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Quantum Optics|429 Article(s)
Laser Ranging Technology Based on Photonic Resonant Tunneling
Zhiyong Wang, Zhiguo Jia, Guangcun Shao, Anran Li, Kaiqiang Zhang, Yukun Ji, and Mingyu Zhong
ObjectivePhotonic tunneling can be regarded as an optical analog for the quantum-mechanical barrier penetration of material particles. As the photon field has no charge and is not subject to the Pauli exclusion principle, some physical problems (such as tunneling time) become easier to study through photonic quantum tunneling, arousing great interest in the study of the quantum tunneling effect of photons. However, up to now, the quantum resonance tunneling phenomena of photons through a double-barrier have not been studied thoroughly. Photons in a state of quantum tunneling correspond to evanescent waves (i.e., surface plasmon polaritons) that are the core concept of nanooptics. Thus, research on photonic resonance tunneling can reveal new physical laws in nanooptics and has potential application value in optical devices (such as optical sensors and optical transistors). Therefore, it is necessary to develop a systematic theory of photonic resonance tunneling through a double-barrier. The application of the resonance tunneling effect of photons in the design of pulse and phase laser ranging systems is an important subject worth studying.MethodsA photonic double-barrier structure is formed by a rectangular waveguide with dielectric discontinuities (Fig. 1). Seeing that the electromagnetic waves propagating along the waveguide satisfy the Helmholtz equation and can be expanded as a superposition of the waveguide modes transverse electric (TE) and transverse magnetic (TM), one can take the TE10 mode as an example. In this case, the electric field component and its first derivative for z are continuous at the boundaries between the two different media inside the waveguide, based on which and using the concept of the Poynting vector one can obtain the quantum tunneling probability formula of photons through the double-barrier. By employing the analytic method and numerical simulation, we can obtain the physical conditions required for the resonance penetration effect of propagating-wave and evanescent-wave photons, respectively. In addition, we can clarify the dependence of the tunneling probability on the geometric size of the double-barrier, the refractive index of the filling medium, and the photon frequency. The parameters in the tunneling probability expression of photons through the double-barrier are related to each other. As a result, the parameter design makes it easy to make a mistake in the numerical analysis, which can be overcome by resorting to the original definitions of these parameters. To explore the potential application of the quantum resonance tunneling effect of photons in optical devices, we provide two new designs for the receivers of pulse and phase laser ranging systems (Figs. 6 and 7). To be specific, the double-barrier structure shown in Fig. 1 is placed in the receiving device of the laser ranging system. Its geometric sizes and the refractive index of the filling media are designed so that the resonant tunneling frequency is equal to the center frequency of the output signal of the laser ranging system.Results and DiscussionsThe quantum tunneling probability of evanescent-wave photons through the double-barrier is given by Eq. (7), and in this case, the double-barrier corresponds to the two cut-off waveguides. The quantum tunneling probability of propagating-wave photons through the double-barrier is given by Eq. (9), and the double-barrier is formed by two normal-sized waveguides. Both Eq. (7) and Eq. (9) show that there are resonant penetration effects, namely that, the tunneling probability can be equal to one and photons can pass through the double-barrier completely. The resonant tunneling conditions of evanescent-wave photons are presented in Eq. (10), while the resonant penetration conditions of propagating-wave photons are provided in Eq. (11) or Eq. (12). The numerical simulation results are given in Figs. 2-5, where the tunneling probability curves containing resonance peaks show that their full widths at half maximum decrease sharply with the variation of parameters (such as the barrier width, the refractive index of the filling medium, and the photon frequency). In particular, when the double-barrier is formed by two cut-off waveguides, a tiny change in frequency or the structure parameters of the double-barrier can make a huge impact on the tunneling probability of photons. As for the laser ranging systems shown by Figs. 6 and 7, the resonant frequency is equal to the center frequency of the output signal. Since the frequency of the echo signal reflected by a static target is basically unchanged, the echo signal can smoothly pass through the double-barrier and enter the next module to complete the timing or phase measurement. Other light waves from the environment, with frequencies usually different from the working frequency of the laser ranging system, will be filtered out by the double-barrier structure. Thus, the received echo signal can be guaranteed to be true. On the other hand, the laser pulse has a non-zero spectral width (there is a frequency distribution around its central frequency). The closer the frequency of a component in the pulse is to the center frequency, the more likely it is to pass through the double-barrier. Therefore, when the echo signal passes through the double-barrier structure, its spectrum becomes narrowed, and its monochromaticity is enhanced. When the object to be measured is a moving object, the influence of the Doppler effect on this design is typically negligible.ConclusionsA photonic double-barrier can be constructed via an electromagnetic waveguide with dielectric discontinuities. For a given frequency, by choosing appropriate parameters, the tunneling probability of photons through the double-barrier structure can be equal to one (resonant penetration effect). When the resonance phenomenon occurs, a small change in the frequency or structural parameters of the double-barrier can significantly influence the tunneling probability of photons through the double-barrier. These physical properties may provide some new-type design principles for some optical devices, such as band-pass filters, optical sensors, and optical transistors. Especially, it can present a new design for the receiving device of a laser ranging system, which is conducive to ruling out spurious returning signals and enhancing the monochromaticity of the true returning signals. ObjectivePhotonic tunneling can be regarded as an optical analog for the quantum-mechanical barrier penetration of material particles. As the photon field has no charge and is not subject to the Pauli exclusion principle, some physical problems (such as tunneling time) become easier to study through photonic quantum tunneling, arousing great interest in the study of the quantum tunneling effect of photons. However, up to now, the quantum resonance tunneling phenomena of photons through a double-barrier have not been studied thoroughly. Photons in a state of quantum tunneling correspond to evanescent waves (i.e., surface plasmon polaritons) that are the core concept of nanooptics. Thus, research on photonic resonance tunneling can reveal new physical laws in nanooptics and has potential application value in optical devices (such as optical sensors and optical transistors). Therefore, it is necessary to develop a systematic theory of photonic resonance tunneling through a double-barrier. The application of the resonance tunneling effect of photons in the design of pulse and phase laser ranging systems is an important subject worth studying.MethodsA photonic double-barrier structure is formed by a rectangular waveguide with dielectric discontinuities (Fig. 1). Seeing that the electromagnetic waves propagating along the waveguide satisfy the Helmholtz equation and can be expanded as a superposition of the waveguide modes transverse electric (TE) and transverse magnetic (TM), one can take the TE10 