Large-scale photonic network with squeezed vacuum states for molecular vibronic spectroscopy

A squeezed vacuum state and a linear optical network for molecular vibronic spectra approximationMolecular vibronic spectra describe the molecular vibronic transition, and the vibronic spectral profile, called the Franck-Condon profile (FCP), which is obtained by computing the corresponding FCF, i.e., the square of the overlap integral of two wave functions in different electronic levels within the Born-Oppenheimer approximation and the Condon approximation31,32,33. In classical computing, solving the classically challenging problem of computing the FCF for a given vibronic molecular transition is equivalent to the loop hafnian of a particular matrix, which is a quantity related to the perfect matchings of a graph with loops6,13,31. The development of quantum computing puts forward a new way, which leads to an efficient solution to the complicated calculation of FCF in the quantum framework. For a molecule with n vibrational modes, as shown in Fig. 1a, the quantum computation of the vibronic spectra needs to consider the transformation between two potential energy surfaces. The normal coordinates of the final and initial electronic states are linearly related by the Duschinsky transformation34, which can be represented by the Doktorov operator \({\widehat{U}}_{{Dok}}\) and further expressed in three quantum operators: squeezing, rotation, and displacement (see detailed in Supplementary Note 1). As shown in Fig. 1b, the previous quantum computation for FCF requires producing squeezed coherent states with m modes. However, the experimental obstacle in squeezed coherent state preparation27 limits the application of this method.Fig. 1: A squeezed vacuum state and a linear photonic network for molecular vibronic spectroscopy.a Molecular structure. b A vibronic transition of a molecule includes displacement, squeezing and rotation operations. c In our algorithm, by extending m vibronic modes to 2 m optical modes and applying straightforward rotation and squeezing operations, it is sufficient to reproduce vibronic spectra. d Photonic quantum circuit model, by translating the Doktorov operator to squeezing operators \(\hat{S}\left(r\right)\) and rotation operator \({\hat{R}}_{V}\). Output photon sampling distribution followed by post-processing can generate the vibronic spectroscopy of the molecule.To perform the quantum computation of FCF efficiently in the experiment, we build a relation between the loop hafnian and the hafnian (the number of perfect matchings of a graph without loops). Since the boson sampling protocol establishes a link between the hafnian matrix function and the output pattern within easily attainable linear optics35,36,37, it is straightforward to implement the FCF experimentally within the quantum frame. Through the mode expansion, the original displacement parameter of the i-th vibrational mode is represented by the covariance between the i-th and (m + i)-th optical modes (see detailed derivation in Supplementary Note 1). The original vibrational transition with m modes is related to squeezing and rotation operations with 2 m optical modes, as shown in Fig. 1c. The established relation between FCF and the hafnian function of matrix B and its extension, matrix A, is$${{{{\rm{FCF}}}}}\left({{{{\bf{n}}}}}\right) ={{{{\mathcal{N}}}}}{\left|{lhaf}\left({\bar{B}}^{\left({{{{\bf{n}}}}}\right)}\right)\right|}^{2}\\ ={{{{\mathcal{N}}}}}{\left|{\sum }_{{{{{\bf{l}}}}}={{{{\bf{0}}}}}}^{{{{{\bf{n}}}}}}\frac{1}{{{{{\bf{l}}}}}!}{haf}\left({A}^{\left({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}}\right)}\right)\right|}^{2},$$
(1)
where \(\bar{B}\) is a matrix generated by replacing the diagonal entries of \(B\) with displacement vector \({{{{\bf{K}}}}}\), in which B is the matrix correlated to the molecular vibronic parameters (detailed in Supplementary Note 1). \({{{{\mathcal{N}}}}}\) is a normalization constant, \({lhaf}({{{{\rm{\cdot }}}}})\) is the loop hafnian function of the matrix, \({haf}(\cdot )\) is the hafnian function of the matrix, and \(({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}})\) is an output pattern. We denote \(({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}})\) by (\({n}_{1},{n}_{2},\,\ldots,\,{n}_{m},\,{l}_{1},{l}_{2},\,\ldots,\,{l}_{m}\)), where \({n}_{k}\) corresponds to the number of photons in the \(k\) th mode and \({l}_{k}\) corresponds to the number of photons in the \(\left(m+k\right)\) th mode. \({A}^{({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}})}\) is a submatrix of \(A\) depending on the measured output pattern \(({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}})\) with the matrix \(A=\left[\begin{array}{cc}B & Z\\ Z & 0\end{array}\right]\), in which \(Z\) is a diagonal matrix associated with the displacement vector between the equilibrium positions of the energy surfaces.Meanwhile, the probability amplitudes of an output pattern are proportional to the square of the hafnian function, \(\Pr \left({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}}\right)=\frac{1}{{{{{\bf{n}}}}}{{{{\boldsymbol{!}}}}}{{{{\bf{l}}}}}{{{{\boldsymbol{!}}}}}\sqrt{{\sigma }_{Q}}}{{|haf}({A}^{({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}})})|}^{2}\), where \({\sigma }_{Q}=\sigma+{I}_{4m}/2\) with \({I}_{4m}\) being the \(4m\times 4m\) identity matrix and \(\sigma\) being the covariance matrix37,38. Moreover, a sign approximation is used such that the sign of the hafnian in the summation of Eq. (1) is ignored (see detailed discussion in Supplementary Note 1). Therefore, the relationship between \(\Pr \left({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}}\right)\) and the FCF can be established, where the sampling results are used to approximate the molecular vibronic spectra. The approximated FCF (\(\widetilde{{{{{\rm{FCF}}}}}}\left({{{{\bf{n}}}}}\right)\)) at \(0K\) is expressed as$$\begin{array}{c}\widetilde{{{{{\rm{FCF}}}}}}\left({{{{\bf{n}}}}}\right){{{{\mathscr{=}}}}}{{{{\mathcal{N}}}}}{\left|{\sum }_{{{{{\bf{l}}}}}{{{{\boldsymbol{=}}}}}{{{{\bf{0}}}}}}^{{{{{\bf{n}}}}}}\frac{1}{{{{{\bf{l}}}}}!}\left|{haf}\left({A}^{\left({{{{\bf{n}}}}}{{{{\boldsymbol{,}}}}}{{{{\bf{l}}}}}\right)}\right)\right|\right|}^{2}\\ {{{{{\mathscr{=}}}}}{{{{\mathcal{N}}}}}}^{{\prime} }{\left|{\sum }_{{{{{\bf{l}}}}}{{{{\boldsymbol{=}}}}}{{{{\bf{0}}}}}}^{{{{{\bf{n}}}}}}{\left(\frac{Pr \left({{{{\bf{n}}}}},{{{{\bf{l}}}}}\right)}{{{{{\bf{l}}}}}!}\right)}^{\frac{1}{2}}\right|}^{2},\end{array}$$
(2)
where \({{{{{\mathcal{N}}}}}}^{{\prime} }\) is a normalization constant and \(\Pr ({{{{\bf{n}}}}},{{{{\bf{l}}}}})\) is the probability of measuring an output pattern \(({{{{\bf{n}}}}},{{{{\bf{l}}}}})\). Subsequently, by sampling many FCFs, the approximated FCP at each given vibrational transition frequency (\({\omega }_{{vib}}\)) is obtained as$$\widetilde{{{{{\rm{FCP}}}}}}\left({{{{{\rm{\omega }}}}}}_{{{{{\rm{vib}}}}}}\right)={\sum }_{{{{{\bf{n}}}}}}^{\infty }\widetilde{{{{{\rm{FCF}}}}}}\left({{{{\bf{n}}}}}\right){{{{\rm{\delta }}}}}\left({\omega }_{{vib}}-{\sum }_{k}{\omega }_{k}^{{\prime} }{n}_{k}\right),$$
(3)
where {\({\omega }_{k}^{{\prime} }\)} are the harmonic angular frequencies of the final and initial states and\({n}_{k}\) corresponds to the number of photons in the \(k\) th mode. Combined with Eq. (2), our algorithm can estimate the FCP by stochastically sampling the known probability distribution for the output modes, and its scaling behavior is described in “Methods” section and Supplementary Note 2. Our algorithm is a straightforward approach that comprises four components, as depicted in Fig. 1d: squeezing operations \({\hat{S}}_{{{{{\boldsymbol{r}}}}}}\), rotation operations \({\hat{R}}_{V}\), Fock basis measurement and post-processing, which can be accomplished without needing a displacement term. The squeezing value \({{{{\bf{r}}}}}\) and unitary matrix \(V\) are given by the Takagi-Autonne decomposition39 of matrix \(A\), resulting in \(A={V}^{T}(\tanh ({{{{\bf{r}}}}}))V\). After the Fock basis measurement40, the molecular vibronic spectra are obtained by post-processing the output photon sampling distribution.Quantum experimental frameworkThe correlation between FCF and the hafnian function allows us to implement the quantum optics simulation of molecular vibronic spectroscopy. An overview diagram of the entire system is shown in Fig. 2, which consists of three subsystems: (1) dual pump preparation, (2) quantum microprocessor, and (3) control and measurement. First, the pump light is prepared with integrated filters based on asymmetric Mach-Zehnder interferometers (MZIs) using the dual-pump scheme (see characteristic source