Dressed state dynamics of two-component Bose-Einstein Condensates in state-dependent potentials

Dressed potentials realized by coupling state-dependent bare potentials with external fields have important applications in trapping and manipulating atoms. Here, we study the dynamics of dressed states for coupled two-component Bose-Einstein condensates (BECs) in state-dependent potentials. Through both analytical and numerical methods, we find that the dressed state dynamics sensitively depend on both the inter-component coupling strength and the initial state. If the inter-component coupling is strong enough and the initial wave packet is located at the potential minimum, the dressed states can be decoupled and the Josephson oscillations and macroscopic quantum self-trapping appear. However, if the initial wave packet is located far away from the potential minimum, the wave packet will acquire a large kinetic energy and Landau-Zener transitiozs between the dressed states occur at the avoided-crossing point. Further, we give the validity ranges and conditions for the formation of adiabatic potentials, where the influences of Landau-Zener transitions can be ignored. Our results give an insight on how the inter-component coupling affects the dressed state dynamics and how to realize adiabatic potentials with BECs in state-dependent potentials.


Results
Model. The system we study is coupled two-component BECs trapped in state-dependent potentials. The state-dependent potentials can be feasibly induced by a spatially varying microwave 22 or a magnetic field gradient 27 . The condensed atoms in two different hyperfine levels experience different harmonic potentials with a spatial separation between the two potential minima. The two hyperfine states are split by an applied magnetic field, and are coupled by a driving field with detuning δ = ω − ω 0 (ω 0 is the transition frequency between the two hyperfine states and ω is the oscillation frequency of the induced field) and coupling strength Ω. Under the rotating wave approximation and treating the atom-atom interaction within the mean-field (MF) theory, the system can be well described by the coupled Gross-Pitaevskii equations (CGPEs): (1) = ω + +ω + ω are the state-dependent potentials with 2x 0 being the spatial separation between the two potential minima.  m 4 a / (i, j 1, 2) ij 2 ij β = π = is the interaction constant with a ij being the s-wave scattering length between i-th and j-th components.
We consider an elongated system of condensed atoms. That is, the atoms are confined by a strong transverse confinement ω y , ω z ≫ ω x . Integrating the transverse coordinates, the elongated system can be described by the one-dimensional (1D) CGPEs: (2) i t (x, t)  , one can obtain the corresponding ranges of the dimensionless parameters: 0 ≤ Ω ≤ 5.73, 0 ≤ x 0 ≤ 4.56 and g 11 = g 22 = g 12 = g = 0.0471. In our calculation, we assume that the intra-component and inter-component interaction strengths are equal. Since the interaction strengths can be tuned by Feshbach resonances in experiments 29 , we will vary g to study the nonlinear effects. In addition, we only consider the on-resonance case, where δ = 0. Dressed potentials. Transforming the bare states to a new basis that diagonalizes the bare potentials V 1 and V 2 and the coupling Ω at each spatial point x, one can obtain a set of spatially dependent eigenstates which are the dressed states. Correspondingly, the spatially varying eigen-energies constitute the so-called dressed potentials. The diagonalization can be done by a unitary matrix U, where After the diagonalization, the dressed potentials can be obtained as For a fixed x 0 , the shape of the above dressed potentials for different Ω are shown in Fig. 1(a). It is obvious that, the gap between the upper and lower dressed potentials becomes larger when the coupling strength Ω increases.
When Ω is small, the lower dressed potential is in the shape of double-well while the upper dressed potential remains a single-well structure. However, as Ω grows larger, the barrier height of the lower double-well decreases. When the coupling strength Ω exceeds the critical value x 0 2 , the lower dressed potential will change from a double-well structure to a single-well structure. It is clear that, to form an adiabatic double-well potential, there is a tradeoff between the gap of the two dressed potentials and the barrier height of the double-well when x 0 is fixed.
After the transformation (3), the CGPEs (2) become  where ∂ x is the partial derivative with respect to x and Ψ ± are the condensate wave functions of the two dressed states. In Eq. (5), one can see that the two dressed states are coupled via the term V c which comes from the kinetic term − ∂ ∂ 1 2 x 2 2 in the bare state basis. The coupling between the dressed states is now determined by the coefficients C 1 , C 2 and ∂ x Ψ − (∂ x Ψ + ). According to Eq. (6), the absolute values of coefficients C 1 and C 2 decrease as Ω increases near the avoided-crossing. The terms x ∂ Ψ − and ∂ Ψ + x are proportional to the velocity of the dressed states. Provided that the coupling strength Ω is strong enough and the velocity of the states is small enough, the coupling term V c becomes sufficiently small to be neglected. In this case, atoms can be trapped in lower (upper) dressed potential and they can hardly be transferred to the upper (lower) dressed states via LZ transitions. Then the dressed potentials can be regarded as the so-called adiabatic potentials and the system can be described by the following decoupled equations: where the dynamics of the atoms in one of the dressed states are solely dependent on the corresponding dressed potential. However, if the coupling between the two dressed states V c is not neglectable, the dynamics of the upper (lower) dressed states will be affected by the lower (upper) dressed state, and one can only use the CGPEs (5) to study the system's dynamics.
