The Unruh Effect in Slow Motion

Abstract

We show under what conditions an accelerated detector (e.g., an atom/ion/molecule) thermalizes while interacting with the vacuum state of a quantum field in a setup where the detector’s acceleration alternates sign across multiple optical cavities. We show (non-perturbatively) in what regimes the probe ‘forgets’ that it is traversing cavities and thermalizes to a temperature proportional to its acceleration, the same as it would in free space. Then we analyze in detail how this thermalization relates to the renowned Unruh effect. Finally, we use these results to propose an experimental testbed for the direct detection of the Unruh effect at relatively low probe speeds and accelerations, potentially orders of magnitude below previous proposals.

1. Introduction

The Unruh effect [1,2,3], one of the fundamental and yet still untested predictions of quantum field theory, tells us that uniformly accelerated observers of the Minkowski vacuum of a quantum field will actually experience a finite temperature proportional to their acceleration [4,5]. Direct detection of the Unruh effect would be a feat that resonates across many fields, ranging from astrophysics [6,7], cosmology [8,9], black-hole physics [10], particle physics [11], and quantum gravity [12,13,14] to the very foundations of QFT. Unsurprisingly, much effort has been made towards finding evidence of the Unruh (and the closely related Hawking) effect, both through direct and indirect observations [15,16,17] as well as in analog systems such as fluids [18], Bose-Einstein condensates [19,20,21], optical fibers [22], slow light [23], superconducting circuits [24] and trapped ions [25,26], to name a few. Despite its fundamental relevance, an uncontroversial direct confirmation of the Unruh effect remains elusive.

In recent times, it has been shown that the Unruh effect is present even when the field state is not KMS (i.e., thermal, see [4,27]) with respect to accelerated observers [27]. This is related to the fact that the only physical Lorentz invariant state of a free field in flat-spacetime is the vacuum, and that any deviations from the vacuum would eventually be red/blue-shifted out of the response window of any physical detector. Moreover, one can see this effect in settings (like optical cavities) where Lorentz invariance is explicitly broken [28]. Indeed, the Unruh effect understood in terms of thermalization of particle detectors is a robust phenomenon.

One commonality of all presently known scenarios exhibiting the (linearly accelerated) Unruh effect is that the probe system becomes ultrarelativistic and therefore travels astronomical distances. This may seem unavoidable since the probe must accelerate for a long time (i.e., long enough to thermalize).

However, as we will demonstrate, ultrarelativistic velocities (from the probe’s initial rest frame) are not necessary in cavity setups to see the Unruh effect. Furthermore, we will show that it can be seen with accelerations orders of magnitude smaller than the best current proposals known to the authors.

2. Motivation

We discussed in the introduction that ultrarelativistic speeds and astronomical distances may be needed to detect the Unruh effect with linear acceleration. Let us expand upon this argument with some scale analysis. It is well known that even very small Unruh temperatures require huge accelerations. For instance, an acceleration of $a=3\times {10}^{19}\mathrm{g}$ where $\mathrm{g}=9.8{\mathrm{m}/\mathrm{s}}^2$ is needed to achieve $T_{\mathrm{Unruh}}=1\mathrm{K}$. A second issue that often receives less comment is that thermalization is a relatively slow process, at least compared to the high accelerations discussed above. In particular, the proper time for a probe to thermalize with its environment, ${\tau }_{\mathrm{thermal}}$, is lower bounded by the probe’s Heisenberg time ${\tau }_{\mathrm{H}}=2\pi /\Omega$ as ${\tau }_{\mathrm{thermal}}\gg {\tau }_{\mathrm{H}}=2\pi /\Omega$ where $\hslash \Omega$ is the typical energy scale of the probe. Think for example of the energy gap between two levels of an atomic transition used as a detector. If, for instance, $\Omega$ is set by the 21-cm Hydrogen transition then ${\tau }_{\mathrm{thermal}}\gg {\tau }_{\mathrm{H}}=4\mathrm{ns}$.

