Physical Review A

Combined experimental and theoretical probe of the lifetime of the 3d2D5/2 state in 40Ca+

Hua Guan, H. Shao, Yuan Qian, Yao Huang, Peiliang Liu, Bian Wu, Chengbin Li, B. K. Sahoo, Kelin Gao

Published Feb 18, 2015• 14 citations

Abstract

In light of the diverse range of reported values for the lifetimes of metastable states of ${}^{40}{\mathrm{Ca}}^{+}$, we have carried out afresh both measurements and theoretical investigation to confirm the lifetime of its $3d\phantom{\rule{0.16em}{0ex}}{}^{2}{D}_{5/2}$ state $({\ensuremath{\tau}}_{3d5/2})$. A high-efficiency quantum state detection method by monitoring the quantum jumps of a laser-cooled single ${\mathrm{Ca}}^{+}$ ion in a miniature ring Paul trap was employed in the measurement. Also, sophisticated calculations were performed considering higher order nonlinear terms in the relativistic coupled-cluster (RCC) method with all possible single and double excitations, but accounting only for the important triple excitations from both the core and the valence orbitals. Systematic factors affecting measurement, such as collision with background gases, heating effects, the power of the 866-nm laser, and state detection errors were carefully analyzed. Our observational and theoretical values for ${\ensuremath{\tau}}_{3d5/2}$ are 1174(10) ms and 1172(3) ms, respectively, which agree well with the experimental results reported by P. A. Barton et al. [Phys. Rev. A 62, 032503 (2000)] and A. Kreuter et al. [Phys. Rev. A 71, 032504 (2005)]. The present theoretical analysis demonstrates that the contributions from the core triples and Breit interaction are notable, as they improve the theoretical results obtained in the previous RCC calculations [B. K. Sahoo, Phys. Rev. A 74, 062504 (2006)].

DOI: 10.1103/physreva.91.022511

Related papers

Measurement of the lifetime of the 3d2D5/2 state in 40Ca+

P. A. Barton, C. J. S. Donald, D. M. Lucas +3 more

arXiv:physics/0002026Physical Review AAug 14, 2000132 citations

Experimental and theoretical study of the 3 d D 2 –level lifetimes of Ca + 40

Axel Kreuter, Christoph Becher, Gavin P. T. Lancaster +8 more

arXiv:physics/0409038Physical Review AMar 11, 2005105 citations

Precision measurement of the 3dD3/22 –state lifetime in a single trapped Ca+40

H. Shao, Yao Huang, H. Guan +2 more

Physical Review AOct 10, 201617 citations

Direct measurement of the 3 d 2 D 3 / 2 to 3 d 2 D 5 / 2 lifetime ratio in a single trapped 40 Ca +

H Shao, Yao Huang, H. Guan +1 more

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Combined experimental and theoretical probe of the branching fractions of the 4P3/2 state in Ca+40

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Physical Review ANov 14, 20196 citations

Lifetime measurement of the metastable 4dD5∕22 state in Sr+88 using a single trapped ion

V. Letchumanan, M. A. Wilson, Patrick Robert Gill +1 more

Physical Review AJul 13, 200539 citations

Physical Review A

Combined experimental and theoretical probe of the lifetime of the 3d2D5/2 state in 40Ca+

Hua Guan, H. Shao, Yuan Qian, Yao Huang, Peiliang Liu, Bian Wu, Chengbin Li, B. K. Sahoo, Kelin Gao

Published Feb 18, 2015• 14 citations

Publisher

Abstract

In light of the diverse range of reported values for the lifetimes of metastable states of

{}^{40}{\mathrm{Ca}}^{+}

, we have carried out afresh both measurements and theoretical investigation to confirm the lifetime of its

3d\phantom{\rule{0.16em}{0ex}}{}^{2}{D}_{5/2}

state

({\ensuremath{\tau}}_{3d5/2})

. A high-efficiency quantum state detection method by monitoring the quantum jumps of a laser-cooled single

{\mathrm{Ca}}^{+}

ion in a miniature ring Paul trap was employed in the measurement. Also, sophisticated calculations were performed considering higher order nonlinear terms in the relativistic coupled-cluster (RCC) method with all possible single and double excitations, but accounting only for the important triple excitations from both the core and the valence orbitals. Systematic factors affecting measurement, such as collision with background gases, heating effects, the power of the 866-nm laser, and state detection errors were carefully analyzed. Our observational and theoretical values for