mode as an example. In this case, the electric field component and its first derivative for z are continuous at the boundaries between the two different media inside the waveguide, based on which and using the concept of the Poynting vector one can obtain the quantum tunneling probability formula of photons through the double-barrier. By employing the analytic method and numerical simulation, we can obtain the physical conditions required for the resonance penetration effect of propagating-wave and evanescent-wave photons, respectively. In addition, we can clarify the dependence of the tunneling probability on the geometric size of the double-barrier, the refractive index of the filling medium, and the photon frequency. The parameters in the tunneling probability expression of photons through the double-barrier are related to each other. As a result, the parameter design makes it easy to make a mistake in the numerical analysis, which can be overcome by resorting to the original definitions of these parameters. To explore the potential application of the quantum resonance tunneling effect of photons in optical devices, we provide two new designs for the receivers of pulse and phase laser ranging systems (Figs. 6 and 7). To be specific, the double-barrier structure shown in Fig. 1 is placed in the receiving device of the laser ranging system. Its geometric sizes and the refractive index of the filling media are designed so that the resonant tunneling frequency is equal to the center frequency of the output signal of the laser ranging system.Results and DiscussionsThe quantum tunneling probability of evanescent-wave photons through the double-barrier is given by Eq. (7), and in this case, the double-barrier corresponds to the two cut-off waveguides. The quantum tunneling probability of propagating-wave photons through the double-barrier is given by Eq. (9), and the double-barrier is formed by two normal-sized waveguides. Both Eq. (7) and Eq. (9) show that there are resonant penetration effects, namely that, the tunneling probability can be equal to one and photons can pass through the double-barrier completely. The resonant tunneling conditions of evanescent-wave photons are presented in Eq. (10), while the resonant penetration conditions of propagating-wave photons are provided in Eq. (11) or Eq. (12). The numerical simulation results are given in Figs. 2-5, where the tunneling probability curves containing resonance peaks show that their full widths at half maximum decrease sharply with the variation of parameters (such as the barrier width, the refractive index of the filling medium, and the photon frequency). In particular, when the double-barrier is formed by two cut-off waveguides, a tiny change in frequency or the structure parameters of the double-barrier can make a huge impact on the tunneling probability of photons. As for the laser ranging systems shown by Figs. 6 and 7, the resonant frequency is equal to the center frequency of the output signal. Since the frequency of the echo signal reflected by a static target is basically unchanged, the echo signal can smoothly pass through the double-barrier and enter the next module to complete the timing or phase measurement. Other light waves from the environment, with frequencies usually different from the working frequency of the laser ranging system, will be filtered out by the double-barrier structure. Thus, the received echo signal can be guaranteed to be true. On the other hand, the laser pulse has a non-zero spectral width (there is a frequency distribution around its central frequency). The closer the frequency of a component in the pulse is to the center frequency, the more likely it is to pass through the double-barrier. Therefore, when the echo signal passes through the double-barrier structure, its spectrum becomes narrowed, and its monochromaticity is enhanced. When the object to be measured is a moving object, the influence of the Doppler effect on this design is typically negligible.ConclusionsA photonic double-barrier can be constructed via an electromagnetic waveguide with dielectric discontinuities. For a given frequency, by choosing appropriate parameters, the tunneling probability of photons through the double-barrier structure can be equal to one (resonant penetration effect). When the resonance phenomenon occurs, a small change in the frequency or structural parameters of the double-barrier can significantly influence the tunneling probability of photons through the double-barrier. These physical properties may provide some new-type design principles for some optical devices, such as band-pass filters, optical sensors, and optical transistors. Especially, it can present a new design for the receiving device of a laser ranging system, which is conducive to ruling out spurious returning signals and enhancing the monochromaticity of the true returning signals.
Acta Optica Sinica
- Publication Date: Apr. 25, 2024
- Vol. 44, Issue 8, 0827001 (2024)
Two-Party Mutual Authentication Semi-Quantum Key Agreement Protocol Based on Bell State
Yefeng He, Xiyuan Liang, and Mingyue Cai
ObjectiveQuantum cryptography uses quantum states as the carriers of information transmission and transmits information between authorized users through quantum channels. Different from that of traditional cryptography, the security of quantum cryptography is guaranteed by the basic principles of quantum mechanics. Therefore, it is theoretically unconditionally secure. In recent years, quantum cryptography has received extensive attention from many researchers engaged in cryptography, and has gradually developed into a popular research direction in the field of cryptography. Specifically, the quantum key agreement is an important research topic in quantum cryptography. It enables all participants to jointly negotiate a session key through a secure quantum channel, and each participants contribution to the negotiated key is the same. On the one hand, due to the high cost and scarce resources, it is difficult for the vast majority of participants to have well-performing quantum devices. Therefore, in order to facilitate the implementation of the protocol, it is necessary to simplify the quantum operations of the participants. In response to this problem, some scholars have proposed a semi-quantum key agreement protocol. The semi-quantum key agreement protocol requires that one of the participants in the protocol has complete quantum capabilities, and the remaining participants only have semi-quantum capabilities. Moreover, the semi-quantum participants can only perform the following two operations: i) reflection operation. No operation is performed on the received particles, and the received particles are returned directly. ii) Measurement operation. Z-based measurement is performed on the received particles, and new particles are prepared according to the measurement results. On the other hand, since participants may be attacked by man-in-the-middle in the process of key agreement, it is necessary to authenticate participants before the key agreement. In recent years, researchers have also proposed some quantum key agreement protocols with mutual authentication. In practical application scenarios, in order to facilitate the implementation of the protocol, it is necessary to design a semi-quantum key agreement protocol with lower requirements for participants ability and equipment. In order to prevent external attackers from counterfeiting authorized users to steal shared keys, the protocol needs to have a mutual authentication function. Therefore, it is necessary to design a semi-quantum key agreement protocol with mutual authentication.MethodsBased on the Bell state, we propose a two-party semi-quantum key agreement protocol with a mutual authentication function, where Alice is a full quantum participant and Bob is a semi-quantum participant. The two sides achieve mutual authentication of identity by preparing and measuring identity information particles. By using the entanglement characteristics of the Bell state, the shared key negotiation was realized. Compared with other entangled states, the