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Coherent nonlinear optical spectroscopy of quantum dots Cameron Nelson Steel Group This work is funded by NSF-CPHOM Old school atomic spectroscopy Figure from Annalen der Physik und der Chemie (Poggendorff), Vol. 110 (1860), pp. 161-189 (dated Heidelberg, 1860) Solar Spectrum Semiconductor quantum dots: artificial atoms Hydrogen atom: Characteristic length is given by Bohr radius MBE Quantum Dot: Can be grown epitaxially within a layer of semiconductor CB e Characteristic length is given by exciton Bohr radius: typically much larger than atomic Bohr radius. Exciton: Coulomb-bound electron-hole pair (quasiparticle) h VB Quantized Energy Levels Dipole moment is big: strong coupling to light! Nonlinear optical spectroscopy is necessary for a complete understanding of the resonant interaction of a laser field and quantum dot. Quantum Harmonic Oscillator Quantum Dot Exciton ON OFF The interaction between a laser field and a two-level system is extremely nonlinear. Compare to a harmonic oscillator, a linear system. Resonant excitation of a linear harmonic oscillator moves it to a higher energy state. This continues indefinitely. In a two-level system, you can only excite the system once. If you try to excite it again, you will drive it back down to the ground state (Rabi oscillations). Applications of optically controlled quantum dots The laser field can be used to coherently control the quantum state of the quantum dot, resulting in a controllable qubit (on-off switch). Quantum information processing Quantum Information Science Quantum optics in solid state X. Xu, et al. Science 317, 929-932 (2007). JQI Website X. Li, et al. Science 301, 809-11 (2002). Coherent control of the quantum state N. H. Bonadeo, et al. Science 282, 5393 (1998). All-optical switching D.A.B. Miller Nat. Photonics. 4, 3 (2010). The world is demanding more bandwidth, so data distributors will become more reliant on all-optical switching Worldwide data usage is increasing every year http://bradhedlund.com/ Currently, most routing of information requires optical to electronic conversion to route signals. Image from Cisco This is slow, and wasteful (electronics generate a lot of heat). Quantum dot-based technology shows some promise for all-optical switching A small pulse of light can be used to switch the quantum state of a quantum dot, thereby allowing another light pulse to either pass or reflect. This offers an extremely low power switch for controlling an information signal. (switching power ~attojoules) Image from NTT photonics Chip-level interconnects also suffer from heat-related issues. All-optical switches may be a good future solution for replacing CMOS transistors. D.A.B. Miller Nat. Photonics. 4, 3 (2010). Many quantum applications requiring multiple qubits will only be realized very far in the future. 1 qubit Even coupling together two quantum dot qubits for applications such as quantum computing is extremely complicated or requires very long integration times for experimental verification. Most importantly, the most well-studied materal for this application, InAs, requires cooling down to 4K to work properly! . The search is still on for new computer hardware… Photoluminescence: useful for characterization of energy level structure of quantum dots Quantum Dot: CB Camera Emission Spectrometer Pump laser (PL) D Photons VB Photoluminescence: Excite high energy charge carriers that decay nonradiatively to the lowest energy emitting state. The resulting spectrum is usually similar to the linear absorption spectrum of the lowest energy states in the system What I study: InGaN/GaN quantum dots for room temperature applications Hanbury Brown-Twiss Experiment D The main advantage of InGaN/GaN systems is a large exciton binding energy compared to other III-V material. This has allowed for quantum dot operation up to room temperature. APD Time-correlation Electronics Quantum Dot Luminescence Excitation Pulse 50:50 beamsplitter D APD The InGaN quantum dot is one of the only stable solid state systems that gives single photon emission behavior up to room temperature. This has immediate applications in quantum cryptography, for example. What do you get from nonlinear absorption that you can’t from linear absorption? You can measure (among other things): Dephasing and population decay rates of exciton Inhomogeneous broadening in the system Coupling between excited states Laser Absorption Quantum Dot: Laser Absorption CB Energy ωL 1.67σW 2γ ω0 Lorentzian VB ωL – ω0 (Homogeneously Broadened) γ is the dipole dephasing rate Gaussian ωL – ω0 Theoretical description of measurements • We utilize the density matrix equations of motion for a two-level system to describe the optical response of a quantum dot exciton. • This theory is extremely well developed and has been used to successfully describe the nonlinear optical response of GaAs/InAs quantum dots. • Most of the following can be found in standard textbooks: Optical Bloch equations for a homogeneously broadened two-level system (Lorentzian absorption) Density matrix: ρ = |ψ><ψ| Density matrix equations of motion: From Schrödinger’s equation, 𝑖ℏ𝝆 = [𝐇, ρ] 𝑯 = 𝑯𝟎 + μ ∙ 𝑬 < 𝑃 > = 𝑁(μ12 ρ21 + μ21 ρ12 ) δ E0 |2> = |X> Pure dephasing rate Maxwell-Bloch equation: relation between absorption and off-diagonal density matrix: EL |1> = |0> Quantum dot systems gain pure dephasing from environmental fluctuations. From S. Rand, Lectures on Light Image by S. Kelly, JQI Univ. of Maryland Fluctuation-dissipation theorem The exciton transition energy often undergoes time-dependent fluctuations due to the solid state background (i.e. Stark shifts from surrounding charges). Optical Bloch equations for a two-level system with multiple laser fields There is no analytical steady-state solution for this problem. To understand it, perturbation