Probing the quantum-classical boundary with optical beam splitters

Abstract

This study explores how optical beam splitters provide a conceptual and operational link between classical and quantum descriptions of light. In the classical picture, they are modeled through energy-conserving field amplitudes and a four-port scattering matrix, which successfully explain interference in systems such as Michelson and Mach–Zehnder interferometers. Moving to the quantum domain, the field amplitudes are replaced by bosonic creation and annihilation operators that obey the Weyl–Heisenberg algebra. Using the Glauber displacement operator, we define coherent states whose expectation values reproduce classical monochromatic fields while exhibiting Poissonian photon statistics. This unified approach shows that beam splitters diagnose the nature of incident light: coherent states yield classical interference, while indistinguishable single-photons reveal quantum signatures such as antibunching and Hong–Ou–Mandel interference. The analysis relies on operator formalism and correlation functions that capture both average-field behavior and nonclassical statistics. As beam-splitting technologies transition from bulk optics to integrated photonics and metasurfaces, this framework offers a consistent basis for modeling optical circuits across the classical-quantum divide. By highlighting the dual role of the beam splitter as a passive device and a quantum probe, we emphasize its importance for quantum optics, quantum information, and hybrid photonic platforms.

Keywords:
Beam splitter; quantum optics; coherent states; Glauber formalism; classical-quantum transition.

1. Introduction

Beam splitters (BSs) occupy a unique position at the interface between classical and quantum optics. Classically, they superpose electromagnetic fields and generate interference fringes; quantum mechanically, they transform bosonic modes and reveal nonclassical correlations, such as the suppression of coincidences in Hong–Ou–Mandel (HOM) interference [¹]. This dual role makes the BS an ideal “bridge” between wave interference and particle statistics. In this article we present a unified treatment that starts from energy-conserving field amplitudes, proceeds to a four-port scattering description, and culminates in an operator formalism that preserves the canonical commutation relations and explains nonclassical effects.

Beam splitters are essential in many optical systems. In classical optics, they serve as core elements of interferometers such as Michelson and Mach–Zehnder, enabling the superposition and interference of optical paths [²]. In quantum optics, they underpin single-photon interference, entanglement generation, and linear optical quantum gate implementation [³]. Quantum-mechanically, a beam splitter is modeled by a unitary transformation on the annihilation and creation operators of the input modes (e.g., a 50:50 splitter), which leads to hallmark non-classical effects such as Hong–Ou–Mandel interference, where two indistinguishable photons exit from the same port [³, ⁴].

2. The Optical Components: Beam Splitter and Interferometer

Consider a beam splitter, an optical component realized with thin metallic layers or multilayer dielectric films on glass, which produces two output beams for each input beam (see Figure 1). To fix notation, we write the (near-monochromatic) incident field as

(1) $${\mathbf{E}}_{\mathrm{in}}(\mathbf{r},t)=A\left[\alpha (\mathbf{r},t)e^{i\omega t}+{\alpha }^{\ast }(\mathbf{r},t)e^{-i\omega t}\right]\mathbf{p}(\mathbf{r},t),$$

where $A$ is a real amplitude, $\alpha (\mathbf{r},t)$ is a slowly varying complex envelope, $\omega$ is the angular frequency (with $\lambda =c/\omega$), and $\mathbf{p}(\mathbf{r},t)$ is the (unit) polarization vector [¹].

Figure 1
Lossless beam splitter (BS): an incident field ${\mathbf{E}}_{\mathrm{in}}$ is split into a reflected field ${\mathbf{E}}_r=r{\mathbf{E}}_{\mathrm{in}}$ and a transmitted field ${\mathbf{E}}_t=t{\mathbf{E}}_{\mathrm{in}}$, with ${\left|r\right|}^2+{\left|t\right|}^2=1$ when absorption is zero ($A=0$, where $R+T+A=1$).

