Quantum Atom Optics by Tim Byrnes & Ebubechukwu O. Ilo-Okeke

Author:Tim Byrnes & Ebubechukwu O. Ilo-Okeke [Byrnes, Tim & Ilo-Okeke, Ebubechukwu]
Language: eng
Format: epub
Published: 2021-06-19T19:00:00+00:00

determine Ck. How is the Fock state representation of the angular momentum state for a two-level atom different from that of a three-level atom?

8.2.2 Evolution of the Initial State

The atoms are initially prepared with N atoms in one of the internal states such as the |F = 1, mf = −1〉 of 87Rb. This is represented by the state

(8.18)

where |0〉 is the vacuum state. To create a linear superposition of two-component atomic condensates, a π/2 pulse is used (see Section 5.5) to couple the bosonic operators and that act on the vacuum state to create an atom in each state, respectively, according to the following rules:

(8.19)

(8.20)

A single-atom state is transformed as

(8.21)

so that the product state of the N-particle system after the π/2 pulse has been applied becomes

(8.22)

where

(8.23)

is the state vector having n1 = n atoms in one of the hyperfine states |F = 1, mf = −1〉 and n2 = N − n atoms in the other hyperfine state |F = 2, mf = 1〉.

The time evolution of the state vector is governed by the Hamiltonian (8.8)

(8.24)

The states |N, n1 = n, n2 = N − n〉 given by (8.23) are eigenstates of the Hamiltonian, with eigenvalues

(8.25)

The state vector of the system at time t = T is

(8.26)

where

(8.27)

is the linear phase difference accumulated due to the relative difference in the nonlinear self-interactions in each component and environment effects that are coupled to the condensate through the chemical potential. Meanwhile,

(8.28)

is the nonlinear phase per atom due to self-interactions in each BEC component and the mutual interactions between the two components.

Exercise 8.2.5 Verify that the state |N, n, N − n〉 is an eigenstate of (8.12) with eigenvalue (8.25). Verify that the state of the system at any time is as given in (8.26).

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