Exploration of the magnetic-field-induced $5s5p$ ³P₀–5s^{2 1}S₀ forbidden transition in bosonic Sr atom

Our experimental setup and the simplified energy levels of ⁸⁸Sr atom were shown in figures 1(a) and (b), individually. Cold atoms preparation was described in detail in [20]. The ¹S₀–${}^{\mathrm{1,3}}{P}_{1}$ transitions were used for first and second stage magnetic-optically trapping which cooled the atoms down to a few μK. The 679 nm and 707 nm lasers were the re-pumping lasers. Atoms in the ${}^{3}{P}_{\mathrm{0,2}}$ metastable states were pumped to the ³S₁ state by using the re-pumping lasers to drive the ³P₀–³S₁ and ³P₂–³S₁ transitions. Eventually, these atoms would decay to the ground state through ³S₁ $\to$ ³P₁ $\to$ ¹S₀ as the spontaneous decay rate from ³P₁ state is much larger than that from ${}^{3}{P}_{\mathrm{0,2}}$ states. After the two-stage cooling, the ultra-cold bosonic Sr atoms were loaded into an optical lattice which was operated at the 'magic wavelength' of ${\lambda }_{L}$ = 813 nm [16, 21]. The 813 nm laser beam was transferred by a single-mode-polarization-maintaining fiber. The 698 nm probe laser was locked to a stable ultra-low expansion (ULE) cavity which was placed on an enclosed vibration isolation platform, and was transferred by a noise canceled fiber link. The line-width of the laser was about 1 Hz and the Allan deviation was 1 × 10⁻¹⁵ at 1 s. This laser was split into two beams, one going to the femtosecond optical frequency comb (OFC, Menlo FC1500) to have its frequency measured. The second one passed through a λ/2 wave plate (HWP) and a mechanical shutter and combined with the lattice laser using a dichroic mirror (DM). The 813 nm laser and the 698 nm laser passed through a Glan–Taylor polarizer to make sure their polarizations, and focused by an achromatic lens (f = 200 mm) into the vacuum chamber and onto the cold atoms. The wave-vector ${\boldsymbol{k}}$ of the two lasers are perpendicular to the gravity. The waist of the 813 nm laser was 38 μm (1/e² radius of intensity), and the 698 nm laser beam was approximately three times larger. After exciting the vacuum chamber, the lattice laser was retro-reflected by another achromatic lens and a mirror to overlapped with the incident laser beam and formed the standing wave. The retro-reflecting mirror was low-reflecting at the 698 nm wavelength. The focusing lens were mounted on two three-dimensional translation stages respectively to optimize the optical lattice and the ¹S₀–³P₁ magneto-optical trap overlapping. The static magnetic field was produced by Helmholtz coils and its direction was parallel to the linear polarization of the 698 nm probe laser.

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About 10⁵ atoms were trapped in the one-dimensional optical lattice. The temperature of these atoms was 8.4 μK and the lifetime of the atoms trapped in the optical lattice was 500 ms. The $5s5p$ ³P₀–5s^{2 1}S₀ forbidden transition was probed in the Lamb–Dicke regime along the lattice longitudinal axis. During the experiment, the 813 nm laser was maintained open. We used the normalized detection method [22] to measure the $5s5p$ ³P₀–5s^{2 1}S₀ excitation fraction. Firstly, we used the 698 nm probe laser pulse (π-pulse) to pump some of these atoms, namely N₁, to the ³P₀ state. The static magnetic field was also applied. Secondly, the remaining atoms, namely N₂, were read out and pushed out of the optical lattice with a 461 nm detection laser pulse. Thirdly, the atoms remained in the ³P₀ state were transferred to the ground state by the 679 nm and 707 nm laser pulses. Finally, the pumped atoms (N₁) were also read out with another 461 nm detection laser pulse. The values of N₁ and N₂ were measured by probing the 461 nm fluorescence intensity. As the lifetime of ³P₀ state is very long, we neglected the population decay of ³P₀ state atoms in the optical lattice after the probe and detection pulses we had chosen. We assumed that atoms in the ³P₀ state were all transferred to the ground state. The excitation fraction was deduced from these two measurements by

$$\begin{eqnarray}&&P=\displaystyle \frac{{N}_{1}}{{N}_{1}+{N}_{2}}.\end{eqnarray} \tag{ 1 }$$

