Rashba realization: Raman with RF

We theoretically explore a Rashba spin-orbit coupling scheme which operates entirely in the absolute ground state manifold of an alkali atom, thereby minimizing all inelastic processes. An energy gap between ground eigenstates of the proposed coupling can be continuously opened or closed by modifying laser polarizations. Our technique uses far-detuned"Raman"laser coupling to create the Rashba potential, which has the benefit of low spontaneous emission rates. At these detunings, the Raman matrix elements that link $m_F$ magnetic sublevel quantum numbers separated by two are also suppressed. These matrix elements are necessary to produce the Rashba Hamiltonian within a single total angular momentum $f$ manifold. However, the far-detuned Raman couplings can link the three XYZ states familiar to quantum chemistry, which possess the necessary connectivity to realize the Rashba potential. We show that these XYZ states are essentially the hyperfine spin eigenstates of $^{87}\text{Rb}$ dressed by a strong radio-frequency magnetic field.


Introduction
Geometric gauge potentials are encountered in many areas of physics [1][2][3][4][5][6][7][8][9]. In atomic gases, the geometric vector and scalar potentials were first considered in the late 90s to fully describe atoms 'dressed' by laser beams [10][11][12]. Atoms that move in a spatially varying, internal state dependent optical field experience geometric vector and scalar potentials. Our understanding of these potentials has been refined, and now allow for the engineered addition of spatially homogeneous geometric gauge potentials [13][14][15]. In many cases, the resulting atomic Hamiltonian is equivalent to iconic models of spin-orbit coupling (SOC): Rashba, Dresselhaus and combinations thereof.
Often, systems with SOC will have multiply degenerate single particle eigenstates with topological character: this suggests that strongly correlated phases will exist in the presence of interactions for both bosonic and fermionic systems. Interesting phenomena such as topological insulating states and the spin-Hall effect include SOC as a necessary component [16,17]. Rashba SOC (present for 2D free electrons in the presence of a uniform perpendicular electric field, such as in asymmetric semiconductor heterostructures) [18,19], is an iconic 2D SOC potential and has maximal ground state symmetry. Indeed, interesting many-body phases [20][21][22] predicted for atomic systems with Rashba SOC include unconventional and fragmented Bose-Einstein condensation [23], composite fermion phases of bosons [24] and anisotropic or topological superfluids in fermionic systems [25].
It is in the context of such potentially fragile many-body states that we propose a scheme that is implemented entirely within the ground hyperfine manifold of an alkali with spin greater than or equal to spin-1. Recently, the Rashba potential was realized with 40 K fermions using lasers coupling the f 7 2 = and f 9 2 = manifolds [26]. In alkali bosonic systems with density n the two-body collisional relaxation lifetime from the f 1 + to the f ground state hyperfine manifold is n 10 cm s 14 3 1 >´-- [27]: a timescale that is potentially too small to observe meaningful many-body physics. Such relaxation may be a lesser, but still pertinent, concern in fermionic systems. We propose an alternative coupling scheme implemented entirely within the ground hyperfine manifold of alkali atoms.
, in terms of an effective vector potential e e y x x y  a s s = -(ˆˆ). The Cartesian components of the vector potential manifestly fail to commute: the vector potential is non-abelian.
An atom that adiabatically traverses a loop about the momentum origin shown in figures 1(b) and (c) acquires a Berry's phase of π. Likewise, an interferometer in which one arm orbits the momentum origin would display destructive interference. It is anticipated that the presence of this phase winding in the ground state potential will result in unusual many-body ground states for both fermionic and bosonic systems [23][24][25].

