Electromagnetically induced transparency in a spin-orbit coupled Bose-Einstein condensate

Zhengfeng Hu; Chengpu Liu; Jin-Ming Liu; Yuzhu Wang

doi:10.1364/OE.26.020122

1. Introduction

The quantum manipulation and control of light by another light in a medium is important in quantum optics and quantum information. A very important example is the electromagnetically induced transparency (EIT) [1–4], which has many interesting phenomena such as slow light [5], light storage [6–9], and very narrow spectroscopy [1–4]. The EIT phenomenon has been observed in several media, including hot atomic vapor [7, 10], cold atomic gas [5, 6, 11–15], solid materials [16–19], and Rydberg atomic gas [20–25]. Lately, the EIT has been theoretically investigated in a strong correlated [26] and a ultracold atomic systems [27]. The quantum many-body effect has been reflected in the EIT spectroscopy [26, 27]. The investigation of the EIT in a different media is very interesting. On the one hand, the particular EIT spectroscopy can be found which is closely related to the properties of the medium, for example strong correlation interaction in ultracold atoms and the nonlocal interaction in optical lattice. On the other hand, the EIT possesses a highly sensitive spectrum. It has been applied to the investigation of frequency standard [4]. This highly accurate spectroscopy can be used as a tool to detect some weak interactions among atoms [28, 29]. Contrary to the destructive absorption measurement, the EIT spectroscopy has the advantage of a non-demolition probe with well-controlled laser parameters, which provides efficient and precise measurements on ultracold atoms in an in-situ way. This EIT spectroscopy can be used as a tool to explore and analyze the magnificent characteristics of a new medium.

Very recently, an artificial magnetic field has been realized in a Bose-Einstein condensate (BEC) by using Raman lasers [30–35] and magnetic field [36–38]. The one dimensional SO coupling has been demonstrated in BEC [39] and in finite-temperature cold bosonic atomic system [40] by suitably arranging the Raman lasers. A space dependent laser-atom coupling is used to simulate an effective artificial magnetic field which produces a Lorentz-like force on neutral atoms. The neutral atom spin Hall effect in a BEC [41] has been observed which can simulate the phenomena in the condensed matter physics with accurately controlled parameters [12, 42–46]. The both static and dynamic properties of the BEC with spin-orbit (SO) coupling have been investigated [42–53]. The realization of the SO coupling can help us understand the basic superfluid physical phenomena and provide a controllable platform to simulate condensed matter states [42–46, 52]. At the same time the experimental realization of the SO coupling in ultracold atoms has provided a new medium which includes additional SO coupling interaction. How to detect the SO coupling strength and to measure the effect of the SO coupling accurately are very interesting. This is the motivation of the present work. As we showed in the following, the EIT with its particular properties can be well suitably to fulfil the sensitive measurement of the SO coupling in a BEC.

2. Formulism of system

In this paper, we study the effects of SO coupling on EIT and find some new properties of the EIT related to the SO coupling, as well as the possibility to detect the interaction of SO coupling by using the EIT spectroscopy.

The system that we investigated is a Λ-configuration atomic structure and the atoms are in the condensed state with SO interaction as shown in Fig. 1. We consider the case of the quasi 1D cold atomic gases. We notice that a pulsed magnetic field can be applied to create both individual half-quantum vortices and vortex lattices [54]. Here SOC was also produced by pulsed inhomogeneous magnetic fields as shown in Fig. 1(a). The ground states that we considered are the Zeeman substates of the hyperfine state of Rubidium atoms, such as the states $| b 〉 = | F_{g} = 1, m_{F_{g}} = 0 〉$ and $| c 〉 = | F_{g} = 1, m_{F_{g}} = - 1 〉$ . At this stage, the effective Hamiltonian is given as [36–38]:

H_{eff} = \sum_{p} \frac{p^{2}}{2 m} (a_{p}^{†} a_{p} + b_{p}^{†} b_{p} + c_{p}^{†} c_{p}) - \sum_{p} γ p (c_{p}^{†} c_{p} + b_{p}^{†} b_{p}),

where the second term is the SOC term. The strength of the SO coupling is γ = (ħk_SO/2m) F_z, ħk_SO = g_F µ_B

