Quantum Theory of Wave Scattering from Electromagnetic Time Interfaces

We start by writing the electric field operator. Unlike in the previous classical treatment, we now extend our analysis to the case where two modes (forward and backward) exist simultaneously before ($t<0$) and after ($t>0$) the temporal jump at $t=0$. The propagation directions of these forward and backward modes are characterized by the arbitrary wavevectors ${\bf k}$ and $-{\bf k}$, respectively. In addition, we consider an arbitrary polarization direction, specified by the unit vector ${\bf e}$, which is orthogonal to the propagation axis. Consequently, the electric field operator $\hat{{\bf E}}({\bf r},t)$ at position ${\bf r}$ is expressed as [34]

$$\begin{split}&\hat{{\bf E}}({\bf r},t<0)=i\Big{(}\frac{\hbar\omega_{1}}{2\epsilon_{1}V}\Big{)}^{1/2}\sum_{{\bf k}^{\prime}}\big{[}\hat{a}_{{\bf k}^{\prime}}e^{i({\bf k}^{\prime}\cdot{\bf r}-\omega_{1}t)}-\hat{a}_{{\bf k}^{\prime}}^{\dagger}e^{-i({\bf k}^{\prime}\cdot{\bf r}-\omega_{1}t)}\big{]}{\bf e},\cr&\hat{{\bf E}}({\bf r},t>0)=i\Big{(}\frac{\hbar\omega_{2}}{2\epsilon_{2}V}\Big{)}^{1/2}\sum_{{\bf k}^{\prime}}\big{[}\hat{b}_{{\bf k}^{\prime}}e^{i({\bf k}^{\prime}\cdot{\bf r}-\omega_{2}t)}-\hat{b}_{{\bf k}^{\prime}}^{\dagger}e^{-i({\bf k}^{\prime}\cdot{\bf r}-\omega_{2}t)}\big{]}{\bf e}.\end{split}$$

(7)

Here, $\hbar$ is the reduced Planck constant, $\omega_{1}$ and $\epsilon_{1}$ ($\omega_{2}$ and $\epsilon_{2}$) are the angular frequency and permittivity for $t<0$ ($t>0$), $V$ is the quantization volume, and ${\bf k}^{\prime}\in\{{\bf k},-{\bf k}\}$. Moreover, and in particular, $(\hat{a},\,\hat{a}^{\dagger})$ and $(\hat{b},\,\hat{b}^{\dagger})$ denote the bosonic annihilation and creation operators before and after $t=0$, respectively. These operators are connected to each other via the electromagnetic boundary conditions which state that the flux density operators must be continuous at $t=0$. In other words, quite similarly to the classical picture [Eq. (4)], we have

$$\begin{split}&\hat{{\bf D}}({\bf r},t=0^{-})=\hat{{\bf D}}({\bf r},t=0^{+}),\cr&\hat{{\bf B}}({\bf r},t=0^{-})=\hat{{\bf B}}({\bf r},t=0^{+}).\end{split}$$

(8)

The electric flux density is readily calculated because $\hat{{\bf D}}=\epsilon\hat{{\bf E}}$ (assuming temporal locality [35] as before), whereas the magnetic flux density is obtained from Faraday’s law $\nabla\times\hat{{\bf E}}=-{\partial\hat{{\bf B}}/\partial t}$. Eventually, by imposing the boundary conditions in Eq. (8) and doing some algebraic manipulations, we conclude that

$$\begin{pmatrix}\hat{b}_{{\bf k}}\\ \hat{b}_{-{\bf k}}^{\dagger}\end{pmatrix}=\begin{pmatrix}\tau&\Gamma\\ \Gamma&\tau\end{pmatrix}\!\begin{pmatrix}\hat{a}_{{\bf k}}\\ \hat{a}_{-{\bf k}}^{\dagger}\end{pmatrix},$$

(9)

where we introduced the two quantities

$$\tau=\frac{1}{2}\Big{(}\sqrt{\frac{n_{2}}{n_{1}}}+\sqrt{\frac{n_{1}}{n_{2}}}\Big{)},\quad\Gamma=\frac{1}{2}\Big{(}\sqrt{\frac{n_{2}}{n_{1}}}-\sqrt{\frac{n_{1}}{n_{2}}}\Big{)}.$$

(10)

Equations (9) and (10) form a central result of our work. They show that the temporal discontinuity is a transformation by which the annihilation operator $\hat{b}_{{\bf k}}$ (creation operator $\hat{b}^{\dagger}_{-{\bf k}}$) of the output forward (backward) mode becomes a mixture of the annihilation (creation) and creation (annihilation) operators of the input forward (backward) and backward (forward) modes, respectively.

