The role of spectator modes in the quantum-logic spectroscopy of single trapped molecular ions

TL;DR

This work investigates how spectator motional modes influence quantum-logic spectroscopy (QLS) of a single trapped molecular ion by coupling to a state-dependent optical-dipole force. It identifies a Debye-Waller-type effect that modulates the two-ion response based on spectator-mode populations and demonstrates that cooling these spectator modes markedly improves state-detection fidelity, achieving over $99.99\%$ with nine experimental repetitions and halving the protocol duration compared with prior results. The study combines detailed cooling protocols (Doppler, EIT, and pulsed sideband cooling) with both classical and quantum simulations to capture how the Debye-Waller effect alters motional excitation and readout signals, enabling reliable rovibrational ground-state detection of N$_2^+$ and access to higher rotational states through extended lattice interaction times. The enhanced sensitivity and reduced measurement time have broad implications for molecular spectroscopy, precision metrology, and quantum information processing with trapped-ion platforms where motional degrees of freedom encode quantum information.

Abstract

Quantum-logic spectroscopy has become an increasingly important tool for the state detection and readout of trapped atomic and molecular ions which do not possess easily accessible closed-cycling optical transitions. In this approach, the internal state of the target ion is mapped onto a co-trapped auxiliary ion. This mapping is typically mediated by normal modes of motion of the two-ion Coulomb crystal in the trap. The present study investigates the role of spectator modes not directly involved in a measurement protocol relying on a state-dependent optical-dipole force. We identify a Debye-Waller-type effect that modifies the response of the two-ion string to the force. We show that cooling the spectator modes of the string allows for the detection of the rovibrational ground state of an N$_2^+$ molecular ion with a computed statistical fidelity exceeding 99.99%, improving on previous experiments by more than an order of magnitude while also halving the experimental time. This enhanced sensitivity enables the simultaneous identification of multiple rotational states with markedly weaker signals.

Figures (7)

• Figure 1: Experimental scheme. a. Schematic of the ion trap with relevant laser beams indicated. b. Partial Zeeman-resolved energy level scheme of $^{40}$Ca$^+$ with spectroscopic transitions used for EIT cooling (blue) and sideband cooling (red) indicated, together with two repumping lasers (pink). c. Typical pulsed sideband cooling sequence. Each box represents repeated pulses on a red sideband of a $(4s)^2S_{1/2}(m_j=-1/2) \rightarrow (3d)^2 D_{5/2}(m_j=-5/2)$ transition in Ca$^+$ or pulses on a $(4s)^2S_{1/2}(m_j=+1/2) \rightarrow (3d)^2D_{5/2}(m_j=-3/2)$ transition to prepare Ca$^+$ in a $S_{1/2}(m_j=-1/2)$ state. The pulses were followed by repumping with a laser beam at 854 nm. d. Typical multi-stage EIT cooling sequence. See text for details.
• Figure 2: Molecular state detection. a. Rabi oscillations on the blue sideband of a narrow $D_{5/2}(m_j=-5/2) \rightarrow S_{1/2}(m_j=-1/2)$ transition in Ca$^+$ after applying an ODF pulse of 1000 $\mu$s duration when N$_2^+$ was in the ground rotational state (orange trace) and in an excited state (violet trace). The population of the spectator mode was $\bar{n}_+\lesssim$ 0.5 phonon. The background signal (blue trace) was obtained without applying an ODF. Uncertainties represent the standard error of the mean. The black dashed lines indicate the interval of Rabi times in which the signal-to-noise ratio is highest for the molecular-state determination. The green dashed line represents a simulation of the Rabi flop. b. Rabi frequencies of the flops on the blue sideband shown in a. as a function of Fock-state quantum number $n_-$. The red and green dashed lines indicate the average Fock state of the motional wavepackets shown in c. generated by motional excitation through the ODF under different experimental conditions. See text for details.
• Figure 3: Simulations of motional excitation. Simulated average motional quantum numbers $\bar{n}_-$ of the ax-IP mode at different lattice interaction times for different ac-Stark shifts. a. No population in the spectator mode, b. with different thermal populations in the spectator mode. In the simulations, the detuning of the travelling lattice with respect to the target ax-IP frequency was set to zero, i.e. $\delta_- = 0$. The dotted lines in (a) correspond to excitation dynamics modelled with the classical model (dotted black trace) and employing a displacement operator (dotted red trace). See text for details. c. Phonon distributions of the target mode after applying an ODF corresponding to an ac-Stark shift of 16 kHz for different interaction times and no spectator mode population, and d. with a thermal population of $\bar{n}_+ =$ 8 phonons obtained after Doppler cooling in the ax-OP spectator mode. The times and colours match the coloured circles in (a) and (b). The solid lines indicate normalised Poissonian distributions expected if the motional excitations produced purely coherent motional states, as for the displacement model and as implied in the current classical treatment.
• Figure 4: Rabi thermometry after motional excitation for different spectator mode populations. Rabi oscillations on the blue sideband of the ax-IP mode on the $D_{5/2}(m_j=-5/2) \rightarrow S_{1/2}(m_j=-1/2)$ transition in Ca$^+$ after applying an ODF pulse for $t_E$ = 500 $\mu$s. The ax-IP target motional mode was initially cooled to $\sim 0.15$ phonons in all experiments. Different cooling methods were used to prepare defined average state populations in the ax-OP spectator mode, $\bar{n}_+$, and the radial modes $\bar{n}_\pm^x, \bar{n}_\pm^y$. $T_D$ indicates thermal populations after Doppler cooling of a motional mode. The green trace for ($\bar{n}_+$, $\bar{n}_{\pm}^x$, $\bar{n}_{\pm}^y$) = $(1,T_D,T_D)$ is the same in both plots. The effect of different temperatures in the radial modes is shown in a., while the influence of the ax-OP mode population is shown in b.. Uncertainties represent the standard error of the mean for 50 experimental repetitions.
• Figure 5: Comparison of simulations with experiment. Rabi oscillations on the blue ax-IP sideband of the $D_{5/2}(m_j=-5/2) \rightarrow S_{1/2}(m_j=-1/2)$ transition in Ca$^+$ after applying a state-dependent ODF on N$_2^+$ for $t_E$ = 500 $\mu$s. The measurements correspond to ax-OP spectator mode populations of a. $<$ 0.5 ph., and b. $\sim 8$ ph. The solid lines correspond to theoretical Rabi flops computed with Eq. (\ref{['eq:probability_excited_state_decoherence']}) and the motional-mode population distributions $P_{n_+,n_-}$ extracted from the simulations. Uncertainties represent the standard error of the mean. See text for details.