Molecule generation<\/p>\n
Our experiment is performed on a mixed-species two-ion crystal consisting of 40Ca+ and 40CaOH+ confined in a linear Paul trap28<\/a>. 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\u22126\u2009mbar. In the process, we apply 397\u2009nm laser that drives the 42S1\/2\u00a0\u2192\u00a042P1\/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<\/a> to verify the generation of CaOH+. In general, it takes around 5\u2009min to form a molecule. After closing the valve, the pressure in the vacuum chamber returns to around 10\u221210\u2009mbar within a few minutes.<\/p>\n 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<\/a>. 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.<\/p>\n Cat-state engineering<\/p>\n 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 \u2212\u03c9z and +\u03c9z from the atomic quadrupole transition between |\u2191\u27e9 and |\u2193\u27e9, with \u03c9z being the in-phase motional frequency of the ion crystal4<\/a>. The Lamb\u2013Dicke parameter of this quadrupole transition in the experiment is given by <\/p>\n $${\\eta }_{{\\rm{a}}}=\\sqrt{\\frac{\\hbar }{2{M}_{{\\rm{a}}}{\\omega }_{z}}}{k}_{z,{\\rm{a}}}{e}_{z,{\\rm{a}}},$$<\/p>\n \n (7)\n <\/p>\n where Ma is the atomic mass, \\({k}_{z,{\\rm{a}}}\\) is the wavevector component of the quadrupole transition laser along the trap axis, and \\({e}_{z,{\\rm{a}}}\\) is the participation factor of the atomic ion in the in-phase mode. In the Lamb\u2013Dicke regime, the light\u2013ion interaction Hamiltonian can be written as38<\/a><\/p>\n $${\\hat{H}}_{{\\rm{int}}}=\\hbar {\\eta }_{{\\rm{a}}}\\frac{{\\varOmega }_{0}}{2}({\\hat{\\sigma }}_{+}\\hat{a}{{\\rm{e}}}^{{\\rm{i}}{\\phi }_{{\\rm{r}}}}+{\\hat{\\sigma }}_{+}{\\hat{a}}^{\\dagger }{{\\rm{e}}}^{{\\rm{i}}{\\phi }_{{\\rm{b}}}}+\\text{h.c.}),$$<\/p>\n \n (8)\n <\/p>\n where \u03a90 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, \u03d5r and \u03d5b are the phases of the frequency components with detuning \u2212\u03c9z and +\u03c9z, respectively, in the light field, and h.c. is Hermitian conjugate. This corresponds to a spin-dependent displacement characterized by \u03d5\u00b1\u00a0=\u00a0(\u03d5b\u00a0\u00b1\u00a0\u03d5r)\/2, given by <\/p>\n $$\\begin{array}{l}{\\hat{H}}_{{\\rm{int}}}=\\hbar {\\eta }_{{\\rm{a}}}\\frac{{\\varOmega }_{0}}{2}({\\hat{\\sigma }}_{x}\\cos {\\phi }_{+}-{\\hat{\\sigma }}_{y}\\sin {\\phi }_{+})\\\\ \\,\\,\\,\\left[{\\rm{i}}(-\\hat{a}+{\\hat{a}}^{\\dagger })\\sin {\\phi }_{-}+(\\hat{a}+{\\hat{a}}^{\\dagger })\\cos {\\phi }_{-}\\right].\\end{array}$$<\/p>\n \n (9)\n <\/p>\n Considering \u03d5+\u00a0=\u00a00, the interaction can be expressed as <\/p>\n $${\\hat{H}}_{{\\rm{int}}}=\\hbar {\\eta }_{{\\rm{a}}}\\frac{{\\varOmega }_{0}}{2}{\\hat{\\sigma }}_{x}(\\hat{a}{{\\rm{e}}}^{-{\\rm{i}}{\\phi }_{-}}+{\\hat{a}}^{\\dagger }{{\\rm{e}}}^{{\\rm{i}}{\\phi }_{-}}).$$<\/p>\n \n (10)\n <\/p>\n 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 |\u00b1\u27e9 for a given pulse duration T.<\/p>\n The generated state is thus a superposition of two wavepackets with different atomic states shown in equation (2<\/a>). By applying the interaction with \u03d5+\u00a0=\u00a0\u03c0 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<\/a>).