Bose-Hubbard dimer: number squeezing in a bosonic Josephson junction#

The two-mode Bose-Hubbard model describes interacting bosons in a double well and provides the minimal quantum model of a bosonic Josephson junction.[1][2][3] At fixed total particle number,

\[ H=-J\left(b_L^\dagger b_R+b_R^\dagger b_L\right) +\frac{U}{2}\sum_{\alpha=L,R}n_\alpha(n_\alpha-1). \]

The competition is between tunneling, which favors phase coherence across the wells, and repulsion, which suppresses number imbalance.

For \(N=8\), the fixed-\(N\) Hilbert space contains \(N+1=9\) Fock states.

Schwinger representation#

Define

\[ S_x=\frac12\left(b_L^\dagger b_R+b_R^\dagger b_L\right), \qquad S_z=\frac12(n_L-n_R). \]

Up to an \(N\)-dependent constant,

\[ H=-2J S_x+U S_z^2. \]

from edinpy import boson as edb

N = 8
J = 1.0
U = 4.0

well = edb.DoF(2, name="well", labels=("left", "right"))
modes = edb.BosonModes(well)
sector = edb.NParticleSector(modes, N=N).build()

H = (
    edb.Hopping(0, 1, -J, modes)
    + edb.Hubbard(0, U, modes)
    + edb.Hubbard(1, U, modes)
)

Sx = edb.SpinX(0, 1, modes)
Sz = edb.SpinZ(0, 1, modes)

Low-energy spectrum#

Lowest excitation energies of the Bose-Hubbard dimer versus U over J — Lowest excitation energies of the \(N=8\) Bose-Hubbard dimer. Repulsion changes the level spacing as fluctuations in the occupation imbalance become increasingly costly.#

First-order coherence#

The normalized transverse coherence is

\[ \mathcal C=\frac{\langle S_x\rangle}{N/2}. \]

At \(U=0\) all particles occupy the bonding orbital and \(\mathcal C=1\).

hamiltonian = edb.Hamiltonian(H, sector)
hamiltonian.eigsolve(k=None)
psi0 = hamiltonian.eigenstate(0)
coherence = np.real(psi0.dag * Sx * psi0) / (N / 2)

Interwell coherence of the Bose-Hubbard dimer versus U over J — Normalized interwell coherence. Repulsive interactions progressively reduce the bonding-orbital coherence of the ground state.#

Number squeezing#

The normalized number-imbalance fluctuation is

\[ \mathcal V_z=\frac{\langle S_z^2\rangle}{(N/2)^2}. \]

imbalance = np.real(psi0.dag * Sz * Sz * psi0) / (N / 2) ** 2

Number-imbalance fluctuations of the Bose-Hubbard dimer versus U over J — Normalized imbalance fluctuations. Increasing \(U/J\) squeezes the relative particle number between the two wells.#

Fock-space redistribution#

The Fock basis is

\[ |n_L,N-n_L\rangle, \qquad n_L=0,1,\ldots,N, \]

with probabilities

\[ P(n_L)=|\langle n_L,N-n_L|\psi_0\rangle|^2. \]

Ground-state occupation probabilities of the Bose-Hubbard dimer for three U over J values — Ground-state Fock probabilities. The broad binomial-like distribution at weak coupling narrows around balanced occupation as repulsion suppresses number fluctuations.#