Abstract
Dressed potentials realized by coupling statedependent bare potentials with external fields have important applications in trapping and manipulating atoms. Here, we study the dynamics of dressed states for coupled twocomponent BoseEinstein condensates (BECs) in statedependent potentials. Through both analytical and numerical methods, we find that the dressed state dynamics sensitively depend on both the intercomponent coupling strength and the initial state. If the intercomponent 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 selftrapping 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 LandauZener transitiozs between the dressed states occur at the avoidedcrossing point. Further, we give the validity ranges and conditions for the formation of adiabatic potentials, where the influences of LandauZener transitions can be ignored. Our results give an insight on how the intercomponent coupling affects the dressed state dynamics and how to realize adiabatic potentials with BECs in statedependent potentials.
Introduction
Laser, radiofrequency (rf) and static external fields are promising tools for trapping and manipulating cold atoms. The control of ultracold atomic systems provides great opportunities for developing quantum technology such as quantum metrology^{1,2,3,4}, quantum engineering^{5,6} and quantum simulation^{7}. Among these investigations, fieldinduced adiabatic dressed potentials created by a versatile combination of rf and static magnetic fields played an important role^{8,9}. Dressed potentials arise when two or more internal spin states that experience different potentials are coupled by an external field. The motions of the trapped atoms are no longer determined by the individual bare potentials but become dominated by the dressed potentials dependent on the bare potentials and the intercomponent couplings. If there is no any LandauZener (LZ) transition induced by the kinetic energy of the atoms, the dressed potentials are referred as adiabatic potentials. Adiabatic potentials can avoid the limitation of magnetic traps and can be used for realizing traps with different geometries, including doublewell traps^{10}, ring traps^{11,12}, lattice potentials^{13,14} and so on. Applications of adiabatic potentials include exploring novel properties of superfluidity^{15,16} and serving as beam splitters in matterwave interferometry^{10}. In atom chips^{17,18}, adiabatic potentials can be tailored more freely for trapping and manipulating atoms on micrometer scale, which can be useful for quantum sensing^{19,20}.
Recently, many theoretical and experimental works focus on how to realize different geometries of adiabatic potentials^{21}. To tailor a desired adiabatic potential, the strength of the induced rf field (i.e., the intercomponent coupling) needs to be carefully controlled according to the geometry of bare potentials^{9}. One of the important tools in atom chips is the combined coherent manipulation of the internal and motional states via statedependent potentials. By dressing different atomic spin states with a spatially varying field, statedependent potentials have been realized with atomic BECs^{22}. The statedependent potentials enable one to control the interstate interactions by tuning the overlap of two internal states, which are a crucial ingredient for entanglement generation^{23}, atomchip quantum gate^{24,25} and spin squeezing^{26}. For atomic BECs in statedependent potentials, the specific adiabatic potential in the doublewell shape is an important element for quantum engineering. If the strength of the rf coupling is too strong, the doublewell structure would be vanished in the dressed potential. However, if the strength of the rf coupling is too weak, the gap between the dressed potentials would be small and nonadiabatic transition between the dressed potentials becomes significant. It is essential to know how the interplay between intercomponent coupling and statedependent potentials influences the formation of adiabatic potentials. Therefore, two natural questions arise: (i) what is the validity ranges for realizing adiabatic potentials with atomic BECs in statedependent potentials? and (ii) how the dressed state dynamics behave under different intercomponent coupling?
In this article, we consider twocomponent BECs in statedependent harmonic potentials, which are coupled by an external field. The separation of the minima of the two different harmonic traps can be tuned by using additional spatially varying microwave or magnetic field in experiments^{22,27}. By applying a coupling field, the two independent singlewell potentials become an upper singlewell potential and a lower doublewell potential in the dressed state picture. We will demonstrate that, the dressed state dynamics sensitively depend on the induced coupling strength and the selection of the initial state.
