Snake States of Neutral Atom from Synthetic Gauge Field in a Ring-Cavity

Snake states of neutral atom from synthetic gauge ﬁeld in a ring-cavity

Poornima Shakya and Sankalpa Ghosh Department of Physics, Indian Institute of Technology Delhi, New Delhi-110016, India

We propose the creation of an atomic analogue of electronic snake states in which electrons move along one-dimensional snake-like trajectory in the presence of a suitable magnetic field gradient. To this purpose, we propose the creation of laser induced synthetic gauge field inside a three-mirror ring cavity and show that under appropriate conditions, the atomic trajectory in such configuration mimics snake-state like motion. We analyse this motion using semi-classical and full quantum mechanical techniques for a single atom. We provide a detailed comparison of the original electronic phenomena and its atomic analogue in terms of relevant energy and length scales and conclude by briefly pointing out the possibility of consequent study of ultra cold condensate in similar ring-cavity configuration.

PACS numbers:

I. INTRODUCTION first experimental confirmation of such states was carried in a 2DEG realised in GaAs-AlGaAs heterostructure by The ability to control the dynamics of an ultra cold K. von Klitzing’s group [29]. These developments were atomic system at nano-kelvin temperature using light- subsequently followed by a substantial number of theo- atom interaction provides a fertile ground for quantum retical and experimental studies [30–37]. simulation [1–3]. Particularly interesting are a wide To realise such light induced synthetic gauge field that range of exotic phenomena that arise in an electronic can create the atomic analogue of such snake states, we condensed matter system coupled to real electromagnetic consider a single two-level atom interacting with two field, can now be quantum simulated at a completely counter-propagating running wave modes in a three- different energy and length scale with the creation of mirror ring cavity (see Fig.1). In a cavity, the optical a wide variety of light-induced synthetic gauge field for field is quantized [38] and there is appreciable back action such charge neutral atoms [4–12]. For example, synthetic of the atomic motion on the cavity modes [39] leading gauge-field driven quantum simulations can validate the to novel quantum phases of many-body atomic system superfluidity in a strongly interacting neutral fermionic through self-organisation [40–43]. Recently, it has been system driven through the BCS-BEC cross-over [13–15]; suggested that ultra cold atomic system inside a single creation of synthetic spin-orbit coupling for charge neu- mode [11] or a multimode cavity [12] can be used to gen- tral bosonic [16] or fermionic atoms [17] can enable the erate synthetic gauge field that are dynamical in nature. realisation of topological phases [18, 19] in such non- Both these studies carried out in a linear cavity. electronic charge neutral quantum materials; it can her- Whereas a linear cavity supports only standing wave ald futuristic quantum technology through the realisa- modes, in a three mirror ring cavity that we consider tion of atomtronic devices such as an atom-squid [20] here, provides a more richer variety of options. The - to name a few. The major focus in this endeavour laser beam can be pumped-in from two different mir- was the creation of quantum Hall phases[21–23] that re- rors and they travel in opposite directions inside the quires uniform synthetic magnetic field or other topologi- cavity [44–57]. Additionally, a laser beam can also be cal phases [24]. In comparison to these studies, much less pumped-in from the transverse direction by directly il- investigation took place to explore such atomic dynam- luminating the atoms. Consequently a ring cavity sup- ics in synthetic magnetic field that is nonuniform in space ports pairs of counter-propagating travelling wave modes and the consequent quantum simulation of any electronic (clockwise and counter-clockwise) apart from the stand- phenomena. In this work, we extend the ambit of such ing wave modes (sine and cosine) [44, 45]. Each of these synthetic magnetic field driven quantum simulation by two counter-propagating modes can be populated sep- proposing the atomic analogue of snake states of elec- arately. The coherent scattering of photons by atoms trons that describes one dimensional snake-like motion results in redistribution of the photons in these two of an electron in a specific form of in-homogenous mag- counter-propagating travelling modes [44, 46]. arXiv:2002.11926v1 [cond-mat.quant-gas] 27 Feb 2020 netic field that changes sign at a given point and has been In this work, we consider such a ring-cavity set up to recently realised in graphene p-n junction [25, 26]. propose a non-uniform synthetic transverse gauge field Snake states are special one-dimensional current car- for a single charge-neutral ultra cold atom that changes rying states that occur in two-dimensional electron gas its sign about a point of symmetry where it vanishes, (2DEG) due to magnetic field gradient, and, exists at and the resulting snake trajectories of the charge-neutral the boundary where the applied transverse magnetic field atom in such gauge field. A brief outline of the paper changes sign (B = 0). They were first theoretically stud- is as follows. In section II, we describe the considered ied by Müller[27] who also pointed out their connection ring cavity system, its corresponding Hamiltonian and with the conventional quantum Hall edge states [28]. The the dressed state energies of the atom in the presence of 2 cavity fields. In section III, we derive the Schrödinger with the cavity field polarized along the y-direction and equation for the atomic wavefunction, and obtain the the pump field polarized along x-direction. Here 0 is equation for the probability amplitude for the atom to the vacuum permittivity, V is the mode volume, w0 is 2π be in the lowest energy dressed state. Due to the adia- the beam waist, k = and λp is the pump wavelength λp batic following of the lowest energy dressed state, we get and ωp is the pump frequency. a vector potential which gives rise to a non-uniform magnetic field. In section IV, we obtain the trajectories of It may be pointed out that the cavity de- the atom moving in the presence of such a non-uniform cay rate, κ, and the atomic spontaneous emis- magnetic field. In section V, we provide an analysis of sion rate, Γ, is included in the master equation the effective potential and the energy spectrum for the for the atom-cavity density matrix, ρ, by using i the Lindblad operators [40]-ρ ˙ = − [HˆSP , ρ] + atom moving in such magnetic fields ~ P † † Lρ, where, Lρ = κ i=1,2 2âiρaî − {aî aî, ρ} + Γ P (2ˆσ ρσˆ − {σˆ σˆ , ρ}). We will be working in II. MODEL SYSTEM AND THE HAMILTONIAN i=1,2 − + + − the good-cavity regime, g0 >> κ [51], where the photon is emitted and reabsorbed a number of times before it The total Hamiltonian describing the coupled atom- leaves the cavity through dissipation processes and gives cavity system [58] depicted in Fig.1 can be written as rise to a coherent evolution of the system. Since we are primarily interested in studying the effect of the atom- HˆSP = HÂ + HˆC + Hînt.. (1) cavity interaction on the atomic dynamics, we neglect the Here the atomic and cavity part of the hamiltonian are the cavity decay rate and the atomic spontaneous decay respectively given as - rate in our current work. ~ 2 Since Ω , g << ω , ω (g ∼ GHz, ω , ω ∼ T Hz) ˆ P ˆ ~ωaσˆz 0 0 a c 0 a c HA = I + (2) [46, 59], we can apply the rotating-wave approximation 2ma 2 (RWA) to Hˆ (see AppendixA) and to Hˆ (see Hˆ = ω (â†aˆ +a ˆ†aˆ ) (3) A−P A−C C ~ c 1 1 2 2 AppendixB). The resulting Hamiltonian for the atom- Here |gi and |ei are respectively the ground and ex- field system after RWA becomes cited state of the two-level atom with energy Eg and E respectively,σ ˆ = |eihe| − |gihg| is the Pauli matrix, e z ˆ ˆ ˆ ˆ ˆ HSP = HA + HC + Hint. Ee − Eg = ~ωa, I = |gihg| + |eihe| is the identity operator ˆ2 in the internal two-dimensional Hilbert space of the atom, P I ~ωaσˆz † † ~ = + + ~ωcaˆ1aˆ1 + ~ωcaˆ2aˆ2 and ~r and P are the atomic centre-of-mass co-ordinate 2ma 2 and momentum. We also introduce |n , n i to denote 1 2 + Ω(y) σˆ+e−iωpt +σ ˆ−eiωpt the number state for the photons of the two counter- ~ + ikx + −ikx propagating running wave modes as shown in Fig.1 re- + ~g(y) σˆ aˆ1e +σ ˆ aˆ2e spectively labelled by the cavity field operatorsa ˆ1 and − † −ikx − † ikxi +σ ˆ aˆ1e +σ ˆ aˆ2e (5) aˆ2 that are degenerate with cavity resonance frequency ωc .The single two-level atom is directly illuminated by a far red-detuned transverse pump beam, at frequency ω . The excited atom scatters the photons into these p −d~·~ex two cavity modes. where Ω(y) = Ω0 cos(ky + φ), Ω0 = E0 is the Rabi frequency, φ is the pump phase, ~g(y) = The third term of HˆSP describes the dipole interaction ~ q 2 2 2 2 ~ q −d·eˆy ωc −y /w −y /w −d·~ey ωc between the atom, the cavity field and the pump field and ~ e 0 = g0e 0 and g0 = ~ . 20V 20V is given as - ~ ~ The time-dependence of the hamiltonian (5) can be re- ˆ ˆ ˆ Hint. = HA−P + HA−C moved through a unitary transformation by the operator ~ ~ = −d · E~P − d · E~C (4)

