Laser-detected magnetic resonance spectra dressed by a radio-frequency field

Zhi Liang; Xu-xing Geng; Pan-li Qi; Kai Jin; Guo-qing Yang; Gao-xiang Li; Gao-xiang Li; Guang-ming Huang; Guang-ming Huang

doi:10.1364/OE.437555

1. Introduction

Electromagnetic field dressing of atoms has been shown to result in many interesting phenomena and applications. In the laser-dressed atoms aspect, the calculations and analyses of the emission spectra [1,2], entanglement and correlation functions of the strong-field-dressed atoms promoted the vigorous development of technologies such as quantum sensing [3], quantum antenna [4], entanglement of qubits [5], and Rydberg-atom-based RF-receiver [6]. In the microwave-dressed atoms aspect, experimental research on the preparation and detection of microwave-dressed systems is helpful to implement high-fidelity entanglement gates with dressed-state qubits and advance the progress of quantum manipulation technology [7,8]. In the radio-frequency-dressed atoms aspect, plentiful researches on cold atoms, for instance, measuring the linear birefringence of the radio-frequency-dressed atomic medium via polarization homodyning [9], multiple radio-frequency-dressed potential for the configurable magnetic confinement of ultracold atoms [10] and realization of the trapping of the cold $^{87}Rb$ atoms in a toroidal geometry in an radio-frequency-dressed quadrupole trap [11] are of great interest both in theory and experiment. Nevertheless, during the investigations into magnetic resonance, especially in the field of radio-frequency-dressed hot atoms, the study of the probe based on the way of a weak field is not very extensive. In this paper, we pursue to study the laser-detected magnetic resonance spectra with radio-frequency-dressed cesium atoms at room temperature.

The optical-radio-frequency double resonance based on atomic alignment [12,13] has been widely used for measuring magnetic field [14–16]. The double resonance alignment magnetometer has a great advantage in obtaining the vectorial information of the magnetic field to be measured in the external radio-frequency field with arbitrary intensity. In most cases, the double resonance configuration is applied to measure slowly varying magnetic field. On the other hand, this configuration can also be used to realize the measurement of radio-frequency magnetic field (RMF) when the RMF is applied as a driving field [17]. Chalupczak et al. [18] have demonstrated a magnetometric scheme that combines indirect optical pumping and spin-exchange collisions (SEC) to measure RMF. The measurements were conducted under the conditions of a nonlinear Zeeman effect, and the scheme based on the combination realized the improvement of signal amplitude but narrower line width of radio-frequency resonance. This work has demonstrated an executable way to measure RMF with a high sensitivity near room temperature, but it is not easy to find the accurate oscillating frequency of RMF in the case of detuning with a linear Zeeman splitting. In our system, the Larmor frequency of the static magnetic field that causes a linear Zeeman splitting is constant, the direction of the driving RMF to be measured is fixed perpendicular to the static magnetic field, and another weak RMF is applied for probe. It is found that the amplitude and oscillating frequency of the driving RMF can be determined by the frequencies of the sidebands of magnetic resonance by applying a weak transverse RMF.

In this paper, on the basis of optical-radio-frequency double resonance, a weak RMF is applied transversely for probing the magnetic resonance of the system, and the transmission for magnetic resonance is discussed in the radio-frequency-dressed representation. In the theoretical part, we first discuss the relaxation process of the system and derive the light pump-probe relation in the lab frame, and give the expression of the absorption coefficient depending on atomic multipole moments. Then we describe the properties of the magnetic resonance of the system from Hamiltonian in dressed-state representation in the rotating frame and give the analytical expression of the transmission for magnetic resonance in the lab frame. In the experimental part, we fit the data with the theoretical curves about the spectral line and its variation trend. Then we demonstrate the linear proportion between the separation of transmission sidebands and the amplitude of the dressing RMF. Finally, we make a conclusion.

2. Theoretical analysis

We consider the $F_{g}=4 \to F_{e}=3$ optical pumping for the $D_{1}$ line of cesium atoms excited by a linearly polarized resonant light. As shown in Fig. 1(a), in the lab frame $xyz$, the polarization of the light field is parallel to the quantization axis ($z$ axis). Both the static magnetic field $B_{0}$ and the propagation direction of light are along $x$ axis. Both the two RMFs $B_{\textit {rf1}}$ and $B_{\textit {rf2}}$ are along $y$ axis. The static magnetic field $B_{0}$ causes the linear Zeeman splitting, then the ground state of the system consists of nine hyperfine Zeeman sublevels and the excited state consists of seven hyperfine Zeeman sublevels. The transverse driving field $B_{\textit {rf1}}$ is used to change the polarization of the ground state of cesium atoms prepared by the laser, while the weak field $B_{\textit {rf2}}$ is used to probe the magnetic resonance signal. In the lab frame $xyz$, one mainly investigates the optical pumping relaxation process and reveal that the absorption coefficient depends on the atomic alignment. To solve atomic alignment conveniently, the quantization axis turns along the direction of $B_0$ by axis rotation for consistency with the direction of the Zeeman splitting, as shown in Fig. 1(b). After the rotation, the two RMFs take along $x'$ axis in the new frame $x'y'z'$.

