Unruh Effect of Detectors with Quantized Center-of-Mass
UUnruh Effect of Detectors with Quantized Center-of-Mass
Vivishek Sudhir,
1, 2, ∗ Nadine Stritzelberger,
3, 4, 5 and Achim Kempf
3, 4, 5, 6 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Department of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
The Unruh effect is the prediction that particle detectors accelerated through the vacuum getexcited by the apparent presence of radiation quanta — a fundamental quantum phenomenon in thepresence of acceleration. Prior treatments of the Unruh effect, that presume a classically prescribedtrajectory, do not account for the quantum dynamics of the detector’s center-of-mass. Here, we studymore realistic detectors whose center of mass is a quantized degree of freedom being accelerated byan external classical field. We investigate the detector’s recoil due to the emission of Unruh quanta.Vice versa, we also study the recoil’s impact on the emission of Unruh quanta and the excitationof the detector. We find that the recoil due to the emission of Unruh quanta may be a relevantexperimental signature of the Unruh effect.
I. INTRODUCTION
An idealized particle detector — a two-level system —accelerated through the vacuum of a quantum field canbecome excited [1–5]. In the case of uniform acceleration,the state of the detector assumes a thermal form withan apparent temperature proportional to the accelera-tion. This is the Unruh effect, and the particle detectoris eponymously referred to as the Unruh-DeWitt (UDW)detector [1, 2]. In the usual treatment, the energy andmomentum required to excite the detector, along withthe subsequent emission of a photon, come from an un-specified external agent which enforces a prescribed clas-sical uniformly accelerated trajectory for the detector.These idealized treatments of the Unruh effect neitherdescribe the accelerating agent, nor the recoil of the de-tector due to photon emission. Both are important, how-ever, to the question of the viability of an experimentalobservation of the Unruh effect. For example, real exper-iments cannot sustain uniform acceleration indefinitely,and also detector recoil makes it unrealistic to consideruniform acceleration. In fact, the recoil could itself be anavenue towards the detection of the Unruh effect. Impor-tantly, however, in this case the sensitivities required toresolve the recoil are likely to be comparable to the quan-tum fluctuations of the detector’s center-of-mass motion.This means that the recoil needs to be calculated withina full quantum mechanical treatment of the detector’scenter-of-mass degree of freedom. From this perspective,the couple of works studying detector recoil from the Un-ruh effect (e.g., [6–8]) remain incomplete.Here, we dynamically account for the acceleration ofthe detector, and self-consistently treat the recoil of thedetector center-of-mass. To this end, the detector’s cen-ter of mass degrees of freedom are treated quantum me- ∗ [email protected] chanically [9], and its acceleration is described by cou-pling the detector to an external classical acceleratingfield. Within this framework, we study the vacuum exci-tation process for the internal and center of mass degreesof freedom of the detector — a process we term the mas-sive Unruh effect , in reference to the detector mass beingassumed finite. By letting the detector mass go to infin-ity, the behavior of classical detector trajectories can berecovered. We also compute the recoil pattern and its re-lationship to the pattern of the emission of Unruh quanta.Both patterns assume characteristic non-isotropic forms.The peak emission probability is proportional to the ac-celeration, which allows us to informally talk of a “tem-perature” for the massive Unruh effect, in analogy withthe conventional Unruh effect. II. UNRUH EFFECT WITH INFINITE-MASSDETECTOR
