## Abstract

We have studied the high resolution spectroscopy of Rydberg state ^{87}Rb in a ladder-type electromagnetically induced transparency (EIT) configuration at room temperature. A highly excited Rydberg atom is nearly degenerate for its hyperfine states but this degeneracy will be broken by an applied magnetic field, resulting in a spectral splitting in a coupled basis. In our ladder-type EIT experiment, we observed the high resolution spectral splitting of Rydberg atoms in an external magnetic field with two different optical polarization combinations of *σ*^{+} − *σ*^{+} and *σ*^{+} − *σ*^{−} for probe and coupling laser beams. A strict theory has been set up to explain the observed spectral line position and intensity accurately considering the Zeeman effects of three EIT-concerned states all in the coupled basis. Specially for the Rydberg state, we can transform its wavefunction back into the decoupled basis for the spectral line assignment.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The optical phenomena based on quantum interference and atomic coherent effects have attracted a lot of attention in recent years. Electromagnetically induced transparency (EIT) is a quantum coherent effect that a weak probe field will experience increased transmission in an absorbing medium when a strong coupling field is added. It has already led to widespread applications in many fields, for example, slowing of the light [1], lasing without inversion [2, 3], quantum memories [4], precision spectroscopy [5] and magnetometry [6], especially in microwave electrometry using the highly excited Rydberg states [7,8]. Up to now, most of these phenomena have been widely studied in simple three-level schemes, such as Λ–type, V-type and ladder type using atomic vapors [9] or solid state materials [10,11].

Recently, Λ–type EIT technique has been widely used in studying the Zeeman effect of atom [12–14]. In a real atomic system, the atomic states usually have complex structure with Zeeman-splitting sublevels [15]. Studying the complex behavior of EIT resonances in the degenerate atomic system in magnetic fields has attracted great interests both theoretically and experimentally [12, 14, 16–19]. The energy levels are Zeeman shifted with different resonant frequencies and their transition probabilities are extremely sensitive to an external magnetic field [20,21]. For example, Gao *et al.* observed the EIT window of ^{87}Rb split into three or four narrower subwindows due to the lifting of magnetic sublevels’ degeneracy [9]. Sargsyan *et al.* have also investigated the EIT of ^{85}Rb and ^{87}Rb in the magnetic field by using a nano-cell, a centimeter-long glass cell and a cell with a neon buffer gas for better spectral resolution [16,17,22,23].

The EIT can also be configured as a ladder type, where the highest energy state of the concerned three levels is a Rydberg state [10, 24–35]. This kind of EIT can explore the fine (hyperfine) structure of Rydberg states, owing to its narrow linewidth and high sensitivity [24,25,36]. In the Λ-type or V-type EIT where the wavelengths of probe and coupling lasers are usually almost the same, the proper choice of co- or counter-propagating beams can reduce the Doppler-broadened vapor to a virtually Doppler free medium. However, for the three-level ladder-type configuration, the coupling and probe wavelengths are usually very different and there exists a wavelength mismatch between them [24, 37, 38], where the multi-intermediate levels lead to a complex spectral structure. Moreover, these hyperfine splittings of the multi-intermediate state are not directly mapped to the observed spectrum, but scaled by a factor of 1 − *λ _{c}/λ_{p}* while the energy level splitting of the upper Rydberg state by

*λ*when scanning the probe laser [39].

_{c}/λ_{p}If a magnetic field applied, this wavelength mismatching effect causes the spectral splitting more complex and it is very difficult to clarify the observed spectral lines [40] extensively compared with those in a Λ–type atomic system [34,41–43]. A splitting of EIT window of ^{133}Cs in the cascade three-level system involved the *n*D or *n*S Rydberg state with a static magnetic field (several tens of Gauss) has been reported [41,42], where the experimental observation is roughly explained based on a simple linear Zeeman effect for the main spectral line positions. Similar spectral assignment for Rb atom in magnetic field has also been reported [35].

