Measuring motion through relativistic quantum effects
MMeasuring motion through relativistic quantum effects
Aida Ahmadzadegan, Robert B. Mann, and Eduardo Mart´ın-Mart´ınez
2, 3, 4 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Applied Math, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, N2L 2Y5, Canada
We show that the relativistic signatures on the transition probability of atoms moving throughoptical cavities are very sensitive to their spatial trajectory. This allows for the use of internal atomicdegrees of freedom to measure small time-dependent perturbations in the proper acceleration of anatomic probe, or in the relative alignment of a beam of atoms and a cavity.
I. INTRODUCTION
Quantum metrology provides techniques to make pre-cise measurements which are not possible with purelyclassical approaches. In quantum metrology protocolssuch as quantum-positioning and clock-synchronization[1, 2], the exploitation of quantum effects such as quan-tum entanglement has allowed for a significant enhance-ment of the precision in estimating unknown parametersas compared to classical techniques [3].On the other hand, there exist metrology settingswhere general relativistic effects play an important role inestablishing the ultimate accuracy of the measurement ofphysical parameters [4]. It is thus pertinent to introducea framework where relativistic effects are considered evenin quantum metrology schemes [5], where it is relevantto study how (or if) incorporating relativistic approachesto quantum metrology may increase the precision andaccuracy of the estimation and measurement of physicalparameters.In this paper we focus on finding suitable quantumoptical regimes where the response of particle detectorsbecomes sensitive to small variations of the parametersgoverning their motion, incorporating relativistic effects.Our goal is to assess the sensitivity of the response ofparticle detectors to such variations, in turn allowing forthe precise measurement of such parameters.In particular, we consider a setting in which an atomicdetector crosses a stationary optical cavity while un-dergoing constant acceleration. Relativistic acceleratingatoms in optical cavities have been considered before inthe context of an enhancement of Unruh-like radiationeffect [6–8], and later, in this context, to analyze thesubtleties of the Unruh effect in the presence of bound-ary conditions [9]. The suitability of such settings astheoretical accelerometers was studied in [10], where itwas shown that a detector’s response is sensitive to vari-ations of its proper acceleration. In this paper, we willshow that near the relativistic regimes, but still, muchbelow the accelerations required for the Unruh effect tobe detectable, the detectors’ response becomes sensitiveto small (and maybe time-dependent) perturbations ineither the parameters that govern their trajectory or inthe alignment of the optical cavity. We will study thissensitivity to determine to what extent it is possible to exploit it for quantum metrological effects.We consider two different scenarios of metrological in-terest. In the first, we study the sensitivity of the re-sponse of the detector to time-dependent variations ofits proper acceleration. Specifically, we consider a uni-formly accelerated atomic detector crossing an opticalcavity with constant proper acceleration that undergoes asmall harmonic time-dependent perturbation. If the sys-tem alignment is tuned, we might wonder how sensitiveit is to the amplitude and frequency of the perturbation.In the second scenario we study the sensitivity of thedetector’s response to variations of its trajectory. To ac-complish this, we consider small harmonic perturbationsof the spatial trajectory of a uniformly accelerated ob-server. We explore how sensitive this setting is to theamplitude and frequency of the perturbation, thus pro-viding a setting to measure the wellness of the atom’strajectory alignment with respect to the cavity frame.To this end, the outline of our paper is as follows. Insec. II, we introduce two physical settings including themethodology for investigating our two scenarios. Sec.III, contains a discussion of our results. Sec. IV containsour concluding remarks.
