Anisotropic quantum scattering in two dimensions
We study the quantum scattering in two spatial dimensions (2D) without the usual partial-wave formalism. The analysis beyond the partial-wave approximation allows a quantitative treatment of the anisotropic scattering with a strong coupling of different angular momenta nonvanishing even at the zero-energy limit. High efficiency of our method is demonstrated for the 2D scattering on the cylindrical potential with the elliptical base and dipole-dipole collisions in the plane. We reproduce the result for the 2D scattering of polarized dipoles in binary collisions obtained recently by Ticknor [Phys. Rev. A 84, 032702 (2011)] and explore the 2D collisions of unpolarized dipoles.
In recent years, the problem of anisotropic quantum scattering in two spatial dimensions (2D) attracts increasing interest. It is stimulated by the spectacular proposals for prospects to create exotic and highly correlated quantum systems with dipolar gases ref1 (); ref2 (). Particularly, there were considered anisotropic superfluidity ref3 (), 2D dipolar fermions ref4 (), and few-body dipolar complexes ref5 (). The recent experimental production of ultracold polar molecules in the confined geometry of optical traps ref6 (); ref7 (); ref8 () has opened up ways to realize these phenomena. Noteworthy also is a rather long history of research of 2D quantum effects in condensed matter physics. One can note superfluid films ref9 (), high-temperature superconductivity ref10 (), 2D materials, such as graphene ref11 (), and even possibilities for topological quantum computation ref12 (). Unique opportunities for modeling these 2D effects in a highly controlled environment have recently appeared with the development of experimental techniques for creating quasi-2D Bose and Fermi ultracold gases ref13 ().
Interest in the processes and effects in 2D-geometry has stimulated the theory of elementary quantum two-body systems and processes in the plane. Special consideration should be given to the anisotropy and long-range character of the dipole-dipole interaction. Actually, usual partial-wave analysis becomes inefficient for describing the dipole-dipole scattering due to the strong anisotropic coupling of different partial-waves in the asymptotic region ref14 (); ref15 (). Recently, considerable progress in the analysis of the 2D and quasi-2D (q2D) scattering of dipoles has been achieved ref16 (); ref17 (); ref18 (); ref19 (); ref20 (). Thus, the 2D dipolar scattering in the threshold and semiclassical regimes was studied in the case of the dipole polarization directed orthogonally to the scattering plane ref16 (). An arbitrary angle of polarization was considered in ref17 ().
In this work, we develop a method for quantitative analysis of the 2D quantum scattering on a long-range strongly anisotropic scatterer. Particularly, it permits the description of the 2D collisions of unpolarized dipoles. Our approach is based on the method suggested in ref21 () for the few-dimensional scattering which was successfully applied to the dipole-dipole scattering induced by an elliptically polarized laser field in the 3D free-space ref15 ().
The key elements of the method are described in Section II. In Section III, we apply the method to the 2D scattering on the cylindrical potential with the elliptical base and the 2D dipole-dipole scattering of unpolarized dipoles. We reproduce the threshold formula ref22b (); ref24 () for the scattering amplitude on the cylinder potential with the circular base and the results of ref16 (); ref17 () for the 2D scattering of polarized dipoles. High efficiency of the method has been found in all problems being considered. The last Section contains the concluding remarks. Some important details of the computational scheme and illustration of the convergence are given in Appendices.
Ii 2D Scattering problem in angular-grid representation
The quantum scattering on the anisotropic potential in the plane is described by the 2D Schrödinger equation in polar coordinates
with the scattering boundary conditions
in the asymptotic region and the Hamiltonian of the system
The unknown wave function and the scattering amplitude are searched for the fixed momentum defined by the colliding energy ( and the direction of the incident wave (defined by the angle and for the scattering angle 111 Hereafter we use the definition of the scattering amplitude introduced in ref22a ().. Here is the reduced mass of the system. In the polar coordinates, the angular part of the kinetic energy operator in has a simple form . The interaction potential can be anisotropic in the general case, i.e. to be strongly dependent on . It is clear that varying the direction of the incident wave can be replaced by the rotation of the interaction potential by the angle for the fixed direction of the incident wave, which we choose to be coincident with the x-axis. Thus, in the case of anisotropic potential the task is to solve the problem (1) with the interaction potential for all possible and fixed with the scattering boundary conditions
If the scattering amplitude is found, one can calculate the differential scattering cross section
where , as well as the total cross section
by averaging over all possible orientations of the scatterer and integration over the scattering angle .
