Hyperfine-mediated electric quadrupole shifts in Al$^+$ and In$^+$ ion   clocks

K. Beloy; D. R. Leibrandt; W. M. Itano

arXiv:1701.09146·physics.atom-ph·April 11, 2017

Hyperfine-mediated electric quadrupole shifts in Al$^+$ and In$^+$ ion clocks

K. Beloy, D. R. Leibrandt, W. M. Itano

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TL;DR

This paper calculates the electric quadrupole shifts in Al$^+$ and In$^+$ ion clocks, revealing a natural cancellation in Al$^+$ that simplifies precision timekeeping.

Contribution

It provides a detailed third-order perturbation theory analysis of hyperfine effects on quadrupole moments in Al$^+$ and In$^+$ ions, highlighting a natural cancellation in Al$^+$.

Findings

01

Al$^+$ has a suppressed quadrupole shift due to cancellation.

02

The analysis extends through third-order perturbation theory.

03

The results support improved clock stability without additional shift control techniques.

Abstract

We evaluate the electric quadrupole moments of the $^{1} S_{0}$ and $^{3} P_{0}$ clock states of $^{27}$ Al $^{+}$ and $^{115}$ In $^{+}$ . To capture all dominant contributions, our analysis extends through third order of perturbation theory and includes hyperfine coupling of the electrons to both the magnetic dipole and electric quadrupole moments of the nucleus. For $^{27}$ Al $^{+}$ , a fortuitous cancellation leads to a suppressed frequency shift. This should allow for continued improvement of the clock without special techniques to control or cancel the shift, such as the averaging schemes that are critical to other optical ion clocks.

Tables3

Table 1. Table 1: Diagonal M 1 𝑀 1 M1 -hyperfine and E 2 𝐸 2 E2 -hyperfine matrix elements for P J 3 superscript subscript 𝑃 𝐽 3 {{}^{3}}\!P_{J} states of Mg, Al + , and In + . Uncorrected matrix elements are inferred from hyperfine intervals Itano et al. ( 2007 ); Lurio ( 1962 ); Larkins and Hannaford ( 1993 ) and nuclear moments Stone ( 2005 ) found in the literature, neglecting higher order effects of the hyperfine interaction. These results are subsequently corrected for second order effects using off-diagonal ab initio CI+MBPT hyperfine matrix elements between the P J 3 superscript subscript 𝑃 𝐽 3 {{}^{3}}\!P_{J} states, together with nuclear moments and fine structure intervals Ralchenko et al. ( 2016 ) . In many cases, the corrections do not or barely change the expressed result. The corrected matrix elements are then compared to ab initio CI+MBPT hyperfine matrix elements. All values are in atomic units aun .

	Mg, $3 s 3 p {}^{3}P_{J}$		Al⁺, $3 s 3 p {}^{3}P_{J}$	In⁺, $5 s 5 p {}^{3}P_{J}$
	$J = 1$	$J = 2$	$J = 1$	$J = 1$	$J = 2$
$⟨ {}^{3}P_{J} ‖ 𝒯_{1} ‖ {}^{3}P_{J} ⟩$ ${\begin{matrix} inferred, uncorrected \\ inferred, corrected \\ CI+MBPT \end{matrix}$	$0.07938 (< 1)$	$0.1573 (< 1)$	$0.1723 (< 1)$	$1.059 (< 1)$	$1.733 (4)$
	$0.07936 (< 1)$	$0.1572 (< 1)$	$0.1723 (< 1)$	$1.059 (< 1)$	$1.733 (4)$
	$0.07847$	$0.1561$	$0.1696$	$1.150$	$1.841$
$⟨ {}^{3}P_{J} ‖ 𝒯_{2} ‖ {}^{3}P_{J} ⟩$ ${\begin{matrix} inferred, uncorrected \\ inferred, corrected \\ CI+MBPT \end{matrix}$	$- 0.482 (7)$	$0.70 (1)$	$- 1.72 (1)$	$- 6.9 (6)$	$7 (2)$
	$- 0.466 (7)$	$0.71 (1)$	$- 1.26 (2)$	$- 6.4 (6)$	$7 (2)$
	$- 0.4621$	$0.7023$	$- 1.199$	$- 6.175$	$8.582$

Table 2. Table 2: Ab initio CI+MBPT results for the five distinct matrix elements appearing in terms ( 3 )–( 5 ), evaluated for Al + and In + . State labels refer to the 3 s 3 p P J 3 3 𝑠 3 𝑝 superscript subscript 𝑃 𝐽 3 3s3p\,{{}^{3}\!P_{J}} and 5 s 5 p P J 3 5 𝑠 5 𝑝 superscript subscript 𝑃 𝐽 3 5s5p\,{{}^{3}\!P_{J}} fine structure manifolds of the respective ion. All values are in atomic units aun .