In fact, even in the magnetic field of tens of Gauss, the Zeeman splitting of the hyperfine levels of the high Rydberg atom is much larger than the hyperfine level gap itself and we have to describe the states in a decoupled basis rather than a coupled one [44], making the analysis of the spectrum more difficult, especially for the line intensity calculation. In this paper, we report an experimental spectral observation of Rydberg atoms in an external magnetic field. A theoretical calculation shows that these transitions are closely related to the hyperfine energy level couplings due to the introduction of the magnetic fields, which has not been extensively discussed in the previous experimental studies [35,41,42].

## 2. Experimental setup

Figure 1(a) shows the energy scheme associated with the three-level ladder-type EIT process of ^{87}Rb atoms. EIT involves a weak probe field with its frequency scanning over transitions from the ground state 5*S*_{1/2}(*F* = 2) to the intermediate state 5*P*_{3/2}(*F′* = 1, 2, 3), and an intense coupling field fixed at frequency of transition from the intermediate state 5*P*_{3/2}(*F′* = 3) to the highly excited Rydberg state *nS*_{1/2}(*F″* = 1, 2), where *n* = 50 in our case.

The schematic diagram of our experimental setup is also presented, as shown in Fig. 1(b). A filter-tuned, cateye refletor feedback external cavity diode laser (Moglabs) with linewidth < 1 MHz is used as a probe beam (*λ _{p}* ∼ 780 nm), which scans across the

^{87}Rb 5

*S*

_{1/2}−5

*P*

_{3/2}transitions. While the coupling field with wavelength 480 nm, driving the transition of 5

*P*

_{3/2}(

*F′*= 3) −

*nS/nD*, is provided by a frequency-doubled laser system (TA-SHG Pro, Toptica) for the 960 nm fundamental light. The frequency of coupling field is measured by a wavelength meter (wavemaster) with resolution about 10 MHz.

The 780 nm laser beam splits into two parts by the beam splitter, where one serves as the probe beam, and the other for a reference signal. The probe and coupling lasers are overlapped almost completely and counter-propagated through a rubidium atom vapor cell with length of only 25 mm at room temperature. The small size avoids the inhomegeneity of the magnetic field generated by the coils to the maximum extent. The probe beam (coupling beam) has a waist of about 0.3 mm (1.5 mm) and power 100 *μ*W (300 mW) in our experiments. The reference laser beam also transmits through the same rubidium vapor cell but without coupling laser overlapped, which is used to record only the absorption spectrum with Gaussian-broadening. The transmissions of the weak probe and reference beams are measured by a balanced photodiode (BPD) and a difference signal can be obtained. Thus we can remove the Gaussian-broadening and make the transmission appear on a flat background while scanning the probe laser frequency, which highlights the Rydberg EIT spectral lines. A current-controlled coil surrounding the cell can supply a magnetic field parallel to the laser beams, which is nearly homogenous in center part. The magnetic field is measured by a Gauss meter (Lakeshore 425), and the magnetic field generated by the coil is homogeneous within the area of a *L* × *d* = 70 mm ×10 mm cylinder in the center. The homogeneous magnetic filed can be up to 230 Gauss with an error less than 0.3%. Both beams pass through a half-wave plate (*λ*/2), a polarizing beam splitter (PBS), and a quarter-wave plate (*λ*/4) before entering the cell. We can configure the probe and coupling laser fields to different polarization combinations such as *σ*^{+} − *σ*^{−}, *σ*^{+} − *σ*^{+}, where *σ*^{+} is defined as the electric field of laser rotating in a left hand sense with respect to the direction of wave traveling while *σ*^{−} in right hand sense.

