Exact Vortex Clusters of Two-Dimensional Quantum

ISSN: 0256 - 307 X

中国物理快报

Chinese Physics Letters

Volume 30 Number 4 April 2013

A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y Institute of Physics PUBLISHING

CHIN. PHYS. LETT. Vol. 30, No. 4 (2013) 040303

Exact Vortex Clusters of Two-Dimensional Quantum Fluid with Harmonic Confinement * CHONG Gui-Shu(崇桂书)1** , ZHANG Ling-Ling(张伶伶)1 , HAI Wen-Hua(海文华)2 1

2

School of Physics and Microelectronics, Hunan University, Changsha 410082 Department of Physics and Key Laboratory of Low-Dimensional Quantum Structure and Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081

(Received 25 December 2012) A family of exact analytical solutions of vortices in quantum fluid governed by a two-dimensional time-dependent Schrödinger equation is presented, which describes different kinds of vortex structures. The dynamics of different vortex clusters, such as the single vortex, vortex pair, vortex dipole and vortex trimer in a two-dimensional quantum fluid are analytically studied based on these exact solutions. The time evolutions of the wave of such vortices are demonstrated, and the orbits of motion of singular points in the vortices are also explored. The interactions of vortices in many-vortex clusters are discussed. A repulsive interaction between vortices with the same topological charge, and inter-annihilation and inter-creation of vortices with opposite topological charge, are shown.

PACS: 03.75.Lm, 67.10.Jn, 67.10.Hk, 47.32.C−

DOI: 10.1088/0256-307X/30/4/040303

Vortices are a type of common dislocation contained in singular wave structures that have a zero point wave function. They have been extensively researched in many branches of physics. Compared to the vortices investigated in cosmology,[1,2] hydrodynamics[3] and nonlinear optics,[4] vortices in quantum mechanics appear to be not in ordinary but in peculiar probability fluid.[5,6] Such quantum vortices have recently been explored in superfluids,[7,8] superconductivity[9] and especially in Bose–Einstein condensation.[10−17] Different kinds of initially vortex states can be prepared by different experimental approaches, such as using Raman transition phaseimprinting methods,[10] rotating the system with a “laser spoon” [11,13,18,19] or applying topological phase engineering,[20] respectively. The Ginzburg–Landau or Gross–Pitaevskii equations have normally been used to study the associated phenomenons in these fields. For such nonlinear partial differential equations, the dynamical behaviors or time evolutions of the quantized vortices are extremely complicated, and it is very difficult to find exact analytical solutions.[21] In particular, there are few analytical solutions that can be found for high dimensional cases. One has to resort either to different kinds of approximations[22−24] to reveal some of the finite features or to extensive numerical calculations.[25] In this Letter, we focus our attention on the standard time-dependent Schrödinger equation with twodimensional harmonic trap, which has many physical similarities, for example, an ideal Bose gas in twodimensional harmonic confinement, to investigate the dynamics of different types of vortex clusters embed-

ded in the probability fluid of quantum particles. A new family of time-dependent exact analytical vortex solutions are obtained. Applying these solutions we can construct different kinds of vortex clusters, such as the single vortex, vortex pair, vortex dipole, vortex trimer, and even vortex lattice. Correspondingly, the time evolution behaviors of these vortex clusters are well studied, and the motion and interaction of these vortices are also investigated. In addition, the motion orbits of the vortices are presented. According to the orbits of motion, we find (especially in comparison with the vortex pair and vortex dipole) a repulsion interaction for vortices with the same topological charge, and inter-annihilation or inter-creation, however, for vortices with opposite topological charge. We consider here a quantum fluid in a twodimensional harmonic trapping potential, for instance, an ideal Bose gas in harmonic confinement, whose dynamics can be governed by a two-dimensional Schrödinger equation. The dimensionless form of such Schrödinger equation can be written as[26,27] 𝑖

𝜕𝜓(𝑥, 𝑦, 𝑡) [︁ 1 (︁ 𝜕 2 𝜕 2 )︁ = − + 𝜕𝑡 2 𝜕𝑥2 𝜕𝑦 2 ]︁ 1 + (𝑘1 𝑥2 + 𝑘2 𝑦 2 ) 𝜓(𝑥, 𝑦, 𝑡), 2

(1)

where 𝜓(𝑥, 𝑦, 𝑡) denotes the quantum fluid density, and 𝑘1 and 𝑘2 characterize the confining frequencies in the 𝑥- and 𝑦-directions, respectively. The eigenvalue problem for the two-dimensional harmonic oscillators in quantum mechanics has been extensively solved in many text books. Here, we use a new ansatz

