High-Precision Spectroscopy of the HD⁺ Molecule at the 1-p.p.b. Level

Abstract

Recently we reported a high-precision optical frequency measurement of the (v, L): (0, 2)$\to$(8, 3) vibrational overtone transition in trapped deuterated molecular hydrogen (HD⁺) ions at 10 mK temperature. Achieving a resolution of 0.85 parts-per-billion (p.p.b.), we found the experimental value [ν ₀ = 383, 407, 177.38 (41) MHz] to be in agreement with the value from molecular theory [ν _th 383, 407, 177.150 (15) MHz] within 0.6 (1.1) p.p.b. (Biesheuvel et al. in Nat Commun 7:10385, 2016). This enabled an improved test of molecular theory (including QED), new constraints on the size of possible effects due to ‘new physics,’ and the first determination of the proton–electron mass ratio from a molecule. Here, we provide the details of the experimental procedure, spectral analysis, and the assessment of systematic frequency shifts. Our analysis focuses in particular on deviations of the HD⁺ velocity distribution from thermal (Gaussian) distributions under the influence of collisions with fast ions produced during (laser-induced) chemical reactions, as such deviations turn out to significantly shift the hyperfine-less vibrational frequency as inferred from the saturated and Doppler-broadened spectrum, which contains partly unresolved hyperfine structure.

This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.

Appendices

Appendix

Appendix 1: Molecular Dynamics Simulations

MD simulations are implemented in Fortran code in order to realistically describe the dynamics of trapped and laser-cooled ions in the presence of the time-dependent trapping field, 313 nm photon scattering by the Be⁺ ions, and fast ionic products from chemical reactions. A cycle of one time step starts by computing the sum of the forces acting on each ion, which consists of the Coulomb force, ${\mathbf{F}}_{\mathrm{C}}$, the time-dependent force from the trapping field, ${\mathbf{F}}_{\mathrm{trap}}$, and an optional rf electric field, ${\mathbf{F}}_{\mathrm{SS}}$, which drives the secular motion:

$$ {\mathbf{F}}_{\mathrm{tot}}=\sum \mathbf{F}={\mathbf{F}}_{\mathrm{C}}+{\mathbf{F}}_{\mathrm{trap}}+{\mathbf{F}}_{\mathrm{SS}}. $$

(22)

The radial and axial part of ${\mathbf{F}}_{\mathrm{trap}}$ are given by

$$ {F}_{\mathrm{trap},x,y}=-\frac{Q{V}_0}{R^2}\left(x\widehat{x}-y\widehat{y}\right)\cos \left(\varOmega t\right)+\frac{1}{2}Q{\omega}_z^2\left(x\widehat{x}+y\widehat{y}\right) $$

(23)

and

$$ {F}_{\mathrm{trap},z}= az+b{z}^3+c{z}^5, $$

(24)

where ${\omega}_z$ is the secular angular frequency in the z-direction. The constants a, b, and c depend on the trap geometry, which are determined through a finite-element analysis performed with the software package SIMION. The forces exerted on each ion are calculated, and trajectories are obtained using the leapfrog method [42]. Doppler-cooling is included at the level of single-photon scattering. Photon momentum kicks are simulated as velocity changes where absorption only takes place in the laser direction. In order to include ion motional heating which occurs in the trap, we implemented additional stochastic velocity kicks with a size of the recoil momentum of a single 313 nm photon with random directions. If an average kick rate of 75 MHz is used, ion temperatures of around 10 mK are obtained.

The processes of elastic and inelastic neutral-ion collisions are simulated as velocity kicks in random directions. For example, simulating reaction 7 in Table 1, a Be⁺ ion is substituted with a BeH⁺ ion at 10 mK, after which its speed is modified so as to give it 0.25 eV of kinetic energy.

Simulation of particles which in some cases have high velocities requires the use of a variable time step size, $\Delta \tau$. If the proper step size is not observed, two particles with a high velocity difference at close distance could 'skip’ each other within one time step instead of colliding. The default step size is $\Delta \tau =0.2$ ns. However, if for any of the trapped particles the condition ${v}_j\Delta \tau >10\times \min \left\{\Delta {x}_{jk}\right\}$ is met, where $\min \left\{\Delta {x}_{jk}\right\}$ is the distance between particle j with velocity ${v}_j$ and the nearest particle k, the time step $\Delta \tau$ is reduced by a factor of 10. Likewise, the step size is increased if the colliding particles separate again and ${v}_j\Delta \tau <100\times \min \left\{\Delta {x}_{jk}\right\}$.

