are shown as dashed black andsolid red curves, respectively. Axial position z (nm) T r a p po t e n ti a l ( m K ) P F =4 S F =3 and 4
FIG. 7. Axial dependence of the potentials of the ground andexcited states of a cesium atom in the nanofiber-based trap.The sublevels of the excited-state hfs level 6 P / F = 4 areshown as solid blue curves, and the sublevels of the ground-state hfs levels 6 S / F = 3 and 6 S / F = 4 are shown asdashed black and solid red curves, respectively. The radialdistance and azimuthal position of the atom are r − a = 224nm and ϕ = 0, respectively. For parameters of the trap, seethe text. x axis, where B fict = 0. The details of the potentials ofthe ground-state hfs levels around the trap minimum at ϕ = 0 are expanded in Fig. 6(b). We observe from Fig.6 that the degeneracy of the sublevels of the hfs levelsof the ground state is lifted at ϕ = 0 , ± π but remainsat ϕ = 0 , ± π , and the energy separation between thesesublevels varies in the azimuthal direction. Such a behav-ior is a consequence of the azimuthal dependence of themagnitude of the vector Stark shift produced by the in-teraction with the quasi-linearly polarized running-waveblue-detuned light field.We also see from Fig. 6(b) that the local minima ofthe potentials of the ground-state hfs sublevels in the az-imuthal direction are slightly displaced from each otherin the vicinities of the intersection points ϕ = 0. Thesame displacements also occur in the vicinity of ϕ = ± π .The reason is that, at ϕ = 0 , ± π , the light-induced fic-titious magnetic field B fict changes its direction from+ˆ x to − ˆ x (see Fig. 8), and so do the differential shifts µ B g nJF B fict ( M − M ′ ) of the sublevels | F M i B fict and | F M ′ i B fict . Here, | F M i B fict are the eigenstates of theangular momentum projection operator F z B fict with thedirection of B fict taken as the quantization axis. We ob-serve from Figs. 5, 6, and 7 that the sublevels M and − M of the hfs levels F = 4 and F = 3, respectively, of theground state have the same vector Stark shift and, con-sequently, the same potential. Meanwhile, the sublevelswith the same quantum number M = 0 of the differenthfs levels F = 4 and F = 3 have opposite vector Starkshifts and, consequently, oppositely displaced potentials. D. Limits of the concept of adiabatic potentials
As outlined, we diagonalize the interaction Hamilto-nian (13) at fixed points R in space. Then, we get a setof local eigenstates | F ( R ) M ( R ) i and local eigenvalues E F M ( R ). We subtract the hfs splittings from the ob-tained eigenvalues E F M ( R ) and add the van der Waalspotential. As a result, we get a set of potential branches U F M ( R ), which are called adiabatic potentials.Such adiabatic potentials are not conventional poten-tials for the translational motion [41–46]. The reasonis that the Hamiltonian (13) acts on both the inter-nal state and the center-of-mass motion of the atom.Because of this, the diagonalization of the Hamilto-nian (13) with respect to only the internal degrees offreedom of the atom is just a partial diagonalization.The full Hamiltonian for the atom is H = T + H int ,where T = P / m is the operator for the kinetic en-ergy, with P and m being the momentum and the massof the atom, respectively. In terms of the local in-ternal eigenstates | F ( R ) M ( R ) i , we can write H int = P F M E F M ( R ) | F ( R ) M ( R ) ih F ( R ) M ( R ) | . We intro-duce the unitary transformation H ′ = R − ( R ) H R ( R )and | Ψ ′ i = R − ( R ) | Ψ i for the full Hamiltonian H andthe full state vector | Ψ i of the atom, where R ( R ) isa position-dependent unitary operator. Such a trans-formation is similar to the use of a position-dependentcoordinate system for describing a spin in an inhomo-geneous magnetic field [44, 45]. We choose R ( R ) = P F MF ′ M ′ h F ′ M ′ | F ( R ) M ( R ) i| F ′ M ′ ih F M | , which actson the internal states such that | F ( R ) M ( R ) i = R ( R ) | F M i . When we perform the above transforma-tion, we obtain H ′ = H ad + ∆ T , where H ad = T + P F M E F M ( R ) | F M ih F M | is the adiabatic Hamiltonianand ∆ T = R − ( R ) T R ( R ) − T = [ A ( R ) · P + P · A ( R ) + A ( R )] / m is the nonadiabatic correction, with A ( R ) = − i ¯ h R − ( R ) ∇R ( R ) [41, 43]. The omission of ∆ T con-stitutes the adiabatic approximation. In this approxima-tion, the local internal eigenstate of the atom is rigidlycoupled to the translational degrees of freedom and, con-sequently, the atom may be considered as having onlytranslational degrees of freedom. Under this approxima-tion, the atom follows a potential branch U F M ( R ) to re-main in an adiabatic state | Ψ( t ) i = | F ( R ) M ( R ) i ψ ( R , t )if it is initially prepared in a local internal eigenstate | F ( R ) M ( R ) i . Here, ψ ( R , t ) is the time-dependent wavefunction for the center-of-mass motion of the atom in thepotential U F M ( R ). The adiabatic approximation holdstrue when the matrix elements h F ′ M ′ | F ( R ) M ( R ) i varyin space slowly enough that ∆ T can be considered as asmall perturbation.Like in the case of magnetic traps, two effects may arisewhen the nonadiabatic term ∆ T is not negligible: (1)the atom can undergo Majorana spin-flip transitions [47](due to nondiagonal matrix elements of ∆ T ) and (2) themotion of the atom in an adiabatic potential is modi-fied by geometric forces [44–46] (due to diagonal matrixelements of ∆ T ). For magnetic traps, the rates of Ma-jorana transitions have been calculated by a perturba-tion method for F = 1 / F = 1 [42] andfor F > ϕ = 0 , ± π . This indicates that the adiabaticapproximation is not valid in this situation. Because ofthis breakdown, the internal and external dynamics ofan atom with a finite center-of-mass motion cannot bedescribed by the calculated adiabatic potentials. How-ever, due to the dominant effect of the scalar Stark shift,all adiabatic potential branches of the ground state aretrapping potentials for the atom. Therefore, if its kineticenergy is low enough, the atom remains in the trap evenwhen undergoing spin-flip transitions.The study of spin-flip transitions and geometric forcesis beyond the scope of this paper. However, in or-der to reach long ground-state coherence times of thenanofiber-trapped atom, e. g., for electromagnetically in-duced transparency (EIT) experiments or the implemen-tation of a quantum memory, spin-flips of the trappedatoms have to be suppressed.These effects have not been considered in the analy-sis of the experiment of Vetsch et al. [9] by the authorsof Ref. [11]. However, we have experimental evidencethat they have to be taken into account when estimatingground-state coherence times [48].
1. Fictitious magnetic fields by nanofiber-guided light fields
In section II, we introduced the scalar, vector, andtensor polarizabilities for the description of the interac-tion of the atom with a light field and pointed out thatthe tensor polarizability is zero J = 1 / et al. [9] and illustrated in Fig. 1. The field in aquasi-linearly polarized running-wave mode can be com-posed as a superposition of the left- and right-handed circular fields, that is, E lin = A [ˆ r e r cos( ϕ − ϕ ) + i ˆ ϕ e ϕ sin( ϕ − ϕ )+ f ˆ z e z cos( ϕ − ϕ )] e ifβz . (14)Here the angle ϕ specifies the principal direction of thepolarization vector in the fiber transverse plane, and theindex f = 1 or − E lin - st = A [ˆ r e r cos( ϕ − ϕ ) cos β ( z − z )+ i ˆ ϕ e ϕ sin( ϕ − ϕ ) cos β ( z − z )+ i ˆ z e z cos( ϕ − ϕ ) sin β ( z − z )] . (15)The parameter z specifies the positions of the nodes andantinodes along the fiber axis.We now study the fictitious magnetic field B fict pro-duced by the light field in a nanofiber guided mode. Asdiscussed in the previous section, the vector Stark shiftof an atom is equivalent to that caused by a fictitiousmagnetic field B fict oriented along the direction of thevector i [ E ∗ × E ] [23–34]. In general, we have i [ E ∗ × E ] = 2Im( E ϕ E ∗ z )ˆ r + 2Im( E z E ∗ r ) ˆ ϕ + 2Im( E r E ∗ ϕ )ˆ z . (16)Note that the spatial dependences of E r , E ϕ , and E z aredetermined by the mode profile functions e r , e ϕ , and e z and the polarization of the field. Unlike plane-wave lightfields, a guided light field may have a nonzero axial com-ponent E z . Therefore, the fictitious magnetic field pro-duced by a guided light field may have a nonzero compo-nent B fict ⊥ = B fict x + B fict y = B fict r + B fict ϕ in the transverseplane as opposed to the fictitious magnetic field createdby a paraxial light field.For quasi-linearly polarized running-wave modes (14),we have Im( E r E ∗ ϕ ) = 0, which leads to B fictlin = α vnJ µ B g nJ J [Im( E ϕ E ∗ z )ˆ r + Im( E z E ∗ r ) ˆ ϕ ] . (17)In this case, B fict lies in the fiber transverse plane ( x, y )(see Fig. 8). It is clear from Eqs. (14) and (17) that theorientation direction of B fict reverses when we changethe propagation direction f of the light field. We notethat, for the azimuthal coordinate ϕ = ϕ ± π/