\int_{0}^{T / 2} B (t) d t = g_{F} μ_{B} B_{\max} T / π

. g_F is the Lande g-factor and µ_B is the Bohr magneton. F_z is the z-component of spin vector F. The production of the EIT spectroscopy was shown in Fig. 1(b). The excited state |a〉 is the Zeeman sublevel of the excited hyperfine state

| F_{e} = 1, m_{F_{e}} = - 1 〉

. The states |b〉 and |a〉 are coupled by a weak probe laser with the angular frequency ω_p and the state |c〉 is coupled to the excited state |a〉 via a stronger coupling laser with the angular frequency ω_c. The one-photon detuning is δ_p = ω_p − (E_a − E_b)/ħ, δ_c = ω_c − (E_a − E_c)/ħ, and the two-photon detuning is δ = δ_p − δ_c = ω_p − ω_c − (E_c − E_b)/ħ, where ħ is the Planck constant. E_a_,_b_,_c are the energies of the states |a〉, |b〉, |c〉. In experiment the two lasers can be two independent lasers with phase locked by a phase-locking loop, or come from one laser device by using an acoustic-optic modulator or electro-optic modulator generating sidebands. The frequency of the coupling laser is fixed with the one-photon detuning and the frequency of the probe laser is scanned by an acoustic-optic modulator. The spectroscopy of the EIT can be obtained by detecting the transmittance of the probe laser or the fluorescence by a detector. The influence of the SO coupling can be obtained by comparing the spectroscopy of the EIT in BEC without the SO coupling and that of the EIT in the SO coupled BEC. For the case of the BEC atoms, we consider the interactions among the atoms which are the s-wave scattering. Under the dipole and rotating wave approximations, the total Hamiltonian is

\hat{H} = {\hat{H}}_{A} + {\hat{H}}_{AL} + {\hat{H}}_{SO} + {\hat{H}}_{U},

{\hat{H}}_{A} = \sum_{p} \frac{p^{2}}{2 m} ({\hat{a}}_{p}^{†} {\hat{a}}_{p} + {\hat{b}}_{p}^{†} {\hat{b}}_{p} + {\hat{c}}_{p}^{†} {\hat{c}}_{p}) + \sum_{p} ℏ (δ_{2} {\hat{c}}_{p}^{†} {\hat{c}}_{p} + δ_{1} {\hat{b}}_{p}^{†} {\hat{b}}_{p}),

{\hat{H}}_{AL} = - ℏ Ω_{p} \sum_{p} {\hat{a}}_{p + p_{p}}^{†} {\hat{b}}_{p} - ℏ Ω_{c} \sum_{p} {\hat{a}}_{p + p_{c}}^{†} {\hat{c}}_{p} + H . c ..,

{\hat{H}}_{SO} = - \sum_{p} ℏ γ p ({\hat{c}}_{p}^{†} {\hat{c}}_{p} + {\hat{b}}_{p}^{†} {\hat{b}}_{p}),

\begin{array}{l} {\hat{H}}_{U} = \frac{1}{2 V} \sum_{p, p^{'}, q} (U_{a a} {\hat{a}}_{p + q}^{†} {\hat{a}}_{p^{'} - q}^{†} {\hat{a}}_{p^{'}} {\hat{a}}_{p} + U_{b b} {\hat{b}}_{p + q}^{†} {\hat{b}}_{p^{'} - q}^{†} {\hat{b}}_{p^{'}} {\hat{b}}_{p} \\ + U_{c c} {\hat{c}}_{p + q}^{†} {\hat{c}}_{p^{'} - q}^{†} {\hat{c}}_{p^{'}} {\hat{c}}_{p} + 2 U_{a b} {\hat{b}}_{p + q}^{†} {\hat{a}}_{p^{'} - q}^{†} {\hat{a}}_{p^{'}} {\hat{b}}_{p} \\ + 2 U_{a c} {\hat{c}}_{p + q}^{†} {\hat{a}}_{p^{'} - q}^{†} {\hat{a}}_{p^{'}} {\hat{c}}_{p} + 2 U_{b c} {\hat{c}}_{p + q}^{†} {\hat{b}}_{p^{'} - q}^{†} {\hat{b}}_{p^{'}} {\hat{c}}_{p}), \\ ≃ \frac{1}{2 V} \sum_{p, p^{'}, q} U_{b b} {\hat{b}}_{p + q}^{†} {\hat{b}}_{p^{'} - q}^{†} {\hat{b}}_{p^{'}} {\hat{b}}_{p} + ρ_{con} U_{a b} {\hat{a}}_{p}^{†} {\hat{a}}_{p} + ρ_{con} U_{b c} {\hat{c}}_{p}^{†} {\hat{c}}_{p} . \end{array}