For a single illuminated input port, the output fields are

(2) $${\mathbf{E}}_r=r{\mathbf{E}}_{\mathrm{in}},$$

(3) $${\mathbf{E}}_t=t{\mathbf{E}}_{\mathrm{in}},$$

with complex amplitude coefficients $r,t$. Energy conservation in the lossless case requires ${\left|r\right|}^2+{\left|t\right|}^2=1$ and energy conservation gives

(4) $${\parallel {\mathbf{E}}_{\mathrm{in}}\parallel }^2={\parallel {\mathbf{E}}_r\parallel }^2+{\parallel {\mathbf{E}}_t\parallel }^2=\left({\left|r\right|}^2+{\left|t\right|}^2\right){\parallel {\mathbf{E}}_{\mathrm{in}}\parallel }^2,$$

so that ${\left|r\right|}^2+{\left|t\right|}^2=1$ in the lossless case. Equations (2)–(3) describe the classical, energy-preserving behavior of this passive element ($A=0$).

In many applications two waves are simultaneously incident on the two input ports (see Figure 2): ${\mathbf{E}}_{\mathrm{in}}$ and ${\mathbf{E}}_u$. For a general, lossless, polarization-independent four-port device,

(5) $${\mathbf{E}}_r=r{\mathbf{E}}_{\mathrm{in}}+t^{\prime }{\mathbf{E}}_u,$$

(6) $${\mathbf{E}}_t=t{\mathbf{E}}_{\mathrm{in}}+r^{\prime }{\mathbf{E}}_u.$$

(7) $$\begin{array}{l}{\Vert {\mathbf{E}}_r\Vert }^2+{\Vert {\mathbf{E}}_t\Vert }^2\\ =\left({\left|r\right|}^2+{\left|t\right|}^2\right){\Vert {\mathbf{E}}_{\mathrm{in}}\Vert }^2+\left({\left|r^{\prime }\right|}^2+{\left|t^{\prime }\right|}^2\right){\Vert {\mathbf{E}}_u\Vert }^2\\ +\left(r{t}^{\prime \ast }+t{r}^{\prime \ast }\right){\mathbf{E}}_{\mathrm{in}}\cdot {\mathbf{E}}_u^{\ast }+\left(t^{\prime }{r}^{\ast }+r^{\prime }{t}^{\ast }\right){\mathbf{E}}_u\cdot {\mathbf{E}}_{\mathrm{in}}^{\ast }.\end{array}$$

Figure 2
Schematic four-port (two-input/two-output) model of a lossless, polarization-independent beam splitter. Reflection/transmission coefficients obey energy conservation and reciprocity (unitarity) and are polarization independent.

For an ideal lossless beam splitter, energy conservation requires that the total output intensity equals the total input intensity, i.e.,

$${\left|E_r\right|}^2+{\left|E_t\right|}^2={\left|E_{\mathrm{in}}\right|}^2+{\left|E_u\right|}^2.$$

In compact matrix form, the scattering relation is

(8) $$\left(\begin{array}{c}{\mathbf{E}}_r\\ {\mathbf{E}}_t\end{array}\right)=\left(\begin{array}{cc}r& t^{\prime }\\ t& r^{\prime }\end{array}\right)\left(\begin{array}{c}{\mathbf{E}}_{\mathrm{in}}\\ {\mathbf{E}}_u\end{array}\right).$$

Losslessness implies ${\left|r\right|}^2+{\left|t\right|}^2={\left|r^{\prime }\right|}^2+{\left|t^{\prime }\right|}^2=1$ and the cross constraint $r{t}^{\ast }+t^{\prime }{r}^{\prime \ast }=0$. We use Figures 1–2 only to illustrate these relations geometrically. Because the beam splitter is polarization independent, the complex coefficients $r,r^{\prime },t,t^{\prime }$ do not depend on $\mathbf{p}(\mathbf{r},t)$.

