Toward direct optical excitation of excitonic many-body effects using intense thermal states

Ryan P. Smith; Eric W. Martin; Mackillo Kira; Steven T. Cundiff; Steven T. Cundiff

doi:10.1364/OSAC.392972

OSA Continuum
Vol. 3,
Issue 5,
pp. 1283-1301
(2020)
•https://doi.org/10.1364/OSAC.392972

Toward direct optical excitation of excitonic many-body effects using intense thermal states

Ryan P. Smith, Eric W. Martin, Mackillo Kira, and Steven T. Cundiff

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Ryan P. Smith, Eric W. Martin, Mackillo Kira, and Steven T. Cundiff, "Toward direct optical excitation of excitonic many-body effects using intense thermal states," OSA Continuum 3, 1283-1301 (2020)

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Abstract

Quantum spectroscopy in solids directly detects nonlinear changes created exclusively by quantum fluctuations of light. So far, it has been realized only by projecting a large set of measurements with a coherent-state laser to a specific quantum-light response. We present two complementary experimental approaches to realize intense and ultrafast thermal-state sources. We investigate the effects of continuous excitation from a superluminescent diode (SLD) as well as an ensemble-averaging technique using phase-modulated pulses. By measuring excitonic nonlinearities in gallium arsenide, we demonstrate that the experimentally realized thermal-state source produces significantly reduced many-body nonlinearities compared to a coherent-state excitation. We also review experimental approaches toward future realization of quantum spectroscopy with thermal states.

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Fig. 1. Comparison of photon number probability distributions for coherent and thermal light sources, both with

$\bar {n} = 10$

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Fig. 2. (a) Probe transmission spectrum through the GaAs sample (gray shaded region) along with the thermal excitation (red region) and the coherent excitation source (blue region). The absorption dip at 1547 meV corresponds to the 1s exciton of GaAs. (b) Nonlinear signal is generated by amplitude modulated pump beam and frequency shifted probe beam. Amplitude modulation can be performed with a mechanical chopper and frequency shifting is performed with acousto-optic modulators (AOMs). A local oscillator (LO) beam, which is frequency shifted by a different frequency than the probe beam, interferes with the signal on a detector. Since the modulation on the detector corresponding to the interference between the differential absorption signal of interest and the LO is unique, we can isolate the signal with a lock-in detector tuned to that modulation frequency.

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Fig. 3. Real part of spectrally resolved nonlinear signal. Here we compare nonlinear signals resulting from thermal (red) and coherent (blue) excitation. We show minimal difference in the induced signal for both low (left) and high (right) temperatures.

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Fig. 4. Time-integrated nonlinear response as a function of pump power for a thermal excitation source (red and maroon represent two separate measurements) and a coherent excitation source (blue). These are also compared to a projected thermal response (black line) calculated using the measured coherent response. These measurements are shown for low temperature (left) and high temperature (right).

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Fig. 5. (a) A spectrum from a mode-locked ti:sapphire laser with a random spectral phase function with spectral bin sizes of

$\Delta \omega _\textrm {bin} =200 \,\mu$

eV. There are

$N_\textrm {bins}=30$

spectral bins across this spectrum. (b) Cross-correlation traces for six different scrambled realizations. All traces are normalized so that the area is unity. For comparison, the normalized transform-limited pulse (flat spectral phase mask) is shown as a dashed red line.

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Fig. 6. (a) Probability distribution function for the normalized cross-correlation values, where the single bin

$P_\textrm {n}$

(10 photons) = 1. (b)

$k^\textrm {th}$

-order coherence values calculated by Eq. (32).

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Fig. 7. (a) Measured second-order coherence function values for several different spectral bin widths. Measured values correspond to the trend predicted in Eq. (31). (b) The average scrambled pump spectrum (shown for

$N_\textrm {ens}=1,14,50$

) converges to the flat mask spectrum as the number of realizations

$N_\textrm {ens}$

is increased.

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Fig. 8. Nonlinear observables as a function of power. (a) Peak heights of

$\beta _\textrm {QW}$

and (b) center resonance positions are shown for varying

$P_\textrm {QW}^\textrm {pump}$

. Fits (solid lines) through each set of points clarify the trends for the observables.