The Lorentz factor (with respect to its initial rest frame) of a probe accelerating at a rate a for a proper time, $\tau$, is $\gamma =\cosh (a\tau /c)$. The lab distance covered in this time is

$$\begin{array}{c}\hfill \Delta x=(c^2/a)(\gamma -1).\end{array}$$

(1)

If the probe becomes ultrarelativistic, $\gamma \gg 1$, then lab time is $\Delta t\approx \Delta x/c$. Please note that each of $\Delta x$ and $\Delta t$ are exponential in the quantity $a\tau /c$. For the Hydrogen probe discussed above we have

$$\begin{array}{c}\hfill a{\tau }_{\mathrm{thermal}}/c\gg a{\tau }_{\mathrm{H}}/c=4000\end{array}$$

(2)

such that in Equation (1), the factor

$$\begin{array}{c}\hfill ({\gamma }_{\mathrm{thermal}}-1)\gg exp\left(4000\right)\approx {10}^{1737}.\end{array}$$

(3)

In this case, the distances and times required for the probe to thermalize are so astronomical that they dwarf cosmological lengthscales.

The above discussion suggests that any feasible direct detection proposal will have $a{\tau }_{\mathrm{thermal}}/c\lesssim 1$. Proposals with these sorts of scales are also problematic. In particular, we then have,

$$\begin{array}{c}\hfill 1\gtrsim \frac{a{\tau }_{\mathrm{thermal}}}{c}\gg \frac{a{\tau }_{\mathrm{H}}}{c}={\left(2\pi \right)}^2\frac{k_{\mathrm{B}}{T}_{\mathrm{Unruh}}}{\hslash \Omega }\end{array}$$

(4)

That is, proposals with $a{\tau }_{\mathrm{thermal}}/c\lesssim 1$ must also have $k_{\mathrm{B}}{T}_{\mathrm{Unruh}}/\hslash \Omega \ll 1$; There must be very few excitations in the post-thermalization probe. Thus, it appears we have a dilemma: either our experiment requires astronomical distances and lab times, or the thermalized probe must be only very weakly excited. However, as we will discuss in this paper, this dilemma can be avoided.

In the above argument that ultrarelativistic motion is unavoidable for the Unruh effect there is a hidden assumption: that the thermalization time is the same as the time that the acceleration is sustained in a single direction. We can circumvent this by separating the two timescales. For instance, we can take the probe to alternate the sign of its acceleration at some regular interval ${\tau }_{\max }$ with ${\tau }_{\max }\ll {\tau }_{\mathrm{thermal}}$. In this way, the probe maintains a constant magnitude of acceleration, $\left|a\right|$, but does not accumulate much speed.

In this regard, our approach has some similarities to several circular Unruh effect proposals [29,30,31,32,33,34,35,36] in which the probe follows a circular trajectory. In these proposals, as in ours, the probe’s acceleration has a constant magnitude but a changing direction. In both cases this prevents the probe from becoming ultrarelativistic. However, our linear setup can overcome some of the limitations of the circular proposals as we will note later. The key difference between such circular proposals and our proposal is that circular trajectories yield final probe temperatures which depend not just on the circular acceleration but also on the probe’s speed and energy gap [36]. In our proposal, as we will see, this does not happen.

Now, the question becomes: will the probe still thermalize to the Unruh temperature when following our alternately accelerated/decelerated trajectory? One may have the intuition that it will since the probe would “see a thermal bath of temperature $T_u=\hslash \left|a\right|/2\pi c{k}_B$” between each acceleration sign-change event. If the probe does not thermalize it must be due to the sudden jerks felt by the probe at each acceleration sign-change event (or due to radiation produced at these events). Contrast this with the circular Unruh effect proposals in which the probe undergoes a slow continuous jerk.

In our proposal, the effect of these jerks can be completely removed by the following alternative setting: we set up a series of adjacent Dirichlet cavities containing quantum fields in their respective vacuums. The walls of each cavity have small (say atom-sized) holes that the probe travels through. We take the probe to switch the sign of its acceleration exactly as it crosses each cavity wall. We note that one can reroute the probe back through old cavities, so long as they have had time to relax back to the ground state before the probe reenters.

The benefits of introducing these cavity walls are two-fold. First, since the probe’s interaction with the field is identical in each two-cavity-cell, we need only simulate the field-probe interaction for a relatively short duration, $\delta \tau =2{\tau }_{\max }\ll {\tau }_{\mathrm{thermal}}$. Indeed, the cavity walls shield the probe from any radiation produced in previous cavities. As we will discuss in detail later, this makes the probe’s dynamics Markovian which allows for efficient non-perturbative calculations.

Secondly, the field’s boundary conditions enforce that the field amplitude vanishes at the cavity walls such that the probe is effectively decoupled from the field at each acceleration sign-change event. This completely eliminates the sudden jerks’ effects on the probe’s dynamics.