~ + − −iω t( σˆz +â†aˆ +â†aˆ ) where d = d(ˆσ +σ ˆ ) is the dipole operator with Uˆ(t) = e p 2 1 1 2 2 (6) + − σˆ = |eihg|, and,σ ˆ = |gihe|. HÂ−P and HÂ−C describe respectively the interaction between the atom with the pump field and the cavity field in one arm of the ring that takes it to the rotating frame of the pump field. The cavity, and their corresponding electric fields are given Hamiltonian (HˆSP ) for the atom-field system after this by - transformation can then be written as [60] (see Appendix C for details) - −iωpt iωpt E~P (~r) =e ˆxE0cos(ky + φ)(e + e ) r ωc 2 2 ~ ~ −y /w0 ikx −ikx EC (~r) =e ˆy e aˆ1e +a ˆ2e + h.c. , ˆ ˆ ˆ 20V HRF = H0 + HI (7) 3

FIG. 1: (Color online) (a) Schematic for a single two-level atom trapped inside a ring cavity. The atom is directly illuminated by a far red-detuned transverse pump laser beam with Rabi frequency, Ω0, which drives the off-resonant atomic transition between the internal states |gi and |ei. The atom scatters the pump photons into either of the two cavity modes, labelled by the operators,a ˆ1 anda ˆ2. κ is the cavity field decay rate. (b) Schematic for snake trajectories of a single atom inside the ring cavity in presence of a non-uniform synthetic magnetic field which arises due to the adiabatic following of the lowest energy dressed state. The magnetic field flips direction as y changes sign and the blue and red trajectories are for two different strengths of the magnetic field.

where where ∆a = ωp − ωa is the atom-pump detuning and ∆c = ωp − ωc is the cavity-pump detuning. In our case Pˆ2 ˆ ˆ † † where only a single pump laser is used, we set ∆c = 0 by H0 = I − ~∆c aˆ1aˆ1 +a ˆ2aˆ2 2ma considering the cavity is in resonance with the pump. ˆ ~∆aσˆz + − HI = − + ~Ω(y) σˆ +σ ˆ ˆ 2 The second and third term of the HI contains all the + ikx + −ikx the terms that represent atom-photon interaction. In the + ~g(y)[ˆσ aˆ1e +σ ˆ aˆ2e space spanned by the atom-photon bare-states, namely +σ ˆ−aˆ†e−ikx +σ ˆ−aˆ†eikx] 1 2 |e, n1, n2i, |g, n1 + 1, n2i and |g, n1, n2 + 1i, HˆI contains (8) oﬀ-diagonal terms and can be written as

√ √  − ~ ∆ g(y)eikx n + 1 g(y)e−ikx n + 1 2 √a ~ 1 ~ 2 Hˆ = g(y)e−ikx n + 1 ~ ∆ 0 (9) I ~ √ 1 2 a  ikx ~ ~g(y)e n2 + 1 0 2 ∆a

Diagonalizing HˆI in these bare states basis, we get They are the following eigenstates that are called dressed states  e−ikx  |D1i, |D2i and |D3i, for the coupled atom-photon system, − (G + ∆a) A1 √ −2ikx with E1, E2 and E3 as their eigenvalues, respectively. ~ 2g(y) n1+1e E1 = − G; |D1i =   (10a) 2  A√1  2g(y) n2+1 A1  −ikx  −e sinφn1,n2 −2ikx = e cosφn1,n2 sinθn1,n2 

cosφn1,n2 cosθn1,n2 4

where

p 2 2 G = ∆a + 4g (y)(n1 + n2 + 2) p p A1 = 2G(G + ∆a); B1 = 2G(G − ∆a) r r n1 + 1 n2 + 1 sinθn1,n2 = ; cosθn1,n2 = n1 + n2 + 2 n1 + n2 + 2 r r G + ∆ G − ∆ sinφ = a ; cosφ = a n1,n2 2G n1,n2 2G r −1 n1 + 1 θn1,n2 = tan n2 + 1 r ! −1 G + ∆a φn1,n2 = tan G − ∆a