To obtain more insight into these relations, let us have a closer look at Eq. (10). First, observing that $\tau+\Gamma=\sqrt{n_{2}/n_{1}}$ and $\tau-\Gamma=\sqrt{n_{1}/n_{2}}$, we conclude that $\tau+\Gamma$ is the inverse of $\tau-\Gamma$. Second, and more importantly,

$$\tau^{2}-\Gamma^{2}=1,$$

(11)

which allows us to parametrize $\tau$ and $\Gamma$ via hyperbolic functions, i.e., $\tau=\cosh r$ and $\Gamma=\sinh r$. Owing to such a representation, we rewrite Eq. (9) as

$$\begin{pmatrix}\hat{b}_{{\bf k}}\\ \hat{b}_{-{\bf k}}^{\dagger}\end{pmatrix}=\begin{pmatrix}\cosh r&\sinh r\\ \sinh r&\cosh r\end{pmatrix}\!\begin{pmatrix}\hat{a}_{{\bf k}}\\ \hat{a}_{-{\bf k}}^{\dagger}\end{pmatrix}.$$

(12)

Very interestingly, from this form, we discover that the time interface acts as a particular Bogoliubov transformation between the mode operators, as in the context of optical parametric amplification and the Unruh effect [36]. Furthermore, it is clear that the transformation matrix in Eq. (12) is real and symmetric, having a determinant equal to one. However, we see that the matrix is not orthogonal, and hence not unitary, which is a reflection of energy nonconservation owing to the external work required to change the refractive index.

The Bogoliubov transformation in Eq. (12) advocates that the unitary evolution of the modes is governed by the two-mode squeeze operator [37, 38]

$$\hat{S}(r)=e^{r(\hat{a}_{\bf k}\hat{a}_{-{\bf k}}-\hat{a}^{\dagger}_{\bf k}\hat{a}^{\dagger}_{-{\bf k}})}.$$

(13)

Indeed, by employing the Baker-Campbell lemma [39], commutators of the $\hat{a}$ operators, and Taylor series of hyperbolic functions, we verify from Eqs. (12) and (13) that

$$\hat{S}(r)\begin{pmatrix}\hat{a}_{{\bf k}}\\ \hat{a}_{-{\bf k}}^{\dagger}\end{pmatrix}\hat{S}^{\dagger}(r)=\begin{pmatrix}\hat{b}_{{\bf k}}\\ \hat{b}_{-{\bf k}}^{\dagger}\end{pmatrix}.$$

(14)

Furthermore, from Eqs. (9), (10), and (12), we find that the squeezing parameter $r$ is purely real and determined by the refractive indices according to

$$r=\operatorname{arctanh}{\Big{(}\frac{\Gamma}{\tau}\Big{)}}=\operatorname{arctanh}{\Big{(}\frac{n_{2}-n_{1}}{n_{2}+n_{1}}\Big{)}}.$$

(15)

In other words, $\hat{b}$ is the unitary transformation of $\hat{a}$ and, accordingly, $\hat{S}(r)$ corresponds to the unitary evolution operator $\hat{U}(t)=\exp(-{i\over\hbar}\hat{H}t)$, where $\hat{H}=i\hbar g(\hat{a}_{\bf k}\hat{a}_{-{\bf k}}-\hat{a}^{\dagger}_{\bf k}\hat{a}^{\dagger}_{-{\bf k}})$ is the Hamiltonian and $gt=r$.

Having derived the mode transformation matrix and the associated unitary evolution operator for the time interface, we now turn to investigate how the initial quantum state and the underlying photon properties are altered due to the temporal jump. To this end, we examine in detail the scenario where the initial state $(t<0)$ is a Fock state $\ket{n,0}$. This means that the forward incident mode contains $n$ photons and the backward incident mode is in the vacuum state. The output state $\ket{\psi}$ $(t>0)$ is found by applying the two-mode squeeze operator $\hat{S}(r)$ in Eq. (13) to the initial state, viz., $\ket{\psi}=\hat{S}(r)\ket{n,0}$. For this purpose, because the $\hat{S}(r)$ operator is in the form of an exponential, it is convenient to employ the disentanglement theorem [40] and rewrite it as

$$\hat{S}(r)=e^{\tanh(r)\hat{K}_{+}}e^{\ln[1-\tanh^{2}(r)]\hat{K}_{0}}e^{-\tanh(r)\hat{K}_{-}},$$

(16)

where $\hat{K}_{-}=\hat{a}_{\bf k}\hat{a}_{-{\bf k}}$, $\hat{K}_{+}=\hat{a}^{\dagger}_{\bf k}\hat{a}^{\dagger}_{-{\bf k}}$, and $\hat{K}_{0}=(1/2)(\hat{a}^{\dagger}_{\bf k}\hat{a}_{\bf k}+\hat{a}^{\dagger}_{-{\bf k}}\hat{a}_{-{\bf k}}+1)$. These Hermitian operators obey the commutation relations $[\hat{K}_{0},\,\hat{K}_{\pm}]=\pm\hat{K}_{\pm}$ and $[\hat{K}_{-},\,\hat{K}_{+}]=2\hat{K}_{0}$, so they define an SU(1,1) algebra [41]. We stress that the decomposition in Eq. (16) is a general property of the two-mode squeeze operator, i.e., it is independent of our chosen initial state $\ket{n,0}$.