{\ensuremath{\tau}}_{3d5/2}

are 1174(10) ms and 1172(3) ms, respectively, which agree well with the experimental results reported by P. A. Barton et al. [Phys. Rev. A 62, 032503 (2000)] and A. Kreuter et al. [Phys. Rev. A 71, 032504 (2005)]. The present theoretical analysis demonstrates that the contributions from the core triples and Breit interaction are notable, as they improve the theoretical results obtained in the previous RCC calculations [B. K. Sahoo, Phys. Rev. A 74, 062504 (2006)].

DOI: 10.1103/physreva.91.022511

Related papers

Measurement of the lifetime of the 3d2D5/2 state in 40Ca+

P. A. Barton, C. J. S. Donald, D. M. Lucas +3 more

We report a measurement of the lifetime of the $3d{}^{2}{D}_{5/2}$ metastable level in ${}^{40}{\mathrm{Ca}}^{+},$ using quantum jumps of a single cold calcium ion in a linear Paul trap. The $4s{}^{2}{S}_{1/2}--3d{}^{2}{D}_{5/2}$ transition is significant for single-ion optical frequency standards, astrophysical references, and tests of atomic structure calculations. We obtain $\ensuremath{\tau}=1.168\ifmmode\pm\else\textpm\fi{}0.007$ s from observation of nearly $64000$ quantum jumps during $\ensuremath{\sim}32$ h. Our result is more precise and significantly larger than previous measurements. Experiments carried out to quantify systematic effects included a study of a previously unremarked source of systematic error, namely, excitation by the broad background of radiation emitted by a semiconductor diode laser. Combining our result with atomic structure calculations yields $1.20\ifmmode\pm\else\textpm\fi{}0.01$ s for the lifetime of $3d{}^{2}{D}_{3/2}.$ We also use quantum jump observations to demonstrate photon antibunching, and to estimate background pressure and heating rates in the ion trap.We report a measurement of the lifetime of the $3d{}^{2}{D}_{5/2}$ metastable level in ${}^{40}{\mathrm{Ca}}^{+},$ using quantum jumps of a single cold calcium ion in a linear Paul trap. The $4s{}^{2}{S}_{1/2}--3d{}^{2}{D}_{5/2}$ transition is significant for single-ion optical frequency standards, astrophysical references, and tests of atomic structure calculations. We obtain $\ensuremath{\tau}=1.168\ifmmode\pm\else\textpm\fi{}0.007$ s from observation of nearly $64000$ quantum jumps during $\ensuremath{\sim}32$ h. Our result is more precise and significantly larger than previous measurements. Experiments carried out to quantify systematic effects included a study of a previously unremarked source of systematic error, namely, excitation by the broad background of radiation emitted by a semiconductor diode laser. Combining our result with atomic structure calculations yields $1.20\ifmmode\pm\else\textpm\fi{}0.01$ s for the lifetime of $3d{}^{2}{D}_{3/2}.$ We also use quantum jump observations to demonstrate photon antibunching, and to estimate background pressure and heating rates in the ion trap.