	Al⁺	In⁺
$⟨ {}^{3}P_{0} ‖ 𝒬_{2} ‖ {}^{3}P_{2} ⟩$	$- 6.271$	$- 6.932$
$⟨ {}^{3}P_{1} ‖ 𝒬_{2} ‖ {}^{3}P_{1} ⟩$	$- 5.428$	$- 5.825$
$⟨ {}^{3}P_{0} ‖ 𝒯_{1} ‖ {}^{3}P_{1} ⟩$	$0.1195$	$0.7556$
$⟨ {}^{3}P_{1} ‖ 𝒯_{1} ‖ {}^{3}P_{2} ⟩$	$- 0.1545$	$- 0.7973$
$⟨ {}^{3}P_{0} ‖ 𝒯_{2} ‖ {}^{3}P_{2} ⟩$	$- 1.382$	$- 7.114$

Table 3. Table 3: Contributions to the E 2 𝐸 2 E2 moments of the ground and excited clock states in 27 Al + and 115 In + . All values are in units of e a B 2 𝑒 superscript subscript 𝑎 𝐵 2 ea_{B}^{2} . The notation x [ y ] 𝑥 delimited-[] 𝑦 x[y] indicates x × 10 y 𝑥 superscript 10 𝑦 x\times 10^{y} .

	²⁷Al⁺	¹¹⁵In⁺
	bare nucleus
$Θ^{(1 + 0)}$	$2.62 [- 9]$	$15 [- 9]$
	ground clock state
$Θ^{(1 + 1)}$	$- 10 [- 9]$
$Θ^{(1 + 2)}$	$- 0.1 [- 12]$
$Θ$ total	$- 8 [- 9]$
	excited clock state
$Θ^{(1 + 1)}$ , term (3)	$- 10.8 [- 6]$	$- 18.5 [- 6]$
$Θ^{(1 + 1)}$ , all other	$9 [- 9]$
$Θ^{(1 + 2)}$ , term (4)¹¹1For ¹¹⁵In⁺, terms (4) and (5) are multiplied by $10 / 9$ , as appropriate for the nuclear spin $I = 9 / 2$ .	$3.9 [- 6]$	$1.1 [- 6]$
$Θ^{(1 + 2)}$ , term (5)¹¹1For ¹¹⁵In⁺, terms (4) and (5) are multiplied by $10 / 9$ , as appropriate for the nuclear spin $I = 9 / 2$ .	$5.1 [- 6]$	$1.7 [- 6]$
$Θ^{(1 + 2)}$ , all other	$- 19 [- 9]$
$Θ$ total	$- 1.7 [- 6]$	$- 15.7 [- 6]$

Equations41

V_{E 2} = E_{2} \cdot Q_{2},

V_{E 2} = E_{2} \cdot Q_{2},

E_{2} \equiv \frac{1}{6} {\nabla \otimes \nabla}_{2} Φ,

E_{2} \equiv \frac{1}{6} {\nabla \otimes \nabla}_{2} Φ,

δ E = ⟨ n F M ∣ V_{E 2} ∣ n F M ⟩ = E_{20} ⟨ n F M ∣ Q_{20} ∣ n F M ⟩,

δ E = ⟨ n F M ∣ V_{E 2} ∣ n F M ⟩ = E_{20} ⟨ n F M ∣ Q_{20} ∣ n F M ⟩,

E_{20} = (- \frac{1}{6} \frac{\partial ^{2}}{\partial x ^{2}} - \frac{1}{6} \frac{\partial ^{2}}{\partial y ^{2}} + \frac{1}{3} \frac{\partial ^{2}}{\partial z ^{2}}) Φ = \frac{1}{2} \frac{\partial ^{2} Φ}{\partial z ^{2}},

E_{20} = (- \frac{1}{6} \frac{\partial ^{2}}{\partial x ^{2}} - \frac{1}{6} \frac{\partial ^{2}}{\partial y ^{2}} + \frac{1}{3} \frac{\partial ^{2}}{\partial z ^{2}}) Φ = \frac{1}{2} \frac{\partial ^{2} Φ}{\partial z ^{2}},

Θ \equiv ⟨ n F F ∣ Q_{20} ∣ n F F ⟩ .

Θ \equiv ⟨ n F F ∣ Q_{20} ∣ n F F ⟩ .

⟨ n F M ∣ Q_{20} ∣ n F M ⟩ =

⟨ n F M ∣ Q_{20} ∣ n F M ⟩ =

\times ⟨ n F F ∣ Q_{20} ∣ n F F ⟩ .

δ E = \frac{1}{2} \frac{\partial ^{2} Φ}{\partial z ^{2}} \frac{3 M ^{2} - F ( F + 1 )}{F ( 2 F - 1 )} Θ.

δ E = \frac{1}{2} \frac{\partial ^{2} Φ}{\partial z ^{2}} \frac{3 M ^{2} - F ( F + 1 )}{F ( 2 F - 1 )} Θ.

V_{E 2} = E_{2} \cdot Q_{2}^{(e)} + E_{2} \cdot Q_{2}^{(n)}, V_{hfi} = μ_{1}^{(n)} \cdot T_{1}^{(e)} + Q_{2}^{(n)} \cdot T_{2}^{(e)} .

V_{E 2} = E_{2} \cdot Q_{2}^{(e)} + E_{2} \cdot Q_{2}^{(n)}, V_{hfi} = μ_{1}^{(n)} \cdot T_{1}^{(e)} + Q_{2}^{(n)} \cdot T_{2}^{(e)} .

