Abstract
As physicists continue to find approaches to achieve quantum squeezing enhancement, a key challenge is to identify the specific parameters of the squeezing operator and their boundary conditions. This paper employs the method of integration within an ordered product of operator (IWOP) in quantum mechanics to solve this problem for two independent parameters. First, the \(q - p\) phase space correspondence for enhanced squeezing is investigated, where \(q,p\) represent the coordinate and momentum, respectively. Then, the squeezing-enhanced state is theoretically obtained by finding the generalized squeezing operator \(S\left( {\lambda ,r} \right) = \exp \left( { - \frac{v}{{2{u^* }}}{a^{\dag 2}}} \right)\exp [\left( {{a^\dag }a + \frac{1}{2}} \right)\ln \frac{1}{{u^* }}]\exp \left( {\frac{{v^* }}{{2{u^* }}}{a^2}} \right)\). It is demonstrated that the characteristic of the phase space transformation is \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda ,\) \(u = \cosh \lambda - i\sinh \lambda \sinh r\). Here, \(\lambda ,r\) are two independent parameters, and they must satisfy \(\tanh \lambda \left( {{{\cosh }^2}r - 1} \right) < \cosh r - 1\) to achieve enhanced squeezing of the quadrature operator. Further, the integral solution of the dissipation master equation is adopted to analyze the attenuation of this type of squeezed field. Meanwhile, the paper presents the canonical decomposition of the enhanced squeezing operator \(S\left( {\lambda ,r} \right)\). Our research provides physicists with more refined insights to enhance the squeezing effect with more precision.
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1 Introduction
Non-commutable optical field operators make it difficult for physicists to enhance the squeezing effect. Motivated by the research work such as electron’s orbit quantization \(\oint {pdq} = n\hbar\) from Sommerfeld and Bohr and the pseudo-probability distribution function for quantum states from Wigner [5,6,7,8,9,10], this paper attempts to leverage the phase-space theory of \(q - p\), which is widely used to connect classical mechanics with quantum mechanics [1,2,3,4]. First, the method of integration within an ordered product of operator (IWOP) in quantum mechanics is employed to investigate phase space correspondence for enhanced squeezing. This approach allows for showing the theoretical representation of two independent parameters and their boundary conditions to achieve enhanced squeezing. Then, the squeezing-enhanced state is theoretically obtained by using the generalized squeezing operator \(S\left( {\lambda ,r} \right) = \exp \left[ {\frac{ - i\lambda }{2}\left( {{Q^2}{e^r} - {P^2}{e^{ - r}}} \right)} \right]\). The characteristic of phase space transformation is demonstrated as \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda ,\) where \(\lambda ,r\) are two independent parameters, and the condition for enhanced squeezing of the quadrature operator is analyzed. Further, the integral solution of the dissipation master equation [11, 12] is utilized to analyze the attenuation of this type of squeezed field. Meanwhile, the method of IWOP [13,14,15,16,17] is employed to accomplish the task. Finally, the canonical decomposition of the enhanced squeezing operator is derived.
2 Quadrature Operator of Squeezed Light
The squeezed light [18,19,20] has non-classical properties, such as the anti-bunching effect [21,22,23], quadrature squeezing effect [24,25,26], and sub-Poissonian distribution [27,28,29], and it has been applied to optical communication [30,31,32], high-precision interferometry [33, 34] and weak signal detection[35,36,37]. The squeezed state of single-mode light [38, 39] is produced by the process of degenerate parameters. The two-photon Hamiltonian in the interaction representation can be represented as:
where \(\left[ {a,{a^\dag }} \right] = 1\), \(\xi\) is the coupling constant between the classical pump light field and the incident light quantum signal in the nonlinear media. Here, \(\hbar = 1\) is set for simplicity. Let \(\xi = i\lambda\), and then the corresponding squeezing operator is
The induced transformation by \(S\left( \xi \right)\) is
Let
Then, \({x_1} + i{x_2} = a\) can be obtained. The amplitude of the rotating light field (the angle is \(\frac{\pi }{4}\)) is introduced as:
It can be seen that
\({Y_1}\) and \({Y_2}\) constitute orthogonal components
The squeezing operator \(S\left( \xi \right)\) transforms \(\left( {{Y_1} + i{Y_2}} \right)\) as
and it changes the two orthogonal components into
Next, this paper finds the representation of the squeezing operator \(S\left( \xi \right)\) in the \(q - p\) phase space.