One may be concerned that these cavity walls will spoil the Unruh effect, for two main reasons: First, the probe creates disturbances in the field that will bounce off the cavity walls and affect the probe in turn. We will see that if the probe spends short times in each cavity, the probe will not have enough time to resolve the backreaction of the probe on the field, becoming blind to those disturbances.

Second, the vacuum in the cavity is not Lorentz invariant: there is a discrete set of field modes, and the probe can notice this difference. Indeed, in the classic Unruh effect, it is relevant that the vacuum state of the field is invariant under Lorentz transformations as well as that the probe accelerates for asymptotically long times for it to thermalize to a temperature proportional to its acceleration [4]. In a cavity setting we do not have Lorentz invariance and one may not expect that an accelerated probe would thermalize if it interacted with the cavity vacuum state. However, it was observed in the past that there is a phenomenon akin to the Unruh effect (thermalization of detectors to a temperature proportional to their acceleration) in cavity setups [28]. We will discuss here that there are indeed regimes where the probe is deprived of the information about the fact that it is flying through a cavity. We will show that the regimes where one finds Unruh effect in cavities (defined as thermalization of the probe to a temperature proportional to its acceleration when interacting with the vacuum) are precisely those regimes where the probe cannot resolve information about the effect of the cavity walls.

In summary, we will show that there are regimes where the probe is blind to the fact that it is in a cavity and so experiences thermalization according to Unruh’s law.

3. Our Setup

Consider a probe which is initially co-moving with the cavity wall at $x=0$ and then begins to accelerate at a constant rate $a>0$ towards the far end of the cavity at $x=L>0$. In terms of the probe’s proper time, $\tau$, this portion of the trajectory is given by

$$\begin{array}{cc}\hfill x\left(\tau \right)& =\frac{c^2}{a}(\cosh (a\tau /c)-1),t\left(\tau \right)=\frac{c}{a}\sinh (a\tau /c),\hfill \end{array}$$

(5)

for $0\le \tau <{\tau }_{\max }=\frac{c}{a}{\cosh }^{-1}(1+aL/c^2)$. The cavity-crossing time in the lab frame is $t_{\max }=\frac{L}{c}\sqrt{1+2c^2/aL}$. The probe exits the first cavity at some speed, $v_{\max }$, relative to the cavity walls with maximum Lorentz factor ${\gamma }_{\max }=\cosh (a{\tau }_{\max }/c)=1+aL/c^2$.

At $\tau ={\tau }_{\max }$ the probe enters the second cavity of the two-cavity cell and begins decelerating with proper acceleration a. The probe reaches the far end of the second cavity, $x=2L$, just as it comes to rest at $\tau =2{\tau }_{\max }$.

Although a full light–matter interaction description would require a $3+1$D setup [37], as proof of principle we will assume that each cavity contains a $1+1$D massless scalar field, $\widehat{\varphi }(t,x)$, with a free Hamiltonian

$$\begin{array}{c}\hfill {\widehat{H}}_{\varphi }=\frac{1}{2}{\int }_0^L\mathrm{d}x{c}^2\widehat{\pi }{(t,x)}^2+{\left({\partial }_x\widehat{\varphi }(t,x)\right)}^2,\end{array}$$

(6)

satisfying $[\widehat{\varphi }(t,x),\widehat{\pi }(t,x^{\prime })]=\mathrm{i}\hslash \delta (x-x^{\prime })\widehat{𝟙}$, where $\widehat{\pi }(t,x)$ is the field’s canonical conjugate momentum. The field obeys Dirichlet boundary conditions at $x=0$ and $x=L$ such that we have the mode decomposition,

$$\begin{array}{cc}\hfill \widehat{\varphi }(t,x)=\sum _{n=1}^{\infty }& \sqrt{\frac{2\hslash c^2}{{\omega }_{n}L}}sin\left(k_{n}x\right)\left({\widehat{a}}_n^{†}{e}^{\mathrm{i}{\omega }_{n}t}+{\widehat{a}}_n{e}^{-\mathrm{i}{\omega }_{n}t}\right),\hfill \end{array}$$

(7)

where mode frequencies and wavenumbers satisfy $c{k}_n={\omega }_n=nc\pi /L$, and ${\widehat{a}}_n^{†},{\widehat{a}}_n$ are the $n^{\mathrm{th}}$-mode’s creation/annihilation operators.

