1. Introduction

Interaction of a light pulse with an atom is determined by the frequency content of the pulse in the atom’s reference frame. If one of the frequencies coincides with the frequency of the atomic transition the atom can become excited by absorbing a photon. A positive-frequency pulse can have negative-frequency content in the frame of atoms moving along certain worldlines leading to emission of a photon by ground-state atoms together with atom’s excitation. This is, e.g., the case if the atom is accelerated through an empty space which yields Unruh acceleration radiation [1–3].

Negative frequency photons describe field in a medium in a moving frame which leads to Cherenkov radiation of neutral atoms which is a quantum effect. The latter can be interpreted as emission of the negative frequency photons by a uniformly moving atom [4–6]. For description of the Cherenkov effect it is convenient to choose mode functions of the field as plane waves which in the medium with refractive index $n$ read (in the lab frame)

In this frame, if $V>c/n$, photons propagating inside the Cherenkov cone $\cos \theta >c/Vn$, where $\theta$ is the angle between $\mathbf {V}$ and $\mathbf {k}$, have negative frequency [7]. This leads to Cherenkov radiation. Namely, an atom which is at rest in the moving frame can become excited by emitting a negative-frequency photon inside the Cherenkov cone. In this process the total energy is conserved because the emitted photon has negative energy in the atom’s frame. Other ground-state atoms which are at rest in the moving frame can not absorb the emitted negative-energy photon, but can emit identical negative-energy photons by the stimulated emission mechanism [6]. Such ensemble of ground-state atoms acts as an inverted medium in a laser which yields light amplification.

An excited atom decays back to the ground state by spontaneous emission of a positive-frequency photon outside the Cherenkov cone. This yields generation of entangled photon pairs, one of the photons in the pair is emitted inside the Cherenkov cone and the other one is emitted outside the cone [8].

Negative frequency is associated with many other physical effects. For example, quantum friction between relatively moving dielectrics stems from the mixing of positive and negative frequency waves in the two materials [9,10]. The behavior of the negative-energy quanta is essential to understanding the kinematics of amplification of waves which can explain the physics of traveling-wave-tube-type amplifiers [11], a resistive-wall amplifier [12], and amplification of ultrasound in semiconductors [13]. A ground-state atom moving above a metal surface can become excited by emitting a surface plasmon which, from the atom’s perspective, has negative frequency [14]. The role of negative frequencies in nonlinear fiber optics was discussed in [15] .

Frequency content of a pulse can change, e.g., due to interaction with an accelerating mirror [16] or due to a sudden change of the medium properties with time. These can lead to appearance of negative frequencies in the pulse spectrum in the lab frame and, as a consequence, to it’s unusual interaction with atoms. For example, a fixed ground-state atom cannot become excited by absorbing a negative-frequency photon because this process would violate energy conservation. The atom can, however, become excited by emitting the negative-frequency photon and this process is stimulated by the photons present in the field.

By changing the basis set of the field modes one can describe the same state of the field in a picture in which photons have only positive frequencies and, thus, interact with atoms in a usual fashion. In this representation the number of photons in the pulse would be different, implying particle creation or annihilation. In particular, in the new representation, a vacuum state is filled with photons. Thus, if a physical process yields appearance of negative frequencies in the pulse spectrum this means generation of particles in a conventional description of the field in terms of the positive-frequency photons.

Let us consider a massless scalar field that obeys the one-dimensional wave equation in Minkowski spacetime

The inner product of two field modes $\phi _{1}(t,x)$ and $\phi _{2}(t,x)$ obeying Eq. (1) is defined as the Klein–Gordon inner product which is a generalization of the Wronksian [16]

The inner product is independent of time and has the following properties

For example, for a plane-wave mode

The situation is different in quantum description because Schrodinger equation is not symmetric under time reversal. For simplicity, let us consider a single-mode field with a mode function $\phi (t,x)$ which has positive norm, $\left \langle \phi,\phi \right \rangle >0$. In quantum description, the positive norm modes are associated with the photon annihilation operators $\hat {a}$. The complex conjugate $\phi ^{\ast }(t,x)$ is a negative-norm mode corresponds to the photon creation operators $\hat {a} ^{\dagger }$. Expansion of the field operator $\hat {\Phi }$ in terms of the modes $\phi$ and $\phi ^{\ast }$ reads

The field quantization procedure breaks the time reversal symmetry of the Eq. (1). Namely, if we make transformation $t\rightarrow -t$ in Eq. (5), the transformed field operator

However, a positive-norm mode function $\phi (t,x)$ can have negative frequency components. For example, a superposition of the right and left-moving Gaussian pulses with opposite carrier frequencies $\nu >0$ and $-\nu$

Classical Maxwell’s equations which do not take into account creation of photons out of vacuum yield that during time reflection a pulse undergoes a frequency shift and splits into two pulses propagating in the opposite directions [17]. The pulse propagating in the direction opposite to that of the incident wave has negative frequency. The time boundary problem conserves the wave momentum but not the energy, in contrast with the usual space boundary problem where the wave energy is conserved but the momentum is not. The difference between reflected and time-reflected pulses in classical description is illustrated in Fig. 1.

 

Fig. 1. Classical electrodynamics picture of reflection and refraction of light from a space boundary at $x=0$ between two media with refractive indices $n_{1}$ and $n_{2}$ in 1+1 dimensions (a) and a time boundary at $t=0$ (b). During reflection and refraction from the space (time) boundary the light energy (momentum) is conserved. When a pulse of frequency $\nu$ passes through a time boundary it splits into two pulses propagating in the opposite directions with frequencies $\pm n_{1}\nu /n_{2}$.

