Quantum radiation by an Unruh-DeWitt detector in oscillatory motion
Quantum radiated power emitted by an Unruh-DeWitt (UD) detector in linear oscillatory motion in (3+1)D Minkowski space, with the internal harmonic oscillator minimally coupled to a massless scalar field, is obtained non-perturbatively by numerical method. The signal of the Unruh-like effect experienced by the detector is found to be pronounced in quantum radiation in the highly non-equilibrium regime with high averaged acceleration and short oscillatory cycle, and the signal would be greatly suppressed by quantum interference when the averaged proper acceleration is sufficiently low. An observer at a fixed angle would see periods of negative radiated power in each cycle of motion, while the averaged radiated power over a cycle is always positive as guaranteed by the quantum inequalities. Coherent high harmonic generation and down conversion are identified in the detector’s quantum radiation. Due to the overwhelming largeness of the vacuum correlators of the free field, the asymptotic reduced state of the harmonics of the radiation field is approximately a direct product of the squeezed thermal states.
Keywords:quantum dissipative system, boundary quantum field theory, thermal field theory.
A uniformly, linearly accelerated point-like detector moving in the Minkowski vacuum will experience thermal fluctuations at a temperature proportional to the proper acceleration of the detector Un76 (). This is called the Unruh effect and the temperature is called the Unruh temperature. While the derivation in time-dependent perturbation theory is well established, direct experimental evidence for the Unruh effect is still lacking. One closely related observation in laboratories is the electron depolarization in storage rings, namely, the Sokorov-Ternov effect ST63 (), which can be connected to the “circular Unruh effect” BL83 (); BL87 (); Un98 (); AS07 (). Nevertheless, the centripetal acceleration in the circular Unruh effect is quite different in nature from the original linear, uniform acceleration in the Unruh effect HJ00 (). To get closer to the original conditions in Unruh’s derivation, there have been existing proposals to look at the quantum correction by the Unruh effect to the radiation emitted by a linearly accelerated charge or atom CT99 (); SSH06 (); SSH08 (), which is called the “Unruh radiation”.
Seeking the evidence of the Unruh temperature in quantum radiation is, however, not as straightforward as it appears. To well define a finite temperature in an atom-field state, the atom should be in equilibrium with the field. Unfortunately a uniformly, linearly accelerated Unruh-DeWitt (UD) detector (analogous to an atom) Un76 (); DeW79 () derivatively coupled to a massless scalar field in the Minkowski vacuum emits no radiation in equilibrium conditions in (1+1)D Minkowski space Gr86 (); RSG91 (); Un92 (); MPB93 (); HR00 (). In (3+1)D Minkowski space there will be radiation by a uniformly accelerated UD detector in steady state at late times (at a constant radiation rate with respect to the proper time of the detector), but the radiated energy is not converted from the one experienced by the detector in the Unruh effect LH06 (). The physical reason for these results is that quantum interference between the vacuum fluctuations driving the detector and the radiation emitted by the driven detector is perfectly destructive in equilibrium conditions.
In laboratories, producing an eternal, constant linear acceleration for a charge or an atom is impossible, anyway. In Ref. CT99 () and similar proposals the charge motion would be driven by an intense laser field, which can make the acceleration linear, but not uniform, thus the radiation corresponding to the Unruh effect may not be totally suppressed by interference. The only concern is that the Unruh temperature is not well defined in these non-equilibrium setups. Fortunately, a time-varying effective temperature whose value is close to the Unruh temperature of the averaged acceleration can be defined for the detectors in oscillatory motion DLMH13 (). As we will demonstrate later, in the regime of high acceleration and short oscillating cycle of the motion, the signal of the effective Unruh temperature can be pronounced in the Unruh radiation 111This paper is partly based on Lin16 ().
