Spatial LandauZenerStückelberg interference in spinor BoseEinstein condensates
Abstract
We investigate the Stückelberg oscillations of a spin1 BoseEinstein condensate subject to a spatially inhomogeneous transverse magnetic field and a periodic longitudinal field. We show that the timedomain Stückelberg oscillations result in modulations in the density profiles of all spin components due to the spatial inhomogeneity of the transverse field. This phenomenon represents the LandauZenerStückelberg interference in the spacedomain. Since the magnetic dipoledipole interaction between spin1 atoms induces an inhomogeneous effective magnetic field, interference fringes also appear if a dipolar spinor condensate is driven periodically. We also point out some potential applications of this spatial LandauZenerStükelberg interference.
pacs:
03.75.Lm, 03.75.Mn, 67.85.FgI introduction
Quantum twolevel systems often exhibit an avoided energylevel crossing which can be traversed using an external control parameter. If one sweeps the external parameter through the avoided crossing, a coherent LandauZener transition occurs landau ; zener . When traversing the avoided crossing twice, by sweeping the external parameter back, the dynamical phase accumulated between transitions may give rise to constructive or destructive interference in the timedomain, known as Stückelberg oscillations stuck ; majorana . When such a twolevel system is subjected to a periodic driving in time, the physical observables of the system exhibit a periodic dependence on some external parameters, which is referred to as LandauZenerStückelberg interferometry nori . Recently, it was shown that LandauZenerStückelberg interferometry is of particular importance to superconducting qubits qubit1 ; qubit2 ; nori3 ; qubit3 ; qubit4 , nitrogen vacancy center nvc , and quantum dots qdots0 ; qdots , as it provides an alternative means to manipulate and characterize the structure of twolevel systems.
In atomic physics, Stückelberg oscillations were observed in the dipoledipole interaction between Rydberg atoms with an externally applied radiofrequency field van . In particular, for ultracold atomic gases, Stückelberg oscillations were demonstrated using the internalstate structure of Feshbach molecules mark ; mark2 and using BoseEinstein condensates in accelerated optical lattices zene . There are numerous theoretical works studying LandauZener tunneling subject to a temporal periodic driving for condensates trapped in doublewell potentials dwell ; dwell1 ; dwell2 . Vasile et al. vasile also proposed an interferometer of spinor condensates using Stückelberg oscillations.
In this work, we study the dynamics of a spin1 condensate subject to a spatially inhomogeneous transverse magnetic field. When the condensate is driven by a temporally periodic longitudinal magnetic field, the Stückelberg phases accumulated at different spatial positions are different, which results in modulations in the density profiles of all spin components. Therefore, by imposing a spatially nonuniform transverse field, we convert the timedomain interference into a spacedomain one, which we refer to as spatial LandauZenerStückelberg interference (SLZSI). In spinor condensates, the spatially inhomogeneous transverse field can also be provided by the magnetic dipoledipole interaction between atoms. We show that the SLZSI occurs even in the absence of any external transverse field. This phenomenon can be used to detect dipolar effects in a spinor condensate.
For a condensate confined in a IoffePritchard trap, Leanhardt et al. ketterle experimentally demonstrated that a vortex can be imprinted by adiabatically inverting the axial magnetic field. Interestingly, they also swept the axial field back to its original direction. Due to the adiabaticity of the whole process, they found that the condensate recovered its original state. The dynamics of a spin1 gas subject to a temporally oscillating field was studied in various papers pu ; sun ; bongs . However, to the best of our knowledge, we are not aware of any work on Stückelberg oscillations subject to a spatially inhomogeneous magnetic field.
This paper is organized as follows. In Sec. II, we consider a continuum of twolevel systems subject to a nonuniform transverse field and a periodic driving along the longitudinal direction. We show that the occupation probabilities are periodically modulated in the spacedomain. Section III studies the SLZSI of a spin1 condensate in a IoffePritchard trap. We show that the positions of the destructive interference points agree with those presented in Sec. II. In Sec. IV, we study the SLZSI induced by the magnetic dipoledipole interaction. Finally, we present our conclusions in Sec. V.
