Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms
We propose a scheme to dynamically synthesize a space-periodic effective magnetic field for neutral atoms by time-periodic magnetic field pulses. When atomic spin adiabatically follows the direction of the effective magnetic field, an adiabatic scalar potential together with a geometric vector potential emerges for the atomic center-of-mass motion, due to the Berry phase effect. While atoms hop between honeycomb lattice sites formed by the minima of the adiabatic potential, complex Peierls phase factors in the hopping coefficients are induced by the vector potential, which facilitate a topological Chern insulator. With further tuning of external parameters, both a topological phase transition and topological flat bands can be achieved, highlighting realistic prospects for studying strongly correlated phenomena in this system. Our Letter presents an alternative pathway towards creating and manipulating topological states of ultracold atoms by magnetic fields.
pacs:37.10.Gh, 67.85.-d, 81.16.Ta, 73.43.-f
Gauge fields lie at the center of our modern understanding of physics in many systems, including those of high energy and condensed matter, as well as of ultracold atoms. Within the gauge-field paradigm, interactions between particles, which enable rich quantum phases of a many-body system, are mediated through gauge fields. For instance, solid-state materials with charged quasiparticles in magnetic fields or with spin-orbit coupling (SOC) show a rich variety of quantum Hall effects and exotic topological superconductivity Hasan and Kane (2010); Qi and Zhang (2011). The interplay between gauge fields and lattice systems is also of great interest (for a pedagogical review, see Bernevig and Hughes (2013)). The spectrum of a charged particle in a square lattice exposed to a strong uniform magnetic field shows a fractal structure, widely known as the Hofstadter butterfly Hofstadter (1976). In another seminal work, Haldane shows that the quantum Hall effect without Landau levels can be realized when a periodically staggered magnetic field is applied to charged particles in a honeycomb lattice Haldane (1988).
Ultracold atoms in lattice systems are considered to be powerful simulators for studying gauge-field physics Dalibard et al. (2011); Goldman et al. (2014); Goldman and Dalibard (2014); Struck et al. (2011); Hauke et al. (2012); Jaksch and Zoller (2003); Celi et al. (2014); Grushin et al. (2014); Zheng and Zhai (2014); Aidelsburger et al. (2013); Miyake et al. (2013); Aidelsburger et al. (2015); Kennedy et al. (); Mancini et al. (2015); Stuhl et al. (2015); Jotzu et al. (2014); Cooper (2011); Jimenez-Garcia et al. (2012). Both the Hofstadter and the Haldane models with cold atoms were theoretically proposed Jaksch and Zoller (2003); Celi et al. (2014); Grushin et al. (2014); Zheng and Zhai (2014) and experimentally demonstrated Aidelsburger et al. (2013); Miyake et al. (2013); Aidelsburger et al. (2015); Kennedy et al. (); Mancini et al. (2015); Stuhl et al. (2015); Jotzu et al. (2014) by making use of novel forms of light-atom interactions Goldman et al. (2014), such as laser-assisted tunneling Aidelsburger et al. (2013); Miyake et al. (2013); Aidelsburger et al. (2015); Kennedy et al. (), the shaking-optical-lattice technique Jotzu et al. (2014), and SOC within a synthetic dimension Mancini et al. (2015); Stuhl et al. (2015). In addition to the optical lattice formed from a space-periodic ac-Stark shift by interfering laser beams, proposals for the generation of a magnetic lattice with a space-periodic Zeeman shift have been put forth Yin et al. (2002); Günther et al. (2005); Singh et al. (2008); Whitlock et al. (2009); Jose et al. (2014); Grabowski and Pfau (2003); Günther et al. (2005); Singh et al. (2008); Whitlock et al. (2009); Leung et al. (2011); Jose et al. (2014); Romero-Isart et al. (2013); Luo et al. (2015) (and some have been realized Günther et al. (2005); Singh et al. (2008); Whitlock et al. (2009); Jose et al. (2014)) using current-carrying wires Yin et al. (2002), microfabricated wires or permanent magnetic structures on atomic chips Grabowski and Pfau (2003); Günther et al. (2005); Singh et al. (2008); Whitlock et al. (2009); Leung et al. (2011); Jose et al. (2014), superconducting vortex lattice shields Romero-Isart et al. (2013), and phase imprinting by gradient magnetic pulses Luo et al. (2015). In contrast to optical lattices, magnetic lattices are free from atomic spontaneous emissions that are always accompanied by heating and loss of atoms. Additionally, they have the potential to reach shorter lattice constants Leung et al. (2011); Romero-Isart et al. (2013) (of a few tens of nanometers as proposed in Ref. Romero-Isart et al. (2013)), leading to improved energy scales and less stringent temperature requirements; the lattice constants can even be continuously tuned Luo et al. (2015). These advantageous features enhance the performance of atomic quantum gases as powerful quantum simulators.
