Bloch Oscillations in Optical and Zeeman Lattices in the Presence of Spin-Orbit Coupling
We address Bloch oscillations of a spin-orbit coupled atom in periodic potentials of two types: Optical and Zeeman lattices. We show that in optical lattices the spin-orbit coupling allows controlling the direction of atomic motion and may lead to complete suppression of the oscillations at specific values of the coupling strength. In Zeeman lattices the energy bands are found to cross each other at the boundaries of the Brillouin zone, resulting in period-doubling of the oscillations. In all cases, the oscillations are accompanied by rotation of the pseudo-spin, with a dynamics that is determined by the strength of the spin-orbit coupling. The predicted effects are discussed also in terms of a Wannier-Stark ladder, which in optical lattices consist of two mutually-shifted equidistant sub-ladders.
pacs:37.10.Jk, 32.60.+i, 03.75.-b
Bloch oscillations (BOs) of electrons in a crystal under the action of a constant electric field (1) is one of the fundamental predictions of quantum mechanics. They were observed in semiconductor superlattices (2); (3), a few years after the observation (4) of a Wannier-Stark ladder (WSL). BOs have been observed in a variety of systems where waves can propagate in periodic environment (5), such as cold atoms (6), Bose-Einstein condensates (BECs) (7) held in optical lattices, and diverse optical settings, including waveguide arrays (8), optically-induced lattices (9), coupled microcavities (10), or plasmonic systems (11).
To date, most studies about BOs in atomic systems have addressed one-component settings. Two-component systems have received less attention. A particularly interesting situation occurs in the presence of linear coupling between components that in steady state locks their energies. Spin-orbit (SO) coupling can be implemented in a multi-level atom (12) for which artificial electric (13) and magnetic (14); (15) fields can be created by managing interactions between different hyperfine states (16). A system involving coupled hyperfine atomic states in a periodic potential and subjected to linear forces induced by a Zeeman field undergo BOs (17) in a static field or quantum walks in a time-periodic field. The dynamics of such a system is strongly influenced by the presence of two oppositely tilted WSL in the eigenmode spectrum (18). Interesting dynamical regimes occur in SO-coupled BECs (19). The interplay of BOs due to a linear force and the spin Hall effect induced by the SO coupling can lead to complex evolution of BECs in a two-dimensional lattice (20). Both, BOs and WSL of a four-component model of a helicoidal molecule with SO coupling have been addressed (21).
In this Letter we report unconventional features that SO coupling brings into the dynamics of BOs in a system held either in an optical lattice (OL) or in a Zeeman lattice (ZL) and subjected to a linear force. We highlight suppression of BOs due to band-flattening in optical lattices, transition from periodic to non-periodic BOs, period-doubling of BOs in Zeeman lattices due to band-crossing at the edges of the Brillouin zone (BZ), and control of the direction of motion of broad wavepackets.
We consider a two-level atom described by the spinor , which solves the Schrödinger equation , where time and position are measured respectively in units of and , with being the mass of the atom and the lattice period. is the Hamiltonian without lattice and external field, is the momentum, is the SO coupling strength, is the Rabi frequency, is the linear force, and are Pauli matrices. The potential is set by an OL that is equal for both spinor components, or by a ZL having opposite signs for the components. Both lattices are -periodic, i.e., ; even functions with respect to the origin ; and odd functions with respect to the quarter-period . In simulations we model OLs and ZLs as and , respectively [see Fig. 1]. Both such potentials are experimentally feasible as described in Refs. (7) and (22) for OLs and ZLs, respectively.
When , the eigenmodes of the system are Bloch waves , i.e., , where , , and is the Bloch momentum in the reduced BZ: . We start with a semi-classical description valid for broad wavepackets (namely, , where is the spectral width of the wavepacket). Similarly to the one-component case (1); (23), one can show (24) that if the linear force is weak, , and the inter-band tunneling is negligible, the center of a wavepacket , initially chosen as a Bloch wave modulated by an envelope ensuring , behaves as . Hence
where and are the initial coordinate and Bloch momentum, respectively. Since the BZ width is , the wavepacket crosses the zone after the time interval .
Important information about the Bloch modes can be obtained from the symmetries of the Hamiltonian (25). Starting with OLs, we recall (26) that the Hamiltonian obeys the Klein four-group symmetry , i.e., , defined by the operators , , and , where is the time reversal operator, , is the parity operator with respect to , i.e., and is the identity operator. Here , where , and are all different. In terms of the Bloch modes , the operators and map a state at to a state at : and . Thus and symmetries imply mapping, while does not affect . At least one of the eigenfunctions of obeys all -symmetries. When at the boundary () [at the center ()] of the BZ energy levels do not cross each other, the eigenvalue  is non-degenerate and its eigenfunction is highly symmetric. Such eigenfunction must satisfy with , and hence . In terms of the average values of the pseudo-spin (or spin, for brevity) components ( for localized nonstationary solutions), this means that at the boundary [center] of the BZ . Such phenomenon is observed in the dynamical simulations shown in Fig. 3.