Ii Spatial quantum interferometry in the singleparticle picture
Let us first briefly summarize the LandauZener tunneling of a single twolevel system under temporally periodic driving nori . This will allow us to introduce quantities that will be used afterwards. Specifically, we consider the model Hamiltonian
(1) 
where is a constant representing the level splitting,
(2) 
is the periodic driving with amplitude and frequency , and are the Pauli operators. For a spin atom, and can be induced by a constant transverse magnetic field and an ac longitudinal field, respectively. Throughout this work, we will assume that the longitudinal periodic driving contains no dc component.
For the model Hamiltonian Eq. (1), we are interested in the timedependence of the occupation probabilities in the upper and lower energy levels at time . Although this problem can be solved by numerous approaches, here we quote the results from the adiabaticimpulse model (see, e.g., Ref. nori ). In Fig. 1, we schematically plot the adiabatic energy levels of the system; namely, the instantaneous eigenvalues of the Hamiltonian (1). The avoidedlevel crossings (with energy splitting ) at times are induced by the transverse field, where . The evolution of the system can be divided into two stages:

Away from the avoided crossings, the system evolves adiabatically by following the evolution matrix
(3) where
and is the dynamical phase.

LandauZener transitions occur at the avoided crossings, which can be represented by the transition matrix
(4) where the transition probability is , with , and the phase jump is , with being the gamma function. In particular, in the fastpassage limit (), the dynamical phase and phase jump become, respectively, and .
After the twolevel system is driven for halfperiods, one may take the measurement of the population either at time or as shown in Fig. 1. Correspondingly, the total evolution matrix becomes or , respectively. One immediately sees that, for the purpose of calculating the transition probability, a very relevant quantity is the transition matrix nori ; nori2
(5) 
where
(6)  
(7)  
(8) 
Assuming that only the upper level is initially populated, the occupation probabilities at the end of the field sweeping become
(9)  
(10) 
Now we generalize the above single twolevel system to a continuum of isolated twolevel systems which are distributed over the range . We further assume that the energy splitting depends linearly on the position , i.e.,
(11) 
where and are two constants. Without loss of generality, we assume that and . Due to the position dependence of the energy splitting, the transition probabilities now must depend on the position of the twolevel system. To proceed further, let us focus on the occupation probability . If is measured after the periodic driving is applied for periods (i.e., ), destructive interference for occurs at , with being an integer. Using Eq. (8), the condition for destructive interference becomes
(12) 
In Fig. 2(a), we schematically plot the left and righthandside of Eq. (12). The coordinates of the intersections of the left and righthandside then represent the positions of different destructive interferences. As a consequence, nontrivial spatial structure forms in the transition probability, representing the SLZSI.
To gain more insight into the spatial structure of the transition probability, we consider the fastpassage limit by assuming that for . The destructive interference condition can be approximated as
(13) 
where is a constant. In the fastpassage regime, we may assume that is sufficiently small, which allows us to focus on the values of in the vicinity of . To this end, we rewrite as , with . By further assuming , equation (13) reduces to
(14) 
Apparently, corresponding to different , the transition probability are spatially modulated with equal interval
(15) 
In addition, the necessary condition for the range to accommodate a destructive interference is , which implies that the temporal periodic driving must be applied for over periods.
In Fig. 2(b), we illustrate an example of the spatial distribution of , which is plotted using Eq. (10) for the set of parameters , , , and . As can be seen there, exhibits nice spatial periodicity. The spatial period read out from Fig. 2(b) is in very good agreement with that predicted by Eq. (15). The nearly sinusoidal dependence of on can be intuitively understood as follows. In the limit , () and have a very weak positiondependence such that is essentially a constant. However, when is sufficiently large, even a weak dependence in is significantly amplified in the function , which is the only term in contributing to the position dependence.