θn1,n2 and φn1,n2 are called the mixing angles, while the laser phase is given by χ = kx. The subscripts n1 and n2 denote the photon number dependence of the mixing angles. The atom-pump detuning gets modified in the presence of atom-field coupling and this modification is FIG. 2: Color online: Dressed-state energies (E1, E2 and directly proportional to the cavity mode-function, g(y) E3), as a function of the atom-pump detuning, ∆a/2π for and to the number of photons in the two cavity modes. g0/2π = 0 (solid lines) and for g0/2π = 50GHz (dotted This dependence of the atom-pump detuning on the cav- lines). E2 < 0 for ∆a < 0 and E2 > 0 for ∆a > 0 ity photon number will later play an important role in for g0/2π = 0 and g0/2π = 50GHz. Refer text for the the simulation of dynamical gauge fields for a single atom parameters used. moving in the presence of two counter-propagating running wave modes inside the ring cavity. When the atom- field coupling, g(y), goes to zero, then the dressed states come back to |g, n1 +1, n2i, |e, n1, n2i, and |g, n1, n2 +1i. We plot the dressed state energies (E1, E2 and E3) as a function of atom-pump detuning ∆a, for different values of the Rabi-frequency, g0, in Fig.(2). E2 is independent of the Rabi-frequency, g , so we get, E ( g0 = 0) = E ( g0 =  0  0 2 2π 2 2π q 50Hz). For g0 = 0, we get a linear plot shown by the ~∆a −e−2ikx n2+1  E2 = ; |D2i =  n1+n2+2  (10b) solid lines. As soon as we include g0 6= 0, the linear 2  q  n1+1 plot becomes non-linear due to the presence of the atom- n1+n2+2 field coupling (shown by dotted lines). The energy gap   0 between the energy levels, E1 and E3, increases as we −2ikx = −e cosθn1,n2  increase the value of the Rabi-frequency.

sinθn1,n2 In the next section, we will solve the time-independent Schrödingerequation to find out the probability amplitude for finding the particle in the lowest energy dressed state, |D1i. The adiabatic evolution of the system in the lowest energy subspace will result in the appearance of a geometric vector potential and a scalar potential which will now depend on the number of photons in the two cavity modes and therefore, the resulting magnetic field will be dynamical.

III. EQUATION OF MOTION FOR THE  e−ikx  ATOMIC WAVE-FUNCTION (G − ∆a) B1 √ −2ikx ~ 2g(y) n1+1e E3 = G; |D3i =   (10c) 2  B√1  Using the dressed state basis (|Dji basis) for the in- 2g(y) n2+1 ternal Hilbert space of the atom at any point ~r, we can B1  −ikx  write the full state vector of the atom as - e cosφn1,n2 −2ikx X = e sinφn1,n2 sinθn1,n2  |Ψ(~r, t)i = ψj(~r, t)|Dji (11)

sinφn1,n2 cosθn1,n2 j=1,2,3 5

Suppose at an initial time, t, the atom is prepared in the Since these dressed states are position-dependent, lowest energy dressed state, |D1i. We deﬁne the recoil therefore, Pˆ will act on both parts of the wave-function, 2k2 diﬀerentiating them with respect to ~r. Thus the action energy ER = ~ , as the kinetic energy of an atom 2ma of the momentum operator on the total wave-function initially at rest when it absorbs or emits a single pho- |Ψ(~r, t)i (see AppendixD)- ton. The frequency scale corresponding to ER is ωR and the frequency scale corresponding to the dressed state   energies (E ,E ,E ) is ω . 1 2 3 D ˆ ~ X Typically the frequency scale of the atomic motion P |Ψ(~r, t)i = −i~∇  ψj(~r, t)|Dji ωR ∼ 4kHz and the frequency scale of the dressed state j=1,2,3 energies ω ∼ 300GHz. Thus the velocity of the atom X h i D = ~pδ − A~ ψ |D i (12) can be considered so small, that it will continue to re- l,j l,j j l j,l main proportional to |D1i at all times. This allows us to do the adiabatic approximation [9, 61–70]. Here, the ~ ~ ~ position of the atom, ~r, corresponds to the slow variables where Al,j = i~hDl|∇|Dji and ~p = −i~∇ = does not of the system, while, the dressed states |D1i, |D2i and act on the spinorial part. The kinetic energy has the |D3i correspond to the fast variables. following form (see AppendixD for details)-

  ~ 2 P 1 ˆ X h ~ i |Ψ(~r, t)i = P  ~pδl,j − Al,j ψj|Dli 2ma 2ma j,l=1,2,3 1 X n h i o = ~pδl,j − A~ l,j ~pδm,l − A~ m,l ψj |Dmi 2ma j,l,m=1,2,3

The Schrödinger equation for the total wave-function in that will finally lead to the equation for the probabil- the dressed state basis ity amplitude, ψ1, to find the atom in the lowest energy dressed state, with energy E (See AppendixE for de- ∂ 1 i |Ψ(~r, t)i = Hˆ |Ψ(~r, t)i (13) tails), ~∂t RF Using the adiabatic approximation, we assume that ψ2, ψ3 = 0 and project Eq.(13) to the eigen space of the lowest energy dressed state, |D1i, to obtain - ∂ hD |i |Ψ(~r, t)i = hD |Hˆ |Ψ(~r, t)i (14) 1 ~∂t 1 RF

2 ∂ h 1 n ~ ~ 2 ~ 2o ~Gi ⇒ i~ ψ1(~r, t) = HSψ1(~r, t) = ~p − A1,1 + |A2,1| + |A3,1| − ψ1(~r, t). (15) ∂t 2ma 2

Here, A~ 1,1 appears with the momentum and so, it acts as governed by the vector potential, A~ 1,1. The scalar poten- a synthetic vector potential while A~ 2,1 and A~ 3,1 will con- tial, the vector potential and the corresponding magnetic tribute to the scalar potential term. The synthetic scalar ﬁeld obtained here depends on the photon number in the 1 2 2 cavity which is a dynamic quantity. potential is given by W = |A~ 2,1| + |A~ 3,1| , which 2ma arises due to the adiabatic following of the dressed state, |D1i. In addition, the last term of Eq.(15), −~G/2, acts as a deep trapping potential for the atomic centre-of- mass motion. The important dynamics of the system are The full expresssion for the vector potential in Eq. 6

