Entanglement-induced quantum radiation

TL;DR

This paper explores how quantum entanglement in the vacuum state leads to entanglement-induced quantum radiation from accelerated objects, expanding understanding of the Unruh effect and energy flux behaviors.

Contribution

It demonstrates that entanglement causes quantum radiation from uniformly accelerated objects, revealing a new consequence of the Minkowski vacuum entanglement.

Findings

Entanglement induces quantum radiation from accelerated objects.

Incoming and outgoing energy fluxes cancel in a thermalized state.

Supports the understanding of the Unruh effect's implications.

Abstract

Quantum entanglement of the Minkowski vacuum state between left and right Rindler wedges generates thermal behavior in the right Rindler wedge, which is known as the Unruh effect. In this letter, we show that there is another consequence of this entanglement, namely entanglement-induced quantum radiation emanating from a uniformly accelerated object. We clarify why it is in agreement with our intuition that incoming and outgoing energy fluxes should cancel each other out in a thermalized state.

Equations

$$S[Q,\varphi ;z]$$

$$z^{\mu }(\tau )=a^{-1}\left(\sinh a\tau ,\cosh a\tau ,0,0\right).$$

$$\varphi (x)={\varphi }_{\mathrm{h}}(x)+{\varphi }_{\mathrm{inh}}(x).$$

$${\varphi }_{\mathrm{inh}}(x)=\lambda \int d\tau Q(\tau )G_{\mathrm{R}}(x-z(\tau )),$$

$$\left({\partial }_{\tau }^2+2\gamma {\partial }_{\tau }+{\Omega }^2\right)Q(\tau )=(\lambda /m){\varphi }_{\mathrm{h}}(z(\tau )).$$

$$⟨\varphi (x)\varphi (y)⟩-⟨{\varphi }_{\mathrm{h}}(x){\varphi }_{\mathrm{h}}(y)⟩=⟨{\varphi }_{\mathrm{inh}}(x){\varphi }_{\mathrm{inh}}(y)⟩$$

$$+⟨{\varphi }_{\mathrm{h}}(x){\varphi }_{\mathrm{inh}}(y)⟩+⟨{\varphi }_{\mathrm{inh}}(x){\varphi }_{\mathrm{h}}(y)⟩.$$

$$⟨{\varphi }_{\mathrm{h}}(x){\varphi }_{\mathrm{h}}(z(\tau ))⟩=-\frac{i}{8{\pi }^2{\rho }_0(x)}{\int }_{-\infty }^{+\infty }d\omega e^{-i\omega \tau }$$

$$\displaystyle~{}~{}~{}~{}~{}~{}\Bigl{(}{e^{\pi\omega/a}\over e^{2\pi\omega/a}-1}e^{i\omega\tau_{+}^{x}}-{1\over e^{2\pi\omega/a}-1}e^{i\omega\tau_{-}^{x}}\Bigr{)},$$

$$\displaystyle|0,{\rm M}\rangle=\prod_{j}\biggl{[}N_{j}\sum_{n_{j}=0}^{\infty}e^{-\pi n_{j}\omega_{j}/a}|n_{j}\rangle_{{\rm R}}\otimes|n_{j}\rangle_{\rm L}\biggr{]},$$

$${\varphi }_{\mathrm{h}}(x)={\int }_0^{\infty }d\omega \int d^2{k}_{\perp }\left({\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{R}}{v}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{R}}(x)+h.c.\right),$$

$$\displaystyle v_{\omega,{\bm{k}}_{\perp}}^{\rm R}(x)=\sqrt{\sinh\pi\omega/a\over 4\pi^{4}a}K_{i\omega/a}\Bigl{(}{\kappa e^{a\xi}\over a}\Bigr{)}e^{i\bm{k}_{\perp}\cdot\bm{x}_{\perp}-i\omega\tau}.$$

$${\varphi }_{\mathrm{h}}(x)={\int }_{-\infty }^{+\infty }d\omega \int d^2{k}_{\perp }\left({\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F}}{v}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F}}(x)+h.c.\right),$$