Iii SLZSI of a spin1 condensate in a magnetic trap
In this section, we study the SLZSI of an optically trapped spin1 condensate subject to a transverse magnetic field of the form of a IoffePritchard trap and a temporally periodic driving along the longitudinal direction. To this end, we first consider the spin dynamics of a single atom in the hyperfine state subject to a constant transverse magnetic field and a timedependent longitudinal field . The Hamiltonian of the system reads
(16) 
where and , with being the Landé gfactor of the atom, the Bohr magneton, and the angular momentum operator. Note that a spin1 atom contains three Zeeman sublevels, corresponding to the magnetic quantum number , , and . Utilizing the Majorana representation majorana ; bloch , the spin dynamics of the Hamiltonian (16) can be easily derived from that of a spin particle. Specifically, the evolution matrix after halfperiods becomes
(17) 
where and are given, respectively, in Eqs (6) and (7). For an initial state with only the spin component populated, the occupation probabilities then become , , and . Obviously, there exists a relation
(18) 
among the occupation probabilities of different spin components.
For an ultracold atomic gas, atoms interact with each other and are free to move. Therefore, to strictly realize the SLZSI obtained from isolated twolevel systems, the effects of the kinetic and interaction energies have to be eliminated. To achieve this, one may load the atoms into an optical lattice potential. If the depth of the optical potential is so high that the system is in the Mott insulator phase, the kinetic energy can be safely ignored. Furthermore, in the case where each lattice site is only occupied by a single atom, the atomatom interaction can also be eliminated. However, as we shall show below, for typical experimental parameters, even for a singly trapped spinor condensate, SLZSI can be well described by the model introduced in Sec. II.
Now we proceed to study the SLZSI of a spin1 condensate. Atoms interact with each other via the shortrange potential ho ; machida
(19) 
where and , with () being the wave scattering length in the combined symmetric channel of total spin . For the sodium atoms considered in this section, and , with being the Bohr radius. Note that the parameters and represent, respectively, the strength of the spinindependent and spinexchange collisional interactions. The spinexchange interaction of the sodium atoms is antiferromagnetic. There also exist atoms whose spinexchange interaction is ferromagnetic (e.g., rubidium). However, for the magnetic field considered in this section, the Zeeman energy is much larger than the spinexchange interaction energy such that the spinexchange interaction is unimportant to the spin dynamics.
We assume that the transverse magnetic field takes the form of a IoffePritchard trap, such that the total external field becomes
(20) 
where is the gradient of the transverse magnetic field. Furthermore, to stably confine the condensate, a spinindependent optical trap
is also applied, where is the radial trap frequency and is the trap aspect ratio. For the numerical simulations presented below, we shall choose , , , and .
Within the framework of meanfield theory, a spin1 condensate of atoms is described by the wave functions (), which satisfy the dynamic equations
where is the kinetic energy term, is the number density of the condensate normalized to the total number of atoms , and
(22) 
is the effective magnetic field which includes the external magnetic field and the contribution originating from the spinexchange interaction
(23) 
with being the spin density. Here we have neglected the magnetic dipoledipole interaction since it is much smaller than the Zeeman energy in a IoffePritchard trap. However, the effect of the dipolar interaction will be addressed in the next section.
We demonstrate the SLZSI in a magnetic trap by numerically evolving Eqs. (LABEL:gpe) for a condensate of sodium atoms. The initial wave functions are taken as the ground state of the spinor condensate under the external field . In the results presented below, we will focus on the behavior of the column density of the different spin components
(24) 
which also corresponds to the absorption image of the atomic gas.
