Efficient transfer of population between rotational levels in deuterated ammonia using THz transitions

Abstract

We demonstrate the efficient transfer of population between rotational levels of ${\mathrm{ND}}_3$ using a commercially available table-top THz source. We use a Stark decelerator to prepare a packet of para-${\mathrm{ND}}_3$ molecules in the $1_1^{-}$ upper inversion component of the rotational ground state, and induce the $2_1^{+}\leftarrow 1_1^{-}$ electric dipole allowed transition both in the presence of an electric field and under field free conditions. Using THz chirps to address all hyperfine components, transfer efficiencies up to 87% are reached, limited by the power of the THz source. Population transfer from the $1_1^{-}$ to the $2_1^{-}$ level is demonstrated either by driving directly the $2_1^{-}\leftarrow 1_1^{-}$ transition in a field, or by a $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ double-resonance excitation scheme. In addition, we present a novel method to produce packets of ${\mathrm{ND}}_3$ ($1_1^{-}$) with pure $M_J=0$ or $|M_J|=1$ character using the Stark decelerator, and a method to probe the $M_J$ composition of a packet of molecules using THz excitation.

GRAPHICAL ABSTRACT

Keywords:

• Molecular beam
• spectroscopy
• microwave
• Stark effect
• Stark decelerator

1. Introduction

The method of using coherent radiation to efficiently transfer population by employing the principle of adiabatic rapid passage (ARP) is well established. Some recent examples include the use of sub-THz radiation to pump significant amounts of population between different rotational levels of CO molecules on a chip [1], the transfer between vibrational levels of CH${}_4$ in molecule-surface scattering experiments [2], the transfer between different electronic states of trapped barium ions using a laser at $1.76\text{μ}\mathrm{m}$ [3], and the use of microwave radiation in electron spin resonance experiments [4]. ARP is a fundamental tool in quantum simulations to manipulate the state of qubits (see e.g. [5, 6]) and theoretical investigations are still carried out to determine best performance schemes under various conditions [7].

In a previous article we demonstrated the efficient transfer of population from the upper $(-)$ to the lower $(+)$ inversion level within the rotational ground state $J_K^p=1_1^{\pm }$ of para-${\mathrm{ND}}_3$ using ARP employing microwave chirps at frequencies near 1.6 GHz [8]. Despite the presence of a complicated 144 state hyperfine structure, population transfer with near unit efficiencies were achieved.

Here, we demonstrate the efficient transfer of population between rotational levels of ${\mathrm{ND}}_3$ using ARP and direct excitation with a commercially available THz source, similar to earlier work on driving pure rotational transitions in CO($a{}^3\mathrm{\Pi }$) [1]. We produce a sample of ${\mathrm{ND}}_3$ molecules in the $1_1^{-}$ upper inversion doublet component using a Stark decelerator, and induce the $2_1^{+}\leftarrow 1_1^{-}$ transition in ${\mathrm{ND}}_3$ at frequencies near 615 GHz. Despite the complicated hyperfine structure of ${\mathrm{ND}}_3$, up to 87% of the initial population can be transferred into the $2_1^{+}$ lower inversion component of the rotationally excited state, limited only by the power of our present THz source. In the presence of an electric field of a few kV/cm, we populate the $2_1^{+}$ level in specific $M_J$ levels by driving $M_J$-resolved Stark-split transitions. We furthermore investigate routes to transfer population from the $1_1^{-}$ state to specific $M_J$ levels within the $2_1^{-}$ upper inversion component, either by direct excitation using the $2_1^{-}\leftarrow 1_1^{-}$ transition in an electric field, or by a $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ double resonance microwave-THz excitation scheme. Finally, we present novel methods to produce samples of ${\mathrm{ND}}_3$ ($1_1^{-}$) molecules with pure $M_J=0$ or $|M_J|=1$ character using the Stark decelerator, and describe methods to directly probe the $M_J$ purity of a sample of ${\mathrm{ND}}_3$ molecules using THz excitations.

Our motivation in investigating these possibilities is mainly given by future low-energy crossed beam collision experiments using ${\mathrm{ND}}_3$ (or other molecules with a similar energy level structure). The ${\mathrm{ND}}_3$ molecule has been important in experiments using cold molecules for more than two decades [9–13]. Interest in this species mainly stems from its ease in production and detection, its relatively large electric dipole moment, and its favourable energy level scheme featuring near-resonant rotational levels of opposite parity. The ability to prepare samples of molecules in (specific $M_J$ components of) the $2_1^{\pm }$ rotationally excited level before the collision, in combination with the ability to probe collision products $M_J$ selectively, offers exciting prospects for detailed collision studies at low collision energies. First, molecular beams with large populations in $2_1^{\pm }$ can be used in collisional de-excitation experiments. The pre-collision added quantum of angular momentum causes the partial wave(s) of the entrance channels to change during the collision. This partial wave dynamics is encoded in the differential cross-section (DCS), which can be probed using velocity map imaging (VMI) [14]. Second, the 20 cm${}^{-1}$ recoil energy of the $2_1\to 1_1$ de-excitation process increases the diameter of the scattering image. In current high-resolution experiments, this 20 cm${}^{-1}$ recoil energy results in a scattering image with a sufficiently large radius to discern structure in velocity-mapped scattering images, even in the limit of near-zero collision energies [14]. Third, the THz tagging pulse can be used to set a ‘time zero’ for the collisions, which is particularly important in merged beam configurations in which beams can overlap for extended periods of time [15]. The ability to prepare either the $2_1^{+}$ or the $2_1^{-}$ state as the initial level gives access to probing both parity-changing and parity-conserving rotational de-excitation collisions, which are governed by different parts of the interaction potential. Last but not least, there currently is large interest to study low-energy collisions in the presence of electric fields [16]. At sufficiently low collision energies, the interaction energy with an external field can become larger than the collision energy itself, offering distinctive possibilities to actively manipulate interaction Hamiltonians and to steer collision cross sections. In an electric field, cross sections depend on the $M_J$ quantum numbers of the initial and final rotational level. The ability to record fully state-resolved $M_J\to M_J^{\mathrm{\prime }}$ cross sections is essential to study field effects in low-energy collisions in the required detail, and can reveal new aspects relating to the stereodynamics of the collision [17]. The study and understanding of fully $M_J$-resolved collisions is also urgently needed to advance experiments that aim to confine (and further cool) ultracold molecules in traps, as inelastic $M_J$-changing collisions could lead to trap loss.

