Unruh-DeWitt detectors as mirrors: Dynamical reflectivity and Casimir effect

We demonstrate that the Unruh-DeWitt harmonic-oscillator detectors in (1+1) dimensions derivative-coupled with a massless scalar field can mimic the atom mirrors in free space. Without introducing the Dirichlet boundary condition to the field, the reflectivity of our detector/atom mirror is dynamically determined by the interaction of the detector's internal oscillator and the field. When the oscillator-field coupling is strong, a broad frequency range of the quantum field can be mostly reflected by the detector mirror at late times. Constructing a cavity model with two such detector mirrors, we can see how the quantum field inside the cavity evolves from a continuous to a quasi-discrete spectrum which gives a negative Casimir energy density at late times. In our numerical calculations, the Casimir energy density in the cavity does not converge until the UV cutoff is sufficiently large, with which the two internal oscillators are always separable.


Introduction
A moving mirror can produce quantum radiation from vacuum [1][2][3][4]; two mirrors at rest can form a cavity with a negative Casimir energy density inside [3][4][5][6][7], and one or two of such cavity mirrors moving in specific ways can create particles in the cavity [8][9][10][11]. All these interesting physics can be obtained by simply modeling a perfect mirror as a Dirichlet boundary condition for the field at the position of the mirror. Nevertheless, such simple models may not always be satisfactory either in theoretical or experimental aspects. Theoretically, a detector or atom inside a cavity of perfect mirrors would experience endless echoes without relaxation if the atom and the field are not started with a steady state of the combined system [12]. The equilibrium approach will never apply if one does not introduce an ad hoc dissipation for the cavity. Experimentally, while the incident waves of the fields at all frequencies get total reflection by a perfect mirror in the Dirichlet boundary condition, a physical mirror is not perfect anyway: The charges of a realistic mirror responding to the incident electromagnetic waves have finite relaxation time and the reflectivity reaches almost 100% only in a finite working range of frequency.
To describe more realistic situations and to see how well the results by simply introducing the Dirichlet boundary conditions can do, there have been some mirror models which are more sophisticated than the simple, strong boundary conditions for the field. For example, Barton and Calogeracos introduced a mass term for the field at the mirror's position which acts like a delta-function potential [13,14], Golestanian and Kardar applied an auxiliary field to constrain the field amplitude around the mirror's position [15,16], and Sopova and Ford replaced perfect conducting mirrors by dispersive dielectrics [17]. Recently Galley, Behunin, and Hu constructed a mirror-oscillator-field (MOF) model with a new internal degree of freedom of the mirror minimally coupled to the field at the mirror's position to mimic the microscopic interaction between the field and the surface charges of the mirror [18]. Such a microscopic treatment captures the mirror-field interaction in a more physically consistent way. The authors of [18] also showed that their MOF model can be connected to the earlier models in refs. [13][14][15][16]19] with different choices of parameters and limits. A similar model with the derivative coupling was considered by Wang and Unruh to study the force exerted on the mirror by vacuum fluctuations [20]. Wang and Unruh further considered a model with the internal oscillator minimally coupled to a massive scalar field [21] to get rid of the divergent effective mass in [20].
In ref. [22] Sinha, Hu and the author of the present paper realized that the mirrors in the MOF models with the minimal and the derivative couplings behave like metal and dielectric mirrors, respectively. They introduced a new coupling to a harmonic-oscillator bath to describe the interaction between the mirror's internal degree of freedom and the mechanical degrees of freedom such as the vibration of the mirror substrate and the environment connected by the suspension of the mirror. They also verified that in the strong coupling regime their results are close to those with the Dirichlet boundary conditions [23][24][25][26].
Since the MOF models are nonlinear due to the mirror motion, one usually needs to make some linear approximations in practical calculations. Among those approximations, restricting the mirror moving along a prescribed worldline may be the simplest one. By doing so, the motion of the mirror is not dynamical, and a derivative-coupling MOF model in ref. [22] reduces to a derivative-coupling Unruh-DeWitt (UD') harmonic oscillator (HO) detector theory [27][28][29][30] with additional HO baths, which is the model we are considering in this paper.
The late-time reflectivity of our "detector mirror" in the weak oscillator-field (OF) coupling regime is similar to the atom mirrors in the cavity and waveguide QED [11,[31][32][33][34][35][36][37], whose reflectivity are peaked in a narrow band around a single frequency of resonance. The cavity of those atom mirrors can only generate few cavity modes inside since the detector/atom mirrors are almost transparent for other harmonics [32]. In the field-theoretical derivation for the Casimir effect [5], however, one needs to sum over all the cavity modes to get the Casimir energy density in a perfect cavity. Thus in this paper we extend our attention to the detector mirrors in the strong OF coupling regime, where the reflectivity of the detector mirror is close to 100% in a very wide frequency range of the field. Later we will see that, while the transient behaviors of the combined system can be significantly different for different coupling strengths, the late-time renormalized energy density of the field inside a cavity of our detector mirrors is always negative even in the weak OF coupling regime where the cavity modes are few.
The paper is organized as follows. The classical theory for our single "detector mirror" is given in section 2, where we examine the relaxation time and the late-time behavior of the system, then derive the late-time reflectivity determined by the interplay between the HO and the field. In section 3 we develop the quantum theory of the single detector mirror and show that the energy density of the field outside the detector mirror is zero at late times while the equal-time correlations of the field amplitudes at different positions are reduced by the mirror. In section 4, we consider a cavity of the detector mirrors, show that there are indeed many cavity modes inside our cavity at late times in the strong OF coupling regime, then calculate the late-time renormalized energy density inside the cavity, which turns out to be negative for all nonzero coupling strengths. After addressing the HO-HO entanglement at late times, a summary of our findings is given in section 5.