Fig. 1. (a) Theoretical configuration in lab frame with light polarization $\vec {E}$ parallel to quantization axis. (b) The rotating frame with $\vec {B_0}$ parallel to quantization axis. The order of coordinate axis rotation is as follows: first rotate 90 degrees counterclockwise around the $y$ axis, then 90 degrees counterclockwise around the $z$ axis.

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2.1 Systematic relaxation process and detection

Since the laser is linearly polarized and the quantization axis takes along the light polarization in the lab frame $xyz$, there is only $\pi$ transition for light pumping, as shown in Fig. 2. The light-atom interaction Hamiltonian is given by

(1)$$H_{I}={-}\sum_{j={-}3}^{3} V_{e_{j}g_{j}} e^{{-}i \omega t} |F_{e},m_{e_{j}}\rangle\langle F_{g},m_{g_{j}}| + H.c. ,$$

where $\omega$ is the oscillating frequency, the symbol $V_{e_{j}g_{j}}$ represents the interaction energy of the atom and the light field. According to Winger-Eckart theorem [19], $V_{e_{j}g_{j}}$ can be given by

(2)$$V_{e_{j}g_{j}}={-}\langle F_{e},m_{e_{j}} |\textbf{d}|F_{g},m_{g_{j}} \rangle \cdot \textbf{E}=({-}1)^{F_{e}-m_{e_{j}}+1} \begin{pmatrix} F_{e} & 1 & F_{g} \\ -m_{e_{j}} & q & m_{g_{j}} \\\end{pmatrix} \Omega_{L},$$

where $\textbf {d}$ is the electric dipole operator, $\textbf {E}$ is the electric-field vector, the symbol $\Omega _{L}=\langle F_{g} || d || F_{e}\rangle E$ is the Rabi frequency of the light-atoms interaction and $\langle F_{g} || d || F_{e} \rangle$ is the reduced electric-dipole matrix element.

Fig. 2. Light pumping and spontaneous decay process between $F_{e}=3$ and $F_{g}=4$ in the lab frame.

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The spontaneous decay of $F_e=3$ $\to$ $F_g=4$ can be expressed as

(3)$$\mathcal{L}_{e}\rho=\Gamma\Big(\frac{2F_{e}+1}{2F_{g}+1}\Big)\sum_{q={-}1}^{1}D[\Sigma_{g_{j+q}e_{j}}]\rho ,$$

where $\Gamma$ is the decay rate from excited state to ground state. The operator $D[\Sigma _{g_{j+q}e_{j}}]\rho \equiv \Sigma _{g_{j+q}e_{j}} \rho \Sigma ^{\dagger }_{g_{j+q}e_{j}} - \frac {1}{2} \rho \Sigma ^{\dagger }_{g_{j+q}e_{j}} \Sigma _{g_{j+q}e_{j}} - \frac {1}{2} \Sigma ^{\dagger }_{g_{j+q}e_{j}} \Sigma _{g_{j+q}e_{j}} \rho$ is the Lindblad superoperator. The transition operator $\Sigma _{g_{j+q}e_{j}}$ with $q=-1,0,1$ represent the total transition operator of decay effect corresponding to the decay process of $m_{g}=m_{e}-1$, $m_{g}=m_{e}$ and $m_{g}=m_{e}+1$, respectively, as shown in Fig. 2.

Considering that the spontaneous emission relaxation is much greater than the pumping rate of the laser, i.e., $\Gamma \gg \vert V_{e_{j}g_{j}} \vert$, the excited states can be adiabatically eliminated. With this condition, the states of describing the system can be reduced to only the ground states with adiabatic approximation [20–22]. The reduced master equation is given by

(4)$$\dot{{\rho}}_{g}={-}i\left[{H}_{B},{\rho}_{g}\right]+\mathcal{L}_{\textrm{laser}}{\rho}_{g}+\mathcal{L}_{\textrm{sec}}{\rho}_{g} .$$

where ${H}_{B}=\mu _{B}g_{F}\textbf {F}\cdot \left [\textbf {B}_{0}+\textbf {B}_{1}\cos \omega _{1}t+\textbf {B}_{2}\cos \left (\omega _{2}t+\phi \right )\right ]$ is the Hamiltonian of the interaction between the magnetic fields and the atom. Here $\mu _{B}$ is the Bohr magneton, $g_{F}$ is the hyperfine Landé $g$-factor, $\textbf {F}$ is the angular momentum operator, $\textbf {B}_{0}$ is the vector of static magnetic field, $\textbf {B}_{1}$ and $\textbf {B}_{2}$, $\omega _{1}$ and $\omega _{2}$ are magnetic-field vectors and oscillating frequencies of $B_{\textit {rf1}}$ and $B_{\textit {rf2}}$, respectively. $\phi$ is a random initial phase of $B_{\textit {rf2}}$ relative to $B_{\textit {rf1}}$. For description convenience, ${H}_{B}$ will be discussed in the rotating frame in the following part. The symbol $\mathcal {L}_{\textrm {laser}}{\rho }_{g}$ represents the effect of population redistribution induced by laser, which is given by