In ZLs only one of the above symmetries remains, but there appears an additional one: , where is the reflection with respect to . Consider an eigenstate at the BZ boundary, , and assume that its eigenvalue is nondegenerate (which implies that no crossing of the energy levels occurs at ). It can be chosen to be either –symmetric , or satisfying . We set the “” sign (the sign “” is analogous). Non-degeneracy implies the linear dependence where is a complex number, and means that because is linear. Thus is simultaneously and -symmetric: . Using Floquet’s theorem we obtain: . The ansatz from the last equality yields for (), which means that . This contradicts the assumption that the energy levels do not cross each other at and that the -symmetric state is nondegenerate. Therefore, the energy levels should cross at the boundary of the BZ. Note that reflects the states with respect to the BZ origin (i.e., maps ) with simultaneous inversion of the spinor components. This means that has opposite signs at the and points.
The spectra for OL and ZL are illustrated in Fig. 1. Panels (a) and (b) show transformation of the first and second bands of the OL spectrum upon increase of the SO coupling. Panel (c) shows the two lowest bands of the ZL at . For both types of lattices, increasing SO coupling leads to the appearance of extrema of in the internal points of the BZ. In OLs, the slope of the dependence may change its sign with increasing . According to (1) this means that the SO coupling may invert the direction of motion of the wavepacket. At a certain [red lines in Fig. 1 (a,b)] either the first or the second bands may become nearly flat. Namely, for the first band, and for the second band, and note that these values only slightly decrease with growth of the lattice depth. Such an extreme band-flattening, discussed also in (27), implies a weakly dispersive propagation and almost complete suppression of BOs. Indeed, according to (1) the amplitude of BOs for an atom in the nth band is given by and it must be a nonmonotonic function of , consistent with Figs. 1(a,b). Band crossing at the edges of the BZ in ZLs is visible in Fig. 1(c).
The trajectory (1) is in excellent agreement with the direct solution of the Schrödinger equation using a Crank-Nicolson method. Calculations were conducted with transverse points and steps , . The input broad wavepacket had the form , where is the Bloch wave at for a given . Fig. 2(a) illustrates the non-monotonic dependence of the maximal displacement , defined over 20 periods of BOs in an OL. becomes zero for specific values of the SO coupling , where we observe nearly complete suppression of BOs [Fig. 3(a)]. In contrast to broad wavepackets, strongly localized initial states do not experience displacement, but exhibit nearly periodic width oscillations in , whose maximal amplitude is shown in Fig. 2(b) as a function of . Zeros of and minima of coincide with the bandwidth mimima from Fig. 2(c).
Figures 3(b,c) show that the SO coupling may change the direction of the wavepacket motion. For and , a wavepacket initially corresponding to the Bloch mode with moves in the opposite direction for a given linear force . While the amplitudes of the BOs in Figs. 3(a,b) differ significantly, the spinor dynamics is similar and is characterized by the integral spin almost parallel to the SO coupling and exhibiting weak oscillations in the -plane: . Increasing the SO coupling results in rotation of the spin direction, which becomes nearly orthogonal to the SO coupling. In Fig. 3(c) the spin is mainly directed along the -axis and . Since flattening of different bands occurs for different values of [Fig. 2(c)], an arbitrary superposition of several band states always exhibits BOs.
The BOs dynamics changes drastically if the initial wavepacket is a superposition of states from more than one band. This phenomenon is illustrated in Fig. 3(d), which was obtained for the same parameters as Fig. 3(b) but now for a superposition of eigenmodes from the two lowest bands. Since a two-level atom is characterized by the Rabi frequency , BOs with frequency are accompanied by rapid oscillations between the spinor components, visible as spin rotation in the plane , , whose frequency exceeds because of the effect of the SO coupling. Such evolution, characterized by two frequencies, can be strictly periodic only if the frequencies are commensurable, a case that corresponds to a fully equidistant WSL as discussed below.