## 3. Theoretical model for the atomic transition

The atom in an external magnetic field can be described by the following Hamiltonian [44,45]

*Ĥ*

_{0}is the coarse atomic structure,

*Ĥ*and

_{fs}*Ĥ*describe the fine and hyperfine interactions, and

_{hfs}*Ĥ*represents the atomic interaction with the external magnetic field. Our optically concerned transition occurs between the hyperfine states in magnetic fields so that the first two terms are kept constant during the transition and we can omit them in a relative measurement for the transition frequency. Since the ground state is well described by the good quantum number

_{B}*F*and

*m*, we will express the whole Hamiltonian in the coupled basis |(

_{F}*JI*)

*Fm*〉. However, in a very strong magnetic field when both angular momenta (

_{F}*J*and

*I*) are totally uncoupled, the most convenient is the uncoupled bases approach where the eigenfunctions of an atomic state can be represented as |

*Jm*〉|

_{J}*Im*〉. The coupled basis is correlated to the decoupled one by [44]

_{I}*jm*symbols.

Specially, the splittings associated with the hyperfine interaction around the zero-detuning energies can be written diagonally as

*A*is the magnetic dipole constant,

_{hfs}*B*is the electric quadrupole constant and

_{hfs}*K*=

*F*(

*F*+ 1) −

*I*(

*I*+ 1) −

*J*(

*J*+ 1). The interaction of atom with the applied external magnetic field has the form where the two terms are expressed in the coupled basis |(

*JI*)

*Fm*〉 as

_{F}*j*symbols and in curled brackets 6

*j*symbols. Note non-zero diagonal part is also included in the matrix for

*i*=

*j*. A diagonalization of the full matrix for the ground state, the intermediate state and the excited Rydberg state, can give the transition frequency and transition amplitude for a one-photon absorption.

The transition probability from the initial state to the final one can be written as *σ* = |ΨΨ′|^{2}, where Ψ and Ψ′ are the eigen-function in the coupled basis for the initial and final states, which has been detailed in the spectral intensity analysis of Rb *D*_{2} line at weak/intermediate fields [46]. It should be noted that for the intermediate states in strong magnetic fields or high Rydberg state even in weak fields, we have to take into account the state-coupling [47,48]. The transition probability between pure states from |(*JI*)*Fm _{F}*〉 to |(

*JI*)

*F′m*〉 is given by

_{F′}*J*||

*er⃗*||

*J′*〉 is a constant in the transitions and all energy levels are expressed in the coupled basis. While for the Rydberg state, it is better to describe it in the decoupled basis since

*m*and

_{J}*m*are closely good quantum numbers there. We can project the calculated wavefunction of the Zeeman levels of high Rydberg atom back to the decoupled basis via Eq. 2 for spectral assignment in a usual way.

_{I}## 4. Results and discussion

There have been some works on the ladder-type EIT with the coupling laser resonant to the highly excited *nD*(*n* > 40) Rydberg state [49,50]. For the highly excited state ^{85}Rb or ^{87}Rb, the hyperfine splittings of the *nD* are neglectable and only the remarkable fine interaction remains (*nD*_{3/2} and *nD*_{5/2}), thus resulting in six two-photon resonant lines, in combination with the hyperfine splittings of the intermediate state 5*P*_{3/2} [49]. For the Rydberg state *nS*, the spectrum is further reduced to three lines as there is no fine splitting for *nS* state. However, even for the case of the coupling laser resonant to the highly excited *nS*_{1/2} Rydberg states, the spectrum will get very complex if an external magnetic field applied. In this paper, we investigate the EIT spectral splitting of ^{87}Rb atoms excited to 50*S*_{1/2} via the intermediate state 5*P*_{3/2}. In zero field, the high Rydberg state has a neglectable hyperfine splitting, but an applied weak magnetic field can induce a strong interaction between different hyperfine energy levels, resulting in a remarkable spectral splitting.

