^{1}and Paul Brumer

^{1,a)}

### Abstract

The response of an arbitrary closed quantum system to a partially coherent electric field is investigated, with a focus on the transient coherences in the system. As a model we examine, both perturbatively and numerically, the coherences induced in a three level V system. Both rapid turn-on and pulsed turn-on effects are investigated. The effect of a long and incoherent pulse is also considered, demonstrating that during the pulse the system shows a coherent response which reduces after the pulse is over. Both the pulsed scenario and the thermally broadened CW case approach a mixed state in the long time limit, with rates dictated by the adjacent level spacings and the coherence time of the light, and via a mechanism that is distinctly different from traditional decoherence. These two excitation scenarios are also explored for a minimal “toy” model of the electronic levels in pigment protein complex PC645 by both a collisionally broadened CW laser and by a noisy pulse, where unexpectedly long transient coherence times are observed and explained. The significance of environmentally induced decoherence is noted.

Z.S. thanks Dr. L. Pachon, Y. Khan, and R. Dinshaw for edifying discussions. P.B. thanks Dr. T. Tscherbul for insightful remarks. This work was partially supported by the (U.S.) Air Force Office of Scientific Research (USAFOSR) under Contract No. FA9550-13-1-0005.

I. INTRODUCTION

II. SHORT TIME RESPONSE OF AN ENERGY LEVEL TO INCOHERENT LIGHT

A. Perturbative treatment (analytic)

B. Perturbative treatment (numerical)

C. Toy PC645 in a collisionally broadened CW source

III. RESPONSE OF AN ISOLATED QUANTUM SYSTEM TO LONG INCOHERENT PULSES

A. Irradiation of toy PC645 using noisy pulses

IV. SUMMARY

### Key Topics

- Excited states
- 19.0
- Coherence
- 17.0
- Electric fields
- 9.0
- Correlation functions
- 5.0
- Proteins
- 5.0

## Figures

Three level system used in our numerical scheme. Population is initially in the ground state |1⟩ and is pumped to two non-degenerate excited states, |2⟩ and |3⟩. The frequency difference between the two excited states ω32 determines a characteristic excited state timescale τ c = 2π/ω32.

Three level system used in our numerical scheme. Population is initially in the ground state |1⟩ and is pumped to two non-degenerate excited states, |2⟩ and |3⟩. The frequency difference between the two excited states ω32 determines a characteristic excited state timescale τ c = 2π/ω32.

(Top) Ensemble averaged excited state coherence (⟨ρ23⟩) plotted versus perturbative coherences for a three level ladder system excited by thermally broadened CW source. (Bottom) Excited state populations for both perturbative and numerical results for the same system. τ c = 60 fs and τ d = 120 fs for both figures.

(Top) Ensemble averaged excited state coherence (⟨ρ23⟩) plotted versus perturbative coherences for a three level ladder system excited by thermally broadened CW source. (Bottom) Excited state populations for both perturbative and numerical results for the same system. τ c = 60 fs and τ d = 120 fs for both figures.

plotted against time for three level ladder system excited by thermally broadened CW source for various level splittings at fixed τ d . The values of τ d used are τ d = 60 fs (top), τ d = 120 fs (middle), and τ d = 240 fs (bottom). For large values of time, the quantity becomes smaller and as t → ∞ the value approaches zero.

plotted against time for three level ladder system excited by thermally broadened CW source for various level splittings at fixed τ d . The values of τ d used are τ d = 60 fs (top), τ d = 120 fs (middle), and τ d = 240 fs (bottom). For large values of time, the quantity becomes smaller and as t → ∞ the value approaches zero.

System purity Tr(ρ^{2}) plotted as a function of time for various excited state periods τ c and radiation coherence times τ d for a three level ladder system excited by thermally broadened CW source. As the excited state period τ c becomes larger, the purity of the system decreases at a faster rate. The following τ d times are shown: 60 fs (top), 120 fs (middle), and 240 fs (bottom).