(5)$$\mathcal{L}_{\textrm{laser}}{\rho}_{g}=\Gamma_{L}\sum_{q={-}1}^{1}D[\Sigma_{g_{j+q}g_{j}}]\rho_{g},$$

with

(6)$$\Sigma_{g_{j+q}g_{j}}=\sum_{j} V^{F_{g},m_{g_{j+q}}}_{F_{e},m_{e_{j}}} V^{F_{e},m_{e_{j}}}_{F_{g},m_{{g_{j}}}} |F_{g},m_{g_{j+q}}\rangle\langle F_{g},m_{g_{j}}|,$$

where $\Gamma _{L}$=$\frac {64\Omega _{L}^{2}}{9\Gamma }$ is the equivalent relaxation coefficient. The transition coefficient $V^{F^{'},m^{'}_{F}}_{F,m_{F}}$ is given by

(7)$$\begin{aligned} V^{F^{'},m^{'}_{F}}_{F,m_{F}}=&({-}1)^{m_{F}+J+I} \sqrt{(2F+1)(2F^{'}+1)(2J+1)}\\ &\times\begin{pmatrix} F^{'} & 1 & F \\ m^{'}_{F} & q & -m_{F} \\\end{pmatrix} \begin{Bmatrix} J & J^{'} & 1 \\ F^{'} & F & I \\\end{Bmatrix}. \end{aligned}$$

The symbol $\mathcal {L}_{\textrm {sec}}{\rho }_{g}$ represents the relaxation induced by the spin-exchange collisions [23,24], which is given by

(8)$$\frac{\mathcal{L}_{\textrm{sec}}{\rho_{g}}}{R_{\textrm{sec}}}={-}\frac{3}{4}{\rho_{g}}+{S_{g}}{\rho_{g}}{S_{g}}+\langle{S_{g}}\rangle\left(\Big \{{S_{g}},{\rho_{g}}\Big \}-2i{S_{g}}\times{\rho_{g}}{S_{g}}\right) ,$$

where $R_{\textrm {sec}}$ is the spin-exchange collisions rate, $S_{g}$ is the spin operator of ground state. In Eq. (8), the first two terms represent linear effect, and the third term ${\langle S_{g}\rangle }( \{{S_{g}},{\rho _{g}} \}-2i{S_{g}}\times {\rho _{g}}{S_{g}})$ represents nonlinear effect, which is related to the effect of the relaxation due to Van der Waals molecules. In our system, the pumping light is appropriately weak, thus the spin polarization of cesium atom is very small, i.e., $\langle {S_{g}}\rangle \ll 1$, in this case, the effect of $-\frac {3}{4}{\rho _{g}}$ is stronger than the nonlinear effect of $\langle {S_{g}}\rangle \{{S_{g}},{\rho _{g}} \}$, meanwhile the effect of $S_{g} \rho _{g} S_{g}$ is more intense than the nonlinear effect of $\langle S_{g} \rangle ( 2i{S_{g}}\times {\rho _{g}}{S_{g}})$. Thus, the total nonlinear terms are weaker than the first two terms of Eq. (8) on the condition of weak light pumping, and can be negligible in our system.

With the condition of $\Gamma \gg \vert V_{e_{j}g_{j}} \vert$, the relationship between the nondiagonal element $\rho _{e_{j}g_{j}}$ and the ground-state diagonal element $\rho _{g_{j}g_{j}}$ is given by

(9)$$\rho_{e_{j}g_{j}}={-}\frac{8i}{3\Gamma}V_{e_{j}g_{j}}\rho_{g_{j}g_{j}} ,\quad j={-}3,-2,\ldots,3.$$

Equation (9) indicates that under the condition of linearly-polarized light pumping in the lab frame $xyz$, the coherent terms of density matrix only depend on the populations of the ground states.

According to [25,26], the detection absorption coefficient of the system in the lab frame $xyz$ is given by

(10)$$\alpha={-}\frac{32\pi \omega_{0} N}{3\hbar c \Gamma}\sum_{e_{j}g_{j}}{\vert \mu_{e_{j}g_{j}} \vert^{2}}\rho_{g_{j}g_{j}},$$

where $\omega _{0}$ is the transition frequency without magnetic field, $N$ is the atomic density, $\mu _{e_{j}g_{j}}$ is the electric dipole moment. Equation (10) manifests the detection absorption coefficient depends on the evolution of the populations of the ground states in the lab frame $xyz$. In our system, due to the weak pumping intensity of linearly polarized light, the multipole moments of the atomic ground state can be reduced to the second order. Therefore, we will express the detection absorption in the representation of the atomic polarization in terms of multipole moments, which has also the complete but concise description for absorption signal of the system. According to the theory of state multipoles [27], the density matrix $\rho$ of an ensemble of polarized atoms with angular momentum can be expressed in the representation of multipole moments $m_{k,q}$, i. e.,

(11)$${\rho}=\sum_{k=0}^{2F}\sum_{q={-}k}^{k}m_{k,q}{T}_{q}^{(k)} ,$$

where ${T}_{q}^{(k)}$ are the standard irreducible tensor operators

(12)$$T_{q}^{(k)}\left(F\right)=\sum_{m={-}F}^{F} \sum_{m^{\prime}={-}F}^{F} \left({-}1\right)^{F-m^{\prime}} \langle F,m,F,-m^{\prime}|{k,q}\rangle|{m}\rangle \langle{m^{\prime}}|$$

constructed from the angular momentum states $|{m}\rangle$ with the corresponding Clebsch-Gordan coefficients $\langle F,m,F,-m^{\prime }|{k,q}\rangle$, and where the multipole moments $m_{k,q}$ are given by

