Molecule generation

Our experiment is performed on a mixed-species two-ion crystal consisting of ⁴⁰Ca⁺ and ⁴⁰CaOH⁺ confined in a linear Paul trap²⁸. To form such a crystal, we first load two ⁴⁰Ca⁺ ions and then leak in water vapour with a leak valve from a gas chamber that contains mainly water vapour with a pressure of around 10⁻⁶ mbar. In the process, we apply 397 nm laser that drives the 4²S_1/2 → 4²P_1/2 dipole transition in ⁴⁰Ca⁺ to cool the ion crystal and monitor the fluorescence from ⁴⁰Ca⁺ ions with a camera. As the molecule does not fluoresce, the generation of a molecular ion leads to one of the ions turning dark on the camera. When this is observed, we switch off the leak valve and perform mass spectrometry on the dark ion by measuring the motional frequency of the ion crystal²⁴ to verify the generation of CaOH⁺. In general, it takes around 5 min to form a molecule. After closing the valve, the pressure in the vacuum chamber returns to around 10⁻¹⁰ mbar within a few minutes.

After generating the mixed-species ion crystal, background gas collisions occur, leading to the atomic and molecular ions swapping positions on the timescale of several seconds. For cat-state spectroscopy measurements, we work with only one of the two-ion crystal configurations because the two configurations exhibit different motional frequencies. This is probably due to imperfect micromotion compensation²⁴. We detect the ion crystal configuration during Doppler cooling before the spectroscopy sequence and attempt to alter the configuration randomly if the wrong configuration is detected. This is done by displacing the ion crystal radially from its trapping position. We repeat this procedure until the ions return to the correct configuration.

Cat-state engineering

The non-classical motional state for probing photon recoil is generated by a bichromatic laser beam applied to the ⁴⁰Ca⁺ ion. The two frequency components of equal intensity in the bichromatic beam are detuned by −ω_z and +ω_z from the atomic quadrupole transition between |↑⟩ and |↓⟩, with ω_z being the in-phase motional frequency of the ion crystal⁴. The Lamb–Dicke parameter of this quadrupole transition in the experiment is given by

where M_a is the atomic mass, \(_\) is the wavevector component of the quadrupole transition laser along the trap axis, and \(_\alpha \rangle \) is the participation factor of the atomic ion in the in-phase mode. In the Lamb–Dicke regime, the light–ion interaction Hamiltonian can be written as³⁸

where Ω₀ is the Rabi frequency, \({\hat{a}}^{\dagger }\) is the creation operator and \(\hat{a}\) is the annihilation operator of the harmonic oscillator describing in-phase motion, ϕ_r and ϕ_b are the phases of the frequency components with detuning −ω_z and +ω_z, respectively, in the light field, and h.c. is Hermitian conjugate. This corresponds to a spin-dependent displacement characterized by ϕ_± = (ϕ_b ± ϕ_r)/2, given by

Considering ϕ₊ = 0, the interaction can be expressed as

This leads to opposite motional displacements of \(\widehat{D}(\alpha )\) and \(\widehat{D}(-\alpha )\) with \(\alpha =-{\rm{i}}{\eta }_{{\rm{a}}}{\varOmega }_{0}T{{\rm{e}}}^{{\rm{i}}{\phi }_{-}}/2\) for the two \({\hat{\sigma }}_{x}\) eigenstates |±⟩ for a given pulse duration T.

The generated state is thus a superposition of two wavepackets with different atomic states shown in equation (2). By applying the interaction with ϕ₊ = π for the same duration, the two wavepackets can be combined again with the inverted spin-dependent displacement. Momentum recoil due to photon absorption is timed to occur between the generation and reversal of the cat state. This leads to a geometric phase difference accumulated between the two wavepackets, which is then detected by measuring the atomic state as shown in equation (3).

Experimental imperfections

Experimental imperfections are described by a physically motivated model, including atomic and motional decoherence, which, in our case, results from fluctuations of the magnetic field and trap voltages, respectively. We simulate the cat-state photon absorption detection using Lindblad master equations, in which we include the two decoherence channels as collapse operators:

where T_spin is the coherence time of atomic two-level system and T_motion is the coherence time of a superposition of the motional ground state and the first excited state. We estimated T_spin = 2.8(3) ms and \({T}_{\mathrm{motion}}=8{5}_{-35}^{+98}\,\mathrm{ms}\), with Ramsey experiments at the time scale of the measurement. The motional coherence time is much longer than the duration of a cat-spectroscopy experiment, which causes the large uncertainty on the estimated value. With these values, we can predict the signal produced by the experiment with various cat-state amplitudes and various kick sizes, as shown in Fig. 3. The purple-shaded region indicates the simulation results considering 1 s.d. of the atomic and motional coherence of the system. The dominating effect of these imperfections is a loss in signal amplitude, which we can also model with a simple heuristic model in equation (5) that multiplies the ideal contrast in equation (4) with a factor \({{\mathcal{S}}}_{\max }\). For a cat-state amplitude of |α| = 6.5(7), we determine the value of \({{\mathcal{S}}}_{\max }\) to be 0.52(3) by fitting equation (5) to the experimental data of various kick sizes.