This article is organised as follows: after describing the experimental methods in Section 2, we will describe in Section 3.1 investigations on the general performance of the THz source with respect to resolution and ARP transfer using the dipole-allowed $2_1^{+}\leftarrow 1_1^{-}$ transition in zero electric field. For measurements inside an electric field, we probe the homogeneity of the field of a capacitor by driving the $1_1^{-}\to 1_1^{+}$ inversion doublet transition using microwaves at various field strengths (see Section 3.2). In Section 3.3, we will then add a bias electric field and investigate the $M_J$-resolved $2_1^{+}\leftarrow 1_1^{-}$ transition. In Sections 3.4 and 3.5 we will discuss the possibilities to transfer ${\mathrm{ND}}_3$ molecules from the $1_1^{-}$ to the $2_1^{-}$ state, either directly inside an electric field or by a double resonance $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ scheme that combines microwaves and THz capabilities. Finally, in Section 3.6 we will discuss how beams with pure $M_J$ composition can be made using the Stark decelerator, and how rotational transitions using THz radiation can be used as an easy-to-implement tool to probe the $M_J$-resolved beam purity.

Throughout the different sections, an understanding of the Stark shift of the involved levels of ${\mathrm{ND}}_3$ is required. The Stark effect can be calculated from the total Hamiltonian matrix representation (1) ${\mathbf{H}}_{\mathrm{total}}={\mathbf{H}}_{\mathrm{vibrot}}-\mathit{\mu }E,$(1) which was obtained from the rovibrational Hamiltonian ${\mathbf{H}}_{\mathrm{vibrot}}$ and the transition dipole matrix $\mathit{\mu }$ as calculated using Pickett's SPCAT [18]. The results for ${}^{14}{\mathrm{ND}}_3$ are shown in Figure 1. The required rovibrational Hamiltonian was derived from semi-rigid rotor parameters as provided on the Cologne database for molecular spectroscopy (CDMS) [19–21] and spin-spin, spin-rotation and quadrupole coupling parameters as provided in [22, 23]. Throughout the Figures in this article we will use a colour convention depending on which states are probed: Blue: $1_1^{+}$, Cyan: $2_1^{+}$, Orange: $1_1^{-}$, Red: $2_1^{-}$. We note that although the scope of our experiments is primarily focussed on the rotational structure and associated quantum numbers J and $M_J$, the hyperfine structure of ${\mathrm{ND}}_3$ is partially resolved in our experiments. A detailed account of the Stark effect including hyperfine structure is given by van Veldhoven et al. [22].

Figure 1. Stark curves for the $1_1^{\pm },M_J$ (lower panel) and $2_1^{\pm },M_J$ (upper panel) states of ${}^{14}{\mathrm{ND}}_3$.

2. Experimental

The experimental setup is schematically shown in Figure 2, and is operated at a repetition rate of $10\mathrm{Hz}$. Details on the Stark decelerator and detector are described elsewhere [24, 25]. A gas mixture of 5% ${\mathrm{ND}}_3$ in Xenon was expanded through a Nijmegen Pulsed Valve [26] at typical backing pressures of $1\mathrm{bar}$ into a vacuum, creating a rotationally cold molecular beam. The molecular beam was skimmed (skimmer radius $1.5\mathrm{mm}$) and passed through a Stark decelerator, operating at $\pm 15\mathrm{kV}$ in the s = 3 guiding mode around $400\mathrm{m}/\mathrm{s}$. The ${\mathrm{ND}}_3$ molecules that exit the decelerator almost exclusively populate the $1_1^{-}$ upper inversion level of the rotational ground state of para-${\mathrm{ND}}_3$ [8]. The detection region was placed $448\mathrm{mm}$ downstream from the decelerator and consists of a VMI spectrometer utilising a state-selective 2 + 1 REMPI scheme via the v = 4 and v = 5 levels of the excited B electronic state [27]. Signal was recorded either as voltage change, picked up on the microchannel plate (MCP) and passed to a Pico-Scope 5444D oscilloscope or visually, using a camera recording the light flashes of a phosphor screen.

Figure 2. Scaled down representation of the experimental setup showing the last section of the Stark decelerator (S), capacitor plates (C) to generate an electric field, microwave wire antenna (M), THz horn antenna (T), and velocity map imaging electrodes with laser used for resonance-enhanced multiphoton ionisation (REMPI-VMI).

