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i l St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.3 Научно-технические ведомости СПбГПУ. Физико-математические науки. 15 (3.3) 2022
THEORETICAL PHYSICS
Conference materials UDC 535.14
DOI: https://doi.org/10.18721/JPM.153.371
Dynamics of the uncertainty value of quadratures for bosonic quantum states
S. S. Medvedeva 1H, A. A. Gaidash 1 2, G. P. Miroshnichenko \ A. D. Kiselev 1 3, A. V. Kozubov 1 2
1 ITMO University, St. Petersburg, Russia;
2 Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia;
3 St. Petersburg State University, St. Petersburg, Russia H ss_medvedeva@itmo.ru
Abstract. In this work we consider the time evolution of the mean values of the first and second moments of the quadrature operators for an arbitrary bosonic quantum state in a single mode transmitted through an optical fiber channel. We utilize the density matrix formalism and the open quantum systems theory and investigate Lindblad master equation in order to derive expressions for the dynamics of mentioned field observables. Obtained expressions contain terms characterized by high frequency oscillations. For the purpose of elimination of these terms we find the envelope functions for the values of the first and second moments of the quadrature operators.
Keywords: quantum optics, open quantum systems theory, quadratures, single mode
Funding: This study was funded by the Ministry of Science and Education of the Russian Federation project title "Quantum dynamics and correlation measurements of multimode photonic systems and topologically nontrivial polarization states" grant No. 2019-0903.
Citation: Medvedeva S. S., Gaidash A. A., Miroshnichenko G. M., Kiselev A. D., Kozubov A. V., Dynamics of the uncertainty value of quadratures for bosonic quantum states. St. Petersburg State Polytechnical University Journal. Physics and Mathematics, 15 (3.3) (2022) 360-364. DOI: https://doi.org/10.18721/JPM.153.371
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)
Материалы конференции УДК 535.14
DOI: https://doi.org/10.18721/JPM.153.371
Динамика неопределенности квадратур бозонных квантовых состояний
С. С. Медведева 1Н, A. A. Гайдаш 1 2, Г. П. Мирошниченко \ A. Д. Киселёв 1 3, A. В. Козубов 1 2
1 Университет ИТМО, Санкт-Петербург, Россия; 2 Математический институт им. В. А. Стеклова РАН, Москва, Россия;
3 Санкт-Петербургский государственный университет, Санкт-Петербург, Россия
н ss_medvedeva@itmo.ru
Аннотация. В работе рассматривается временная эволюция первых и вторых моментов квадратур одномодовых фотонных квантовых состояний, передаваемых по оптоволоконному каналу. При использовании формализма матриц плотности и теории открытых квантовых систем выведены выражения, описывающие динамику указанных выше полевых наблюдаемых. Для исключения из анализируемых выражений членов, характеризующихся высокочастотными осцилляциями, были найдены огибающие значений.
© Medvedeva S. S., Gaidash A. A., Miroshnichenko G. M., Kiselev A. D., Kozubov A. V., 2022. Published by Peter the Great St.Petersburg Polytechnic University.
Ключевые слова: квантовая оптика, теория открытых квантовых систем, квадратуры, одномодовый случай
Финансирование: Работа выполнена в рамках Государственного задания «Квантовая динамика и корреляционные изменения многомодовых фотонных систем и топологически нетривиальных поляризационных состояний» (код темы 2019-0903).
Ссылка при цитировании: Медведева С. С., Гайдаш А. А., Мирошниченко Г. П., Киселёв А. Д., Козубов А. В. Динамика неопределенности квадратур бозонных квантовых состояний // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 15. № 3.3. C. 360-364. DOI: https://doi.org/10.18721/JPM.153.371
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)
Introduction
One of the main constraints on the technological utilization of the unique quantum features such as superposition or squeezing lies in decoherence: the detrimental influence of environment leads any quantum system to the loss of its beneficial quantum features [1]. A theory that may be employed to investigate the evolution of quantum systems considering decoherence is open quantum systems approach [2]. Within this theory different methods are being used; in our research we focus on solving a master equation for a density matrix of a quantum state.
Materials and Methods
In order to give a description of a nonunitary dynamics of a bosonic quantum state study the Liouville master equation that is a special case of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [3]:
д г ~ i ~
-p(t) = -i H,p(t) +Гр(t)
dt L J ,
p( t )| t=0 =p0
(1)
where Hamiltonian of the system is
H = ш| ста + —
I 2,
(2)
here a, (cfi ) are the annihilation (creation) operator, ra is the frequency of the optical mode, and relaxation superoperator acting on a density matrix of a state has the form:
Гр (t) = --(пт +!) (Cap (t) + p (t) аtа - 2ар (t) tf ) - (аа^р (t) + p (t) аа^ - 2<3tp (t) а), (3)
where у denotes the thermalization rate, nT denotes the mean number of thermal photons.
The explicit solution to this equation may be found, for example, with the use of SU(1,1) algebra formalism [4], Jordan mapping [5]. In this investigation we act by the quadrature operators [6]:
q = ~!=(аt + а),
V2 (4)
p=^2(а f - а),
of interest on the master equation, apply the trace operation and then solve the resulting equation to obtain the time-dependence of a mean value of an operator [7, 8].
© Медведева С. С., Гайдаш А. А., Мирошниченко Г. П., Киселёв А. Д., Козубов А. В., 2022. Издатель: Санкт-Петербургский политехнический университет Петра Великого.