Single detector mirror: Classical theory
A mirror moving along the worldline z µ with its internal degree of freedom Q coupled with a quantum field Φ in (1+1)D Minkowski space may be described by the action given in eq.
(1) of ref. [22], with Z there replaced by z 1 , and with the mechanical damping Γ and noise ξ in eq. (35) there introduced. Since the position of the mirror z 1 is not considered as a dynamical variable in this paper, we can write down the reduced action as which is a derivative-coupling Unruh-DeWitt (UD') detector theory [27][28][29][30]. Here the natural unit with the speed of light c = 1 is adopted, (t, x) are the Minkowski coordinates, τ is the proper time of the detector mirror, Q is a harmonic oscillator(HO) of mass m = 1 living in an internal space of the detector, and λ(τ ) is the switching function of the oscillator-field(OF) coupling, assumed to be vanishing before the initial time τ I . The derivative coupling is chosen for its well behaved radiation reaction term, which is the first derivative of the proper time of the detector (e.g. eq. (2.13)). The functionλ(τ ) corresponds to the coupling between the internal HO and the environmental oscillatorbath responsible for the mechanical damping and noise. It can be switched on at a different initial moment τ I = τ I . In the strong OF coupling regime, the absolute value of the OF coupling |λ| is much greater than the oscillator-environment(OE) coupling |λ| so that the former interaction dominates and the detail of the environment would not be important. Thus for simplicity, we model the environmental oscillator-bath by another massless scalar field Z y (τ ) in another internal space y ∈ R 1 , and assume that the internal HO of the mirror also acts as an UD' detector located at y = ϑ in that internal space 1 . Then the action (2.1) is quadratic and the combined system is linear and solvable. When considering two or more mirrors, the internal space y, the phase parameter ϑ, and the couplingλ of each mirror will be considered independent of those of the other detector mirrors. From (2.1) the conjugate momenta of the detector, the field, and the mechanical environment read respectively, with which the Hamiltonian on a t-slice is given by where τ t is obtained by solving t = z 0 (τ t ), and v µ (τ ) ≡ ∂ τ z µ (τ ) is the two-velocity 2 . Suppose the detector is at rest at x = 0 in the external Minkowski space, so that z 1 (τ ) = 0, z 0 (τ ) = τ = t, v 1 (τ ) = 0 and v 0 (τ ) = 1. Then the value of the Hamiltonian (2.5) equals 6) which appears no cross term between different kinds of the degrees of freedom. Anyway, the Euler-Lagrange equations in this case, x (t) + Φ [1] x (t), (2.10) where Φ [0] x (t) is the homogeneous solution satisfying 2Φ [0] = 0, and Φ [1] x (t) is the inhomogeneous solution given by Φ [1] after an integration by part. Here the retarded Green's function for a massless scalar field is the Heaviside step function with the convention θ(0) = 1/2. The surface terms in (2.11) have been dropped since lim τ →∞ G ret (t, x; z 0 (τ ), z 1 (τ )) = 0 for all finite t and x, and we assume lim τ →−∞ λ(τ ) = 0 long before the coupling is switched on. Similarly, the general solution of Z in (2.8) is y (t) is the homogeneous solution. Inserting the solutions of the field Φ and the mechanical environment Z into (2.9), one obtains ϑ (t), (2.13) which shows that Q behaves like a driven, damped HO with a time-dependent frequency.

Late-time solutions
Introducing a right-moving wave Φ x (t) = e −iωt+ikx , ω = k > 0, as the driving force in (2.14) and assuming Z y (t) = 0 for simplicity, once the OF coupling λ has become a positive constant for a sufficiently long time for relaxation (t − t 0 t rlx ), one has Q(t) ∝ e −iωt at late times according to (2.14). Suppose the time scale of switching-on the OF coupling is much shorter than t rlx . Inserting the late-time ansalz Q(t) →Q ω e −iωt into (2.13), one can solveQ ω and find the late-time solution with the susceptibility function x (t) (2.20) from (2.11).

Reflectivity
In (2.19), the first term and the second term in the x < 0 region can be interpreted as the incident and reflected waves Φ x and Φ [R] x , respectively, while the superposition of Φ x and Φ [1] x in the x > 0 region can be interpreted as the transmitted wave Φ [T ] x , as in (2.20). Thus at late times we can define the reflectivity as and the transmittivity as (2.22) Figure 1. The late-time reflectivity |R| 2 (black) and the sum of the reflectivity and transmittivity |R| 2 +|T | 2 (red) against ω = |k|, given in (2.21) and (2.22).
Here Ω 0 = 1,γ = 0.05, γ = 0.2 (dashed) and 10 (solid or dotted). We plot the reflectivity of the minimal-coupling model (eq.(17) in ref. [22]) with the same parameters (Green) for comparison. One can see that the derivative-coupled and the minimal-coupled detectors act like a dielectric and a metal mirrors, respectively, in the regime of ω → 0 An example of the above late-time reflectivity and transmittivity is shown in figure  1. One can see that the reflectivity is peaked around ω = Ω 0 where the internal HO of the detector mirror and the incident wave of the field are resonant. Observing that |R(k)| 2 = [γ/(γ +γ)] 2 and |T (k)| 2 = [γ/(γ +γ)] 2 at ω = Ω 0 , the UD' detector will be a perfect mirror (|R(k)| 2 = 1) for the incident monochrome wave if the internal HO is decoupled from the mechanical environment (γ = 0). When the OE couplingγ is not negligible, however, the energy of the field Φ around the resonant frequency will be significantly absorbed by the environment Z, so that |R(k)| 2 + |T (k)| 2 becomes lower than 1 around ω ≈ Ω 0 .
In figure 1 one can also see that when the frequency of the incident wave is far offresonance, the reflectivity is small and so the mirror is almost transparent for that incident wave. With weak couplings (γ,γ < Ω 0 ), large reflectivity occurs only in a narrow frequency range of the width O(γ +γ) around the resonance (dashed curves). This feature is similar to the usual dielectric mirrors and atom mirrors [11,[31][32][33][34]. The cavities of these kind of mirrors can produce only one or few pairs (k = ±ω) of resonant modes inside [32], since the detector mirrors are nearly transparent for other harmonics. In constructing a cavity model for comparing with the conventional approach to the Casimir effect, one may need a detector mirror with a very wide working range of frequency to form an effective Dirichlet boundary condition Φ x 0 (t) ≈ 0 at the mirror's position x = x 0 . This could be done by carefully arranging a collection of detectors/atoms to form the mirror [35,36]. Alternatively, one can simply raise the OF coupling of a single detector all the way to the over-damping regime for the internal HO (γ Ω 0 ,γ) to achieve it. As shown by the solid curve in figure 1, the reflectivity of a detector mirror in this over-damping regime will go to approximately 1 in a wide frequency range at late times, though it may take a very long relaxation time to reach this stage, as discussed in section 2.1. Later we will see explicitly in the quantum theory that a cavity of the detector mirrors in the over-damping regime can indeed generate many cavity modes and the field spectrum inside the cavity is quasi-discrete at late times.
Note that the definition of reflectivity (2.21) makes sense only at late times. |Φ [R] x (t)| 2 / |Φ [0] x (t)| 2 can be greater than 1 in transient when the initial zero-point fluctuations of the detector burst out right after the OF coupling is switched on (see, e.g., the left plots of figure 4).

