Quantum trajectories replace one deterministic master equation by many stochastic conditioned histories. Each history follows the information available in a particular measurement record; averaging sufficiently many histories recovers the unconditional density matrix.
Guide 52 owns Lindblad master equations, Guide 54 owns continuous measurement and filtering, and Guide 74 owns non-Hermitian conditional evolution. This guide connects them algorithmically: Monte Carlo wave functions, quantum jumps, no-jump evolution, diffusive homodyne trajectories, waiting-time sampling, ensemble recovery and the freedom to unravel one Lindblad equation in different physically meaningful ways.
Unconditional Lindblad equation → choose monitored environment → derive measurement record → evolve conditioned state stochastically → average trajectories → recover the same master equation.
1. Start from a Lindblad master equation
Consider
dρ/dt=−i[H,ρ]+Σ_k[L_kρL_k†−(1/2){L_k†L_k,ρ}].
This equation gives the state when environmental measurement results are ignored or averaged over.
An unraveling specifies one stochastic pure- or mixed-state process whose ensemble mean equals ρ(t).
2. Effective non-Hermitian Hamiltonian
Define
H_eff=H−(i/2)Σ_k L_k†L_k.
Between detected jumps, an unnormalised state evolves according to
d|ψ̃⟩/dt=−iH_eff|ψ̃⟩.
The decreasing norm records the probability that no monitored jump has occurred.
3. Jump probability in a short interval
For a normalised conditioned state |ψ⟩, the probability of jump channel k during dt is
dp_k=⟨ψ|L_k†L_k|ψ⟩dt.
The total jump probability is dp=Σ_k dp_k.
If a jump of type k occurs, update
|ψ⟩→L_k|ψ⟩/√⟨L_k†L_k⟩.
4. No-jump update
If no jump occurs during dt, evolve with I−iH_eff dt and renormalise.
To first order, the unnormalised squared norm after the step is
1−Σ_k⟨L_k†L_k⟩dt=1−dp.
Thus the norm loss exactly matches the probability assigned to jumps.
5. Why averaging jumps reproduces Lindblad evolution
For one short step, combine:
- the no-jump branch with probability 1−dp;
- jump branch k with probability dp_k.
Expanding the ensemble density matrix to first order in dt gives
ρ+dρ=ρ−i[H,ρ]dt−(1/2)Σ_k{L_k†L_k,ρ}dt+Σ_kL_kρL_k†dt.
This is exactly the Lindblad increment. The trajectory algorithm is therefore an unbiased stochastic representation of the master equation under the chosen unraveling.
6. Amplitude damping example
Take a two-level atom with jump operator
L=√κ |g⟩⟨e|=√κ σ_-
and H=0.
The effective Hamiltonian is
H_eff=−iκ|e⟩⟨e|/2.
An initially excited unnormalised no-jump state is exp(−κt/2)|e⟩, whose squared norm is exp(−κt).
7. Waiting-time distribution
For the initially excited atom, the no-jump survival probability is
S(t)=e^{-κt}.
The jump-time density is
w(t)=−dS/dt=κe^{-κt}.
This is an exponential waiting-time distribution with mean 1/κ.
8. Exact inverse-transform sampling
Draw uniform random number r∈(0,1). Set survival probability equal to r:
e^{-κt_j}=r.
Then
t_j=−ln r/κ.
For constant-rate amplitude damping, sampling jump times directly is more accurate and efficient than taking extremely small fixed time steps.
9. Driven atom has state-dependent hazard
If H drives population between |g⟩ and |e⟩, the instantaneous jump rate becomes
λ(t)=κ⟨e|ρ_c(t)|e⟩.
The waiting-time distribution is no longer a simple exponential because the conditioned excited population changes between jumps.
One can integrate the unnormalised no-jump state until its norm crosses a random threshold.
10. Monte Carlo wave-function algorithm
- Prepare a normalised pure state |ψ⟩.
- Draw random threshold r∈(0,1).
- Evolve under H_eff without renormalising.
- Stop when the norm squared reaches r.
- Select jump channel k with probability proportional to ⟨L_k†L_k⟩.
- Apply L_k and renormalise.
- Draw a new threshold and continue.
- Repeat many trajectories and average observables.
Dalibard, Castin and Mølmer developed this Monte Carlo wave-function approach as an efficient alternative to density-matrix propagation for suitable open systems.
11. Memory advantage in simulation
A pure state on Hilbert-space dimension d requires O(d) complex amplitudes. A dense density matrix requires O(d²).
Quantum trajectories can therefore reduce memory substantially for large sparse systems, at the cost of statistical sampling over many realisations.
The crossover depends on the number of trajectories required, sparsity, observable variance and whether the state remains approximately pure under the monitored unraveling.
12. Statistical error
For an observable estimate from M independent trajectories, ordinary Monte Carlo standard error scales as
σ_mean=σ_traj/√M.
