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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

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SLIDE 1

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

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SLIDE 2

Outline

1

Introduction Excitons in quantum dots as qubits State preparation by resonant excitation Adiabatic rapid passage

2

Adiabatic control in many-particle systems Theoretical models and approaches Pairwise-coupled dots 1D chains Mean-field limit

3

Conclusions

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 2 / 21

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SLIDE 3

Excitons in quantum dots as qubits?

Island of reduced bandgap in optically active semiconductor, e.g. InGaAs in GaAs. H = Egsz+g

  • s+E(t) + E∗(t)s−

Why not? Decoherence? Lifetimes typically 1ns . . . but E(t) fast – 1ps Inhomogeneity 1/(∆Eg) ∼ 0.01 ps (best 0.3 ps?)

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 4 / 21

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SLIDE 4

State preparation by resonant excitation

H = Egsz + g

  • s+E(t) + E∗(t)s−

How to prepare an initial state | ↑? Resonant excitation E(t) = eiEg(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, 3π, 5π, . . ..

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 5 / 21

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SLIDE 5

Chirped adiabatic rapid passage

Inhomogeneous ensemble: dot-to-dot fluctuations in Eg, g ⇒ resonant excitation unusable. Use chirped pulse E(t) = eiω(t)t|E(t)| ω(t) = Eg + αt H = [Eg − ω(t)] sz + 2g|E(t)|sx 1 − P↑ ∼ e−g2|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

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SLIDE 6

Chirped adiabatic rapid passage in ensembles

Works in ensembles despite variation in Eg, 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

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SLIDE 7

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

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SLIDE 8

Theoretical models

H =

  • i

Eg,isz

i + gi

  • s+

i E(t) + E∗(t)s− i

  • −
  • ij

Jij(s+

i s− j )

1

Pairwise coupling – Stacked quantum dots + F¨

  • rster

coupling/wavefunction overlap

2

1D chain – Coupled cavity-QED?

3

Mean-field limit – Many quantum dots + optical cavity?

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 10 / 21

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SLIDE 9

ARP to populate pairwise-coupled dots

Solve equations of motion for pair w/coupling jT , with model pulse, – duration τ, chirp rate α, centre frequency Eg, peak Rabi frequency g0. Large g0: fully occupied regions, separated by lines of fringes Moderate g0: finite jT improves adiabaticity

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 11 / 21

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SLIDE 10

Interpretation: Pairwise-coupled dots

H = −α(t − t0)sz + 2g|E(t)|sx + jT s+s− | ↑↑ | ↑↓ − | ↓↑ | ↑↓ + | ↓↑ | ↓↓ A: all crossings inside pulse and adiabiatic. |T− → |T0 → |T+. B: |T− crosses |T0 outside pulse ∴ |T0 unoccupied, but perturbatively couples |T±, recovering adiabaticity. Diagonal fringes: |T−, |T0 crossing becoming non-adiabatic.

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 12 / 21

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Pairwise-coupled dots: creating entangled states

Could populate (entangled) state |T0 – centre pulse on |T−, |T0 crossing, pulse off before |T0|T+ crossing

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 13 / 21

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SLIDE 12

Pairwise-coupled dots: creating entangled states

  • 10
  • 5

5 10 15 20 25 30

  • 50
  • 40
  • 30
  • 20
  • 10

10 20 30 40 50

eigenvalues

− +

|S

pump

|T

−

|T0

> > >

(a)

A B |S |T0 |T |T

> > > > |T+> |T0

>

|T >

−

j T

(b)

  • 10
  • 5

5 10 15 20 25 30

  • lution 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

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SLIDE 13

1D chains

H =

  • i

−αtsz

i + 2g|E(t)|sx i + 4J(s+ i s− i+1 + h.c.)

Diagonalize with Jordan-Wigner transform sz

i = c† ici − 1

2 s−

i = 1

2eiπ

j<i c† jcjci = Tici

H = −

  • k

[αt 2 + J cos k]c†

kck + 2g|E(t)|

  • i

sx

i

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 15 / 21

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SLIDE 14

1D chains

H = −

  • k

[αt 2 + J cos k]c†

kck + 2g|E(t)|

  • i

sx

i

Energy levels for N = 4 sites

J<0 J>0 J/α

Colors– N + 1 “bands” labelled with n = c†c (Sz/population) In each band, set of levels from n fermions in N k-states (S2) Uniform field conserves S2.

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 16 / 21

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SLIDE 15

Mean-field limit

Numerically solve equations of motion in mean-field approx : H =

  • i

[−αt]sz

i + g|E(t)|

  • s+

i + s− i

  • −
  • i=j

Jij(s+

i s− j ),

→ −

  • i=j

Jeff[s+

i s− j + h.c.]

– Exact for Jij = J/N2, N → ∞; LMG model for finite N. Final occupation for g0τ = 3 Loss of adiabaticity for fast chirp Fan of finite occupation with sharp boundaries J ≷ 0 increases (reduces)

  • ccupation/adiabaticity

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 17 / 21

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Mean-field limit: interpretation

Final occupation for g0τ = 3

J<0 J>0 J/α

Loss of adiabaticity for fast chirp Fan of finite occupation with sharp boundaries J ≷ 0 increases (reduces)

  • ccupation/adiabaticity

Paul Eastham (Trinity College Dublin) Adiabatic control of many-particle states 18 / 21

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SLIDE 17

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

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Future directions

Experimental implementations of entanglement generation, non-equilibrium condensation Theoretical modelling of tolerance to fluctuations in Eg,i, gi, 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