What exactly are "quantum rotors"?

Question

The Wikipedia entry on the subject is rather short. I am also curious about generalizations of quantum rotors in n-dimensions. An introductory explanation with at least one resource for further reading would be greatly appreciated.

Answer 1

Quantum rotor models are quantum systems based on the quantization of systems with rotational configuration spaces. For example, a particle moving on a ring or a pendulum are rotors whose configuration spaces are circular $S^1$, while a rigid body a system whose configuration space is the three-dimensional sphere $S^3$ (or equivalently, the group manifold $SU(2)$).

The dynamics of these systems depend in addition on the angular momenta of the configuration space coordinates, whose values are not confined. Thus, for the case of the pendulum, the phase space (spanned by coordinates and angular momenta) is the two-dimensional cylindrical manifold $S^1 \times \mathbb{R}$ and in the rigid body case $S^3 \times \mathbb{R}^3$.

Once, an appropriate Hamiltonian function on the phase space is defined, there are quantization rules which allow writing the Schrödinger equation to treat the system quantum mechanically.

The cylindrical phase space quantum rotor is especially relevant to quantum computation as it enters in the description of the condensate dynamics of Josephson junction arrays which can be used to implement superconducting qubits; Please see the following review by: Devoret, Wallraff and Martinis. In this case a generic Hamiltonian (of a single isolated qubit) has the form:

$$H = \frac{E_C}{2} (n – n_0)^2 + \frac{E_J}{2} \cos\theta$$

Where $E_C$ is the charging energy and $E_J$ is the total junction energy and, $n$ is the number of Cooper pairs and $n_0$ is an offset proportional to the junction residual charge and: $$\theta = 2 \pi \frac{\Phi}{\Phi_0} \mod 2 \pi$$ is the phase across the junction ($\Phi$ is the magnetic flux, and $\Phi_0$ is the flux quantum), and most importantly is that the cooper pair number is conjugate to the phase: $$n = \frac{1}{E_C}\frac{\partial \theta }{\partial t }$$ This is a consequence of Faraday's law. (Thus this model is defined on a cylindrical phase space since the angular momentum is proportional to the angular velocity).

After quantization, the Cooper pair number operator is replaced by: $$\hat{n} =i \hbar \frac{\partial }{\partial \theta }$$

The Schrödinger equation can be solved exactly in terms of Mathieu functions. However, the qubit dynamics can be obtained if we restrict the dynamics to the two lowest eigenfunctions of the number operator $|0\rangle$ and $|1\rangle$, in this case, the kinetic and the potential terms of the Hamiltonian can be approximated for $n_0 = \frac{1}{2}$ as:

$$ E_C (n – n_0)^2 \approx E_C \sigma_z$$ And:

$$E_J cos\theta \approx E_J \sigma_x$$ Thus, the quantum Hamiltonian restricted to the lowest two-dimensional number subspace: $$\hat{H} = \frac{E_J}{2}(\sigma_x + \frac{E_C}{E_J} \sigma_z)$$ and we can use the ratio $\frac{E_C}{E_J}$ to control the quantum evolution.

Remarks:

While the authors of the main references given above don't use the name quantum rotors in relation to qubits, other authors do, for example: Girvin, Devoret, and Schoelkopf

A description of the cylindrical phase and other phase spaces from the point of view of quantum computation can be found in: Albert,. Pascazio and Devoret .

I don't know of an application of three dimensional rotors to quantum computation, they especially appear in quantum models of molecules.

Answer 2

First of all, quantum rotors generally appear in the context of quantum rotor models, which are lattice models, analogous to quantum spin systems on a lattice - this is, identical quantum systems arrange on a lattice and interacting via some (natural) interaction. One wouldn't usually talk of an isolated quantum rotor.

So what are quantum rotors? They are quantum mechanical versions of classical rotors, just as quantum spins can be regarded as quantum versions of classical two-level systems.

So what is a classical rotor? It is basically a tiny magnet which can rotate freely in a certain number $d$ of dimensions, such as a compass needle ($d=2$, this would be a O(2) rotor model). It is thus characterized by a unit vector $\vec n$ in $d$ dimensions. The natural interactions of these models would be dipole-dipole interactions (i.e. $\vec n_i\cdot \vec n_j$), and the natural on-site energy would be a kinetic term $\vec{L}^2/2$ with $L$ the angular momentum.

These interactions can be turned quantum mechanical in the canonical way for conjugate variables, as explained on the Wikipedia page.