$$\displaystyle v_{\omega,{\bm{k}}_{\perp}}^{\rm F}(x)={-ie^{i\omega\zeta}\over 2\pi\sqrt{4a\sinh(\pi|\omega|/a)}}J_{-i|\omega|/a}\Bigl{(}{\kappa e^{a\eta}\over a}\Bigr{)}e^{i\bm{k}_{\perp}\cdot\bm{x}_{\perp}}.$$

$${\varphi }_{\mathrm{h}}(x)={\varphi }_{\mathrm{F}}^{\mathrm{d}}(x)+{\varphi }_{\mathrm{F}}^{\mathrm{s}}(x),$$

$$\displaystyle\phi_{\rm F}^{\rm d}(x)=\int_{0}^{+\infty}d\omega\int d^{2}k_{\perp}\Bigl{(}{\hat{a}}_{\omega,{\bm{k}}_{\perp}}^{\rm F,d}v_{\omega,{\bm{k}}_{\perp}}^{{\rm F},\rm d}(x)+{\rm h.c.}\Bigr{)},$$

$$\displaystyle\phi_{\rm F}^{\rm s}(x)=\int_{0}^{+\infty}d\omega\int d^{2}k_{\perp}\Bigl{(}{\hat{a}}_{\omega,{\bm{k}}_{\perp}}^{\rm F,s}v_{\omega,{\bm{k}}_{\perp}}^{{\rm F},\rm s}(x)+{\rm h.c.}\Bigr{)},$$

$$\displaystyle|0,{\rm M}\rangle=\prod_{j}\Bigl{[}N_{j}\sum_{n_{j}=0}^{\infty}e^{-\pi n_{j}\omega/a}|n_{j},{{\rm s}}\rangle_{\rm F}\otimes|n_{j},{{\rm d}}\rangle_{\rm F}\Bigr{]},$$

$$⟨0,\mathrm{M}|{\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F},\mathrm{s}†}{\hat{a}}_{{\omega }^{\prime },{\mathbf{k}}_{\perp }^{\prime }}^{\mathrm{R}}|0,\mathrm{M}⟩=\frac{\delta (\omega -{\omega }^{\prime }){\delta }^2({\mathbf{k}}_{\perp }-{\mathbf{k}}_{\perp }^{\prime })}{e^{2\pi \omega /a}-1},$$

$$⟨0,\mathrm{M}|{\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F},\mathrm{s}}{\hat{a}}_{{\omega }^{\prime },{\mathbf{k}}_{\perp }^{\prime }}^{\mathrm{R}†}|0,\mathrm{M}⟩=\frac{\delta (\omega -{\omega }^{\prime }){\delta }^2({\mathbf{k}}_{\perp }-{\mathbf{k}}_{\perp }^{\prime })}{1-e^{-2\pi \omega /a}},$$

$$⟨0,\mathrm{M}|{\varphi }_{\mathrm{F}}^{\mathrm{s}}(x){\varphi }_{\mathrm{R}}(z(\tau ))|0,\mathrm{M}⟩={\int }_0^{\infty }d\omega \int d^2{k}_{\perp }$$

$$\displaystyle\Bigl{(}{v^{{\rm F},{\rm s}*}_{\omega\bm{k}_{\perp}}(x)v^{{\rm R}}_{\omega\bm{k}_{\perp}}(z(\tau))\over e^{2\pi\omega/a}-1}+{v^{{\rm F},{\rm s}}_{\omega\bm{k}_{\perp}}(x)v^{{\rm R}*}_{\omega\bm{k}_{\perp}}(z(\tau))\over 1-e^{-2\pi\omega/a}}\Bigr{)}.$$

$$\displaystyle\int_{0}^{\infty}d\kappa\kappa J_{-i\omega/a}\bigl{(}{\kappa\over a}e^{a\eta}\bigr{)}K_{-i\omega/a}\bigl{(}{\kappa\over a}\bigr{)}J_{0}(\kappa x_{\perp})={ae^{i\omega\tau_{+}^{x}-i\omega\zeta}\over 2\rho_{0}(x)}.$$

