Zeeman interaction in $^3\Delta_1$ state of HfF$^+$ to search for the electron electric-dipole-moment

TL;DR

This paper presents a theoretical study on the Zeeman interaction in HfF$^+$ to optimize conditions for electron EDM measurements, focusing on g-factor suppression and electronic structure calculations.

Contribution

It provides detailed calculations of g-factors and dipole moments in HfF$^+$, combining electronic structure methods with experimental data fitting to improve electron EDM search accuracy.

Findings

Minimum g-factor difference at 7 V/cm electric field

Calculated g-factor $G_{\parallel}$ matches experimental fit

Molecular dipole moment agrees with experimental value

Abstract

We report the theoretical investigation of the suppression of magnetic systematic effects in HfF$^+$ cation for the experiment to search for the electron electric dipole moment. The g-factors for $J = 1$, $F=3/2$, $|M_F|=3/2$ hyperfine levels of the $^3\Delta_1$ state are calculated as functions of the external electric field. The lowest value for the difference between the g-factors of $\Omega$-doublet levels, $\Delta g = 3 \times 10^{-6}$, is attained at the electric field 7 V/cm. The body-fixed g-factor, $G_{\parallel}$, was obtained both within the electronic structure calculations and with our fit of the experimental data from [H. Loh, K. C. Cossel, M. C. Grau, K.-K. Ni, E. R. Meyer, J. L. Bohn, J. Ye, and E. A. Cornell, Science {\bf 342}, 1220 (2013)]. For the electronic structure calculations we used a combined scheme to perform correlation calculations of HfF$^+$ which includes both the direct 4-component all-electron and generalized relativistic effective core potential approaches. The electron correlation effects were treated using the coupled cluster methods. The calculated value $G_{\parallel}=0.0115$ agrees very well with the $G_{\parallel}=0.0118$ obtained in the our fitting procedure. The calculated value $D_{\parallel}=-1.53$ a.u. of the molecule frame dipole moment (with the origin in the center of mass) is in agreement with the experimental value $D_{\parallel}=-1.54(1)$ a.u. [H. Loh, Ph.D. thesis, Massachusetts Institute of Technology (2006)].

Equations

$$E_{\mathrm{Zeeman}}=-g{\mu }_B\mathcal{B}{M}_F,$$

$$g=-G_{\parallel }\frac{F(F+1)+J(J+1)-3/4}{2F(F+1)J(J+1)}+$$

$$g_F\frac{{\mu }_N}{{\mu }_B}\frac{F(F+1)-J(J+1)+3/4}{2F(F+1)}$$

$${\Psi }_{\Omega }{\theta }_{M,\Omega }^J(\alpha ,\beta )U_{M_I}^{\mathrm{F}}.$$

$${\hat{\mathbf{H}}}_{\mathrm{mol}}={\hat{\mathbf{H}}}_{\mathrm{el}}+{\hat{\mathbf{H}}}_{\mathrm{rot}}+{\hat{\mathbf{H}}}_{\mathrm{hfs}}+{\hat{\mathbf{H}}}_{ext}.$$

$${\hat{\mathbf{H}}}_{\mathrm{rot}}=B_{\mathrm{rot}}{\mathbf{J}}^2-2B_{\mathrm{rot}}(\mathbf{J}\cdot {\vec{\mathbf{J}}}^e)$$

$${\hat{\mathbf{H}}}_{\mathrm{hfs}}=g_{\mathrm{F}}\mathbf{I}\cdot \sum _i\left(\frac{{\mathbf{\alpha }}_i\times {\mathbf{r}}_i}{r_i^3}\right)$$

$${\hat{\mathbf{H}}}_{\mathrm{ext}}={\mu }_{\mathrm{B}}({\mathbf{L}}^e-g_S{\mathbf{S}}^e)\cdot \mathbf{B}-g_{\mathrm{F}}\frac{{\mu }_N}{{\mu }_B}\mathbf{I}\cdot \mathbf{B}-\mathbf{D}\cdot \mathbf{E}$$

$${}^3{\Delta }_1$$

$${}^3{\Delta }_2$$

$${}^3{\Pi }_{0^{-}}$$

$${}^3{\Pi }_{0^{+}}$$

$$G_{\perp }^{(1)}$$

$$D_{\perp }^{(1)}$$

$${\Delta }^{(1)}$$

$$D_{\Vert }$$

$$D_{\perp }^{(2a)}={⟨}^3{\Delta }_1|{\hat{D}}_{+}{|}^3{\Pi }_{0^{+}}⟩=0.457a.u.,$$

