Spatiotemporal Bloch states of a spin–orbit coupled Bose–Einstein condensate in an optical lattice
1. IntroductionSpin–orbit coupling (SOC), the interaction between the spin and momentum of a quantum particle, is crucial for many important condensed matter phenomena.[1] Due to the ground breaking work of Dresselhaus and Rashba, and the further theories and experiments they initiated,[2–8] the physical relevance of SOC can be found. These results were extended to ultracold atomic systems, where various synthetic SOC can be induced and managed by an external laser field. In particular, SOC has been experimentally realized[9] and theoretically investigated[10–14] for binary mixtures of Bose–Einstein condensate (BEC). The cold atomic systems confined in an optical lattice with SOC have also been widely studied. In the mean-field theory, as a many-body system, a BEC governed by Gross–Pitaevskii equation (GPE) provides an important basis for studying the corresponding physical properties. It is important to find exact solutions[15,16] and analytically perturbed solutions[17–20] to the GPE, which can be used to discuss many physical properties such as the macroscopic quantum (or semi-classical) chaos,[17–22] BEC stability,[23–28] the superfluid velocity and flow density,[29] and the generation of solitons.[26] At present, researchers have obtained some exact stationary solutions to the BEC systems in the quasi-one-dimensional (quasi-1D) Kronig–Penny potential,[30] the optical lattice potential,[24,25,31,32] and the two-dimensional optical lattice potential.[33,34] However, the GPE with nonlinear interactions is not easy to solve in general, since the exact solution can exist only under certain parameter conditions which are hard to find. The BEC system with SOC[35–37] is a more complex nonlinear system which poses a challenge to finding the exact solutions. The high frequency approximation method has been widely used because the corresponding approximate systems can be some exactly solvable quasistationary ones,[38–40] which is called the rotating-wave approximation (RWA) for historical reasons.[38] Recently, by means of the high frequency approximation, the SOC tunability has been investigated theoretically and experimentally.[41,42] The SOC strength can also be adjusted by the magnitude and direction of the Raman laser wave-vector. Here, we are motivated in generation of the exact solutions to such a quasistationary system with SOC, by using a high-frequency external field to adjust and reconstruct the system parameters. Consequently, it is possible to control the physical properties of the system and to make quantum transitions between the different analytical quantum states.
The spin–motion entangled state of a single atom has been defined well and was employed in many previous works,[43–45] which contains the well-known Schrödinger cat state[46]
with the spin-up and spin-down internal states
and
and the coherent motional states
. For a spin–orbit (SO) coupled BEC trapped in a superlattice, the similar spin–motion entangled states can be precisely defined by using two linearly independent motional states to replace the coherent motional states
.[47] In the full-quantum treatment, these spin–motion entangled states may be implemented to encode the qubits,[43–45,48] and the quantum information can be transported through the spin-dependent currents.[49,50] Therefore, we are also interested in how to generate the spin–motion entanglement and to manipulate the spin-dependent currents by tuning the SOC strength and other parameters.[41,42]
In this paper, we consider a SO-coupled BEC driven by a high-frequency field and confined in an optical lattice, through which we study the spatiotemporal Bloch states[51] and the associating physical properties. First, based on the RWA, such a driven system is approximated by a quasistationary one and the relating exact spatiotemporal Bloch states to the latter system are obtained. We analytically show that the SOC strength can be adjusted by the high-frequency field, such that the existence conditions of the exact particular solutions are met. Meanwhile, we find that the parameter regions on the SOC-strength vs. lattice-strength plane for the existence of the exact solution enlarge with the increase in Rabi coupling strength. Then, we show the periodic distribution of the atomic number density, which contains the discrete zero points, corresponding to the instability.[28] Furthermore, we analytically and numerically demonstrate several novel features related to the exact states: (i) the SOC leads to that the spatiotemporal Bloch states become the spin–motion entangled states; (ii) the SOC influences the population imbalance between two BEC components and may result in that all the atoms completely occupy one of the hyperfine states; and (iii) the SOC can be applied to manipulate the stable atomic flow, which is conducive to avoid instability and to control quantum transport of the BEC for different application purposes.[52,53]
2. Exact solutions to the quasi-stationary systemWe consider a BEC confined in a quasi-1D optical lattice potential
oriented in the longitudinal x direction, and the transverse dynamics of the condensate is assumed to be frozen to the respective ground states of the harmonic traps. In the case of high-frequency driving, the macroscopic quantum state of the BEC is
with the spin-up and spin-down internal states
and
and the macroscopic wave function
, which is governed by the nonlinear GP equations[54]
where the optical lattice potential
. All the terms in Eq. (
1) are the same as the corresponding terms in
[54], except for the driving term. The SOC and Rabi coupling have been realized experimentally,
[9] and the driving can be realized experimentally by using a periodic magnetic field gradient applied along the
x direction.