$$⟨0,\mathrm{M}|{\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F},\mathrm{d}}{\hat{a}}_{{\omega }^{\prime },{\mathbf{k}}_{\perp }^{\prime }}^{\mathrm{R}}|0,\mathrm{M}⟩=⟨0,\mathrm{M}|{\hat{a}}_{\omega ,{\mathbf{k}}_{\perp }}^{\mathrm{F},\mathrm{d}†}{\hat{a}}_{{\omega }^{\prime },{\mathbf{k}}_{\perp }^{\prime }}^{\mathrm{R}†}|0,\mathrm{M}⟩$$

$$=\frac{e^{\pi \omega /a}}{e^{2\pi \omega /a}-1}\delta (\omega -{\omega }^{\prime }){\delta }^2({\mathbf{k}}_{\perp }-{\mathbf{k}}_{\perp }^{\prime }).$$

$$\displaystyle\langle 0,{\rm M}|\phi_{\rm F}^{\rm d}(x)\phi_{\rm R}(z(\tau))|0,{\rm M}\rangle=\int_{0}^{\infty}d\omega\int d^{2}k_{\perp}\Bigl{(}v^{{\rm F},{\rm d}}_{\omega\bm{k}_{\perp}}(x)$$

$$\displaystyle v^{{\rm R}}_{\omega\bm{k}_{\perp}}(z(\tau))+v^{{\rm F},{\rm d}*}_{\omega\bm{k}_{\perp}}(x)v^{{\rm R}*}_{\omega\bm{k}_{\perp}}(z(\tau))\Bigr{)}{e^{\pi\omega/a}\over e^{2\pi\omega/a}-1}.$$

Full text

††preprint: KEK-TH-1970, HUPD-1703

Introduction.— A major outstanding question in quantum field theory is whether an Unruh-de Witt detector (which is a uniformly accelerated object coupled to a radiation field) would emit radiation. A detection of such radiation, if it exists, would have a huge impact upon fundamental physics, and various experimental proposals toward detecting the radiation have been made ChenTajima ; Schutzhold ; Schutzhold2 ; ELI ; LH ; IYZ ; Lin . After an infinitely long time, the Unruh-de Witt detector would become thermalized to the Unruh temperature Unruh , and, in analogy with an ordinary equilibrium system in a thermal bath, one may intuitively expect that the incoming and outgoing energy fluxes from the accelerated object should cancel out, resulting in a net flux of zero.

This issue has been debated for a long time. In a toy model with $d=1+1$ dimensions, HuRaval showed that there is no net flux, which is consistent with our intuition. For a charged particle model in $d=3+1$ dimensions, Refs. IYZ ; OYZ15 ; OYZ16 have shown that, although most of the terms cancel out when interference effects are carefully considered, there remains some radiation. However, these calculations could be performed only approximately, leaving it unclear whether this radiation exists. The authors of LH ; IOTYZ showed that, in a toy model in $d=3+1$ dimensions where we can solve the equation of motion exactly, the cancellation is not exact and quantum radiation emanates; thus, there must be an error in our intuition.

In our previous paper IOTYZ , we presented some circumstantial evidence that this radiation is responsible for a nonlocal correlation of the quantum field in the Minkowski vacuum state, which originates in the entanglement between the left (L) and right (R) Rindler wedges. In this letter, we explicitly show that this is indeed the case and explain why it does not contradict to our intuition that net radiation should cancel out in a thermalized system. More specifically, the origin of the radiation is entanglement between the wave functions (Rindler modes) in the R-region and the right-moving waves (Kasner modes) in the F-region in Fig 1.

Radiation from an Unruh-de Witt detector.— We consider a toy model in $d=3+1$ dimensions, a coupled system of a harmonic oscillator, $Q$, accelerating uniformly and a massless scalar field, $\phi(x)$. The action is given by

[TABLE]

with the world-line trajectory $z^{\mu}(\tau)$ in the R-region (see Fig. 1) at a uniform acceleration $a$,

[TABLE]

The equations of motion can be solved exactly LH ; IYZ , and after an infinitely long time, the system relaxes to an equilibrium state.