$$D_{\perp }^{(2b)}={⟨}^3{\Delta }_1|{\hat{D}}_{+}{|}^3{\Pi }_{0^{-}}⟩=0.447a.u..$$

$$G_{\perp }^{(2a)}$$

$$G_{\perp }^{(2b)}$$

$${\Delta }^{(2a)}$$

$${\Delta }^{(2b)}$$

$$A_{\parallel }=g_{\mathrm{F}}⟨{\Psi }_{{}^3{\Delta }_1}|\sum _i{\left(\frac{{\mathbf{\alpha }}_i\times {\mathbf{r}}_i}{r_i^3}\right)}_{\zeta }|{\Psi }_{{}^3{\Delta }_1}⟩=-58.1\mathrm{MHz}$$

$$G_{\parallel }$$

Full text

I Introduction

Search for the electron electric dipole moment (eEDM), $d_{e}$, is important test of the standard model and its extensions Commins (1998); Chupp and Ramsey-Musolf (2015). The best current limit on the electron EDM, $|d_{\mathrm{e}}|<9\times 10^{-29}$ $e{\cdotp}{\rm cm}$  was set with a molecular beam of the thorium monoxide (ThO) molecules by ACME collaboration Baron et al. (2014) using the theoretical data from Refs. Skripnikov et al. (2013a); Skripnikov and Titov (2015a); Skripnikov (2016). A number of other systems are considered to search for the eEDM and other manifistations of effects of time-reversal (T) and spatial parity (P) symmetries violation of the fundamental interactions: ThO Flambaum et al. (2014); Skripnikov et al. (2013a, 2014a); Skripnikov and Titov (2015a); Skripnikov (2016), TaNFlambaum et al. (2014); Skripnikov et al. (2015a), ThF+ Loh et al. (2013); Skripnikov and Titov (2015b), PbF McRaven et al. (2008); Skripnikov et al. (2014b); Petrov et al. (2013); Skripnikov et al. (2015b), WC Lee et al. (2013, 2009), RaO Flambaum (2008); Kudashov et al. (2013), RaF Isaev and Berger (2012); Kudashov et al. (2014), PtH+ Meyer et al. (2006); Skripnikov et al. (2009a), etc.), TlF Skripnikov et al. (2009b); Petrov et al. (2002); Hinds and Sandars (1980); Laerdahl et al. (1997) molecules and cations.

E. Cornell’s group has suggested to use the trapped molecular ions for the eEDM search Meyer et al. (2006); Meyer and Bohn (2008). One of the most promising systems for the search is the HfF+ cation Cossel et al. (2012); Loh et al. (2013); Petrov et al. (2007, 2009); Fleig and Nayak (2013); Meyer et al. (2006); Le et al. (2013) which is also of interest for other fundamental experiments Skripnikov et al. (2008); Skripnikov et al. (2017a). It has the long-lived metastable ${}^{3}\Delta_{1}$ electronic state with lifetime $\approx 2s$ Petrov et al. (2009); Cossel et al. (2012) which means a very large coherence time is achievable in the experiment. The other main feature of the ${}^{3}\Delta_{1}$ state is that it has a very small g-factor (zero in nonrelativistic limit in approximation with free-electron g-factor, gS, equal to -2.0) which leads to the suppression of the magnetic systematic effects. It was shown that further suppression of systematics is possible due to existence of the $\Omega$-doublet structure of molecules in the the ${}^{3}\Delta_{1}$ electronic state DeMille et al. (2001); Petrov et al. (2014); Vutha and DeMille (2009); Petrov (2015). For preparation and implementation of the eEDM experiment one should investigate the dependence of upper and lower $\Omega$-doublet states g-factors on the strength of the laboratory electric field. And this is the goal of the present paper.

II Theory

We define the $g$-factors such that Zeeman shift is equal to

[TABLE]

where $\mu_{B}$ is the Bohr magneton, $M_{F}$ is the projection of the total angular momentum on the lab $z$ axis, ${\bf B}=\mathcal{B}\hat{z}$ is the external magnetic field. This definition matches the ones in the papers Petrov et al. (2014); Loh et al. (2013). Using the angular momentum algebra Landau and Lifshitz (1977), one can calculate that in the adiabatic approximation and in the limit of zero hyperfine interaction $g$-factors of hyperfine sublevels of the ${}^{3}\Delta_{1}$ state of HfF+ are determined by

[TABLE]

Here $g_{F}=5.25773$ is 19F nucleus $g-$factor, $\mu_{N}$ is the nuclear magneton. The first term in the right hand side of Eq. (3) is the electronic contribution Petrov (2011) and the second term is contribution from the magnetic moment of 19F nucleus.

