Quantum Phase Transitions and Entanglement in the Detuned Dicke Model

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Quantum Phase Transitions and Entanglement in the
Detuned Dicke Model

• Ru-Fen Liu
• Phys. NCKU
• 11/21/2005

Ru-Fen Liu and Chia-Chu Chen, e-print
quant-ph/0510071
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Outlines

• Quantum phase transition(QPT) and the
  discontinuity of the concurrence
• Dicke model
• Results
• Discussion and conclusions

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Quantum phase(ground state) transition

• Caused by quantum fluctuations near absolute zero
  temperature

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What is the difference?
The ground state coherence has been changed
Magnetic
Paramagnetic
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Quantum coherence Entanglement

• There exists another quantum coherence ?
  Entanglement
• Does the appearance of QPTs accompany any change
  of entanglement??
• Is there any one-one correspondence between QPTs
  and discontinuity of entanglement?
• Is it possible to control the ground state
  transition of the system and then gain
  entanglement automatically?

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References
Spin system

• Osterloh A, Amico L, Falci G, Fazio R, Nature,
  416, 608 (2002) Scaling of entanglement close to
  a quantum phase transition
• Shi-Jian Gu, Hai-Qing Lin, and You-Quan Li, Phys.
  Rev. A 68, 042330 (2003) Entanglement, quantum
  phase transition, and scaling in the XXZ chain
• H. M. Rønnow et al., Science, 308, 389 (2005)
  Entanglement and Quantum Phase Transition in the
  Extended Hubbard Model
• Gu et al., Phys. Rev. Lett. 93, 086402 (2004)
  Entanglement and Quantum Phase Transition in the
  Extended Hubbard Model
• L.-A. Wu, M.S. Sarandy, and D. A. Lidar, Phys.
  Rev. Lett., 93, 250404(2004), Quantum Phase
  Transitions and Bipartite Entanglement
• T. Brandes et al., 92, 073602 (2004),
  Entanglement and the Phase Transition in the
  Single-Mode Superradiance
• V. Buzek et al., Phys. Rev. Lett., 94, 163601
  (2005) Instability and entanglement of the
  ground state of the Dicke model
• S. Schneider and G. J. Milburn, Phys. Rev. A 65,
  042107 (2002) Entanglement in the steady state
  of a collective-angular-momentum (Dicke) model

two-atom case
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Physical picture
Single mode cavity
R. H. Dicke, Phys. Rev. 93, 99, (54)
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Single mode Dicke model
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Diagonalization

• Conserved quantities? Let

H is block diagonalized in j
Eigen-energy
Eigen-vector
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Degenerate spectrum
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N3, j3/2, on-resonance
1/2
-1/2
-3/2
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Spectrum
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N12, on resonance (r0)
Sequential QPTs ?Instability of ground state
transitions (V. Buzek et al., Phys. Rev. Lett.,
94, 163601 (2005) )
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Pair-wise entanglement
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Instability of the ground state discontinuity
of concurrence --- on resonance case
Discontinuity of concurrence when QPT occurs (V.
Buzek et al., Phys. Rev. Lett., 94, 163601 (2005)
)
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Full spectrum
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Sequential quantum phase transitions --- N2, r1
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Infinite sequential quantum phase transitions
The N2 case is proved
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(No Transcript)
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Concurrence ---
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Strong coupling
Strong coupling
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Discussion
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Extra discontinuity of the concurrence
Extra discontinuity (derivative) without any QPTs
? M. F. Yang, PRA, 71, 030302(R) (2005)
By definition of Concurrence
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N2
No any discontinuity of the concurrence but QPTs
are still there
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Discussion
N2

• On resonance ? Eigenvectors are independent of
  the coupling constant ? the discontinuity of
  concurrence always happens accompanying with QPT
  (infinite)
• Off resonance ? Eigenvectors are the coupling and
  the detuning dependent

• Extra discontinuity of derivative of concurrence
• without QPTs

2. The continuity of concurrences with QPTs
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N3
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11 modes Dicke Model

• On-resonance off-resonance

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Critical coupling with 11 mode
Weak coupling
Faster QPT
Back to single mode, on-resonance result ?
decoupling
With extra off-resonant mode, the QPT can happen
in weak coupling region
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Concurrence
Maximal
Back to single mode, on-resonant result ?
decoupling
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Two off-resonance modes
C0 with QPTs
Better than the 11 case with one on resonance
mode
Back to single mode result ? decoupling
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Discussions and conclusions

• We give a rigorous proof for the sequential
• quantum phase transitions for N2.

2.The extra discontinuity of derivative
concurrence have been found without QPTs in
single off-resonance mode and 11 mode Dicke
Model. ? Finite atom number? (N4 case) large
N limit? ? pair-wise entanglement is not
enough? ? On-resonance cases gives special
results
3.The entanglement of ground state seems to be
controllable by tuning the on/off-resonance of
photon modes in cavity. ? near T0