This paper is organized as follows. To get non-perturbative, time-dependent results of the radiation field emitted by a point-like detector in oscillatory motion, in Section 2 we introduce the UD harmonic-oscillator (HO) detector model considered in Refs. LH06 (); DLMH13 () and then address some technical issues. We determine the radiation in the radiation zone defined in the Minkowski coordinates for a laboratory observer Hi02 (); Ja98 (); LH06 (). Then we present our numerical results of the radiated power in Section 3. We show that when observed at a fixed angle, there will be negative radiated power in some periods during each cycle of the oscillatory motion OYZ15 (); Lin16 (). This indicates that the Unruh radiation observed at the null infinity may correspond to a multi-mode squeezed state of the field KF93 (); UW84 (); SSH06 (); SSH08 () or something similar. So we further study the correlations in the radiation field in Section 4, where we identify the nonlinear optical effects such as the coherent high-harmonic generation and down-conversion SSH08 (). Then in Section 5 the asymptotic reduced state of the field harmonics is constructed using the correlators of the field in the radiation zone. Our analysis shows that this asymptotic reduced state of the radiation field looks like a direct product of the squeezed thermal states. Finally, a summary of our results is given in Section 6. A few analytic results for the two-point correlators are given in Appendix A, which can help to control the singularities in our numerical calculation.
2 Renormalized expectation values of stress-energy tensor
Consider an Unruh-DeWitt detector with its internal degree of freedom acting as a harmonic oscillator and minimally coupled to a massless scalar field in (3+1)D Minkowski space, described by the action
where , is the worldline of the detector parametrized by its proper time , is the bare natural frequency of the internal HO, and is the coupling constant of the detector and the field. Here we take . From (1) one can derive the conjugate momenta and of the detector and the field, respectively. Below we set for simplicity.
As we discussed in Ref. LH06 (), by virtue of the linearity of this UD detector theory, the field operator after the detector-field coupling is switched on will become a linear combination of the mode functions each associated with a creation () or annihilation operator () of the free field mode with wave vector , or a raising () or lowering operator () of the free internal HO of the detector . Also due to the linearity, a mode function of the field has the form (; and are associated with and , respectively), which is the superposition of the homogeneous solution corresponding to vacuum fluctuations of the free field and the inhomogeneous solution sourced from the point-like detector. One can group the homogeneous solutions of the mode functions with the associated operators into and the inhomogeneous solutions into such that the field operator is in the form .
Suppose the detector-field coupling is switched on at , when the combined system is initially in the factorized state
which is a product of the ground state of the free UD detector and the Minkowski vacuum of the field . The wavefunction (or density matrix) is thus in a Gaussian form. In the Heisenberg picture, the correlators of the field amplitude at different spacetime points and after are given by
where with respect to the initial state in (2). Then the expectation value of the stress-energy tensor (minimal, ) can be written as
where is contributed by .
is the Green’s function of the free field, it diverges as , so does . Nevertheless, there is no physical effect from this part of the stress-energy tensor in Minkowski space, and so it can be subtracted in the spirit of the normal ordering in obtaining the vacuum energy in the conventional quantum field theory. We thus define the renormalized stress-energy tensor as . Doing this is nothing but setting the zero point of vacuum stress-energy.
Suppose a UD detector is oscillating about the spatial origin of the Minkowski coordinates, which is chosen as the laboratory frame. Suppose a set of the radiation-detecting apparatus are located at a large constant radius at different angles from the spatial origin, namely, at in the Minkowski coordinates. Then the differential radiated power per unit solid angle measured in laboratories can be written as
where is the element of the solid angle and the component is defined by
In this paper we are considering the cases with the UD detector in linear oscillatory motion in -direction, namely, , and the radiation will be independent of the azimuth angle by symmetry.
The whole calculation will be started with the subtracted two-point correlators of the field. However, as many quantities for quantum fields with infinite degrees of freedom, each of , and is still singular in the coincidence limit. One has to control the singularities with the hope that some of them could cancel in the measurable quantities while others could be tamed by introducing physical cutoffs. To identify the problem, let us look into more details of the correlators.