(13)$$m_{k,q}=\langle{T^{(k)\dagger}_{q}}\rangle=Tr\left(\rho{T^{(k)\dagger}_{q}}\right).$$

According to Eqs. (9)–(13), the absorption coefficient [12,28,29] is expanded using multipole moments as

(14)$$\alpha={-}\frac{4\pi \omega_{0} N}{9\hbar c}\left(14m_{0,0}-\sqrt{77}m_{2,0}\right),$$

where the multipole moment $m_{0,0}$ describes the total population of the ground state, the multipole moment $m_{2,0}$ describes the alignment of population of the ground state. We assume the optical transition to be closed and the light intensity to be so weak that excited state populations remain negligible, so that $m_{0,0}$ does not depend on time, which can be treated as a constant. The only time dependent component of absorption coefficient is therefore proportional to $m_{2,0}$.

2.2 Analytical solution for magnetic resonance detection

Based on the above discussion, we continue to study Eq. (4) in the representation of atomic polarization in terms of multipole moments in the rotating frame $x'y'z'$. Within the rotating wave approximation, the Hamiltonian of the interaction between the magnetic fields and the atom in the spherical basis in the rotating frame $x'y'z'$ is given by

(15)$${H}_{B}=\Omega_{0}{F}_{0}+\Omega_{1}\left (e^{i\omega_{1}t}{F}_{{-}1}-e^{{-}i\omega_{1}t}{F}_{{+}1}\right)+\Omega_{2}\left [e^{i\left(\omega_{2}t+\phi\right)}{F}_{{-}1}-e^{{-}i\left(\omega_{2}t+\phi\right)}{F}_{{+}1}\right] ,$$

where $\Omega _{0}=\mu _{B}g_{F}B_{0}$ represents the Zeeman splitting of $F_{g}=4$, $\Omega _{1}=\frac {\mu _{B}g_{F}}{2\sqrt {2}}B_{1}$ and $\Omega _{2}=\frac {\mu _{B}g_{F}}{2\sqrt {2}}B_{2}$ are Larmor frequency of $B_{\textit {rf1}}$ and $B_{\textit {rf2}}$, respectively. In our system, the operators ${F}_{0}$, ${F}_{-1}$ and ${F}_{+1}$ are hyperfine angular momentum components in the spherical basis [30], which is defined by

(16)$$F_{\mu}=\sum_{m} 2\sqrt{5} C^{4m+\mu}_{4m1\mu} |m+\mu\rangle \langle m|,$$

with $\mu =0,\pm 1$. Here, $m$ $(m=-4,\ldots ,4)$ is magnetic quantum number of the ground state of $F_{g}=4$, $C^{\cdot \cdot }_{\cdot \cdot \cdot \cdot }$ is Clebsch-Gordan coefficient of the ground state of $F_{g}=4$, $|m+\mu \rangle \langle m|$ is the transition operator from the state $|m\rangle$ to the state $|m+\mu \rangle$ in the bare-state representation.

To understand how the magnetic fields impact on the transmission of the system conveniently, we study the system in the dressed-state representation. The system satisfies the condition $\Omega _{0}>\Omega _{1}\gg \Omega _{2}$, therefore, we perform the diagonalization operation on the Hamiltonian containing the fields $B_{0}$ and $B_{\textit {rf1}}$, i.e., $(\Omega _0-\omega _{1})F_0+\Omega _1(F_{-1}-F_{+1})$ in the rotating frame with frequency $\omega _1$, which satisfies the eigenvalue equation $[(\Omega _0-\omega _{1})F_0+\Omega _1(F_{-1}-F_{+1})]|m'\rangle =\lambda |m'\rangle$. The state vector $|m'\rangle$ is the eigenvector in the dressed-state representation. Eigenvalue $\lambda =m'\Omega$ ($m^\prime =-4,\ldots ,4$) with $\Omega =\frac {1}{2}\sqrt {\Delta ^2+2\Omega _{1}^2}$ and $\Delta =\Omega _{0}-\omega _{1}$. Eigenvector $|m'\rangle$ in the dressed-state representation is given by

(17)$$|m^{\prime}\rangle = \sum_{m} \left[ D^{(4)}_{mm^{\prime}}\left(\beta\right)\right] ^{*} |m\rangle,$$

where function $D^{(4)}_{mm^{\prime }}(\beta )$ is the Wigner D-functions [30] with the Euler angle $\beta$ , which satisfies $\cos \beta =\frac {\Delta }{2\Omega }$ and $\sin \beta =\frac {\Omega _{1}}{\sqrt {2}\Omega }$. Similarly, the components of the angular momentum operator $F'_{\mu }$ in the dressed-state representation is defined by

(18)$$F^{\prime}_{\mu}=\sum_{m^{\prime}} 2\sqrt{5} C^{4m^{\prime}+\mu}_{4m^{\prime}1\mu} |m^{\prime}+\mu\rangle \langle m^{\prime}|,$$

where $|m^\prime +\mu \rangle \langle m^\prime |$ is the transition operator from $|m^\prime \rangle$ to $|m^\prime +\mu \rangle$. According to Eq. (16–18), the relation of angular momentum components between the bare-state representation and the dressed-state representation is given by