2, we have E z = 0, which leads to the exact linear polarization and,consequently, to B fict = 0, that is, to the vanishing ofthe vector Stark shift [11]. Without loss of generality, wechoose ϕ = π/
2. For this particular choice, we can showthat B fictlin ∝ | A | α vnJ K ( qr ) sin ϕ { ˆ x [(1 − s ) K ( qr ) − (1 + s ) K ( qr ) cos 2 ϕ ] − ˆ y (1 + s ) K ( qr ) sin 2 ϕ } . (18)Since s ≃ − K ( qr ) > K ( qr ) >
0, we have | B fict x | ≫ | B fict y | . Thus, the fictitious magnetic field B fict is oriented mainly along the axis x , which is perpendic-ular to the axis z of the fiber and to the principal axis y (a) xz y B fict (b) (c) ϕ y - a (nm) π−π B f i c t & B f i c t ( G ) x y E β FIG. 8. (a) Azimuthal vector profile (blue arrows) and (b)radial and (c) azimuthal dependences of the components B fict x (green curves) and B fict y (pink curves) of the fictitiousmagnetic field B fict produced by a quasi-linearly polarizedrunning-wave guided light field. The fiber radius is a = 250nm. The wavelength and power of light are λ = 780 nm and P = 25 mW, respectively. The field propagates along the + z direction and is polarized along the y direction. In (a) and(c), the distance to the fiber surface is r − a = 224 nm. In(b), the azimuthal angle is ϕ = π/ y axis). Other coordinates are arbitraryif not specified. of the polarization of the quasi-linearly polarized guidedlight field. The calculations in Fig. 8 show, that the mag-nitude of the fictitious magnetic field for r − a ≈
200 nm(typical atom-surface separation for atoms in a two-colortrap) reaches up to a few Gauss. This is comparableto the magnitude of real magnetic fields present in con-ventional magnetic traps for cold atoms. In contrast,the gradient of the fictitious magnetic field for the casein Fig. 8 goes beyond what is possible with most con-ventional magnetic traps, reaching about 40 000 G/cm.These large gradients of the fictitious magnetic field inconjunction with the steep potentials of optical near fieldtraps can pronounce effects that are beyond what is ex-pected from the adiabatic approximation.For standing-wave modes with quasi-linear polariza-tion (15), the polarization is linear at every point inspace. In this case, the fictitious magnetic field vanishes,that is, B fictlin - st = 0 . (19)For completeness, we discuss other nanofiber-guidedlight field configurations and the associated fictitiousmagnetic fields in the appendix. ϕ T r a p po t e n ti a l ( m K ) π−π r - a (nm) z (nm)(a) (b) (c) FIG. 9. Radial (a), azimuthal (b), and axial (c) dependencesof the potentials of the ground-state hfs levels 6 S / F = 3(dashed black curves) and 6 S / F = 4 (solid red curves) of acesium atom in the nanofiber-based trap with a homogeneousmagnetic field oriented along the z axis. The magnitude ofthe offset magnetic field is B = 5 G. The coordinates of theatom are ϕ = 0 and z = 0 in (a), r − a = 224 nm and z = 0in (b), and r − a = 224 nm and ϕ = 0 in (c). For parametersof the trap, see the text. IV. TWEAKING THE STATE-DEPENDEDPOTENTIALS OF A NANOFIBER-BASEDTWO-COLOR TRAP
Inspired by the physics of magnetic traps for coldatoms, we study the effect of an external magnetic offsetfield on the two-color trap. As main results we find, thatthe degeneracy point in Fig. 6 can be removed and, thus,spin flips can be suppressed. Moreover, we show thatthe Zeeman-state-dependent displacement of the trap-ping potential depends on the magnitude and directionof the offset field. The option of microwave cooling inthe displaced potentials is suggested and Franck-Condonfactors are calculated.