Θ^{(1 + 0)} = \frac{1}{2} Q,

Θ^{(1 + 0)} = \frac{1}{2} Q,

Θ^{(1 + 1)} = \frac{1}{5} Q [Q_{2}, T_{2}]_{1},

Θ^{(1 + 2)} = \frac{2}{15} μ^{2} A_{11; I} [Q_{2}, T_{1}, T_{1}] + \frac{1}{3} μ^{2} A_{11; I} [T_{1}, Q_{2}, T_{1}] + \frac{1}{5} μ Q A_{12; I} [Q_{2}, T_{1}, T_{2}] + \frac{1}{15} μ Q A_{12; I} [Q_{2}, T_{2}, T_{1}] - \frac{1}{15} μ Q A_{12; I} [T_{1}, Q_{2}, T_{2}] + \frac{1}{10} Q^{2} A_{22; I} [Q_{2}, T_{2}, T_{2}] + \frac{1}{20} Q^{2} A_{22; I} [T_{2}, Q_{2}, T_{2}] + \frac{1}{6} μ^{2} Q B_{1; I} [T_{1}, T_{1}]_{2} + \frac{1}{40} Q^{3} B_{2; I} [T_{2}, T_{2}]_{2},

\displaystyle A_{k_{1}k_{2};I}=(-1)^{2I}\frac{\left(\begin{array}[]{ccc}I&2&I\\ -I&0&I\end{array}\right)\left\{\begin{array}[]{ccc}k_{1}&k_{2}&2\\ I&I&I\end{array}\right\}}{\left(\begin{array}[]{ccc}I&k_{1}&I\\ -I&0&I\end{array}\right)\left(\begin{array}[]{ccc}I&k_{2}&I\\ -I&0&I\end{array}\right)},

\displaystyle A_{k_{1}k_{2};I}=(-1)^{2I}\frac{\left(\begin{array}[]{ccc}I&2&I\\ -I&0&I\end{array}\right)\left\{\begin{array}[]{ccc}k_{1}&k_{2}&2\\ I&I&I\end{array}\right\}}{\left(\begin{array}[]{ccc}I&k_{1}&I\\ -I&0&I\end{array}\right)\left(\begin{array}[]{ccc}I&k_{2}&I\\ -I&0&I\end{array}\right)},

\displaystyle B_{k;I}=(-1)^{2I}\frac{\left\{\begin{array}[]{ccc}I&I&k\\ I&I&2\end{array}\right\}-\left\{\begin{array}[]{ccc}I&I&k\\ I&I&0\end{array}\right\}}{\left(\begin{array}[]{ccc}I&k&I\\ -I&0&I\end{array}\right)^{2}}.

[X_{k}, Y_{k}]_{r} = n^{'} \sum \frac{⟨ n ∣∣ X _{k} ∣∣ n ^{'} ⟩ ⟨ n ^{'} ∣∣ Y _{k} ∣∣ n ⟩}{( E _{n} - E _{n^{'}} ) ^{r}},

[X_{k}, Y_{k}]_{r} = n^{'} \sum \frac{⟨ n ∣∣ X _{k} ∣∣ n ^{'} ⟩ ⟨ n ^{'} ∣∣ Y _{k} ∣∣ n ⟩}{( E _{n} - E _{n^{'}} ) ^{r}},

[X_{k_{1}}, Y_{k_{2}}, Z_{k_{3}}] = n^{'} n^{''} \sum \frac{⟨ n ∣∣ X _{k_{1}} ∣∣ n ^{'} ⟩ ⟨ n ^{'} ∣∣ Y _{k_{2}} ∣∣ n ^{''} ⟩ ⟨ n ^{''} ∣∣ Z _{k_{3}} ∣∣ n ⟩}{( E _{n} - E _{n^{'}} ) ( E _{n} - E _{n^{''}} )},

- \frac{1}{5} \frac{Q}{Δ _{20}} ⟨^{3} P_{0} ∣∣ Q_{2} ∣∣^{3} P_{2} ⟩ ⟨^{3} P_{2} ∣∣ T_{2} ∣∣^{3} P_{0} ⟩,

- \frac{1}{5} \frac{Q}{Δ _{20}} ⟨^{3} P_{0} ∣∣ Q_{2} ∣∣^{3} P_{2} ⟩ ⟨^{3} P_{2} ∣∣ T_{2} ∣∣^{3} P_{0} ⟩,

\frac{8 2}{75} \frac{μ ^{2}}{Δ _{20} Δ _{10}} ⟨^{3} P_{0} ∣∣ Q_{2} ∣∣^{3} P_{2} ⟩ ⟨^{3} P_{2} ∣∣ T_{1} ∣∣^{3} P_{1} ⟩ ⟨^{3} P_{1} ∣∣ T_{1} ∣∣^{3} P_{0} ⟩,

\frac{4}{15} \frac{2}{15} \frac{μ ^{2}}{Δ _{10}^{2}} ⟨^{3} P_{0} ∣∣ T_{1} ∣∣^{3} P_{1} ⟩ ⟨^{3} P_{1} ∣∣ Q_{2} ∣∣^{3} P_{1} ⟩ ⟨^{3} P_{1} ∣∣ T_{1} ∣∣^{3} P_{0} ⟩,

Φ_{T} = \frac{V _{0}}{2} cos (Ω t) \frac{X ^{2} - Y ^{2}}{R ^{2}} + U_{0} \frac{Z ^{2} - α X ^{2} - ( 1 - α ) Y ^{2}}{d ^{2}},

Φ_{T} = \frac{V _{0}}{2} cos (Ω t) \frac{X ^{2} - Y ^{2}}{R ^{2}} + U_{0} \frac{Z ^{2} - α X ^{2} - ( 1 - α ) Y ^{2}}{d ^{2}},

\frac{\partial ^{2} Φ _{T}}{\partial z ^{2}} = \frac{2 U _{0}}{d ^{2}} [\frac{3 cos ^{2} θ - 1}{2} - (α - \frac{1}{2}) sin^{2} θ cos (2 ϕ)],