The expression for $\hat{S}(r)$ in Eq. (16) involves three terms. The last term preserves the initial state, because it contains only annihilation operators and the backward incident mode is in the vacuum state. Regarding the middle term, we obtain that

$$e^{\ln[1-\tanh^{2}(r)]\hat{K}_{0}}\ket{n,0}=[1-\tanh^{2}(r)]^{n+1\over 2}\ket{n,0}.$$

(17)

In other words, $[1-\tanh^{2}(r)]^{n+1\over 2}$ is the eigenvalue of the operator in the middle with the eigenstate $\ket{n,0}$. In consequence, we observe that the last and middle terms do not change the state and $\ket{n,0}$ is conserved. However, the situation is drastically different for the first term because it includes the creation operators. Indeed, if we apply the standard operator identities $(\hat{a}^{\dagger})^{m}\ket{0}=\sqrt{m!}\ket{m}$ and $\hat{a}^{\dagger}\ket{m}=\sqrt{m+1}\ket{m+1}$ [42], we infer the output state to be [41]

$$\ket{\psi}=\big{[}1-\tanh^{2}(r)\big{]}^{1+n\over 2}\sum_{m=0}^{\infty}\sqrt{{(n+m)!\over n!m!}}\tanh^{m}(r)\ket{n+m,m}.$$

(18)

This expression explicitly demonstrates that the first term in Eq. (16) transforms the initial state into an entangled superposition state of higher photon numbers specified by $m$. In terms of the material parameters $\Gamma=\sinh r$ and $\tau=\cosh r$, we find that Eq. (18) takes the form

$$\ket{\psi}=\Big{(}\frac{1}{\tau}\Big{)}^{1+n}\sum_{m=0}^{\infty}\sqrt{{(n+m)!\over n!m!}}\Big{(}\frac{\Gamma}{\tau}\Big{)}^{m}\ket{n+m,m}.$$

(19)

Thus, the time interface gives rise to photon-pair generation, and the initial state $\ket{n,0}$ is only a part of the distribution of the output state $\ket{\psi}$ that contributes with the probability of $[1-\tanh^{2}(r)]^{1+n}=(1/\tau^{2})^{1+n}$. Photon-pair generation, characterized by the integer $m$ in Eqs. (18) and (19), serves as a quantum optical analogue to the time reflection phenomenon observed in classical wave dynamics. At a more fundamental level, this process is in fact rooted in the principle of wavenumber conservation (or linear momentum conservation), which governs the creation of a negative frequency component, as detailed in Section 2 [see, Eq. (3)]. Remember that in the classical picture, the concept of negative frequency corresponds to time reflection.

When not only the backward incident mode but also the forward incident mode is in the vacuum state ($n=0$), Eq. (18) shows that the output state becomes the two-mode squeezed vacuum [37, 38]:

$$\ket{\psi}=\frac{1}{\cosh r}\sum_{m=0}^{\infty}\tanh^{m}(r)\ket{m,m}=\frac{1}{\tau}\sum_{m=0}^{\infty}\Big{(}\frac{\Gamma}{\tau}\Big{)}^{m}\ket{m,m}.$$

(20)

In this case merely the paired states $\ket{m,m}$ appear in the superposition, which makes Eq. (20) the strongest entangled state of two light modes with given total energy [43, 44]. After the interface, our system consists of two distinguishable but correlated subsystems: The output forward and backward propagating modes that are distinguished by opposite propagation directions. Due to the superposition in Eq. (20), we do not precisely know the number of photons in each mode. However, upon measuring the photon count in one mode, we instantaneously ascertain the number of photons in the other, highlighting the quantum entanglement between these two modes.

According to Eqs. (18) and (19), the probability-distribution function of the output state is

$$\begin{split}P(n,m)&=\big{[}1-\tanh^{2}(r)\big{]}^{1+n}\tanh^{2m}(r){(n+m)!\over n!m!}\cr&=\Big{(}\frac{1}{\tau^{2}}\Big{)}^{n+1}\Big{(}\frac{\Gamma}{\tau}\Big{)}^{2m}\frac{(n+m)!}{(n!m!)}.\end{split}$$

(21)

From the theory of series, we know that $1/(1-x)^{n+1}=\sum_{m}x^{m}(n+m)!/(n!m!)$ for $|x|<1$. Replacing $x$ by $(\Gamma/\tau)^{2}$ we confirm that $\sum_{m}{P(n,m)}=1$, corroborating that $P(n,m)$ is a proper probability-distribution function. Let us inspect the scenario in which the initial state is a vacuum state $(n=0)$ and, as a result, the output state is in the two-mode squeezed vacuum as given by Eq. (20). Accordingly, the expression in Eqs. (21) is reduced to

$$P(m)={\sinh^{2m}(r)\over\cosh^{2m+2}(r)}={\Gamma^{2m}\over\tau^{2m+2}}=4\beta^{2}{(\beta^{2}-1)^{2m}\over(\beta^{2}+1)^{2m+2}},$$

