Relativistic Unruh-DeWitt detectors with quantized center of mass

In this paper, we extend the Unruh-DeWitt (UDW) model to include a relativistic quantized center of mass (c.m.) for the detector, which traditionally has a classical c.m. and follows a classical trajectory. We develop a relativistic model of an inertial detector following two different approaches, starting from either a first- or second-quantized treatment, which enables us to compare the fundamental differences between the two schemes. In particular, we find that the notion of localization is different between the two models and leads to distinct predictions, which we study by comparing the spontaneous emission rates for the UDW detector interacting with a massless scalar field. Furthermore, we consider the UDW system in both a vacuum and medium and compare our results to existing models describing a classical or quantized c.m. at low energies. We find that the predictions of each model, including the two relativistic cases, can in principle be empirically distinguished, and our results can be further extended to find optimal detector states and processes to perform such experiments. This would clarify both the role of a quantized c.m. for interactions with an external field and the differing localizations between the first- and second-quantized treatments.

Figure 1

Template functions for the first- and second-quantized localizations in a vacuum ($\nu =c=1$), alongside their semi- and nonrelativistic limits. Results are given with respect to the Compton scale of the detector, where $m$ is the detector’s rest mass. For small energy gap (a), all cases coincide for small $p/m$ (i.e., for small momenta or sufficiently massive detectors); for large energy gap (b), results no longer agree between the relativistic, classical, and semi- and nonrelativistic c.m., even for small $p/m$. The distinction between the different template functions suggests that these cases can be empirically distinguished.

Figure 2

Template functions for energy gap $E/m=0.001$ between the first- and second-quantized localizations in a medium, alongside their semi- and nonrelativistic limits. Results are given with respect to the Compton scale of the detector, where $m$ is the detector’s rest mass. In comparison to the vacuum case, one observes new transient behavior for small $p/m$; in (a), the relativistic template functions closely follow their semi- and nonrelativistic limits, while in (b), the relativistic template functions initially peak, reaching a local maximum dependent on the propagation speed of the field.

Figure 3

Spontaneous emission rates between the relativistic first- and second-quantized localizations for a medium ($\nu =0.1$) and vacuum ($\nu =c=1$). Results are given with respect to the Compton scale of the detector, where $m$ is the detector’s rest mass, ${\lambda }_c\equiv m^{-1}$ is the Compton wavelength, and $L$ is the spread of the Gaussian wave function in position space. For propagation in a medium, the distributions of the transition rates retain the same form as that in a vacuum, with the two localizations converging for large energy gap $E/m$, albeit at a lower rate. For (a) Gaussian widths below the Compton wavelength, the first- and second-quantized localizations can be distinguished, although they would not be observable in practice given the inability to empirically access this regime; for (b) Gaussian widths above the Compton wavelength, the two localizations coincide.

Figure 4

Spontaneous emission rates between the first- and second-quantized localizations in a vacuum, alongside their semi- and nonrelativistic limits. Results are given with respect to the Compton scale of the detector, where $m$ is the detector’s rest mass, ${\lambda }_c\equiv m^{-1}$ is the Compton wavelength, and $L$ is the spread of the Gaussian wave function in position space. We compare the effect of different Gaussian widths on the transition rate: for small Gaussian widths (a), the relativistic transition rates converge for large energy gap but do not in the case of a classical c.m., while the semi- and nonrelativistic cases are approximately constant compared to the other cases. For large Gaussian widths (b), the spontaneous emission rates for all cases converge for small energy gaps $E/m$, and remarkably the relativistic first- and second-quantized localizations coincide for all $E/m$.

Figure 5

Spontaneous emission rates for relativistic (a) first- and (b) second-quantized localizations in a vacuum. Results are given with respect to the Compton scale of the detector, where $m$ is the detector’s rest mass, ${\lambda }_c\equiv m^{-1}$ is the Compton wavelength, and $L$ is the spread of the Gaussian wave function in position space. In (a) the first-quantized model, smaller spreads amplify the transition rate with respect to the energy gap of the detector, while the inverse occurs in (b) the second-quantized model, where the rate of spontaneous emission is suppressed for smaller Gaussian widths.