Tunable entangled photon-pair generation in a liquid crystal

Material and sample preparation

The material used in this study is FNLC-1751 supplied by Merck Electronics KGaA. FNLC-1751 shows a stable ferroelectric nematic phase at room temperature, with the phase sequence Iso 87 °C, N 57° C, M2 45 °C, NF on cooling, where Iso refers to the isotropic phase, N to the non-polar nematic phase, M2 to the so-described splay modulated antiferroelectric nematic phase^2,44,45 and NF to the ferroelectric nematic phase.

The material was confined in glass LC cells filled by capillary forces at 100 °C in the isotropic phase. We used both commercially available and homemade cells. In the latter case, soda-lime-glass square 2 cm × 2 cm plates coated with a transparent indium-tin-oxide conductive layer were assembled with plastic bead (EPOSTAR) spacers to achieve a variety of cells with different thicknesses ranging from 7 μm to 8 μm. In the bottom glass, indium-tin-oxide electrodes with a 500-μm gap prepared by etching created the applied in-plane fields. In addition, both plates were treated with a 30% solution of polyimide SUNEVER 5291 (Nissan) film and rubbed to achieve orientational in-plane anchoring (planar alignment) of the LC. Combinations of different relative rubbing directions of the top and bottom glass plates (0, π/2 or π) result in different twist structures of the LC sample in the ferroelectric nematic phase^5,13,46. In the three cases, the electrode glass was rubbed along the gap. In-plane switching LC cells purchased from Instec were used for switching experiments on π-twisted structures. The cells had interdigitated electrodes in one of the substrates with alternating polarity; both the electrode width and the gap between them were 15 μm. The surfaces had antiparallel rubbing along the electrodes (aligning agent KPI300B).

After filling at 95 °C, the sample was brought to room temperature by controllably decreasing the temperature at a rate of 0.5 °C min⁻¹. In the case of initially large domains breaking down into smaller ones after applying large voltages, the initial configuration was restored by reheating the samples and subsequent controlled cooling back to room temperature. The quality of the achieved alignment was inspected via polarizing optical microscopy and comparison with transmission spectra simulations (Berreman 4 × 4 matrix method performed with the open software package ‘dtmm’⁴⁷, cell thickness 7.6 μm, and ordinary and extraordinary refractive indices as given in Extended Data Fig. 2).

Photon-pair generation and detection

The scheme of the experimental set-up used for photon-pair generation and detection is depicted in Extended Data Fig. 5. As a pump, we used a continuous-wave pigtailed single-mode fibre diode laser with a central wavelength of 685 nm. After the power and polarization control, the pump beam was focused into the LC cell with a focusing spot size of 5 μm. The maximum delivered pump power did not exceed 10 mW. A function generator applied to the cell an electric field with different time profiles (Fig. 2). The generated photon pairs were collected with a lens with a numerical aperture of 0.69. Then a set of long-pass filters with a cut-on wavelength no longer than 1,250 nm cut the pump and short-wavelength photoluminescence from the sample and the optical elements of the set-up. Photon pairs were further sent into a Hanbury Brown–Twiss-like (HBT) set-up comprising a non-polarizing beamsplitter and two superconducting nanowire single-photon detectors (SNSPDs). At each output of the non-polarizing beamsplitter, we placed a set of a half-wave plate, a quarter-wave plate and a polarizing beamsplitter, which acted as a polarization filter. The arrival time differences between the pulses of both SNSPDs were registered by a time-tagging device.

Two-photon spectrum measurement

As the SPDC radiation from an 8-μm layer is extremely weak, direct measurement of the two-photon spectrum (that is, with a spectrometer or optical spectrum analyser) is nearly impossible. Therefore, we measured the spectrum of the detected photon pairs via single-photon fibre spectroscopy²⁸. Before one of the SNSPDs, we inserted a 2-km-long dispersion-shifted fibre with a zero-dispersion wavelength at 1.68 μm. Owing to the dispersion of the fibre, the photon wavepacket stretched in time, resulting in a spread of the coincidence peak, which then inherited the spectrum’s features and the spectral losses of the set-up. We acquired the coincidence histogram with different sets of spectral filters (Extended Data Fig. 6a) to map the arrival time differences to the corresponding wavelengths of the dispersed photon. The calibration curve (Extended Data Fig. 6b) was obtained by fitting the reference points with a quadratic polynomial function. However, the spectrum is strongly affected by the spectral losses of the set-up and the dispersive fibre. For that reason, we additionally measured the spectrum of photon pairs generated in a thin (7 μm) layer of LiNbO₃ (Extended Data Fig. 6c), where the generated two-photon spectrum is mostly flat, up to a modulation by the Fabry–Pérot effect inside the layer. We then used the spectrum of photon pairs from the LiNbO₃ wafer as a reference spectrum.

