Solid state realisation of werner quantum states via kondo spins
This presentation is the property of its rightful owner.
Sponsored Links
1 / 29

Solid state realisation of Werner quantum states via Kondo spins PowerPoint PPT Presentation


  • 82 Views
  • Uploaded on
  • Presentation posted in: General

Solid state realisation of Werner quantum states via Kondo spins. Ross McKenzie Sam Young Cho. Reference: S.Y. Cho and R.H.M, Phys. Rev. A 73, 012109 (2006) . Thanks to . Discussions with Briggs (RKKY in nanotubes) Doherty and Y.-C. Liang (Werner states)

Download Presentation

Solid state realisation of Werner quantum states via Kondo spins

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Solid state realisation of werner quantum states via kondo spins

Solid state realisation of Werner quantum states via Kondo spins

Ross McKenzie

Sam Young Cho

Reference: S.Y. Cho and R.H.M,

Phys. Rev. A 73, 012109 (2006)


Thanks to

Thanks to

Discussions with

  • Briggs (RKKY in nanotubes)

  • Doherty and Y.-C. Liang (Werner states)

  • Dawson, Hines, and Milburn (decoherence and entanglement sharing)


Big goals for quantum nano science

Big goals for quantum nano-science

  • Create and manipulate entangled quantum states in solid state devices

  • Understand the quantum-classical boundary, e.g., test quantum mechanics versus macro-realism (Leggett)

  • Understand the competition between entanglement and decoherence


Entanglement vs decoherence

Entanglement vs. decoherence

  • Interaction of a qubit with its environment leads to decoherence and entanglement of qubit with environment.

  • Interactions between qubits entangles them with one another.

  • We will also see that the environment can entangle the qubits with one another.


Outline

Outline

  • Classical correlations vs. entanglement vs. violation of Bell inequalities (Werner states)

  • Experimental realisations of two impurity Kondo model

  • Competition between Kondo effect and RKKY interaction

  • Entanglement between the two Kondo spins

  • How to create Werner states in the solid state.


Quantum correlations in different regions of hilbert space

Quantum correlations in different regions of Hilbert space

Entangled states

No correlations

Violate

Bell

inequalities

Correlations but no

entanglement


Solid state realisation of werner quantum states via kondo spins

0

0

5

7

8

<

<

p

p

p

s

s

s

:

:

Werner states

Mixed states of two qubits

In the Bell basis

Reduced density matrix

is probability of a singlet

No entanglement

Bell-CSSH inequalities satisfied


Model system two kondo spins interact with metallic environment via heisenberg exchange interaction

Two impurity

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Model system: two Kondo spins interact with metallic environment via Heisenberg exchange interaction


Experimental realisation i

Two impurity

Kondo system

Experimental realisation I

N. J. Craig et al., Science 304, 565 (2004)

2DEG between spins

in quantum dots

induces an

RKKY interaction

between spins.

Gates vary J


Experimental realisation ii

Two impurity

Kondo system

Experimental realisation II

  • Endohedral fullerenes inside nanotubes

A. Khlobystov et al. Angewandte Chemie International Edition

43, 1386-1389 (2004)


Solid state realisation of werner quantum states via kondo spins

Single impurity

Kondo system

Single impurity Kondo model

Hamiltonian

Conduction electrons

J is the spin exchange coupling

Conduction-electron spin density at impurity site R = 0

Low temperature properties determined by single energy scale

. Kondo temperature

Band width D and the single particle density of state at the Fermi surface


Solid state realisation of werner quantum states via kondo spins

Single impurity

Kondo system

Tuneable quantum many-body states:

Kondo effect in quantum dots

For a review, L. Kouwenhoven and L. Glazman, Physics World 14, 33 (2001)

Conduction

electron spin

Impurity spin

Kondo temperature can be varied

over many orders of magnitude


Solid state realisation of werner quantum states via kondo spins

RKKY interaction

Two impurity

Kondo system

Two impurity Kondo model

Hamiltonian

To second order J, the indirect RKKY (Ruderman Kittel-Kasuya-Yosida)interaction is

Ground state determined by competition

between Kondo of single spins and RKKY


Solid state realisation of werner quantum states via kondo spins

Single impurity

Kondo system

Impurity spin A

Spin-rotational invariant!