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Pulses propagating backward in time (having negative frequency) can be produced in the lab frame using superluminal boundaries between two media or if properties of the host material depend on time [17,18]. Such boundaries have been demonstrated experimentally. For example, superluminal ionization fronts have been used to generate electromagnetic radiation [19]. In a nonlinear medium, one can make refractive index inhomogeneities propagate faster than light by creating interference between two laser beams with differing frequencies (flying mirrors) [20]. Recently temporal reflection at photonic time interfaces has been observed experimentally [21,22]. Negative frequency pulses have been produced experimentally by optical solitons through a resonant emission process [23].

As we mentioned above, appearance of the negative frequency pulses can be interpreted as particle generation or annihilation. Namely, by changing the basis set of the photon mode functions one can describe the state of the field in terms of the positive frequency photons but with a different particle content. In the latter description, the superluminal interface or an abrupt change of the refractive index yield generation of positive-frequency entangled photon pairs propagating in the opposite directions [24]. This is analogous to the creation of entangled particle pairs in the dynamical Casimir effect and Hawking radiation. In Section 5 we show that during the process of particle pair production, entanglement of vacuum is transferred to the entanglement of the generated photon pairs.

If initially an electromagnetic pulse is present in the medium the photon generation is stimulated into the pulse mode, and since photons are created as entangled pairs the counter-propagating photon partners produce a positive-frequency pulse moving in the opposite direction. Thus, pulse splitting (time reflection) under an abrupt change of the refractive index occurs due to stimulated generation of particle pairs and no photons in the original pulse are in fact reflected back.

The moving boundary mimics an event horizon of a black hole (BH) if it’s speed is subluminal for the medium 1 (analog of BH exterior) and superluminal for the medium 2 (BH interior). In Section 6 we show that in this case the moving boundary generates entangled photon pairs, with one photon propagating into the medium 1 and the other into the medium 2, which is analogous to Hawking radiation. Production of photon pairs by the event horizon can be stimulated by electromagnetic field present in the system.

2. Interaction of light pulses with a superluminal boundary: classical description

Let us assume that a Gaussian pulse with electric field

 

Fig. 2. (a) A right-moving Gaussian pulse passes through a superluminal boundary between two dielectric media. The latter are at rest and only the interface is moving. (b) The interface splits the pulse into two transmitted Gaussian pulses moving in the opposite directions, and there is no reflected pulse (that is there is no pulse on the left side of the boundary after interaction). In the classical description the pulse $\nu _{3}$ has negative frequency and propagates backward in time.

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At the interface, the pulse splits into two transmitted Gaussian pulses moving in the opposite directions with the speed $\pm c/n_{2}$, and there is no reflected pulse, that is there is no pulse on the left side of the boundary after interaction (see Fig. 2(b)). As we show in Appendix A, classical Maxwell’s equations yield for the field in the medium $2$

If the pulse in the medium $1$ is moving to the right (lower sign in Eq. (10)), and velocity of the interface $V>c/n_{1,2}$, the carrier frequency of the left-moving pulse $A_{3}$ is negative $(\nu _{3}<0)$, while $\nu _{2}>0$. Thus, pulse $A_{3}$ propagates backward in time and it’s mode function is obtained from the pulse moving forward in time

Thus, superluminal boundary produces pulses with negative frequency (time reflection). In quantum description this implies particle creation which we discuss next.

3. Stimulated generation of entangled photon pairs by a superluminal boundary

As we mentioned above, one can change the basis set of the field mode functions such that photons in the medium 2 have only positive frequencies in their Fourier spectrum. To do this we consider two independent orthogonal Gaussian modes of the field propagating perpendicularly to the moving boundary (we assume that motion of the interface is superluminal for both media, $V>c/n_{1,2}$)

Since $\nu$, $\nu _{1}$, $\nu _{2}$ and $\nu _{3}$ are positive the mode functions (11) and (12) have a positive norm. In the medium $2$ ($x>-Vt$) the mode functions (11) and (12) are linear combinations of Gaussian pulses with the opposite sign of the carrier frequencies. Photon operators in the modes (11) and (12) we denote as $\hat {a}$ and $\hat {a}_{1}$, while operators of photons in the modes

Equation (20) guarantees that operators $\hat {a}$, $\hat {a}_{1}$, and $\hat {b}_{2}$, $\hat {b}_{3}$ obey the same bosonic commutation relations.

In the medium 1, the mode functions (11) and (12) have only positive frequency term. Therefore, if state of the field is such that a fixed atom doesn’t become excited in the medium 1 (Minkowski vacuum) then it is a vacuum state for the photons $\hat {a}$ and $\hat {a}_{1}$. Number of photons in the modes $\hat {a}$ and $\hat {a}_{1}$ does not change with time and, thus, no $\hat {a}$ and $\hat {a}_{1}$ photons are present in both media. However, according to Eqs. (11) and (12), in the medium 2 the fixed atom can become excited by emitting photons into the modes $\hat {a}$ and $\hat {a}_{1}$. This is the case because in the medium 2 the modes (11) and (12) have negative frequency components.

If we look at the same state of the field from the perspective of photons $\hat {b}_{2}$ and $\hat {b}_{3}$, then media 1 and 2 are filled with particles. Using Bogoliubov transformations (25) and (26) we obtain for the average number of photons in the modes $\hat {b}_{2}$ and $\hat {b}_{3}$

According to Eqs. (21) and (22), in the medium 2 the mode functions of photons $\hat {b}_{2}$ and $\hat {b}_{3}$ have only positive frequency components, and, thus, a fixed atom can become excited only by absorbing photons $\hat {b}_{2}$ and $\hat {b}_{3}$. If we describe the state of the field by positive-frequency photons in both media (that is by $\hat {a}$ and $\hat {a}_{1}$ in the medium 1 and by $\hat {b}_{2}$ and $\hat {b}_{3}$ in the medium 2), in this description the superluminal interface generates photons out of vacuum with the average photon number per mode given by Eq. (29).