Two-photon-state reconstruction

We performed quantum tomography to reconstruct the two-photon polarization state generated in the LC. The procedure is analogous to measuring the Stokes parameters for classical light or a single photon. By measuring the pair detection rates for different polarization states filtered in the two arms of the HBT set-up, we were able to reconstruct the density matrix of the two-photon state. As there is no prior assumption about the generated two-photon state, we performed all 9 required measurements for the reconstruction of the 3 × 3 density matrix. The full protocol is described in Extended Data Fig. 7. The values in the table refer to the orientation of the fast axis of each wave plate with respect to the horizontal direction. It is worth mentioning that the described protocol does not take into account the mirroring effect of polarization in the reflected arm of the HBT set-up. Therefore, either the angles of the wave plates in the reflected arm must be changed to the opposite values, or an odd number of mirrors must be used in the reflected arm of the HBT set-up. The protocol used for the qutrit state reconstruction is the reduced version of the protocol for the reconstruction of the two-photon polarization state with two distinguishable photons (ququart state)³⁰.

To avoid systematic errors in the density-matrix reconstruction, we additionally post-processed the measured data using the maximum likelihood method. The maximum likelihood method aims to find the density matrix closest to the measured one that satisfies all basic physical properties of a density matrix. We used a procedure similar to the one described in ref. ³⁰ with minor modifications (Supplementary Information Section 4).

Theoretical model of SPDC in LCs

We developed a theoretical model to predict the polarization two-photon state generated via SPDC in a nonlinear LC with an arbitrary but linear molecular orientation twist along the cell. The goal is to determine the complex amplitudes of the polarization two-photon state C₁, C₂ and C₃ from equation (1). We assumed a single-mode, collinear and frequency-degenerate photon-pair generation in the plane-wave approximation for simplicity. However, the model can be further extended towards the multi-mode regime of SPDC with realistic angular and frequency spectra, as well as for the case of a non-gradual molecular twist.

Owing to weak interaction, we can use perturbation theory for the unitary transformation of the state vector⁴⁸. The state can be written as

where \({a}_{{\rm{s}}}^{\dagger }\) and \({a}_{{\rm{i}}}^{\dagger }\) are the photon creation operators for signal and idler photons, respectively, each of them defined in some polarization eigenmode, χ⁽²⁾(z) is the second-order nonlinear tensor, and the polarization vectors e_s,i,p(z) also encode the phase accumulation during the propagation along the crystal of length L. Variable z marks the direction of propagation. The constant C contains only the information about the overall generation efficiency and, therefore, is of no interest to us.

For convenience, we use two polarization bases instead of the polarization eigenmodes. The first basis is a standard linear polarization basis with horizontal and vertical polarizations determined with respect to the laboratory coordinate system, {H, V}. In this basis, the two-photon polarization state can be expressed as a qutrit state (1) as two photons are assumed to be indistinguishable in all other Hilbert spaces apart from polarization³². The final goal of the calculations is to determine complex amplitudes C_1,2,3 from equation (1). As the molecular orientation changes along the crystal and implies the spatial modulation of the nonlinearity, it is more convenient to calculate the convolution of the χ⁽²⁾ tensor with the polarization vectors of the interacting photons in the second basis aligned with the instant orientation of the molecules, {e, o}. We denote the corresponding projections with indices e and o for the linear polarization along and orthogonal to the instant molecular orientation, respectively. Instead of the χ⁽²⁾ tensor, we use the standard notation of the Kleinman d tensor. Therefore, the convolution is written as

where the polarization basis vectors and the tensor components are functions of z, and the convolution is defined in the local coordinate system of the molecules. The z direction is defined in the same way for both bases and denotes the photon propagation direction along the crystal.

To calculate the polarization two-photon state, we consider an LC with a uniform rotation of the molecules along the crystal (Extended Data Fig. 8). At an arbitrarily chosen layer of thickness dz at position z, the pump polarization is modified by all the previous layers it has passed through. The polarization state of photon pairs generated from the corresponding layer dz is further modified by all subsequent layers of the LC. The final state at the output of the crystal is the superposition of all polarizations generated along the crystal. Therefore, to calculate the output two-photon polarization state, we integrate the contribution of each layer of the LC taking into account the corresponding polarization transformations of both the pump and the incremental photon-pair state generated from each layer.