We used a photomultiplier tube (PMT) to measure the 461 nm fluorescence intensity for evaluating the numbers of the atoms in the ground state. In order to obtain a high signal-to-noise ratio of the fluorescence intensity, a 461 nm filter, a lens group and an aperture were used to eliminate the background noise. The measured spectrums were presented in figures 2(a)–(d) under the circumstance of the magnetic field strength B = 1.0, 1.2, 1.7, 2.2 mT, respectively. During the measurements, the power of the 698 nm laser was maintained constant. The spectrums were generated by repeating the detection cycle and stepping the frequency of the 698 nm laser using an acousto-optic modulator. The experimental data were fitted with Lorentzian function. From this figure, we can see that the intensity of the spectrum peak decreased and the line-width shrinked with the magnetic field strength decreasing. When B = 1.0 mT, the line-width is 180 Hz which is larger than the Fourier limited width (20 Hz) due to 45 ms interrogation pulse. The line-width is broadened by many factors, for example, the atomic collisions. Since in this paper the intensity of the spectrum peak is the focus of the discussion, the line-width broadening is neglected.

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3.1.1. Magnetic-field-induced transition rate

In the presence of an external magnetic field ${\bf{B}}$, the atomic Hamiltonian is [23]

$$\begin{eqnarray}&&H={H}_{{\rm{fs}}}+{H}_{m},\end{eqnarray} \tag{ 2 }$$

where H_fs is the relativistic fine-structure Hamiltonian which include the Breit interaction and the main part quantum electrodynamical (QED) effects and H_m is the Hamiltonian for the interaction between the external magnetic field and the atom. We choose the direction of the magnetic field as the quantization axis and assume the magnetic field does not vary throughout the atomic system, the interaction Hamiltonian H_m is expressed as [24]

$$\begin{eqnarray}&&{H}_{m}=({N}^{(1)}+{\rm{\Delta }}{N}^{(1)}){B}.\end{eqnarray} \tag{ 3 }$$

Here, ${\rm{\Delta }}{N}^{(1)}$ is the so-called Schwinger QED correction [24]. For an N-electron atom, these two operators are

$$\begin{eqnarray}&&{N}^{(1)}=\displaystyle \sum _{j=1}^{N}{n}^{(1)}(j)=\displaystyle \sum _{j=1}^{N}-{\rm{i}}\displaystyle \frac{\sqrt{2}}{2\alpha }{r}_{j}{({{\boldsymbol{\alpha }}}_{j}{{\boldsymbol{C}}}^{(1)}(j))}^{(1)},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{N}^{(1)}=\displaystyle \sum _{j=1}^{N}{\rm{\Delta }}{n}^{(1)}(j)=\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{g}_{s}-2}{2}{\beta }_{j}{{\boldsymbol{\Sigma }}}_{j}.\end{eqnarray} \tag{ 5 }$$

Here, ${\rm{i}}$ is the imaginary unit, r_j is the radial coordinate of the jth electron, ${{\boldsymbol{C}}}^{(1)}(j)$ is the spherical tensor operator of rank 1, ${{\boldsymbol{\Sigma }}}_{j}$ is the relativistic spin-matrix, g_s is the g-factor of the electron spin corrected by the QED effects, α is the fine-structure constant and ${\boldsymbol{\alpha }}$ and β are the Dirac matrices.

