Quantum Dots

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					Quantum Dots
   Timothy Paik
 Marcus Dahlstrom
   Michael Nip
      Implementing Quantum
• Many implementations for quantum

• Why solid state?
  – Scalability
  – Decoherence is less of a problem
      What is a quantum dot?
• In two words, a
• Easily tunable by
  changing the size and
  composition of the
Gallium Arsenide Quantum Dots
• Gallium arsenide is a III-V semiconductor
  – Higher saturated electron velocity and higher
    electron mobility than silicon
  – Gallium arsenide can emit and absorb light,
    unlike silicon
     • No silicon laser is possible (or has been made yet)
            Energy Band Levels
• Electrons exist in discrete
  energy levels in bulk
  semiconductor material.
   – There exists a forbidden
     range of energy levels in
     any material called the
     band gap.
            Energy Band Levels
• By absorbing some sort
  of stimulus (in light or
  heat form), an electron
  can rise to the conduction
  band from the valence
   – This action leaves behind a
     “hole” in the valence band.
     The hole and the electron
     together are called an
          Energy Band Levels
• The average distance
  between an electron and
  a hole in a exciton is
  called the Excited Bohr
• When the size of the
  semiconductor falls below
  the Bohr Radius, the
  semiconductor is called a
  quantum dot.
    Tuning Quantum Dots

• By changing size,
  shape, and
  quantum dots can
  change their
  absorptive and
  emissive properties
     Manufacturing methods
• Electron beam lithography
• Molecular beam epitaxy
Electron Beam Lithography
             • Electrons are accelerated
               out of an electron gun
               and sent through
               condenser lens optics
               directly onto a wafer
             • λ = (12.3 Å / √V)
             • Advantages:
                – generation of micron and
                  submicron resist geometries
                – greater depth of focus than
                  optical lithography
                – masks are unnecessary
                – Optical diffraction limit is not a
                  real concern
Electron Beam Lithography
             • Disadvantage(s):
               – The lithography is serial
                 (masks aren’t used; instead
                 the beam itself sweeps
                 across the wafer) =>
                 Comparatively low
                 throughput ~5 wafers per
                 hour at less than 1
                 micrometer resolution
               – The proximity effect:
                 Electrons scatter because
                 they are relatively low in
                 mass, reducing the
                  • Heavy ion lithography has
                    been proposed, but still is
                    in development stages
      Molecular Beam Epitaxy
• Molecular beam epitaxy (MBE) is the deposition
  of one or more pure materials onto a single
  crystal wafer one layer of atoms at a time in
  order to form a perfect crystal
  – This is done by evaporating each of the elements to
    combine, then condensing them on top of the wafer.
  – The word “beam” means that the evaporated atoms
    only meet each other on the wafer
 Spin Quantum Computing

Qubit information is stored in the spin state of an
          electron in an artificial atom

               Long decoherence time

                   Future Scalabilty

Artifical atoms are bigger than regular atoms therefore
                  easier to manipulate
    Decoherence time ~ 100ns
• Time before the quantum mechanical system starts
  acting in a classical way with it's complex
• The state of the system has not yet collapsed due
  to (unwanted) environmental effects
• Spin - DT are 100 as long as for the Exciton
• Need to SWITCH 104 during DT for reliable error
  correction. This requirement is met.
            Artificial Atom
• Double Barrier
• Dot: In0.05Ga0.95As
• Source &Drain : GaAs
• 2D Electron Gas
• Confine with gate bias
• D ~ Fermi wavelength
  → Discrete energy
Adding Electrons, changing Vgate
                • 2D-Harmonic
                • Shell structure as in
                • Magic Numbers: 2, 6,
                • To add “even”
                  electron requires only
                  additional Coulomb
 Comparison with Hydrogen
• Artificial Atom:   • Hydrogen:

 Energy levels ~      Energy levels ~ 1eV
 1meV                 Size ~ 1Å

 Size ~ 10μm          Only strong magnetic
                      fields can perturb
 Weak magnetic        energy levels
 fields can affect
 energy levels
                      Factor 1000...
      Tuning the Quantum Dot
• Tune so we have one        Energy
                                      Unoccupied state
  valence electron
• Initial state can be set
  by applying                                  Gate bias

  magnetic field → |0>                Spin up - electron

• Low temperature:                               position
  kT < ΔE (state gap)
• Now we have defined
  our single qubit
     Single Qubit manipulation
• Unitary operations can
  be made by applying
  a local magnetic field:
  HZE = -μ·B = g μB S·B
• MF microscope
• AF microscope
• Sub grid of current
• Magnetic dots
• Etc...
                            (Magnetic force microscope tip)
      Two Qubit Manipulation
• Complete set of logic
  requires a CNOT
• Dots are placed so
  close that they overlap
  and interact:
• Hspin = J(t)S1·S2

  Exchange coupling:              (4:th order Harmonic Oscillator)
  J(t,E,B) = Etriplet -Esinglet
 Ground State Splitting (J = Et –
• 2 coupled fermions must have an total anti-
  symmetric wave function
• Lowest coupled state is the singlet. It has a
  symmetric spatial wave function and an anti
  symmetric spin (Coulomb dominates):
  |ψs> ~ (|12> + |21>) (|↓↑> - |↑↓>)
• The triplet states are:
  |ψt> ~ (|12> - |21>) (|↓↑> + |↑↓>)
• <1|2> ≠ 0, |i> is spatial w.f. Coulomb dominates
     Solving J(B(t)): Exchange
• Different solutions:
  * Heitler-London
  * Hund-Mulliken
  * Hubbard
• Important conclusion:
  We can control
  coupling from zero to
  non-zero by changing
  the magnetic field →
  We can perform two
  qubit operations.
           SWAP - gate
• Assume J can be pulsed:
  J(t) = {0, J0}