Conduction

electrons C

Spin singlet

Entanglement in single impurity Kondo model

[T. A. Costi and R. H. McKenzie, Phys. Rev. A 68, 034301 (2003)]

J

S=1/2

Subsystem A

Subsystem B

Total system A+B

Ground state

Reduced density matrix for the impurity

von Neumann entropy

The impurity spin is maximally entangled with the conduction electrons

c.f., Yosida’s variational wavefunction

[K. Yosida, Phys. Rev. 147, 233 (1966)]


Entanglement between the two kondo spins

~

~

S

S

<

>

¢

A

B

Entanglement between the two Kondo spins

  • Given by concurrence of the reduced density matrix for the two localised spins (Wootters)

  • Ground state is a total spin singlet (S=0) and thus invariant under global spin rotations

  • Entanglement is determined by

.

$ \langle \vec{S}_A . \vec{S}_B \rangle $


Solid state realisation of werner quantum states via kondo spins

Two impurity

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Reduced density matrix for the impurities

In the Bell basis


Solid state realisation of werner quantum states via kondo spins

Non Fermi-liquid behaviour

Low temperature behaviour of two impurity Kondo model

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]

Numerical renormalization group calculation shows that

Left:

the staggered susceptibility and the specific heat coefficients diverge.

The spin-spin correlation is continuously varying and approaches at

the critical value of around the divergence of susceptibility.

Right:


Solid state realisation of werner quantum states via kondo spins

Entanglement & Quantum Phase transition


Unstable fixed point

I

T

2

2

'

K

:

Unstable fixed point

  • At the fixed point

    [Gan, Ludwig, Affleck, and Jones]

  • Thus, for the critical coupling there is no entanglement between two qubits.


Questions for future

Questions for future

  • Can the competition between Kondo and RKKY be better understood in terms of entanglement sharing?

  • Why does the entanglement between Kondo spins vanish at the quantum critical point?

  • What effect does temperature have?


Conclusions

Conclusions

  • Two spin Kondo model provides a model system to study competition between entanglement of two qubits with each other and entanglement of each qubit with environment

  • Entanglement between the two Kondo spins vanishes at the unstable fixed point.

  • Varying system parameters will produce all the Werner states

    S.Y. Cho and RHM, Phys. Rev. A 73, 012109 (2006)


Solid state realisation of werner quantum states via kondo spins

Non Fermi-liquid behaviour

Low temperature behaviours of two impurity Kondo model

[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]

Numerical renormalization group calculation shows that

Left:

the staggered susceptibility and the specific heat coefficients diverge.

The spin-spin correlation is continuously varying and approaches at

the critical value of around the divergence of susceptibility.

Right:


Solid state realisation of werner quantum states via kondo spins

Unstable fixed point

[B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989)]

Renormalization group flows


Solid state realisation of werner quantum states via kondo spins

Two impurity

spins A and B

Two impurity

Kondo system

Impurity spin A

Impurity spin B

Conduction

electrons C

Conduction

electrons C

One impurity

spin A

Three types of entanglements

(i)

and

(ii)

and

(iii)

and

Subsystem A

Subsystem B


Solid state realisation of werner quantum states via kondo spins

spin-spin correlation

singlet state

triplet state

Probabilities for spin singlet/triplet states

for singlet state

for triplet state

For P(S)=P(T)=1/2, the state for the two spins can be regarded as

an equal admixture of the total spin of impurities Simp=0 and Simp=1.

at ps=1/2

spin-spin correlation


Solid state realisation of werner quantum states via kondo spins

Two impurity

Kondo system

Impurity spin A

Impurity spin B

Entanglement (ii) between the impurities

(ii)

and

Total system A+B+C

Although the total system is in a pure state,

the two impurity spins are in a mixed state.

Need to calculate the concurrence

as a measure of entanglement

[W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)]


Solid state realisation of werner quantum states via kondo spins

Concurrence

Critical correlation

Concurrence & Critical Correlation

In terms of the Werner state

Hence, at ps=1/2, there exists a critical value of the spin-spin correlation

separating entangled state from disentangled state.


Solid state realisation of werner quantum states via kondo spins

Comparison of criteria

singlet fidelity

[42] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995)

[48] S. Popescu, Phys. Rev. Lett. 72, 797 (1994)


Solid state realisation of werner quantum states via kondo spins

Two impurity

spins A and B

Two impurity

Kondo system

Conduction

electrons C

Entanglement (iii)

S=1/2

Subsystem A and B

Subsystems C

Total system A+B+C

von Neumann entropy


  • Login