In the presence of the external magnetic field, only the magnetic quantum number M_J remains the good quantum number. The atomic states with the same magnetic quantum number and parity are mixed due to the interaction between the external magnetic field and the atom [23]. Therefore, the atomic state wave function $| {M}_{J}\rangle$ can be written as

$$\begin{eqnarray}&&| {M}_{J}\rangle =\displaystyle \sum _{{\rm{\Gamma }}J}{d}_{{\rm{\Gamma }}J}| {\rm{\Gamma }}{{JM}}_{J}\rangle ,\end{eqnarray} \tag{ 6 }$$

where the atomic state wave functions $| {\rm{\Gamma }}{{JM}}_{J}\rangle$ are eigenstates of the Hamiltonian H_fs, J and M_J are the total and magnetic quantum numbers. According to the first-order perturbation theory, the expansion coefficient ${d}_{{\rm{\Gamma }}J}$ is given by

$$\begin{eqnarray}&&{d}_{{\rm{\Gamma }}J}=\displaystyle \frac{\langle {\rm{\Gamma }}{{JM}}_{J}| {H}_{m}| {\rm{\Gamma }}^{\prime} J^{\prime} {M}_{J}\rangle }{E({\rm{\Gamma }}^{\prime} J^{\prime} )-E({\rm{\Gamma }}J)},\end{eqnarray} \tag{ 7 }$$

where $| {\rm{\Gamma }}^{\prime} J^{\prime} {M}_{J}\rangle$ is the reference atomic state.

Using the equation (6), we deduced the transition matrix element and submitted it into the formula of the electric dipole transition rate [25]. Therefore, the magnetic-field-induced transition (MIT) rate can be obtained by [23, 26]

$$\begin{eqnarray}&&{A}_{{\rm{MIT}}}=\displaystyle \frac{2.026\,13\times {10}^{18}}{{\lambda }^{3}}\\ \quad \times \,\displaystyle \sum _{q}{\left|\displaystyle \sum _{{\rm{\Gamma }}J}\displaystyle \sum _{{\rm{\Gamma }}^{\prime} J^{\prime} }{d}_{{\rm{\Gamma }}J}d{{\prime} }_{{\rm{\Gamma }}^{\prime} J^{\prime} }{(-1)}^{J-{M}_{J}}\left(\begin{array}{ccc}J & 1 & J^{\prime} \\ -{M}_{J} & q & {M}_{J^{\prime} }^{{\prime} }\end{array}\right)\langle {\rm{\Gamma }}J\parallel {{\bf{P}}}^{(1)}\parallel {\rm{\Gamma }}^{\prime} J^{\prime} \rangle \right|}^{2},\end{eqnarray} \tag{ 8 }$$

where λ is the wavelength in $\mathring{\rm A}$ and ${{\bf{P}}}^{(1)}$ is the electric dipole transition operator.

3.1.2. MCDHF method

According to the MCDHF method [27], the atomic state wave function (ASF) ${\rm{\Psi }}({\rm{\Gamma }}{{JM}}_{J})$ is a linear combination of a number of configuration state functions (CSFs) ${{\rm{\Phi }}}_{j}({\gamma }_{j}{{JM}}_{J})$ with the same parity P, total angular momentum J and its component along z direction M_J

$$\begin{eqnarray}&&{\rm{\Psi }}({\rm{\Gamma }}{{JM}}_{J})=\displaystyle \sum _{j}^{N}{c}_{j}{{\rm{\Phi }}}_{j}({\gamma }_{j}{{JM}}_{J}),\end{eqnarray} \tag{ 9 }$$

where c_j stands for the mixing coefficient, γ represents the other quantum numbers to uniquely define the state. The CSFs ${{\rm{\Phi }}}_{j}({\gamma }_{j}{{JM}}_{J})$ are constructed as linear combinations of Slater determinants, each of which is a product of one-electron Dirac orbitals.

