Driven Two-Level Systems

Driven two-level systems provide compact but demanding tests of time-dependent quantum propagation. They expose frequency conventions, rotating-frame signs, time ordering, and phase error without a spatial discretization. The Rabi-oscillation treatment explains resonance and coherent transfer; here we use these results to check numerical methods.

Helpful background. The numerical Schrödinger equation supplies state and observable error measures.

Laboratory and rotating-frame Hamiltonians

For a linearly polarized drive, choose

$$H_{\rm lab}(t)=\frac{\hbar\omega_0}{2}\sigma_z +\hbar\Omega\cos(\omega_dt)\sigma_x.$$

Take $\omega_0,\omega_d>0$ and choose the drive phase so that $\Omega\geq0$. The coefficient $\Omega$ is the on-resonance angular Rabi frequency in the rotating-wave approximation (RWA) with this convention. Other amplitude conventions differ by a factor of two, so write the Hamiltonian before quoting a “Rabi frequency.”

Let $\psi_{\rm lab}=R(t)\psi_{\rm rot}$ with $R(t)=e^{-i\omega_dt\sigma_z/2}$. Then

$$H_{\rm rot}=R^\dagger H_{\rm lab}R -i\hbar R^\dagger\dot R.$$

Dropping the terms oscillating at $2\omega_d$ gives

$$H_{\rm RWA}=\frac{\hbar}{2} (\Delta\sigma_z+\Omega\sigma_x), \qquad \Delta=\omega_0-\omega_d.$$

The RWA is a model approximation, appropriate under a near-resonant weak-drive regime with suitable envelope variation. Tightening a numerical solver does not remove its discrepancy from the laboratory Hamiltonian.

Exact constant-frame propagation

Since $(\Delta\sigma_z+\Omega\sigma_x)^2 =(\Delta^2+\Omega^2)I$, define $\Omega_R=\sqrt{\Delta^2+\Omega^2}$ and obtain

$$U_{\rm RWA}(t)= \cos(\Omega_Rt/2)I -i\sin(\Omega_Rt/2) \frac{\Delta\sigma_z+\Omega\sigma_x}{\Omega_R}.$$

The zero-generator case $\Delta=\Omega=0$ gives $U=I$ by continuity. Otherwise, starting from a $\sigma_z$ eigenstate, the probability to occupy the other one is

$$P(t)=\frac{\Omega^2}{\Omega_R^2} \sin^2(\Omega_Rt/2).$$

On resonance, a pulse of duration $\pi/\Omega$ gives full inversion. Off resonance, the maximum is below one. With $\Delta=\Omega$, the maximum is $1/2$ and the oscillation frequency is $\sqrt2\,\Omega$.

A circularly rotating drive, $\hbar\Omega[\cos(\omega_dt)\sigma_x+\sin(\omega_dt)\sigma_y]/2$, transforms exactly to the same constant rotating-frame Hamiltonian. It is therefore a particularly clean test of frame transformations without RWA error.

A comparison that separates two errors

First propagate the constant $H_{\rm RWA}$ and compare the complete state with the exact formula. Next propagate the circular drive in the laboratory frame and transform back; the trajectories should agree after numerical refinement. Finally use the linearly polarized laboratory drive and compare it with RWA predictions.

In the last comparison, vary solver tolerance at fixed drive strength to isolate numerical error, then vary $\Omega/\omega_0$ to explore the approximation error. A persistent difference after numerical convergence can be a physical counter-rotating effect. A difference in the circular-drive test instead points to propagation or frame conventions.

Lab-frame integration must resolve rapid carrier oscillation as well as the slower envelope. Adaptive solvers can miss narrow features if their initial steps are too large; impose a justified maximum step and align discontinuity boundaries.

Bloch-vector diagnostics and phases

Compute $s_a=\langle\sigma_a\rangle$. For a pure normalized state, $|\mathbf s|=1$. Under $H=\hbar\mathbf b\cdot\boldsymbol\sigma/2$,

$$\dot{\mathbf s}=\mathbf b\times\mathbf s.$$

Compare this independently integrated real vector equation with the spinor calculation. Population alone is insufficient: two states with equal populations can have different transverse Bloch components and respond differently to the next pulse.

A frame transformation rotates observables as well as state vectors. Compare laboratory and rotating-frame quantities only after transforming both consistently.

Exercises

For a resonant pulse with duration $(\pi+\delta)/\Omega$, compute the leading inversion error.

Solution

$P=\sin^2[(\pi+\delta)/2]=\cos^2(\delta/2)$, so $1-P=\delta^2/4+O(\delta^4)$. The pulse-area error produces a quadratic population error near ideal inversion, although the state-vector error is first order.

References

• Allen, L., and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, 1975 — driven two-level dynamics and rotating frames.
• Shirley, J. H., “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979–B987 (1965), doi:10.1103/PhysRev.138.B979 — periodic drive treatment beyond a constant-frame approximation.