Moreover, Bogoliubov transformations (25) and (26) yield that in terms of photons $\hat {b}_{2}$ and $\hat {b}_{3}$ the state of the field is a two-mode squeezed state, namely,

If modes $\hat {b}_{2}$ and $\hat {b}_{3}$ are considered separately, then tracing over or absorbing one of the modes leaves the remaining mode in a thermal state. Namely, if we trace over the mode $\hat {b}_{3}$, the reduced density operator for the photons $\hat {b}_{2}$ is thermal

Equation (29) is obtained under the assumption that media have no dispersion. If this is the case the number of generated photons in each mode is independent of frequency as well. In reality, however, the medium has dispersion and $n_{1,2}\rightarrow 1$ at high frequencies. In this limit the interface is no longer superluminal and, thus, there is no generation of high frequency photons. This is analogous to Cherenkov radiation for which high-frequency photons are not emitted because particle motion is not superluminal relative to the speed of high-frequency photons in the medium.

One can use Eqs. (27) and (28) to calculate pulse amplification by the superluminal interface. A right-moving initial pulse described by the mode $\hat {a}$ carries energy $W_{0}=\hslash \nu N$. The interface splits the pulse into two pulses into the modes $\hat {b}_{2}$ and $\hat {b}_{3}$ with the net energy $W=\hslash \nu _{2}N_{2}+\hslash \nu _{3}N_{3}$. Plug here Eqs. (27) and (28) with $N_{1}=0$ and, using Eqs. (18 ) and (19), in the limit $N\gg 1$ we obtain

Equation (33) shows that pulse amplification is large if velocity of the interface is close to the speed of light in the medium 2, $V\approx c/n_{2}$. In this case the frequency and the duration of the counter-propagating pulse $A_{3}$ go as

That is pulse undergoes substantial frequency up-conversion and shortening. This property can be used for generation of intense short pulses with high frequency.

Equations (27) and (28) provide an interesting insight on the mechanism of the counter-propagating pulse generation (time reflection). If initially there is only right-moving pulse with the photon number $N$, Eqs. (27) and (28) reduce to

Equations (34) and (35) show that pair production is stimulated by the pulse present in the medium. Namely, spontaneous generation term $C_{3}^{2}$ in Eq. (34) is enhanced by the factor $N+1$. That is generation of the right-moving photon is stimulated by the pulse. However, since photons are generated as entangled pairs, the counter-propagating entangled photon partners yield a pulse moving in the opposite direction. Thus, the counter-propagating pulse appears due to stimulated generation of the entangled photon pairs by the moving boundary. This is different from the mechanism of pulse splitting by a subluminal interface or by spatial inhomogeneities for which no photons are generated.

4. Photon generation and time reflection by a sudden change of the medium refractive index

According to Eq. (29), the number of generated photons in each mode is independent of the velocity of the superluminal interface. This can be understood from the following argument. By making Lorentz transformation one can choose an inertial frame in which the superluminal boundary has the worldline $t=t_{0}$. In this frame, the boundary corresponds to a sudden change of the medium refractive index.

Such Lorentz transformation exists if the boundary moves at a speed greater than speed of light in vacuum $c$. For example, a worldline

In this Section we consider a non-moving dielectric medium for which the refractive index suddenly changes from $n_{1}$ to $n_{2}$ (see Fig. 3). As a result of such change, the Gaussian pulse (8) splits into two Gaussian pulses (9) moving in the opposite directions. The carrier frequencies of these pulses and the pulse widths can be obtained from Eqs. (10) and (65) by taking the limit $V\rightarrow \infty$. This is the case because previous calculations are valid for any speed of the interface, including those faster than light in vacuum. As a verification we performed calculations for the case of a sudden change of the medium refractive index separately. In either way we obtain

 

Fig. 3. A pulse is moving through a medium whose refractive index suddenly changes from $n_{1}$ to $n_{2}$. This change splits the pulse into two pulses propagating in the opposite directions.

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However, in the conventional quantum treatment of the problem we choose the positive-frequency modes (15) and (16) to quantize the field. In this picture, both generated pulses have the same positive frequency $\nu n_{1}/n_{2}$ but different particle content. Namely, the average number of photons in the two pulses is given by Eq. (34) for the co-moving pulse, and Eq. (35) for the counter-propagating pulse.

The last term in Eqs. (34) and (35) describes stimulated generation of particles by the sudden change of the refractive index. As we mentioned in the previous Section, appearance of the counter-propagating pulse (35) is caused by the stimulated generation of the entangled photon pairs. That is no photons in the original pulse are in fact being reflected and the energy of the backward-propagating pulse comes from the particle generation. This is the quantum mechanism of time reflection of light.

It has been shown that time refraction/reflection can be described by a squeezing transformation [24]. The total momentum of the field is conserved during time reflection since the medium is spatially uniform (there is space translation symmetry). However, the energy is not conserved because there is no time translation symmetry. In the limit $N\gg 1$ Eqs. (34) and (35) give that sudden change of the refractive index from $n_{1}$ to $n_{2}$ changes energy of the field by a factor $(1+n_{1}^{2}/n_{2}^{2})/2$. This is different from the ordinary reflection from stationary spatial inhomogeneities of the refractive index for which the field energy is conserved but the momentum does not (there is time but no space translation symmetry in the problem).

