Optically Driven Spins in Semiconductor Quantum Dots: Toward III-V Based Quantum Computing

1. Optically Driven Spins in Semiconductor Quantum Dots:Toward III-V Based Quantum Computing Duncan Steel - Lecture 1 DPG Physics School on "Nano-Spintronics” Bad Honnef 2010

2. Requirements to build a QC(Divincenzo Criteria) • Well defined qubits • Universal set of quantum gates (highly nonlinear) • Initializable • Qubit specific measurements • Long coherence time (in excess of 104 operations in the coherence time)

3. InAs GaAs Coupled QD’s [001] Coupled QD’s GaAs 72 nm x 72 nm Cross sectional STM Boishin, Whitman et al. Quantum Dots:Atomic Properties ButEngineerable • Larger oscillator strength (x104) • High Q (narrow resonances) • Faster • Designable • Controllable • Using ultrafast light, we have fast (200 GHz) switching with no ‘wires’. • Integratablewith direct solid state photon sources (no need to up/down convert) • Large existing infrastructure for nano-fabrication • 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

4. KEY REQUIREMEMT: CONTROLA logic device is highly nonlinear Requires a two state system: 0 and 1 Semiconductor with periodic lattice

5. The Principle Physics for Optically Driven Quantum Computing in semiconductors is the Exciton Semiconductor with periodic lattice

6. Semiconductor with periodic lattice Can the Exciton be Controlled in High Dimensional crystals? hole With coulomb coupling, the e-h pair forms an exciton: Extended state of the crystal 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

7. Bloch Theorem: for a periodic potential of the form The solution to Schrödinger’s equation has the form hole Is the Exciton a Well defined qubit in 1, 2, or 3 Dimensional Cystal? electron The exciton in higher dimensional cyrstals is not a well defined qubit.

8. Recall the spin paradigm for quantum computing: Can the Exciton be Controlled in High Dimensional cyrstals: i.e., can you build a universal set of quantum gates? Rabi Oscillations: Qubit Rotations

9. Coherent optical control z z 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. y y Precession Rabi x x

10. Simple Coherent Control in an Atom – Rabi Flops z Laser Pulse y x Controlling t and/or ΩR allows control of the switching between up and down, creating states like:

11. Rabi Oscillations 6     7  0  Pulse Area

12. 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

13. 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

14. Nearly Degenerate Differential Transmission Quantum Dot Spectrum Pump Pump excitation reduces absorption on excited transition Differential Probe Tuning

15. RF electronics Lock-in amplifier CW Nonlinear Spectroscopy Experimental Set-up Frequency stabilized lasers E probe Detector E probe E signal E pump Acousto-optic Modulators ƒ≈100 Mhz

16. Many-Body Effects in High Dimensional Semiconductors Excitation Wavelength Wang et al. PRL 1993

17. To Suppress Extended State Wave Function, consider a zero dimensional system: a Quantum Dot Still a complex manybody system Exciton Electron based qubit Trion Spin based qubit |1> |i> |1> |0> |0> e e 300 A 300 A h h 4 6 2 4 Figure of merit ~10 -10 Figure of merit ~10 -10 -9 -9 Dephasing time >>10 sec Dephasing time ~10 sec (in SAD’s)

18. 1630 1628 1626 1624 1622 1621 1622 1623 1624 1625 1626 1627 Quantum Dot Photoluminescence as a Function of Laser Excitation Energy . Excitation energy (meV) Detection energy (meV) Atomic-like spectrum – Discrete states followed by continuum

19. 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 PL Intensity Nonlinear Signal Intensity

20. Use a Quantum Dot to Build a 2-Qubit Computer? Empty Conduction band Filled valence band 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

21. |1> |1> B-Field + Coulomb Interaction |0> |0> How to Build a Two-Bit Quantum Computer Two spin-polarized excitons Need two quantum bits Need coupling Coulomb interaction Need coherent control Resonant polarization- dependent optical coupling |11> s- s+ |01> |10> |00>

22. The Two-Bit System Optical Field AlGaAs GaAs AlGaAs |00>

23. The Two-Bit System Optical Field AlGaAs GaAs s+ AlGaAs |01>

24. The Two-Bit System Optical Field AlGaAs GaAs s- AlGaAs |10>

25. Formation of the |11> state Optical Field AlGaAs GaAs s- s+ AlGaAs |11> Biexciton

26. Do quantum dots experience pure dephasing? Detection of coherence is made by measuring an observable proportional to where The equation of motion for the coherence is arises from either loss of probability amplitude or pure dephasing due to a randomly fluctuating phase between the two probability amplitudes: Relationship to NMR language

27.   0   10  ph ph rel - 2 - 1 0 1 2 - 2 - 1 0 1 2 Probe detuning (  u n i t s ) Calculated Coherent Wavelength-Resolved Differential Transmission from a Two Level System No pure dephasing Strong pure dephasing • The coherent contribution 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). Nonlinear Signal Intensity (a.u.)