k −1 (15) is given as - where B0 = ~ and has the dimension of [MT ]. It w0 defines the unit and the dimension of the synthetic mag- ~ ~ A1,1 = i~hD1|∇|D1i netic field and the synthetic magnetic length is given by q 2 ~ G + ∆a 8g (y)(n1 + 1) lB = B . The spatially varying magnetic field in the ⇒ Ax(y)ˆx = ~k + xˆ(16) 0 2G 2G(G + ∆a) unit of B0 is plotted in Fig.(4b). The magnetic field flips direction as y changes sign. The strength of the mag- The vector potential is similar like one in Landau gauge netic field depends on the difference between the number having only one component, but symmetric in y and of photons in the two cavity modes, n1 and n2. As the hence gives rise to an anti-symmetric magnetic field that difference between n1 and n2 changes sign, the result- is shown in Fig.(4a). It may be pointed out that such ing magnetic field also flips the direction from −zˆ to +ˆz. inhomogeneous synthetic gauge field can be created us- The dependence of the magnetic field on the atom-field ing different methods [71]. However in the presence of coupling, g(y), gives it a Gaussian variation and it at- a ring cavity the strength of such gauge field depends tains its peak value B ∼ ±0.03 B0 at y ∼ ∓20 lB for on the photon numbers, n and n and the atom-photon 1 2 g0 = 2π × 120 GHz. coupling through G and thus can be made more tunable. It is calculated in units of B0lB which has dimensions of recoil momentum, ~k. The magnetic field corresponding to this vector po- The accompanying synthetic scalar potential, W, tential is directed along the z-axis and has the form - which appears in the equation for the probability am- ~ ~ plitude, ψ1, has a dependence on y. It is measured in ∇ × A1,1 = B(y)ˆz 2 ~ units of energy, E0 = 2 and is shown in Fig.(4c). 2 2 2mal ∆ag0 y 2y B with, B(y) = 4(n1 − n2)B0 3 exp(− 2 )(17). The profile is given by G w0 w0

2 2 2 2 2 ! ~ k g (y) 2 ∆a 2y 2 8G(n1 + 1)(n2 + 1) W (y) = 2 (n1 + n2 + 2) 2 + (n2 − n1) + (18) 2ma G (n1 + n2 + 2) k Gw0 (G + ∆a)

In the subsequent sections, we shall discuss the dynam- where the effective potential Vˆ (~r) is given as ics of a single atom in the presence of magnetic fields ˆ X ˆ arising from such vector potentials first using a semi- V (~r) = Ej(~r)Qj(~r) (20) classical method followed by a full quantum mechanical j=1,2,3 treatment. with Qˆj(~r) = |Dj(~r)ihDj(~r)| is the projector onto the jth-eigenstate of the atom-field coupling. We can define an ~r dependent force operator in this Hilbert space as Fˆ(~r) = −∇~ Vˆ (~r) IV. TRAJECTORIES OF A SINGLE ATOM X h~ ˆ ~ ˆ i MOVING IN PRESENCE OF A NON-UNIFORM = − ∇Ej(~r) Qj(~r) + Ej(~r) ∇Qj(~r) MAGNETIC FIELD j (21) In this section, we treat the equations of motion for a single atom moving in the presence of the non-uniform The expectation value hFî is then given by hFî = ˆ P magnetic field, Bz(y)ˆz, which arises due the vector poten- hΨ|F (~r)|Ψi where |Ψ(~r, t)i = j=1,2,3 ψj(t)|Dj(~r(t))i tial in Eq.(15) using semiclassical method. This enables comes out to be [72]- us to find the trajectories of the atom moving in such a ˆ X 2 ~ X ∗ ~ magnetic field and to identify them as snake orbits. In hF (~r, t)i = − |ψj| ∇Ej+ ψj ψk(Ek−Ej)hDk|∇Dji the next section we provide a full quantum mechanical j j,k treatment of the same problem. The Hamiltonian for the The semi-classical equation of motion for the atom in system can be written as - the lowest energy dressed state can now be given as (see AppendixF and [72] for details)- ˆ2 ˆ P ˆ d~v H = + V (~r) (19) ma = −∇~ E1 − ∇~ W (~r) + ~v × B~ (~r) (22) 2ma dt 7

Quantity For electrons[27, 32] For atoms −23 −1 Magnetic ﬁeld B0 = 2 T B0 ∼ 3.4 × 10 kgs ; Bpeak ∼ ±0.03 B0 Dimensions of magnetic ﬁeld [MA−1T −2] [MT −1] q q q w0λ Magnetic length lB = ~ ∼ 18 nm lB = ~ = ∼ 2 µm eB0 B0 2π eB0 B0 Bpeak Cyclotron frequency ωc = ∼ 352 GHz ωc0 = ∼ 232 Hz; ωcp = ∼ 7 Hz me ma ma ωc ~ωc0 ~ωcp Temperature T0 = ~ ∼ 3 K T0 = ∼ 2 nK; Tp = ∼ 0.05 nK kB kB kB Radius of the particle trajectory re ∼ 0.5 nm ra ∼ 30 µm −11 Reference energy E0 ∼ 0.12 meV E0 ∼ 7.67 × 10 meV

Energy gap between Landau levels δe ∼ 0.13 E0 δa ∼ 0.03 E0

TABLE I: Comparison of the numerical values of quantities measured in typical condensed matter experiments and the numerical values obtained in our system.

−25 10 kg, λp = 780.5nm, w0 = 25.3µm, lB ∼ 2µm and −23 −1 B0 ∼ 3.4×10 kgs . Here, we have plotted the figures for minimum photon number difference, n1 − n2 = 1. For these parameters, the first two terms provide a deep trapping potential for the confinement of the atom in lowest energy dressed state, |D1i. This potential can therefore be approximated by a square well potential with a constant depth, V0, which in turn sets a box potential condition along the y− direction. Under such conditions, the first two terms of Eq.(23b) are (∇~ V0) inside this box and we get -

d2x dy m = B(y) (24a) a dt2 dt d2y dx m = −B(y) (24b) a dt2 dt We solve the Eqs.(24a-24b) to obtain the values of x(t) FIG. 3: Color online: Snake states for a single atom in and y(t) and plot the trajectory of the atom in the x − y the presence of a non-uniform synthetic magnetic field, plane as shown in Fig.(3), for 2 different values of the in the vicinity ofy ¯ = 0. The magnetic field B(¯y) changes atom-photon coupling, g0. At y = 0, B(y) = 0 and the direction from −zˆ toz ˆ asy ¯ changes sign. Blue (solid atom experiences magnetic field with alternating direc- line) and red ( dashed lines) trajectories correspond to tions on both sides of this line and consequently it moves two magnetic field strengths which depend on the atom- in a snake trajectory which drifts along the −xˆ direction, photon coupling strengths, g0/2π. Refer text for the pa- i.e., in a direction perpendicular to the synthetic field rameters used in the figures. gradient [27, 74]. The radius of curvature for the particle trajectory is inversely proportional to the synthetic 1 magnetic field (r ∝ B(y) ). Therefore, for large(small) where W is the scalar potential that appears in the Eq. values of the magnetic field, the particle will trace a tra- (15) . In component form, the equations in the synthetic jectory with a small(large) radius of curvature and this electric and magnetic field [73] for such atom will look results in the particle scanning a finite region in the x−y like plane. The appearance of snake-states gives rise to a non-

