arXiv:2603.10740

Zero crossings of the differential scalar polarizability of Ba$^+$ clock transition

N Jayjong, M D K Lee, K J Arnold, M D Barrett

physics.atom-phquant-phSubmitted Mar 11, 2026• Announced Mar 12, 2026

Abstract

The differential scalar polarizability $Δα_0(ω)$ of the Ba$^+$ S$_{1/2}$-to-D$_{5/2}$ clock transition has a zero crossing near 481nm, which is measured to be 623.603\,13(17)\,THz. From this measurement, we infer a ratio of reduced matrix elements $\langle P_{3/2}\|r\|S_{1/2}\rangle/\langle P_{1/2}\|r\|S_{1/2}\rangle=1.411\,81(13)$, which provides a stringent test of atomic structure calculations and experimental determination of matrix elements. Additionally, it enables the construction of an accurate approximation to $Δα_0(ω)$, valid for frequencies up to 450\,THz, with only one reduced matrix element, $\langle P_{1/2}\|r\|S_{1/2}\rangle$, appearing in the model's parameterization. We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation (BBR) shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.

Journal reference: Phys. Rev. A 114, 012804, 2026
DOI: 10.1103/wdgl-2vhf
Comments: 8 pages, 5 figures

Related papers

Magic wavelength of the Ba+138 6s S1/22–5d D5/22 clock transition

Sapam Ranjita Chanu, V. P. W. Koh, K. J. Arnold +5 more

arXiv:1910.09736Physical Review AApr 22, 20209 citations

Measurement of the reduced dipole matrix element in Ba$^+$

N. Jayjong, M. D. K. Lee, K. J. Arnold +1 more

physics.atom-phCross-listedarXiv:2607.20920Jul 24, 2026

Polarizability assessments of ion-based optical clocks

M. D. Barrett, K. J. Arnold, Marianna S. Safronova

arXiv:1905.04976Physical Review AOct 24, 201923 citations

Polarizability and magic wavelength of Yb+clock transition2S1/2 →2D3/2

Atish Roy, Subhadeep De, Bindiya Arora +1 more

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)Jul 1, 20171 citation

Measurement of the tune-out wavelength for Cs133 at 880 nm

Apichayaporn Ratkata, Philip D. Gregory, Andrew D. Innes +6 more

arXiv:2110.00043Physical Review ANov 22, 202115 citations

Blackbody radiation shift assessment for a lutetium ion clock

K. J. Arnold, R. Kaewuam, Atish Roy +2 more

arXiv:1712.00240Nature CommunicationsApr 19, 201874 citations

arXiv:2603.10740

Zero crossings of the differential scalar polarizability of Ba $^+$ clock transitionZero crossings of the differential scalar polarizability of Ba$^+$ clock transition

N Jayjong, M D K Lee, K J Arnold, M D Barrett

physics.atom-phquant-phSubmitted Mar 11, 2026• Announced Mar 12, 2026

arXiv HTML PDF

Abstract

The differential scalar polarizability

Δα_0(ω)

of the Ba

^+

_{1/2}

-to-D

_{5/2}

clock transition has a zero crossing near 481nm, which is measured to be 623.603,13(17),THz. From this measurement, we infer a ratio of reduced matrix elements

\langle P_{3/2}\|r\|S_{1/2}\rangle/\langle P_{1/2}\|r\|S_{1/2}\rangle=1.411\,81(13)

, which provides a stringent test of atomic structure calculations and experimental determination of matrix elements. Additionally, it enables the construction of an accurate approximation to

Δα_0(ω)

, valid for frequencies up to 450,THz, with only one reduced matrix element,

\langle P_{1/2}\|r\|S_{1/2}\rangle

, appearing in the model's parameterization. We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation (BBR) shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.

Journal reference: Phys. Rev. A 114, 012804, 2026
DOI: 10.1103/wdgl-2vhf
Comments: 8 pages, 5 figures

Related papers

Magic wavelength of the Ba+138 6s S1/22–5d D5/22 clock transition

Sapam Ranjita Chanu, V. P. W. Koh, K. J. Arnold +5 more

The zero crossing of the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ clock transition in $^{138}\mathrm{Ba}^{+}$ has been determined to be $459.1614(28)\phantom{\rule{0.28em}{0ex}}\mathrm{THz}$ . Together with previously determined matrix elements and branching ratios, this tightly constrains the dynamic differential scalar polarizability of the clock transition over a large wavelength range ($\ensuremath{\gtrsim}700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). In particular, it allows an estimate of the blackbody radiation shift of the clock transition at room temperature.The zero crossing of the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ clock transition in $^{138}\mathrm{Ba}^{+}$ has been determined to be $459.1614(28)\phantom{\rule{0.28em}{0ex}}\mathrm{THz}$. Together with previously determined matrix elements and branching ratios, this tightly constrains the dynamic differential scalar polarizability of the clock transition over a large wavelength range ($\ensuremath{\gtrsim}700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). In particular, it allows an estimate of the blackbody radiation shift of the clock transition at room temperature.