(19)$$\begin{aligned} {F}_{0}&=\cos\beta {F}^{\prime}_{0}+\frac{\sin\beta}{\sqrt{2}} {F}^{\prime}_{{+}1}-\frac{\sin\beta}{\sqrt{2}} {F}^{\prime}_{{-}1} ,\\ {F}_{{+}1}&={-}\frac{\sin\beta}{\sqrt{2}} {F}^{\prime}_{0}+\frac{1+\cos\beta}{2} {F}^{\prime}_{{+}1}+\frac{1-\cos\beta}{2} {F}^{\prime}_{{-}1} ,\\ {F}_{{-}1}&=\frac{\sin\beta}{\sqrt{2}} {F}^{\prime}_{0}+\frac{1-\cos\beta}{2} {F}^{\prime}_{{+}1}+\frac{1+\cos\beta}{2} {F}^{\prime}_{{-}1} . \end{aligned}$$

Therefore, the Hamiltonian (15) in the rotating frame with frequency $\omega _{1}$ in the dressed-state representation is given by

(20)$$\begin{aligned} {H}^{\prime}_{B}=\Omega {F}^{\prime}_{0}&+ \Omega_{2} \left[ e^{i\left(\delta t+\phi \right)} \left( \frac{\sin\beta}{\sqrt{2}}F^{\prime}_{0} + \frac{1-\cos\beta}{2}F^{\prime}_{{+}1} + \frac{1+\cos\beta}{2}F^{\prime}_{{-}1} \right) \right]\\ &- \Omega_{2} \left[ e^{{-}i\left(\delta t+\phi \right)} \left( -\frac{\sin\beta}{\sqrt{2}}F^{\prime}_{0} + \frac{1+\cos\beta}{2}F^{\prime}_{{+}1} + \frac{1-\cos\beta}{2}F^{\prime}_{{-}1} \right) \right] , \end{aligned}$$

where the detuning $\delta =\omega _{2}-\omega _{1}$. With the radio-frequency magnetic field $B_{\textit {rf1}}$ dressing the ground state of $F_{g}=4$, the each of the sublevels of the ground state can be dressed to nine new sublevels, as shown in Fig. 3. The Hamiltonian (20) indicates there are three kinds of transitions when $B_{\textit {rf2}}$ is scanned for probing the magnetic resonance in the dressed-state representation. When the frequency of $B_{\textit {rf2}}$ is scanned at $\omega _{1}-\Omega$, the radio-frequency photons at this frequency are absorbed, corresponding to the transitions of $\Delta {m^{\prime }}=-1$ with the coupling strength $\Omega _{2}\left (\frac {1-\cos \beta }{2}\right )$ in the dressed states ( as the green solid lines shown in Fig. 3). When it is scanned at $\omega _{1}+\Omega$, the radio-frequency photons at this frequency are absorbed, corresponding to the transitions of $\Delta {m^{\prime }}=+1$ with the coupling strength $\Omega _{2}\left (\frac {1+\cos \beta }{2}\right )$ in the dressed states (red solid lines in Fig. 3). For the transitions of $\Delta {m^{\prime }}=\pm 1$, both of them will cause changes in atomic alignment. However, for the transition of $\Delta {m^{\prime }}=0$, i.e., when it is scanned at $\omega _{1}$, the absorption of photons with frequency $\omega _{1}$ will not cause changes in atomic alignment (black dashed lines in Fig. 3).

Fig. 3. Magnetic resonance in the dressed-state representation. In the process of scanning $B_{\textit {rf2}}$ to probe the magnetic resonance of the ground state, there are three kinds of radio-frequency photons with difference frequencies that will be absorbed, corresponding to the transitions with $\Delta m^{\prime }=-1$, $\Delta m^{\prime }=0$ and $\Delta m^{\prime }=+1$ as the green, black and red lines shown respectively in the dressed-state representation.

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In the dressed-state representation, bring Eq. (11) and Eq. (20) into the Liouville equation [31], the evolution of alignment is given by

(21)$$\begin{aligned} \dot{{m}^{\prime}}_{2,0}&=\left(\Gamma _{20}+\Gamma _{g}\right) {m}^{\prime}_{2,0}-\sqrt{3}iC_{{-}1} {m}^{\prime}_{2,1}+\sqrt{3}iC_{{+}1} {m}^{\prime}_{2,-1}+\Gamma _{0} {m}^{\prime}_{0,0} ,\\ \dot{{m}}^{\prime}_{2,\pm1}&=\left({\mp} iC_{0}+\Gamma _{1}+\Gamma _{g}\right) {m}^{\prime}_{2,\pm1}\pm\sqrt{3}iC_{\pm1} {m}^{\prime}_{2,0}\mp\sqrt{2}iC_{\mp1} {m}^{\prime}_{2,\pm2} ,\\ \dot{{m}^{\prime}}_{2,\pm2}&=\left(\mp2 iC_{0}+\Gamma _{2}+\Gamma _{g}\right) {m}^{\prime}_{2,\pm2}\pm\sqrt{2}iC_{\pm1} {m}^{\prime}_{2,\pm1} , \end{aligned}$$