When $r$ and $t$ are taken as real (a common idealization), one convenient unitary parametrization is

(9) $$\left(\begin{array}{c}{\mathbf{E}}_r\\ {\mathbf{E}}_t\end{array}\right)=\left(\begin{array}{cc}\sqrt{\epsilon }& \sqrt{1-\epsilon }\\ \sqrt{1-\epsilon }& -\sqrt{\epsilon }\end{array}\right)\left(\begin{array}{c}{\mathbf{E}}_{\mathrm{in}}\\ {\mathbf{E}}_u\end{array}\right),0\le \epsilon \le 1,$$

where the minus sign corresponds to a $\pi$ (${180}^{\circ }$) phase shift upon one of the reflections; for a single dielectric interface this arises when reflecting from the higher-index side, though different mirror designs can implement different overall phases [⁵].For a non-ideal (absorbing or scattering) beam splitter with power loss $A$ ($0\le A<1$), the throughput becomes

(10) $${\parallel {\mathbf{E}}_t\parallel }^2+{\parallel {\mathbf{E}}_r\parallel }^2=\left(1-A\right)\left({\parallel {\mathbf{E}}_{\mathrm{in}}\parallel }^2+{\parallel {\mathbf{E}}_u\parallel }^2\right).$$

The special case $\epsilon =\frac{1}{2}$ describes a balanced $50:50$ beam splitter.

2.1. Quantized-field description and commutation relations

To incorporate quantum features, we quantize the field and replace classical amplitudes with bosonic annihilation/creation operators obeying the Weyl-Heisenberg algebra [⁶, ⁷].These operators allow us to describe not only the mean behavior of light fields but also their underlying quantum statistics and coherence properties:

(11) $$[{\widehat{a}}_i,{\widehat{a}}_j]=0,[{\widehat{a}}_i^{†},{\widehat{a}}_j^{†}]=0,[{\widehat{a}}_i,{\widehat{a}}_j^{†}]={\delta }_{ij}.$$

Heuristically, the field/operator correspondence is

(12) $${\mathbf{E}}_{\mathrm{in}}\propto {\widehat{a}}_{\mathrm{in}},{\mathbf{E}}_r\propto {\widehat{a}}_r,{\mathbf{E}}_t\propto {\widehat{a}}_t.$$

Unlike the classical case, the unused BS input cannot be discarded in the quantum description: vacuum fluctuations enter that port and must be transformed by the same unitary as any other field. Including the second input mode ${\widehat{a}}_u$ is therefore mandatory to preserve unitarity and the canonical commutation relations, $[{\widehat{a}}_r,{\widehat{a}}_r^{†}]=[{\widehat{a}}_t,{\widehat{a}}_t^{†}]=1$ and $[{\widehat{a}}_r,{\widehat{a}}_t^{†}]=0$.

If only one input is considered classically, naive relations like ${\widehat{a}}_r=r{\widehat{a}}_{\mathrm{in}}$ and ${\widehat{a}}_t=t{\widehat{a}}_{\mathrm{in}}$ would not preserve the commutators in (11). The correct quantum model must include the vacuum entering the unused port. Let ${\widehat{a}}_u$ denote the second (vacuum) input mode. The most general lossless, polarization-independent, linear transformation is

(13) $${\widehat{a}}_r=r{\widehat{a}}_{\mathrm{in}}+t^{\prime }{\widehat{a}}_u,$$

(14) $${\widehat{a}}_t=t{\widehat{a}}_{\mathrm{in}}+r^{\prime }{\widehat{a}}_u,$$

or, in matrix form,

(15) $$\left(\begin{array}{c}{\widehat{a}}_r\\ {\widehat{a}}_t\end{array}\right)=\left(\begin{array}{cc}r& t^{\prime }\\ t& r^{\prime }\end{array}\right)\left(\begin{array}{c}{\widehat{a}}_{\mathrm{in}}\\ {\widehat{a}}_u\end{array}\right).$$