(22)

where $\beta=\sqrt{n_{2}/n_{1}}$. We observe from Eq. (22) that the probability of remaining in the vacuum state $m=0$ after the temporal jump is $P(0)=4\beta^{2}/(\beta^{2}+1)^{2}$. In particular, when $\beta\rightarrow 0$ (which corresponds to changing to a zero-index material or low-index material compared to the initial medium) or when $\beta$ is very large (jumping to a high-index material) such possibility is negligible. To create a one-photon pair ($m=1$), the corresponding probability equals to $P(1)=4\beta^{2}(\beta^{2}-1)^{2}/(\beta^{2}+1)^{4}$. This probability is smaller than for the vacuum state ($m=0$), but it can still be reasonably large for specific values of $\beta$. The maximum of $P(1)$ is found where the derivative with respect to $\beta$ vanishes, i.e., $\beta=\sqrt{3\pm 2\sqrt{2}}$ or $n_{2}=(3\pm 2\sqrt{2})n_{1}$. In this scenario, the probability of generating a one-photon pair is maximally 25$\%$ for both positive and negative signs. On substituting these particular values of $\beta$ into the expression for $P(0)$ we find that there is a 50% chance to remain in the vacuum state (regardless of those signs), which is only 2 times larger than for $P(1)$. These results are illustrated in Fig. 2 in which the probability function $P(m)$ described by Eq. (22) is plotted as a function of $\beta$ for different values of $m$. The figure shows that for $\beta=1$ the probability $P(0)=1$, and it is zero for $m\neq 0$. Figure 2 also shows that when the value of $\beta$ is large enough, the probability distribution becomes more uniform regarding different excited states. Indeed, if $\beta\gg 1$, Eq. (22) reduces to $P(m)\approx 4/\beta^{2}$, confirming that the probability function becomes independent of $m$.

The probability distribution with respect to $n$ and $\beta$ for several values of $m$ is presented in Fig. 3. We see from Fig. 3(a) that the probability of no photon creation ($m=0$) is high when the value of $\beta$ is close to unity. However, when $\beta$ deviates from unity, the probability $P(n,0)$ decreases. At the same time, the probabilities $P(n,1)$, $P(n,2)$, and $P(n,3)$ to excite one-, two-, and three-photon pairs increase and even surpass $P(n,0)$ for certain values of $n$, as indicated by Figs. 3(b)–3(d). Regarding the generation of a one-photon pair, Fig. 3(b) illustrates that there is the possibility of roughly 35% to achieve $m=1$. The exact maximum for a given $n$ is obtained analytically by considering the derivative with respect to $\beta$ of

$$P(n,1)=\Big{(}\frac{\beta^{2}-1}{\beta^{2}+1}\Big{)}^{2}(n+1)\bigg{[}1-\Big{(}\frac{\beta^{2}-1}{\beta^{2}+1}\Big{)}^{2}\bigg{]}^{n+1}.$$

(23)

Accordingly, we deduce that $[(\beta^{2}-1)/(\beta^{2}+1)]^{2}=1/(n+2)$ which results in $P_{\rm{max}}(n,1)=[(n+1)/(n+2)]^{n+2}$. The lower limit of $25{\%}$ is associated with $n=0$, as we discussed below Eq. (22), and the upper limit of $36{\%}$ corresponds to high values of $n$ [recall that $(1-x)^{1/x}=1/e\approx 0.36$ when $x$ approaches zero].

Next, we assess the photon statistics of the forward and backward output modes by studying the expectation values and variances of their corresponding number operators $\hat{n}_{b_{{\bf k}}}=\hat{b}_{{\bf k}}^{\dagger}\hat{b}_{{\bf k}}$ and $\hat{n}_{b_{-{\bf k}}}=\hat{b}_{-{\bf k}}^{\dagger}\hat{b}_{-{\bf k}}$. According to Eq. (9),

$$\begin{split}\hat{n}_{b_{{\bf k}}}&=\tau^{2}\hat{a}_{{\bf k}}^{\dagger}\hat{a}_{{\bf k}}+\Gamma^{2}\hat{a}_{-{\bf k}}\hat{a}_{-{\bf k}}^{\dagger}+\Gamma\tau\big{(}\hat{a}_{{\bf k}}^{\dagger}\hat{a}_{-{\bf k}}^{\dagger}+\hat{a}_{-{\bf k}}\hat{a}_{{\bf k}}\big{)},\cr\hat{n}_{b_{-{\bf k}}}&=\Gamma^{2}\hat{a}_{{\bf k}}\hat{a}_{{\bf k}}^{\dagger}+\tau^{2}\hat{a}_{-{\bf k}}^{\dagger}\hat{a}_{-{\bf k}}+\Gamma\tau\big{(}\hat{a}_{{\bf k}}\hat{a}_{-{\bf k}}+\hat{a}_{-{\bf k}}^{\dagger}\hat{a}_{{\bf k}}^{\dagger}\big{)}.\end{split}$$

(24)