3 Representation of \(S\left( \xi \right)\) in \(q - p\) Phase Space
The Hermitian coordinate \(Q\) and momentum operator \(P\) are introduced as
and \(\left[ {Q,P} \right] = i,\) thus
Therefore, \(S\left( {\xi = i\lambda } \right)\) can be represented as
and then we have
and
The coherent state [40,41,42,43,44] \(\left| z \right\rangle\) is introduced as
Let \(z = \frac{q + ip}{{\sqrt 2 }}\) and rewrite \(\left| z \right\rangle\) in canonical form \(\left\vert {p,q} \right\rangle\)
Then, the corresponding completeness relation is expressed as
By Eq. (15) and Eq. (16), its classical correspondence is
Thus, the representation of \(S\left( {\xi = i\lambda } \right)\) in the \(q - p\) phase space is
where \(\sqrt {\cosh \lambda }\) is introduced for the unitarity of \(S\left( {\xi = i\lambda } \right)\). Based on Eq. (18) and the normally ordered form [45,46,47,48] of the vacuum state \(\left| 0 \right\rangle \left\langle 0 \right|\) (\(:\begin{array}{*{20}{c}} : \end{array}\) denotes the normal ordering)
and using the IWOP method, we have
Then, using
Equation (23) can be represented as
and the squeezed vacuum state[49,50,51,52,53] is
Therefore, the fluctuation of the rotation amplitude of the light field in the squeezed vacuum state is
which demonstrates standard squeezing and can be experimentally implemented in degenerate parametric amplifiers.
4 Enhanced Squeezing by Finding a Generalized Squeezing Operator
This section explores the new state that can enhance squeezing (see Eq. (47) below). For this purpose, the degenerate parametric amplifier should be improved to relate two different parameters. The squeezing-enhanced operator [54,55,56] is introduced as
which involves two asymmetric parameters. Let \({\lambda_1} = \lambda {e^r},{\lambda_2} = \lambda {e^{ - r}}\), and we have
Then, the corresponding squeezed state is derived. Using the transformation properties
its classical correspondence is \(p \to p\cosh \lambda - q\sinh \lambda {e^r},\) \(q \to q\cosh \lambda - p{e^{ - r}}\sinh \lambda\). Similar to Eq. (21), the squeezing-enhanced operator is proposed as
Considering that it is complex to perform this double integration in \(S\left( {\lambda ,r} \right)\), the canonical coherent state \(\left| {p,q} \right\rangle\) is expressed as
Then, the ket in Eq. (33) is expressed as
with
and the integration in Eq. (33) becomes
By performing this integration using the IWOP method and based on Eq. (22) and \(\left| z \right\rangle = \exp \left[ { - \frac{{|z{|^2}}}{2} + z{a^\dag }} \right]\left| 0 \right\rangle\), we have
and the new squeezed vacuum state is
By introducing the quadrature operators
with
we have
Then, the following expectation values in state \({\left| 0 \right\rangle_s}\) can be calculated
and it follows
and
The enhancement of squeezing means
and \({(\Delta {\widehat X_1})^2} > \frac{1}{4}{e^{2\lambda }}\) indicates
When \(\lambda > 0,\) \(\sinh \lambda > 0,\) we have
and \(\tanh \lambda \left( {1 + \cosh r} \right) > - 1\). Since \(1 - \cosh r < 0\),
Similarly, from the inequality \({(\Delta {\widehat X_2})^2} < \frac{1}{4}{e^{ - 2\lambda }},\) we have
which also leads to Eq. (50) and indicates that the squeezing enhancement condition is \(\tanh \lambda < \frac{1}{1 + \cosh r}\).
