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Problem Sheet 3 - Assessed Problems - Autumn 2016/17
To be handed in in the School Office, due 14:00 Wed 30th Novebember 2016 3.1 A two-level atom is placed in a cavity supporting a single mode of the electromagnetic field of frequency ?21 (resonant with the atom). The atom-cavity coupling
strength is h
? g. Neglect damping and radiative decay. (2) a. Assume that initially the atom is in the excited state, |2i, and the cavity is in
the ground state. Consider the basis |1, ni, |2, ni, corresponding to the atom
in states |1i, |2i, and n photons in the cavity. Calculate the quantum state of
the system as a function of time in this basis. (2) b. Under the conditions of question 2.a, calculate the probability of finding the
atom in the ground state as a function of time. (2) c. Under the conditions of question 2.a, calculate the mean value of the quadrature
operator X = a + a? as a function of time. (3) d. Assume now that initially the atom is in the superposition state |?i = ?12 (|1i + i|2i)
and the cavity is in the ground state. Calculate the quantum state of the system as a function of time in the basis |1, ni, |2, ni. (3) e. Under the conditions of question (d), calculate the mean value of the quadrature operator X = a + a? as a function of time. 3.2 Consider a system of N atoms interacting with a single mode of the electromagnetic
field (Tavis-Cummings model). We assume that the mode is resonant with the
2 ?z,j + h
? ?a? a + h
j=1,...,N X ?j+ a + ?j? a? . (1) j=1,???N The initial state of the system has n = 0 photons, and it is a linear combination
of quantum states with only one atom excited in state |2i. Each term in the linear
superposition corresponds to a different atom being excited (spin-wave excitation),
|?(0)i = X
? (|1i1 . . . |2ij . . . |1iN ) |n = 0i,
N (2) where |1ij , and |2ij correspond to atom j in levels 1 and 2, respectively.
(4) a. Calculate the quantum state of the system as a function of time (4) b. Calculate the evolution of the number of photons in the cavity ha? ai. How
does the oscillation frequency scale with the number of atoms, N ?
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