In the region between the end of the decelerator and laser detection point, transfer of ${\mathrm{ND}}_3$ molecules from the $1_1^{-}$ to either the $1_1^{+}$, $2_1^{-}$ or $2_1^{+}$ can be induced using microwave or THz radiation. To study these transitions in an electric field, a $85\mathrm{mm}$ long capacitor is placed $27\mathrm{mm}$ downstream from the decelerator. This capacitor consisted of two aluminum plates separated by $10\mathrm{mm}$, with one operated at voltages up to $5\mathrm{kV}$ and the other grounded. Microwaves were provided using a Rohde & Schwarz SMA103B signal generator with chirp functionality, as described in our recent paper [8]. The signal generator was connected to a custom made $\lambda /4$ monopole antenna, positioned near the exit of the Stark decelerator. The resulting microwave radiation was not directional, effectively creating microwave fields in the entire vacuum chamber.

THz radiation was generated according to the circuit shown in Figure 3. The $10\mathrm{GHz}$ output of a National Instruments QuickSyn Signal generator at $15\mathrm{dBm}$, was passed as ‘local oscillator’ (LO) into a mixer (Mini-Circuits ZX05-153MH-S+). The LO signal was mixed with an ‘intermediate frequency’ (IF) provided by a second SMA103B, which was varied around $2.8\mathrm{GHz}$ and passed through a $-10\mathrm{dB}$ attenuator (to limit the possible power input) before entering the mixer. The mixer creates a ‘radio frequency’ (RF) signal composed of the difference signal at $7.2\mathrm{GHz}$ and the sum signal at $12.8\mathrm{GHz}$. The mixed signal was then passed through a bandpass filter for $11.7-14.5\mathrm{GHz}$ (Mini-Circuits ZVBP-13R1G-S+) for signal purification and the elimination of one side band. Keeping the QuickSyn generator running at all time, the control over output and pulse characteristics – such as chirp parameters – in this setup was provided by the SMA103B. The signal – typically around $12.8\mathrm{GHz}$ – was then passed into a VDI (Virginia Diodes) amplifier multiplier chain (VDI, Model WR1.5AMC-615-5mWp) providing a multiplication factor of 48 and a typical output power of $5\mathrm{mW}$ in a range from 605 to $625\mathrm{GHz}$.

Figure 3. Circuit diagram for the generation of THz radiation around $615\mathrm{GHz}$. Signals from a QuickSyn and SMA103B generator are mixed, passed through a bandpass filter, multiplied by a factor of 48 using a VDI chain, and are transmitted using a horn antenna.

The THz radiation was highly directional and passed either through fused silica viewports (on CF100 or CF40 flanges with a window thickness of $6.35\mathrm{mm}$ and $4.04\mathrm{mm}$ and a transmission of 50% and 72%, respectively) or through a special polymer (ZEONEX, obtained from TYDEX) window (CF40 – window thickness $4.04\mathrm{mm}$, transmission of 88%). Depending on the experiment a polymer (TPX) collimator lens with a focal length of $10\mathrm{cm}$ was used (THORLABS TPX100). The THz horn antenna was positioned such that the THz beam intercepted the ${\mathrm{ND}}_3$ molecules within the capacitor, see Figure 2.

3. Results and discussion

3.1. $2_1^{+}\leftarrow 1_1^{-}$ THz excitation in zero field

To test the resolution of our THz source and its ability to drive pure rotational transitions in ${\mathrm{ND}}_3$, we positioned the THz source relatively far away from the chamber (about $70\mathrm{cm}$) and worked with unfocused radiation passing over a mirror and through a 100CF fused silica viewport into the machine (transmission 50%). Single frequency pulses of $30\text{μ}\mathrm{s}$ duration were used, while the population in the $2_1^{+}$ state was probed using the laser. The recorded experimental THz spectrum is shown in the left panel of Figure 4, together with a simulation of the spectrum based on a low power approximation. The intensity in the spectrum is proportional to the population transferred into the probed $2_1^{+}$ state. The simulations were performed using the codes as described in Ref. [8]. Intensities were predicted using Pickett's SPCAT [18] program, while the linewidth (Full Width at Half Maximum (FWHM) $=21\mathrm{kHz}$) was determined from the pulse duration $t=30\text{μ}\mathrm{s}$ of a rectangular pulse, using the relation FWHM $=1/(1.568\mathrm{t}$). As our experiment suffers from inaccuracies in absolute frequency due to the absence of a good frequency standard, the experimental spectrum was shifted by $30\mathrm{kHz}$ to match the simulated spectrum. It is seen that the experimental spectrum is well reproduced by the simulations, and the hyperfine structure pertaining to the $F_1$ quantum number is fully resolved. Note that all hyperfine levels that correlate to the $M_J=0$ and $|M_J|=1$ quantum numbers are populated. This is a consequence of the scrambling of $M_J$ states in the zero field of the free flight section between the exit of the decelerator and interaction region, as discussed in detail in Ref. [8].

Figure 4. Left: Hyperfine resolved spectrum for the $2_1^{+}\leftarrow 1_1^{-}$ rotational transition in ${\mathrm{ND}}_3$ using single frequency pulses of $30\text{μ}\mathrm{s}$ duration. The $F_1^{\prime }\leftarrow F_1^{″}$ quantum numbers referring to coupled rotational angular momentum J and nitrogen spin $I_{\mathrm{N}}$ by $\overrightarrow{F_1}=\overrightarrow{J}+\overrightarrow{I_{\mathrm{N}}}$ are indicated. Right: Measured (orange solid lines) and predicted (orange dashed lines) depletion in $1_1^{-}$ population (orange) and corrsponding derived gain in $2_1^{+}$ population (cyan) throughout a $40\text{μ}\mathrm{s}$ THz chirp covering $3.6\mathrm{MHz}$ from $\mathrm{614,969.418}\mathrm{MHz}$ to $\mathrm{614,965.818}\mathrm{MHz}$.