We briefly recall the Unruh effect for a detector acceler-ated on a prescribed trajectory through a scalar quantumfield. The detector-field system is described by the freeHamiltonian ˆ H = Ω | e (cid:105) (cid:104) e | + (cid:90) d k ck ˆ a † k ˆ a k , (1)where Ω denotes the energy gap of the detector’s ground( | g (cid:105) ) and excited ( | e (cid:105) ) states, and ˆ a † k (ˆ a k ) is the creation(annihilation) operator of the scalar quantum field modeof momentum k . Following Unruh and DeWitt [1, 2],and in analogy with realistic models of light-matter in-teraction in electrodynamics [10], we assume that the de-tector’s internal state couples to the field through theinteraction Hamiltonian,ˆ H int = q ˆ µ ( t ) ⊗ ˆ φ ( x ( t )) . (2)Here, q is the coupling strength, ˆ µ = | e (cid:105) (cid:104) g | + | g (cid:105) (cid:104) e | isthe detector’s “monopole” moment, and ˆ φ ( x ( τ )) is the a r X i v : . [ qu a n t - ph ] F e b scalar field operator along the detector’s trajectory. Welimit ourselves to the regime of non-relativistic detectorvelocity, which allows us to identify the detector’s propertime τ with the coordinate time t .The crux of the Unruh effect is that the detector can beexcited by accelerating it through the quantum vacuumof the scalar field. Given the structure of the interaction,the detector can be excited only if the quantum field issimultaneously excited to (at least) the single particlequantum state, ˆ a † k | (cid:105) . That is, contrary to resonanceeffects such as absorption, the Unruh effect is the resultof counter-rotating-wave terms in ˆ H int [11]. To elucidatethis, we compute the probability that the initial state, | ψ i (cid:105) = | g (cid:105) ⊗ | (cid:105) – the joint ground state of the system –transitions to the final state, | ψ f (cid:105) = | e (cid:105) ⊗ ˆ a † k | (cid:105) , whereboth the detector and the field are excited. Working inthe interaction picture defined by the free Hamiltonian,in which,ˆ µ ( t ) = e i Ω t | e (cid:105) (cid:104) g | + h . c . ˆ φ ( x , t ) = (cid:90) d k (2 π ) / (cid:114) c k (cid:2) e − ickt + i k · x ˆ a k + h . c . (cid:3) , the probability amplitude (to first order in perturbationtheory) for the afore-mentioned excitation process is, A U ( k ) = (cid:104) ψ f | (cid:90) ∞−∞ dt ˆ H int ( t ) | ψ i (cid:105) = qc π √ πk I , where, I = (cid:90) ∞−∞ dt e it ( ck +Ω) − i k · x ( t ) . (3)We now define the probabilities for: the detector to getexcited and a field quantum of momentum k to be emit-ted, P U ( k ) = |A U ( k ) | ;and the excitation of the detector, irrespective of the mo-mentum of the emitted photon, P U = (cid:90) d k |A U ( k ) | . When the detector is in inertial motion, x ( t ) = x + v t ,the excitation amplitude is zero: A U = qc π √ πk (cid:90) ∞−∞ dt e it ( ck +Ω − k · v ) − i k · x = qc √ πk e − i k · x δ (Ω + ck − k · v )= 0 , essentially because Ω+ ck − k · v (cid:54) = 0, owing to the fact theenergy gap is positive (Ω > k > | v | < c ). That is, in inertial motion through thevacuum, the detector does not get excited. In contrast, consider the detector in non-inertial mo-tion with a uniform acceleration a for time T along the z − direction, i.e., a ( t ) = a Θ( t )Θ( T − t ) e z , where Θ is the Heaviside step function. Assuming thatthe detector’s initial position coincides with the origin ofthe coordinate system, its trajectory is, x ( t ) = (cid:20) at t ) Θ( T − t ) + aT t − T ) Θ( t − T ) (cid:21) e z . (4)In the following, we restrict the time duration T sothat the velocity developed in that time with an accel-eration a is well within the non-relativistic regime; inparticular, we will always take, | v ( T ) | = aT (cid:46) . c .Within this non-relativistic regime, and for the spa-tial trajectory in Eq. (4), we obtain for the time in-tegral given in Eq. (3) [here we define, ω = Ω + ck , ω (cid:48) = ω − ak z T , and, k z the z − component of the mo-mentum