* Supported the National Natural Science Foundation of China under Grant Nos 10904035 and 11175064, the Foundation from the Ministry of Education of China under Grant No 757204003, and the Foundation of Central High University. ** Corresponding author. Email: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd

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to find the exact time-dependent solutions to the timedependent equation Eq. (1). It is well known that the two-dimensional solutions of Eq. (1) can be constructed by the solutions in the 𝑥- and 𝑦-dimensions, separately. We write 𝜓(𝑥, 𝑦, 𝑡) = 𝜓1 (𝑥, 𝑡)𝜓2 (𝑦, 𝑡) and insert it into Eq. (1), a one-dimensional standard Schrödinger equation with a harmonic potential is satisfied by 𝜓(𝑥, 𝑡) and 𝜓(𝑦, 𝑡) in the 𝑥- and 𝑦-directions, respectively. Now, our task is firstly to find the time-dependent solutions for the one-dimensional Schrödinger equation. Following the similar method as in our previous work,[28] we write the one-dimensional wave functions in the form of 𝜓𝑛,𝑖 = 𝑎𝑛,𝑖 𝐻𝑛 (𝜉𝑖 ) exp[𝑏𝑖 (𝑡)𝑥𝑖 − 𝑐𝑖 (𝑡)𝑥2𝑖 − 𝑓𝑖2 (𝑡)/2] (2) with 𝜉𝑖 = 𝑒𝑖 (𝑡)𝑥𝑖 − 𝑓𝑖 (𝑡) and 𝑎𝑛,𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑒𝑖 , 𝑓𝑖 being complex time-dependent functions. The subindex 𝑖 = 1 or 2 indicates a wave function in the 𝑥- or 𝑦direction, respectively, such that 𝑥1 = 𝑥, 𝑥2 = 𝑦. 𝐻𝑛 denotes the Hermit polynomial. Substituting Eq. (2) into Eq. (1), and noticing the relation 𝜕 2 𝐻𝑛 (𝜉𝑖 ) 𝜕𝐻𝑛 (𝜉𝑖 ) + 2𝑛𝐻𝑛 (𝜉𝑖 ) = 0, − 2𝜉𝑖 2 𝜕𝜉𝑖 𝜕𝜉𝑖

(3)

which is satisfied by 𝐻𝑛 , we can find that the set of time functions 𝑎𝑛,𝑖 (𝑡), 𝑏𝑖 (𝑡), 𝑐𝑖 (𝑡), 𝑒𝑖 (𝑡) and 𝑓𝑖 (𝑡) should satisfy the following set of differential equations: 𝑑𝑐𝑖 (𝑡) 𝑑𝑡 𝑑𝑒𝑖 (𝑡) 𝑖 𝑑𝑡 𝑑𝑓𝑖 (𝑡) 𝑖 𝑑𝑡 𝑖 𝑑𝑎𝑛,𝑖 (𝑡) 𝑎𝑛,𝑖 (𝑡) 𝑑𝑡 𝑖

= 2𝑐2𝑖 (𝑡) −

𝑘𝑖2 2

, 𝑖

𝑑𝑏𝑖 (𝑡) = 2𝑏𝑖 (𝑡)𝑐𝑖 (𝑡), 𝑑𝑡

= 2𝑐𝑖 (𝑡)𝑒𝑖 (𝑡) − 𝑒3𝑖 (𝑡), = 𝑏𝑖 (𝑡)𝑒𝑖 (𝑡) − 𝑓𝑖 (𝑡)𝑒2𝑖 (𝑡), = 𝑛𝑒2𝑖 (𝑡)+𝑖𝑓𝑖 (𝑡)

𝑑𝑓𝑖 (𝑡) 𝑏2 (𝑡) +𝑐𝑖 (𝑡)− 𝑖 . 𝑑𝑡 2 (4)

