The single-photon quantum filtering problems have been investigated recently with applications in quantum computing. In practice, the detector responds with a quantum efficiency of less than unity since there exists some mode mismatch between the detector and the system and the single-photon signal may be corrupted by quantum white noise. Consequently, quantum filters based on multiple measurements are designed in this paper to improve estimation performance. More specifically, the filtering equations for a 2-level quantum system driven by a single-photon input state and under multiple measurements are presented in this paper. Four scenarios, ie, (1) 2 diffusive measurements with Q-P quadrature form, (2) 2 diffusive measurements with Q-Q quadrature form, (3) diffusive plus Poissonian measurements, and (4) 2 Poissonian measurements, are considered. It is natural to compare the filtering results, ie, measuring a single channel or both channels, which one is better? By the simulation where we use a single photon to excite an atom, it seems that multiple measurements enable us to excite the atom with higher probability than only measuring a single channel. In addition, a measurement back-action phenomenon is revealed by the simulation results. KEYWORDShomodyne detection, photon counting, quantum filtering, quantum trajectories, single-photon state Over the past few decades, quantum filtering has drawn researchers' lots of attention and has been rapidly developed. 1-3 Its modern form and foundational framework were firstly studied by Belavkin. 4,5 Particularly in quantum optics, quantum filtering is known as a master equation and a stochastic master equation (SME). The latter represents the stochastic evolution of the conditional density operator when the system interacts with the field. The quantum trajectory theory, which can describe this stochastic process, was developed by Carmichael 6 and was widely applied in quantum filtering and quantum control. [7][8][9][10][11][12][13] The framework of quantum filtering for a system driven by Gaussian input fields, such as coherent state, squeezed state, thermal state, and vacuum state, were well treated in a series of articles. 14-17 A 2-level atom driven by a single-photon state was considered in the work of Gheri et al, 18 and master equations were derived in detail to illustrate the formalism presented is applicable to N-photon wave packets. Filtering equations for systems driven by single-photon states or 528 ;32:528-546. Recently, based on the general framework for single-photon filtering, 9 the SMEs for quantum systems driven by a single-photon input state that is contaminated by quantum vacuum noise have been presented in our other work. 19 The composite state is prepared as |1 ⟩⊗|0⟩, where |1 ⟩ is the single-photon state and |0⟩ is the vacuum noise. By input-output formalism, 6,15,24,25 the output field state would be a superposition state | out ⟩ = s 11 |1 ⟩ ⊗ |0⟩ + s 21 |0⟩ ⊗ |1 ⟩, where is the output pulse shape, and s 11 and s 21 are parameters of the beam spli...
The purpose of this paper is to propose quantum filters for a twolevel atom driven by two continuous-mode counter-propagating photons and under continuous measurements. Two scenarios of multiple measurements, 1) homodyne detection plus photodetection, and 2) two homodyne detections, are discussed. Filtering equations for both cases are derived explicitly. As demonstration, the two input photons with rising exponential and Gaussian pulse shapes are used to excite a two-level atom under two homodyne detection measurements. Simulations reveal scaling relations between atom-photon coupling and photonic pulse shape for maximum atomic excitation.
The purpose of this paper is to study the interaction between a two-level system (qubit) and two continuous-mode photons. Two scenarios are investigated: Case 1, how a two-level system changes the pulse shapes of two input photons propagating in a single input channel; and Case 2, how a two-level system responds to two counter-propagating photons, one in each input channel. By means of a transfer function approach, the steady-state output field states for both cases are derived analytically in both the time and frequency domains. For Case 1, two examples are presented. In Example 1 a two-photon input state of Gaussian pulse shape is used to excite a two-level atom. The joint probability distribution in the time domain and the joint spectra of the output two-photon state are plotted. The simulation demonstrates that in the time domain the atom tends to stretch out the two photons. Moreover, the prominent difference between the joint probability distribution of the output two-photon state and that of the input two-photon state occurs exactly under the setting when the two-level atom is most efficiently excited. In Example 2, a two-photon input state of rising exponential pulse shape is used to excite a two-level atom. Strong anti-correlation of the output two-photon state is observed, which is absent in Example 1 for the Gaussian pulse shape. Such difference indicates that different pulse shapes give rise to drastically different frequency entanglement of the output two-photon state. Example 3 is used to illustrate Case 2, where two counter-propagating single photons of rising exponential pulse shapes are input to a two-level atom. The frequency-dependent Hong-Ou-Mandel (HOM) interference phenomenon is observed. Moreover, when the two output photons are in the same channel, they are anti-correlated. The simulation results base on the analytic forms of output twophoton states are consistent with those based on quantum master or filter equations [43], [11]. Similar physical phenomena have been observed in physical settings such as cavity opto-mechanical systems and Keer nonlinear cavities.
In this paper, the problem of quantum filtering with two homodyne detection measurements for a two-level system has been considered. The quantum system is driven by two input light field channels, each of which contains a single photon. A quantum filter based on multiple measurements is designed; both the master equations and stochastic master equations are derived. In addition, numerical simulations for master equations with various pulse shape parameters are also compared.
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