In the presence of the harmonic oscillator (which is our Unruh-de Witt detector) $Q(\tau)$, the field $\phi(x)$ in the Heisenberg picture is given by a sum of homogeneous $\phi_{\rm h}(x)$ and inhomogeneous $\phi_{\rm inh}(x)$ terms,

[TABLE]

The homogeneous term $\phi_{\rm h}(x)$ is nothing but the vacuum fluctuation of the field at the position $x$, while $\phi_{\rm inh}(x)$ can be solved in terms of the $Q$ operator as

[TABLE]

where $G_{R}(x-y)$ is the retarded Green’s function of the d’Alembertian operator. $\phi_{\rm inh}(x)$ can be further written in terms of the field $\phi_{h}(z(\tau))$ on the trajectory $x=z(\tau)$ in the R-region, by using the solution of the equation of motion for $Q(\tau)$

[TABLE]

Here $\gamma=\lambda^{2}/8\pi m$ and $\Omega$ is the renormalized frequency (see Ref. LH ; IYZ ).

Thus, when we calculate the energy flux of radiation in the F-region, namely the two-point correlation function of fields $\langle\phi(x)\phi(y)\rangle$ at $x,y\in\text{F-region}$; it is necessary to calculate the two-point correlation functions of the field $\phi_{\rm h}(x)$ in the F-region and the field $\phi_{\rm h}(y)$ on the trajectory $y=z(\tau)$ in the R-region (or both in the R-region).

Let us look at this more explicitly. The quantity we are interested in is the two-point correlation function $\langle\phi(x)\phi(y)\rangle$ in the F-region ($x,y\in\text{F}$), since the future light-cone of the detector is almost in the F-region in the the large distance limit and non-vanishing radiation flux is expected to appear there. By subtracting the vacuum correlation function $\langle\phi_{\rm h}(x)\phi_{\rm h}(y)\rangle$, it is given by the sum of three terms,

[TABLE]

We call the first term on the right-hand side a naive radiation term; its detection was the target of the original experimental proposals. The second and the third terms relate to the interference. The existence or non-existence of the Unruh radiation is determined by how much of the naive radiation is canceled by the interference terms (Refs. IYZ ; IOTYZ ).

As we noted, $\phi_{\rm inh}(x)$ is determined by the vacuum fluctuation $\phi_{\rm h}(z(\tau))$ on the trajectory in the R-region. Thus, the naive radiation term is calculated in terms of the correlation function of $\phi_{\rm h}(z(\tau))$ and $\phi_{\rm h}(z(\tau^{\prime}))$, both of which are in the R-region. On the other hand, the interference terms are calculated in terms of $\langle\phi_{\rm h}(x)\phi_{\rm h}(z(\tau))\rangle$, where $x\in{\text{F}}$ and $z(\tau)$ is the trajectory of the detector in the R-region. Thus, the interference terms reflect the correlations between the vacuum fluctuations in the F- and R-regions. A straightforward calculation shows that IYZ ; IOTYZ

[TABLE]

where $\rho_{0}(x)$ is the distance between the position $x$ and the detector defined in Eq.(3.1) in IOTYZ , and $\tau_{\pm}^{x}$ are the proper times depicted in Fig. 1.

The second term in parentheses in Eq.(7), which depends on $\tau_{-}^{x}$ on the real trajectory, is shown to completely cancel the naive radiation term IOTYZ . Thus, our intuition concerning the absence of naive radiation in a thermalized system seems to be correct so far. However, the interference terms have an additional contribution, namely the first term in parentheses in Eq.(7). We showed that this term generates a new type of quantum radiation from the accelerated object in the R-region. This term differs from the second one in two ways. First, it is a function of $\tau_{+}^{x}$, which is a proper time on the imaginary trajectory in the L-region. Second, the first term in parentheses in Eq.(7) is multiplied by an additional factor, $e^{\pi\omega/a}$. In IOTYZ , we discussed how these two points indicate that a new type of radiation is induced by entanglement of the vacuum state. We call it entanglement-induced quantum radiation. In the rest of this letter, we make this radiation more transparent by scrutinizing the Hilbert space structure in the F-region.

