CoCalc -- 41.sagews

All published worksheets from http://sagenb.org

Project: sagenb.org published worksheets

Views: ¹⁶⁸⁷³⁸
Image: ubuntu2004

# Define some symbolic variables
var('a k_eff T v m h gamma')

\left(a, k_{\mbox{eff}}, T, v, m, h, \gamma\right)

# Expression of the phase shift. Attention: k_eff is the effective Raman wave vector.
phi = a*k_eff*T^2 + (h * k_eff^2 * T^3 * gamma)/(2*m) + k_eff*T^2 * (7*a*T^2/12 - T*v)*gamma + k_eff*T^4*(31*a*T^2/360 - h*k_eff*T/(2*m) - T*v/4)*gamma^2
phi

<html><span class="math">{{{k_{\mbox{eff}} {T}^{4} } \left( \frac{{{31 a} {T}^{2} }}{360} - \frac{{v T}}{4} - \frac{{{h k_{\mbox{eff

T}}{{2 m}} \right)} {\gamma}^{2} } + \frac

}}}

{{{id=11|
# Truncate phi to the first order value (this is expression 64 of Borde, Metrologia 2002, 39, 435-463)
phi = taylor(phi, gamma, 0, 1).expand()
phi

<html><span class="math">\frac{{{{{7 a} k_{\mbox{eff

{T}^{4} } \gamma}}{12} -

}}}

{{{id=13|
# Divide the phase shift by the scaling factor
var('S')
assume(S > 0)
phi = phi.subs_expr(T==sqrt(S/k_eff))
Phi = (phi/S).expand()
Phi

<html><span class="math">\frac{{{{7 a} S} \gamma}}{{12 k_{\mbox{eff

- \frac

}}}

{{{id=9|
var('epsilon kappa delta_a')
Phi_Rb = Phi.subs(k_eff=k_eff*(1+epsilon), m=m*kappa, a=a+delta_a)

delta_Phi = Phi - Phi_Rb
# Remove the UFF component, to keep only the gradient component
delta_Phi = delta_Phi
delta_Phi.simplify()

<html><span class="math">\frac{{{{-7 \left( \delta_{a} + a \right)} S} \gamma}}{{{12 \left( \epsilon + 1 \right)} k_{\mbox{eff

+ \frac

}}}

{{{id=21|
delta_Phi = taylor(delta_Phi, epsilon, 0, 1)
(delta_Phi/(gamma*S)).expand().simplify().factor().simplify()

<html><span class="math">\frac{{{{{{{{7 \delta_{a}} \epsilon} \kappa} \sqrt{ k_{\mbox{eff}} }} m} {S}^{\frac{3}{2}} } \gamma} + {{{{{{{7 a} \epsilon} \kappa} \sqrt{ k_{\mbox{eff}} }} m} {S}^{\frac{3}{2}} } \gamma} - {{{{{{7 \delta_{a}} \kappa} \sqrt{ k_{\mbox{eff}} }} m} {S}^{\frac{3}{2}} } \gamma} - {{{{{{{6 \epsilon} \kappa} k_{\mbox{eff

m} v} S} \gamma} +

}}}

{{{id=14|
# 1 second estimate
numerics = dict(
  a = 9.81,
  v = 900/3.6,
  k_eff = 4*3.14/767e-9,
  epsilon = 1/60.,
  S = (4*3.14/767e-9)*1^2,
  m = 1.49e-25,
  h = 1e-34,
  kappa = 1/2.18,
)
delta_Phi.subs(numerics).expand()

{{-0.573611111111111 \delta_{a}} \gamma} - {1.994542415538619 \gamma} - {1.00000000000000 \delta_{a}}

print latex(delta_Phi.subs(numerics).expand())

{{-0.573611111111111 \delta_{a}} \gamma} - {1.994542415538619 \gamma} - {1.00000000000000 \delta_{a}}

# 2 seconds estimate
numerics2 = numerics.copy()
numerics2['S'] = (4*3.14/767e-9)*2^2
delta_Phi.subs(numerics2).expand()

{{-2.294444444444445 \delta_{a}} \gamma} - {3.798334831077238 \gamma} - {1.00000000000000 \delta_{a}}

# 1 second estimate, with very small velocity
numerics3 = numerics.copy()
numerics3['v'] = 900./3600
delta_Phi.subs(numerics3).expand()

{{-0.573611111111111 \delta_{a}} \gamma} + {0.0867075844613810 \gamma} - {1.00000000000000 \delta_{a}}