Single detector mirror: Quantum theory
The Heisenberg equations of motion in the quantum theory of our model (2.1), which is a linear system, have the same form as the Euler-Lagrange equations (2.7)-(2.9): One can see that each operator will gradually evolve to other operators whenever the couplings are on. To deal with, we write the operators of the dynamical variables at finite t in terms of the linear combinations of the free operators defined before the couplings are switched on, each multiplied by a time-dependent c-number coefficient called the "mode function", namely, where κ runs over A, {k}, and {k}, which are the indices for the free HO labeled A, the free field mode of wave-number k, and the free mechanical environment mode of wave-number k, respectively. Here we have renamedQ toQ A to be consistent with the multi-detector cases later in this paper, and we denote k ≡ dk/(2π), k ≡ dk/(2π), Ω A ≡ Ω 0 , Ω k ≡ w ≡ |k|, and Ωk ≡w ≡ |k|. The raising and lowering operators of the free internal HO have the commutation relation [â A ,â † A ] = 1, while the creation and annihilation operators for the free massless scalar field and the free environment Applying these commutation relations ofâ andâ † to the Heisenberg equations (3.1)-(3.3), one obtains the equations of motion for the mode functions, Again they have the same form as the Euler-Lagrange equations, while the initial conditions will be different from those in the classical theory. The solutions for ϕ and ζ are similar to (2.11) and (2.12): Thus, similar to (2.13), eq.(3.9) becomes after including the back reactions of the field and the environment.

Mode functions for internal HO
Suppose the OE couplingλ(t) was switched on at t =t 0 −γ −1 < 0 and settled to a constantλ, while the OF coupling λ(t) is not switched on until t = 0. Suppose the combined system started with a factorized state, which is a product of the ground state of the free internal HO |0 Q , the vacuum state of the free environment |0 Z , and the vacuum state of the free field |0 Φ . Then, right before t = 0, the quantum state of the combined system has become where ρ QZ is the late-time state of the HO-environment subsystem and ρ Φ = |0 Φ 0| is still the vacuum state of the field. Between t =t 0 and t = 0, q A A (t) follows the equation of motion and behaves like a damped harmonic oscillator, while qk A (t) follows the equation Thus, ast 0 → −∞, ρ QZ in (3.14) is characterized by the two-point correlators with the late-time solutions, q A A (0) = ∂ t q A A (0) = 0 for (3.15), and A (0) = 0, and those from (3.17) for qk A and ∂ t qk A around t = 0, the solutions of (3.12) for t > 0 are found to be and q A A (t) = 0. Here η ≡ t − t 0 ≥ 0 with t 0 = 0, and the propagator K(s) has been given in (2.15). The integral in the first line of (3.19) can be worked out to get an expression similar to the second line of (3.18).
In our numerical calculation, we replace θ(t) in λ(t) by a C 1 function to regularize the delta function δ(t) = ∂ t θ(t). Then we find q κ A (t) are always continuous, and our numerical results do approach to (3.18) and (3.19) in the small T limit. Note that our θ T (t) is not smooth or normalizable ( ∞ −∞ θ T (t) diverge), thus our results are not restricted by the quantum inequalities for smooth and normalizable switching functions [40][41][42].

Detector energy and HO-field entanglement
With the operator expansion (3.4) and the initial state (3.13), the symmetric two-point correlators of the internal oscillator of the detector read For t > t 0 , q A A = 0 and so only the integrals in the above expressions contribute. The closed form of these integrals can be obtained straightforwardly after the mode functions are inserted. For example, by inserting (3.18) we get for real Γ in the over-damping cases. Here Ei(s) is the exponential integral function, If the environment is excluded in our consideration, (3.23) will be identical to the vpart of the detector correlator Q 2 (t) v defined in refs. [12,39], where their closed forms in the under-damping regime have been given. Indeed, (3.23) withγ = 0 can be obtained from eq.(A9) in ref. [39] with Re f there written as Re [f + f * ]/2, then replace the renormalized natural frequency Ω r there for the minimal-coupling Unruh-DeWitt HO detector theory in (3+1)D Minkowski space by Ω 0 here for the derivative-coupling detector model in (1+1)D (also see appendix of ref. [12]), and finally replace every iΩ there by Γ here while notice that Re{Γ(0, s)} = −Re{Ei(−s)} with the incomplete gamma function Γ(0, s). Note that in this paper we have changed the definitions of Λ 0 and Λ 1 , corresponding to the UV cutoffs, from −γ e − ln Ω(· · · ) in our earlier works to −γ e − ln Ω 0 (· · · ) here since the latter is more convenient in the over-damping regime (one cannot simply replace Ω in the former by −iΓ, which leads to complex values of Λ 0 and Λ 1 ). From now we will use these new definitions for Λ 0 and Λ 1 even in the under-and critical-damping cases. Associated with this change, the ln[(γ/Ω) ± i] = ln[(γ ± iΩ)/Ω] terms in (A9)-(A12) of ref. [39] should be replaced by ln[(γ ± iΩ)/Ω 0 ] here.
The closed form of the integral dk 2π 2w qk A (t)qk * A (t ) is much more lengthy than (3.23) due to the second line of (3.19). Fortunately all these extra terms decay out at late times, and the late-time result of the integral with qk A in the over-damping regime is simply (3.24) with the overall factor γ replaced by γ. Summing these two integrals together we find at late times, which also applies to the under-and critical-damping cases for Γ = iΩ and Γ → 0, respectively. In the latter case, Q 2 A → /(πγ 2 ) at late times. The late-time result for the correlators can also be obtained by inserting the late-time mode functions with the susceptibility function χ ω given in (2.18), into the integrals in (3.21) and (3.22).
at late times. Note that Λ 1 has to be large enough to make P 2 A positive. From the detector sector of (2.6), the expectation value of the energy of the internal oscillator of the UD' detector is at late times from (3.25) and (3.27). It also depends on Λ 1 and will be positive if Λ 1 is sufficiently large. The HO-field entanglement will be strong if the direct coupling γ between them is strong. In this case the linear entropy A can be very large in the strong OF coupling limit with a sufficiently large Λ 1 .