of the emitted Unruh photon, k = ( k x , k y , k z ) =( k sin( θ ) cos( φ ) , k sin( θ ) sin( φ ) , k cos( θ ))], I = 1 iω + πδ ( ω ) + e iT ( ω (cid:48) + ak z T/ ) (cid:18) iω (cid:48) + πδ ( ω (cid:48) ) (cid:19) + (cid:115) πe iω akz iak z (cid:20) Erf (cid:18) iω √ iak z (cid:19) − Erf (cid:18) iω (cid:48) √ iak z (cid:19)(cid:21) Since both the energy gap of the detector and the abso-lute value of the momentum of the emitted photon arestrictly positive (Ω > k > δ ( ω ) can be omitted. Similarly, since the detector’svelocity is strictly smaller than the speed of light, we find ω (cid:48) >
0, and so δ ( ω (cid:48) ) can also be omitted. Thus, I = 1 iω − e iT ( ω (cid:48) + ak z T/ iω (cid:48) + (cid:115) πe iω /ak z iak z Erf (cid:18) iω √ iak z (cid:19) − (cid:115) πe iω /ak z iak z Erf (cid:18) iω (cid:48) √ iak z (cid:19) ; (5)which is in general non-zero in contrast to the case of in-ertial motion. Therefore, the total excitation probability,defined by, P U = q c π (cid:90) − dz (cid:90) ∞ dk k |I| , where, z = cos( θ ) ∈ [ − , k irrespec-tive of direction, becomes, P U ( k ) = q c k π (cid:90) − dz |I| . FIG. 1. Emission probability for an accelerated UDW detec-tor. (a) The total (i.e. angle-integrated) probability P U ( k ) forthe emission of an Unruh quantum with momentum k , for var-ious values of the energy gap Ω. (b) Angle-resolved emissionprobability P U ( k, θ ) (in units of 10 − cT q ), for energy gapΩ = 0 . /T . Both plots are for acceleration, a = 8 · − ( c/T ).Note that the oscillations are in Figure 1a arise from acceler-ating the detector for a time interval that is compact. The symmetry of the problem along the z − axis meansthat the photon emission is azimuthally symmetric, sothat it is useful to consider the probability density, P U ( k, z ) = q c k π |I| corresponding to a photon of momentum k emitted alongthe polar angle, θ = cos − z .Figure 1a shows the probability P U ( k ) for the excita-tion of the UDW detector by the emission of an Un-ruh quantum of momentum k ; by momentum conser-vation, this is equivalent to the angle-integrated prob-ability of finding an Unruh quantum of momentum k .Note that for the case we consider here, where the de-tector is not eternally accelerated, the emission is not isotropic (see Fig. 1b), precluding complete analogy withblackbody radiation. In fact, radiation is preferentiallyemitted along/against the direction of acceleration. (Forrealistic scenarios involving charges or atoms as UDWdetectors, the correct vacuum would not be that of thescalar quantum field — which is what we consider here —but the vacuum of the full vector electromagnetic field;the vectorial character of the latter is expected to pro-duce radiation transverse to the acceleration, in analogywith classical synchrotron radiation [12].) However thereis one aspect of blackbody radiation that is reflected inFig. 1a: as the energy gap gets smaller the peak of theemission shifts to lower momenta, an observation thatcan be put in correspondence with Wien’s displacementlaw for blackbody radiation (i.e. k peak ∝ temperature)if we associate a temperature proportional to the energygap of the detector. (Note that one can always assigna temperature for a two-level system whenever its den-sity matrix is diagonal: the ratio of the diagonal ele-ments can be compared to that of the canonical thermalstate, and so an effective temperature can be defined.)In this sense, we may formally associate a temperatureto the Unruh process, even in the case where the detec-tor is accelerated only for a finite time interval. Finally,in the non-relativistic regime, when the emitted photonmomentum is “small”, we have that, β − γ = aT k z (cid:28) k z (cid:28)