While the simulation of gauge-field physics and the manipulation of topological states in optical lattices have shown fruitful results Dalibard et al. (2011); Goldman et al. (2014); Goldman and Dalibard (2014); Struck et al. (2011); Hauke et al. (2012); Jaksch and Zoller (2003); Celi et al. (2014); Grushin et al. (2014); Zheng and Zhai (2014); Aidelsburger et al. (2013); Miyake et al. (2013); Aidelsburger et al. (2015); Kennedy et al. (); Mancini et al. (2015); Stuhl et al. (2015); Jotzu et al. (2014); Cooper (2011); Jimenez-Garcia et al. (2012), it remains to show whether this is also the case for magnetic lattices. This Letter provides an affirmative first answer to this question.
This Letter presents a scheme for synthesizing a time-independent effective Hamiltonian with nontrivial band topology for atomic gases with internal spin degrees of freedom, based on the phase imprinting technique Burger et al. (1999); Denschlag et al. (2000). A two-dimensional (2D) magnetic lattice with triangular geometry emerges in the effective Hamiltonian. In the limit when an atom is confined in the lowest Zeeman level, an adiabatic scalar potential and a geometric vector potential are simultaneously generated for the center-of-mass motion Dalibard et al. (2011); Cooper (2011); Jimenez-Garcia et al. (2012). The adiabatic scalar potential surface can form a honeycomb lattice, while the associated geometric vector potential provides complex phases for next-nearest-neighbor (NNN) hopping coefficients in realizing the Haldane model Haldane (1988); Shao et al. (2008). With the flexibility and tunability of magnetic fields, our scheme can be extended to produce a set of effective Hamiltonians whose lowest energy bands undertake a topological phase transition from a topological (Chern) insulator to a trivial one. Moreover, models possessing topological quasiflat bands are realized near the phase-transition point.
The protocol.—We consider a pancake-shaped quasi-2D ultracold atomic gas of spin confined in the - plane (at ). In the presence of a bias magnetic field , the single-particle Hamiltonian is given by
where is the 2D kinetic momentum, is the atomic mass, is the reduced Planck constant, is the third component of the atomic spin vector (in unit of ) and is the Larmor frequency at , where is the Land factor for the spin- hyperfine state manifold and is the Bohr magneton.
A short gradient-magnetic-field pulse of duration imprints a space-dependent phase factor Burger et al. (1999); Denschlag et al. (2000); Xu et al. (2013); Anderson et al. (2013); Luo et al. (2015, ) onto the wave function as , where is the SOC strength Xu et al. (2013); Anderson et al. (2013) with the magnetic gradient. After a free evolution time , a second magnetic field pulse in the opposite direction imprints an opposite phase. The two pulses combined together enact a unitary transformation
which rotates the magnetic field to a space-periodic form . Similarly, an opposite uniform-field pulse pair with a pulse area inverts the magnetic field to as . More generally, a gradient magnetic field pulse along an arbitrary direction in the - plane imprints a phase factor , where and are, respectively, the coordinate vector and the spin vector projected to the direction. Following a period of free evolution and a second pulse from an opposite gradient field, an expression analogous to Eq. (2) generates a magnetic field with spatial periodicity along the direction.