Bell state used in this protocol is easier to prepare, and the protocol only uses two quantum measurement operations, namely Z-based measurement and Bell measurement, which are easier to implement in existing technology. In addition, we proved that the proposed scheme can effectively resist participant attacks and external attacks, and that the protocol is equipped with a wavelength quantum filter and a photon number separator on both sides of Alice and Bob to avoid Trojan horse attacks. In the performance analysis of this protocol, the Cabello qubit efficiency was used to measure the performance of the quantum key agreement protocol.Results and DiscussionsFirst of all, in the previous research on quantum key agreement protocols, some scholars focus on how to simplify the quantum operation of participants, so as to better apply to the actual scene of resource scarcity, while others pay attention to how to prevent the man-in-the-middle attacks that may be encountered during the key agreement process and further improve the security of the protocol. The two-party mutual authentication semi-quantum key agreement protocol based on the Bell state proposed in this paper can not only reduce the requirements for participants capabilities and devices, but also realize mutual authentication between participants before key agreement to prevent the protocol from being attacked by man-in-the-middle. Finally, a security analysis shows that the protocol can effectively resist participant and external attacks. In addition, the performance analysis shows that the protocol can improve the quantum bit efficiency compared with some quantum key agreement protocols that meet a single function under the condition of satisfying two functional characteristics at the same time.ConclusionsIn this study, a two-party mutual authentication semi-quantum key agreement protocol based on the Bell state is proposed. The protocol not only ensures that the shared key can be fairly negotiated between the full quantum party, Alice, and the semi-quantum party, Bob, but more importantly, the two parties need to authenticate each others identity before the key agreement, so as to resist external attackers posing as legitimate users to steal the shared key. Security analysis shows that this semi-quantum key agreement protocol can resist both participant and external attacks. Finally, through a performance analysis and comparison with existing quantum key agreement protocols, it is found that the protocol has certain advantages in terms of its function and performance. ObjectiveQuantum cryptography uses quantum states as the carriers of information transmission and transmits information between authorized users through quantum channels. Different from that of traditional cryptography, the security of quantum cryptography is guaranteed by the basic principles of quantum mechanics. Therefore, it is theoretically unconditionally secure. In recent years, quantum cryptography has received extensive attention from many researchers engaged in cryptography, and has gradually developed into a popular research direction in the field of cryptography. Specifically, the quantum key agreement is an important research topic in quantum cryptography. It enables all participants to jointly negotiate a session key through a secure quantum channel, and each participants contribution to the negotiated key is the same. On the one hand, due to the high cost and scarce resources, it is difficult for the vast majority of participants to have well-performing quantum devices. Therefore, in order to facilitate the implementation of the protocol, it is necessary to simplify the quantum operations of the participants. In response to this problem, some scholars have proposed a semi-quantum key agreement protocol. The semi-quantum key agreement protocol requires that one of the participants in the protocol has complete quantum capabilities, and the remaining participants only have semi-quantum capabilities. Moreover, the semi-quantum participants can only perform the following two operations: i) reflection operation. No operation is performed on the received particles, and the received particles are returned directly. ii) Measurement operation. Z-based measurement is performed on the received particles, and new particles are prepared according to the measurement results. On the other hand, since participants may be attacked by man-in-the-middle in the process of key agreement, it is necessary to authenticate participants before the key agreement. In recent years, researchers have also proposed some quantum key agreement protocols with mutual authentication. In practical application scenarios, in order to facilitate the implementation of the protocol, it is necessary to design a semi-quantum key agreement protocol with lower requirements for participants ability and equipment. In order to prevent external attackers from counterfeiting authorized users to steal shared keys, the protocol needs to have a mutual authentication function. Therefore, it is necessary to design a semi-quantum key agreement protocol with mutual authentication.MethodsBased on the Bell state, we propose a two-party semi-quantum key agreement protocol with a mutual authentication function, where Alice is a full quantum participant and Bob is a semi-quantum participant. The two sides achieve mutual authentication of identity by preparing and measuring identity information particles. By using the entanglement characteristics of the Bell state, the shared key negotiation was realized. Compared with other entangled states, the Bell state used in this protocol is easier to prepare, and the protocol only uses two quantum measurement operations, namely Z-based measurement and Bell measurement, which are easier to implement in existing technology. In addition, we proved that the proposed scheme can effectively resist participant attacks and external attacks, and that the protocol is equipped with a wavelength quantum filter and a photon number separator on both sides of Alice and Bob to avoid Trojan horse attacks. In the performance analysis of this protocol, the Cabello qubit efficiency was used to measure the performance of the quantum key agreement protocol.Results and DiscussionsFirst of all, in the previous research on quantum key agreement protocols, some scholars focus on how to simplify the quantum operation of participants, so as to better apply to the actual scene of resource scarcity, while others pay attention to how to prevent the man-in-the-middle attacks that may be encountered during the key agreement process and further improve the security of the protocol. The two-party mutual authentication semi-quantum key agreement protocol based on the Bell state proposed in this paper can not only reduce the requirements for participants capabilities and devices, but also realize mutual authentication between participants before key agreement to prevent the protocol from being attacked by man-in-the-middle. Finally, a security analysis shows that the protocol can effectively resist participant and external attacks. In addition, the performance analysis shows that the protocol can improve the quantum bit efficiency compared with some quantum key agreement protocols that meet a single function under the condition of satisfying two functional characteristics at the same time.ConclusionsIn this study, a two-party mutual authentication semi-quantum key agreement protocol based on the Bell state is proposed. The protocol not only ensures that the shared key can be fairly negotiated between the full quantum party, Alice, and the semi-quantum party, Bob, but more importantly, the two parties need to authenticate each others identity before the key agreement, so as to resist external attackers posing as legitimate users to steal the shared key. Security analysis shows that this semi-quantum key agreement protocol can resist both participant and external attacks. Finally, through a performance analysis and comparison with existing quantum key agreement protocols, it is found that the protocol has certain advantages in terms of its function and performance.
