Spatially extended Unruh-DeWitt detectors for relativistic quantum information

Unruh-DeWitt detectors interacting locally with a quantum field are systems under consideration for relativistic quantum information processing. In most works, the detectors are assumed to be pointlike and, therefore, couple with the same strength to all modes of the field spectrum. We propose the use of a more realistic detector model where the detector has a finite size conveniently tailored by a spatial profile. We design a spatial profile such that the detector, when inertial, naturally couples to a peaked distribution of Minkowski modes. In the uniformly accelerated case, the detector couples to a peaked distribution of Rindler modes. Such distributions are of special interest in the analysis of entanglement in noninetial frames. We use our detector model to show the noise detected in the Minkowski vacuum and in single particle states is a function of the detector’s acceleration.

Figure 1

$(1+1)$-dimensional example of a frequency distribution peaked around $\pm \lambda$ given by Eq. (16) for $\sigma =1$ and $\lambda =5$. This frequency distribution peaks around the desired frequency $\lambda$ but has a double peaking due to the two exponential terms. In the $(1+1)$ massless case, the field is expanded as an integral over $\omega >0$ and so the peak in the $\omega <0$ region does not contribute.

Figure 2

Comparison of exact coupling strength (as modeled with a quantum harmonic oscillator) and our approximate coupling strength. The choices are quantum numbers $(n,m)=(0,1)$ and parameters $(\lambda ,\sigma )=(1.66,1/\sqrt{0.89})$.

Figure 3

Comparison of exact coupling strength (as modeled with a quantum harmonic oscillator) and our approximate coupling strength. The choices are quantum numbers $(n,m)=(0,3)$ and parameters $(\lambda ,\sigma )=(2.5,1)$.

Figure 4

The transition rate for an inertial, Dirac delta spatial profile (i.e. pointlike) detector probing a $(3+1)$ massive field. We have plotted the transition rate for mass values $m=0$, 1, 1.5. We observe that if the detector is in its excited state ($-\mathrm{\Delta }>0$), its energy gap needs to be larger than the mass of the field to undergo spontaneous emission. Also, for all energy gaps $-\mathrm{\Delta }<m$, the detector cannot undergo spontaneous emission, i.e., it is stable in its excited state.

Figure 5

The transition rate for an inertial, double peaked Gaussian spatial profile detector [see Eq. (11)] probing a $(3+1)$ massive field. We have plotted the transition rate for mass values $m=0$, 1, 1.5. We observe that if the detector is in its excited state ($-\mathrm{\Delta }>0$), for high energy gaps the detector’s transition rate is negligibly small. This implies that for large energy gaps, a nonpointlike detector is stable, i.e., it will not undergo spontaneous emission.

Figure 6

The transition rate for an accelerated, Dirac delta spatial profile detector probing a ($1+1$) massless field. We have plotted the transition rate for acceleration values $a=0.1$, 1, 1.5. We see that the transition rate is zero only for asymptotically large values of $\mathrm{\Delta }$. The divergent region $\mathrm{\Delta }=0$ is due to the nature of the massless ($1+1$) field.

Figure 7

The transition rate for an accelerated, double peaked Gaussian spatial profile detector probing a ($1+1$) massless field. We have plotted the transition rate for acceleration values $a=0.1$, 1, 1.5. We see that there are resonant values for $\mathrm{\Delta }$ where probability of the detector undergoing spontaneous emission (or absorption) is increased. This can be traced back to the Gaussian spatial profile which causes the detector to interact more strongly with modes with energy $\sim |\mathrm{\Delta }|$.