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Frontiers in Photonics ·

Volume 6 - 2026 | https://doi.org/10.3389/fphot.2025.1685128

Mach–Zehnder interferometers incorporating electrochromic molecules for controlled single-photon detection

  • Francesco Scotognella1*corresponding author
  1. Department of Applied Science and Technology, Politecnico di Torino, Torino, Italy
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Abstract

One particularly fruitful research in the fields of integrated photonics, carried out by a good number of physicists and engineers, concerns the study of different types of materials to be used to control the detection of photons, if not just single photons, in interferometers. In a Mach–Zehnder interferometer, which consists of two beam-splitters, two mirrors, and two detectors, a material that can cause a controlled change in the phase of light in one of the two arms of the interferometer consequently allows control of the probability of detection at the two detectors. In this work, we use an electrochromic molecule, N,N′-bis(cysteine)pyromellitic diimide (BCPD), that has a refractive index dependent on the applied electric field. We simulate the single-photon detection probability in a Mach–Zehnder interferometer with direct light transmission and a waveguide-based Mach–Zehnder interferometer, consisting of two 3-dB couplers connected by two optical channel waveguides. With the employment of the non-equilibrium Green’s function formalism, we have simulated the conductance of BCPD. The results could be of interest in quantum communication.

Keywords

Introduction

Electrochromic molecules are materials that can reversibly change their optical properties, such as color or transparency, through redox reactions triggered by an external electrical potential. This phenomenon, known as electrochromism, has gained great interest due to its potential applications in energy-efficient smart windows, displays, wearable electronics, and multifunctional energy-storage devices (Pradhan et al., 2025; Chen et al., 2025; Cho et al., 2025). Among the electrochromic molecules, viologens and viologen-based materials are the most intensely studied species (Bird and Kuhn, 1981; Stolar, 2020; Ambrose et al., 2024; Otaegui et al., 2025). Other noteworthy electrochromic molecules are graphene molecules, also called polycyclic aromatic hydrocarbons (Ji et al., 2015; Stec et al., 2017; Chapkin et al., 2020). Moreover, the small molecule N,N′-bis(cysteine)pyromellitic diimide (BCPD) is a good example of a short molecule that exhibits a distinct and reversible color change (Stolar, 2020; Abdinejad et al., 2017; Abdinejad et al., 2018; Guo et al., 2014).

The color change, which is related to a change in the complex refractive index of the material, can also be exploited as a phase shift component in a Mach–Zehnder interferometer (Rioux, 2019). Mach–Zehnder interferometers with photochromic molecules or polymers—and with vanadium dioxide exploiting its insulating-to-metallic phase change—have been reported (Scotognella, 2024a; Scotognella, 2024b). Mach–Zehnder interferometers that are based on direct transmission of light are composed of a light source, two beam-splitters, two mirrors, and two detectors (Rioux, 2019). Mach–Zehnder interferometers that are based on channel waveguides, e.g., in a glass slide, are composed of a light source, two 3-dB couplers that have the same function as the two beam-splitters, and two detectors (De Ridder et al., 2006; Ng et al., 2012).

In this study, we use an electrochromic molecule, namely, BCPD (Stolar, 2020; Abdinejad et al., 2017; Abdinejad et al., 2018), as the material in both arms of a Mach–Zehnder interferometer for direct transmission of light and in a Mach–Zehnder interferometer composed of channel waveguides, e.g., buried in a glass slab. We simulate the application of the electric field on BCPD via the addition of an electron charge to the molecule, with a subsequent change of its complex refractive index. The electric field-induced change in the BCPD refractive index in one of the arms of the interferometers allows control of the photon detection in the two detectors of the apparatus. Moreover, by using the non-equilibrium Green’s function formalism, we found the conductance of BCPD.