In our scheme, discussed below, repeated pulse pairs are concatenated. A complete cycle of the evolution period contains three gradient pulse pairs along directions separated by an angle of , together with a pulse pair along the direction as shown in Fig. 1(a)-1(b). The total evolution over one complete cycle (of period ) is then given by
where is the 2D spin operator, is the SOC frequency, and is an effective magnetic field whose three components are given by
The first two terms in Eq. (4) arise from the unitary transformations by gradient pulse pairs applied to the momentum operator Xu et al. (2013); Luo et al. (2015). The third term describes a magnetic (Zeeman) lattice that couples the atomic spin to the effective magnetic field , as shown in Fig. 1(c).
Geometric potentials and energy spectrum.—The above protocol for the generation of a triangular magnetic lattice is general, and can be applied to atoms with arbitrary spins. For concreteness, we choose a specific atomic species, fermionic Li, with electron spin , nuclear spin , and we consider the total hyperfine spin ground-state manifold. The Land factor can be evaluated according to the Breit-Rabi formula Woodgate (1980) to be . The spin operator reduces to , where is the vector of Pauli matrices. To be more specific, in all numerical calculations, we assume a set of fixed parameters unless otherwise noted. They are , ?, which correspond to and for the manifold of Li. With these parameters, the lattice term in Eq. (4) dominates during time evolution.
We denote the space-dependent eigenstates of the magnetic lattice by , which satisfy
where is the adiabatic potential for atomic center-of-mass motion in the lower-energy eigenstate . For an atom adiabatically moving in this space-periodic Zeeman level, a vector potential emerges Sup (); Sca (),
with . Associated with the vector potential is the flux density , which shares the same spatial periodicity as and can be considered in general as a type of flux lattice Cooper (2011).
The adiabatic potential , vector potential , and the flux density for our magnetic lattice are shown in Figs. 2(a) and 2(b) Vec (). As shown in Fig. 2(a), the local minima of are located at the corners of the unit cell, forming a honeycomb lattice. When an atom hops between these honeycomb sites, the vector potential contributes a complex Peierls phase factor to the hopping coefficient Hofstadter (1976); Haldane (1988); Shao et al. (2008), with the integration evaluated along the corresponding hopping path. As vanishes along the edges of the hexagon, the nearest-neighboring (NN) phase factor is a trivial unity. While along the NNN hopping paths [dashed lines in Fig. 2(a)], the accumulated phases are always nonzero. Thus the adiabatic scalar potential together with the geometric vector potential realizes the Haldane model in the tight-binding limit. As a caveat, our flux pattern shown in Fig. 2(b) is not the same as that suggested by Haldane Haldane (1988), where the staggered flux density gives a vanishing net flux over a unit cell. The flux distribution shown in Fig. 2(b) is non-negative everywhere, and the net flux over one unit cell is unity rather than zero; this can be checked by integrating the following over a unit cell: , with Cooper (2011). Thus, the nontrivial winding pattern of shown in Fig. 1(c) leads to a quantized net flux . A nonzero net flux generally leads to larger Peierls phases (of order unity). It also facilitates simulation of charged particles in strong magnetic field with nondispersive Landau levels Cooper (2011); Cooper and Dalibard (2011).
To quantitatively confirm that our model indeed maps onto the Haldane model, we numerically study the spectrum and Berry curvature Xiao et al. (2010) of the effective Hamiltonian Eq. (4) using the plane-wave expansion method Ashcroft and Mermin (1976); Sup (). The typical band structure and the Berry curvature for the lowest band are shown, respectively, in Figs. 2(c) and 2(d). A band gap opens at the corners of the first Brillouin zone ( points), where the Berry curvature is at a maximum. Both the eigenenergies and the Berry curvatures are even functions of quasimomentum, so the spectrum at is not shown. The Chern numbers Thouless et al. (1982) for the lowest two bands are , respectively. The spectrum and the Berry curvature thus resemble the ones from the Haldane model. To further validate this correspondence, we adopt the method used in Ref. Ibanez-Azpiroz et al. (2014) to get the NN hopping constant and the complex NNN hopping constant of the Haldane model from the calculated band structure. We find and with . Using these three parameters together with an overall energy shift, the tight-binding band structure of the Haldane model is plotted with dashed lines in Fig. 2(c).