Reducing error by a factor of ten therefore needs roughly one hundred times as many statistically independent trajectories unless variance reduction is used.
13. Rare jumps create high-variance estimators
If the quantity of interest is dominated by very rare trajectories, naive sampling can converge slowly even when each trajectory is cheap.
Importance sampling, cloning/tilted-ensemble methods or specialised rare-event techniques may be needed. Trajectory simulation does not make rare events statistically free.
14. Unravelling is not unique
The same Lindblad master equation can arise from different continuous measurements of the environment.
Direct photon counting produces discrete jumps. Homodyne detection mixes the output with a strong local oscillator and produces a continuous noisy current. Heterodyne detection measures two field quadratures and yields another diffusion process.
All can average to the same unconditional ρ(t) while producing different conditioned histories.
15. Jump unraveling and measurement record
For photon counting, introduce Poisson increments dN_k∈{0,1} with conditional mean
E[dN_k|ρ_c]=Tr(L_k†L_kρ_c)dt.
The conditioned stochastic master equation contains deterministic no-jump drift plus terms proportional to dN_k−E[dN_k].
The innovation is the surprise relative to the expected count rate.
16. Diffusive homodyne unraveling
For one monitored channel L with homodyne phase φ and efficiency η, a standard conditioned equation is
dρ_c=−i[H,ρ_c]dt+D[L]ρ_c dt+√η H[e^{-iφ}L]ρ_c dW,
where dW is a Wiener increment with mean zero and variance dt, D[L]ρ=LρL†−(1/2){L†L,ρ}, and
H[c]ρ=cρ+ρc†−Tr[(c+c†)ρ]ρ.
The measurement current has schematic form
dY=√η Tr[(e^{-iφ}L+e^{iφ}L†)ρ_c]dt+dW.
17. Ito calculus matters
Wiener increments obey dW²=dt in Ito calculus, while products such as dt dW and dt² are neglected at first stochastic order.
Treating dW as an ordinary infinitesimal and setting dW²=0 produces wrong drift terms and can destroy normalisation.
Guide 54 develops continuous filtering in more detail; here the emphasis is the trajectory interpretation and ensemble recovery.
18. Pure-state stochastic Schrödinger equations
For unit-efficiency monitoring and an initially pure state, the conditioned state can remain pure. One can then evolve a stochastic state vector instead of a density matrix.
The exact stochastic Schrödinger equation depends on the chosen unraveling. Diffusive and jump equations are different sample-path descriptions of the same average channel.
19. Ensemble average removes the innovation
For a correctly derived filter, E[dW]=0 and the compensated Poisson innovation also has zero conditional mean.
Averaging the stochastic equation therefore removes record-dependent terms and returns the deterministic Lindblad equation.
This is a powerful implementation test: simulate many trajectories and compare their mean with an independent master-equation solver.
20. Measurement efficiency
If η<1, only part of the environmental record is observed. The conditioned state is generally mixed because unobserved channels continue to carry unknown information.
One can split a physical channel into observed and unobserved components, for example √ηL and √(1−η)L. The unconditional dissipator remains D[L].
21. Dark counts and detector dead time
Real photon-counting records can include dark counts, missed photons and dead-time effects. These modify the likelihood of the record and therefore the conditioned state.
A trajectory model that assumes perfect detection but is fitted to imperfect records can attribute detector noise to quantum dynamics.
22. Gauge freedom of Lindblad operators
Lindblad representations are not unique. Unitary mixing among jump operators leaves the dissipator invariant, and shifts by constants can be compensated by Hamiltonian terms under appropriate transformations.
Different physical measurement schemes select different unravelings from this freedom.
Therefore a plotted jump history is not an invariant property of the unconditional master equation alone.
23. Trajectories are conditioned knowledge states
A quantum trajectory usually represents the observer’s state conditioned on a measurement record.
Whether one interprets it as an underlying objective pure-state path is an additional interpretive claim not required for the predictive formalism.
Two observers with different access to the record can assign different conditioned states while agreeing on all observable probabilities appropriate to their information.
24. Quantum jumps are not instantaneous Hamiltonian kicks
The state-update L_k|ψ⟩ arises from conditioning on a detected environmental event in the Markovian measurement model.
It need not represent a literal instantaneous force on the system. The microscopic system-environment interaction can be continuous while the observed record is a discrete detection event.
25. Trajectories and non-Hermitian physics
Between jumps, H_eff is non-Hermitian. Guide 74 explains why this does not by itself define a complete trace-preserving quantum theory.
The missing norm is precisely restored statistically by jump branches. Ignoring jumps but renormalising the state changes the ensemble to a postselected no-jump experiment.
26. Trajectories and non-Markovian environments
The standard Monte Carlo wave-function derivation assumes a Markovian Lindblad equation or an equivalent memoryless monitored bath.