Methods

To study the optical response of BCPD, we employed the Hückel theory. Specifically, with this model, we simulated the absorption bands of the BCPD molecule in its neutral its charged states. The Hückel theory is an established molecular orbital theory in which the molecular orbitals are written as the linear combination of atomic orbitals, in which the nuclei positions are fixed (this is a strong approximation, and more accurate calculations can go beyond this approximation (Janoš et al., 2025), but such approaches are not considered in this work) and the molecular orbitals are linear combinations of carbon p *z* orbitals (other elements can also be considered), neglecting electron–electron interactions (Guy and Troy, 2022). Equation 1 presents the time-independent Schrödinger equation:

H ψ = E ψ . (1)

In this study, the HuLiS package (Huckel Theory and HuLiS, 2013) was employed to determine the Hamiltonian for BCPD (neutral and charged). Then, eigenvalues and eigenvectors were calculated. To find the transition probabilities, we employed the Fermi golden rule, as reported in Equation 2:

Γ i → j = 2 π ℏ ψ j H ψ i 2 . (2)

Using Equation 2, the transition probabilities were determined between the highest occupied molecular orbital (HOMO) and the unoccupied orbitals. The simulated absorption spectra were built using a set of Gaussian peaks, with a peak linewidth c of 0.1 eV. Considering the transitions from the ground ith state to the different j th excited states, the absorption coefficient can be written as shown in Equation 3:

α E = ∑ j Γ i → j ⁡ exp E − E j − E i 2 2 c 2 . (3)

For clarity, we emphasize that Hückel’s theory finds six unoccupied states for BCPD. Therefore, the simulated absorption spectrum is given by the six transitions from HOMO to the six unoccupied states. For the calculations based on the density functional theory (DFT), the following procedure was followed. The molecules were designed using the Avogadro package (Hanwell et al., 2012). We optimized the geometries and calculated the electronic transitions via density functional theory with the ORCA package developed by Neese (2012). In these calculations, we used the B3LYP functional (Lee et al., 1988). Moreover, we employed the Ahlrichs split valence basis set (Schäfer et al., 1992) together with the all-electron nonrelativistic basis set SVPalls1 (Schäfer et al., 1994; Eichkorn et al., 1997). Finally, for these calculations, we used the Libint library (Valeev, 2014) and the Libxc library (Lehtola et al., 2018; Marques et al., 2012).

Using Equation 4, modified from the equation reported by Kohandani and Saini (2022), the imaginary part of the complex refractive index can be extracted from the absorption coefficient:

k E = A α E 4 π . (4)

In Equation 4, A is a parameter (its value is 1 × 10 15). Another formalism for determining the imaginary part of the refractive index was reported by Dresselhaus (2017). Notably, the parameter A has been chosen to reproduce the experimental data. The real part of the complex refractive index can be derived via the Kramers–Kronig relations (Pankove, 1975) (Equation 5):

n ν ¯ − 1 = 2 π P ∫ 0 ∞ ω k ω ω 2 − ν ¯ 2 d ω . (5)

The Kramers–Kronig relations are actually Hilbert transforms, as in the formalism reported by Ogilvie and Fee (2013). For the real part of the refractive index, a constant offset of 1.5 has been used, as detailed by Wiebeler et al. (2014) and Scotognella (2020). Such an offset is used to consider the experimental factors, such as light scattering. Thus, Equation 6 can be used to describe the complex refractive index of BCPD in the neutral (n *n*) and charged (n *c*) forms:

n n , c c o m p l e x E = n n , c E + i k n , c E . (6)

In the model, we simulated the contribution of the electric field applied to the molecule by studying the optical properties of the charged molecule, i.e., the molecule to which an electron has been added. The change in the electronic configuration of the molecule, i.e., the different filling of the molecular orbitals in the charged molecule compared to that in the neutral molecule, gives rise to different absorption bands, with a consequent variation in the complex refractive index of the molecule.