Topological phase transition and quasiflat bands.—Our protocol allows for the easy tuning of two parameters: the SOC strength and the bias magnetic field . Both can be tuned continuously, and can be turned on adiabatically to reach the ground state for our model system Eq. (4) Goldman and Dalibard (2014); Jotzu et al. (2014) (see Sup () for details). Once the ground state is achieved, we can apply an additional weak optical gradient field (which commutes with all the pulse manipulation operations) in the - plane to drive Bloch oscillations and then measure the perpendicular center-of-mass drift to extract the topological properties for the lowest energy band Dauphin and Goldman (2013); Jotzu et al. (2014); Aidelsburger et al. (2015). With unequal durations between subsequent pulse pairs, or allowing for specific and values for different subperiods, several variants of the effective Hamiltonian can be synthesized. A topological phase transition for the lowest energy band can be achieved by a simple tuning of the bias magnetic field. For this to occur, we set the field strength to be for the first three subperiods and switch to for the fourth subperiod; our protocol then leads to a change for the component of the effective magnetic field in Eq. (5) as
The case corresponds to the original proposal with topological bands, while the case describes a system of trivial energy bands with zero Chern numbers. By continuously tuning from to , a topological phase transition with band touching and reopening takes place, as summarized in Fig. 3.
Figure 3(a) presents the changing Chern number and, hence, the band topology, for the lowest band with increasing . The lattice geometry of the adiabatic potential is found to undergo a structural transformation first from a simple triangular lattice to a decorated triangular lattice Jo et al. (2012), and then to a honeycomb lattice. The corresponding tight-binding descriptions for orbitals involve , , and bands, respectively, for the three cases. As increases, the band originating from hopping between orbitals located at unit cell centers crosses the two bands from orbitals located at the corners. Their corresponding Chern numbers change after band touching and reopening. Figure 3(b) shows the behavior of the gap between the lowest two bands as well as the band width for the lowest one. Gap closing occurs at the point when , and the gap-over-width ratio is found to be quite large over a limited range after a gap opening with a peak value as large as when , as shown in the inset of Fig. 3(b). The energy spectrum at is shown in Fig. 3(c). The lowest band is a Landau-level-like topological quasiflat one Sørensen et al. (2005). Such a nondispersive topological band also persists beyond the adiabatic limit Sup (). It is a promising candidate for simulating the fractional quantum Hall effect when suitable interactions are included Yang et al. (2012); Parameswaran et al. (2013); Cooper and Dalibard (2013); Bergholtz and Liu (2013). It is perhaps worth pointing out, also, that a flat band can emerge as the second excited band in a Kagome lattice Jo et al. (2012), or as the first excited band in a Lieb lattice Taie et al. (). The properties of the localized states in the flat band of a Lieb lattice have been investigated in a recent experiment Taie et al. ().
We focus in this Letter on discussing single-particle physics of a fermionic spin- system, though our magnetic lattice generation protocol can be equally applied to a higher-spin atom, be it a boson or fermion. When local momentum-independent ( wave) interaction is taken into account, it can be simply added to the effective Hamiltonian because it commutes with all the pulse manipulation operations (see also Xu et al. (2013); Anderson et al. (2013)). The topological phase is expected to be stable for weak interactions due to the presence of a gap. However, stronger interaction can drive the system to new phases, in which the physics may be dominated by the interplay between the correlation and band topology. A detailed study of the interaction effects for this system deserves further efforts.
In conclusion, we propose an experimentally feasible protocol to realize a synthetic magnetic field with real magnetic field pulses. The synthetic magnetic field forms a lattice with nontrivial band topology, and under certain limits can be mapped to the Haldane model. The high tunability of our scheme makes it possible to design a topological phase transition as well as quasiflat energy bands with nontrivial topology, which could push the effective model into the strongly correlated regime.
