1. Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China 2. Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China 3. School of Information Engineering, Nanchang Institute of Technology, Nanchang 330099, China
A new type of entangled fractional squeezing transformation (EFrST) has been theoretically proposed for 2D entanglement [Front. Phys. 10, 100302 (2015)]. In this paper, we shall extend this case to that of 3D entanglement by introducing a type of three-mode entangled state representation, which is not the product of three 1D cases. Using the technique of integration within an ordered product of operators, we derive a compact unitary operator corresponding to the 3D fractional entangling transformation, which is an entangling operator that presents a clear transformation relation. We also verified that the additivity property of the novel 3D EFrST is of a Fourier character by using its quantum mechanical description. As an application of this representation, the EFrST of the three-mode number state is calculated using the quantum description of the EFrST.
E. U. Condon, Immersion of the Fourier transform in a continuous group of functional transformations, Proc. Natl. Acad. Sci. USA 23(3), 158 (1937)
https://doi.org/10.1073/pnas.23.3.158
2
V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Appl. Math. 25(3), 241 (1980)
https://doi.org/10.1093/imamat/25.3.241
3
D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation (I), J. Opt. Soc. Am. A 10(9), 1875 (1993)
https://doi.org/10.1364/JOSAA.10.001875
4
H. M. Ozakatas and D. Mendlovic, Fractional Fourier transforms and their optical implementation (II), J. Opt. Soc. Am. A 10(12), 2522 (1993)
https://doi.org/10.1364/JOSAA.10.002522
5
H. M. Ozaktas and D. Mendlovic, Fourier transforms of fractional orders and their optical interpretation, Opt. Commun. 101(3-4), 163 (1993)
https://doi.org/10.1016/0030-4018(93)90359-D
6
Y. B. Karasik, Expression of the kernel of a fractional Fourier transform in elementary functions, Opt. Lett. 19(11), 769 (1994)
https://doi.org/10.1364/OL.19.000769
7
R. G. Dorsch and A. W. Lohmann, Fractional Fourier transform used for a lens-design problem, Appl. Opt. 34(20), 4111 (1995)
https://doi.org/10.1364/AO.34.004111
8
A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, J. Opt. Soc. Am. A 10(10), 2181 (1993)
https://doi.org/10.1364/JOSAA.10.002181
9
D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform, Appl. Opt. 33(26), 6188 (1994)
https://doi.org/10.1364/AO.33.006188
10
H. Y. Fan and L. Y. Hu, Correpondence between quantumoptical transform and classical-optical transform explored by developing Dirac’s symbolic method, Front. Phys. 7(3), 261 (2012)
https://doi.org/10.1007/s11467-011-0206-z
11
H. Y. Fan, H. L. Lu, and Y. Fan, Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations, Ann. Phys. 321(2), 480 (2006)
https://doi.org/10.1016/j.aop.2005.09.011
H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics, Shanghai: Shanghai Scientific and Technical publishers, 1997 (in Chinese)
15
K. M. Zheng, S. Y. Liu, H. L. Zhang, C. J. Liu, and L. Y. Hu, A generalized two-mode entangled state: Its generation, properties, and applications, Front. Phys. 9(4), 451 (2014)
https://doi.org/10.1007/s11467-014-0419-z
16
H. Y. Fan, Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation, Opt. Lett. 28(22), 2177 (2003)
https://doi.org/10.1364/OL.28.002177
17
H. Y. Fan, J. H. Chen, and P. F. Zhang, On the entangled fractional squeezing transformation, Front. Phys. 10(2), 100302 (2015)
https://doi.org/10.1007/s11467-014-0457-6
18
H. Y. Fan and J. H. Chen, On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations, Front. Phys. 10(1), 100301 (2015)
https://doi.org/10.1007/s11467-014-0445-x
19
C. H. Lv, H. Y. Fan, and D. W. Li, From fractional Fourier transformation to quantum mechanical fractional squeezing transformation, Chin. Phys. B 24(2), 020301 (2015)
https://doi.org/10.1088/1674-1056/24/2/020301
20
H. Y. Fan, L. Y. Hu, and J. S. Wang, Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics, J. Opt. Soc. Am. A. 25(4), 974 (2008)
https://doi.org/10.1364/JOSAA.25.000974
21
S. Xu, L. Y. Hu, and J. H. Huang, New fractional entangling transform and its quantum mechanical correspondence, Chin. Opt. Lett. 13(3), 030801 (2015)
https://doi.org/10.3788/COL201513.030801
22
H. Y. Fan, H. L. Lu, and Y. Fan, Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations, Ann. Phys. 321(2), 480 (2006)
https://doi.org/10.1016/j.aop.2005.09.011
23
H. Y. Fan, S. Wang, and L. Y. Hu, Evolution of the single-mode squeezed vacuum state in amplitude dissipative channel, Front. Phys. 9(1), 81 (2014)
https://doi.org/10.1007/s11467-013-0367-z
24
H. Y. Fan and S. Y. Lou, Studying bi-partite entangled state representations via the integration over ket–bra operators in Q-ordering or P-ordering, Front. Phys. 9, 464 (2014)
https://doi.org/10.1007/s11467-013-0397-6
25
C. H. Lv and H. Y. Fan, Optical entangled fractional Fourier transform derived via non-unitary SU(2) bosonic operator realization and its convolution theorem, Opt. Commun. 284(7), 1925 (2011)
https://doi.org/10.1016/j.optcom.2010.12.046
26
F. T. Arrechi, C. Eric, G. Robert, and T. Harry, Atomic coherent states in quantum optics, Phys. Rev. A 6(6), 2211 (1972)
https://doi.org/10.1103/PhysRevA.6.2211