Reduction of late-time field correlations
A perfect mirror placed at x = 0 forces a Dirichlet boundary condition Φ x=0 (t) = 0 at its position. This would cut the equal-time correlations of the field amplitudes on different sides of the mirror, namely, Φ x (t)Φ x (t) = 0 for xx < 0. Our detector mirror is not perfect, but it still can reduce the correlations of the field on different sides.
From (3.5) and (3.13), the two-point correlators of the field is given by At late times, in the presence of the detector mirror at x = 0, one has ϕ A x = 0 and from (3.10), (3.11), and (3.26). Inserting these mode functions into (3.29), one obtains a sum of two integrals of dummy variablesk and k. One can rename both k andk to k to get with ω = |k| > 0, by applying the identity straightforwardly from (2.18), which has the form of the fluctuation-dissipation relation. The first term in the integrand of (3.32) gives the correlator of the free field, thus the late-time renormalized two-point correlator of the field reads then express both terms in ∞ 0 dω(· · · ). The above integral can be done analytically, which yields In the strong OF couplings, over-damping regime, γ Ω 0 ,γ, one has Γ ≈ γ, and with s ∼ Ω 2 0 |∆ ± |/γ and s ∼ γ|∆ ± | (given e s Ei(−s) → ln s + γ e + O(s ln s) as s → 0 and e s Ei(−s ) → −1/s + O(s −2 ) as s → ∞). On the other hand, the two-point correlator of the free massless scalar field in (1+1)D Minkowski space is given by up to a complex constant C.  (3.37), one finds that the real part of the full equal-time correlation of the field amplitudes on different sides of the mirror, Re Φ x (t)Φ x (t) with xx < 0, will indeed be suppressed for small |x| and |x | at late times (figure 2 (upper-right)). However when |x| or |x | gets greater, the correlation would not be largely corrected since Φ x (t)Φ x (t) ren goes to zero as |x| + |x | → ∞ while x (t ) does not (figure 2 (upper-left) and (upper-middle)). Actually the real part of the equal-time correlator of the field amplitudes on the same side of the mirror (xx > 0) are also reduced since the real part of (3.35) for t = t is a negative function of |x| + |x | only. This may be interpreted as a consequence of the image "point-charge" in the Green's function of the field in the presence of the detector mirror.
Regarding to the imaginary part of the field correlator, the renormalized correlator simply adds the effect of the mirror to the retarded and advanced Green's functions of the field in free space. In figure 2 (lower right) one can see the reflected and transmitted fields generated by the detector mirror at x = 0. In the presence of the detector mirror, the translational symmetry of the system is broken.
Anyway, compare (3.36) and (3.37), one can see that for x and x fixed at finite values with xx < 0, which implies (|x| + |x |) 2 = (x − x ) 2 , one has the full correlator Φ x (t)Φ x (t) → 0 as γ → ∞ (such that s → 0 and s → ∞ in (3.36)). This is exactly the property we mentioned: A perfect mirror will suppress the correlations of the field amplitudes on different sides of the mirror.

Field spectrum
From (3.29) we define the field spectrum F k x by looking at the full correlators of the field in the coincidence limit, with ω ≡ |k| such that in the presence of our single detector mirror. Note that k is simply a dummy variable in the integral of (3.38) and F k x is not only contributed by the vacuum fluctuations of the field Φ. At late times, the last term in (3.39) decays out and the field spectrum becomes which is independent of t, from (3.32). An example in the over-damping regime is shown in figure 3. For kx < 0, the factor e i(ω|x|−kx) = e −2ikx produces the ripple structure. For kx > 0, F k x = 1 − γ(χ ω + χ * ω ) is independent of x (figure 3 (right), in particular). In this case, forγ γ, one has F k x ≈ |T (k)| 2 , which is the transmittivity defined in (2.22) with Z y (t) = 0. Thus one may interpret that F k x for kx > 0 is small in our example because the low-|k| modes are almost totally reflected in the over-damping regime, while the ripple structure of F k x for kx < 0 is due to the interference of the incident and the reflected waves. The minimum values in the valleys of the ripple in the low-|k| regime can be very close to zero, which is significantly deviated from the value 1 for the field vacuum in free space. In contrast, the field spectrum at fixed x goes to 1 as |k| → ∞, so the detector mirror is almost transparent to the short-wavelength fluctuations (figure 3 (middle)).

Renormalized energy density of the field
The expectation value of the energy density of the field is given by While the above expression formally diverges in the coincident limit (t , x ) → (t, x), we are only interested in the renormalized energy density of the field with the contribution by the free field subtracted, which can be obtained from (3.41) with the full correlator of the field Φ x (t),Φ x (t ) replaced by the renormalized one, Φ x (t),Φ x (t ) ren . When substituted into (3.41) and (3.42), the late-time correlator Φ x (t),Φ x (t ) ren in (3.34) is always a function of t − t and x + x since x and x must have the same sign in the coincidence limit for ,Φ x (t ) ren at late times, and thus T 00 (t, x) ren → 0 for x = 0, namely, the late-time energy density of the field outside the detector is the same as the vacuum energy density, though the field spectra are quite different. This is not surprising: It is well known that the late-time stress energy tensor of the field for a uniformly accelerated UD' detector (without coupling to Z) is exactly zero [43].
Right at the position of the detector x = 0, if we choose the regularization |x| = √ x 2 + 2 , = 0+, then ∂ x |x| will vanish at x = 0 for any finite regulator and we will end up with T 00 (t, x = 0) ren → 1 2 lim t →t ∂ t ∂ t Φ 0 (t),Φ 0 (t ) ren = −(γ/2) P 2 A at late times, with the late-time result of P 2 A given in (3.27).