arXiv:1910.09736Physical Review AApr 22, 20209 citations

Measurement of the reduced dipole matrix element in Ba $^+$ Measurement of the reduced dipole matrix element in Ba$^+$

N. Jayjong, M. D. K. Lee, K. J. Arnold +1 more

We present a high-precision measurement of the reduced electric-dipole matrix element $\langle P_{1/2}\|r\|S_{1/2}\rangle$ in $^{138}\mathrm{Ba}^+$ . By comparing off-resonant scattering rates with dispersive Stark-shift measurements, we determine the matrix element to be $3.322\,7(12)$ , corresponding to a $P_{1/2}$ excited-state radiative lifetime of $7.866\,3(56)$ ~ns. Combining our experimental results with a prior model for the differential scalar polarizability $Δα_0(ω)$ , we extract the static value $Δα_0(0) = -73.09(12)$ a.u. This determination directly improves the evaluation of the blackbody radiation (BBR) shift, minimizing a prominent systematic limitation in room-temperature $\mathrm{Ba}^+$ optical clock error budgets. These results provide a stringent benchmark for atomic-structure calculations and advance high-precision applications in quantum metrology and tests of fundamental physics.We present a high-precision measurement of the reduced electric-dipole matrix element $\langle P_{1/2}\|r\|S_{1/2}\rangle$ in $^{138}\mathrm{Ba}^+$. By comparing off-resonant scattering rates with dispersive Stark-shift measurements, we determine the matrix element to be $3.322\,7(12)$, corresponding to a $P_{1/2}$ excited-state radiative lifetime of $7.866\,3(56)$~ns. Combining our experimental results with a prior model for the differential scalar polarizability $Δα_0(ω)$, we extract the static value $Δα_0(0) = -73.09(12)$a.u. This determination directly improves the evaluation of the blackbody radiation (BBR) shift, minimizing a prominent systematic limitation in room-temperature $\mathrm{Ba}^+$ optical clock error budgets. These results provide a stringent benchmark for atomic-structure calculations and advance high-precision applications in quantum metrology and tests of fundamental physics.

physics.atom-phCross-listedarXiv:2607.20920Jul 24, 2026

Polarizability assessments of ion-based optical clocks

M. D. Barrett, K. J. Arnold, Marianna S. Safronova

It is shown that the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ transition in $^{138}\mathrm{Ba}^{+}$ can be determined to an inaccuracy below $0.5%$ across a wide wavelength range ($\ensuremath{\lambda}>700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). This can be achieved using measurements for which accurate determination of laser intensity is not required, and most of the required measurements are already in the literature. Measurement of a laser-induced ac-Stark shift of the clock transition would then provide an in situ measurement of the laser's intensity to the same $0.5%$ level of inaccuracy, which is not easily achieved by other means. This would allow accurate polarizability measurements for clock transitions in other ions, through comparison with $^{138}\mathrm{Ba}^{+}$ . The approach would be equally applicable to ${\mathrm{Sr}}^{+}$ and ${\mathrm{Ca}}^{+}$ , with the latter being immediately applicable to ${\mathrm{Al}}^{+}/{\mathrm{Ca}}^{+}$ quantum logic clocks.It is shown that the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ transition in $^{138}\mathrm{Ba}^{+}$ can be determined to an inaccuracy below $0.5%$ across a wide wavelength range ($\ensuremath{\lambda}>700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). This can be achieved using measurements for which accurate determination of laser intensity is not required, and most of the required measurements are already in the literature. Measurement of a laser-induced ac-Stark shift of the clock transition would then provide an in situ measurement of the laser's intensity to the same $0.5%$ level of inaccuracy, which is not easily achieved by other means. This would allow accurate polarizability measurements for clock transitions in other ions, through comparison with $^{138}\mathrm{Ba}^{+}$. The approach would be equally applicable to ${\mathrm{Sr}}^{+}$ and ${\mathrm{Ca}}^{+}$, with the latter being immediately applicable to ${\mathrm{Al}}^{+}/{\mathrm{Ca}}^{+}$ quantum logic clocks.

