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arXiv:1810.03225

Regimes of Coherent Intermittency in the Next Quantum Jump of a Multilevel System

Massimo Porrati B, Nicholas E. Penthorn, John Rooney, H. W. Jiang, Seth Putterman

Published Oct 7, 2018• 1 citation
arXivHTMLPDF

Abstract

In an externally driven multilevel quantum system observation that the NEXT jump has not yet happened affects its future development. In previous work [Phys. Rev. A36, 929 (1987)] it was shown that this class of measurement makes it possible to observe remarkably long dark intervals -- or intermittency -- in the atomic fluorescence of an atom with 3 or more levels. Those calculations were carried out when the driven oscillations or Rabi flopping between the ground state and a strongly fluorescing state were fast compared to its lifetime. In systems with solid state Qubits the accessible parameter space is generally limited to the regime where oscillations are slower than the lifetime. In this paper we evaluate intermittency in atomic transitions, due to measurements with a null result, in this limit. During the dark periods the wave function of the continuously measured multilevel system is coherent.

DOI
10.48550/arxiv.1810.03225

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In an externally driven multilevel quantum system observation that the NEXT jump has not yet happened affects its future development. In previous work [Phys. Rev. A36, 929 (1987)] it was shown that this class of measurement makes it possible to observe remarkably long dark intervals -- or intermittency -- in the atomic fluorescence of an atom with 3 or more levels. Those calculations were carried out when the driven oscillations or Rabi flopping between the ground state and a strongly fluorescing state were fast compared to its lifetime. In systems with solid state Qubits the accessible parameter space is generally limited to the regime where oscillations are slower than the lifetime. In this paper we evaluate intermittency in atomic transitions, due to measurements with a null result, in this limit. During the dark periods the wave function of the continuously measured multilevel system is coherent.
Time correlations in the fluorescence of a multilevel atom are studied in terms of the quantum statistics which describe the emission of the next photon. The next photon equations are substantially simpler than the a photon equations (or so-called optical Bloch equations) yet they contain phase information lost in the averaging procedure that yields the a photon equations. Employing the next photon equations we present a quantum formalism in which the time development of the wave function of the atom is dramatically modified by the observation of a dark period. This ``collapse'' of the quantum state due to measurements with a null result (such as the failure to record events in a photodetector) is the cause of intermittent atomic fluorescence even when the exciting field is arbitrarily coherent. The next photon and a photon formalisms are contrasted and applied to photon antibunching. The phenomenon of photon bunching which characterizes sideband correlations is also calculated. Observable effects particular to the coherent intermittency are emphasized.
For pt.I see Phys. Rev. A, vol.37, p.4303 (1988). The principle of wavefunction collapse by non-detection of photons leads to simple but rigorous expressions for the average dark and bright periods, T D and T B , of intermittent quantum jump fluorescence from a V-configuration three-level atom. Here further consequences of this principle are explored, including: the derivation of some new correlation functions which exhibit the physical characteristics of extended dark and bright periods; the expressions for T D and T B when one transition is driven coherently and the other incoherently; and a general expression for the characteristic time to be sure that a quantum jump has occurred for any type of driving field.
Early results of Agarwal and Jha [Z. Phys. B 35, 391 (1979)] are used to examine the question of quantum jumps in a Raman (\ensuremath{\Lambda}) system driven by two coherent fields. We discuss both short-time (${\ensuremath{\gamma}}_{1}$t\ensuremath{\lesssim}1) and long-time (${\ensuremath{\gamma}}_{2}$t\ensuremath{\approxeq}1) behavior of the intensity correlations assuming that the Stokes transition is very weak, ${\ensuremath{\gamma}}_{2}$\ensuremath{\ll}${\ensuremath{\gamma}}_{1}$. We calculate the counting statistics of the dark periods using the higher-order intensity correlations. The results differ from those obtained using arguments based on population trapping which predict only a very rare ``jump'' event. The intermittence in fluorescence produced by a coherently driven \ensuremath{\Lambda} system is interpreted in terms of an equivalent two-state Markov process.
A measurement of the time between quantum jumps implies the capability to measure the next jump. During the time between jumps the quantum system is not evolving in a closed or unitary manner. While the wave function maintains phase coherence it evolves according to a non-Hermitian effective Hamiltonian. So under null measurement the timing of the next quantum jump can change by very many orders of magnitude when compared to rates obtained by multiplying lifetimes with occupation probabilities obtained via unitary transformation. The theory developed in 1987 for atomic fluorescence is here extended to transitions in transmon qubits. These systems differ from atoms in that they are read out with a harmonic cavity whose resonance is determined by the state of the qubit. We extend our analysis of atomic fluorescence to this infinite level system by treating the cavity as a quantum system. We find that next photon statistics is highly nonexponential and when implemented will enable faster readout, such as on timescales shorter than the decay time of the cavity. Commonly used heterodyne measurements are applied on timescales longer than the cavity lifetime. The overlap between the next photon theory and the theory of heterodyne measurement which are described according to the stochastic Schr\"odinger equation is elucidated. In the limit of large dispersion the intrinsic error for next jump detection, at short time, tends to zero. Whereas for short-time dyne detection the error remains finite for all values of dispersion.
Recently we predicted a random blinking, i.e., macroscopic quantum jumps, in the fluorescence of a laser-driven atom-cavity system [Metz et al., Phys. Rev. Lett. 97, 040503 (2006)]. Here we analyze the dynamics underlying this effect in detail and show its robustness against parameter fluctuations. Whenever the fluorescence of the system stops, a macroscopic dark period occurs and the atoms are shelved in a maximally entangled ground state. The described setup can therefore be used for the controlled generation of entanglement. Finite photon detector efficiencies do not affect the success rate of the state preparation, which is triggered upon the observation of a macroscopic fluorescence signal. High fidelities are achieved even in the vicinity of the bad cavity limit due to the inherent role of dissipation in the jump process.