The non-equilibrium Green’s function formalism was employed for the calculation of the elastic transmission, following Solomon et al. (2011). Assuming that only a single site of the molecule couples to each electrode, for BCPD, the vector for the left electrode is provided in Equation 7:

V L = γ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . (7)

On the other hand, the vector for the right electrode is provided in Equation 8:

V R = 0 0 0 0 0 0 0 0 0 0 γ 0 0 0 0 0 . (8)

Following the sketch of BCPD in Figure 1, the non-zero element of VL is the first one, representing the connection between the nitrogen atom of BCPD indexed “1” and the electrode (which will replace the hydrogen atom). Similarly, the non-zero element of VR is the 11th one, representing the connection between the nitrogen atom of BCPD indexed “11” and the electrode (which also replaces the hydrogen atom next to the nitrogen atom). The value of γ is −1 eV, as reported by Solomon et al. (2011) for different studied molecule/electrode junctions. A similar value was used for another molecular/electrode system by Chang et al. (2014). The broadening function is provided in Equation 9:

Γ L R = 2 π ρ V L R V L R † . (9)

Chemical sketch of N,N′-bis(cysteine)pyromellitic diimide. The atoms are numbered as in the Hückel Hamiltonian (the four hydrogen atoms are not numbered).
Figure 1. Chemical sketch of N,N′-bis(cysteine)pyromellitic diimide. The atoms are numbered as in the Hückel Hamiltonian (the four hydrogen atoms are not numbered).

Here, ρ is the density of state of the electrode, which is set to 1/2π (eV)−1 (following the formalism presented by Solomon et al. (2011)). With the determination of the broadening function from Equation 9, we are able to find the tunneling self-energy, which is purely of imaginary value, using Equation 10:

Σ T L R = − i 2 Γ L R . (10)

In Equation 11, the energy-dependent retarded Green’s function is obtained as follows:

G r E = E − H m o l − Σ T L − Σ T R − 1 . (11)

It is noteworthy that the advanced Green’s function G a E is the conjugated transpose of the retarded Green’s function. Using the broadening functions and Green’s functions, it is possible to determine the energy-dependent elastic transmission, as provided in Equation 12:

T E = T r Γ L G r E Γ R G a E . (12)

The transmission is related, through Equation 13, to the conductance by the relation (Chang et al., 2014).Here, G 0 is the quantum of conductance (with value 7.748 × 10 − 5 S).

G = G 0 T E F . (13)

For the Mach–Zehnder interferometer with direct transmission of light, the state vectors and operators are implemented in agreement with Rioux (2019). Equation 14 describes the photon moving along the x-axis as the state vector:

x = 1 0 . (14)

Meanwhile, in Equation 15, the photon moving on the y-axis is represented by the state vector:

y = 0 1 . (15)

The two mirrors are represented by the operator M (Equation 16):

M = 0 1 1 0 . (16)

Equation 17 describes the two beam-splitters that are represented by the operator BS:

B S = 1 2 1 i i 1 . (17)

A 90° phase shift is assigned to the reflection. The BCPD-based electrochromic (BCPD) layer is represented by the operator (Equation 18):

B C P D = exp i 2 π d n n , c c o m p l e z λ 0 0 exp i 2 π d n n , c c o m p l e z λ . (18)

Here, d is the thickness of the layer. The thickness of the BCPD layers is 2 µm. This value for the thickness was chosen to obtain a significant phase difference between the BCPD film in the neutral form and BCPD in the charged form.

With the BCPD-based electrochromic layers in the Mach–Zehnder interferometer, the probability of measuring the photon using detector D1 is provided in Equation 19:

D 1 B C P D = x T B S M B C P D B S x 2 . (19)

Meanwhile, the probability of measuring the photon using detector D2 is obtained as shown in Equation 20:Here, x T is the transpose of vector x, while y T is the transpose of vector y.

D 2 B C P D = y T B S M B C P D B S x 2 . (20)

In the waveguide-based Mach–Zehnder interferometer, the phase shift for two branches of equal length L is provided in Equation 21 (De Ridder et al., 2006):

Δ φ = β 2 L − β 1 L . (21)

The parameters β 1 and β 2 in Equation 21 can be defined in the following way (Equation 22):

β 1 , 2 = 2 π n 1 , 2 λ . (22)

In Equation 22, n 1 , 2 is the complex refractive index for the materials in the two branches of the Mach–Zehnder interferometer. For this study, n 1 = n n c o m p l e x and n 2 = n c c o m p l e x. The output powers at ports 1 and 2 (Figure 2b) are obtained using Equations 23 and 24, respectively:

P o u t , 1 = sin 2 Δ φ 2 P i n , 1 + cos 2 Δ φ 2 P i n , 2 , (23)

