Since the realization of Bose–Einstein condensates (BECs) in dilute and ultracold gases, lots of theoretical and experimental studies on BECs have been reported, such as the creation of solitons,^[1–5] vortices,^[6,7] and symmetry breaking.^[8,9] Since BECs have long coherence time and high controllability, as ideal coherent sources they are widely used to study matter–wave interference.^[10–12] In the mean-field approximation, the condensates can be described by the macroscopic wavefunction. This theory paves a way toward studying the interference phenomena of condensates theoretically.

Interference between two condensates was first observed experimentally in 1997.^[13] Wallis et al.^[14] calculated theoretically the macroscopic interference of two independent condensates released from a double-well potential trap. The double-well potential used to form two condensates was created by adding a Gaussian barrier into the center of a harmonic potential.^[15] The interference pattern was aperiodic, while dark solitons were generated when two condensates overlapped.^[16]

A recent experiment^[17] reports a novel pattern of interference between two condensates, which is different from the previous works.^[13–15] Motivated by this experiment, the periodic interference pattern of two condensates is demonstrated by numerical simulation in this paper. We also give a theoretical explanation on the generation mechanism of the periodic interference pattern, which has been mentioned by little research up to now.

In the zero-temperature limit, single-component BECs trapped in a harmonic potential are described by the nonlinear Schrödinger equation, known as Gross–Pitaevskii (GP) equation^[18]

We consider the condensates in a quasi-one-dimensional system, where the harmonic potential V_h(x,y,z) is a cigar-shaped potential with ω_z = ω_y ≫ ω_x. The three-dimensional (3D) GP equation (1) can be reduced to a 1D GP equation.^[19] After a dimensionless process (we denote the order parameter with the same ψ), we have

As is known to all, two condensates without interaction can form periodic interference.^[16] However, there is little research on the periodic interference between two condensates with interaction. Here we first demonstrate the periodic interference between two condensates with repulsive interaction and give a theoretical explanation of this periodic interference pattern. The condensates are initially trapped in a double-well potential, which consists of truncated harmonic wells centered at positions x_i with i = 1,2 and ^[20] Here we choose x₁ = −x₂ = x_a, where x_a related to the position of the two minima, and then the potential in Eq. (3) is ^[21] In the initial system, the condensates are divided into two parts without being overlapped spatially. When g_s = 0, the ground state wavefunction can be described by

Table 1.

Table 1.

Table 1. The relations between the physical quantities |ϕg(x = 0)|2, Eg, Ng and xa. . xa 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 |ϕg(x = 0)|2 1.13 0.42 0.02 1.39×10−4 1.27×10−7 1.57×10−11 2.62×10−16 5.92×10−22 Eg 1.0 1.0 2.4725 4.9995 8.50 13.0 18.5 25.0 Ng 2.0 1.3679 1.0183 1.0001 1.0 1.0 1.0 1.0

Table 1. The relations between the physical quantities |ϕg(x = 0)|2, Eg, Ng and xa. .

From Table 1, we can see that when x_a = 5, ϕ_g is normalized to unit 1 with corresponding energy Besides, |ϕ_g(x = 0)|² = 1.57×10⁻¹¹ shows that the condensates are divided into two parts without overlap.

When g_s ≠ 0, the ground state wavefunction of condensates ψ(x;t = 0), chosen as the initial state wavefunction for the following interference, is obtained by propagating in imaginary time starting with the test wavefunction ϕ₀(x) = π^−1/4e^−x2/2. With x_a = 5, the ground state density |ψ (x;t = 0)|² varying with g_s is shown in Fig. 1. Figure 1 shows that when the atom interaction is repulsive, the condensates are divided into two parts without overlap.

Fig. 1.

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	Fig. 1. The ground density of the system with xa = 5 and varying gs.

As is known to all, when atom interaction is too strong, the condensates are trapped in one well (i.e., self-trapping), so we choose g_s = 20 and x_a = 5 to demonstrate the interference between two condensates with repulsive interaction. We aim to analyze the interference between these two condensates. After preparing the initial state, the trap V(x) is suddenly changed into at t = 0. The time evolution of the GP equation (2) is studied numerically by using the time-splitting spectral method,^[22] which was used to solve the GP equation.^[3,23] We discretize Eq. (2) with time step Δt = 0.001 and space step Δx = 0.025. The evolution of condensates’ density distribution is shown in Fig. 2.

From Fig. 2(a), it is surprising that the time evolution of density distribution with g_s = 20 is periodic, and the dark and bright stripes appear in the overlapped regions. This interference between two condensates is similar to a Young’s double slit experiment.^[17] Each condensate executes its own harmonic motion. They have the same kinetic energy and make relative motions. This harmonic motion is similar to that in Refs. [24] and [25], but there are some differences between interference patterns in Fig. 2(a) and that in Refs. [16] and [24]. The most important difference is that the interference pattern here is periodic. Maybe this motion can be described by one-dimensional harmonic oscillator wavefunctions, which will be discussed in the following paper. In the overlapped regions, the density minima shown in Fig. 2(a) along with phase jumps are identified as dark solitons, which is confirmed by the density and phase distributions of two condensates at t = 1.59 as shown in Fig. 2(b). From Fig. 2, we can also see that the number and position of dark solitons are not changed at the points of maximum overlap of two condensates. It is noticeable that the 2π phase jumps in Fig. 2(b) are artifacts of the modular arithmetic.^[3,26]

Fig. 2.