Fig. 1 (a) The scheme for generating 1D spin-orbit coupling. A cold atom ensemble is prepared in a dipole trap in a cigar-like shape. A periodically modulated gradient magnetic field $B (t) z \hat{z}$ imparts opposite forces (black arrows or green dash arrows at different times) to the m_F = 1 (red disk and arrow) and m_F = −1 (blue disk and arrow) states of the F = 1 hyperfine. (b) The system diagram for producing electromagnetically induced transparency in a cold spin-orbit coupled ensemble. The state $| a 〉 = | F_{e} = 1, m_{F_{e}} = - 1 〉$ , $| b 〉 = | F_{g} = 1, m_{F_{g}} = 0 〉$ , $| c 〉 = | F_{g} = 1, m_{F_{g}} = - 1 〉$ . A weak probe laser light with Rabi frequency Ω_p and another intense control laser light with Rabi frequency Ω_c drive the transitions |a〉 ↔ |b〉 and |a〉 ↔ |c〉. δ_p, δ_c are the one-photon detunings of the two lasers.

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Here ${\hat{H}}_{A}$ is the kinetic and internal state energy of the atoms, where the operator of atomic field satisfies the bosonic commutation relation, $[a_{p}, {\hat{a}}_{p^{'}}^{†}] = δ_{p p^{'}}$ . The laser-atom interactions are included in ${\hat{H}}_{AL}$ , where Ω_p and Ω_c are the Rabi frequencies associated with the probe and coupling lasers. H.c. denotes the Hermitian conjugate. p_p_,_c are the recoil momenta of the probe and control fields, respectively. ${\hat{H}}_{SO}$ represents the SO coupling induced by pulsed magnetic fields. We mainly consider the effects of SO coupling on EIT. ${\hat{H}}_{U}$ is the interaction of the atoms which is the s-wave scattering. p is the atom momentum along the x direction and m is the mass of an atom. U (i.j = a, b, c) = 4πħ²a_ij/m (i, j = a, b, c) are the interactions of atoms which are considered as the s-wave scattering interactions and a_ij are the s-wave scattering lengths for atoms. The volume of the condensate is V and its density is ρ_con. Here the interaction picture has been used to remove the free parts of the Hamiltonian.

Because the probe laser is weak, the interaction of the probe laser is treated as a perturbation and all the other interactions are kept. Because of the weak probe field, most of the atoms are in the state |b〉 and a few atoms are in the states |c〉. In the basis $B_{p} = {[{\hat{b}}_{p}, c_{p + p_{r}}, {\hat{a}}_{p + p_{p}}]}^{T}$ , with p_r ≡ p_p − p_c and $\hat{H} = \sum_{p} B_{p}^{†} {\hat{M}}_{p} B_{p}$ , where the matrix ${\hat{M}}_{p}$ is

\begin{array}{l} {\hat{M}}_{p} = (\begin{matrix} ℏ δ_{p} + \frac{p^{2}}{2 m} + ℏ γ p & 0 & - ℏ Ω_{p} \\ 0 & ℏ δ_{c} + \frac{{(p + p_{r})}^{2}}{2 m} - ℏ γ (p + p_{r}) + ρ_{con} U_{b c} & - ℏ Ω_{c} \\ - ℏ Ω_{p} & - ℏ Ω_{c} & \frac{{(p + p_{p})}^{2}}{2 m} + ρ_{con} U_{a b} \end{matrix}), \\ = ρ_{con} U_{a b} \hat{I} + (\begin{matrix} ℏ {\bar{δ}}_{p} + \frac{p^{2}}{2 m} + ℏ γ p & 0 & - ℏ Ω_{p} \\ 0 & ℏ {\bar{δ}}_{c} + \frac{{(p + p_{r})}^{2}}{2 m} - ℏ γ (p + p_{r}) & - ℏ Ω_{c} \\ - ℏ Ω_{p} & - ℏ Ω_{c} & \frac{{(p + p_{p})}^{2}}{2 m} \end{matrix}), \end{array}

where the common mean-field energy ρ_conU_ab has been extracted and

ℏ {\bar{δ}}_{p} = ℏ δ_{p} - ρ_{c o n} U_{a b}

,

ℏ {\bar{δ}}_{c} = ℏ δ_{c} + ρ_{c o n} (U_{b c} - U_{a b})

