⇦Chinese Physics Letters, 2017, Vol. 34, No. 10, Article code 103302⇨ Zeeman Effect of the Rovibronic Ground State of I$^{35}$Cl at Hyperfine Level Kai Chen(陈凯), Zheng Hu(胡正), Qing-Hui Wang(王庆辉), Xiao-Hua Yang(杨晓华)** Affiliations School of Science, Nantong University, Nantong 226019 Received 31 March 2017 ^**Corresponding author. Email: xhyang@ntu.edu.cn Citation Text: Chen K, Hu Z, Wang Q H and Yang X H 2017 Chin. Phys. Lett. 34 103302 Abstract Zeeman effect at the hyperfine level of the rovibronic ground state of I$^{35}$Cl are determined on the basis of $|I_{1}JF_{1}I_{2}FM_{F}\rangle$ via an effective Hamiltonian matrix diagonalization method. Perturbations of the Zeeman sublevels are observed and the perturbation selection rules are summarized as well. Several potential applications of such Zeeman effect are suggested. DOI:10.1088/0256-307X/34/10/103302 PACS:33.15.Pw, 32.60.+i © 2017 Chinese Physics Society Article Text Molecular hyperfine structures are caused by the nuclear spin interacting with the total angular momentum and the nuclear quadrupole interacting with the electric field gradient of electrons at the nuclei. Thus the nuclear information can be studied by ultrahigh resolution molecular spectroscopy. However, the interactions due to nuclei are so weak that the spectral splitting is often merged in the Doppler broadening. Typically, for example, the hyperfine splitting of I$^{35}$Cl of about 1 MHz corresponds to the spectral Doppler broadening of its $A^{3}{\it \Pi}_{1}$–$X^{1}{\it \Sigma}^{+}$ transition at the temperature of about 1.9 mK, which means that the hyperfine structure can only be observed at the temperature below 1 mK by means of the laser spectroscopy. Owing to the developments of cold molecules,[1,2] single mode laser and wide range optical comb,[3,4] molecular hyperfine structures can be precisely determined experimentally,[5] which will help us to realize such weak interactions and to apply them in related fields. In addition, the molecular Zeeman effect at hyperfine level provides a new way to manipulate cold molecules and further cooling cold molecules to quantum (ultralow temperature) regime.[6,7] The rotational level has neither the Stark effect nor the Zeeman effect due to the $^{1}{\it \Sigma}$ symmetry electronic ground state of the heteronuclear dihalogen molecule, ICl, which makes its rotational resolved spectrum often used for optical frequency calibration in the near infrared region. Additionally, the nuclear spins of Cl and I atoms are 3/2 and 5/2, respectively, and the nuclear quadrupoles of the two atoms are both relatively large, resulting in complex hyperfine structures, which make ICl a good candidate for molecular hyperfine studying. However, its hyperfine level has both the Stark[8] and the Zeeman effects due to the nuclear quadrupoles and nuclear spins. The Zeeman effect at hyperfine level of ICl are studied in the present work, which provides fundamental information for further investigations and applications, such as evaporative cooling of cold molecules in a magnetic-optical trap (MOT).[6,9] The Hamiltonian of ICl in a magnetic field consists of five parts: electronic term, vibrational term, rotational term, hyperfine term and the Zeeman term. When studying the rovibronic ground state, the first two terms can be treated as a constant, thus only the latter three terms are left. The rotational and the hyperfine terms have been given in our previous work.