Is “enhanced squeezed light” can be realized by some physical mechanism? To answer this question, this paper analyzes the operator in Eq. (30), which can be expressed as
where \(\exp \left[ { - \frac{i\lambda }{2}\left( {{Q^2} - {P^2}} \right)} \right]\) describes the physical mechanism of a degenerate parametric amplifier, while the operator \({e^{i\left( {QP - \frac{i}{2}} \right)\sqrt r }}\) represents the stretching or contract size of the crystal with susceptibility. Thus, using a stretched or contracted crystal may realize squeezing enhancement[57,58,59].
5 Dissipation of Enhanced Squeezed State
No system in nature can be completely isolated from its environment, the coupling between the system and the environment always generates noise, and dissipation is an irreversible process in the dynamic evolution of the system. In the framework of quantum statistical mechanics, dissipation is described by the master equation of density operator [60,61,62]. In this section, the IWOP method and coherent state representation are employed to derive the master equation of quantum amplitude attenuation and the corresponding power series solutions. Meanwhile, the attenuation law of the squeezed vacuum light field is obtained.
Since pure coherent state is the quantum state closest to the classical one, any density operator can be represented by coherent state representation
where \(P\left( \alpha \right)\) is referred to as \(P -\) representation [63,64,65]. A typical example of the evolution of light field in the amplitude attenuation channel is the amplitude attenuation of the pure coherent state density operator \(\left| \alpha \right\rangle \left\langle \alpha \right|\)
where \(\kappa\) denotes the attenuation rate. The following discusses the equations that govern this evolution. By using the normal product properties and
we have
Let \(\left| {\alpha {e^{ - \kappa t}}} \right\rangle \left\langle {\alpha {e^{ - \kappa t}}} \right| = \rho \left( t \right)\), and then Eq. (56) is equivalent to
Thus, the quantum amplitude attenuation equation is obtained. Its infinite and power series solutions are discussed below.
Considering that
we have
Since any final density operator can be expressed as
by using Eq. (59), we have
where
and \({M_n}\) and \(T\) are defined as
Now, the IWOP method is utilized to find the integral solution of the quantum attenuation equation. By using Eq. (63), Eq. (61) can be rewritten as
where
so
The inverse relation of Eq. (62) is
where \(\left| \beta \right\rangle\) is also a coherent state. Substituting Eq. (67) into Eq. (66), the integral solution of the amplitude attenuation master equation is obtained:
This equation has an advantage: With a given initial density operator \({\rho_0}\), when the matrix element \(\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle\) is calculated and Eq. (68) is integrated by the IWOP method, the final state density operator \(\rho \left( t \right)\) can be derived easily.
At this point, an approach for finding the solution to the master equation can be found in the research work of Mattos's team [66]. The method allows for obtaining analytical expressions for the time-evolved P-function of the system without needing to solve the corresponding master equation.
Now, the attenuation of the enhanced squeezed light field \({\rho_0}\) is discussed
The matrix element \(\left\langle { - \beta } \right|{\rho_0}\left| \beta \right\rangle\) is
Substituting Eq. (70) into Eq. (68) and using the integral formula
we have
here
Particularly, when \(r = 0\), from Eq. (73) we have
and
with
In this case, Eq. (72) reduces to
or
where
It can be further proved that
when \(t = 0,T = 0\), \(\rho (t)\) will go back to the initial state.
6 Decomposition of the Enhanced Squeezing Operator
Based on Eq. (37) of the squeezing-enhancing operator
where
the symplectic matrix can be decomposed as
Since the symplectic transformations constitute a symplectic group product, its quantum mechanical correspondence also supports the following decomposition
where
Therefore, the enhanced squeezing operator is decomposed as
One can check its correctness by evaluating
then
Further, from
the final transform result is
which agrees with Eq. (31).
7 Conclusion
To sum up, by using the IWOP method and following the quantum phase space theory, this paper analyzes the properties of the quadrature field operator of squeezed light and the representation of the squeezing operator \(S\left( \xi \right)\) in the \(q - p\) phase space. Based on this, the phase-space evolution of squeezing-enhanced light, the new squeezing state, and its attenuation can be theoretically represented. These representations give physicists more refined insights for squeezing enhancement. This paper demonstrates the approach for two parameters, and we hope our work will inspire further research that scales to multi-parameters.
