Coherent phase control of orbital-angular-momentum light-induced torque in a double-tripod atom-light coupling scheme

TL;DR

The paper demonstrates phase-controlled optical torque in a five-level double-tripod atomic system driven by four strong control fields and two weak OAM-carrying probe beams. By solving steady-state optical Bloch equations, it shows that the induced torque and resulting atomic rotation are highly sensitive to the relative phases, enabling reconfiguration between coupled Λ and double-Λ subsystems as φ and θ are varied. Key findings include phase-dependent torque spectra that align with EIT-like features and can be tuned via δ, with explicit cases for φ = π, φ = 0, and intermediate φ values. The work offers a versatile framework for optically steering atomic motion and currents in annular geometries, with potential applications in quantum control and information processing, and suggests experimental routes in cold 87Rb systems using structured light.

Abstract

We investigate a phase-controllable mechanism for generating optical torque in a five-level double-tripod (DT) atom-light coupling scheme interacting with four strong coherent control fields as well as two weak optical vortex probe beams carrying orbital angular momentum (OAM). The spatial phase gradients of the OAM-carrying probes induce a quantized torque that is transferred to the atoms, rotating them and generating a directed atomic flow within an annular geometry. Analytical solutions of the optical Bloch equations under steady-state conditions show that the induced torque and resulting rotational motion exhibit high sensitivity to phase variations. We show that the DT system coherently reconfigures into either coupled Λ or double-Λ schemes depending on the relative phases, with each configuration exhibiting distinct quantized torque characteristics. This enables precise phase control of the atomic current flow, with potential applications in quantum control, precision measurement, and quantum information processing.

Figures (6)

• Figure 1: (a) Schematic of the double-tripod (DT) configuration, in which three ground states $|0\rangle$, $|1\rangle$, $|2\rangle$ are coupled to two excited states $|A\rangle$, $|B\rangle$ via two probe fields $\varepsilon_{A}$ and $\varepsilon_{B}$, along with four strong control fields $\Omega_{A1}$, $\Omega_{A2}$, $\Omega_{B1}$, $\Omega_{B2}$. (b) Representation of two coupled $\Lambda$-systems with a common effective Rabi frequency $\sqrt{2}\Omega$. This model is equivalent to the DT system, when all four coupling fields have equal amplitude $\Omega$, the relative phase is $\phi=\pi$, and the additional phase is $\theta=0$. The two-photon detuning $\delta$ establishes a coupling between the two effective $\Lambda$-systems. (c) Equivalent transition diagram for the DT system at $\phi=0$ and $\theta=0$, where the structure combines a double-$\Lambda$ system with a two-ground-state system. The coupling between the subsystems is mediated by ground-state coherences.
• Figure 2: The induced torque function $\tau$ as a function of probe detuning $\Delta$ at nonzero two-photon detuning. Subplots (a), (b), and (c) correspond to $\delta=\Gamma,2\Gamma,$ and $3\Gamma$, while subplots (d), (e), and (f) correspond to $\delta=-\Gamma,-2\Gamma,$ and $-3\Gamma$. The strength of probe fields are $|\Omega_{A0}|=|\Omega_{B0}|=0.1\Gamma$. The amplitudes of the individual control fields are $|\Omega_{A1}|=|\Omega_{A2}|=|\Omega_{B1}|=|\Omega_{B2}|=|\Omega|=\Gamma$. The phases of the individual control fields are set as $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(\pi,0,0,0)$ leading to $\phi=\pi$ and $\theta=0$, which simplifies the DT system to two coupled $\Lambda$ systems. All the relevant parameters are scaled with $\Gamma$.
• Figure 3: The induced torque function $\tau$ as a function of probe detuning $\Delta$ at nonzero two-photon detuning ($\delta=\Gamma$) and for different values of additional phase $\theta$. The strength of probe fields are $|\Omega_{A0}|=|\Omega_{B0}|=0.1\Gamma$. The amplitudes of the individual control fields are $|\Omega_{A1}|=|\Omega_{A2}|=|\Omega_{B1}|=|\Omega_{B2}|=|\Omega|=\Gamma$. The phases of the individual control fields are set as $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(\frac{\pi}{2},0,0,\frac{\pi}{2})$ leading to $\phi=\pi$ and $\theta=-\pi/2$ (a), $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(\frac{\pi}{6},0,0,\frac{5\pi}{6})$ leading to $\phi=\pi$ and $\theta=-5\pi/6$ (b), $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(\frac{\pi}{3},0,0,\frac{2\pi}{3})$ leading to $\phi=\pi$ and $\theta=-2\pi/3$ (c), and $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(\frac{5\pi}{6},0,0,\frac{\pi}{6})$ leading to $\phi=\pi$ and $\theta=-\pi/6$ (d).
• Figure 4: The induced torque function $\tau$ as a function of probe detuning $\Delta$ at nonzero two-photon detuning ($\delta=\Gamma$). The strength of probe fields are $|\Omega_{A0}|=|\Omega_{B0}|=0.1\Gamma$. The amplitudes of the individual control fields are $|\Omega_{A1}|=|\Omega_{A2}|=|\Omega_{B1}|=|\Omega_{B2}|=|\Omega|=\Gamma$. Subplots (a), (b), (c) and (d) correspond to $\delta=\Gamma,2\Gamma,3\Gamma$ and $4\Gamma$. The phases of the individual control fields are set as $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})=(0,0,0,0)$ leading to $\phi=\theta=0$, which simplifies the DT system to a DL system combined with two ground levels with no light (degenerate DT).
• Figure 5: The induced torque function $\tau$ as a function of probe detuning $\Delta$, evaluated at nonzero two-photon detuning $\delta=\Gamma$ for different values of the relative phase $\phi$. The strengths of the probe fields are $|\Omega_{A0}|=|\Omega_{B0}|=0.1\Gamma$, and the amplitudes of the individual control fields are $|\Omega_{A1}|=|\Omega_{A2}|=|\Omega_{B1}|=|\Omega_{B2}|=|\Omega|=\Gamma$. The phases of the control fields $(\phi_{A1},\phi_{B1},\phi_{A2},\phi_{B2})$ are set as follows: (a) $(\frac{\pi}{6},0,0,0)$, (b) $(\frac{\pi}{4},0,0,0)$, (c) $(\frac{\pi}{2},0,0,0)$, and (d) $(\frac{5\pi}{6},0,0,0)$, corresponding to different values of $\phi=\phi_{A1}$ with $\theta=0$.