Decay Rate Measurement of the First Vibrationally Excited State of ${\mathrm{MgH}}^{\mathbf{+}}$ in a Cryogenic Paul Trap

We present a method to measure the decay rate of the first excited vibrational state of polar molecular ions that are part of a Coulomb crystal in a cryogenic linear Paul trap. Specifically, we have monitored the decay of the $|\nu =1,J=1{⟩}_X$ towards the $|\nu =0,J=0{⟩}_X$ level in ${\mathrm{MgH}}^{+}$ by saturated laser excitation of the $|\nu =0,J=2{⟩}_X\mathrm{\text{−}}|\nu =1,J=1{⟩}_X$ transition followed by state selective resonance enhanced two-photon dissociation out of the $|\nu =0,J=2{⟩}_X$ level. The experimentally observed rate of $6.32(0.69){\mathrm{s}}^{-1}$ is in excellent agreement with the theory value of $6.13(0.03){\mathrm{s}}^{-1}$ (this Letter). The technique enables the determination of decay rates, and thus absorption strengths, with an accuracy at the few percent level.

Figure 1

(a) Level scheme of ${\mathrm{MgH}}^{+}$ depicting the lowest relevant rovibrational levels $|\nu ,J{⟩}_X$ in the $X^1{\Sigma }^{+}$ electronic potential. States $|{\nu }^{\prime },J^{\prime }{⟩}_X$ are coupled with rate constants ${\Gamma }_{{\nu }^{\prime },J^{\prime }}^{\nu ,J}$. In the decay rate measurement, a two-step REMPD process state-selectively dissociates molecules in the $|\nu =0,J=2{⟩}_X$ level via $|\nu =0,J=1{⟩}_A$ in the $A^1{\Sigma }^{+}$ potential, and a quantum cascade laser (QCL) couples the $|\nu =0,J=2{⟩}_X$ and $|\nu =1,J=1{⟩}_X$ levels. Note that the energy scale is discontinuous. (b) Simplified effective decay rate scheme when assuming ${\Gamma }_{0,2}^{0,J^{\prime }}$, ${\Gamma }_{0,J^{\prime }}^{0,2}\ll {\Gamma }_{0,0}^{1,1}$ and ${\Gamma }_{0,2}^{1,1}$, ${\Gamma }_{0,0}^{1,1}\ll {\Omega }_{\mathrm{QCL}}$, where ${\Omega }_{\mathrm{QCL}}$ is the Rabi frequency induced by the QCL (see text for more details).

Figure 2

(a) Sketch of the setup: Cryogenic Paul trap electrodes (yellow), parts of the $\sim 5\mathrm{K}$ inner and $\sim 35\mathrm{K}$ outer shields (gray), laser beams for dissociation (purple), photoionization (lavender), and laser cooling at 280 nm (dark green), the QCL beam (red), the Mg source (black spheres), and the ${\mathrm{H}}_2$ source (green spheres). (b) Front view of the trap, including the imaging system consisting of a plano-convex spherical lens ($f=40\mathrm{mm}$), and an image intensified CCD (charge-coupled device) camera ($3.2\times \mathrm{\text{magnification}}$) mounted below the trap. (c) Projection images of Coulomb crystals during an experimental sequence: I, pure ${\mathrm{Mg}}^{+}$ crystal consisting of 6630(200) ions (after photoionization loading). II, crystal composed of 1250(100) fluorescing ${\mathrm{Mg}}^{+}$ ions and 5200(200) ${\mathrm{MgH}}^{+}$ non-fluorescent ions (after reactions). III, after a pair of REMPD pulses addressing the P(2) transition of the resonant step was applied. The dashed-dotted ellipses indicate the outer boundary of the initial ion Coulomb crystal, while the dotted one indicates the reduced sized crystal after a set of REMPD pulses have been applied. All images correspond to a CCD exposure time of 500 ms.

Figure 3

Main figure: Refilling dynamics of the population of the $|\nu =0,J=2{⟩}_X$ level obtained by measuring the resonantly dissociated molecular fraction after initial depletion versus time. The red curve represents an exponential fit to the data with rate $0.088(11){\mathrm{s}}^{-1}$. Inset: Measured equilibrium rotational state distribution depicted by red bars ($1\sigma$ standard deviations shown). The black bars display a (36 K) thermal distribution for comparison.

Figure 4

Main figure: Relative population of the $|\nu =0,J=2{⟩}_X$ level of the ${\mathrm{MgH}}^{+}$ molecules after application of the QCL for a time $t_{\mathrm{QCL}}$. The solid red curve represents an exponential fit to the data points (black squares). Inset: Close-up of the data for the first 20 ms, including an exponential fit to this data selection (blue dotted line).