analysis in Fig. S2) as shown in Fig. 2a. To increase the input light power for enlarging the squeezing level of the squeezed vacuum state, an erbium-doped fiber amplifier is connected to amplify the laser power, and a pair of wavelength division multiplexer (WDM) is used to filter the pump signal. The integrated quantum microprocessor for vibronic transition profile reconstruction is illustrated in Fig. 2b. The squeezing source is applied via a degenerate spontaneous four-wave mixing process from the spiral lines, which is a single-pass squeezer. The unitary matrix is then performed in the evolution through the MZI network. As the programmable gate network embedded in the chip can be reconfigured as large as 16 × 16 dimensions, the implemented unitary operation can be tuned to match a particular target molecule. Besides, a pseudo-number-resolving detection scheme (up to 4 photons) is applied to achieve the photon number resolution at the photon counter. This scheme can be achieved by adding the 50:50 on-chip beam splitters and detecting all single outputs with the superconducting single-photon detectors. This approach could probabilistically split them at three beam splitters when four photons exist in a single output mode. Subsequently, the output coincidence photon numbers are also probabilistic. The detailed analysis of the probabilistic photon number-resolving is described in Supplementary Note 3. Finally, as shown in Fig. 2c, a fiber array is used to couple light out of the chip, and the output photons pass through the polarization controllers and the filters. Single-photon detectors finally collect them with a time tagger. The server computer controls the operating conditions for calculating the molecular vibrational spectra, including a thermoelectric controller and water-cooling temperature controller under the chip, the electronic control of squeezing level of the quantum sources and linear optical circuit, and the photon number resolving detection.Fig. 2: Schematic of a quantum microprocessor chip and experimental setup.a Setup to generate single-mode squeezing using dual pumps from 2-ps laser pulses, consisting of an optical pulse compressor, an erbium-doped fiber amplifier (EDFA), wavelength-division multiplexers (WDM), and a polarization controller (PC). The two pumps at the wavelength of 1546 nm and 1553 nm are selected and combined using WDM and coupled into the chip through a grating coupler. b Photograph of the microprocessor chip. Spiral sources are used to produce a vacuum-squeezed state. Programmable interferometer network is designed to achieve an arbitrary unitary matrix, and on-chip beam splitters are used for pseudo-number-resolving detection. Input single pump light is coupled to the chip by a one-dimensional subwavelength grating coupler. Output photons are coupled by edge couplers to fiber arrays. The scale bar denotes 100 \(\mu m\). c An overview of off-chip control and measurement devices, including temperature controller (TEC), electrical modulators, output photon detection (Time Tagger), Analog-to-Digital Converter (DAQ), and data processing with a server computer.The quantum microprocessor chip is 13 × 4 mm2, monolithically comprising mainly 16 photon sources, 272 thermo-optic phase shifters, 319 multimode interferometer beam splitters, and 65 optical couplers. The chip is packaged optically, electronically, and thermally, featuring high-density electrodes mounted to a printed circuit board through double-line wire bonding. The details of the experimental setup are discussed in Supplementary Note 3. The reconfigurability and control of the unitary transformation are detailed in Supplementary Note 4, and the achieved fidelities of the reconstructed matrix are presented in Supplementary Fig. 3. Some basic quantum characterizations of the circuit are shown in Supplementary Fig. 4, which manifests that the quantum photonic chip could be used to study molecular vibronic spectra.Molecular vibronic spectraThe quantum microprocessor chip is employed to compute the FCPs of two molecules: formic acid (CH2O2) and thymine (C5H6N2O2), which possess a considerable number of vibrational modes, specifically 9 modes for formic acid and 39 modes for thymine. Due to the chip’s size limitation, we deliberately exclude specific modes with relatively small FCFs. More details of the chosen