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**Optically Driven Spins in Semiconductor Quantum Dots**

Duncan Steel - Lecture 2 DPG Physics School 2010 on "Nano-Spintronics"

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**Semiconductor Quantum Coherence Engineering**

The qubit for real systems is the electron or hole spin: The key to optically driven quantum computing in semiconductors is the negatively charged exciton (trion) in a quantum dot |1> |1> |0> |0> Optical Bloch Vector Qubit Electronic Spin Qubit Semiconductor Quantum Coherence Engineering Successful coherent optical manipulation of the optical Bloch vector necessary to manipulate the spin vector

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**The electron spin vector**

AlGaAs (GaAs) GaAs (InAs) AlGaAs (GaAs) |1> |0>

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**The electron spin vector**

AlGaAs (GaAs) GaAs (InAs) AlGaAs (GaAs) l |1> |0>

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**The electron spin vector**

AlGaAs (GaAs) GaAs (InAs) AlGaAs (GaAs) l |1> |0>

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**The electron spin vector**

Long coherence time AlGaAs (GaAs) GaAs (InAs) AlGaAs (GaAs) |1> |0>

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**Optical Excitation of Spin Coherence: Two-photon stimulated Raman**

Circularly polarized pump pulse creates coherent superposition of spin up and down state. Raman coherence oscillates at frequency of the Zeeman splitting due to electron in-plane g-factor and decays with time. With the dark transitions now allowed, our system resembles a standard L-system from atomic physics. Starting with one spin state,the pump pulse in 1st order creates a optical dipole coherence between the ground state and the trion state. Then in second order it creates a coherent superposition between the spin up and down states. This raman coherence oscillates at the frequency of the zeeman splitting between the states which is governed by the electron in-plane g-factor. The decay of this coherence T2 is the spin coherence time.

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**Single Electron Spin Coherence: Raman Quantum Beats**

Charged Exciton System X - CNOS (a. u.) G G Neutral Exciton System X G G hgs (meV) T2* >10 nsec at B=0 Phys. Rev. Lett

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**Anomalous Variation of Beat Amplitude and Phase**

Standard Theory (a) (b) Plot of beat amplitude and phase as a function of the splitting.

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**Anomalous Variation of Beat Amplitude and Phase**

Standard Theory (a) Plot of beat amplitude and phase as a function of the splitting.

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**Spontaneously Generated Coherence (SGC)**

Trion G G Coupling to electromagnetic vacuum modes can create coherence* !! Modeled in density matrix equations by adding a relaxation term: Normally forbidden in atomic systems or extremely weak.

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Anomalous Variation of Beat Amplitude and Phase: The result of spontaneously generated Raman coherence Standard Theory (a) Plot of beat amplitude and phase as a function of the splitting. Phys. Rev. Lett

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Two-Photon Spin Rabi Trion Trion Laser Pulse

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Initialization

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**Phase Gate - Demonstration of Geometric Phase (Aharonov & Anandan)**

Optical Control of Trion Optical Bloch Vector Optical Control of Spin Bloch Vector

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**Coherent Generation of a Geometrical Phase**

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**Demonstration of the Phase Control**

Modulation effect clearly seen Frequency of the modulations depends on the strength of the CW field Phase change after modulation points consistent with theory for 0.2, 5 and 10 mW scans Action of CW field can be likened to a spin phase gate 17

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**The Mollow Absorption Spectrum, AC Stark effect, and Autler Townes Splitting: Gain without Inversion**

Dressed State Picture Mollow Spectrum: New physics in absorption Autler Townes Splitting S. H. Autler, C. H. Townes, Phys. Rev. 100, 703 (1955) B. R. Mollow, Phys. Rev. 188, 1969 (1969). B. R. Mollow, Phys. Rev. A. 5, 2217 (1972)..

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**Power Spectrum of the Rabi Oscillations: Gain without inversion The Mollow Spectrum of a Single QD**

|3> |2> Driving the dipole transition in a single quantum dot with a strong optical field results in Rabi oscillations of the electron. The Rabi oscillations can be measured in the frequency domain (basically a Fourier spectrum of the Rabi oscillation) by probing the resultant absorption with a weak probe. The spectrum is more complicated because of quantum coherence effects in the measurement. The dots in the data on the left represent the power spectrum for various values of the pump field, the solid line is the theory as first developed by Mollow resulting in the so-called Mollow absorption spectrum. The absorption spectrum goes below zero, representing gain without inversion. The inset on the right shows the expected power dependence (SQRT (intensity)) of the peaks in the data on the left. Measurements are made on a single quantum dot and show that it GHz switching between electronic states can be achieved with about 10 photons per switching event. Weak probe Strong pump X. Xu, B. Sun, P. R. Berman, D.G. Steel, A. Bracker, D. Gammon, L. J. Sham, “Coherent optical spectroscopy of a strongly driven quantum dot,” Science, 317 p 929 (2007).

