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CritPt / Challenge_29_main / Fermionic atoms trapped in an optical lattice are used to simulate the Hubbard…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

code template

Code

import sympy as sp
import sympy.physics.quantum.constants

lambda_ = sp.symbols('lambda')
E       = sp.symbols('E')
W       = sp.symbols('W')
alpha   = sp.symbols('alpha')
m       = sp.symbols('m')
a_s     = sp.symbols('a_s')
hbar    = sp.physics.quantum.constants.hbar

def answer(lambda_, E, W, alpha, m, a_s, hbar):
    r"""
    Return the expressions of tunneling energy t and contact interaction U in Sympy format.

    Inputs
    ----------
    lambda_:      sympy.Symbol, wavelength, $\lambda$
    E      :      sympy.Symbol, beam amplitude, $E$
    W      :      sympy.Symbol, beam waist, $W$
    alpha  :      sympy.Symbol, atomic polarizability, $\alpha$
    m      :      sympy.Symbol, atomic mass, $m$
    a_s    :      sympy.Symbol, s-wave scattering length, $a_s$
    hbar   :      sympy.Symbol, reduced Planck constant, $\hbar$

    Outputs
    ----------
    t      :      sympy.Expr, the expression of tunneling energy t associated with the Hubbard model
    U      :      sympy.Expr, the expression of contact interaction U associated with the Hubbard model
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    t = ...  # a SymPy expression for the tunneling energy
    U = ...  # a SymPy expression for the contact interaction
    # ---------------------------------------------------------------

    return t, U

problem description

# Problem setup: Fermionic atoms trapped in an optical lattice are used to simulate the Hubbard model. Specifically, two pairs of counter-propagating beams intersect at 90 degrees. The beams have equal wavelength

\lambda

, amplitude

E

, beam waists

W

, and are polarized normal to the plane in which they intersect. The atoms are trapped in the resulting interference pattern because of their polarizability

\alpha

at a wavelength of

\lambda

. The atoms have a mass

m

and an

s

-wave scattering length of

a_s

. # Main problem: Compute the tunneling energy

t

and contact interaction

U

associated with the resulting Hubbard model. To obtain analytical answers, assume that the Wannier functions can be approximated by the harmonic-oscillator eigenstates and that

W\gg\lambda

and

V_0\gg E_R

Plain-text mathematical notation (without MathML)

# Problem setup:

Fermionic atoms trapped in an optical lattice are used to simulate the Hubbard model. Specifically, two pairs of counter-propagating beams intersect at 90 degrees. The beams have equal wavelength λ, amplitude E, beam waists W, and are polarized normal to the plane in which they intersect. The atoms are trapped in the resulting interference pattern because of their polarizability α at a wavelength of λ. The atoms have a mass m and an s-wave scattering length of a_(s).

# Main problem:

Compute the tunneling energy t and contact interaction U associated with the resulting Hubbard model.

To obtain analytical answers, assume that the Wannier functions can be approximated by the harmonic-oscillator eigenstates and that W≫λ and V₀≫E_(R).

Original LaTeX notation

# Problem setup:

Fermionic atoms trapped in an optical lattice are used to simulate the Hubbard model. Specifically, two pairs of counter-propagating beams intersect at 90 degrees. The beams have equal wavelength $\lambda$, amplitude $E$, beam waists $W$, and are polarized normal to the plane in which they intersect. The atoms are trapped in the resulting interference pattern because of their polarizability $\alpha$ at a wavelength of $\lambda$. The atoms have a mass $m$ and an $s$-wave scattering length of $a_s$.

# Main problem:

Compute the tunneling energy $t$ and contact interaction $U$ associated with the resulting Hubbard model.

To obtain analytical answers, assume that the Wannier functions can be approximated by the harmonic-oscillator eigenstates and that $W\gg\lambda$ and $V_0\gg E_R$.

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Source and history

upstream_id / Original problem ID: Challenge_29_main
task_id: 0f0a7e02-ffb4-54cd-ae1d-0b7b13d7285b
task_key: train--Challenge~5f29~5fmain
task_revision_id: 1
config / split: / train

Official source

initial import