The value of \({{\mathcal{S}}}_{\max }\) shows the effect of decoherence and varies for different cat state amplitudes and different durations of the measurement sequence. When investigating the effective photon absorption probability as a function of the number of femtosecond laser pulse n_pulse, as presented in Fig. 4a, the variation in n_pulse leads to the change in the time interval between the generation and the reversal of the cat state. This results in the variation in \({{\mathcal{S}}}_{\max }\) as a function of n_pulse. We investigated this effect with the electric kick measurement and model it as

for |α| = 6.5(7), where τ_d = 0.92(6) ms. We thus adjust the measured signals for different numbers of laser pulses accordingly.

Another potential source of imperfections besides decoherence is the timing jitter of the photon absorption events with respect to the ion crystal motion. The magnitude of the recoil-induced geometric phase difference Φ = 2η_mRe(α) depends on the phase difference between the cat state and the displacement operator from the photon absorption recoil. Therefore, it is influenced by the free evolution time between the generation of the cat state and the photon absorption event T_wait. In our case, the photon absorption event is synchronized with the femtosecond laser pulses. To achieve deterministic timing, we trigger the cat-state generation sequence on the periodic femtosecond laser pulses with a timing jitter of less than 10 ns. This jitter is considerably shorter than the oscillation period of the ions and ensures a constant free evolution time T_wait during the measurement. Moreover, the values of the trap frequency ω_z and the repetition rate of the laser f_rep influence the phase. We set the trap frequency to be an integer multiple of the repetition rate ω_z/2π = 4 ⋅ f_rep, so that we expect the same Re(α) for photon recoil from each of the femtosecond laser pulses. A variation of the trap frequency or the repetition rate, therefore, results in a varying phase for the signal obtained from each pulse. The variation of ω_z/2π is measured to be around 50 Hz, and the fluctuation of f_rep is identified to be below 10 Hz. The variation in arg(α) is thus less than 3.2 × 10⁻² for the maximum T_wait and can be neglected.

Quantum-chemical calculations

The vibrational transition frequencies and intensities are calculated ab initio using state-of-the-art quantum-chemical methods. We focus on the observed O–H stretching mode, which forms the most intense spectral band. Initially, an accurate potential energy surface is computed using electronic structure coupled-cluster methods, from which the harmonic frequencies and normal coordinates of the stretching modes are obtained. The resulting best estimate for the O–H harmonic frequency is 3,950 cm⁻¹. Subsequently, we neglect couplings between different normal modes, reducing the multidimensional anharmonic vibrational problem to a set of one-dimensional anharmonic vibrational problems along the normal coordinates. We solve the one-dimensional Schrödinger equation for nuclear motion along the normal coordinate dominated by the O–H stretching using the discrete variable representation method³⁹. The computed vibrational energies and wavefunctions give the fundamental transition frequency of 3,792 cm⁻¹ and oscillator strength of 3.7 × 10⁻⁵. Finally, couplings of the fundamental O–H stretching mode with the Ca–O stretching and Ca–O–H bending modes are reintroduced using vibrational second-order perturbation theory (VPT2)⁴⁰, which lowers the O–H fundamental frequency by 9.5 cm⁻¹ and results in its final value of 3,783 cm⁻¹.

The electronic structure computations use the range of correlation-consistent polarized weighted core–valence orbital basis sets augmented with diffuse functions (aug-cc-pwCVnZ-PP, n = T, Q, 5) (refs. ^41,42,43). The 10 core electrons of calcium are replaced by the ECP10MDF effective core potential to account for relativistic effects⁴⁴. The potential energy surface is obtained using a composite scheme in which the total energy is expressed as the sum of three contributions: (1) the Hartree–Fock mean-field energy; (2) the leading part of the correlation energy obtained using the coupled-cluster method restricted to single, double and noniterative triple excitations (CCSD(T))⁴⁵; and (3) the contribution from triple excitations in CCSDT neglected by CCSD(T)⁴⁶. The latter term is computed using a triple-ζ basis set, whereas the first two terms are extrapolated to the complete basis set limit using consecutive basis sets up to quintuple-ζ. The three-point exponential⁴⁷ and two-point 1/n³ (ref. ⁴⁸) extrapolation formulas are used for the Hartree–Fock and correlation energies, respectively. The permanent electric dipole moment is calculated using the analytical-derivative technique at the CCSD(T)/aug-cc-pwCV5Z level. Its value for the ground vibrational level is 6.2 D.

The potential energy surface of linear CaOH⁺ is computed as a function of the Ca–O and O–H bond lengths on a uniform two-dimensional grid with a spacing of 0.002 bohr around the equilibrium geometry²⁸ and interpolated with a natural cubic spline. After determining the normal coordinate, 33 electronic energies are calculated along this coordinate. The VPT2 correction is estimated from anharmonic constants obtained at the CCSD(T)/aug-cc-pwCVTZ level.