Acta Optica Sinica
- Publication Date: Mar. 10, 2024
- Vol. 44, Issue 5, 0527002 (2024)
Microcavity-Assisted Spin Dynamics Characteristics and Superradiant Phase Transition
Chao Cui, and Yanlin Feng
ObjectiveCompared with the traditional method of changing spin freedom through external magnetic fields, the spin-orbit coupling, which utilizes the coupling between the spin freedom and the motion freedom of atoms, is a new method for regulating spin. With the continuous realization of artificial spin-orbit coupling in cold atomic systems in experiments, many novel physical phenomena based on spin-orbit coupling have been widely promoted. In addition, since the realization of the superradiant quantum phase transition in experiments in 2010, the system of coupling ultracold atomic gas and cavity quantum electrodynamics has become an ideal platform for exploring novel many-body physics, which has aroused a research boom among theoretical scientists and experimental scientists. This coupling system couples ultracold atoms into a high-precision optical microcavity. Under specific electromagnetic boundary conditions, light interacts with ultracold atoms and induces novel many-body quantum properties. In this coupling system, one can not only explore the complex quantum behavior induced by the long-range interaction among atoms mediated by cavity photons but also understand the collective dynamical properties of cavity photons and ultracold atoms at the single-photon level. At the same time, the optical microcavity has both driving and dissipation, and it is a natural non-equilibrium system, which allows one to study the non-equilibrium steady-state dynamical properties. However, the time-dependent cavity-assisted spin dynamics has not been considered experimentally and theoretically. On the one hand, the time-dependent Schr?dinger equation is difficult to obtain an exact analytical solution mathematically, and on the other hand, the physical process expressed by the time-dependent Schr?dinger equation involves complex energy changes, time evolution, and interaction problems, which makes it difficult to solve. In view of these problems, we proposed a method for realizing the superradiant phase transition with the assistance of an optical microcavity. This method coupled the optical microcavity system with a Bose-Einstein condensate trapped in a harmonic potential that oscillates with time to obtain a new model, which could be used to study the self-organized phase transition and spin dynamics of Bose-Einstein condensates in microcavities and provide a reference for studying other Bose-Einstein condensates based on spin.MethodsWe considered the preparation of Bose-Einstein condensates using a magneto-optical trap and the coupling of these Bose-Einstein condensates bound in an oscillatory harmonic potential field with a high-precision optical microcavity, thereby establishing a one-dimensional coupled system where the Bose-Einstein condensates only moved in the x direction. The atoms we considered were those with four internal energy levels, and under conditions of large detuning, the excited states of the Bose-Einstein condensates were removed adiabatically, and the resulting Hamiltonian was then quantized. Through mean-field calculations, we obtained the coupled mean-field equations, which were then specialized for the time-dependent part, thereby transforming the problem with time dependence into a classification discussion without time dependence.Results and DiscussionsWe studied the steady-state properties of matter, obtained the relationship diagram of the order parameter with the coupling strength and explored the influence of the external magnetic field strength and the harmonic potential field vibration strength on the critical point of the superradiant phase transition. The results show that the effective magnetic field mz experienced by the atoms and the vibration strength ξ0 of the harmonic potential well will affect the phase transition. Specifically, the coupling strength corresponding to the critical point of the superradiant phase transition increases monotonically with the increase in mz. When ξ0 /1 /πmω≤31.5, the coupling strength corresponding to the critical point of the superradiant phase transition decreases with the increase in ξ0. When ξ0 /1 /πmω>31.5, the coupling strength corresponding to the critical point of the superradiant phase transition increases with the increase in ξ0(Fig. 4). In addition, we also analyzed the non-trivial spin dynamics induced by the interaction between light and atoms and the influence of the vibration strength of the harmonic potential field on the dynamical properties. It was found that when the system does not undergo superradiance, the oscillation of σxt at zero over time is still symmetric but not smooth, and the value of ξ0 /1 /πmω affects the atomic spin resonance effect (Fig. 5).ConclusionsIn this study, we propose a feasible method for realizing optical microcavity-assisted superradiant phase transition and spin dynamics and explore the superradiant quantum phase transition and non-trivial spin dynamics that oscillate with time. We adopt the mean-field approximation method for the cavity field and the matter field and treat the time-dependent system, so as to obtain the superradiant phase transition of the system and give the complete phase diagram of the phase transition. On this basis, we study the non-trivial spin dynamics of the system by qualitatively analyzing the average value of the Pauli operator. We find that the coupling strength corresponding to the occurrence of the superradiant phase transition increases with the increase in the external magnetic field, and it decreases first and then increases with the increase in the vibration intensity of the external harmonic potential field. The vibration intensity of the harmonic potential field affects the spin dynamics effect of the system, because the vibration intensity of the harmonic potential field changes the coupling strength corresponding to the critical point of the superradiant phase transition, thus resulting in changes in the spin dynamics effect of the system. ObjectiveCompared with the traditional method of changing spin freedom through external magnetic fields, the spin-orbit coupling, which utilizes the coupling between the spin freedom and the motion freedom of atoms, is a new method for regulating spin. With the continuous realization of artificial spin-orbit coupling in cold atomic systems in experiments, many novel physical phenomena based on spin-orbit coupling have been widely promoted. In addition, since the realization of the superradiant quantum phase transition in experiments in 2010, the system of coupling ultracold atomic gas and cavity quantum electrodynamics has become an ideal platform for exploring novel many-body physics, which has aroused a research boom among theoretical scientists and experimental scientists. This coupling system couples ultracold atoms into a high-precision optical microcavity. Under specific electromagnetic boundary conditions, light interacts with ultracold atoms and induces novel many-body quantum properties. In this coupling system, one can not only explore the complex quantum behavior induced by the long-range interaction among atoms mediated by cavity photons but also understand the collective dynamical properties of cavity photons and ultracold atoms at the single-photon level. At the same time, the optical microcavity has both driving and dissipation, and it is a natural non-equilibrium system, which allows one to study the non-equilibrium steady-state dynamical properties. However, the time-dependent cavity-assisted spin dynamics has not been considered experimentally and theoretically. On the one hand, the time-dependent Schr?dinger equation is difficult to obtain an exact analytical solution mathematically, and on the other hand, the physical process expressed by the time-dependent Schr?dinger equation involves complex energy changes, time evolution, and interaction problems, which makes it difficult to solve. In view of these problems, we proposed a method for realizing the superradiant phase transition with the assistance of an optical microcavity. This method coupled the optical microcavity system with a Bose-Einstein condensate trapped in a harmonic potential that oscillates with time to obtain a new model, which could be used to study the