Preservation of the canonical commutation relations (CCR) requires

(16) $${\left|r\right|}^2+{\left|t\right|}^2=1,{\left|r^{\prime }\right|}^2+{\left|t^{\prime }\right|}^2=1,r{t}^{\ast }+t^{\prime }{r}^{\prime \ast }=0,$$

and reciprocity implies $\left|r^{\prime }\right|=\left|r\right|$, $\left|t^{\prime }\right|=\left|t\right|$ (phases may differ). With (16), the output modes satisfy $[{\widehat{a}}_r,{\widehat{a}}_r^{†}]=1$, $[{\widehat{a}}_t,{\widehat{a}}_t^{†}]=1,[{\widehat{a}}_r,{\widehat{a}}_t^{†}]=0$ so the Weyl–Heisenberg algebra is preserved.

A convenient $50:50$ realization (up to overall phases) is

(17) $$\left(\begin{array}{c}{\widehat{a}}_r\\ {\widehat{a}}_t\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ i& 1\end{array}\right)\left(\begin{array}{c}{\widehat{a}}_u\\ {\widehat{a}}_{\mathrm{in}}\end{array}\right),$$

which, component-wise, yields

(18) $${\widehat{a}}_r={\displaystyle \frac{1}{\sqrt{2}}}\left({\widehat{a}}_u+i{\widehat{a}}_{\mathrm{in}}\right),$$

(19) $${\widehat{a}}_t={\displaystyle \frac{1}{\sqrt{2}}}\left(i{\widehat{a}}_u+{\widehat{a}}_{\mathrm{in}}\right).$$

Equivalently, using $e^{i\pi /2}=i$,

$${\widehat{a}}_r=\frac{1}{\sqrt{2}}\left({\widehat{a}}_u+e^{i\pi /2}{\widehat{a}}_{\mathrm{in}}\right),{\widehat{a}}_t=\frac{1}{\sqrt{2}}\left(e^{i\pi /2}{\widehat{a}}_u+{\widehat{a}}_{\mathrm{in}}\right).$$

or, in matrix form,

$$\left(\begin{array}{c}{\widehat{a}}_r\\ {\widehat{a}}_t\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& e^{i\pi /2}\\ e^{i\pi /2}& 1\end{array}\right)\left(\begin{array}{c}{\widehat{a}}_{\mathrm{in}}\\ {\widehat{a}}_u\end{array}\right).$$

2.2. Fock-state transformations and Hong–Ou–Mandel interference

With vacuum inputs one has

(20) $${|0⟩}_{\mathrm{in}}{|0⟩}_u\stackrel{BS}{\to }{|0⟩}_r{|0⟩}_t.$$

Now, let us consider a beam splitter that mixes two photons at the inputs.

(21) $${|1⟩}_{\mathrm{in}}{|1⟩}_u={\widehat{a}}_{in}^{†}{|0⟩}_{\mathrm{in}}{\widehat{a}}_u^{†}{|0⟩}_u$$

(22) $$\begin{array}{cc}& {|1⟩}_{\mathrm{in}}{|1⟩}_u\stackrel{BS}{\to }\frac{1}{\sqrt{2}}\left(i{\widehat{a}}_r^{†}+{\widehat{a}}_t^{†}\right)\frac{1}{\sqrt{2}}\left({\widehat{\widehat{a}}}_r^{†}+i{\widehat{a}}_t^{†}\right){|0⟩}_r{|0⟩}_t\hfill \\ & =\frac{i}{\sqrt{2}}\left({|2⟩}_r{|0⟩}_t+{|0⟩}_r{|2⟩}_t\right)\hfill \end{array}$$

so coincidence events ${|1⟩}_r{|1⟩}_t$ vanish—the hallmark Hong–Ou–Mandel dip [⁶, ⁸].