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Fig. 9. (a) The average spectral response from the scrambled realizations reveals the HH

$1s$

-resonance as less saturated than from coherent excitation with the same

$P_\textrm {IN}^\textrm {pump}$

. (b) Similarly, the projected thermal response determined from applying a set of coherent data using Eq. (18) also reveals less saturation compared with coherent excitation for the same pump excitation density (

$4.1\times 10^9$

electron-hole pairs / cm

$^2$

/layer).

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Equations (32)

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\begin{aligned} \hat{E} (r, t) & = {\hat{E}}^{+ (r, t)} + {\hat{E}}^{- (r, t)}, with {\hat{E}}^{+ (r, t)} \equiv \sum_{q} i E_{q} u_{q} (r) B_{q}, \\ {\hat{E}}^{- (r, t)} \equiv {[{\hat{E}}^{+ (r, t)}]}^{†}, E_{q} \equiv \sqrt{\frac{ℏ ω_{q}}{2 ϵ_{0}}}, \end{aligned}

[B_{q}^{†}, B_{q^{'}}] = δ_{q, q^{'}}, [B_{q}, B_{q^{'}}] = 0 = [B_{q}^{†}, B_{q^{'}}^{†}],

\hat{E} (t) = i E [B (t) - B^{†} (t)],

g^{(1)} (τ) \equiv \frac{⟨ {\hat{E}}^{(-)} (t) {\hat{E}}^{(+)} (t - τ) ⟩}{\sqrt{I (t) I (t - τ)}} = \frac{⟨ B^{†} (t) B (t - τ) ⟩}{\sqrt{n (t) n (t - τ)}},

g^{(2)} (τ) \equiv \frac{⟨ {\hat{E}}^{(-)} (t) {\hat{E}}^{(-)} (t) {\hat{E}}^{(+)} (t - τ) {\hat{E}}^{(+)} (t - τ) ⟩}{I (t) I (t - τ)} = \frac{⟨ B^{†} (t) B^{†} (t) B (t - τ) B (t - τ) ⟩}{n (t) n (t - τ)},

B | β ⟩ = β | β ⟩,

\hat{H_{q}} | n ⟩ = ℏ ω (n + \frac{1}{2}) | n ⟩, n = 0, 1, 2, \dots,

| β ⟩ = e^{- | β |^{2} / 2} \sum_{n = 0}^{\infty} \frac{β^{n}}{\sqrt{n!}} | n ⟩,

{\hat{ρ}}_{thermal} = \sum_{n = 0}^{\infty} P_{n} | n ⟩ ⟨ n |, with P_{n} = \frac{1}{1 + \bar{n}} {(\frac{\bar{n}}{1 + \bar{n}})}^{n} .

{\hat{ρ}}_{thermal} = \int d^{2} β \frac{1}{π \bar{n}} e^{- \frac{| β |^{2}}{\bar{n}}} | β ⟩ ⟨ β | .

R_{thermal} = \int d^{2} β \frac{1}{π \bar{n}} e^{- \frac{| β |^{2}}{\bar{n}}} R (β) .

P_{n}^{coherent} = e^{- \bar{n}} \frac{{\bar{n}}^{n}}{n!},

⟨ {[B^{†}]}^{J} B^{J} ⟩_{coherent} = ⟨ B^{†} B ⟩^{J} = ⟨ B^{†} ⟩^{J} ⟨ B ⟩^{J}

⟨ {[B^{†}]}^{J} B^{J} ⟩_{thermal} = J! ⟨ B^{†} B ⟩^{J}

Δ n^{2} \equiv ⟨ (\hat{n} - ⟨ \hat{n} ⟩)^{2} ⟩ = ⟨ {\hat{n}}^{2} ⟩ - {⟨ \hat{n} ⟩}^{2} = ⟨ B^{†} B^{†} B B ⟩ + ⟨ B^{†} B ⟩ - {⟨ B^{†} B ⟩}^{2} .