To investigate the maximum population transfer efficiency that can be achieved, we moved the THz source closer to the window, collimated the beam using a TPX lens and radiated through a low loss ZEONEX polymer window. To transfer population from all hyperfine components, a $40\text{μ}\mathrm{s}$ chirp of $3.6\mathrm{MHz}$ bandwidth that covers the complete hyperfine structure was applied. The chirp was stopped at different timings to monitor the development of the $1_1^{-}$ population (with the $2_1^{+}$ population derived from it) and the experimental results are plotted together with predictions in the right panel of Figure 4. These predictions follow from our model that includes the full hyperfine structure of ${\mathrm{ND}}_3$ and numerical integration of the von Neumann equation as detailed in Ref. [8]. It is seen that a maximum population transfer of 87% was achieved. In the simulations, we chose the power of the THz radiation to best match our experimental observations ($3\mathrm{W}/{\mathrm{m}}^2$). However, increasing the power in the simulation would still lead to higher expected transfers (up to $97\mathrm{\%}$ at $12.8\mathrm{W}/{\mathrm{m}}^2$; data not shown). This suggests that the maximum transfer observed in our current experiments is limited by the power of the THz source, but that near-unit transfer efficiencies are inherently possible.

3.2. Calibration of electric field by $1_1^{-}\to 1_1^{+}$ microwave transfer

In our previous study our capabilities to carry out $M_J$ specific transitions were limited by working in inhomogeneous fringe fields [8]. In the present study we use a dedicated electrostatic capacitor consisting of two parallel aluminum plates to create homogeneous fields. For the long wavelengths of the microwave radiation, the capacitor plates have the additional benefit to act as a polariser, only allowing radiation polarised parallel to the direction of the static electric field. This enforces a $\mathrm{\Delta }{M}_J=0$ selection rule for microwave transitions, greatly facilitating the interpretation of the spectra.

To calibrate the strength and test the homogeneity of the electric field of the capacitor, we simulated and measured the hyperfine resolved spectrum of the $1_1^{-},|M_J|=1\to 1_1^{+},|M_J|=1$ transition of ${}^{14}{\mathrm{ND}}_3$ using a voltage difference of $1.2\mathrm{kV}$ applied to the capacitor plates. The results are shown in the left panel of Figure 5. From the spectrum it is evident that the field of the capacitor must be fairly homogeneous since the components of the hyperfine structure arising from the electric quadrupole interaction of the ${}^{14}\mathrm{N}$ nucleus can be resolved. Details arising from the deuterium nuclei are completely blurred out. The measured spectrum is reproduced well by the simulations, although the observed linewidth was only reproduced by assuming a reduced pulse duration of $9\text{μ}\mathrm{s}$ (compared to the rectangle pulse of $20\text{μ}\mathrm{s}$ as applied in the experiment). Such deviations can reflect significant inhomogeneities in the microwave field, such as a frequency dependence of the amplitude or a variation of amplitude over the distance travelled by the molecular packet. Small scale electric field inhomogeneities on the order of a few $\mathrm{V}/\mathrm{cm}$ can also cause broadening. A slightly inhomogeneous microwave field can be expected, since the antenna is placed relatively far from the capacitor, and not in line of sight of the molecules. Despite this broadening on the kHz scale, the spectrum is consistent with a homogeneous electric field of $1.2\mathrm{kV}/\mathrm{cm}$.

Figure 5. Left: Experimental (blue) and simulated (black) hyperfine resolved microwave spectrum for the $1_1^{-}\to 1_1^{+}$ transition in ${}^{14}{\mathrm{ND}}_3$ in an electric field of 1.2 kV/cm. Right: Measured (orange solid lines) and predicted (orange dashed lines) depletion in $1_1^{-}$ population and corresponding derived gain in $1_1^{+}$ population (blue) throughout a $20\text{μ}\mathrm{s}$ microwave chirp covering $5\mathrm{MHz}$ from $1820.1\mathrm{MHz}$ to $1825.1\mathrm{MHz}$.

Similar to our previous study [8], we tested the maximum population transfer to specific $M_J$ final states that can be achieved in the electric field. For this purpose we used a $5\mathrm{MHz}$ bandwidth $20\text{μ}\mathrm{s}$ microwave pulse centred at $1822.6\mathrm{MHz}$ increasing frequency with time (up direction). We stopped the microwave chirps at different timings to measure the population transfer throughout the pulse, as is illustrated in the right panel of Figure 5. The orange line represents the $1_1^{-}$ signal depletion, the blue line represents the derived increase in $1_1^{+}$ population, and the dashed lines symbolise simulated results. We measured a maximum population transfer of 93%, consistent with our earlier findings [8] for ${}^{14}{\mathrm{ND}}_3$. Similar results, also with 93% efficiency, were achieved for ${}^{15}{\mathrm{ND}}_3$. The high efficiency for the ${}^{15}{\mathrm{ND}}_3$ population transfer is in contrast to the 77% efficiency achieved previously in zero field [8]. We conclude that the preservation of pure $|M_J|$ states caused by the presence of an electrical bias field and driving transitions exclusively between the $|M_J|=1$ states is beneficial in the context of transfer efficiency.

