Multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multicomponent Bose-Einstein condensates

Axel U. J. Lode

Phys. Rev. A 93, 063601 – Published 1 June 2016

Abstract

In this paper, the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is derived for the case of $N$ identical bosons with internal degrees of freedom. The theory for bosons with internal degrees of freedom constitutes a generalization of the MCTDHB method that substantially enriches the many-body physics that can be described. We demonstrate that the numerically exact solution of the time-dependent many-body Schrödinger equation for interacting bosonic particles with internal degrees of freedom is now feasible. We report on the MCTDHB equations of motion for bosons with internal degrees of freedom and their implementation for a general many-body Hamiltonian with one-body and two-body terms, both of which may depend on the internal states of the considered particles and time. To demonstrate the capabilities of the theory and its software implementation integrated in the mctdh-x software, we apply MCTDHB to the emergence of fragmentation of parabolically trapped bosons with two internal states: we study the ground state of $N = 100$ bosons as a function of the separation between the state-dependent minima of the two parabolic potentials. To quantify the coherence of the system, we compute its normalized first-order correlation function. We find that the coherence within each internal state of the atoms is maintained, while it is lost between the different internal states. This is a hallmark of a kind of fragmentation absent in bosons without internal structure. We term the emergent phenomenon “composite fragmentation.”

Received 18 February 2016

DOI:https://doi.org/10.1103/PhysRevA.93.063601

Physics Subject Headings (PhySH)

Bose gases Bose-Einstein condensates

Nonperturbative methods

Atomic, Molecular & Optical

Authors & Affiliations

Axel U. J. Lode^*

Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

^*axel.lode@unibas.ch

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Vol. 93, Iss. 6 — June 2016

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Images

Figure 1
State-dependent potentials. In each internal state ( $α = +$ and $α = -$ ) of the atoms, the potential is harmonic. The minima of the potentials are displaced by the separation parameter $Δ$ . All quantities shown are dimensionless.
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Figure 2
Ground-state density and fragmentation as a function of the separation $Δ$ . The top panel shows the composite density $ρ^{+ / -} (x)$ of the two internal states $α = +$ and $α = -$ . The second and third panels depict the component densities $ρ^{α} (x)$ of the system in the respective internal state $α = +$ and $α = -$ . The bottom panel shows the fragmentation of the system. Fragmentation is energetically favorable as soon as the overlap of the densities of the internal states becomes small (cf. top and bottom panels). All quantities shown are dimensionless.
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Figure 3
Signatures of composite fragmentation in the one-body correlation function as a function of the separation $Δ$ . The rows of panels correspond to the separations $Δ = 0, Δ = 1$ , and $Δ = 5.5$ from top to bottom. The values of fragmentation are, $F = 0.004, F = 0.006, F = 0.490$ , respectively. The first column shows the composite correlation function of both internal states, $| g^{(1), + / -} |^{2}$ , the middle column shows the correlation function of the $α = +$ state, $| g^{(1), +} |^{2}$ , and the right column shows the correlation function of the $α = -$ state, $| g^{(1), -} |^{2}$ . The correlations are only plotted for coordinates $(x, x^{'})$ if the component (composite) one-body density at these coordinates is larger than $0.05$ , to avoid analyzing component (composite) correlations where there are no particles. While the component correlations exhibit full coherence, i.e., $| g^{(1), α} |^{2} \approx 1$ in the middle and left columns, the composite correlation function shows a quick loss of coherence between the components, i.e., $| g^{(1), + / -} |^{2} \approx 0$ on the off-diagonals in the left column: the fragmentation in the system is of “composite” type. All quantities shown are dimensionless; see text for further discussion.
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Figure 4
Natural orbitals ${\vec{φ}}_{1}^{(NO)} (x)$ and ${\vec{φ}}_{2}^{(NO)} (x)$ as a function of the separation $Δ$ . The top row shows the composite natural orbitals $ϕ_{k}^{+ / -, (NO)} (x) = \sum_{α} 1_{x}^{α} ϕ_{k}^{α, (NO)} (x)$ of the two internal states $α = +$ and $α = -$ . The second and third rows depict the component natural orbitals $ϕ_{k}^{α, (NO)} (x)$ of the system in the respective internal state $α = +$ and $α = -$ . For the separations, where the fragmentation increases to its maximal value, the composite natural orbital densities are localized in both potential wells (compare top panels and Fig. 2, lower panel). For separations $Δ ≳ 4.5$ , the composite natural orbital densities become localized (top panels) because the component first (second) natural orbital densities in the state $α = -$ ( $α = +$ ) become zero (middle and lower panels). This localization is accompanied by the fragmentation reaching its maximum. All quantities shown are dimensionless.
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Figure 5
Coefficients as a function of the separation $Δ$ . The magnitude of the coefficients $| C_{n} |^{2}$ is plotted (see Ref. [53] for the formula to compute the index $n$ from the vector $\vec{n}$ ) for the separations $Δ = 1, 4, 5.5$ in the top, middle, and bottom panels, respectively. The thick vertical gray line through all panels shows the configuration $| \frac{N}{2}, \frac{N}{2}, 0 〉$ for which the bosons are equally distributed in the first two one-particle basis states. For small separations, the system is essentially described by a single coefficient. Encompassing its fragmentation, the distribution of coefficients centers itself around the equally partitioned configuration $| \frac{N}{2}, \frac{N}{2}, 0 〉$ and broadens significantly. All quantities shown are dimensionless.
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