To integrate the problem (1),(2), we use the method suggested in ref21 () to solving a few-dimensional scattering problem and applied in ref15 () for the dipole-dipole scattering in the 3D free-space. Following the ideas of these works we choose the eigenfunctions
of the operator as a Fourier basis for the angular-grid representation of the searched wave-function . We introduce the uniform grid ) over the and -variables and search the wave function as expansion
where is the inverse matrix to the square matrix defined on the angular grid222 To calculate the inverse matrix , we use the completeness relation for the Fourier basis , which in our grid representation reads ..
In the representation (7) the unknown coefficients are defined by the values of the searched wave function on the angular grid , any local interaction is diagonal
and the angular part of the kinetic energy operator has a simple form
Note that the presence in the interaction potential of the “nonlocal” angular part (i.e. the integration or differentiation over angular variable) leads to destroying the diagonal structure in (8).
Since the wave function must be finite at the origin , the “left-side” boundary condition for the functions reads as
In the asymptotic region the scattering boundary condition (3) accepts the form
By using the Fourier expansion for the plane wave and the scattering amplitude 333 Here are the first kind Bessel functions of integer order. Their asymptotic behavior ref23 ():
we eliminate the angular dependence from the asymptotic equation (II) and represent the “right-side” boundary condition for the functions in the form
To solve the boundary-value problem (II),(11) and (II), we introduce the grid over the and reduce the system of differential equations (II) by using the finite-difference approximation of the sixth order to the system of algebraic equations
with the band-structure of the matrix with the width of the band. By using the asymptotic equations (II) in the last two points and one can eliminate the unknown vector from equation (II) and rewrite the “right-side” boundary condition in the form
which can be efficiently solved with standard computational techniques such as the sweeping method ref25 () or the LU-decomposition ref26 (). The detailed structure of the matrix of the coefficients is discussed in Appendix A. After the solving of Eq.(19) and finding the wave function the scattering amplitude is constructed according to Eqs.(II) and (14).
Iii Results and Discussion
iii.1 Scattering on anisotropic scatterer
First, we have analyzed the 2D scattering on the cylindrical potential barrier with the elliptical base
was obtained at the zero-energy limit . Here and is the Euler constant. We have analyzed the scattering on the potential barrier with circular base for arbitrary momentum . The results of calculation presented in Figs. 1 and 2 confirm the convergence of the scattering amplitude to the analytical value (21) at . In this Subsection all the calculations were performed in the units .
In the limiting case of the infinitely high potential barrier (20) with the circular base the asymptotic formula (21) becomes exact for arbitrary . This is confirmed by investigation presented in Table 1 which illustrates the convergence of the numerical values with increasing () and narrowing () of the potential barrier to the analytic result (21). In the limit case and we obtain for the scattering length extracted from the calculated amplitude by the formula (21), what is in agreement to the estimate given in ref22b (). The range of applicability of Eq. (21) was investigated recently in ref22h ().
|10||-0.42692 + i0.08772||-0.42693 + i0.08772||-0.19832 + i0.05304||-0.19837 + i0.05304|
|10||-0.47960 + i0.11141||-0.47961 + i0.11141||-0.22999 + i0.07362||-0.23016 + i0.07362|
|10||-0.49055 + i0.11674||-0.49056 + i0.11674||-0.23673 + i0.07853||-0.23696 + i0.07853|
|10||-0.49400 + i0.11846||-0.49402 + i0.11846||-0.23895 + i0.07988||-0.23920 + i0.07988|
|10||-0.49408 + i0.11850||-0.49409 + i0.11849||-0.23899 + i0.07992||-0.23925 + i0.07992|
|Eq.(21)||-0.49486 + i0.11430||-0.49486 + i0.11430||-0.23901 + i0.07952||-0.23901 + i0.07952|
Then, we have applied our scheme for calculation of the scattering cross section for the isotropic and anisotropic scattering. In Fig. 3 the differential cross section, calculated for the circular base of the scatter (20), is given as a function of and . The dependence of the cross section on disappears with decreasing momentum and the dependence on is absent for any due to the spherical symmetry of the potential (20) if . Further, the analysis was extended to more general case of elliptical base of the potential barrier (20). In Figs. 4 and 5 the calculated differential cross sections on the anisotropic scatter (20) are presented for the cases of weak () and strong anisotropy (). Here, we observe more sharp dependence on and in the cross section with increasing anisotropy of the scatterer. The anisotropy in the scattering cross sections appears with increasing earlier for the anisotropic potential barrier than for the barrier with circular base.