Cavity of two detector-mirrors
With the knowledge about single detector mirrors, we are ready to model a cavity with two detector mirrors coupled to a common scalar field in (1+1)D Minkowski space while each detector mirror couples to its own mechanical environment. Our model is described by the action Similar to the cases of single detectors, inserting the solutions for (4.2) and (4.3), into (4.4), one obtains whereĀ ≡ B, andB ≡ A.

Relaxation and resonance
Suppose the combined system is going through a process similar to the one in section 3.1: It is started with the product of the ground states of the free internal HOs and the vacuum states of the free field and of the free mechanical environments, the OE couplings of both detector mirrors have been switched on for a long time (t 0 → −∞) when their OF couplings are switched on at t = t 0 = 0. In (4.5) and (4.7), one can see that only half of the retarded field emitted by one detector mirror of the cavity in (1+1)D Minkowski space will reach the other detector mirror of the cavity. The other half will go all the way to the null infinity and never return. Carried by the retarded field, it seems that all the initial information in the internal HO and the switching function of the OF coupling would eventually dissipate into the deep Minkowski space, so that there would be no initial information around t = 0 kept in our cavity at late times. Nevertheless, as we will see below, in the absence of the OE coupling (γ = 0), there can exist late-time non-steady states of the combined system which may depend on the initial conditions around t = 0, if the internal HOs of the detector mirrors are resonant with their mutual influences via the field. Let q κ ± = (q κ A ± q κ B )/ √ 2. Then (4.7) can be rewritten as where the driving force is defined as f κ )/ √ 2. Now q κ + and q κ − decouple and each is driven by a nonlocal force.
Suppose q κ ± (t) = Ω α κ ± (Ω)e −iΩt for t L > 0 t 0 and T L in (3.20), so that γ andγ have become constants of time. Since f κ ± (t) are zero for κ = A, B and simple harmonic for κ = {k}, {k A }, {k B } (cf. the expressions below eq. (3.11)), for those Ω = w(≡ |k|) for κ = k, or Ω =w d (≡ |k d |) for κ =k d , eq. (4.8) requires for nonzero α κ ± (Ω). Let Ω = R + iI with R, I ∈ R. Then the real and imaginary parts of (4.9) read R 2 − I 2 − Ω 2 0 − 2I γ + γ 1 ± e −IL cos RL ∓ 2γR e −IL sin RL = 0 (4.10) 2RI + 2R γ + γ 1 ± e −IL cos RL ∓ 2γI e −IL sin RL = 0 (4.11) The real solutions for Ω, if they exist, will have I = 0 and so (4.11) implies ∓ cos RL = 1 +γ γ , (4.12) which will not be true unlessγ = 0 since | cos RL| ≤ 1 and γ,γ ≥ 0. Forγ = 0, the real solution for (4.9) is Ω = Ω 0 for q κ − when Ω 0 = 2nπ/L for some positive integer n, or for q κ + when Ω 0 = (2n − 1)π/L. When one of these happens, the internal HOs in the detector mirrors are resonant with their mutual influences, while q κ ± (t) will never both settle down to steady states of constant amplitudes. This makes the late-time field spectrum (∼ |ϕ k x (t)| 2 , see section 4.2) inside the cavity restless forever in a range of frequency |k| of the driving force f k ± (t) (k = k,k A ,k B ), due to the mixing of the driving and the resonant frequencies. Outside the cavity, the late-time field spectrum at the same frequencies will never settle down, either, though the changes in time are less significant in magnitude than those inside the cavity. These time-varying patterns of the field spectrum at late times may depend on the initial conditions such as the time-scale and the functional form of the switching function γ(t) for the OF coupling.
If there exist purely imaginary solutions, which has R = 0, then (4.11) will be trivial (0 = 0) and (4.10) will become (4.13) which implies that I = 0 and I[γ+γ(1±e −IL )] must be negative. If I > 0, then 1±e −IL > 0 and so I[γ + γ(1 ± e −IL )] > 0, which contradicts (4.13). Thus I must be negative here. Similarly, when both R and I are nonzero, (4.10) and (4.11) yield which implies that the expression in the curly brackets must be negative. If I > 0, then 1 + (2/I)[γ + γ(1 ± e −IL cos RL)] > 0 and (4.14) can not hold. So I must be negative here, too. Therefore, the imaginary parts of the complex solutions for Ω = w,w d , or Ω 0 if Ω 0 = nπ/L for some positive integer n, must be negative, and the corresponding modes e −iΩt = e −|I|t e −iRt will decay out as t → ∞. At late times, only the oscillations of Ω = w andw d for all values of Ω 0 , and additionally Ω = Ω 0 when Ω 0 happens to be nπ/L for some positive integer n, will survive. Longer relaxation times would occur in the cases with Ω ≈ Ω 0 ≈ nπ/L for some positive integer n. In these near-resonance cases, one may write where | n |, |I n | nπ/L and I n < 0. Assuming | n | and |I n | are roughly the same order and | n L|, |I n L| 1, then they can be approximated by n ≈ Ω 2 0 − (nπ/L) 2 2(nπ/L)(1 + γL) , I n ≈ 1 γL 2 J n − J 2 n + 2γL 2 γ + γ 2 n L 2 2 , J n ≡ 1 + γL + (−1) n γ n L 2 nπ , (4. 16) from (4.10) and (4.11). To keep the above approximate expression of n small, one should take a large value of γL, and/or Ω 0 should be very close to nπ/L with some positive integer n. This can be achieved more easily when the separation of the mirrors L is large, since |Ω 0 − nπ/L| ≤ π/(2L) will be small for a general Ω 0 and the integer n closest to the value of Ω 0 L/π. For a very large L the approximation can be good even for |Ω 0 − n π/L| being a few times of π/L for some n = n. Note that I n | n=0 vanishes forγ = 0, when we return to the resonant cases. As we have known in (4.13), besides Ω ≈ nπ/L with positive integer n, there may exist purely imaginary solutions Ω = iI 0 for q κ + (in general) and for q κ − (in some particular parameter ranges). According to (4.13), indeed, when |I 0 L| 1, one has for q + , which is always closer to zero than the counterpart for q − (if any) is. These would be clear by arranging (4.13) into I 2 + 2I(γ + γ) + Ω 2 0 = ∓2Ie −IL for q ± , and then observing that the left-hand side is a concave-up parabola with the minimum at some negative I while the right-hand-side is zero at I = 0 and monotonically decreasing (increasing) for q κ + (q κ − ) as I approaches to 0 from a negative value. The relaxation time for our cavity with a not-too-small separation of the mirrors could be estimated by the inverse of the minimal |I n | (n = 0, 1, 2, 3, · · · ) among the above solutions. In the cases with the minimal |I n | = |I 0 | (namely, n > 0), when the separation is sufficiently large so that γL γ, and Ω 0 is close enough to nπ/L, one has for the HO pair in the weak OE coupling and strong OF coupling, over-damping regime. Compared with (2.16) for the HO in a single detector mirror in the same regime, we see that a stronger OF coupling still makes the relaxation time longer and t rlx ∼ γ for very large γ in both cases, but a stronger HO-environment here plays the opposite role to those in the single mirror cases and shorten the relaxation time of the cavity near resonance. Note that, unlike the (3+1)D case in ref. [44], there is no instability in the small L limit here since the retarded field is independent of the distance L from the source in (1+1)D, while it is proportional to 1/L in [44]. As L → 0, the equations of motion in (4.8) simply become regular, ordinary differential equations without delay.

