Optically Driven Spins in Semiconductor Quantum Dots: Toward III-V Based Quantum Computing Duncan Steel - Lecture 1 DPG Physics School on "NanoSpintronics” Bad Honnef 2010 Requirements to build a QC (Divincenzo Criteria) 1. Well defined qubits 2. Universal set of quantum gates (highly nonlinear) 3. Initializable 4. Qubit specific measurements 5. Long coherence time (in excess of 104 operations in the coherence time) Quantum Dots: Atomic Properties But Engineerable • • • • • • • • • • Larger oscillator strength (x104) High Q (narrow resonances) Faster Designable Controllable Using ultrafast light, we have fast (200 GHz) switching with no ‘wires’. Integratable with direct solid state photon sources (no need to up/down convert) Large existing infrastructure for nanofabrication High temperature operation – Compared to a dilution refrigerator CHALLENGE: spatial placement and size heterogeneity AFM Image of Al0.5Ga0.5As QD’s formed on GaAs (311)b substrate. Figure taken from R. Notzel GaAs InAs Coupled QD’s Coupled QD’s 72 nm x 72 nm GaAs Cross sectional STM Boishin, Whitman et al. KEY REQUIREMEMT: CONTROL A logic device is highly nonlinear Requires a two state system: 0 and 1 Semiconductor with periodic lattice The Principle Physics for Optically Driven Quantum Computing in semiconductors is the Exciton Semiconductor with periodic lattice Semiconductor with periodic lattice Can the Exciton be Controlled in High Dimensional crystals? hole electron Excitons in high dimenisonal crystals do not have a simple atomic like nonlinearity: Quantum gates are hard to imagine Rabi oscillations in quantum wells Cundiff et al. PRL 1994 Schulzgen et al., PRL 1999 With coulomb coupling, the e-h pair forms an exciton: Extended state of the crystal Is the Exciton a Well defined qubit in 1, 2, or 3 Dimensional Cystal? hole electron Bloch Theorem: for a periodic potential of the form V r d V rd The solution to Schrödinger’s equation has the form r eik r ur where ur d ur The exciton in higher dimensional cyrstals is not a well defined qubit. Can the Exciton be Controlled in High Dimensional cyrstals: i.e., can you build a universal set of quantum gates? Recall the spin paradigm for quantum computing: Rabi Oscillations: Qubit Rotations Coherent optical control •Coherent optical control of an electronic state means controlling the state of the spin or pseudo- spin Bloch vector on the Bloch sphere. •It is a highly nonlinear optical process and is achieved with a combination of Rabi oscillations and precession. or excited or excited z z y Rabi x or ground y Precession x or ground Simple Coherent Control in an Atom – Rabi Flops h o 1 0 h 0 H * 2 0 1 2 R r r E R h R cost 0 z o Laser Pulse y x Ω allows control Controlling t and/or R of the switching between up and down, creating states like: 1 2 Rabi Oscillations ih H 0 E 0 sin t t H 0 un E n un u1 er u2 n 1,2; 2 1 C t 2 2 0 Pulse Area 6 7 ht E 0 tdt 20 Can the Exciton be Controlled in High Dimensional cyrstals: i.e., can you build a universal set of quantum gates? Excitons in high dimenisonal crystals do not have a simple atomic like nonlinearity: Quantum gates are hard to imagine What does an atomic like nonlinearity look like in the laboratory: Saturation (Spectral Hole Burning) Spectroscopy Quantum computing is a highly nonlinear system (intrinsic feature of a two level system in contrast to a harmonic oscillator. Nonlinear spectroscopy quantifies the behavior. Absorption Saturated absorption Differential absorption o 0 2 2 I 2 1 I sat Nearly Degenerate Differential Transmission Quantum Dot Spectrum Pump Pump excitation reduces absorption on excited transition Differential Probe Tuning CW Nonlinear Spectroscopy Experimental Set-up Frequency stabilized lasers Eprobe Eprobe Detector Esignal Epump Acousto-optic Modulators ƒ≈100 Mhz RF electronics Lock-in