By Kurt Jacobs
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15) We note that σn (t)w may be written as a function of τ = kt, being the time scaled by the measurement constant. Hence, as we expect, increasing the measurement time has the same effect on σn (t)w as increasing the measurement constant. 2. 1: The conditional uncertainty in photon number averaged over all trajectories, σn (t)w , is plotted here against the dimensionless scaled time, τ = kt. The dotted line corresponds to an initial coherent state, and the solid line to an initial thermal state.
1). 5) combine trivially and we obtain ˜ |ψ(t) w = lim eAN ∆t exp B ∆t→0 ∆Wn n |ψ(0) = eAt eBW (t) |ψ(0) . 10) 38 CHAPTER 3. EVOLUTION OPERATORS FOR LINEAR QUANTUM TRAJECTORIES As the Wiener process, W (t), is a sum of independent Gaussian distributed random variables, Wn , it is naturally Gaussian distributed, the mean and variance of W (t) being zero and t respectively. 2), the Wiener process will have a particular value at each time t, and as we mentioned above, the set of all these values corresponds to the trajectory that is taken by that particular realisation.
This is useful because, as we show in Chapter 3, it may be generalised to equations containing multiple variables, and hence to quantum trajectories. 21) where the last equality is easily shown by expanding the exponential to first order in ∆t and using ∆W 2 = ∆t. The exponential now acts as an operator which propagates x(t) forward in time by ∆t. 22) which is indeed the solution obtained previously. This concludes our discussion of stochastic equations driven by the Wiener process. For a comprehensive treatment of stochastic equations and stochastic calculus the reader is referred to reference [42].
Topics in quantum measurement and quantum noise by Kurt Jacobs
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