Multiconfigurational Hartree-Fock theory for identical bosons in a double well

Multiconfigurational Hartree-Fock theory is presented and implemented in an investigation of the fragmentation of a Bose-Einstein condensate made of identical bosonic atoms in a double-well potential at zero temperature. The approach builds in the effects of the condensate mean field and of atomic correlations by describing generalized many-body states that are composed of multiple configurations which incorporate atomic interactions. Nonlinear and linear optimization is utilized in conjunction with the variational and Hylleraas-Undheim theorems to find the optimal ground and excited states of the interacting system. The resulting energy spectrum and associated eigenstates are presented as a function of double-well barrier height. Delocalized and localized single configurational states are found in the extreme limits of the simple and fragmented condensate ground states, while multiconfigurational states and macroscopic quantum superposition states are revealed throughout the full extent of barrier heights. Comparison is made to existing theories that either neglect mean field or correlation effects and it is found that contributions from both interactions are essential in order to obtain a robust microscopic understanding of the condensate’s atomic structure throughout the fragmentation process.

Figure 1

(Color online) Parametrized model ground and excited state energies for $g=0.1\hslash \omega {\beta }^3$ and $N=20$ as a function of $\alpha$. Note the energy level mergings and resulting ridge structure separating BEC (small $\alpha$) and fragmented states (large $\alpha$). The Fock states below the ridge are delocalized and nondegenerate while the states above the ridge are localized and doubly degenerate. The alternation of line style has been chosen to aid in visualization.

Figure 2

(Color online) Schrödinger model ground and excited state energies for $g=0.1\hslash \omega {\beta }^3$ and $N=20$ as a function of barrier height. Note the qualitative similarities of this energy level correlation diagram to that of Fig. 1. The solid curve beginning at $20\hslash \omega$ is the single-particle Schrödinger energy while the remaining curves are the eigenvalues of the Hamiltonian (6) built from the Schrödinger solutions at each barrier height. Note that the Schrödinger energy $N_1{\epsilon }_1+N_2{\epsilon }_2$ differs from the many-body eigenvalues $E^{\nu }$ because it includes no atom-atom interaction. The alternation of line style has been chosen to aid in visualization.

Figure 3

(Color online) The BHF energy reduces to that of the GP ground state when $N_1^{\prime }∕N=1$ and to that of the first GP excited state when $N_1^{\prime }∕N=0$. This corresponds to all particles in the symmetric GP ground state and the first antisymmetric GP excited state, respectively. A dimensionless interaction strength of ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)] for 100 atoms in a single-well potential was used in this calculation of $E^{\mathrm{BHF}}$.

Figure 4

(Color online) BHF single-particle wave functions ${\chi }_1$ and ${\chi }_2$ versus coordinate $z$ at four different barrier heights 0, 3, 9, $13\hslash \omega$. A dimensionless interaction strength of ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)] was used. In each panel, ${\chi }_1$ corresponds to the BHF configuration $\mid N,0⟩$, while ${\chi }_2$ corresponds to the BHF configuration $\mid 0,N⟩$. The BHF energies associated with $\mid N,0⟩$ and $\mid 0,N⟩$ at zero barrier height appear in Fig. 3. Both wave functions have been set at the chemical potentials ${\mu }_{11}$ and ${\mu }_{22}$ associated with each configuration. At each barrier height, we plot the corresponding Thomas-Fermi wave function as a solid black curve.

Figure 5

(Color online) Schrödinger single-particle wave functions ${\chi }_1$ and ${\chi }_2$ versus coordinate $z$ at four different barrier heights 0, 3, 9, $13\hslash \omega$. These solutions have no dependence on the number of atoms in each single-particle state. Both wave functions have been set at the same BHF chemical potentials used in Fig. 4. At each barrier height, we plot the Thomas-Fermi wave functions of Fig. 4 as a solid black curve. Little resemblance is seen between the Schrödinger and Thomas-Fermi results.

Figure 6

(Color online) MCBHF ground and excited state energies and BHF ground state energy for $N=100$ atoms in a single-well potential (or zero barrier limit of a double well) with a dimensionless coupling of ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)] ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)] versus BHF reference configuration. The solid black curve represents the BHF ground state energy. Bullets are placed at the minima of BHF ground and MCBHF ground and excited energies. The line style is alternated to aid in visualization. This figure is a zoomed in and more detailed version of the first panel in Fig. 9.