Let the probe’s internal degree of freedom be a quantum harmonic oscillator with some energy gap, $\hslash {\Omega }_P$. The probe is characterized by dimensionless quadrature operators ${\widehat{q}}_P$ and ${\widehat{p}}_P$ obeying $[{\widehat{q}}_P,{\widehat{p}}_P]=\mathrm{i}\widehat{𝟙}$. In these terms the probe’s free Hamiltonian is ${\widehat{H}}_P=\hslash {\Omega }_P({\widehat{q}}_P^2+{\widehat{p}}_P^2-1)/2$. In the interaction picture ${\widehat{q}}_P\left(\tau \right)$ evolves with respect to $\tau$ as ${\widehat{q}}_P\left(\tau \right)={\widehat{q}}_P\left(0\right)cos\left({\Omega }_P\tau \right)+{\widehat{p}}_P\left(0\right)sin\left({\Omega }_P\tau \right)$.

We take the probe to couple to the field via the Unruh-DeWitt interaction Hamiltonian [4,5,38],

$$\begin{array}{c}\hfill {\widehat{H}}_I\left(\tau \right)=\lambda {\widehat{q}}_P\left(\tau \right)\widehat{\varphi }\left(t\left(\tau \right),x\left(\tau \right)\right),\end{array}$$

(8)

where $\lambda$ is the coupling strength. This Hamiltonian captures the fundamental features of the light–matter interaction when exchange of angular momentum is not relevant [39,40,41,42]. Please note that $x\left(\tau \right)$ and $t\left(\tau \right)$ are given by Equation (5) while the probe accelerates through the first cavity. The trajectory in the second cavity of the cell is a straightforward reversed-translation of this trajectory.

4. Non-Perturbative Time-Evolution

We next compute the probe’s dynamics in the first cell. In the interaction picture the time-evolution operator for the probe-field system in the $n^{\mathrm{th}}$ cavity is,

$$\begin{array}{c}\hfill {\widehat{U}}_n^{\mathrm{I}}=\mathcal{T}exp\left(\frac{-\mathrm{i}}{\hslash }{\int }_{(n-1){\tau }_{\max }}^{n{\tau }_{\max }}\mathrm{d}\tau {\widehat{H}}_I\left(\tau \right)\right).\end{array}$$

(9)

The probe’s reduced dynamics is given by,

$$\begin{array}{c}\hfill {\Phi }_n^{\mathrm{I}}\left[{\widehat{\rho }}_P\right]={\mathrm{Tr}}_{\varphi }\left({\widehat{U}}_n^{\mathrm{I}}\left(\right|0\rangle \langle 0\left|\right){\widehat{U}}_n^{\mathrm{I}}{}^{†}\right).\end{array}$$

(10)

Composing the cases $n=1$ and $n=2$ (where the probe accelerates and decelerates respectively) we can build the interaction picture update map for the first cell, ${\Phi }_{1,2}^{\mathrm{I}}={\Phi }_2^{\mathrm{I}}\circ {\Phi }_1^{\mathrm{I}}$.

Analogously, one can find the update map for the second cell, ${\Phi }_{3,4}^{\mathrm{I}}={\Phi }_4^{\mathrm{I}}\circ {\Phi }_3^{\mathrm{I}}$, but unfortunately this map is different for every cell (${\Phi }_{3,4}^{\mathrm{I}}\ne {\Phi }_{1,2}^{\mathrm{I}}$). However in the Schrödinger picture the update map is in fact the same for each cell, ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}={\Phi }_{1,2}^{\mathrm{S}}={\Phi }_{3,4}^{\mathrm{S}}=\cdots$. We can build ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}$ from the above discussed update maps as ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}={\mathcal{U}}_0^2\circ {\Phi }_2^{\mathrm{I}}\circ {\Phi }_1^{\mathrm{I}}$ where ${\mathcal{U}}_0\left[{\widehat{\rho }}_P\right]=U_0{\widehat{\rho }}_P{U}_0^{†}$ and $U_0=exp(-\mathrm{i}{\tau }_{\max }{\widehat{H}}_P/\hslash )$ (see Appendix A for auxiliary technical details).