arXiv:physics/0002026Physical Review AAug 14, 2000132 citations

Experimental and theoretical study of the 3 d D 2 –level lifetimes of Ca + 40

Axel Kreuter, Christoph Becher, Gavin P. T. Lancaster +8 more

We report measurements of the lifetimes of the $3d\phantom{\rule{0.3em}{0ex}}^{2}D_{5∕2}$ and $3d\phantom{\rule{0.3em}{0ex}}^{2}D_{3∕2}$ metastable states of a single laser-cooled $^{40}\mathrm{Ca}^{+}$ ion in a linear Paul trap. We introduce a measurement technique based on high-efficiency quantum state detection after coherent excitation to the ${D}_{5∕2}$ state or incoherent shelving in the ${D}_{3∕2}$ state, and subsequent free, unperturbed spontaneous decay. The result for the natural lifetime of the ${D}_{5∕2}$ state of 1168(9) ms agrees excellently with the most precise published value. The lifetime of the ${D}_{3∕2}$ state is measured with a single ion and yields 1176(11) ms which improves the statistical uncertainty of previous results by a factor of four. We compare these experimental lifetimes to high-precision ab initio all order calculations [ ${D}_{3∕2}$ state: 1196(11) ms; ${D}_{5∕2}$ state: 1165(11) ms] and find a very good agreement. These calculations represent an excellent test of high-precision atomic theory and will serve as a benchmark for the study of parity nonconservation in ${\mathrm{Ba}}^{+}$ which has similar atomic structure.We report measurements of the lifetimes of the $3d\phantom{\rule{0.3em}{0ex}}^{2}D_{5∕2}$ and $3d\phantom{\rule{0.3em}{0ex}}^{2}D_{3∕2}$ metastable states of a single laser-cooled $^{40}\mathrm{Ca}^{+}$ ion in a linear Paul trap. We introduce a measurement technique based on high-efficiency quantum state detection after coherent excitation to the ${D}_{5∕2}$ state or incoherent shelving in the ${D}_{3∕2}$ state, and subsequent free, unperturbed spontaneous decay. The result for the natural lifetime of the ${D}_{5∕2}$ state of 1168(9) ms agrees excellently with the most precise published value. The lifetime of the ${D}_{3∕2}$ state is measured with a single ion and yields 1176(11) ms which improves the statistical uncertainty of previous results by a factor of four. We compare these experimental lifetimes to high-precision ab initio all order calculations [${D}_{3∕2}$ state: 1196(11) ms; ${D}_{5∕2}$ state: 1165(11) ms] and find a very good agreement. These calculations represent an excellent test of high-precision atomic theory and will serve as a benchmark for the study of parity nonconservation in ${\mathrm{Ba}}^{+}$ which has similar atomic structure.

arXiv:physics/0409038Physical Review AMar 11, 2005105 citations

Precision measurement of the 3dD3/22 –state lifetime in a single trapped Ca+40

H. Shao, Yao Huang, H. Guan +2 more

We present a high-precision measurement of the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$ --state lifetime in a single trapped $^{40}\mathrm{Ca}^{+}$ . The measurement was performed using a high-efficiency quantum-state detection technique to monitor quantum jumps and a high-precision and highly synchronous measurement sequence for laser control. A feature in our measurement is the pumping rate of the 729-nm laser that was corrected in a real-time way. The $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$ --state lifetime was obtained through the measurement of the spontaneous decay rate after incoherent shelving of the ion to the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$ state with a wait time. Systematic errors, such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates, and state detection, were carefully analyzed and estimated. We determined an improved value of the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$ --state lifetime to be ${\ensuremath{\tau}}_{3/2}=1.195(8)$ s. Furthermore, the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}\ensuremath{\rightarrow}4s\phantom{\rule{4pt}{0ex}}^{2}S_{1/2}$ quadrupole transition matrix element was measured to be ${S}_{ki}=7.936(26)e{a}_{0}^{2}$ , and the ratio between the lifetimes of $3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{3/2}$ and $3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{5/2}$ was determined to be 1.018(11). Our method can be universally applied to lifetime measurements of other single ions and atoms with a similar structure.We present a high-precision measurement of the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$--state lifetime in a single trapped $^{40}\mathrm{Ca}^{+}$. The measurement was performed using a high-efficiency quantum-state detection technique to monitor quantum jumps and a high-precision and highly synchronous measurement sequence for laser control. A feature in our measurement is the pumping rate of the 729-nm laser that was corrected in a real-time way. The $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$--state lifetime was obtained through the measurement of the spontaneous decay rate after incoherent shelving of the ion to the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$ state with a wait time. Systematic errors, such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates, and state detection, were carefully analyzed and estimated. We determined an improved value of the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}$--state lifetime to be ${\ensuremath{\tau}}_{3/2}=1.195(8)$ s. Furthermore, the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}\ensuremath{\rightarrow}4s\phantom{\rule{4pt}{0ex}}^{2}S_{1/2}$ quadrupole transition matrix element was measured to be ${S}_{ki}=7.936(26)e{a}_{0}^{2}$, and the ratio between the lifetimes of $3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{3/2}$ and $3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{5/2}$ was determined to be 1.018(11). Our method can be universally applied to lifetime measurements of other single ions and atoms with a similar structure.