Unlike the Λ-type EIT [45, 51,52], there is a serious wavelength mismatching effect due to the large discrepancy between the probe and coupling lasers in wavelength. Figure 2 shows the observed absorption spectra of the probe laser as a function of probe frequency shift at different coupling detunings. Owning a nearly degeneracy for the state 50*S*_{1/2}, two spectral lines are observed, corresponding to the hyperfine-splitting of the intermediate state 5*P*_{3/2}, namely, 5*S*_{1/2}, *F* = 2 → 5*P*_{3/2}, *F′* = 2 → 50*S*_{1/2} and 5*S*_{1/2}, *F* = 2 → 5*P*_{3/2}, *F′* = 3 → 50*S*_{1/2}. Due to the wavelength mismatch between the probe and coupling lasers, the hyperfine splitting of the 5*P*_{3/2} state is scaled by a factor of 1 − *λ _{c}/λ_{p}* = 0.38, and the new interval between

*F′*= 2 and

*F′*= 3 is 0.38 × 267 MHz ≈ 101 MHz, which is in good agreement with the experiment.

As observed in the previous work [53–57], the spectral line amplitude is strongly dependent on the detuning of the coupling laser field. The transmittance spectrum of the 5*S*_{1/2}(*F* = 2) → 5*P*_{3/2}(*F′* = 2, 3) → *nS* transition is a double-resonance optical pumping process. At the coupling detuning Δ = 0 MHz shown in Fig. 2(c), where the coupling laser is resonant to the 5*P*_{3/2}, *F′* = 2 → 50*S*_{1/2} transition, this spectral line has a maximum amplitude. When the coupling detuning increases from Δ = −180 MHz up to Δ = 215 MHz as shown in Figs. 2(a)–2(e), with the coupling laser approaching the 5*P*_{3/2}, *F′* = 3 → 50*S*_{1/2} resonant transition corresponding to an energy interval of 267 MHz between *F′* = 2 and *F′* = 3, the associated spectral line gets stronger and stronger significantly while the one with *F′* = 2 vanishes. The variable detuning is accomplished by locking the coupling laser to an interferometric frequency control (IScan system, Germany). The maximum detuning value in Fig. 2(e) is about 215 MHz, still less than the energy interval 267 MHz. The theoretical simulation agrees well with the experimental result.

The introduction of magnetic field induces an energy level splitting for 50*S*_{1/2}, very different from those for the intermediate state 5*P*_{3/2} and ground state 5*S*_{1/2}. The energy diagram is shown in Fig. 3. For the ground state 5*S*_{1/2}, the hyperfine splitting is about 6.8 GHz, far larger than the interaction induced by the applied magnetic field, which leads to a paramagnetic energy shift for every *m _{F}* component. For every

*m*energy level, there is no remarkable interaction between different hyperfine levels and

_{F}*F*is approximately a good quantum number. This approximation is also valid for the intermediate state 5

*P*

_{3/2}since its hyperfine splittings are also far larger than the Zeeman energy level shift in our case. The energy gap between

*F′*= 0 and

*F′*= 1 is rather small, about 72 MHz, comparable with the applied maximum magnetic field induced interaction, which is estimated to be on the order of 40 MHz at

*B*= 40 Gauss. However, unfortunately, these levels do not participate in our ladder-EIT observation for weak transition probabilities. Therefore, principally, only the splittings for 50

*S*

_{1/2}should be considered seriously. The ground state and the intermediate state can be nearly diagonal in the coupled basis |(

*JI*)

*Fm*〉. While for the 50

_{F}*S*

_{1/2}term, the energy gap between hyperfine levels is almost ignorable, only 0.25 MHz [37], which makes even a weak magnetic field introduce a remarkable interaction between the degenerated hyperfine states, where the states are well described and then classified in the decoupled basis |

*Jm*〉|

_{J}*Im*〉.