Figure 3 shows the column densities of all spin components for and after the driving field being applied for , , , and periods. Due to the axial symmetry of the column densities, they are plotted as functions of . Initially, only the component is populated for the given parameters. As we start to drive the condensate with an ac longitudinal field, other spin states also become occupied. In particular, ripples start to develop in the density profiles of all spin components. If the condensate is driven for a longer time, more ripples will appear.
The positions of the destructive interference in the component is also determined by Eq. (12). Here, instead, we will focus on since the component contains the majority of the atoms when the system is only driven for a short period of time. For the given parameters, we have . Therefore, the positions of the destructive interference in are mainly determined by the first term on the righthandside of Eq. (9). Following the same analysis as that presented in Sec. II, destructive interference in occurs at the positions determined by the equation
(25) 
with being integer, where is the number of periods that the driving field has been applied for. For the parameters given in Fig. 3, the location of the destructive interference points predicted by Eq. (25) are shown as black vertical arrows in the figure. As can be seen, they agree well with those obtained through the full numerical simulations.
To gain more insight into the SLZSI of a spin1 condensate, let us verify the relation Eq. (18), which holds rigorously in the singleparticle picture. For the column densities of the condensates, Eq. (18) becomes . In our numerical simulation, we find that and agree with each other for small . However, as shown in Fig. 4(a), a prominent discrepancy appears for . Since Eq. (18) is derived from the Majorana representation, its violation indicates a deviation from the singleparticle picture. To identify which source causes the discrepancy, one may evolve Eq. (LABEL:gpe) with the kinetic energy term removed. This treatment is equivalent to taking the ThomasFermi approximation. The result presented in Fig. 4(b) shows that is visually identical to after the kinetic energy term is dropped, which suggests that the discrepancy in Fig. 4(a) is mainly caused by the center of mass motion of the atoms.
Vengalattore et al. kurn have demonstrated that spinor condensates can be regarded as highresolution magnetometers. Potentially, due to its sensitivity to the inhomogeneity of the magnetic field, the SLZSI in a spinor condensate can also be used to measure the gradient of the magnetic field.
Iv SLZSI in a dipolar spin1 condensate
Now, we turn to study the SLZSI in a dipolar spinor condensate. In addition to the shortrange interaction , there also exists a longrange magnetic dipoledipole interaction between two spin1 atoms. The interaction potential between two magnetic dipoles takes the form
(26) 
where the strength of the dipolar interaction is characterized by , with being the vacuum magnetic permeability and is a unit vector. Within the framework of meanfield theory, the dipolar interaction generates an effective magnetic field of the form zhang
(27) 
Consequently, the total effective magnetic field becomes
(28) 
We note that the magnitude of the transverse component of can be formally expressed as
(29) 
As we will show, is nonuniform. Therefore, SLZSI may be induced in spinor condensates even in the absence of an external transverse magnetic field.
To proceed further, we consider a concrete example of a spin1 condensate containing rubidium atoms. The Landé factor of the atoms is . Moreover, the wave scattering lengths between rubidium atoms are and . The optical trap has the same parameters as those adopted in Sec. III. Finally, to emphasize the effect of the dipolar interaction, we assume that the external field only contains a longitudinal component, i.e.,
The value of is typically around several hundreds G, under which the spins of the atoms are fully polarized along the axis.
Before we turn to study the dynamics of the condensate under an external driving field, it is instructive to examine the structure of the magnetic field induced by the dipolar interaction. For simplicity, we assume that only the spin component is populated, which is essentially the ground state under the external field . Figure 5(a) shows the magnitude of the transverse component of for a spinpolarized condensate. Due to the cylindrical symmetry of the system, reduces to a function of and . Clearly, is nonuniform and takes a butterfly shape in the plane. For this specific example, the maximum value of is around . In particular, as can be deduced from Eq. (27), the effective transverse field vanishes in the plane.
To reveal more details about the transverse field, we plot for various ’s in Fig. 5(b). On a given plane, is roughly a linear function when is small. However, the gradient of the transverse field sensitively depends on the value of . Moreover, becomes a timedependent function once the external driving field is applied. Therefore, one should not use Eq. (25) to predict the positions of the destructiveinterference points.