In the following, we treat the system as multiple coupled modes of macroscopic matter waves. If the barrier of the lower dressed potential is sufficiently high, we can expand the condensate wave function of the lower dressed state and the upper dressed state as: Where φ 1 (φ 2 ) is the localized state for the right (left) well of the lower dressed potential and φ 3 is the localized state for the well of the upper dressed potential. ψ j (j = 1,2,3) is the corresponding time-dependent complex amplitude. Substituting the variational wave function (8) into the decoupled equations (7), one can get the nonlinear three-mode dynamical equations: and the intra-mode and inter-mode exchange collision interaction strength In the exchange collision denoted by ↔ ij kl, an atom from the k-th mode collides with an atom from the l-th mode and then these two atoms are transferred into an atom in the i-th mode and an atom in the j-th mode. For sufficiently deep wells, these three localized states are well localized at the corresponding well-centers, so that the high-order overlaps between the localized states are very small, namely U U , (i, j, k, l 1, 2, 3 and at least one of {j, k, l} not equal i) ij kl ii ii = .
↔ ↔  Therefore the inter-mode exchange collision terms ⁎ U ij kl j k l ψ ψ ψ ↔ in the three-mode dynamical equations (9) can be ignored, and Eq. (9) can be further simplified to be: in which the intra-mode exchange collision interaction = ↔ U (i 1, 2, 3) ii ii has been expressed as U ii for simplicity.
In the following, we will consider both the Bose-Josephson (BJ) dynamics and LZ dynamics and study the impact of the coupling between dressed states on the dressed state dynamics in our system. Bose-Josephson dynamics. BJ effect is a peculiar tunneling phenomenon in the physical system of two BECs linked by Josephson coupling [30][31][32][33] . In our coupled two-component BEC system, when the inter-component coupling has modest strength, the lower dressed potential has a double-well structure. In analogy with the conventional BJ junctions, BJ dynamics will occur in the dressed state picture. When considering a weak interacting BEC (g < g c ), the Josephson oscillations in the lower dressed state are observed. When taking into account a BEC with stronger atom-atom interaction (g > g c ), the MQST in the lower dressed state will take place. Here, g c is the critical interaction strength between the normal phase (i.e. the Josephson oscillations) and the MQST phase of the lower dressed state. Remarkably, different from BJ dynamics in an external BJ junction or an internal BJ junction 30,33 , in our system the BEC bulk oscillates not only between two external modes but also two internal states. In the following, we will discuss the Josephson oscillations and MQST dynamics for the dressed states and determine the critical value g c by both analytic and numerical methods.
At first, for non-interacting BEC (g = 0), we simulate the dressed state dynamics with different coupling strength Ω by solving the CGPEs (5). We set the initial state as the located state of the right well of the lower dressed potential: 1 see Fig. 1(b) (I). The wave packet of the lower dressed state will tunnel from the right well to the left well and vice versa due to the BJ effect. With fixed x 0 , the barrier between two wells become lower as Ω increases, and the period of the Josephson oscillations will decrease quickly, from T ≈ 21000 (for Ω = 1) to T ≈ 180 (for Ω = 5), see Fig. 2(a-c). The average population imbalance between the two wells over a long time-evolution tends to be zero. When the gap between the upper and lower dressed potentials is large enough, the coupling between the upper and lower dressed states will be so weak that can be neglected. Therefore, the BJ dynamics for the lower dressed state can be captured by the decoupled equations (7). In Fig. 2(d-f), we provide the simulation results based on the decoupled equations (7) for Ω = 1, 3, 5. Comparing with the numerical results based on the CGPEs (5) [ Fig. 2(a-c)], the decoupled equations (7) are in accordance when Ω is large (e.g., Ω = 3, 5). While for small Ω (e.g., Ω = 1), the time-evolution of the density profile for lower dressed state obtained by the decoupled equations (7) is quite different from the original one calculated by the CGPEs (5). The main difference reflects on the period of the Josephson oscillations T. The decoupled equations (7) predict smaller period when Ω is small.
