The Collapse Before a Quantum Jump Transition

doi:10.1016/S0034-4877(20)30011-2

Reports on Mathematical Physics

Volume 85, Issue 1, February 2020, Pages 69-76

https://doi.org/10.1016/S0034-4877(20)30011-2 Get rights and content

We may infer a transition $| n 〉 \to | m 〉$ between energy eigenstates of an open quantum system by observing the emission of a photon of Bohr frequency $ω_{m n} = (E_{n} - E_{m}) / ℏ$ . In addition to the “collapses” to the state $| m 〉$ , the measurement must also have brought into existence the pre-measurement state $| n 〉$ . As quantum trajectories are based on past observations, the condition state will jump to $| m 〉$ , but the state $| n 〉$ does not feature in any essential way. We resolve this paradox by looking at quantum smoothing and derive the time-symmetric model for quantum jumps.

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 $| a 〉$ . 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 ${\begin{matrix} | n 〉 & : n = 0,1,2,… \end{matrix}}$ of energy eigenstates with corresponding energy eigenvalues {E_n: n = 0, 1, 2, …}; when the atom emits a photon its frequency must be one of the Bohr frequencies ω_mn = E_n - E_m. Furthermore let us assume that the set of Bohr frequencies is nondegenerate so that we observe a photon with the frequency ω_mn, then we know that the electron has undergone a transition $| n 〉 \to | m 〉$ .

Conventional theory tells us that the state collapses to $| m 〉$ 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 $| n 〉$ : 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 $H_{sys} = \sum_{n = 0}^{N - 1} E_{n} | n 〉〈 n |$ with energy eigenvalues E₀ < E₁ < ··· < E_N₋₁. The set, F, of Bohr frequencies is then the collection ω_mn = E_n - E_m. The positive Bohr frequencies are then ω_mn 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 ω_mn at time τ > 0, then we have Y_nm(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 ${\hat{p}}_{τ} + = {\hat{p}}_{τ} - + {| m 〉〈 m | - {\hat{p}}_{τ} -} \equiv | m 〉〈 m | .$ 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 $| n 〉 \to | m 〉$ causing the conditioned state to jump from $\hat{ρ} (τ^{-})$ to the eigenstate $| m 〉$ , which ignores that state must have transitioned from $| n 〉$ at the τ.

Knowledge of the conditioned state $\hat{ρ} (τ^{-})$ 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 H_λ = H₀ + λH_int where $\begin{matrix} H_{0} = H_{sys} \otimes ∥_{bath} + ∥_{sys} \otimes \int ω (k) a (k) * (a) (k) d^{3} k, \\ H_{int} = \int Θ (k) \otimes a (k) * d^{3} k + H . c . \end{matrix}$ 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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