When the coupling strength is sufficiently strong, we observe Josephson oscillations of the dressed states. As the coupling strength becomes weaker, Josephson oscillations of the dressed states gradually change. We show that, the periods and the critical condition of macroscopic quantum selftrapping (MQST) are dramatically influenced by the coupling strength. In addition, even for a strong coupling strength, if the initial wave packet located at a position far away from the minimum, the dressed state will acquire a large kinetic energy, which leads to the occurrence of LZ transitions. We find that the transition rate will become smaller as the initial location getting closer to the potential minimum, which can be well explained by the LZ formula. Under certain coupling strengths, we give the corresponding validity ranges and conditions for the formation of adiabatic potential where the influences of LZ transitions can be avoided.
Results
Model
The system we study is coupled twocomponent BECs trapped in statedependent potentials. The statedependent 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 atomatom interaction within the meanfield (MF) theory, the system can be well described by the coupled GrossPitaevskii equations (CGPEs):
Here, Ψ_{1,2}(r, t) are the condensate wave functions of the two components and m is the mass of the atom. \({{\rm{V}}}_{1}({\bf{r}})=\frac{1}{2}{\rm{m}}[{{\rm{\omega }}}_{{\rm{x}}}^{2}{({\rm{x}}{{\rm{x}}}_{0})}^{2}+{{\rm{\omega }}}_{{\rm{y}}}^{2}{{\rm{y}}}^{2}+{{\rm{\omega }}}_{{\rm{z}}}^{2}{{\rm{z}}}^{2}]\) and \({{\rm{V}}}_{2}({\bf{r}})=\frac{1}{2}{\rm{m}}[{{\rm{\omega }}}_{{\rm{x}}}^{2}{({\rm{x}}+{{\rm{x}}}_{0})}^{2}+{{\rm{\omega }}}_{{\rm{y}}}^{2}{{\rm{y}}}^{2}+{{\rm{\omega }}}_{{\rm{z}}}^{2}{{\rm{z}}}^{2}]\) are the statedependent potentials with 2x_{0} being the spatial separation between the two potential minima. \({{\rm{\beta }}}_{{\rm{ij}}}=4{\rm{\pi }}{\hslash }^{2}{{\rm{a}}}_{{\rm{ij}}}/m\,({\rm{i}},{\rm{j}}=1,2)\) is the interaction constant with a_{ij} being the swave scattering length between ith and jth 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 onedimensional (1D) CGPEs:
in which \({{\rm{V}}}_{1}=\frac{1}{2}{({\rm{x}}{{\rm{x}}}_{0})}^{2},{{\rm{V}}}_{2}=\frac{1}{2}{({\rm{x}}+{{\rm{x}}}_{0})}^{2}\) and \({{\rm{g}}}_{{\rm{ij}}}=2{{\rm{a}}}_{{\rm{ij}}}{({{\rm{\omega }}}_{{\rm{y}}}{{\rm{\omega }}}_{{\rm{z}}})}^{1/2}{{\rm{\omega }}}_{{\rm{x}}}^{1}{{\rm{a}}}_{{\rm{H}}}^{1}\,({\rm{i}},{\rm{j}}=1,2)\). The condensate wave functions are normalized to give the populations \({{\rm{N}}}_{{\rm{i}}}({\rm{t}})=\int {\rm{dx}}{{{\rm{\Psi }}}_{{\rm{i}}}({\rm{x}},{\rm{t}})}^{2}({\rm{i}}=1,2)\), where the total number N = N_{1} + N_{2} is constant. Here, time is in units of \({{\rm{t}}}_{{\rm{s}}}={{\rm{\omega }}}_{{\rm{x}}}^{1}\), the spatial distance is in units of \({{\rm{x}}}_{{\rm{s}}}={(\hslash /{m{\rm{\omega }}}_{{\rm{x}}})}^{1/2}\), the wave function amplitude is in units of \({{\rm{x}}}_{{\rm{s}}}^{3/2}\,\)and the energy is in units of \({{\rm{E}}}_{{\rm{s}}}=\hslash {{\rm{\omega }}}_{{\rm{x}}}\). To relate the dimensionless parameters with some typical experiments, we take ^{87}Rb atoms for an example. The mass of the atom m = 1.44 × 10^{−25} kg, the scattering lengths a_{11} = a_{12} = a_{22} = 5.3 nm, and the trapping frequencies ω_{x} = 2π × 109 Hz, ω_{y} = ω_{z} = 2π × 500 Hz. For hyperfine states \({\rm{F}}=1,{{\rm{m}}}_{{\rm{F}}}=1\rangle \) and \({\rm{F}}=2,{{\rm{m}}}_{{\rm{F}}}=1\rangle \), the statedependent potentials split along x with the distance in the range of 0–9.4 μm^{22} and the Raman coupling strength varies between zero and 2π × 625 Hz in experiments^{28}. Setting \(\hslash ={\rm{m}}={{\rm{\omega }}}_{{\rm{x}}}=1\), 