“Quantum computation with quantum dots and terahertz cavity quantum

electrodynamics”Sherwin, et al. Phys. Rev A. 60, 3508 (1999)

Norm Moulton

LPS

The hook...

• Other proposed QC architectures involving quantum dots utilize only nearest-neighbor interactions

J(t)s3•s4

•At the time of publication, this was the first proposal for which gate operations might be performed for an arbitrary pair of dots in the QC.

The approach is analogous to the Cirac-Zoller approach using laser-cooled trapped ions:

Phonons THz Resonant Cavity Photons

Laser Pulses

Voltage pulses applied to QDots

Both use an “Auxiliary State” to affect quantum gate operations

Proposed system:

•Array of GaAs/AlxGa1-xAs triple-well nanostructures with electrical gates•Each QD is charged with 1 and only 1 electron

•CW laser with fixed l introduced into the side of the cavity

•Dots are in a sharply resonant THz cavity, >>Ldot

GaAs

InAsGaAs

Stacked self-assembled quantum dots

GaAs

Etched quantum well structure

AlxGa1-xAs

GaAs

AlxGa1-xAs

GaAs

AlxGa1-xAs

GaAs

AlxGa1-xAs

Superconducting electrodes

Effective axial potential in the dot

System Hamiltonian

aac

cω 22201110 σσ

eEeE

cc aaeg 100101 σσ

titie lll expσexpσ 100101,

cc aaeg 122112 σσ

tiatiae lclcl ωexpσωexpσ 211212,

H

Cavity Photons ProjectionsRabi Oscillations driven by cavity

photons (0-1)

Rabi Oscillations driven by laser+cavity photons (1-2)

Rabi Oscillations driven by cavity

photons (1-2)

Rabi Oscillations driven by laser photons (0-1)

tiatiaeH lclc expσexpσ 2002

~

photontwo

Auxiliary state (|2> )driven by two-photon processes

c

l

l

l

e

eg

e

ege

ωωωω 21

01,12

21

12,01~

Where:

Transition Energies vs. Applied Field

e (MV/m)

ecelel+c

0 0.5 1.0 1.5 2.05

10

15

20

25

Ene

rgy

(meV

)

E20

E10

t

c

c

c

t

-pulse at ec

State vector picks up phase of i

2-pulse at el+c

State vector picks up phase of -1

-pulse at ec

State vector picks up phase of i

c

c

E10(ec)

E10(el)

E20(el+c)

CNOT Gate Operation: 001tc100tc

i 100tc

i 001tc

c

c

c

-pulse at ec

State vector picks up phase of i

2-pulse at el+c

Not on resonance with E12 so no flopping.

-pulse at ec

State vector picks up phase of i

c

c

E10(ec)

E10(el)

E20(el+c)

CNOT Gate Operation: 011tc110tc

i 110tc

i 001tc

t

Requirements for Quantum Computation

•Initializing the computer For kBT<<E10 a wait of less than 1 sec will ensure that all qubits are in state |0>.

•Inputting initial data

Arbitrary one-bit rotations are effected using Rabi oscillations induced by laser field.•Readout

Propose to integrate new quantum well detector into the cavity. Detector is tuned to cavity resonance at the readout phase of the calculation.•Error correction

Enlarge the cavity to create several cavity modes in the QD tunable level-spacing range. This slows things down by reducing e in the cavity resulting in lower

Make a hybrid device that uses nearest-neighbor concepts in the cavity.

Requirements for Quantum Computation

•Decoherence

•Cavity Photons

•Electronic StateNo experimental data exists on these dots

Sources:

–Emission of freely propagating photons

Prevented by high-Q 3-D cavity

–Interaction with fluctuating gate potential (x-talk, Johnson noise)

Frequencies lower than E10/, cause adiabatic changes to the energy levels En, leading to phase errors.

tdteEt Nn

1

When QD not addressed: SC electrodes, SC path, SC ground-> No dissipation so no thermal fluctuations.

Noise during switching will cause errors and will have to be addressed (“in a future publication”).

Requirements for Quantum Computation

Sources of decoherence:

–Interaction with metastable traps in the semiconductor

Traps far from electrode are shielded by the electrode

Traps in the volume between gate electrodes pose a problem

Rely on future advances in production technology

–Inhomogeneity in the dots

Calibrate each quantum dot prior to computation

–Cavity photon lifetime

Engineer ultra-low loss THz cavity

Make cavity from Ultrapure Si (finite two-phonon losses)

Use QDs with E01 smaller than the gap of an s-wave superconductor, make cavity from superconducting transmission line

Requirements for Quantum Computation

Sources of decoherence:

–Coupling between radial and and axial wave functions

Calculations for assumed dot dimensions and properties show that the Eradial,10=30meV, larger than the highest electron energy during a CNOT (26.5meV).

Interactions with Acoustic Phonons

• Electron relaxation via acoustic phonon emission

– T1 processes: e- scattering from potential fluctuations arising from

local volume compressions and dilations induced by the phonons.(Deformation-potential approximation)

kkiffifi EEEW

21 2

Relaxation rate (Fermi’s Golden Rule)

Numerical calculation based on all previous assumptions yields =150 s

– T2 processes:

•Pure dephasing of quantum confined excitons is dominated by radiative lifetime of exciton at low temperatures

•Polaronic couping to excitons gives DOS peaks nonzero width

•Polaronic effects on electrons in QDs will be more like the effects of hydrogenic donors.

•Work with CdTe showed that phonon-induced linewidths of transitions of hydrogenic donors much smaller than those of excitons.

•Sherwin et al. Speculate that the phonon-induced linewidths will be sufficiently small as to not limit operation of the quantum computer.

CNOT Execution Time

meV5.11c

mn

c

cc

30

nc=3.6

mVevac 49max

33

min 272

mmVol c

mkVel 7.30

=25ns

=3.3ns

single-bit=few ps (with laser attenuated so Rabi frequency can be low enough.