3.3. $2_1^{+}\leftarrow 1_1^{-}$ THz excitation in electric fields

To test the ability to also drive pure rotational $M_J$-resolved transitions in the presence of an electric field with our THz source, we measured the $2_1^{+}\leftarrow 1_1^{-}$ transition as a function of the electric field strength. The THz source was operated using pulses of varying bandwidth (dependent on chosen stepsize), while the spectrum was scanned downwards from the zero field transition frequency (see Figure 4(a)). The THz source was oriented in a way that the electric field vector of the THz radiation was perpendicular to the capacitor field, resulting in a $\mathrm{\Delta }{M}_J=\pm 1$ selection rule. The obtained spectra for fields up to 3 kV/cm are shown in Figure 6. Lines are labelled according to the $M_J$ quantum numbers in the $1_1^{-}$ and $2_1^{+}$ levels, that formally only become good quantum numbers in fields above $\sim 1\mathrm{kV}/\mathrm{cm}$. This is directly visible in the oberved spectra. At fields of 2 and 3 kV/cm, only transitions with $\mathrm{\Delta }{M}_J=\pm 1$ are observed, consistent with the selection rules imposed by the geometry of the experiment. At lower fields, where $M_J$ loses its meaning and descriptions in terms of the hyperfine quantum numbers F and $M_F$ are more appropriate, transitions with $\mathrm{\Delta }{M}_J=0$ are visible that become weaker as the field is increased.

Figure 6. Stark THz spectrum of the $2_1^{+}\leftarrow 1_1^{-}$ rotational transition in ${\mathrm{ND}}_3$ as a function of electric field strength. The detuning with respect to the most intense line in the zero field hyperfine spectrum ($F_1^{\prime }=3\leftarrow F_1^{″}=2$, see left panel of Figure 4) is given. Lines are labelled according to quantum numbers $|M_J|$. Transitions that follow or violate the $\mathrm{\Delta }{M}_J=\pm 1$ selection rule are indicated in green and red, respectively. The dashed coloured lines that connect spectral features are manually drawn to guide the eye.

From the measured line intensities in comparison to the zero field spectrum, we conclude that significant population transfer into $M_J$-selected levels of the $2_1^{+}$ can be achieved using our THz source. From simulations and degeneracies of the involved levels, we expect that maximum transfer efficiencies close to unity can in principal be achieved for the $|M_J|=1\to |M_J|=1$ and $|M_J|=1\to |M_J|=2$ transitions, whereas the maximum transfer efficiency is close to 50% for $|M_J|=1\to |M_J|=0$.

3.4. Direct $2_1^{-}\leftarrow 1_1^{-}$ THz excitation in electric fields

As a second objective, we studied the possibilities to transfer population from the $1_1^{-}$ state into specific $M_J$ levels of the $2_1^{-}$ state. This is inherently more difficult than populating the $2_1^{+}$ state, as the direct $2_1^{-}\leftarrow 1_1^{-}$ transition is electric dipole forbidden in zero field. Yet, in the presence of an electric field, the inversion doublet components of both the $1_1$ and the $2_1$ levels mix, such that the direct transition becomes allowed and gains intensity.

We measured the $2_1^{-}\leftarrow 1_1^{-}$ transition of ${}^{14}{\mathrm{ND}}_3$ and ${}^{15}{\mathrm{ND}}_3$ in a field of $1.2\mathrm{kV}/\mathrm{cm}$, focussing on the $M_J=0\leftarrow |M_J|=1$ and $|M_J|=2\leftarrow |M_J|=1$ transitions. The THz source was again oriented in a way that the electric field vector of the THz radiation was perpendicular to the capacitor field, resulting in a $\mathrm{\Delta }{M}_J=\pm 1$ selection rule. A frequency chirp of 1.44 MHz bandwidth was used when scanning the frequency of the THz source. The resulting Stark shifted spectra are shown in Figure 7 for ${}^{14}{\mathrm{ND}}_3$ and ${}^{15}{\mathrm{ND}}_3$. The $|M_J|=2\leftarrow |M_J|=1$ transitions are observed at smaller detunings than the $M_J=0\leftarrow |M_J|=1$ transitions, as the Stark shifts of the low-field seeking $2_1^{-},|M_J|=2$ and $1_1^{-},|M_J|=1$ levels partially cancel. A small signature of the $|M_J|=1\leftarrow |M_J|=1$ transition is observed in the spectrum of ${}^{15}{\mathrm{ND}}_3$, just right of the $M_J=0\leftarrow |M_J|=1$ peak. The appearance of this peak is likely related to an imperfect orientation of the THz source, resulting in weak transitions with $\mathrm{\Delta }{M}_J=0$ character.

Figure 7. Stark shifted spectra of the zero field forbidden $2_1^{-}\leftarrow 1_1^{-}$ transition of (left) ${}^{14}{\mathrm{ND}}_3$ and (right) ${}^{15}{\mathrm{ND}}_3$, using $1.44\mathrm{MHz}$ chirps. Peaks are labelled by $M_J$ quantum numbers, and the maximum observed transfer efficiency is given for each transition. The region of the $M_J=0\leftarrow M_J=1$ transition in ${}^{14}{\mathrm{ND}}_3$ was recorded with additional samples to improve the signal-to-noise ratio.

Population transfer efficiencies were determined by probing the depletion of the $1_1^{-}$ population. The maximum transfer efficiencies that could be achieved for the different transitions are indicated by numbers at the respective peaks in Figure 7. It is seen that in particular the $M_J=0\leftarrow |M_J|=1$ transitions are driven with poor efficiency, and transfer efficiencies of only 6% and 17% are achieved for ${}^{14}{\mathrm{ND}}_3$ and ${}^{15}{\mathrm{ND}}_3$, respectively. Much higher efficiencies exceeding 50% are achieved for the $|M_J|=2\leftarrow |M_J|=1$ transitions.

