Universal transformation of displacement operators and its application to homodyne tomography in differing relativistic reference frames

In this paper, we study how a displacement of a quantum system appears under a change of relativistic reference frame. We introduce a generic method in which a displacement operator in one reference frame can be transformed into another reference frame. It is found that, when moving between noninertial reference frames, there can be distortions of phase information, modal structure, and amplitude. We analyze how these effects affect traditional homodyne detection techniques. We then develop an in-principle homodyne detection scheme which is robust to these effects, called the ideal homodyne detection scheme. We then numerically compare traditional homodyne detection with this in-principle method and illustrate regimes when the traditional homodyne detection schemes fail to extract full quantum information.

Figure 1

A ($1+1$)-dimensional representation of the trajectories followed by an observer accelerated to the right (Rob) and left (anti-Rob). These observers live in parts of space-time known as the right and left Rindler wedges.

Figure 2

We write the trajectories that are followed by Rob, anti-Rob and the Minkowski-delayed Rob (Rob’) and anti-Rob (anti-Rob’). World lines of the red line are the ones followed by Rob’ and anti-Rob’. Anti-Rob and Rob are causally disconnected, as well as anti-Rob’ and Rob’.

Figure 3

A schematic diagram which demonstrates how balanced-homodyne detection works.

Figure 4

A schematic diagram which demonstrates how self-homodyne detection works.

Figure 5

The unitary has been moved to the frame of the observer by introducing the basis transformation operator. In this decomposition, all interactions are conducted in one reference frame; thus, the standard self-homodyne detection scheme can be implemented.

Figure 6

Utilizing the technique of the circuit model and universal transformation of displacement operators, we can move all the operators to the signaller’s reference frame.

Figure 7

A plot which demonstrates how the local phase at the horizon affects the variance. We have utilized the following settings: $a=1$, $V_0=1$, $k_0=1$, $\sigma =0.2$, $|{\beta }_f|=1$, $\psi =\pi /3$, and ${\omega }_{\min }={10}^{-3}$.

Figure 8

A plot which compares the balanced- and self-homodyne detection schemes to the idealized homodyne detection scheme. We have utilized the following settings: $a=1$, $V_0=1$, $k_0=1$, $\sigma =0.2$, $|{\beta }_f|=1$, $\psi =\pi /3$, and ${\omega }_{\min }={10}^{-3}$.

Figure 9

A plot which compares the balanced- and self-homodyne detection schemes to the idealized homodyne detection scheme. We have utilized the following settings: $a=1$, $v_0=2.5$, ${\omega }_0=0.5$, $\sigma =0.2$, $|{\beta }_g|=1$, $\psi =\pi /4$, and ${\omega }_{\min }={10}^{-3}$.

Figure 10

A plot which compares the balanced- and self-homodyne detection scheme to the ideal homodyne detection scheme. We have utilized the following settings: $a=1$, $v_0=2.5$, ${\omega }_0=0.5$, $\sigma =0.2$, $|{\beta }_g|=1$, $\psi =\pi /4$, and ${\omega }_{\min }=1$.