Minkowski vacuum as an entangled state.— The Minkowski vacuum state of $\phi(x)$ can be written as an entangled state in terms of the L and R Rindler states, as shown by UnruhWald

[TABLE]

with $j=(\omega,\bm{k}_{\perp})$ and $N_{j}=\sqrt{1-e^{-2\pi\omega/a}}$. The quantized field in the R-region is expanded Higuchi as

[TABLE]

where $x\in\text{R-region}$ and

[TABLE]

Here, $\kappa=|\bm{k}_{\perp}|$, and the Rindler coordinates $(\tau,\xi)$ are defined by $t={a^{-1}}e^{a\xi}\sinh a\tau$,  $x={a^{-1}}e^{a\xi}\cosh a\tau$. The line element of a flat Minkowski spacetime is given by $ds^{2}=e^{2a\xi}(d\tau^{2}-d\xi^{2})-d\bm{x}_{\perp}^{2}$ in Rindler coordinates. The coordinates cover the R-region, a quarter of the Minkowski spacetime in Fig. 1 and Fig. 2. The state $|n_{j}\rangle_{\text{R}}$ is an $n_{j}$-th excited state created by the operator ${\hat{a}}_{\omega,{\bm{k}}_{\perp}}^{\rm R\dagger}$. The state $|n_{j}\rangle_{\text{L}}$ is similarly defined in the L-region with the sign of its momentum reversed for convenience. Then, as in (8), the Minkowski vacuum state is written as the entangled state of the excited states in the L and R regions.

The Minkowski vacuum can be similarly written in the F-region. The F-region ($t>|x|$) is described by the expanding degenerate Kasner universe with coordinates $(\eta,~{}\zeta)$. $x$ and $t$ are respectively given by $t={a^{-1}}e^{a\eta}\cosh a\zeta$ and  $x={a^{-1}}e^{a\eta}\sinh a\zeta$, and the line element is $ds^{2}=e^{2a\eta}(d\eta^{2}-d\zeta^{2})-d\bm{x}_{\perp}^{2}$. In the F-region, the quantized field $\phi_{\rm h}(x)$ can be expanded Higuchi as

[TABLE]

where $x\in\text{F}$ and

[TABLE]

It is important to note that $\omega$ corresponds to the momentum in the $\zeta$ direction and takes either a positive or a negative value. A positive $\omega$ represents a right-moving Kasner mode, whereas a negative $\omega$ represents a left-moving Kasner mode. We then separate the field $\phi_{\rm h}(x)$ in the F-region into $\phi_{\rm F}^{\rm d}(x)$ with $\omega>0$ (right-moving) modes and $\phi_{\rm F}^{\rm s}(x)$ with $\omega<0$ (left-moving) modes as

[TABLE]

where $x\in\text{F}$. Here, the indices ”d” and ”s” indicate dexter (right) and sinister (left) in Latin, respectively. The field $\phi_{\rm F}^{\rm d}$ is defined by $\phi_{\rm h}(x)$, whose $\omega$ is restricted to $\omega>0$:

[TABLE]

where $v_{\omega,{\bm{k}}_{\perp}}^{{\rm F},{\rm d}}(x)=v_{\omega,-{\bm{k}}_{\perp}}^{\rm F}(x)$ and $\hat{a}^{\rm F,\rm d}_{\omega,\bm{k}_{\perp}}=\hat{a}^{\rm F}_{\omega,-\bm{k}_{\perp}}.$ The transverse momenta are reversed for later convenience. Another field, $\phi_{\rm F}^{\rm s}(x)$, is defined as