Cavity modes at late times
With a non-vanishing coupling to the environmentγ, one can get rid of the late-time nonsteady states described in section 4.1. After the OF coupling is switched on, if we look at the field amplitudes only in the cavity, the field spectrum will appear to evolve from continuous to nearly discrete in the neighborhood the resonant frequency.
For t > 0, the field spectrum defined in (3.38) can be read off from the coincidence limit of the symmetrized two-point correlator of the field, (4.19) in the presence of the cavity. An example on the time evolution of the field modes is given in figure 4, where we consider a case with a larger value ofγ, namely, Ω 0 , L −1 <γ γ to reach the late-time steady states sooner while a wide range of the cavity modes can still be generated. In this example, the evolution of each single field mode from the initial moment to late times can roughly be divided into four stages: 1) At very early times, the shock waves produced by the switching-on of the OF coupling propagate freely in space; 2) After the waves produced by two different mirrors collide, violent changes of the field-amplitudesquared occur; 3) After a timescale comparable with the relaxation time of the cavity, the interference pattern of the cavity mode is basically built up, but the field-amplitudesquared keeps ringing down with small oscillations in time; 4) After a longer timescale the shape of the field spectrum against x gets into late-time steady state. The resonant modes rlx , respectively.  figure  4 for k = 2.01π/L (black) and k = −1.5π/L (red). (Right) F k x at the cavity center x = L/2 = 20 shows that the field spectrum in the cavity is nearly discrete in the low-|k| regime (inset), while the sharpness and the contrast of the comb teeth around |k| = (2n − 1)π/L, n = 1, 2, 3, · · · , decrease as |k| increases.
At late times, the mode functions in (4.19) become The field spectrum against x for the cavity mode of k = 5.0628π/L ≈ 0.3976 ≈ Ω 0 −γ (black) and the field mode of k = −6.5345π/L ≈ −(7 − (1/2))π/L (red, resonant transmission from right to left). (Right) F k x at the cavity center x = L/2 = 20 (blue) and x = 2L = 80 (red dashed) for k > 0. The blue curve shows that the only significant cavity mode for k > 0 is peaked at k ≈ 5.0628L/π. The red dashed curve indicates that the transmittivity through the cavity is suppressed around the cavity mode, and close to 1 around the resonant transmissions at k ≈ (n − 1/2)π/L, n = 1, 2, 3, · · · . with E ± ω (x) ≡ e iω|x| ± e iω|x−L| , (4.23) , (4.24) such that q k ± = χ ± ω (−λϕ , ω = |k|, from (4.8). Then the coincidence limit (t , x ) → (t, x) gives the late-time field spectrum, as defined in (3.38). Here we have used the identity similar to (3.33). Note that the odd functions of k in the integrand for the late-time Φ x (t),Φ x (t ) do not contribute to the k-integral and so they are not included in the above F k x . Examples of the late-time field spectra in the over-and under-damping regimes are shown in figures 5 and 6, respectively. Figure 5 is the late-time result of the case considered in figure 4. One can see that there are indeed many cavity modes inside the cavity (0 < x < L) in the strong OF coupling, over-damping regime. The standing waves due to the interference of the incident and reflected waves outside the cavity, similar to those in the single detector mirror case in figure 3, can also be seen. Sampling at the center of the cavity x = L/2, the field spectrum F k L/2 looks discrete in the low-|k| regime. In this example, Ω 2 0 2γπ/L and so the peak values of the comb teeth of F k L/2 with small n are about 2γ/γ, while in the high-|k| regime F k L/2 ≈ 1 + 4(γ/ω) sin ωL looks continuous and goes to the free-space value 1 as ω = |k| → ∞. The working range of this detector mirror is about 0 < k < 150π/L from figure 5 (right).
When our attention is restricted in the cavity, it appears that all the two-point correlators of a off-resonant mode in the cavity, Φ k , Φ −k , Π k , Π −k , and Φ k , Π −k , are suppressed in the strong OF coupling regime, and the uncertainty relation of that mode would be violated. This is not true since in looking at those correlators in the k space we have to consider the field spectrum outside the cavity as well.
As we discussed in section 2.3 and illustrate in figure 6, there are only one or few pairs of significant cavity modes at late times in the weak OF coupling, under-damping regime. In figure 6 the only significant cavity modes are peaked around |k| ≈ 5π/L, which is nearly resonant with the natural frequency Ω 0 of the internal HO in this example. The reflectivity in the vicinity of the resonant frequency is high enough to suppress the transmitted wave on the other side of the cavity, while the detector mirrors become almost transparent for the field modes away from this narrow resonance. Outside the cavity, one can see the interference pattern of the incident wave and the reflected waves by the two detector mirrors if the reflectivity of the mirror for that field mode is not too small or too large. The interferences of the waves reflected by the two detector mirrors are destructive for k ≈ ±(n − (1/2))π/L, n = 1, 2, 3, · · · , where the resonant transmission occurs, and constructive for k ≈ ±nπ/L, which is the basis of Bragg reflection [35,36,45,46]. The result in the over-damping regime in figure 5 (left) does not show this feature because the reflectivity of the detector mirrors in the plot is so close to 1 that the waves (say, from x < 0) transmitted through the first mirror (at x = 0) and reflected by the second mirror (at x = L), then transmitted through the first mirror again to the incident region (x < 0), are negligible. In the same conditions as those in figure 5 but now going to the high-|k| regime where the reflectivity is lower, similar destructive and constructive interferences of the incidence and reflected waves outside the cavity can also be observed.

