Probing geometric information using the Unruh effect in the vacuum
Aida Ahmadzadegan, Fatemeh Lalegani, Achim Kempf, Robert B. Mann
PProbing geometric information using the Unruh effect in the vacuum
Aida Ahmadzadegan,
1, 2
Fatemeh Lalegani, ∗ Achim Kempf,
1, 2, 4, 5 and Robert B. Mann
5, 2 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
We present a new method by which, in principle, it is possible to “see in absolute darkness”, i.e.,without exchanging any real quanta through quantum fields. This is possible because objects modifythe mode structure of the vacuum in their vicinity. The new method probes the mode structure ofthe vacuum through the Unruh effect, i.e., by recording the excitation rates of quantum systemsthat are accelerated.
Any quantum system that can act as a detector offield quanta must couple to the field, i.e., it must con-tain a charge. The Unruh effect then arises because,as the detector is accelerated, so is its charge and thiswill generally excite the quantum field. Crucially, at thesame time, through the same interaction Hamiltonian,the quantum field can then also excite the detector. Forexample, a uniformly accelerated detector coupled to aquantum field in its Minkowski vacuum will get excitedin this way as if exposed to a thermal bath of temper-ature T = α/ π , where α is the magnitude of the de-tector’s proper acceleration [1–6]. The Unruh effect hasbeen predicted and derived in a broad variety of contexts,and it has been extended to fields confined within cavi-ties [7, 8] and to non-uniformly accelerated trajectories[9–11]. In particular, it has been shown in [12] that theUnruh effect is highly sensitive to non-uniformity of theacceleration.Here, we explore the possibility that the sensitivity ofthe Unruh effect can be further exploited, namely to ‘see’neutral objects in complete darkness, i.e., without theuse of real photons. Seeing in complete darkness shouldbe possible because objects influence the structure of thevacuum around them by effectively setting boundary con-ditions on field modes or, more generally, by leading toa dressing of the vacuum around the objects though vir-tual photons. The dressing is known to arise because theground state of a composite system consisting of a lo-calized system of first quantized matter and a quantumfield is generally not the tensor product of their respec-tive ground states, due to the presence of their interac-tion Hamiltonian. In principle, any method for detectingthe dressing could be used to see in the dark, i.e., to seewithout real photons, for example, by using the Casimireffect, or perhaps by using the dark port of a quantumhomodyne detector to register modulations of the statis-tics of vacuum fluctuations.Here we show that, in principle, seeing in completedarkness can be realized elegantly by using just a single ∗ Work done while this visitor was at the physics department ofthe University of Waterloo. non-uniformly accelerated qubit, i.e., by using the Unruheffect.The general detector model we employ is an Unruh-DeWitt detector (UDW) [4, 13, 14], an idealized modelof a real particle detector that encompasses all fundamen-tal features of the light-matter interaction when there isno angular momentum exchange involved [15]. It con-sists of a localized two-level quantum system (a qubit)that linearly couples to a scalar field. For examples ofstudies of the response of UDW detectors in Minkowskiand curved spacetimes, see, e.g., [16–22].Our goal is to determine the response of such detectorsundergoing various non-uniform acceleration regimes in-side an optical cavity of proper length L with reflect-ing boundary conditions. As we show, the detector’s re-sponse can be used to infer the location of the boundaryof the cavity. In other words, the presence and structureof the cavity can be inferred without any exchange ofreal quanta, and so can be seen, in this sense, in com-plete darkness. For the trajectories of the detector, thereare of course many choices. The literature on UDW de-tectors, apart from the standard uniformly acceleratedand asymptotically null cases [23], considers for exampletrajectories for which a constant energy flux is emitted[24]. There are also several asymptotically inertial tra-jectories [25] that have been considered, and yet othertrajectories possess the virtue that their cases are exactlysolvable and exhibit interesting physical features [26]. I. SETTINGS