self-organized phase transition and spin dynamics of Bose-Einstein condensates in microcavities and provide a reference for studying other Bose-Einstein condensates based on spin.MethodsWe considered the preparation of Bose-Einstein condensates using a magneto-optical trap and the coupling of these Bose-Einstein condensates bound in an oscillatory harmonic potential field with a high-precision optical microcavity, thereby establishing a one-dimensional coupled system where the Bose-Einstein condensates only moved in the x direction. The atoms we considered were those with four internal energy levels, and under conditions of large detuning, the excited states of the Bose-Einstein condensates were removed adiabatically, and the resulting Hamiltonian was then quantized. Through mean-field calculations, we obtained the coupled mean-field equations, which were then specialized for the time-dependent part, thereby transforming the problem with time dependence into a classification discussion without time dependence.Results and DiscussionsWe studied the steady-state properties of matter, obtained the relationship diagram of the order parameter with the coupling strength and explored the influence of the external magnetic field strength and the harmonic potential field vibration strength on the critical point of the superradiant phase transition. The results show that the effective magnetic field mz experienced by the atoms and the vibration strength ξ0 of the harmonic potential well will affect the phase transition. Specifically, the coupling strength corresponding to the critical point of the superradiant phase transition increases monotonically with the increase in mz. When ξ0 /1 /πmω≤31.5, the coupling strength corresponding to the critical point of the superradiant phase transition decreases with the increase in ξ0. When ξ0 /1 /πmω>31.5, the coupling strength corresponding to the critical point of the superradiant phase transition increases with the increase in ξ0(Fig. 4). In addition, we also analyzed the non-trivial spin dynamics induced by the interaction between light and atoms and the influence of the vibration strength of the harmonic potential field on the dynamical properties. It was found that when the system does not undergo superradiance, the oscillation of σxt at zero over time is still symmetric but not smooth, and the value of ξ0 /1 /πmω affects the atomic spin resonance effect (Fig. 5).ConclusionsIn this study, we propose a feasible method for realizing optical microcavity-assisted superradiant phase transition and spin dynamics and explore the superradiant quantum phase transition and non-trivial spin dynamics that oscillate with time. We adopt the mean-field approximation method for the cavity field and the matter field and treat the time-dependent system, so as to obtain the superradiant phase transition of the system and give the complete phase diagram of the phase transition. On this basis, we study the non-trivial spin dynamics of the system by qualitatively analyzing the average value of the Pauli operator. We find that the coupling strength corresponding to the occurrence of the superradiant phase transition increases with the increase in the external magnetic field, and it decreases first and then increases with the increase in the vibration intensity of the external harmonic potential field. The vibration intensity of the harmonic potential field affects the spin dynamics effect of the system, because the vibration intensity of the harmonic potential field changes the coupling strength corresponding to the critical point of the superradiant phase transition, thus resulting in changes in the spin dynamics effect of the system.
Acta Optica Sinica
- Publication Date: Mar. 10, 2024
- Vol. 44, Issue 5, 0527001 (2024)
Secure Continuous Variable Quantum Cloning Based on EPR Steering
Jun Wang, and Shuqin Zhai
ObjectiveQuantum communication is based on the three principles of uncertainty, measurement collapse, and no-cloning in quantum mechanics. Compared with traditional classical communication methods, quantum communication features security and high efficiency and has great application significance and prospect in information security. In recent years, domestic and international scientists have conducted a lot of research on theories and experiments and made outstanding achievements in long-distance transmission and practical network of quantum communication. Quantum teleportation and quantum cloning have caught extensive attention as important protocols in quantum communication. With the help of quantum entanglement and classical communication, the transmission of any unknown quantum state from one location to another can be realized. As important resources of quantum information, quantum entanglement and EPR steering are widely adopted in various quantum communication tasks. The natural asymmetry of EPR steering makes quantum steering a helpful resource in various quantum information processes. In the tasks of single-side device-independent quantum-key distribution, secure quantum teleportation, and subchannel discrimination, quantum steering can improve key acquisition rate, and enhance the protocol efficiency and security. In 2000, Cerf N J et al. proposed quantum cloning of Gaussian states with continuous variables and gave the fidelity boundary of quantum cloning as 2/3. In 2001, the Grangier P group presented the quantum and classical fidelity boundary of coherent state continuous variable quantum cloning under Heisenberg representation. For coherent state input, quantum teleportation is achieved when the fidelity exceeds the classical limit of 1/2, which is the best value that can be obtained without entanglement. However, it is necessary to have certain requirements for entangled beams to realize quantum teleportation with a fidelity greater than 2/3. In 2004, the Furusawa group applied three single-mode OPOs to obtain a continuous variable quantum teleportation network with an optimal fidelity of 0.64, and then they utilized four OPOs to achieve quantum teleportation with a fidelity of 0.7. In 2012, Pan J W group experimentally realized long-distance quantum teleportation. In 2018, Wei J H et al. put forward a quantum teleportation scheme using non-maximum entangled states for measurement. In 2018, Wang K et al. studied teleportation by partially entangled GHZ states. The analysis based on quantum cloning shows that for coherent state inputs, secure teleportation is guaranteed if the teleportation fidelity is greater than 2/3. To sum up, the research on remote transmission security is still a long-term important topic.MethodsBased on the basic idea of quantum teleportation, we employ the method of combining quantum channel and classical channel to design a 1→2 quantum cloning scheme with continuous variables by partially disembodied transport. The relationship between the fidelity of a partially disembodied transport cloning scheme and EPR entanglement source is studied theoretically. Firstly, the fidelity of two output modes in 1→2 cloning scheme, the entanglement and steering of EPR shared entanglement source are analyzed. Secondly, the relationship between the fidelity of the output mode Clone 1 and the steering characteristics under the optimal gain is studied. Thirdly, the fidelity of the output mode Clone 2 varies with the reflectance and squeezing parameters under the optimal gain of the output mode Clone 1.Results and DiscussionsFirst, we analyze the variation of the steering between entanglement sources b^1 and b^2 and optimal gain with η1 and η2. Only if η1>0.5 there is a steering of b^2 by b^1, and if η2>0.5 there is a steering of b^1 by b^2. The results are as follows: when η1>0.5 and η2>0.5, there is a two-way steering between b^1 and b^2, and the entanglement amount between the sources increases with the improving transmission efficiency η1 and η2. The range of optimal gain gopt=maxgb2|b1,gb1|b2 is 2≤gopt<5, and the optimal gain corresponds to the optimal gain of output mode Clone 1, which is not optimal for output mode Clone 2. Second, the fidelity of output modes Clone 1 and Clone 2 varies with η1 and η2 under different reflectance when the optimal gain gopt is taken. The fidelity F1>23 should be in the two-way steering region, but the fidelity of the two-way steering region may not always meet F1>23. Meanwhile, the fidelity of output mode Clone 1 decreases with the increasing