_{I}We firstly present a spectral observation for the probe and coupling laser configured as *σ*^{+} and *σ*^{−} polarized at a magnetic field *B* = 37 Gauss as shown in Fig. 4. In the measurement, we lock the coupling laser to the resonance of 5*P*_{3/2}, *F′* = 3 to 50*S*_{1/2} for a better signal to noise ratio according to the discussion on the effect of laser detuning on the spectral line as shown in Fig. 2. We define *ω*_{0p} and *ω*_{0c} to denote the resonant frequency of transition from *F* = 2 to *F′* = 3 and *F′* = 3 to 50*S*_{1/2} state respectively at zero magnetic field for convenience. As discussed previously, the magnetic field induced interaction breaks the degeneracy of the Rydberg state 50*S*_{1/2} and splits the energy level. In this case, *F* is not a good quantum number any longer and the Zeeman interaction can not be taken as a perturbation. The coupled hyperfine states will mix seriously and the Zeeman energy level of Rydberg state can be expressed in the coupled basis as

*C*is the expansion coefficient of the wavefunction Ψ in the coupled basis |

_{F}*Fm*〉 while

_{F}*m*is a good quantum number and has a given value. We can project this state back into the decoupled basis |

_{F}*Jm*〉|

_{J}*Im*〉 since the eigen-state is well described by the quantum number

_{I}*m*and

_{J}*m*. An example of the analysis for the spectral line A and A′ is shown in Fig. 5. All transition assignments are listed in Table 1, where the spectral lines are assigned in the coupled and decoupled basis at the same time for the Rydberg state. For the spectral line A(A′), B(B′) and C(C′), each pair is the result of interaction between different component

_{I}*F*in the coupled basis for high Rydberg state, thus the spectral line should be a linear combination of different |

*Fm*〉 as given in Eq. 8. Therefore, we have to assign the spectral lines with probabilities in the coupled basis. The only exception is for the spectral lines D and E where there is no interaction between

_{F}*F*for the Rydberg state. As

*m*is a good quantum number, the magnetic field induced interaction can happen only between the same

_{F}*m*but all values of

_{F}*F*satisfying

*F*≥ |

*mF*|. For the state

*m*(|

_{F}*m*| =

_{F}*F*), there is only one

*F*value, corresponding to the cases of spectral lines D and E with

*F*= 2,

*m*= −2 and

_{F}*F*= 2,

*m*= 2, respectively. All states can be projected back into decoupled basis |

_{F}*Jm*〉

_{J}*Im*〉, which is also labeled in Table 1. In the decoupled basis, a pair of (

_{I}*m*,

_{J}*m*) is enough for assignment of the spectral line as all other projection components are nearly zero, making little contribution.

_{I}We also investigate EIT spectrum via polarization combination *σ*^{+} and *σ*^{+}, respectively for probe and coupling beams at magnetic field *B* = 40 Gauss. It is shown in Fig. 6. An analysis similar to Fig. 4 is made considering the state mixture. The assignment of the spectral lines is shown in Table 2 where the quantum number in the decoupled basis and the mixing coefficients in the coupled basis are also listed.

It should be noted that the spectra of Rydberg EIT for different polarization combinations of *σ*^{+}−*σ*^{+} and *σ*^{+}−*σ*^{−} are completely different even at the same magnetic fields. We calculate these spectra at *B* = 40 Gauss as shown in Fig. 7, where the coupling laser has the same detuning as that in Fig. 6. In the combination of *σ*^{+} and *σ*^{−}, the initial and final states in the Raman process have the same *m _{F}* while

*m*is changed by 2 in the combination of

_{F}*σ*

^{+}and

*σ*

^{+}, corresponding to very different energy levels, therefore resulting in different spectral line positions.

The dependence of Rydberg EIT spectrum on the applied magnetic fields is also investigated as shown in Fig. 8. The polarizations of the probe and coupling fields are configured as *σ*^{+} and *σ*^{+}, respectively. Figures 8(a)–8(b) are the normalized experimental and calculated spectra, respectively. To clarify the spectral lines clearly, a narrow line broadening (3.5 MHz) is used for the calculated transitions. In the experiment, a high power of the coupling laser has to be used to record a Rydberg EIT spectrum with high signal-to-noise ratio, but at the same time it causes the broadening of the peaks (4.3 MHz). We define the frequency shifts of the Zeeman levels in 5*S*_{1/2}, 5*P*_{3/2} and 50*S*_{1/2} states relative to their frequencies at zero magnetic field as Δ_{1}, Δ_{2} and Δ_{3}, respectively. Then the frequency detunings of probe and coupling lasers in magnetic fields relative to *ω*_{0p} and *ω*_{0c} are Δ* _{p}* = Δ