Physical Review AOct 10, 201617 citations

Direct measurement of the 3 d 2 D 3 / 2 to 3 d 2 D 5 / 2 lifetime ratio in a single trapped 40 Ca +

H Shao, Yao Huang, H. Guan +1 more

Abstract We present for the first time a direct measurement of the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 metastable states in a single trapped 40 Ca + . A high-efficiency quantum state detection technique is adopted to monitor the quantum jumps, and a high precision and synchronized measurement sequence is used for laser control to study the rule of spontaneous decay. Our method shows that the lifetime ratio is a constant and is irrelevant to the dwell time; it is only determined by the spontaneous decay probabilities of the two metastable states at one random decay time, independent of the lifetimes of the two metastable states. Systematic errors such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates and the photon counts are analyzed, and the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 states is directly measured to be 1.0257(43) with an uncertainty of 0.42%. Our result is in good agreement with the most precise many-body atomic structure calculations. Our method can be used to obtain the lifetime of a state which is usually difficult to measure and can be used to determine the magnetic dipole transition matrix elements in heavy ions such as Ba + and Ra + , and can also be universally applied to lifetime ratio measurements of other single atoms and ions with a similar structure.Abstract We present for the first time a direct measurement of the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 metastable states in a single trapped 40 Ca + . A high-efficiency quantum state detection technique is adopted to monitor the quantum jumps, and a high precision and synchronized measurement sequence is used for laser control to study the rule of spontaneous decay. Our method shows that the lifetime ratio is a constant and is irrelevant to the dwell time; it is only determined by the spontaneous decay probabilities of the two metastable states at one random decay time, independent of the lifetimes of the two metastable states. Systematic errors such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates and the photon counts are analyzed, and the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 states is directly measured to be 1.0257(43) with an uncertainty of 0.42%. Our result is in good agreement with the most precise many-body atomic structure calculations. Our method can be used to obtain the lifetime of a state which is usually difficult to measure and can be used to determine the magnetic dipole transition matrix elements in heavy ions such as Ba + and Ra + , and can also be universally applied to lifetime ratio measurements of other single atoms and ions with a similar structure.

Journal of Physics B Atomic Molecular and Optical PhysicsDec 13, 20175 citations

Combined experimental and theoretical probe of the branching fractions of the 4P3/2 state in Ca+40

Hong-Fang Song, Yong-Bo Tang, Shao‐Long Chen +4 more

In this work, we carry out experimental measurements and theoretical calculations of the branching fractions for the $4{P}_{3/2}$ state of $^{40}\mathrm{Ca}^{+}$ . The measurements are based on counting photon numbers emitted from hundreds of ions trapped in a linear Paul trap, as their being affected by lasers. The calculations are performed using the relativistic coupled-cluster method at different approximations. The measured values for the decays of the $4{P}_{3/2}$ state into the $4{S}_{1/2}$ , $3{D}_{5/2}$ , and $3{D}_{3/2}$ states are, respectively, 0.9350(4), 0.0587(4), and 0.0063(2), which agree well with our recommended theoretical values 0.9344(13), 0.0590(12), and 0.0066(1). Our measured values are also consistent with the experimental values of Gerritsma et al. [R. Gerritsma, G. Kirchmair, F. Z\"ahringer, J. Benhelm, R. Blatt, and C. F. Roos, Eur. Phys. J. D 50, 13 (2008)] using a completely different method. Furthermore, we also calculate the branching fractions of the $4{P}_{1/2}$ state using the same method, and compare to the available experimental and theoretical data.In this work, we carry out experimental measurements and theoretical calculations of the branching fractions for the $4{P}_{3/2}$ state of $^{40}\mathrm{Ca}^{+}$. The measurements are based on counting photon numbers emitted from hundreds of ions trapped in a linear Paul trap, as their being affected by lasers. The calculations are performed using the relativistic coupled-cluster method at different approximations. The measured values for the decays of the $4{P}_{3/2}$ state into the $4{S}_{1/2}$, $3{D}_{5/2}$, and $3{D}_{3/2}$ states are, respectively, 0.9350(4), 0.0587(4), and 0.0063(2), which agree well with our recommended theoretical values 0.9344(13), 0.0590(12), and 0.0066(1). Our measured values are also consistent with the experimental values of Gerritsma et al. [R. Gerritsma, G. Kirchmair, F. Z\"ahringer, J. Benhelm, R. Blatt, and C. F. Roos, Eur. Phys. J. D 50, 13 (2008)] using a completely different method. Furthermore, we also calculate the branching fractions of the $4{P}_{1/2}$ state using the same method, and compare to the available experimental and theoretical data.