Casimir effect
Inserting the results (4.19)-(4.24) into (3.41) and (3.42), one obtains the late-time renormalized field energy density in the presence of the cavity mirrors, and ω M is the UV cutoff, which should be identical to the ones for the internal HOs of our detector mirrors (will be introduced in section 4.4) since (4.27) has included the back  7 (left)). The amplitude of this oscillation remains large until ω M gets much greater than γ,γ, and Ω 0 , when the ω 4 term dominates the denominator of the integrand in (4.29) for ω close to ω M and makes the integral evolving on top of the lower-UV-cutoff result, so that T 00 (t, x) ren in the cavity oscillates roughly about the constant ρ Φ with the amplitude decreasing as ω −1 M . One cannot see whether the value of the renormalized energy density is negative or positive if the UV cutoff is not large enough. If ρ Φ < 0, one should take the value of ω M much greater than 2 γ 2 /(πL|ρ Φ |) to resolve the negativity of ρ Φ . This reminds us the fact that the Casimir effect is a finite-size effect of constraints on quantum fluctuations [47], which is not a purely IR or UV phenomenon. It depends not only on the field modes of long-wavelengths comparable with the scale of the background geometry. One has to sum over all the cavity modes in a perfect cavity to obtain the conventional result of the Casimir energy density [5].
If one introduces a normalizable, smooth switching function such as Gaussian or Lorentzian function of time for the coupling of an apparatus to the cavity field, it will suppress the contribution from the short-wavelength modes [12] and makes the "observed" energy density not so negative [40][41][42]. In our model the spectrum of the short-wavelength modes is closer to the ones in free space than those in a perfect cavity. One may wonder if there exists some choice of the parameter values which leads to a non-negative late-time energy density in our cavity for ω M sufficiently large. To answer this question, one needs to know the exact sign of ρ Φ , which looks very hard in calculating (4.29) numerically when ρ Φ is extremely close to zero.
Fortunately, the poles in the integrand of (4.29) are all located in the lower half of the complex plane. Thus the integral along a closed contour from ω = 0 → ∞ → i∞ → 0 in the upper complex plane (figure 7 (right)) gives zero. Since L > 0 in the factor e 2iLω in the numerator of the integrand in (4.29), which suppressed the contribution around ω ∼ i∞ (the dotted part of the contour in figure 7 (right)), we have which is Wick-rotated from (4.29) by letting ω = iβ [23][24][25][26]. Eq. (4.30) converges much faster than (4.29) in numerical calculations. Further, the integrand in (4.30) is positive definite for β ≥ 0, so ρ Φ must be negative for all regular, non-resonant choices of the parameter values in our model (in the resonant case withγ = 0 and Ω 0 = nπ/L for some positive integer n, the system will never settle down to the late-time steady state with (4.29); see section 4.2). Note that we did not take the strong OF coupling limit in obtaining (4.29) and (4.30). Even in the weak OF coupling regime where the working range of our detector mirrors is narrow (recall figures 1 and 6), the Casimir energy density in our cavity with sufficiently large ω M is still negative, though it may be very close to zero. In the example in figure 6, indeed, one has ρ Φ ≈ −6.9096 × 10 −10 < 0 in the cavity, while only one pair of the cavity modes are significant in the under-damping regime there. It is obvious in (4.29) and (4.30) that the Casimir energy density goes to zero as the OF coupling γ → 0. Going to the other extreme, if one takes the limit γ → ∞ before doing integration [23][24][25][26], then and one recovers the conventional result for a perfect cavity in (1+1)D [4]. In the above calculation a regularization L → L + i with → 0+ is understood. For L = 40, ρ Φ ≈ −0.0000818123 in (4.31), which is the same order of magnitude as the Casimir energy density in figure 7.
Right at the position of a detector mirror (z 1 A = 0 or z 1 B = L), one has the late-time renormalized energy density of the field which appears to have logarithmic divergence in the first term if we did not introduce a UV cutoff ω M for P 2 d (t) (see (4.37) and below). With a finite ω M , while the above energy density of the field has a large negative value, its contribution to the field energy is about {−(γ/2) P 2 d (t) + T 00 ren | 0<x<L }dx, which is small compared with the detector energy E d = ( P 2 d (t) + Ω 2 0 Q 2 d (t) )/2. Thus the total energy around is still positive. Also the total Casimir energy of the field is still x) ren = L T 00 ren 0<x<L (4.33) since the contribution by the finite T 00 (z µ d ) ren at x = 0 and x = L are infinitesimal in the integral.
When L → 0, the conventional result for the Casimir energy diverges like L×(−L −2 ) = −L −1 from (4.31) and (4.33). In contrast, ρ Φ in (4.30) behaves like ln L when L is small, so the total Casimir energy E Φ ∼ L × ln L goes to zero as the separation L → 0 in our model. The total energy of our HO-field system (with the field energy radiated in transient ignored) is thus finite and cutoff-dependent, and would be positive when the UV cutoff is sufficiently large.