iii.2 Dipole-dipole scattering in plane
Here we analyze the 2D quantum scattering on the long-range anisotropic scatterer defined by the dipole-dipole interaction. This problem simulates the collisions of polar molecules in pancake optical traps. The interaction potential between two arbitrarily oriented dipoles reads
where – dipole moments and – their projections onto the collision axis. The expression (22) can be written in the polar coordinates
where the angles and define the tilt of dipoles to the scattering plane and the angle denotes the mutual orientation of the dipole polarization planes and in Fig. 6.
If we consider the scenario when the polarization of colliding molecules is orthogonal to the plane of motion , interaction is fully isotropic and repulsive
For dipoles oriented in the plane , anisotropy arises and the interaction potential reads
A particular case of parallel dipoles with the polarization axis tilted to the plane of motion ( with short-range interaction modeled by a hard wall at the origin
with the width
was considered in paper ref17 (). We have investigated this case with our approach and have obtained good agreement with the results of paper ref17 (). This is illustrated by Fig. 7, where the calculated total cross section (5) is given in the units of . Here is the dipolar length and is the value of the total scattering cross section in the eikonal approximation that is valid in the high-energy regime, ref17 (). All calculations in this section were performed for the following parameters: and ; the number of grid points on was .
Then, we have analyzed how the found “resonant” structure for the polarized dipoles (see Fig. 7) in the calculated dependence of the scattering cross section on the dipole tilt angle varies with destroying the polarization. Depolarization was simulated by rotating the angle between the dipole polarization planes and (see Fig. 6). We found progressive narrowing of the “resonance” area with a simultaneous decrease of the amplitudes of the “resonance” oscillations with increasing angle from to (see Fig. 8). When approaching the point the “resonant” structure disappears, the cross section becomes smooth relative to and reaches its maximum value. This effect is due to the fact that when approaching the angle repulsive feature of the dipole-dipole interaction becomes dominant (see Fig. 9). With decreasing from to the attractive part appears for some and . It leads to appearing the “resonant” part in the scattering cross section. Note, that the presented cross sections were obtained for distinguishable particles. The effect of symmetrization/antisymmetrization (i.e. transition to identical particles) is shown in Fig. 10 for and varying from to , where we observe the strong dependence of the total cross sections on the angle with the maximal enhancement at . The cross sections are symmetric with respect to this point. We have to note that the curves and describing the scattering of bosonic and fermionic particles exactly repeat the behavior of the curve for distinguishable particles.
Finally, we have analyzed the scattering of arbitrarily oriented dipoles in the case of mutual orthogonality of their polarization planes and (). Here also we found a strong “resonant” structure by the tilt angle of one dipole, if the other dipole is oriented in the scattering plane () (see Fig. 11), that appears due to the attractive feature of the dipole-dipole interaction strength with increasing of the tilt angle .
iii.3 Advantages and prospects of the angular-grid representation
Let us discuss here the advantages of our angular-grid representation (7) in comparison with the traditional partial-wave formalism, which was used in particular by Ticknor in the analysis of the 2D scattering of polarized dipoles ref17 ().
First, for any potential the estimate of the residual term in the expansion of the desired wave function on partial waves is unknown. This fact becomes crucial in the case of strongly anisotropic and long-range potential considered in the above Subsection III.B, where the partial-wave expansion becomes very slow due to the strong coupling between different angular momenta remaining even at the zero-energy limit ref14 (); ref15 (). Opposite, our angular-grid representation belongs to a class of mesh methods admitting an estimate of an approximation error. Thus, since the representation (7) can be considered, following ref21 (), as a Fourier interpolation of the order in the variable , the error of the approximation (7) can be estimated as ref29 ()
where and is the number of existing continuous and bounded derivatives over . Due to this estimate one can await a rapid convergence of our angular-grid representation (7) with increasing on the sequence of compressed grids . It was confirmed in all the computations we performed (see also illustration of the convergence of the angular-grid representation given in Table.II of Appendix B). The convergent results were obtained even in the regions of “resonant” scattering of polarized as well as unpolarized dipoles (see Figs.7,8 and 11). Slight shifts of the positions of the “resonances” in the scattering of polarized dipoles calculated with our method relative to the values obtained by Ticknor (see Fig.7) can be explained by the error due to the truncation of his partial-wave sum.