Data Availability
No datasets were generated or analysed during the current study.
References
H.Y. Fan, Q. Guo, Phys. Lett. B 21, 1831 (2007)
H.Y. Fan, J. Vanderlinde, Phys. Rev. A 40, 4785 (1989)
L.Y. Hu, H.Y. Fan, Phys. Rev. A 80, 2554 (2008)
M.J. Zhang, J.H. Fang, P. Lin, K. Lu, T. Pang, Commun. Theor. Phys. 52, 561 (2009)
E.P. Wigner, Phys. Rev. A 40, 749 (1932)
P. Patra, Braz. J. Phys. 54, 24 (2024)
V.V. Dodonov, Phys. Lett. A 364, 368 (2007)
J.P. Dowling, G.S. Agarwal, W.P. Schleich, Phys. Rev. A 49, 4101 (1994)
H. Dessano, R.A.S. Paiva, R.G.G. Amorim et al., Braz. J. Phys. 49, 715 (2019)
R.G.G. Amorim, S.C. Ulhoa, A.E. Santana, Braz. J. Phys. 43, 78 (2013)
Q.S. Zhu, X.Y. Kuang, Phys. Lett. A 366, 367 (2007)
B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)
H.Y. Fan, H.Q. Li, X.L. Xu, Commun. Theor. Phys. 55, 415 (2011)
K. Zhang, L.L. Li, P.P. Yu, Y. Zhou, D.W. Guo, H.Y. Fan, Chin. Phys. B 32, 040302 (2023)
H.Y. Fan, H.R. Zaidi, Phys. Lett. A 124, 303 (1987)
H.J. Yu, H.Y. Fan, Braz. J. Phys. 52, 8 (2022)
H.Y. Fan, H.L. Lu, W.B. Gao, X.F. Xu, Ann. Phys. 321, 2116 (2006)
P. Grangier, Phys. Rev. Lett. 59, 2153 (1987)
J. Appel, E. Figueroa, D. Korystov, M. Lobino, A.I. Lvovsky, Phys. Rev. Lett. 100, 093602 (2008)
E.S. Polzik, J. Carri, H.J. Kimble, Phys. Rev. Lett. 68, 3020 (1992)
A. Belfakir, E.M.F. Curado, Y. Hassouni, Ann. Phys. 423, 168331 (2020)
X.W. Wu, M. Gong, C.H. Dong, J.M. Cui, Z.F. Han, Opt. Express 18, 6340 (2010)
Q.Y. Yang, J.W. Sun, Acta. Photon. Sin. 34, 1745 (2005)
M. Bartkowiak, L.A. Wu, A. Miranowicz, J. Phys. B: At. Mol. Opt. 47, 18824 (2014)
J.Q. Liao, C.K. Law, Phys. Rev. A 83, 033820 (2011)
Z.X. Yu, Z.Y. Jiao, J.Z. Sun, Commun. Theor. Phys. 41, 593 (2004)
Z.Y. Jiao, J.M. Ma, Y.T. Shang, N. Li, X. Fu, Commun. Theor. Phys. 50, 971 (2008)
Z.X. Yu, Z.Y. Jiao, Commun. Theor. Phys. 38, 81 (2002)
L.Z. Wang, S.Y. Zhu, J. Bergou, Phys. Rev. A.43, 2436 (1991)
S. Mitra, Braz. J. Phys. 52, 28 (2022)
H.P. Yuen, J.H. Shapiro, IEEE. T. Inform. Theory 24, 657 (1978)
C. Paterson, Phys. Rev. Lett. 94, 153901 (2005)
H. Katsuki, K. Hosaka, H. Chiba, K. Ohmori, Phys. Rev. A 76, 693 (2007)
J. Bethge, C. Grebing, G. Steinmeyer, Opt. Express 15, 14313 (2007)
A.V. Kozlovskii, Quantum Electron. 37, 74 (2007)
M. Xiao, L.A. Wu, H.J. Kimble, Opt. Lett. 13, 476 (1988)
W. Li, Y.B. Jin, X.D. Yu, J. Zhang, Phys. Rev. A 96, 023808 (2017)
J.X. Zhang, C.D. Xie, K.C. Peng, Chin. Phys. Lett. 22, 3005 (2005)
J.X. Zhang, J.T. Jing, C.D. Xie, K.C. Peng, Chin. Phys. Lett. 22, 2751 (2005)
J. Lin, X.L. Gao, K.L. Wang, Commun. Theor. Phys. 47, 161 (2007)
H.Y. Fan, T.N. Ruan, Commun. Theor. Phys. 2, 1563 (1983)
H.Y. Fan, J.H. Chen, Commun. Theor. Phys. 38, 147 (2002)
