generation of continuous variable entangled light department of physics dalian university of...
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Generation of continuous variable entangled light
Department of Physics
Dalian University of Technology
Dalian, 116024, the People's Republic of China
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Outline
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Introduction of continuous variable (CV) entanglement1
Recent works2
Our scheme of generation CV entangled lights3
Conclusion4
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1.Introduction of CV entanglement
infinite
2N-1
3
2
CV system
Spin N system
Spin 1 system
Spin ½ system
|0> and |1>
|-1>, |0> and |1>
|-N>… |0> …|N>
|0>,|1>,|2>,|3>……
Dimension of the Hilbert Space
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1.Introduction of CV entanglement
Continuous variable entanglement brings many applications such as,
CV
Quantum computation
Quantum teleportation
Error correction
Quantum cloning
Quantum optics
Quantum cryptography
• Samuel L. Braunstein et al. Review of Modern Physics, 77, 513(2005)
• 连续变量的量子信息处理 与非定域性 逯怀新,郁司夏,杨洁,陈增兵,张永德 《量子力学新进展》(第三辑)
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2.Recent work
Bouwmeester, D. et al. Nature 390, 575–579 (1997).
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Xiong H, Scully M O , and Zubairy M S Phys. Rev. Lett. 94, 023601
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2.Recent work
Kiffner M, Zubairy M S, Evers J, Keitel C H Phys. Rev. A 75 033816
Alebachew E Phys.Rev. A 76 023808 (2007)
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2.Recent work
1g
2g
| a
| b
| c1
2
Zhou L, Xiong H, and Zubairy M S Phys. Rev. A 74 022321 (2006)
Cassemiro K N and Villar A S Phys. Rev. A 77 022311 (2008)
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3.1 Our Scheme on generation three-mode entangled light
This is our scheme of generation multimode entangled lights. Three cavity modes resonantly interact with atomic transitions |a>↔|b>, |b>↔|c>, and |c>↔|d> withcoupling constants g1, g2, and g3, respectively. Two classical fields drive the atomic level resonantly between|a>↔|c> and |b>↔|d> with Rabi frequencies Ωac and Ωbd
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22 Quantum Computation over Continuous Variables
Seth Lloyd et al. Phys. Rev. Lett. 82, 1784 (1999)
Applications of multimode entanglement:
3.1.1 Why multimode
33 Secret sharing
Tripartite Quantum State SharingAndrew M. Lance et al. Phys. Rev. Lett. 92, 177903 (2004)
11 Quantum teleportation based on CVE
Multipartite Entanglement for Continuous Variables: A Quantum Teleportation NetworkP. van Loock et al. Phys. Rev. Lett. 84, 3482
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3.1.2 DERIVATION OF THE MASTER EQUATION
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3.1.3 Multimode entanglement criterion
Duan et al. proposed the summation of the quantum fluctuations
It has been often used to measure entanglement between two modes.
(Duan L M, Giedke G ,Cirac J I ,Zoller P Phys. Rev. Lett. 84 2722)
Recently, other criteria are employed to test entanglement in many models. (Phys. Rev. A 77, 062308 )
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Multimode entanglement criterion
However, we need a criterion to test n-mode entanglement. Here, we employ the PPT (positivity of partial transpose) criterionConsider n-mode Gaussian states with annihilation and creation operators aj and a†
j
with
Define a covariance matrix,
It must satisfy
Robertson-Schrödinger uncertainty principlePhys. Rev. 46, 794 (1934)
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Multimode entanglement criterion
partial transpose
If the transposed part can be separated from the other parts, then,
It indicates that all the eigenvalues of 2)~
( V are bigger than 1
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Multimode entanglement criterion
To calculate the smallest eigenvalue, we rewrite the covariance matrix V in term of . Then, all the elements of the variance matrix are composed of a series of mean values,
Using the relation
We can get all of these eigenvalues, then we can test the entanglement.
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Numerical results
For all the three modes, the smallest eigenvalue is smaller than 1, it will be a sufficient evidence for the existence of the quantum entanglement between the transposed mode and the other modes.
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Numerical results
We assume that the atoms in state
are injected into the cavity with rate ra. The following picture shows the entanglement and the photon number various with
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Numerical results
The effect of two classical driven field
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Entanglement generation in double-Λsystem
Scully and Zubairy, Phys. Rev. A 35 752, 1987
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Entanglement generation in double-Λsystem
2
3
1
Traditional method of generating CVE--Parametric down conversion
Cascade configuration--Creating and annihilating a photon in two modes at the same time
Our scheme--Annihilating a photon in one mode and creating one in another mode, similar with “quantum beat”
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Entanglement generation in double-Λsystem
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Entanglement generation in double-Λsystem
Following the standard procedure in laser theory developed by Scully and Zubairy, we get the master equation
Unless cascade configuration, our scheme is similar with “quantum beat” leaser (Scully and Zubairy, Phys. Rev. A 35 752, 1987) It contains
Recently, they investigate the entanglement in quantum beat. (Phys. Rev. A 77, 062308 )
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Cascade VS Double- Λ
In cascade model, both of the two modes will be created or annihilated one photon in one loop. So, it contains the term
Annihilating a photon in one mode and creating one in another mode, similar with “quantum beat”
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Entanglement criterion
Although the criterion- the sum of the quantum fluctuations was widely used in our previous work, this criterion can not be applied to measure some special coherent state.
Here is an example given by E. Shchukin and W. Vogel in PRL 95, 230502 (2005)
According to their results, the sum of the quantum fluctuations “fail to demonstrate the entanglement of this state”.
We also find that this criterion is not suitable for measure entanglement in V type configuration.
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Entanglement criterion
We recall that the criterion proposed by Hillery and Zubairy which can be used for non- Gaussionian state The criterion say if
the two-mode field is entangled.
Here is an example
With these equations, we can calculate
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Numerical results
The quantum fields are in ”V” form. If the photon number in two mode only oscillatebecause of the symmetry.
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Numerical results
Effect of classical field on entanglement
With the increasing of the classical field, the entangled time will be shorten.
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Numerical results
Effect of classical field on photon numbers
With the increasing of the classical field, the photon numbers will be amplified more quickly.
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Numerical results
Effect of classical field on overcoming the cavity loss
With a large cavity loss we can get entanglement with a stronger classical field. But at the same time, the time entanglement exist will be shorten.
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Conclusion
1
We generate three- mode
entanglement by using the interaction of
atom and cavity field
2
In our scheme we need an pure
initial state of the atom rather
than a mixed state. That will be more easier
to realize in experiment.
3
Our scheme can be extend to multimode
by using multi-level atoms.
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Our study is helpful in understanding the entanglement characteristic when the master equation contains such as quantum beats laser and Hanle e ect laser system. ffDi erent from similar ffNPD, the scheme is another way to produce CVE.
Conclusion
Our scheme
We derive the theory of this system and
analyze the available entanglement criterion for double-Λ system. When the atoms are
injected in the ground state |d>, the
entangled laser can be achieved under the
condition of suitable parameters.
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