For a 50:50 BS with ${\widehat{a}}_r^{†}=\left({\widehat{a}}_u^{†}+i{\widehat{a}}_{\mathrm{in}}^{†}\right)/\sqrt{2}$ and ${\widehat{a}}_t^{†}=\left(i{\widehat{a}}_u^{†}+{\widehat{a}}_{\mathrm{in}}^{†}\right)/\sqrt{2}$, the two-photon input ${\widehat{a}}_{\mathrm{in}}^{†}{\widehat{a}}_u^{†}|0⟩$ transforms as

(23) $$\begin{array}{l}{\widehat{a}}_{\mathrm{in}}^{†}{\widehat{a}}_u^{†}|0⟩\mapsto {\displaystyle \frac{1}{2}}\left({\widehat{a}}_r^{†}-i{\widehat{a}}_t^{†}\right)\left(-i{\widehat{a}}_r^{†}+{\widehat{a}}_t^{†}\right)|0⟩\\ ={\displaystyle \frac{1}{2}}\left(-i{\widehat{a}}_r^{†}{\widehat{a}}_r^{†}+{\widehat{a}}_r^{†}{\widehat{a}}_t^{†}-{\widehat{a}}_t^{†}{\widehat{a}}_r^{†}-i{\widehat{a}}_t^{†}{\widehat{a}}_t^{†}\right)|0⟩.\end{array}$$

Because bosonic creation operators commute, ${\widehat{a}}_r^{†}{\widehat{a}}_t^{†}={\widehat{a}}_t^{†}{\widehat{a}}_r^{†}$, the two cross terms cancel: ${\widehat{a}}_r^{†}{\widehat{a}}_t^{†}+{\widehat{a}}_t^{†}{\widehat{a}}_r^{†}=0$. Only the “bunched” terms remain, giving ${|2⟩}_r{|0⟩}_t$ or ${|0⟩}_r{|2⟩}_t$ (up to a global phase). This destructive interference between indistinguishable paths is the essence of the HOM effect.

We illuminate the beam splitter with the two-single-photon input state ${|1⟩}_{\mathrm{in}}{|1⟩}_u$, meaning that a single photon is present at each input within the same time window. The key result from Eq. (22) is that the output contains two photons either in the reflected or in the transmitted arm, while the terms ${|1⟩}_r{|1⟩}_t$ and ${|1⟩}_t{|1⟩}_r$ – one photon in each output – cancel due to the indistinguishability of the input photons. Thus, for a well-balanced beam splitter with good spatial/temporal mode overlap, coincidence events with one photon in each arm are strongly suppressed. This is a genuinely quantum interference effect (Hong–Ou–Mandel coalescence), not a classical one, and it is revealed experimentally with a coincidence counter that only triggers when both outputs contain one photon [⁶, ⁸].

These two treatments – classical wave optics and quantum particle optics – describe markedly different phenomena depending on whether the electromagnetic field is modeled as a continuous wave or as quanta. However, they can be unified by quantizing the field and adopting the Glauber formalism [⁹, ¹⁰]. In this approach, classical field amplitudes arise as expectation values of field operators, while normally ordered correlation functions account for photon statistics, quantum interference, and nonclassical effects such as antibunching and entanglement. Hence, both the fringe patterns of wave optics and the particle-like coincidences of quantum optics emerge as limiting manifestations of the same underlying quantum field theory.

2.3. Unifying quantum and classical approach

The Glauber formalism establishes the foundation of quantum optical coherence by treating the electromagnetic field as a collection of quantized harmonic modes described by creation and annihilation operators. Rather than relying solely on classical field amplitudes, Glauber showed that interference and intensity-correlation properties can be computed from normally ordered correlation functions,

$$G^{\left(n\right)}({\mathbf{r}}_1,t_1;{\mathbf{r}}_2,t_2;\dots ;{\mathbf{r}}_n,t_n),$$

which connect directly to photon-detection statistics and account for phenomena such as antibunching and super-/sub-Poissonian statistics that have no explanation within purely classical optics [⁹, ¹⁰].

