**Numerical Simulations of Matter Wave Interference**

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Monte Carlo wave function (MCWF)
simulations can be used to understand many interesting signatures of a single
state atom interferometer. Examples include the characteristics of the echo
signal from the interferometer, the relative population of momentum states, the
influence of spontaneous emission on the echo signal and the behaviour of the
interferometer in the Bragg regime. The MCWF method is a well-known approach
for solving dissipation problems in quantum optics. The method is equivalent to
a master equation approach, but the random nature of quantum jumps is simulated
more directly using a Monte Carlo treatment.

**Key Papers**

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(a)
Evolution of the position space probability density as a function of time after
the first excitation pulse for a T ~ 10 nK sample. The pulse parameters are
Ω_{0} = 1 Γ_{n}, Δ =10 Γ_{n } and δt_{1} = 4 τ_{n}.

(b)
Evolution of 2k_{L} Fourier component of density distribution shown in
(a), which represents the experimental signal. A fit to the signal, shown in
gray, gives a recoil frequency ω_{q} = 2¹ × 15.430(1) kHz
and temperature T ~ 7 nK, which is consistent with the input temperature.

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(a) Evolution of the position space
probability density after two sw excitation pulses for a T = 10 μK sample.
The pulse parameters are Ω_{0} = 2 Γ_{n}, Δ = 4
Γ_{n}, δt_{1} = δt_{2} = 4 τ_{n}
and T = 2 τ_{q} = 64.77 μs.

(b)
Evolution of 2k_{L} Fourier component of density distribution shown in
(a). The signal decays exponentially after each excitation pulse due to Doppler
dephasing on a timescale consistent with t_{coh} ~ 4 μs. In the
vicinity of t = 2T, interference between momentum states creates a modulation
in the density distribution and the grating echo is formed.

Probabilities of momentum states associated with the ground
state versus τDP; probabilities extracted from fits using g_{f }(δT),
Δ = 39 MHz and Ω ~ 18 MHz. The points are joined using a 5^{th}
order interpolation function.

b) MCWF simulations neglecting the effects of spontaneous emission and spatial profile showing probabilities versus τDP; Δ = 39 MHz and Ω = 18 MHz.

c) MCWF simulations including the effects of spontanoues emission and spatial profile for the same conditions as in b).