06 Jun 2026
III: A Fixed-Confinement Angular-Momentum Quench
A quantum quench is a rapid, controlled change of a Hamiltonian parameter. The state is continuous through an ideal sudden switch, but from that instant onward it evolves with a different Hamiltonian. This chapter turns the stable trap and exact product states already constructed in the first two chapters into one precise nonequilibrium protocol.
The new physics is deliberately narrow: the confining potential is held fixed while a signed angular-momentum coupling is switched on. This lets us identify exactly which part of the Hamiltonian causes the two preparation coordinates to interact.
What is carried forward
The stability and coordinate reduction fix the units, coordinates, and stable parameter domain. We use $\hbar=2M=1$ and
\[[q_\alpha,p_\beta]=i\delta_{\alpha\beta}, \qquad [q_\alpha,q_\beta]=[p_\alpha,p_\beta]=0, \qquad \alpha,\beta\in\{+,-\}.\]The labels $+$ and $-$ are fixed coordinate directions, not creation and annihilation operators. Their confining scales already contain the intended diamagnetic dressing:
\[\Omega_\pm^2 =\frac{\omega_x^2+\omega_y^2+2\omega_c^2}{2} \pm\frac12\sqrt{(\omega_x^2-\omega_y^2)^2+4g^2}.\]The stable regime is
\[\omega_x^2>0,\qquad \omega_y^2>0,\qquad \omega_x^2\omega_y^2-g^2>0,\]or, equivalently in the rotated coordinates,
\[\Omega_+^2>\omega_c^2, \qquad \Omega_-^2>\omega_c^2.\]The exceptional-Hermite preparation supplies the one-coordinate Hamiltonians and their exact eigenstates:
\[h_{m_\alpha}=p_\alpha^2+U_\alpha(q_\alpha), \qquad U_\alpha =V^-_{m_\alpha}(q_\alpha;\Omega_\alpha) -\epsilon_{m_\alpha}(\Omega_\alpha).\]Here $m_\alpha$ is even, so the rational potential is regular on the real line. Before the quench the Hamiltonian and state are
\[H_0=h_{m_+}+h_{m_-},\] \[|\Psi_0\rangle =|m_+,\ell_+\rangle\otimes|m_-,\ell_-\rangle, \qquad H_0|\Psi_0\rangle=E_0|\Psi_0\rangle.\]Each factor is real and has definite parity. The only initial-state identities needed in this chapter are
\[\langle q_\alpha\rangle_0 =\langle p_\alpha\rangle_0=0, \qquad \frac12\langle q_\alpha p_\alpha+p_\alpha q_\alpha\rangle_0=0.\]These compact equations are the complete input. The construction and normalization of $U_\alpha$ and \(\lvert m_\alpha,\ell_\alpha\rangle\) are not repeated here.
The new task: define the sudden angular quench
In the fixed coordinates, the angular-momentum generator is
\[L_z=q_+p_- -q_-p_+.\]It is Hermitian because operators from different coordinates commute. The quench protocol is
\[H(t)= \begin{cases} H_0, & t<0,\\[2pt] H_f=H_0-\omega_cL_z, & t\geq0. \end{cases}\]The coupling
\[\omega_c=\frac{q_{\mathrm e}B}{2Mc}\]is signed: reversing the charge or the field sends $\omega_c\mapsto-\omega_c$. In a synthetic rotating trap, $\omega_c$ can instead be regarded directly as an engineered rotation strength.
For an ideal sudden quench,
\[|\Psi(0^+)\rangle=|\Psi(0^-)\rangle=|\Psi_0\rangle, \qquad |\Psi(t)\rangle=e^{-iH_ft}|\Psi_0\rangle \quad(t>0).\]The state does not jump. Its generator of time evolution does.