We thank Professors W. Vincent Liu and Kun Yang for valuable discussions. This work is supported by the MOST Grant No. 2013CB922004 of the National Key Basic Research Program of China and by the NSFC (Grants No. 91121005, No. 91421305, and No. 11374176), as well as by U.S. AFOSR (Grant No. FA9550-12-1-0079), ARO (Grant No. W911NF-11-1-0230), the Charles E. Kaufman Foundation, and the Pittsburgh Foundation (J.Y. and Z.-F. X.). R.L. also wants to acknowledge support from the NSFC (Grant No. 11274195).
Supplemental Material for:
Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms
Jinlong Yu, Zhi-Fang Xu, Rong Lü, and Li You
State Key Laboratory of Low Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing 100084, China
MOE Key Laboratory of Fundamental Physical Quantities Measurements,
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
This supplementary material provides more details for the various points mentioned in the main article.
.1 Derivation of the effective Hamiltonian
As explained in the main text, a gradient magnetic field pulse along in the - plane prints a phase factor onto the single particle wave function. Under the action of an opposite pulse pair, the atomic momentum and spin transform according to,
where is the spin operator perpendicular to . Thus the evolution including the pulse pair gives rise to
with the corresponding Hamiltonian
A pulse pair along the direction flips the direction of the magnetic field seen by atom without inducing coupling to momentum. The evolution operator from such a pulse pair is given by
with the corresponding Hamiltonian
A complete evolution period contains three gradient pulse pairs separated by a mutual angle of in the - plane, together with a pulse pair along the direction, the corresponding evolution operator for one period is given by
with . We use to denote the Hamiltonian for each subperiods as
To lowest order of the Trotter’s expansion for , the effective Hamiltonian is the average of ,
with for the current case. Expanding out the terms in this Hamiltonian explicitly, we arrive at the effective Hamiltonian as shown in the main text after recombining them.
The higher order corrections of the effective Hamiltonian can also be derived through the Magnus expansion method, which to first order of gives Slichter (1990)
From Eq. (S6) we see that, our protocol can also be viewed as a periodically driven 4-step sequence, where the dynamics of the driven system can be understood in terms of the effective Hamiltonian and the associated micromotion Goldman and Dalibard (2014). We now provide the details for how our system fits into this formulism. The Hamiltonian for a general N-step sequence is described by , with the constraint . Thus we find
which is just the time-averaged effective Hamiltonian Eq. (S8), and correspondingly
The evolution operator for such a driven system can be partitioned in the following as introduced in Ref. Goldman and Dalibard (2014)
where the effective Hamiltonian and the initial-kick operator of the N-step sequence to first order of are given by Goldman and Dalibard (2014)
with . For a full evolution cycle (), the evolution operator is found to be
where the Hamiltonian to first order of is given by [using ]
This shows that, the micromotion of the N-step periodically driven system gives the first order correction to the effective Hamiltonian. This calls for the use of the time-averaged effective Hamiltonian Eq. (S8) to approximate the evolution in the short limit. This approximation will be validated numerically in the later sections.
.2 Solving the effective Hamiltonian with the plane wave expansion method
For spin-1/2 with , the effective Hamiltonian takes the following form
with after omitting a constant energy shift . As in Eq. (S16) is space-periodic, its eigenstates can be labeled with quasimomentum in the first Brillouin zone as good quantum numbers. Expanded in the plane wave basis Ashcroft and Mermin (1976), the eigenstates take the form
where is the band index, are the two reciprocal unit vectors, are integers taking values etc. The expansion coefficients are to be determined. The plane wave expansion Eq. (S17) together with the eigenvalue equation
determines the energy spectrum, as well as the Bloch wave functions as eigenfunctions.
In our numerical calculations, we use a cutoff as the maximal value for and . And we have checked that, for cases, the band structures shown in the main text are independent of the choice of . Thus in the plane wave basis, the Hamiltonian is expressed as a matrix for each quasimomentum within the first Brillouin zone. The eigenvalues and corresponding eigenvectors give respectively the energy spectrum and Bloch wave functions.
Once the Bloch wave functions are obtained, we can calculate their Berry curvatures and (first) Chern numbers for each energy bands according to Xiao et al. (2010)
where is the cell-periodic part of the Bloch function.