We shall adopt these various trajectories to addressthe problem of interest, analyzing the excitation prob-ability of a UDW detector along a given non-uniformlyaccelerating trajectory whilst inside a cavity. Though wework in (1+1) dimensions, our results can be straight-forwardly extended to higher dimensions. We comparethe result of each case with the excitation probability ofa detector moving on the Rindler trajectory. We furtherclassify the motions into two broad categories: trajecto-ries with vanishing asymptotic flux and trajectories witha finite asymptotic flux; we depict the respective velocity a r X i v : . [ g r- q c ] O c t profiles of these trajectories in Figs. 1(a) and 1(b). Foreach case we analyze the motion of the detector from thecavity frame as it travels through the cavity. Through-out, Minkowski coordinates in the cavity’s rest frame aredenoted as ( x, t ), and τ denotes the proper time of thedetector; we follow the convention of setting c = (cid:126) = 1.For any trajectory x = x ( t ), it is straightforward to de-fine the following quantities associated with the motionof the detector, v = d x d t coordinate velocity τ = (cid:82) √ − v d t detector proper time u µ = (cid:16) √ − v , v √ − v (cid:17) detector 2-velocity a µ = d u µ d τ = (cid:16) v (1 − v ) , − v ) (cid:17) d v d t detector 2-acceleration α = √ a µ a µ = √ − v ) d v d t proper accelerationWe are particularly interested in comparing detectorresponses between various trajectories, each of which hasthe detector entering the optical cavity at (cid:0) t , x ( t ) (cid:1) .Therefore, we must calibrate the motions of the detectorsas they enter the cavity so that they all begin with thesame initial velocity (so as to remove spurious Doppler ef-fects) and the same initial acceleration (so as to properlycompare to the uniformly accelerated case). Imposingthese constraints fixes the initial time parameter t andthe acceleration parameter for each of the motions un-der consideration. For each trajectory, the ratio of itsacceleration α ( t ) relative to the uniform Rindler case is amonotonically increasing function of t . Further descrip-tion of all the trajectories is given in the appendix.Our comparisons of the detector responses are withineach category of trajectories, since each produces quali-tatively distinct responses at late times. For each trajec-tory, we measure the excitation probability P of the de-tector for a period of time T as it traverses the full properlength L of the optical cavity. They are all calibrated tohave the same initial velocities and accelerations as theRindler case. The time evolution of the system followsthe atom-field Hamiltonian that generates evolution forthe entire system with respect to the time coordinate t of the cavity’s proper frame [8] given byˆ H ( t ) = d τ d t ˆ H (d)free [ τ ( t )] + ˆ H (f)free ( t ) + d τ d t ˆ H int [ τ ( t )] , (1)where ˆ H I [ τ ( t )] = λχ ( t )ˆ µ ( t ) ˆ φ [ x ( t )] models the detector-field interaction [1, 3, 13, 16, 27]. The constant λ isthe coupling strength, χ ( t ) ≥ µ ( t ) is the monopolemoment of the detector, and ˆ φ [ x ( t )] is the massless scalarfield that the detector is interacting with along its tra-jectory. The monopole moment operator takes the usualform of ˆ µ ( t ) = (cid:0) σ + e iΩ τ ( t ) + σ − e − iΩ τ ( t ) (cid:1) , in which Ω is theproper energy gap between the ground state, | g (cid:105) and theexcited state, | e (cid:105) of the detector and σ ± are ladder oper-ators ( σ + | g (cid:105) = | e (cid:105) , σ − | e (cid:105) = | g (cid:105) ). Working in this frameand expanding the field in terms of an orthonormal set ●● ● ● ● ● ● ● ● ● ●□ □ □ □ □ □ □ □□ □ □■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ □ Rindler Proex ● Darcx ■ CV * * * * * * * * * * * * * * * ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ● ● ● ●● ● ● ● ● ● ● ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx ○ DF ● CW ■ Logex Omex
Figure 1. Velocities for the various trajectories of a) van-ishing flux: Rinder, Costa-Villalba (CV), Darcx, and Proexand b) finite flux: Davies-Fulling (DF), Arcx, Omex, Logex,and Carlitz-Willey (CW). Initial velocities of all trajectoriesin (a) and (b) are chosen to be v = 0 .
