Generation of continuous variable entangled light Department of Physics Dalian University of Technology Dalian, , the People's Republic of China

Outline 4 5 Introduction of continuous variable (CV) entanglement 1 Recent works 2 Our scheme of generation CV entangled lights 3 Conclusion 4

1.Introduction of CV entanglement infinite 2N 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 1.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). Xiong H, Scully M O, and Zubairy M S Phys. Rev. Lett. 94,

2.Recent work Kiffner M, Zubairy M S, Evers J, Keitel C H Phys. Rev. A Alebachew E Phys.Rev. A (2007)

2.Recent work Zhou L, Xiong H, and Zubairy M S Phys. Rev. A (2006) Cassemiro K N and Villar A S Phys. Rev. A (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> with coupling constants g 1, g 2, and g 3, respectively. Two classical fields drive the atomic level resonantly between|a>↔|c> and |b>↔|d> with Rabi frequencies Ω ac and Ω bd

2 ◆ Quantum Computation over Continuous Variables Seth Lloyd et al. Phys. Rev. Lett. 82, 1784 (1999) Applications of multimode entanglement: Why multimode 3 ◆ Secret sharing Tripartite Quantum State Sharing Andrew M. Lance et al. Phys. Rev. Lett. 92, (2004) 1 ◆ Quantum teleportation based on CVE Multipartite Entanglement for Continuous Variables: A Quantum Teleportation Network P. 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 ) Recently, other criteria are employed to test entanglement in many models. (Phys. Rev. A 77, )

Multimode entanglement criterion However, we need a criterion to test n-mode entanglement. Here, we employ the PPT (positivity of partial transpose) criterion Consider n-mode Gaussian states with annihilation and creation operators a j and a † j with Define a covariance matrix, It must satisfy Robertson-Schrödinger uncertainty principle Phys. 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 ofare 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 r a. 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 , 1987

Entanglement generation in double-Λsystem 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

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 , 1987) It contains Recently, they investigate the entanglement in quantum beat. (Phys. Rev. A 77, )

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, (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 oscillate because 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 ff ect laser system. Di ff erent from similar NPD, 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.