To simulate EMCCD images, we made use of a simpler MD implementation, which treats the motion of the ions in the pseudopotential approximation and which does not include high-energy ions. This allows for a larger integration time step (10 ns) and, thus, faster MD simulations.

Appendix 2: Derivation of the Nonlinear Fluorescence Function

The relative HD⁺ loss during REMPD, $\epsilon$, is related to the spectroscopic signal S through the nonlinear function ${f}_{\mathrm{NL}}$, which we derive here. The fluorescence yield during a secular scan depends on the Be⁺ temperature, T, and is described by the scattering rate formula integrated over a Maxwell–Boltzmann velocity distribution. Neglecting micromotion effects, we take the scattering rate ${R}^{\mathrm{MB}}={R}^{\mathrm{MB}}\left(T,\Delta, I/{I}_{\mathrm{sat}}\right)$ defined in Eq. (16). During a secular scan in the experiment, we use the values $\Delta$ = 2$\pi \times -$300 MHz and $I/{I}_{\mathrm{sat}}=67$, and in what follows we drop these variables from the function argument of ${R}^{\mathrm{MB}}$. While performing the secular scan, the temperature T varies, which leads to a fluorescence peak as described by Eq. (16). The spectroscopic signal S is the relative difference between the areas under the fluorescence peaks (see Eq. 7). We may rewrite the area, A, as

$$ A=C\left[{R}^{\mathrm{MB}}\left(\overline{T}\right)\Delta t-{R}^{\mathrm{MB}}\left({\overline{T}}_{\mathrm{bl}}\right)\Delta t\right], $$

(25)

where C is a constant taking into account the collection and quantum efficiencies of the PMT or EMCCD imaging system, $\Delta t$ denotes the duration of the secular scan (10 s), and $\overline{T}$ stands for the ‘effective’ value of T during a secular scan. The effective temperature $\overline{T}$ is defined through Eq. (25) and can be interpreted as follows. If we would fix the secular excitation field frequency and amplitude at a certain value during a scan, the Be⁺ temperature and the fluorescence level would remain constant. $\overline{T}$ is the constant temperature that leads to the same area under the fluorescence trace as during a true secular scan. Likewise, ${\overline{T}}_{\mathrm{bl}}$ stands for the effective baseline temperature during $\Delta t$ (corresponding to the fluorescence level that results if no HD⁺ ions are present). In the experiment, the baseline may have a small slope due to the wing of the secular resonance of particles with mass 4 and 5 amu (see, for example, Fig. 5). This slope is detected and removed by the Mathematica code we use to analyze the PMT signal traces.

Both in the experiment and in the MD simulations described below, we observe the area A under experimental or simulated fluorescence traces, to which we subsequently can assign an effective temperature $\overline{T}$ through Eq. (25). We emphasize that no attempt is made to derive $\overline{T}$ directly from, for example, the simulated velocities of Be⁺ ions. Also note that in practice, we only use Eq. (25) to assign effective temperatures to simulated fluorescence traces, as in this case the constant C is known ($C=1$).

Inserting Eq. (25) into Eq. (7), we obtain

$$ S=\frac{R^{\mathrm{MB}}\left({\overline{T}}_i\right)-{R}^{\mathrm{MB}}\left({\overline{T}}_f\right)}{R^{\mathrm{MB}}\left({\overline{T}}_i\right)-{R}^{\mathrm{MB}}\left({\overline{T}}_{\mathrm{bl}}\right)}, $$

(26)

which is a relationship between the spectroscopic signal S and the effective ion temperatures during the initial and final secular scan, ${\overline{T}}_i$ and ${\overline{T}}_f$, respectively.

