Coherence and optical electron spin rotation in a quantum dot. Sophia Economou NRL. L. J. Sham, UCSD R-B Liu, CUHK Duncan Steel + students, U Michigan

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1 Coherence and optical electron spin rotation in a quantum dot Sophia Economou Collaborators: NRL L. J. Sham, UCSD R-B Liu, CUHK Duncan Steel + students, U Michigan T. L. Reinecke, Naval Research Lab

2 Outline Part I Background: QC with quantum dots, Λ system Spontaneously generated coherence: theory Experimental results Part II Background: Rabi oscillations, hyperbolic secant pulses Single-qubit rotations

3 Quantum Ψ> = cos(θ/) 0> + sin(θ/) e iφ 1> computing Requirements Qubit/Scalability Operations: arbitrary qubit-rotations and -qubit conditional operations Initialization Readout Qubit-specific measurement α Long = β coherence = ½ times ρ = α 0 ><0 + β 1 ><1 Completely mixed (Unpolarized) Two-level Qubit candidates system Electron spins in QDs Nuclear spins Atomic levels Superconducting qubits Bloch vector Two-level QM systems can be represented by a vector on/in a unitradius sphere

Quantum dots Semiconductor nanostructures with

, PRL 76, 305 (1996) J. P. Reithmaier et al.

4 Quantum dots Semiconductor nanostructures with 3D nanometer confinement for electrons/holes Atomic-like energy levels Fluctuation dots, SADs, gated dots Growth axis z D. Gammon et al., PRL 76, 305 (1996) J. P. Reithmaier et al., Nature 43, 197 (004) J. M. Elzerman et al., Nature 430, 431 (004)

5 QIP with optically controlled electron spins trapped in QDs Quantum dot with single excess electron e spin carries quantum information Operations: optically by Raman transitions via trion Trion: bound state of electron and exciton Inter-dot coupling: With common cavity mode (Imamoglu et al. PRL 99 ) Optical RKKY (C. Piermarocchi et al. PRL 0)

6 Energy levels & HH-LH splitting Bulk Bands of III-V compounds e,±1/ 1 J= Quantum dot e,±1/ 1 J= h,±3/ h,±1/ 3 J= h,±3/ h,±1/ Confinement- induced H-L hole splitting 3 J=

7 Lambda system in QD Without B field, no Raman transitions possible: cannot implement qubit operations: B = 0 Perpendicular B field mixes spin states, enables Raman transitions B x 0 3/> -3/> 3/> -3/> σ + σ - σ + σ + σ - σ - 1/>+-1/> +x> 1/> -1/> 1/>- -1/> -x> Choosing eg σ + light yields a Lambda system

8 Decay & decoherence Decay equations of generic Λ system known from atomic physics Can be derived from a Master equation. Basic idea: Start with total system dynamics, ignore (trace out) the bath End up with non unitary evolution for system Wavefunction Density matrix Decay & decoherence come from ignoring a part ( bath ) of the total system

9 Example: Spontaneous emission of generic Λ system initially excited Ψ> = e> e> Γ 1 Γ 1> Finally: ρ = 0.5 1>< >< > Common wisdom : spontaneous emission always produces decoherence.

10 Spontaneously generated coherence (SGC) Theoretically predicted in atoms: Spontaneous decay may result in superposition (coherence) of recipient states, i.e. a term ( t ρ 1 ) sp = Γρ ee (Javanainen 9) Has not been observed in atoms d 1 d e> Conditions E 1 small d 1.d 0 1> >

11 Features of the QD Λ-type system light propagation (z) e e h t> σ+ Γ Γ σ+ B (x) x> ω L x> Small Zeeman splitting transitions have same polarization Fluctuation QDs: HH trion splitting B 3 g x,hh 0 (J. G. Tischler et al.) trion does not precess! SGC requirements are fulfilled

12 Origin of SGC: Intuitive Picture Instead of energy eigenstates ±x> consider the ±z> states =>two-level system ( -z> decoupled by selection rules) B(x) t > φ +z> z> -z Limits

13 Experimental setup (theorist s view) Pump-probe experiment Detector τ 1 Sample τ Differential transmission as fn of delay timet d =τ τ 1 T +z σ+ z z Bloch sphere

14 Experimental setup (experimentalist s view) Picks up nonlinear response (DTS) For low excitation power 3 rd order dominant DTS(σ-) - DTS(σ+) ~ S z Dutt et al., PRL 94, 7403 (005)

