Consistent histories for tunneling molecules subject to collisional decoherence

The decoherence of a two-state tunneling molecule, such as a chiral molecule or ammonia, due to collisions with a buffer gas is analyzed in terms of a succession of quantum states of the molecule satisfying the conditions for a consistent family of histories. With $\hslash \omega$ the separation in energy of the levels in the isolated molecule and $\gamma$ a decoherence rate proportional to the rate of collisions, we find for $\gamma \gg \omega$ (strong decoherence) a consistent family in which the molecule flips randomly back and forth between the left- and right-handed chiral states in a stationary Markov process. For $\gamma <\omega$ there is a family in which the molecule oscillates continuously between the different chiral states, but with occasional random changes of phase, at a frequency that goes to zero at a phase transition $\gamma =\omega$. This transition is similar to the behavior of the inversion frequency of ammonia with increasing pressure, but will be difficult to observe in chiral molecules such as D${}_2$S${}_2$. There are additional consistent families both for $\gamma >\omega$ and for $\gamma <\omega$. In addition, we relate the speed with which chiral information is transferred to the environment to the rate of decrease of complementary types of information (e.g., parity information) remaining in the molecule itself.

Figure 1

(a) The steady-state solutions for $\phi$ correspond to the intersections of $\sin 2\phi$ (solid curve) with $\pm \omega /\gamma$ (dashed lines), shown here for $\gamma >\omega >0$ (“strong decoherence” regime). (b) These steady-state solutions are plotted schematically on the Bloch sphere, as if the $z$ axis is going into the page.

Figure 2

Time evolution of the Bloch-sphere angles $\phi$ and $\theta$ for the consistent description associated with the (a) forward and (b) backward conditions. In this case, $\phi \left(0\right)=0$, $\theta \left(0\right)=0.2$, $\omega =1$, and $\gamma =0.5,1.2,4$, respectively, for the solid, dashed, and dotted curves. The angles $\phi$ and $\theta$ are in units of radians, and the time $t$ is in units of $1/\omega$.

Figure 3

Information flows in terms of the $\widehat{\chi }$ information measure (see text for definition), (a) in the strong decoherence regime with $\gamma =2$ and $\omega =0.8$, and (b) in the weak decoherence regime with $\gamma =1$ and $\omega =20$. The measure $\widehat{\chi }$ is in units of bits, and the time $t$ is in units of $0.8/\omega$ in (a) and $20/\omega$ in (b).