Adiabatic control of many-particle states in coupled quantum dots - PowerPoint PPT Presentation

Adiabatic control of many-particle states in coupled quantum dots Paul Eastham Trinity College Dublin R. T. Brierley, C. Creatore, R. T. Phillips (University of Cambridge) P. B. Littlewood (Argonne National Lab) easthamp@tcd.ie http://www.tcd.ie/Physics/People/Paul.Eastham Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 1 / 21

Outline Introduction 1 Excitons in quantum dots as qubits State preparation by resonant excitation Adiabatic rapid passage Adiabatic control in many-particle systems 2 Theoretical models and approaches Pairwise-coupled dots 1D chains Mean-field limit Conclusions 3 Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 2 / 21

Excitons in quantum dots as qubits? Island of reduced bandgap in optically active semiconductor, e.g. InGaAs in GaAs. H = E g s z + g s + E ( t ) + E ∗ ( t ) s − � � Why not? Decoherence? Inhomogeneity Lifetimes typically � 1ns 1 / (∆ E g ) ∼ 0 . 01 ps (best 0 . 3 ps ?) . . . but E(t) fast – � 1ps Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 4 / 21

State preparation by resonant excitation H = E g s z + g s + E ( t ) + E ∗ ( t ) s − � � How to prepare an initial state | ↑� ? Resonant excitation E ( t ) = e iE g ( t ) t | E ( t ) | , H → UHU † − iU † dU dt = g | E ( t ) | ( s + + s − ) , d� s dt = ( g | E ( t ) | , 0 , 0) × � s � � | ↑� after pulse when g | E ( t ) | dt = π, 3 π, 5 π, . . . . Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 5 / 21

Chirped adiabatic rapid passage Inhomogeneous ensemble: dot-to-dot fluctuations in E g , g ⇒ resonant excitation unusable. Use chirped pulse E ( t ) = e iω ( t ) t | E ( t ) | ω ( t ) = E g + αt H = [ E g − ω ( t )] s z + 2 g | E ( t ) | s x 1 − P ↑ ∼ e − g 2 | E | 2 /α . PRE and R. T. Phillips, Phys. Rev. B 79 165303 (2009); E. R. Schmigdall, PRE and R. T. Phillips, Phys. Rev. B 81 195306 (2010) Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 6 / 21

Chirped adiabatic rapid passage in ensembles � Works in ensembles despite variation in E g , g , for all those dots satisfying adiabatic criterion ∼ ps pulse creates a population equivalent to thermal equilibrium at 0.6 K Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 7 / 21

Experimental implemetations Single quantum dot in photodiode, pulsed laser excitation ← − 1 exciton/pulse Chirped excitation Resonant excitation [Wu et al., Phys. Rev. Lett. 106 067401 (2011); Simon et al., Phys. Rev. Lett. 106 166801 (2011).] Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 8 / 21

Theoretical models � � E g,i s z s + i E ( t ) + E ∗ ( t ) s − J ij ( s + i s − � � H = i + g i − j ) i i � ij � Pairwise coupling 1 – Stacked quantum dots + F¨ orster coupling/wavefunction overlap 1D chain 2 – Coupled cavity-QED? Mean-field limit 3 – Many quantum dots + optical cavity? Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 10 / 21

ARP to populate pairwise-coupled dots Solve equations of motion for pair w/coupling j T , with model pulse, – duration τ , chirp rate α , centre frequency E g , peak Rabi frequency g 0 . Large g 0 : fully occupied regions, separated by lines of fringes Moderate g 0 : finite j T improves adiabaticity Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 11 / 21

Interpretation: Pairwise-coupled dots H = − α ( t − t 0 ) s z + 2 g | E ( t ) | s x + j T s + s − | ↑↑� | ↑↓� − | ↓↑� | ↑↓� + | ↓↑� | ↓↓� A: all crossings inside pulse and adiabiatic. | T − � → | T 0 � → | T + � . B: | T − � crosses | T 0 � outside pulse ∴ | T 0 � unoccupied, but perturbatively couples | T ± � , recovering adiabaticity. Diagonal fringes: | T − � , | T 0 � crossing becoming non-adiabatic. Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 12 / 21

Pairwise-coupled dots: creating entangled states Could populate (entangled) state | T 0 � – centre pulse on | T − � , | T 0 � crossing, pulse off before | T 0 �| T + � crossing Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 13 / 21

Pairwise-coupled dots: creating entangled states -10 -5 0 5 10 15 20 25 30 50 |T + > |S > |T 0 > (b) 40 > |T |T > − − (a) 30 |T > + 20 eigenvalues 10 |S > j T 0 A B |T 0 > |T 0 > -10 -20 -30 |T > − -40 pump -50 -10 -5 0 5 10 15 20 25 30 olution of the energy eigenvalues of an inter [R. G. Unanyan, N. V. Vitanov and K. Bergmann, Phys. Rev. Lett. 87 137902 (2001)] Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 14 / 21

1D chains � − αts z i + 2 g | E ( t ) | s x i + 4 J ( s + i s − H = i +1 + h.c.) i Diagonalize with Jordan-Wigner transform i c i − 1 i = c † s z 2 i = 1 j<i c † 2 e iπ � j c j c i = T i c i s − [ αt � 2 + J cos k ] c † � s x H = − k c k + 2 g | E ( t ) | i k i Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 15 / 21

1D chains [ αt � 2 + J cos k ] c † � s x H = − k c k + 2 g | E ( t ) | i k i Energy levels for N = 4 sites Colors– J>0 J<0 N + 1 “bands” labelled with n = � c † c ( S z /population) In each band, set of levels from n fermions in N k-states ( S 2 ) J/ α Uniform field conserves S 2 . Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 16 / 21

Mean-field limit Numerically solve equations of motion in mean-field approx : � [ − αt ] s z s + i + s − � J ij ( s + i s − � � H = i + g | E ( t ) | − j ) , i i i � = j � J eff [ s + i � s − → − j � + h.c.] i � = j – Exact for J ij = J/N 2 , N → ∞ ; LMG model for finite N . Final occupation for g 0 τ = 3 Loss of adiabaticity for fast chirp Fan of finite occupation with sharp boundaries J ≷ 0 increases (reduces) occupation/adiabaticity Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 17 / 21

Mean-field limit: interpretation Final occupation for g 0 τ = 3 Loss of adiabaticity for fast chirp Fan of finite occupation with sharp boundaries J ≷ 0 increases (reduces) occupation/adiabaticity J>0 J<0 J/ α Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 18 / 21

Conclusions Can adiabatically populate a single quantum dot by driving with chirped laser pulses In models (anti-)ferromagnetic x-y coupling initially enhances (suppresses) populations . . . but too strong coupling J ∼ ατ → no mixing at critical level crossing → scheme fails Virtual transitions allow population even for large J in small systems Straightforward extensions to generate entangled states Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 20 / 21

Future directions Experimental implementations of entanglement generation, non-equilibrium condensation Theoretical modelling of tolerance to fluctuations in E g,i , g i , J (random-field models) Decoherence due to acoustic phonons, Johnson-Nyquist noise Approaches to probing decoherence, interaction strengths (cf. NMR!) Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 21 / 21