Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 60 / 5 / d

Solution

The Heisenberg update is

d s = U^{†} s U - s

; the left side of the printed first update relation must be this increment, rather than

s (t)

itself. For

s = a

, the linear noise term in the supplied stochastic equation is

γ [a^{†} \otimes d B - a \otimes d B^{†}, a] = - γ d B,

[a, a^{†}] = I

The two thermal drift expressions simplify separately to

2 a^{†} aa - a a^{†} a - a^{†} aa = - a, 2 aa a^{†} - aa a^{†} - a a^{†} a = a .

The anomalous terms vanish because

[a^{†}, [a^{†}, a]] = 0

and

[a, [a, a]] = 0

. Hence the factor inside the drift braces is

- (N + 1) a + N a = - a

, independent of

N

and

M

. Therefore

d a = - \frac{γ}{2} a d t - γ d B_{in} .

Equivalently, in the formal white-noise notation of the cavity-reservoir interaction,

\overset{a}{˙} (t) = - \frac{γ}{2} a (t) - γ b_{in} (t) .

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