2 vanishing flow of atoms along the −xˆ direction. d x dy The snake trajectories of electrons had been widely ma = B(y) (23a) dt2 dt studied in various condensed matter systems consisting d2y dG dW of two-dimensional electron gas [27, 30, 32, 37], graphene m = ~ − a dt2 2 dy dy p − n junctions [25, 26], etc. It may be noted that dx the dimensions of the magnetic field for an electron, − B(y) (23b) −1 −2 dt [MA T ], includes the dimensions of charge, ([q] = [AT ]). Since an atom is charge neutral, the dimension The parameters considered in the plots for the magnetic of synthetic magnetic field for an atom, [MT −1]. For field and the effective potential in the later sections are the typical system parameters it will be therefore use- ∆a = −2π × 128GHz, g0 = 2π × 120GHz, ma = 1.4 × ful to compare the values of the various relevant quan- 8 tities in these two completely different system. As we GHz, n1 − n2 = 1, λp = 780.5 nm, w0 = 25.3 µm and −23 −1 demonstrate in the TableI in the relevant experimental B0 ∼ 3.4×10 kgs . The magnetic length lB ∼ 2µm, range the synthetic magnetic field on the atom is rel- gives the spatial extent of the atomic wave-function in atively weaker, making the size of the trajectory much the considered parameter regime. Length and momen- larger than the electron trajectory in a condensed matter tum values are, therefore, expressed in terms of lB and −1 system. In the next section, we shall do a full quantum lB , respectively, and the energy values in the unit of mechanical treatment to understand the single particle ~ E0/~ = 2 ∼ 116 Hz. The total trap potential which 2malB energy spectrum for such snake orbits. is responsible for trapping the atom in the lowest energy dressed state can be approximated by a box potential with a constant depth, |V0| ∼ 4 GHz, and a strong con- V. ENERGY SPECTRUM FOR AN ATOM IN A finement along the y-direction at y = ±35 l and can be NON-UNIFORM MAGNETIC FIELD B eliminated for the purpose of studying the dynamics in the synthetic magnetic field (∼ 100Hz) represented by We will now analyse the behaviour of an atom mov- V¯ (¯y). ing in the presence of a synthetic non-uniform mag- kx Therefore, the dynamics is almost entirely governed netic field given by Eq.(17). The related energy spec- by V¯ (¯y) as shown in Fig.5 where we plot the energy trum of an electron in the presence of a uniform mag- kx ~ eigenvalues for an atom and relate them to the shape netic field (B = B0zˆ), was studied by Landau almost ¯ change of such effective kx -dependent potential [27]. For a century back, using a particular choice of vector po- ¯ a given kx the potential is symmetric functiony ¯. But the tential called Landau gauge. The eigenfunctions of the potential profile changes in a nontrivial and asymmetric resulting Schrödingerequation are labelled by magnetic ¯ way as one changes kx. This may be contrasted with the quantum number, n, and one of the momentum com- behavior of the corresponding potential for the Landau ponents, say px = kx. The magnetic length, lB = ~ , ~ eB0 problem in the presence of a uniform magnetic field (in gives the spatial variation of the electronic wave-function. Landau gauge) which is a symmetric function of both k¯ The energy levels known as Landau levels have energy x 1 andy ¯. En = n + 2 ~ωc which are independent of kx and hence (c) 1 ¯ ¯ At a given value kx = kx , Vk¯x (¯y) changes its shape. degenerate with the degree of degeneracy given by 2πl2 . B ¯(c) The presence of a non-uniform magnetic field lifts the For the values used in our representative plot kx = ¯ degeneracy of the Landau levels and results in a finite 14.76. At further higher value of kx the double well potential structure gets converted in a barrier potential. dispersion, En(kx) and this dispersion is plotted in Fig. We plot the potentials V¯ (¯y) as a function ofy ¯ for sev- 5. Here we provide an analysis of this dispersion. kx ~ ¯(1) ¯(c) In Eq.(16) Ax(y) is independent of x, ⇒ [~px,HS] = eral representative values ((kx (14.7) < kx (14.76) < (2) (3) (4) 0. Hence the wave-function for the atom in the lowest k¯x (14.9) < k¯x (15.0) < k¯x (15.1)) to demonstrate this ¯ energy (|D1i) subspace can be written as - change in the profile of the potential with changing kx. This potential is symmetrical abouty ¯ = 0. For k¯ < k¯(c), ψ (x, y) = eikxxφ(y) (25) x x 1 it forms a single well with the minima of the potential ¯ ¯(c) We substitute the expression for ψ1(x, y) in Eq.(15), to lying aty ¯ = 0. For kx ≥ kx , there are two symmet- obtain (AppendixG for details)- rically located minima aty ¯ = ±y¯1 around a maxima at ¯ y¯ = 0. At the two minima, V¯ (kx, y¯1) = 0. And as 2 kx ∂ φ(¯y) E ¯ mentioned for higher values of kx, the central maxima 2 + − Veff (¯y) φ(¯y) = 0 (26) ∂y¯ E0 aty ¯ = 0 grows and effectively turning the potential in a potential barrier. The locations of the double minima where ¯ pointsy ¯1 as a function of kx are plotted in Fig.6.

Veff (¯y) = Vk¯x (¯y) + Vtr(¯y) (27) The energy spectrum for the atom is shown in Fig.5. We only plot few lowest energy levels that brings forth the with quantum mechanical behaviour of a neutral atom in such synthetic non-uniform magnetic field. Energy bands of ¯ Ax(¯y) 2 V¯ (¯y) = (kx − ) (28) kx B l the magnetic problem in Eq.26 are labelled by the band 0 B index ”n”. We continue to call each such bands as Lan- W − ~G 2 dau levels even though they have now free particle like Vtr(¯y) = (29) E0 dispersion. The minima of the lowest energy band lies at k¯ = 14.76 and as the band index increases for higher q q 2 x ~ w0λp ~ ¯ with lB = = ; E0 = 2 ; kx = excited states, the minima of the energy band shifts to a B0 2π 2malB y ¯ kxlB;y ¯ = . This is the Schrödingerequation higher value of kx. lB for a single atom moving in the presence of an effec- To understand these energy bands better, we note that ¯ tive potential, Veff (¯y). Numerical values of the param- these energy bands show a strong asymmetry in kx which eters used are ∆a = −2π × 128, GHz, g0 = 2π × 120 corresponds to the asymmetry in Vk¯x (¯y) as a function of 9

a b c

FIG. 4: Color online: Geometric potentials and the resulting magnetic ﬁeld, for two values of the atom-photon coupling strengths, arising due to the adiabatic following of the lowest energy dressed state, |D1i. Fig.(4a) Vector k potential Ax(¯y) (in units of B0lB) as a function ofy ¯. Fig.(4b) Magnetic ﬁeld Bz(¯y) (in units of B0 = ~ ) as a function w0 ofy ¯. Fig.(4c) Scalar potential W (¯y) (in units of E0) as a function ofy ¯. Refer text for the parameters used.

