Traveling microwave lattice deceleration of polar molecules: power utilization, finite-waveguide boundary effects, small-detuning losses, and chirp-schedule optimization

Hu and co-workers proposed decelerating any polar molecule in its rotational ground state by trapping it in the moving beat lattice formed by two counter-propagating, slightly detuned microwave fields near a rotational transition inside a circular waveguide, exploiting ac Stark (dipole-force) potentials rather than the optical cycling (repeated photon absorption and spontaneous emission) that makes laser cooling applicable to only a few molecular species. Their analysis left four quantitative issues unresolved: (1) only half of the microwave power from the two sources is usable in the proposed waveguide design; (2) the simulations omit boundary conditions at the ends of the finite-length waveguide and at the U-shaped rectangular-to-circular mode transformers; (3) the small-detuning regime that strengthens the Stark force also raises the probability of driving real transitions out of the ground state, whose quantitative impact on trapping and deceleration efficiency was not computed; and (4) time-dependent chirp schedules were not optimized against constant acceleration. The problem of quantitatively validating the traveling microwave lattice decelerator remains open.

Background

Producing slow, cold neutral molecules is a prerequisite for cold-collision studies, precision spectroscopy, and tests of fundamental symmetries. The classic tool, Stark deceleration, switches static electric fields of tens of kV/cm and therefore requires molecules in low-field-seeking states, i.e. rotational states whose energy increases with field; this excludes many chemically stable species. Microwave-based manipulation offers an alternative that works for essentially every polar molecule in its rotational ground state. It exploits the ac Stark shift — the energy shift of a molecular level in an oscillating electric field. When a microwave field of amplitude $E$ is tuned near the transition between the rotational ground state and the first excited rotational state (transition dipole moment $\mu_{01}$, transition frequency $\omega_{01}$, detuning $\Delta=\omega-\omega_{01}$), the induced shift of the ground state is of order $\mu_{01}^{2}E^{2}/(4\hbar\Delta)$ at large detuning and saturates toward $\mu_{01}E/2$ near resonance. Because rotational transition dipoles are comparable to permanent dipole moments, this near-resonant shift is first order in the field amplitude, and a few kV/cm of microwave field gives potential depths of order 0.1 K for typical species — the reference design of Hu et al. for $^{24}\mathrm{Mg}^{16}\mathrm{O}$ yields a moving-well depth of about 62 mK at a 7 kV/cm peak on-axis field — and, unlike static-field Stark potentials, the microwave lattice can confine the high-field-seeking rotational ground-state molecules that electrostatic traps cannot. Enomoto and Momose (2005) proposed a microwave Stark decelerator; Kuma and Momose (2009) simulated dipole-force deceleration; Odashima et al. (2010) demonstrated a microwave lens; and Merz et al. (2012) demonstrated the first experimental microwave deceleration and acceleration of a molecular beam using a switched standing-wave cavity. In a standing wave the field nodes sweep across the molecules as the cavity is switched, and at each node crossing molecules can make nonadiabatic transitions into untrapped states, so acceptance falls at high deceleration rates. Hu, Yi, Zhao, Deng, and Luo (Phys. Rev. A 89, 053428, 2014) proposed eliminating the node crossings entirely: two microwave fields of slightly different frequencies counter-propagate in a circular waveguide (fed through U-shaped transformers converting the rectangular $\mathrm{TE}_{10}$ feed mode to the circular $\mathrm{TE}_{11}$ mode — transverse-electric waveguide modes, whose electric field is transverse to the propagation direction, with subscripts labeling the field pattern) and beat to form a lattice of microwave field amplitudes that travels at the velocity set by the frequency difference; molecules confined in the near-resonant ac Stark potential wells of this moving lattice are carried along and decelerated (or accelerated) by chirping the difference frequency. Their trajectory simulations assumed ideal sinusoidal fields and a constant lattice acceleration, and identified four issues that limit immediate experimental implementation — usable microwave power, unmodeled finite-waveguide and transformer boundary effects, unquantified transition losses in the small-detuning regime (detuning $\Delta$ much smaller than the resonant coupling scale $\mu_{01}E/\hbar$), and unexplored nonlinear chirp schedules. Quantifying these four issues is what is needed to judge whether the traveling-lattice decelerator is experimentally feasible.