A. Two-color trap in the presence of ahomogeneous magnetic offset field
Let us consider the effect of a weak external magneticfield B on the ground-state potential of the atom in thetwo-color nanofiber-based trap (see Fig. 2). We calcu-late numerically the adiabatic potential of the atom inthe presence of the magnetic field using the formalismoutlined in Sec. II. We plot in Figs. 9 and 10 the po-tentials of the ground-state hfs levels 6 S / F = 3 and6 S / F = 4 of a cesium atom in the nanofiber-basedtrap with a homogeneous magnetic field oriented alongthe z and x axes, respectively. The magnitude of themagnetic field is B = 5 G.The figures show that the degeneracy of the Zeemansublevels is lifted in all directions, as opposed to the situ-0 ϕ T r a p po t e n ti a l ( m K ) π−π r - a (nm) z (nm)(a) (b) (c) FIG. 10. Same as Fig. 9 but the applied magnetic field isoriented along the x axis. ation of the red curves in Figs. 5–7, by the light-inducedfictitious magnetic field B fict and the applied magneticfield B , that is, by the total effective (fictitious + real)magnetic field B total = B fict + B . The splittings of theZeeman sublevels of the ground state depend on the mag-nitudes of the components B fict and B and on their rel-ative orientation. It is clear that the differential shifts ofthe sublevels vary in space. Note that, unlike Fig. 9(b),the two peaks in Fig. 10(b) are not symmetric. The dif-ference between the peaks in Fig. 10(b) is due to the factthat, according to Figs. 8(a) and 8(c), at the peak posi-tions ϕ = − π/ ϕ = π/
2, the vector B fict is orientedalong the direction ˆ x (parallel to B ) and − ˆ x (opposite to B ), respectively, leading to differing magnitudes of thetotal effective magnetic field B total and, consequently, todiffering Zeeman-state shifts. Note that Figs. 10(a) and10(c) are identical to Figs. 9(a) and 9(c), respectively.The reason is that, for ϕ = 0 , ± π , that is, along the x axis, the fictitious magnetic field B fict vanishes and,therefore, the energy shifts of the sublevels of the groundstate is determined only by the scalar polarizability andthe applied magnetic field B . Such shifts do not dependon the direction of B .Figure 11 shows clearly that the residual degeneracy ofthe ground-state sublevels, observed at the points with ϕ = 0 or ± π in the situation of Fig. 6(b), is lifted in thesituations of Figs. 9(b) and 10(b). This is due to theapplied magnetic field. When the splitting between theZeeman sublevels is large enough, the atom can adiabati-cally follow a given trapping potential thereby maintain-ing its M state.We observe from Fig. 11(b) that, when the externalmagnetic field is oriented along the x direction, the min-ima of the state-dependent trapping potentials are dis-placed from each other in the azimuthal direction. Inorder to get deep insight into the displacement of the po-tentials, we consider the close vicinity of the central trapposition ( r = r m , ϕ = 0). In this region, the component Azimuthal position ϕ T r a p po t e n ti a l ( m K ) π /5 (a) (b) π /5 −π /5 −π /50 0 FIG. 11. Expanded view of Figs. 9(b) and 10(b). B fict x of the fictitious magnetic field B fict can be consid-ered as a linear function of ϕ while the component B fict y can be neglected [see Fig. 8(c) and Eq. (18)]. Since theapplied magnetic field B in the case of Figs. 10 and 11 isoriented along the x direction and is significant enough,the fictitious magnetic field B tot is oriented mainly alongthe x direction and can therefore be approximated by the x component B tot x = B + B fict x ≃ B + B ′ fict ϕ. (20)Here the parameter B ′ fict is defined as B ′ fict =( ∂B fict x /∂ϕ ) | ϕ =0 . Hence, the azimuthal potential of asublevel | F M i of the ground state can be written as U F M ( ϕ ) ≡ U ( ϕ ) = U s ( ϕ )+ µ B g