The relationship between the number of trapped HD⁺ molecules and $\overline{T}$ can be obtained from MD simulations. A Coulomb crystal is simulated containing 750 trapped Doppler-cooled Be⁺ ions and with HD⁺ numbers varying from 0 to 100. This is done once using a number of additional H${}_2$D⁺ and HD${}_2^{+}$ ions equal to that of scenario a, and once using the numbers of H${}_2$D⁺ and HD${}_2^{+}$ of scenario b (see Sect. 4.2). The simulated secular scans over the HD⁺ secular resonance frequency produce fluorescence peaks which agree qualitatively with those obtained in the laboratory as shown in Fig. 18.

Fig. 18

A simulated (yellow) and a real (blue) secular scan peak plotted on the same frequency axis. The signals are scaled vertically to achieve matching peak heights

In the experiment, secular scans are acquired over a time span of 10 s. The dynamics due to the time-varying frequency of the ac electric field take place at a timescale much longer than the timescale of fluorescence dynamics during laser cooling, which takes place at timescales of the order of 10 μs [43], and also longer than typical ion oscillation periods and the frequency of the ac electric field itself (1–20 μs). However, due to limited computational resources, the simulated duration of a secular scan is approximately 100 ms, which implies that the simulated dynamics take place at a considerably faster rate than in the experiment, typically on the scale of milliseconds. However, this still is much longer than the timescale of fluorescence dynamics and the motional dynamics. Therefore, we assume that the simulated secular scan peaks provide a reliable model of the experimentally observed secular scans.

The MD simulations reveal a linear relationship between the number of trapped HD⁺ ions and $\overline{T}$. This agrees with the intuitive picture of Be⁺ ions with frictionally damped motion (because of the laser cooling), whose temperature rise during secular excitation is directly proportional to the number of HD⁺ ions. Figure 19 shows the (${N}_{{\mathrm{HD}}^{+}},\overline{T}$) relationship for the two scenarios a and b. Having established that $\overline{T}$ is a linear measure of the number of trapped HD⁺ molecules, we now combine the relations $\epsilon =\left({N}_i-{N}_f\right)/{N}_i$, ${\overline{T}}_i={c}_1{N}_i+{c}_2$ and ${\overline{T}}_f={c}_1{N}_f+{c}_2$, where ${c}_1$ and ${c}_2$ are constants derived from MD simulations (Fig. 19), to obtain

$$ {\overline{T}}_f\left(\epsilon \right)={\overline{T}}_i\left(1-\epsilon \right)+{c}_2\epsilon . $$

(27)

${\overline{T}}_i$ can also be defined as the effective temperature with zero HD⁺ loss (${\overline{T}}_i={\overline{T}}_f\left(\epsilon =0\right)\equiv {\overline{T}}_0$), while the term ${c}_2$ can be considered as the effective baseline temperature (${c}_2={\overline{T}}_f\Big(\epsilon$ = 1)$={\overline{T}}_{\mathrm{bl}}$). Inserting Eq. (27) into Eq. (26) results in the nonlinear function

$$ {f}_{\mathrm{NL}}\left({\overline{T}}_0,\epsilon \right)\equiv \frac{R^{\mathrm{MB}}\left({\overline{T}}_0\right)-{R}^{\mathrm{MB}}\left(\left({\overline{T}}_{\mathrm{bl}}-{\overline{T}}_0\right)\epsilon +{\overline{T}}_0\right)}{R^{\mathrm{MB}}\left({\overline{T}}_0\right)-{R}^{\mathrm{MB}}\left({\overline{T}}_{\mathrm{bl}}\right)}, $$

(28)

which is plotted for scenario a in Fig. 20. Note that the nonlinear dependence on $\epsilon$ originates from the nonlinear dependence of ${R}^{\mathrm{MB}}$ on T in Eq. (16).

Fig. 19

Results from MD simulations showing the effective Be⁺ ion temperature during a secular scan $\overline{T}$ versus the number of HD⁺ ions ${N}_{{\mathrm{HD}}^{+}}$, assuming numbers of H${}_2$D⁺ and HD${}_2^{+}$ ions as in scenario a (blue dots) and scenario b (yellow dots). The blue and yellow lines represent least-squares fits, revealing a linear relationship between $\overline{T}$ and ${N}_{{\mathrm{HD}}^{+}}$

Fig. 20

3D plot of the function ${f}_{\mathrm{NL}}\left({\overline{T}}_0,\epsilon \right)$ (here plotted for scenario a) which connects the raw measurement signal S to the actual fractional loss of HD⁺, $\epsilon$

In the analysis ${\overline{T}}_0$ is treated as a free fit parameter. ${\overline{T}}_{\mathrm{bl}}$ is kept at a fixed value which is obtained from MD simulations. From Fig. 19, it can be seen that ${\overline{T}}_{\mathrm{bl}}\simeq 0.5$ K for scenarios a and b. This indicates that the rf field used for secular excitation already induces heating of Be⁺ while the field is still far away from the Be⁺ resonance (at $\sim 300$ kHz). This effect is also seen in the experiment.

