Transcript Document

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)
• 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 macrorealism (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
No correlations
Violate
Bell
inequalities
Correlations but no
entanglement
Entangled states
Werner states
Mixed states of two qubits
In the Bell basis
Reduced density matrix
ps
ps< 0:5
ps< 0:78
is probability of a singlet
No entanglement
Bell-CSSH inequalities satisfied
Model system: two Kondo spins
interact with metallic environment
via Heisenberg exchange interaction

SA

SB

Two impurity
Kondo system


Two impurity
spins A and B
 AB
Conduction
electrons C
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
Two impurity
Kondo system
Experimental realisation II
• Endohedral fullerenes inside nanotubes
A. Khlobystov et al. Angewandte
Chemie International Edition
43, 1386-1389 (2004)
Single impurity Kondo model
Hamiltonian
Single impurity
Kondo system


 J sC (0 )  S A
H  HC
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 T  D J  exp[  1 / J  ]
K
F
F
Band width D and the single particle density of state  F at the Fermi surface
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
Single impurity
Kondo system
Kondo temperature can be varied
over many orders of magnitude
Two impurity Kondo model
Two impurity
Kondo system
Hamiltonian
To second order J, the indirect RKKY (Ruderman
Kittel-Kasuya-Yosida) interaction is
RKKY interaction
I
Ground state determined by competition
between Kondo of single spins and RKKY
H
RKKY


 I ( R ) S1  S 2
R
Entanglement in single impurity Kondo model
[T. A. Costi and R. H. McKenzie, Phys. Rev. A 68, 034301 (2003)]

SA
S=1/2
J



SA
Single impurity
Kondo system
Total system A+B
Ground state
Subsystem A

 A  Tr B   
Spin singlet
Reduced density matrix for the impurity
A
Conduction
electrons C
Impurity spin A
1
  I   r  
2 
  x, y,z





I
2
Subsystem B
 B  Tr A   
Spin-rotational invariant!
r     0
von Neumann entropy
E (  A )   Tr  A log
2
A 1
The impurity spin is maximally entangled with the conduction electrons
c.f., Yosida’s variational wavefunction
[K. Yosida, Phys. Rev. 147, 233 (1966)]
G 
1
2

C  A  C  A

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 impurities

SA

SB

Two impurity
Kondo system
In the Bell basis


Two impurity
spins A and B
 AB
Conduction
electrons C
Low temperature behaviour of two impurity Kondo model
[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]
Left:
Numerical renormalization group calculation shows that
the staggered susceptibility and the specific heat coefficients  diverge.
Non Fermi-liquid behaviour
Right: The spin-spin correlation is continuously varying and approaches at
the critical value of
around the divergence of susceptibility.
Entanglement & Quantum Phase transition
Unstable fixed point
• At the fixed point
I ' 2:2TK
[Gan, Ludwig, Affleck, and Jones]
• Thus, for the critical coupling there is no
entanglement between two qubits.
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)
Low temperature behaviours of two impurity Kondo model
[B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988)]
Left:
Numerical renormalization group calculation shows that
the staggered susceptibility and the specific heat coefficients  diverge.
Non Fermi-liquid behaviour
Right: The spin-spin correlation is continuously varying and approaches at
the critical value of
around the divergence of susceptibility.
Unstable fixed point
[B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989)]
Renormalization group flows
Three types of entanglements

SA
(i)

SB
and
Conduction
electrons C
One impurity
spin A

SA
(ii)

SB
and
Two impurity
Kondo system
Impurity spin A
(iii)

SA
Impurity spin B

SB
and
Two impurity
spins A and B
Conduction
electrons C
Subsystem A
Subsystem B
Probabilities for spin singlet/triplet states
spin-spin correlation
 
3
1
  SA SB 
4
4
P ( S )  p S for singlet state
P (T )  3 p t for triplet state
P ( S )  P (T )  p S  3 p t  1
singlet state
S imp  0
triplet state
S imp  1
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.
spin-spin correlation
 
1
SA SB  
4
at ps=1/2
Entanglement (ii) between the impurities
(ii)

SA

SB
and
Two impurity
Kondo system
Total system A+B+C
Impurity spin A

Impurity spin B
 AB  Tr 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)]
Concurrence & 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 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)
Entanglement (iii)
S=1/2

SA

SB


Two impurity
Kondo system
Total system A+B+C

2
p S  (1  p S ) log
Conduction
electrons C
Subsystem A and B
Subsystems C
 AB  Tr C   
von Neumann entropy
E (  AB )   p S log
Two impurity
spins A and B
1  pS
2
3