Ground-state OH molecule in combined electric and magnetic fields: Analytic solution of the effective Hamiltonian

Abstract

The OH molecule is currently of great interest from the perspective of ultracold chemistry, quantum fluids, precision measurement, and quantum computation. Crucial to these applications are the slowing, guiding, confinement, and state control of OH, using electric and magnetic fields. In this article, we show that the corresponding eight-dimensional effective ground-state Stark-Zeeman Hamiltonian is exactly solvable and explicitly identify the underlying chiral symmetry. Our analytical solution opens the way to insightful characterization of the magnetoelectrostatic manipulation of ground-state OH. Based on our results, we also discuss a possible application to the quantum simulation of an imbalanced Ising magnet.The OH molecule in its ground X 2 3/2 state is presently widely employed in investigations of ultracold chemistry [1-4], precision measurements [5,6], and quantum computation [7]. Particularly interesting is the recently implemented evaporative cooling of OH close to Bose-Einstein condensation [8]. With such experiments under way, the exploration of quantum degeneracy and molecular optics [9] with OH should shortly become reality.A substantial reason behind the suitability of OH as a workhorse for these experiments is the fact that it is a polar paramagnetic molecule; i.e., it carries both electric and magnetic dipole moments. Electric and magnetic fields can therefore be used to slow, guide, confine, and generally manipulate OH [10][11][12][13][14]. It follows that a quantitative as well as qualitative understanding of the corresponding Stark-Zeeman spectrum is of great relevance.In this article we present the exact solution of the eightdimensional Stark-Zeeman Hamiltonian of OH in its X 2 3/2 ground state [13] and identify the intriguing underlying symmetry. This molecular Hamiltonian is an effective one, neglects hyperfine structure, and has been used to numerically model experimental data accurately [8,12,13]. However, there is interest in analytic solutions also: during the preparation of this article, the field-dependent part of the Hamiltonian was diagonalized exactly in an insightful article by Bohn and Quemener [15].Based on our analysis, we suggest that the OH molecule may be used to simulate a mixed spin Ising magnet, which is of interest in condensed matter physics [16]. Another use for our results is a realistic theory of nonadiabatic processes in traps, which so far has relied on a simplified four-dimensional model of the OH ground state [17]. Our work may also be of relevance to atmospheric [18], interstellar [19], and combustion physics [20], where OH plays an important role. Lastly, we hope that our results will usefully add to the handful of exact solutions available for molecules, especially in strong fields [21].We begin with the Stark-Zeeman Hamiltonian for OH in the X 2 3/2 state, as presented earlier [13],where H o is the field-free Hamiltonian, μ e and μ b are the electric and magnetic dipole moments of the molecule, respectively, and E [ B] is the electric [magnetic] field impose...

Hyperfine structure of the hydroxyl free radical (OH) in electric and magnetic fields

“…We first examine the Stark and Zeeman effects for the spectrum of = J 3 2 states in the Π 2 3 2 manifold separately, and then investigate the energy spectra in the presence of combined electric and magnetic fields. While the spectroscopy of OH in combined fields has been studied by several authors [8,[12][13][14][15][16][17]45], our results are distinguished in two important aspects. First, our study is the first to include all effects, including hyperfine structure and centrifugal distortion, down to few kiloHertz accuracy, which leads to observable spectral differences, as we shall show.…”

“…Finally, it has been shown in a recent paper [17] that an effective Hamiltonian restricted to the lowest rotational level of OH without hyperfine structure has the property that its spectrum is reflection symmetric about zero energy, and this property enables the spectrum to be determined analytically due to the characteristic polynomial being of degree four. This property has further been shown to be linked to a 'chiral symmetry' generated by a rotation operator which anticommutes with the Hamiltonian.…”

“…Although there are many studies of the single-particle spectrum of the OH molecule in the context of chemistry and radio-astronomy as mentioned above, in the presence of electric and magnetic fields the energy spectra of OH have been calculated previously only to energy scales of milliKelvin temperatures (or frequencies of megaHertz) [8][9][10][11][12][13][14][15][16][17], far from the sub-microKelvin temperatures at which OH molecules will become quantum degenerate. Therefore, as a necessary step towards a correct description of a quantum degenerate molecular dipolar gas of OH molecules at sub-microKelvin temperatures, we investigate its hyperfine structure in the lowest electronic and rovibrational states under combined electric and magnetic fields.…”

Spin squeezing a cold molecule

“…The domain of validity of this Hamiltonian and the details of the states involved can be found in several articles [4,14,74,76], and will not be repeated here. This effective Hamiltonian was recently diagonalized analytically following the detection of an underlying chiral symmetry [75]. In the process of identifying that symmetry, it was found that the OH matrix Hamiltonian could be re-expressed in terms of two interacting spins (to avoid notational clutter we set = 1; to make contact with laboratory parameters, requisite factors of can be supplied to any formula in this article by inspection, see the example supplied below),…”

“…The advantage of using static, rather than optical or microwave [15], fields for squeezing is that damping and decoherence due to spontaneous emission can be avoided. Our starting point will be an effective eightdimensional matrix Hamiltonian that has been shown to model recent OH experiments quite well [14,73,74], and has also been diagonalized analytically [75,76]. We demonstrate that for Zeeman and Stark shifts small compared to the Lambda-doublet splitting of the OH ground state, this Hamiltonian can yield spin-squeezing of the types considered by Kitagawa and Ueda [18], Law et al [30] and Agarwal and Puri [21].…”