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Yangchao
Shen
^{a},
Yao
Lu
^{a},
Kuan
Zhang
^{a},
Junhua
Zhang
^{a},
Shuaining
Zhang
^{a},
Joonsuk
Huh
*^{b} and
Kihwan
Kim
*^{a}
^{a}Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, P. R. China. E-mail: kimkihwan@mail.tsinghua.edu.cn
^{b}Department of Chemistry, Sungkyunkwan University, Suwon 16419, Korea. E-mail: joonsukhuh@skku.edu

Received
24th October 2017
, Accepted 19th November 2017

First published on 1st December 2017

Molecules are one of the most demanding quantum systems to be simulated by quantum computers due to their complexity and the emergent role of quantum nature. The recent theoretical proposal of Huh et al. (Nature Photon., 9, 615 (2015)) showed that a multi-photon network with a Gaussian input state can simulate a molecular spectroscopic process. Here, we present the first quantum device that generates a molecular spectroscopic signal with the phonons in a trapped ion system, using SO_{2} as an example. In order to perform reliable Gaussian sampling, we develop the essential experimental technology with phonons, which includes the phase-coherent manipulation of displacement, squeezing, and rotation operations with multiple modes in a single realization. The required quantum optical operations are implemented through Raman laser beams. The molecular spectroscopic signal is reconstructed from the collective projection measurements for the two-phonon-mode. Our experimental demonstration will pave the way to large-scale molecular quantum simulations, which are classically intractable, but would be easily verifiable by real molecular spectroscopy.

(1) |

Fig. 1 A pictorial description of the photoelectron spectroscopy of SO_{2} and the trapped-ion simulator. (a) The photoelectron process of SO_{2} → SO_{2}^{+}. The molecule is initially at the vibrational ground state of the symmetrical stretching and scissoring modes. After absorbing a photon, an electron is removed from the molecule and the molecule finds a new equilibrium structure for SO_{2}^{+}, where the new potential energy surface is displaced, squeezed, and rotated from the original one. The transition of SO_{2}^{−} → SO_{2} can be described in a similar manner. (b) The trapped-ion simulator performing Gaussian transformation for molecular vibronic spectroscopy. The two vibrational modes of SO_{2} are mapped to the two radial modes (X and Y) of a single trapped-ion. The photoelectron process is simulated by applying a series of quantum optical operations, which are implemented by Raman laser beams (see section C in ESI†). Generally, the photoelectron process of more complicated molecules with N vibrational modes can be mapped to the collective motional modes of the N ions with similar operations by the Raman laser beams. |

Doktorov et al.^{14} decomposed Û_{Dok} in terms of the elementary quantum optical operators as follows:

Û_{Dok} = _{N}(α)Ŝ^{†}_{N}(ζ′)_{N}(U)Ŝ_{N}(ζ) | (2) |

The process of molecular vibronic spectroscopy can be understood as a modified boson sampling with Gaussian input states, such as thermal and squeezed vacuum states. Gaussian boson sampling, which is classified as a classically hard problem from a computational complexity perspective,^{8,9} requires more quantum optical operations on top of beam splitting and phase shifting operations for standard boson sampling. Boson sampling, however, is challenging in an optical system^{2–5} because of the difficulties in preparing the initial states: single Fock states for original boson sampling and squeezed coherent states for molecular simulation. Non-optical boson sampling devices, such as trapped-ion devices^{16,17} and superconducting circuits,^{18} have been suggested theoretically for the scalable boson sampling machine to overcome the difficulties of optical implementation in preparing the single photon states. Moreover, these non-optical devices can handle the squeezed states with relative ease. In this article, we present the first quantum simulation of molecular vibronic spectroscopy with the particular example of photoelectron spectroscopy of sulfur dioxide (SO_{2}).^{19,20}

Accordingly, for the first step of the molecular spectroscopy simulation, we prepare the ion in the ground state |n_{X} = 0, n_{Y} = 0〉 by Doppler cooling and resolved sideband cooling methods.^{21,22} Next, we perform the required displacement, squeezing, and rotation operations by converting the molecular parameters to the corresponding device parameters. The molecular parameters, α, ζ′, U and ζ, can be obtained via conventional quantum chemical calculations with the available program packages (e.g., ref. 23). See section B in the ESI† for the details of the parameter conversion for SO_{2}.