Physical Review ANov 14, 20196 citations

Lifetime measurement of the metastable 4dD5∕22 state in Sr+88 using a single trapped ion

V. Letchumanan, M. A. Wilson, Patrick Robert Gill +1 more

The natural lifetime of the metastable $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state in $^{88}\mathrm{Sr}^{+}$ has been measured using a single laser-cooled, trapped ion. The ion was prepared in the $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state by laser excitation of the $5s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}\ensuremath{-}4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ quadrupole transition at 674 nm. The decay time was measured by using Dehmelt's electron shelving method to monitor the ion's electronic state. Two separate investigations were conducted under slightly different experimental conditions. 160 000 decay times from the $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state were observed in total, and two different methods for determining the statistical value of the measured lifetime have been compared. Systematic effects have also been estimated, and including these effects yields a value of $\ensuremath{\tau}=390.8(1.6)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$ ($1\ensuremath{\sigma}$ uncertainty) for the natural lifetime.The natural lifetime of the metastable $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state in $^{88}\mathrm{Sr}^{+}$ has been measured using a single laser-cooled, trapped ion. The ion was prepared in the $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state by laser excitation of the $5s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}\ensuremath{-}4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ quadrupole transition at 674 nm. The decay time was measured by using Dehmelt's electron shelving method to monitor the ion's electronic state. Two separate investigations were conducted under slightly different experimental conditions. 160 000 decay times from the $4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state were observed in total, and two different methods for determining the statistical value of the measured lifetime have been compared. Systematic effects have also been estimated, and including these effects yields a value of $\ensuremath{\tau}=390.8(1.6)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$ ($1\ensuremath{\sigma}$ uncertainty) for the natural lifetime.

Physical Review AJul 13, 200539 citations

In light of the diverse range of reported values for the lifetimes of metastable states of

{}^{40}{\mathrm{Ca}}^{+}

, we have carried out afresh both measurements and theoretical investigation to confirm the lifetime of its

3d\phantom{\rule{0.16em}{0ex}}{}^{2}{D}_{5/2}

state

({\ensuremath{\tau}}_{3d5/2})

. A high-efficiency quantum state detection method by monitoring the quantum jumps of a laser-cooled single

{\mathrm{Ca}}^{+}

{\ensuremath{\tau}}_{3d5/2}

We report a measurement of the lifetime of the

3d{}^{2}{D}_{5/2}

metastable level in

{}^{40}{\mathrm{Ca}}^{+},

using quantum jumps of a single cold calcium ion in a linear Paul trap. The

4s{}^{2}{S}_{1/2}--3d{}^{2}{D}_{5/2}

transition is significant for single-ion optical frequency standards, astrophysical references, and tests of atomic structure calculations. We obtain $\ensuremath{\tau}=1.168\ifmmode\pm\else\textpm\fi{}0.007$ s from observation of nearly

64000

quantum jumps during $\ensuremath{\sim}32$ h. Our result is more precise and significantly larger than previous measurements. Experiments carried out to quantify systematic effects included a study of a previously unremarked source of systematic error, namely, excitation by the broad background of radiation emitted by a semiconductor diode laser. Combining our result with atomic structure calculations yields $1.20\ifmmode\pm\else\textpm\fi{}0.01$ s for the lifetime of

3d{}^{2}{D}_{3/2}.

We also use quantum jump observations to demonstrate photon antibunching, and to estimate background pressure and heating rates in the ion trap.