\frac{\partial ^{2} Φ _{T}}{\partial z ^{2}} = \frac{2 U _{0}}{d ^{2}} [\frac{3 cos ^{2} θ - 1}{2} - (α - \frac{1}{2}) sin^{2} θ cos (2 ϕ)],

Φ_{I} = \frac{e}{( X - X _{i} ) ^{2} + ( Y - Y _{i} ) ^{2} + ( Z - Z _{i} ) ^{2}},

Φ_{I} = \frac{e}{( X - X _{i} ) ^{2} + ( Y - Y _{i} ) ^{2} + ( Z - Z _{i} ) ^{2}},

\frac{\partial ^{2} Φ _{I}}{\partial z ^{2}} = \frac{2 U _{0}}{d ^{2}} \frac{3 cos ^{2} θ - 1}{2} .

\frac{\partial ^{2} Φ _{I}}{\partial z ^{2}} = \frac{2 U _{0}}{d ^{2}} \frac{3 cos ^{2} θ - 1}{2} .

\frac{U _{0}}{d ^{2}} = \frac{m _{1} ω ^{2}}{2 e} \frac{μ}{1 + μ - 1 - μ + μ ^{2}},

\frac{U _{0}}{d ^{2}} = \frac{m _{1} ω ^{2}}{2 e} \frac{μ}{1 + μ - 1 - μ + μ ^{2}},

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Full text

Hyperfine-mediated electric quadrupole shifts in Al+ and In+ ion clocks

K. Beloy

National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA

D. R. Leibrandt

National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA

Department of Physics, University of Colorado, 440 UCB, Colorado 80309, USA

W. M. Itano

National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA

Abstract

We evaluate the electric quadrupole moments of the ${{}^{1}}\!S_{0}$ and ${{}^{3}}\!P_{0}$ clock states of 27Al+ and 115In+. To capture all dominant contributions, our analysis extends through third order of perturbation theory and includes hyperfine coupling of the electrons to both the magnetic dipole and electric quadrupole moments of the nucleus. For 27Al+, a fortuitous cancellation leads to a suppressed frequency shift. This should allow for continued improvement of the clock without special techniques to control or cancel the shift, such as the averaging schemes that are critical to other optical ion clocks.

I Introduction

State-of-the-art optical ion clocks owe their low systematic uncertainties, in part, to the external electric fields that provide strong confinement to the ion being probed. Electric field gradients are an inevitable byproduct of this trapping and, when coupled with the electric quadrupole ( $E2$ ) moments of the clock states, induce substantial frequency shifts in clocks based, for example, on Hg+ Rosenband et al. (2008), Sr+ Madej et al. (2012); Barwood et al. (2014), and Yb+ Godun et al. (2014); Huntemann et al. (2016). Exploiting the rotational symmetry of the $E2$ interaction allows cancellation of the shift by averaging over different magnetic substates or bias magnetic field orientations Itano (2000); Dubé et al. (2005). The additional operational burden of these averaging schemes is accompanied by distinct technical challenges, such as maintaining stringent control over the bias magnetic field direction.

In contrast to the clocks above, the Al+ clock Rosenband et al. (2008); Chou et al. (2010) employs two $J=0$ clock states, where $J$ specifies the total electronic angular momentum. Given the spherical symmetry implied by $J=0$ , the electrons nominally do not contribute to an $E2$ moment. Rather, the $E2$ moments of the clock states reduce to the negligibly small, state-independent nuclear $E2$ moment. To-date, this has allowed the Al+ clock to operate without special regard for electric field gradients, including those attributed to the co-trapped Be+ or Mg+ logic ion.

Under closer scrutiny, the $J=0$ symmetry of the clock states is weakly broken by the coupling of the electrons to the electromagnetic moments of the nucleus, i.e. the hyperfine interaction. This gives rise to state-dependent “hyperfine-mediated” corrections to the $E2$ moments, which may readily overshadow the bare nuclear $E2$ moment. Estimates of these effects have given justification for their prior neglect Itano et al. (2007). However, given the continued improvement of the Al+ clock Chen et al. , a more refined analysis is now warranted. In this paper, we evaluate the $E2$ moments of the Al+ clock states using methods of many-body atomic structure theory and discuss implications for the Al+ clock going forward. We further extend our analysis to In+, motivated by the pursuit of a many-ion clock based on this species in Braunschweig Herschbach et al. (2012).

Hyperfine-mediated $E2$ moments have recently been analyzed for ground-state alkali-metal atoms, with potential relevance to microwave atomic clocks Derevianko (2016). As one critical difference to that work, here we find it necessary to proceed through third order of perturbation theory (first order in the external field, second order in the hyperfine interaction) to capture all dominant effects.

Gaussian electromagnetic expressions are employed throughout. We let $e$ and $a_{B}$ denote the elementary charge and the Bohr radius, respectively.