The quantum optical operations (displacement , squeezing Ŝ, and rotation ), which preserve the phase coherence amongst themselves, are implemented by the σ^{+}-polarized Raman laser beams from a pico-second pulse laser with a wavelength of 375 nm (see section C in the ESI†). In the trapped-ion experiment, the quantum optical operations with the desired parameters can be performed by controlling the applied laser frequency, duration, intensity, and phase. With different Raman laser configurations (see section C in the ESI†), we can realize the displacement, squeezing, and rotation operations, respectively.^{24,25}Fig. 2a shows the performance of the experimental displacement and squeezing operations, where a_{X} and a^{†}_{X} are the annihilation and creation operators of the bosonic mode X, respectively. The amount of displacement α and the squeezing parameter ζ are controlled by the duration of the corresponding Raman beams with rates of 0.042 μs^{−1} and 0.004 μs^{−1}, respectively. We examine the trapped-ion implementation of the rotation operation between modes X and Y with two sets of initial states, as indicated in Fig. 2b. The rotation angle θ is also controlled by the duration of the operation with a rate of 0.005 rad μs^{−1}. The oscillations in Fig. 2b of the initial state |n_{X} = 1, n_{Y} = 0〉 (orange and green) are two times slower than those from state |1,1〉 (blue, black, and red), as expected. We note that at t = 157 μs, the near zero probability of 〈1,1||1,1〉 originates from the Hong–Ou–Mandel interference.^{25}

Fig. 3a depicts a scheme for reconstructing the spectroscopy at zero Kelvin from the output measurements of the trapped-ion simulator. The transition intensities from the ground state to the excited states are aligned according to the transition frequencies. Fig. 3b illustrates the transition between the two-dimensional Fock spaces resulting from the two-dimensional harmonic oscillators. Finally, we perform the collective quantum-projection measurement of the final state |n_{X}, n_{Y}〉 advanced from the measurement scheme of ref. 26 and 27: first, we transfer the population of a target state |n_{X}, n_{Y}〉 to the |0,0〉 state by a sequence of π-pulse transitions. Then, we measure the state population by applying three sequential fluorescence detections combined with the uniform red sideband technique (see section D in the ESI†). Our quantum projection measurement is limited by the imperfection of the state transfer and the fluorescence detection efficiency. We plot the fidelity of the collective projection measurement of the |n_{X}, n_{Y}〉 state in Fig. 3c. Based on the fidelity analysis, we perform measurement-error corrections for the experimental raw data (see section E in the ESI† for detailed information).

Fig. 3 The construction scheme for the Franck–Condon profile of the photoelectron process with the trapped-ion simulator. (a) A generic diagram for the molecular transition process at T = 0 K. The lower bar indicates the initial state and the upper bar shows the final states after the process. The vibronic spectroscopy is constructed by measuring the transition probabilities from |n_{X} = 0, n_{y} = 0,…〉 to |n_{X′},n_{Y′},…〉. (b) The transition process of a molecule in the two-dimensional Fock space. The process begins with the lower plane and ends at the upper plane. The points in the grid represent the phonon number states. The transition probability is obtained by the collective projection measurements of the two phonon modes (see section D in the ESI†). (c) The fidelity analysis of the collective projection measurements. The fidelity of measuring the transition probability of the state |n_{X′}, n_{Y′}〉 is experimentally examined from |0,0〉 to |9,9〉. The fidelity is measured by applying the measurement sequence twice, starting from |0,0〉 to |n_{X′}, n_{Y′}〉, and then bringing back to |0,0〉. The square root of the remaining population represents the fidelity. |

Fig. 4 presents the photoelectron spectra, SO_{2} → SO_{2}^{+} and SO_{2}^{−} → SO_{2}, obtained from our trapped-ion quantum simulation; these are compared with the theoretical classical calculations. Fig. 4 shows good agreement between the theory calculations and the trapped-ion simulations of the two photoelectron processes of sulfur dioxide, and the required molecular parameters are described in the figure caption. In Fig. 4a, the photoelectron spectroscopy of SO_{2} is dominated by the ω′_{2} transitions due to the significant large displacement along the second mode: α = (−0.026,1.716). The photoelectron spectroscopy of SO_{2}^{−} in Fig. 4b shows tiny combination bands of the first and second modes regardless of the dominant contribution of the first mode (α = (1.360,−0.264)). We note that the observation of the tiny band combinations indicates reliable performance of the trapped-ion simulation.

Fig. 4 The trapped-ion simulation of the photoelectron spectra of SO_{2} and SO_{2}^{−} with measurement-error correction. The two vibrational frequencies of the harmonic potentials for SO_{2}^{+}, SO_{2}, and SO_{2}^{−} are (1112.7, 415.0), (1178.4, 518.9) and (989.5, 451.4) cm^{−1}, respectively. (a) The displacement vector α is (−0.026, 1.716) and the rotation angle θ is 0.189. (b) The displacement vector α is (1.360, −0.264) and the rotation angle θ is 0.065. The theoretical lines are intentionally broadened by convoluting with a Gaussian function with a width of 50 cm^{−1} (ref. 20) for comparison. Here N^{i}_{0} denotes the i-phonon excitation on the N-th mode from the vibrational ground state |0〉, and accordingly, 0^{0}_{0} located at the off-set energy ω_{0–0} = 0. |

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## Footnote |

† Electronic supplementary information (ESI) available: Trapped ion implementation of quantum optical operations, an experiment detection scheme, and experimental data error analysis. See DOI: 10.1039/c7sc04602b |

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