In the self-consistent field (SCF) procedure, the coefficients c_j and the one-electron relativistic orbitals are optimized by solving the MCDHF equations, which are derived from the variational principle. The Breit interaction

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{ij}}=-\displaystyle \frac{1}{2{r}_{{ij}}}\left[{{\boldsymbol{\alpha }}}_{i}\cdot {{\boldsymbol{\alpha }}}_{j}+\displaystyle \frac{({{\boldsymbol{\alpha }}}_{i}\cdot {r}_{{ij}})({{\boldsymbol{\alpha }}}_{j}\cdot {r}_{{ij}})}{{r}_{{ij}}^{2}}\right]\end{eqnarray} \tag{ 10 }$$

and QED effects are included in the subsequent relativistic configuration interaction (RCI) calculation, where only the mixing coefficients are variable.

In our calculations, we used the active space method to capture the electron correlation. The $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}5{s}^{2}$ and $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}5s5p$ configurations were treated as the reference configuration for the ground state (5s^{2 1}S₀) and the excited states ($5s5p$ ${}^{\mathrm{1,3}}P$), respectively, where the $5s$ and $5p$ electrons are the valence electrons and the others the core. The configuration expansions were generated by single (S) and double (D) excitations from the reference configuration to the active set. Starting with the ground state, the occupied spectroscopic orbitals in the reference configurations were optimized in the Dirac–Hartree–Fock (DHF) approximation and kept frozen in the following computations, and others in the active set as correlation orbitals. The valence–valence (labeled as VV) and core–valence (labeled as CV) correlations between the cores with n = 3, 4 and the valence electrons were systematically considered in the subsequent SCF calculation procedure. The SD excitations were restricted that at most one electron may be promoted from the core–shells. The active sets were expanded as

$$\begin{eqnarray}n5 & = & \{3s,3p,3d,4s,4p,4d,4f,5s\},\\ n6 & = & n5+\{5p,5d,5f,5g,6s\},\\ n7 & = & n6+\{6p,6d,6f,6g,7s\},\\ n8 & = & n7+\{7p,7d,7f,7g,8s\},\\ n9 & = & n8+\{8p,8d,8f,9s\},\\ n10 & = & n9+\{9p,9d,10s\},\\ n11 & = & n10+\{10p,11s\}.\end{eqnarray} \tag{ 11 }$$

The seven layers of virtual orbitals were added to make sure the convergence of the atomic parameters under investigation. Only the added orbitals in each layer of the active set were varied. Calculations for the excited states were performed in the same way except for the first layer of the active set $n5^{\prime}$ = {3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p}.

To consider the core–core (labeled as CC) correlation of the n = 4 core–shell, the CSFs by unrestricted SD excitations from the reference configuration to the active set of orbitals with n = 8 were generated. Furthermore, the higher-order electron correlations among the n = 4 and n = 5 shells were taken into account by the multi-reference (marked as MR) SD model. The final MR computational model contains the higher-order electron correlations as well as the VV, CV and CC correlations. In this step, the {$4{s}^{2}4{p}^{6}5{p}^{2};$ $4{s}^{2}4{p}^{5}4d5s5p$} and {$4{s}^{2}4{p}^{6}4d5p;$ $4{s}^{2}4{p}^{6}5s6p;$ $4{s}^{2}4{p}^{6}5p6s;$ $4{s}^{2}4{p}^{6}4d6p$} configurations were added to the single reference configuration set. The configuration space was expanded by replacing one or two electrons in the reference configurations with the ones in the active set n8. Finally, the Breit interaction and the QED correlations were evaluated. These calculations were performed in the RCI computation. We used the GRASP2K package [28] to accomplish our calculations.

3.3.1. Excitation energies and rates of ¹S₀–${}^{\mathrm{1,3}}{P}_{1}$ E1 transition

3.3.2. Magnetic-field-induced transition

For a bosonic Sr atom, the external magnetic field mixes the ³P₁ and ¹P₁ states with ³P₀, and thus opens up a one-photon E1 transition channel from the ³P₀ state to the ground state. Since the magnetic field strength is weak and the energy separation between the states belonging to different configurations is large, the reference ³P₀ state can be approximately expressed as [23]

$$\begin{eqnarray}| 5s5p{}^{3}{P}_{0}^{{\prime} },M=0\rangle & = & | 5s5p{}^{3}{P}_{0},M=0\rangle \\ & & +\displaystyle \sum _{s=1,3}{d}_{s;J=1}| 5s5p{}^{s}{P}_{1},M=0\rangle .\end{eqnarray} \tag{ 12 }$$