Because the initial state before the temporal jump is taken to be $\ket{n,0}$, Eq. (24) implies that the expectation values of the output number operators are $\langle\hat{n}_{b_{{\bf k}}}\rangle=n\tau^{2}+\Gamma^{2}$ and $\langle\hat{n}_{b_{-{\bf k}}}\rangle=n\Gamma^{2}+\Gamma^{2}$. In addition, since the respective photon numbers of the input modes are $\langle\hat{n}_{a_{{\bf k}}}\rangle=n$ and $\langle\hat{n}_{a_{-{\bf k}}}\rangle=0$, the difference in the average number of photons after and before the temporal discontinuity equals to

$$\langle\hat{n}_{b_{{\bf k}}}+\hat{n}_{b_{-{\bf k}}}\rangle-\langle\hat{n}_{a_{{\bf k}}}+\hat{n}_{a_{-{\bf k}}}\rangle=2\Gamma^{2}(1+n).$$

(25)

This equation shows that the initial photon number $n$ is not conserved at the time interface, but instead the temporal jump creates on average $2\Gamma^{2}(1+n)$ additional photons. The nonconservation of the photon number is a manifestation of the Bogoliubov transformation in Eq. (9) and a consequence of the external work when changing the refractive index. For example, Eq. (25) states that under the condition of $\Gamma^{2}=1/[2(n+1)]$, which in terms of the refractive indexes translates into $n_{2}=[(2+n\pm\sqrt{3+2n})/(1+n)]n_{1}$, we generate on average a single photon. In this case, if the initial number of photons is zero ($n=0$), the refractive index of the medium after $t=0$ should be $2\pm\sqrt{3}$ times the refractive index of the initial medium. However, although the average photon number is not preserved under the time jump, we note that the average photon number difference between the modes remains invariant:

$${\langle\hat{n}_{b_{{\bf k}}}-\hat{n}_{b_{-{\bf k}}}\rangle=\langle\hat{n}_{a_{{\bf k}}}-\hat{n}_{a_{-{\bf k}}}\rangle}=n.$$

(26)

Having derived the number operators and their expectation values of the output modes, we calculate the corresponding photon number fluctuations in terms of the variances $\Delta n^{2}=\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}$. In accordance, after some algebraic steps, we deduce that

$$\begin{split}\Delta n_{b_{{\bf k}}}^{2}&=\Gamma^{2}\tau^{2}(n+1)=\Gamma^{2}(\langle\hat{n}_{b_{{\bf k}}}\rangle+1),\cr\Delta n_{b_{-{\bf k}}}^{2}&=\Gamma^{2}\tau^{2}(n+1)=\tau^{2}\langle\hat{n}_{b_{-{\bf k}}}\rangle.\end{split}$$

(27)

The first equalities on the two lines in Eq. (27) reveal that the time interface engenders the same amount of nonzero photon number fluctuations in both output modes, even though the fluctations are strictly zero for the input modes due to the Fock state $\ket{n,0}$. However, and more importantly, the second equalities on the two lines in Eq. (27) underline that the quantum light characters of the forward and backward output modes are drastically different. First, since $\tau>1$, we observe that for the backward mode the variance is always larger than the expectation value. Hence, this mode corresponds to super-Poissonian light. Second, because of the two possibilities $\Gamma<1$ or $\Gamma>1$, we discover that for the forward mode the variance can be smaller or larger than the expectation value. This means that the forward mode after the temporal jump can have the character of either sub-Poissonian or super-Poissonian light. By using the ratio parameter $\beta=\sqrt{n_{2}/n_{1}}$ and requiring $(\Delta n_{b_{{\bf k}}}^{2}/\langle\hat{n}_{b_{{\bf k}}}\rangle)>1$ in Eq. (27), the condition for having super-Poissonian statistics is

$$\beta>\Big{(}{n\over n+1}\Big{)}^{1\over 4}+\bigg{[}\Big{(}{n\over n+1}\Big{)}^{1\over 2}+1\bigg{]}^{1\over 2}\quad\mathrm{or}\quad\beta<-\Big{(}{n\over n+1}\Big{)}^{1\over 4}+\bigg{[}\Big{(}{n\over n+1}\Big{)}^{1\over 2}+1\bigg{]}^{1\over 2}.$$

(28)

Based on the above expressions, we see that if the initial number of photons is considerable ($n\gg 1$), the aforementioned conditions are simplified to $\beta>1+\sqrt{2}$ and $\beta<-1+\sqrt{2}$, in other words $n_{2}>(3+2\sqrt{2})n_{1}$ or $n_{2}<(3-2\sqrt{2})n_{1}$. In the opposite case, if the initial photon number is zero ($n=0$), Eq. (28) states that the forward mode displays always super-Poissonian statistics regardless of the ratio parameter $\beta$.