II The $E2$ energy shift

Here we present formulas for the $E2$ energy shift for an atom in a hyperfine state $|nFM\rangle$ , where $n$ identifies the electronic state (inclusive of $J$ ), nuclear spin $I$ is implicit, and $F$ and $M$ specify the total angular momentum and its projection onto the $z$ -axis, respectively. To clarify, here $|nFM\rangle$ represents an eigenstate of the full atomic Hamiltonian, inclusive of nuclear spin and the hyperfine interaction. Generally, we are interested in a scenario in which static and radiofrequency electric fields simultaneously perturb the atom. Here we limit our concern to effects first order in the field, in which case only the static component needs to be considered. Describing this field with the electric potential $\Phi$ , the $E2$ interaction can be written as the scalar product

[TABLE]

where $\mathcal{E}_{2}$ is shorthand for the field-gradient tensor

[TABLE]

understood to be evaluated at the center-of-mass position of the atom. Here $\left\{\bm{\nabla}\otimes\bm{\nabla}\right\}_{2}$ represents the second rank tensor formed by coupling $\bm{\nabla}$ with itself Varshalovich et al. (1988). $\mathcal{Q}_{2}$ is a rank-2, even-parity operator that acts on both electronic and nuclear coordinates. It is given by $\mathcal{Q}_{2}=\sum_{i}q_{i}r_{i}^{2}C_{2}\left(\mathbf{\hat{r}}_{i}\right)$ , with $q_{i}$ the charge and $\mathbf{r}_{i}$ the position of the $i$ -th electron or nucleon ( $\mathbf{r}_{i}=r_{i}\mathbf{\hat{r}}_{i}$ ) and with $C_{2}(\mathbf{\hat{r}})$ being the conventional rank-2 $C$ -tensor (normalized spherical harmonic).

The first order $E2$ energy shift to the state $|nFM\rangle$ reads

[TABLE]

where, following from elementary angular selection rules, the matrix element $\langle nFM|\mathcal{Q}_{20}|nFM\rangle$ is non-zero only for $F\geq 1$ . Generally, $V_{E2}$ mixes states of different $M$ , suggesting the need to diagonalize a $(2F+1)$ -dimensional matrix. Here contributions off-diagonal in $M$ are assumed to be negligible. In practice this is ensured by application of a sufficiently large bias magnetic field, which defines the $z$ -axis and lifts the degeneracy in the $M$ states. This permits us to take $M$ as a “good” quantum number and has the consequence of isolating the zero-component of the scalar product in Eq. (1). The zero-component of $\mathcal{E}_{2}$ is

[TABLE]

where we invoked $\nabla^{2}\Phi=0$ to arrive at the last expression.

We proceed to define the $E2$ moment $\Theta$ as the expectation value of $\mathcal{Q}_{20}$ for the “stretched” state,

[TABLE]

Meanwhile, from the Wigner-Eckart theorem,

[TABLE]

Evaluating the 3 $j$ -symbols and combining the preceding expressions, we arrive at the energy shift expression

[TABLE]

Note that the $M$ -dependence appears explicitly in this last expression, with $\Theta$ itself being independent of $M$ .

The definition of the $E2$ moment $\Theta$ is chosen to be consistent with previous works for $J\geq 1$ atomic states, where nuclear structure and hyperfine effects are typically neglected. Expressions in the following section make explicit reference to the nuclear $E2$ moment $Q$ . For this quantity, we adhere to the conventional nuclear physics definition, which incorporates an additional factor of two. Specifically, in a complete absence of electrons, $\Theta$ equates to $Q/2$ . In any case, our formulas are self-consistent and unambiguously specify the physical meaning of $\Theta$ .

III Hyperfine-mediated $E2$ moments for $J=0$ states

The expressions in the previous section are valid for generic hyperfine states. In the remainder, we limit our attention to $J=0$ atomic states, with the immediate consequence that $F=I$ ( $I\geq 1$ assumed throughout). To arrive at practical expressions for evaluating the $E2$ moments, we treat the $E2$ interaction and the hyperfine interaction on equal footing as perturbations,

[TABLE]

Subscripts appearing on the right-hand-side of the expressions specify tensor rank, while superscripts identify operators acting exclusively on electronic ( $e$ ) or nuclear ( $n$ ) coordinates, inclusive of spin. Note that the operator $\mathcal{Q}_{2}$ of the preceding section has been partitioned into electronic and nuclear parts. $V_{\mathrm{hfi}}$ accounts for coupling of the electrons to the magnetic dipole and electric quadrupole moments of the nucleus. We will refer to the respective terms as the $M1$ -hyperfine interaction and $E2$ -hyperfine interaction, where the latter is not to be confused with the $E2$ interaction with the external field, $V_{E2}$ . In the limit of a point nucleus, $\mathcal{T}^{(e)}_{1}=-e\sum_{i}(1/r_{i}^{2})\left(\bm{\alpha}_{i}\times\mathbf{\hat{r}}_{i}\right)$ and $\mathcal{T}^{(e)}_{2}=-e\sum_{i}(1/r_{i}^{3})C_{2}\left(\mathbf{\hat{r}}_{i}\right)$ , where $\bm{\alpha}$ is composed of the conventional Dirac matrices ( $\alpha_{x},\alpha_{y},\alpha_{z}$ ) and the summations run over all electrons. We neglect coupling of the electrons to higher-multipolar moments of the nucleus ( $M3$ -hyperfine, $E4$ -hyperfine, etc.).