Results and Discussion
Firstly, utilizing the method described in the above, we obtain evolutionary equations for the first moment of quadratures operators:
-Y t
{q) = e 2 (<q0>cosrat + <p0>sinrat),
-Y t
(P> = e 2 (<Po>cosrat -<qo>sinrat), where q0, p0 are the mean values of the quadrature operators at the initial time moment:
< qo > = Tr {qPo}, (6)
<Po> = Tr {ppo}.
Eqs. (5) show that the time dependence of both quadrature operators' mean values has the form of high frequency (optical frequency ra) damped oscillations. However, the part of steady-state oscillations does not provide the essential information concerning the dynamics of the quadratures. Thus we find the envelope function:
I- -Yt
fenv. (t) = ±l<q0 >2 +<Po >2 e 2 , (7)
that has an identical form for both q(t) and p(t).
Secondly, keeping in mind that some quantum states, for example, squeezed vacuum states, possess unique qualities which can be observed through the use of the second order of an operator, we derive the equations for the squares (q2), (p2):
<q2 > = 1 (e~Jt (co + aocos2ra t + bosin2ra t - do) + do),
<p2> = 1 (e~Jt (co - aocos2rat - bosin2rat - do) + do), and variances Aq(t), Ap(t) of the quadratures:
Aq = 1 (e~Jt (Co + ^ocos2ra t + Bosin2ra t - do) + do),
Ap = 1 (e~Jt (Co - 4)cos2ra t - Bosin2ra t - do) + do),
where coefficients are
ao = <qo2> -<Po2X 4> = Aqo -APo, bo =<qPo > + <pqo > Bo =<qPo > + <pqo > - 2<qo ><Po > co = <qo2 > + <Po2 X Co = Aqo + ^o, do = 2 nT +1,
(8)
(9)
(10)
Aq0, Ap0 are the mean value variances of the quadrature operators at the initial moment of time:
Aqo = <qo2 >-<qo )2, APo = <Po2 >-<Po >2,
and <qo2 >, <qpo > and <pqo > , <p^ > are the constituents of the covariance matrix:
(11)
"< qo2 > < qpo >
_< pqo > < Po2 > _
Tr {?2Po} Tr {» } Tr {pqpo} Tr {p2po}
It can be seen that the dynamics of both the squares and variances incorporates high frequency (optical frequency) damped oscillations likewise. Thus, we proceed to determine the envelope function of the dynamics. We find bend points for multiplier of e-Yt from, for example, Eqs. (9):
d
—(C0 + A0cos2rat + £0sin2rat - d0) = 0,
dt (13)
d
—(C0 - A0cos2rat - B0sin2rat - d0) = 0,
After a simplification we obtain the following envelope functions of the dynamics of the squares F :
i env
Few. (t) = 2 ((c0 WC02 + V - d0) + d0), (14)
and variances Af of the quadratures:
env
¥env.(t) = 1 ((C0 WA02 + B02 -d0)e"* + d0). (15)
Obtained expressions are of a more utility considering technical realisation of the quadratures detection [9].
Conclusion
Utilizing the method of solving GKSL master equation for a mean value of a particular operator we obtain the expressions of time evolution for the first and second moments of the quadrature operators for an arbitrary bosonic quantum state in a single mode transmitted through an optical fiber channel. Moreover, we find the envelope functions for the obtained expressions for the purpose of detection that are of a more utility considering technical realisation.
REFERENCES
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3. Carmichael H., An open systems approach to quantum optics: lectures presented at the Université Libre de Bruxelles, October 28 to November 4, 1991. Springer Science & Business Media. 18 (2009).
4. Gaidash A. A., Kozubov. A. V., Miroshnichenko G. P., Dissipative dynamics of quantum states in the fiber channel. Physical Review A. 102 (2) (2020) 023711.
5. Gaidash A. A., Kozubov A.V., Miroshnichenko G.P., Kiselev A.D., Quantum dynamics of mixed polarization states: effects of environment-mediated intermode couplings. Optical Society of America. 38 (9) (2021) 2603-2611.
6. Scully M. O., Zubairy M. S., Quantum optics. American Association of Physics Teachers. (1999).
7. Gaidash A. A., Kozubov A. V., Medvedeva S. S., Miroshnichenko G. P., The Influence of Signal Polarization on Quantum Bit Error Rate for Subcarrier Wave Quantum Key Distribution Protocol. Multidisciplinary Digital Publishing Institute. 22 (12) (2020) 1393.
8. Medvedeva S. S., Gaidash A. A., Kozubov A. V., Miroshnichenko G. P., Dynamics of field observables in quantum channels. IOP Publishing. 1984 (1) (2021) 012007.
9. Laudenbach F., Pacher C., Fung C. H. F., Poppe A., Peev M., Schrenk B., Hübel H. Continuous-variable quantum key distribution with Gaussian modulation—the theory of practical implementations. Advanced Quantum Technologies, 1 (1) (2018) 1800011.
THE AUTHORS
MEDVEDEVA Svetlana S.
ss_medvedeva@itmo.ru ORCID: 0000-0001-5249-5955
KISELEV Alexei D.
adkiselev@itmo.ru ORCID: 0000-0002-1023-3284
GAIDASH Andrei A.
KOZUBOV Anton V.
avkozubov@itmo.ru ORCID: 0000-0002-4468-5406
andrei_gaidash@itmo.ru ORCID: 0000-0001-9870-9285
MIROSHNICHENKO George M.
gpmiroshnichenko@itmo.ru ORCID: 0000-0002-4265-8818
Received 18.07.2022. Approved after reviewing 18.07.2022. Accepted 18.07.2022.
© Peter the Great St. Petersburg Polytechnic University, 2022