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 intracomponent and intercomponent 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 onresonance 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 eigenenergies constitute the socalled 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 doublewell while the upper dressed potential remains a singlewell structure. However, as Ω grows larger, the barrier height of the lower doublewell decreases. When the coupling strength Ω exceeds the critical value \({{\rm{x}}}_{0}^{2}\), the lower dressed potential will change from a doublewell structure to a singlewell structure. It is clear that, to form an adiabatic doublewell potential, there is a tradeoff between the gap of the two dressed potentials and the barrier height of the doublewell when x_{0} is fixed.
After the transformation (3), the CGPEs (2) become
with
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 \(\frac{1}{2}\frac{{\partial }^{2}}{\partial {{\rm{x}}}^{2}}\) 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 avoidedcrossing.
The terms \({\partial }_{{\rm{x}}}{{\rm{\Psi }}}_{}\) and \({\partial }_{{\rm{x}}}{{\rm{\Psi }}}_{+}\) 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 socalled 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 timedependent complex amplitude. Substituting the variational wave function (8) into the decoupled equations (7), one can get the nonlinear threemode dynamical equations:
with the Josephson coupling strength
the zeropoint energy for atoms in jth mode
and the intramode and intermode exchange collision interaction strength
In the exchange collision denoted by \({\rm{ij}}\leftrightarrow {\rm{kl}}\), an atom from the kth mode collides with an atom from the lth mode and then these two atoms are transferred into an atom in the ith mode and an atom in the jth mode. For sufficiently deep wells, these three localized states are well localized at the corresponding wellcenters, so that the highorder overlaps between the localized states are very small, namely
Therefore the intermode exchange collision terms \({{\rm{U}}}_{{\rm{ij}}\leftrightarrow {\rm{kl}}}{{\rm{\psi }}}_{{\rm{j}}}^{\ast }{{\rm{\psi }}}_{{\rm{k}}}{{\rm{\psi }}}_{{\rm{l}}}\) in the threemode dynamical equations (9) can be ignored, and Eq. (9) can be further simplified to be:
in which the intramode exchange collision interaction \({{\rm{U}}}_{{\rm{ii}}\leftrightarrow {\rm{ii}}}\,({\rm{i}}=1,2,3)\) has been expressed as U_{ii} for simplicity.
In the following, we will consider both the BoseJosephson (BJ) dynamics and LZ dynamics and study the impact of the coupling between dressed states on the dressed state dynamics in our system.
BoseJosephson 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 twocomponent BEC system, when the intercomponent coupling has modest strength, the lower dressed potential has a doublewell 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 atomatom 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 noninteracting 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:
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 timeevolution 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 timeevolution 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 threemode 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 atomatom 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 atomatom 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 threemode equations (14). The complex amplitudes in the threemode equations (14) can be expressed as \({{\rm{\psi }}}_{{\rm{j}}}({\rm{t}})=\sqrt{{{\rm{n}}}_{{\rm{j}}}}\exp ({{\rm{i}}{\rm{\theta }}}_{{\rm{j}}})\), 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:
in which \({{\rm{E}}}_{{\rm{c}}}={{\rm{U}}}_{11}+{{\rm{U}}}_{22}\).