[55,56] However, the system of units that we have adopted is different. Here, the driving frequency
ω and energies (including the lattice depth
V0 and Rabi-coupling strength
) have been normalized in units of the recoil frequency
and recoil energy
, with
m and
k being the atomic mass and the standing wave vector. The spatial coordinate
x, time
t, and atomic number density
have also been normalized in units of
,
, and
k, respectively. The lattice tilt
ξ and SOC strength
are in units of
and
, respectively, and the constants
g and
g12 are the corresponding dimensionless intra- and inter-species interaction strengths which can be adjusted independently by the optical and magnetic Feshbach-resonance techniques
[57,58] in actual experiments. By the high-frequency limit and strong field we mean that the driving frequency and strength obey
and
. Due to the particle-number conservation, we let the average total number
of particles in each well be a constant with
Nj being the average atomic number per well of the
j-th component. To make use of the RWA, we seek the wave functions in the form
with
being treated as a slowly varying function of time.
[38–40] Inserting such wave functions into Eq. (
1) yields the
without driving, where we have replaced the fast-varying functions
and
by their average values 1/2 and 3/8, respectively, according to the RWA. By substituting the stationary solutions
with chemical potential
μ into Eq. (
2), we obtain the time-independent equations of
with
and
being the recombined SOC strengths and chemical potentials, which are tuned by the driving parameter ratio
.
It is worth noting that for γ=0 and Γ=0, equation (4) is reduced to the basic equations in [31], and the corresponding exact solutions have been constructed. The presence of SOC and Rabi coupling makes the system more complicated and more difficult to solve. However, according to the famous Floquet theorem, there exists an exact Bloch solution to a spatially periodic system.[51] In fact, if we adjust the driving parameter ratio
and SOC strength γ to satisfy the following conditions:
then the exact particular solutions to the spatially periodic Eq. (
4) can be obtained by inserting the trial Bloch solutions
with real undetermined constants
aj and
bj into Eq. (
4), and by using the normalization condition. Firstly, the normalization and equation (
6) give the average number of particles per well in Eq. (
5) as
[24,31]
Then, by combining Eqs. (
6) and (
5) with Eq. (
4), we derive three pairs of equations
for
j=1,2. Solving these equations yields the four squares of the undetermined constants. Hereafter, we take the positive constants in the forms
for
j=1,2. Making use of Eqs. (
5) and (
7), we can easily prove that equation (
6) is just a pair of exact solutions to Eq. (
4). For the fixed parameters
and
γ, the first condition of Eq. (
5) can be satisfied by the undetermined chemical potential and the second condition is adjusted by the driving parameter ratio
. The latter condition means that the system (
4) with nonzero SOC and/or Rabi strengths has exact solution (6), if the ratio
is equal to
. The result is similar to the well-known result that when the driving parameter ratio coincides with a zero of the zero-order Bessel function, a high-frequency driven two-state model arrives at a particular localized state; namely, the coherent destruction of tunneling.
[39,40]Rewriting Eq. (6) in its exponential form
and applying Eq. (
7) to Eq. (
8) leads to the atomic number densities
Clearly, the exact solutions in Eq. (
8) have the similar forms with those in
[24] and
[31] and in
[24–26] and
[28] with zero elliptic modulus for the spatially periodic nonlinear systems without SOC. However, to construct such a solution to system (4) with SOC, the high-frequency driving with the parameters obeying Eq. (
5) must be applied. Given Eq. (
8), we can calculate the average atomic numbers per well as
[24,31]
To keep the positive semidefinite property of the atomic number densities,
for any
x value, from Eq. (
9) we have the necessary and sufficient conditions
[31]
The exact solutions Eqs. (
6) and (
7) to the quasistationary Eq. (
4) are valid if and only if the depth
V0 of the lattice potential obeys the constraint relation Eq. (
11). According to Eq. (
10), we know that the quantity in the square brackets is greater than zero due to
for any
j, which gives a limit to the maximal value of
γ for a set of fixed other parameters. Therefore, for
or
, equation (
11) gives
or
, respectively. Without loss of generality, we can suppose
. Then, based on Eq. (
11), taking the parameters
,
, and
, we plot the boundary curves
for the parameter regions of the exact solution under the RWA as a function of
γ for Γ=0.6, 0.8, and 1.4, respectively, as shown in Fig.