reflectivity, and that of output mode Clone 2 reduces with the rising reflectance. Third, the fidelity of output modes Clone 1 and Clone 2 varies with η1 and η2 under different squeezing parameters when the optimal gain gopt is taken. The fidelity of output mode Clone 1 in the two-way steering region is greater than 2/3, and the fidelity beyond the no-cloning threshold can also be achieved by two-way steering under smaller squeezing parameters. The fidelity of the output mode Clone 2 decreases with the increase in squeezing parameters.ConclusionsIn summary, we theoretically investigate the relationship between the fidelity of cloning and EPR steering based on the partially disembodied transport continuous variable 1→2 quantum cloning scheme. Meanwhile, we explore the fidelity variation with the reflectance of the beam-splitter and squeezing parameters at a given gain. The results show that for the output mode Clone 1, when the optimal gain is obtained, the two-way steering of the entanglement source should be shared when the fidelity exceeds the no-cloning threshold, but not all two-way steering resources can make the cloning fidelity greater than 2/3. The fidelity of output mode Clone 1 decreases with the rising reflectance and decreasing squeezing parameters, and the two-way steering can also achieve fidelity beyond the no-cloning threshold under smaller squeezing parameters. Additionally, the fidelity of the output mode Clone 2 reduces with the increasing reflectance and squeezing parameters. Therefore, high cloning fidelity does not require significant squeezing and high reflectivity. Therefore, we can employ the combination of quantum channel and classical channel to improve the cloning fidelity. The two-way quantum steering state is the necessary resource for secure quantum cloning of the coherent states. The research results provide certain references for the security of quantum communication networks. ObjectiveQuantum communication is based on the three principles of uncertainty, measurement collapse, and no-cloning in quantum mechanics. Compared with traditional classical communication methods, quantum communication features security and high efficiency and has great application significance and prospect in information security. In recent years, domestic and international scientists have conducted a lot of research on theories and experiments and made outstanding achievements in long-distance transmission and practical network of quantum communication. Quantum teleportation and quantum cloning have caught extensive attention as important protocols in quantum communication. With the help of quantum entanglement and classical communication, the transmission of any unknown quantum state from one location to another can be realized. As important resources of quantum information, quantum entanglement and EPR steering are widely adopted in various quantum communication tasks. The natural asymmetry of EPR steering makes quantum steering a helpful resource in various quantum information processes. In the tasks of single-side device-independent quantum-key distribution, secure quantum teleportation, and subchannel discrimination, quantum steering can improve key acquisition rate, and enhance the protocol efficiency and security. In 2000, Cerf N J et al. proposed quantum cloning of Gaussian states with continuous variables and gave the fidelity boundary of quantum cloning as 2/3. In 2001, the Grangier P group presented the quantum and classical fidelity boundary of coherent state continuous variable quantum cloning under Heisenberg representation. For coherent state input, quantum teleportation is achieved when the fidelity exceeds the classical limit of 1/2, which is the best value that can be obtained without entanglement. However, it is necessary to have certain requirements for entangled beams to realize quantum teleportation with a fidelity greater than 2/3. In 2004, the Furusawa group applied three single-mode OPOs to obtain a continuous variable quantum teleportation network with an optimal fidelity of 0.64, and then they utilized four OPOs to achieve quantum teleportation with a fidelity of 0.7. In 2012, Pan J W group experimentally realized long-distance quantum teleportation. In 2018, Wei J H et al. put forward a quantum teleportation scheme using non-maximum entangled states for measurement. In 2018, Wang K et al. studied teleportation by partially entangled GHZ states. The analysis based on quantum cloning shows that for coherent state inputs, secure teleportation is guaranteed if the teleportation fidelity is greater than 2/3. To sum up, the research on remote transmission security is still a long-term important topic.MethodsBased on the basic idea of quantum teleportation, we employ the method of combining quantum channel and classical channel to design a 1→2 quantum cloning scheme with continuous variables by partially disembodied transport. The relationship between the fidelity of a partially disembodied transport cloning scheme and EPR entanglement source is studied theoretically. Firstly, the fidelity of two output modes in 1→2 cloning scheme, the entanglement and steering of EPR shared entanglement source are analyzed. Secondly, the relationship between the fidelity of the output mode Clone 1 and the steering characteristics under the optimal gain is studied. Thirdly, the fidelity of the output mode Clone 2 varies with the reflectance and squeezing parameters under the optimal gain of the output mode Clone 1.Results and DiscussionsFirst, we analyze the variation of the steering between entanglement sources b^1 and b^2 and optimal gain with η1 and η2. Only if η1>0.5 there is a steering of b^2 by b^1, and if η2>0.5 there is a steering of b^1 by b^2. The results are as follows: when η1>0.5 and η2>0.5, there is a two-way steering between b^1 and b^2, and the entanglement amount between the sources increases with the improving transmission efficiency η1 and η2. The range of optimal gain gopt=maxgb2|b1,gb1|b2 is 2≤gopt<5, and the optimal gain corresponds to the optimal gain of output mode Clone 1, which is not optimal for output mode Clone 2. Second, the fidelity of output modes Clone 1 and Clone 2 varies with η1 and η2 under different reflectance when the optimal gain gopt is taken. The fidelity F1>23 should be in the two-way steering region, but the fidelity of the two-way steering region may not always meet F1>23. Meanwhile, the fidelity of output mode Clone 1 decreases with the increasing reflectivity, and that of output mode Clone 2 reduces with the rising reflectance. Third, the fidelity of output modes Clone 1 and Clone 2 varies with η1 and η2 under different squeezing parameters when the optimal gain gopt is taken. The fidelity of output mode Clone 1 in the two-way steering region is greater than 2/3, and the fidelity beyond the no-cloning threshold can also be achieved by two-way steering under smaller squeezing parameters. The fidelity of the output mode Clone 2 decreases with the increase in squeezing parameters.ConclusionsIn summary, we theoretically investigate the relationship between the fidelity of cloning and EPR steering based on the partially disembodied transport continuous variable 1→2 quantum cloning scheme. Meanwhile, we explore the fidelity variation with the reflectance of the beam-splitter and squeezing parameters at a given gain. The results show that for the output mode Clone 1, when the optimal gain is obtained, the two-way steering of the entanglement source should be shared when the fidelity exceeds the no-cloning threshold, but not all two-way steering resources can make the cloning fidelity greater than 2/3. The fidelity of output mode Clone 1 decreases with the rising reflectance and decreasing squeezing parameters, and the two-way steering can also achieve fidelity beyond the no-cloning threshold under smaller squeezing parameters. Additionally, the fidelity of the output mode Clone 2 reduces with the increasing reflectance and squeezing parameters. Therefore, high cloning fidelity does not require significant squeezing and high reflectivity. Therefore, we can employ the combination of quantum channel and classical channel to improve the cloning fidelity. The two-way quantum steering state is the necessary resource for secure quantum cloning of the coherent states. The research results provide certain references for the security of quantum communication networks.