2.1 Two-point correlators at late times
At late times, the retarded field has carried the initial information in the detector away to the null infinity, so the behavior of the combined system around the detector is dominated by vacuum fluctuations of the field as well as the detector’s response to them. From Ref. LH06 () and OLMH12 (), one has
where the v-part of the correlator is defined by the mode functions associated with the field operators and the initial data in the field state only. Here is the coupling strength, is the natural frequency with renormalized from the bare natural frequency of the detector , is the propagator, is the proper time of the detector at the initial moment, and
is the positive-frequency Wightman function of the free massless scalar field in the Minkowski vacuum state, with a proper choice of the integration contour understood BD82 (). While one could get rid of the divergence of the integrand as in (7) by choosing the integration contour, the divergence of in the coincidence limit is unavoidable.
From (A1) in Ref. LH06 (), denoting and , one has
with the singular behaviors of (7) and other correlators of the detector. Also at late times,
whose integrands and integrals diverge as and , respectively. Here we denote , , , the retarded time is defined by subject to with Synge’s world function , and is the retarded distance determined by the local frame of the detector as
and , while and . Note that the retarded distance and the retarded proper time for a uniformly accelerated detector in Ref. LH06 () has been generalized to and for a detector in oscillatory motion here 222In Eq. (A1) in Ref. LH06 (), only counts the contribution by the v-parts of the correlators of the detector . The expression for the a-part, , has the same form as Eq. (A1) in Ref. LH06 () except the v-parts of the detector-detector correlators are replaced by the a-parts . Since in the cases we are considering, the expression for is simply the same expression as Eq. (A1) in Ref. LH06 () except all the v-part of the detector-detector correlators there are replaced by the complete one, namely, .. For an observer at the null infinity the more the 4-velocity of the detector is pointing towards the observer, the smaller is.
2.2 Controlling the singularities
From the correlators of the detector , , , and in (10) and thus (6), one could extract the Unruh or the effective temperature experienced by the detector LH07 (). So we call the all-retarded-field part of the differential radiated power as the naive Unruh radiation. It diverges when one takes the coincidence limit on the two-point correlators of the detector, as one can see from (6), (7) and (10). When the trajectory of the detector is not as simple as those in uniform motion or uniform acceleration, setting consistent cutoffs in the double integral for the correlators such as (7) is not easy. In Refs. DLMH13 () and OLMH12 () we have dealt with these singularities carefully. We subtract the integral for the two-point correlators of the detector in oscillatory motion by those for a uniformly accelerated detector. The subtracted integral gives a finite result. Then we add the analytic results for the uniformly accelerated detector back, whose singular behavior are well understood and under control once the UV cutoff is introduced.
For the interference terms of the differential radiated power, , the situation is similar. As we mentioned, in the integrand of (12),
diverges as and . When is positive and non-zero, expanding , one finds
with . To subtract out the divergent and terms, one needs to introduce a reference worldline, with , , and . For a general worldline at a specific moment , the simplest reference worldline for subtraction is again the one for a uniformly accelerated detector, and luckily, we have also obtained the analytic results of the interference terms for the uniformly accelerated detector in closed form in Ref. LH06 (). Similar to what we did for , after we get the finite result for the subtracted interference terms, we add the analytic result back in the final step to get the complete result with the divergences well controlled.
In the cases with , the reference worldline and the analytic result to be added reduce to those for the detector in uniform motion with and . Some analytic expressions of the correlators for an UD detector in uniform acceleration and uniform motion are given in Appendix A, for adding back to the subtracted numerical results.
For the reference worldlines either in uniform acceleration or in uniform motion, the UV divergence ( in Section A) in will be exactly canceled by the ones in the interference terms LH06 (). Thus, after combining the numerical result of the subtracted power and the exact analytic result from the reference worldlines, the final result will be regular and independent of the UV cutoff for the detector.
2.3 On-resonance case
When the period of a cycle of oscillatory motion in the proper time of the UD detector is integer times of the natural period of the internal HO (, integer), it is possible to get the late-time result with a finite domain of integration to make the numerical calculation more economic. Our experience in calculating the effective temperature in a UD detector in oscillatory motion shows that such kind of the resonance condition is not catastrophic DLMH13 (). In these cases, since and for a detector in oscillatory motion of period in proper time and in the coordinate time, and with integer , when and at late times, (7) becomes
The integrand can be written in closed form by noting that , which is the Hurwitz-Lerch transcendent math10 (). This reduces the domain of the integral from to a finite square, though the integrand diverges at for some integer and have to be treated in the way given in Section 2.2.
whose domains are reduced to finite intervals, though the integrands also diverge as for some positive integer and should be properly treated.