. Because Ω_p is treated as perturbation, the unperturbed Hamiltonian is obtained by setting Ω_p to be zero in the Hamiltonian

\hat{H} = \sum_{p} B_{p}^{†} {\hat{M}}_{p} B_{p}

which can be diagonalized. The eigenvalues of Eq. (6) when Ω_p ≃ 0 are

E_{0} (p) = ℏ {\bar{δ}}_{p} + \frac{p^{2}}{2 m} + ℏ γ p,

E_{\pm} (p) = \frac{ℏ δ_{c}^{'} + \frac{{(p + p_{p})}^{2}}{2 m} \pm \sqrt{{[ℏ δ_{c}^{'} - \frac{{(p + p_{p})}^{2}}{2 m}]}^{2} + {(2 ℏ Ω_{c})}^{2}}}{2},

where

ℏ δ_{c}^{'} = ℏ {\bar{δ}}_{c} + {(p + p_{r})}^{2} / (2 m) - ℏ γ (p + p_{r})

and

\cos ϕ_{p} = {(\frac{E_{+} (p) - \frac{{(p + p_{p})}^{2}}{2 m}}{E_{+} (p) - E_{-} (p)})}^{1 / 2} .

The angle ϕ_p is the mixing of the ground and the excited states. These eigenvalues which are dependent on the SO coupling are also the energies of the dressed states. E₀ is the energy of the dark state and E_± are the energies of the bright states. The gap of the energies of the two bright states is also dependent on the SO coupling. The variations of the eigenvalues with the SO coupling are calculated and are shown in figure 2. The energy of the dark state is increasing with the increase of the SO coupling, but the energies of the bright states and their gap are decreasing with the increase of the SO coupling. By using the dressed state and Hartree−Fock−Bogoliubov, Popov approximations for the first term in Eq. (6), the total Hamiltonian can be diagonalized by Bogoliubov transformation. By introducing the chemical potential $μ = ℏ {\bar{δ}}_{p} + ρ_{con} U_{bb}$ and considering the conservation of the total atomic number, there has

H - μ N = E_{g} - μ N_{con} + \sum_{q \neq 0} E (q) ψ_{q}^{†} ψ_{q},

where N is the total number of atoms. E_g is the ground state energy including the mean-field energy and energy correction. E(q) is the energy of the quasi-particle. N con is the number of condensed atoms. The ground state energy is given as

E_{g} = E_{0} (p = 0) N_{con} + \frac{U_{bb}}{2 V} (N_{con}^{2} + 2 N_{ex}^{2}) + \frac{N_{con} N U_{ab}}{V} + \sum_{q > 0} (E - E_{1} (q)),

where N_ex is the number of the thermal excited atoms. The eigenvalue and eigenstates of the Hamiltonian of Eq. (11) are

E (q) = \sqrt{E_{1}^{2} (q) - E_{2}^{2}}

,

ψ_{q} = \cosh θ (q) Ψ_{q} + \sinh θ (q) Ψ_{- q}^{†}

, where Ψ_q is the annihilation operator for a boson at momentum q.

\tanh 2 θ (q) = \frac{E_{2}}{E_{1} (q)}

and E₁(q) = E₀(q) − µ + 2ρ_conU_bb, E₂ = ρ_conU_bb.

Fig. 2 Energies of the dressed states with SO coupling. (a) The energies of the dressed states with the change of the SO coupling. The solid (blue) line is the eigenvalue E₀. The dashed (red) line is the energy E₊ and the dash-doted (green) line is the energy E₋. (b) The difference of the energies of the two bright states. The parameters which we used are: ρ = 10¹⁴cm⁻³, T_c = 0.01µK, ρ_c = 0.9ρ, δ_c = 5.0Γ, Ω_c = 1.0Γ, ρ_cU_bb = 0.9Γ, Γ is the spontaneous decay rate of the excited state.

