Non-Hermitian Effective Hamiltonians

A no-jump effective Hamiltonian propagates an unnormalized conditional state. Its decreasing squared norm is a survival probability, not a numerical defect. This distinction is essential when using non-Hermitian evolution inside quantum trajectories. For resonance calculations with outgoing boundary conditions, see complex-energy resonances.

Helpful background. Lindblad solvers defines the collapse operators used here.

No-jump propagation and norm loss

For Hermitian $H$ and collapse operators $L_\mu$, define

$$H_{\rm eff}=H-\frac{i\hbar}{2}\sum_\mu L_\mu^\dagger L_\mu, \qquad i\hbar\,\dot{\tilde\psi}=H_{\rm eff}\tilde\psi.$$

Differentiating the squared norm gives

$$\frac{d}{dt}\|\tilde\psi\|^2 =-\sum_\mu\|L_\mu\tilde\psi\|^2\leq0.$$

Starting immediately after a known normalized preparation or jump, $S(t)=\|\tilde\psi(t)\|^2$ is the probability of no subsequent detection by time $t$ in this unraveling. Conditional expectations divide by $S(t)$, but the stored survival weight must be retained.

For the normalized conditional state $\psi=\tilde\psi/\sqrt S$, the hazard is $\lambda(t)=\sum_\mu\|L_\mu\psi\|^2$ and $S(t)=\exp[-\int_0^t\lambda(s)\,ds]$. It need not be a single exponential when the conditional state changes.

Event-driven jump times

Draw $r$ uniformly in $(0,1)$ and propagate until $S(t)=r$. Bracket the crossing, refine the event time, choose channel $\mu$ with probability proportional to $\|L_\mu\tilde\psi\|^2$, and reset to $L_\mu\tilde\psi/\|L_\mu\tilde\psi\|$. Draw a new threshold after the jump.

A threshold crossing should use the integrator’s dense output or a converged local solve. Linear interpolation of a coarse norm history adds event-time bias even when the endpoint state is accurate. Track event-location tolerance separately from state-propagation tolerance.

For very long intervals it can be preferable to integrate a normalized conditional state together with a log survival weight. This avoids underflow; renormalizing without updating that weight loses the waiting-time distribution.

Writing $C=\sum_\mu L_\mu^\dagger L_\mu$ and $\ell=\log S$, the equivalent equations are

$$\dot\psi=-\frac{i}{\hbar}H\psi-\frac12(C-\lambda I)\psi, \qquad \dot\ell=-\lambda, \qquad \lambda=\langle\psi|C|\psi\rangle.$$

The added scalar drift exactly compensates the norm loss. Stop at $\ell=\log r$ and reset $\ell=0$ after each jump. This is deterministic nonlinear evolution between jumps, not a diffusive stochastic equation. Monitor both the normalization error and the integrated hazard.

Decaying qubit with a dark component

Let $H=0$, $L=\sqrt\gamma|0\rangle\langle1|$, and $\psi(0)=a|0\rangle+b|1\rangle$. Then

$$\tilde\psi(t)=a|0\rangle+b e^{-\gamma t/2}|1\rangle,\qquad S(t)=|a|^2+|b|^2e^{-\gamma t}.$$

If $r\leq|a|^2$, no jump occurs at any finite time. Otherwise

$$t_{\rm jump}=-\frac1\gamma \log\frac{r-|a|^2}{|b|^2}.$$

This benchmark is more informative than a fully excited state: it tests the nonzero probability of an infinite waiting time and the change of the conditional state toward the dark state. A solver must not force a jump merely because it has integrated for a long time.

Event conditioning also changes near the dark limit. A survival error $\delta S$ near a simple crossing gives approximately $|\delta t|=|\delta S|/(\lambda S)$. A small absolute norm residual can therefore correspond to a large time error when the hazard is tiny. Report event-time convergence for thresholds close to $|a|^2$, and treat trajectories surviving beyond a finite observation window as censored records rather than invented late jumps.

Stability and interpretation

Use general exponential or ODE algorithms; $H_{\rm eff}$ is not Hermitian. Check monotonically decreasing survival probability to the numerical tolerance and compare its derivative with the collapse rate. Nonnormal matrices can make eigenvector diagonalization ill-conditioned, so a residual-checked exponential action is often preferable.

Conditional no-jump evolution alone does not give the unconditioned density matrix. Omitting jump branches removes the probability transferred into final states. Also, a general phenomenological imaginary potential need not admit the same monitored-system interpretation unless a consistent set of collapse channels is supplied.

Exercise

For an initially excited qubit, derive the mean waiting time and a sampling formula.

Solution

$S(t)=e^{-\gamma t}$, so the density is $\gamma e^{-\gamma t}$ and the mean is $1/\gamma$. The inverse survival construction gives $t=-\log r/\gamma$. Histograms and empirical survival curves test the random-number path separately from the propagator.

References

• Dalibard, J., Castin, Y., and Mølmer, K., “Wave-function approach to dissipative processes in quantum optics,” Physical Review Letters 68, 580–583 (1992), doi:10.1103/PhysRevLett.68.580.
• QuTiP developers, “Monte Carlo Solver”, QuTiP documentation — norm thresholds and collapse selection.