When with integer , the situation is similar to the above case with . Only minor modifications on the above integrals are needed.
3 Radiated power
As an example, let us consider a detector moving along the worldline given by Chen and Tajima in Ref. CT99 (),
which is the trajectory of a charge at a nodal point of magnetic field in a cavity. The effective temperature of the detector in this worldline has been studied in DLMH13 (). Let the coupling is switched on at (when ). Then . Since does not go to , we are not really at late times here, and in the results of this section we have actually included the a-parts of the correlators LH06 () in addition to the v-parts discussed in Section 2.1, though the contributions by the a-parts are small in the figures we are going to present. Suppose the observer is located at . For , one can compute of the retarded time by solving
from . Then and in the radiation zone,
where and are the four-velocity and four-acceleration of the detector moving along the worldline , respectively, while is the elliptic integral of the first kind DLMH13 (). (22) can be quickly derived by partially differentiate the equation . Let the period of the oscillatory motion be in the coordinate time and in the proper time of the detector. We define the directional proper acceleration where is the proper acceleration DLMH13 (). Then one has for in (19), so that , around and in the radiation zone. In the cases with large , can dominate over at most of the observing angles. Also the reference worldline to control the singularities for an observer at would be, for ( mod or (when ), , otherwise
3.1 Negative radiated power
In Figure 1, we demonstrate the time evolution of the full differential radiated power in one cycle at large times at fixed angles. One can see that the time evolution of the full differential radiated power at each fixed angle with the interference terms included has a double-peak structure. Each pulse consists of two main peaks of positive flux, and valleys of negative flux in-between.
In the moving-mirror model in (1+1)D, Fulling and Davies found that a negative radiated power will be observed when the acceleration of the mirror is time varying while the mirror is moving towards the observer FD76 (). Our results at and (on the plane of the oscillatory motion of the detector) are consistent with this observation. However, when observed far off the oscillation plane, the negative radiated power in our examples does not occur around the moment when the observed proper acceleration has the most significant change. This is clear in the plots with or in Figure 1, where the double-peak pulse occurs around the maximum of the observed proper acceleration , while the value of can be small in the period of negative flux between the peaks. Actually, the double-peak structure in Figure 1 has been obvious in the naive term (green dashed curve). We find this behavior in our example is dominated by the factor in (10) (here in , see the statement below (23).) When the scaled retarded distance for the observer at some fixed angle become very small, the observed radiated power will be amplified and a pulse emerges. However, around the moment that the retarded distance reaches the minimum, one has (i.e. by (23)) and so a valley between two positive main peaks is formed in a pulse. On the other hand, the negative correction from the interference terms become the most negative at some moment a little bit ahead of the valley of the naive term, so the total differential radiated power around the valley become negative (also see the inset of Figure 4 (left)).
At each fixed time, there will always exist negative radiated power around some observing angle, as shown in Figure 2. Nevertheless, the averaged radiated power over a cycle of oscillation at each fixed angle must be positive (Figure 2 (lower-right)), as guaranteed by the quantum inequalities Fo91 ().
is sufficiently small ( in Figure 1), and the observing angle is in the vicinity of or , there may be a longer period of negative radiated power between the second positive peak of one pulse and the first positive peak of the successive pulse, while the value of the negative radiated power is very close to zero. In some cases the period of this negative radiated power can be longer than a half of the period of a cycle (e.g. Figure 4).
In our examples the period of the oscillatory motion are small compared to the time scale of the relaxation of the detector . Under this condition, if we increase the coupling strength with other parameters fixed, the time evolution of the angular distribution of the differential radiated power will be similar to Figures 1 and 2, except that the peaks will be roughly amplified as . The ratio of the maximum amplitudes of the negative radiated flux to the positive one does not change significantly as we increase or but keep fixed.