1(a). Any curve in Fig.
1(a) with a fixed Γ value gives a boundary corresponding to the parameter regions for the existence and nonexistence of the exact solution (
6). Noticing the sign “
” in Eq. (
11), the parameter regions of exact solution (
6) are
A, (
A+
B), and (
A+
B+
C) for Γ=1.4, 0.8, and 0.6, respectively, which means that the stronger Rabi coupling increases the size of the considered parameter region. In Fig.
1(b) based on
, we also show the existence regions of the exact solution for
and different nonlinear interaction strengths
g=0.1 and
. Similar to Fig.
1(a), the parameter regions for the existence of the exact solution still are
A, (
A+
B), and (
A+
B+
C) for Γ=1.4, 0.8, and 0.6, respectively. The existence regions enlarge with the increase in Rabi coupling strength Γ. For any set of the considered parameters, in the two regions labeled
D, the exact solution (
6) cannot exist. All the curves in Fig.
1 monotonically decrease such that for any fixed Γ value, the maximal lattice strength associated with the exact solution has to decrease with increasing SOC strength. Comparing Fig.
1(a) with Fig.
1(b), we can see that increasing the average atomic number
Nt per well can enlarge the region area existing exact solution for a set of fixed parameters.
From Eq. (9), we show the spatial distributions of the atomic number density
and the lattice potential V(x) for the parameters g=0.6, g12=0.2, γ=0.1, Nt=5, Γ=0.8, and
in Fig. 2(a), V0=1.57839=Vc in Fig. 2(b). When
is set, although the atomic number densities have different spatial distributions in Fig. 2(a), their maxima always align to the center points of the potential wells, and the atomic number density is greater than zero at any spatial position. This means that more atoms are distributed around the centers of the potential wells, corresponding to the higher stability of the BEC system compared to that of the case where the density peaks align to potential barrier sites and some zero density points exist. In the case of V0=Vc, Figure 2(b) shows that
maybe zero at the peak sites of the potential.
3. Physics related to the spatiotemporal Bloch statesBy applying the existence conditions of the considered exact solutions to Eq. (4) and the corresponding parameter regions, we have established the exact particular solutions (6) and (7), and illustrated the associated atomic number densities. In this section, we will demonstrate that the obtained states are just the spatiotemporal Bloch states and reveal some physical properties of the BEC system in such states.
We first demonstrate that the obtained states are just the spatiotemporal Bloch states. From Eq. (6), we rewrite the exact solutions as
for j=1,2. Clearly, these are two Bloch solutions to Eq. (4) with the Bloch wave vector −1 and Bloch state functions
which possess the same period with the potential V(x). Then from the macroscopic quantum state
and Eq. (3), we have the space–time dependent state
in the coordinate representation. Obviously, equation (
12) is a Floquet solution to Eq. (
1) with the Floquet quasienergy −
μ and Floquet state
of the same period with the driving force. At any fixed time, equation (
12) is an usual Bloch state. Because the considered system is a time–space periodic system, we call Eq. (
12) the spatiotemporal Bloch state.
[51] The motional state functions
contain the constants
aj and
bj with some arbitraries, so varying these constants can produce different spatiotemporal Bloch states.
We then show that SOC leads to generation of spin–motion entanglement. Quantum entanglement is a universal but very special kind of quantum state in the multi-particle system. In previous work, for a SO-coupled BEC trapped in a superlattice, Kong et al. have studied the spatially chaoticity-dependent spin–motion entanglement.[47] Here we study the effects of SOC on the chaoticity-independent spin–motion entanglement between internal hyperfine (pseudospin) states and external motional (orbit) states. From Eq. (7), we observe that the constants in exact solution (6) to the quasistationary Eq. (4) obey
and
only if γ=0. Consequently, the two motional states in Eq. (12) become the same,
, such that the quantum state
can be separated as the direct product between the spin part and motional part and become an unentangled state. The nonzero SOC strength makes the different and linearly independent motional states in Eq. (12) and leads to the spin–motion entanglement. The orbital part of the spin–motion entangled state can be used to manipulate the qubits for quantum information processing with spins.
