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Chinese Control and Decision Conference (CCDC)

Parameter estimation of spontaneous emission by performing projective measurements

Xi Chen, Weiwei Zhou, Hong‐Yi Dai, Ming Zhang

Published May 1, 2012• 0 citations
Publisher

Abstract

The identification problem of open quantum systems was explored in this paper. We proposed an algorithm to testify the feasibility of a framework for dynamical generator identification through numerical simulations on the specific quantum process - the hydrogen atomic spontaneous emission induced by the two-level electronic transition from a high energy state to a low energy state. We demonstrated how to estimate spontaneous emission parameters by repeated projective measurements through the identification algorithm, and discussed the sampling frequency, the required least quantity of initial states and projective measurements and this algorithm's the scope and performance.

DOI
10.1109/ccdc.2012.6244258

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The identification problem of open quantum systems was explored in this paper. We proposed an algorithm to testify the feasibility of a framework for dynamical generator identification through numerical simulations on the specific quantum process - the hydrogen atomic spontaneous emission induced by the two-level electronic transition from a high energy state to a low energy state. We demonstrated how to estimate spontaneous emission parameters by repeated projective measurements through the identification algorithm, and discussed the sampling frequency, the required least quantity of initial states and projective measurements and this algorithm's the scope and performance.
By considering the identification problem of unknown but fixed Hamiltonian H = S 0 σ 0 + Σ i = x, y, z S i σ i where σ i (i = x, y, z) are pauli matrices and σ 0 = I, we explore the feasibility and limitation of empirically determining the Hamiltonian parameters for quantum systems under experimental conditions imposed by projective measurements and initialization procedures. It may be unsurprising to physicists that one can not obtain the knowledge of So no matter what kind of projective measurements and initialization are permitted, but the observation draws our attention to the importance of the parameter identifiability under different experimental condition. It has also been revealed that one can obtain the knowledge of |S z | and S x 2 + S y 2 at most when only the projective measurement {|0⟩⟨0|, |1⟩⟨1|} is permitted to perform on and initialize the qubit. Further more, we demonstrated that it is feasible to distinguish |S x |, |S y |, and |S Z | even without any a priori information about Hamiltonian if at least two kinds of projective measurement or initialization procedures are permitted. It should be emphasized that both projective measurements and initialization procedures play an important role in quantum system identification.
According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
This technical note studies the identification of unknown decoherence rates for a class of two-level quantum systems undergoing spontaneous emission. Our previous work shows that estimates of the unknown decoherence rates can be obtained from the ensemble averages by imposing constant control and monitoring a sequence of identical systems continuously. Inspired by the work, this technical note further investigates the identification based on a single quantum system. It is shown that a mean-ergodic solution exists for a single system under constant control and convergence to an ergodic process can be guaranteed by introducing some noise into the control signals. The ensemble average can thus be replaced by the time average of a single system, and the parameter estimate can be obtained from a single implementation.
An observer-based Hamiltonian identification algorithm for quantum systems has been recently proposed by two of the authors to estimate the dipole moment matrix of a quantum system requiring the measurement of the populations on all states. This could be experimentally difficult to achieve. We propose here an extension to a 3-level quantum system, having access to the population of the ground state only. By a suitable choice of the control field, we show that a continuous measurement of this population, alone, is enough to identify the field coupling parameters (dipole moment). Simulations with 20% of noise on the measured population illustrate the robustness of the proposed estimation algorithm and confirm the convergence analysis.
The spontaneous emission is investigated for an effective atomic two-level system in an intense coherent field with frequency lower than the vacuum-induced decay width. As this additional low-frequency field is assumed to be intense, multiphoton processes may be induced, which can be seen as alternative transition pathways in addition to the simple spontaneous decay. The interplay of the various interfering transition pathways influences the decay dynamics of the two-level system and may be used to slow down the spontaneous decay considerably. We derive from first principles an expression for the Hamiltonian including up to three-photon processes. This Hamiltonian is then applied to a quantum mechanical simulation of the decay dynamics of the two-level system. Finally, we discuss numerical results of this simulation based on a rubidium atom and show that the spontaneous emission in this system may be suppressed substantially.
Hamiltonian identification plays a key role in learning the dynamics of closed quantum systems. A novel strategy for Hamiltonian identification is developed in the paper by applying repeated projective measurements with a fixed period. The corresponding measurement outcome sequence forms a time-homogeneous Markov chain, the transition matrix of which is a quadratic function of the Hamiltonian. As a result, the identification method is achieved by two steps: 1) learning the transition matrix of the observed Markov chains; 2) identifying the Hamiltonian using the function relating it to the transition matrix, which is transformed to a manifold optimization problem. A simulation example is provided to show the efficacy of the proposed method.