¯ FIG. 6: Color online:y ¯1 as a function of kx which shows ¯(2) ¯(3) ¯(4) the double well minima at ±y¯1 for kx and kx . At kx , FIG. 5: Color online: Energy, E (in units of E0 = a barrier is formed which separates minima points. Since 2 ~ ¯ ¯ 2 ), as a function of kx. The eﬀective potential, the barrier width does not increase with increase in kx, 2malB ¯ therefore, the minima points remain at a constant sepa- V¯ (¯y), as a function ofy ¯ is shown for 4 values of kx kx ration with increasing k¯ . Refer text for the parameters referred in text and shows the transition of the eﬀective x used. potential from a single well to a double-well to a barrier. Refer text for the parameters used.

at the minima of each well form their own Landau level ¯ En(kx) with parabolic dispersion like a free particle and ¯ ¯ ¯(c) ¯ kx. For kx > kx , the eﬀective potential has a dou- are degenerate. With decreasing kx, the central barrier ble well structure that is symmetrically located about a is getting lowered and there will be tunnelling across this central potential barrier aty ¯ = 0. The states localised barrier. It is because of this tunnelling the degeneracy 10

¯ of the two branches of En(kx) will be lifted splitting it a suitably loaded ultra cold condensate leads to the for- into symmetric and antisymmetric states separated by a mation of a supersolid as has been confirmed by exper- gap. This explains the splitting of the energy levels for imental and theoretical studies. Thus it will be what if ¯ ¯(c) kx < kx . there is any connection between the exploration of the ¯ As we go to higher values of kx, the potential forms a inhomogenous gauge field formation at the single atom strong barrier and this is visible in the energy dispersion level in the current work and supersolidity in a crossed where the energy bands become degenerate in pairs. An or ring cavity system for an ultra cold condensate. All approximate expression for the energy can be given as these promises a rich future for such atom-photon system and the resulting inhomogenous synthetic gauge field. ¯ 1 ~ We thank P. Mondal, N. Dogra, K. Khare, M. Bhat- En(kx) ' n + B(¯y1) 2 ma tacharya, for a number of helpful discussion at various stages of this work. SG thanks the ETH group of T. ¯ Ax(¯y) Esslinger and P. Ohberg at Herriot-Watt University for wherey ¯1 is the solution of kx − = 0. In the B0lB helpful discussions during his visits there.This work is ¯ expression for En(kx), we have replaced the magnetic supported by a BRNS ( DAE, Govt. of India) Grant Bz (¯y1) field in the cyclotron frequency, ωcp = , by the No. 21/07/2015-BRNS/35041 ( DAE-SRC Outstanding ma expression obtained in Eq.(17). The spacing between the Investigator Scheme). PS is supported by a UGC, Govt. energy levels in this case is δa and is of the order of of India fellowship. 0.03E0 (corresponds to a temperature of 0.03 nK) which is much smaller than the corresponding energy difference Appendix A: Atom-pump field interaction in an electron system, denoted by δe and has a value of 0.13E0 (corresponds to a temperature of 0.2 K). In this section, we derive the expression for the atom-pump field interaction Hamiltonian, HÂ−P in the VI. CONCLUSIONS rotating-wave approximation as given in Eq.(5). The pump field is polarized along the x-direction and the ex- In this work, we have used atom-photon coupling inside pression for the electric field is given by - a ring-cavity and adiabatic approximation to eliminate ~ −iωpt iωpt higher energy dressed states to propose a non-uniform EP =e ˆxE0cos(ky + φ)(e + e ) (A1) geometric gauge field that supports the atomic analogue Then, Hˆ takes the following form - of electronic snake states. The current study is done in a A−P non-interacting picture and we primarily investigate the Hˆ = −d~ · E~ single atom-dynamics for such atomic snake states and A−P P ~ − + −iωpt iωpt compare with the typical value for the electronic snake = −d(ˆσ +σ ˆ ) · eˆxE0cos(ky + φ)(e + e ) states through TableI. In future, this work can be ex- h + −iω t + iω t = Ω(y) σˆ e p +σ ˆ e p tended to Bose-Einstein condensates trapped in one arm ~ i of the ring cavity and study their behavior in presence +σ ˆ−e−iωpt +σ ˆ−eiωpt (A2) of such inhomogenous dynamical gauge fields. It will be particularly interesting to see if we can get similar prop- −d~·~ex where Ω(y) = Ω0cos(ky + φ), Ω0 = is the Rabi agating states for an ultra cold condensate that can be ~ used to study the flow of an atomic Bose-Einstein con- frequency and φ is the pump phase. Since the atom- densate along an atomic waveguide [75], which can have pump coupling strength Ω0 is of the order of ∼ GHz, applications in the designing of devices for guided mat- ωa, ωp ∼ T Hz [46, 59], and |ωa − ωp| << |ωa + ωp|, ter wave optics. Moreover, changing the direction of the we can use the rotating wave approximation. So we magnetic field is equivalent to flipping the charge degrees move to the interaction picture and the time evolution