Indirect Measurements of a Harmonic Oscillator

Abstract

The measurement of a quantum system becomes itself a quantum-mechanical process once the apparatus is internalized. That shift of perspective may result in different physical predictions for a variety of reasons. We present a model describing both system and apparatus and consisting of a harmonic oscillator coupled to a field. The equation of motion is a quantum stochastic differential equation. By solving it, we establish the conditions ensuring that the two perspectives are compatible, in that the apparatus indeed measures the observable it is ideally supposed to.

Appendix

We shall derive (1.4, 1.5). The commutation relation $[a, a^*]=1$ implies ${{\,\mathrm{i}\,}}[H_0,a]=-{{\,\mathrm{i}\,}}\omega a$ and thus ${{\,\mathrm{e}\,}}^{{{\,\mathrm{i}\,}}H_0 t}a{{\,\mathrm{e}\,}}^{-{{\,\mathrm{i}\,}}H_0 t}=a{{\,\mathrm{e}\,}}^{-{{\,\mathrm{i}\,}}\omega t}$. Since the relation remains true upon replacing a by $a-\alpha$, and $a^*$ accordingly, we also have

$$\begin{aligned} {{\,\mathrm{e}\,}}^{{{\,\mathrm{i}\,}}H t} a {{\,\mathrm{e}\,}}^{-{{\,\mathrm{i}\,}}H t}=\alpha +(a-\alpha ){{\,\mathrm{e}\,}}^{-{{\,\mathrm{i}\,}}\omega t}. \end{aligned}$$

By (1.1) that expression has to be multiplied from the left by its adjoint and then time averaged in order to obtain ${\overline{N}}_T$. As a result

$$\begin{aligned} {\overline{N}}_{\infty }=|\alpha |^2+(a^*-{\overline{\alpha }})(a-\alpha ), \end{aligned}$$

because terms $\sim {{\,\mathrm{e}\,}}^{\pm {{\,\mathrm{i}\,}}\omega t}$ do not contribute to the limit. This proves (1.4), which in turn implies

$$\begin{aligned} \langle n |\, {\overline{N}}_{\infty }^2 \,| n \rangle =\langle n |\, (N+2|\alpha |^2)^2 \,| n \rangle +|\alpha |^2\langle n |\, a^*a+aa^* \,| n \rangle , \end{aligned}$$

because monomials $(a^*)^la^m$ with $l\not =m$ have vanishing expectation. The first term on the r.h.s. equals $\langle n |\, {\overline{N}}_{\infty } \,| n \rangle ^2$ and (1.5) follows.