The first one in Eq. (4) can be transformed into the form of 𝑑2 𝜙𝑖 /𝑑𝑡2 = −𝑘𝑖2 𝜙𝑖 (𝑡) by setting 𝑐𝑖 = 𝑑𝜙𝑖 (𝑡)/𝑑𝑡/(2𝑖𝜙𝑖 (𝑡)). Thus, it is easy to obtain the complex solutions of 𝜙𝑖 (𝑡) for time-independent harmonic confinement as 𝜙𝑖 (𝑡) = 𝐴𝑖 cos(𝑘𝑖 𝑡 + 𝛼𝑖 ) + 𝑖𝐵𝑖 cos(𝑘𝑖 𝑡 + 𝛽𝑖 ) = 𝜌𝑖 (𝑡)𝑒−𝑖𝜃𝑖 (𝑡)

integral constants 𝑐0,𝑖 = 𝜌𝑖 𝑑𝜃𝑖 /𝑑𝑡 = 𝐴𝑖 𝐵𝑖 𝑘𝑖 sin(𝛼𝑖 −𝛽𝑖 ) and 𝑐′𝑖 = (𝑑𝜌𝑖 /𝑑𝑡)2 + 𝑐20,𝑖 /𝜌2𝑖 − 𝜌2𝑖 .[28] Consequently, solutions to the time-dependent functions in Eq. (4) can be solved and expressed in the form of 𝜌𝑖 (𝑡) and 𝜃𝑖 (𝑡) as 𝑑𝜃𝑖 𝑖 𝑑𝜌𝑖 exp(−𝑖𝜃𝑖 ) − , 𝑏𝑖 (𝑡) = 𝑏0,𝑖 , 2𝑑𝑡 2𝜌𝑖 𝑑𝑡 𝜌𝑖 √︂ √ 𝑐0,𝑖 𝑑𝜃𝑖 𝑏0,𝑖 cos(𝜃𝑖 ) 𝑒𝑖 (𝑡) = = , 𝑓𝑖 = , √ 𝑑𝑡 𝜌𝑖 𝑐0,𝑖 {︁ [︁ 𝑏20,𝑖 sin(2𝜃𝑖 ) ]︁}︁ 𝐴𝑛,𝑖 1 . 𝑎𝑛,𝑖 = √ exp − 𝑖 (𝑛 + )𝜃𝑖 − 𝜌𝑖 2 4𝑐0,𝑖 (6) 𝑐𝑖 (𝑡) =

Here, the integral constants 𝑏0,𝑖 are also determined by the preparation of initial wave packet. Applying the normalization condition we can yield the constant √ √ 𝐴𝑛,𝑖 = ( 𝑐0,𝑖 / 𝜋2𝑛 𝑛!)1/2 . Therefore, we now finally obtain the exact time-dependent solutions of the wave functions in a single dimension: 𝜓𝑖 (𝑥𝑖 , 𝑡) = 𝑅𝑛,𝑖 (𝜉𝑖 ) exp[𝑖Θ𝑖 (𝑥𝑖 , 𝑡) − 𝑖𝑛𝜃𝑖 ] with (︁ −𝜉 2 )︁ 1 𝑖 𝑅𝑛,𝑖 (𝜉𝑖 ) = 𝐴𝑛,𝑖 √ 𝐻𝑛 (𝜉𝑖 ) exp , 𝜌𝑖 2 Θ𝑖 (𝑥𝑖 , 𝑡) =

𝑥2𝑖 𝑑𝜌𝑖 𝑏0,𝑖 sin(𝜃𝑖 )𝑥𝑖 𝑏20,𝑖 sin(2𝜃𝑖 ) 1 − + − 𝜃𝑖 . 2𝜌𝑖 𝑑𝑡 𝜌𝑖 4𝑐0,𝑖 2 (8)

Combining these one-dimensional solutions, we obtain the exact time-dependent solutions for the twodimensional Schrödinger equation (1) as √ [︁ ]︁1/2 𝑐0,1 𝑐0,2 𝜓𝑛𝑥 ,𝑛𝑦 = 𝑛𝑥 +𝑛𝑦 𝐻𝑛𝑥 (𝜉1 )𝐻𝑛𝑦 (𝜉2 ) 2 𝑛𝑥 !𝑛𝑦 !𝜋𝜌1 𝜌2 (︁ 𝜉 2 + 𝜉 2 )︁ 2 · exp − 1 exp[−𝑖(𝑛𝑥 𝜃1 + 𝑛𝑦 𝜃2 )] 2 · exp[𝑖(Θ1 + Θ2 )]. (9) After possessing the above linear independent solutions of Eq. (1), it is interesting that we can use different kinds of linear combinations of these solutions to construct the exact vortex solutions for different vortex clusters. If the vortices are initially located at positions 𝑟𝑗 = (𝑥𝑗 , 𝑦𝑗 ), then the corresponding wave functions of the initial quantum fluid can be set as

