Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space

Elke Fasshauer and Axel U. J. Lode

Phys. Rev. A 93, 033635 – Published 21 March 2016

Abstract

We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on, and exploits the recursive implementation of MCTDH for bosons (r-mctdhb) package. Together with r-mctdhb, the present implementation of MCTDHF forms the mctdh-x package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate that MCTDHF is capable of solving the time-dependent many-fermion Schrödinger equation to an arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. We study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: The escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.

Received 10 October 2015
Revised 2 November 2015

DOI:https://doi.org/10.1103/PhysRevA.93.033635

Physics Subject Headings (PhySH)

Fermions

Nonperturbative methods

Atomic, Molecular & Optical

Authors & Affiliations

Elke Fasshauer^*

Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø—The Arctic University of Norway, N-9037 Tromsø, Norway

Axel U. J. Lode^†

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

^*elke.fasshauer@uit.no
^†axel.lode@unibas.ch

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Vol. 93, Iss. 3 — March 2016

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Images

Figure 1
Convergence of the MCTDHF for the TDHIM model for $N = 2$ and $N = 7$ . The upper panel shows the time-dependent parameter $f (t)$ modulating both interactions and the frequency of the parabolical trap [cf. Eqs. (14) and (15)]. The lower panel depicts the computed results for the time-dependent energy $ε_{CM}^{'} (t)$ for the exact solution (solid red line) and MCTDHF( $M$ ) for increasing numbers of orbitals $M$ for two (blue dashed lines) and seven fermions (black dashed-dotted line with points). Fast convergence of the MCTDHF( $M$ ) predictions for $ε_{CM}^{'} (t)$ to the exact value is observed for the present case of almost erratically time-dependent one- and two-body parts in the Hamiltonian when $M$ is increased. We find that the MCTDHF prediction for $ε_{CM}^{'} (t)$ is indistinguishable from the exact energy for $M = 10$ in the case of $N = 2$ and for $M = 15$ for the case of $N = 7$ . All quantities shown are dimensionless.
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Figure 2
Scheme and model for the fermion tunneling to open space process. (a) The setup used to study how initially parabolically confined (black dashed line) fermions tunnel to open space when the potential is transformed to its open form (solid black line). For reference, an initial density is sketched (blue line). After the transformation of the potential, the system becomes unbound and tunnels through the barrier to open space. (b) Model for the tunneling process. The one-dimensional Hilbert space is split into IN and OUT regions at the maximum of the barrier (vertical red line). The many-fermion process can be built up from concurrently happening single-fermion tunneling processes: Each fermion escapes into OUT with a characteristic momentum. This characteristic momentum is in turn defined by the chemical potentials $μ_{i}$ of the harmonically trapped, interacting system in the IN region before the ejection. Neglecting interactions in the OUT region, one obtains momenta $k_{i} = \sqrt{2 m μ_{i}}$ . (The figure is adapted from Refs. [38, 41], and all quantities shown are in dimensionless units.)
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Figure 3
Comparison of characteristic momenta of two-fermion tunneling to open space to model predictions. Panel (a) shows the overall momentum distributions $ρ (k)$ at times $t = 0, 30, 100$ in the time evolution of the tunneling process for $N = 2$ and $λ_{0} = 1.0$ . Centered around $k = 0$ , the momentum distributions have a broad feature corresponding to the trapped fraction of the system (cf. harmonic oscillator [HO] background label). The particles that escape to open space correspond to the peaks which emerge in the momentum distributions with time. These peaks' momentum distributions are enlarged for $λ_{0} = 1.0$ in panel (b) and for $λ_{0} = 0.5$ in panel (c). According to the model in