Time-frequency approach to relativistic correlations in quantum field theory

Moving detectors in relativistic quantum field theories reveal the fundamental entangled structure of the vacuum, which manifests, for instance, through its thermal character when probed by a uniformly accelerated detector. In this paper, we propose a general formalism inspired from both signal processing and correlation functions of quantum optics to analyze the response of pointlike detectors following a generic, nonstationary trajectory. In this context, the Wigner representation of the first-order correlation of the quantum field is a natural time-frequency tool to understand single-detection events. This framework offers a synthetic perspective on the problem of detection in relativistic theory and allows us to analyze various nonstationary situations (adiabatic, periodic) and how excitations and superpositions are deformed by motion. It opens up an interesting perspective on the issue of the definition of particles.

Figure 1

Decomposition of the Fourier plane into four quadrants: the upper quadrant corresponds to absorption processes, the lower quadrant to emission processes, and the side quadrants to the coherences between emission and absorption processes. While this particle-like interpretation makes sense for inertial observers, it does not necessarily hold as such for any trajectory.

Figure 2

Representation of the universal functions of the thermal distribution coming as corrections to the pure thermal response $W[a_{\tau }]$ in the derivative expansion. Their form is independent of the trajectory.

Figure 3

Comparison between a correction to $W[a_{\tau }]$ at a given order $W_{ij}$ and the exact one at the same order ${\mathrm{\Delta }}_{ij}W=W-\sum _{(k,l)<(i,j)}{W}_{kl}$. In the adiabatic regime, at low frequency $f$, the derivative expansion is meaningful, each correction being an order of magnitude lower than the previous one.

Figure 4

Wigner function representation of an oscillatory acceleration $a(\tau )=a_0+a_1\sin (2\pi f\tau )$ in different regimes controlled by the expansion parameters $(a_0/2\pi f)$ and $(a_1/a_0)$. In the regime of small frequency $f$ and amplitude $a_1$ compared to $a_0$, the adiabatic response works globally. Outside this regime, the adiabatic expansion breaks down, which is witnessed by the appearance of inner oscillations, but can still be meaningful locally.

Figure 5

Wigner (left) and Page (right) distributions of the vacuum for a finite-duration uniform acceleration between two inertial phases. After a transition time of the order of $a^{-1}$, the thermal behavior at temperature $a/2\pi$ settles down. The decreasing oscillating high-frequency parts are solely controlled by the discontinuity of the acceleration.

Figure 6

Evolution of the Wigner function of the Gaussian Fock state (top row) and coherent state (bottom row) emitted at different positions $x_0$, with a frequency $p_0/a=4$ and a width $a{\sigma }_x=1/2$ in the inertial frame. The signal is centered around a spot at $({\tau }_r,{\omega }_r)$ given by Eqs. (45) and (47), which are the special relativistic reception time and frequency, respectively, and follow the instantaneous frequency curve for different $x_0$. As the emission gets closer to the horizon, the spot flattens and gets strongly redshifted.

Figure 7

Wigner function representations of a Gaussian Fock state (top row) and Gaussian coherent state (bottom row) probed by a detector following a uniformly accelerated twinlike trajectory: the spot is chirped by the accelerated motion with a rate $-a(\tau )\omega (\tau )$ but follows the instantaneous frequency curve. The total time of the accelerating phase is $a{\tau }_{\mathrm{acc}}=4$, with transitions at $\tau =-2$, $-1$, 1, 2. The frequency of the wave packet in the inertial frame is $p_0/a=4$, and its width is $a{\sigma }_x=1/3$.

Figure 8

Wigner function representation of a superposition of a Gaussian photon wave packet probed by a detector following a uniformly accelerated twinlike trajectory for different ${\tau }_{r,1}$ and ${\tau }_{r,2}$ reception times associated with the first and second components of the superposition, respectively. The total time of the accelerating phase is $a{\tau }_{\mathrm{acc}}=4$ with transitions at $\tau =-2$, $-1$, 1, 2. The wave packet is emitted at frequency $p_0/a=4$ in the inertial frame with a width $a{\sigma }_x=1/3$.

Figure 9

Wigner function representations of a Gaussian superposition received at different times for a uniformly accelerated observer. The coherences are spread and eventually lost when a member of the superposition gets close to or crosses the horizon. The wave packet is emitted at frequency $p_0/a=10$ in the inertial frame with a width $a{\sigma }_x=1/5$.

Figure 10

Gaussian average of the Wigner function over a window given by ${\tau }_{\mathrm{s}}$ such that $|a/\delta a|\le 1/20$ for the sine case, where $a_0=a_1$, and $2\pi f/a_0=1/5$. To have a consistent time-frequency representation, we need to perform another Gaussian average of typical spread, given by the shaded area which depends on ${\tau }_{\mathrm{s}}$. A particle picture is meaningful only above this frequency threshold.