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**Autler-Townes Splitting in a Single Quantum Dot**

Dressed state Picture Probe Frequency (GHz) 321591 321594 0 Io 10 Io 20 Io 30 Io 40 Io 50 Io 5 Io Probe Abosorption as a Function of the Pump Intensity (on resonance) Pump intensity (Io=0.03w/cm ) 2 |b(N)> |1> |3> } WR |a(N)> |b(N-1)> |2> } WR |a(N-1)> Absorption (a.u.) 1 Rabi Splitting (GHz) 4 8 Pump Field Strength( )

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**Probe Absorption as a Function of Pump Frequency Detuning**

Experimental Data Theoretical Plot Pump Intensity 30Io Pump Detuning (GHz) 1.7 Absorption (a.u.) 0.6 0.3 0.0 -0.3 -0.6 -1.7 321591 321594 -5.0 -2.5 2.5 5.0 Probe Frequency (GHz) Probe Detuning (G units)

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**Thy Physical Model of the Dark State Experiment**

Bx |T+> |T-> V1 V2 H1 H2 |T-> |X+> |X-> H1 V2 Wp Wd The Quartet Transition Pattern V1 V2 H1 H2 Laser Detuning (GHz) -8 8 Theoretical plot of the CPT including electron spin dephasing B=1.32 T 1 DT/T (10-4) Laser Detuning (G units) -3

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**The Observation of the Coherent Population Trapping of an Electron Spin**

The probe absorption spectrum scanning across transition H1 5 DT/T (10-4) Wd/2p(GHz) 0.56 0.78 0.83 1.26 1.38 Probe Detuning (GHz) -5 1 |T-> |X+> |X-> H1 V2 Ωp Ωd Solide lines are the fits, which yield electron spin T2* of 4 ns. Nature - Physics, 2008

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**Probing Dynamic Nuclear Spin Polarization by Dark State Spectroscopy**

h e |T-> Probe absorption spectra by varying the laser scan rate Ωpump Ωprobe e |X+> e |X-> Now I will show you the probing of dynamic nuclear spin polarization using the dark state spectrum. The is a dark state spectrum by scanning the probe laser from X- to T-. It has broadened spectrum with round top, nothing look like a normal dark state spectrum. Even more interesting, we reverse the laser scan direction and observe a hysteresis effect of the spectrum between the forward and backward scan. Strikingly, the spectral positions of backward scan is drastically shifted from the forward scan. As we just talked, the spectral position of the dark state represents the electron spin Zeeman splitting. The shift of the dark state spetral position indicates the change of the Zeeman splitting. Since we do not change the applied magnetic field in the scan, then the cause of the Zeeman splitting change is due to the change of hyperfine field. For example, in the forward scan, nuclear spin has this type of configuration, then in the backward scan, the nuclear spin changes configuration and changes the hyperfine field, so as to affect the Zeeman splitting. Also, the round top indicates large trion excitation is a favorable system for the nuclear fields. We also perform dark sate spectrum by chansing the laser scan rate. the dynamical control of the nuclear field by the laser frequency scans on a timescale comparable to the nuclear spin relaxation time, which is of the order of a second12,14,16 . Broadened & rounded trion peak Large trion excitation (absorption) is favored Scan direction dependence: hysteresis & dark state shift (Dark state position reflect Zeeman Splitting) Dynamic control of nuclear field

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**Time Dependent Probe Absorption Spectrum**

B=2.6 T In order confirm the maximum trion excitation is the stable configuration of nuclear field, we performed a set of measurements by fixing the frequencies of both lasers and recording the optical response as a function of time. This figure shows the probe absorption spectra with forward and backward scans. We begin by scanning the laser backward and stopping the laser just before the sharp rising edge of the trion peak, then We record the absorption signal as a function of time with the laser frequency fixed. the system remains in low absorption for a minute and then abruptly switches into high absorption where it remains. This signifies that the nuclear field switches to a stable value that maximizes the trion excitation. e h Ωprobe Ωpump |T-> |X+> |X->