(5)

with 𝜌𝑖 (𝑡) = (𝐴2𝑖 cos2 (𝑘𝑖 𝑡 + 𝛼𝑖 ) + 𝐵𝑖2 cos2 (𝑘𝑖 𝑡 + 𝛽𝑖 ))1/2 and 𝜃𝑖 = arctan[𝐵𝑖 cos(𝑘𝑖 𝑡 + 𝛽𝑖 )/𝐴𝑖 cos(𝑘𝑖 𝑡 + 𝛼𝑖 )]. Here, 𝐴𝑖 , 𝐵𝑖 , 𝛼𝑖 , 𝛽𝑖 are arbitrary constants that are determined by the initial conditions in the 𝑥- and 𝑦directions. In this study, for simplicity to construct the latter vortex solutions, we choose the initial conditions for 𝜃𝑖 (0) = 0, i.e., 𝛽𝑖 = 𝜋/2. Meanwhile, the initial conditions of 𝜌𝑖 and 𝜃𝑖 also determine another two

(7)

Ψ (𝑟, 0) = 𝐿𝜓gs

𝑛 ∏︁

[(𝑥 − 𝑥𝑗 ) + 𝑖𝑝(𝑦 − 𝑦𝑗 )]

𝑗=1

(10)

with 𝐿 being a normalized constant and 𝜓gs being the ground state wave function which provides a background; 𝑝 indicates the topological charge of the 𝑗th vortex. In this study, we set 𝑝 = 1 or −1. Such an initial state can be readily prepared experimentally by different methods.[11,13,18−20]

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Due to the linearity of the Schrödinger equation, an arbitrary linear superposition of solution (9) should also be an exact solution to the original Eq. (1). Therefore, considering the initial state in Eq. (10) we can construct different solutions with different vortex cluster structures. In the following, taken as examples, we will present four kinds of vortex clusters, the single vortex, vortex pair, vortex dipole and vortex trimer. (a)

(b)

(c)

y

5

(d)

0

-5 -5

0

5

x

Fig. 1. Time evolutions of the vortex clusters for the single vortex (top panel), pair of vortices (second panel), vortex trimer (third panel) and vortex dipole (bottom panel). From left to right, the time is chosen as 0, 2, 4, 6, 8. 1.5

(d)

y

0

-0.5 -1

1

y

0.5 0

-0.5 -1 0

x

0.5

1

1 Ψ2𝑣 = √ (𝜓20 − 𝜓02 ) + 𝑖𝜓11 − 𝜉1 (0)𝜓00 2 1 1 = {(𝜉12 − ) exp(−2𝑖𝜃1 ) − (𝜉22 − ) exp(−2𝑖𝜃2 ) 2 2 + 2𝑖𝜉1 𝜉2 exp[−𝑖(𝜃1 + 𝜃2 )] − 𝜉1 (0)2 }𝜓00 . (12)

√ 1 Ψ𝑣𝑑 = √ (𝜓02 +𝜓20 )+𝑖 2𝜉1 (0)𝜓01 +(1−𝜉12 (0))𝜓00 2 1 1 = {(𝜉22 − ) exp(−2𝑖𝜃2 ) + (𝜉12 − ) exp(−2𝑖𝜃1 ) 2 2 + 𝑖2𝜉1 (0)𝜉2 exp(−𝑖𝜃2 ) + (1 − 𝜉12 (0))}𝜓00 . (13)

0.5

-1.5 -1.5 -1 -0.5

Obviously, it is an exact solution to Eq. (1) and is consistent with the initial state form in Eq. (10). In this single vortex solution, we can easily find the time evolution behaviors of the quantum fluid and see the movement of the single vortex point. The orbit of motion for the single vortex can be calculated from the term before 𝜓00 in Eq. (11) and has an analytical form of 𝜉1 = 𝜉1 (0) cos 𝜃2 / cos(𝜃1 − 𝜃2 ), 𝜉2 = 𝜉1 (0) sin(𝜃1 )/ cos(𝜃1 − 𝜃2 ). Vortex pair and vortex dipole : Imagining that there are two vortices with the same topological charge 𝑝 = 1 in the quantum fluid, which occupy the positions of (±𝜉1 (0), 0) initially, we can make the vortex solution the same way as the vortex pair in the form of

(b)

(a)

(c)

