Qubit Physics

Canonical quantization is a method for introducing quantum structure into classical systems by replacing the Poisson structure of the Hamiltonian with commutator relations. In the previous section, we derived the Lagrangian. Now, the Hamiltonian can be obtained via the Legendre transform.

The canonical momentum \(\Phi_\#\) is defined as: \[ \Phi_\# \equiv \frac{\partial \mathcal{L}}{\partial \dot{Q}} = \mathcal{L}\dot{Q} \leftrightarrow \dot{Q} = \frac{\Phi_\#}{L}. \] Here, \(\Phi_\#\) has the same unit as magnetic flux and can be interpreted as the flux inside the inductor. The Hamiltonian \(H(\Phi_\#, Q)\) is defined as: \[ \mathcal{H} \equiv \Phi_\# \dot{Q} - \mathcal{L} = \frac{\Phi_\#^2}{2L} + \frac{Q^2}{2C}. \]

In a quantum system, canonical coordinates \(q\) and canonical momentum \(p\) satisfy the commutator relation: \[ [x, p] = i\hbar. \] For the parameters in this paper, the quantization condition becomes: \[ [Q, \Phi_\#] = i\hbar. \]

For this discussion, we use \(\Phi_\#\) as the notation because it aligns with the traditional literature where the roles of \(\Phi_\#\) and \(Q\) in the Hamiltonian are symmetric. By convention, this paper reverses the traditional approach and treats flux as the generalized coordinate, denoted as \(\Phi\), while \(Q\) becomes the generalized momentum. Consequently, the quantization condition becomes: \[ [\Phi, Q] = i\hbar, \] with a sign difference: \[ \Phi = -\Phi_\#. \] This sign can be further explained by the antisymmetric nature of the Lagrangian.

To proceed, we introduce new variables \(a\) and \(a^\dagger\): \[ a = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L}{C}}} \left( Q + i\sqrt{\frac{C}{L}} \Phi_\# \right), \quad a^\dagger = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L}{C}}} \left( Q - i\sqrt{\frac{C}{L}} \Phi_\# \right). \] Conversely, these can be expressed as: \[ Q = \sqrt{\frac{\hbar}{2} \sqrt{\frac{C}{L}}} (a^\dagger + a), \quad \Phi_\# = i\sqrt{\frac{\hbar}{2} \sqrt{\frac{L}{C}}} (a^\dagger - a). \]

It can be shown that \([a, a^\dagger] = 1\): \[ [a, a^\dagger] = aa^\dagger - a^\dagger a = \frac{1}{2\hbar} \sqrt{\frac{L}{C}} \left[ Q, \Phi_\# \right] = 1. \] The Hamiltonian \(H(\Phi_\#, Q)\) can be rewritten as: \[ \mathcal{H} = \frac{\Phi_\#^2}{2L} + \frac{Q^2}{2C} = \frac{\hbar \omega}{4} \left[ - (a^\dagger - a)^2 + (a^\dagger + a)^2 \right], \] where \(\omega = \frac{1}{\sqrt{LC}}\). Expanding yields: \[ \mathcal{H} = \frac{\hbar \omega}{2} \left[ a^\dagger a + a^\dagger a + 1 \right] = \hbar \omega \left[ a^\dagger a + \frac{1}{2} \right]. \]

Finally, the quantized Hamiltonian is obtained, with each energy level separated by \(\hbar \omega\).

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.