We now investigate the population imbalance of the two-component BEC. Given Eq. (10), the difference between the average atomic numbers reads
, which is called the population imbalance of the two-component BEC. The zero SOC strength γ=0 results in balance with the same average number per well
and causes the entanglement loss. In the presence of SOC. for
, we have
so equation (10) implies that only N2 may reach zero with a large enough SOC strength. In contrast, the case
means that only N1 may reach zero. In Fig. 3, based on Eq. (10), we take Rabi coupling strength Γ=0.1 and average atomic number Nt=5 to plot the population imbalance
versus SOC strength γ for g=0.6, g12=0.2 (solid curve), and g=0.2, g12=0.6 (dotted curve). Clearly, the population imbalance monotonically increases or decreases with the increase in SOC strength for the case
or the case
, respectively. When the SOC strength arrives at
, we obtain
or
associated with N2=0 or N1=0. Thus, all the atoms are completely populated on one of the hyperfine states, as shown in Fig. 3. The results reveal the population transfer between the two BEC components by adjusting the SOC strength.
Finally, we discuss the establish of the stable superfluid density. Applying Eqs. (7) and (9) to the definitions of the superfluid velocity and density,
,
, after adopting our dimensionless parameters we obtain the superfluid densities and flow velocities
The constant superfluid densities describe two uniform atom currents for the two BEC components, and can be conveniently manipulated by tuning any one of the experimental parameters
Nt,
g12,
g,
V0 and recombined
γ. The signs of
Jj determine the directions of the atomic currents, which can be initially set
[43–45] when the two species of atoms are loaded in the optical lattices and provide the persistent incident currents.
[52] In Fig.
4, we plot the flow density
Jj versus the SOC strength for the parameters Γ=0.1,
Nt=5,
V0=1,
g=0.6, and
g12=0.4. When
Jj take the positive sign of Eq. (
13), the two arrays of atomic quasi-clusters flow toward the right; when
Jj take the negative sign, the atomic quasi-clusters of the two arrays flow to the left. However, when
J1 and
J2 have different signs, the two arrays of atomic quasi-clusters flow with opposite directions. As shown in Fig.
4, the flow density
J1+ increases with increasing SOC strength
γ, while
J2+ decreases with the increase in SOC strength. Conversely,
have different increasing trends with
for
j=1,2. Interestingly, when we change the SOC strength to
γ=0.702682, only the superfluid density
J1 exists and the second component vanishes,
. From the known research results,
[59] when the atomic number density
does not contain a zero, the BEC system may be stable. In contrast, when
has a zero point, the system may be unstable for the given parameters.
[28,32] This stability is revealed from a different viewing angle in Eq. (
13) where the zero density
is associated with the infinite superfluid velocity resulting in the instability. Therefore, we can establish the stable superfluid currents by avoiding the critical parameter point
and adopting the parameters in regions obeying
.
In addition, under RWA, we can use the exact solution (6) and the corresponding parameter conditions to come up with some other interesting physical phenomena and properties.
4. Discussion and conclusionIn summary, we have researched the spatiotemporal Bloch states and the relating physical properties for a SO-coupled BEC trapped in an optical lattice and driven by a high-frequency field. Based on the widely used RWA method, we obtain the exact solution (6) to the quasistationary equation (4) and the analytical spatiotemporal Bloch state (12). We establish a different method for tuning SOC via external field such that the existence conditions of the exact particular solutions are fitted. From the existence conditions, we find that the parameter region on the plane of lattice-strength vs. SOC-strength can be adjusted by the Rabi coupling strength. As the Rabi coupling strength increases, the regions of the parameters existing exact solutions increase. Then, we show the periodic distributions of the atomic number density which are associated with the stability and instability of the system. When
is set, more atoms are distributed around the centers of the potential wells without zero point of the atomic number density, corresponding to higher stability of the BEC system. When
is selected, the atomic number density contains zero points corresponding to the instability. In particular, several new physical features related to the exact states are revealed, as follows. (i) The SOC leads to generation of the spin–motion entanglement in the obtained spatiotemporal Bloch states. In contrast, zero SOC strength results in disappearance of the spin–motion entanglement. The orbital part of the spin–motion entangled state can be used to manipulate the spin qubits for quantum information processing. (ii) Tuning SOC changes the population imbalance between two BEC components and may cause all of the atoms to completely concentrate on one of the hyperfine states. (iii) The SOC can be applied to manipulate the stable atomic flow. For a set of fixed parameters, one component of the superfluid density decreases with the increase in SOC strength, and another component consequently increases.
BEC is a well-controlled collection of atoms that can be used to study new aspects of quantum optics, many-body physics, and superfluidity. It is highly desirable to transport BEC from a stable source to a designated place.[52] The results of this paper provide a method that avoids instability of the system and controls BEC motion via the modulated optical lattice and the tunable SOC, which is particularly conducive to control quantum transport of BEC for different application purposes.[52,53]