Figure 7

(Color online) MCBHF expansion coefficients $C_{N_1}^{51}$ of the 51st excited state ${\mid {\Psi }^N;\{{56}^{•},{44}^{•}\}⟩}_{51}$ versus the occupation number in each GCS. The label {56, 44} in the state vector signifies that the minimal 51st MCBHF excited state energy has been attained in the BHF configuration $\mid N_1^{\prime }=56,N_2^{\prime }=44⟩$. This probability amplitude corresponds to 100 atoms in a single-well trapping potential with ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)].

Figure 8

(Color online) BHF single-particle wave functions ${\chi }_1$ and ${\chi }_2$ versus coordinate $z$ associated with the BHF configuration $\mid N_1^{\prime }=56,N_2^{\prime }=44⟩$, which underlies the optimal 51st MCBHF excited state at a barrier height of $0\hslash \omega$. A dimensionless coupling of ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ [see (47)] for 100 atoms was used. Both wave functions have been set at the chemical potentials ${\mu }_{11}$ and ${\mu }_{22}$ associated with this configuration. The corresponding Thomas-Fermi wave function is plotted as a solid black curve.

Figure 9

(Color online) MCBHF ground and excited state energies and BHF ground state energy for ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ and $N=100$ versus BHF configuration. Four different barrier heights 0, 3, 9, and $13\hslash \omega$ are displayed from left to right and top to bottom. The solid black curve in each panel represents the BHF ground state energy at each barrier height and for each BHF configuration. Bullets are placed at the minima of the BHF ground state and all MCBHF energies. These are the variationally optimal states at each energy level. To aid in visualization, we alternate line style and only display every fifth excited state. The box in the first panel corresponds to Fig. 6.

Figure 10

(Color online) MCBHF energy level correlation diagram versus barrier height for ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ and $N=100$ atoms. Every fifth energy level is plotted in the left-hand panel and the line styles are alternated in both to aid in visualization. The solid black curve in the left-hand panel denotes the BHF ground state energy. Energy level mergings, which lead to a pronounced ridge structure separating nondegenerate oscillator-like states from doubly degenerate macroscopic superposition states, are focused on in the second panel. This panel is a zoomed and more detailed version of the boxed region in the first, where every line has been displayed.

Figure 11

(Color online) Schrödinger energy level correlation diagram versus barrier height for ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ and $N=100$ atoms. Every fifth energy level is plotted in the left-hand panel and the line styles are alternated in both to aid in visualization. The second panel is a zoomed and more detailed version of the boxed region in the first, where every line has been displayed.

Figure 12

(Color online) Left-hand side two panels, expansion coefficients of the optimal MCBHF ground state and first and second excited states versus occupation number in each GCS as well as expansion coefficients of the optimal 63rd, 64th, and 65th excited states versus occupation number in each GCS. The optimal BHF reference configurations $\mid N_1^{\prime },N_2^{\prime }⟩$ have been provided for each state separately within the text. Right-hand side two panels, expansion coefficients of the Schrödinger ground state and first and second excited states versus occupation number as well as expansion coefficients of the Schrödinger 63rd, 64th, and 65th excited states versus occupation number. These probability amplitudes correspond to $N=100$ atoms at a barrier height of $3\hslash \omega$. The left-hand column has the dimensionless coupling ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ in both the many-body Hamiltonian and BHF equations, while the right-hand column has ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$ only in the many-body Hamiltonian.

Figure 13

(Color online) BHF single-particle wave functions ${\chi }_1$ and ${\chi }_2$ versus coordinate $z$ at a barrier height of $3\hslash \omega$. The first panel corresponds to the BHF configuration that optimizes the MCBHF ground state. That is, the first panel depicts the BHF configuration ∣99,1⟩ associated with ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$. The second panel correspond to the BHF configuration that optimizes the 63rd MCBHF excited state. That is, the second panel depicts the BHF configuration ∣50,50⟩ associated with ${\alpha }_{\mathrm{Q}1\mathrm{D}}=40$. All wave functions have been set at the chemical potentials ${\mu }_{11}$ and ${\mu }_{22}$ associated with each configuration. At each barrier height, we plot the corresponding Thomas-Fermi wave function as a solid black curve.