The dynamics of the condensate can be simulated by numerically evolving Eqs. (LABEL:gpe) with the effective magnetic field in Eq. (28). The initial wave function is taken as the ground state under the external field . In Fig. 6, we present the column densities of all spin components after the driving field is applied for various periods. The parameters of the driving field are and . As can be seen, even though the total density remains unchanged, ripples start to develop on the density profiles of all spin components after the driving field is applied for a few periods, which indicates SLZSI occurs in a spin1 condensate subject to a periodic driving. More ripples will appear if one evolves the system for a longer time.
The structure of the density ripples also depends sensitively on the parameters of the driving field. As it can be seen from Fig. 7(a), the ripples in are significantly suppressed if we increase the amplitude of the driving field to . This can be intuitively understood as follows. A destructive interference on occurs when a considerable number of atoms in the component are transferred to other spin components. If one increases while keeping unchanged, the transition probability decreases. Therefore, in order to gain visible density ripples, one has to lower the frequency of the driving field. Indeed, as shown in Fig. 7(b), density ripples appear again by lowering the driving frequency to .
From Eq. (23), it is tempting to think that, even in the absence of the magnetic dipoledipole interaction, the spinexchange interaction can also induce density ripples. This turns out to be untrue for the initial states considered in this section. To show this, we consider an initial state with spins being polarized to an arbitrary direction. For such a state, the total spin of the condensate is and the number of atoms in all spin components are uniquely determined pu2 ; yi
where is the component of the total spin. Now we assume that an magnetic field is applied along the axis. Since the total spin () and its component () are conserved in the absence of the dipolar interaction ho , the contact spinexchange interaction will not cause any spinmixing. Therefore, for an initially polarized spin1 condensate subject to a periodic driving, the appearance of density ripples is an unambiguous manifestation of the dipolar interaction in a spinor condensate.
In a previous work zhang , we proposed to detect the effect of dipolar interaction in spinor condensates by adiabatically inverting the longitudinal field, where the strength of the longitudinal field is only around several tens of microGauss. Compared to that scheme, the obvious advantage of using the dipolarinteractioninduced SLZSI is that the strength of the longitudinal field can be much higher.
V Conclusion
We have proposed a method to convert the timedomain Stückelberg oscillations to an interference pattern in the spacedomain by imposing a spatially nonuniform transverse magnetic field. For a continuum of twolevel systems, we showed that the occupation probabilities are periodically modulated in space. In addition, we also obtained a relation between the spatial period and the system parameters in the fastpassage limit. We then demonstrated the SLZSI for a spin1 condensate subject to a transverse field of the form of a IoffePritchard trap. We found that the kinetic and interaction energies only slightly modify the interference patterns obtained from the singleatom model if the system is not driven for a long time. Finally, we showed that the SLZSI can also be induced by the magnetic dipoledipole interaction, even in the absence of an external transverse field. Potential applications of the SLZSI include the measurement of the spatial inhomogeneity of the magnetic field and the detection of weak magnetic dipoledipole interactions in spinor condensates. Finally, we want to point out that SLZSI is a singleparticle property, therefore, a fermionic gas should also exhibit the phenomena described in this work.
Acknowledgements.
We thank S. Ashhab for valuable comments on the manuscript. FN acknowledges partial support from the Laboratory of Physical Sciences, National Security Agency, Army Research Office, National Science Foundation under Grant No. 0726909, DARPA, AFOSR, JSPSRFBR under Contract No. 090292114, GrantinAid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the Funding Program for Innovative R&D on S&T (FIRST). CPS acknowledges the supports from the NSFC under Grant Nos 10935010. SY acknowledges the supports from the NSFC (Grant Nos. 11025421 and 10974209) and the “Bairen” program of the Chinese Academy of Sciences.References
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