Rashba SOC in cold atoms
One of several methods for producing SOC in ultracold neutral atoms uses lasers to impart a discrete momentum kick whenever they induce a spin flip. This SOC is strictly 1D, e.g. SOC in the analogous electron system would have the form p x x s µ , motivating the addition of a third atomic ground state and two added Raman couplings, each with a distinct orientation of momentum kick, to produce the desired 2D Rashba SOC. The necessary coupling configuration is either a laser scheme that links all three ground states to a common state [28] or, when the excited state(s) are adiabatically eliminated, a closed group of N-states where each constituent state is coupled to exactly two others (this may be visualized as N-states coupled in a loop) [29]. The two configurations can overlap in the case of N=3, which is the configuration adopted in this paper.
The problem encountered with the simplest possible implementation of this scheme is that direct Raman coupling of spins within the ground hyperfine manifold of alkali atoms cannot couple differences in spin greater than 1 unit of angular momentum [30]. Coupling as shown in figure 2(a) is possible, while coupling as shown in figure 2(b) does not link 1 + ñ | and 1 -ñ | . Detuning the lasers near to a transition with the excited electronic hyperfine states as proposed in [31] lifts the angular momentum restriction sufficiently to realize a coupling scheme similar to figure 2(b) but the spontaneous emission rate increases and atomic ensemble lifetimes become much shorter than typical equilibration times. Many theoretical results, and so far the only experimental result, use one or more states from both hyperfine f manifolds to complete the minimum set of three states [26,28]. Although feasible, collisions that change f are expected to lead to atom-loss and heating, potentially de-cohering fragile many-body phases. Our approach, illustrated in figure 3, uses a primary then a secondary coupling. First, a rf coupling is applied, much stronger than the intended Raman coupling, to produce a set of eigenstates. These rf eigenstates are themselves Raman coupled by lasers to produce a set of eigenstates that include the Rashba subspace. We perform a Floquet analysis showing the viability of this approach when a rf coupling strength in excess of 100 kHz is achieved.

Overview
In the following section we build the Rashba potential by Raman coupling arbitrary states. We then find the form of the Raman coupling in the rf eigenbasis. With these building blocks, the eigenenergies may be calculated using Floquet theory. From the Floquet Hamiltonian we pick a set of three states that are resonantly coupled to one another, together with their resonant couplings, and construct an effective 3×3 Hamiltonian. Here, we learn that it is not necessary to phase lock the rf to the Raman couplings while, by contrast, the laser polarizations are constrained to a particular geometry. In the appendix we detail the parameters of an experiment that could produce the Rashba potential using this technique.
. This approach is practical when The 3D laser geometry shown in panel (c) produces the necessary Raman couplings as well as the necessary momentum kicks, panels (d) and (e), to realize the Rashba potential. Each laser is labeled by its electric field amplitude E j | | and wavevector k R . When the coupling strengths are set equal ), the dispersion of the Rashba potential is obtained for a slice taken along the x-axis as shown in panel (f).

Building the Rashba potential by Raman coupling arbitrary states
We consider a subspace of three long-lived states jñ | for j 1, 2, 3 Î { }within a potentially much larger pool of available states. We illuminate these states with three coherent lasers that are indexed by , } . Each of these lasers has distinct wavevector k b , each with magnitude k R , and frequency The possible two-photon Raman frequencies differences are given by , Figure 3(d) illustrates the relationship between laser momentum recoil k  b , with magnitude k R  , and Raman recoil k ,  b b¢ . The Hamiltonian describing Raman coupling in this general form is where E j is the eigenenergy of state jñ | in the absence laser-coupling. We shall make the simplifying assumption that each pair of lasers uniquely Raman couples a pair of states, greatly simplifying the form of the coupling amplitude in equation (3): j j , W ¢ , which is potentially complex. This configuration can be realized by requiring that the lasers resonantly Raman couple pairs of states where we have linked each Raman coupling to a state with this resonance condition (recall that the laser frequencies are given by equation (2). We also apply the rotating wave approximation (RWA) to eliminate terms that are With these constraints on equation (3) it is always possible to apply a unitary transformation that eliminates the complex exponentials from the Hamiltonian and also applies a state-dependent momentum displacement to the momentum operator in equation (3). In the rotating frame, where we have made a simple substitution of variables, k q  , indicating that the momentum term is a quasimomentum.