28. 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 robustagainst decoherence) Nonlinear Signal Intensity (a.u.)

29. The Two-Bit System Optical Field AlGaAs GaAs s+ AlGaAs

30. The Two-Bit System Optical Field AlGaAs GaAs s- AlGaAs

31. s- polarized exciton state s+ polarized exciton state s- c+ c- + s+ + 1 3 1 3 - - - - 1 3 3 1 + + + + 2 2 2 2 2 2 2 2 First Step Towards Semiconductor Based Quantum Computing:Two Exciton-State Quantum Entanglement Quantum wave function shows entanglement of two exciton-states. 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.

32. The Exciton Based Two Qubit System Bloch Spin Vector Basis (Feynman, Vernon, Hellwarth)

33. + - Probe s+ Pump s- g Turn off the Coulomb Correlation Turn on the Coulomb Correlation + - Probe s+ Pump s- g s- s+ Ground state depletion Pump: s- Entanglement No Signal !! Total Signal 3 -2 -1 0 1 2 4 5 6 7 8 9 3 -2 -1 0 1 2 4 5 6 7 8 9 Probe ( g ) Probe ( g )

34. Experiment : Coulomb Correlation Quantum Entanglement of two exciton-states

35. C g C C C b y = + s + s + 0 + + - - b b b DE s+ s- s+ s- g g Entanglement of Two Exciton States: Non Factorizable Wavefunction Non-interacting Case Factorizable wavefunction: With Coulomb Correlation How small Cb is depends on linewidth of state b and DE

36. The Two (Exciton) Qubit System Optical Field AlGaAs GaAs AlGaAs |00>

37. The Two (Exciton) Qubit System Optical Field AlGaAs GaAs s+ AlGaAs |01>

38. The Two (Exciton) Qubit System Optical Field AlGaAs GaAs s- AlGaAs |10>

39. The Two (Exciton) Qubit System Optical Field AlGaAs GaAs s- s+ AlGaAs NOTE: In semiconductor systems the “Dipole Blockade” is a naturally occuring phenomena, but much stronger, usually, than the dipole term (Coulomb Blockade). |11> Biexciton

40. Photoluminescence and Coherent Nonlinear Optical Spectra • Superlinear excitation intensity dependence of photoluminescence from the biexciton-to-exciton transition

41. DE=biexciton binding energy m=1/2 m=-1/2 m=-3/2 m=3/2 The Bound Biexciton (Positronium Molecule) • 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

42. C C C C + - b g b DE s+ <<0.005 s- 0.3 0.9 0.3 g Quantification of Entanglement: Entropy* 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, * C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A54, 3824 (1996) *E~0.2 measured beyond chi-3 limit. Now up to E~1

43. s- 1 1 3 3 - - - - 3 1 3 1 + + + + 2 2 2 2 2 2 2 2 Creation of the Bell State unexcited state Biexciton state Quantum wave function shows entanglement of the ground state and the biexciton. c+- c0 + s+ +

44. The Two (Exciton) Qubit SystemRabi Oscillations Optical Field AlGaAs GaAs AlGaAs |00>

45. The Two (Exciton) Qubit SystemRabi Oscillations Optical Field AlGaAs GaAs s+ AlGaAs |01>

46. Rabi Oscillations - qubit rotations      0  Pulse Area

47. Epump or One Qubit Rotation in a Single Quantum Dot The Exciton Rabi Oscillation Excitonic energy levels Rabi oscillations • Rabi oscillations demonstrate an arbitrary coherent superposition of exciton and ground states, • A pulse area of p gives a complete single bit rotation, p-pulse p/2-pulse 2p-pulse or population: Time (ps) Time (ps) Time (ps) final quantum state (before decoherence): “Damping” is due to excitation induced increase in T1

48. Physics for Optically Driven Spin Neutral Exciton Negative Exciton |X> |0> Electronic Spin Qubit Semiconductor Quantum Coherence Engineering Successful coherent optical manipulation of the optical Bloch vector necessary to manipulate the spin vector

49. Raman coherence oscillates at frequency of the Zeeman splitting due to electron in-plane g-factor and decays with time. • Circularly polarized pump pulse creates coherent superposition of spin up and down state. Optical Excitation of Spin Coherence:Two-photon stimulated Raman

50. 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. - 2005