54 and v = 0 . α = 1. Initial accelerations of theother trajectories are normalized to this value. of solutions to the Schr¨odinger equation inside the cavityyields the following Hamiltonianˆ H I ( t ) = λ d τ d t ∞ (cid:88) n =1 ˆ µ ( t ) √ ω n L (cid:0) ˆ a † n u n [ x ( t ) , t ] + ˆ a n u ∗ n [ x ( t ) , t ] (cid:1) (2)in the interaction picture. We consider Dirichlet (re-flective) boundary conditions φ [0 , t ] = φ [ L, t ] = 0, andso the field modes take the form of the stationary waves u n [ x ( t ) , t ] = e iω n t sin[ k n x ( t )]. Here, ω n = k n + m where k n = nπ/L . In our study, we work with scalar fields,therefore, m = 0.To characterize the vacuum response of a particle de-tector undergoing different trajectories, we initially pre-pare the detector in its ground state and the cavity inthe vacuum state, so that its initial density matrix is ρ = | g (cid:105) (cid:104) g | ⊗ | (cid:105) (cid:104) | . The time evolution of the sys-tem is governed by the interaction Hamiltonian (2) in thetime interval 0 < t < T and is given by ˆ U ≡ ˆ U ( T,
0) = T e − i (cid:82) d τ ˆ H I ( τ ) . We consider the coupling constant, λ , tobe a small parameter so we can work within the valid-ity of perturbation theory. Therefore, using the Dysonperturbative expansion up to second order in λ , we canwrite [28], ρ T = (cid:2) U (1) + ˆ U (2) + O ( λ ) (cid:3) ρ (cid:2) U (1) + ˆ U (2) + O ( λ ) (cid:3) † (3)whereˆ U (1) = λi ∞ (cid:88) n =1 (cid:2) σ + a † n I + ,n + σ − a n I ∗ + ,n + σ − a † n I − ,n + σ + a n I ∗− ,n (cid:3) I ± ,n = (cid:90) T d τ d t e i[ ± Ω τ ( t )+ ω n t ] sin (cid:2) k n (cid:0) x ( t ) − x ( t ) (cid:1)(cid:3) dt. (4)We compute the density matrix ρ T, (d) for the detectorby taking the partial trace over the field degrees of free-dom [28]. The first order contribution to the transitionprobability vanishes, so the leading contribution comesfrom second order in the coupling strength. Therefore,the excitation probability of the detector is P = λ ∞ (cid:88) n =1 | I + ,n | (5)We work with this quantity rather than the transitionrate as there is no formal or computational advantagein the latter given the absence of time translation in-variance in our setting; both quantities contain the sameinformation. II. RESULTS
In obtaining our results, there are a few factors thatdetermine the response of the detector inside the cav-ity. We keep the coupling constant small ( λ = 0 .
01) andchoose the gap of the detector to be in resonance with oneof the field modes inside the cavity. By changing the res-onance mode (choosing a different gap for the cavity) thebehaviour of the excitation probability P changes. Forexample, in Fig. 2, P for a detector moving on the Omextrajectory is given as a function of the cavity length forthree different values of resonance mode. As we can see,the excitation probability of a detector in resonance withlower modes of the field shows more sensitivity to thechange in length of the optical cavity, and so is a pre-ferred choice for inferring the location of its boundary. Note that in (1 + 1) dimensions, the coupling constant has unitsof inverse length in natural units. Here, small coupling strengthmeans the dimensionless quantity, λσ is small, where σ = 1 isthe fiducial unit length of the cavity; all length scales are in unitsof σ . ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● × - × - × - × - ● mode = ■ mode = ○ mode = Figure 2. Plots of the excitation probability of the detectormoving on the Omex trajectory with initial acceleration ratio α = 1. The detector gap is chosen such that it resonateswith the third mode of the field (empty circle), with the sixthmode of the field (square), and with the tenth mode of thefield (full circle). ● ● ●● ● ● ● ● ● ● ●■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ● ● ●● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin
Figure 3. The three acceleration ratios are CV (pink square),Darcx (blue circle), and Proex (green star) to Rindler. Eachplot depicts the behaviour of these ratios as a function of thecavity length for initial accelerations: a) α = 0 .
01 and b) α = 1. In the following plots we depict the ratio of excitationprobabilities of a UDW detector moving on different non-uniform trajectories relative to the uniformly acceleratedRindler case. We find that varying choices of acceleratedtrajectories are differently suited for the task of seeing inthe dark. In Fig. 4 we plot as a function of the cavity’s ● ● ●● ● ● ● ● ● ● ● ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ●● ● ● ● ● ● ● ● ● ● ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin
Figure 4. The three transition probability ratios are CV(pink square), Darcx (blue circle), and Proex (green star)to Rindler. Each plot represents the behaviour of these ra-tios as a function of the cavity length that they are travelingthrough with different initial accelerations: a) α = 0 .