.3 Validity of the effective Hamiltonian
We use the effective Hamiltonian to describe the evolution of ultracold atoms under the magnetic field pulse sequence of the proposed protocol. The use of Trotter expansion limits its validity to short evolution periods. In this subsection we use two complimentary methods to check for this approximation.
The first method replies on evolving an eigenstate, whereby the initial state is prepared in an eigenstate of the effective Hamiltonian with a given band index and quasimomentum . The actual magnetic field pulse sequence is then used to evolve it to a later time where the wave function is denoted by . The overlap between these two states should be identically equal to unity at all times for the ideal case when the effective Hamiltonian exactly represents the evolution by the pulse sequence.
We arbitrarily choose a quasimomentum point in the first Brillouin zone, and the eigenstate for the lowest band , for the case and , is calculated according to the method described in the last section. The absolute values of the overlap after evolving for for , and , are respectively calculated to be , and , indicating improved level of approximations with shorter periods.
The second method relies on wave packet evolution. For this case, an external trapping potential is included into the Hamiltonian, and the initial state is prepared as a wave packet in the harmonic trap. Since the trap potential commutes with all the pulses, it is simply added into the effective Hamiltonian. The initial wave packet is then evolved respectively through the effective Hamiltonian and through the actually pulse sequence. We can then calculate some physical observables, e.g., population imbalance, by the corresponding time-dependent states, and compare their respective results. Figure S1 shows an example of this comparison. We can see that, they coincide with each other better for shorter evolution periods.
These studies indicate that for sufficiently short evolution periods, the effective Hamiltonian faithfully describes the evolution under the actual magnetic field pulse sequence. For the cases we discuss, is short enough to validate the effective Hamiltonian approximation.
.4 Derivation of the emergent gauge fields
The evolution of a single-particle state is governed by the effective Hamiltonian according to
with given by Eq. (S16) for the spin case. Expanded in the adiabatic basis, with corresponding center-of-mass wave functions labeled as , the wave function takes the form
which when acted upon by the spin-dependent shift leads to
In the adiabatic limit , if the initial state is prepared in the dressed state , then the probability amplitude for the particle to be in the orthogonal state remains zero at all time. Thus by projecting Eq. (S20) to the dressed state and taking , we get a closed equation for
is the geometric vector potential that couples to the center-of-mass motion, and
is the geometric scalar potential.
The resulting scalar potential consists of two terms. One is the adiabatic potential , whose characteristic energy scale is . The other is the geometric scalar potential , which scales as . Thus makes a negligible contribution to the total scalar potential when compared to , in the limit. In Fig. S2, geometric potentials for , , and at are shown. Compared to the corresponding adiabatic potentials shown respectively in Fig. 3(a) in the main text, with the corresponding potential (minima, maxima) in units of being , and respectively, the geometric potentials only give small corrections to the final total scalar potentials. The qualitative understanding of the Haldane model as well as the topological phase transition enabled by our model is not affected by including these corrections.
.5 Energy spectrum beyond the limit of adiabatic approximation
Although the discussion in the main text on the generation and understanding of the topological energy bands is based on models under the adiabatic as well as tight-binding approximation, the nontrivial topology of the energy bands is found to exist beyond these two approximations Cooper (2011); Cooper and Dalibard (2011). As an example, we take , and reduce the original value by a factor of five to . The term of a Zeeman level is used instead of the adiabatic potential when the adiabatic approximation is not satisfied. At with , the (minima,maxima) of the two Zeeman levels of the magnetic lattice are respectively given by and in units of . The separation between the corresponding two levels is , and the lattice depth of the lower Zeeman level is a mere . As the energy scale for the spatial uniform term in Eq. (S16) is of order , and SOC in this term can flip the spin, the lattice term does not dominate during the evolution; the adiabatic approximation fails for this case.
The lowest six energy bands of the effective Hamiltonian for this case are shown in Fig. S3(a). Their Chern numbers are all found to be , which invalidates directly a tight-binding description, because the sum of Chern numbers for a set of tight-binding bands should be equal to zero. The band width for the lowest energy band is , and the lowest band gap is , which gives a gap-over-width ratio of . Thus these Chern numbers for the lowest few bands resemble the ones for a charged particle in a magnetic field in a weak lattice background Ashcroft and Mermin (1976).