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	Fig. 2. (a) Time evolution of density distributions of two condensates with gs = 20. (b) The density (solid line) and phase (dashed line) distributions of the condensates at t = 1.59.

Comparing Fig. 2 with the interference between two condensates without interaction in Refs. [16] and [27], we find that (i) the time evolution of density and phase distributions of these two scenarios are both periodic, but with different periods. The period of interference between two condensates with repulsive interaction is greater than 2π as shown in Fig. 3; (ii) dark solitons appear in the overlapped regions.

In order to explain this periodic behavior, we assume that the two condensates execute harmonic motions with their periods T_g, and their momentums satisfy

The period T_g in Eq. (4) can be obtained from the time evolution of total condensates density |ψ(x;t)|² in x = 0. The time evolution of |ψ(x = 0;t)|² and |Φ(x = 0;t)|² is shown in Fig. 3. It is noted that the total condensate density |ψ(x = 0;t)|² has a maximum value at the point of maximum overlap of two condensates (i.e., at t = kT_g, k = 1/4,3/4,5/4,…). Figure 3 shows that the period of interference pattern shown in Fig. 2 is T_g = 6.36, which is greater than 2π, the period of interference between two condensates without interaction. This is due to the existence of the repulsive interaction. At the first quarter of period (i.e., 0 ≤ t ≤ T_g/4), the evolution processes arising from interactions produce a resistance force on the condensate tending to reduce its velocity, so the period of interference becomes longer. Figure 3 also demonstrates that during the first quarter of the period (i.e., 0 ≤ t ≤ T_g/4), |ψ(x = 0;t)|² and |Φ(x = 0;t)|² are almost overlapped at t ≤ 1.33, however, there is obvious difference at 1.33 < t ≤ T_g/4 for the impact of interaction on condensate becomes more obvious.

Then we focus on analyzing the impact of interaction on the density distribution in ψ(x;t) for 0 ≤ t ≤ T_g/4. The density distributions in ψ(x;t) and Φ(x;t) at t = 0,1.0,1.2, and 1.4 are shown in Fig. 4. From Figs. 4(a) and 4(b), the density distribution in ψ(x;t) and Φ(x;t) at t = 0 and 1.0 are almost overlapped, because for 0 ≤ t ≤ 1.0, the centers of two condensates execute simple harmonic motions and the repulsive interaction plays little effect on the momentum of two condensates. From Figs. 4(c) and 4(d), we can see that at t = 1.2 and 1.4, the numbers of peak density in ψ(x;t) and Φ(x;t) are the same, however, there are differences in the strength and position of peak density. In this time interval, the effect of repulsive interaction on the momentum of two condensates becomes more obvious, especially at t = 1.59. We will modify the condensates’ momentum P(t) and c_n(t) ≠ 0 (n = 0,2,4) to reduce the differences between Φ(x;t) and ψ (x;t) at t = 1.59.

Fig. 3.

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	Fig. 3. The time evolution of |ψ(x = 0;t)|2 (solid line) and |Φ(x = 0;t)|2 (dashed line) described by Eqs. (5) and (7), respectively.

Fig. 4.

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	Fig. 4. The density distribution in ψ(x;t) (solid line) and Φ(x;t) (dashed line) described by Eqs. (5) and (7) at (a) t = 0, (b) t = 1.0, (c) t = 1.2, and (d) t = 1.4.

The density distribution in ψ(x;t) and Φ(x;t) at t = 1.59 is shown in Fig. 5. The coefficient c_n(t) (n = 0,2,4) at t = 0,1.0,1.2,1.4, and 1.59 can be given as follows:

At t = 1.59, the momentum of Φ(x;t = 1.59) is |P(1.59)| = 5.00. There is obvious difference between the density in Φ(x;t = 1.59) and ψ(x;t = 1.59). However, when the momentum is changed into P₀ = 4.41, and d₀ = 0.9223, d₂ = 0.3681, d₄ = 0.1178, the |Φ_t0|² can coincide well with |ψ(x;t = 1.59)|². This implies that even when two condensates are maximally overlapped, the condensates wavefunction ψ(x;t) can also be expanded with It is also noted that there is a small difference between |Φ_t0|² and |ψ(x;t = 1.59)|², which is due to a numerical error in our simulation and the expansion of ψ(x;t) only with . In summary, the condensates wavefunction ψ(x;t) can be well described by with optimal parameters, seeing Figs. 4(a), 4(b), and 5.

Fig. 5.

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	Fig. 5. The density distribution in ψ (x;t = 1.59) (solid line), Φ (x;t = 1.59) (dashed line), and Φt0 (dashed-dotted line) described by Eqs. (5), (7), and (8), respectively.

In this paper, we demonstrate that two condensates with repulsive interaction execute approximately harmonic motions. They form a periodic interference pattern, which can be expressed in terms of one-dimensional harmonic oscillator wavefunctions with optimal parameters. It is also noticeable that during time evolution of condensates, a small amount of atoms are transferred among three states ϕ_n (n = 0,2,4), which is due to the existence of interaction leading to the internal excitations of atoms. This atom transfer coincides with the energy change, for which the repulsive interaction tends to hold off atoms. Besides, dark solitons are generated when two condensates overlap. Our method provides a way to study the interaction-leading internal excitations of atoms.