arXiv:1905.04976Physical Review AOct 24, 201923 citations

Polarizability and magic wavelength of Yb+clock transition2S1/2 →2D3/2

Atish Roy, Subhadeep De, Bindiya Arora +1 more

Differential static polarizability and black body radiation (BBR) shift for Ytterbium ion (171Yb+) clock transition from ground state2S1/2to metastable state2D3/2corresponding to the wavelength 435.5 nm are calculated accurately for frequency standard applications. The values are found to be −7.5(5) × 10−40Jm2/V and −0.4(1) Hz, respectively, which are well in agreement with pre-available data and more accurate than the previous theoretical results. We further determine the values of dynamic polarizability of these states to find out the magic and tune-out wavelengths for the above mentioned clock transition.Differential static polarizability and black body radiation (BBR) shift for Ytterbium ion (171Yb+) clock transition from ground state2S1/2to metastable state2D3/2corresponding to the wavelength 435.5 nm are calculated accurately for frequency standard applications. The values are found to be −7.5(5) × 10−40Jm2/V and −0.4(1) Hz, respectively, which are well in agreement with pre-available data and more accurate than the previous theoretical results. We further determine the values of dynamic polarizability of these states to find out the magic and tune-out wavelengths for the above mentioned clock transition.

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)Jul 1, 20171 citation

Measurement of the tune-out wavelength for Cs133 at 880 nm

Apichayaporn Ratkata, Philip D. Gregory, Andrew D. Innes +6 more

We perform a measurement of the tune-out wavelength, ${\ensuremath{\lambda}}_{0}$, between the ${D}_{1}$ , ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{1/2}$, and ${D}_{2}$ , ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{3/2}$, transitions for $^{133}\mathrm{Cs}$ in the ground hyperfine state $(F=3,{m}_{F}=+3)$ . At ${\ensuremath{\lambda}}_{0}$, the frequency-dependent scalar polarizability is zero leading to a zero scalar ac Stark shift. We measure the polarizability as a function of wavelength using Kapitza-Dirac scattering of a $^{133}\mathrm{Cs}$ Bose-Einstein condensate in a one-dimensional optical lattice, and determine the tune-out wavelength to be ${\ensuremath{\lambda}}_{0}=880.21790{(40)}_{\text{stat}}{(8)}_{\text{sys}}$ nm. From this measurement we determine the ratio of reduced matrix elements to be ${|\ensuremath{\langle}6{P}_{3/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}/{|\ensuremath{\langle}6{P}_{1/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}=1.9808(2)$. This represents an improvement of a factor of 10 over previous results derived from excited-state lifetime measurements. We use the present measurement as a benchmark test of high-precision theory.We perform a measurement of the tune-out wavelength, ${\ensuremath{\lambda}}_{0}$, between the ${D}_{1}$, ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{1/2}$, and ${D}_{2}$, ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{3/2}$, transitions for $^{133}\mathrm{Cs}$ in the ground hyperfine state $(F=3,{m}_{F}=+3)$. At ${\ensuremath{\lambda}}_{0}$, the frequency-dependent scalar polarizability is zero leading to a zero scalar ac Stark shift. We measure the polarizability as a function of wavelength using Kapitza-Dirac scattering of a $^{133}\mathrm{Cs}$ Bose-Einstein condensate in a one-dimensional optical lattice, and determine the tune-out wavelength to be ${\ensuremath{\lambda}}_{0}=880.21790{(40)}_{\text{stat}}{(8)}_{\text{sys}}$ nm. From this measurement we determine the ratio of reduced matrix elements to be ${|\ensuremath{\langle}6{P}_{3/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}/{|\ensuremath{\langle}6{P}_{1/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}=1.9808(2)$. This represents an improvement of a factor of 10 over previous results derived from excited-state lifetime measurements. We use the present measurement as a benchmark test of high-precision theory.

arXiv:2110.00043Physical Review ANov 22, 202115 citations

Blackbody radiation shift assessment for a lutetium ion clock

K. J. Arnold, R. Kaewuam, Atish Roy +2 more

Abstract The accuracy of state-of-the-art atomic clocks is derived from the insensitivity of narrow optical atomic resonances to environmental perturbations. Two such resonances in singly ionized lutetium have been identified with potentially lower sensitivities compared to other clock candidates. Here we report measurement of the most significant unknown atomic property of both transitions, the static differential scalar polarizability. From this, the fractional blackbody radiation shift for one of the transitions is found to be −1.36(9) × 10−18 at 300 K, the lowest of any established optical atomic clock. In consideration of leading systematic effects common to all ion clocks, both transitions compare favorably to the most accurate ion-based clocks reported to date. This work firmly establishes Lu+ as a promising candidate for a future generation of more accurate optical atomic clocks.Abstract The accuracy of state-of-the-art atomic clocks is derived from the insensitivity of narrow optical atomic resonances to environmental perturbations. Two such resonances in singly ionized lutetium have been identified with potentially lower sensitivities compared to other clock candidates. Here we report measurement of the most significant unknown atomic property of both transitions, the static differential scalar polarizability. From this, the fractional blackbody radiation shift for one of the transitions is found to be −1.36(9) × 10−18 at 300 K, the lowest of any established optical atomic clock. In consideration of leading systematic effects common to all ion clocks, both transitions compare favorably to the most accurate ion-based clocks reported to date. This work firmly establishes Lu+ as a promising candidate for a future generation of more accurate optical atomic clocks.