1, which, in the dimen-sioned units of Fig. 1a reads, k z (cid:28) ( cT ) − ); explicitlyexpanding the amplitude integral in Eq. (5) in the smallparameter β − γ ∝ a , one can also show that P U ∝ a .Thus we are able to formally establish that the temper-ature is proportional to the acceleration even in the casewhere the detector is accelerated only for a finite dura-tion. III. UNRUH EFFECT WITH FINITE-MASSDETECTOR
Experimentally producing the large accelerations re-quired to observe the Unruh effect call for low mass UDWdetectors. Any such detector will experience significantrecoil once an Unruh quantum is emitted. Once the pos-sibility of recoil is admitted, it becomes unphysical toconsider an externally prescribed acceleration, even fora finite time interval (especially if we are interested inmeasuring the random recoil over an ensemble of multi-ple emission events). We also envision the possibility ofinferring the Unruh effect through a direct experimen-tal measurement of the recoil; the required measurementsensitivities are expected to be comparable to the quan-tum fluctuations of the detector center-of-mass degree offreedom. For these reasons, we must treat the detector’scenter-of-mass in a full quantum framework, and, self-consistently incorporate external agency that acceleratesthe system.To this end, we consider a massive detector with aquantized center-of-mass degree of freedom which couplesto a quantum scalar field ˆ φ [9, 13]. The detector’s centerof mass is coupled to a classical “electric” field E , whichallows us to dynamically model the acceleration of thedetector. This scenario is modeled by the Hamiltonian,ˆ H = ˆ p M − q E · ˆ x + Ω | e (cid:105) (cid:104) e | + (cid:90) d k ck ˆ a † k ˆ a k + q (cid:90) d x ˆ P ( x ) ⊗ ˆ µ ⊗ ˆ φ ( x )with M the mass of the detector, and, ˆ P ( x ) = | x (cid:105) (cid:104) x | is the projector onto the center-of-mass position eigen-states. In the interaction picture, the Hamiltonian reads, H int ( t ) = q (cid:90) d x ˆ P ( x , t ) ⊗ ˆ µ ( t ) ⊗ ˆ φ ( x , t ) , (6)with ˆ P ( x , t ) = | x ( t ) (cid:105) (cid:104) x ( t ) | . In order to model a scenariocomparable to the situation considered in the previoussection, we assume an electric field, E ( t ) = E Θ( t ) Θ( T − t ) e z , with a non-zero strength E in the time interval t ∈ [0 , T ],and zero elsewhere. It models the detector’s center-of-mass being uniformly accelerated in that interval, whileit evolves freely for t / ∈ [0 , T ].Before we can study the vacuum excitation process forthe massive detector, we need the time evolved operatorsˆ P ( x , t ) = | x ( t ) (cid:105) (cid:104) x ( t ) | . To this end, we write the Heisen-berg equation for the detector’s position and momentum,˙ˆ x ( t ) = ˆ p ( t ) M , ˙ˆ p ( t ) = q E ( t ) e z , which produces the time-dependent position operator,ˆ x ( t ) = ˆ x (0) + ˆ p (0) t/M + f ( t ) e z , where, f ( t ) = a (cid:2) t Θ( t ) Θ( T − t ) + T (2 t − T )Θ( t − T ) (cid:3) , with, a = q E /M , being the uniform acceleration due tothe electric field. We note that f ( t ) is of the same formas the z − component of the classical trajectory which weprescribed in Eq.(4) for the UDW detector with classicalcenter of mass. Since the position and momentum oper-ators coincide at time t = 0 between the Heisenberg andSchr¨odinger pictures, we have that, ˆ x (0) ψ ( x ) = x ψ ( x )and ˆ p (0) ψ ( x ) = − i ∇ ψ ( x ). Next, to find the time de-pendent position eigenfunction ψ ξ ( x , t ) = (cid:104) x | ψ ξ ( t ) (cid:105) fora given position eigenvalue ξ , we solve the Schr¨odingerequation, (cid:18) x + f ( t ) e z − itM ∇ (cid:19) ψ ξ ( x , t ) = ξ ψ ξ ( x , t ) , with the initial condition | ψ ξ (0) (cid:105) = | ξ (cid:105) , and enforcingthe normalization condition, (cid:90) d x ψ ∗ ξ ( x , t ) ψ ξ (cid:48) ( x , t ) = δ (3) ( ξ − ξ (cid:48) ) . The required wavefunction is, | ψ ξ ( t ) (cid:105) = (cid:90) d p (2 π ) / exp (cid:20) it p M − i p · ξ + ip z f ( t ) (cid:21) | p (cid:105) . Putting all this together, we obtain the time evolved pro-jection operator,ˆ P ( x , t ) = (cid:90) d pd q (2 π ) exp (cid:104) it q − p M − i ( q − p ) · x + i ( q z − p z ) f ( t ) (cid:105) | q (cid:105) (cid:104) p | , that fully determines the interaction hamiltonian inEq. (6). A. Transition amplitude, probability andprobability densities
