Atom position measurement in a four-level Lambda-shaped scheme with twofold lower-levels

Abstract

The two-dimensional position dependent probe absorption spectrum of a driven four-level Λ-type atomic system with twofold lower-levels interacting with two orthogonal standing wave laser fields and one microwave field is investigated. It is found that due to the position dependent nature of atom–field interaction, the spatial distribution of the atom can be controlled by scanning the resulting absorption spectra of the weak probe field. The effect of different controlling parameters of the system such as detunings, the intensity of standing wave fields as well as the relative phase of applied fields on the position measurement uncertainty of the atom is then discussed. We show that by properly adjusting the system parameters, different spatial structures of localization as ‘∞’-like, spot-like, deltoid-like and elliptic like patterns can be designed, so that high-precision and high-resolution atom localization can be engineered.

Appendix

Taking into account only the strong driven fields, the effective Hamiltonian in an interaction picture can be expressed as (Sahrai et al. 2005; Kapale and Zubairy 2006)

$$ H = - \hbar \left[ {\begin{array}{*{20}c} { - \Delta_{m} } & {\Omega_{m} e^{i\phi } } & {\Omega_{d} } \\ {\Omega_{m} e^{ - i\phi } } & { - \Delta_{c} } & {\Omega_{c} } \\ {\Omega_{d} } & {\Omega_{c} } & 0 \\ \end{array} } \right] $$

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which is in the basis {$\left| c \right\rangle$, $\left| d \right\rangle$, $\left| b \right\rangle$}. The above equation leads to the secular equation

$$ \begin{aligned} f(\lambda ) & = \lambda^{3} + (\Delta_{c} + \Delta_{m} )\lambda^{2} + (\Delta_{c} \Delta_{m} - (\Omega_{c}^{2} + \Omega_{d}^{2} + \Omega_{m}^{2} ))\lambda \\ & \quad + 2\Omega_{c} \Omega_{d} \Omega_{m} \cos \phi - \Delta_{c} \Omega_{d}^{2} - \Delta_{m} \Omega_{c}^{2} , \\ \end{aligned} $$

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with λ being the eigenenergies of the Hamiltonian. The positions of dressed sublevels $\left| {\lambda}_{1} \right\rangle$, $\left| {\lambda}_{2} \right\rangle$ and $\left| {\lambda}_{3} \right\rangle$ generated by the driven fields can be obtained by the dressed state eigenvalues λ. Probing $\left| a \right\rangle$ ↔ $\left| {\lambda}_{i} \right\rangle$(i = 1, 2, 3) by the weak probe field, the resonances will occur at the points where the probe frequency matches the energy-level difference between the levels $\left| a \right\rangle$ and $\left| {\lambda}_{i} \right\rangle$. If the probe detuning Δ _p is chosen to be in resonance with one of the dressed states then it experiences absorption maxima.

Equation 14 shows that the eigenvalues of the three dressed sublevels are dependent on the relative phase ϕ through term 2Ω _c Ω _d Ω _m  cos ϕ. Here, for simplicity, we take Δ _c  = Δ _m  = 0 and Ω _c  = Ω _d  = Ω _m  = Ω and then, Eq. 14 reads

$$ f(\lambda ) = \lambda^{3} - 3\Omega^{2} \lambda + 2\Omega^{3} \cos \phi . $$

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As a result, for example, when ϕ = 2 mπ, the eigenvalues become λ ₁ = λ ₂ = Ω and λ ₃ = −2Ω, while for ϕ = (2 m + 1)π, they are λ ₁ = 2Ω and λ ₂ = λ ₃ = −Ω. Moreover, for $\phi = \left( {m + \frac{1}{2}} \right)\pi$, the eigenvalues are $\lambda_{1} = \sqrt 3 \Omega ,\lambda_{2} = 0,\lambda_{3} = - \sqrt 3 \Omega$ (m integer). Consequently, different contribution of three bare-state levels to the dressed states can lead to different localization patterns and precision of an atom.