Measurement-induced localization of relative degrees of freedom

We present a comprehensive study, using both analytical and numerical methods, of measurement-induced localization of relational degrees of freedom. Looking first at the interference of two optical modes, we find that the localization of the relative phase can be as good for mixed states—in particular, for two initially Poissonian or thermal states—as for the well-known case of two Fock states. In a realistic setup the localization for mixed states is robust and experimentally accessible, and we discuss applications to superselection rules. For an ideal setup we show how a relational Schrödinger cat state emerges and investigate circumstances under which such a state is destroyed. In our second example we consider the localization of relative atomic phase between two Bose Einstein condensates, looking particularly at the build up of spatial interference patterns, an area which has attracted much attention since the work of Javanainen and Yoo. We show that the relative phase localizes much faster than was intimated in previous studies focusing on the emerging interference pattern itself. Finally, we explore the localization of relative spatial parameters discussed in recent work by Rau, Dunningham, and Burnett. We retain their models of indistinguishable scattering but make different assumptions. In particular we consider the case of a real distant observer monitoring light scattering off two particles, who records events only from a narrow field of view. The localization is only partial regardless of the number of observations. This paper contributes to the wider debate on relationism in quantum mechanics, which treats fundamental concepts—reference frames and conservation laws—from a fully quantum and operational perspective.

Figure 1

Photon number states leak out of their cavities and are combined on a 50:50 beamsplitter. The two output ports are monitored by photodetectors. The variable phase shift $\tau$ is initially fixed at 0.

Figure 2

The evolution of $C_{l,r}(\theta ,\varphi )$. In (a) localization about ${\Delta }_0=\pi$ after 1, 5, and 15 counts when photons are recorded in the left photodetector only. (b) Localization about ${\Delta }_0=\pm 2\arccos (1∕\sqrt{3})\sim 1.9$ after 3, 6, and 15 counts when twice as many photons are recorded in the left detector as the right one. The symmetry properties of the Kraus operators $K_L$ and $K_R$ cause $C_{l,r}$ to have multiple peaks.

Figure 3

A plot of the exact values of the probabilities $P_{l,r}$ for all the possible measurement outcomes to the procedure a finite time after the start, against the absolute value of the relative phase which is evolved. The initial state is ∣20⟩ ∣20⟩ and the leakage parameter $ϵ$, corresponding roughly to the time, has a value of 0.2. Each spot corresponds to a different measurement outcome with $l$ and $r$ counts at detectors $D_l$ and $D_r$, respectively. The value ${\Delta }_0$ of the relative phase which evolves in each case is given by $2\arccos \sqrt{r∕(r+l)}$.

Figure 4

$I\left(\tau \right)$ is the intensity at the left output port after the second mode undergoes a phase shift of $\tau$ and is combined with the first at a 50:50 beam splitter. This intensity is evaluated for all possible settings of the phase shifter. Extremizing over $\tau$, the visibility for the two mode state is defined as $V=(I_{\max }-I_{\min })∕(I_{\max }+I_{\min })$.

Figure 5

Expected visibilites for (a) an initial product of two Poissonian states (plusses) and (b) an initial product of two thermal states (crosses), with an average photon number $\overline{N}$ for both cavities.

Figure 6

Photons with momentum $k$ pass through a “rubber cavity”—Mach-Zehnder interferometer in which two of the mirrors are mounted on “quantum springs” and are initially delocalized along an axis. Two photodetectors monitor the output channels.

Figure 7

Plane wave photons with momentum $k$ scatter off two free particles, delocalized in a region of length $R$, and are either deflected at an angle $\theta$ or continue in the forward direction. The observer can “see” photons which forward scatter or which are deflected only by a small amount.

Figure 8

Probability densities between $L_{\text{lower}}$ and $L_{\text{upper}}$ for the relative separation of two free thermal particles after five photons, each with momentum $k=5$, have scattered off them, either being deflected into some large angle or continuing in the forward direction. The spatial spread parameter $d=\sqrt{1∕2m{k}_{B}T}$ is set to 0.2 (units $2\pi ∕k$).

Figure 9

Probability densities for the relative separation of two free thermal particles after five thermal wavepackets have scattered off them, either being deflected into some large angle or continuing in the forward direction. The momentum parameter $k=5$ and the spatial spread parameter $d=0.2$ (units $2\pi ∕k$). The thermal wavepackets have mean photon number $\overline{n}=5$.