We are now equipped to study what we refer to asthe massive Unruh effect , that is, the excitation processboth of a UDW detector with a finite mass, initially inthe ground state of its internal degree of freedom coupledto a scalar quantum field initially in its vacuum state,and accelerated by an external electric field. That is, weconsider initial and final states of the form, | ψ i (cid:105) = | ϕ (cid:105) ⊗ | g (cid:105) ⊗ | (cid:105)| ψ f (cid:105) = | r (cid:105) ⊗ | e (cid:105) ⊗ ˆ a † k | (cid:105) where, | ϕ (cid:105) = (cid:82) d p ˜ ϕ ( p ) | p (cid:105) is the initial center of massstate, | p (cid:105) the center of mass momentum eigenstates, ˜ ϕ ( p )the initial center of mass wave function in the momentumrepresentation, and r the detector’s recoil momentum.The transition amplitude for the process where the de-tector gets excited, its center-of-mass recoils with mo-mentum r , and emits an Unruh quantum of momentum k , is (upto a phase factor), A M = qc π √ πk ˜ ϕ ( r + k ) J ( r ) , where we define, J ( r ) = (cid:90) ∞−∞ dt exp (cid:20) it (cid:18) k M − r · k M + ck + Ω (cid:19) − ik z f ( t ) (cid:21) . The corresponding transition probability density is, P M ( k , r ) = |A M | = q c (2 π ) πk | ˜ ϕ ( r + k ) J ( r ) | To study the recoil of the detector, we consider theexcitation probability density as a function of the recoilmomentum r (and irrespective of the momentum of theemitted photon), P M ( r ) = (cid:90) d k q c (2 π ) πk | ˜ ϕ ( r + k ) J ( r ) | ;and the total excitation probability for the massive Un-ruh process, P M = (cid:90) d k (cid:90) d p q c (2 π ) πk | ˜ ϕ ( p ) J ( p − k ) | . To resolve the angular dependence of the emis-sion and recoil, we write k = ( k x , k y , k z ) =( k sin( θ ) cos( φ ) , k sin( θ ) sin( φ ) , k cos( θ )), with φ the az-imuthal angle and with θ the polar angle, that is, theangle between the momentum of the emitted photon andthe direction of the electric field lines (and therefore, ofthe acceleration); as before, we also define, z = cos( θ ) (inthe following we will refer to both z and θ as the polarangle of the emitted photon). The excitation probabilitydensity for the process to happen while emitting an Un-ruh quantum of momentum k in magnitude, irrespectiveof direction, is P M ( k ) = (cid:90) − dz (cid:90) π dφ (cid:90) d k (cid:90) d p q c k (2 π ) π × | ˜ ϕ ( p ) J ( p − k ) | (7)Similarly, we define the excitation probability density asa function of both the magnitude k and polar angle z ofthe emitted photon: P M ( k, z ) = (cid:90) π dφ (cid:90) d p q c k (2 π ) π | ˜ ϕ ( p ) J ( p − k ) | (8)All the above expressions depend on the time integral J ,which can be explicitly evaluated, J ( p − k ) = 1 iω M − e iT ( ω (cid:48) M + ak z T/ iω (cid:48) M + (cid:115) πe iω /ak z iak z Erf (cid:18) iω M √ iak z (cid:19) − (cid:115) πe iω /ak z iak z Erf (cid:18) iω (cid:48) M √ iak z (cid:19) (9)in terms of, ω M = ω + k M − p · k M , and, ω (cid:48) M = ω M − ak z T . In writing the above expression, we have omittedtwo terms each involving delta distributions δ ( ω M ) and δ ( ω (cid:48) M ), which is justified for the following reasons. First,let us write p · k = pk cos( κ ), with κ the angle between p and k , and define, ω = ω + k M for brevity. Thedelta distribution δ ( ω M ) then peaks only for cos( κ ) = Mω pk . Furthermore, the delta distribution δ ( ω (cid:48) M ) peaksonly for cos( κ ) = Mpk ( ω − aT k z ). But since cos( κ ) ∈ [ − , δ ( ω M ) to peak is p ≥ M c , which translates to saying that the initial virtualcenter of mass velocities would have to be superluminal,which is ruled