nJF M ( B + B ′ fict ϕ ), where U s ( ϕ ) is the scalar Stark shift. In the close vicinityof the trap minimum, the scalar shift can be approx-imated as U s ( ϕ ) ≃ (1 / U ′′ s ϕ + const, where U ′′ s =( ∂ U s ( ϕ ) /∂ϕ ) | ϕ =0 . Then, we obtain the azimuthal po-tential U F M ( ϕ ) = (1 / U ′′ s ϕ + µ B g nJF M B ′ fict ϕ + const,which is the potential of a displaced harmonic oscilla-tor. The equilibrium position of this potential is dis-placed from the central trap position ( r = r m , ϕ =0) in the azimuthal direction by the angle ∆ ϕ F M = − µ B g nJF M B ′ fict /U ′′ s . Thus, the displacement ∆ ϕ F M isproportional to the quantum number M , amounting toa relative displacement of 0 .
01 rad or about 4 . M = 1. B. Franck-Condon factors
When we apply a microwave field near to resonancewith the hyperfine splitting of the ground state ( ∼ . | F ′ M ′ ν ′ i ≡ | F ′ M ′ i| ν ′ i of the atom in the hfs sub-level | F ′ = 4 , M ′ i to a state | F M ν i ≡ |
F M i| ν i of theatom in the hfs sublevel | F = 3 , M i and vice versathrough a magnetic dipole transition. Here, ν ′ and ν T r a p po t e n ti a l ( µ K ) Azimuthal position r m ϕ (nm) S F =4 M =46 S F =3 M =3 (a)(b) FIG. 12. Eigenfunctions (thin lines) for the center-of-massmotion of a cesium atom in the azimuthal potentials (thick redlines) of the sublevels 6 S / F = 4 , M = 4 (a) and 6 S / F =3 , M = 3 (b) in the vicinity of the trap minimum at r m − a =224 nm. The offset magnetic field is B = 5 G and is alignedalong the x axis. For parameters of the trap, see the text. ν F νν ’ ν = 0 ν ’ = 0 ν ’ (a) (b) FIG. 13. Franck-Condon factor F νν ′ for the overlap betweenthe motional states | ν i and | ν ′ i of the Zeeman sublevels | F =3 , M = 3 i and | F ′ = 4 , M ′ = 4 i , respectively, of the cesiumground state 6 S / . The offset magnetic field is B = 5 G andis aligned along the x axis. For parameters of the trap, seethe text. are quantum numbers for the center-of-mass motion ofthe atom in the trapping potentials of the atomic in-ternal states | F ′ M ′ i and | F M i , respectively, and | ν ′ i and | ν i are the corresponding center-of-mass eigenfunc-tions. Since the magnetic dipole operator acts only onthe internal states, the transition probability dependson the Franck-Condon factor F νν ′ ≡ F ( F M ) ν ( F ′ M ′ ) ν ′ ≡|h ν | ν ′ i| , which characterizes the spatial overlap betweenthe atomic center-of-mass wave functions | ν i and | ν ′ i .In the absence of the vector Stark shift, the trappingpotentials of the sublevels | F ′ M ′ i and | F M i of the hfs levels of the ground state are identical to each other be-cause they are determined by the scalar Stark shift, whichis the same for all of these sublevels. In this case, we have h ν | ν ′ i = δ νν ′ , which leads to F νν ′ = δ νν ′ . This formulaindicates that, in the absence of the vector Stark shift, amicrowave transition between the sublevels | F ′ M ′ i and | F M i of the ground state cannot lead to a change inthe motional state. In the presence of the vector Starkshift, the trapping potentials of the sublevels | F ′ M ′ i and | F M i of the hfs levels F ′ = I + 1 / F = I − / M ′ + M = 0. They are split from eachother by an integer multiple M ′ + M of the quantity µ B g nJ B total / (2 I + 1), which varies in space. When theapplied magnetic field B is oriented in the x direction,the state-dependent trapping potentials are, as shown inthe previous subsection, displaced from each other in theazimuthal direction. In this case, a microwave transitionbetween the atomic electronic states | F ′ M ′ i and | F M i of two different hfs levels F ′ and F the ground state maylead to a change in the atomic center-of-mass motionalstate.It is computationally difficult to calculate the Franck-Condon factors between the states of the center-of-massmotion of the atom in a three-dimensional trappingpotential. Therefore, we limit ourselves to the one-dimensional motion of the atom along the azimuthal di-rection. To be specific, we consider the motion of theatom along the azimuthal line ϕ going through the trapminimum at r m − a ≃