of of freedom in an uni-directional magnetic field, as has the atomic field operators is found by solving the Heisen- ± 2 dσˆ ± ± P~ ωa berg equation i = [ˆσ , HÂ] = [ˆσ , + ~ σˆz] been demonstrated recently in the electronic snake states ~ dt 2ma 2 ± ± ±iωat in Graphene [25, 26]. Creation of atomic analogue of such and we obtain -σ ˆ (t) =σ ˆ (0)e - where ωa is the snake states in a neutral atomic system can therefore be atomic resonance frequency. So, the atom-pump field also viewed as introducing effective charge degrees of free- Hamiltonian in the interaction picture is - dom in an atomic trajectory in a restricted sense. Such h quantum simulation in atomic system is therefore potent ˆ I + i(ωa−ωp)t + i(ωa+ωp)t HA−P = ~Ω(y) σˆ (0)e +σ ˆ (0)e with the possibilities of new device application for fields i like atom-tronics [76, 77]. It is also worth noting that the +σ ˆ−(0)e−i(ωa+ωp)t +σ ˆ−(0)e−i(ωa−ωp)t (A3) atom-photon hamiltonian for our system, HˆRF of Eq.(7) is similar to the hamiltonian for an atom trapped at the In the interaction picture, we can see that there are two intersection of a crossed-cavity system, used in [42] or a types of terms with frequencies ωa + ωp and ωa − ωp. For similar ring cavity system [40]. In either of these cases, ωa ∼ ωp, the fast oscillating terms with frequency ωa +ωp 11 can be neglected as compared to the slow oscillating tion [78–80]- terms with frequency ωa − ωp. This is called the rotat- ˆ + ikx + −ikx ing wave approximation (RWA). We transform back HA−C = ~g(y) σˆ aˆ1e +σ ˆ aˆ2e from the interaction picture to the Schrödingerpicture − † −ikx − † ikxi to get back the time-independent atomic field operators, +ˆσ aˆ1e +σ ˆ aˆ2e ± σˆ , which appear in HÂ−P . Therefore, the final form of the atom-pump field interaction in the Schrödingerpic- ture is - Appendix C: Going to the rotating frame of the pump field ˆ + −iωpt − iωpt HA−P = ~Ω(y) σˆ e +σ ˆ e (A4) In this section, we provide the detailed steps to derive Eq.(7). We show how the observables, the states and the Schrödingerequation transform under the unitary trans- Appendix B: Atom-cavity field interaction formation given by Eq.(6) of the main text and we also find out the expression for the single particle Hamilto- In this section, we find out the form of the atom-cavity nian, Hˆ , in the rotating frame of the pump field. The ˆ SP field interaction, HA−C in the rotating wave approxi- unitary operator mentioned in Eq.(6) is - mation as geiven in Eq.(5). HÂ−C describes the interaction between the atoms and the cavity field. The cavity −iω t( σˆz +â†aˆ +â†aˆ ) Uˆ(t) = e p 2 1 1 2 2 field is polarized along the y-direction and the expression for the electric field in one arm of the ring cavity is given Under this unitary transformation, the observables trans- by - form as - r ˆ ˆ † ˆ ˆ ~ωc −y2/w2 ikx −ikx ORF = U OU (C1) E~C (~r, t) =e ˆy e 0 aˆ1e +a ˆ2e 20V r The states transform as - ωc 2 2 ~ −y /w0 † −ikx † ikx +e ˆy e aˆ1e +a ˆ2e |Ψ i = Uˆ †|Ψi (C2) 20V RF (B1) In the rotating frame, the Schrödingerequation trans- forms as - So again to get a clearer picture, we move to the interaction picture. The atomic field operators are given as ∂ ∂ ˆ † ˆ i~ |ΨRF i = i~ U |Ψi = HRF |ΨRF i σˆ±(t) =σ ˆ±(0)e±iωat. The time evolution of the cavity ∂t ∂t field operators is found by solving the Heisenberg equa- daˆ1,2 ˆ † † where tion i~ dt = [â1,2, HC ] = [â1,2, ~ωcaˆ1aˆ1 + ~ωcaˆ2aˆ2] and † −iωct − ωpσˆz † † we obtain -a ˆ1,2(t) =a ˆ1,2(0)e . Similarly, fora ˆ1,2, ˆ ~ ˆ † ˆ ˆ HRF = − ~ωpaˆ1aˆ1 − ~ωpaˆ2aˆ2 + U HSP U (C3) † † iωct 2 we get -a ˆ1,2(t) =a ˆ1,2(0)e , where ωc is the cavity resonance frequency. Therefore, the atom-cavity field in- is the single-particle Hamiltonian in the rotating frame † teraction in the interaction picture is - of the pump field. We now find Uˆ HˆSP Uˆ. We use the Baker-Hausdorff formula to find out the transformation ˆ I ~ ~ ˆ HA−C = −d · EC of the operators appearing in HSP . Baker-Hausdorff formula h + ikx + −ikx = ~g(y) σˆ (t)â1(t)e +σ ˆ (t)â2(t)e +σ ˆ+(t)â†(t)e−ikx +σ ˆ+(t)â†(t)eikx h i i2λ2 h h ii 1 2 eiGλˆ Aeˆ −iGλˆ = Aˆ + iλ G,ˆ Aˆ + G,ˆ G,ˆ Aˆ − ikx − −ikx +σ ˆ (t)â1(t)e +σ ˆ (t)â2(t)e 2! i inλn h h h h iii i +σ ˆ−(t)â†(t)e−ikx +σ ˆ−(t)â†(t)eikx (B2) + ... + G,ˆ G,ˆ G,ˆ ... G,ˆ Aˆ ... + ... 1 2 n! ˆ ~ q 2 2 2 2 where G is Hermitian and λ is a real parameter. The −d·eˆy ωc −y /w −y /w where g(y) = ~ e 0 = g0e 0 . If ωa ∼ ˆ ~ 20V operators in HSP transform as - ±i(ωa−ωc)t ωc, then the terms with e will have small transi- † −iωpt 1 Uˆ aˆ1,2Uˆ =a ˆ1,2e tion amplitudes that are proportional to 2 . There- (ωa+ωc) ˆ † † ˆ † iωpt fore, the fast oscillating terms with frequency ωa +ωc can U aˆ1,2U =a ˆ1,2e be neglected as compared to the slow oscillating terms Uˆ †σˆ+Uˆ =σ ˆ+eiωpt with frequency ωa − ωc. This is called the rotating wave ˆ † − ˆ − −iωpt approximation. We transform back to the Schrödinger U σˆ U =σ ˆ e † picture to obtain the final form of the atom-field interac- Uˆ σˆzUˆ =σ ˆz 12