We report measurements of the lifetimes of the

3d\phantom{\rule{0.3em}{0ex}}^{2}D_{5∕2}

and

3d\phantom{\rule{0.3em}{0ex}}^{2}D_{3∕2}

metastable states of a single laser-cooled

^{40}\mathrm{Ca}^{+}

ion in a linear Paul trap. We introduce a measurement technique based on high-efficiency quantum state detection after coherent excitation to the

{D}_{5∕2}

state or incoherent shelving in the

{D}_{3∕2}

state, and subsequent free, unperturbed spontaneous decay. The result for the natural lifetime of the

{D}_{5∕2}

state of 1168(9) ms agrees excellently with the most precise published value. The lifetime of the

{D}_{3∕2}

state is measured with a single ion and yields 1176(11) ms which improves the statistical uncertainty of previous results by a factor of four. We compare these experimental lifetimes to high-precision ab initio all order calculations [

{D}_{3∕2}

state: 1196(11) ms;

{D}_{5∕2}

state: 1165(11) ms] and find a very good agreement. These calculations represent an excellent test of high-precision atomic theory and will serve as a benchmark for the study of parity nonconservation in

{\mathrm{Ba}}^{+}

which has similar atomic structure.

We present a high-precision measurement of the

3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}

--state lifetime in a single trapped

^{40}\mathrm{Ca}^{+}

. The measurement was performed using a high-efficiency quantum-state detection technique to monitor quantum jumps and a high-precision and highly synchronous measurement sequence for laser control. A feature in our measurement is the pumping rate of the 729-nm laser that was corrected in a real-time way. The

3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}

--state lifetime was obtained through the measurement of the spontaneous decay rate after incoherent shelving of the ion to the

3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}

state with a wait time. Systematic errors, such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates, and state detection, were carefully analyzed and estimated. We determined an improved value of the

3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}

--state lifetime to be ${\ensuremath{\tau}}_{3/2}=1.195(8)$ s. Furthermore, the $3d\phantom{\rule{4pt}{0ex}}^{2}D_{3/2}\ensuremath{\rightarrow}4s\phantom{\rule{4pt}{0ex}}^{2}S_{1/2}$ quadrupole transition matrix element was measured to be

{S}_{ki}=7.936(26)e{a}_{0}^{2}

, and the ratio between the lifetimes of

3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{3/2}

and

3d{\phantom{\rule{4pt}{0ex}}}^{2}{D}_{5/2}

was determined to be 1.018(11). Our method can be universally applied to lifetime measurements of other single ions and atoms with a similar structure.

In this work, we carry out experimental measurements and theoretical calculations of the branching fractions for the

4{P}_{3/2}

state of

^{40}\mathrm{Ca}^{+}

. The measurements are based on counting photon numbers emitted from hundreds of ions trapped in a linear Paul trap, as their being affected by lasers. The calculations are performed using the relativistic coupled-cluster method at different approximations. The measured values for the decays of the

4{P}_{3/2}

state into the

4{S}_{1/2}

3{D}_{5/2}

, and

3{D}_{3/2}

states are, respectively, 0.9350(4), 0.0587(4), and 0.0063(2), which agree well with our recommended theoretical values 0.9344(13), 0.0590(12), and 0.0066(1). Our measured values are also consistent with the experimental values of Gerritsma et al. [R. Gerritsma, G. Kirchmair, F. Z\"ahringer, J. Benhelm, R. Blatt, and C. F. Roos, Eur. Phys. J. D 50, 13 (2008)] using a completely different method. Furthermore, we also calculate the branching fractions of the

4{P}_{1/2}

state using the same method, and compare to the available experimental and theoretical data.

The natural lifetime of the metastable

4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}

state in

^{88}\mathrm{Sr}^{+}

has been measured using a single laser-cooled, trapped ion. The ion was prepared in the

4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}

state by laser excitation of the $5s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}\ensuremath{-}4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ quadrupole transition at 674 nm. The decay time was measured by using Dehmelt's electron shelving method to monitor the ion's electronic state. Two separate investigations were conducted under slightly different experimental conditions. 160 000 decay times from the

4d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}

state were observed in total, and two different methods for determining the statistical value of the measured lifetime have been compared. Systematic effects have also been estimated, and including these effects yields a value of $\ensuremath{\tau}=390.8(1.6)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$ ($1\ensuremath{\sigma}$ uncertainty) for the natural lifetime.