Berry Phase from the Entanglement of Future and Past Light Cones: Detecting the Timelike Unruh Effect

The Unruh effect can not only arise out of the entanglement between modes of left and right Rindler wedges, but also between modes of future and past light cones. We explore the geometric phase resulting from this timelike entanglement between the future and past, showing that it can be captured in a simple $\mathrm{\Lambda }$ system. This provides an alternative paradigm to the Unruh-deWitt detector. The Unruh effect has not been experimentally verified because the accelerations needed to excite a response from Unruh-deWitt detectors are prohibitively large. We demonstrate that a stationary but time-dependent $\mathrm{\Lambda }$-system detects the timelike Unruh effect with current technology.

Figure 1

A spacetime diagram separated into four quadrants: the left and right Rindler wedges, and the future and past light cones. The vacuum state can be written as an entangled state between the Rindler wedges, or between the light cones. For an observer in one of these quadrants (e.g., the future), tracing out the unobserved modes (e.g., in the past) leads to the (timelike) Unruh effect. The arrow represents the spacetime trajectory of the detector.

Figure 2

(a) A three-level $\mathrm{\Lambda }$ system with two degenerate ground states ($|g_1⟩$, $|g_2⟩$) which couple to the excited state ($|e⟩$), with time-dependent transition angular frequency $\omega (t)$. (b) The $\mathrm{\Lambda }$ configuration can be represented in an $L$ configuration in the $|\pm ⟩\equiv (|g_1⟩\pm |g_2⟩)/\sqrt{2}$ basis. Here $|-⟩$ is a dark state that does not couple to the other states of the system $\{|+⟩,|e⟩\}$ or the environment. As the $|+⟩$ does couple to the environment, a relative GP can arise in the system. This GP is used to detect the thermal state of the environment.

Figure 3

$\mathrm{\Delta }\beta$ (normalized to $\mathrm{\Gamma }/\omega$) as a function of $a/\omega$, after one quasicycle (${\tau }_c=2\pi /\omega$).

Figure 4

Shift in the ground state population probability due to the Unruh effect, over ten quasicycles. Parameters: $a/\omega =1$, $\mathrm{\Gamma }/\omega ={10}^{-4}$.