<\/p>\n Experimental imperfections<\/p>\n 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: <\/p>\n $${\\hat{C}}_{{\\rm{spin}}}=\\sqrt{\\frac{1}{2{T}_{{\\rm{spin}}}}}{\\hat{\\sigma }}_{z},\\,{\\hat{C}}_{{\\rm{motion}}}=\\sqrt{\\frac{2}{{T}_{{\\rm{motion}}}}}{\\hat{a}}^{\\dagger }\\hat{a}$$<\/p>\n \n (11)\n <\/p>\n where Tspin is the coherence time of atomic two-level system and Tmotion is the coherence time of a superposition of the motional ground state and the first excited state. We estimated Tspin\u00a0=\u00a02.8(3)\u2009ms 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<\/a>. 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<\/a>) that multiplies the ideal contrast in equation (4<\/a>) with a factor \\({{\\mathcal{S}}}_{\\max }\\). For a cat-state amplitude of |\u03b1|\u00a0=\u00a06.5(7), we determine the value of \\({{\\mathcal{S}}}_{\\max }\\) to be 0.52(3) by fitting equation (5<\/a>) to the experimental data of various kick sizes.<\/p>\n 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 npulse, as presented in Fig. 4a<\/a>, the variation in npulse 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 npulse. We investigated this effect with the electric kick measurement and model it as <\/p>\n $${{\\mathcal{S}}}_{\\max }(t)={{\\mathcal{S}}}_{\\max }{{\\rm{e}}}^{-t\/{\\tau }_{d}},$$<\/p>\n \n (12)\n <\/p>\n for |\u03b1|\u00a0=\u00a06.5(7), where \u03c4d\u00a0=\u00a00.92(6)\u2009ms. We thus adjust the measured signals for different numbers of laser pulses accordingly.<\/p>\n 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 \u03a6\u00a0=\u00a02\u03b7mRe(\u03b1) 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 Twait. 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\u2009ns. This jitter is considerably shorter than the oscillation period of the ions and ensures a constant free evolution time Twait during the measurement. Moreover, the values of the trap frequency \u03c9z and the repetition rate of the laser frep influence the phase. We set the trap frequency to be an integer multiple of the repetition rate \u03c9z\/2\u03c0\u00a0=\u00a04\u00a0\u22c5\u00a0frep, so that we expect the same Re(\u03b1) 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 \u03c9z\/2\u03c0 is measured to be around 50\u2009Hz, and the fluctuation of frep is identified to be below 10\u2009Hz. The variation in arg(\u03b1) is thus less than 3.2\u00a0\u00d7\u00a010\u22122 for the maximum Twait and can be neglected.<\/p>\n Quantum-chemical calculations<\/p>\n The vibrational transition frequencies and intensities are calculated ab initio using state-of-the-art quantum-chemical methods. We focus on the observed O\u2013H 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\u2013H harmonic frequency is 3,950\u2009cm\u22121. 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 Schr\u00f6dinger equation for nuclear motion along the normal coordinate dominated by the O\u2013H stretching using the discrete variable representation method39<\/a>. The computed vibrational energies and wavefunctions give the fundamental transition frequency of 3,792\u2009cm\u22121 and oscillator strength of 3.7\u00a0\u00d7\u00a010\u22125. Finally, couplings of the fundamental O\u2013H stretching mode with the Ca\u2013O stretching and Ca\u2013O\u2013H bending modes are reintroduced using vibrational second-order perturbation theory (VPT2)40<\/a>, which lowers the O\u2013H fundamental frequency by 9.5\u2009cm\u22121 and results in its final value of 3,783\u2009cm\u22121.<\/p>\n The electronic structure computations use the range of correlation-consistent polarized weighted core\u2013valence orbital basis sets augmented with diffuse functions (aug-cc-pwCVnZ-PP, n\u2009=\u2009T,\u2009Q,\u20095) (refs.