Mathematically, fixed points for (θ, z) are determined by \(\frac{d{\rm{\theta }}}{{\rm{dt}}}=0\) and \(\frac{{\rm{dz}}}{{\rm{dt}}}=0\). In the linear regime, namely E_{c} = 0, there are two normal fixed points (θ^{*} = 0, z^{*} = 0) and (θ^{*} = π, z^{*} = 0), and the general solutions are sinusoidal oscillations around one of these fixed points with frequencies ω_{R} = 2J. In the nonlinear regime, i.e., E_{c} ≠ 0, as the nonlinear interaction strength increases and exceeds a threshold
there exist two new fixed points: \(({{\rm{\theta }}}^{\ast }={\rm{\pi }},{{\rm{z}}}^{\ast }=\pm \sqrt{1\frac{16{{\rm{J}}}^{2}}{{{\rm{E}}}_{{\rm{c}}}^{2}{{\rm{N}}}^{2}}})\) when E_{c} < 0; \(({{\rm{\theta }}}^{\ast }=0,{{\rm{z}}}^{\ast }=\pm \sqrt{1\frac{16{{\rm{J}}}^{2}}{{{\rm{E}}}_{{\rm{c}}}^{2}{{\rm{N}}}^{2}}})\) when E_{c} < 0. 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 normaltoMQST 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 threemode 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 threemode approximation is regarded as valid. For fixed potential separation x_{0} = 3, we find the reasonable range of the threemode 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 noninteracting and interacting BEC, only when the gap between the upper and lower dressed potentials is sufficiently large, the lower dressed doublewell 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 threemode approximation is suitable to describe the system.
LandauZener dynamics
In our coupled twocomponent 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}:
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 avoidedcrossing 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 nonadiabatic transition probability of the LZ transition depends on the twocomponent coupling strength Ω and the wave packet’s initial position. The nonadiabatic transition probability P can be obtained numerically from the simulation results of the CGPEs (5). The results of nonadiabatic transition probability P for different coupling strength Ω and initial wave packet position are shown in Fig. 6(c). One can find that the nonadiabatic 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 avoidedcrossing and the terms \({\partial }_{{\rm{x}}}{{\rm{\Psi }}}_{}\), \({\partial }_{{\rm{x}}}{{\rm{\Psi }}}_{+}\) in the dressed state coupling terms (6) will decrease, resulting in the decrease of P.
One can also derive the nonadiabatic transition probability P by the LZ formula. The transition only occurs in the small region near the avoidedcrossing, 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 avoidedcrossing 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 nonadiabatic transition probability for each momentum components is
Therefore, the total nonadiabatic transition probability can be obtained by integrating the momentum space:
The nonadiabatic 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 doublewell. 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 nonadiabatic 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 twocomponent BECs in a statedependent potential. In our system, a doublewell structure can be formed in the lower dressed potential. The dressed state dynamics sensitively depend on both the intercomponent coupling strength and the choice of initial state. We discuss how the BJ dynamics and LZ dynamics are affected by the intercomponent 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 intercomponent 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 threemode approximation.
If the initial state is far away from the minimum of the lower dressed potential, there will appear several LZ transitions at the avoidedcrossing 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 intercomponent coupling strength should be strong enough but also the initial wave packet should not locate too far away from the potential minima.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants No. 11374375, No. 11574405 and No. 11704420). J. H. is partially supported by National Postdoctoral Program for Innovative Talents of China (Grant No. BX201600198).
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All authors discussed the results and reviewed the manuscript. Q.Y., J.H. and C.L. developed the theoretical scheme. Q.Y. and J.H. implemented the analytical calculation and the numerical calculation. Q.Y. and J.H. prepared the first draft. M.Z. and H.Z. discussed the results and reviewed the manuscript. C.L. conceived and supervised the project and prepared the final manuscript.
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Ye, Q., Huang, J., Zhuang, M. et al. Dressed state dynamics of twocomponent BoseEinstein Condensates in statedependent potentials. Sci Rep 8, 4484 (2018). https://doi.org/10.1038/s41598018225825
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