Within this framework, coherent states$|\alpha ⟩$ play a central role. They are defined as eigenstates of the annihilation operator,

(24) $$\widehat{a}\mathrm{}|\alpha ⟩=\alpha |\mathrm{}\alpha ⟩,$$

and realize minimum-uncertainty states (the product of quadrature variances attains the Heisenberg limit). When $\alpha$ has well-defined modulus and phase, the expectation value of the field operator reproduces a monochromatic classical field with amplitude proportional to $\alpha$, while the photon-number distribution remains Poissonian.

A convenient representation expands $|\alpha ⟩$ in the Fock basis:

(25) $$|\alpha ⟩=e^{-{\left|\alpha \right|}^2/2}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\alpha }^n}{\sqrt{n!}}}|n⟩=e^{-\mu /2}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(\sqrt{\mu }{e}^{i\vartheta }\right)}^n}{\sqrt{n!}}}|n⟩,\mathrm{}\alpha =\sqrt{\mu }{e}^{i\vartheta },$$

so that the mean photon number is $\mu ={\left|\alpha \right|}^2$ and the optical phase is $\vartheta$.

Equivalently, coherent states are generated by the displacement operator,

(26) $$|\alpha ⟩=D\left(\alpha \right)|0⟩=\exp \left(\alpha {\widehat{a}}^{†}-{\alpha }^{\ast }\widehat{a}\right)|0⟩,$$

$D\left(\alpha \right)$ acts as a unitary phase space shift, taking the vacuum $|0⟩$ and translating its quadrature amplitudes by $\alpha$. Now we consider a situation where a coherent state enters in the beam splitter:

(27) $$\begin{array}{cc}& {|\alpha ⟩}_{\mathrm{in}}{|0⟩}_u(=e^{\alpha {\widehat{a}}_{in}^{†}-{\alpha }^{\ast }{\widehat{a}}_{\mathrm{in}}}{|0⟩}_{\mathrm{in}}{|0⟩}_u\stackrel{BS}{\to })\hfill \\ & \cdot e^{\left[\alpha /\sqrt{2}\left(i{\widehat{a}}_r^{†}+{\widehat{a}}_t^{†}\right)-{\alpha }^{\ast }/\sqrt{2}\left(-i{\widehat{a}}_r+{\widehat{a}}_t\right)\right]}{|0⟩}_r{|0⟩}_t\hfill \end{array}$$

(28) $$={|i\alpha /\sqrt{2}⟩}_r{|\alpha /\sqrt{2}⟩}_t,$$

A phase shift can be observed in the reflected coherent state

(29) $${|i\alpha /\sqrt{2}⟩}_r{|\alpha /\sqrt{2}⟩}_t={|\left(\alpha e^{i\pi /2}\right)/\sqrt{2}⟩}_r{|\alpha /\sqrt{2}⟩}_t,$$

the new state is not a superposition, as in the single-photon setup. What happens when two coherent states cross the beam splitter?

(30) $$\begin{array}{l}{|\alpha ⟩}_{\mathrm{in}}{|\beta ⟩}_u\stackrel{BS}{\to }{e}^{\left[\alpha /\sqrt{2}\left(i{\widehat{a}}_r^{†}+{\widehat{a}}_t^{†}\right)-{\alpha }^{\ast }/\sqrt{2}\left(-i{\widehat{a}}_r+{\widehat{a}}_t\right)\right]}\\ \cdot e^{\left[\beta /\sqrt{2}\left(i{\widehat{\widehat{a}}}_r^{†}+{\widehat{a}}_t^{†}\right)-{\beta }^{\ast }/\sqrt{2}\left(-i{\widehat{a}}_r^{†}+{\widehat{a}}_t\right)\right]}{|0⟩}_r{|0⟩}_t,\end{array}$$

