Structure and dynamics of binary Bose−Einstein condensates with vortex phase imprinting

doi:10.1007/s11467-023-1316-0

Front. Phys.

2023, Vol. 18

Issue (6) : 62302 https://doi.org/10.1007/s11467-023-1316-0

RESEARCH ARTICLE

Structure and dynamics of binary Bose−Einstein condensates with vortex phase imprinting

Jianchong Xing¹, Wenkai Bai¹, Bo Xiong², Jun-Hui Zheng^1,³, Tao Yang^1,³(

)

¹. Shaanxi Key Laboratory for Theoretical Physics Frontiers, Institute of Modern Physics, Northwest University, Xi’an 710127, China
². School of Science, Wuhan University of Technology, Wuhan 430070, China
³. Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China

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Abstract

The combination of multi-component Bose−Einstein condensates (BECs) and phase imprinting techniques provides an ideal platform for exploring nonlinear dynamics and investigating the quantum transport properties of superfluids. In this paper, we study abundant density structures and corresponding dynamics of phase-separated binary Bose−Einstein condensates with phase-imprinted single vortex or vortex dipole. By adjusting the ratio between the interspecies and intraspecies interactions, and the locations of the phase singularities, the typical density profiles such as ball-shell structures, crescent-gibbous structures, Matryoshka-like structures, sector-sector structures and sandwich-type structures appear, and the phase diagrams are obtained. The dynamics of these structures exhibit diverse properties, including the penetration of vortex dipoles, emergence of half-vortex dipoles, co-rotation of sectors, and oscillation between sectors. The pinning effects induced by a potential defect are also discussed, which is useful for controlling and manipulating individual quantum states.

Keywords Bose−Einstein condensates phase separation angular momentum energy competition

Corresponding Author(s): Tao Yang

About author:

^* These authors contributed equally to this work.

Issue Date: 06 July 2023

Cite this article:

Jianchong Xing,Wenkai Bai,Bo Xiong, et al. Structure and dynamics of binary Bose−Einstein condensates with vortex phase imprinting[J]. Front. Phys. , 2023, 18(6): 62302.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1316-0
https://academic.hep.com.cn/fop/EN/Y2023/V18/I6/62302

Fig.1 Density profiles of binary condensates with single vortex phase imprinting in component-1. (a, b) The typical ball-shell structures of the density profiles with respect to different

λ

. The vortex phase singularity is in the trap center [

b = 0

, see the inset between (a) and (b)]. (c, d) Crescent-gibbous density profiles are realized when an off-center phase imprinting [

b ≠ 0

, see the inset between (c) and (d)] is applied. The corresponding velocity fields for component-1 are indicated by the white arrows in the top row. Here, we set

g 12 / g 22 = 1.5

Fig.2 (a) Angular momentum variation dependents on the off-center-distance of vortex for fixing

g 12 / g 22 = 1.5

. The inset shows the value of red dot-line minus blue dot-line. (b) Density profiles of component-1 along

x

axis. (c) Schematic illustration of component-1 of BEC to analyze the angular momentum of the system. Point

o

and

s

represent the center of the trap and vortex singularity, respectively. In (b) and (c), the red solid line and blue dashed line correspond to the parameter

λ = 1.1

and

0.9

, respectively.

Fig.3 Dynamical evolution of the density profiles after imprinting a vortex phase at position

(x 0, 0)

of component-1 and applying a weak inverted Gauss potential

V G

for component-2. The white (red) curved arrow indicates that the condensate rotates counterclockwise (clockwise), while the case without the arrow indicates that the condensate is hardly rotating. Here, the parameters used are

λ = 1.1

and

g 12 / g 22 = 1.5

Fig.4 (a) Variation of angular momentum with time after the trap adding the Gaussian potential under the parameter

λ = 1.1

. Here,

g 12 / g 22 = 1.5

. (b) A typical density profile of condensates when it interacts with the Gaussian defect in dynamics. (c, d) The center-of-mass trajectories of component-1 with the blue dot start point for

A = ? 3.6

and

A = ? 4.4

, respectively. (e) The amplitude of the change of the total angular momentum

Δ L z

within the first half period with respect to the amplitude

A

of the Gaussian defect.

Fig.5 (a) Schematic illustration of the phase imprinting in component-1 while the phase of component-2 is uniform. The singularity point of the negative vortex phase is located in

(? d / 2, 0)

and the positive one is located in

(d / 2, 0)

. (b) Six kinds of density profiles of the condensates are realized by adjusting the value of

d

and

g 12 / g 22

for

λ = 0.9

via using the vortex phase imprinting of (a). The insets in (b) indicates the total density of the two components. The phase diagrams of the density profile of the two-component BECs with vortex dipole phase imprinting with respect to

g 12 / g 22

and

d

for (c)

λ = 0.9

and (d)

λ = 1.1

. The inset in (d) indicates the density distribution VII.