The negative radiated power in the above result does not imply absorption, or radiation in the opposite direction. It can excite an UD detector at a rate lower than the case in zero energy density Gr88 (), and produce no decrease of entropy DOS82 (). Our result reveals another resemblance between the detector theory and the moving-mirror models in quantum field theory in curved spacetime. Actually, some Unruh-DeWitt detector theories in (1+1)D have been used to describe mirrors in a more realistic way than those simply introducing boundary conditions for the fields at the mirror’s position GBH13 (); WU14 (); WU15 (); SLH15 ().
3.2 Evidence of Unruh effect in radiation
In Ref. DLMH13 () we observed that, at a lower (higher) value of the averaged proper acceleration , the effective temperature of an UD detector in oscillatory motion tends to be higher (lower) than the naive Unruh temperature experienced by a uniformly accelerated detector at the proper acceleration . Indeed, the effective temperature in the example shown in Figures 1 and 2 is about to , which is higher than the naive Unruh temperature (), while in Figure 3 (right), the effective temperature is about , which is lower than (). Anyway, in Figure 3 one can see that the deviation of the effective temperature from the naive Unruh temperature due to non-uniform acceleration is negligible in the radiated power, compared with the correction from the interference terms.
A detector at rest still has non-zero correlators and contributed by vacuum fluctuations at zero temperature, such that the naive Unruh radiation from (10) is positive even at zero averaged acceleration. Since we expect that the radiation by the detector should cease as its averaged proper acceleration , the negative interference terms should be able to cancel the naive differential radiated power in this case. Indeed, we find the radiated power tends to be suppressed larger by the interference terms when or gets smaller (Figure 3 (left)). Here, for a fixed , a smaller on the one hand gives a smaller averaged proper acceleration , on the other hand it implies a longer period of oscillatory motion, so that the detector has more time to approach to the equilibrium conditions studied in Ref. LH06 (). Both suppress the radiated power.
In contrast, as or increases, the importance of the interference terms decreases, and the full result of the differential radiated power get much closer to the naive result at the effective or Unruh temperature than the naive result at zero temperature (Figure 3 (middle) to (right)). This suggests that the Unruh-like effect experienced by a UD detector could be observed in the Unruh radiation in highly non-equilibrium conditions, with a very short period of oscillatory motion and a very high averaged proper acceleration.
4 Correlations in radiation
To obtain the late-time two-point correlators more efficiently, we consider the on-resonance case below.
4.1 High harmonic generation
The frequency spectrum of the differential radiated power with respect to the observer’s clock shows that there are quanta at high harmonics generated in a broad range of frequencies. Indeed, in Figure 4 (right) one can see two main humps in the frequency spectrum of the differential radiated power at , the left one has the maximum around , and the right one around . The former is the frequency of the oscillatory motion of the detector, while the latter corresponds to the time-scale of the double-peak pulse around , when the detector reaches its maximum speed toward the observer at . The major part of the right hump around the maximum is contributed by the interference terms . The two humps are quite broad, which is similar to the high harmonic generation (HHG) in atomic and plasma physics. The HHG here, however, is caused by the relativistic motion MK05 (); LKK05 (); TW07 (), which is not simple-harmonic in spacetime, rather than the bremsstrahlung during the recombination in the three-stage model KSK93 (); Co93 (); LBIHC (). The pulses are compressed in time and concentrated in a narrow angular distribution by the Lorentz boost of the radiated field, similar to the Larmor radiation of a classical charge in relativistic motion in electrodynamics Ja98 ().
While the origin of the HHG sounds classical in our linear system, the harmonics generated here are quantum coherent. Later in the scaled two-point correlators of the field in the radiation zone, we will see that the harmonics are correlated in the frequency domain. Thus the two short pulses of the radiated energy around and in the upper-left and lower-right plots of Figure 1 in each cycle of the motion is associated with the coherent HHG. In this aspect the highly compressed pulses we found are similar to the attosecond laser.
Interesting enough, a wide range of the right slope of the right hump in Figure 4 (right) behaves like