Here, the state with prime describes the dominant component of the eigenvector. Isolated from other states, the ground state is given as

$$\begin{eqnarray}&&| 5{s}^{2}{}^{1}{S}_{0}^{{\prime} },M=0\rangle =| 5{s}^{2}{}^{1}{S}_{0},M=0\rangle .\end{eqnarray} \tag{ 13 }$$

The expansion coefficient defined in equation (7) is simplified as

$$\begin{eqnarray}&&{d}_{s}=\displaystyle \frac{{\langle }^{s}{P}_{1}\parallel {N}^{(1)}+{\rm{\Delta }}{N}^{(1)}{\parallel }^{3}{P}_{0}\rangle {B}}{E{(}^{3}{P}_{0})-E{(}^{s}{P}_{1})}={d}_{s}^{R}{B},\end{eqnarray} \tag{ 14 }$$

where d^R_s is the reduced mixing coefficient and the off-diagonal reduced magnetic interaction matrix element is obtained by using the HFSZEEMAN package [24].

Inserting equations (12) and (13) into equation (8), we obtain the magnetic-field-induced the $5s5p$ ³P₀–5s^{2 1}S₀ transition rate

$$\begin{eqnarray}&&{A}_{{\rm{MIT}}}({}^{1}{S}_{0},{}^{3}{P}_{0})=\displaystyle \frac{2.026\,13\times {10}^{18}}{3{\lambda }^{3}}{\left|\displaystyle \sum _{s=\mathrm{1,3}}{d}_{s}^{R}\langle 5{s}^{2}{}^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}\parallel 5s5p{}^{s}{P}_{1}\rangle \right|}^{2}\cdot {{B}}^{2}.\end{eqnarray} \tag{ 15 }$$

In table 2 we present the off-diagonal reduced magnetic interaction matrix elements W (in a.u.) and the reduced mixing coefficients d_s^R (in T⁻¹) for various computational models. According to equation (14), d_s^R is obtained by the magnetic interaction matrix elements W and the computed transition energies. And its unit is obtained by a conversion factor 2.353 × 10⁵ a.u. T⁻². To obtain an accurate value of the MIT rate, we include both the ¹P₁ and ³P₁ perturbations. The MIT rate (in s⁻¹) for the ⁸⁸Sr atom in an external magnetic field with the strength B (in T) can be expressed as,

$$\begin{eqnarray}&&{A}_{{\rm{MIT}}}{(}^{1}{S}_{0},{}^{3}{P}_{0})={A}_{{\rm{MIT}}}^{R}{(}^{1}{S}_{0},{}^{3}{P}_{0})\cdot {{B}}^{2}=0.198\cdot {{B}}^{2},\end{eqnarray} \tag{ 16 }$$

where the reduced transition rate A^R_MIT(¹S₀,³P₀) = 0.198 s⁻¹ T⁻².

Table 2.  Off-diagonal reduced magnetic interaction matrix elements W (in a.u.) and reduced mixing coefficients d_s^R (in T⁻¹) for various computational models. The numbers in square brackets are the expansion in base 10.

Model	(3P1, ${}^{3}{P}_{0})$		(1P1,3P0)
	W	d3R	W	d1R
DHF	0.40913	−2.090[−3]	−0.40914[−2]	2.507[−7]
n5	0.40899	−2.062[−3]	−0.11304[−1]	1.313[−6]
n6	0.40895	−2.011[−3]	−0.12762[−1]	1.611[−6]
n7	0.40896	−2.016[−3]	−0.12614[−1]	1.589[−6]
n8	0.40895	−1.998[−3]	−0.12622[−1]	1.585[−6]
n9	0.40895	−1.991[−3]	−0.12624[−1]	1.594[−6]
n10	0.40895	−1.991[−3]	−0.12531[−1]	1.581[−6]
n11	0.40895	−1.991[−3]	−0.12507[−1]	1.577[−6]
CC	0.40908	−2.077[−3]	−0.76010[−2]	6.502[−7]
MR	0.40898	−2.141[−3]	−0.11295[−1]	1.407[−6]

The line-width of the forbidden transition depends on the magnetic field strength. To obtain an ultra-narrow line-width, the static magnetic field strength should be extremely small [18, 19]. In this experiment, the external magnetic field strength is set to be below 5 mT.