_{2}− Δ

_{1}and Δ

*= Δ*

_{c}_{3}− Δ

_{2}. Considering the wavelength mismatch, the detunings satisfy the relation ${\mathrm{\Delta}}_{p}+{\mathrm{\Delta}}_{c}=({\mathrm{\Delta}}_{2}-{\mathrm{\Delta}}_{1})+\frac{{\lambda}_{c}}{{\lambda}_{p}}({\mathrm{\Delta}}_{3}-{\mathrm{\Delta}}_{2})$. When the coupling laser is on the resonant transition of 5

*P*

_{3/2},

*F′*= 3 →

*nS*

_{1/2}, the EIT subpeaks exist at ${\mathrm{\Delta}}_{p}=({\mathrm{\Delta}}_{2}-{\mathrm{\Delta}}_{1})+\frac{{\lambda}_{c}}{{\lambda}_{p}}({\mathrm{\Delta}}_{3}-{\mathrm{\Delta}}_{2})=-{\mathrm{\Delta}}_{1}+\left(1-\frac{{\lambda}_{c}}{{\lambda}_{p}}\right){\mathrm{\Delta}}_{2}+\frac{{\lambda}_{c}}{{\lambda}_{p}}{\mathrm{\Delta}}_{3}$. Our calculated results explains the experimental peculiarities well.

As discussed previously, the ground state 5*S*_{1/2} has a large energy gap (6.8 GHz) between the hyperfine levels and the interaction between them can be neglected so that the Zeeman effect shows a linear behavior vs. the magnetic field below 1000 Gauss. On the other extreme that the hyperfine levels are nearly degenerated for the high Rydberg state 50*S*_{1/2}, the Zeeman is completely linear beyond a nonzero magnetic field (> 10 Gauss) as well. This feature implies that the dependence of the spectral transition frequency on the applied magnetic field relies on the intermediate state 5*P*_{3/2} mainly. For the intermediate state 5*P*_{3/2}, the hyperfine structure splittings are about several hundreds of MHz, comparable with the magnetic field induced interaction, which results in a non-linear dependence of the total transition frequency on the magnetic field. However, for the cases whether in a weak magnetic field or in a strong one, our calculation based on the Hamiltonian in the coupled basis works well, even for the line intensity. Specially, for the Rydberg state, we can project the wavefunction back into the decoupled basis for spectral line assignment in a usual way.

## 5. Conclusion

In summary, we have investigated the double-resonance in a 5*S*_{1/2} → 5*P*_{3/2} → 50*S*_{1/2} ladder-type EIT system for ^{87}Rb atom in magnetic fields. For the ground state 5*S*_{1/2} and intermediate state 5*P*_{3/2}, the hyperfine levels are well separated by an energy gap much larger than the Zeeman interaction and the Zeeman interaction can be taken as a perturbation for the hyperfine energy levels in the coupled basis. However, for the nearly degenerated high Rydberg state 50*S*_{1/2}, this Zeeman interaction can not be taken as a perturbation any more and the coupled hyperfine states will mix seriously. The eigen state of 50*S*_{1/2} in magnetic field has to be viewed as an linear combination of eigen wavefunction |*Fm _{F}*〉 in the coupled basis. For a better understanding for the Zeeman-splitting of Rydberg state in a usual way, we project the wavefunction back into the decoupled basis where the spectral line is well assigned by a pair of (

*m*,

_{J}*m*). All the experimental observations are well explained by the combination of coupled and decoupled basis.

_{I}## Funding

National Natural Science Foundation of China (NSFC) (91421305, 91121005, 11674359); Program 973 (2013CB922003).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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