P o u t , 2 = cos 2 Δ φ 2 P i n , 1 + sin 2 Δ φ 2 P i n , 2 . (24)

**(a)** Sketch of the Mach–Zehnder interferometer with direct transmission of light, showing the photon source, the two mirrors M, the two 50/50 beam-splitters BS1 and BS2, the two electrochromic layers BCPDx (along the x-direction transmitted by BS1) and BCPDy (along the y-direction reflected by BS1), and the two detectors D1 and D2. **(b)** Waveguide-based Mach–Zehnder interferometer with inputs In1 and In2, two 3-dB couplers, and two outputs Out1 and Out2. In one of the branches of the interferometer, the electric field induces a change in the refractive index of BCPD.
Figure 2. (a) Sketch of the Mach–Zehnder interferometer with direct transmission of light, showing the photon source, the two mirrors M, the two 50/50 beam-splitters BS1 and BS2, the two electrochromic layers BCPDx (along the x-direction transmitted by BS1) and BCPDy (along the y-direction reflected by BS1), and the two detectors D1 and D2. (b) Waveguide-based Mach–Zehnder interferometer with inputs In1 and In2, two 3-dB couplers, and two outputs Out1 and Out2. In one of the branches of the interferometer, the electric field induces a change in the refractive index of BCPD.

Results and discussion

Figure 1 shows the chemical sketch of the molecule BCPD. The carbon (black), oxygen (red), and nitrogen (blue) atoms are indexed as the Hückel Hamiltonian.

In the Hamiltonian of BCPD reported in Equation 25, the diagonal elements correspond to the different atoms of the molecules, while the non-null off-diagonal elements correspond to nearest neighbor elements:

H = α N β N 0 0 β N 0 0 0 0 0 0 0 0 0 0 0 β N α β 0 0 0 0 0 0 0 0 0 β O 0 0 0 0 β α β 0 β 0 0 0 0 0 0 0 0 0 0 0 0 β α β 0 0 0 β 0 0 0 0 0 0 0 β N 0 0 β α 0 0 0 0 0 0 0 0 0 0 β O 0 0 β 0 0 α β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β α β 0 0 0 β 0 0 0 0 0 0 0 0 0 0 β α β β 0 0 0 0 0 0 0 0 0 β 0 0 0 β α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β 0 α β N 0 0 0 β O 0 0 0 0 0 0 0 0 0 0 β N α N β N 0 0 0 0 0 0 0 0 0 0 β 0 0 0 β N α 0 β O 0 0 0 β O 0 0 0 0 0 0 0 0 0 0 α O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β O 0 α O 0 0 0 0 0 0 0 0 0 0 0 β O 0 0 0 0 α O 0 0 0 0 0 β O 0 0 0 0 0 0 0 0 0 0 α O . (25)

Here, the on-site energy α is set to 0 eV, and the nearest neighbor element β is set to −3 eV, following the publication of Solomon et al. (2011). Equation 26 reports the other matrix elements:

α N = 0.51 β α O = 0.97 β β N = 1.02 β β O = 1.02 β . (26)

In the literature, the absorption spectrum of BCPD has been studied (Abdinejad et al., 2018). Neutral BCPD shows two absorption peaks at approximately 320 and 395 nm, while charged BCPD shows an additional peak at 605 nm. It is worth noting that the spectrum of charged BCPD is related to photocoloration, i.e., exposure to light at 365 nm. However, the authors show that the effects of photocoloration are very similar to the effects of electrocoloration in terms of the optical response of BCPD (Abdinejad et al., 2018). In Supplementary Figure S1 of Supplementary Material, the calculated light absorption spectra via the Hückel theory and via DFT have been reported for a comparison. Both the calculations are in fair agreement with the experimental data, considering that both predict the formation of the peak in red upon electrocoloration. Supplementary Material contains the transitions with the corresponding oscillator strengths and the geometry of BCPD related to the DFT calculations. Finally, in Supplementary Material, we list the weight of the individual excitations for each transition with which we have built the absorption spectrum employing density functional theory. In this work, although all single-particle transitions contribute in principle to the absorption spectrum, only a selected subset has been considered (i.e., the transitions between the HOMO and the six lowest unoccupied orbitals).