15 Analytical expressions Amplitude Phase Economou, Liu, Sham and Steel, PRB 71, (005) Γ = L L c arctan - - arctan ω γ ω γ φ ( ) L c L 4 4 ω γ ω γ + Γ + A ( ) d t d t d e B e t A Γ + Τ L cos γ φ ω

16 Calculated & experimental results Ensemble experiment Dutt, Cheng, Li, Xu, Li, Berman, Steel, Bracker, Gammon, Economou, Liu, and Sham, PRL 94, 7403 (005)

17 Outline Part I Background: QC with quantum dots, Λ system Spontaneously generated coherence: theory Experimental results Part II Background: Rabi oscillations, hyperbolic secant pulses Single-qubit rotations

18 Review of proposals for optical spin rotations in QDs Chen,Piermarocchi,Sham,Steel (PRB 04): No explicit frequency selectivity, but ω L >>Ω (weak pulses) Adiabatically eliminate trion Implicitly requires long pulses Kis & Renzoni (PRA 03): Stimulated Raman adiabatic passage Requires auxiliary lower level Adiabaticity will slow down operations Calarco,Datta,Fedichev,Pazy,Zoller (PRA 03): π pulse to populate trion/wait/ π pulse to de-excite trion Suffers from trion decay rate z rotations only

19 Rabi oscillations Two-level system with energy splitting ω ο Driven by laser with central frequency ω Define detuning =ω ο ω Laser can be CW Rabi oscillations in time Pulsed Rabi oscillations as fn of pulse area e> g> ρ ee Ω R = de o A two-level system can be mapped onto a spin (pseudospin). SU() dynamics π rotation: back to - g>

20 Review of sech pulses in -level systems e> V ge = Ω sech(βt) e i t = detuning g> β = bandwidth Exact solution (Rosen & Zener Phys. Rev. 3) Pulse area can be defined for any When Ω = β π pulse Population returns to g> with an acquired phase: Global for lvl sys Useful in presence of A third level Economou et al. PRB 74 (006)

21 Use of π sech pulses for rotations: Strategy outline By choice of polarization, decouple different two level systems: T z z σ + z T z Each time the ground state is a spin state along A phase is induced, which is a function of the detuning nˆ nˆ Phase φ on spin nˆ is a rotation about by φ By changing we can span the whole space T x x π x π x T x x nˆ

22 I. Small Zeeman splitting: z rotations Broadband σ + pulse means β >> ω e Spin precession ~ frozen during pulse -level system + uncoupled levels Tz Tz σ + z z (in the z basis) Ultra fast z rotations Economou, Sham, Wu, Steel, PRB 74, (006)

23 I. Small Zeeman splitting: x rotations Use of linearly polarized light decouples the 4-level system to two -level systems: Detunings for transitions 1, Bandwidth β x π sech pulse induces a different phase in x and x. The difference of phases is angle of rotation Splitting (uev) g h We have designed rotations about two axes, z and x By combining them we can make any rotation! Field (T)

24 Example: π rotation about y axis Fidelity 99.8% z rotation x rotation z rotation Parameters for InAs QDs used Economou & Reinecke, cond-mat/

25 II. Large Zeeman splitting Above scheme requires large bandwidths for z rotations For QDs with large Zeeman splittings such lasers may not be available Modification of proposal Use narrowband pulses to select a Λ system Total laser field Choosing equal detuning and same f(t) creates a coherently trapped state Bright/dark states determined by phase and relative strength of two lasers

26 Coherent population trapping + π sech pulses: analytic sln to Λ system Energy eigenstates x, x are related to bright/dark B D, by Tx Tx VB T, x where Bright state coupling to trion is B D where We want the total pulse acting on bright state to have area π :

27 Example: π/ rotation about z axis Fidelity 98.84% Parameters for CdSe QDs used Economou & Reinecke, cond-mat/

28 Summary-I SGC has important effect on quantum beats in QDs First observation of SGC in QDs (not atoms)

29 Summary-II π sech pulses to decouple level systems Phase Small Zeeman splitting T z σ + T z z z Large Zeeman splitting: CPT scheme Tx B VB, Tx Tx D Simple for experimental demonstration

arxiv:cond-mat/ v1 [cond-mat.mes-hall] 3 Mar 2007

arxiv:cond-mat/ v1 [cond-mat.mes-hall] 3 Mar 2007 Theory of fast optical spin rotation in a quantum dot based on geometric phases and trapped states arxiv:cond-mat/0703098v1 [cond-mat.mes-hall] 3 Mar 2007 Sophia E. Economou and T. L. Reinecke Naval Research