To calculate the propagation of the pump, the initial pump polarization is represented by a Jones vector (Extended Data Fig. 8) in the {H, V} basis. The angle φ₀ is defined as the angle between both coordinate systems at the beginning of the sample, that is, the angle between the global coordinate H direction and the extraordinary molecule axis e at the beginning of the sample. The first step is to bring the pump from the global basis to the local basis at the beginning of the sample via rotating the pump Jones vector by φ₀:

where R is the standard rotation matrix. The polarization transformation of light propagating through a twisted nematic LC (TLC) with a uniform twist is described by the corresponding Jones matrix^49,50,51

Here, \(\phi =\widetilde{k}l\) is the average phase acquired by both polarizations, with \(\mathop{k}\limits^{ \sim }=\frac{1}{2}({k}^{{\rm{e}}}+{k}^{{\rm{o}}})\) being the average k vector and l being the thickness of the TLC layer performing polarization transformation; φ is the twist angle; \(\beta ={\rm{\pi }}l({n}_{{\rm{e}}}-{n}_{{\rm{o}}})/\lambda =gl\) characterizes birefringence, where we introduce notation \(g=\frac{1}{2}({k}^{{\rm{e}}}-{k}^{{\rm{o}}})\), and n_o and n_e are ordinary and extraordinary refractive indices of the sample at optical wavelength λ, respectively. The additional parameter X is defined as \(X=\sqrt{{\varphi }^{2}+{\beta }^{2}}\).

At a certain chosen position z, the pump polarization is transformed by the part of the LC from −L to z, with the effective length of this layer being z + L. The pump polarization vector in the local basis at position z then has the form

where φ denotes the full twist of the sample. We intentionally leave the pump polarization defined in the local basis as it is convenient for calculating its convolution with \({\widehat{\chi }}^{(2)}\). We explicitly write the pump polarization vector at position z in the local basis as a function of the input pump polarization in the {H, V} basis

where

By inserting these expressions into equation (3), we can find the polarization state of photon pairs generated from a unit layer at position z in the local basis. However, as we are interested in the output polarization state, the polarization of both signal and idler photons must be propagated from z to the end of the crystal in a similar way. This transformation can be written as

where the photons are propagating from z to 0. The explicit form of the output polarization for the signal and idler photons generated at z is

with similar notation as before

To perform convolution (3), equation (10) needs to be reversed to express \({e}_{{\rm{s}},{\rm{i}}}^{{\rm{e}},{\rm{o}}}(z)\) as functions of the outcome polarizations \({e}_{{\rm{s}},{\rm{i}}}^{{\rm{H}},{\rm{V}}}\). With this transformation, alongside equations (3) and (7) the convolution is written as

where further notation shortening was introduced via

To find the state, we have to substitute the components of the Jones vectors \({{\bf{e}}}_{{\rm{s}},{\rm{i}}}^{{\rm{H}},{\rm{V}}}\) with the corresponding photon creation operators. In this case, transformations (7) and (10) are equivalent to the unitary transformations of a beamsplitter with two input and two output polarization modes. Substituting (12) into (2) and grouping the components with the same pair of the creation operators, we can finally find the two-photon polarization state in the qutrit form (1) with the complex amplitudes

The polarization state vector has to be further normalized with the norm \(\sqrt{| {C}_{1}{| }^{2}+| {C}_{2}{| }^{2}+| {C}_{3}{| }^{2}}\). Although we use the normalized values of the complex amplitudes for the analysis of the two-photon polarization state (Extended Data Fig. 9), the norm itself shows the relative generation efficiency for different parameters of the LC, such as length and twist (Fig. 4c,d).

Further development of the model involves more strict quantum-optical calculations, with the real angular and spectral distributions of the generated photons, as well as the spatial properties of the pump beam, internal reflections of both the pump and the generated photons, and so on. Furthermore, the approximation of a non-depleted pump is valid only in the low-gain regime of SPDC, while such a source is incredibly promising for generating squeezed vacuum and twin beams. Finally, we assume a perfect uniform twist of the molecules, which is hard to achieve experimentally, especially for twists not multiple to π. Although this model is significantly simplified, it proved to be reliable and provides a great insight into the physics of this type of material.