Next we discuss the validity of the effective Hamiltonian for this case. In previous subsection and in Fig. S1, we show the validity of the effective Hamiltonian for , with respectively evolution periods , , and . In the present case of , i.e., reduced by a factor of five, the suitable evolution period is expected to be increased by a similar factor. We can repeat the previous method by scanning state evolution period to illustrate the validity of adiabatic approximation, or alternatively, we can calculate the Floquet spectrum of quasienergies by evaluating the evolution operator Eq. (S6) directly for different evolution periods, with the results as shown in Fig. S3(b-d). We see that the spectrum of the quasienergies strongly resembles the one from the effective Hamiltonian at least for . The topological properties of the lowest energy band is expected to be preserved for , because no signature of band closing is found. For even larger evolution period, e.g. , the quasienergies for the lower energy bands are found to be smeared out due to the wrapping of the higher Floquet sectors with quasienergies at multiples of . Thus we conclude the suitable evolution period required for the validity of the effective Hamiltonian approximation is of the order of for the present choice of .
.6 Adiabatic preparation of the ground state
As is verified in the previous sections, one can use the effective Hamiltonian Eq. (S16) to describe the state evolution under the pulse sequence for short evolution periods. In this section, we describe how can the ground state of the effective Hamiltonian Eq. (S16) be reached through an adiabatic loading approach. For a non-interacting many-body fermionic system, the ability of preparing its ground state is essentially equivalent to the statement that all the (single-particle) eigenstates in the lowest energy band of in Eq. (S16) can be reached appropriately.
As shown in the main text, one complete cycle of the evolution period consists of four subperiods. To prepare the ground state, we will keep the fourth subperiod intact, while adiabatically ramp up (a) the strength of the gradient pulses in complete cycles and then (b) the bias magnetic field for the first three subperiods in complete cycles.
A quantitative analysis is provided as follows. We first introduce an auxiliary Hamiltonian
where the three components of the vector are respectively , , and . The effective Hamiltonian before the aforementioned ramping process is ; the effective Hamiltonian at the end of the ramping stage (a) is ; and the effective Hamiltonian after ramping stage (b) is , which recovers the effective Hamiltonian in Eq. (S16).
As a proof-of-principal illustration, suppose the initial state for the ramping stage (a) is prepared as a plain wave in the spin up state
which is the eigenstate of in the lower branch (we always consider positive ). The target state for the ramping stage (a) is the corresponding eigenstate of :
Here the spin state is not normalized. We denote the wavefunction during stage (a) under the pulse sequence as . The fidelity between this state and the target state is then given by
We notice that, the initial fidelity for stage (a) is
which is close to unity for small momentums and large . For instance, for the momentum point in the lower energy branch, with and , we get . After linearly ramping up from zero to one in cycles with period , the final fidelity is numerically found to be . This shows that the low energy eigenstates of can be reached reliably.
We then consider the ramping process for stage (b). We choose the initial state for this stage as the lowest eigenstates of . The vanishingly small lattice term introduced in our numerical simulation accounts for the effect of the Bragg reflections at the edges of the first Brillouin zone, which turns the good quantum number from momentum (denoted as ) to quasimomentum (denoted as ) restricted in the first Brillouin zone Ashcroft and Mermin (1976). The fidelity between the target state [i.e., eigenstate of Eq. (S16) with the form of Eq. (S17) with band index ] and the time-dependent state evolved by pulse sequence with increased magnetic lattice strength (which is proportional to ) is defined as:
We suggest ramping up the strength of the magnetic lattice as a tangent function (rather than a linear one), which corresponds to a time-dependent function with the form
where the parameter is taken as . The typical results for the time-dependent fidelity functions for different quasimomentums, together with the function, are shown in Fig. S4. As can be seen in this figure, the fidelities after ramping up the magnetic lattice approach unity for all tested quasimomentum points. We thus conclude that the eigenstates in the lowest energy band of the effective Hamiltonian in Eq. (S16) can be prepared appropriately through the adiabatic approach we propose.
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