We studied the complex refractive index of BCPD in the neutral form and with the addition of a charge. We focused on a wavelength of 800 nm since commercially available single-photon sources are present in this wavelength.

The extinction coefficient k for the neutral BCPD is k n = 1.1788 × 10 − 23, while the extinction coefficient k for the charged BCPD is k c = 1.1078 × 10 − 16. The real part of the refractive index for the neutral BCPD is n *n* = 1.5773, while the real part of the refractive index for the charged BCPD is n *c* = 1.4901.

We have simulated the Mach–Zehnder interferometer with direct transmission of light, taking into account the 2-μm-thick slabs of BCPD. Examples of electrochromic organic films are reported in the literature (Rendón-Enríquez et al., 2023). Given that the possible formation of different types of molecular aggregates in thin films is an important aspect (Zhang et al., 2020), we would like to emphasize that in this work, we have not studied in depth the possibility of having BCPD aggregates in this slab, but this could be an interesting topic for future research. With BCPD in the neutral form along the x-axis and BCPD in the charged form along the y-axis (Figure 2a), the probability of detecting a photon at detector D1 is 0.8056, while the probability of detecting a photon at detector D2 is 0.1944.

We then simulated the waveguide-based Mach–Zehnder interferometer in a glass slab, with 2-μm-long arms. Considering the input power at port 2 equal to 0 (P i n , 2 = 0, Figure 2b) and considering branches of the interferometer with length L = 2 μm, the probability of detecting a photon at output port 1 is 0.4001, while the probability of detecting a photon at output port 2 is 0.5999.

Finally, with the non-equilibrium Green’s function formalism, we determined the conductance of BCPD. In Figure 3, we show the elastic transmission spectrum (in the logarithm scale) of BCPD. A node close to −3 eV is evident due to destructive interference (Solomon et al., 2011). Other nodes occurring in the spectrum are higher-order nodes, i.e., supernodes, at −2 eV and 2 eV, which can be ascribed to cross-conjugation in BCPD—the delocalization of π electrons branches into two or more non-continuous paths (Solomon et al., 2011; Zhao et al., 2025). When setting the Fermi energy at 0 eV in Equation 13, the conductance of BCPD is 0.127 µS.

Simulated elastic transmission spectrum as a function of the energy of N,N′-bis(cysteine)pyromellitic diimide.
Figure 3. Simulated elastic transmission spectrum as a function of the energy of N,N′-bis(cysteine)pyromellitic diimide.

Conclusion

In this work, we studied an electrochromic molecule, BCPD. Employing a simple method such as the Hückel theory, we were able to investigate the optical properties of BCPD in the neutral and charged forms, with good agreement with the experimental findings. The electrochromism of BCPD is related to the refractive index of the molecule, which depends on the applied electric field. Taking advantage of this refractive index control, we simulated the probability of single-photon detection on a Mach–Zehnder interferometer with direct light transmission and a waveguide-based Mach–Zehnder interferometer—for example, buried in a glass slab. The waveguide-based Mach–Zehnder interferometer can be useful in integrated photonic platforms. We also studied the conductance of BCPD with the non-equilibrium Green’s function formalism. The results of this study could be of interest for quantum communication.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphot.2025.1685128/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Alessandro Alabastri, Rice University, United States

Reviewed by: Luca Bursi, Dipartimento di scienze fisiche informatiche e matematiche, Italy

Andrea Schirato, Politecnico di Milano, Italy

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 816313) and the EIC LEAF (grant agreement No. 101186701).

Data availability

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

FS: Conceptualization, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review and editing.

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Citation

Scotognella, F. (2026). Mach–Zehnder interferometers incorporating electrochromic molecules for controlled single-photon detection. Frontiers in Photonics, Volume 6. https://doi.org/
Received
Accepted
Published
Volume
6 - 2026

Copyright

Copyright © 2026 Scotognella.

License: https://creativecommons.org/licenses/by/4.0/

*Correspondence: Francesco Scotognella

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