To investigate the validity of the decoupling approximation, we compare the period of the Josephson oscillations obtained from the CGPEs (5) and the decoupled equations (7) under different Ω, see Fig. 3. It is shown that, the results of the decoupled equations (7) accord well with the original one of the CGPEs (5) when Ω is large. As Ω decreases, the results of the decoupled equations (7) gradually deviate from the original one, which is due to the increasing coupling between the two dressed states. Specifically, we assume that if the difference between the periods obtained from the CGPEs (5) and those obtained from the decoupled equations (7) is below a threshold 5%, the decoupling approximation is judged to be valid. According to the criteria, we find the decoupling approximation is valid when 3.3 ≤ Ω ≤ 5 for fixed x 0 = 3.
In addition, the period of Josephson oscillation T can also be obtained by solving the three-mode equations (14) in linear regime, namely T = π/J. In Fig. 3, we also plot the results and it agrees perfectly with the results of the decoupled equations (7) when 0 ≤ Ω ≤ 5.
Interestingly, when we consider the effects of atom-atom interaction, the nonlinearity will gradually affect the Josephson oscillations. Still, we locate the initial wave packet at the center of the right well of the lower dressed potential: To see the impact of atom-atom interaction, as an example, we plot the time evolution of density profiles for the dressed states with x 0 = 3 and Ω = 3 under gN = 0, 0.018 and 0.02, see Fig. 4(a-c). For gN = 0.018, MQST occurs in the lower dressed state with the average population imbalance between the two wells becoming nonzero. The lower dressed state prefers to stay in the initially populated well as the interaction plays a role. When gN increases, MQST begins to dominate and the oscillations between the two wells are gradually suppressed.
The coupling between the dressed states will influence the critical condition from Josephson oscillations to MQST. Here, we also use the decoupled equations (7) to simulate the BJ dynamics for the lower dressed state in the nonlinear regime. The results of decoupled equations (7) are slightly different from the original ones. For example, when gN = 0.018, MQST appears in the simulation results of the CGPEs (5) while the corresponding results of the decoupled equations (7) still show Josephson oscillations. For decoupled equations (7), the MQST occurs when gN ≈ 0.02, which is larger than the original critical point.
The critical point of gN for the transition from Josephson oscillations to MQST obtained by the CGPEs (5) and the decoupled equations (7) for different Ω are shown in Fig. 5. Also, we assume that if the difference between the results obtained from the CGPEs (5) and those obtained from the decoupled equations (7) is below a threshold 5%, the decoupling approximation is regarded as valid. According to the criteria, it is shown that the decoupled equations (7) can give a relatively correct critical interaction strength when 3.4 ≤ Ω ≤ 5, see Fig. 5. However, when Ω becomes small enough, the results deviate from the results of the CGPEs (5).
The MQST can also be explained by the solution of the stationary states of the three-mode equations (14). The complex amplitudes in the three-mode equations (14) can be expressed as ψ = θ (t) n exp(i ) j j j , where n j is the atom number and θ j is the phase. For the two dressed down modes, we introduce two new variables, the relative population imbalance z = (n 1 − n 2 )/N and the phase difference θ = θ 1 − θ 2 . By inserting them into (14), (θ, z) obey: The corresponding general solutions are oscillations enclosing these fixed points. Different from the general solutions in the linear regime, they have nonzero average population imbalances, which indicate the MQST. In Fig. 5, it is shown that the critical interaction strength of normal-to-MQST transition calculated analytically with Eq. (18) is well consistent with the numerical results of decoupled equations (7) when Ω is not too large. However, when Ω is large enough, the wells in the lower dressed potential are too shallow to ensure the accuracy of the three-mode approximation. We assume that if the difference between the critical interaction strength obtained by the decoupled equations (7) and the analytical results (18) is below a threshold 5%, the three-mode approximation is regarded as valid. For fixed potential separation x 0 = 3, we find the reasonable range of the three-mode approximation is 0 ≤ Ω ≤ 4.5, see Fig. 5. It is obvious that, the decoupled equations (7) are valid when the coupling strength is large enough. Based on our analytic and numerical results, for both non-interacting and interacting BEC, only when the gap between the upper and lower dressed potentials is sufficiently large, the lower dressed double-well potential can be treated as adiabatic potential and the BJ dynamics can be characterized by the decoupled equations (7). In addition, if the wells in the dressed potential is not too shallow (i.e. Ω is not too large), the three-mode approximation is suitable to describe the system. Landau-Zener dynamics. In our coupled two-component BEC system, even when the coupling strength Ω is large enough, the atoms may still transfer from an adiabatic potential to the other so that the adiabaticity is not guaranteed. In general, this can happen when an atom accumulates large enough kinetic energy so that it cannot follow the adiabatic potential. Therefore, the dynamics of the dressed states depend not only on the structure of the dressed potentials but also on the initial state. Since the boundaries of the potentials can be sufficiently high, the initial wave packet that far away from the minimum of the lower dressed potential would acquire a large kinetic energy. The kinetic energy enables the transition between the lower and upper dressed states at the crossing point x = 0, which is the LZ transitions. It is necessary to determine the conditions in which the LZ transition is negligible since the technique of adiabatic potential is always used in the limits of adiabaticity.
