^{1}and J. Knoester

^{1,a)}

### Abstract

Based on the generating function formalism, we investigate broadband photon statistics of emission for single dimers and trimers driven by a continuous monochromatic laser field. In particular, we study the first and second moments of the emission statistics, which are the fluorescence excitation line shape and Mandel's *Q* parameter. Numerical results for this line shape and the *Q* parameter versus laser frequency in the limit of long measurement times are obtained. We show that in the limit of small Rabi frequencies and laser frequencies close to resonance with one of the one-exciton states, the results for the line shape and *Q* parameter reduce to those of a two-level monomer. For laser frequencies halfway the transition frequency of a two-exciton state, the photon bunching effect associated with two-photon absorption processes is observed. This super-Poissonian peak is characterized in terms of the ratio between the two-photon absorption line shape and the underlying two-level monomer line shapes. Upon increasing the Rabi frequency, the *Q* parameter shows a transition from super- to sub- to super-Poissonian statistics. Results of broadband photon statistics are also discussed in the context of a transition (frequency) resolved photon detection scheme, photon tracking, which provides a greater insight in the different physical processes that occur in the multi-level systems.

We are grateful to Professor R. J. Silbey and Dr. V. A. Malyshev for helpful discussions.

I. INTRODUCTION

II. THEORETICAL FRAMEWORK

A. Model Hamiltonian

B. Generating function formalism and photon statistics

C. Photon tracking

III. NUMERICAL ANALYSIS

A. Homogeneous dimer

1. Small Rabi frequency limit

2. Intermediate Rabi frequencies

B. Inhomogeneous dimer

C. Linear homogeneous trimer

IV. CONCLUSIONS

### Key Topics

- Photons
- 69.0
- Polymers
- 30.0
- Multiphoton processes
- 20.0
- Photon statistics
- 20.0
- Signal generators
- 12.0

## Figures

Level diagram of the dimer. In the special case of a homogeneous dimer, the dotted anti-symmetric state |−⟩ is optically dark and the dimer can be regarded as an effective three-level system.

Level diagram of the dimer. In the special case of a homogeneous dimer, the dotted anti-symmetric state |−⟩ is optically dark and the dimer can be regarded as an effective three-level system.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* in the small Rabi frequency limit for the homogeneous dimer. Chosen parameters are ω_{0} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, Ω_{0} = −1 × 10^{−3} *J*. Calculated parameters based on Eqs. (A3) and (A4) are Γ_{ g +} = 5.3 × 10^{−2} *J*, Γ_{+e } = 2.9 × 10^{−2} *J*, Ω_{+} = −1.4 × 10^{−3} *J*. (Inset) Our numerical results for the super-Poissonian peak (solid line) compared to the results derived from Eq. (20).

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* in the small Rabi frequency limit for the homogeneous dimer. Chosen parameters are ω_{0} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, Ω_{0} = −1 × 10^{−3} *J*. Calculated parameters based on Eqs. (A3) and (A4) are Γ_{ g +} = 5.3 × 10^{−2} *J*, Γ_{+e } = 2.9 × 10^{−2} *J*, Ω_{+} = −1.4 × 10^{−3} *J*. (Inset) Our numerical results for the super-Poissonian peak (solid line) compared to the results derived from Eq. (20).

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) for the homogeneous dimer as a function of (ω_{ L } − ω_{0})/*J* for intermediate Rabi frequencies, Ω_{0} = −4 × 10^{−2} *J*, −8 × 10^{−2} *J*, and −1.2 × 10^{−1} *J*. Parameters are the same as those of Fig. 2, leading to Ω_{+} = −5.7 × 10^{−2} *J*, −1.1 × 10^{−1} *J*, and −1.7 × 10^{−1} *J*, respectively. (Inset) Transition from super- to sub-Poissonian behavior in more detail.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) for the homogeneous dimer as a function of (ω_{ L } − ω_{0})/*J* for intermediate Rabi frequencies, Ω_{0} = −4 × 10^{−2} *J*, −8 × 10^{−2} *J*, and −1.2 × 10^{−1} *J*. Parameters are the same as those of Fig. 2, leading to Ω_{+} = −5.7 × 10^{−2} *J*, −1.1 × 10^{−1} *J*, and −1.7 × 10^{−1} *J*, respectively. (Inset) Transition from super- to sub-Poissonian behavior in more detail.