 Formula 1

 Formula 2

• Now we can put many qubits on a line
  and move them so that they all can
  interact [not all at once though]
              XOR ~ CNOT
• Formula 3

• Requirements:
  * Spin rotations about the z-axis
  * Squareroot of Uswap
         Read out / Memory
• Assume dot with an electron with some
  information stored in spin-state
• Connect two leads to dot
• Apply a small bias (ΔV) → Current (i)?!
                     Unoccupied state

                              Gate bias
                     Spin up - electron

Another Spin up electron enters
• Pauli principle forces electrons with spin
  up to occupy the higher energy state
• Negligible chance of tunneling

     i=0             Higher energy level
                     (forbidden classically)

                              Gate bias
                     Spin up - electron

 Spin down electron enters dot
• Pauli principle allows the new electron to join the
  same energy level as the original electron
• Coulomb interaction perturbs the ground-state so
  that it is raised above the right bias and current
  will flow
                         Unoccupied state

                                  Gate bias
                         Spin up - electron

          Read out / Memory
• We have a way of measuring the spin state of an
  electron in a quantum dot
• The first electron that passes though measures
  the spin-state in the dot and other electrons that
  follow will all have the same spin properties
• To be able to predict the original state of the dot,
  the state has to be prepared again and then
  measured using the same technique
• The electron current can be specialized (we can
  aim it's spin to make measurement efficient)
         5 DiVincenzo QC Criteria
1.   A scalable physical system with well-characterized
2.   The ability to initialize the state of the qubits to a
     simple fiducial state.
3.   Relatively long decoherence times compared to gate-
     operation times.
4.   A universal set of quantum gates.
5.   Qubit-specific measurement capability.
     The Physical System: Excitons
    Trapped in GaAs Quantum Dots
•   Exciton - a Coulomb correlated
    electron-hole pair in a semiconductor,
    a quasiparticle of a solid.
•   Often formed when photons excite
    electrons from the valence band into
    the conduction band.
•   Wavefunctions are “hydrogen-like” i.e.
    an “exotic atom” though the binding
    energy is much smaller and the extent
    much larger than hydrogen because of
    screening effects and the smaller
    effective masses
•   Decay by radiating photons. Decay
    time ~50ps-1ns
•   Hence can define the computational
    basis as absence of an exciton |0>, or
    existence of an exciton |1>
• Register relaxes to the |00…0> state within 50ps-1ns
  due to radiative decay
   – Experimental systems are cooled to liquid helium temps ~4K to
     prevent thermal excitations
• Hence initialization with such a system is relatively easy
• Other states can be initialized by applying gates to the
     Relatively Long Decoherence
• Mechanisms:
  – Radiative Decay ~10ps-1ns
     • Can be lengthened by electron-hole separation
  – Background Electromagnetic fluctuations
     • Less of a problem than in other systems since the
       exciton and III-V heterostructure is on average
       electrically neutral.

• Gate times are determined by energy band spacing, i.e.
  creation and annihilation energies.
   – Gate operations for GaAs QDs are estimated at ~1ps
     or less
 A Universal set of Quantum Gates
• Single Qubit Rotations through laser induced
  Rabi Oscillations
• CNOT operations through dipole interactions
  and laser excitation
Single Qubit Gates: Rabi Flopping
                 • Light-particle interaction is
                   characterized by the product of
                   the dipole moment and the
                   electric field:

                            μ•E(t)= ħR(t)

                 Where R(t) is the Rabi frequency
                   and the pulse area is given by:


                 and the state at time t is then
                   given by:
                         Stufler et al.
Large wafer containing InGaAs
   QD was placed between a bias
   voltage and exposed to
   ultrafast laser pulses.


|1> => electric charge

=>Photocurrent (PC)

π-pulse corresponds to a
   population inversion
        CNOT: Dipole Coupling
Nearest neighbor interactions alter the energy states:

Effective energy:   E’i = Ei + ∑j≠i ∆Eij nj

Hence, a coherent π-pulse with energy E’t(nc) results in a
  state flop iff the control state is occupied.
Overcoming Short Interaction
              •   Electrostatic Dipole fields fall off
                  as 1/R^3 hence the CNOT gate
                  can only be used for closely
                  neighboring QDs.
              •   Solution: Use a sequence of
                  CNOTs on nearest neighbors to
                  swap the desired qubits until they
                  are contained in adjacent dots.
              •   Optical Cavity coupling and fiber
                  optical interconnects have also
                  been proposed.
Read Out of Specified Qubit States
• Optical readout:
Excitons decay spontaneously and the resulting radiation can be

Alternatively, an excitation/probe beam spot can be physically
   positioned in the region of the desired QD.

Due to the statistical distribution of QD shape and size variations,
  individual QDs can be more accurately identified and addressed
  through frequency discrimination.

In either case, repeated measurements have to be made. A single shot
    readout still needs to be developed.
         5 DiVincenzo QC Criteria
1.   A scalable physical system with well-characterized
2.   The ability to initialize the state of the qubits to a
     simple fiducial state.
3.   Relatively long decoherence times compared to gate-
     operation times.
4.   A universal set of quantum gates.
5.   Qubit-specific measurement capability.

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