Fig.6 (a) The dynamical evolution of the Matryoshka-like density profile [see Fig.5(b)(I)] for

λ = 0.9

and

g 12 / g 22 = 3.0

with the initial vortex spacing being

d = 1.2 x 0

. The insets in the red solid oblongs magnify the corresponding regions of the density profiles.(b) The dynamical evolution of the Matryoshka-like with component-2 being the bull’s eye profile [see Fig.5(b)(II)] while the initial vortex spacing is

d = 1.7 x 0

for

λ = 0.9

and

g 12 / g 22 = 3.0

. The insets for

t = 13 t 0

and

t = 49.5 t 0

are the total density of the two components and the phase distribution of the region surround by the white oval circle in the density distribution, respectively. For both (a) and (b), the first row is for component-1 and the other one is for component-2.

Fig.7 Dynamics of the transposed sandwiched density profile [see Fig.5(b)(V)] while the initial vortex spacing is

d = 6.5 x 0

for

λ = 0.9

and

g 12 / g 22 = 3.0

. The first row is for the component-1 and the other one is for the component-2. The insets in the red solid oblongs magnify the corresponding regions of the density profiles.

Fig.8 (a) Dynamics of the sector?sector density profile [see Fig.5(b)(III)] while the initial vortex spacing is

d = 1.8 x 0

for

λ = 0.9

and

g 12 / g 22 = 3.0

. (b) The center-of-mass trajectories of the sector?sector density profile [see (a)] where the red(blue) solid line indicates the trajectory of component-1(-2) with the start point red (blue) dot. (c) Dynamics of the sandwiched density profile [see Fig.5(b)(IV)] while the initial vortex spacing is

d = 2.3 x 0

for

λ = 0.9

and

g 12 / g 22 = 3.0

. (d) Dynamics of the ball-shell density profile [see Fig.5(b)(VI)] while the initial vortex spacing by the phase imprinting is

d = 14 x 0

for

λ = 0.9

and

g 12 / g 22 = 3.0

. Here, for (a), (c) and (d), the first row is for the component-1 and the other one is for the component-2.

Type of phase-imprinting	Position of the phase singularity in component-1	$λ$	Total density structure	Density profiles of two components	Dynamics

Single vortex	$b = 0$ (Centered phase singularity)	$λ < 1$	Ball-shell	Component-1: ball with vortex Component-2: shell	Stable
	$b = 0$ (Centered phase singularity)	$λ > 1$	Ball-shell	Component-1: shell Component-2: ball without vortex	Stable
	$b ≠ 0$ (Off-center phase singularity)	$λ < 1$	Crescent-gibbous	Component-1: Gibbous Component-2: Crescent	Rotating with interface deform (amplitude of $V G : \| A \| < \| A c \|$ ) Pinning (amplitude of $V G : \| A \| = \| A c \|$ )Oscillating (amplitude of $V G : \| A \| > \| A c \|$ )
	$b ≠ 0$ (Off-center phase singularity)	$λ > 1$	Crescent-gibbous	Component-1: Crescent Component-2: Gibbous
Vortex dipole	Varing $d$ and $g 12 / g 22$	$λ < 1$	Ball-shell (I)	Component-1: ball with vortex dipole Component-2: shell	Transition from ball-shell structure to sector?sector structure with formation of half-vortex dipole
			Matryoshka-like (II)	Component-1: ball with density dip Component-2: bull’s eye	Transition from Matryoshka-like structure to sector?sector structure with penetration of vortex dipole
			Sector-sector (III)	Component-1: sector Component-2: sector	Rotating with interface deform
			Sandwich-type (IV)	Component-1: two separated gibbous Component-2: central part	Oscillating periodically
			Transposed sandwich-type (V)	Component-1: central part Component-2: two separated gibbous	Transition from sandwich-type structure to sector?sector structure
			Ball-shell (VI)	Component-1: ball Component-2: shell	Transition from ball-shell structure to oscillating asymmetric sandwich structure
	Varing $d$ and $g 12 / g 22$	$λ > 1$	Ball-shell (VII)	Component-1: shell Component-2: ball	Transition from ball-shell structure to oscillating asymmetric sandwich structure
	Varing $d$ and $g 12 / g 22$	$λ > 1$	Sandwich-type (IV)	Component-1: two separated gibbous Component-2: central part	Oscillating periodically

Tab.1 Comparison of initial structure and dynamic behavior for two-component BEC.

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