Fig. 2, the characteristic fermion ejection momenta are defined by the chemical potentials $μ_{i}$ of the harmonically trapped, interacting system. The obtained momenta $k_{i} = \sqrt{2 m μ_{i}}$ are shown as black arrows in panels (b) and (c). The position of the main peaks agrees very well with the prediction of the model, $k_{1}, k_{2}$ . The additional structure in the case of the stronger interactions in panel (b) can be attributed to a correlated two-body escape process (see text for further discussion). The momentum associated with this process is $k_{red} = \sqrt{2 κ (μ_{1} + μ_{2})} = 1.613$ , where $κ = \frac{1}{2}$ is the reduced mass of the two-fermion system [see gray arrow in panel (b)]. For the weaker interactions $λ_{0} = 0.5$ in panel (c), the two-fermion tunneling or escape seems to be less favorable, since no clear features are seen at the respective momentum $k_{red} = 1.538$ . All quantities shown are dimensionless. See text for further discussion.
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Figure 4
Spatial and momentum one-body correlation functions of fermion tunneling to open space. We plot the correlation functions $| g^{(1)} (x^{'}, x; t) |^{2}$ and $| g^{(1)} (k^{'}, k; t) |^{2}$ in real and momentum space in the top and bottom rows, respectively, for the $λ_{0} = 1$ case. In comparison to the bosonic case (see Refs. [41, 54]), we observe a generally more strongly correlated behavior in both real and momentum space for the ground states depicted in the left column. In the momentum correlations (top left and top right panels), a line structure emerges at the momenta corresponding to the escaping fermions, $k_{1}, k_{2}$ with time. While the fermions that still reside in the well at $k \approx k^{'} \approx 0$ are almost uncorrelated ( $| g^{(1)} |^{2} \approx 1$ ), the fermion escaping with momentum $k_{1}$ is uncorrelated with those at rest ( $| g^{(1)} (k^{'} \approx k_{1}, k \approx 0; t = 200) |^{2} \approx 0$ , see upper right panel). Interestingly, the fermion escaping with $k_{2}$ is correlated with the one at rest ( $| g^{(1)} (k^{'} \approx k_{2}, k \approx 0; t = 200) |^{2} \approx 1$ , see black arrows in top right panel). However, the $k_{2}$ velocity is embedded by thin lines where $| g^{(1)} (k^{'} = k_{2} \pm ε, k; t = 200) |^{2} < 1$ (see dark lines next to the arrows in the top right panel). The periodic structure in the real-space correlation in the bottom right panel reflects the momentum correlations in the top right panel. To guide the eye, we mark the top of the barrier by the black lines. The difference in the escape velocities $k_{1}, k_{2}$ leads to a strong spatial correlation (see rectangular areas with $| g^{(1)} (x, x^{'}; t = 200) |^{2} \approx 0$ on the off-diagonal of the bottom right panel). All quantities shown are dimensionless; see text for further discussion.
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Figure 5
Momentum two-body correlation functions of fermion tunneling to open space. We plot the correlation functions $g^{(2)} (p_{1}, p_{2}, p_{1}, p_{2}; t)$ in momentum space, for the $λ_{0} = 1$ case for times $t = 0$ (left) and $t = 200$ (right). In both panels, the Pauli exclusion principle is manifest, since $g^{(2)} (p, p, p, p) = 0$ . For the initial state in the left panel, we find large values of $g^{(2)} (p_{1}, p_{2}, p_{1}, p_{2})$ , where ${|g^{(1)} (p_{1}, p_{2})|}^{2}$ is small and vice versa (cf. top left panel of Fig. 4). This also vaguely holds for the $t = 200$ plot (cf. top right panel of Fig. 4). In the right panel at $t = 200$ , the black arrows mark the escape momenta obtained from our model and the momentum distributions (cf. Fig. 3). The $p_{1 / 2} = k_{2} = 1$ tunneling at $p_{1 / 2} = k_{2} = 1.0$ is anticorrelated or antibunched ( $g^{(2)} < 1$ ), whereas the tunneling at $k = 2.103$ is correlated or bunched ( $g^{(2)} > 1$ ). The found correlated pair tunneling process emerges as a strongly bunched structure on the diagonal around $k_{red} = 1.613$ , i.e., $g^{(2)} (k_{red}, k_{red}, k_{red}, k_{red}) ≫ 1$ . All quantities shown are dimensionless; see text for further discussion.
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