\u200941<\/a>,42<\/a>,43<\/a>). The 10 core electrons of calcium are replaced by the ECP10MDF effective core potential to account for relativistic effects44<\/a>. 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\u2013Fock 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<\/a>; and (3) the contribution from triple excitations in CCSDT neglected by CCSD(T)46<\/a>. The latter term is computed using a triple-\u03b6 basis set, whereas the first two terms are extrapolated to the complete basis set limit using consecutive basis sets up to quintuple-\u03b6. The three-point exponential47<\/a> and two-point 1\/n3 (ref.\u200948<\/a>) extrapolation formulas are used for the Hartree\u2013Fock 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\u2009D.<\/p>\n The potential energy surface of linear CaOH+ is computed as a function of the Ca\u2013O and O\u2013H bond lengths on a uniform two-dimensional grid with a spacing of 0.002\u2009bohr around the equilibrium geometry28<\/a> 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.<\/p>\n All electronic structure and VPT2 calculations are performed with the CFOUR v.2.1 program49<\/a>.<\/p>\n Molecular excitation dynamics<\/p>\n We model the excitation dynamics of the O\u2013H 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\u27e9 and |e\u27e9. 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 \u03c4\u03c3 is the 1-sigma pulse duration. The molecule then experiences the following time-dependent electric field:<\/p>\n $${\\bf{E}}(t)={{\\bf{E}}}_{0}{{\\rm{e}}}^{-{t}^{2}\/4{\\tau }_{\\sigma }^{2}}({{\\rm{e}}}^{-{\\rm{i}}(\\omega t-\\phi )}+\\,\\text{c.c.}),$$<\/p>\n \n (13)\n <\/p>\n where \u03c9 and \u03d5 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 Iavg as \\({E}_{0}=\\sqrt{{I}_{\\mathrm{avg}}\/({f}_{\\mathrm{rep}}\\tau \\sqrt{{\\rm{\\pi }}\/\\mathrm{ln}2}\\cdot {\\epsilon }_{0}c)}\\). The light\u2013molecule interaction can then be described as <\/p>\n $${\\hat{H}}_{{\\rm{m}}}(t)=-{\\bf{d}}\\cdot {\\bf{E}}(t)={\\mu }_{{eg}}E(t)\\cos \\theta ,$$<\/p>\n \n (14)\n <\/p>\n where the transition dipole moment \u03bceg can be related to the oscillator strength as \\({\\mu }_{eg}=\\sqrt{{f}_{eg}\\cdot 3\\hbar {e}^{2}\/{m}_{e}{\\omega }_{0}}\\) and \u03b8 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 <\/p>\n $${\\hat{H}}_{\\,\\text{m}}^{\\text{I}\\,}(t)=\\hbar \\frac{{\\Omega }(t)}{2}(|e\\rangle \\langle g|{{\\rm{e}}}^{{\\rm{i}}\\phi }+\\,\\text{h.c.})+\\hbar \\Delta |g\\rangle \\langle g|,$$<\/p>\n \n (15)\n <\/p>\n where \u0394\u00a0=\u00a0\u03c9\u00a0\u2212\u00a0\u03c90 is the detuning of the laser from the transition frequency \u03c90 and \u03a9(t) is the Rabi rate given by <\/p>\n $$\\varOmega (t)=\\frac{2{\\mu }_{{eg}}{E}_{0}\\cos \\theta }{\\hbar }{{\\rm{e}}}^{-{t}^{2}\/4{\\tau }_{\\sigma }^{2}}.$$<\/p>\n \n (16)\n <\/p>\n We aim to describe the experimental results using this model with no free parameters with a Monte Carlo simulation of the light\u2013molecule interaction expressed in equation (15<\/a>). In each trial of the simulation, for each laser pulse in the train, we randomly assign the phase \u03d5 of the light field, which is not stabilized from pulse to pulse for our laser, and the angle \u03b8, 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 \u2018Quantum-chemical calculations<\/a>\u2019) and the laser parameters described below.