The dramatic difference in performance for $M_J=0$ and $|M_J|=2$ transfer is rationalised by the small transition dipole moments of these types of transitions when $M_J=0$ states are involved. For the observed transitions to become allowed, the states of different parity $1_1^{\pm }$ and $2_1^{\pm }$ must each mix with their counterparts. The $M_J=0$ states do not mix in first order, such that the overlap is determined only by the mixing of the $1_1^{\pm },|M_J|=1$ states.

From Figure 7 it is clear that higher transfer efficiencies were observed for ${}^{15}{\mathrm{ND}}_3$ than for ${}^{14}{\mathrm{ND}}_3$. This is related to the smaller inversion splitting of ${}^{15}{\mathrm{ND}}_3$ compared to ${}^{14}{\mathrm{ND}}_3$ resulting in a more pronounced parity mixing of the levels in an electric field, and to the slightly lower rotational spacing around $\mathrm{614,150}\mathrm{MHz}$ for ${}^{15}{\mathrm{ND}}_3$, which is closer to the optimal output frequency of the THz source near $615\mathrm{GHz}$. However, we speculate that mostly the more narrow hyperfine structure of ${}^{15}{\mathrm{ND}}_3$ is beneficial. The optimum chirps that were found empirically were of $1.44\mathrm{MHz}$ bandwidth. Comparison with Figure 4 shows that these chirps only cover a part of the hyperfine structure for ${}^{14}{\mathrm{ND}}_3$, while for ${}^{1}5{\mathrm{ND}}_3$ the hyperfine structure is dominated by the quadrupole coupling of the deuterium nuclei and only extends over a few $100\mathrm{kHz}$.

3.5. $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ microwave-THz double resonance excitation in electric fields

One of our objectives for future experiments is to populate the $M_J=0$ component of the $2_1^{-}$ state, as this state is (almost) immune to electric fields, potentially affording advanced beam merging protocols. The low transfer efficiency observed when inducing the direct $2_1^{-}\leftarrow 1_1^{-}$ transition in a field is clearly not ideal to achieve this goal. We therefore designed an alternative double-resonance scheme to populate this level. In this scheme, first the microwave transition $1_1^{-}\to 1_1^{+}$ is induced populating the $1_1^{+},|M_J|=1$ level with 93% transfer efficiency (see Figure 5). Subsequently, the zero-field dipole-allowed $2_1^{-},M_J=0\leftarrow 1_1^{+},|M_J|=1$ transition is induced using a THz pulse. The quantum mechanical limit for the transfer efficiency of this second transition, involving transfer from a doubly degenerate $|M_J|=1$ level into a single $M_J=0$ level, is 50%. Together, the double resonance scheme should allow for a total transfer efficiency of population from the $1_1^{-}$ state into the $M_J=0$ level of the $2_1^{-}$ state of about 46%.

We investigated this double resonance scheme in a field of $1.2\mathrm{kV}/\mathrm{cm}$, using different velocities for the ${\mathrm{ND}}_3$ beam ranging between $360\mathrm{m}/\mathrm{s}$ and $450\mathrm{m}/\mathrm{s}$. As before, we used chirps of $1.44\mathrm{MHz}$ bandwidth to achieve optimal transfer. The observed Stark shifted $2_1^{-}\leftarrow 1_1^{+}$ spectrum corresponding to the second step of the double-resonance scheme is shown for ${}^{14}{\mathrm{ND}}_3$ in the left panel of Figure 8. The $2_1^{-},M_J=0\leftarrow 1_1^{+},|M_J|=1$ and the $2_1^{-},|M_J|=2\leftarrow 1_1^{+},|M_J|=1$ transitions, that are both allowed by the $\mathrm{\Delta }{M}_J=\pm 1$ selection rule, are clearly visible.

Figure 8. Left: Experimental Stark shifted THz spectrum of the $2_1^{-}\leftarrow 1_1^{+}$ rotational transition in ${}^{14}{\mathrm{ND}}_3$ in an electric field of $1.2\mathrm{kV}/\mathrm{cm}$, subsequent to a microwave pulse transferring 93% of the original $1_1^{-}$ population into the $1_1^{+}$ level (as shown in the right panel of Figure 5). Peaks are labelled with the $M_J$ quantum numbers, and the transfer efficiencies of the chirp are given for each peak (estimates flagged with ‘≈’). THz chirps of $30\text{μ}\mathrm{s}$ duration covering $1.44\mathrm{MHz}$ were used. Right: measured depletion in $1_1^{+}$ population (blue) and corresponding derived gain in $2_1^{-}$ population (red) throughout the THz chirp from $\mathrm{618,242.62}\mathrm{MHz}$ to $\mathrm{618,241.18}\mathrm{MHz}$.

From a comparison of Figures 7 and 8, it is apparent that the $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ double resonance scheme offers a (much) larger transfer efficiency to populate the $2_1^{-},M_J=0$ level than the direct zero-field dipole forbidden $2_1^{-}\leftarrow 1_1^{-}$ transition. This is quantified further by measuring the depletion of $1_1^{+}$ population while applying a $30\text{μ}\mathrm{s}$ chirp of decreasing frequency centred at $\mathrm{618,241.900}\mathrm{MHz}$ (${}^{14}{\mathrm{ND}}_3$) or $\mathrm{615,805.152}\mathrm{MHz}$ (${}^{15}{\mathrm{ND}}_3$), again using $1.44\mathrm{MHz}$ bandwidth. The recorded evolution of $1_1^{+}$ population during the THz chirp is shown in the right panel of Figure 8.

Accurate values for the maximum population depletion were acquired by applying the full chirp, and repeating the experiment 1000 times (at $10\mathrm{Hz}$ repetition) toggling between measurements with and without radiation applied. We went for this level of precision only for transitions populating the $2_1^{-},M_J=0$ level; transfer efficiencies for transitions that populate the $2_1^{-},|M_J|=2$ level were estimated from the relative peak intensities in the Stark spectrum of Figure 8, to be twice as high as the population in $M_J=0$.