Therefore, we get - momentum operator, Pˆ, on the atomic wave-function, |Ψ(~r, t)i. † Uˆ HˆSP Uˆ = HˆSP   Therefore, the single particle Hamiltonian, HˆRF , in the ˆ ~ X rotating frame of the pump field is - P |Ψ(~r, t)i = −i~∇  ψj(~r, t)|Dji j=1,2,3 Pˆ2 ∆ σˆ Hˆ = Î − ~ a z − ∆ aˆ†aˆ +a ˆ†aˆ X h i RF ~ c 1 1 2 2 = −i (∇~ ψj(~r, t))|Dji + ψj(~r, t)(∇|~ Dji) 2ma 2 ~ + − j + ~Ω(y) σˆ +σ ˆ " + ikx + −ikx X ~ X + ~g(y) σˆ aˆ1e +σ ˆ aˆ2e = −i~∇ψj(~r, t) |DlihDl|Dji +σ ˆ−aˆ†e−ikx +σ ˆ−aˆ†eikx (C4) j l 1 2 # X ~ − i~ ψj(~r, t) |DlihDl|∇|Dji where ∆a = ωp − ωa is the atom-pump detuning and ∆ = ω − ω is the cavity-pump detuning. l c p c X h i = ~pδl,j − A~ l,j ψj|Dli (D1) j,l=1,2,3 Appendix D: Action of momentum operator on the atomic wave-function ~ ~ where Al,j = i~hDl|∇|Dji is the vector potential and In this section, we provide the detailed derivation of ~p = −i~∇~ = does not act on the spinorial part. The Eq.(12) of the main text to find out the action of the kinetic energy term can be written as -

We can write down a matrix A~ whose components are given as -

~ ~ Al,j = i~hDl|∇|Dji

  hD1|∇|~ D1i hD1|∇|~ D2i hD1|∇|~ D3i ~  ~ ~ ~  A = i~ hD2|∇|D1i hD2|∇|D2i hD2|∇|D3i hD3|∇|~ D1i hD3|∇|~ D2i hD3|∇|~ D3i √  2 2 2 2  G+∆a 8g (y)(n1+1) 4~kg(y) (n1+1)(n2+1) G −∆a ∆a −2y 4g (y)(n2−n1) k + xˆ − √ xˆ −i 2 yˆ − ik xˆ ~ 2G 2G(G+∆a) A n +n +2 ~ A1B1 G w A1B1  √ 1 1 2 √0  4 kg(y) (n1+1)(n2+1) 2 k(n +1) 4 kg(y) (n1+1)(n2+1) =  − ~ √ xˆ ~ 2 xˆ − ~ √ xˆ   A1 n1+n2+2 n1√+n2+2 B1 n1+n2+2   2 2 2 2  G −∆ 4g (y)(n −n ) 4 kg(y) (n1+1)(n2+1) 8g (y)(n +1) a ∆a −2y 2 1 ~ √ G−∆a 1 i~ 2 yˆ + ik xˆ − xˆ ~k + xˆ A1B1 G w0 A1B1 B1 n1+n2+2 2G 2G(G−∆a)

RHS of SchrödingerEq.(E1) is - The first term of Eq.(E3) gives - " ! # Pˆ2 hD1|HˆRF |Ψ(~r, t)i = hD1| Î + HÎ |Ψ(~r, t)i 2ma (E3)

P~ 2 1 X h i hD1| |Ψ(~r, t)i = { ~pδl,j − A~ l,j ~pδm,l − A~ m,l ψj }hD1|Dmi (E4) 2ma 2ma j=1;l,m=1,2,3 2 1 n 2 2o = ~p − A~ 1,1 + |A~ 2,1| + |A~ 3,1| ψ1(~r, t) (E5) 2ma

The second term of Eq.(E3), gives - From Eq.(E2) and Eq.(E3), we obtain the equation for the probability amplitude, ψ1, to ﬁnd the atom in the G hD |Hˆ |Ψ(~r, t)i = −~ ψ (~r, t) lowest energy dressed state, E1 - 1 I 2 1

2 ∂ h 1 n ~ ~ 2 ~ 2o ~Gi i~ ψ1(~r, t) = HSψ1(~r, t) = ~p − A1,1 + |A2,1| + |A3,1| − ψ1(~r, t) (E6) ∂t 2ma 2 (E7)

−iE1t/ Appendix F: Equations of motion for the snake At order zero, we get ψ1 = e ~ and ψ2, ψ3 = 0. trajectories At ﬁrst order in ~v, the equation of motion for ψ1 is -

In this section, we provide a detailed derivation of ˙ ~ Eq.(22). The expectation value of the force operator is - ψ1 = −i E1 − ~v · A ψ1/~ (F5) ˆ X 2 ~ X ∗ ~ hF (~r, t)i = − |ψj| ∇Ej+ ψj ψk(Ek−Ej)hDk|∇Dji j j,k whose solution is a number of modulus 1. For (F1) j 6= 1 (assuming adiabatic motion of the atom, i.e., To obtain the force at first order in ~v, we have to find T (Ej − E1) /~ >> 1), we get - out all the coefficients, ψj at first order. The Schrödinger equation for |Ψ(~r, t)i is [72]- ~v · hD |∇~ D i ˙ ˆ j 1 −iE1T/ i~|Ψ(~r, t)i = V (~r, t)|Ψ(~r, t)i (F2) ψj(T ) ' i~ e ~ (F6) Ej − E1 and   ˙ d X ˆ |Ψ(~r, t)i =  ψj(t)|Dj(~r(t))i We see that first term of hF i in Eq.(F1), has no first- dt 0 j order component in ~v, since the contributions of ψjs for j 6= 1 are at least of order 2 and the contribution of ψ X ˙ X ~ 1 = ψj|Dji + ψj~v · |∇Dji (F3) is independent of ~v. In the second term of Eq.(F1), the j j relevant terms are the ones where one of two indices j or k is 1. Applying the closure property, and keeping terms We get the corresponding equation of motion for ψ as - j ˆ ~ ~ upto first order in ~v, we get hF i = i~h∇D1| ~v · |∇D1i + ˙ ψj X ~ ψj = −iEj + ψj~v · hDj|∇~ Dki (F4) c.c.. This expression gives the Lorentz force ~v × B in the ~ k equation of motion. 14

Appendix G: Schrödingerequation for a particle in tion for an atom moving in the presence of a non-uniform ikxx a non-uniform magnetic field magnetic field. We substitute ψ1(x, y) = e φ(y), in Eq.(15), to obtain - In this section, we provide a detailed derivation of Eq.(26) of the main text, which is the Schrödingerequa-

" 2 # 1 ∂ 1 2 ~G −i~ + (~kx − Ax(y)) + W − φ(y) = Eφ(y) 2ma ∂y 2ma 2 2 2 ∂ φ(y) Ax(y) 2ma ~G ⇒ − kx − φ(y) + E − W + φ(y) = 0 ∂y2 ~ ~2 2 2 2 2 2 ∂ φ(y) Ax(y)lB 2malB ~G ⇒ lB − lBkx − φ(y) + E − W + φ(y) = 0 ∂y2 ~ ~2 2 ∂2φ(¯y) E ⇒ 2 + − Veff (¯y) φ(¯y) = 0 (G1) ∂y¯ E0

This is the Schr¨odingerequation of a single atom moving in the presence of a non-uniform magnetic ﬁeld.

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