theory has to be used. We are particularly interested in third order terms. ks = kµ - k ν + kσ δs = δµ - δ ν + δσ For simplicity, we use two fields, Epump = E1 and Eprobe = E2. This gives a total of 8 terms. Isolating two nonlinear terms along the probe beam direction Sample C.W. Pump (ω1) C.W. Probe (ω2) Pump and probe are crossed at the center of the sample A detector can be aimed along the probe beam propagation direction. In this way, we can isolate the detection to nonlinear terms that propagate along the probe beam direction. Recall that the third order nonlinear density matrix element goes like: 𝐼 (3) ρ21 ~𝑒 −𝑖(𝑘𝑠 𝑅+ δ𝑠 𝑡) We therefore select ks = kµ - kν + kσ = k2 This requires μ = ν = 1, σ = 2 and μ = 1, ν = 2, σ = 1. δs = δ µ - δν + δσ = δ2 1 ρ𝐼21 (3) = 2𝑖𝑒 −𝑖(𝑘2 𝑅+ δ2 𝑡) |χ1 |2 χ2 { 1 (𝛾 + 𝑖δ2 )2 (𝛾 − 𝑖δ1 ) 2 1+ 2Γ γ2 + 𝑖 ω1 − ω2 +[ 1 𝛾 2 + 𝑖δ1 2 𝛾 + 𝑖δ2 ] 1+ 2Γ } γ2 Imaginary component of total nonlinear response No pure dephasing Some dephasing Γdeph = 0 1. Population pulsation term: follows the probe frequency and has a FWHM ~γ2 Γdeph = 10 γ2 Im(ρ21(3)) 2. Saturation term: does not follow the probe term. Has a FWHM ~γ γ2 δ2 δ2 The nonlinear optical spectrum reveals both the pure dephasing rate and the exciton decay rate. This cannot be easily obtained using linear techniques. Note: the previous slides only applied to the case of homogeneous broadening. • In general, the Bloch equations have to be modified to account for inhomogeneous broadening (Gaussian lines) properly. The final results are a lot more complicated, usually. Inhomogeneously broadened response Homogeneously broadened response Nonlinear response of inhomogeneously broadened transitions: spectral hole burning • The third order spectra look similar to the homogeneously broadened case, except the saturation term tracks with the pump laser. Linear Absorption, Inhomogeneously Broadened Transition σW = 20 γ2 Pump Wavelength 20 10 0 10 20 γ2 Nonlinear Absorption Nonlinear Absorption Γ= 0 γ2 20 10 Γ= 3γ2 10 20 20 10 10 20 γ2 • The nonlinear spectrum can be used to distinguish inhomogeneous and homogeneous broadening. Measuring nonlinear signals with differential transmission Optical Chopper Sample C.W. Pump (ω1) C.W. Probe (ω2) dT Tprobe with pump on Tprobe with pump off T Tprobe with pump off Lock-in Amplifier Function Gen The lowest order contribution comes from the third order nonlinear signal ~χ(3)|Epu|2|Epr|2 Positive dT/T: Pump beam modifies sample absorption so that the probe beam absorbs less → saturation Negative dT/T: Pump beam modifies samples absorption so that the probe beam absorbs more Our study: Coherent nonlinear optical spectrum of InGaN disks in GaN nanowires Sample grown by Zetian Mi’s group at McGill University Current studies show evidence of a compact quantum dot in the center of the disk. Disk-in-nanowires show single photon emission behavior, which allows for unique quantum applications. Quantum cryptography requires quantum key generation. For this, can use polarization of photons from a single quantum dot, for example. A novel quantum key distributor can be made using single quantum dot a single electrically injected quantum disk LED 1 μm Polarized Emission Background states can cause a significant spectral diffusion (dephasing) for the InGaN exciton Charges captured by traps may cause a Stark shift of the exciton resonance M. Holmes et al. Phys Rev. B 115447 (2015) Stark Shifts from background states in InGaN dots-in-nanowires can cause extremely large optical linewidths. -Very small nonlinear signal The photoluminescence spectrum of a single InGaN quantum dot shows fluctuations in the transition energy over time. J.H. Rice et al. Appl. Phys. Lett. 84, 4110 (2004). InGaN materials are known to be very messy (lots of background disorder states) Optical Setup Preliminary sample data: bright PL from quantum confined excitons Photoluminescence The PL data shows evidence for radiative recombination from quantum confined exciton states: there is no continuous blue shift as a function of excitation intensity, a signature of quantum confinement. The PL originates from disksin-nanowires with different diameters, and therefore different emission wavelengths. The broad emission comes from inhomogeneous broadening. Ongoing work: Analyzing the full coherent nonlinear spectrum Exciton excited states Emissive Exciton States 10 K Nondegenerate nonlinear optical spectrum reveals evidence of saturation resonances with ~100 fs dephasing rates, this is at least 10,000x faster dephasing than similar QD systems. Coherent population pulsations of excitons reveal that excitons are stable up to room temperature Total nonlinear spectrum Pump-probe detuning (GHz) Pump-probe detuning (GHz) We observe strong coherent population pulsation resonances in the sample that reveal the population decay time (T1) of the exciton states all the way up to room temperature. We find that the nonlinear signal is relatively robust against temperature, but fast dephasing is still present, likely due to very fast Stark shifting from background states. 400 200 0 200 400 400 200 0 200 400 400 200 0 200 400 400 200 0 200 400 Δpu,pr (γ2 units) Conclusions • We observe coherent population pulsation resonances that allow us to extract fundamental decay parameters of the exciton states and to verify a low density of metastable trap states. • For the first time, we show that the quantum dot exciton is likely homogeneously broadened, and the dipole dephasing rate is ~100 fs, probably due to background disorder states. • Our results show that excitons can be stably excited up to room temperature, InGaN is typically grown along polar axes, which leads to internal electric fields InGaN has a wurtzite crystal structure The internal electric fields changes the band diagram and pushes the electron and hole apart along the growth direction