In summary, as the probe travels through many cells it is repeatedly updated by ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}$. Noting that ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}$ depends on the cell-crossing time, $\delta \tau =2{\tau }_{\max }$, we have,

$$\begin{array}{c}\hfill {\widehat{\rho }}_P\left(n\delta \tau \right)={\left({\Phi }_{\mathrm{cell}}^{\mathrm{S}}\left(\delta \tau \right)\right)}^n\left[{\widehat{\rho }}_P\left(0\right)\right].\end{array}$$

(11)

This dynamics is Markovian and time-independent: the same update map is applied each time-step.

There are powerful tools to analyze the dynamics of such repeated-update systems. One such tool is the Interpolated Collision Model formalism, ICM [43,44,45], which allows us to rewrite the discrete update Equation (11) as a differential equation with no approximation and without needing to take $\delta \tau \to 0$ unlike in other common approaches [46,47,48,49,50,51,52,53,54,55,56,57].

Additionally, we take advantage of the fact that our setup is Gaussian: all the states involved have Gaussian Wigner functions and interact through quadratic Hamiltonians. This enables us to simplify our description of the probe’s state from an infinite-dimensional density matrix, ${\widehat{\rho }}_P\left(n\delta \tau \right)$, to just a $2\times 2$ covariance matrix, ${\sigma }_P\left(n\delta \tau \right)$, for the probe’s quadrature operators; see [58,59,60,61].

Using recent results on Gaussian ICM [43,62] we can efficiently calculate the fixed points and convergence rates of repeated application of ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}$. This is achieved by straightforward application of the formalism developed in [62]. For the convenience of the reader, we provide a quick summary particularized to our setup in Appendix B.

5. Results

As we have discussed above, we can efficiently compute the probe’s final covariance matrix, ${\sigma }_P(\infty )$, after it has traveled through many cells. ${\sigma }_P(\infty )$ is the unique fixed point of ${\Phi }_{\mathrm{cell}}^{\mathrm{S}}$. To characterize this state, we write it in standard form,

$$\begin{array}{c}\hfill {\sigma }_P(\infty )=\mathcal{R}\left(\theta \right)\left(\begin{array}{cc}\nu exp\left(r\right)& 0\\ 0& \nu exp(-r)\end{array}\right)\mathcal{R}{\left(\theta \right)}^{⊺},\end{array}$$

(12)

for some symplectic eigenvalue $\nu \ge 1$, squeezing parameter $r\ge 0$ and angle $\theta \in [-\pi /2,\pi /2]$ where $\mathcal{R}\left(\theta \right)$ is the $2\times 2$ rotation matrix. The questions that we will answer next are: (a) is the probe’s final state thermal? and if so, (b) how does the probe’s final temperature depend on the parameters of our setup?

The free parameters are: (1)—the cavity length, L, (2)—the probe’s proper acceleration, a, (3)—the probe’s proper frequency ${\Omega }_P$, and (4)—the coupling strength, $\lambda$. The relevant dimensionless variables are $a_0=aL/c^2,{\Omega }_0={\Omega }_{P}L/c,\mathrm{and}{\lambda }_0=\lambda L/\sqrt{\hslash c}$. We fix ${\lambda }_0=0.01$ to be in the ultrastrong coupling regime [63], but our results are independent of the coupling strength provided ${\lambda }_0\lesssim 1$.

We next investigate for what values of $a_0$ and ${\Omega }_0$ the final probe state is approximately thermal. From (12), if the probe state is not squeezed (i.e., $r=0$) then it is in a thermal state with temperature $k_{B}T=\hslash {\Omega }_P/2\mathrm{arccoth}\left(\nu \right)$. It is intuitive that if r is “small enough” then we can say the state is approximately thermal. The question is then “how small is small enough?” For the interested reader, we consider several different temperature estimates and measures of thermality in Appendix C. Over the parameter range considered in this manuscript these measures of thermality all indicate that the probe’s final state is effectively indistinguishable from thermal. Thus, as we show in Appendix C, our various temperature estimates all take on essentially the same values.

Since the probe is indistinguishable from thermal, we next ask how its (dimensionless) final temperature, $T_0=k_{B}TL/\hslash c$, depends on $a_0$ and ${\Omega }_0$. A clear signature of the Unruh effect would be finding $T\propto a$. We thus search for regimes where $\mathrm{d}{T}_0/\mathrm{d}{a}_0$ is constant (i.e., independent of both $a_0$ and ${\Omega }_0$). Figure 1a shows $\mathrm{d}{T}_0/\mathrm{d}{a}_0$ for a wide range of accelerations and probe gaps. Please note that we approach a constant value of $\mathrm{d}{T}_0/\mathrm{d}{a}_0$ in the bottom-right of the figure.

