^{1,a)}and William W. Kennerly

^{1}

### Abstract

We investigate theory of single-photon control from a two-level single-molecule source irradiated by laser pulses of various shapes and pulse durations in terms of quantum trajectories which link stochastic dynamics of the radiating source with quantum measurement theory. Using Monte Carlo wave function simulation, we analyze the detailed dissipative dynamics of the single-molecule source and the photon statistics as revealed by repeated *Gedanken*photonmeasurement on the single radiating source. We show that much of the photon statistics from the two-level single-molecule single-photon sources, including few-photon emission probability, waiting time distribution, and two-time correlation function of the fluorescent light, can be understood qualitatively from the simple picture of Rabi nutation and pulse in terms of pulse areas.

This work was supported partially by the Army Research Office through the DURINT program and the Semiconductor Research Corporation through the FCRP and NRI programs.

I. INTRODUCTION

II. THEORETICAL MODEL

A. Quantum trajectory and single emitter dynamics

B. Monte Carlo wave function simulation

III. RESULTS AND THEIR INTERPRETATION

A. Dissipative dynamics of the single-molecule source

B. Few-photon emission probability

C. Waiting time distribution

D. Two-time correlation functions

IV. CONCLUSION

### Key Topics

- Photons
- 112.0
- Excited states
- 20.0
- Wave functions
- 17.0
- Quantum measurement theory
- 15.0
- Correlation functions
- 13.0

## Figures

MCWF simulation of the 2LS source, plotted as the excited state population vs time, for several pulse shapes and detunings at fixed pulse area of . The thick solid line shows the average excited state population over 100 000 quantum trajectories. The thin solid and dashed lines are two sample quantum trajectories showing the conditional excited state probability of the pulse-driven 2LS source . The profiles of the square , isosceles triangle, left triangle, and right triangle pulses are shown in red at right figure (in arbitrary units).

MCWF simulation of the 2LS source, plotted as the excited state population vs time, for several pulse shapes and detunings at fixed pulse area of . The thick solid line shows the average excited state population over 100 000 quantum trajectories. The thin solid and dashed lines are two sample quantum trajectories showing the conditional excited state probability of the pulse-driven 2LS source . The profiles of the square , isosceles triangle, left triangle, and right triangle pulses are shown in red at right figure (in arbitrary units).

Probability that a single 2LS source irradiated by a square resonant laser pulse will emit exactly zero, one, and two photons, as a function of Rabi frequency and pulse duration , in the long measurement time limit. The red curves in the figure represent laser pulses of pulse area for , 2, 3, 4, and 5.

Probability that a single 2LS source irradiated by a square resonant laser pulse will emit exactly zero, one, and two photons, as a function of Rabi frequency and pulse duration , in the long measurement time limit. The red curves in the figure represent laser pulses of pulse area for , 2, 3, 4, and 5.

Probability of emitting exactly one photon in the long measurement time limit as a function of pulse duration and maximum Rabi frequency for four different pulse shapes. The axis for the square pulse is scaled by half compared to those of the triangular pulses, so the pulse areas match, point for point, between different panels. Also shown in the figure are the contour lines representing for each pulse shape.

Probability of emitting exactly one photon in the long measurement time limit as a function of pulse duration and maximum Rabi frequency for four different pulse shapes. The axis for the square pulse is scaled by half compared to those of the triangular pulses, so the pulse areas match, point for point, between different panels. Also shown in the figure are the contour lines representing for each pulse shape.

Waiting time distribution of photon emission for a ground state 2LS source irradiated by four differently shaped resonant laser pulses. The envelope profiles of the square , isosceles triangle, left triangle, and right triangle pulses of duration are shown in the left figure, where we also show the waiting time distribution compiled from all 100 000 simulation runs. The right figures show waiting time distribution compiled selectively from those simulation runs where exactly one, two, and three photon emissions were recorded, the fractions of which are shown at the top panel. Also shown in the square pulse plot (upper-left corner) is the analytical result for waiting time distribution result in the long pulse duration limit (Ref. 20), which is superposed on the histogram plot of the square pulse.

Waiting time distribution of photon emission for a ground state 2LS source irradiated by four differently shaped resonant laser pulses. The envelope profiles of the square , isosceles triangle, left triangle, and right triangle pulses of duration are shown in the left figure, where we also show the waiting time distribution compiled from all 100 000 simulation runs. The right figures show waiting time distribution compiled selectively from those simulation runs where exactly one, two, and three photon emissions were recorded, the fractions of which are shown at the top panel. Also shown in the square pulse plot (upper-left corner) is the analytical result for waiting time distribution result in the long pulse duration limit (Ref. 20), which is superposed on the histogram plot of the square pulse.

Fluorescence spectrum of a 2LS source irradiated by the same pulses of different shapes as those shown in the left figure of Fig. 4. For each pulse shape, we have shown three sample plots corresponding to the spectral density of radiated light as would be measured by a narrow-band photodetector at times (solid line), (short dashed line), and (long dashed line). The upper half of the spectra plots has been cut off in order to show more clearly the Mollow triplet structure.

Fluorescence spectrum of a 2LS source irradiated by the same pulses of different shapes as those shown in the left figure of Fig. 4. For each pulse shape, we have shown three sample plots corresponding to the spectral density of radiated light as would be measured by a narrow-band photodetector at times (solid line), (short dashed line), and (long dashed line). The upper half of the spectra plots has been cut off in order to show more clearly the Mollow triplet structure.

Left figure shows the normalized second-order correlation functions for a 2LS source irradiated by the same resonant pulses of four shape as those shown in Fig. 4. The normalized correlation functions are plotted as a function of time delay for three initial times (solid line), (short dashed line), and (long dashed line). The right figure shows the Mandel parameter plotted as a function of pulse duration for four different pulse shapes with for the square pulse and for the triangular pulses. In the case of square pulses (upper-right figure), the analytic result for Mandel parameter of He and Barkai (Ref. 17) has also been shown in red curve, which is virtually undistinguishable from the MCWF simulation results.

Left figure shows the normalized second-order correlation functions for a 2LS source irradiated by the same resonant pulses of four shape as those shown in Fig. 4. The normalized correlation functions are plotted as a function of time delay for three initial times (solid line), (short dashed line), and (long dashed line). The right figure shows the Mandel parameter plotted as a function of pulse duration for four different pulse shapes with for the square pulse and for the triangular pulses. In the case of square pulses (upper-right figure), the analytic result for Mandel parameter of He and Barkai (Ref. 17) has also been shown in red curve, which is virtually undistinguishable from the MCWF simulation results.

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