Rashba subspace
We apply a discrete Fourier transform to equation (7) n j n j This is a useful diagonalization tool when all the off-diagonal matrix elements are nearly equal in amplitude and larger than the energy scale of any of the three two photon recoils k m 2 j j 2 , 2  ¢ . We specify our discussion to equal amplitudes j j , W = W ¢ | |for each matrix element. We also define a phase i ln Applying the discrete Fourier transform in equation (8) to the Hamiltonian in equation (7) we find the diagonal elements in this transformed Hamiltonian (which are nearly the eigenenergies) are E n 2 cos 2 3 , 9 n The phase sum f adds the phase contributions from nearest neighbor matrix elements that sequentially chains all three states together. f is an example of a phase that is not simply the result of our choice of basis: it cannot be eliminated by the transformation in equation (6). If 0 f = the states n 1 = ñ | and n 2 = ñ | are degenerate in energy.
We define an effective vector k where 1 is the identity for a two state system. The last term in equation (12) describes a gap opening at q 0 = between the ground eigenstates for small values of f .

Phase considerations
As made evident by its presence in equation (3) and absence in equation (7) the phase of each laser does not contribute to the steady state Hamiltonian. This symmetry is absent when there are more than three Raman frequency differences for a three state subsystem or in ring coupling geometries with N 3 > [29]. This consideration is very compelling from an experimental perspective since small variations in the pathlength of each laser could otherwise produce dramatic changes in the potential. The Raman matrix elements j j , W ¢ acquire a sign from the lasers' detuning 1 1 1 3 2 1 2 D = D -D from the P 3 2 and P 1 2 lowest electronic fine structure. A phase of π is contributed to f when 1 D is negative and 0 otherwise. Recently, an experiment realized the Rashba dispersion using positive 1 D [26]. In most schemes, laser detuning is the primary consideration but with our approach both the detuning of the Raman relative to the rf and laser polarization will additionally contribute to f .

Physical implementation
Raman coupling in the spin basis We introduce the local electric field of linearly polarized lasers impinging upon an atomic system. The vector light shift of the local electric field acting upon the ground state hyperfine spin manifold in an alkali atom can be cast in the form of a time and position dependent effective magnetic field [30,32]. This gives a coupling in the ground hyperfine manifold of an alkali atom, where g S is the gyromagnetic ratio of the electron spin, g F is the Landé g-factor for the hyperfine states, B m is the Bohr magneton, and the two-photon vector light shift matrix element is The far off-resonant Wigner-Eckart reduced matrix element is given by d d l l 0 1 á ñ = á = = ñ || || || || where l=0, 1 is the orbital angular momentum quantum number for the ground and excited electronic states, respectively.
We compute the pairwise product of components of the local electric field in the effective Hamiltonian equation (14) and retain terms that have in the argument of the complex exponentials. When laser polarizations are linear we may rearrange terms and obtain the effective coupling between the ground electronic hyperfine spin projections is the vector of spin-1 operators. We tune some Raman frequency differences to near resonance , where θ is the Heaviside function and F F F i x y = + +ˆˆ. The matrix elements , W b b¢ in the RWA are  produces an off-diagonal coupling.