[TABLE]

where $v_{\omega,{\bm{k}}_{\perp}}^{{\rm F},{\rm s}}(x)=v_{-\omega,{\bm{k}}_{\perp}}^{\rm F}(x)$ and $\hat{a}^{\rm F,s}_{\omega,\bm{k}_{\perp}}=\hat{a}^{\rm F}_{-\omega,\bm{k}_{\perp}}.$

The vacuum state with respect to these annihilation operators is defined by $|0,{\rm s}\rangle_{\rm F}\otimes|0,{\rm d}\rangle_{\rm F}$, and their $n_{j}$-th excited states are represented by $|n_{j},{{\rm s}}\rangle_{\rm F}$ and $|n_{j},{{\rm d}}\rangle_{\rm F}$. In terms of these Kasner modes, the Minkowski vacuum can be written as

[TABLE]

which expresses the cosmological quantum entanglement in the F-region EMM . Notice that in this expression, $\omega$ in $j=(\omega,\bm{k}_{\perp})$ is restricted to positive values.

In order to understand the physical origin of entanglement-induced quantum radiation given by the first term in parentheses in Eq.(7), we need to know how the quantum fields $\phi_{\rm h}(x)$ of the F and R regions are correlated. Namely, we need to relate two expressions of the Minkowski vacuum: Eq.(8) in the R-region and Eq.(14) in the F-region. For this purpose, we use the following properties that relate the wave functions of the Rindler and Kasner modes. As discussed in Ref. Higuchi , the Rindler-mode function in the R-region, $v_{\omega,{\bm{k}}_{\perp}}^{\rm R}(x)$, can be identified with the left-moving mode $v_{\omega,\bm{k}_{\perp}}^{\rm F,s}(x)$ in the F-region by a continuation through the Minkowski-mode functions, which properly describe the evolution across the horizons. Similarly, the Rindler-mode function in the L-region, $v_{\omega,{\bm{k}}_{\perp}}^{\rm L}(x)$, is continued to the right-moving mode $v_{\omega,\bm{k}_{\perp}}^{\rm F,\rm d}(x)$ in the F-region. The observation of continuations of the wave functions suggest that a state created by the operator $({\hat{a}}_{\omega,{\bm{k}}_{\perp}}^{\rm R})^{\dagger}$ in the R-region propagates into the F-region and becomes identified with a state created by the operator $(\hat{a}^{\rm F,s}_{\omega,\bm{k}_{\perp}})^{\dagger}$ in the F-region. Intuitively, this is quite natural. Further detailed analysis is under investigations.

Entanglement-induced quantum radiation.— We now calculate the two-point correlation function (7) of fields in the R and F regions using the operator formalism based on the Hilbert-space structure discussed above. Since $x\in\text{F}$, we decompose $\langle\phi_{\rm h}(x)\phi_{\rm h}(z(\tau))\rangle$ into the terms $\langle\phi_{\rm F}^{\rm s}(x)\phi_{\rm R}(z(\tau))\rangle$ and $\langle\phi_{\rm F}^{\rm d}(x)\phi_{\rm R}(z(\tau))\rangle$ using (13).

First, we evaluate $\langle\phi_{\rm F}^{\rm s}(x)\phi_{\rm R}(z(\tau))\rangle$. The field $\phi_{\rm F}^{\rm s}(x)$ contains the left-moving modes in the F-region, which are identified with the modes in the R-region as discussed above. Thus, we can obtain the relations

[TABLE]

and

[TABLE]

which reflect the fact that the left-moving modes in the F-region are in thermal equilibrium at temperature $T=a/2\pi$. Then, $\langle\phi_{\rm F}^{\rm s}(x)\phi_{\rm R}(z(\tau))\rangle$ can be evaluated as

[TABLE]

The integrations can be performed using the relation $\int_{0}^{2\pi}d\varphi e^{i\kappa x_{\perp}\cos\varphi}=2\pi J_{0}(\kappa x_{\perp})$ and the mathematical formulae in Tables ; Magnus as follows:

[TABLE]