1 Ψ1𝑣 = √ (𝜓10 + 𝑖𝜓01 ) − 𝜉1 (0)𝜓00 2 = [𝜉1 exp(−𝑖𝜃1 ) + 𝑖𝜉2 exp(−𝑖𝜃2 ) − 𝜉1 (0)]𝜓00 . (11)

On the other hand, if these two vortices have opposite topological charges 𝑝 = ±1, then we can obtain a different vortex cluster, which is called the vortex dipole. The analytical solution in this case can be written as

1

-1.5 1.5

structed in the form of

1.5

-1 -0.5 0

0.5

1

1.5

x

Fig. 2. Vortex trajectories for the (a) single vortex, (b) pair of vortices, (c) vortex dipole, and (d) vortex trimer cases discussed in the text. Here, the evolutions happen in the duration of 𝑡 = 0–20.

Single vortex : Suppose there is a single vortex with a positive topological charge 𝑝 = 1 localized initially at the position of (𝜉1 (0), 0), in other words, at (𝜉1 (0)/𝑒1 (0) + 𝑓1 (0), 0) in the framework of the 𝑥- and 𝑦-coordinates. A single vortex solution can be con-

Vortex trimer : More complexly, we can also consider three vortices that have the same topological charge 𝑝 = 1, and initially lie in the positions of (±𝜉1 (0), 0), (0, 𝜉2 (0)). Consequently, the evolutions of such vortex trimers can be described exactly by the solution √ 3 3 Ψ3𝑣 = (𝜓30 − 𝑖𝜓03 ) + (𝑖𝜓21 − 𝜓12 ) 2 2 𝜉2 (0) + 𝑖 √ (𝜓02 − 𝜓20 ) − 𝑖𝜉2 (0)𝜓11 2 2 𝜉 (0) − 1√ (𝜓10 + 𝑖𝜓01 ) + 𝑖𝜉12 (0)𝜉2 (0)𝜓00 . 2 (14) Obviously, if the initial quantum fluid contains many vortices and has a vortex lattice pattern, using the same method, we can also construct the exact

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dynamical solution of this vortex lattice and demonstrate its dynamical behavior and the interaction between each pair of vortices. Now we study the dynamical features of the above vortex clusters. For simplicity, we only consider the isotropic situation. For the anisotropic trap or initial preparation, the behaviors of the vortices become more complex and will be discussed elsewhere. By applying the analytical results in Eqs. (11)–(14) and taking the parameters that are determined by the initial conditions 𝐴1 = 𝐴2 = 1, 𝐵1 = 𝐵2 = −1, 𝑐01 = 𝑐02 = 1, 𝛼1 = 𝛼2 = 0, 𝑏01 = 𝑏02 = 1, and considering the isotropic case 𝑘1 = 𝑘2 = 1, we plot the evolutions of the quantum fluid that have different vortex cluster structures of the single vortex, vortex pair, vortex trimer and vortex dipole that initially lie in the positions of (0.5, 0), (±0.5, 0), (±0.5, 0), (0, 0.4) and (±0.5, 0) in Fig. 1, as seen from the first panel to the fourth panel, respectively. From Fig. 1, we can find that each vortex in a pair rotates with respect to the center of the vortices’ clusters but does not move exactly in a circle orbit. For the clusters where each vortex has the same topological charge, the singular point is conserved during the rotation. For the vortex dipole in the fourth row, however, the vortices are annihilated and created again and again because of the interaction between the vortices. Since the total topological charge in the vortex cluster is conserved during the time evolutions, there is no vortex that can be annihilated or created, which will break the conservation of the total topological charge. On the other hand, the orbits of motion for the vortices in each pair of clusters are plotted in Fig. 2. We can see clearly that the vortices in different vortex clusters all rotate in the confinement potential. In particular, we can observe the difference in the interaction between vortices with the same topological charge and the opposite one. It is obvious that, for a vortex pair, two vortices repel each other just like two elastic particles, as shown in Fig. 2(b). Otherwise, for the case of the vortex dipole in Fig. 2(c), we plot the orbits of motion for a vortex with a positive (negative) one topological charge in green (blue) points. Clearly, these two orbits are not closed, which means that these two vortices would annihilate each other and cause their orbits of motion to disappear. After a while, these two vortices would be recreated again at the same time, and their orbits of motion appear again. In summary, we have studied a quantum probability fluid in two-dimensional confinement. A new family of exactly analytical time-dependent solutions for the system are obtained, which can be used to describe the motion of the vortices for different vortex

clusters, such as the single vortex, vortex pair, vortex dipole, vortex trimer and even vortex lattices. The dynamical time evolutions of such vortex clusters and the vortices motion in each structure were researched, and the orbits of motion for different vortices were also explored. Furthermore, the different interactions of the vortices in the vortex pair and vortex dipole are discussed.