If initial state of the field is vacuum the sudden change of the refractive index yields generation of entangled pairs of photons with equal frequencies moving in the opposite directions. The photons in the pair are in the two-mode squeezed state given by Eq. (30). Tracing over the entangled partner leaves the remaining photon in a thermal state (32) with the average number of photons per mode given by Eq. (29).

Particle generation by superluminal boundaries or by a sudden change of the medium properties is a quantum effect. Quantum vacuum emission from a moving refractive-index step in 1+1 dimensions for realistic dispersive media has been developed in [25,26]. Hawking-like radiation can be produced in this setup [26]. Time-varying media have inspired the discussion of new quantum optical phenomena. For example, a sudden change in the medium anisotropy allows us to control the angular distribution of the photons generated via vacuum amplification effects [27]. By matching temporal layers one can suppress generation of backward waves in temporal boundaries (antireflection temporal coatings) [28,29]. Transluminal gratings can also yield spontaneous photon generation [30].

In the next section we will make a connection between generation of entangled photon pairs by a sudden change of the medium properties and vacuum entanglement.

5. Transfer of Minkowski vacuum entanglement to the entanglement of the generated photon pairs by a sudden change of refractive index

Minkowski vacuum is a ground state of free field in Minkowski spacetime. However, particle content of Minkowski vacuum depends on the choice of mode functions to quantize the field. For example, if we choose plane-wave modes as a basis set the ground-state of the free-field Hamiltonian

However, if, e.g., we choose the basis set as a combination of two plane-waves with opposite frequencies and equal momenta

Plug Eq. (38) into Eq. (36) yields the free-field Hamiltonian in terms of operators $\hat {b}_{\mathbf {k}}$ which upto an irrelevant constant reads

The last term in the Hamiltonian (39) shows that photons $\hat {b}_{ \mathbf {k}}$ are created and annihilated out of vacuum in pairs with opposite momenta. Similarly to the interacting Bose-Einstein condensate the ground state of the Hamiltonian (39) is filled with excitations. Namely, in terms of photons $\hat {b}_{\mathbf {k}}$, Minkowski vacuum is a product of two-mode squeezed states (see Appendix E in [31])

However, photons $\hat {b}_{\mathbf {k}}$ are “invisible” for stationary atoms. This is clear because stationary atoms cannot become excited in Minkowski vacuum. One can also show this directly by calculating the excitation probability of a stationary atom interacting with photons $\hat {b}_{\mathbf {k}}$. Since mode functions (37) have both positive and negative frequency components the atom can become excited by absorbing or emitting photons $\hat {b}_{\mathbf {k}}$. Calculations yield that the process in which the atom becomes excited by absorbing (emitting) photon $\hat {b}_{\mathbf {k}}$ destructively interferes with the process in which the atom becomes excited by emitting (absorbing) photon $\hat {b}_{-\mathbf {k}}$, and the corresponding probability amplitudes cancel each other. Entanglement between photons $\hat {b}_{\mathbf {k}}$ and $\hat {b}_{-\mathbf {k}}$, described by Eq. (40), is crucial for such cancellation.

Next we show that sudden change of the medium refractive index transfers entanglement of Minkowski vacuum into entanglement of the generated photon pairs. According to Eq. (21) taken in the limit $V\rightarrow \infty$, under sudden change of the refractive index from $n$ to $n_{2}$ the mode function

Thus, if we choose $n_{2}$ such that $(n_{2}-n)/(n_{2}+n)=\gamma$, that is

6. Optical analog of Hawking radiation

General relativity predicts existence of black holes (BHs). These hypothetical objects possess an event horizon - the boundary under which no particles, at least if they are treated classically and moving forward in time, can escape. In BH spacetime, particles under the event horizon always move away from the horizon toward the BH center. In 1974 S. Hawking showed that entangled particle-antiparticle pairs can be generated near the event horizon of a BH, with one carrying positive energy to infinity and the other carrying negative energy into the BH [32,33]. This is known as Hawking radiation which leads to BH evaporation.

Hawking predicted that BH radiation is thermal. However, if nonunitary absorption of the negative energy photons near the BH center is taken into account, this alters the state of the outgoing entangled photon partner. As a result, radiation of the evaporating BH is not thermal; it carries information about BH interior, and entropy is preserved during evaporation [34].

In this Section we discuss an optical analog of Hawking radiation in which a superluminal boundary between two dielectric media plays the role of the event horizon. In Sections 2 and 3 we assumed that the boundary moves faster than the speed of light in both media 1 and 2, and showed that the moving boundary generates entangled photon pairs out of vacuum in the medium 2 (see Fig. 2). To make connection with the BH event horizon the speed of the boundary must be subluminal for the medium 1 and superluminal for the medium 2 ($c/n_{1}>V>c/n_{2}$). In this case, medium 2 is analogous to the BH interior region - light from the medium 2 can not escape into the medium 1 (exterior region). Light always move away from the moving boundary in the medium 2. In contrast, in the medium 1, light can move both toward and away from the boundary which is analogous to the BH exterior region.

Next we show that under these conditions the moving boundary generates entangled photon pairs, with one photon propagating into the medium 1 and the other photon into the medium 2. In addition, the boundary generates pairs of photons propagating in the medium 2 in the opposite directions.