The nonlinear function ${f}_{\mathrm{NL}}$ is used to map the relative HD⁺ loss $\epsilon$ onto the spectroscopic signal S. However, for the correction of background signals (Sect. 4.5) we need to map S to $\epsilon$, which requires the inverse nonlinear function ${f}_{\mathrm{NL}}^{-1}$. This inverse function is obtained numerically by use of Mathematica.

Appendix 3: Micromotion Fit Function

The time-dependent electric field of the trap, ${\mathbf{E}}_{\mathrm{t}}$, can be expressed as [31]

$$ {\displaystyle \begin{array}{cc}\hfill {\mathbf{E}}_{\mathrm{t}}\left(x,y,z,t\right)\cong & -\frac{V_0}{R^2}\left(x\widehat{x}-y\widehat{y}\right)\cos \left(\varOmega t\right)\hfill \\ {}\hfill & -\frac{\kappa {U}_0}{Z_0^2}\times \left(2z\widehat{z}-x\widehat{x}-y\widehat{y}\right),\hfill \end{array}} $$

(29)

where R is half the distance between two diagonally opposing electrodes, ${U}_0$ is the endcap voltage, ${Z}_0$ stands for half the distance between the end caps, and $\kappa$ is a shielding factor. The Be⁺ micromotion amplitude can be written as

$$ {\mathbf{x}}_0=\frac{Q}{m_{\mathrm{Be}}{\varOmega}^2}{\mathbf{E}}_{\mathrm{t}}\left(x,y,z,0\right). $$

(30)

The measured micromotion amplitude can be written as

$$ {x}_{0,k}=\frac{\mathbf{k}\cdotp {\mathbf{x}}_0}{\left\Vert \mathbf{k}\right\Vert }, $$

(31)

which is the projection of the 313 nm laser direction onto ${\mathbf{x}}_0$. From simulations of the rf trap circuitry with the simulation software Spice, we find a small possible phase difference ${\phi}_{\mathrm{ac}}$ of 4 mrad in between the rf electrodes, which has a negligible effect on the ion micromotion and is ignored here. Using the program SIMION, we calculate the shielding factor $\kappa$ and the static electric field ${\mathbf{E}}_{\mathrm{dc}}$ as a function of the dc voltages applied to the trap electrodes. From the static electric field, the radial ion displacement ${\mathbf{r}}_d$ is obtained by balancing the ponderomotive force and static E-field in the radial direction,

$$ {m}_{\mathrm{Be}}{\omega}_r^2{\mathbf{r}}_d=-Q{\mathbf{E}}_{\mathrm{dc}}, $$

(32)

where ${\omega}_r$ is the radial secular trap frequency. By inserting the x- and y-components of ${\mathbf{r}}_d$ into Eq. (29), the field vector ${\mathbf{E}}_{\mathrm{t}}$ at the location of the ions is obtained.

A geometric imperfection of the trap could lead to an axial rf field, which can be written as (here we ignore the small modification of the radial rf field of the trap due to the same imperfection):

$$ {E}_{\mathrm{ax},{\mathrm{HD}}^{+}}\left({V}_0,t\right)=\frac{1}{Q}\frac{V_0}{V_{0,\mathrm{e}}}{m}_{\mathrm{HD}}{x}_{\mathrm{HD}}{\varOmega}^2\cos \left(\varOmega t\right), $$

(33)

where ${x}_{\mathrm{HD}}$ is the HD⁺ micromotion amplitude along the trap z axis, ${m}_{\mathrm{HD}}$ is the mass of HD⁺, and ${V}_{0,\mathrm{e}}$ is the rf voltage used during the spectroscopic measurements, which is 270 V.