The $E2$ moment $\Theta$ is attributed to energy shifts first order in $V_{E2}$ and all orders in $V_{\mathrm{hfi}}$ . In practice, however, we only retain the leading contributions in $V_{\mathrm{hfi}}$ . The notation $\Theta^{(1+m)}$ is used to denote the $(1+m)$ -th order contribution, with $m$ being the number of hyperfine interactions involved. Applying conventional perturbation theory through third order, we arrive at the following expressions specific to $J=0$ states,

[TABLE]

where $\mu$ and $Q$ are the conventional nuclear $M1$ and $E2$ moments. The $I$ -dependent angular factors $A_{k_{1}k_{2};I}$ and $B_{k;I}$ are given in terms of $3j$ - and $6j$ -symbols by

[TABLE]

The string of operators appearing in square brackets is compact notation for the pure electronic factors

[TABLE]

where we have dropped the superscript $(e)$ on the operators involved. Here $|n\rangle$ and $E_{n}$ , with arbitrary number of primes attached to the $n$ , denote electronic states and energies. That is, these are eigensolutions to the atomic Hamiltonian in absence of nuclear spin and the hyperfine interaction. Reduced matrix elements and energies are independent of the magnetic substates, and it is understood that the sum-over-states expressions above exclude summation over magnetic quantum numbers. The unprimed $n$ designates the $J=0$ state of concern, while $n^{\prime}$ and $n^{\prime\prime}$ run over all other electronic states (having $J^{\prime}$ and $J^{\prime\prime}$ , respectively). Selection rules insist that $J^{\prime}=k$ for the top expression and $J^{\prime}=k_{1}$ and $J^{\prime\prime}=k_{3}$ for the bottom expression. Moreover, since only even-parity operators play a role, all states involved must have identical parity.

The first order $\Theta^{(1+0)}$ represents the $E2$ moment in absence of the hyperfine interaction. In accordance with the Introduction, this reduces to the bare nuclear value, with a factor of $1/2$ bridging the atomic physics and nuclear physics definitions of the $E2$ moment. At second order, a correction $\Theta^{(1+1)}$ arises due to the $E2$ -hyperfine interaction. Selection rules preclude the $M1$ -hyperfine interaction from having effect at this order, prompting further progression to third order. The expression for $\Theta^{(1+2)}$ includes several terms. The first two terms involve two $M1$ -hyperfine interactions, while the remaining terms can be regarded as higher-order in interactions that already contribute to $\Theta^{(1+0)}$ and $\Theta^{(1+1)}$ . For the specific problem at hand, we further note that the excited clock state is part of a ${{}^{3}\!P_{J}}$ fine structure manifold. The hyperfine-mediated effects are consequently expected to be dominated by hyperfine-mixing of the ${}^{3}\!P_{0}$ clock state with the neighboring ${}^{3}\!P_{1}$ and ${}^{3}\!P_{2}$ states, with corresponding terms in the sum-over-states expressions entering with small energy denominators. From the preceding arguments, we identify one second order term and two third order terms to be of principal interest,

[TABLE]

with fine structure splittings $\Delta_{10}\equiv\left(E_{{}^{3}\!P_{1}}-E_{{}^{3}\!P_{0}}\right)$ and $\Delta_{20}\equiv\left(E_{{}^{3}\!P_{2}}-E_{{}^{3}\!P_{0}}\right)$ . Terms (4) and (5) are specific to $I=5/2$ (27Al+). For other $I$ , these terms should be multiplied by an additional factor $(5/8)(2I-1)/I$ . For the sake of completeness, below we evaluate all first through third order contributions to the $E2$ moments of both clock states. Terms (3)–(5) are found to be comparable to one another and dominate significantly over all other contributions. Given that the fine structure splittings and nuclear moments are known, the critical task reduces to the accurate determination of the five distinct matrix elements appearing in these three terms.

IV Numerical Analysis and Results

To evaluate electronic properties, we employ the method of configuration interaction plus many-body perturbation theory (CI+MBPT) Dzuba et al. (1996); Savukov and Johnson (2002). In brief, we start with a Dirac-Hartree-Fock description of the atomic core (nuclear charge plus core electrons). The CI procedure treats the strongly-interacting valence electrons in the presence of the “frozen” core. The MBPT extension accounts for additional perturbative effects of the valence electrons on the core. Further details of our CI+MBPT implementation, with only minor modification, can be found in Ref. Beloy et al. (2008).

The CI+MBPT method has been applied to numerous divalent systems, including heavy systems such as Ra Dzuba and Ginges (2006). Al+ is comparatively simple. To best gauge the accuracy of our calculations, in particular the matrix elements appearing in terms (3)–(5), we first identify relevant data in the literature for comparison. Hyperfine intervals for the ${{}^{3}}\!P_{1}$ state of 27Al+ Itano et al. (2007) and the ${{}^{3}}\!P_{1,2}$ states of 25Mg Lurio (1962) have been measured. Mg has similar electronic structure to Al+, justifying its inclusion here. Neglecting higher order effects, the hyperfine intervals may be combined with nuclear moments Stone (2005) to infer diagonal $M1$ -hyperfine and $E2$ -hyperfine matrix elements, $\langle{{}^{3}}\!P_{J}||\mathcal{T}_{1}||{{}^{3}}\!P_{J}\rangle$ and $\langle{{}^{3}}\!P_{J}||\mathcal{T}_{2}||{{}^{3}}\!P_{J}\rangle$ , for the respective states. These results are presented in Table 1 with the label “inferred, uncorrected.” To these values, we add second order corrections attributed to hyperfine mixing between the neighboring ${{}^{3}}\!P_{J}$ states. The corrections are evaluated using off-diagonal ab initio CI+MBPT hyperfine matrix elements, together with the nuclear moments and fine structure intervals Ralchenko et al. (2016). The results are presented in Table 1 with the label “inferred, corrected.” Finally, these values are compared to the respective diagonal ab initio CI+MBPT matrix elements, labeled “CI+MBPT.”