In the present case, the total field consists of the incident and reflected waves in the medium 1, and two refracted waves in the medium 2. Thus, the boundary-value problem is degenerate and can be solved uniquely only if we make additional assumptions about the structure of the interface, nonlinearities, etc. [35]. Keeping in mind analogy with BH radiation, we assume that the interface structure yields no reflected wave, as shown in Fig. 4(a). The corresponding positive-norm mode function is given by

 

Fig. 4. Interface between two static media moving with a constant velocity that is subluminal (superluminal) for the medium 1 (2) mimics a BH event horizon. Orthogonal mode functions of the field obeying the proper boundary conditions at the interface with one ingoing and two transmitted waves (Eq. (refk1a)), and three outgoing waves (Eq. (refk1d)) are sketched in (a) and (b) respectively.

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We assume that state of the field is vacuum for photons $\hat {a}$, that is there are no photons falling on the interface. Then Eq. (43) gives

A mode function orthogonal to (42) reads

It is sketched in Fig, 4(b). The mode function (47) consists of three outgoing waves and the left-moving pulse $f_{4}$ propagating at $x\rightarrow \infty$ which is not shown in the figure. Photon operator of the mode $E_{1}(t,x)$ we denote as $\hat {a}_{1}$. It can be expressed in terms of $\hat {b}_{2}$ and $\hat {b}_{3}$ as

There is no reason to believe that state of the field is vacuum for photons $\hat {a}_{1}$. The latter assumption would yield that there are photons in the outgoing mode $f_{1}$ even if there is no interface ($n_{1}=n_{2}$).

To find the average number of photons $N_{1,2,3}$ in the outgoing modes we will use physical arguments. Consider outgoing plane waves moving in the media 1 and 2 that have positive frequencies given by Eqs. (17)–(19). In the reference frame of the moving interface all three outgoing waves have the same absolute value of frequency, however, $\nu _{3}<0$, while $\nu _{1,2}>0$. Thus, the interface degrees of freedom can become excited by emitting a photon into the mode $\nu _{3}$ and then spontaneously decay back to the ground state by emitting the positive-frequency photon $\nu _{1}$ or $\nu _{2}$. This process generates photons in pairs, namely, a pair of $\nu _{3}$ and $\nu _{1}$ photons, and $\nu _{3}$ and $\nu _{2}$ photons. For a small change of the dielectric constant across the interface ($C_{3}^{2}\ll 1$) the probabilities of emission of the $\nu _{1}$ and $\nu _{2}$ photons are approximately the same which gives

Energy conservation in the moving frame yields an additional equation

Combining Eqs. (48) and (49) with Eq. (46), we obtain in the limit $C_{3}^{2}\ll 1$

Equation (50) yields $N_{1}=(N_{2}+N_{3})/3$, that is number of generated photons propagating into the exterior region (medium 1) is three times smaller than that generated in the interior region (medium 2). This is the case because there is an additional channel of generation the particle pairs in the medium 2.

7. Discussion

In 1970, G. Moore predicted that an accelerated mirror can excite the quantum vacuum, generating photons [16]. This effect is nowadays known as the dynamical Casimir effect and was investigated, during the 1970s, in other pioneering articles by DeWitt [36], Fulling and Davies [37,38], Candelas and Deutsch [39], among others. In the dynamical Casimir effect the particle creation from the vacuum occurs when a quantized field is submitted to time-dependent boundary conditions, with moving mirrors being a particular case. A fundamental property of the effect is that particles are created out of vacuum as entangled pairs.

The mechanism of the dynamical Casimir effect is somewhat similar to that of Cherenkov radiation of a neutral atom if both are viewed from the negative frequency perspective [6]. As we discussed in Introduction, if an atom is moving through a medium faster than light in the medium, in the reference frame of the atom the frequencies of the plane-wave photons propagating inside the Cherenkov cone are negative. Thus, the moving atom can become excited by emitting the negative-frequency photon inside the Cherenkov cone followed by spontaneous decay back to the ground state by emitting a positive-frequency photon outside the Cherenkov cone. As a result, the superluminal motion of the atom through a medium is accompanied by generation of entangled photon pairs [8]. In the lab frame, frequencies of both generated photons are positive.

One should mention that Cherenkov radiation of a neutral atom is a quantum effect. It is different from the classical Vavilov-Cherenkov radiation of an electric charge moving at a superluminal speed in a medium in the same way as spontaneous emission of an atom differs from radiation of a classical electric dipole.

Now let us assume that a mirror is uniformly accelerated through Minkowski vacuum in 1+1 dimension along the trajectory

In Eq. (51), $\tau$ is the proper time of the mirror and $a>0$ is the proper acceleration. The left-moving positive-norm Unruh-Minkowski modes of a scalar field are defined as [40]

In Rindler space, the Unruh-Minkowski mode functions are plane waves

Equations (52) show that in the Rindler space the mode function $F_{1\Omega }$ has a negative frequency $\nu =-a\Omega /c$, while $F_{2\Omega }$ has a positive frequency $a\Omega /c$.

Since the mirror is composed of particles, particles at the mirror’s surface can become excited by emitting Unruh-Minkowski photon with the negative frequency into the mode $F_{1\Omega }$ and then decay back to the ground state by emitting a photon into the positive-frequency mode $F_{2\Omega }$. Thus, mirror acceleration yields generation of entangled pairs of Unruh-Minkowski photons, which is the dynamical Casimir effect.