01 andb) α = 1. proper length L (in units of inverse gap frequency) theexcitation probability ratio (P-ratio) of a detector mov-ing on CV, Darcx, and Proex trajectories relative to theexcitation probability of a detector moving on a Rindlertrajectory, with initial velocity v = 0 .
54 and two differentinitial proper accelerations α ( t = t ) ≡ α over the range0 . < L <
5. Similarly, Fig. 5 presents the P-ratio ofa detector moving on Omex, Logex, CW, DF, and Arcxtrajectories relative to the Rindler trajectory, with initialvelocity v = 0 .
71. For each trajectory we calibrate thedetector so that it enters the cavity at t with the sameinitial acceleration and velocity as that of the Rindlertrajectory; for each case the gap of the detector is in res-onance with the sixth mode of the field. For small initialacceleration, there is a correlation between the acceler-ation ratio as shown in Fig. 3(a) and detector responseratio illustrated in Fig. 4(a).However, as the initial acceleration increases, interest-ing structure emerges in the detector response ratio asa function of cavity length, indicated in Fig. 4(b). Themonotonic behaviour of α ( t ) in Fig. 3(b) does not yieldmonotonicity of the detector response ratio – indeed, wesee that it oscillates, decreasing over certain ranges of L despite the increase in α ( t ). This behaviour is the resultof choosing the energy gap of the detector to resonate ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ● ● ● * * * * * * * * * * * * * * * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ * * * * * * * * * * * * * * * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin
Figure 5. The five transition probability ratios are Omex (ingreen flake), Logex (in pink square), CW (in blue circle), DF(in red circle), and Arcx (in yellow star) to Rindler. Each plotrepresents the behaviour of these ratios as a function of thecavity length that they are traveling through with differentinitial accelerations: a) α = 0 .
01 and b) α = 1. with a specific field mode for all trajectories. Similar be-haviour in the finite flux case is illustrated in Figs. 5(a)and 5(b). More examples with different initial accelera-tions are given in the appendix. In general, a detectorgoes out of resonance with the field mode at a differenttime (position along the cavity) than for the Rindler tra-jectory, leading to a distinct signature for a given trajec-tory with given boundary conditions. Furthermore, onecan see the sensitivity to the non-uniformity of accelera-tion [12], with the Omex trajectory indicating the great-est sensitivity for the mode in question. Given a specificnon-uniform trajectory, its value of P is sensitive to thelength of the cavity and thus sensitive to the location ofeach of its boundaries. This sensitivity can be exploitedto detect the cavity boundaries, without any exchange ofreal quanta, by only measuring the relative response rate(the P-ratio) of the detector. In other words, we can ‘seein absolute darkness’ by only probing the vacuum field,without sending any signal or radiation. III. OUTLOOK
Our results point towards generalization to sharp vi-sion in all directions in complete darkness, leading to anintriguing close relationship to the field of spectral geom-etry. One branch of spectral geometry asks, for example,to what extent the geometry of a Riemannian manifoldcan be inferred from the spectrum of differential oper-ators on the manifold [29, 30]. A related but differentbranch of spectral geometry asks, for example, to whichextent the shape of a drum is encoded in the spectra ofthe sound it makes [31].In our context here, let us consider an optical cav-ity of arbitrary (e.g., convex) shape. This cavity thenpossesses a corresponding normal mode decompositionof standing waves of the quantum field, with the shapeof the cavity determining the pattern of these standingwaves. By sending in multiple detectors with differentenergy gaps moving on varying accelerated trajectories,their excitation rates will provide information about thecavity boundaries in various directions, and could there-fore allow the detection, in complete darkness, of the fullgeometry of the cavity. In this way, an equivalence couldbe established between the geometry of the cavity andthe quantum fluctuations of a quantum system. The es-tablishment of any equivalence between curved shapesor geometries as they occur in general relativity on one hand, and quantum phenomena, such as excitation rates,on the other hand, could ultimately be useful for quan-tum gravity.Finally we note that our work may have longer-termapplications for short-range sensing. Indeed, in the ab-sence of a cavity the P -ratio of trajectories will be sen-sitive to the proximity of objects in free space. This isbecause each object will furnish a boundary conditionfor the field, or more generally, it will create a dressedquantum vacuum around it. We showed that this changeof the dressing of the vacuum can be sensed by acceler-ated detectors. It will be very interesting to determinethe type of trajectories that possess the optimally suited P -ratios for such sensing tasks, also in higher dimensionsand for massive fields. ACKNOWLEDGMENTS
A.K. and R. B. M. are supported in part by the Dis-covery Grant Program of the Natural Sciences and Engi-neering Research Council of Canada.