Only $-\omega_cL_z$ is switched. The functions $U_\pm$, their scales $\Omega_\pm$, and hence the coordinate confinement are identical on the two sides of $t=0$. A literal magnetic-field switch would generally change both the signed angular term and a term proportional to $B^2$. The present protocol is therefore most naturally a synthetic angular coupling added to a trap that has already been prepared with its target confinement.
A finite experimental ramp of duration $\tau_{\mathrm{ramp}}$ approaches the sudden idealization when it is short compared with the internal periods $\Omega_\pm^{-1}$. That statement is a scale criterion, not an assertion that any nonzero ramp is exactly sudden.
Stability is retained across the quench
The stability conditions carried forward above remain the correct ones after activation. A compact check is obtained by completing the kinetic squares:
\[\begin{aligned} H_f={}& \left(p_++\frac{\omega_c}{2}q_-\right)^2 +\left(p_--\frac{\omega_c}{2}q_+\right)^2\\ &+\left[U_+(q_+)-\frac{\omega_c^2q_+^2}{4}\right] +\left[U_-(q_-)-\frac{\omega_c^2q_-^2}{4}\right]. \end{aligned}\]For a regular exceptional-Hermite extension,
\[U_\alpha(q_\alpha) =\frac{\Omega_\alpha^2q_\alpha^2}{4}+O(1) \qquad(|q_\alpha|\to\infty).\]Thus $\Omega_\alpha^2>\omega_c^2$ leaves positive asymptotic confinement along both axes. The equality boundary is excluded because the bounded rational correction alone does not supply the same quadratic control.
Why the preparation coordinates become coupled
The statement “the final Hamiltonian is coupled” should be proved rather than inferred from its appearance. We use
\[[p_\alpha^2,q_\alpha]=-2ip_\alpha, \qquad [f(q_\alpha),p_\alpha]=if'(q_\alpha).\]The two kinetic contributions cancel:
\[[p_+^2,q_+p_-]=-2ip_+p_-, \qquad [p_-^2,-q_-p_+]=+2ip_-p_+.\]The potential contributions are
\[[U_+(q_+),L_z]=-iq_-U_+'(q_+),\] \[[U_-(q_-),L_z]=+iq_+U_-'(q_-).\]Adding them gives the fixed-coordinate result
\[\boxed{ [H_0,L_z] =i\left[q_+U_-'(q_-)-q_-U_+'(q_+)\right]. }\]For unequal or rationally deformed $U_+$ and $U_-$, the right-hand side is generically nonzero. Therefore $H_0$ and $L_z$ do not share a complete eigenbasis, and $H_f$ is not additive on the fixed preparation split $\mathcal H_+\otimes\mathcal H_-$.
The scope of this conclusion matters. It does not rule out every possible canonical transformation, and a nonzero operator commutator does not prove that every individual eigenstate of $H_0$ evolves. A state-specific test is needed.
The state-specific stationarity test
A normalized state is an eigenstate of a time-independent Hamiltonian if and only if its energy variance vanishes. Because \(H_0\lvert\Psi_0\rangle=E_0\lvert\Psi_0\rangle\),
\[\operatorname{Var}_0(H_f) =\omega_c^2\operatorname{Var}_0(L_z).\]For the real definite-parity product preparation, $\langle L_z\rangle_0=0$. Define the local variances
\[V_{q,\alpha}=\langle q_\alpha^2\rangle_0, \qquad V_{p,\alpha}=\langle p_\alpha^2\rangle_0.\]The canonical commutator and the vanishing symmetrized $qp$ expectation give
\[\langle q_\alpha p_\alpha\rangle_0=\frac{i}{2}, \qquad \langle p_\alpha q_\alpha\rangle_0=-\frac{i}{2}.\]Now expand
\[L_z^2 =q_+^2p_-^2+q_-^2p_+^2 -q_+p_+p_-q_- -p_+q_+q_-p_-.\]Expectations factorize between the two initial factors, producing
\[\boxed{ \langle L_z^2\rangle_0 =V_{q,+}V_{p,-}+V_{q,-}V_{p,+}-\frac12. }\]Hence
\[\boxed{ \operatorname{Var}_0(H_f) =\omega_c^2 \left( V_{q,+}V_{p,-}+V_{q,-}V_{p,+}-\frac12 \right). }\]A positive value proves that the selected preparation is not stationary under $H_f$. A zero value means that this particular state is unchanged apart from a phase; it does not make the full operator additive.