out. Similarly, a necessary condition for δ ( ω (cid:48) M ) to peak is p + M aT ≥ M c , which would requirethe virtual center of mass velocities to be superluminalby the end of the accelerated phase. Physically, the delta distributions δ ( ω M ) and δ ( ω (cid:48) M ) have their origin in thevirtual inertial motion of the detector, respectively forthe times t < t > T during which the electric fieldis switched off. Inertial virtual motion (just like inertialreal motion) should not cause excitation of the detectorand the field, which is reflected in the vanishing of thesedelta distributions. B. The Unruh effect as a limiting case of themassive Unruh effect
Before proceeding further, let us see how to recover theconventional Unruh effect (of Section II) — i.e. a UDWdetector with a prescribed classical trajectory — fromthe “massive Unruh effect” studied above.In order to recover the traditional Unruh effect for a de-tector experiencing a uniform acceleration a , let us con-sider the limit of infinite detector mass — so that thecenter of mass wave function delocalizes infinitely slowlyand so it effectively behave classically. A classical particleof charge q and mass M in a constant electric field E ex-periences an acceleration a = q E /M . Defining M = mγ and E = εγ and letting γ → ∞ allows us to keep the ac-celeration a experienced by the detector constant, whileconsidering the infinite mass limit:lim γ →∞ P M = P U . The above equation holds true for all the probabilitiesdefined above; in this sense, the massive Unruh effectsubsumes the conventional (infinite-mass) Unruh effect.
C. Example of a Gaussian center of mass wavepacket
In order to apply the formalism developed in Sec-tion III A for the massive Unruh effect, we need to specifyan initial wave function, ˜ ϕ ( p ), for the detector’s centerof mass. We consider a Gaussian initial center of mass(momentum) wave packet of the form,˜ ϕ ( p ) = (cid:18) L π (cid:19) / e − p L / , assumed to exist at time t = 0, so that when the elec-tric field is switched on, the detector’s center of mass inposition space is localized at the origin with width L .Since we work in the non-relativistic regime, it is nec-essary to choose the parameters L , M , T and a in a waythat ensures that the detector’s virtual center of mass ve-locities are much less than the speed of light. The initialmomentum of the detector in the z − direction, i.e., par-allel to the electric field, is Gaussian distributed around p z = 0, with a standard deviation of √ /L . Initial mo-menta which are σ standard deviations away from themean then correspond to initial virtual center of massspeed, | v z (0) | = 2 σ · √ / ( LM ). If we require that thesetails of the wavepacket, after having the accelerating field E turned on for a time T , be still less than 1% the speedof light, then we need to maintain an electric field thatsatisfies, v z ( T ) = | v z (0) | + aT (cid:46) . c . In the following,we choose electric field strengths weak enough that thisis satisfied for virtual velocities that are σ = 3 . v/c = p/ ( M c ) (cid:28)
1; thus |J ( p − k ) | (fromEq. (9)) can be approximated around p / ( M c ) = 0 byusing only its first two Taylor coefficients: J = (cid:104) |J ( p − k ) | (cid:105) ω = ω ,J = 12 (cid:20) ∂ ∂A |J ( p − k ) | (cid:21) ω = ω . Expressing J , in terms of the variables k and z , wethen obtain for the basic excitation probability densitiesintroduced above (valid to O (( LM c ) − )): P M ( k, z ) ≈ q c k π (cid:20) J ( k, z ) + k J ( k, z )( M L ) (cid:21) P M ( r, ζ ) ≈ L q c π ) / (cid:90) ∞ dk (cid:90) − dz ke − ( r + k +2 rkζ ) L / × (cid:20) J ( k, z ) + ( rkζ + k ) M J ( k, z ) (cid:21) Here, r is the magnitude of the recoil momentum and ζ = r · k / ( rk ) is the cosine of the angle between therecoil momentum ( r ) and Unruh photon momentum ( k ).Figure 2a depicts the difference, P U ( k ) − P M ( k ), be-tween the angle-integrated emission probabilities of theconventional infinite-mass UDW detector and a UDWdetector of finite mass, as a function of the momen-tum k of the Unruh quantum. As the mass increases,the finite mass case approaches the standard case (i.e. P U − P M →