In our previous experimental work, we measured the Rabi oscillation with different Rabi frequencies. The data were fitted with the function $P=a(1-\cos (2\pi {\rm{\Omega }}t)\exp (-t/\tau ))$ [20, 48] where τ is the decoherent time scale which remain constant. The Rabi frequency Ω of the MIT is defined as,

$$\begin{eqnarray}{\rm{\Omega }} & = & \displaystyle \frac{1}{{{\hslash }}^{2}}[{d}_{1}^{R}\langle {}^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}\parallel {}^{1}{P}_{1}\rangle +{d}_{3}^{R}\langle {}^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}\parallel {}^{3}{P}_{1}\rangle ]({\bf{E}}\cdot {\bf{B}})\\ & = & \alpha \sqrt{{I}}{B}{\rm{c}}{\rm{o}}{\rm{s}}\theta .\end{eqnarray} \tag{ 17 }$$

Here, ${\bf{E}}$ describes the electric vector of the 698 nm laser, ${\langle }^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}{\parallel }^{1}{P}_{1}\rangle$ and ${\langle }^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}{\parallel }^{3}{P}_{1}\rangle$ are the reduced matrix elements for the ¹S₀–${}^{\mathrm{1,3}}{P}_{1}$ transitions. The values of the two matrix elements and the reduced mixing coefficients were accurately determined in previous calculations. The Rabi frequency is a function of the magnetic field strength B as the intensity of 698 nm laser I is kept constant. For a two-level atom, the excitation fraction P is obtained as

$$\begin{eqnarray}P & = & \displaystyle \frac{{\rm{\Omega }}{B}\tau }{1+{{\rm{\Omega }}}^{2}\tau /{A}_{{\rm{MIT}}}}\\ & & \times [{d}_{1}^{R}\langle {}^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}\parallel {}^{1}{P}_{1}\rangle +{d}_{3}^{R}\langle {}^{1}{S}_{0}\parallel {{\bf{P}}}^{(1)}\parallel {}^{3}{P}_{1}\rangle ]\sin ({\rm{\Omega }}\pi t).\end{eqnarray} \tag{ 18 }$$

In this experiment, we used the π-pulse (t = 1/(2Ω)) laser to excite atoms to the excited state and the intensity of 698 nm laser is 3.6 W cm⁻². In figure 3, we present the experimental measurements and the theoretical calculations of the excitation fraction. The solid squares are the weighted average of several measured excitation fractions with the magnetic field strength B = 1.0, 1.2, 1.7, 2.2 mT and the error bars are the statistical uncertainty of our measurements. The red line is the calculated excitation fraction as a function of the magnetic field strength. As can be seen from this figure, our calculated results agree with the measurements when B = 1.2, 1.7, 2.2 mT. The deviation in the small magnetic field strength results from the low signal-to-noise ratio of the spectrum due to the fluctuation of the cold atom number in the optical lattice. As the magnetic field strength is small, the measured excitation fraction is very small. The atom number fluctuation effect is comparable to the measured excitation fraction, resulting in the low signal-to-noise ratio of the spectrum. What's more, in our experiment, the power fluctuation of the 461 nm laser is one of the factors to induce the uncertainty. And we assume that the excitation is substantially faster than the dominant loss mechanisms, for example, the frequency and amplitude fluctuation of the 813 nm laser and the vibration of the achromatic lens and the retro-reflecting mirror.

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