All electronic structure and VPT2 calculations are performed with the CFOUR v.2.1 program⁴⁹.

Molecular excitation dynamics

We model the excitation dynamics of the O–H stretching vibrational transition of CaOH⁺ in the experiment as follows. The molecule is considered here as a two-level system consisting of the ground and first excited states, |g⟩ and |e⟩. The intensity of the laser pulses applied to the molecule is assumed to have a Gaussian temporal profile with a full width at half maximum (FWHM) pulse duration of \(\tau =2{\tau }_{\sigma }\sqrt{2\mathrm{ln}2}\), where τ_σ is the 1-sigma pulse duration. The molecule then experiences the following time-dependent electric field:

where ω and ϕ are the centre frequency and the phase of the laser and E₀, and c.c. is the complex conjugate. The peak electric field strength can be calculated from the average laser intensity I_avg as \({E}_{0}=\sqrt{{I}_{\mathrm{avg}}/({f}_{\mathrm{rep}}\tau \sqrt{{\rm{\pi }}/\mathrm{ln}2}\cdot {\epsilon }_{0}c)}\). The light–molecule interaction can then be described as

where the transition dipole moment μ_eg can be related to the oscillator strength as \({\mu }_{eg}=\sqrt{{f}_{eg}\cdot 3\hbar {e}^{2}/{m}_{e}{\omega }_{0}}\) and θ is the angle between the molecular axis and the direction of the polarization of the light field. In the interaction picture under the rotating-wave approximation, the interaction can be described by

where Δ = ω − ω₀ is the detuning of the laser from the transition frequency ω₀ and Ω(t) is the Rabi rate given by

We aim to describe the experimental results using this model with no free parameters with a Monte Carlo simulation of the light–molecule interaction expressed in equation (15). In each trial of the simulation, for each laser pulse in the train, we randomly assign the phase ϕ of the light field, which is not stabilized from pulse to pulse for our laser, and the angle θ, assuming that the molecule is randomly oriented in the experiment. The Rabi rate of a single laser pulse is calculated based on ab initio calculations on the transition frequency and oscillator strength (see section ‘Quantum-chemical calculations’) and the laser parameters described below.

In our setup, the femtosecond laser pulses are produced by an ORPHEUS-HP optical parametric amplifier (OPA) pumped by a CARBIDE-CB5 femtosecond laser, both produced by Light Conversion. The light applied to the ions is linearly polarized in the vertical plane and with our magnetic field geometry, this results in an equal superposition of σ₊ and σ₋ polarizations. The average intensity at the position of the ions is 1.1(1) × 10⁴ W cm⁻². The spectrum of the light field is measured with a Redstone OSA305 optical spectrum analyser to determine the centre wavelength and linewidth for each frequency measured in the presented absorption spectrum. We observe absorption lines because of the presence of water vapour in the beam path, but we assume that this does not affect the absorption spectroscopy experiment because of power broadening.

The measurement presented in Fig. 4 is performed with a laser centre frequency of ν = 3,703.3(2) cm⁻¹ and an FWHM linewidth of 126.7(4) cm⁻¹. The simulation is based on these parameters and the assumption that the laser pulses are Fourier-transform-limited with an FWHM pulse duration of 116.2(4) fs. For the absorption spectrum shown in Fig. 4d, the experimental results are compared with the simulation performed with the Fourier-transform-limited pulse durations given by the measured centre frequency and linewidth at each of the OPA wavelength settings.

Non-destructive state detection

Photon absorption detection can provide an effective non-destructive state detection acting on a specific subspace of the molecular degrees of freedom. We sketch an example of such a measurement method, acting on the two lowest states of a vibrational mode in a molecule. Consider the three lowest energy vibrational states |0⟩, |1⟩ and |2⟩. We assume that we have the ability to perform selective operations on these states that can be spectroscopically addressed because the anharmonicity of the molecule yields unique transition frequencies. The proposed technique provides a projective measurement in the {|0⟩, |1⟩} manifold. We assume that the system is in a superposition of the vibrational states, the shared motion and the atom are in their ground state

We will then create the cat state using the atom–motion interaction, yielding the atom–motion state

We assume that the cat-state spectroscopy is set up to completely flip the atomic electronic state if a photon absorption has occurred Re(α) = π/(4η_m). The combined state of the molecular, atomic and motional degrees of freedom is then

We then assume that we can perfectly transfer the population from the |1⟩ state to the state |2⟩ with a laser pulse that applies a displacement operator on the motion of the ions:

After reversing the cate state generation, the state is then

representing an entangled state between the molecular vibrational degree of freedom and the electronic state of the atom. A projective measurement on the atomic electronic state will thus also project the molecular state into either |0⟩ or |2⟩. If the state is projected into |2⟩, the population can be brought back to |1⟩, which concludes the method and provides a non-destructive state detection.