The fact that both of the output modes exhibit super-Poissonian photon statistics when $n=0$ can be understood by considering the two-mode squeezed vacuum state in Eq. (20). By tracing over either the forward or backward mode results exactly in the same mixed state:

$$\hat{\rho}_{b_{{\bf k}}}=\hat{\rho}_{b_{-{\bf k}}}=\frac{1}{\tau^{2}}\sum_{m=0}^{\infty}\Big{(}\frac{\Gamma}{\tau}\Big{)}^{2m}\ket{m}\!\bra{m}.$$

(29)

On then making the association $(\Gamma/\tau)^{2}=e^{-\hbar\omega_{2}/k_{\mathrm{B}}T}$, where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is the temperature, one observes that Eq. (29) translates into

$$\hat{\rho}_{b_{{\bf k}}}=\hat{\rho}_{b_{-{\bf k}}}=\big{(}1-e^{-\hbar\omega_{2}/k_{\mathrm{B}}T}\big{)}\sum_{m=0}^{\infty}e^{-m\hbar\omega_{2}/k_{\mathrm{B}}T}\ket{m}\!\bra{m}.$$

(30)

This is the standard expression for a thermal state, with the photon fluctuations obeying super-Poissonian statistics in terms of the Bose–Einstein probability distribution [37, 38].

After determining the super-Poissonian and sub-Poissonian photon statistics of the output modes, it is sensible to examine also the bunched and antibunched photon character of the modes. To this end, we consider the degree of second-order coherence at a single space–time point, which for a generic number operator $\hat{n}$ reads [45]

$$g^{(2)}(0)=\frac{\langle\hat{n}(\hat{n}-1)\rangle}{\langle\hat{n}\rangle^{2}}=1+\frac{\Delta n^{2}-\langle\hat{n}\rangle}{\langle\hat{n}\rangle^{2}}.$$

(31)

On using the previously calculated expectation values and variances of the number operators in Eq. (24), we can determine from Eq. (31) the conditions under which the photons of the forward and backward modes after the temporal jump exhibit bunching [$g^{(2)}(0)>1$] or antibunching [$g^{(2)}(0)<1$].

After performing the calculations, we end up with

$$g^{(2)}_{b_{{\bf k}}}(0)=1+{1+(n+1)(\Gamma^{4}-1)\over(n+\Gamma^{2}+n\Gamma^{2})^{2}},\quad g^{(2)}_{b_{-{\bf k}}}(0)=1+{1\over n+1}.$$

(32)

First, we explicitly see that the $g^{(2)}_{b_{-{\bf k}}}(0)$ function for the backward mode does not depend on the variable $\Gamma$. It is a function of only $n$. Second, Eq. (32) indicates that $1<g^{(2)}_{b_{-{\bf k}}}(0)\leq 2$, with the lower and upper limits met when $n\rightarrow\infty$ and $n=0$, respectively. Hence, regardless of the initial photon number $n$, the backward output mode corresponds to bunched light. Note that the condition $g^{(2)}_{b_{-{\bf k}}}(0)=2$ when $n=0$ is a consequence of the fact that in this case the backward mode is in the thermal state given by Eqs. (29) and (30). However, concerning the $g^{(2)}_{b_{{\bf k}}}(0)$ function of the forward mode, we see from Eq. (32) that it is strongly dependent on both $\Gamma$ and $n$. If $\Gamma=0$, we arrive at $g^{(2)}_{b_{{\bf k}}}(0)=1-1/n<1$. This is expected, because for $\Gamma=0$ there is no temporal discontinuity ($n_{2}=n_{1}$), whereupon the forward output mode merges with the forward input mode whose specific antibunched light character arises from the Fock state $\ket{n}$ [see Eq. (31)]. On the other hand, if $\Gamma$ tends to infinity, $g^{(2)}_{b_{{\bf k}}}(0)=1+1/(n+1)>1$. Thus, there is a particular value of $\Gamma$ for which the $g^{(2)}_{b_{{\bf k}}}(0)$ function becomes unity. So depending on the value range of $\Gamma$, the photons in the forward mode can display either antibunching [$g^{(2)}_{b_{{\bf k}}}(0)<1$] or bunching [$g^{(2)}_{b_{{\bf k}}}(0)>1$)]. We have thereby deduced exactly the same conditions for antibunching and bunching that were derived respectively for sub-Poissonian and super-Poissonian statistics in the previous section. This is consistent with the conventional understanding that antibunched light is sub-Poissonian and bunched light is super-Poissonian. We have summarized all these key conclusions explained above about photon statistics in Table 1 (see the first three rows).