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Under the dressed state which is related to the eigenvalues of Eq. (9) and a constant coupling field, the susceptibility is given by

\begin{array}{l} χ (ω_{p}) = \frac{d_{a b}^{2}}{ℏ Ω_{p} A} \int d^{2} r_{⊥} 〈 Ψ_{b}^{†} (r) Ψ_{a} (r) 〉 e^{- i p_{p} \dot{r} / ℏ}, \\ = \frac{ρ d_{a b}^{2}}{ℏ N} \sum_{p} F_{p} n_{β_{p}}, \end{array}

where F_p is the kernel function given as

\begin{array}{l} F_{p} = - F_{p} [\frac{\sin^{2} ϕ_{p}}{E_{0} (p) - E_{+} (p)} + \frac{\cos^{2} ϕ_{p}}{E_{0} (p) - E_{-} (p)}] \\ = \frac{ℏ {\bar{δ}}_{p} - ℏ {\bar{δ}}_{c} + \frac{p^{2}}{2 m} - \frac{{(p + p_{r})}^{2}}{2 m} + ℏ γ (2 p + p_{r})}{{(ℏ Ω_{c})}^{2} - [ℏ {\bar{δ}}_{p} + \frac{p^{2}}{2 m} - \frac{{(p + p_{p})}^{2}}{2 m} + ℏ γ p + i ℏ Γ] [ℏ {\bar{δ}}_{p} - ℏ {\bar{δ}}_{c} + \frac{p^{2}}{2 m} - \frac{{(p + p_{r})}^{2}}{2 m} + ℏ γ (2 p + p_{t})]} . \end{array}

Ψ_a(r) and Ψ_b(r) are the atomic field operators in real space. r_⊥ is the direction perpendicular to p_p. The distribution of quasi-particle $n_{β_{p}} (p) = 1 / (e^{(E_{0} (p) - μ) / k_{B} T} - 1)$ [55]. d_ab is the dipole moment of the atomic transition. A is the cross-section area perpendicular to the propagating direction of the probe light. The relaxation rate Γ is added phenomenologically by considering the spontaneous emission of the excited state. By using the ground state wave function, the susceptibility is given as

\begin{array}{l} χ (ω_{p}) = χ_{con} (ω_{p}) + χ_{qd} (ω_{p}) + χ_{th} (ω_{p}) \\ = \frac{ρ d_{a b}^{2}}{ℏ} [\frac{ρ_{con}}{ρ} F_{p = 0} + \frac{1}{N} \sum_{p \neq 0} F_{p} \sinh^{2} θ + \frac{1}{N} \sum_{p \neq 0} F_{p} n_{b} \cosh 2 θ], \end{array}

where the susceptibility is divided into three parts that are the condensate, quantum depletion, and thermal depletion. Bogoliubov quasi-particle distribution is

n_{b} (q) = 1 / [e^{ϵ (q) / (k_{B} T)} - 1]

. That the susceptibility is affected by the SO coupling is included in Eq. (15) through the kernel function.

3. Numerical results of susceptibility

Further, we numerically solved susceptibility and obtained the results, as shown in Fig. 3. The real and imaginary parts are related to the dispersion and the absorption. The temperature of the BEC can be obtained through the formula of critical temperature [55]. It can be seen from the Fig. 3 that the EIT still exists although the SO coupling is included in the system. Comparing with the coupling laser atom interactions, the SO coupling is weak. At the same time to reduce the influence of the spontaneous emission on the BEC, lasers are applied in a far-off resonant way. Therefore we see that the transparent window and the peak of the imaginary part of the susceptibility are displaced from the resonant point when there is the one-photon detuning. The line shapes of the real and imaginary parts of the susceptibility are a little different from those of the usual EIT [2–4] because of the existence of the SO coupling. In order to compare with the case of no SO coupling, we made the calculations when there was the SO coupling and there was no SO coupling and the results were plotted in Figs. 3(b) and 3(c). From Figs. 3(b) and (c) we see that the lines of the real part and the imaginary part have been moved toward the left side compared to the case of without SO coupling. This can also be seen from the expression of the susceptibility of Eq. (13) and Eq. (14). Indeed the SO coupling has added additional interaction to the Hamiltonian and this interaction has made the shifts of the atomic levels. Therefore the susceptibility has a frequency shift. For ⁸⁷Rb BEC, considering the condensate’s temperature T = 0.01µK, atomic cloud density ρ = 10¹⁴cm⁻³, condensate’s density ρ_c = 0.9ρ, the strength of the SO coupling γP = 9 × 10⁻⁴Γ, the frequency shift will reach several hundreds kHz.