0) as expected.Although the total emission probability, and hence theemitted flux of Unruh quanta, is seen to decrease withmass, each individual emission causes greater recoil of thedetector center-of-mass for a lower mass detector. Thisis shown in Fig. 2b, where the angular recoil probabilityis plotted as a function of the magnitude of the recoilmomentum. Recoil with larger momenta happen in thedirection opposite to the acceleration (0 ◦ in the polarplot) — a consequence of momentum conservation. IV. CONCLUSION
We analyzed the behavior of UDW detectors whichpossess a quantum mechanical center of mass and thatare finitely accelerated through the vacuum of a scalarquantum field. We found a characteristic interplay be-tween the acceleration-induced excitation of the UDWdetector, the non-isotropic patterns of the flux of the
FIG. 2. Emission and recoil probability for a massive UDWdetector. (a) Plot shows the difference of the emission proba-bilities between the cases where the detector center-of-mass isinfinitely heavy (i.e. P U ( k )) and the case where the detectorcenter-of-mass has a finite mass (i.e. P M ( k )). (b) Recoil prob-ability density P M ( r, ζ ) (in units of 10 − ( cT q ) − ) as a func-tion of recoil momentum angle ζ , for a few values of the mag-nitude of the recoil momentum magnitude r . The detectormass is M = 10 / ( c T ), while in both panels, it’s internal en-ergy gap is Ω = 0 . /T , acceleration a = 8 · − ( c/T ), and thecenter-of-mass wavepacket has an initial width, L = 100( cT ). emitted Unruh quanta and the corresponding quantumrecoil. This makes the quantum recoil a potentially ex-perimentally relevant signature of the Unruh effect.In practice, trapped and accelerated (sub-)atomic par-ticles with internal states can act as UDW detectors. Theability to precisely measure forces that act on them maythen provide a route to a direct detection of the Unruheffect on a single particle. Indeed, electron bunches instorage rings have long been suspected of being probesof the Unruh effect [14, 15]. However, space charge effectsand other systematics have prevented a decisive measure-ment of the Unruh effect (nor does the accelerator envi-ronment offer an ideal venue to study more fundamentalpredictions of the Unruh effect).Although any realistic experiment along these lines willhave to consider the coupling of the internal degree offreedom to the electromagnetic vacuum, the scalar field scenario considered here allows a first qualitative glimpseof what to expect. Further studies along these directionsare in progress. [1] W. G. Unruh, Phys. Rev. D , 870 (1976).[2] B. S. DeWitt, ”Quantum gravity: The new synthesis”, General Relativity, an Einstein Centenary Survey (eds.S. W. Hawking and W. Israel) (Cambridge UniversityPress, 1979).[3] G. W. Gibbons and S. W. Hawking, Phys. Rev. D ,2738 (1977).[4] N. D. Birrell and P. C. W. Davies, Quantum Fields inCurved Space , Cambridge Monographs on MathematicalPhysics (Cambridge University Press, 1982).[5] L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev.Mod. Phys. , 787 (2008).[6] R. Parentani, Nuclear Physics B , 227 (1995).[7] R. Casadio and G. Venturi, Physics Letters A , 33(1995).[8] B. Reznik, Physical Review D , 2403 (1998). [9] N. Stritzelberger and A. Kempf, Phys. Rev. D ,036007 (2020).[10] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applica-tions (Wiley, 2004).[11] M. O. Scully, S. Fulling, D. M. Lee, D. N. Page, W. P.Schleich, and A. A. Svidzinsky, Proceedings of the Na-tional Academy of Sciences , 8131 (2018).[12] J. D. Jackson,
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