Appendix A: Vanishing Flux Mirror Trajectories
We begin with a description of the trajectories given in Fig. 6 that have vanishing flux. Their velocities are presentedin the main text in Fig. 1(a). In all the following trajectories, T is measured in the frame of the cavity ( x, t ). ● ● ● ● ● ● ● ● ● ● ●□ □ □ □ □ □ □ □ □ □ □■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ □ Rindler Proex ● Darcx ■ CV Figure 6. Various trajectories of vanishing flux: Rinder, CV, Darcx, and Proex are shown. Initial position of all trajectories ischosen to be x = 1 so that all trajectories and their parameters stay positive and physical. The acceleration in the Rindlercase has been normalized to α = 1. Initial accelerations of the other trajectories are normalized to this value.
1. Rindler trajectory
This is the most commonly studied detector trajectory, x ( t ) = (cid:114) t + 1 a , τ ( t ) = 1 a arcsin( at ) , (A1)where α = a is the proper acceleration. It is known that the response rate corresponds to that of a detector in athermal bath of scalar radiation with temperature T = a/ π . The detector enters the cavity with an initial velocity v . Given its initial velocity and acceleration, the time T that it takes for the detector to travel the whole length ofthe cavity L is T = (cid:115) L (cid:112) a t + L + t a (A2)obtained by setting x ( t ) = x ( t ) + L in (A1).
2. Costa-Villalba trajectory
This trajectory is one for which the detector has inertial motion in the distant past and increasingly accelerates toattain uniform acceleration in the distant future along an asymptotically null trajectory [32]. The parametrization ofthe CV trajectory is given by x ( t ) = tw + √ w t w , (A3) τ ( t ) = arcsinh( tw + √ t w ) w − (cid:113) tw + √ t w ) w , where w is a positive constant in CV trajectory whose value is proportional to the uniform acceleration asymptoticallyattained at late times. Choosing the same initial acceleration and velocity as for the Rindler case, we obtain T = 12 L w + 2 Lt w − (cid:16)(cid:113) t w ( L w + Lt w ) (A4)+ 2 L w + 3 L t w + Lt w − L − t (cid:17) . for the time that it takes for the detector to travel the length L of the cavity.The proper acceleration for this trajectory is α ( t ) = w (cid:16) tw + √ t w ) (cid:17) / , (A5)and as t → ∞ it is straightforward to show that α → w .
3. Proex
The Proex trajectory x ( t ) = W (e σt ) σ (A6) τ ( t ) = 2 (cid:112) W (e σt ) + 1 + ln (cid:20) √ W (e σt )+1 − √ W (e σt )+1+1 (cid:21) σ , where W is the product log or Lambert-W function, is a trajectory for which there is a finite number of particlesoccupying each mode. Its mirror trajectory has a finite, nonzero energy flux that vanishes at late times. Both itsproper acceleration and acceleration vanish in the distant past and future and in the future the magnitude of thevelocity approaches the speed of light.The proper acceleration for this trajectory is given by α ( t ) = σW (e σt )(2 W (e σt ) + 1) / , (A7)and it takes time T = log (cid:16) e Lσ + W (e σt ) ( Lσ + W (e σt )) (cid:17) σ (A8)to travel the full length of the cavity. Both σ and t are fixed by choosing the initial acceleration and velocity of theProex detector to be equal to that of the Rindler detector.