with

(22)$$\begin{aligned} C_{0}&=\frac{\Omega_{2}}{\sqrt{2}}\left[e^{i\left(\delta t+\phi \right)}+e^{{-}i\left(\delta t+\phi \right)}\right]\sin\beta+\Omega ,\\ C_{{+}1}&=\Omega_{2}\left[\sin^{2}\left(\frac{\beta}{2}\right)e^{i\left(\delta t+\phi \right)}-\cos^{2}\left(\frac{\beta}{2}\right)e^{{-}i\left(\delta t+\phi \right)}\right] ,\\ C_{{-}1}&=\Omega_{2}\left[\cos^{2}\left(\frac{\beta}{2}\right)e^{i\left(\delta t+\phi \right)}-\sin^{2}\left(\frac{\beta}{2}\right)e^{{-}i\left(\delta t+\phi \right)}\right] , \end{aligned}$$

where $\Gamma _{20}=-\frac {1823+345\cos \left (2\beta \right )}{9072}\Gamma _{L}$, $\Gamma _{0}=- \frac {7\sqrt {77}\left [1+3\cos \left (2\beta \right )\right ]}{1296}\Gamma _{L}$, $\Gamma _{1}=-\frac {1177+115\cos \left (2\beta \right )}{6048}\Gamma _{L}$, $\Gamma _{2}=\frac {-531+115\cos \left (2\beta \right )}{3024}\Gamma _{L}$, and $\Gamma _{g}=-\frac {3R_{SEC}}{64}$.

The multipole moment ${m}^{\prime }_{2,q}$ can be expressed with Floquet method approach [31]

(23)$${m}^{\prime}_{2,q}\left(t\right)=\sum_{l={-}\infty}^{\infty} {m}_{2,q}^{{\prime}{\left(l\right)}}\left(t\right)e^{il\delta t} ,$$

where ${m}^{\prime }_{2,q}$ is truncated to the second order, since the higher-order multipole moments are negligible in the weak light field [32]. Considering $B_{\textit {rf2}}$ is much weaker than $B_{\textit {rf1}}$, it implies that Eq. (21) can be solved using the perturbation approximation. The time-independent component of $m_{2,0}$ in the lab-frame representation is given by

(24)$$\begin{aligned} m^{(\textit{DC})}_{2,0}=&\frac{3\cos^{2}\beta -1}{4} \Bigg(\frac{\Gamma_{0}}{3\gamma_{0}}-\frac{2\Gamma_{0}\gamma_{1}\Omega_{2}^2}{\gamma_{0}^{2}}\Big[\frac{\sin^{4}\left(\frac{\beta}{2}\right)}{\gamma_{1}^{2}+\left(\delta+\Omega\right)^{2}}+\frac{\cos^{4}\left(\frac{\beta}{2}\right)}{\gamma_{1}^{2}+\left(\delta-\Omega\right)^{2}}\Big]-\\ &\frac{4\Gamma_{0}\Omega_{2}^4}{\gamma_{0}^{3}}\Big \{\frac{\sin^{8}\left(\frac{\beta}{2}\right)\left[\Lambda_{1}+\left(\delta+\Omega\right)^{2}\Lambda_{2}\right]}{\left[\gamma_{1}^{2}+\left(\delta+\Omega\right)^{2}\right]^{2}\left[4\left(\delta+\Omega\right)^{2}+\gamma^{2}_{2}\right]}+\frac{\cos^{8}\left(\frac{\beta}{2}\right)\left[\Lambda_{1}+\left(\delta-\Omega\right)^{2}\Lambda_{2}\right]}{\left[\gamma_{1}^{2}+\left(\delta-\Omega\right)^{2}\right]^{2}\left[4\left(\delta-\Omega\right)^{2}+\gamma^{2}_{2}\right]}\Big \} \Bigg) , \end{aligned}$$

where $\gamma _{0}=\Gamma _{20}+\Gamma _{g}$, $\gamma _{1}=\Gamma _{1}+\Gamma _{g}$, $\Lambda _{1}=-\gamma ^{2}_{1}\gamma _{2}\left (\gamma _{0}+3\gamma _{2}\right )$ and $\Lambda _{2}=-12\gamma ^{2}_{1}+\gamma _{0}\left (4\gamma _{1}+\gamma _{2}\right )$. Back to Eq. (14), under the system conditions, the absorption coefficient proportionally depends on $m_{2,0}$, therefore, $m^{(\textit {DC})}_{2,0}$ can represent the DC signal of magnetic resonance transmission spectra. In Eq. (24), the first term in the bigger brace is the constant term that has no effect on the information carried by the transmission spectra. This constant term simply means that the interaction of the laser with the atom causes the atomic system to establish a new dynamic equilibrium to reach a steady state. In the following content, all fitting curves ignore this term. The second term in the bigger brace represents the DC signal of transmission. The information about the amplitude and oscillating frequency of the dressing field is contained in the separation of dressed sublevels, i.e., there are two resonant sidebands corresponding to the resonant frequencies of $\omega _{1}-\Omega$ and $\omega _{1}+\Omega$ based on the condition of linear Zeeman splitting. The third term in the bigger brace is an amendment to the second term. This amendment term represents the splitting effect on the transmission spectra as the amplitude of the probing field $B_{\textit {rf2}}$ increases. This splitting effect is mainly reflected in the splitting of the resonant sidebands, but this splitting effect does not change the frequency position of the resonant sidebands. In general, the terms containing $\delta +\Omega$ in Eq. (24) represent the absorption of photons at this frequency, which correspond to the transition from $|m^\prime \rangle$ to $|m^\prime -1\rangle$ in the dressed-state representation (the green solid lines in Fig. 3) and correspond to the dip at the frequency of $\omega _{1}-\Omega$ of the spectra line; the terms containing $\delta -\Omega$ in Eq. (24) represent the absorption of photons at this frequency, which correspond to the transition from $|m^\prime \rangle$ to $|m^\prime +1\rangle$ in the dressed-state representation (the red solid lines in Fig. 3) and correspond to the dip at the frequency of $\omega _{1}+\Omega$ of the spectra line. Eq. (24) does not depend on initial phase $\phi$. It can be seen that the arbitrary phase difference between the two RMFs does not affect the alignment of the system. Apparently, the frequency difference between the two signal sidebands is $2\Omega$, the center frequency of the two sidebands is $\omega _{1}$, which can be utilized to determine the dressing RMF $B_{\textit {rf1}}$ experimentally.