Fig. 3 Variations of the real and imaginary parts of the susceptibility with SO coupling. (a) The susceptibility of the system with the change of the detuning δ_p of the probe laser when the SO coupling was included. The solid (blue) line is the real part of the susceptibility and dashed (red) line is the imaginary part of the susceptibility. (b) The imaginary part of the susceptibility of the system vs the detuning δ_p of the probe laser. The solid (blue) line is the imaginary part of the susceptibility when there is the SO coupling and dashed (red) line is the imaginary part of the susceptibility without the SO coupling. There was an additional red shift due to the SO coupling. (c) The real part of the susceptibility of the system vs the detuning δ_p of the probe laser. The solid (blue) line is the real part of the susceptibility with the SO coupling and the dashed (red) line is the real part of the susceptibility without the SO coupling. It is evident that the line has a red shift. The parameters which we used are: ρ = 10¹⁴cm⁻³, T_c = 0.01µK, ρ_c = 0.9ρ, δ_c = 5.0Γ, Ω_c = 1.0Γ, ρ_cU bb = 0.9Γ, and γP = 0.015Γ.

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In order to further check the influence of the SO coupling in detail, we have also calculated the variations of the real and imaginary parts of the susceptibility with the change of the strength of the SO coupling and the detuning of the probe laser to see how the SO coupling affects the EIT. The results are shown in Fig. 4. We see not only the real part but also the imaginary part depend on the SO coupling linearly. Because the SO coupling is usually weak, therefore the frequency shift of the transparent point is small. Although the frequency shift is relative small, the EIT spectroscopy is sensitive enough to detect the frequency shift by using the techniques of the frequency standard [4,56]. Therefore by exploiting the techniques of the atomic frequency standard, we may measure the strength of the SO coupling and detect the effects of the SO coupling. For example, the procedure of the detection of the SO coupling can be as following. First after the BEC has been obtained, a coupling and a probe laser lights are used to the BEC to obtain the spectral line of the EIT without the SO coupling. Then the coupling and probe lights are turned off and pulsed magnetic fields are used as stated above to generate the SO coupling in the condensate. Finally the coupling and the probe lasers are turned on again and applied to the SO coupled BEC to obtain the EIT spectroscopy. By comparing the two EIT spectroscopies, the SO coupling can be detected. By comparing the EIT spectroscopies produced with different SO coupling strengthes, the strength of the SO coupling can also be measured. Usually the detection of the SO coupling is performed by using the time of flight (TOF) signal [39]. The TOF method is a destructive measurement and this method can not measure a SO coupled BEC repeatedly. But by using EIT the detection and measurement of the strength of the SO coupling can be carried out nondestructively and repeatedly.

Fig. 4 Variations of the real and imaginary parts of the susceptibility with the strength of the SO coupling and the detuning of the probe laser light. The real and imaginary parts of the susceptibility with the changes of the strength of the SO coupling and the detuning δ_p of the probe laser. The results show that the transparent window and the line of the imaginary part of the susceptibility are shifted linearly with the strength of the SO coupling. (a) The real part of the susceptibility with the strength of the SO coupling and detuning δ_p of the probe laser. (b) The imaginary part of the susceptibility with the strength of the SO coupling and the detuning δ_p of the probe laser. All the parameters are the same as those of the figure 2, but γP changes from 0.015Γ to 0.135Γ.

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4. Conclusion

The EIT has been investigated in the new SO coupled BEC medium, which is interesting in two aspects: some new EIT phenomena and applications. Compared with the EIT in usual BEC media, both the analytical expression and the numerical calculations have shown that the real and imaginary parts of the susceptibility have an additional red frequency shift which is linearly proportional to the strength of the SO coupling. This red frequency shift can be easily detected by using the sensitive EIT spectroscopy. As an alternative nondestructive method, its EIT spectral lines can be used to accurately measure the strength of the SO coupling in an in-situ approach and to detect some transient phenomena in the SO coupled BEC media precisely and repeatedly.

An interesting application is that SO coupling can be used as a switch to control the laser light propagation in a SO coupled BEC which has been shown in the properties of the susceptibility. For example, if initially the medium is transparent for the probe laser light, the medium can absorb the probe laser light by changing the strength of the SO coupling. This switch parameter is dependent on the properties of both atomic medium and magnetic field. All of those properties will be studied in detail in the future.

Funding

State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University (KF201707); National Natural Science Foundation of China (NSFC) (91536220, 11674312, 11174081).

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Electromagnetically induced transparency in a spin-orbit coupled Bose-Einstein condensate

Abstract

1. Introduction

2. Formulism of system

3. Numerical results of susceptibility

4. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (15)

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