Abstract

We analyze the entanglement dynamics in a nonlinear Kerr-like coupler interacting with external environment. Whenever the reservoir is in a thermal vacuum state, the entanglement (measured by concurrence for a two-qubit system) exhibits regular oscillations of decreasing amplitude. In contrast, for thermal reservoirs we can observe dark periods in concurrence oscillations (which can be called a sudden death of the entanglement) and the entanglement rebuild (which can be named the sudden birth of entanglement). We show that these features can be observed when we deal with two-qubit system as well as qubit–qutrit system.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
    [CrossRef]
  2. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
    [CrossRef] [PubMed]
  3. Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74, 024304 (2006).
    [CrossRef]
  4. Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77, 054301 (2008).
    [CrossRef]
  5. M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
    [CrossRef]
  6. A. Al-Qasimi and D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A 77, 012117 (2008).
    [CrossRef]
  7. J. P. Paz and A. J. Roncaglia, “Dynamics of entanglement between two oscillators in the same environment,” Phys. Rev. Lett. 100, 220401 (2008).
    [CrossRef] [PubMed]
  8. A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
    [CrossRef]
  9. S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
    [CrossRef]
  10. F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689-2699 (2006).
    [CrossRef]
  11. A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and entanglement generation for a nonlinear coupler,” Phys. Rev. A 73, 042318 (2006).
    [CrossRef]
  12. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
    [CrossRef]
  13. S. Li and J. Xu, “Stationary entanglement and nonlocality of two qubits or qutrits collectively interacting with the thermal environment: The role of Bell singlet state,” arXiv:quant-ph/0505216v2.

2008 (3)

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77, 054301 (2008).
[CrossRef]

A. Al-Qasimi and D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A 77, 012117 (2008).
[CrossRef]

J. P. Paz and A. J. Roncaglia, “Dynamics of entanglement between two oscillators in the same environment,” Phys. Rev. Lett. 100, 220401 (2008).
[CrossRef] [PubMed]

2007 (2)

M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
[CrossRef]

S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
[CrossRef]

2006 (3)

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689-2699 (2006).
[CrossRef]

A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and entanglement generation for a nonlinear coupler,” Phys. Rev. A 73, 042318 (2006).
[CrossRef]

Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74, 024304 (2006).
[CrossRef]

2004 (2)

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef] [PubMed]

2001 (1)

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

1998 (1)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
[CrossRef]

Al-Qasimi, A.

A. Al-Qasimi and D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A 77, 012117 (2008).
[CrossRef]

Benatti, F.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689-2699 (2006).
[CrossRef]

De Siena, S.

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Eberly, J. H.

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef] [PubMed]

Ficek, Z.

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77, 054301 (2008).
[CrossRef]

Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74, 024304 (2006).
[CrossRef]

Floreanini, R.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689-2699 (2006).
[CrossRef]

Horodecki, M.

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

Horodecki, P.

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

Horodecki, R.

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

Ikram, M.

M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
[CrossRef]

Illuminati, F.

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

James, D. F. V.

A. Al-Qasimi and D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A 77, 012117 (2008).
[CrossRef]

Kowalewska-Kudlaszyk, A.

A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and entanglement generation for a nonlinear coupler,” Phys. Rev. A 73, 042318 (2006).
[CrossRef]

Leonski, W.

A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and entanglement generation for a nonlinear coupler,” Phys. Rev. A 73, 042318 (2006).
[CrossRef]

Li, F.

M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
[CrossRef]

Li, S.

S. Li and J. Xu, “Stationary entanglement and nonlocality of two qubits or qutrits collectively interacting with the thermal environment: The role of Bell singlet state,” arXiv:quant-ph/0505216v2.

Maniscalco, S.

S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
[CrossRef]

Olivares, S.

S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
[CrossRef]

Paris, M. G. A.

S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
[CrossRef]

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Paz, J. P.

J. P. Paz and A. J. Roncaglia, “Dynamics of entanglement between two oscillators in the same environment,” Phys. Rev. Lett. 100, 220401 (2008).
[CrossRef] [PubMed]

Roncaglia, A. J.

J. P. Paz and A. J. Roncaglia, “Dynamics of entanglement between two oscillators in the same environment,” Phys. Rev. Lett. 100, 220401 (2008).
[CrossRef] [PubMed]

Serafini, A.

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Tanas, R.

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77, 054301 (2008).
[CrossRef]

Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74, 024304 (2006).
[CrossRef]

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
[CrossRef]

Xu, J.

S. Li and J. Xu, “Stationary entanglement and nonlocality of two qubits or qutrits collectively interacting with the thermal environment: The role of Bell singlet state,” arXiv:quant-ph/0505216v2.

Yu, T.

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef] [PubMed]

Zubairy, M. S.

M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
[CrossRef]

Zyczkowski, K.