II. THE SETTING
In this section we consider two different scenarios inwhich we want to precisely measure different parametersof the trajectory of an atomic probe. For the first sce-nario, which we will call the accelerometer setting , weconsider an atomic probe following a constantly accel-erated trajectory, but whose proper acceleration under-goes a harmonically time-dependent perturbation. In thesecond scenario, which we will refer to as the alignmentmetrology setting , we consider that the atomic probe’strajectory undergoes small harmonic perturbations asseen from the lab frame, so as to be able to measurethe precision of the alignment of a cavity with a beam ofatomic detectors.In both scenarios we model the light-atom interactionby means of the Unruh-DeWitt model. Although simple,this model captures the fundamental features of the cou-pling between atomic electrons and the EM field involv-ing no exchange of orbital angular momentum [11, 12]. a r X i v : . [ qu a n t - ph ] N ov A. A quantum accelerometer
Particle detectors with time dependent accelerationshave been previously studied in [13, 14], where the re-sponse of an Unruh-DeWitt detector with time depen-dent acceleration in the long time regimes has been con-sidered in a flat spacetime with no boundary conditions.We would like to study how sensitive the detector re-sponse is to time-dependent perturbations of its properaccelerations in the short-time regime and in optical cav-ity settings.In order to analyze this accelerometer setting, let usfirst consider the parametrization of the trajectory of anatomic probe for a general time dependent trajectory interms of the probe’s proper time τ [15]: x ( τ ) = x + (cid:90) ττ dτ (cid:48) sinh (cid:2) ξ ( τ (cid:48) ) (cid:3) , (1) t ( τ ) = t + (cid:90) ττ dτ (cid:48) cosh (cid:2) ξ ( τ (cid:48) ) (cid:3) , (2)where ξ ( τ ) = ξ + (cid:90) ττ dτ (cid:48) a ( τ (cid:48) ) (3)represents the atom’s instantaneous speed, and a ( τ ) isthe instantaneous proper acceleration of the probe.For our purposes, we consider that the probe undergoesa constant acceleration, which is disturbed by a smallharmonic perturbation: a ( τ ) = a (cid:2) (cid:15) sin( γτ ) (cid:3) (4) (cid:15) and γ are the respective relative amplitude and fre-quency of the harmonic perturbation.The general form of the trajectories for both perturbedand constant accelerations is shown in Fig. 1.In our setting, to find the transition probability of thedetector, we let it cross a cavity of length L with an initialvelocity ξ and we measure its excitation probability forthe period of time T that it spends traveling the fulllength of the cavity. The Hamiltonian that describes oursystem generates translations with respect to time τ inthe detector’s proper frame. This Hamiltonian consists ofthree terms: ˆ H (d)free , the free Hamiltonian of the detector,ˆ H (f)free , the free Hamiltonian of the field, and the field-detector interaction Hamiltonian ˆ H int :ˆ H ( τ ) = ˆ H (d)free + ˆ H (f)free + ˆ H int ( τ ) . (5)We model the detector-field interaction with the well-known Unruh-DeWitt interaction [16, 17]ˆ H int = λχ ( τ )ˆ µ ( τ ) ˆ φ [ x ( τ )] , (6)where the constant λ is the coupling strength, χ ( τ ) isthe switching function or time window function control-ling the smoothness of switching the interaction on andoff. ˆ µ ( τ ) is the monopole moment of the detector and Figure 1. (Color online) The non-perturbed (blue-dashedcurve) and perturbed (green-solid curve) trajectory for theaccelerometer scenario. The trajectory is parameterized interms of the proper time, τ of the detector. ˆ φ [ x ( τ )] is the massless scalar field to which the detectoris coupling. We consider the coupling constant to be asmall parameter so we can work with perturbation the-ory to second order in λ . In our setting, the switchingfunction is nonvanishing only during the time the atomspends in the cavity, i.e., χ ( τ ) = 1 during 0 ≤ τ ≤ T .The monopole moment operator of the detector has theusual form in the interaction picture [18–20],ˆ µ d ( τ ) = (ˆ σ + e iΩ d τ + ˆ σ − e − iΩ d τ ) , (7)in which, Ω d is the proper energy gap between the groundstate, | g (cid:105) and the excited state, | e (cid:105) of the detector andˆ σ − and ˆ σ + are ladder operators.Expanding