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**Time Dependent Probe Absorption Spectrum**

Laser frequency parked here Partial backward scan e h Ωprobe Ωpump |T-> Stable configuration: maximum trion excitation (absorption) |X+> |X->

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**Time Dependent Probe Absorption Spectrum**

We also examined the dynamics of the nuclear field by monitoring the dark state. Here L indicates the nuclear field configuration right before the dark state, D means the dark state and R is the nuclear field configuration on the right shoulder of the dark state. We first scan the laser forward and stop the laser right before the dark state, and then fix the laser frequency and monitor the probe absorption as a function of time. The nuclear field stays at L configuration, indicated by the signal level. Then evolves to the value which match the conditions to form the dark state. Finally, it goes back to the configuration R, which maximize the trion absorption. This measurement indicates the dark state is a metastable configuration for the nuclear field. In the experiment, we have noticed that the nuclear field stays at the dark state configuration from a few seconds to a few mints. e h Ωprobe Ωpump |T-> Dark State is a meta-stable state for nuclear field |X+> |X->

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**Trion Induced Dynamic Nuclear Spin Polarization**

anisotropic hyperfine from hole |T> nuclear Zeeman << trion linewidth Flip up rate: The previous data tell us that maximum trion absorption is a stable configuration for the nuclear field. When a trion is excited, its two constituent electron spins form an inert singlet, leaving its hole to interact with the nuclear spins. This is a unique element for optical control of the nuclear spin environment while manipulating the electron spin, and it accounts for the experimental observations. The hyperfine interaction between the hole spin and nuclear spin is strongly anisotropic, and has a term like, it can flip a nuclear spin without flipping a hole spin. Since nuclear Zeeman energy is on the order of tens of MHz, much smaller than trion linewidth, this flipping process stands out from various DNP process. There are two process, either flip up, or flip down. The nuclear flip rate can be solved from Fermi’s Gold rule. The flip rate is proportional to the trion population. Since the initial trion population is the same for both process, then the flip process which ends up with larger trion population always wins. The net DNP rate is proportional to this term. Flip down rate: DNP rate Whichever increases rt dominates! Nuclear field dynamics:

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**Dynamic Nuclear Spin Polarization Induced Spectral Servo**

Probe laser frequency Two photon detuning Absorption Here I am going to show how the weird dark state spectrum forms. Nuclear field Probe detuning ( = 2-ph detuning - nuclear field )

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**Numerical Simulation Results : Slow Scan**

Theory Experiment Nuclear field dynamics: Parameters: Nuclear T1

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**Numerical Simulation Results : fast Scan**

Theory Experiment Parameters: Nuclear T1 Microscopic theory: Weng Yang et al., Q ;

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**Nuclear Field Locking Effect**

Metastable configurations Stable configurations for DNP DNP rate: We can understand this dynamic nuclear spin polarization by calculating the probe absorption spectrum without the nuclear field effect, shown in the top figure. It has two trion maximum, correspond to the stable configuration of the nuclear field. The bottom figure shows the dynamic nuclear spin polarization rate. It shows at trion maximum, the rate is zero. The slope of the DNP rate is negative, which means the DNP act as a restoring force and lock the nuclear field to a value which maximize the trion excitation. We notice that the DNP rate at the dark state is also zero but with a positive slope, which agrees with the metastable feature of the dark state. Two-photon detuning Nuclear field locked to stable value

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**C. Latta et al., Nature Phys. 5, 758 (2009) **