For non-Markovian dynamics, naive reuse of the same jump rules may fail because the environment carries memory and intermediate conditioned states may not obey a closed Markov process.
Extended pseudomode, reaction-coordinate or non-Markovian stochastic Schrödinger methods can restore a trajectory-like description in an enlarged space. Guide 76 owns the memory framework.
27. Full counting statistics
Trajectories naturally record numbers and times of jumps. Introduce a counting field χ and weight each jump by e^{iχ}. The dominant eigenvalue of the resulting tilted generator can encode long-time cumulants of the count distribution.
This connects open quantum dynamics with large-deviation theory, dynamical phase transitions and rare-event sampling.
28. Example: count statistics of spontaneous decay
A single initially excited two-level atom with no repumping can emit at most one photon. The trajectory count N(t) is therefore Bernoulli:
P[N=0]=e^{-κt};P[N=1]=1−e^{-κt}.
The mean count is 1−e^{-κt}, not κt. A Poisson count process arises only when repeated excitation/steady emission makes multiple events possible.
29. Common misconceptions
“A trajectory is the same as the density matrix.” One trajectory is a conditioned stochastic history; the density matrix is the ensemble average when the record is ignored.
“The Lindblad equation determines a unique trajectory.” Different monitored observables can unravel the same equation differently.
“Norm loss under H_eff is missing probability.” It is the probability weight transferred to jump outcomes.
“A jump must be a literal instantaneous microscopic event.” It is an update conditioned on a detector record within the effective measurement model.
30. Worked synthesis problem
An excited atom decays with rate κ.
H_eff=−iκ|e⟩⟨e|/2.- No-jump amplitude:
e^{-κt/2}. - No-jump probability:
e^{-κt}. - Jump-time density:
κe^{-κt}. - Sampling rule: draw r and set
t_j=−lnr/κ. - After the jump, the state becomes |g⟩.
Averaging many such trajectories gives excited-state population e^{-κt}, exactly matching the Lindblad solution.
31. Numerical validation checklist
- Verify jump probabilities are nonnegative and total O(dt).
- Check no-jump norm loss equals total jump probability to first order.
- Compare ensemble averages with an independent master-equation solver.
- Decrease time step and confirm convergence when fixed-step sampling is used.
- For diffusive equations, use an Ito-consistent integrator.
- Track statistical confidence intervals over trajectories.
- For inefficient detection, compare against the correctly mixed conditional state.
32. Practice set
- Write H_eff for a Lindblad equation.
- What is the probability of jump k in dt?
- What is the post-jump state?
- Why does no-jump norm decrease?
- How does the MCWF ensemble recover the master equation?
- What is the waiting-time density for simple amplitude damping?
- Why can trajectories reduce memory cost?
- Why is an unraveling not unique?
- What distinguishes a jump record from a homodyne record?
- Why can naive Markovian jumps fail for a non-Markovian bath?
Answers
H−(i/2)ΣL_k†L_k.⟨L_k†L_k⟩dt.L_k|ψ⟩/√⟨L_k†L_k⟩.- The lost norm equals the probability that a monitored jump occurred.
- Weighted no-jump and jump branches reproduce every Lindblad term to first order.
κe^{-κt}.- Each pure trajectory stores O(d) amplitudes instead of O(d²) density-matrix entries.
- The same unconditional channel can correspond to photon counting, homodyne, heterodyne or other monitored-environment schemes.
- Counting gives discrete Poisson increments; homodyne gives a continuous noisy current with Wiener increments.
- The environment’s hidden state carries memory, so the conditioned system alone may not form a closed Markov process.
Sources and further study
[1] Jean Dalibard, Yvan Castin and Klaus Mølmer, Wave-function approach to dissipative processes in quantum optics, Physical Review Letters 68, 580 (1992). Foundational Monte Carlo wave-function method.
[2] R. Dum, P. Zoller and H. Ritsch, Monte Carlo simulation of the atomic master equation for spontaneous emission, Physical Review A 45, 4879 (1992).
[3] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Reviews of Modern Physics 70, 101 (1998). Comprehensive trajectory review.
[4] Howard M. Wiseman and Gerard J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). Continuous measurement, jump and diffusive stochastic master equations.
[5] H. Carmichael, An Open Systems Approach to Quantum Optics, Springer Lecture Notes in Physics (1993). Quantum trajectory and photodetection framework.
Continue through Quantum Mathematics — Batch 20
Guide 77: Quantum Zeno Effect, Anti-Zeno Effect, Survival Probabilities and Zeno Subspaces develops repeated-measurement constraints. Guide 78: Quantum Speed Limits, Mandelstam–Tamm, Margolus–Levitin, Bures Geometry and Control Times develops minimum-time bounds. Guide 79: Quantum Quasiprobabilities, Wigner Functions, Negativity and Discrete Phase Space develops quantum phase-space representations.