3. Experimental setup and results discussion

The experimental setup is schematically shown in Fig. 4. The cesium gas is contained in a cylindrical cell coated with paraffin inside with the length of 30 mm and diameter of 25 mm. The coating material of the atomic cell is Alpha Olefin Fraction C20-24 from Chevron Phillips (CAS Number 93924-10-8). The relaxation rate of the collision between cesium atoms and the wall is usually less than 1 Hz. The cell is put in the middle of a pair of Helmholtz coils where $B_{\textit {rf1}}$ and $B_{\textit {rf2}}$ can be generated in the same direction. The size of the Helmholtz coils former matches the solenoid with the inner diameter of 120mm, and the angle of the slanted slot of the former is $26.6^{\circ }$, which ensures that the coils corresponding to each turn in the slanted slot meet the requirement that the ratio of distance to coil diameter is 1/2, i.e., $\tan 26.6^{\circ }$. The solenoid is used to generate a static magnetic field $B_{0}$, where the Helmholtz coils with the cell can be placed inside with $B_{0}$ perpendicular to the two RMFs. The solenoid is non-moment with double-layer-coil structure, which can avoid the risks of magnetizing the shields and freezing in large field gradients. $B_{0}$ in the range of zero to $5\times 10^{4}$ nT produced by the solenoid is driven by a precision current source (Keysight B2912A). The two RMFs produced by the Helmholtz coils are driven by a function waveform generator with double output channels (RIGOL DG4162). To isolate the system from environmental magnetic fields in the laboratory, the solenoid with the Helmholtz coils is placed in a magnetic shield with four layers of $\mu$-metal cylinders, whose fluctuation of magnetic field in the middle zone is about 2nT/cm when $B_{0}$ is about $43500$ nT. A Toptica DL pro 894 nm laser is used as the light source with a spot diameter of 5 mm. The Toptica DigiLock 110 is utilized to lock the laser to the $F_{g}=4 \to F_{e}=3$ transition of the $D_{1}$ line of cesium atom. The specific frequency value of the laser light is 335.111961349 THz, and the accuracy of laser saturation absorption and frequency stabilization (realized by the Digilock 110) in the experiment is about 500 kHz. The laser power can be tuned with a half wave plate and a polarized beam splitter, by which the laser beam is split into two beams: one beam is used for saturated absorption frequency stabilization, while the other beam is used for the magnetic resonance. The latter beam passes through a Glan-Taylor polarizer, with which a pure, linearly polarized light can be obtained. This pure light prepares cesium atoms with incident power of 30 $\mu$W and then received by a photodetector (Newport optical receiver 2031). Next, the transition spectra will be showed and analyzed by a digital oscilloscope (Keysight DSOX6004A).

Fig. 4. Experimental setup: DL pro, grating stabilized tunable single-mode diode laser; HWP, half-wave plate; PBS, polarization beam splitter; SAS, saturated absorption frequency stabilization; Digilock, feedback controller digilock 110; GT, Glan-Taylor prism; PD, photodetector; OSC, digital storage oscilloscope. All details can be found in the text.

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In the experiment of the Fig. 5, the static magnetic field $B_{0}$ is about 43500 nT, the dressing field $B_{\textit {rf1}}$ is about 200 nT, while the probing field $B_{\textit {rf2}}$ is about 20 nT. With the presence of $B_{\textit {rf2}}$, firstly, $\omega _{1}$ is tuned resonant with the Zeeman splitting of $B_{0}$, the resonant frequency is about 152.3 kHz. By locking the resonant frequency point of $\omega _{1}$, the measured transmission spectrum versus $\omega _{2}$ (circles) is plotted in Fig. 5(a). Then, we change the detuning between $B_{0}$ and $B_{\textit {rf1}}$ by turning the frequency $\omega _{1}$. For the conditions of $\Delta =-120$ Hz and $\Delta =120$ Hz, we obtain two transmission spectra with opposite sideband distribution (circles) shown in Fig. 5(b) and Fig. 5(c), respectively. These two opposite sideband distributions can be understood from Eq. (24). When the amplitude of the probing field $B_{\textit {rf2}}$ is very small, the spectral line of the transmission is mainly determined by the second term on the right hand of Eq. (24). When $\Delta =0$ is met, it means $\beta =\pi /2$, and the numerators of the two subterms of the second term of Eq. (24) are equal, which represents the two sidebands have equal amplitudes. When $\Delta <0$ is met, it means $\beta >\pi /2$, and the left numerator is bigger than the right one, which means the left sideband is bigger than the right one. And it is just the opposite situation when $\Delta >0$. For theoretical parameters, we uniformly take $\gamma _{1}=95$ Hz according to the experimental line width, take $\Omega _{0}=0$ Hz to set the center frequency as zero, and take $\Omega _{1}=710$ Hz. Corresponding to the experimental data in Fig. 5(a), (b) and (c), $\Delta$ theoretically also takes 0 Hz, $-120$ Hz and 120 Hz, respectively. Other theoretical parameters are not taken uniform values because of the shifting baseline of transmission signal in the experiment. The theoretical curve of Eq. (24) shows a good fitting with experimental data, from which one can obtain the information of the oscillating frequency and amplitude of the dressing field according to