224 nm in the situation of Figs.10(b) and 11(b). We plot in Fig. 12 the wave func-tions of the eigenstates of a cesium atom in the poten-tials of the electronic sublevels 6 S / F = 4 , M = 4 and6 S / F = 3 , M = 3. Our numerical calculations showthat the trap frequencies in the azimuthal direction areabout 157 kHz for both the upper and lower hfs levels ofthe ground state. The characteristic size of the groundmotional states of the potentials is about 16 nm whichis comparable to the displacements of the azimuthal po-tential minima for different Zeeman states.We plot in Fig. 13 the Franck-Condon factor F νν ′ as functions of ν ′ and ν . We observe that F νν ′ = 0for a number of transitions with ν = ν ′ . This indi-cates the possibility of a change in the motional statewhen the atom changes its hfs state in a microwave tran-sition. Comparison between parts (a) and (b) of Fig.13 shows that F νν ′ ≃ F ν ′ ν , that is, F ( F M ) ν ( F ′ M ′ ) ν ′ ≃ F ( F M ) ν ′ ( F ′ M ′ ) ν . The reason is that, in the close vicinityof the trapping minimum, the potentials U F M ( ϕ ) for dif-ferent sublevels | F M i have approximately the same formbut are displaced in the ϕ direction.We note that along the x axis, the fictitious magneticfield is zero. The one-dimensional potentials U F M of dif-ferent sublevels | F M i along these directions are identicalup to a constant, namely U F M = U s + µ B g nJF M B . Thisleads to F νν ′ = δ νν ′ .2 V. SUMMARY
We have calculated the adiabatic trapping potentialsof cesium atoms in a nanofiber-based two-color trap withand without an externally applied static magnetic field.This external field can simply be added to the fictitiousmagnetic field which is produced by the combined actionof the vector polarizability and the ellipticity of the po-larization of the guided light field. In the case of opticalnanofibers, the ellipticity and intensity of the guided fieldand, consequently, the fictitious magnetic field varies ona nanoscale. We showed that this results in a remark-ably high magnetic field gradient on the order of severalG/ µ m for the nanofiber-based two-color trap in [9]. Thisleads to a Stern-Gerlach-type mechanical force on thetrapped atoms which, in the absence of an external mag-netic field, mutually displaces the adiabatic potentials ofdifferent Zeeman sublevels | F M i of the atomic groundstate.The quantitative understanding of the light-inducedfictitious magnetic field allowed us to devise and to an-alyze trap configurations that make use of a magneticoffset field in order to suppress the state-dependent dis-placement of the potentials and / or to overcome itsnegative effects on the coherence of nanofiber-trappedatoms [11, 48]. Moreover, our work suggests that it ispossible to employ the fictitious magnetic field for tai-loring trapping potentials and for controlling and ma-nipulating the internal and external atomic states. Asan example, we show that for an appropriately appliedexternal magnetic field, the state dependent potentialsresult in nonzero Franck-Condon factors for the over-lap between different motional states of different hfs lev-els of the electronic ground state of the trapped atoms.This then opens the possibility of microwave cooling ofnanofiber-trapped atoms. ACKNOWLEDGMENTS
We thank R. Grimm and H. J. Kimble for helpful dis-cussions. Financial support by the Wolfgang Pauli Insti-tute and the Austrian Science Fund (FWF; Lise Meitnerproject No. M 1501-N27 and SFB NextLite project No. F4908-N23) is gratefully acknowledged.