The upward sloping lines in Figure 1a indicate the parameters for which the probe’s free Hamiltonian rotates through a phase of $\Theta ={\Omega }_P{\tau }_{\max }=n\pi /2$ inside each cavity. The $\Theta =\pi /2$ line is dashed. There are fundamental limits to the energy resolution that detectors can achieve coming from energy-time uncertainty principles [64]. To resolve the cavity into discrete energy levels any detector would need to interact for a time long enough to allow its internal energy uncertainty to decrease to a point where it can confidently distinguish between two different discrete levels. For our detector this means $\Theta \gg 2\pi$. Please note that the regime where $\mathrm{d}{T}_0/\mathrm{d}{a}_0\approx \mathrm{constant}$ is located below the $\Theta =\pi /2$ line such that in this regime the probe cannot fully resolve the cavity into discrete levels, i.e., if the probe does not spend much time in each cavity, the energy levels as seen by the probe are ‘blurred’ and hence some information about the cavity is obscured.

Resolving the cavity’s discrete spectrum is not the only way that the probe could learn that it is in a cavity. Indeed, the probe may learn of the cavity walls by bouncing a signal off them. Consider the disturbances that the detector is effecting on the field as it goes along its trajectory. If initially right-moving (left-moving), these disturbances cross paths with the probe an odd (even) number of times. In each case the minimum number of crossings is achieved for $M\le 3$ where $M=c{t}_{\max }/L$ is the ratio of the probe’s cavity-crossing time, $t_{\max }$, to the cavity’s light-crossing time, $L/c$. The vertical lines in Figure 1a correspond to $M=3$ (dashed) and $M=4,5,6,\cdots$. Please note that the regime where $\mathrm{d}{T}_0/\mathrm{d}{a}_0\approx \mathrm{constant}$ is located to the right of the $M=3$ line (i.e., for $a_0>1/4$). In this region the probe does not spend long in each cavity (less than three light-crossing times) and therefore interacts minimally with any reflected signals.

Summarizing, Figure 1a,b show that above $a_0=1/4$ and below $\Theta =\pi /2$ we have $\mathrm{d}{T}_0/\mathrm{d}{a}_0\approx 1/2$. The detector thermalizes to a temperature which is proportional to its acceleration and independent of ${\Omega }_0$, (and ${\lambda }_0$ and L) the hallmark of the Unruh effect. Please note that unlike the circular Unruh effect (where the probe’s temperature depends on the probe gap and its orbit speed/radius [36]), in our proposal the temperature only depends on the probe’s acceleration. The only difference between our proposal and the continuum Unruh effect is the factor of $\pi$.

6. The Missing Pie

Undoubtedly this mismatch of slopes ($1/2\mathrm{vs}.1/2\pi$) is a glaring difference between this Unruh effect in many cavities and the canonical one in the continuum. We account for this difference by noting that there is no limit in which our setup returns the canonical Unruh effect scenario. It is critical in our setup that the probe does not have time to resolve the cavity into discrete energy levels, i.e., that $\Theta ={\Omega }_P{\tau }_{\max }\lesssim 2\pi$. This precludes the probe from thermalizing within a single cavity, since ${\tau }_{\max }\lesssim 2\pi /{\Omega }_P$ is less than the probe’s Heisenberg time. Thus, in our setup the probe’s thermalization is necessarily a multi-cavity phenomenon, making it unachievable in the $L\to \infty$ limit and hence difficult to compare with the continuum.

The exact magnitude of the slope may be capturing geometric factors (that are dimension dependent, yielding missing $\pi$’s) and/or the scales we have fixed e.g., the probe’s initial velocity. However, we will still argue along the lines of [27,65] that the most fundamental part of the Unruh effect is that an accelerated detector interacting with the ground state of a quantum field thermalizes in the long time limit to a temperature proportional to its acceleration regardless of its internal energy-gap and the coupling strength.