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Number 4

April 2013

GENERAL 040201 Parameter Extension and the Quasi-Rational Solution of a Lattice Boussinesq Equation NONG Li-Juan, ZHANG Da-Jun, SHI Ying, ZHANG Wen-Ying 040301 A Quantum Communication Protocol Transferring Unknown Photons Using Path-Polarization Hybrid Entanglement Jino Heo, Chang Ho Hong, Jong In Lim, Hyung Jin Yang 040302 From the Anti-Yang Model to the Anti-Snyder Model and Anti-De Sitter Special Relativity QI Wei-Jun, REN Xin-An 040303 Exact Vortex Clusters of Two-Dimensional Quantum Fluid with Harmonic Confinement CHONG Gui-Shu, ZHANG Ling-Ling, HAI Wen-Hua 040304 Long-Lived Rogue Waves and Inelastic Interaction in Binary Mixtures of Bose–Einstein Condensates LIU Chong, YANG Zhan-Ying, ZHAO Li-Chen, YANG Wen-Li, YUE Rui-Hong 040305 Improvement of Controlled Bidirectional Quantum Direct Communication Using a GHZ State YE Tian-Yu, JIANG Li-Zhen 040501 Semiclassical Ballistic Transport through a Circular Microstructure in Weak Magnetic Fields ZHANG Yan-Hui, CAI Xiang-Ji, LI Zong-Liang, JIANG Guo-Hui, YANG Qin-Nan, XU Xue-You 040502 Generalized Chaos Synchronization of Bidirectional Arrays of Discrete Systems ZANG Hong-Yan, MIN Le-Quan, ZHAO Geng, CHEN Guan-Rong 040601 A Potassium Atom Four-Level Active Optical Clock Scheme ZHANG Sheng-Nan, WANG Yan-Fei, ZHANG Tong-Gang, ZHUANG Wei, CHEN Jing-Biao 040701 A New Mach–Zehnder Interferometer to Measure Light Beam Dispersion and Phase Shift YANG Xu-Dong 040702 A High Sensitivity Index Sensor Based on Magnetic Plasmon Resonance in Metallic Grating with Very Narrow Slits XU Bin-Zong, LIU Jie-Tao, HU Hai-Feng, WANG Li-Na, WEI Xin, SONG Guo-Feng 040703 Principal Component Analysis and Minimum Description Length Criterion Based on Through-Wall Image Enhancement Muhammad Mohsin Riaz, Abdul Ghafoor

THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS 041201 Finding a Way to Determine the Pion Distribution Amplitude from the Experimental Data HUANG Tao, WU Xing-Gang, ZHONG Tao

NUCLEAR PHYSICS 042501 Observation of New Isotope 131 Ag via the Two-Step Fragmentation Technique WANG He, N. Aoi, S. Takeuchi, M. Matsushita, P. Doornenbal, T. Motobayashi, D. Steppenbeck, K. Yoneda, K. Kobayashi, J. Lee, LIU Hong-Na, Y. Kondo, R. Yokoyama, H. Sakurai, YE Yan-Lin 042502 The Influence of the Dependence of Surface Energy Coefficient to Temperature in the Proximity Model M. Salehi, O. N. Ghodsi

ATOMIC AND MOLECULAR PHYSICS 043101 The Hydrogen Molecular Ion in Strong Fields Using the B-Spline Method ZHANG Yue-Xia, LIU Qiang, SHI Ting-Yun 043102 First Principles Study of Single Wall TiO2 Nanotubes Rolled by Anatase Monolayers ZHANG Hai-Yang, DONG Shun-Le

043201 The Probe Transmission Spectra of 87 Rb in an Operating Magneto-Optical Trap in the Presence of an Ionizing Laser LIU Long-Wei, JIA Feng-Dong, RUAN Ya-Ping, HUANG Wei, LV Shuang-Fei, XUE Ping, XU Xiang-Yuan, DAI Xing-Can, ZHONG Zhi-Ping

FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) 044201 The Collins Formula Applied in Optical Image Encryption CHEN Lin-Fei, ZHAO Dao-Mu, MAO Hai-Dan, GE Fan, GUAN Rui-Xia 044202 Femtosecond Laser Pulses for Drilling the Shaped Micro-Hole of Turbine Blades JIA Hai-Ni, YANG Xiao-Jun, ZHAO Wei, ZHAO Hua-Long, DU Xu, YANG Yong 044203 Two-Mode Steady-State Entanglement in a Four-Level Atomic System PING Yun-Xia, ZHANG Chao-Min, CHEN Guang-Long, ZHU Peng-Fei, CHENG Ze 044204 Time-Grating for the Generation of STUD Pulse Trains ZHENG Jun, WANG Shi-Wei, XU Jian-Qiu 044205 The High Quantum Efficiency of Exponential-Doping AlGaAs/GaAs Photocathodes Grown by Metalorganic Chemical Vapor Deposition ZHANG Yi-Jun, ZHAO Jing, ZOU Ji-Jun, NIU Jun, CHEN Xin-Long, CHANG Ben-Kang 044206 Experimental Demonstration of a Low-Pass Spatial Filter Based on a One-Dimensional Photonic Crystal with a Defect Layer SONG Dong-Mo, TANG Zhi-Xiang, ZHAO Lei, SUI Zhan, WEN Shuang-Chun, FAN Dian-Yuan 044207 Controlling the Spectral Characteristics of Bismuth Doped Silicate Glass Based on the Reducing Reaction of Al Powder WANG Yan-Shan, JIANG Zuo-Wen, PENG Jing-Gang, LI Hai-Qing, YANG Lu-Yun, LI Jin-Yan, DAI Neng-Li 044208 Double Grating Expanders for Fourth-Order Dispersion Compensation in Chirped Pulse Amplifiers WANG Cheng, LENG Yu-Xin 044209 A Tunable Blue Light Source with Narrow Linewidth for Cold Atom Experiments ZHAI Yue-Yang, FAN Bo, YANG Shi-Feng, ZHANG Yin, QI Xiang-Hui, ZHOU Xiao-Ji, CHEN Xu-Zong 044210 The Multi-Scale and the Multi-Fractality Properties of Speckles on Rough Screen Surfaces ZHANG Mei-Na, LI Zhen-Hua, CHEN Xiao-Yi, LIU Chun-Xiang, TENG Shu-Yun, CHENG Chuan-Fu 044301 The Existence of Simultaneous Bragg and Locally Resonant Band Gaps in Composite Phononic Crystal XU Yan-Long, CHEN Chang-Qing, TIAN Xiao-Geng 044501 Patterns in a Two-Dimensional Annular Granular Layer CAI Hui, CHEN Wei-Zhong, MIAO Guo-Qing 044701 The Effect of Micro-ramps on Supersonic Flow over a Forward-Facing Step ZHANG Qing-Hu, YI Shi-He, ZHU Yang-Zhu, CHEN Zhi, WU Yu

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES 045201 Particle-in-Cell Simulations of Fast Magnetic Reconnection in Laser-Plasma Interaction ZHANG Ze-Chen, LU Quan-Ming, DONG Quan-Li, LU San, HUANG Can, WU Ming-Yu, SHENG Zheng-Ming, WANG Shui, ZHANG Jie 045202 The Angular Distribution of Optical Emission Spectroscopy from a Femtosecond Laser Filament in Air SUN Shao-Hua, LIU Xiao-Liang, LIU Zuo-Ye, WANG Xiao-Shan, DING Peng-Ji, LIU Qing-Cao, GUO Ze-Qin, HU Bi-Tao

CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES 046101 In situ XAFS Investigation on Zincblende ZnS up to 31.7 GPa YANG Jun, ZHU Feng, ZHANG Qian, WU Ye, WU Xiang, QIN Shan, DONG Jun-Cai, CHEN Dong-Liang 046102 High Quality Pseudomorphic In0.24 GaAs/GaAs Multi-Quantum-Well and Large-Area Transmission Electro-Absorption Modulators YANG Xiao-Hong, LIU Shao-Qing, NI Hai-Qiao, LI Mi-Feng, LI Liang, HAN Qin, NIU Zhi-Chuan 046103 Dislocation Multiplication by Single Cross Slip for FCC at Submicron Scales CUI Yi-Nan, LIU Zhan-Li, ZHUANG Zhuo 046201 Dynamic Behaviors of Hydrogen in Martensitic T91 Steel Evaluated by Using the Internal Friction Method HU Jing, WANG Xian-Ping, ZHUANG Zhong, ZHANG Tao, FANG Qian-Feng, LIU Chang-Song 046202 CuO Nanoparticle Modified ZnO Nanorods with Improved Photocatalytic Activity WEI Ang, XIONG Li, SUN Li, LIU Yan-Jun, LI Wei-Wei 046401 A Diblock-Diblock Copolymer Mixture under Parallel Wall Confinement PAN Jun-Xing, ZHANG Jin-Jun, WANG Bao-Feng, WU Hai-Shun, SUN Min-Na 046601 A Dynamic-Order Fractional Dynamic System SUN Hong-Guang, SHENG Hu, CHEN Yang-Quan, CHEN Wen, YU Zhong-Bo 046801 The Enhancement of Laser-Induced Transverse Voltage in Tilted Bi2 Sr2 Co2 Oy Thin Films with a Graphite Light Absorption Layer YAN Guo-Ying, ZHANG Hui-Ling, BAI Zi-Long, WANG Shu-Fang, WANG Jiang-Long, YU Wei, FU Guang-Sheng

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 047101 An Effective Description of Electron Correlation in the Green Function Approach LIU Yu-Liang 047201 Novel Transport Properties in Monolayer Graphene with Velocity Modulation SUN Li-Feng, FANG Chao, LIANG Tong-Xiang 047202 Graphene Quantum Wells and Superlattices Driven by Periodic Linear Potential YAN Wei-Xian 047301 Characteristics of an Indium-Rich InGaN p–n Junction Grown on a Strain-Relaxed InGaN Buffer Layer YANG Lian-Hong, ZHANG Bao-Hua, GUO Fu-Qiang 047401 Electronic Band Structure and Optical Response of Spinel SnX2 O4 (X = Mg, Zn) through Modified Becke–Johnson Potential A. Manzar, G. Murtaza, R. Khenata, S. Muhammad, Hayatullah 047501 Room-Temperature Ferromagnetism in Fe/Sn-Codoped In2 O3 Powders and Thin Films JIANG Feng-Xian, XI Shi-Bo, MA Rong-Rong, QIN Xiu-Fang, FAN Xiao-Chen, ZHANG Min-Gang, ZHOU Jun-Qi, XU Xiao-Hong 047502 Electric and Magnetic Properties and Magnetoelectric Effect of the Ba0.8 Sr0.2 TiO3 /CoFe2 O4 Heterostructure Film by Radio-Frequency Magnetron Sputtering WANG Ye-An, WANG Yun-Bo, RAO Wei, GAO Jun-Xiong, ZHOU Wen-Li, YU Jun 047503 Dzyaloshinskii–Moriya Interaction in Spin 1/2 Antiferromagnetic Rings with Nearest Next Neighbor Coupling LI Peng-Fei, CAO Hai-Jing, ZHENG Li 047504 Robust Half-Metallicity and Magnetic Properties of Cubic Perovskite CaFeO3 Zahid Ali, Iftikhar Ahmad, Banaras Khan, Imad Khan 047701 A Metal Oxide Heterostructure for Resistive Random Access Memory Devices LIAO Zhao-Liang, CHEN Dong-Min 047702 Direct Piezoelectric Potential Measurement of ZnO Nanowires Using a Kelvin Probe Force Microscope WANG Xian-Ying, XIE Shu-Fan, CHEN Xiao-Dong, LIU Yang-Yang

047703 First Principle Study on the Influence of Sr/Ti Ratio on the Atomic Structure and Dislocation Behavior of SrTiO3 GUAN Li, JIA Guo-Qi, ZUO Jin-Gai, LIU Qing-Bo, WEI Wei, GUO Jian-Xin, DAI Xiu-Hong 047901 The Effect of an Incident Electron Beam on the I–V Characteristics of a Au-ZnSe Nanowire-Au Nanostructure TAN Yu, WANG Yan-Guo

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 048101 The Nature of Stresses in a Giant Static Granular Column GE Bao-Liang, SHI Qing-Fan, RAM Chand, HE Jian-Feng, MA Shao-Peng 048201 Strong Coupling of Light with Extremely Thin Non-Close-Packing Colloidal Crystals: Experimental and Theoretical Studies HUANG Zhong, DONG Wen 048501 Optically Modulated Bistability in Quantum Dot Resonant Tunneling Diodes WENG Qian-Chun, AN Zheng-Hua, HOU Ying, ZHU Zi-Qiang