vibrational frequencies and local basis transformations in our experiment can be found in Supplementary Note 5. The theoretical FCPs are calculated with the vibronic structure program package hotFCHT32,41. The experimental encoding and calculation details are listed in Supplementary Note 6.In the experiment, for formic acid (CH2O2) with 9 vibrational modes, only 4 quantum harmonic oscillators are selected and then effectively doubled and encoded to an 8-mode network. The incoming photons from the 1st and 5th modes are probabilistically split into four waveguides via three \(1\times 2\) beam splitters. The theoretical and experimental FCPs for formic acid are presented in Fig. 3a as vertical gray bars above and below the x-axis, respectively. The red and blue curves represent the Lorentzian broadening of the bars, depicting the theoretical and experimental spectra, respectively. The similarity between the experimentally reconstructed and theoretical FCPs is characterized by computing the fidelity of two sequences, \(F={\sum}_{i}\sqrt{{p}_{i}{q}_{i}}\), where \({{{{\boldsymbol{\{}}}}}p{{{{\boldsymbol{\}}}}}}\) and \({{{{\boldsymbol{\{}}}}}q{{{{\boldsymbol{\}}}}}}\) are the normalized theoretical and experimental FCF sequences of the molecules, respectively42,43,44. A fidelity \(F\) of 92.9% is achieved in this case, which is limited by the inevitable flaws in circuit fabrication and operation, photon noise, and photon loss. In particular, the loss affects the FCF values at high frequencies. Because of the photon loss, only a maximum of four-photon coincidence is achievable. Therefore, events with more than four photons cannot be stored, and the FCF values are further reduced at high frequencies (see a more detailed discussion in Supplementary Note 7). We further calculate the difference between the fidelity of the reconstructed FC profile \({F}_{Q}\) and the optimal fidelity obtainable from a classical strategy \({F}_{C}(C={F}_{Q}-{F}_{C})\) to benchmark against the classical simulation25,42. The result shows an improvement in the fidelity of the experimentally reconstructed FCP to the ideal FCP over the classical strategy (C = 6.8%) for formic acid molecules at the experimental maximum squeezing level (~0.3). The simulated theoretical fidelity difference between quantum and classical methods versus squeezing values under different loss values for formic acid molecules is shown in Supplementary Fig. 8, which shows that C is ~10% for the case without considering any losses (at ~0.3 squeezing level).Fig. 3: Vibronic spectra reconstruction.Franck−Condon profiles are obtained from chip distributions programmed according to the vibronic transitions of formic acid (a, with the structure shown in the inset) and thymine (b, with the structure shown in the inset). Gray bar graphs depict the histogram of energies, whereas red and blue continuous curves show the Lorentzian broadening of the bars.Calculating the FCP of thymine (C5H6N2O2) presents a more intricate task due to its vibronic boson sampling involving 39 modes. To manage this complexity, we chose to work with a subset of 7 modes, which have been meticulously encoded into our chip. Furthermore, the selected 1st, 6th, and 7th modes are probabilistically split into two waveguides via a 1 × 2 beam splitter. Figure 3b depicts the theoretical and experimental spectra, including the line-broadening effects (red and blue curves). The fidelity is 97.4%, corresponding to C = 7.3%. The fidelity is limited to the experimental mode number, leading to the loss of some frequencies upon detection. Moreover, the truncation of the FCP at 4-photon contributions also ignores the values in some frequencies. In addition, photon loss, leading to the misclassification of specific multiphoton events as single photons, inflates the count of single-photon events. A detailed discussion of these errors is found in Supplementary Note 7. Nonetheless, it can reconstruct most of the peaks in this profile. Details of the selected vibrational modes can be found in Supplementary Note 5. In conclusion, the vibronic spectroscopies of two molecules are simulated experimentally on the quantum microprocessor chip. While the classical computation of FCP is easier by relying on the Condon approximation, it is essential to acknowledge that, for many large systems, FC-forbidden or weakly