amplifier ENL Im[ ] E probe I pump Idetected ENL E*probe 3 Many-Body Effects in High Dimensional Semiconductors DT/T (a.u.) Absorbance (a.u.) Excitation Wavelength 1.508 hh lh 1.516 Energy (meV) 0 1.5075 1.5125 1.5175 Energy (eV) Wang et al. PRL 1993 To Suppress Extended State Wave Function, consider a zero dimensional system: a Quantum Dot Still a complex manybody system Exciton Electron based qubit |1> |0> |0> e 300 A h Trion Spin based qubit |i> |1> e 300 A h 2 4 4 6 Figure of merit ~10 -10 Figure of merit ~10 -10 -9 Dephasing time ~10 sec Dephasing time >>10 -9 sec (in SAD’s) Quantum Dot Photoluminescence as a Function of Laser Excitation Energy . Excitation energy (meV) 1630 1628 1626 1624 1622 1621 1622 1623 1624 1625 Detection energy (meV) 1626 Atomic-like spectrum – Discrete states followed by continuum 1627 Nonlinear Signal Intensity PL Intensity Photoluminescence and Nonlinear Spectra Comparison • The luminescence and nonlinear spectra have many lines in common • The luminescence and nonlinear techniques do not measure the same optical properties • The nonlinear response is resonant and highly isolated Use a Quantum Dot to Build a 2-Qubit Computer? Empty Conduction band Filled valence band 1 2 3 2 1 2 j 12,m j 3 2 j 32,m j Ground and first excited states for neutral quantum dot First break with atom picture: Lack of spherical symmetry means angular momentum is not a good quantum number How to Build a Two-Bit Quantum Computer Two spin-polarized excitons Coulomb interaction Resonant polarizationdependent optical coupling Need two quantum bits Need coupling Need coherent control |1> + |0> |1> |0> |11> B-Field Coulomb Interaction |01> |10> |00> The Two-Bit System Optical Field AlGaAs GaAs AlGaAs |00> The Two-Bit System Optical Field AlGaAs GaAs AlGaAs |01> The Two-Bit System Optical Field AlGaAs GaAs AlGaAs |10> Formation of the |11> state Optical Field AlGaAs + - GaAs AlGaAs |11> Biexciton Do quantum dots experience pure dephasing? Detection of coherence is made by measuring an observable C proportional to 2C1 where C1 1 C2 2 The equation of motion for the coherence is d C 2C1 C 2C1 other terms dt arises from either loss of probability amplitude or pure dephasing due to a randomly fluctuating phase between the two probability amplitudes: i 0 t R t C 2C1 c 2c1e 2 1 1 1 Relationship to NMR T1 ; T2 2 puredephasing language Calculated Coherent Wavelength-Resolved Differential Transmission from a Two Level System No pure dephasing Strong pure dephasing ph 0 • The coherent contribution ph 10 rel Nonlinear Signal Intensity (a.u.) leads to an asymmetric lineshape in the absence of extra dephasing processes. • In the presence of strong extra dephasing processes the lineshape develops into a sharp resonance on top of a broader resonance (Prussian helmet). -2 -1 0 1 2 -2 -1 0 Probe detuning ( units) 1 2 Nonlinear Signal Intensity (a.u.) Measured Coherent Differential Transmission from a Single Quantum Dot: No extra dephasing =>quantum coherence is robust • “Coherent” and “incoherent” contributions •Homogeneously broadened • T1~ 19ps and T2~ 32ps (i.e. T2 ~ 2 T1 , absence of significant extra dephasing shows dots are robust against decoherence) The Two-Bit System Optical Field AlGaAs GaAs AlGaAs The Two-Bit System Optical Field AlGaAs GaAs AlGaAs First Step Towards Semiconductor Based Quantum Computing: Two Exciton-State Quantum Entanglement - polarized exciton state c- 1 2 1 2 3 2 3 2 + polarized exciton state + c+ 1 2 1 2 3 2 3 2 Quantum wave function shows entanglement of two excitonstates. e e c e 3 e 1 c e 1 e 3 2 2 2 2 + Quantum entanglement in the wave function is a key feature in quantum computers. This is the property which allows them to surpass classical computers in computational ability. The Exciton Based Two Qubit System Bloch Spin Vector Basis (Feynman, Vernon, Hellwarth) Turn off the Coulomb Correlation Turn on the Coulomb Correlation + Pump g + - Probe Pump g Probe Ground state depletion Pump: Entanglement No Signal !! 