<\/p>\n 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 \u03c3+ and \u03c3\u2212 polarizations. The average intensity at the position of the ions is 1.1(1)\u00a0\u00d7\u00a0104\u2009W\u2009cm\u22122. 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.<\/p>\n The measurement presented in Fig. 4<\/a> is performed with a laser centre frequency of \u03bd\u00a0=\u00a03,703.3(2)\u2009cm\u22121 and an FWHM linewidth of 126.7(4)\u2009cm\u22121. 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)\u2009fs. For the absorption spectrum shown in Fig. 4d<\/a>, 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.<\/p>\n Non-destructive state detection<\/p>\n 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\u27e9, |1\u27e9 and |2\u27e9. 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\u27e9,\u2009|1\u27e9} 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 <\/p>\n $$(a|0\\rangle +b|1\\rangle )\\otimes |\\downarrow \\rangle \\otimes {|0\\rangle }_{z}.$$<\/p>\n We will then create the cat state using the atom\u2013motion interaction, yielding the atom\u2013motion state <\/p>\n $${|\\varPsi \\rangle }_{{\\rm{cat}}}=\\frac{1}{\\sqrt{2}}|+\\rangle {|\\alpha \\rangle }_{z}+|-\\rangle {|-\\alpha \\rangle }_{z}.$$<\/p>\n We assume that the cat-state spectroscopy is set up to completely flip the atomic electronic state if a photon absorption has occurred Re(\u03b1)\u00a0=\u00a0\u03c0\/(4\u03b7m). The combined state of the molecular, atomic and motional degrees of freedom is then <\/p>\n $$(a|0\\rangle +b|1\\rangle )\\otimes {|\\varPsi \\rangle }_{{\\rm{cat}}}.$$<\/p>\n We then assume that we can perfectly transfer the population from the |1\u27e9 state to the state |2\u27e9 with a laser pulse that applies a displacement operator on the motion of the ions:<\/p>\n $$a|0\\rangle \\otimes {|\\varPsi \\rangle }_{{\\rm{cat}}}+b|2\\rangle \\otimes \\hat{D}({\\eta }_{{\\rm{m}}}){|\\varPsi \\rangle }_{{\\rm{cat}}}.$$<\/p>\n After reversing the cate state generation, the state is then <\/p>\n $$(a|0\\rangle \\otimes |\\downarrow \\rangle +b|2\\rangle \\otimes |\\uparrow \\rangle )\\otimes {|0\\rangle }_{z},$$<\/p>\n 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\u27e9 or |2\u27e9. If the state is projected into |2\u27e9, the population can be brought back to |1\u27e9, which concludes the method and provides a non-destructive state detection.<\/p>\n","protected":false},"excerpt":{"rendered":"Molecule generation Our experiment is performed on a mixed-species two-ion crystal consisting of 40Ca+ and 40CaOH+ confined in…\n","protected":false},"author":2,"featured_media":872901,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[1366,49,48,1793,1099,1100,314,18481,66],"class_list":["post-872900","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-atomic-and-molecular-interactions-with-photons","tag-ca","tag-canada","tag-chemical-physics","tag-humanities-and-social-sciences","tag-multidisciplinary","tag-physics","tag-quantum-metrology","tag-science"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/872900","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/comments?post=872900"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/872900\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media\/872901"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media?parent=872900"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/categories?post=872900"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/tags?post=872900"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}