Acta Optica Sinica
- Publication Date: Feb. 10, 2024
- Vol. 44, Issue 3, 0327002 (2024)
Intracavity-Squeezed Cooling in Double-Laguerre-Gaussian-Cavity Optomechanical System
Qinghong Liao, Haiyan Qiu, Shaoping Cheng, Hongyu Zhu, and Yongqiang Zeng
ObjectiveCooling of mechanical oscillators is an important direction of cavity optomechanics research. Cooling the mechanical oscillators to their quantum ground state is a prerequisite for a wide range of applications based on cavity optomechanics. Therefore, ground-state cooling of mechanical oscillators is the focus of cavity optomechanics at present, which attracts a large number of scholars to study it. However, due to noise interference from external environments, the mechanical oscillators cannot enter the quantum regime. The hybrid system-coupled optical parametric amplifier provides a unique platform to solve the above problem.MethodsThe hybrid optomechanical system consists of two fixed mirrors FM including a rotational mirror RM mounted on the support S which can rotate around the Z axis, and an OPA medium. Cavity 1 which couples the OPA medium is made up of partially transparent FM1 and perfectly reflecting RM while cavity 2 is composed of FM1 and another perfectly reflecting FM2. The cavity 1 is driven by the transmitted beam with charge 0 and a Laguerre-Gauss beam G of charge 0 is incident on FM1. The charge 0 beam reflected from the RM is charged to +2l and then returns to FM1, where a mode with charge 0 is generated and enters cavity 2. After the reflection of FM2, it is also charged to +2l. We study the problem of the intracavity-squeezed cooling in the optical parametric amplifier coupled by a double Laguerre-Gaussian-cavity optomechanical system by calculating the optical force noise spectrum and the steady-state final phonon number. In the weak coupling regime, the optical force noise spectrum of the system is obtained by the perturbation approximation method, and the analytical expression of the final phonon number is calculated by the Fermi Golden Rule theory.Results and DiscussionsWhen the OPA medium is considered in the hybrid optomechanical system, the heating rate of the optical noise spectrum SFFω at ω=-ω? is reduced to 0, with an unaffected cooling rate. In other words, A+ drops while A- remains the same, the net cooling rate Γ=A--A+ naturally becomes larger, and the cooling effect is improved (Fig. 2). Next, we proceed to study how the optical noise spectrum SFFω is affected by the coupling strength J between two cavities. The value of SFFω at ω/ω?=1 is greater in the presence of the auxiliary cavity (Fig. 3). We depict the variations of the optical noise spectrum SFFω with ω/ω? for a given coupling strength J when Δc1=-ω?, Δc1=-2ω?, Δc1=-2.5ω?, and Δc1=-3ω?. The right-hand peak of the optical noise spectrum SFFω is observed to move rightward with the decreasing effective detuning Δc1. As a result, a suitable set of effective detuning Δc1 and coupling strength J can be chosen to make sure that the location of the right peak of the optical noise spectrum is at ω=ω?, which can greatly enhance the cooling process as much as possible (Fig. 4). Fig. 5(a) illustrates the optical noise spectrum SFFω as a function of ω/ω? for three different decay rates κ2. As shown in Fig. 5(a), the value of the optical noise spectrum SFFω at ω=ω? notably rises with the reducing κ2, which means that the decay rate decrease of the auxiliary optical cavity helps promote the cooling process. Meanwhile, SFFω goes down to zero at ω=-ω?, which indicates that the heating is completely suppressed whether the decay rate κ2 is changed or not. As exhibited in Fig. 6, the influence of the different optical coupling strengths J on the net cooling rate Γ is plotted. With the increasing coupling strength J, the net cooling rate Γ first rises to a maximum value and then decreases. Additionally, the net cooling rate Γ is significantly reinforced when the OPA medium is added. Subsequently, we investigate the final phonon number nf versus the coupling strength J with or without the OPA medium. With the increasing coupling strength J, the final phonon number nf will first decrease and then increase. Notably, as the coupling strength J rises, the final phonon number nf of RM drops to markedly less than 1 in the presence of an OPA medium (Fig. 7). Meanwhile, the final phonon number can be less than 1 by regulating the detuning of the auxiliary cavity (Fig. 8) and the decay rate of the cavity field (Fig. 9) respectively.ConclusionsWe propose an intracavity-squeezed cooling scheme to achieve a quantum ground state of RM in a double-Laguerre-Gaussian cavity optomechanical system comprising of an OPA medium. We demonstrate that the quantum backaction heating can be completely suppressed by adding OPA and the cooling efficiency is improved by coupling the auxiliary cavity. Further, the perfect cooling effect can be remarkably accomplished by selecting appropriate coupling strength, effective detuning, and decay rate, respectively. The restriction on the auxiliary cavity in the hybrid system is considerably loosened with the help of OPA. These results may have potential applications for achieving the quantum ground-state of mechanical resonators and greatly promote the study of various quantum phenomena in mechanical systems. ObjectiveCooling of mechanical oscillators is an important direction of cavity optomechanics research. Cooling the mechanical oscillators to their quantum ground state is a prerequisite for a wide range of applications based on cavity optomechanics. Therefore, ground-state cooling of mechanical oscillators is the focus of cavity optomechanics at present, which attracts a large number of scholars to study it. However, due to noise interference from external environments, the mechanical oscillators cannot enter the quantum regime. The hybrid system-coupled optical parametric amplifier provides a unique platform to solve the above problem.MethodsThe hybrid optomechanical system consists of two fixed mirrors FM including a rotational mirror RM mounted on the support S which can rotate around the Z axis, and an OPA medium. Cavity 1 which couples the OPA medium is made up of partially transparent FM1 and perfectly reflecting RM while cavity 2 is composed of FM1 and another perfectly reflecting FM2. The cavity 1 is driven by the transmitted beam with