Now, we turn to the case of a linear string of Be⁺ ions, which is the configuration used to determine the axial rf field amplitude. Adding ${E}_{\mathrm{ax},{\mathrm{HD}}^{+}}$ to the z-component of ${\mathbf{E}}_{\mathrm{t}}$ gives a new expression for ${\mathbf{E}}_{\mathrm{t}}$ which is inserted into Eq. (30). We then obtain the following expression for ${\mathbf{x}}_0$:

$$ {\mathbf{x}}_0=\left(\frac{2{E}_{\mathrm{dc},x}Q{R}^2{V}_0{Z}_0^2}{Q{V}_0^2{Z}_0^2-2{m}_{\mathrm{Be}}{R}^4{U}_0\kappa {\varOmega}^2},\frac{2{E}_{\mathrm{dc},y}Q{R}^2{V}_0{Z}_0^2}{Q{V}_0^2{Z}_0^2-2{m}_{\mathrm{Be}}{R}^4{U}_0\kappa {\varOmega}^2},\frac{m_{\mathrm{HD}}{V}_0{x}_{\mathrm{HD}}}{m_{\mathrm{Be}}{V}_{0,\mathrm{e}}}\right), $$

(34)

which is subsequently inserted into Eq. (31), together with the wavevector, which is written as

$$ \mathbf{k}=\frac{2\pi }{\lambda}\left(\sin \left(\theta \right)\cos \left(\phi \right),\sin \left(\theta \right)\sin \left(\phi \right),\cos \left(\theta \right)\right). $$

(35)

Here $\theta$ is the angle between $\mathbf{k}$ and the trap z axis, and $\phi$ is the angle between $\mathbf{k}$ and the trap y axis, which is very close to $\pi /4$ in our setup. The value of $\theta$ lies between $\pm 10$ mrad and is treated as a free fit parameter. We insert Eqs. (35) and (34) into Eq. (31) and then expand the expression in powers of $\theta$. This gives us the following fit function:

$$ {\displaystyle \begin{array}{cc}\hfill {x}_{0,k}\left({V}_0\right)& =\frac{m_{\mathrm{HD}}}{m_{\mathrm{Be}}}\frac{V_0{x}_{\mathrm{HD}}}{V_{0,\mathrm{e}}}\hfill \\ {}\hfill & \kern1em -\frac{8\left({E}_{\mathrm{h},\mathrm{dc}}-{E}_{\mathrm{v},\mathrm{dc}}+\delta {E}_{\mathrm{h}}-\delta {E}_{\mathrm{v}}\right){Q}^2{V}_0}{m_{\mathrm{Be}}^2{R}^2{\varOmega}^4\left(2{a}_{\mathrm{M}}+{q}_{\mathrm{M}}^2\right)}\theta \hfill \\ {}\hfill & +\kern1em \mathcal{O}\left({\theta}^2\right).\hfill \end{array}} $$

(36)

Here ${E}_{\mathrm{h},\mathrm{dc}}$, ${E}_{\mathrm{v},\mathrm{dc}}$ are the applied static electric fields (corresponding to E _dc) in the horizontal and vertical directions, respectively, and $\delta {E}_{\mathrm{h}}$, $\delta {E}_{\mathrm{v}}$ are the unknown offset electric fields (due to e.g. charging of electrodes). The Mathieu parameters ${a}_{\mathrm{M}}$ and ${q}_{\mathrm{M}}$ are given by

$$ {a}_{\mathrm{M}}=\frac{-4 Q\kappa {U}_0}{m_{\mathrm{Be}}{Z}_0^2{\varOmega}^2},\kern1em {q}_{\mathrm{M}}=\frac{2Q{V}_0}{m_{\mathrm{Be}}{R}^2{\varOmega}^2}. $$

(37)

The displacement of the Be⁺ string in the vertical direction can be accurately determined with images of the EMCCD camera, and therefore $\delta {E}_h$ can be zeroed (for example, by minimizing the displacement of the Be⁺ string while the radial confinement of the trap is modulated by varying the rf amplitude). However, the displacement in the horizontal direction (i.e., perpendicular to the EMCCD image plane) is not accurately known, and therefore, we treat $\delta {E}_{\mathrm{h}}$ as another free fit parameter. In summary, we use Eq. (36) as a fit function with ${x}_{{\mathrm{HD}}^{+}}$, $\theta$ and $\delta {E}_{\mathrm{h}}$ as free fit parameters while neglecting higher orders of $\theta$. The fitted curves and the result for ${x}_{0,k}$ are shown in Sect. 4.3.

