Quantum theory of molecular orientations

TL;DR

This work develops a symmetry-centered quantum phase space for molecular rotations and nuclear spins by leveraging induced representations of the molecular symmetry group, linking angular-momentum states to angular-position states via a generalized Fourier transform on coset spaces $SO(3)/G$. It reveals two key phenomena: (i) symmetry- and spin-statistics–driven rotation-spin entanglement with Schmidt rank $rak d$ for perrotational isomers, and (ii) fiber degrees of freedom that yield holonomic (Berry) and non-Abelian monodromies enabling robust, topological quantum operations and fiber-based quantum codes. The authors classify all isomers, quantify entanglement fractions across representative molecules (e.g., BF$_3$, $^{13}$C$_{60}$, NH$_3$), and derive conditions under which position-state holonomy is Abelian or non-Abelian, including analytical results for several groups and numerical evidence for others. They also outline how fiber codes can protect information against intra-isomer noise and momentum kicks, and propose experimental schemes to observe monodromy via interferometry, stroboscopic pulses, or internal-state methods, thereby bridging molecular physics and quantum information science with potential practical quantum technologies.

Abstract

We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed

Figures (10)

• Figure 1.1: We use techniques from molecular spectroscopy Rasetti_incoherent_1929Wilson_statistical_1935Wilson_symmetry_1935longuet-higgins_symmetry_1963Van_Vleck_coupling_1951hougen_interpretation_1971louck_eckart_1976harter_orbital_1977Watson_aspects_1977berger_classification_1977ezra_symmetry_1982Papousek_molecular_1982bunker_molecular_1998bunker_fundamentals_2004brown_rotational_2003biedenharn_angular_2010Herzberg_spectra_2013Herzberg_infrared_1987Herzberg_electronic_1966yurchenko2023computational to develop a rigid-body-based casimir1931rotationLandau_mechanics_1976Littlejohn_gauge_1997chirikjian_engineering_2000wormer_rigid_nodateLynch_modern_2017 classification of molecular angular-momentum states. We show that each molecular symmetry isomer corresponds to a particular quantization isham_topological_1984Landsman_geometry_1991Tanimura_reduction_2000levay_canonical_1996 of the configuration space, yielding angularposition states and a Fourier transform (5).
• Figure 1.2: Molecular rotational states can be characterized by their behavior under orientation-preserving rotations. Orientations, or angular positions, of asymmetric molecules are the same as those of any rigid body (e.g., an airplane) whose center of mass is fixed. While every rotation moves an asymmetric molecule to a different position, rotationally symmetric molecules remain in the same position under some rotations. Such molecules can have nonzero nuclear spin (marked by "$\nearrow$"), but any rotations that permute identical spinful nuclei must also rotate the rest of the molecule into a different position. Molecule-frame rotations that permute spinful nuclei and do leave the rest of the molecule invariant have to produce the molecule's nuclear-spin statistics. We call any molecule admitting such permutation-rotations a perrotationally symmetric molecule. The rotational state space of such molecules has no classical analogue and is the main subject of this work. Perrotationally symmetric molecules with non-commutative symmetries can admit separable or entangled nuclear spin isomers, with the latter exhibiting complete rotation-spin entanglement due to a combination of symmetry and nuclear-spin statistics.
• Figure 2.1: With the molecular frame collapsed due to identification of positions related by molecule-based symmetries, we are still free to rotate the lab frame. (a) Lab-based rotations, $\overrightharpoon{X}_{\mathsf{g}}$ for $\mathsf{g}$ in a molecule's symmetry group, map the molecule to the same position. For example, a rotation rotates the methane molecule by 120$^{\circ}$, with the path traversed by the forward-most nucleus depicted by two arrows. (b) A position state $|\mathsf{a}=\mathsf{e}\rangle$ (with $\mathsf{a}$ set to the identity element $\mathsf{e}$ for simplicity) of a perrotationally symmetric molecule carries with it a fiber space, spanned by the basis $\{|\mu\rangle\}$. Symmetry rotations transform the internal state of fiber via the monodromy matrix ${\color{red}\Gamma_{\!\textnormal{rot}}(\mathsf{g}^{-1})}$. (c) In a cartoon depiction, the fiber can be thought of as a vertical line at each point $\mathsf{a}$ in the position-state space $\mathsf{SO(3)}/\mathsf{G}$, with a non-contractible closed path resulting in a monodromy action on the fiber state.
• Figure 2.2: (a) Molecular code states albert_robust_2020 are superpositions of several position states of an asymmetric molecule, related by rotations in a subgroup $\mathsf{G}$. (b) Fiber codes $\{|\mathsf{e},\mu\rangle\}_{\mu=1}^{\mathfrak{d}}$\ref{['eq:fourier']} encode in a single molecular position, $\mathsf{a} = \mathsf{e}$, of a perrotationally $\mathsf{G}$-symmetric entangled isomer (here, $\mathsf{O}$-symmetric SF6). The former induces the trivial induced representation, $\textsc{a}\uparrow\mathsf{SO(3)}$, on the asymmetric molecular state space, while the latter forms a non-trivial induced representation, $\Gamma_{\!\textnormal{rot}}\uparrow\mathsf{SO(3)}$, with a fiber degree of freedom of dimension $\mathfrak{d} = \dim \Gamma_{\!\textnormal{rot}}$. Both encodings are comparable in performance against shifts in the molecules' orientation and kicks in their momenta (see Sec. \ref{['fig:qec']}).
• Figure 2.3: Sketch of the evolution of a homonuclear diatomic evolving under its intrinsic rigid rotor Hamiltonian, $B\hat{\mathbf{J}}^2$ with rotational constant $B$, while undergoing a stroboscopic re-orientation from the $\mathbf{z}$ to the $-\mathbf{z}$ axis via a series of ultrafast pulses. The first pulse aligns the molecule along the desired $\mathbf{z}$-axis, initializing it in what is a close approximation to the corresponding position state. Subsequent Hamiltonian evolution induces phases, $\exp(-iBJ(J+1)t)$, on the molecule's rotational states $|^{J}_m\rangle$, but all phases disappear at times $t$ that are multiples of the rotational period $T_{\text{rev}}=2\pi/B$. At the rotational period, a second pulse, with slightly rotated polarization relative to the first, is incident on the molecule. Subsequent pulses can be applied at multiples of $T_{\text{rev}}$, with each pulse incrementally edging the position state closer to that aligned along the $-\mathbf{z}$ axis. In the impulsive limit, where the pulse length is much shorter than $T_{\text{rev}}$, relative phases are imprinted that cause the molecular wavepacket to rephase at the new, rotated polarization. As long as the tilt between polarizations of successive laser pulses is not too large, a sufficiently discretized set of pulses should incrementally re-orient the molecule from $\mathbf{z}$ to $-\mathbf{z}$.

Theorems & Definitions (52)

• Theorem
• Example 1: calcium hydrosulfide
• Example 2: methylamine
• Example 3: semi water
• Example 4: C59$^{13}$C doped fullerene
• Example 5: conical intersection toy model
• Example 6: $\mathsf{G}=\mathsf{C}_{\infty}$ isotypic decomposition
• Example 7: calcium monohydrosulfide
• Example 8: hydrogen chloride
• Example 9: disulfur