solid state realisation of werner quantum states via kondo spins
Post on 12-Feb-2016
37 Views
Preview:
DESCRIPTION
TRANSCRIPT
Solid state realisation of Werner quantum states via Kondo spins
Ross McKenzieSam 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)• Dawson, Hines, and Milburn (decoherence
and entanglement sharing)
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
• 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
• 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
Entangled states
No correlations
Violate Bell
inequalitiesCorrelations but no
entanglement
Reduced density matrix Reduced density matrix
In the Bell basis
Werner states
ps is probability of a singlet
Mixed states of two qubits
No entanglementBell-CSSH inequalities satisfied
ps<0:5ps<0:78
Model system: two Kondo spins interact with metallic environment
via Heisenberg exchange interaction
Two impurityKondo system
Two impurity spins A and B
Conduction electrons C
AS
BS
AB
Experimental realisation ITwo impurityKondo system
N. J. Craig et al., Science 304, 565 (2004)
2DEG between spinsin quantum dots induces an RKKY interactionbetween spins.Gates vary J
Experimental realisation II
• Endohedral fullerenes inside nanotubes
Two impurityKondo system
A. Khlobystov et al. Angewandte Chemie International Edition43, 1386-1389 (2004)
Single impurity Kondo modelSingle impurity Kondo model
ACC SsJHH
)0( Hamiltonian
Conduction electrons
Conduction-electron spin density at impurity site R = 0J is the spin exchange coupling
Low temperature properties determined by single energy scale. Kondo temperature ]/1exp[ FFK JJDT
Band width D and the single particle density of state at the Fermi surfaceF
Single impurityKondo system
Single impurityKondo system
For a review, L. Kouwenhoven and L. Glazman, Physics World 14, 33 (2001)
Conduction electron spin
Impurity spin
Tuneable quantum many-body states: Kondo effect in quantum dots
Kondo temperature can be variedover many orders of magnitude
Two impurity Kondo modelTwo impurity Kondo modelTwo impurityKondo system
Hamiltonian
To second order J, the indirect RKKY (Ruderman Kittel-Kasuya-Yosida) interaction is
R
I RKKY interaction
21)( SSRIHRKKY
Ground state determined by competitionbetween Kondo of single spins and RKKY
221
,,
IrIzyx
A
c.f., Yosida’s variational wavefunction ACACG 2
1
Entanglement in single impurity Kondo modelEntanglement in single impurity Kondo model
[K. Yosida, Phys. Rev. 147, 233 (1966)]
[T. A. Costi and R. H. McKenzie, Phys. Rev. A 68, 034301 (2003)]
Impurity spin A
AS
Conduction electrons C
Subsystem A Subsystem B
Single impurityKondo system
AS
AB Tr BA TrTotal system A+B
1logTr)( 2 AAAE
S=1/2
Ground state Spin singlet
0 r
Spin-rotational invariant!
The impurity spin is maximally entangled with the conduction electrons
Reduced density matrix for the impurity
von Neumann entropy
J
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 < ~S A ¢ ~S B >.
Reduced density matrix for the impuritiesReduced density matrix for the impurities
Two impurityKondo system
Two impurity spins A and B
Conduction electrons C
AS
BS
AB
In the Bell basis
[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]Low temperature behaviour of two impurity Kondo model
the staggered susceptibility and the specific heat coefficients diverge. Numerical renormalization group calculation shows that
The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.
Left:
Right:
Non Fermi-liquid behaviour
Entanglement & Quantum Phase transitionEntanglement & Quantum Phase transition
Unstable fixed point
• At the fixed point
[Gan, Ludwig, Affleck, and Jones]• Thus, for the critical coupling there is no
entanglement between two qubits.
I ' 2:2TK
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• 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)
[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]Low temperature behaviours of two impurity Kondo model
the staggered susceptibility and the specific heat coefficients diverge. Numerical renormalization group calculation shows that
The spin-spin correlation is continuously varying and approaches at the critical value of around the divergence of susceptibility.
Left:
Right:
Non Fermi-liquid behaviour
Unstable fixed pointUnstable fixed point[B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989)]
Renormalization group flows
Three types of entanglementsThree types of entanglements
Two impurity spins A and B
AS
BS
Conduction electrons C
One impurity spin A
AS
Conduction electrons C
BS
Two impurityKondo system
and
and
and
(i)
(ii)
(iii)
Subsystem A Subsystem B
Impurity spin A
AS
Impurity spin B
BS
Probabilities for spin singlet/triplet statesProbabilities for spin singlet/triplet states
13)()( tS ppTPSP
41
43
BA SS
spin-spin correlation
for singlet stateSpSP )(
tpTP 3)( for triplet state
singlet state
0impS triplet state
1impS
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.
41
BA SS
spin-spin correlation at ps=1/2
Entanglement (ii) between the impuritiesEntanglement (ii) between the impurities
CAB TrTotal system A+B+C
Two impurityKondo system
and(ii)
Impurity spin A
AS
Impurity spin B
BS
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)]
Concurrence & Critical CorrelationConcurrence & Critical Correlation
In terms of the Werner state
Concurrence
Hence, at ps=1/2, there exists a critical value of the spin-spin correlation separating entangled state from disentangled state.
Critical correlation
Comparison of criteriaComparison of criteria
[42] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995)
[48] S. Popescu, Phys. Rev. Lett. 72, 797 (1994)
singlet fidelity
Entanglement (iii)Entanglement (iii)
Subsystem A and B Subsystems C
CAB TrTotal system A+B+C
31log)1(log)( 22
SSSSAB
ppppE
S=1/2
von Neumann entropy
Two impurityKondo system
Two impurity spins A and B
AS
BS
Conduction electrons C
top related