In the previous subsections, we studied the situation where the forward input mode is in the number state $\ket{n}$. Here, we concisely look into the scenario in which the forward mode before the temporal discontinuity is in the coherent state

$$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}{\alpha^{n}\over\sqrt{n!}}|n\rangle,$$

(33)

where $\alpha$ is the eigenvalue of the annihilation operator $\hat{a}_{{\bf k}}$. In this case, by using Eqs. (24) and (33), we find that the expectation values of the number operators for the forward and backward output modes are $\langle\hat{n}_{b_{{\bf k}}}\rangle=\tau^{2}|\alpha|^{2}+\Gamma^{2}$ and $\langle\hat{n}_{b_{-{\bf k}}}\rangle=\Gamma^{2}(|\alpha|^{2}+1)$, respectively. The difference between the average photon numbers before and after the temporal jump is therefore $(\langle\hat{n}_{b_{{\bf k}}}\rangle+\langle\hat{n}_{b_{-{\bf k}}}\rangle)-(\langle\hat{n}_{a_{{\bf k}}}\rangle+\langle\hat{n}_{a_{-{\bf k}}}\rangle)=2\Gamma^{2}(1+|\alpha|^{2})$, which is similar to Eq. (25) in the context of the number state.

We continue by deducing the variances of the number operators. This allows us to determine the photon number fluctuations and the photon statistics class (sub-Poissonian or super-Poissonian) of the created modes. Recall that the coherent state is associated with a Poissonian distribution because the variance of the number operator is equal to the mean value. However, as we observed for the number state, the temporal discontinuity may essentially change the situation. By again employing Eqs. (24) and (33), we infer that in the case of an input coherent state

$$\begin{split}&\Delta n_{b_{{\bf k}}}^{2}=\Gamma^{2}\tau^{2}\Big{(}|\alpha|^{2}+1+{\tau^{2}\over\Gamma^{2}}|\alpha|^{2}\Big{)}=\Gamma^{2}\Big{(}\langle\hat{n}_{b_{{\bf k}}}\rangle+1+{\tau^{4}\over\Gamma^{2}}|\alpha|^{2}\Big{)},\cr&\Delta n_{b_{-{\bf k}}}^{2}=\Gamma^{2}\tau^{2}\Big{(}|\alpha|^{2}+1+{\Gamma^{2}\over\tau^{2}}|\alpha|^{2}\Big{)}=\tau^{2}\Big{(}\langle\hat{n}_{b_{-{\bf k}}}\rangle+{\Gamma^{4}\over\tau^{2}}|\alpha|^{2}\Big{)}.\end{split}$$

(34)

Comparing these results with those in Eq. (27) that we achieved for the number state, we see from the first expressions in Eq. (34) that this time the photon number variances of the forward and backward modes are not the same. There is instead a difference which equals $\Delta n_{b_{{\bf k}}}^{2}-\Delta n_{b_{-{\bf k}}}^{2}=(\tau^{2}+\Gamma^{2})|\alpha|^{2}$. From the second expressions in Eq. (34) we notice that for the backward mode the variance is larger than the expectation value because $\tau>1$. Consequently, it corresponds again to super-Poissonian light. Concerning the forward mode, one might initially think that similarly to the number state it can have sub- or super-Poissonian statistics, because the expression inside the parentheses in Eq. (34) is multiplied by $\Gamma^{2}$ which can be zero. However, a quick calculation reveals that the variance can be cast into the form $\Delta n_{b_{{\bf k}}}^{2}=\tau^{2}(\langle\hat{n}_{b_{{\bf k}}}\rangle+\Gamma^{2}|\alpha|^{2})>1$, which shows that the forward mode is also following super-Poissonian photon statistics.

As a final point, similarly to what we did for the input number state, we deduce the degree of second-order coherence [Eq. (31)] of the output modes when the input state is a coherent state. Having the associated expectation values and the variances, we derive that

$$g^{(2)}_{b_{{{\bf k}}}}(0)=2-{\tau^{4}|\alpha|^{4}\over(\tau^{2}|\alpha|^{2}+\Gamma^{2})^{2}},\quad g^{(2)}_{b_{-{{\bf k}}}}(0)=2-{|\alpha|^{4}\over(|\alpha|^{2}+1)^{2}}.$$

(35)

Akin to the input number state [Eq. (32)], we observe that $g^{(2)}_{b_{-{{\bf k}}}}(0)$ of the backward mode in Eq. (35) is completely independent of the material parameters, while for the forward mode $g^{(2)}_{b_{{{\bf k}}}}(0)$ depends on $\tau$ and $\Gamma$. In addition, due to the two latter terms in Eq. (35) we conclude that $1<g^{(2)}_{b_{{{\bf k}}}}(0)\leq 2$ and $1<g^{(2)}_{b_{{-{\bf k}}}}(0)\leq 2$, which means that both modes correspond to bunched light. Remember that, in contrast, for the input number state the forward output mode can exhibit photon antibunching under some conditions. The main conclusions about the output photon statistics for the input coherent state are summarized in Table 1.