One can also explain the effect by making a connection with Glauber’s coupled oscillators one of which is inverted, that is it has negative frequency (a negative potential and kinetic energy). Such a system works as an amplifier which leads to excitation of the oscillators out of the vacuum fluctuations [42]. Indeed, in the rotating wave approximation, two coupled harmonic oscillators with frequencies $\omega$ and $-\omega$ are described by the Hamiltonian

In this paper we consider two static media separated by a boundary moving with a constant velocity $\mathbf {V}$ (Secs. 2 and 3). If the boundary moves from the medium 2 to the medium 1 with a speed exceeding the speed of light in both media this yields generation of entangled pairs of photons out of vacuum in the medium 2 in a two-mode squeezed state. Such effect is quantum and is analogous to the creation of entangled particle pairs in the dynamical Casimir effects. Indeed, in the reference frame of the moving boundary the frequency of plane-wave photons propagating inside the Cherenkov cone in the medium 2 is negative, while photons propagating outside the Cherenkov cone have positive frequency. The moving boundary couples these positive and negative frequency modes which leads to their excitation as in the case of Glauber oscillators.

The limit $V\rightarrow \infty$ corresponds to a sudden change of the medium properties with time. In the absence of the medium dispersion, the number of generated photons in each mode is independent of the frequency and the boundary velocity. That is in the limit $V\rightarrow \infty$ the number of generated photons is given by the same formula (29) as in the case of finite $V$ (see Sec. 4.). In Section 5 we show that during the process of particle pair production by a sudden change of the refractive index, entanglement of vacuum is transferred to the entanglement of the generated photon pairs.

If an electromagnetic pulse crosses the superluminal boundary or there is an abrupt change of the refractive index with time the pulse stimulates photon generation into the pulse mode. Since photons are created as entangled pairs the counter-propagating photon partners produce a pulse moving in the opposite direction. Thus, pulse splitting by the superluminal boundary or by a sudden change of the refractive index (time reflection) occurs due to stimulated generation of the entangled photon pairs. That is during time reflection no photons in the original pulse are actually being reflected and the energy of the backward-propagating pulse comes from the particle generation. This is different from the mechanism of pulse splitting by a subluminal interface or spatial inhomogeneities for which no photons are generated.

The moving boundary mimics an event horizon of a black hole if it’s speed is subluminal for the medium 1 (analog of BH exterior) and superluminal for the medium 2 (BH interior). In this case, the boundary couples three outgoing waves $\nu _{1,2,3}$. One of them (mode $\nu _{3}$) propagates inside the Cherenkov cone in the medium 2 and has negative frequency in the reference frame of the moving boundary. This is analogous to three coupled oscillators one of which is inverted, and yields generation of photons out of vacuum in these three modes. Photons are produced in pairs, namely, a pair of $\nu _{3}$ and $\nu _{1}$ photons, and $\nu _{3}$ and $\nu _{2}$ photons. One of the generated photons ($\nu _{1}$) propagates into the medium 1 and photons $\nu _{2}$ and $\nu _{3}$ propagate into the medium 2. Thus, the optical event horizon created by the moving boundary generates photons into the exterior region (medium 1) similarly to Hawking radiation of a BH (see Sec. 6.). The radiation mechanism is analogous to that of the dynamical Casimir effect which, in turn, can be explained by means of the Glauber oscillators [43].

One should mention that analogs of gravitational effects in various systems, including quantum emission of particles, have been previously discussed in the literature in different settings, see, e.g., [44–56] and references therein. Stimulated Hawking-like radiation was measured using light pulses in nonlinear fiber optics to establish artificial event horizon [57]. Our paper is an attempt to provide an additional insight on Hawking radiation and make a connection with the Cherenkov and the dynamical Casimir effects.

It is worth mentioning that the latter non necessarily requires an accelerated mirror. For instance, particles can be also generated by a motionless mirror in vacuum whose internal properties rapidly vary in time [58]. In this context, Wilson et al. [59] observed experimentally the particle creation from vacuum in a transmission line with a superconducting quantum interference device (SQUID) at one end. The SQUID forms a tunable inductance, and by changing the magnetic flux threading it, it can be tuned from almost a short to a highly inductive state. A time-dependent magnetic flux applied to the SQUID changes the SQUID inductance, yielding a time-dependent boundary condition. Schneider et al. demonstrated broadband entanglement of microwave photon pairs generated by the dynamical Casimir effect in a superconducting circuit which consists of a semi-infinite transmission line, terminated by two parallel SQUIDs [60].

Physics behind generation of photon pairs in these experiments is the sum combination resonance which occurs when system’s parameters are modulated with frequency $\nu _{d}$ equal to the sum of two normal mode frequencies $\nu _{1}$ and $\nu _{2}$

In both experiments mentioned above, the boundary condition was modulated harmonically with a pump frequency $\nu _{d}$ in the GHz range. This creates a pair of microwave photons, whose frequencies add up to $\nu _{d}$. Other experiments have also been done in metamaterials [61] and fibers [62].

Optical parametric amplification (OPA) [63–66] which transfers energy from the driving field to the signal and idler waves is another example of the sum combination resonance caused by the nonlinear light-matter interaction. OPA led to development of tunable radiation sources throughout the infrared, visible, and ultraviolet spectral regions [67]. Such sources yield amplification of coherent radiation at lower-frequencies $\nu _{1}$ and $\nu _{2}$ by pumping nonlinear optical crystals with intense laser light at the sum frequency $\nu _{d}$.

One should mention that Glauber’s oscillators, described by the Hamiltonian (53), are a special case of the sum combination resonance for which $\nu _{1}=-\nu _{2}=\omega$. In this case the modulation frequency $\nu _{d}=0$.