First, we examine the $M1$ -hyperfine matrix elements in Table 1. For all three Mg and Al+ states, the second order corrections are small, with the resulting “inferred, corrected” values being accurate to within the displayed digits. The corresponding CI+MBPT results exhibit agreement in the range 0.7–1.6%. This is indicative of the CI+MBPT accuracy for the $M1$ -hyperfine matrix elements between the different ${{}^{3}}\!P_{J}$ states. Next, we examine the $E2$ -hyperfine matrix elements for the three Mg and Al+ states. Here, non-negligible uncertainty in the “inferred, uncorrected” values stems from uncertainty in the nuclear $E2$ moments. At the same time, the second order corrections are more prominent. These corrections are dominated by terms involving off-diagonal $M1$ -hyperfine matrix elements. Our preceding assessment of the CI+MBPT performance for these matrix elements allows us to ascribe a fair uncertainty to the corrections. This is only of significance for the ${{}^{3}}\!P_{1}$ state of Al+, where the correction leads to a doubling of the uncertainty. Finally, for the ${{}^{3}}\!P_{1,2}$ states of Mg, the diagonal CI+MBPT matrix elements are found to be within $1\sigma$ ( $\sim\!1.5\%$ ) of the “inferred, corrected” values. Meanwhile, for the ${{}^{3}}\!P_{1}$ state of Al+, the CI+MBPT result is found to be within $3\sigma$ ( $\sim\!5\%$ ) of the “inferred, corrected” value. This provides us with a measure of the CI+MBPT accuracy for the $E2$ -hyperfine matrix elements between the different ${{}^{3}}\!P_{J}$ states. Unfortunately, literature data is lacking to directly assess CI+MBPT performance for $E2$ matrix elements ( $\mathcal{Q}_{2}$ operator) between the ${{}^{3}}\!P_{J}$ states. However, we have no reason to expect the $E2$ matrix elements to be any less accurate than the $M1$ -hyperfine or $E2$ -hyperfine matrix elements between these states. We will briefly return to the discussion of accuracy below.

Table 2 presents our CI+MBPT results for the five distinct matrix elements appearing in terms (3)–(5). Together with nuclear moments Stone (2005) and fine structure splittings Ralchenko et al. (2016), these matrix elements yield the results $-1.08\times 10^{-5}\,ea_{B}^{2}$ for term (3), $3.92\times 10^{-6}\,ea_{B}^{2}$ for term (4), and $5.13\times 10^{-6}\,ea_{B}^{2}$ for term (5). The third order $M1$ -hyperfine-mediated terms (4) and (5) are seen to be comparable in magnitude and add constructively. Together they are close in magnitude to the second order $E2$ -hyperfine-mediated term (3) but of opposite sign. This results in significant cancellation, with a suppressed cumulative result of $-1.7\times 10^{-6}\,ea_{B}^{2}$ for these three terms. Clearly, it would have been erroneous to limit our analysis to second order perturbation theory or to account for hyperfine-mediated effects using only the $M1$ -hyperfine interaction.

Despite the cancellation, the terms (3)–(5) provide a correction to the $E2$ moment of the excited clock state three orders of magnitude larger than the bare nuclear value of $2.62\times 10^{-9}\,ea_{B}^{2}$ . Remaining second and third order contributions were evaluated using CI+MBPT matrix elements and energies, with experimental nuclear moments. For both clock states, these contributions amount to $-1\times 10^{-8}\,ea_{B}^{2}$ , verifying the dominance of the terms (3)–(5). Table 3 provides a breakdown of the contributions to the $E2$ moments of both clock states.

To estimate uncertainty of $\Theta$ for the excited clock state, we ascribe a 3% uncertainty to each of the five distinct matrix elements entering terms (3)–(5). We assume uncorrelated error in these matrix elements and propagate uncertainty accordingly, rendering a final value $\Theta=-1.7(6)\times 10^{-6}\,ea_{B}^{2}$ .

In Ref. Itano et al. (2007), the $E2$ moment of the excited clock state was given as $\Theta\approx-1.2\times 10^{-5}\,ea_{B}^{2}$ , this being an order-of-magnitude larger than the present evaluation. After inspecting notes from that work, a sign error was discovered for the term (5) contribution. A large relative error in $\Theta$ resulted from an absence of the cancellation discussed above. Correcting the sign error, the previous result is brought into agreement with the present result.

V Implications for the Al+ ion clock

Al+ ion clocks to-date have operated with a single Al+ ion and a single logic ion (Be+ or Mg+) simultaneously confined in a linear RF Paul trap Raizen et al. (1992). For this configuration, there are two contributions to the electric quadrupole field at the position of the Al+ ion: that due to the static axial confining trap potential and that due to the logic ion. Following Ref. Wübbena et al. (2012), we write the total trap electric potential as

[TABLE]

where $V_{0}/2$ and $\Omega$ are the amplitude and angular frequency of the voltage applied to the rf electrodes, $U_{0}$ is the voltage applied to the endcap electrodes, $R$ and $d$ are characteristic radial and axial dimensions of the trap, and $\alpha$ parameterizes the radial asymmetry of the static field. Above, $X$ , $Y$ , and $Z$ are spatial coordinates in the trap coordinate frame, while $t$ denotes time. Dropping the rf term, transforming to the quantization coordinate frame, and taking the second derivative with respect to $z$ , we get