In our previous discussion, we explored the output state when the initial state was set to $|n,0\rangle$, demonstrating the remarkable phenomenon of photon-pair generation. This occurrence is primarily attributed to the first exponential term of the two-mode squeeze operator in Eq. (16), which symmetrically incorporates creation operators. Conversely, we see that the third exponential term of the squeeze operator features annihilation operators in its argument, but even so, we did not observe the destruction of photons. The explanation for this lies in our selection of the initial state; we assumed that the incident backward mode is in the vacuum state $|0\rangle$. Consequently, this particular term in Eq. (16) exerts no influence on the state $|n,0\rangle$. However, if the incident backward mode contains photons, the last term in Eq. (16) may destroy photons, and thus, the outcome can be drastically different. Indeed, if we apply it on the state $|n,\Pi\rangle$ where $0<\Pi\leq n$ and utilize the Taylor expansion, we obtain that

$$e^{-\tanh(r)\hat{K}_{-}}|n,\Pi\rangle=\sum_{m=0}^{\Pi}(-1)^{m}{\sqrt{n!\Pi!}\over m!\sqrt{(n-m)!(\Pi-m)!}}\tanh^{m}(r)|n-m,\Pi-m\rangle,$$

(36)

which explicitly illustrates the phenomenon of photon-pair destruction.

To derive the output state, we must also consider the other terms of the squeeze operator. The action of the second term in Eq. (16) on the state $|n-m,\Pi-m\rangle$ in Eq. (36) results in

$$e^{\ln[1-\tanh^{2}(r)]\hat{K}_{0}}|n-m,\Pi-m\rangle=[1-\tanh^{2}(r)]^{{n+\Pi-2m+1\over 2}}|n-m,\Pi-m\rangle,$$

(37)

indicating that $|n-m,\Pi-m\rangle$ is an eigenstate. Next, we must consider the first term of the squeeze operator, whose operation on the state $|n-m,\Pi-m\rangle$ yields

$$e^{\tanh(r)\hat{K}_{+}}|n-m,\Pi-m\rangle=\sum_{l=0}^{\infty}{\sqrt{(n-m+l)!(\Pi-m+l)!}\over l!\sqrt{(n-m)!(\Pi-m)!}}\tanh^{l}(r)|n-m+l,\Pi-m+l\rangle.$$

(38)

By eventually combining Eqs. (36), (37), and (38), we deduce a closed-form solution for the output state which is given by

$$\begin{split}&|\Psi\rangle=\Lambda\sum_{l=0}^{\infty}\sum_{m=0}^{\Pi}{(-1)^{m}\over l!m!}{\sqrt{(n-m+l)!(\Pi-m+l)!}\over(n-m)!(\Pi-m)!}{\tanh^{(m+l)}(r)\over[1-\tanh^{2}(r)]^{m}}|n-m+l,\Pi-m+l\rangle,\end{split}$$

(39)

where $\Lambda=\sqrt{n!\Pi!}[1-\tanh^{2}(r)]^{n+\Pi+1\over 2}$. As a sanity check, we choose $\Pi=0$ and confirm that Eq. (39) reduces to Eq. (18), which we discussed earlier. Equation (39), which provides a general framework, simultaneously encapsulates a combination of two effects: Photon-pair generation and destruction. In particular, the symmetric scenario $n=\Pi$ where $n=m-l$ offers the possibility to generate the vacuum state $|0,0\rangle$ within the superposition of Eq. (39). Since the integer $m$ is equal to or lower than $n$, the only conditions that need to be met are $m=n$ and $l=0$. Accordingly, the state $|0,0\rangle$ is produced with a probability $P=[1-\tanh^{2}(r)]\tanh^{2n}(r)$.

We also want to highlight another interesting property concealed within Eq. (39). For simplicity, we consider a symmetric scenario with single-photon illumination in each mode, specifically setting $n=\Pi=1$. In this case, Eq. (39) takes the form

$$|\Psi\rangle=[1-\tanh^{2}(r)]^{1\over 2}\bigg{\{}-\tanh(r)|0,0\rangle+\sum_{l=1}^{\infty}\tanh^{l}(r)\Big{[}{1-\tanh^{2}(r)\over\tanh(r)}l-\tanh(r)\Big{]}|l,l\rangle\bigg{\}}.$$

(40)

Firstly, we note that our general expression for the probability of the state $|0,0\rangle$ aligns with the above equation when $n=1$. More importantly, we observe that the coefficient inside the square brackets in the series is proportional to the integer $l$. This observation opens up the possibility of eliminating a specific state from the superposition in Eq. (40). To achieve such an elimination, we require the following criterion to hold:

$$\tanh^{2}(r)=\Big{(}{\Gamma\over\tau}\Big{)}^{2}={l\over l+1},$$

(41)

which in terms of the refractive indices translates to two possible relationships: $n_{2}/n_{1}=\big{[}1+\sqrt{l/(l+1)}\big{]}/\big{[}1-\sqrt{l/(l+1)}\big{]}$ or $n_{2}/n_{1}=\big{[}1-\sqrt{l/(l+1)}\big{]}/\big{[}1+\sqrt{l/(l+1)}\big{]}$. For instance, to eliminate the state $|1,1\rangle$, the refractive index ratio should be either $n_{2}/n_{1}=5.8$ or $n_{2}/n_{1}=0.17$. By designing the time interface to match one of these specific values, we ensure that repeated measurements will never yield a scenario where one photon is detected in the forward direction and another single photon in the backward direction. This phenomenon represents a form of unambiguous state discrimination.