Particles can also be created in entangled pairs with zero net momentum from the vacuum by the curved space-time of an expanding universe [68,69]. The process of cosmological particle creation is accompanied by the generation of quantum entanglement at large distances and played an important role in the early Universe. Together with other effects, it is considered to be responsible for the reheating of our Universe after the inflationary period. Furthermore, a very similar process based on the tearing apart of quantum vacuum fluctuations by an expanding space-time during cosmic inflation explains the creation of the seeds for structure formation.

The expansion of the universe stretches all length scales, including the wavelengths of the particle modes. Thus, modes evolve with time leading to appearance of the negative frequency components. This implies that the modes at early and late times are related by a Bogoliubov transformation [70]. If this stretching process becomes too fast, the cosmic expansion tears apart the initial quantum vacuum fluctuations and turns them into pairs of particles with opposite momenta. This yields spontaneous production of cosmological particles, including the primordial density fluctuations which led to the acoustic peaks in the cosmic microwave background radiation spectrum [71–73].

A. Interaction of light with superluminal interface between two media

Here we consider a plane interface between two media moving with a constant velocity $\mathbf {V}$, while both media are at rest (see Fig. 5). In the absence of surface charges and currents, the boundary conditions for the electromagnetic field at the interface read

(54)$$\hat{n}\times (\mathbf{E}_{2}-\mathbf{E}_{1})=\frac{V}{c}(\mathbf{B}_{2}- \mathbf{B}_{1}),$$

(55)$$\hat{n}\times (\mathbf{H}_{2}-\mathbf{H}_{1})={-}\frac{V}{c}(\mathbf{D}_{2}- \mathbf{D}_{1}),$$

where $V$ is the projection of the velocity of the interface along the normal $\hat {n}$ to it, and the index $1$ ($2$) refers to the field at the left (right) side of the interface. Speeds of light in the two media are given by

(56)$$c_{1}=\frac{c}{\sqrt{\varepsilon _{1}\mu _{1}}},\quad c_{2}=\frac{c}{\sqrt{ \varepsilon _{2}\mu _{2}}}$$

respectively. In the absence of currents, the Maxwell’s equations and the constitutive relations for linear materials read

$$\nabla \times \mathbf{E}={-}\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla \times \mathbf{H}=\frac{1}{c}\frac{\partial \mathbf{D}}{ \partial t},$$

(57)$$\mathbf{D}=\varepsilon \mathbf{E},\quad \mathbf{B}=\mu \mathbf{H}.$$

We will assume that the media are not magnetic, that is $\mu _{1}=\mu _{2}=1$. Then, using Eqs. (54), (55) and (57), we obtain

$$\hat{n}\times (\hat{n}\times (\mathbf{E}_{2}-\mathbf{E}_{1}))={-}\frac{V^{2}}{ c^{2}}(\mathbf{D}_{2}-\mathbf{D}_{1}),$$

or

(58)$$\hat{n}\left( \hat{n}\cdot (\mathbf{E}_{2}-\mathbf{E}_{1})\right) -(\mathbf{E }_{2}-\mathbf{E}_{1})={-}\frac{V^{2}}{c^{2}}(\mathbf{D}_{2}-\mathbf{D}_{1}).$$

 

Fig. 5. Ingoing wave $A_{1}$ propagates through a superluminal interface between two dielectric media and splits into two transmitted waves $A_{2}$ and $A_{3}$ without reflection.

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If a plane electromagnetic wave propagates through the interface at the normal incidence, the boundary condition (58) reduces to

$$\mathbf{E}_{2}-\mathbf{E}_{1}=\frac{V^{2}}{c^{2}}(\mathbf{D}_{2}-\mathbf{D} _{1}).$$

Using the constitutive relations (57), we find

(59)$$\left( 1-\frac{V^{2}}{c^{2}}\varepsilon _{1}\right) \mathbf{E}_{1}=\left( 1- \frac{V^{2}}{c^{2}}\varepsilon _{2}\right) \mathbf{E}_{2}.$$

Assuming that the $x-$axis is normal to the surface, $\mathbf {E}$ is polarized along the $z-$axis, and $\mathbf {B}$ is along the $y-$axis (see Fig. 5), Eq. (54) yields the second boundary condition in the form

(60)$$E_{2}-E_{1}+\frac{V}{c}\left( B_{2}-B_{1}\right) =0.$$

The wave equation for the field in a dielectric medium

(61)$$\left( \frac{\partial ^{2}}{\partial t^{2}}-\frac{c^{2}}{\varepsilon }\frac{ \partial ^{2}}{\partial x^{2}}\right) \mathbf{E}=0$$

has the following plane wave solutions

$$\mathbf{E}=\hat{z}e^{i\nu \left( t\pm \frac{\sqrt{\varepsilon }}{c}x\right) }$$

describing right and left moving waves. Maxwell’s equations yield for the magnetic field

$$\mathbf{B}={\mp} \sqrt{\varepsilon }\hat{y}e^{i\nu \left( t\pm \frac{\sqrt{ \varepsilon }}{c}x\right) }.$$

We will assume that a right or left-moving plane wave falls into the interface, that is

$$E_{1}=A_{1}e^{i\nu \left( t\pm \sqrt{\varepsilon _{1}}\frac{x}{c}\right) },\quad B_{1}={\mp} \sqrt{\varepsilon _{1}}E_{1},$$

there is no reflected wave, and there are two waves $A_{2}$ and $A_{3}$ moving in the other side of the interface (see Fig. 5)

$$E_{2}=A_{2}e^{i\nu _{2}\left( t-\frac{\sqrt{\varepsilon _{2}}}{c}x\right) }+A_{3}e^{i\nu _{3}\left( t+\frac{\sqrt{\varepsilon _{2}}}{c}x\right) }, \qquad B_{2}=\sqrt{\varepsilon _{2}}A_{2}e^{i\nu _{2}\left( t-\frac{\sqrt{ \varepsilon _{2}}}{c}x\right) }-\sqrt{\varepsilon _{2}}A_{3}e^{i\nu _{3}\left( t+\frac{\sqrt{\varepsilon _{2}}}{c}x\right) }.$$