Plots of *I*(ω_{ L }) and *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* for the dimer in both limits of inhomogeneity. Panels (a) and (c) present the data for small inhomogeneity, with parameter choices ω_{1} = 10*J*, ω_{2} = 10.2*J* (i.e., σ = 0.1*J*), Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −2 × 10^{−3} *J*. From this it follows that Γ_{ g +} = 5.3 × 10^{−2} *J*, Γ_{+e } = 2.9 × 10^{−2} *J*, Γ_{ g −} = 7.2 × 10^{−5} *J*, Γ_{−e } = 1.3 × 10^{−4} *J*, Ω_{ g +} = Ω_{+e } = −2.8 × 10^{−3} *J*, and Ω_{ g −} = Ω_{−e } = −1.4 × 10^{−4} *J*. Panels (b) and (d) present data for large inhomogeneity, with parameter choices ω_{1} = 10*J*, ω_{2} = 30*J* (i.e., σ = 10*J*), Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −2 × 10^{−3} *J*. From this we have Γ_{ g +} = 7.5 × 10^{−2} *J*, Γ_{+e } = 2.7 × 10^{−3} *J*, Γ_{ g −} = 2.2 × 10^{−3} *J*, Γ_{−e } = 6.1 × 10^{−2} *J*, Ω_{ g +} = Ω_{+e } = −2.1 × 10^{−3} *J*, and Ω_{ g −} = Ω_{−e } = −1.9 × 10^{−3} *J*. Inset (a): details of the |*g*⟩ → |−⟩ transition line shape. Inset (d): observation of the small super-Poissonian peak for ω_{ L } ≈ ω_{0}.

Plots of *I*(ω_{ L }) and *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* for the dimer in both limits of inhomogeneity. Panels (a) and (c) present the data for small inhomogeneity, with parameter choices ω_{1} = 10*J*, ω_{2} = 10.2*J* (i.e., σ = 0.1*J*), Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −2 × 10^{−3} *J*. From this it follows that Γ_{ g +} = 5.3 × 10^{−2} *J*, Γ_{+e } = 2.9 × 10^{−2} *J*, Γ_{ g −} = 7.2 × 10^{−5} *J*, Γ_{−e } = 1.3 × 10^{−4} *J*, Ω_{ g +} = Ω_{+e } = −2.8 × 10^{−3} *J*, and Ω_{ g −} = Ω_{−e } = −1.4 × 10^{−4} *J*. Panels (b) and (d) present data for large inhomogeneity, with parameter choices ω_{1} = 10*J*, ω_{2} = 30*J* (i.e., σ = 10*J*), Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −2 × 10^{−3} *J*. From this we have Γ_{ g +} = 7.5 × 10^{−2} *J*, Γ_{+e } = 2.7 × 10^{−3} *J*, Γ_{ g −} = 2.2 × 10^{−3} *J*, Γ_{−e } = 6.1 × 10^{−2} *J*, Ω_{ g +} = Ω_{+e } = −2.1 × 10^{−3} *J*, and Ω_{ g −} = Ω_{−e } = −1.9 × 10^{−3} *J*. Inset (a): details of the |*g*⟩ → |−⟩ transition line shape. Inset (d): observation of the small super-Poissonian peak for ω_{ L } ≈ ω_{0}.

Maximum value of the observed super-Poissonian peak *Q* _{ max } as a function of the disorder parameter σ. Numerical results (squares) are compared with those obtained from Eq. (22) (solid line). Chosen parameters are ω_{1} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, Ω_{0} = −1 × 10^{−3} *J*. (Inset) Same as the main plot, but now the energy dependence of the spontaneous decay rates Γ_{ ij } is neglected.

Maximum value of the observed super-Poissonian peak *Q* _{ max } as a function of the disorder parameter σ. Numerical results (squares) are compared with those obtained from Eq. (22) (solid line). Chosen parameters are ω_{1} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, Ω_{0} = −1 × 10^{−3} *J*. (Inset) Same as the main plot, but now the energy dependence of the spontaneous decay rates Γ_{ ij } is neglected.

Level diagram of the linear homogeneous trimer with all molecules having equal transition dipole vectors. The arrows correspond to the optically allowed transitions.

Level diagram of the linear homogeneous trimer with all molecules having equal transition dipole vectors. The arrows correspond to the optically allowed transitions.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* for the homogeneous trimer in the limit of small Rabi frequency. Chosen parameters are ω_{0} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −1 × 10^{−3} *J*.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) versus (ω_{ L } − ω_{0})/*J* for the homogeneous trimer in the limit of small Rabi frequency. Chosen parameters are ω_{0} = 10*J*, Γ_{0} = 2 × 10^{−2} *J*, and Ω_{0} = −1 × 10^{−3} *J*.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) for the trimer plotted against (ω_{ L } − ω_{0})/*J* for Ω_{0} = −0.05*J* and −0.1*J*. Chosen parameters are the same as those of Fig. 7. The range for ω_{ L } is chosen to reflect the characteristics of multi-exciton influences in more detail. (Inset) Detailed behavior of the frequency regime near , which is the transition frequency between the ground state and the |1; 1⟩ one-exciton state.

(a) *I*(ω_{ L }) and (b) *Q*(ω_{ L }) for the trimer plotted against (ω_{ L } − ω_{0})/*J* for Ω_{0} = −0.05*J* and −0.1*J*. Chosen parameters are the same as those of Fig. 7. The range for ω_{ L } is chosen to reflect the characteristics of multi-exciton influences in more detail. (Inset) Detailed behavior of the frequency regime near , which is the transition frequency between the ground state and the |1; 1⟩ one-exciton state.

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