7. Towards Experimental Detection

Our proposed setting can achieve the Unruh effect for dimensionless accelerations as small as $a_0=aL/c^2=1/4$ where L is the cavity length (maximum Lorentz factor of ${\gamma }_{\max }=1+a_0=5/4$). For a tabletop setup with $L=1\mathrm{m}$ this is an acceleration of $a=2.3\times {10}^{15}\mathrm{g}$. This matches the lowest-acceleration experimental proposals for direct detection known to the authors [15,16,17]. For the largest cavity on Earth (LIGO, $L=4\mathrm{km}$) we can lower the required acceleration way below any previous proposal to $a=5.7\times {10}^{11}\mathrm{g}$. Example parameters for experimental realizations at these scales are shown in

Table 1. Our proposed setup with $a_0=1/4$, ${\Omega }_0=\pi /16$ and ${\lambda }_0=0.01$ realized at two different scales, $L=1\mathrm{m}$ (tabletop) and $L=4\mathrm{km}$ (LIGO-sized). $t_{\mathrm{thermal}}$ estimates the lab-time needed for the probe to thermalize. $N_{\mathrm{cells}}$ is the number of cells crossed in this time. Note these can be substantially decreased by increasing ${\lambda }_0$. See Appendix B for details on $t_{\mathrm{thermal}}$ and $N_{\mathrm{cells}}$.

	Tabletop	LIGO-Sized
L	$1\mathrm{m}$	$4\mathrm{km}$
a	$2.3\times {10}^{15}\mathrm{g}$	$5.7\times {10}^{11}\mathrm{g}$
${\Omega }_p$	$60\mathrm{MHz}$	$15\mathrm{kHz}$
$t_{\max }$	$10\mathrm{ns}$	$40\mathsf{\mu }\mathrm{s}$
$\lambda \sqrt{\hslash c}/\hslash {\Omega }_P$	0.051	0.051
T	$280\mathsf{\mu }\mathrm{K}$	$71\mathrm{nK}$
$k_{\mathrm{B}}T/\hslash {\Omega }_P$	$0.64$	$0.64$
$t_{\mathrm{thermal}}$	$14\mathrm{ms}$	$56\mathrm{s}$
$N_{\mathrm{cells}}$	$7\times {10}^5$	$7\times {10}^5$

, as realized at two different scales: $L=1\mathrm{m}$ (tabletop) and $L=4\mathrm{km}$ (LIGO-sized). It is worth noting that at either of these scales the lab-time needed for the probe to thermalize, $t_{\mathrm{thermal}}$, are not unreasonably large. One may argue that the number of cavities is too large to be considered realistic. However, it is worth noting that as discussed in Section 2, a much smaller number of cavities would be required in practice if we let the cavities rethermalize with the environment after the probe crosses them (a process that is much faster than the time it takes to do the experiment) and reverse the polarity of the accelerating force so that the probe may revisit old cells assuming they have had enough time to relax back to their ground state. As such the number of cavities actually needed may be much less than $N_{\mathrm{cells}}$.

The theoretical setting proposed in this manuscript is general and independent of any particular implementation, paving the way for future experimental proposals. In particular, there is freedom in picking the mechanism which accelerates the probe. Two possibilities are laser pulses and voltage differences (see Figure 2). In either case, we can estimate the kinetic energy that the probe needs to gain/lose across each cavity. For an electron this is $128\mathrm{keV}$; for a hydrogen atom this is $235\mathrm{MeV}$. The laser technology needed to provide the sustained accelerations needed are already available [66,67]. The voltages needed are also not outside of the realm of possibility: the largest voltages produced in a lab are ∼10${}^2\mathrm{MV}$ [68]. Although not exempt from technical difficulties, the experimental challenges involved in the proposed setups are expected to be solvable with near-future technology.

8. Conclusions

We have presented a setup which displays the Unruh effect (thermalization of a particle detector to a temperature proportional to its acceleration) without the detector becoming ultrarelativistic. Moreover, this setup has the potential to provide an experimental testbed for the Unruh effect orders of magnitude lower than previous proposals. We achieved this by having the probe alternate between accelerating and decelerating at regular intervals.

Despite the departures from the canonical Unruh effect scenario (the vacuum of a free field in a cavity is not Lorentz invariant) we still see the Unruh effect (as in [28]) and further discuss that when the Unruh effect is present it is because the probe does not have enough time to learn that it is in cavity (either by resolving the cavity’s discrete energy levels or by bouncing a signal off the walls). In this regime, the probe thermalizes to an Unruh temperature with the cavities collectively despite not having time to thermalize with each one individually.

Finally, we provided two possible concrete examples of experimental implementation of our proposal and discussed whether the scales of such implementations are in principle experimentally feasible.