(31) $$\begin{array}{l}=e^{\left[\left(i\alpha /\sqrt{2}+i\beta /\sqrt{2}\right){\widehat{a}}_r^{†}-{\left(i\alpha /\sqrt{2}+i\beta /\sqrt{2}\right)}^{\ast }{\widehat{a}}_r\right]}\\ \cdot e^{\left[\left(\alpha /\sqrt{2}+\beta /\sqrt{2}\right){\widehat{a}}_t^{†}-{\left(\alpha /\sqrt{2}+\beta /\sqrt{2}\right)}^{\ast }{\widehat{a}}_t\right]}{|0⟩}_r{|0⟩}_t\end{array}$$

(32) $$={|i\left(\alpha +\beta \right)/\sqrt{2}⟩}_r{|\left(\alpha +\beta \right)/\sqrt{2}⟩}_t.$$

Again, in this situation, we have the classical behavior described in quantum formalism, paying attention to the relationship between $\alpha$ or $\beta$ and the total photon number in a coherent state:

(33) $$⟨\alpha \left|{\widehat{a}}_r^{†}\widehat{a}\right|\alpha ⟩=⟨\alpha \left|\widehat{n}\right|\alpha ⟩={\left|\alpha \right|}^2,$$

(34) $$⟨i\alpha /\sqrt{2}|{\widehat{a}}_r^{†}{\widehat{a}}_r{|i\alpha /\sqrt{2}⟩}_r=⟨i\alpha /\sqrt{2}|{\widehat{n}}_r{|i\alpha /\sqrt{2}⟩}_r=\frac{{\left|i\alpha \right|}^2}{2},$$

(35) $$⟨\alpha /\sqrt{2}|{\widehat{a}}_r^{†}{\widehat{a}}_t{|\alpha /\sqrt{2}⟩}_t=⟨\alpha /\sqrt{2}|{\widehat{n}}_t{|\alpha /\sqrt{2}⟩}_t=\frac{{\left|\alpha \right|}^2}{2}.$$

which reproduces classical field addition at the outputs within the quantum formalism. Physically, a coherent state incident on a balanced beam splitter divides, on average, its photon number equally between the two outputs; the reflection phase ($\pi /2$) does not change the photon number, and each output remains coherent (Poisson distribution).

3. Outlook and Experimental Implications

The unified framework presented here is not only of conceptual value but also has direct implications for the design and interpretation of modern quantum optical experiments. In particular, beam splitters play a central role in quantum key distribution (QKD) protocols, boson sampling, and linear-optical quantum computing architectures, where input-output correlations must be precisely engineered and measured [¹¹, ¹², ¹³].

In such contexts, the beam splitter acts as a diagnostic tool. When the input states are classical (e.g., coherent states), the output intensities are statistically independent and follow predictable power-splitting ratios. In contrast, nonclassical input states, such as single-photon Fock states or squeezed states, lead to output behavior that violates classical inequalities; for example, the suppression of coincidence counts at the outputs (Hong–Ou–Mandel interference) is a clear signature of quantum indistinguishability and photon interference.

These features can be used to benchmark the quantumness of a photonic circuit, validate the fidelity of quantum logic gates, and detect decoherence and mode mismatch. Moreover, as beam-splitting elements are increasingly integrated on-chip with metasurface control or tunable splitting ratios, the formalism developed in this study can guide the design of circuits that transition dynamically between the classical and quantum regimes, depending on the desired functionality.

Over recent decades, beam-splitting functionality has migrated from bulk optics into both fiber-optic and integrated-chip platforms:

• Fiber-optic beam splitters are produced via directional couplers or fused biconical-taper techniques, fusing and tapering two fibers to allow evanescent-field coupling [¹⁴, ¹⁵]. These inline couplers feature low insertion loss ($\underset{˜}{\mathrm{<}}\mathrm{}0.2\mathrm{}\text{dB}$) and wide bandwidth, with standard split ratios of 50:50, 75:25, 90:10, 99:1, and 99.9:0.1, and custom ratios (e.g., 80:20) available from international manufacturers such as Thorlabs [¹⁶].