Eq. (3) does not take into account the hyperfine interaction between different rotational levels and nonadiabatic interaction with other electronic states. To take these effects into account, following Refs. Petrov (2011); Petrov et al. (2014), the $g$-factors are obtained by numerical diagonalization of the molecular Hamiltonian (${\rm\bf\hat{H}}_{\rm mol}$) in external electric ${\bf E}=\mathcal{E}\hat{z}$ and magnetic ${\bf B}=\mathcal{B}\hat{z}$ fields over the basis set of the electronic-rotational wavefunctions

[TABLE]

Here $\Psi_{\Omega}$ is the electronic wavefunction, $\theta^{J}_{M,\Omega}(\alpha,\beta)=\sqrt{(2J+1)/{4\pi}}D^{J}_{M,\Omega}(\alpha,\beta,\gamma=0)$ is the rotational wavefunction, $\alpha,\beta,\gamma$ are Euler angles, $U^{F}_{M_{I}}$ is the F nuclear spin wavefunctions and $M$ $(\Omega)$ is the projection of the molecule angular momentum on the lab $\hat{z}$ (internuclear $\hat{n}$) axis, $M_{I}=\pm 1/2$ is the projection of the nuclear angular momentum on the same axis. Note that $M_{F}=M_{I}+M$.

We represent the molecular Hamiltonian for 180Hf19F+ as

[TABLE]

Here ${\rm\bf\hat{H}}_{\rm el}$ is the electronic Hamiltonian,

[TABLE]

is the rotational Hamiltonian,

[TABLE]

is the hyperfine interaction between electrons and flourine nuclei,

[TABLE]

describes the interaction of the molecule with external magnetic and electric fields, $B_{\mathrm{rot}}=0.2989$ Cossel et al. (2012) is the rotational constant, $g_{S}=-2.0023$ is a free$-$electron $g$-factor, D is the dipole moment operator.

For the current study we have considered the following low-lying electronic basis states: ${}^{3}\Delta_{1}$, ${}^{3}\Delta_{2}$, ${}^{3}\Pi_{0^{+}}$ and ${}^{3}\Pi_{0^{-}}$. ${\rm\bf\hat{H}}_{\rm el}$ is diagonal on the basis set (4). Its eigenvalues are transition energies of these states. They were calculated and measured in Ref. Cossel et al. (2012):

[TABLE]

Other terms of molecular Hamiltonian ${\rm\bf\hat{H}}_{\rm mol}$ are determined by parameters given by Eqs. (10)— (21) below. We have performed electronic calculations for the following matrix elements of the basis electronic states:

[TABLE]

[TABLE]

Matrix element (14) is in a good agreement with the value $D_{\perp}^{(2a)}=0.467$ a.u. calculated in Ref. Petrov et al. (2009). Calculated permanent dipole moment $D_{\|}$ is in a good agreement with the experimental value $D_{\|}=-1.54(1)$ a.u. Loh (2006). Matrix elements

[TABLE]

were chosen in such a way to reproduce the experimental value $0.369\cdot J(J+1)$ MHz for $\Omega$ doubling of ${}^{3}\Delta_{1}$. Matrix element

[TABLE]

were taken from Ref. Petrov et al. (2009). Hyperfine structure only of the ${}^{3}\Delta_{1}$ state was taken into account. G*∥* is given by the following formula:

[TABLE]

To perform electronic structure calculations of the diagonal matrix elements (13) and (21) we have used the combined computational scheme similar to that used in Refs. Skripnikov (2016); Skripnikov et al. (2017b, a) which includes electronic structure treatment within the generalized relativistic effective core (GRECP) potential approach Mosyagin et al. (2010, 2016) and the direct relativistic 4-component Dirac-Coulomb(-Gaunt) approach. This scheme includes the following stages: (i) 2-component 52-electron relativistic correlation calculation using the coupled cluster with single, double and noniterative triple cluster amplitudes, CCSD(T), method. For this we have used the semilocal version of the 44-electron GRECP operator Mosyagin et al. (2010, 2016). The 28 inner core $1s..3d$ electrons of Hf have been excluded from the correlation treatment by the GRECP operator and all other (outer core and valence) electrons were included in the correlation calculation. (ii) To treat the correlation contribution from the inner core electrons we have performed direct 4-component calculations at the level of the coupled cluster with single amplitudes (CCS) method as the difference in the calculated properties within the 80-electron (i.e. all-electron) CCS versus the 52-electron CCS. (iii) Calculation of vibration correction for $G_{\parallel}$.. (iv) Calculation of the correction on high order correlation effects.