FC-allowed transitions cannot be excluded, and the FC assumption is known to break down in some cases (most notably in aromatic molecules). In such instances, it becomes imperative to introduce corrections to ensure a meaningful interpretation of the observed transitions.Non-Condon effects in molecular spectroscopyThe Condon approximation, which assumes a constant transition dipole moment, may no longer suffice in simulating some molecules45. To improve the accuracy, it is necessary to include the non-Condon approach. Reference46 shows that a non-Condon approximation could be considered and implemented on linear optical quantum hardware. A non-Condon profile is approximated with Gaussian states through the first- and second-order Herzberg−Teller expansions of the transition dipole moment operator. Then, following our previous derivation, all these Gaussian states can be implemented on the integrated quantum photonic chip. The derivation details can be found in Supplementary Note 8.As a proof of principle, the vibronic spectra of naphthalene (C10H8), phenanthrene (C14H10), and benzene (C6H6) for the first-order Herzberg−Teller expansion are experimentally demonstrated in Fig. 4. The measured fidelities with the vibronic spectra of naphthalene, phenanthrene, and benzene are 98.4%, 98.4%, and 93.4%, respectively. To illustrate the non-Condon approximation, the computed FCPs are compared to non-Condon scenarios by omitting the transition dipole moment operator and exclusively considering the Doktorov operators. The molecular characteristic parameters, including vibrational frequencies and transformation matrices, are summarized in Supplementary Note 5. The \(\tau\) resulting from the quadratic truncation error \(O({\tau }^{2})\) for approximating the nonunitary operator as a linear combination of unitary operators in the experiment is set to 1. Though this signifies that a smaller \(\tau\) leads to a more minor error, the smallest possible \(\tau\) is dictated by the sampling error bars in our integrated chip device. For naphthalene and phenanthrene, only the two vibrational modes of the molecules that are relevant to demonstrate our method are considered, and photons from the two modes are probabilistically split into four channels. As shown in Fig. 4a, naphthalene does not exhibit vibronic spectral progression in the Condon regime due to the absence of displacement in its vibronic transition. In contrast, when considering the linear HT operator, the vibronic spectral progression is obtained (see red bars located above the x-axis in Fig. 4a). Non-negligible non-Condon effects in the FCP of phenanthrene are also observed, resulting in neighboring peaks at higher wavenumbers due to the Duschinsky mode mixing effect (Fig. 4b). The experimental results (red bars) match well with the theoretical ones (blue bars).Fig. 4: Vibronic spectra with non-Condon effects.Non-Condon and Franck−Condon profiles are obtained from chip distributions programmed according to the vibronic transitions of naphthalene (a, with structure shown in the inset), phenanthrene (b, with the structure shown in the inset), and benzene (c, with the structure shown in the inset). Red bar graphs depict the histogram of experimental energies, whereas blue bars show the theoretical results. Insets are enlarged parts of small peaks.After the obtained simulation for naphthalene and phenanthrene by restricting to two modes, it is crucial to evaluate the precision of our experimental platform in a broader scenario. Figure 4c shows that the experimental vibronic spectrum of benzene corresponds to the \({e}_{2g}\) symmetry block for the first-order HT expansion (detailed parameter information in Supplementary Note 5 and Ref46). The experiment considers five vibrational modes, with the number of photons per mode limited to two. Figure 4c shows that the generated profile from our experiment (red bars above the x-axis) matches the profile computed by the classical algorithm (blue bars). In addition, the non-Condon spectrum with the FCP (red bars below the x-axis) reveals a discernible contrast. As a result, integrating the new algorithm with the linear optical quantum simulation platform enables the experimental simulation of intricate molecular processes beyond the Condon regime, thereby broadening the range of potential applications for boson sampling.