System purity Tr(ρ^{2}) plotted as a function of time for various excited state periods τ c and radiation coherence times τ d for a three level ladder system excited by thermally broadened CW source. As the excited state period τ c becomes larger, the purity of the system decreases at a faster rate. The following τ d times are shown: 60 fs (top), 120 fs (middle), and 240 fs (bottom).

Excited state purity plotted against time for three level ladder system excited by thermally broadened CW source for various level splittings, τ c at various τ d . The following τ d times are shown: 60 fs (top), 120 fs (middle), and 240 fs (bottom).

Excited state purity plotted against time for three level ladder system excited by thermally broadened CW source for various level splittings, τ c at various τ d . The following τ d times are shown: 60 fs (top), 120 fs (middle), and 240 fs (bottom).

Plot of , red regions represent highest intensity and blue regions represent lowest intensity.

Plot of , red regions represent highest intensity and blue regions represent lowest intensity.

(Top) Coherences between the |DBV ^{−}⟩ and |DBV ^{+}⟩ in the toy PC645 model as a function of time for the sudden turn-on case. (Bottom) Populations of |DBV ^{−}⟩ and |DBV ^{+}⟩ in the toy model of PC645 as a function of time for the sudden turn-on case. Excitation frequency of the laser is in the middle of the two transitions.

(Top) Coherences between the |DBV ^{−}⟩ and |DBV ^{+}⟩ in the toy PC645 model as a function of time for the sudden turn-on case. (Bottom) Populations of |DBV ^{−}⟩ and |DBV ^{+}⟩ in the toy model of PC645 as a function of time for the sudden turn-on case. Excitation frequency of the laser is in the middle of the two transitions.

Plot of for a toy model of PC645 upon excitation by a collisionally broadened CW source, for parameters indicated in the text.

Plot of for a toy model of PC645 upon excitation by a collisionally broadened CW source, for parameters indicated in the text.

Plot of for various excited state splittings, τ c for long incoherent pulses incident on a three level ladder system. The pulse used is 1 ps in duration. τ d = 120 fs. The frequency center of the pulse is chosen to be in the center of the two transitions.

Plot of for various excited state splittings, τ c for long incoherent pulses incident on a three level ladder system. The pulse used is 1 ps in duration. τ d = 120 fs. The frequency center of the pulse is chosen to be in the center of the two transitions.

, the magnitude of the post pulse excited state coherence fraction of a toy model of PC645, plotted as a function of pulse duration, τ p . For long pulses, it is evident that post pulse coherence is extremely small.

, the magnitude of the post pulse excited state coherence fraction of a toy model of PC645, plotted as a function of pulse duration, τ p . For long pulses, it is evident that post pulse coherence is extremely small.

Comparison of numerically generated (top) and numerical [Eq. (1) ] (bottom) ⟨ɛ(t ^{′})ɛ(t ^{″})⟩. Both plots share the same color legend. The times are in units of femtosecond.

Comparison of numerically generated (top) and numerical [Eq. (1) ] (bottom) ⟨ɛ(t ^{′})ɛ(t ^{″})⟩. Both plots share the same color legend. The times are in units of femtosecond.

(Top) Exact contour plot of the correlation function in Eq. (2) and (bottom) numerical reproduction of correlation for the noisy pulsed source. Both plots share the same color legend. The times are in units of femtosecond.

(Top) Exact contour plot of the correlation function in Eq. (2) and (bottom) numerical reproduction of correlation for the noisy pulsed source. Both plots share the same color legend. The times are in units of femtosecond.

Plot of |F|^{2} defined in Eq. (B8) using the parameters of toy PC645. Solid line is the stationary coherence as a function of τ d while the dashed line represents the saturation level from Eq. (B10) .

Plot of for a three level system irradiated with white noise, as a function of dimensionless time, .

Plot of for a three level system irradiated with white noise, as a function of dimensionless time, .

## Tables

Parameters used in perturbative calculations of PC645.

Parameters used in perturbative calculations of PC645.

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