G.S. Agarwal, Phys. Rev. A 24, 2889 (1981)
X.Y. Zhang, J.S. Wang, L. Wang, X.G. Meng, B.L. Liang, Chin. Phys. B 31, 054205 (2022)
X.B. Tang, X.F. Xu, H.Y. Fan, Commun. Theor. Phys. 54, 67 (2010)
H.Y. Fan, Commun. Theor. Phys. 12, 219 (1989)
H.Y. Fan, H.Q. Li, Chin. Phys. Lett. 24, 3322 (2007)
H.Y. Fan, X.L. Xu, H.Q. Li, Commun. Theor. Phys. 53, 261 (2010)
X.G. Meng, J.S. Wang, B.-L. Liang, Commun. Theor. Phys. 52, 487 (2009)
G.J. Milburn, S.L. Braunstein, Phys. Rev. A 60, 937 (1999)
X.G. Meng, J.S. Wang, H.Y. Fan, Phys. Lett. A 361, 183 (2007)
Z.X. Zhang, H.Y. Fan, Phys. Lett. A 165, 14 (1992)
J.S. Wang, C.Y. Sun, Int. J. Theor. Phys. 37, 1213 (1998)
S.M. Xu, X.L. Xu, H.Q. Li, Mod. Phys. Lett. B 22, 1673 (2008)
H.Y. Fan, J.H. Chen, Commun. Theor. Phys. 40, 589 (2003)
H.Y. Fan, H.L. Lu, Int. J. Theor. Phys. 45, 627 (2006)
J.R. Klauder, E.C.G. Sudarshan, M.M. Miller, Phys. Today 22, 79 (1969)
D.F. Walls, Nat 306, 141 (1983)
D.F. Walls, G.J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1991)
L.M. Duan, G.C. Guo, Chin. Phys. Lett. 12, 589 (1995)
H.Y. Fan, L.Y. Hu, Mod. Phys. Lett. B 22, 2435 (2008)
M.S. Ashrafi, R.M. Bazrafkan, Chin. Phys. B 23, 090303 (2014)
H.Y. Fan, X.L. Xu, Commun. Theor. Phys. 48, 67 (2007)
K.A. Al-Hassanieh, V.V. Dobrovitski, E. Dagotto, B.N. Harmon, Phys. Rev. Lett. 97, 037204 (2006)
H.Y. Fan, H.L. Cheng, Commun. Theor. Phys. 37, 655 (2002)
E.P. Mattos, A. Vidiella-Barranco, Ann. Phys. 422, 168321 (2020)
Acknowledgements
This research project is supported by the Foundation for Young Talents in College of Anhui Province, China (Grant Nos. gxyq2021210 and gxyq2019077), and the Natural Science Foundation of the Anhui Higher Education Institutions of China (Grant Nos.2022AH051580 and 2022AH051586).
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Zhang is responsible for the overall calculation and writing of the paper, Li has calculated the content of some chapters, and Fan has provided guidance for the paper and all authors reviewed the manuscript.
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Appendix
Appendix
It will be proved in this section that Eq. (38) is unitary. From Eq. (38), we have
and
By using the completeness of coherent states
we have
Then, using the following integral formula
it can be derived that
From Eqs. (1) and (7), we have
so Eq. (38) is unitary.
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Zhang, K., Li, LL. & Fan, HY. Phase-Space Evolution of Squeezing-Enhanced Light and Its Attenuation. Braz J Phys 54, 222 (2024). https://doi.org/10.1007/s13538-024-01587-8
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DOI: https://doi.org/10.1007/s13538-024-01587-8