Gerry & Knight Book

Chapter 7.7

pp 174-182

Assume for all problems that alpha is real.

1. Derive Eq. 7.113 for the cat normalization and fix the typo.

2. Derive the photon number Pn for all three cat states, Eq.7.122, 7.123, 3.25 and verify that Pn for the Y.S. cat is the same for the ordinary coherent state and the statistical mixture of Eq.7.120.

3. Compute the Mandel Q parameter, Eq.7.99, for the three cats (dead and alive) and the statistical mixture (dead or alive). Plot these as a function of nbar=alpha^2. What can we conclude about the cats as nbar becomes larger?

4. Compute the quadrature variances for all three cats and the statistical mixture and verify (or fix the typos in) Eqs.7.126–7.133. Plot the variances as a function of nbar=alpha^2. What can you conclude about the cats as nbar becomes larger?

5. Compute the Husimi-Bopp Q function, Eq.3.112, for all three cats and the statistical mixture. Plot the Q function in 3D for all four cases and compare to the results in the notes.

6. Using the results from Chapter 3.8, computer the characteristic function CA(lambda) for all four Q functions, using Eq.3.128c. Then compute the characteristic function for each corresponding Wigner function using Eq.3.129. Finally use Eq. 3.136 to compute all four Wigner functions (one for each cat and another for the mixture). Compare your results to Eqs.7.136-7.139 in the text and then plot the four Wigner functions using the same values of alpha you used in the Q(beta) function. Compare the plots. What can you conclude in the regimes nbar>>1 and nbar<<1?