The BOs dynamics in ZLs is substantially different than in OLs, due to two main features of the spectrum: (i) band crossing at BZ edges and (ii) vanishing of the spin-projection at and inversion of at . Such features have a direct impact on the BOs: the period of BOs in ZLs doubles in comparison to that encountered in OLs, and amounts to . Indeed, a broad wavepacket starting its motion around and moving along the lowest band still crosses the BZ after a time period , but it arrives to the opposite edge of the BZ at with an inverted component. Subsequently, it keeps moving along the second band, intersecting with the lowest band at and only after passing the BZ again, but now along a new path, it returns to the original state after the total time interval of (see Fig. 4). As in the case of OLs, for a moderate SO coupling, the spin dynamics in ZLs is bound to the plane (. When grows the spin becomes orthogonal to the SO coupling [ in Fig. 4(b)]. Thus the amplitude of the BOs in ZLs is determined by the combined width of the two lowest bands. This is confirmed by Fig. 2(d), which compares the BOs amplitude obtained numerically with the predictions of (1), but taking the combined width of the two lowest bands as a total range of variation of . The BOs amplitude in ZLs becomes independent of for a sufficiently strong SO coupling, because one of the spinor components can be strongly suppressed at and the dynamics resembles that of the one-component system, as explained in (28)).
An alternative approach to analyze the BOs is based on the calculation of the localized Wannier-Stark modes of the total Hamiltonian on numerically large windows with zero boundary conditions. For OLs such a spectrum ( vs eigenvalue number ) is shown in Fig. 5(a). It reveals a specific WSL with two characteristic steps between neighboring eigenvalues. Due to the spinor character of the system the total WSL is a combination of two equidistant sub-ladders, namely and , shifted along . At the mutual shift of the sub-ladders amounts to . The Wannier-Stark modes from each sub-ladder have identical first and second components, and they are in-phase for one sub-ladder and out-of-phase in the other sub-ladder. If BOs and Rabi periods are commensurable, i.e., at , the two sub-ladders overlap exactly, the separation of the neighboring eigenvalues becomes equal to , and any excitation is recovered after the period . At the entire ladder becomes equidistant with , and the period of BOs for any wavepacket exciting modes from both sub-ladders increases to . At the SO coupling results in an additional mutual shift of the two sub-ladders shown in Fig. 5(b). Due to such a shift the dynamics is in general non-periodic for inputs that excite both sub-ladders unless the SO coupling is selected such that () and the equidistance of the entire spectrum is restored. If only one sub-ladder is excited the dynamics may still be periodic. In contrast to the behavior observed in OLs, the WSL for ZLs is equidistant for any with , which indicates BOs with a period .
Summarizing, we have shown that SO coupling brings important new features into Bloch oscillations. It allows controlling the direction of atomic motion and enables nearly complete suppression of oscillations in OLs. In ZLs, oscillations feature period-doubling due to crossing of energy bands at the boundary of the Brillouin zone.
Acknowledgements.The work was supported by the FCT (Portugal) grant UID/FIS/00618/2013, by the Severo Ochoa Excellence program (Spain), and by Fundacio Cellex Barcelona.
Supplemental Material for “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling”
which leads to Eq. (1) from the main text. In fact, we will address a slightly more general case: we consider the situation when the linear force depends on time: . The particular case obviously corresponds to the situation addressed in the main text. The derivation is valid for state described by a ket-vector of any dimension (i.e. having one, two or more components) and for the arbitrary strength of the SO-coupling (provided the Conditions formulated below are not violated).
We use the complete set of Bloch states of the Hamiltonian : , where is the energy of the state, the Bloch quasi-momentum belongs to the reduced Brillouin zone (BZ): , and is the band number ( being the number of the lowest band). Following Houston’s approach (23), we introduce adiabatically-varying Bloch states where
and is the initial value of the quasi-momentum (or central quasi-momentum in the case of localized Bloch wavepackets). The spinor can be expanded in terms of the adiabatically-varying states as:
and, for the sake of convenience, the spectral coefficients will be represented as
with the functions to be determined latter. The normalization condition (in the direct and Fourier spaces)
is also imposed.
We assume that the following conditions hold:
Bloch states of only one band, say of the band , are initially excited, i.e. for .
A wavepacket is a Bloch wave with a smooth and broad envelope (compared to the lattice period) and its spectrum centered at a quasi-momentum in the reduced BZ is much narrower than the BZ zone, so that the approximation
is valid [here we take into account the normalization (6)].
The linear force is weak, i.e., , so that inter-band tunneling is negligible.
Let us start with the case . Using the expansion
from the Schrödinger equation one gets
Using Floquet theorem, the Bloch states can be expressed in the form of the expansion
where are vector Fourier coefficients. The Bloch states are orthonormal, which implies
Now, using (11) we get
and using the relation we obtain
On the other hand, the same expression is computed as
Equation (17) is exact, no approximations were made so far. Using now the Conditions 1 and 2 formulated above, one arrives at the formula
By direct differentiating and using that at each instant , are the eigenfunctions of with we obtain
Now we require , which, so far, are arbitrary functions of time, to ensure . To this end we require for all , which yields the set of equations
As we already did above, now we use the Fourier expansion (11) with the substitution and obtain
This is an exact formal result representing an infinite number of equations. If however all three conditions introduce above hold, Eq. (22) for a given band can be approximated by
Since at we have the conservation (12), we conclude that
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