Problem Statement

Determine, quantitatively, whether the traveling microwave lattice decelerator proposed by Hu et al. (Phys. Rev. A 89, 053428, 2014) is experimentally feasible, by resolving four interlinked issues that the source paper itself raises: (1) microwave power utilization — waveguide or source configurations that use more of the available microwave power than the source paper's two-source design, and the resulting gains in lattice trap depth and phase-space acceptance (the set of starting positions and velocities that remain trapped); (2) boundary effects — the modifications of modal purity (the degree to which the field remains in the desired propagation mode), of the field amplitude at the molecules, and of the lattice transport efficiency produced by the finite length of the waveguide and by the U-shaped transformers that couple the rectangular $\mathrm{TE}_{10}$ feed mode to the circular $\mathrm{TE}_{11}$ mode; (3) small-detuning losses — the quantitative impact of nonadiabatic and resonant-transition processes, which become more probable as the detuning $\Delta$ is reduced below the resonant coupling scale $\mu_{01}E/\hbar$, on the ac Stark shifts, trap loss, and deceleration efficiency; and (4) chirp optimization — time-dependent (nonlinear) acceleration schedules that maximize the number of decelerated molecules for a given waveguide length and microwave power, relative to constant acceleration.

All four issues (1)–(4) are required, and none is optional: the design counts as quantitatively validated only when each of them has been resolved against the reference design below. The objective is a quantitatively validated traveling-lattice decelerator design that resolves these source-identified issues. Experimental demonstration is not part of the problem as posed here, although the source envisions a working decelerator as "an ideal starting point for subsequent experiments like a molecular fountain, spectroscopy and scattering studies, or further sympathetic and evaporative cooling".

Feasibility, power-utilization gains, loss impact, and chirp optimization are to be assessed relative to the source paper's reference design, which fixes the parameters otherwise left open: test molecule $^{24}\mathrm{Mg}^{16}\mathrm{O}$ ($\mu = 6.2$ D, $\omega_{01}/2\pi = 34.290\ \mathrm{GHz}$, no hyperfine structure), predecelerated by a rotating jet to an average velocity of $\sim 100\ \mathrm{m/s}$; two microwave sources of power $P = 2\ \mathrm{kW}$ each; detuning $\Delta = 2\pi\times 0.1\ \mathrm{GHz}$; waveguide radius $R = 2.9\ \mathrm{mm}$; on-axis field maximum $E_{\max} = 7.0\ \mathrm{kV/cm}$, each source transmitting only half of its power to the deceleration area; synchronous phase $\phi_s = 40^\circ$, corresponding to constant deceleration $a = 1.4\times 10^{4}\ \mathrm{m/s^2}$ and chirp rate $\alpha = 2\pi\times 0.75\ \mathrm{MHz/s}$; deceleration to standstill in $t = 7.3\ \mathrm{ms}$ over $0.36\ \mathrm{m}$; limiting moving-well depth $62\ \mathrm{mK}$; and $2.2\%$ of molecules decelerated to standstill with a six-dimensional acceptance of about $[3\ \mathrm{mm}\times 10\ \mathrm{m/s}]$ (x), $[5\ \mathrm{mm}\times 10\ \mathrm{m/s}]$ (y), and $[4\ \mathrm{mm}\times 10\ \mathrm{m/s}]$ (z). Thus "the available microwave power" in item (1) and "a given waveguide length and microwave power" in item (4) are anchored to this design ($2\ \mathrm{kW}$ per source, $0.36\ \mathrm{m}$ stopping distance); alternative waveguide or source configurations may be proposed, as item (1) invites, but are to be compared quantitatively against this baseline.