Worked control: an undeformed anisotropic ground state
Before using rational states, every implementation should pass a Gaussian control. Set $m_+=m_-=0$. The one-coordinate potential, ground state, and variances are
\[U_0(q;\Omega)=\frac{\Omega^2q^2}{4}-\frac{\Omega}{2},\] \[\phi_{0,0}(q;\Omega) =\left(\frac{\Omega}{2\pi}\right)^{1/4}e^{-\Omega q^2/4},\] \[V_q=\frac1{\Omega}, \qquad V_p=\frac{\Omega}{4}.\]Choose
\[\Omega_+=2,\qquad \Omega_-=1,\qquad \omega_c=\frac12.\]The stability margins are
\[\Omega_+^2-\omega_c^2=\frac{15}{4}, \qquad \Omega_-^2-\omega_c^2=\frac34,\]and
\[\begin{aligned} \langle L_z^2\rangle_0 &=\frac14\left( \frac{\Omega_-}{\Omega_+} +\frac{\Omega_+}{\Omega_-}-2 \right)\\ &=\frac18. \end{aligned}\]Therefore
\[\operatorname{Var}_0(H_f)=\frac1{32}>0.\]The anisotropic ground state must evolve. In contrast, if $\Omega_+=\Omega_-$, the product is the rotationally symmetric two-dimensional Gaussian ground state and \(L_z\lvert\Psi_0\rangle=0\). That isotropic case is a useful null test for signs and numerical propagation.
What is exact, and what still requires approximation
Three logically different claims now coexist:
- \(\lvert\Psi_0\rangle\) is an exact eigenstate of the additive $H_0$.
- The definition of $H_f$, its stability test, the commutator, and the stationarity variance above are operator-level results with no basis truncation.
- A finite-time calculation of \(e^{-iH_ft}\lvert\Psi_0\rangle\) generally requires either a controlled short-time expansion or a converged numerical representation.
Knowing the spectrum of $H_0$ does not give the spectrum of $H_f$. Likewise, representing $H_f$ in finitely many product eigenstates is useful but is not an exact finite-dimensional reduction.
Compact symbolic check of the control case
The Gaussian calculation also exposes the anisotropy dependence in closed form:
\[\langle L_z^2\rangle_0 =\frac{(\Omega_+-\Omega_-)^2} {4\Omega_+\Omega_-}.\]The following Maxima input reduces the variance and evaluates the stated control parameters:
kill(all)$
L2 : Om/(4*Op) + Op/(4*Om) - 1/2$
factor(L2);
ev([L2, wc^2*L2, Op^2-wc^2, Om^2-wc^2],
Op=2, Om=1, wc=1/2);
It returns
(Om-Op)^2/(4*Om*Op)
[1/8, 1/32, 15/4, 3/4]
The first line proves non-negativity for positive scales and makes the isotropic null case explicit. In a numerical study, normalization, final energy conservation, convergence with representation size, and the limits $\omega_c=0$ and $\Omega_+=\Omega_-$ are the most informative accompanying checks; they follow directly from the physics above rather than from an assumed finite-dimensional model.
The quench is now fully specified. The next chapter derives the initial observable response from $H_f$ instead of assuming a closed dynamics.
Series navigation
- Magnetic oscillator stability and reduction
- Exceptional-Hermite preparation
- Fixed-confinement angular quench — current chapter
- Exact covariance response
- Survival curvature
- Spectral propagation and synchronization
- Non-Gaussian mutual information
Discussion