• On-chip photonic beam splitters leverage lithographically defined waveguides on platforms such as III–V materials (e.g., InAs or InP), silicon-on-insulator (SOI), silicon nitride (${\mathrm{Si}}_3{N}_4$), and lithium niobate on insulator (LNOI). Implementations include Y-branch splitters, multimode interference (MMI) couplers, and directional couplers, offering compact footprints (tens of microns), high extinction ratios, and broad spectral operation – key components of large-scale quantum photonic circuits and optoelectronic hybrids [¹⁷].

• Next-generation metasurface-based splitters monolithically integrate vectorial metasurfaces with standard photonic platforms, enabling tunable splitting ratios, polarization control, and submicrometer footprints. Recent studies have achieved on-chip integration with variable split ratios and enhanced functionality for advanced photonic systems [¹⁸, ¹⁹].

4. Conclusion

By analyzing the beam splitter using both classical wave optics and quantum field theory, we have shown that a single compact formalism can simultaneously accommodate energy-conserving field amplitudes, four-port scattering matrices, and the operator language of bosonic modes. In this unified picture, classical interference patterns emerge as the expectation values of field operators, whereas distinct quantum phenomena – antibunching, sub-Poissonian statistics, and Hong–Ou–Mandel coalescence – arise from higher-order, normally ordered correlation functions. The Glauber displacement operator links these regimes by demonstrating that coherent states produced from the vacuum via a phase-space shift possess classical mean fields while retaining intrinsic photon-number fluctuations.

Crucially, the beam splitter itself acts as a diagnostic interface:

• Classical or quasi-classical inputs (coherent states) yield two output modes whose intensities follow deterministic power-splitting ratios and reproduce the familiar interference fringes of a Mach–Zehnder interferometer.

• Nonclassical inputs (single-photon or photon-number states) produce output correlations that violate classical bounds, enabling direct observation of quantum statistics and entanglement.

This dual sensitivity positions the beam splitter as a uniquely powerful probe at the interface between classical and quantum optics, revealing the point at which classical descriptions of light become insufficient and quantum effects emerge. As beam-splitting technologies evolve toward fiber-based, planar, and metasurface-integrated architectures with increasingly compact footprints, the unified theoretical framework presented herein provides a consistent foundation for optimizing both classical performance metrics (insertion loss, bandwidth, polarization independence) and quantum figures of merit (visibility, photon indistinguishability, entanglement fidelity). This integrated approach paves the way for photonic circuits that can dynamically transition between classical signal processing and quantum information processing. As quantum photonic platforms mature, the theoretical tools discussed here furnish a basis for modeling and interpretation, as well as for the design and validation of experiments that probe the nonclassical features of light.

Acknowledgments

This work was supported by FINEP (Project QUANTUM – Pesquisa e Desenvolvimento de Tecnologias Quânticas para Segurança e Defesa Nacional), the Brazilian National Council for Scientific and Technological Development (CNPq) as part of the research program CNPq/MCTI 26/2023 on quantum communication technologies. Additional support was provided by the São Paulo Research Foundation (FAPESP; process number 2022/00209-6). The authors acknowledge the institutional support of the Institute of Physics at the University of Brasília (UnB) and the Military Institute of Engineering (IME, Rio de Janeiro), which provided a collaborative environment for the theoretical development of this study.

Data Availability

All data that support the findings of this study are included within the article.

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Edited by

• Editor-in-Chief:
  Marcello Ferreirahttps://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  09 Jan 2026
• Date of issue
  2025

History

• Received
  14 Aug 2025
• Reviewed
  06 Nov 2025
• Accepted
  07 Nov 2025