Adiabatic passage of $^{205}$TlF with microwaves in a cryogenic beam

TL;DR

This paper demonstrates microwave-driven adiabatic passage (AP) in TlF molecules within a spatially varying electric field to coherently transfer population between selected hyperfine sublevels of the ground state. By implementing two sequential AP steps—$|J=0,m_J=0 angle o |J=1,m_J=0 angle$ and $|J=1,m_J=0 angle o |J=2,m_J=0 angle$—in a cryogenic TlF beam, the authors achieve a total population transfer efficiency of $epsilon_{02}=0.97(8)$. The measured AP1 and AP2 efficiencies are $epsilon_{01}=0.92(6)$ and $epsilon_{12}=1.05(5)$, respectively, with careful management of stray microwave leakage and detuning to maximize fidelity. The results validate high-fidelity, robust state preparation essential for CeNTREX’s nuclear time-reversal symmetry-violation search, and indicate minimal dependence of transfer efficiency on the nuclear spin manifold when switching between $S=0$ and $S=1$ states. Overall, the work provides a practical, scalable method for preparing specific TlF rotational/hyperfine states in a cryogenic molecular beam, advancing the experimental capabilities for precision Schiff-moment measurements.

Abstract

We present a hyperfine-resolved state preparation scheme for thallium fluoride (TlF) molecules based on microwave-driven adiabatic passage (AP) in a spatially varying electric field. This method enables efficient and robust population transfer between selected $\left|J,m_J=0\right\rangle$ hyperfine sublevels of the $X\,^1Σ^+_0$ ground state in a cryogenic molecular beam, a key requirement for the CeNTREX search for nuclear time-reversal symmetry violation. Two sequential stages of AP are implemented. The first transfers population from $J=0$ to $J=1$ at a local field of $173~\mathrm{V/cm}$, and the second transfers from $J=1$ to $J=2$ at $110~\mathrm{V/cm}$. Transfer efficiencies are quantified through laser-induced fluorescence, and accounting for residual population in excited rotational levels after a prior stage of rotational cooling. We achieve state transfer efficiencies of $0.92(6)$ and $1.05(5)$ for the first and second states of AP, respectively. This corresponds to a total efficiency of $0.97(8)$ for population transfer from $J=0$ to $J=2$. These results demonstrate robust and high-fidelity preparation of specific rotational/hyperfine states in TlF.

Figures (7)

• Figure 1: a) A two-level system with energy splitting $\omega_0$, where the states are coupled by an oscillating field with frequency $\omega(t)$ and a Rabi rate $\Omega(t)/2$. In the rotating frame, the Hamiltonian simplifies to the form shown in Eqn. \ref{['eqn:LZ_hamiltonian']}, incorporating time-dependent Rabi rate $\Omega(t)$ and non-linear detuning $\Delta(t)$. Under the rotating wave approximation, this Hamiltonian also describes microwave-driven rotational transitions in TlF. b) Eigenenergies of the Landau-Zener system as the detuning $\Delta(t)$, here due to a time-varying E$_z$(t), evolves from $\Delta \to -\infty$ ($t \to -\infty$) to $\Delta \to +\infty$ ($t \to +\infty$). The ground state transitions from $\Psi(t \to -\infty) = \left|\uparrow\right\rangle$ to $\Psi(t \to +\infty) = \left|\downarrow\right\rangle$. Under adiabatic conditions ($\frac{d\Delta}{dt} \ll \Omega^2$), the system follows the instantaneous eigenstates, remaining in the ground state if initially prepared there.
• Figure 2: Schematic of the experimental apparatus. Molecules emerge from the cryogenic buffer-gas beam source and first pass through the rotational cooling region, where optical pumping and microwave transitions transfer population from the $J=1,2,3$ rotational levels to $J=0$. The molecules then enter the state preparation region, where they can be transferred to $J=1$ or $J=2$ depending on the microwave settings. Finally, the molecules are detected in the detection region by collecting laser-induced fluorescence (LIF) on a PMT. The detection laser can excite different rotational/hyperfine states, allowing determination of their relative populations under different preparation conditions. For convenience in these measurements we operate in the triplet state ($S=1$), pumping into $\left|J=0,F=1\right\rangle$ in RC. CeNTREX will operate in the singlet state ($S=0$), pumping into $\left|J=0,F=0\right\rangle$ in RC.
• Figure 3: Schematic overview of the measurement setup. A $271.75\,\mathrm{nm}$ laser, phase-modulated on the $P(2)\,\tilde{F}^\prime_1=3/2\,F^\prime=2$ transition, makes 13 passes through the rotational cooling (RC) region, where it intersects two focused Gaussian microwave beams (inset shows frontal view). An aluminum honeycomb mesh suppresses microwave leakage from RC into SPA. In the SPA chamber, two ring electrodes at $+5\,\mathrm{kV}$ and $+450\,\mathrm{V}$ generate a spatially varying electric field, nominally parallel to the molecular beam. The molecular beam then encounters two orthogonal microwave fields that drive successive adiabatic passages. A grounded plate rapidly drives the electric field to zero at the exit of the SPA region. In the detection chamber, an aperture of $13\,\mathrm{mm}\,w\times 3\,\mathrm{mm}\,h$ restricts the transverse velocity and beam spread. Populations are measured via laser-induced fluorescence (LIF) from a single $271.75\,\mathrm{nm}$ phase-modulated probe beam, collected by two opposing photomultiplier tubes (PMTs) located above and below the plane of the page.
• Figure 4: a) Applied electric field and Stark-shifted detunings of the $\left|J=0,m_J=0\right\rangle\leftrightarrow\left|J=1,m_J=0\right\rangle$ (blue) and $\left|J=1,m_J=0\right\rangle\leftrightarrow\left|J=2,m_J=0\right\rangle$ (orange) transitions ($\Delta_{01}$ and $\Delta_{02}$, respectively), as functions of the axial position $z$ in the state preparation region. The two microwave frequencies are chosen to be on resonance at the positions corresponding to the maxima of their intensities ($I_{01}$ and $I_{12}$
• Figure 5: Adiabatic evolution of $\left|J,m_J=0,S\right\rangle$ states from high field ($|E|\approx100\,\mathrm{V/cm}$) to zero field. Blue arrows indicate the $\textsc{AP1}$ ($J=0\to1$) transition, and orange arrows the $\textsc{AP2}$ ($J=1\to2$) transition. Nuclear singlet transitions are indicated with dotted lines and nuclear triplet transitions with solid lines. At zero field, the nuclear spin triplet states map to states (in the basis of fully coupled angular momenta) of the form $\left|J,F_1=J+1/2,F=J+1,m_F=-1,0,1\right\rangle$, while the singlet state maps to $\left|J,F_1=J+1/2,F=J,m_F=0\right\rangle$. Population in the highest $F$ manifold of each $J$ (green) state is detected via LIF.