Construction of fully coupled basis states
For the remainder of this manuscript we narrow our discussion to the f=1 ground hyperfine manifold of Rb 87 and adopt the simplified labels m F ñ | , where m 1, 0, 1 F Î -+ { } label hyperfine (spin) projections and E m F label spin eigenenergies. We divide the overall Zeeman shift into a scalar part which we neglect, a linear part given by ) and a quadratic part given by , ñ = ñ ñ | | | and Zñ | eigenstates, which consist of linear combinations of m F ñ | states in the f=1 hyperfine manifold , and 0 . 26 }. The Raman couplings from the previous subsection have a spin dependence F F F , , or x y z µˆˆˆand may therefore couple any pair of X Y Z , , ñ | states. This observation was made recently by [33] in the context of producing optical flux lattices.
A set of atomic eigenstates which approach the XYZ states can be produced by an oscillating magnetic field B t cos rf rf rf ) that is orthogonal to B e z dc . The rf coupling is described by The rf is chosen to be resonant with the average of the two hyperfine spin transitions in the ground manifold, In the rotating frame of the rf and applying the RWA, the complex exponential t exp i rf w ( ) can be eliminated from equation (28). Together with the atomic hyperfine energy levels the Hamiltonian with rf coupling is ) has an equivalent magnitude rf W and phase i ln rf x ( ). The rf eigenenergies E j of the Hamiltonian in the presence of the rf magnetic field are plotted verses B dc in figure 5(b). For large rf W , the rf eigenvalues change weakly as a function of B dc . We therefore set Z r f for the remainder of this document. When we write down the rf eigenstates We see that these adiabatically approach the X Y Z , , ñ | states as  W  -¥. Here we defined 4 2 rf 2 *  W = + W | | . We resonantly link these eigenstates with the Raman coupling of the form described in the previous subsection and operate in the limit where the Raman is much smaller than the rf coupling, We define a rf eigenstate coupling matrix while defining the matrix elements, and we incorporate the rf phases into the definition of the total coupling in the next section. These rf phases ultimately cancel in our coupling scheme.
The matrix elements j D j l á ¢ ñ |ˆ| of D l , linking rf-eigenstate pairs are We can transform between the rf-eigenbasis and the m F basis using the rf-eigenstate coupling matrices, e.g. F D x x ˆ.
Numerically calculating the eigenstates of the Raman and rf coupling The rf and Raman couplings produce a time-periodic effective Hamiltonian. Using Floquet theory we decompose the states of our Hamiltonian [ˆˆ]  Figure 4 depicts a quasi-energy unit cell. When 0 f = the two lowest quasi-energies meet at some quasimomentum q  . For the parameters used in figure 4, noticeable drift in q  due to close spacing of Floquet unit cells occurs when either 30 rf W W < or E 2 3 0 R rf W < . Adjusting the balance of Raman matrix elements, e.g. adjusting laser intensities, can compensate for this drift and return the quasimomentum at which the quasienergies meet to q 0  = . This is a configuration where the degeneracy of the ground state dispersion is maximized.
. These quantities are defined in the preceding sections. The least (solid) and next least (dashed) energetic states are closely spaced while the most energetic (dashed dotted) state is separated by 3 W when 0 f = . The point at which the ground Floquet states meet is slightly displaced from the q 0 = at these couplings due to the presence of nearby Floquet states.  depending upon the laser geometry; E 2 R is based on a laser geometry where all the lasers are perpendicular to one another. To produce the Rashba potential using our laser scheme and laser geometry the Raman coupling strength is bounded E 2 . In alkali atoms, dc magnetic field fluctuations often limit the long-term stability of an experiment that optically couples two or more magnetically split internal states. This is partly the case because the splitting between internal states is nominally linear with magnetic field. At resonance, the rf eigenstates respond quadratically to magnetic field fluctuations: When the rf coupling is strong h 200 kHz rf  W =| | compared to the Zeeman splitting amplitude fluctuations of a lab without active field control h 1 kHź , the resulting impact of the magnetic fluctuations is reduced E h 5 Hz j D »´. Hence, rf eigenstates produced by sufficiently strong rf coupling become engineered clock states.

Raman laser frequencies, intensity and geometry
We illuminate a cloud of Rb 87 atoms using three linearly polarized lasers, all with wavelength very near 790.024 nm l = . At this wavelength, the two-photon vector light shift matrix element u is negative, while the scalar light shifts are zero. The frequencies of these beams are