[8] The Zeeman term of the Hamiltonian consists of two parts from rotation and nuclei as[10,11] $$\begin{alignat}{1} H_{\rm Z} =-g_{\rm r} \mu _{\rm N} {\boldsymbol J}\cdot {\boldsymbol B}-\sum\limits_{i=1}^2 {g_i \mu _{\rm N} {\boldsymbol I}_i \cdot {\boldsymbol B}(1-\sigma _i )},~~ \tag {1} \end{alignat} $$ where ${\boldsymbol J}$ is the molecular total angular momentum excluding the nuclear spin, $I$ is the nuclear spin, $g_{\rm r}$ is the rotational $g$-factor, $\mu_{\rm N}$ is the nuclear magnetic moment, $g_{i}$ is the nuclear $g$-factor, $\sigma _{i}$ is the shielding coefficient of electrons to nuclear spin, $i=1$ or 2 denotes iodine (I) or chlorine (Cl) atom, respectively, and $B$ is the applied magnetic field. In the weak field limit, the momentum coupling sequence is that the spin of iodine ${\boldsymbol I}_{1}$ first couples with ${\boldsymbol J}$ to form intermediate momentum ${\boldsymbol F}_{1}$ (${\boldsymbol F}_{1}={\boldsymbol J}+{\boldsymbol I}_{1}$), then ${\boldsymbol F}_{1}$ couples with the spin of chlorine ${\boldsymbol I}_{2}$ to form the total angular momentum ${\boldsymbol F}$ (${\boldsymbol F}={\boldsymbol F}_{1}+{\boldsymbol I}_{2}$), and finally ${\boldsymbol F}$ projects to the applied magnetic field along the $z$ axis and $M_{F}$ denotes the projection quantum number. Thus in the $|I_{1}JF _{1}I_{2}FM _{F}\rangle$ couple basis, the Zeeman Hamiltonian matrices take the form of [12,13] $$\begin{align} &\langle {J}'I_1 {F'}_1 I_2 {F}'{M'}_F |-g_{\rm r} \mu _{\rm N} J \cdot B|JI_1 F_1 I_2 FM_F \rangle \\ =\,&-\delta _{J,{J}'} \delta _{F,{F}'} g_{\rm r} \mu _{\rm N} B_{\rm Z}\\ &\cdot(-1)^{{F}'-{M'}_F +2{F'}_1 +J+F+I_1 +I_2 +1} \\ &\times [(2F+1)(2{F}'+1)(2F_1 +1)(2{F'}_1 +1)]^{1/2}\\ &\cdot[J(J+1)(2J+1)]^{1/2} \\ &\times \left\{\begin{matrix} {{J}'}& {{F'}_1 } & {I_1}\\ {F_1 }& J & 1 \\ \end{matrix} \right\}\left\{\begin{matrix} {{F}'}& {{F'}_1 }& {I_2 }\\ {F_1 } &F & 1\\ \end{matrix}\right\}\\ &\cdot\left(\begin{matrix} {{F}'} & 1 & F \\ {-{M'}_F } & 0 & {M_F } \\ \end{matrix}\right),~~ \tag {2}\\ \end{align} $$ $$\begin{align} &\langle {J}'I_1 {F'}_1 I_2 {F}'{M'}_F |-g_1 \mu _{\rm N} I_1\\ &\cdot B(1-\sigma _1 )|JI_1 F_1 I_2 FM_F \rangle \\ =\,&-\delta _{F,{F}'} g_1 \mu _{\rm N} B_{\rm Z} (1-\sigma _1 )\\ &\cdot(-1)^{{F}'-{M'}_F +{F'}_1 +{J}'+F+F_1 +I_1 +I_2 } \\ &\times [(2F+1)(2{F}'+1)(2F_1 +1)(2{F'}_1 +1)]^{1/2}\\ &\cdot[I_1 (I_1 +1)(2I_1 +1)]^{1/2} \\ &\times \left\{\begin{matrix} {I_1 } & {{F'}_1 } & {{J}'}\\ {F_1 } &{I_1 }& 1 \\ \end{matrix}\right\}\left\{\begin{matrix} {{F'}_1 } & {{F'}_1 }& {I_2 } \\ F & {F_1 } & 1 \\ \end{matrix}\right\}\\ &\cdot\left(\begin{matrix} {{F}'} & 1& F \\ {-{M'}_F } & 0 & {M_F } \\ \end{matrix}\right),~~ \tag {3} \end{align} $$ $$\begin{align} &\langle {J}'I_1 {F'}_1 I_2 {F}'{M'}_F|-g_2 \mu _{\rm N} I_2 \\ &\cdot\cdot B(1-\sigma _2)|JI_1 F_1 I_2 FM_F \rangle \\ =\,&-\delta _{F_1,{F'}_1} g_2 \mu _{\rm N} B_{\rm Z} (1-\sigma _2 )\\ &\cdot(-1)^{{F}'-{M'}_F +{F'}_1 +{F}'+I_2 +1} \\ &\times [(2F+1)(2F'+1)(2F_1 +1)]^{1/2}\\ &\cdot[I_2 (I_2+1)(2I_2 +1)]^{1/2} \\ &\times \left\{\begin{matrix} {I_2 } & F & {{F'}_1 } \\ {{F}'} & {I_2 }& 1 \\ \end{matrix}\right\}\left(\begin{matrix} {{F}'} & 1 & F \\ {-{M'}_F } & 0 & {M_F }\\ \end{matrix} \right),~~ \tag {4} \end{align} $$ where symbol $\{~\}$ is the Wigner 6-$j$ symbol and $(~)$ is the Wigner 3-$j$ symbol. Therefore, the Zeeman sublevels at hyperfine level can be determined by diagonalizing the Hamiltonian matrix, and the assignment of the sublevels can be accomplished according to their eigenvectors. Two parameters $g_{i}$ and $\mu_{\rm N}$ listed in the above equations can be expressed as[12] $$\begin{align} g_i =\,&\frac{\mu _i }{\mu _{\rm N} I},~~ \tag {5} \end{align} $$ $$\begin{align} \mu _{\rm N} =\,&\frac{e\hbar }{2M_{\rm p} },~~ \tag {6} \end{align} $$ where $\mu_{\rm I}= 2.81\mu_{\rm N}$, $\mu_{\rm Cl}=0.82\mu_{\rm N}$, $e$ is the charge of a proton, $\hbar$ is the reduced Planck constant, and $M_{\rm p}$ is the mass of proton. The rotational $g$-factor and the shielding coefficients of electrons to the nuclear spin can be calculated by employing the Dalton program.[14] We perform the calculation under the basis of Sadlej-pVTZ,[15,16] and the results are listed in Table 1.