For the stage (i) we have generated the uncontracted basis set for Hf that includes 25 $s-$, 25 $p-$, 21 $d-$, 14 $f-$, 10 $g-$, 5 $h-$ and 5 $i-$ type Gaussians For fluorine the (13,7,4,3,2)/[6,5,4,3,2] aug-ccpVQZ basis set Dunning, Jr (1989); Kendall et al. (1992) was used. Note that the reduction of the basis set on Hf to 15 $s-$, 10 $p-$, 8 $d-$, 7 $f-$, 4 $g-$, 2 $h-$ and 1 $i-$ type Gaussians (only $g-$, $h-$ and $i-$ type basis functions were contracted using the code from Ref. Skripnikov et al. (2013b)) leads only to slight changes in the calculated values.

For the stage (ii) the CVDZ Dyall (2007, 2012) basis set for Hf and the ccpVDZ Dunning, Jr (1989); Kendall et al. (1992) basis set for F were used. In the stage (iv) the high order correlation effects were considered as a difference in the values of considered properties calculated within the coupled cluster with single, double, triple and noniterative quadruple amplitudes and the CCSD(T) method. In the calculations 20 valence and outer core electrons of HfF+ were correlated.

To calculate off-diagonal matrix elements (10), (11) and (12) we have used 12-electron version of the GRECP operator for Hf used earlier in Refs.Petrov et al. (2007, 2009); Skripnikov et al. (2008) to perform 2-component 20-electron correlation calculations. For the calculations we have used the [12,16,16,10,8]/(6,5,5,3,1) basis set for Hf and [14,9,4,3]/(4,3,2,1) ANO-I basis set for F Roos et al. (2005). Calculations of the matrix elements (10), (11) and (12) were performed within the linear-response coupled cluster with single and double cluster amplitudes, LR-CCSD, method.

Electronic calculations were performed within the DIR and MRC codes. The code to calculate matrix elements of the g-factor operator over the 4-component molecular bispinors has been developed in the present paper.

III Results and discussions

$G_{\parallel}$ obtained from the electronic structure calculation is equal to 0.0115 and is in avery good agreement with the value $G_{\parallel}=0.011768$ obtained by fitting the $g_{\rm fit}=-0.00306$ value. In Ref. Loh et al. (2013) the experimental value $g_{\rm exp}=+0.00306$ obtained in the external electric field $\cal{E}=$11.6 V/cm is given. The electronic structure calculation is in agreement with the experiment only if the sign of $g-$factor will be changed. Thus, for consistency with the experiment, in this work we furhter use the g-factor value $g_{\rm fit}=-g_{\rm exp}$ with the sign reversed from that in Ref. Loh et al. (2013). Only $G_{\parallel}$ parameter was optimized in the fitting procedure. Eq. (3) gives $G_{\parallel}$ = 0.012043.

In Fig. 1 the calculated $g$-factors for the $J=1,F=3/2,M_{F}=3/2$ levels of HfF+ ${}^{3}\Delta_{1}$ state are shown as functions of the laboratory electric field. The calculated difference $\Delta g=g^{u}-g^{l}=3.4\times 10^{-6}$ between the g-factors of the upper ($g^{u}$) and lower ($g^{l}$) levels of $\Omega$ doublets is in a good agreement with the experimental value $-1(2)\times 10^{-5}$ Loh et al. (2013). Note, that the difference is zero in the adiabatic approximation. The lowest value for the difference, $\Delta g=3\times 10^{−6}$, is attained at the electric field E=7 V/cm. The smaller $\Delta g$, the smaller are the systematics $\sim\mu_{B}\Delta g\tilde{B}$ coming from a spurious magnetic field $\tilde{B}$.

IV Acknowledgement

Molecular calculations were partly performed on the Supercomputer “Lomonosov”. The development of the code for the computation of the matrix elements of the considered operators as well as the performance of all-electron calculations were funded by RFBR, according to the research project No. 16-32-60013 mol_a_dk. GRECP calculations were performed with the support of President of the Russian Federation Grant No. MK-7631.2016.2 and Dmitry Zimin “Dynasty” Foundation. The calculations of $g-$factor dependence on electric field were supported by the grant of Russian Science Foundation No. 14-31-00022.