Then, noting the relation $e^{i\omega\tau_{+}^{x}-i\omega\zeta}=e^{-i\omega\tau_{-}^{x}+i\omega\zeta}$, we can show that $\langle\phi_{\rm F}^{\rm s}(x)\phi_{\rm R}(z(\tau))\rangle$ is reduced to the second term in parentheses in Eq. (7), canceling out the naive radiation produced by the correlation function $\langle\phi_{\rm inh}(x)\phi_{\rm inh}(y)\rangle$. This result is consistent with our intuition that the net radiation should cancel out in an equilibrium state in a thermal bath. The naive radiation is completely canceled out by the interference term of the left-moving modes in the F-region. The modes come from the R-region, and the cancellation shows that if we restrict the mode functions of the field $\phi(x)$ in the F-region to those coming from the R-region, no net radiation will emanate from an accelerating object.

Note, however, that there is an additional contribution. We next evaluate $\langle\phi_{\rm F}^{\rm d}(x)\phi_{\rm R}(z(\tau))\rangle$; this provides an additional contribution to the correlation function in the F-region. The calculations can be similarly performed. In this case, we use the relations

[TABLE]

These relations reflect the identification of the left-moving modes $v_{\omega,\bm{k}_{\perp}}^{\rm F,s}(x)$ in the F-region and the modes $v_{\omega,{\bm{k}}_{\perp}}^{\rm R}(x)$ in the R-region, and can be understood either as the entangled correlation between the L and R Rindler modes in (8) or as the same kind of the correlation between left-moving and right-moving modes in the F-region in (14). Using Eq. (16), we have

[TABLE]

Then we can show that $\langle\phi_{\rm F}^{\rm d}(x)\phi_{\rm R}(z(\tau))\rangle$ is reduced to the first term in parentheses in Eq. (7).

Remembering that the field $\phi_{\rm F}^{\rm d}(x)$ comprises the right-moving modes in the F-region coming from the L-region, we can definitely say that the entanglement-induced quantum radiation is responsible for the entanglement between the modes in the L-region and those in the R-region or the entanglement between the left-moving and right-moving modes in the F-region.

Summary.— Now it is clear why the existence of a net radiation flux from an Unruh-de Witt detector is in agreement with our intuition that flux should be canceled out in an equilibrium state in a thermal bath. The Unruh-de Witt detector is described as a thermal system only when we can integrate out the modes in the L Rindler wedge; a typical example is the Unruh effect, the thermal behavior observed by a uniformly accelerated object in the R-region. In the F-region, however, this is not the case. The quantum field $\phi(x)$ in the F-region contains both the modes coming from the L and R regions. If we neglect $\phi_{\rm F}^{\rm d}(x)$, the rest of the modes come from the R-region, and we can safely integrate these modes in the L-region. But in order to calculate a two-point correlation function in the F-region, we cannot neglect such right-moving modes, $\phi_{\rm F}^{\rm d}(x)$, in the F-region; indeed, entanglement-induced quantum radiation is generated owing to the entanglement of $\phi_{\rm F}^{\rm d}(x)$ and the field $\phi_{\rm R}(z(\tau))$ on the trajectory in the R-region. The quantum radiation produced by a charged particle undergoing uniformly accelerated motion OYZ15 ; OYZ16 also has the same origin. Therefore, detection of the non-vanishing quantum radiation will be a test of the entanglement of the quantum field. In the case of a uniformly accelerated charged particle in which the classical Larmor radiation is dominant, the difference in the spectrum and the angular distribution OYZ16 ; IOTYZ may make it possible to separate the quantum contribution from the classical one. A detection of such radiation will clarify nature of the entangled quantum vacuum.

Acknowledgments.— This work is supported by MEXT/JSPS KAKENHI Grant Number 15H05895 (KY), 23540329 and 16H06490 (SI). We thank J. Soda, Y. Nambu, M. Hotta, F.-L. Lin, R. Schutzhold, and W. G. Unruh, for useful comments. K.Y. thanks A. Higuchi for crucial comments and discussions and for his hospitality during the stay at York University.