**Table 1.** Nuclear properties and coupling constants for the I$^{35}$Cl molecule.
	$\sigma _{\rm I}$ (ppm)	$\sigma _{\rm Cl}$ (ppm)	$g_{\rm I}$	$g_{\rm Cl}$	$g_{\rm r}$	$\mu_{\rm N}$ (J/T)
I$^{35}$Cl	4314.11	1658.08	1.12	0.53	$-$0.36	5.05$\times$10$^{-27}$

The Zeeman effects of the hyperfine levels of the rovibronic ground state of I$^{35}$Cl are calculated by diagonalizing the effective Hamiltonian matrix in the basis of $|I_{1}JF_{1}I_{2}FM_{F}\rangle$ based on our previous work.[8] Figure 1 shows the Zeeman sublevels at a fixed applied field of 500 G, and also the quantum number and values of the sublevels are labeled. The Zeeman splitting is generated by the nuclear spins and molecular rotational momentum as expressed in Eq. (1). Each hyperfine level splits into ($2F+1$) sublevels, which are denoted by $M_{F}$ ($M_{F}=-F, -F+1,{\ldots}, F$). The Zeeman splitting of ICl is at the order of MHz, much less than its Stark splitting.[8] To verify our results, we try to calculate the Zeeman splitting of the hyperfine levels in the rovibronic ground state of $^{39}$K$^{85}$Rb, and the results are in good agreement with those of Aldegunde et al.,[17] suggesting the validity of our method.

Fig. 1. (Color online) Zeeman sublevels (not scaled), denoting with quantum numbers and values, of the hyperfine states of the rovibronic ground state of I$^{35}$Cl at a fixed external magnetic field of 500 G.

Fig. 2. (Color online) Zeeman sublevels of the $|J=0, F_{1}=2.5, F\rangle$ hyperfine states in the rovibronic ground state of I$^{35}$Cl varying with an applied magnetic field increasing from 0 through 1000 G.

The pattern of the Zeeman sublevels of ICl is very complex and the sublevels are severely staggered, as shown in Fig. 1. The assignment of the sublevels is basically according to their eigenvectors. To further verify the assignment, we also calculate the sublevels varying with the applied magnetic field increasing from 0 through 1000 G, and Fig. 2 plots such evolution of $|J,F_{1},F\rangle=|0,2.5,F\rangle$ hyperfine states. Figure 2 also shows the perturbation of the Zeeman sublevels clearly. The perturbation selection rules are summarized as follows: $\Delta J=0, \pm1$, $\Delta F_{1}=0, \pm1$, $\Delta F=0, \pm1$, and $\Delta M_{F}=0$. The perturbation repulsion, for example, occurs between the pairs of $|F,M_{F}\rangle=|1,-1\rangle$ and $|2,-1\rangle$, which results in the $|1, -1\rangle$ sublevel changing from weak-field-seeking (WFS) into high-field-seeking (HFS) as the applied magnetic field increases, as shown in Fig. 2.

Fig. 3. Frequency of the $| 2, 0\rangle \leftarrow |1, -1\rangle$ transition in the rovibronic ground state of I$^{35}$Cl varying with an applied magnetic field.

