Predictability sieve, pointer states, and the classicality of quantum trajectories

We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states—the most predictable states of an open quantum system selected by the predictability sieve—are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).

Figure 1

(Color online) Purity as a function of time under unconditional evolution at $T=0$ starting from an initial pure Gaussian state parametrized by a squeezing parameter $\xi =\ln \kappa$. Purity is not lost for $\xi =0$, i.e., the coherent state with $(\alpha ,\beta )=(1,1)$. Coherent states are perfect pointer states at $T=0$, and optimize the predictability sieve criterion. By contrast, purity is initially lost on a decoherence time scale for other states with $\xi \ne 0$. Time is measured in units of the spontaneous emission time.

Figure 2

Purification time as a function of the measurement scheme parametrized by $s∊[0,1]$: ${\eta }_x=\eta {\sin }^2\left[s\pi ∕2\right]$ and ${\eta }_y=\eta {\cos }^2\left[s\pi ∕2\right]$. The initial state is a fat Gaussian state ${\alpha }_0={\beta }_0={10}^{-8}$. The temperature of the bath is zero. We note that while there is a clear optimum in Fig. 1, the very best state $s=0.5$ is not all that much better than $s\sim 0.2$ or 0.8. Thus, there is a large “quantum halo” [26] of states that maintain purity well (and in this sense are reasonably classical).

Figure 3

The minimal efficiency threshold ${\eta }_{\mathrm{thr}}$ needed to reach the threshold purity $P_{\mathrm{thr}}=0.5$ after the time $t_{\mathrm{thr}}=2.5$ as a function of the measurement scheme parametrized by $s∊[0,1]$: ${\eta }_x={\eta }_{\mathrm{thr}}{\sin }^2\left[s\pi ∕2\right]$ and ${\eta }_y={\eta }_{\mathrm{thr}}{\cos }^2\left[s\pi ∕2\right]$. The initial state and temperature are the same as in Fig. 1. Again, $s=0.5$ is surrounded by a sizable “halo” of (nearly) pointer states [26].

Figure 4

(Color online) Purity as a function of time under unconditional evolution at finite temperature starting from an initial pure Gaussian state parametrized by the squeezing parameter $\xi =\ln \kappa$. The temperature of the bath is $k_{\mathrm{B}}T∕\hslash {\omega }_0={10}^6$, where ${\omega }_0$ is the frequency of the system oscillator. Again, coherent states $(\xi =0)$ are selected by predictability sieve, as they lose purity most slowly. For large times, all initial states lead to the same stationary state with purity $P\left(\infty \right)=1∕(1+2n)$, where $n\approx {10}^6$ for this temperature. Time is measured in units of the spontaneous emission time.

Figure 5

(Color online) Purity under unconditional evolution at $T={10}^6$ after a short time $t=0.1∕4T=2.5\times {10}^{-8}$ starting from an initial pure Gaussian state parametrized by two numbers $(A,C)$. The parameter $A$ is related to the squeezing of the probability distribution of the initial state, and $C$ is related to its tilt in phase space. Purity is maximized for an initial state that is approximately parametrized by $(A,C)=(1,0)$, that is, the eigenstate of the squeezed annihilation operator $c$ with $(\alpha ,\beta ,\gamma )=(4T,1∕4T,0)$, which is the most predictable state after this short time of unconditional evolution. However, the vertical range of the plot is only (0.999 999 99, 1)—at this early time the unconditional evolution is still very predictable no matter what its initial pure state is. The most predictable eigenstates of $c$ are not well distinguished from the rest. Rescaled units are used.

Figure 6

(Color online) Purity after unconditional evolution at $T={10}^6$ for a time $t=2∕\sqrt{4T}={10}^{-3}$ when the unconditional evolution starting from the most robust initial state loses approximately one-half of its initial purity. Purity is maximal when $(A,C)=(1.75,1.75)$ or $(\alpha ,\beta ,\gamma )=(1.75\sqrt{4T},2.32∕\sqrt{4T},1.75)$. At this time the most predictable initial states are well distinguished from (are much more pure than) other states, which already have negligible purity. Rescaled units are used.

Figure 7

Purification time as a function of the homodyne angle $\varphi$ $(r=1)$ for an initial thermal state with temperature $T$. Purification time is the shortest for the $x$ homodyne, i.e., when $\varphi =0,\pi$. By contrast, the $y$ homodyne $(\varphi =\pi ∕2)$ does not purify the state at all. Rescaled units are used.

Figure 8

Efficiency threshold as a function of the homodyne angle $\varphi$ for an initial thermal state with temperature $T$. The threshold is the lowest for the $x$ homodyne $(\varphi =0,\pi )$ and the highest for the $y$ homodyne $(\varphi =\pi ∕2)$. Rescaled units are used.

Figure 9

Purity loss time needed by unconditional evolution to reduce purity from the initial value $P=1$ to $P=0.5$, which is half way to the stationary value $P=0$ in the thermal state. The homodyne angle that maximizes the purity loss time is $\varphi =1.35\pi ∕2$. On the other hand, the purity loss time is zero for the $y$ homodyne with $\varphi =\pi ∕2$. Rescaled units are used.

Figure 10

(Color online) The Wigner functions of the stationary conditional states selected at high temperatures $T={10}^6$ by different robustness criteria: (a) purification time and efficiency threshold (homodyne angle $\varphi =0$) and (b) purity loss time (homodyne angle $\varphi =1.35\pi ∕2$). The two selected states are very similar (scalar product equal to 0.97), and Gaussians squeezed to “approximate” position eigenstates. Rescaled units are used.

Figure 11

(Color online) The Wigner functions of the stationary conditional states selected at low temperatures $T=1$ by different robustness criteria: (c) purification time and efficiency threshold (homodyne angle $\varphi =0$), and (d) purity loss time (homodyne angle $\varphi =1.35\pi ∕2$). The two selected states are very similar (scalar product equal to 0.99), and approximately equal to coherent states with $\alpha =\beta =1$, $\gamma =0$ (scalar product with a coherent state is equal to 0.99 and 0.98, respectively). Rescaled units are used.