To observe the LZ dynamics for the dressed states, one can set the initial state as a wave packet in the lower dressed state centering at a position away from the minimum of the lower dressed potential with a distance x d : 1 d See Fig. 1(b) (II). The time evolution of the density profile for the dressed states given by the CGPEs (5) is shown in Fig. 6(a). One can find that the dynamics are dominated by the harmonic oscillations and the LZ transitions of the dressed states. During the evolution, when the bulk of BECs reaches the avoided-crossing at x = 0, the LZ transitions occur and part of the atoms in the lower (upper) dressed state are transferred to the upper (lower) dressed state, see Fig. 6(b).
The non-adiabatic transition probability of the LZ transition depends on the two-component coupling strength Ω and the wave packet's initial position. The non-adiabatic transition probability P can be obtained numerically from the simulation results of the CGPEs (5). The results of non-adiabatic transition probability P for different coupling strength Ω and initial wave packet position are shown in Fig. 6(c). One can find that the non-adiabatic transition probability P decreases as the coupling strength Ω increases, which is due to the decreasing of the coupling coefficients C 1 and C 2 . On the other hand, if the initial wave packet approaches the potential minima, which corresponds to a lower potential energy, the wave packet will get a smaller velocity near the avoided-crossing and the terms ∂ Ψ − x , ∂ Ψ + x in the dressed state coupling terms (6) will decrease, resulting in the decrease of P. One can also derive the non-adiabatic transition probability P by the LZ formula. The transition only occurs in the small region near the avoided-crossing, so that we may treat the velocity of the wave packet v as constant during the LZ transition. The initial wave packet can be decomposed to different momentum components by Fourier transforms: By treating the BEC atoms classically, the magnitude of the velocity v at the avoided-crossing for each momentum components can be determined by: in which x m is the position of the right minima of the lower dressed potential. According to the LZ formula 34 , the non-adiabatic transition probability for each momentum components is Therefore, the total non-adiabatic transition probability can be obtained by integrating the momentum space:

LZ
The non-adiabatic transition probabilities calculated from the LZ formula with different initial location are shown by solid lines in Fig. 6(c), which agree well with the numerical results obtained from the CGPEs (5) [shown by the markers in Fig. 6 (c)].
Therefore, to guarantee the adiabaticity of the dressed potential and study solely the dynamics of lower dressed state, not only the coupling strength Ω should be large enough, but also the initial wave packet should be located near the minima of the double-well. The valid range of adiabaticity can be obtained from both the numeric results given by the CGPEs (5) and the analytical results given by the LZ formula (23). Specifically, we assume that if the non-adiabatic transition probability P is below a threshold 1%, the LZ transition can be neglected during the dynamics. Therefore, to avoid the LZ transition in practice, the distance x d between the center of initial wave packet and the potential minima should be less than 3.0, 4.5, 6.3 when Ω = 3, 4 and 5 respectively, see Fig. 6(c). That is, as the coupling strength increases, the LZ transition is suppressed and the valid distance x d increases. These results give an instruction for experimental realization of adiabatic potentials to avoid the influences of LZ transitions.

Summary and Discussion
In summary, we have investigated BJ dynamics and LZ dynamics for the dressed states in coupled two-component BECs in a state-dependent potential. In our system, a double-well structure can be formed in the lower dressed potential. The dressed state dynamics sensitively depend on both the inter-component coupling strength and the choice of initial state. We discuss how the BJ dynamics and LZ dynamics are affected by the inter-component coupling strength and the initial state.
If the initial state is prepared at the minimum of the lower dressed potential, one can observe BJ dynamics in the dressed state. When the inter-component coupling strength is large enough, the dressed states can be treated as decoupled. As the nonlinear interaction comes into play, Josephson oscillations would gradually transit to MQST, and the critical condition for the transition can be well described by the three-mode approximation.
If the initial state is far away from the minimum of the lower dressed potential, there will appear several LZ transitions at the avoided-crossing point even though the gap is large. The LZ transitions between the dressed states well obey the LZ formula. To ensure the adiabaticity of the dressed potential, not only the inter-component coupling strength should be strong enough but also the initial wave packet should not locate too far away from the potential minima.