Therefore, the cold collision cross section can be tuned by varying the applied magnetic field, and this Feshbach-like resonance behavior has many potential applications in the field of cold molecular physics. The Feshbach spectra of the HFS state are much more dense and more intense than those of the WFS state.[18] Thus, for example, after the WFS cold molecules at about 10 mK are trapped in an MOT, they will transit into the HFS state by turning off the MOT and applying a strong magnetic field. As a result, the HFS molecules will collide more frequently and the evaporative cooling process would be accomplished in a much shorter time. Then turning off the strong applied magnetic field and returning on the MOT, we could quickly obtain ultracold molecules at about (even below) 1 μK. Another interesting potential application is for magnetic field calibration using the radio-frequency magnetic field resonance spectrum of the Zeeman sublevels at the hyperfine level. Figure 3 plots the frequency of the spectrum of the $|2, 0\rangle \leftarrow |1, -1\rangle$ transition. By the way, the magnetic dipole transition selection rules at hyperfine level are $\Delta F=0, \pm1$ and $\Delta M_{F}=\pm1$. There is a magic magnetic field, $B= 363.6$ G, at which the spectral frequency reaches its minimum value. This magic magnetic field can be used as a standard field for calibration. In summary, the Zeeman effect of the hyperfine levels of the rovibronic ground state of I$^{35}$Cl has been studied at the basis of $|I_{1} JF _{1}I_{2} FM _{F}\rangle$ by diagonalizing the effective Hamiltonian matrix. The perturbations of the Zeeman sublevels are observed and the perturbation selection rules are $\Delta J=0, \pm1$, $\Delta F_{1}=0, \pm1$, $\Delta F=0, \pm1$, and $\Delta M_{F}=0$. Some Zeeman sublevels can transit from WFS into HFS as the applied field increases, showing the Feshbach-like resonance behavior of molecules, which has many potential applications in the field of cold molecular physics as well as cold chemistry. References New frontiers for quantum gases of polar moleculesManipulation and Control of Molecular BeamsOptical frequency combs: From frequency metrology to optical phase controlHighly selective terahertz optical frequency comb generatorHyperfine interactions and perturbation effects in the B0_u^+(^3Î _u) state of ^127I_2Evaporative cooling of the dipolar hydroxyl radicalSubmillikelvin Dipolar Molecules in a Radio-Frequency Magneto-Optical TrapStark effect of the hyperfine structure of ICl in its rovibronic ground state: Towards further molecular coolingMagneto-optical trapping of a diatomic moleculeTheory of Molecular Hydrogen and Deuterium in Magnetic FieldsMicrowave Spectroscopy and Nuclear Magnetic Resonance SpectroscopyWhat Is the Connection?Hyperfine energy levels of alkali-metal dimers: Ground-state homonuclear molecules in magnetic fieldsOrbital energies and negative electron affinities from density functional theory: insight from the integer discontinuitySelf-consistent perturbation theoryBenchmark calculations with correlated molecular wave functions. III. Configuration interaction calculations on first row homonuclear diatomicsHyperfine energy levels of alkali-metal dimers: Ground-state polar molecules in electric and magnetic fieldsMagnetic field modification of ultracold moleculeâmolecule collisions

[1]	Moses S A, Covey J P, Miecnikowski M T, Jin D S and Ye J 2016 Nat. Phys. 13 13
[2]	Van de Meerakker S Y, Bethlem H L, Vanhaecke N and Meijer G 2012 Chem. Rev. 112 4828
[3]	Ye J, Schnatz H and Hollberg L W 2003 IEEE J. Sel. Top. Quantum Electron. 9 1041
[4]	Ye J, Ma L S, Daly T and Hall J L 1997 Opt. Lett. 22 301
[5]	Chen L, Cheng W Y and Ye J 2004 J. Opt. Soc. Am. B 21 820
[6]	Stuhl B K, Hummon M T, Yeo M, Quéméner G, Bohn J L and Ye J 2012 Nature 492 396
[7]	Norrgard E B, Mccarron D J, Steinecker M H, Tarbutt M R and DeMille D 2016 Phys. Rev. Lett. 116 063004
[8]	Wang Q H, Shao X P and Yang X H 2016 Chin. Phys. B 25 013301
[9]	Barry J F, Mccarron D J, Norrgard E B, Steinecker M H and DeMille D 2014 Nature 512 286
[10]	Ramsey N F 1952 Phys. Rev. 85 60
[11]	Bryce D L and Wasylishen R E 2003 Acc. Chem. Res. 36 327
[12]	Brown J M and Carrington A 2003 Rotational Spectroscopy of Diatomic Molecules (Cambridge University Press)
[13]	Aldegunde J and Hutson J M 2009 Phys. Rev. A 79 013401
[14]	Aidas K, Angeli C, Bak K L et al 2014 Wires. Comput. Mol. Sci. 4 269
[15]	Dodds J L, Mcweeny R and Sadlej A J 1977 Mol. Phys. 34 1779
[16]	Peterson K A, Kendall R A and Dunning T H 1993 J. Chem. Phys. 99 9790
[17]	Aldegunde J, Rivington B A, Żuchowski P S and Hutson J M 2008 Phys. Rev. A 78 033434
[18]	Tscherbul T, Suleimanov Y V, Aquilanti V and Krems R V 2009 New J. Phys. 11 055021