[TABLE]

where $\theta$ and $\phi$ are the polar and azimuthal angles of the bias magnetic field as referenced from the trap frame. This result is independent of the position of the Al+ ion in the trap. The electric potential due to the other ion can be written as

[TABLE]

where $(X_{i},Y_{i},Z_{i})$ is the position of the logic ion. Transforming to the quantization coordinate frame, taking the second derivative with respect to $z$ , and substituting the equilibrium ion positions $(X,Y,Z)=(0,0,\pm(ed^{2}/8U_{0})^{1/3})$ , we get

[TABLE]

The trap dc potential satisfies

[TABLE]

where $m_{1}$ and $m_{2}=\mu m_{1}$ are the masses of the logic and clock ions and $\omega$ is the angular frequency of the axial common secular mode. Given that $\omega$ is a readily-accessible experimental parameter, $U_{0}/d^{2}$ can be immediately determined from the expression above.

For the Al+ clock reported in Ref. Chou et al. (2010), with $\omega/2\pi=3.00$ MHz, $\alpha=1.65$ , and $\theta=\phi=45^{\circ}$ , we calculate an $E2$ clock shift of $-28$ $\mu$ Hz, or $-2.5\times 10^{-20}$ fractionally. The magnetic field orientation with respect to the trap axes was not characterized with high accuracy in this experiment, but due to geometric constraints we can bound the uncertainty of $\theta$ and $\phi$ to be better than $\pm 5^{\circ}$ . Taking this field-gradient uncertainty together with uncertainty in $\Theta$ , we find an uncertainty in the clock shift of $19$ $\mu$ Hz, or $1.7\times 10^{-20}$ fractionally. As expected, both the shift and its uncertainty are negligible in comparison with the total clock uncertainty of Ref. Chou et al. (2010). Even as the Al+ clock continues to improve Chen et al. , this shift will likely remain small with respect to the uncertainty budget for the immediate future.

VI Estimates for an In+ many-ion clock

Following Refs. Pyka et al. (2014) and Keller (2015), we consider a many-ion clock based on 115In+. We evaluate terms (3)–(5), with appropriate scaling for the nuclear spin $I=9/2$ , and neglect all other contributions. Results are tabulated alongside 27Al+ results in Tables 1–3 above. Unlike 27Al+, there is not significant cancellation among the terms (3)–(5). The second order $E2$ -hyperfine-mediated term (3) dominates, with a final value of $\Theta=-1.6\times 10^{-5}ea_{B}^{2}$ . Through inspection of Table 1, this result is expected to be accurate to within $\sim\!20\%$ .

To estimate the clock shift, we assume a linear crystal of eight In+ clock ions and two Yb+ sympathetic cooling ions, with the Yb+ ions residing in the center. We suppose that the trap strength is such that the axial secular frequency of a single 172Yb+ ion is 330 kHz and that $\alpha\approx 1/2$ and $\theta=25^{\circ}$ . By generalizing the above formulas to the many-ion case, we calculate the mean of the $E2$ shift of each of the eight In+ ions to be $-490\,\mu$ Hz, or $-3.9\times 10^{-19}$ fractionally. The full width of the associated inhomogeneous broadening is $530\,\mu$ Hz, which given the lifetime limited transition linewidth of 820 mHz Becker et al. (2001) will not limit the spectroscopy linewidth. We note that at some level it might be necessary to take into account anharmonicity in the trapping potential over the extended size of the ion crystal for a more accurate estimate of the $E2$ shift.

VII Conclusion

Here we have derived expressions for hyperfine-mediated electric quadrupole moments of $J=0$ atomic states and have used these expressions to evaluate the corresponding clock shift in optical ion clocks based on 27Al+ and 115In+. Interestingly, for 27Al+, the second order $E2$ -hyperfine-mediated contribution is nearly equal but opposite to the third order $M1$ -hyperfine-mediated contributions, leading to a substantial suppression of the already small shift.

While the magnitudes of the $E2$ shifts for the clocks considered here are small with respect to current total systematic uncertainties, the $E2$ shift may be more significant for future clocks with lower total systematic uncertainties, clocks with stronger confinement, or clocks based on other atomic species. For these cases, in addition to the established techniques of cancelling the shift by averaging over different magnetic substates or bias magnetic field orientations, it is worth noting that the shift can also be suppressed by setting $\theta=\arccos(1/\sqrt{3})\approx 54.7^{\circ}$ and $\phi=45^{\circ}$ . Finally, while the quadrupole moments are rather small, it may be possible to do a direct experimental measurement using the technique of Ref. Shaniv et al. (2016).

Acknowledgements.

The authors thank S. Brewer and D. Nicolodi for their careful reading of the manuscript. This work is the contribution of NIST, an agency of the US government, and is not subject to US copyright.

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TL;DR

Contribution

Findings

Abstract

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Hyperfine-mediated electric quadrupole shifts in Al+ and In+ ion clocks

Abstract

I Introduction

II The E2E2E2 energy shift

III Hyperfine-mediated E2E2E2 moments for J=0J=0J=0 states

IV Numerical Analysis and Results

V Implications for the Al+ ion clock

VI Estimates for an In+ many-ion clock

VII Conclusion

Acknowledgements.

II The $E2$ energy shift

III Hyperfine-mediated $E2$ moments for $J=0$ states