I (ν, T) = \frac{2 h ν^{3}}{c^{2}} \bar{n} (h ν), \bar{n} (h ν) \frac{1}{e^{h ν / k_{B} T} - 1},

\begin{aligned} E_{pump} (t) & = E_{pump, no mod.} sgn(sin (ω_{pump} t)) e^{- i (ω_{light}) t} \\ E_{probe} (t) & = | E_{probe} | e^{- i (ω_{laser} + ω_{probe}) t} \\ E_{LO} (t) & = | E_{LO} | e^{- i (ω_{laser} + ω_{LO}) t}, \end{aligned}

R_{th} (I_{th}) = \frac{\int_{0}^{\infty} d I_{coh} e^{- I_{coh} / I_{th}} R_{coh} (I_{coh})}{\int_{0}^{\infty} d I_{coh} e^{- I_{coh} / I_{th}}},

\begin{aligned} ⟨ q ⟩ = \int d q q P (q) = \begin{array}{cc} Lim \\ N \to \infty \end{array} \frac{1}{N} \sum_{j = 1}^{N} q_{j}^{meas} |_{ide}, \end{aligned}

⟨ ⟨ q ⟩ ⟩ \equiv \frac{1}{N_{ens}} \sum_{r = 1}^{N_{ens}} ⟨ q ⟩_{r}^{ens},

⟨ ⟨ q ⟩ ⟩ = \int d q q \frac{1}{N_{ens}} \sum_{r = 1}^{N_{ens}} P_{r}^{ens} (q),

P_{eff} (q) \equiv lim_{N_{ens} \to \infty} \frac{1}{N_{ens}} \sum_{r = 1}^{N_{ens}} P_{r}^{ens} (q),

{\hat{ρ}}_{eff} \equiv lim_{N_{ens} \to \infty} \frac{1}{N_{ens}} \sum_{r = 1}^{N_{ens}} {\hat{ρ}}_{r}^{ens},

{\hat{ρ}}_{eff}^{†} = {\hat{ρ}}_{eff}, Tr [{\hat{ρ}}_{eff}] = 1, {\hat{ρ}}_{eff} is positively valued .

⟨ ⟨ S (ω) ⟩ ⟩ = \frac{1}{N_{ens}} \sum_{r = 1}^{N_{ens}} S_{r} (ω),

E (ω) = \sum_{j = 1}^{N_{bins}} E_{j}^{0} (ω) e^{i Δ ϕ_{j} (r)},

\hat{B} (r) = \sum_{j = 1}^{N_{bins}} e^{i Δ ϕ_{j} (r)} {\hat{B}}_{j},

{\hat{B}}_{j} | β_{j} ⟩ = β_{j} | β_{j} ⟩ .

⟨ ⟨ \hat{B} ⟩ ⟩ = ⟨ ⟨ \sum_{j = 1}^{N_{bins}} e^{i Δ ϕ_{j} (r)} {\hat{B}}_{j} ⟩ ⟩ = \sum_{j} ⟨ ⟨ e^{i Δ ϕ_{j}} ⟩ ⟩ β_{j} = 0,

⟨ ⟨ {\hat{B}}^{†} \hat{B} ⟩ ⟩ = \sum_{j, k} ⟨ ⟨ e^{i [Δ ϕ_{k} - Δ ϕ_{j}]} ⟩ ⟩ β_{j}^{*} β_{k} = \sum_{j} {| β_{j} |}^{2},

⟨ ⟨ {[B_{L}^{†}]}^{J} {[B_{L}]}^{K} ⟩ ⟩ = δ_{J, K} J! ⟨ ⟨ B_{L}^{†} B_{L} ⟩ ⟩^{J} [1 + O (N_{bin}^{- 1})], when N_{bin} ≫ J .

g^{(k)} = ⟨ : {\hat{n}}^{k} : ⟩ / {⟨ \hat{n} ⟩}^{k} = \frac{1}{{⟨ \hat{n} ⟩}^{k}} \sum_{m = k}^{\infty} \frac{m!}{(m - k)!} P_{m},