(25)$$\begin{aligned} &\omega_{1}=\frac{\omega_{\textit{left}}+\omega_{\textit{right}}}{2} ,\\ &\Omega_{1}=\sqrt{\frac{\left(\frac{\omega_{\textit{right}}-3\omega_{\textit{left}}}{2}+\Omega_{0}\right)\left(\frac{3\omega_{\textit{right}}-\omega_{\textit{left}}}{2}-\Omega_{0}\right)}{2}}, \end{aligned}$$

where $\omega _{\textit {left}}$ and $\omega _{\textit {right}}$ are resonant frequencies of left sideband and right sideband, respectively.

Fig. 5. Measurements (circles) of the transmission spectra under resonance and detuning with Zeeman splitting . The short dashed lines with colors of red, green and blue are theoretical fittings of $\Delta =0$ Hz, $\Delta =-120$ Hz and $\Delta =120$ Hz, respectively. (a) The transmission under resonance; (b) the transmission under detuning of $\Delta =-120$ Hz; (c) the transmission under detuning of $\Delta =120$ Hz.

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The relative ratio of the amplitude of the two transmission sidebands at different $\Delta$ is consistent with the theoretical prediction, as shown in Fig. 6.

Fig. 6. The relative amplitude ratio of left sideband versus right sideband dependence on $\Delta$. The purple solid dots correspond to the experimental data with $\Delta$ from $-260$ Hz to 260 Hz by step of 20 Hz. The red solid line is the theoretical fitting according to Eq. (24).

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The separation between two sidebands under resonance shows a great linear relationship with the amplitude of $B_{\textit {rf1}}$, as shown in Fig. 7. Since the amplitude of the detected signal mainly depends on the amplitude of $B_{\textit {rf2}}$, the latter is set a variably appropriate value according to the amplitude of $B_{\textit {rf1}}$ but always less than the third of $B_{\textit {rf1}}$ to insure $B_{\textit {rf2}}$ is a weak field relative to $B_{\textit {rf1}}$. As the amplitude of the driving field $B_{\textit {rf1}}$ decreases, the DC signal collected by the digital oscilloscope will show a severe distortion in waveform. Based on this reason, in our experiment, the minimum effective amplitude of $B_{\textit {rf1}}$ that the function waveform generator can generate is about 13 nT (the corresponding amplitude of $B_{\textit {rf2}}$ is about 4 nT). By the error-bar analysis on each measured experimental data, the precision of the experimental measurement is below 1.5 nT and attests the reliable linear relation between the separation of the two sidebands and the amplitude of $B_{\textit {rf1}}$. Essentially, this linear relationship is a reflection that the separation of the sublevels in the dressed-state representation is linear with the Larmor frequency of the dressing field $B_{\textit {rf1}}$.

Fig. 7. The separation of two sidebands dependence on the amplitude of $B_{\textit {rf1}}$ under resonance. The blue solid dots correspond to the experimental data of the amplitude of $B_{\textit {rf1}}$ from about 16 nT to about 407 nT. The red solid line is the linear fitting of the data.

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4. Summary

In conclusion, we theoretically and experimentally study the optical-radio-frequency double resonance with the radio-frequency-dressed cesium atoms under a resonant linearly polarized light field when another weak RMF is applied for probe. Under linear Zeeman splitting, as the driving RMF dresses cesium atoms, the each of the sublevels of the ground state can be dressed to nine new sublevels with equal separation. It is explained from the point of view of the dressed state that the weak transverse RMF can probe the magnetic resonance by coupling the neighboring sublevels of dressed states with satisfying $\Delta {m}^{\prime }=\pm 1$. It is found that the information of the dressing field can be acquired from the resonant frequencies of two transmission sidebands. Different from the traditional method of measuring RMF by measuring the signal amplitude [33,34], the method proposed in this paper to measure RMF by determining the frequencies of resonant sidebands can achieve the accurate measurement of the resonant sidebands frequencies even when the resonance signal is so weak that the waveform is slightly distorted. The separation between two sidebands under resonance shows a reliable linear relationship with the amplitude of the dressing field, which may provide a practical scheme for designing a radio-frequency tunable atomic magnetometer.

Funding

Fundamental Research Funds for the Central Universities (CCNU18CXTD01, CCNU19GF003, CCNU19GF005); National Natural Science Foundation of China (11774118).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Laser-detected magnetic resonance spectra dressed by a radio-frequency field

Abstract

1. Introduction

2. Theoretical analysis

2.1 Systematic relaxation process and detection

2.2 Analytical solution for magnetic resonance detection

3. Experimental setup and results discussion

4. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (25)

Optics Express