Generation of continuous variable entangled light

Department of Physics

Dalian University of Technology

Dalian, 116024, the People's Republic of China

Outline

4

5

Introduction of continuous variable (CV) entanglement1

Recent works2

Our scheme of generation CV entangled lights3

Conclusion4

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

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）

2.Recent work

Bouwmeester, D. et al. Nature 390, 575–579 (1997).

2

2

Xiong H, Scully M O , and Zubairy M S Phys. Rev. Lett. 94, 023601

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)

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)

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

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

3.1.2 DERIVATION OF THE MASTER EQUATION

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 )

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)

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

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.

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.

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

Numerical results

The effect of two classical driven field

Entanglement generation in double-Λsystem

Scully and Zubairy, Phys. Rev. A 35 752, 1987

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”

Entanglement generation in double-Λsystem

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 )

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”

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.

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

Numerical results

The quantum fields are in ”V” form. If the photon number in two mode only oscillatebecause of the symmetry.

Numerical results

Effect of classical field on entanglement

With the increasing of the classical field, the entangled time will be shorten.

Numerical results

Effect of classical field on photon numbers

With the increasing of the classical field, the photon numbers will be amplified more quickly.

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.

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.

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.