References

1. Zhong-Kun Hu, Yi Ke, Jin-Bo Zhao, Xiao-Bing Deng, Jun Luo, 'Decelerating polar molecules using traveling microwave lattices', Physical Review A 89, 053428 (2014). DOI: 10.1103/physreva.89.053428, https://doi.org/10.1103/physreva.89.053428
2. Katsunari Enomoto, Takamasa Momose, 'Microwave Stark decelerator for polar molecules', Physical Review A 72, 061403 (2005). DOI: 10.1103/physreva.72.061403, https://doi.org/10.1103/physreva.72.061403
3. Susumu Kuma, Takamasa Momose, 'Deceleration of molecules by dipole force potential: a numerical simulation', New Journal of Physics 11, 055023 (2009). DOI: 10.1088/1367-2630/11/5/055023, https://doi.org/10.1088/1367-2630/11/5/055023
4. Hitoshi Odashima, Simon Merz, Katsunari Enomoto, Melanie Schnell, Gerard Meijer, 'Microwave Lens for Polar Molecules', Physical Review Letters 104, 253001 (2010). DOI: 10.1103/physrevlett.104.253001, https://doi.org/10.1103/physrevlett.104.253001
5. Simon Merz, Nicolas Vanhaecke, Wolfgang Jäger, Melanie Schnell, Gerard Meijer, 'Decelerating molecules with microwave fields', Physical Review A 85, 063411 (2012). DOI: 10.1103/physreva.85.063411, https://doi.org/10.1103/physreva.85.063411
6. Simon Merz, Claudia Brieger, Nicolas Vanhaecke, Gerard Meijer, Melanie Schnell, 'Manipulating the motion of polar molecules with microwave radiation', Molecular Physics 111, 1855-1864 (2013). DOI: 10.1080/00268976.2013.810791, https://doi.org/10.1080/00268976.2013.810791
7. Samuel A. Meek, Maxwell F. Parsons, Georg Heyne, Viktor Platschkowski, Henrik Haak, Gerard Meijer, Andreas Osterwalder, 'A traveling wave decelerator for neutral polar molecules', Review of Scientific Instruments 82, 093108 (2011). DOI: 10.1063/1.3640413, https://doi.org/10.1063/1.3640413
8. Paul Jansen, Marina Quintero-Pérez, Thomas E. Wall, Joost E. van den Berg, Steven Hoekstra, Hendrick L. Bethlem, 'Deceleration and trapping of ammonia molecules in a traveling-wave decelerator', Physical Review A 88, 043424 (2013). DOI: 10.1103/physreva.88.043424, https://doi.org/10.1103/physreva.88.043424
9. S. Chervenkov, X. Wu, J. Bayerl, A. Rohlfes, T. Gantner, M. Zeppenfeld, G. Rempe, 'Continuous Centrifuge Decelerator for Polar Molecules', Physical Review Letters 112, 013001 (2014). DOI: 10.1103/physrevlett.112.013001, https://doi.org/10.1103/physrevlett.112.013001
10. Katsunari Enomoto, Nagisa Hizawa, Yuji Furuta, Naoyuki Hada, Takamasa Momose, 'Focusing of a cold PbO molecular beam with a superconducting microwave resonator', Journal of Physics B: Atomic, Molecular and Optical Physics 52, 035101 (2019). DOI: 10.1088/1361-6455/aaf666, https://doi.org/10.1088/1361-6455/aaf666
11. S. C. Wright, T. E. Wall, M. R. Tarbutt, 'Microwave trap for atoms and molecules', Physical Review Research 1, 033035 (2019). DOI: 10.1103/physrevresearch.1.033035, https://doi.org/10.1103/physrevresearch.1.033035
12. Fatemeh S. Tahsildaran, Manish Vashishta, Amir Hossein Farahbod, Rasoul Malekfar, Pavle Djuricanin, Katsunari Enomoto, Takamasa Momose, 'A superconducting Fabry–Perot cavity for trapping cold molecules', Journal of Physics B: Atomic, Molecular and Optical Physics 54, 015101 (2021). DOI: 10.1088/1361-6455/abcb51, https://doi.org/10.1088/1361-6455/abcb51
13. Simon Merz, 'Manipulating the motion of neutral polar molecules with microwave fields', doctoral thesis, Freie Universität Berlin (2015). URN: urn:nbn:de:kobv:188-fudissthesis000000099430-0, https://refubium.fu-berlin.de/handle/fub188/7619 (DNB record 1073868737)