4. Darcx
The Darcx trajectory, given below, is asymptotically inertial in the past and future, but is not necessarily asymp-totically static in the future. A finite amount of particles and energy is produced by this mirror trajectory, but withvanishing asymptotic flux. x ( t ) = κ arcsinh(e ζt ) ζ , (A9) τ ( t ) = t + √ − κ ln (cid:104) − κ )e tζ + 2 √ − κ (cid:112) (1 + (1 − κ )e tζ )(1 + e tζ ) + 2 − κ (cid:105) ζ − ln (cid:104) tζ ) − κ e tζ + 2 (cid:112) (1 + (1 − κ )e tζ )(1 + e tζ ) (cid:105) ζ (A10)The detector enters the cavity at x = κ arcsinh(1) ζ (A11)and the setting of the parameters differs from the previous two cases. We fix t and ζ by choosing the initialacceleration and initial velocity to be equal to that of the Rindler case. However κ is a free parameter 0 < κ < T = ln (cid:104) sinh (cid:16) Lζ + κ arcsinh( e t ζ ) κ (cid:17) (cid:105) ζ . (A12)to travel the full length of the cavity, and α ( t ) = κζ e ζt (1 − ( κ − ζt ) / (A13)is its proper acceleration.We see that this quantity vanishes at late times. Appendix B: Finite Flux mirror trajectories
These trajectories have the common feature that at late times the energy flux from the mirror trajectory asymptotesto a constant value. This value can be calibrated to be equal for all such trajectories, and we shall do so here. Webegin with a description of these trajectories shown in Fig. 7. The velocities of these trajectories are illustrated inFig. 1(b) in the main text.
1. Davies-Fulling
This is one of the earliest mirror trajectories studied, and was used to demonstrate that a Planck spectrum from amoving mirror can be obtained that is analogous to that found for black hole evaporation. The Davies-Fulling mirrortrajectory has a time-dependent acceleration that is asymptotically null. The parametrization of this trajectory isgiven by [26] x ( t ) = ln[cosh( ξt )] ξ , τ ( t ) = 2arctan(tanh( ξt )) ξ (B1)where ξ is a positive constant whose relationship to the proper acceleration is α ( t ) = ξ (cid:112) sech( ξt ) . (B2)Setting the initial acceleration and velocity of the detector to be that of the corresponding Rindler detector, we fixboth ξ and t , obtaining T = arccosh (cid:16) ξt ξ ( t − L ) (cid:17) ξ , (B3) * * * * * * * * * * * * * * * ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx ○ DF ● CW ■ Logex Omex
Figure 7. Various trajectories of finite flux: DF, Arcx, Omex, Logex, and CW are shown. Initial position of all trajectories ischosen to be x = 1 so that all trajectories and their parameters stay positive and physical. The acceleration in the Rindlercase has been normalized to α = 1. Initial accelerations of the other trajectories are normalized to this value. for time this detector spends in the cavity.
2. Carlitz-Willey
This trajectory, the CW trajectory, is of physical interest insofar as it simulates an eternal black hole that evaporatesthermally at fixed temperature. The mirror trajectory has constant energy flux (and thus a divergent amount of totalenergy). There is a thermal spectrum at all times and the Bogoliubov coefficients can be computed exactly andanalytically. It does not make use of any late time approximations.The CW trajectory is parametrized as x ( t ) = t + W (e − kt ) k , τ ( t ) = − (cid:112) W (e − kt ) k (B4)and as t → −∞ is asymptotically null with zero proper acceleration [26]. The quantities k and t are fixed as beforeby requiring equality of the initial acceleration and velocity with the Rindler case, and T = ( L + t ) (B5)+ W ( − e − k ( L − t ) W (e − kt ) ) + W (e − kt ) k , is time that it takes for the detector to traverse the cavity and α ( t ) = k (cid:112) W (e − kt ) , (B6)is its proper acceleration.
3. Arcx
This trajectory is analogous to the Davies-Fulling trajectory, with the advantage that it has a static start butwith velocity and acceleration continuous at all times, allowing for a solution that is valid globally [26]. The mirrortrajectory has thermal late time emission and infinite acceleration, and an energy flux that also asymptotes to aconstant value in the distant future. It is given by x ( t ) = arcsinh(e kt ) k , τ ( t ) = − arctanh (cid:18) √ kt (cid:19) k (B7)where α ( t ) = k e kt , (B8)and T = log (cid:2) sinh (cid:0) kL + arcsinh(e kt ) (cid:1)(cid:3) k (B9)are the respective proper acceleration and time spent by the detector in the cavity, with k and t calibrated to theRindler case as before.