The wave $A_{2}$ is right-moving and $A_{3}$ is left-moving. If the interface is moving along the trajectory $x=-Vt$, the boundary conditions (59) and (60) yield the following equations

$$\left( 1-\varepsilon _{1}\frac{V^{2}}{c^{2}}\right) A_{1}e^{i\nu t\left( 1\mp \sqrt{\varepsilon _{1}}\frac{V}{c}\right) }= \left( 1-\varepsilon _{2} \frac{V^{2}}{c^{2}}\right) \left( A_{2}e^{i\nu _{2}t\left( 1+\sqrt{ \varepsilon _{2}}\frac{V}{c}\right) }+A_{3}e^{i\nu _{3}t\left( 1-\sqrt{ \varepsilon _{2}}\frac{V}{c}\right) }\right) ,$$

$$\left( 1\mp \sqrt{\varepsilon _{1}}\frac{V}{c}\right) A_{1}e^{i\nu t\left( 1\mp \sqrt{\varepsilon _{1}}\frac{V}{c}\right) }= \left( 1+\sqrt{\varepsilon _{2}}\frac{V}{c}\right) A_{2}e^{i\nu _{2}t\left( 1+\sqrt{\varepsilon _{2}} \frac{V}{c}\right) } +\left( 1-\sqrt{\varepsilon _{2}}\frac{V}{c}\right) A_{3}e^{i\nu _{3}t\left( 1-\sqrt{\varepsilon _{2}}\frac{V}{c}\right) }.$$

Solution of these equations gives the following result for the wave frequencies and amplitudes

(62)$$\nu _{2}=\frac{1\mp n_{1}\frac{V}{c}}{1+n_{2}\frac{V}{c}}\nu ,\quad \nu _{3}= \frac{1\mp n_{1}\frac{V}{c}}{1-n_{2}\frac{V}{c}}\nu ,$$

(63)$$A_{2}=\frac{1}{2}\left( 1\mp \frac{n_{1}}{n_{2}}\right) \frac{\nu _{2}}{\nu } A_{1},$$

(64)$$A_{3}=\frac{1}{2}\left( 1\pm \frac{n_{1}}{n_{2}}\right) \frac{\nu _{3}}{\nu } A_{1},$$

where $n_{1,2}=\sqrt {\varepsilon _{1,2}}$ are the media refractive indices.

Equations (62), (63) and (64) show that the boundary conditions for the electromagnetic field at the moving interface can be satisfied assuming no reflected wave and two waves propagating at the other side of the interface in the opposite directions. This is the case for any speed of the interface (superluminal or subluminal). For the subluminal motion, the wave $A_{3}$ is interpreted as ingoing, while for the superluminal interface, $A_{3}$ is interpreted as transmitted.

Next we assume that the ingoing field is a Gaussian pulse with a carrier frequency $\nu$, that is

$$E_{1}=A_{1}e^{{-}i\nu \left( t\pm \frac{n_{1}}{c}x\right) }e^{-\Omega ^{2}\left( t\pm \frac{n_{1}}{c}x\right) ^{2}},\quad B_{1}={\mp} n_{1}E_{1},$$

and there are two transmitted pulses

$$E_{2}=A_{2}e^{{-}i\nu _{2}\left( t-\frac{n_{2}}{c}x\right) }e^{-\Omega _{2}^{2}\left( t-\frac{n_{2}}{c}x\right) ^{2}} +A_{3}e^{{-}i\nu _{3}\left( t+ \frac{n_{2}}{c}x\right) }e^{-\Omega _{3}^{2}\left( t+\frac{n_{2}}{c}x\right) ^{2}},$$

$$B_{2}=\sqrt{\varepsilon _{2}}A_{2}e^{{-}i\nu _{2}\left( t-\frac{n_{2}}{c} x\right) }e^{-\Omega _{2}^{2}\left( t-\frac{n_{2}}{c}x\right) ^{2}} -\sqrt{ \varepsilon _{2}}A_{3}e^{{-}i\nu _{3}\left( t+\frac{n_{2}}{c}x\right) }e^{-\Omega _{3}^{2}\left( t+\frac{n_{2}}{c}x\right) ^{2}}.$$

The boundary conditions (59) and (60) yield the same carrier frequencies (62) and the amplitudes (63), (64) for the transmitted Gaussian pulses as for the plane waves, and the following relations between the pulse widths

(65)$$\Omega _{2}=\frac{1\mp n_{1}\frac{V}{c}}{1+n_{2}\frac{V}{c}}\Omega ,\quad \Omega _{3}=\frac{1\mp n_{1}\frac{V}{c}}{1-n_{2}\frac{V}{c}}\Omega .$$

Reflection and transmission of a light pulse when passing through superluminal inhomogeneities of the refractive index have been investigated theoretically using classical Maxwell’s equations in [35,74–77].

Funding

U.S. Department of Energy (DE-SC-0023103, FWP-ERW7011, DE-SC0024882); Welch Foundation (A-1261); National Science Foundation (PHY-2013771); Air Force Office of Scientific Research (FA9550-20-1-0366).

Acknowledgments

I am very grateful to Marlan Scully and Bill Unruh for stimulating discussions.

Disclosures

The author declares no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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