Late-time entanglement between mirror oscillators
For our cavity of two detector mirrors, the symmetric two-point correlators of the internal HOs of the detectors can be formally represented as and so on. After some algebra, the late-time correlators of the oscillators are found to be and Q d (t),P d (t) = 0. Here with the UV cutoff ω M and the susceptibility functions χ ± ω defined in (4.24). The above late-time results are actually constants of t and very similar to eqs. (48)(49)(50)(51)(52) in ref. [44] except the oscillating term (∝ γe iωL in the denominator of χ ± ω ) due to the differences in the coupling and the number of spatial dimensions. Unlike its counterpart in [44], the oscillating term here keeps the denominator of the integrand of F c± regular as L → 0 for every finite ω.
For L = 0, the integrals of F c± can be done analytically to get with γ ± ≡γ + γ(1 ± 1) and Γ ± ≡ γ 2 ± − Ω 2 0 , which can be real (over-damping) or imaginary (under-damping). Here we set ω M = Ω 0 e Λ 1 to recover eq. (A12) in ref. [39] after the Λ 1 there is redefined as Λ 1 = −γ e − ln Ω r |τ − τ |, as we discussed in section 3.2 3 . While Re F 2± | L=0 is UV divergent as ω M → ∞, when the UV cutoff ω M and so Λ 1 are set to be finite and not too large, the internal HOs of the two UD' detectors can be entangled. For example, when Λ 1 = 100, γ = 10,γ = 0.01, Ω 0 = 0.1, and c = = 1, we find c 2 is positive. This implies that the reduced state of the oscillator pair, which is a Gaussian state, is well behaved and the oscillators are entangled (with the logarithmic negativity E N = max{0, − log 2 (2c − / )} ≈ 0.94) [44,[48][49][50]. If we increase the value of Λ 1 while keeping all other parameters unchanged, the oscillators will be entangled until Λ 1 exceeds about 400. For L > 0, the integrals of F c± deviate significantly from those with L = 0 for ω M > O(π/L) (figure 8). When we fixγ, Ω 0 , and L, the unphysical negative-Υ region in which the uncertainty relation Υ ≥ 0 is violated looks like a wedge in the ω M γ-plane in our examples with either ω M or γ not too large (gray regions in figure 9). The angle and the slopes of the two boundaries of the wedge decrease as L increases (compare the upper-left, upper-right, and lower-left plots in figure 9). Around the boundary of the negative-Υ region there are islands of parameter values in which one has Σ < 0 while the uncertainty relation Υ ≥ 0 3 Below eq. (A9) in Appendix A of [12], ω M is put as 2πΩe Λ 0 +γe or 2πΩe Λ 1 +γe . To exactly recover eqs.
(A9-A12) in [39], they should be corrected to ω M = Ωe Λ 0 or Ωe Λ 1 .   holds (dark regions). The late-time quantum entanglement between the oscillators of the two mirrors only occurs when the point (ω M , γ) with the fixed values ofγ, Ω 0 , and L, are located in one of these islands in the parameter space. The islands look disconnected in the ω M γ-plane because Υ(ω M ) and Σ(ω M ) are alternating when ω M ∼ O(γ), namely, if Υ(ω M ) > Σ(ω M ) for some ω M = Λ then Υ(ω M ) < Σ(ω M ) for ω M ≈ Λ ± π/L, as shown in figure 8 (right). This is due to the alternating nature of the γ(1 ± e iωL ) term in the denominators of χ ± ω in F c± . Asγ increases, the projections of the islands on the ω M -axis is roughly invariant, while the whole wedge of the Υ < 0 region shifts along the +ω M direction (from left to right in the lower row of figure 9). The width of those islands in ω M is about O(π/L), thus the larger L would give a smaller scale of the islands in the ω M γγ-space.
For any UV cutoff ω M , no matter how large it is, our results suggest that one still has a chance to find a OF coupling strength γ ∼ O(ω M ) while adjusting the UV cutoff around ω M ± π/L (withγ, Ω 0 , and L fixed) to make the two internal HOs entangled at late times. However, when γ and ω M have the same order of magnitude whileγ, Ω 0 , and 1/L are relatively small, the denominator of the integrand in (4.29) is approximately ω 4 + 4iγω 3 + 4γ 2 ω 2 (e 2iωL − 1), whose three terms are roughly the same order of magnitude, namely O(γ 4 ), so the energy density of the field in the cavity around this parameter range oscillates largely between positive and negative values as ω M increases. Indeed, in figure 7 (left) one can see that the maximum amplitude of the oscillating value of the field energy density occurs around ω M ≈ 2γ, and the oscillation will not be suppressed until ω M is much larger. It turns out that quantum entanglement between the HOs of the cavity mirrors and the convergence of the Casimir energy density in the same cavity never co-exist at late times in the parameter range of our results.

Summary
We employed the derivative-coupling Unruh-DeWitt(UD') HO detector theory in (1+1) dimensions to model the atom mirror interacting with a massless quantum field (OF coupling) and an environment of mechanical degrees of freedom (OE coupling). The reflectivity of our atom/detector mirror is dynamically determined by the interplay of the detector's internal oscillator and the field. In the strong OF coupling regime, the effect of the mechanical environment is negligible and the detector acts like a perfect mirror at late times, when the energy density of the field outside the detector vanishes while the field spectrum is nontrivial. Compared with the field correlators in free space, in the presence of a detector mirror the late-time correlators are reduced both for the field amplitudes on the same side and those on two different sides of the mirror.
A pair of such UD' detector mirrors can form a cavity. If both oscillators are decoupled from the environment, the system will not settle to a steady state at late times if the two internal HOs of the cavity mirrors are on-resonance, namely, the natural frequency of the oscillator is integer times of the frequency for the massless scalar field in the cavity traveling from one detector mirror to the other.
If the OE coupling is nonvanishing, the field in this cavity will evolve into a steady, quasi-discrete spectrum at late times. Then there will be many cavity modes in the strong OF coupling, over-damping regime but only one or few pairs of significant cavity modes in the weak OF coupling, under-damping regime. With the UV cutoff sufficiently large, the late-time renormalized field energy density in the cavity converges to a negative value for all positive OF coupling strengths. In the infinite OF coupling limit, the negative field energy density goes to the conventional result in the Casimir effect. In contrast to the conventional result with the perfect mirrors, however, the total energy density in our cavity does not diverge as the separation of the detector mirrors goes to zero. Outside the cavity the renormalized field energy density is again vanishing while the field spectrum is nontrivial.
Our result suggests that the internal oscillators of the two mirrors of our cavity can have late-time entanglement when the OF coupling strength is roughly the same order as the UV cutoff for the two identical HOs. In this regime, however, the field energy density in the cavity does not converge but very sensitive to the choice of the UV cutoff. The HOs in the parameter range of our results are always separable when the UV cutoff is large enough to obtain a convergent value of the Casimir energy density.