4. Logex
Unlike the other mirror trajectories, Logex emits a pulse of energy flux before asymptoting to the CW value [26].This trajectory that starts off asymptotically static is always accelerating and is given by x ( t ) = ln(1 + e kt )2 k , (B10) τ ( t ) = 12 k (cid:32) (cid:16)(cid:112) kt (cid:17) + ln (cid:34) √ kt − √ kt + 1 (cid:35)(cid:33) where T = ln(e k ( L + t ) + e kL − k (B11)is the time spent in the cavity and α ( t ) = 2 k e kt (1 + e kt )(1 + 2e kt ) / (B12)is the proper acceleration, which diverges at late times. As before, calibration with the Rindler trajectory fixes k and t .
5. Omex
The last trajectory we study is the Omex mirror trajectory. This one is of considerable interest since its Bogoliubovcoefficients are identical to those of a Schwarzschild black hole truncated to two spacetime dimensions [26]. Its energyflux asymptotes to a constant value in the distant future.The Omex trajectory is of similar form to that of the Carlitz-Willey trajectory, but is asymptotically static in thedistant past, and is given by x ( t ) = t + W (e − kt )2 k , (B13) τ ( t ) = − k (cid:32)(cid:113) (2 + W (e − kt )) W (e − kt )+ ln (cid:20) W (e − kt ) + (cid:113) (2 + W (e − kt )) W (e − kt ) (cid:21)(cid:33) , where k and t are determined from calibration with the Rindler trajectory as before. The detector spends a time T = 2 k ( L + t ) + W (e − kt )(1 − e − kL )2 k , (B14)travelling the proper length L of the cavity and α ( t ) = 2 k (cid:112) W (e − kt )(2 + W (e − kt )) (B15)is its proper acceleration.0 Appendix C: Detector responses: Vanishing and Non-vanishing responses
In Figs. 8 and 9 we plot the responses of detectors traveling along each of the trajectories listed above for increasingacceleration parameters, alongside plots showing how the acceleration increases as a function of time. The Omextrajectory provides the greatest contrast with the Rindler case. ● ● ●● ● ● ● ● ● ● ● ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ● ● ●● ● ● ● ● ● ● ●■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ● ● ●● ● ● ● ● ● ● ● ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ● ● ● ● ● ● ● ● ● ● ●■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ●● ● ● ● ● ● ● ● ● ● ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin ● ● ● ● ● ● ● ● ● ● ●■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ Proex / Rin ● Darcx / Rin ■ CV / Rin
Figure 8. A comparison of the excitation probability (left column) and proper acceleration (right column) of detectors travelingon non-uniformly accelerated trajectories with vanishing flux with the excitation probability of a uniformly accelerated detector.The three ratios are Costa-Villalba to Rindler (pink square), Darcx to Rindler (blue circle), and Proex to Rindler (green star).Each plot represents the behaviour of these ratios as a function of the length of the cavity that they are traveling through fordifferent initial accelerations: a) α = 0 .
01, c) α = 0 .
1, and e) α = 1. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ● ● ● * * * * * * * * * * * * * * * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * * * * * * * * * * * * * * * ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■● ●● ● ● ● ● ● ● ● ● * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ● ● ● * * * * * * * * * * * * * * * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * * * * * * * * * * * * * * * ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■●● ● ● ● ● ● ● ● ● ● * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ * * * * * * * * * * * * * * * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ * * * * * * * * * * * * * * * ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■●● ● ● ● ● ● ● ● ● ● * Arcx / Rin ○ DF / Rin ● CW / Rin ■ Logex / Rin Omex / Rin
Figure 9. Comparing the excitation probability of detectors traveling on non-uniformly accelerated trajectories with non-vanishing flux with the excitation probability of a uniformly accelerated detector. The five ratios are Omex to Rindler (in greenflake), Logex to Rindler (in pink square), CW to Rindler (in blue circle), DF to Rindler (in red circle), and Arcx to Rindler (inyellow star). Each plot represents the behaviour of these ratios as a function of the length of the cavity that they are travelingthrough for different initial accelerations: a) α = 0 .
01, c) α = 0 .
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