The Collapse Before a Quantum Jump Transition
Introduction
In the Copenhagen interpretation, an observable A does not possess an actual value until we measure it, in which case we observe an eigenvalues a and the state collapses into the eigenstate . This assumes a direct measurement. In practice, we perform indirect measurements and make inferences on what the state must be.
Consider an atomic electron with a complete orthonormal basis of energy eigenstates with corresponding energy eigenvalues {En: n = 0, 1, 2, …}; when the atom emits a photon its frequency must be one of the Bohr frequencies ωmn = En - Em. Furthermore let us assume that the set of Bohr frequencies is nondegenerate so that we observe a photon with the frequency ωmn, then we know that the electron has undergone a transition .
Conventional theory tells us that the state collapses to at the time of measurement: but in fact we learn more! We are not measuring the Hamiltonian H of the electron, but instead measuring a transition between its eigenstates, and so also infer that the state of the system immediately before measurement was : despite making no assumptions on the initial state!
There has been much interest in using the full data recorded through continual quantum measurement of an open system to estimate, for instance, indirect measurement made during the monitoring period. This had lead to a time-symmetric theory. interventions in quantum measurement [1], as well as recent experimental tests [2]. In Section 2 we outline the theory, and in Section 3 derive the time-symmetric form. Here we must derive the result for photon counting as opposed to homodyne measurement of quadratures from previous papers.
Section snippets
Estimating Bohr transitions
Let us now describe the model. We assume, for simplicity, that we have an N-level system with Hamiltonian with energy eigenvalues E0 < E1 < ··· < EN−1. The set, F, of Bohr frequencies is then the collection ωmn = En - Em. The positive Bohr frequencies are then ωmn with m < n and we assume that they are nondegenerate.
The system couples to a bath (the quantum electromagnetic field) and the evolution is described by the quantum stochastic differential equation (QSDE)
Observing a transition event
Having set up the model, we now look at what typically happens. From an experimental point of view, we are monitoring the output light field and watching for photons resonant with one of the Bohr frequencies. If our first observation is a photon of frequency ωmn at time τ > 0, then we have Ynm(t) jumping from value 0 to 1 at τ, with the other channels all registering zero counts. From the SME (9) we therefore have jump This just says that the state
Conclusion
We have derived the time-symmetric estimation for indirect measurements made during a continuous monitoring of photons emitted from an open quantum system. The observation of a photon at time τ reveals that there was a transition causing the conditioned state to jump from to the eigenstate , which ignores that state must have transitioned from at the τ.
Knowledge of the conditioned state however is insufficient if we wish to estimate the
Appendix A: Derivation of the QSDE
We now give a microscopic derivation of the QSDE (1) from the weak coupling limit formalized as a quantum central limit [9]. The total Hamiltonian for the system and bath is taken to be Hλ = H0 + λHint where Here a (k) is the annihilator for a photon of wave-number k (we ignore polarizations), ω (k) = c|k| is the associated energy, and Θ (k) is an operator on the system space (due to dipole moments it may have a k
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