arXiv:1712.00240Nature CommunicationsApr 19, 201874 citations

Zero crossings of the differential scalar polarizability of Ba

^+

clock transition

The differential scalar polarizability

Δα_0(ω)

of the Ba

^+

_{1/2}

-to-D

_{5/2}

clock transition has a zero crossing near 481nm, which is measured to be 623.603,13(17),THz. From this measurement, we infer a ratio of reduced matrix elements

\langle P_{3/2}\|r\|S_{1/2}\rangle/\langle P_{1/2}\|r\|S_{1/2}\rangle=1.411\,81(13)

, which provides a stringent test of atomic structure calculations and experimental determination of matrix elements. Additionally, it enables the construction of an accurate approximation to

Δα_0(ω)

, valid for frequencies up to 450,THz, with only one reduced matrix element,

\langle P_{1/2}\|r\|S_{1/2}\rangle

The zero crossing of the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ clock transition in

^{138}\mathrm{Ba}^{+}

has been determined to be

459.1614(28)\phantom{\rule{0.28em}{0ex}}\mathrm{THz}

. Together with previously determined matrix elements and branching ratios, this tightly constrains the dynamic differential scalar polarizability of the clock transition over a large wavelength range ($\ensuremath{\gtrsim}700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). In particular, it allows an estimate of the blackbody radiation shift of the clock transition at room temperature.

Measurement of the reduced dipole matrix element in Ba

^+

We present a high-precision measurement of the reduced electric-dipole matrix element

\langle P_{1/2}\|r\|S_{1/2}\rangle

^{138}\mathrm{Ba}^+

. By comparing off-resonant scattering rates with dispersive Stark-shift measurements, we determine the matrix element to be

3.322\,7(12)

, corresponding to a

P_{1/2}

excited-state radiative lifetime of

7.866\,3(56)

~ns. Combining our experimental results with a prior model for the differential scalar polarizability

Δα_0(ω)

, we extract the static value

Δα_0(0) = -73.09(12)

a.u. This determination directly improves the evaluation of the blackbody radiation (BBR) shift, minimizing a prominent systematic limitation in room-temperature

\mathrm{Ba}^+

optical clock error budgets. These results provide a stringent benchmark for atomic-structure calculations and advance high-precision applications in quantum metrology and tests of fundamental physics.

It is shown that the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ transition in

^{138}\mathrm{Ba}^{+}

can be determined to an inaccuracy below

0.5%

across a wide wavelength range ($\ensuremath{\lambda}>700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). This can be achieved using measurements for which accurate determination of laser intensity is not required, and most of the required measurements are already in the literature. Measurement of a laser-induced ac-Stark shift of the clock transition would then provide an in situ measurement of the laser's intensity to the same

0.5%

level of inaccuracy, which is not easily achieved by other means. This would allow accurate polarizability measurements for clock transitions in other ions, through comparison with

^{138}\mathrm{Ba}^{+}

. The approach would be equally applicable to

{\mathrm{Sr}}^{+}

and

{\mathrm{Ca}}^{+}

, with the latter being immediately applicable to

{\mathrm{Al}}^{+}/{\mathrm{Ca}}^{+}

quantum logic clocks.

We perform a measurement of the tune-out wavelength, ${\ensuremath{\lambda}}_{0}$, between the

{D}_{1}

, ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{1/2}$, and

{D}_{2}

, ${6}^{2}{S}_{1/2}\ensuremath{\rightarrow}{6}^{2}{P}_{3/2}$, transitions for

^{133}\mathrm{Cs}

in the ground hyperfine state

(F=3,{m}_{F}=+3)

. At ${\ensuremath{\lambda}}_{0}$, the frequency-dependent scalar polarizability is zero leading to a zero scalar ac Stark shift. We measure the polarizability as a function of wavelength using Kapitza-Dirac scattering of a

^{133}\mathrm{Cs}

Bose-Einstein condensate in a one-dimensional optical lattice, and determine the tune-out wavelength to be ${\ensuremath{\lambda}}_{0}=880.21790{(40)}_{\text{stat}}{(8)}_{\text{sys}}$ nm. From this measurement we determine the ratio of reduced matrix elements to be ${|\ensuremath{\langle}6{P}_{3/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}/{|\ensuremath{\langle}6{P}_{1/2}\ensuremath{\parallel}d\ensuremath{\parallel}6{S}_{1/2}\ensuremath{\rangle}|}^{2}=1.9808(2)$. This represents an improvement of a factor of 10 over previous results derived from excited-state lifetime measurements. We use the present measurement as a benchmark test of high-precision theory.