We determined a transfer efficiency for the $2_1^{-},M_J=0\leftarrow 1_1^{+},|M_J|=1$ transition of 36% and 39% for ${}^{14}{\mathrm{ND}}_3$ and ${}^{15}{\mathrm{ND}}_3$, respectively. These transfer efficiencies were averaged for performance over the 360−450 m/s velocity range probed, varying only by a few % for the different velocities. For the $2_1^{-},M_J=2\leftarrow 1_1^{+},|M_J|=1$ transition, transfer efficiencies of 72% and 78% were estimated for ${}^{14}{\mathrm{ND}}_3$ and ${}^{15}{\mathrm{ND}}_3$, respectively.

To obtain the total population transfer from $1_1^{-},|M_J|=1$ to $2_1^{-},M_J=0$, the efficiencies of microwave and THz pulse have to be multiplied, yielding in total $93\mathrm{\%}\cdot 36\mathrm{\%}=33.5\mathrm{\%}$ for ${}^{14}{\mathrm{ND}}_3$ and $36.3\mathrm{\%}$ for ${}^{15}{\mathrm{ND}}_3$. Clearly, the double resonance scheme is much more effective in populating the $2_1^{-},M_J=0$ level than the direct excitation. But the scheme is also beneficial for population transfer into the $2_1^{-},|M_J|=2$ level, in particular for ${}^{14}{\mathrm{ND}}_3$, for which the achieved direct transfer from the $1_1^{-}$ to the $2_1^{-}$ state was low as seen in Figure 7. We note that transitions to the $2_1^{-},|M_J|=1$ level were not recorded but could in principle be observed if the THz source was rotated by ${90}^{\circ }$ to change the selection rules.

3.6. THz radiation as a tool to probe $M_J$-resolved rotational populations

Referring again to Figure 6, rotational THz transitions in the presence of an electric field provide the option to directly probe the $M_J$-resolved population in a rotational level. To fully exploit this possibility in future $M_J\to M_J^{\mathrm{\prime }}$ resolved collision experiments, methods to prepare molecules in a specific $M_J$ level prior to the collision are required as well. For population transfer into the $2_1^{\pm }$ levels, this is possible by driving THz transitions in an electric field, as discussed in the previous sections. Here, we present a novel method that also allows for the production of samples of ${\mathrm{ND}}_3$ molecules in the $1_1^{-}$ state with either pure $M_J=0$, pure $M_J=1$, or mixed $M_J$ character. For this, some changes were made to the setup presented in Figure 2. No microwave radiation was used, the THz source was moved to radiate at the centre of the VMI region, the capacitor plates were removed, and the distance between the Stark decelerator and detection region changed to $530\mathrm{mm}$. To generate the THz radiation, the LO frequency was changed to $10.5\mathrm{GHz}$ since an improved performance was observed for this setting. Furthermore, the gas mixture was changed to ∼5% ${\mathrm{ND}}_3$ in Neon, producing beams with a mean speed of $890\mathrm{m}/\mathrm{s}$.

Let's first discuss the possibilities to produce packets of ${\mathrm{ND}}_3$ ($1_1^{-}$) molecules with well-defined $M_J$ composition. In principle, the Stark decelerator only selects those ${\mathrm{ND}}_3$ molecules from the molecular beam pulse that reside in the $1_1^{-},|M_J|=1$ level, as this level is low-field seeking. Throughout the deceleration process, the $|M_J|=1$ projection is preserved, as the molecules experience a sufficiently large electric field strength at all times and at all positions. However, upon exiting the decelerator, preservation fields may not be present, and reprojections into the $M_J=0$ can occur. Indeed, in default operation with the Stark decelerator being switched off with subsequent free flight towards the detector, as described in [8], the ${\mathrm{ND}}_3$ molecules are distributed over the $M_J=0$ (25%) and $|M_J|=1$ (75%) levels. This non-statistical population ratio can be understood from the hyperfine levels of ${\mathrm{ND}}_3$ that are only partially populated when ${\mathrm{ND}}_3$ redistributes over the $M_J=0$ and $|M_J|=1$ levels [8].

It is relatively straightforward to keep the $|M_J|=1$ projection upon exiting the decelerator, by ensuring that the molecules keep experiencing a sufficiently strong bias field throughout their path towards the detector [28, 29]. To allow for experiments with molecules in a pure $|M_J|=1$ state, a rectangular electrode was placed parallel to the beam axis, starting $57\mathrm{mm}$ downstream from the decelerator and reaching to the entrance of the VMI detector. This electrode will be referred to as ‘preservation electrode’, and was operated at $+1\mathrm{kV}$. The preservation electrode secured a bias field towards the machine ground over the full length of the electrode, as well as in the regions where the molecules exit the decelerator or enter the VMI detector. Fringe fields in the region between the exit of the decelerator and the preservation electrode were enhanced by leaving the last electrodes of the decelerator on for elongated times.