Appendix A: Fictitious magnetic fields ofnanofiber-guided light fields – continued
In general, an arbitrary guided mode of a single-modefiber at a given frequency can be composed as a super-position of HE modes with different polarizations andpropagation directions. To complete the discussion offictitious magnetic fields generated by fiber-guided lightfields as started in the main text, we present resultsfor the following light field configurations: (a) a quasi-circularly polarized running-wave mode, (b) a pair of (a) xz y (b) r - a (nm) B fict B f i c t & B f i c t ( G ) ϕ z E β FIG. 14. Azimuthal vector profile (a) and radial dependence(b) of the tangential component B fict ϕ (blue arrows and bluecurve) and the axial component B fict z (red arrows and redcurve) of the fictitious magnetic field produced by a quasi-circularly polarized running-wave guided light field. The fieldpropagates along the + z direction and is counterclockwise po-larized. The fiber radius is a = 250 nm. The wavelength andpower of light are λ = 780 nm and P = 25 mW, respectively.In (a), the distance to the fiber surface is r − a = 224 nm.Other coordinates are arbitrary if not specified. counter-propagating modes with the same quasi-circularpolarization, (c) a pair of counter-propagating modeswith the opposite quasi-circular polarization, and (d) apair of counter-propagating modes with opposite quasi-linear polarization. A combination of (a) and (b) wouldform a nanofiber-based trap with toroidal trapping po-tential whereas a combination of (a) and (c) is relevantwhen realizing a nanofiber-based helical trap [49].(a) The electric component of the guided light field ina quasi-circularly polarized running-wave mode is givenby E circ = A (ˆ r e r + l ˆ ϕ e ϕ + f ˆ z e z ) e ifβz + ilϕ . (A1)The index l = 1 or − f = 1 or − A is determined fromthe propagating power of the light field in the axial di-rection.According to Eqs. (9), the function e r is imaginary-valued and the functions e ϕ and e z are real-valued.Therefore, for quasi-circularly polarized running-wavemodes (A1), we have Im( E ϕ E ∗ z ) = 0, and consequentlythe fictitious magnetic field B fict , given by Eq. (4), re-duces to B fictcirc = α vnJ µ B g nJ J [Im( E z E ∗ r ) ˆ ϕ + Im( E r E ∗ ϕ )ˆ z ] . (A2)In this case, B fict has a tangential component B fict ϕ and an3axial component B fict z (see Fig. 14). It is clear from Eqs.(A1) and (A2) that the orientation direction ± ˆ ϕ of thetangential component B fict ϕ depends on the propagationdirection f = ± l = ±
1, while the orientation direction ± ˆ z ofthe axial component B fict z depends on the polarizationcirculation direction l = ± f = ± E circ - st = A [(ˆ r e r + l ˆ ϕ e ϕ ) cos β ( z − z )+ i ˆ z e z sin β ( z − z )] e ilϕ . (A3)In this case, we have Im( E z E ∗ r ) = 0, which leads to B fictcirc - st = α vnJ µ B g nJ J [Im( E ϕ E ∗ z )ˆ r + Im( E r E ∗ ϕ )ˆ z ] . (A4)The resulting B fict thus has a radial component B fict r andan axial component B fict z (see Fig. 15). Equations (A3)and (A4) and Fig. 15(c) show that the magnitudes of B fict r and B fict z oscillate when we vary the axial position z . The modulations of B fict r and B fict z along the fiber axisare governed by the functions cos β ( z − z ) sin β ( z − z )and cos β ( z − z ), respectively. Hence, the sign ofthe radial component B fict r varies with varying the ax-ial position z while the sign of the axial component B fict z does not. Moreover, at the axial