Hyperfine Coupling in Double Quantum Dots by MikeJenny

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									Hyperfine Coupling in Double
      Quantum Dots
                  Journal Club 19/11/2007
                     Carlos López-Monís
             Supervisors: G. Platero & J. Iñarrea
                ICMM – CSIC (Madrid-Spain)


19/11/2007               HF Coupling in DQD
             Outline
• Introduction to Open quantum
  Systems Theory.
• The Ono & Tarucha Experiment.
• Hyperfine Coupling in Double
  Quantum Dots.




19/11/2007    HF Coupling in DQD
     Introduction to Open
    Quantum Systems Theory




19/11/2007   HF Coupling in DQD
                 Preliminary Ideas: Familiar Example
                 Revisiting the Stern-Gerlach (SG)
                             Experiment

O. Stern                                                   W. Gerlach
           Ag atoms with random spin orientation

Environment




                                                 Ag atoms with
                                      System      definite spin
                                                   orientation
    19/11/2007              HF Coupling in DQD
                 Preliminary Ideas: Familiar Example
                 Revisiting the Stern-Gerlach (SG)
                             Experiment

O. Stern                                                           W. Gerlach
   Once the Ag atoms have passed through the magnetic field
   their spin orientation can be written as a linear combination
   of the states         :

                                                   Coherent superposition
                                                             or
                                                        Pure state


  “...the phase relation between          and         contains vital
  information on the spin orientation...” J.J. Sakurai
  We can’t use a pure state to describe spin orientation before the
  Ag atoms pass through the magnetic field because it’s random !!!

    19/11/2007                HF Coupling in DQD
               Preliminary Ideas: In Short


 Open quantum systems theory deals with the
dynamics of a quantum system coupled to its
environment.
                                                      Ugo Fano
 Main difference between the system and the environment:
number of degrees of freedom (dof).  Finite (system) //
“Infinite” (environment).

 U. Fano (1957): The existence of a set of experiments (for
which the results can be predicted with certainty) gives a
necessary and sufficient characterization for a state of
“maximum knowledge”.  Pure states.

 Quantum systems that can’t be describe by pure states are said
to be in a incoherent or mixed state.


  19/11/2007                HF Coupling in DQD
              Density Operator Formalism

Developed in 1927 by John von Neumann to deal
with both pure and mixed ensembles.

Definition:
                                                         J. von Neumann
                                   ;

              probability of finding the system in the pure state


Expectation value of an operator:


The density operator contains all the relevant information about
the system !!!

                                                        (Blackboard)
 19/11/2007                HF Coupling in DQD
               Density Operator Formalism

SG teaching:

    System–Environment are coupled
                    +                            The environment places the
                                                  system in a mixed state.
     Only system dof are measured

The description of a physical system by means of pure states or
mixed states depends on the amount of information of the system
we are able to measure !!!

Principle of nonseparability (B. d’Espagnat, 1976):             B. d’Espagnat
If two systems have interacted in the past it is, in
general, not possible to assign a single state to either
of the two subsystems. (Blackboard)

Reduced density operator:
Contains all the relevant information about the system !!!
  19/11/2007                HF Coupling in DQD
                              The Redfield Equation

                      In 1957 A. G. Redfield wrote down the equation for
                      the reduced density operator using two important
A. G. Redfield        assumptions:

1. The Fano (Born) approximation: the density operator factors
approximately at all times into                  where      is
independent of time. The environment has so many dof that the
effects of the interaction with the system dissipate away quickly and
will not react back onto the system  the environment remains
described by a thermal equilibrium distribution at constant
temperature, irrespective of the amount of energy and polarization
diffusing into it from the system.                             Max Born


 Weak coupling between the system and the environment

 Basic condition of irreversibility !!! (relaxation theory)


    19/11/2007                HF Coupling in DQD
                             The Redfield Equation

                      In 1957 A. G. Redfield wrote down the equation for
                      the reduced density operator using two important
A. G. Redfield        assumptions:

2. The Markov approximation: The time evolution of the density
operator doesn’t depend on its past history, i.e., the system loses
memory of its past.

Reasonable assumption: 1.  the reaction of the environment on the
system is damped  damping destroys the knowledge of the past
behaviour of the system.
                                                         A. A. Markov
  Redfield equation




                System + Environment Hamiltonian
   19/11/2007                HF Coupling in DQD
       The K. Ono & S. Tarucha
             Experiment




19/11/2007     HF Coupling in DQD
             Experiment




19/11/2007   HF Coupling in DQD
                Experiment: Perliminary Ideas

                                                 Control Parameters


                                                 Gate Voltage (VG)
   Double
 Quantum
Dot Device
                                                 Source-Drain Voltage (V)




2DEG + 2D harmonic potential harmonic gate  0D confinement
Quantum Dot (QD)  0D system embedded in a solid  Artificial atom
Double Quantum Dot (DQD)  Artificial molecule
In natural atoms electrons see 1 nucleus //
In QD/DQD electrons see 102 – 104 nuclei (lattice) !!!
   19/11/2007               HF Coupling in DQD
                  Experiment: Perliminary Ideas
                      Spin blockade regime (SB)



Electron transport through a generic
two-site system with one electron
trapped permanently on site 2.

Electron transport is blocked (no
current throught the DQD) due
to Pauli exclusion principle !!! 
Spin Blockade (SB).

                                                    Dot 1   Dot 2


       The number of electrons in DQD oscillates between 1 and 2
          Our dynamical process: (0,1)  (1,1)  (0,2)  (0,1)

     19/11/2007                HF Coupling in DQD
                         Experiment: Setup
       Current voltage characteristic measured zero magnetic field.


 Current                                                 Leakage
 through                                                 Current !!!
 the DQD




                                                          (SB) region




Vertical (DQD)

     19/11/2007              HF Coupling in DQD
                  Experiment: Results
Magnetic field dependence of the leakage current in the middle
of the SB region (Vs = 3.0mV) as a function of in-plane magnetic
field for sweep up (black) and sweep down (grey).

                                                    Hysteretical
                                                    behaviour




                                                   Strong
                                                   oscillations




19/11/2007              HF Coupling in DQD
Hyperfine Coupling in Double
      Quantum Dots




19/11/2007   HF Coupling in DQD
                      HF Coupling in DQD

Questions:

          Why is there a leakage current in the SB regime?

          Why does it present hysteretical behaviour?

          Why do we see oscillations?

Possible answers:
          Phonon mediated transitions.
          (J. Iñarrea, G. Platero and A. H. MacDonald; PRB 76, 085329 (2007))

          Cotunneling processes, virtual transitions, non-conservative
          transitions.

          Hyperfine coupling

         Others ...

  19/11/2007                 HF Coupling in DQD
               HF Coupling in DQD: Contact

Dof of the DQD  7




Dof of the nuclear bath  Infinite

We are only able to measure the dof of the DQD !!!  current (I)

DQD  Quantum system (Ag atom - SG)
                                               Open quantum system
Nuclear bath  Environment: (Oven - SG)
                                  Trace out
Hamitonian:
                      H = HDQD + HN + VHF


 19/11/2007               HF Coupling in DQD

								
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