Another important advantage of the angular-grid representation is its flexibility connected with the lack of the time-consuming procedure of calculating the matrix elements of the interaction potential. As was mentioned above, the potential matrix in this representation is diagonal and consists of the values of the potential in the angular grid nodes. This circumstance permitted us to perform extended detail computations of the 2D dipole-dipole scattering of the different mutual orientations. This also makes the method very perspective in the case of nonseparable interactions and generalization to higher dimensions.
We have developed a computational scheme for quantitative analysis of the 2D quantum scattering on the long-range anisotropic potentials. High efficiency of the method was demonstrated in the analysis of scattering on the cylindrical potential with the elliptical base and dipole-dipole collisions in the plane. In the last case we found the strong dependence of the scattering cross section on the mutual orientation of dipoles.
The method can be applicable for analyzing the collisional dynamics of the polarized as well as unpolarized polar molecules in 2D and quasi-2D traps. A natural application is the quantitative analysis of the confinement-induced resonances (CIR) in quasi-2D traps. In this problem the crucial element is an inclusion of the transverse confinement what we suppose to explore in our future work with generalization of the developed approach to this 3D case. Particularly, this analysis can resolve the puzzle with the position of the 2D CIR measured recently ref27 (), which is under intensive discussions.
Acknowledgements.The authors thank V. V. Pupyshev and V. B. Belyaev for fruitful discussions and comments. The authors acknowledge the support by the Russian Foundation for Basic Research, grant 14-02-00351.
|1||0.94212 - i0.36446||-0.33027 + i1.24225||-0.95099 - i0.52976||0.36649 + i1.04884|
|2||0.65275 – i0.30331||-0.61946 + i1.30336||-1.48979 + i0.79336||-0.50755 + i2.02677|
|3||0.68693 – i0.30254||-0.65392 + i1.30275||-0.31229 + i0.79336||-1.01013 + i1.68129|
|4||0.68794 – i0.30507||-0.65902 + i1.29149||-0.28629 + i0.18596||-0.90989 + i1.50458|
|5||0.68799 – i0.30506||-0.65940 + i1.29164||-0.50155 + i0.05523||-0.87486 + i1.52425|
|10||-0.60611 + i0.07641||-0.89428 + i1.53586|
|20||-0.61098 + i0.07836||-0.89552 + i1.53816|
|30||-0.61173 + i0.07971||-0.89572 + i1.53885|
Appendix A Finite-difference approximation for boundary-value problem (10),(11) and (15)
where and . In this representation the angular dependence is built into the matrix and the interaction is included into the diagonal matrix of values of the potential in the angular grid nodes. The constant matrix couples all equations in a system and does not depend on the radial variable. There is no need to compute any matrix elements of the potential, what essentially minimizes the computational costs.
For solving boundary value problem (28) the seven-point finite-difference approximation for second derivatives of sixth order
where the coefficients are the square matrices, and the elements of the right-side of (30) are the dimensional vectors. After employing the “right-side” boundary condition in the form (II) in the last three grid points and, analogously, the “left-side” boundary condition in the form (II) in the first two points , the detailed structure of the matrix is represented as
The block structure of the system (31) provides several significant advantages. The block matrix can be stored in a packaged form, which allows the use of optimal resource. The system (31) can be efficiently solved by a fast implicit matrix algorithm based on the idea of the block sweep method ref25 ().
Appendix B Convergence of computational scheme
In the Table 2 we illustrate the convergence of the calculated scattering amplitude over the number of angular grid points for the scatterers with weak and essential anisotropy at in the potential barrier (20). For the case we reach the accuracy of four significant digits in the scattering amplitude on the angular grids with . For stronger anisotropy the accuracy of two significant digits was reached at .
The number of radial grids and the border of integration were chosen to keep the accuracy of four significant digits in the calculated amplitudes.
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