quasi-node positions β ( z − z ) = ( n + 1 / π , where n is an arbitrary integer,both components B fict r and B fict z and consequently thefictitious magnetic field B fict become zero everywhere inthe fiber transverse plane. At these specific axial posi-tions, the transverse component E ⊥ of the electric field E vanishes, that is, the field E becomes linearly polar-ized along the fiber axis z . We note that the signs of thecomponents B fict r and B fict z depend on the polarizationcirculation direction l = ± E σ l - σ − l = A { ˆ r e r cos[ βz + l ( ϕ − ϕ )]+ il ˆ ϕ e ϕ sin[ βz + l ( ϕ − ϕ )]+ i ˆ z e z sin[ βz + l ( ϕ − ϕ )] } . (A5)In this case, the polarization is linear at every point inspace. In this case, the fictitious magnetic field vanishes,that is, B fict σ l - σ − l = 0 . (A6)(d) For a pair of counter-propagating modes with op- (a) xz y B fict (b) (c) r - a (nm) z - z (nm) B f i c t & B f i c t ( G ) r z E FIG. 15. Azimuthal vector profile (a) and radial (b) andaxial (c) dependences of the components B fict r (blue arrowsand curves) and B fict z (red arrows and curves) of the fictitiousmagnetic field B fict produced by a quasi-circularly polarizedstanding-wave guided light field. The field is counterclockwisepolarized. The fiber radius is a = 250 nm. The wavelengthand power of light are λ = 780 nm and P = 2 ×
25 mW,respectively. The radial position is r − a = 224 nm in (a) and(c). The axial position is z − z = 100 nm in (a) and (b).Other coordinates are arbitrary if not specified. posite quasi-linear polarizations, the field is E lin ⊥ lin = A n ˆ r e r { i sin[ β ( z − z ) + ϕ − ϕ ]+ cos[ β ( z − z ) − ϕ + ϕ ] } + ˆ ϕ e ϕ { cos[ β ( z − z ) + ϕ − ϕ ] − i sin[ β ( z − z ) − ϕ + ϕ ] } + ˆ z e z { cos[ β ( z − z ) + ϕ − ϕ ]+ i sin[ β ( z − z ) − ϕ + ϕ ] } o . (A7)In this case, the radial, azimuthal, and axial componentsof the fictitious magnetic field are, in general, nonzeroand are given by the expression B fictlin ⊥ lin = α vnJ µ B g nJ J [Im( E ϕ E ∗ z )ˆ r + Im( E z E ∗ r ) ˆ ϕ + Im( E r E ∗ ϕ )ˆ z ] . (A8)It is clear from the above expression and from Fig. 16that both the axial component B fict z and the transversecomponent B fict ⊥ = q | B fict x | + | B fict y | of the fictitiousmagnetic field can be nonzero. Figure 16 shows thatthe magnitude and orientation of the fictitious magneticfield vary in space in a complicated manner. We observefrom Fig. 16 that both the axial component B fict z andthe transverse component B fict ⊥ become zero along somespecific radial directions at some specific axial positions.4 (a) xz y (b) (c) r - a (nm) B f i c t & B f i c t ( G ) z B fict lin lin z - z (nm) FIG. 16. Azimuthal vector profile (a) and radial (b) and axial(c) dependences of the transverse component B fict ⊥ (blue ar-rows and curves) and the axial component B fict z (red arrowsand curves) of the fictitious magnetic field B fict produced bya pair of counter-propagating guided light fields with orthog-onal quasi-linear polarizations. The parameters z = 0 and ϕ = 0 are chosen. The fiber radius is a = 250 nm. The wave-length and power of light are λ = 780 nm and P = 2 × r − a = 224 nm in(a) and (c). The azimuthal position is ϕ = π/ z = 20 nm in (a) and z = 0 in (b).Other coordinates are arbitrary if not specified. A close inspection of Eqs. (A7) and (A8) reveals that B fict vanishes at the points with the coordinates ϕ − ϕ = π/ nπ/ β ( z − z ) = π/ mπ/
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