The generation of a pure sample of molecules in $M_J=0$ is unfortunately less straightforward. We found that the Stark decelerator itself can be used to first produce a sample of ${\mathrm{ND}}_3$ ($1_1^{-}$) molecules with mixed $M_J=0$ and $|M_J|=1$ composition by switching all electrodes off when the packet has reached a position close to the exit of, but still within, the decelerator. An off-time of approximately $10\text{μ}\mathrm{s}$ was found sufficient to fully reproject the molecules onto the $M_J=0$ and $|M_J|=1$ levels. Molecules in the $|M_J|=1$ level can then subsequently be deflected from the beam axis by operating the remaining stages of the decelerator assymmetrically. This so-called ‘kicker operation’ is achieved by applying 13 kV to only a single electrode from the horizontal pairs, while keeping all other electrodes (the other horizontal electrodes and both vertical ones) at ground. The effectiveness of this kicker operation depends on the forward velocity of the ${\mathrm{ND}}_3$ packet, but we found that typically 10 stages of kicker operation in our experiment was sufficient to fully eliminate molecules in $|M_J|=1$ from the beam for speeds up to 900 m/s.

To probe the $M_J$ composition of the packet of ${\mathrm{ND}}_3$ ($1_1^{-}$) molecules, we recorded $2_1^{+}\leftarrow 1_1^{-}$ THz spectra in the presence of an electric field, similar to the spectra presented in Figure 6. For this, we used the THz source in single frequency mode. We used the repeller and extractor plates of the VMI detector to produce a field of $4.6\mathrm{kV}/\mathrm{cm}$, and crossed the THz radiation with the laser beam in the centre of the VMI detector. We then recorded Stark spectra with the REMPI laser tuned to detect population in the $2_1^{+}$ level. The voltages on the VMI electrodes were switched back to their default values a few $\text{μ}\mathrm{s}$ prior to the laser ionisation in order to detect the ions.

Figure 9 shows the resulting THz spectra when packets of ${\mathrm{ND}}_3$ are produced with different $M_J$ composition as outlined above. In default operation – switching off the Stark decelerator when the molecules exit the decelerator, while the preservation electrode is kept grounded – three peaks are observed. These three peaks correspond to all possible $\mathrm{\Delta }{M}_J=\pm 1$ transitions and are a signature of a mixed $M_J=0$ and $|M_J|=1$ population (see also again Figure 6). Keeping the Stark decelerator on for an extended period of time to produce a fringe field upon exiting the decelerator, and combining this with the preservation field, only the two peaks remain that originate from $1_1^{-},|M_J|=1$ states. This demonstrates that a pure $|M_J|=1$ population is kept throughout the experiment, provided that bias fields are present at all times. Switching the Stark decelerator off 10 stages early and then using the last electrodes in kicker operation, followed by the preservation field with the preservation electrode at $1\mathrm{kV}$, only a single peak in the spectrum remained. This peak originates from molecules in $M_J=0$, demonstrating the ability to produce samples of ${\mathrm{ND}}_3$ molecules that are exclusively in the $1_1^{-},M_J=0$ state.

Figure 9. THz spectrum of the $2_1^{+}\leftarrow 1_1^{-}$ transition of ${}^{14}{\mathrm{ND}}_3$ in an electric field of $4.6\mathrm{kV}/\mathrm{cm}$. Packets of ${\mathrm{ND}}_3$ $(1_1^{-})$ molecules are produced with different $M_J$ compositions. Top: default operation of the decelerator resulting in a mixed composition (25% in $M_J=0$ and 75% in $|M_J|=1$). Center: preservation fields downstream from the decelerator enable a pure sample of $|M_J|=1$ molecules. Bottom: operation of the decelerator with kicker to deflect molecules in $|M_J|=1$ in combination with preservation fields results in a pure sample of $M_J=0$ molecules. Spectra are recorded with the THz source in single frequency mode, using a stepsize of 10 kHz while scanning the frequency.

4. Conclusions

We have demonstrated the direct transfer of population between rotational levels of ${\mathrm{ND}}_3$ molecules by driving the $2_1^{+}\leftarrow 1_1^{-}$ transition using a commercially available THz radiation source at frequencies near 615 GHz. Using chirps to address all hyperfine components, transfer efficiencies up to 87% were achieved, limited by the maximum available power of the THz source. In the presence of an electric field, fully $M_J$-resolved transitions were measured for both the zero-field dipole allowed $2_1^{+}\leftarrow 1_1^{-}$ and dipole forbidden $2_1^{-}\leftarrow 1_1^{-}$ transitions. Population transfer from the $1_1^{-}$ to the $2_1^{-}$ level was also achieved with a superior transfer efficiency using a zero-field dipole allowed $1_1^{-}\to 1_1^{+}\to 2_1^{-}$ microwave-THz double resonance excitation scheme.

We presented a novel method to produce packets of ${\mathrm{ND}}_3$ ($1_1^{-}$) molecules with well defined $M_J$ composition. Beams with near-perfect $M_J=0$ or $|M_J|=1$ purity can be produced using novel operation modes of the Stark decelerator in combination with preservation fields, whereas beams with a $M_J=0$ (25%) and $|M_J|=1$ (75%) population distribution can be produced by propagating the beam in zero field downstream from the decelerator. The $M_J$ purity is conveniently probed by inducing $M_J$-resolved $2_1^{+}\leftarrow 1_1^{-}$ THz transitions in the presence of an electric field. The different routes to produce beams with pure $M_J$ composition in either the rotational ground or $2_1^{\pm }$ excited states, in combination with methods to record $M_J$ resolved populations, allows for state-to-state inelastic (rotational de-excitation) collision experiments with full $\mathrm{\Delta }{M}_J$ resolution. This offers interesting prospects in scattering experiments with cold and ultracold molecules in the presence of external electric fields.

Acknowledgments

This work is part of the research program of the Netherlands Organisation for Scientific Research (NWO). We thank Niek Janssen and André van Roij for expert technical support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

S.Y.T.v.d.M. acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program (Grant Agreement No. 817947 FICOMOL) and Dutch Research Council (NWO).