1 Total Signal -2 -1 0 1 2 3 4 5 6 Probe ( ) 7 8 9 -2 -1 0 1 2 3 4 5 6 7 Probe ( ) 8 9 Experiment : Coulomb Correlation Quantum Entanglement of two exciton-states Entanglement of Two Exciton States: Non Factorizable Wavefunction C0 g C C Cb b Non-interacting Case Factorizable wavefunction: b Cb CC C0 + g DE With Coulomb Correlation Cb 0 How small Cb is depends on linewidth of state b and DE - b + g The Two (Exciton) Qubit System Optical Field AlGaAs GaAs AlGaAs |00> The Two (Exciton) Qubit System Optical Field AlGaAs GaAs AlGaAs |01> The Two (Exciton) Qubit System Optical Field AlGaAs GaAs AlGaAs |10> The Two (Exciton) Qubit System Optical Field AlGaAs + - GaAs AlGaAs |11> Biexciton NOTE: In semiconductor systems the “Dipole Blockade” is a naturally occuring phenomena, but much stronger, usually, than the dipole term (Coulomb Blockade). Photoluminescence and Coherent Nonlinear Optical Spectra • Superlinear excitation intensity dependence of photoluminescence from the biexciton-to-exciton transition The Bound Biexciton (Positronium Molecule) DEbiexciton binding energy m=-1/2 m=-3/2 m=1/2 m=3/2 •Higher order Coulomb correlations lead to 4-particle correlations and the bound biexciton •An essential feature of optically induced entanglement and a quantum controlled not gate Quantification of Entanglement: Entropy* Cg C+ C- Cb DE - 0.9 0.3 0.3 <<0.005 b + g For two-particle system, the entropy of entanglement goes between 0 and 1. Zero entropy means product state. Non-zero entropy indicating entanglement. From our experiment, using the upper limit for Cb, E 0.08 0.02* C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A 54, 3824 (1996) *E~0.2 measured beyond chi-3 limit. Now up to E~1 Creation of the Bell State c0 unexcited state Biexciton state 1 2 1 2 1 2 1 2 3 2 3 2 + c+- + 3 2 3 2 Quantum wave function shows entanglement of the ground state and the biexciton. The Two (Exciton) Qubit System Rabi Oscillations Optical Field AlGaAs GaAs AlGaAs |00> The Two (Exciton) Qubit System Rabi Oscillations Optical Field AlGaAs GaAs AlGaAs |01> Rabi Oscillations - qubit rotations i H 0 E0 sin t t H0 un En un n 1, 2; u1 er u2 2 1 C t 2 2 0 Pulse Area ht E0 tdt 20 One Qubit Rotation in a Single Quantum Dot The Exciton Rabi Oscillation Excitonic energy levels Rabi oscillations Epump •Rabi oscillations demonstrate an arbitrary coherent superposition of exciton and ground states, c c -pulse -pulse or c c •A pulse area of gives a complete single bit rotation, /2-pulse or “Damping” is due to excitation induced increase in T1 population: Time (ps) final quantum 1 1 state (before 2 2 decoherence): Time (ps) Time (ps) Physics for Optically Driven Spin |X> |0> Neutral Exciton Negative Exciton T : trion Semiconductor Quantum Coherence Engineering Successful coherent optical manipulation of the optical Bloch vector necessary to manipulate the spin vector Electronic Spin Qubit 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 gfactor and decays with time. Single Electron Spin Coherence: Single Charged Exciton Raman Quantum Beats Charged Exciton System X- Neutral Exciton System CNOS (a. u.) Ensemble Charged Excitons Single Neutral Exciton X Phys. Rev. Lett. - 2005 hs (eV) 0 500 1000 1500 Delay (ps) 2000 2500 T2* >10 nsec at B=0 Anomalous Variation of Beat Amplitude and Phase Standard Theory (a) • Plot of beat amplitude and phase as a function of the splitting. (b) Anomalous Variation of Beat Amplitude and Phase Standard Theory (a) • Plot of beat amplitude and phase as a function of the splitting. Spontaneously Generated Coherence (SGC) Trion • 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. 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.