Dynamic Nuclear Spin Feedback Suppresses Fluctuations DNP by trion Nuclear field self-focus to stable value Nuclear field unstable against DNP Medium trion excitation Maximum trion excitation CW laser excitation Stable-config nuclear field locked to frequencies Single QD arbitrary nuclear spin config The demonstrated self-locking effect of nuclear spin leads to the suppression of the nuclear field fluctuations. Initially the nuclear spin in a arbitrary configuration, laser comes in and excites the trion state, which interacts with the nuclear spin. Only maximum trion excitation is a stable configuration, dynamical nuclear spin polarization will adjust this hyperfine field and maximize the trion excitation. If there is any flucutations of the nuclear field, which switch the trion excitation from maximum to a medium level, then the DNP feedback process with acting as a restoring force to correct this mistake. Therefore, Once the system has switched to a configuration of maximum trion excitation, the electron spin Zeeman energy and hence the nuclear field are determined and controlled only by the instantaneous laser frequencies. Recently, there are two different measurements also observe the nuclear field locking effects. One experiment is done by ESP tehcnique at electrostatic gate confined quantum dot system. And the other one is trion absorption measurements with the magnetic field applied in the Fraday geometry, and no coherent dark state involved. Although both experiments indicate the nuclear field locking effects could potentially lead to the enhancement of the electron spin coherence, the effect can not be directly demonstrated in those experimental configurations. As I discussed in the beginning, we generate coherent dark state between the spin ground states and it represents the electron spin coherent time. In the following, I will show you the experimen results that the observed nuclear field locking effect 2-photon resonance shifts Nuclear spin fluctuation C. Latta et al., Nature Phys. 5, 758 (2009) Ivo T. Vink et al, Nature Phys. 5, 764 (2009)

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**Suppression of Nuclear Field Inhomogeneous Broadening**

More enhancement on spin T2* with larger pump strength larger pump larger slope in tighter locking Pump intensity 70 90 60 20 40 Absorption spin T2* peak-to-dip ratio Probe detuning

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**Suppression of Nuclear Field Inhomogeneous Broadening**

Thermal value Spin decoherence rate extracted from dip-to-peak ratio e h Ωprobe Ωpump |T-> |X-> |X+> T2* extended well above thermal value Deficiency: locking position changes with probe scan

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**(fixed freq) (fixed freq) (freq scan)**

Coherent Spin Manipulations without Hyperfine Induced Dephasing Pump >> Pump 2 >> Probe (fixed freq) (fixed freq) (freq scan) Pump 1 + pump 2 locks nuclear field to a constant value Now we know that we have a dynamic nuclear spin polarization process, which will suppress the nuclear spin fluctuations. Therefore, we narrow down the inhomogeneous broadening of the nuclear field, so as to increase the electron spin coherence time. This observation answers the part of the question we asked at the begin. Now the question is if we can coherently manipulate the electron spin without the concern of the hyperfine induced decoherence. In order to do this, we design a three beam measurements. The stronger pump 1 remains near-resonant with transition H1 and the weaker pump 2 is tuned to transition V2 and fixed at the spectral position that maximizes the trion absorption. The two pumps lock the nuclear field to a constant value and suppress nuclear fluctuations for the duration of the experiment. We use the weak probe beam to generate the dark-state spectrum, or coherently manipulate the electron spin. Pump 1 + probe measures spin T2*

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**Three Beam Measurement**

The experimental results confirms the spin manipulation with the suppression of nuclear spin fluctuations. This is the reslut. It has a very clean lineshape, no broaden lineshape compare to the two beam case. Since in the three beam, the nuclear field is locked to a constant value by the two pumps and does not change during the spin manipulation. The striking results is that the dip at the two photo resonance almost approaches zero. As we talk about, the dip strength represents spin coherence. The observation indicates the spin docoherence is suppressed. A fit indicates the spin decoherence is 1 MHz, which shows a factor of 400 ehancement of electron spin cohrence time compared without DNP effect. Clean line shape Spin decoherence rate ~ 1 MHz, reduced by a factor of 400 Xu, X. et al., Nature 59, (2009)

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Where’s the Frontier? Engineering coupled dot system with one electron in each dot with nearly degenerate excited states. Demonstration of optically induced entanglement. Integration into 2D photonic bandgap circuits. Understanding of decoherence. Possible exploitation of nuclear coupling.

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**Semiconductor Nano-Optics: An Interdisciplinary Collaboration**

Dan Gammon Naval Research Lab Lu Sham UC-San Diego Paul Berman Luming Duan Roberto Merlin U. Mich.

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**Outstanding Graduate Students****

Nicolas Bonadeo (graduated) Jeff Guest (graduated) Gang Chen (graduated) Todd Stievater (graduated) Anthony Lenihan (graduated) Elizabeth Tabak (graduated) Elaine Li (graduated) Gurudev Dutt (graduated) Jun Cheng (graduated) Yanwen Wu (graduated) Qiong Huang (graduated) Xiaodong Xu Erik Kim Katherine Smirl Bo Sun John Schaible Vasudev Lai **Alberto Amo - Autonoma University of Madrid

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