the field in terms of an orthonormal set ofsolutions inside the cavity yields the Hamiltonian in theinteraction picture [19]ˆ H int ( t ) = λ ∞ (cid:88) n =1 ˆ µ d ( τ ) √ ω n L (cid:0) ˆ a † n u n [ x ( τ ) , t ( τ )]+ˆ a n u ∗ n [ x ( τ ) , t ( τ )] (cid:1) (8)We will consider Dirichlet (reflective) boundary condi-tions for our cavity, φ [0 , t ] = φ [ L, t ] = 0 (9)and since we are in the Minkowski background, the fieldmodes take the form of stationary waves u n [ x ( τ ) , t ( τ )] = e i ω n t ( τ ) sin [ k n x ( τ )] , (10)where ω n = | k n | = nπ/L .To characterize the vacuum response of the particledetector undergoing trajectory (1), we initially preparethe detector in the ground state and the field in the op-tical cavity in a coherent state | α (cid:105) . We choose the coher-ent state to be in the j -th cavity mode with frequency ω j = jπ/L , while the rest of the cavity modes are in theground state. This way the main effects will not comefrom vacuum fluctuations but will instead be amplifiedby the stimulated emission and absorption of the atomcoupled to the coherent state [21, 22]. Therefore the ini-tial state of the system will be ρ = | g (cid:105)(cid:104) g | ⊗ | α j (cid:105)(cid:104) α j | (cid:79) n (cid:54) = j | n (cid:105)(cid:104) n | . (11)While passing through the cavity, the detector spendsa period of time T inside the cavity. Time evolution ofthe system is governed by the interaction Hamiltonian(8) in the proper frame of the detector. We define a timeevolution operator for the detector inside the cavity tobeˆ U ( T,
0) = − i (cid:90) T dτ ˆ H int ( τ ) (cid:124) (cid:123)(cid:122) (cid:125) ˆ U (1) − (cid:90) T dτ (cid:90) τ dτ (cid:48) ˆ H int ( τ ) ˆ H int ( τ (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) ˆ U (2) + ... (12)Therefore the system’s density matrix at the time Twould be evaluated as [21] ρ T = (cid:2)
1+ ˆ U (1) + ˆ U (2) + O ( λ ) (cid:3) ρ (cid:2)
1+ ˆ U (1) + ˆ U (2) + O ( λ ) (cid:3) † . (13)Using the interaction Hamiltonian and the time evo-lution operator we defined above, the first order term ofthe perturbative expansion takes the following formˆ U (1) = λ i ∞ (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) , (14)where I ± ,n is I ± ,n = (cid:90) T dτ e i[ ± Ω d τ + ω n t ( τ )] sin [ k n x ( τ )] , (15)To compute the density matrix for the detector, ρ (d) T ,we need to take the partial trace over the field degreesof freedom [21]. The leading contribution comes fromsecond order in the coupling strength, λ and the finalform of the detector density matrix will be [23] ρ T, (d) = Tr (f) (cid:104) ρ + ˆ U (1) ρ ˆ U (1) † + ˆ U (2) ρ + ρ ˆ U (2) † (cid:105) , (16)which yields ρ T, (d) = Tr f ρ T = (cid:20) − P α P α (cid:21) . (17) P α is the transition probability of the detector from theground state to the first excited state to leading order inperturbation theory, given by [22] P α ( (cid:15), γ ) = λ L (cid:34) α k α (cid:0) | I + ,j | + | I − ,j | (cid:1) + ∞ (cid:88) n =1 | I + ,n | (cid:35) , (18) where α is the amplitude of the coherent state. Noticethat the probabilities P α ( (cid:15), γ ) depend on γ and (cid:15) throughthe integrals I ± ,n , given in (15) as functions of x ( τ ) and t ( τ ). x ( τ ) and t ( τ ) dependence on a , γ, (cid:15) is obtained bysubstituting (3) and (4) into (1). B. Alignment metrology
In the alignment metrology setting, we study the sen-sitivity of the response of a detector to small harmonicspatial perturbations of its otherwise constantly accel-erated trajectory, and analyze its possible use as a wit-ness of the relative alignment of an optical cavity witha beam of atomic detectors. In this setting, the atomicprobes move along a constantly accelerated trajectorywhich undergoes a spatial perturbation that is harmonicin the cavity’s reference frame, ( x, t ): x ( t ) = 1 a (cid:104)(cid:112) a t − (cid:105) + (cid:15) sin( γt ) (19)where (cid:15) and γ are characterizing the amplitude and fre-quency of the perturbation, respectively. In this case,since the motion is analyzed from the lab’s frame, weneed to find the (rather non-trivial) relationship betweenthe proper time of the accelerated atom and the cav-ity frame. The relationship between the cavity frame’sproper time and the atomic probe’s