Classical analog of the Unruh effect

In the Unruh effect an observer with constant acceleration perceives the quantum vacuum as thermal radiation. The Unruh effect has been believed to be a pure quantum phenomenon, but here we show theoretically how the effect arises from the correlation of noise, regardless of whether this noise is quantum or classical. We demonstrate this idea with a simple experiment on water waves where we see the first indications of a Planck spectrum in the correlation energy.

Figure 1

Principal idea. A container is filled with water subject to noise creating ripples on the water surface. (Top) A movable mirror guides a laser beam over the water surface illuminating a sharp spot recorded by a video camera. (Bottom) Video of the water surface and space-time diagram of the illuminated spot following the trajectory of the accelerated observer (Fig. 2).

Figure 2

Space-time diagram. The accelerated observer follows a hyperbola (curve) in space time. The observer comes in from $\infty$ with asymptotically $-c$, gets slower due to the acceleration in positive direction until coming to rest for a fleeting moment at $z=\xi$, and changing direction. Then the observer gains speed, asymptotically approaching $+c$ at $\infty$. The dotted lines indicate the causal cones straddled by the accelerated observer. The trajectory obeys Rindler's formula, Eq. (2), for constant $\xi$.

Figure 3

Mirrors. Two reflecting boundaries or nodes, acting as mirrors, confine the waves between 0 and $L$. In this case, instead of tracing the full Rindler trajectory (Fig. 2) it is sufficient to scan its mirror images with the appropriate signs indicated. The pulsation along the space-time trajectory indicates the changing measure of time experienced by the accelerated observer. As time flows exponentially slowly for velocities approaching $c$, an exponentially large laboratory would be required to trace a sufficiently long trajectory, were it not for the mirrors.

Figure 4

Plane waves. (a) The accelerated observer (Fig. 2) traces a single running plane wave; on the side panel the real part (solid) and imaginary part (dotted) of the signal are plotted as functions of $\eta$. One sees exponentially rapid oscillations for $\eta \ll -1$ and an exponential freeze for $\eta \gg 1$. (b) The observer traces a standing wave. The real part (solid) is an even function in $\eta$, the imaginary part (dotted) is odd in $\eta$, both oscillate exponentially for $\left|\eta \right|\gg 1$.

Figure 5

Waves produced through Faraday instability. (Top) Measured wave amplitudes $A$ in arbitrary units along $z$ in units of wavelength for the first 100 cycles of wave propagation. The wave pattern continues to the left and right of the figure, but with decreasing amplitude. We selected two nodes of the standing waves as our mirrors (dashed lines). One sees that the waves are not perfectly harmonic—Fourier analysis (not shown here) reveals that anharmonicities contribute to about $10\%$ of the amplitude. (Bottom) Decline of the average amplitude $\overline{A}$ over the runs of the experiment. Each run comprises 100 cycles with randomized initial phase in which the accelerated observer traverses a Rindler trajectory (Fig. 2). The quantity $\overline{A}$ we obtain for each run as the root-mean square of the entire wave amplitude between the mirrors with respect to space and time. We corrected for this decline in our data analysis.

Figure 6

Experimental results. The dots show the real and imaginary half-odd Fourier coefficients in the units of $A$ (Fig. 5) for each run of the experimental data: black, $\nu =1/4$; gray, $\nu =3/4$; light gray, $\nu =5/4$; $Q=\mathrm{Re}\tilde{A}$, $P=\mathrm{Im}\tilde{A}$. The Fourier coefficients are taken according to Eq. (31) along the space-time trajectory of an accelerated observer (Fig. 2). The ellipses represent the theory, assuming the squeezing of noise with fixed amplitude (matched with the data) and random phase; the squeezing parameter is given by Eq. (27).

Figure 7

Planck curve. (Dots) Squeezing energy and excess noise $\overline{n}$ calculated from the data (Fig. 6) for $\nu =1/4$ and $\nu =3/4$. To obtain the two dots shown here, centered ellipses are fitted to the two corresponding data sets of Fig. 6. From the ellipses the squeezing energy is calculated. (Curve) Theoretical prediction of a Planck curve according to Eq. (29). The experimental points lie remarkably close to the theoretical curve, despite clear deviations of the waves from harmonicity (Fig. 5), which illustrates the robustness of the Unruh effect against experimental imperfections.

Figure 8

Space-time diagram of conjugate accelerated observers. The observer on the right (R) is on a Rindler trajectory with constant positive $\xi$ as in Fig. 2 and described by Eq. (2). The observer on the left (L) moves on the conjugate trajectory with $-\xi$ instead of $\xi$. In this case the $\eta$ parameter must run backward for the proper time to be running forward; see Eq. (5). The two conjugate observers turn out to be correlated by the wave noise; see Eq. (A9).

Figure 9

Half-odd Fourier analysis. The signal [Fig. 4, $m=10$, $\xi =0.001L]$ was Fourier–transformed in the minimal observation window $\eta \in \{-2\pi ,+2\pi \}$. The Fourier-transformed signals were compared for the mode (solid curve) and its complex conjugate divided by the characteristic exponential $e^{-\pi \nu }$ (dotted). One sees that the two curves only agree, to a very good approximation, at the half-odd Fourier components (points) of Eq. (B9). For these Fourier components one can measure with minimal observation time a nearly perfect Planck spectrum with Unruh temperature [Eq. (1)].