Comment on “Quantum measurement and decoherence”

Ford, Lewis, and O’Connell [Phys. Rev. A 64, 032101 (2001)] have recently discussed a thought experiment in which a Brownian particle is subjected to a double-slit measurement . Analyzing the decay of the emerging interference pattern, they derive a decoherence rate that is much faster than previous results and even persists in the limit of vanishing dissipation. This result is based on the definition of a certain attenuation factor, which they analyze for short times. In this note, we point out that this attenuation factor captures the physics of decoherence only for times larger than a certain time $t_{\mathrm{mix}}$, which is the time it takes until the two emerging wave packets begin to overlap. Therefore, the strategy of Ford et al. of extracting the decoherence time from the regime $t<t_{\mathrm{mix}}$ is in our opinion not meaningful. If one analyzes the attenuation factor for $t>t_{\mathrm{mix}}$, one recovers familiar behavior for the decoherence time; in particular, no decoherence is seen in the absence of dissipation. We confirm the latter conclusion by calculating the off-diagonal elements of the reduced density matrix.

Figure 1

(a) The probability density $P(x,t)$ for finding the particle at time $t$ at coordinate $x$ is plotted for $T=E,\gamma =0.3E$. An interference fringe is seen to appear at time $t_{\mathrm{mix}}$. (b) $P(x,t=t_{\mathrm{mix}})$ is plotted for the same parameters as in (a) (dashed line), and for $T=\gamma =0$ (solid line). The interference fringe in the former curve is seen to be somewhat suppressed with respect to the latter, but to be qualitatively very similar.

Figure 2

$P(x,t)$ is shown at the times $t=0.1\times t_{\mathrm{mix}}$; $t=0.3\times t_{\mathrm{mix}}$; and $t=t_{\mathrm{mix}}$ (from top to bottom), for the parameters $T=E$, $\gamma =0.3E$ (thick line). Also shown: The noninterfering contribution $P_{\mathrm{cl}}(x,t)$ (thin line) and the envelope $P_{\mathrm{cl}}(x,t)\pm P_{\mathrm{int}}(x,t)$ of the interference pattern (dashed line) around $P_{\mathrm{cl}}(x,t)$.

Figure 3

Upper part: $P_{\mathrm{int}}(x=0,t)$ is shown for $\gamma =0$ and $T=0$ (solid; this line also coincides with that for $P_{\mathrm{cl}}(x=0,t)$ for all $T$ shown), $T=E$ (dashed), and $T=10E$ (dotted). For small $T$, both $P_{\mathrm{cl}}$ and $P_{\mathrm{int}}$ grow rapidly after time $t_{\mathrm{spread}}$ (indicated by the arrow), after which broadening of the wave packets begins. The reduction of the interference pattern $P_{\mathrm{int}}$ with increasing $T$ is due to the increasingly mixed-state nature of the initial state, not to decoherence, and has no time scale associated. In particular, at high temperatures $\left(T=10E\right)$ $P_{\mathrm{int}}$ practically vanishes for all times. Lower part: the attenuation factor $a_{\mathrm{FLO}}=P_{\mathrm{int}}∕P_{\mathrm{cl}}$ is plotted. Its decay from 1 down to $a^{\infty }$ takes place only for $t<t_{\mathrm{spread}}$. Also, for $T=E$ the case of no dissipation $\left(\gamma =0\right)$ is compared to weak dissipation ($\gamma =0.3E$, thin solid line); a decoherence-induced decay of $a_{\mathrm{FLO}}$ for long times is seen in the latter case only.