proper time can beworked out from (cid:18) dτdt (cid:19) = 1 − (cid:18) dxdt (cid:19) . (20)Solving this differential equation for dτ /dt together with(19) yields τ ( t ) = arcsinh( at ) a − a(cid:15) (cid:16) cos( γt ) + tγ sin( γt ) (cid:17) γ + O ( (cid:15) ) . (21)The general form of this trajectory is shown in Fig. 2for both the perturbed and the non-perturbed cases.While crossing the cavity, the detector spends a periodof time T in traversing the full length along its trajectory.In order to find the time evolution of the system, we firstneed to find the form of the atom-field Hamiltonian thatgenerates evolution for the entire system with respectto the time coordinate of the lab frame, t . The way toobtain this is explained in detail in [19]. The correcttime-reparametrization of (5) in terms of t is given byˆ H ( t ) = dτdt ˆ H (d)free [ τ ( t )] + ˆ H (f)free ( t ) + dτdt ˆ H int [ τ ( t )] . (22)The monopole moment operator takes the usual formˆ µ d ( t ) = (cid:0) σ + e iΩ d τ ( t ) + σ − e − iΩ d τ ( t ) (cid:1) , (23)and interaction Hamiltonian becomes Figure 2. (Color online) The non-perturbed (blue-dashedcurve) and perturbed (green-solid curve) trajectory for thealignment metrology detting. The trajectory is parameter-ized in the lab’s frame ( x, t ). ˆ H int ( t ) = λ dτdt ∞ (cid:88) n =1 ˆ µ d ( t ) √ ω n L (cid:0) ˆ a † n u n [ x ( t ) , t ] + ˆ a n u ∗ n [ x ( t ) , t ] (cid:1) , (24)with u n [ x ( t ) , t ] = e i ω n t sin [ k n x ( t )] . (25)Working in this frame, the form of the function I ± ,n inthe time evolution operator (14) turns into I ± ,n = (cid:90) T dt e i[ ± Ω d τ ( t )+ ω n t ] sin [ k n x ( t )] . (26)Using the same approach as in the accelerometer set-ting, for characterizing the vacuum response of the parti-cle detector undergoing trajectory (19), we prepare a co-herent state (11) for the scalar field to which the groundstate of the detector is coupled and find the transitionprobabilities from (18). III. RESULTS
The transition probabilities of atomic detectors cross-ing the cavity contain information about the parame-ters characterizing the detectors’ motion. Of course, wewould not want to use perturbations of the Unruh tem-perature as a means to characterize the the trajectoryof the detector. This would be a rather futile endeav-our since the the Unruh temperature itself is somethingextremely difficult to measure, let alone small perturba-tions of it. Instead, we will operate in a non-equillibriumregime where the detector will not have enough time tothermalize with the ‘modified’ Unruh radiation. There-fore, we let the detector spend a small amount of timeinside the cavity such that it does not thermalize withits environment. On top of that, and as discussed above, we consider a coherent state background which helps am-plify the signal. This is the reason why we may expectour system to show more sensitivity to the atom’s tra-jectory. In this section, we analyze the sensitivity of theresponse of the detectors to perturbations in the kinemat-ical parameters of the detectors’ trajectory that we wantto measure, both in the accelerometer and the alignmentsettings.We pause to remark that our choice of switching func-tion χ ( t ) (shown above equation (7)) removes the inter-action between the field and the atom is off when theatom is outside of the cavity. This assumption needssome justification since one cannot just ‘switch off’ theinteraction of the atom with the electromagnetic fieldwhen it is outside the cavity. The rationale of this as-sumption is twofold. On one hand we assume that theatomic state preparation happens at the entrance of thecavity, when the atom’s speed is zero. Equivalently, weare considering a situation in which the atom is post-selected to be in its ground state prior to entering thecavity, and so pre-existing excitations as may be presentoutside of the cavity are not relevant. On the other hand,the main effects on the atomic state responsible for theresults reported here are provoked by the variation of theboundary conditions and the perturbation of the atomictrajectory, which are amplified by the fact that the tra-jectory is relativistic. As we discussed above, the signa-ture of the Unruh effect itself is small as compared tothe non-equilibrium effects coming from the time depen-dence of the trajectory perturbations. Therefore if theflight of the atom includes some small segments of freeflight (outside the cavity), since the Unruh noise wouldbe in these cases arguably negligible it should not modifyour results. A. A quantum accelerometer