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

J. Phys. A (1)

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689-2699 (2006).
[CrossRef]

Phys. Rev. A (8)

A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and entanglement generation for a nonlinear coupler,” Phys. Rev. A 73, 042318 (2006).
[CrossRef]

Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74, 024304 (2006).
[CrossRef]

Z. Ficek and R. Tanaś, “Delayed sudden birth of entanglement,” Phys. Rev. A 77, 054301 (2008).
[CrossRef]

M. Ikram, F. Li, and M. S. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A 75, 062336 (2007).
[CrossRef]

A. Al-Qasimi and D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A 77, 012117 (2008).
[CrossRef]

K. Życzkowski, P. Horodecki, M. Horodecki, and R. Horodecki, “Dynamics of quantum entanglement,” Phys. Rev. A 65, 012101 (2001).
[CrossRef]

A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef] [PubMed]

J. P. Paz and A. J. Roncaglia, “Dynamics of entanglement between two oscillators in the same environment,” Phys. Rev. Lett. 100, 220401 (2008).
[CrossRef] [PubMed]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245-2248 (1998).
[CrossRef]

Other (1)

S. Li and J. Xu, “Stationary entanglement and nonlocality of two qubits or qutrits collectively interacting with the thermal environment: The role of Bell singlet state,” arXiv:quant-ph/0505216v2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

The concurrence C ( t ) for the coupler initially prepared in a MES state and later left inside the cavity for n a = n b = 0 . χ a = χ b = χ = 25 , γ a = γ b = 0.001 , α = 0 , and ϵ = π χ . The inset presents the same but in a different time scale. Time is scaled in 1 χ units.

Fig. 2
Fig. 2

The concurrence C ( t ) for the coupler initially prepared in a MES state and later left inside the cavity for various values of n a and n b and for γ a = γ b = π ( 20 χ ) . The remaining parameters are chosen the same as in Fig. 1. The inset presents the same but in a different time scale.

Fig. 3
Fig. 3

Maps showing the dependence of C ( t ) on time and value of n a for various values of n b . The other parameters are the same as in Fig. 2. Black areas correspond to C ( t ) = 0 , whereas the white regions correspond to the positive values of C ( t ) .

Fig. 4
Fig. 4

The concurrence C ( t ) for the coupler initially prepared in a MES state and later left inside the cavity for n b = 3 and various values of n a . The remaining parameters chosen are the same as in Fig. 3.

Fig. 5
Fig. 5

The concurrence C ( t ) for the coupler initially prepared in a MES state and after that left inside the cavity for n a = n b = 0 . The parameters χ a = χ b = χ = 25 , γ a = γ b = 0.001 , α = π χ , and ϵ = π χ . The inset ( a 1 ) shows the same but in a different time scale, whereas ( a 2 ) is plotted for γ a = γ b = 0.01 .

Fig. 6
Fig. 6

Maps showing concurrence C ( t ) for various moments of time and values of external pumping α. The initial state is B 1 ⟩, n a = n b = 1 . The remaining parameters are chosen the same as in Fig. 2.

Fig. 7
Fig. 7

The concurrence C ( t ) for the coupler initially prepared in a MES state. The other parameters are χ a = χ b = χ = 25 , γ a = γ b = π ( 20 χ ) , ϵ = π χ , and n a = n b = 1 .

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

H ̂ = H ̂ NL + H ̂ int + H ̂ ext ,
H ̂ NL = χ a 2 ( a ̂ ) 2 a ̂ 2 + χ b 2 ( b ̂ ) 2 b ̂ 2 ,
H ̂ int = ϵ ( a ̂ ) 2 b ̂ 2 + ϵ * ( b ̂ ) 2 a ̂ 2 ,
H ̂ ext = α a ̂ + α * a ̂ .
d ρ ̂ d t = 1 i ( ρ ̂ H ̂ H ̂ ρ ̂ ) + k = 1 2 [ C ̂ k ρ ̂ C ̂ k 1 2 ( C ̂ k C ̂ k ρ ̂ + ρ ̂ C ̂ k C ̂ k ) ] + 2 k = 1 2 [ C ̂ k n ρ ̂ C ̂ k n + C ̂ k n ρ ̂ C ̂ k n C ̂ k n C ̂ k n ρ ̂ ρ ̂ C ̂ k n C ̂ k n ] ,
C ̂ 1 = 2 γ a a ̂ ,
C ̂ 2 = 2 γ b b ̂ ,
C ̂ 1 n = γ a n a a ̂ ,
C ̂ 2 n = γ b n b b ̂ .
C ( t ) = max ( λ 1 λ 2 λ 3 λ 4 , 0 ) .
ρ ̃ ̂ c = σ y σ y ρ ̂ c * σ y σ y .
ρ ̂ c = Π 0 , 2 Π 0 , 2 ρ ̂ Π 0 , 2 Π 0 , 2 ,
Π 0 , 2 Π 0 , 2 ( 0 0 + 2 2 ) ( 0 0 + 2 2 ) .
B 1 = 1 2 ( 2 a 0 b + i 0 a 2 b ) ,
B 2 = 1 2 ( 2 a 0 b i 0 a 2 b ) .
B 3 = 1 2 ( 2 a 0 b + 1 a 2 b ) .

Metrics