The question arises what happens if the 782 nm laser propagates at a small angle with respect to the trap axis, while the HD⁺ ions form a shell structure around the trap axis. In this case, a small fraction of the radial micromotion is projected onto the wavevector. However, from Eqs. (30–29) it follows that the sign of this additional micromotion alternates for each quadrant in the (x, y) plane. As long as the radial micromotion component does not exceed the axial micromotion amplitude (which is the case here), the former averages out to zero given the radial symmetry of the HD⁺ crystal.

Appendix 4: Stark Shift Calculations

Here, we summarize the formulas that are used to calculate the ac Stark shift of a ro-vibrational transition $\left(v,L\right)\to \left({v}^{\prime },{L}^{\prime}\right)$ in the HD⁺ molecule induced by a laser with intensity I and polarization state p. A general expression for the second-order energy shift depending on the angle $\theta$ between the polarization direction and the quantization axis is:

$$ {\displaystyle \begin{array}{cc}\hfill \Delta E& =-\frac{1}{2}\frac{I}{c}\Big[{\alpha}_{vL}^{(0)}\left(\omega \right)\hfill \\ {}\hfill & \kern1em +{P}_2\left(\cos \theta \right)\frac{3{M}^2-L\left(L+1\right)}{L\left(2L-1\right)}{\alpha}_{vL}^{(2)}\left(\omega \right)\Big],\hfill \end{array}} $$

(38)

where ${P}_2(x)=\frac{1}{2}\left(3{x}^2-1\right)$ is a Legendre polynomial. This expression contains the scalar and tensor polarizabilities

$$ {\displaystyle \begin{array}{cc}\hfill {\alpha}_{vL}^{(0)}\left(\omega \right)& =4\pi {a}_0^3{Q}_s,\hfill \\ {}\hfill {\alpha}_{vL}^{(2)}\left(\omega \right)& =4\pi {a}_0^3\sqrt{\frac{L\left(2L-1\right)}{\left(L+1\right)\left(2L+3\right)}}{Q}_t,\hfill \end{array}} $$

(39)

where ${a}_0$ is the Bohr radius and ${Q}_s$ and ${Q}_t$ stand for the two-photon scalar and tensor matrix elements:

$$ {\displaystyle \begin{array}{cc}\hfill {Q}_s& =\frac{\left\langle vL\left\Vert {Q}^{(0)}\right\Vert {v}^{\prime }L\right\rangle }{\sqrt{2L+1}}\hfill \\ {}\hfill {Q}_t& =\frac{\left\langle vL\left\Vert {Q}^{(2)}\right\Vert vL\right\rangle }{\sqrt{2L+1}}.\hfill \end{array}} $$

(40)

Here ${Q}^{(0)}$ and ${Q}^{(2)}$ are the irreducible scalar and tensor components that belong to the two-photon operator (in atomic units):

$$ {Q}_{pp}(E)=\mathbf{d}\cdotp {\boldsymbol{\epsilon}}_p\frac{1}{H-E}\mathbf{d}\cdotp {\boldsymbol{\epsilon}}_p, $$

(41)

with Hamiltonian H, dipole moment operator, $\mathbf{d}$ and polarization vector ${\boldsymbol{\epsilon}}_p$. The matrix elements ${Q}_s$ and ${Q}_t$ were calculated numerically using the three-body variational wave functions described in [8].

Since the hyperfine structure is partially resolved in this spectrum, we also have to consider the contribution of the Stark shifts to off-resonant coupling to hyperfine levels in $v=0$ and $v=8$ by the 782 nm laser during spectroscopy. Here, the situation is more complicated as the 782 nm laser also nonresonantly couples $v=8$ states to continuum states above the dissociation limit of the 1s$\sigma$ electronic ground state. The 782-nm contribution to the Stark shift was calculated at the hyperfine level, and will be published elsewhere. The corresponding shifts turn out to be negligible for our experiment, contributing only at the level of a few Hertz.