We focus first on the accelerometer setting, in whichthere might be small fluctuations of the probe’s acceler-ation of the detector in its own proper frame. We willmodel this by assuming that the proper acceleration ofa set of uniformly accelerated detectors is perturbed bya small harmonic function. One possible way to thinkabout these time dependent oscillations is associate themwith possible inexactnesses in the measure of the acceler-ation in the proper frame of the detector, so that throughrelativistic quantum effects we may expect to be able touse the internal degree of freedom of the atomic probe toincrease the accuracy in exactly determining this properacceleration.With this aim, we study the sensitivity of the transi-tion probability of the detector to the amplitude of theharmonic perturbations and characterize the spectral re-sponse of the setting to the specific frequency range ofthe perturbations. The detector’s trajectory (with a har-monically perturbed acceleration) is given by inserting(4) in (1).To study how sensitive the setting is to the parame-ters of the perturbation, we will analyze the followingsensitivity estimator: S ( (cid:15), γ ) = | P α ( (cid:15), γ ) − P α (0 , γ ) | P α (0 , γ ) (27)where P α ( (cid:15), γ ) is the transition probability of the detectorwith a perturbed acceleration given by (4), and P α (0 , γ )is the transition probability for a constantly accelerateddetector whose trajectory is unperturbed.Fig. 3 shows the explicit dependence of the sensitiv-ity estimator (27) on the parameters characterizing theperturbation,. Namely, it shows the sensitivity of theresponse of the detector to the amplitude (cid:15) of the pertur-bations for different values of acceleration, whereas thespectral response of the sensitivity to different values ofthe perturbation frequency ( γ ) is shown in Fig. 4. Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù
Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê
Á Á Á Á Á Á Á Á Á Á ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï - - e S H e , g = L Ù a = Ï a = ‡ a = Á a = Ê a = Figure 3. (Color online) Behaviour of the sensitivity of thedetector’s transition probability as a function of the ampli-tude of a perturbed proper acceleration for different initialaccelerations.
As one can observe in Fig. 3, for small accelerations,closer to the regimes where the atom does not attain rel-ativistic speeds while crossing the cavity, the sensitivity(to acceleration perturbations) of the detector’s transi-tion probability is monotonic on the amplitude of theperturbations. However, for large accelerations the sen-sitivity does not behave monotonically, and there appearspecific amplitudes for which the sensitivity dips. Thespectral response displayed in Fig. 4 shows that the re-sponse of the detector is always more sensitive to thelower frequencies. The behaviour for higher frequenciesdepends on the energy gap of the atomic probe. For afixed gap, the sensitivity of the probe seems to be ex-ponentially suppressed as the frequency of the perturba-tions grows. One possible way to understand this is thatwhen the frequency of the harmonic acceleration pertur-bation is much higher than the frequency associated withthe transition of the atom, the the atomic probe is pri-marily responsive to its average constant acceleration;the perturbations are much faster than the dynamics ofthe atom and so become invisible to it. However, as wesee in Fig. 4b), it is possible to adjust the gap of the atomic transition used as a probe to tune out to a spe-cific frequency range of the perturbations.In Fig. 4c), we show how sensitive the response ofthe atomic probes is to the length of the cavity they’retransversing. This in turns also determines how muchrelativistic the probes are when existing the cavity forconstant acceleration. These curves also suggest that itmay be possible to use similar settings as a means todetermine the length of an optical cavity.
Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï
Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê
Á Á Á Á Á Á Á Á Á Á ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù g S H e = . , g L Ù a = Ï a = ‡ a = Á a = Ê a = a Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï
Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
Á Á Á Á Á Á Á Á Á Á
Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê g S H e = . , g L Ù a= Ï a= ‡ a= Á a= Ê a= b Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù
Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á
Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê g S H e = . , g L Ù L = Ï L = ‡ L = Á L = Ê L = c Figure 4. (Color online) Spectral response of the detectorfor a) both relativistic and nonrelativistic accelerations, b)different modes of the field which are in coherent states andcoupled to the ground state of the detector and c) differentlengths of the cavity.
Of course, the sensitivity estimator we studied onlygives us an idea of the potentiality of these settings forthe measurement of the parameters of the perturbation.A more realistic practical implementation of such set-tings would require considerable effort. For example thismight be implemented by comparing one setting whereall the parameters are known with another setting wherethe parameters are not known. The comparison of thetransition rates of beams of atoms in these two settingsmay reveal the information about the parameters to bedetermined. In such a comparison the estimator builthere becomes relevant.
B. Alignment metrology
In the alignment setting we assume that the trajectoryof uniformly accelerated detectors is perturbed by a smallharmonic motion, that we could, for instance, ascribe tooscillations of the trajectory of the detector in the cavityframe. These can be understood as time dependent im-precisions in the alignment of the setting with the opticalcavity.Here we study the sensitivity of the detector’s responseto the amplitude of the harmonic perturbations and char-acterize the spectral response of the setting to the fre-quency of perturbations. We consider the spatial per-turbation as expressed in equation (19). Since in thederivation of the parametrization of the detector’s worldline (21) we linearized in the amplitude of the perturba-tion (cid:15) , we only consider small amplitudes 0 < (cid:15) < . (cid:15) for different values of ac-celeration and different frequencies are shown in Fig. 5a)and b) respectively. We estimate this sensitivity by us-ing the same quantity (27) as in the accelerometer settingwith the only difference that P α ( (cid:15), γ ) represents transi-tion probability of the detector with a spatially perturbedtrajectory which is otherwise constantly accelerated.As shown in Fig. 5a) for small accelerations wherethe system is closer to nonrelativistic regimes, the detec-tor’s response shows more sensitivity to the perturbationof its trajectory than in the case of higher accelerations(relativistic regimes). In contrast to the previous case ofperturbations in the probe’s proper acceleration, we seefrom Fig. 5b) that the detector’s response is less sensitiveto low frequency perturbations of its spatial trajectory.This is again reasonable, considering that higher the fre-quency of perturbations of the spatial trajectory in thelab frame, the more of an effective change they will haveon the detector’s proper acceleration; a high frequencyspatial perturbation in the lab frame corresponds to alarge instantaneous change of the proper acceleration ofthe detector. This in turn affects the response of the de-tector more dramatically than if the perturbation of thespatial trajectory is slow. As expected, the sensitivityincreases monotonically as the amplitude of fluctuationsgrows, as seen in the figures.We display in Fig. 6 the spectral response of the sen- ·· · · · · · · · · ‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ÙÙ Ù Ù Ù Ù Ù Ù Ù Ù
Ï ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï
ŸŸ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ
ÁÁ Á Á Á Á Á Á Á Á e S H e , g = L Ù a = Ï a = Ÿ a = · a = ‡ a = Ê a = a ŸŸ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ
ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê
ÁÁ Á Á Á Á Á Á Á Á ‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ÏÏ Ï Ï Ï Ï Ï Ï Ï Ï - e S H e , g = L Ÿ g= Ï g= ‡ g= Á g= Ê g= b Figure 5. (Color online) The sensitivity of the excitation prob-ability of the detector to the amplitude (cid:15) of the trajectoryperturbations for a) different constant accelerations and forb) different frequencies of perturbation. sitivity of the probe’s excitation probability for a fixedamplitude of the perturbation for different values of thesetting parameters: proper accelerations Fig. 6a), cavitylengths Fig. 6b) and detector gaps Fig. 6c).The general trend in all cases is that the transitionprobability of the detector presents dips for specific val-ues of the perturbation frequency γ . In other words,there are some specific perturbation frequencies for whichthe sensitivity of the setting goes down abruptly, beingthe position of these dips is a function of the system pa-rameters. This resonance-like effect that may be relatedwith the spatial distribution of the cavity modes as seenfrom the reference frame of the atom whose trajectory isperturbed, but it seems to depend non-trivially on thesystem parameters and we have not been able to identifyits exact origin through numerical analysis. IV. CONCLUSIONS
We have analyzed the sensitivity of the response of aconstantly accelerated atomic probe, transversing an op-tical cavity, when its trajectory is perturbed. We showedthat the probe’s transition probability is, in principle,sensitive to small deviations from constant acceleration.
Ù Ù Ù Ù Ù Ù Ù Ù Ù ÙÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ · · · · · · · · · ·Ï Ï Ï Ï Ï Ï Ï Ï Ï ÏÁ Á Á Á Á Á Á Á Á Á g S H e = . , g L Ù a = Ï a = Ÿ a = · a = ‡ a = Á a = Ê a = a Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù
Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï
Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ · · · · · · · · · ·
Á Á Á Á Á Á Á Á Á Á
Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê g S H e = . , g L Ù L = Ï L = Ÿ L = · L = Á L = Ê L = b · · · · · · · · · · Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê
Á Á Á Á Á Á Á Á Á Á
Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ
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Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù g S H e = . , g L Ê a= Ê a= Ê a= Ê a= Ê a= Ê a= c Figure 6. (Color online) Spectral response of the detector fora) different accelerations from nonrelativistic regimes ( a =0 . , . , .
05) to relativistic regimes ( a = 0 . , . , . , We conclude that the transition rate of a beam of atomstransversing optical cavity can provide information aboutits past spatial trajectory.We have theoretically studied the potential of the useof an atomic internal quantum degree of freedom to de-sign novel quantum metrology settings. In particular weconsidered two scenarios: one where the probe undergoessmall time-dependent perturbations of its proper accel-eration, and another one when the probe’s trajectory ex-periences small spatial time-dependent perturbations asseen from the laboratory’s frame.The first scenario could correspond to an accelerome-ter setting where we use the internal degree of freedom ofthe atom to identify small time-dependent forces actingon the probe that will cause it to deviate from constantproper acceleration. The second scenario could corre-spond to an alignment measurement setting where weuse the internal atomic degree of freedom to characterizesmall vibrations or imperfections of the alignment of anoptical cavity with a beam of atoms transversing it.While an analysis of a proper experimental implemen-tation goes beyond the scope of this paper, these findingshave a potential use in quantum metrology of optical se-tups. For instance one could compare one setting whereall the parameters are known with another setting wherethey are not known. In practice, however, the ratio ofthe probabilities will be subject to significant statisticalfluctuations that could mask the effects we have obtained.To achieve the sensitivity levels that are potentially avail-able, the implementation of our scheme will require ac-cumulation of statistics over a number of identical ex-periments by sending a large number of atomic probesthrough the cavity. Thus, by analyzing the transitionrates of different atomic beams, one could in principlededuce the specific form of the trajectory of such beamsor infer the parameters of the optical cavities they aretraversing.
V. ACKNOWLEDGMENTS
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