charge 0 and a Laguerre-Gauss beam G of charge 0 is incident on FM1. The charge 0 beam reflected from the RM is charged to +2l and then returns to FM1, where a mode with charge 0 is generated and enters cavity 2. After the reflection of FM2, it is also charged to +2l. We study the problem of the intracavity-squeezed cooling in the optical parametric amplifier coupled by a double Laguerre-Gaussian-cavity optomechanical system by calculating the optical force noise spectrum and the steady-state final phonon number. In the weak coupling regime, the optical force noise spectrum of the system is obtained by the perturbation approximation method, and the analytical expression of the final phonon number is calculated by the Fermi Golden Rule theory.Results and DiscussionsWhen the OPA medium is considered in the hybrid optomechanical system, the heating rate of the optical noise spectrum SFFω at ω=-ω? is reduced to 0, with an unaffected cooling rate. In other words, A+ drops while A- remains the same, the net cooling rate Γ=A--A+ naturally becomes larger, and the cooling effect is improved (Fig. 2). Next, we proceed to study how the optical noise spectrum SFFω is affected by the coupling strength J between two cavities. The value of SFFω at ω/ω?=1 is greater in the presence of the auxiliary cavity (Fig. 3). We depict the variations of the optical noise spectrum SFFω with ω/ω? for a given coupling strength J when Δc1=-ω?, Δc1=-2ω?, Δc1=-2.5ω?, and Δc1=-3ω?. The right-hand peak of the optical noise spectrum SFFω is observed to move rightward with the decreasing effective detuning Δc1. As a result, a suitable set of effective detuning Δc1 and coupling strength J can be chosen to make sure that the location of the right peak of the optical noise spectrum is at ω=ω?, which can greatly enhance the cooling process as much as possible (Fig. 4). Fig. 5(a) illustrates the optical noise spectrum SFFω as a function of ω/ω? for three different decay rates κ2. As shown in Fig. 5(a), the value of the optical noise spectrum SFFω at ω=ω? notably rises with the reducing κ2, which means that the decay rate decrease of the auxiliary optical cavity helps promote the cooling process. Meanwhile, SFFω goes down to zero at ω=-ω?, which indicates that the heating is completely suppressed whether the decay rate κ2 is changed or not. As exhibited in Fig. 6, the influence of the different optical coupling strengths J on the net cooling rate Γ is plotted. With the increasing coupling strength J, the net cooling rate Γ first rises to a maximum value and then decreases. Additionally, the net cooling rate Γ is significantly reinforced when the OPA medium is added. Subsequently, we investigate the final phonon number nf versus the coupling strength J with or without the OPA medium. With the increasing coupling strength J, the final phonon number nf will first decrease and then increase. Notably, as the coupling strength J rises, the final phonon number nf of RM drops to markedly less than 1 in the presence of an OPA medium (Fig. 7). Meanwhile, the final phonon number can be less than 1 by regulating the detuning of the auxiliary cavity (Fig. 8) and the decay rate of the cavity field (Fig. 9) respectively.ConclusionsWe propose an intracavity-squeezed cooling scheme to achieve a quantum ground state of RM in a double-Laguerre-Gaussian cavity optomechanical system comprising of an OPA medium. We demonstrate that the quantum backaction heating can be completely suppressed by adding OPA and the cooling efficiency is improved by coupling the auxiliary cavity. Further, the perfect cooling effect can be remarkably accomplished by selecting appropriate coupling strength, effective detuning, and decay rate, respectively. The restriction on the auxiliary cavity in the hybrid system is considerably loosened with the help of OPA. These results may have potential applications for achieving the quantum ground-state of mechanical resonators and greatly promote the study of various quantum phenomena in mechanical systems.
Acta Optica Sinica
- Publication Date: Feb. 10, 2024
- Vol. 44, Issue 3, 0327001 (2024)
Integrated Resonant Photodetector with High Signal-to-Noise Ratio
Wei Li, Zhixue Wu, Qingwei Wang, Ruixin Li, Qinghui Li, Mingjian Ju, Zichao Gao, Xin Shang, Long Tian, and Yaohui Zheng
ObjectiveIn squeezing-enhanced system, the stability and squeezing level of the squeezed states directly affect the improvement of quantum enhancement sensitivity and signal-to-noise ratio (SNR). Squeezed states can be generated by an optical parametric oscillator (OPO) based on second-order nonlinearity. At present, Pound-Drever-Hall (PDH) is the most commonly employed method for locking the OPO cavity, and the photodetector plays a key role in extracting extremely weak signals. For PDH locking systems, the useful signals coupled to photodetectors are narrowband signals at the modulated frequency, while the traditional wideband photodetectors amplify signals and noise in the whole frequency band, which is not conducive to improving the SNR. It is worth noting that the seed light is employed in the active stable parametric cavity in the preparation of the bright squeezed state, and the increased seed power will lead to the coupling of the pump noise into the bright squeezed state, thereby resulting in the reduced squeezing level. However, the increased optical power of extracting signal can improve the error signal of the locked parameter cavity. Thus, it is important to design photodetectors with high gain and SNR. Photodiodes have certain junction capacitance, and combined with variable inductance, inductance and capacitance (LC) resonance circuit can be formed to enhance the resonance of specific frequency signals. The detector is named resonant photodetector (RPD). The LC resonance circuit can be equivalent to the parallel resonance circuit and is regarded as a bandpass filter, which only amplifies the required frequency band and suppresses the noise of unnecessary frequency bands. However, the quality factor Q directly characterizes the suppression effect on the external noise signal of the resonant frequency, and the SNR of the error signal directly affects the minimum jitter of the cavity length and phase after locking.MethodsTo evaluate the newly designed RPD, this paper builds a test platform to evaluate transfer functions and error signals, as shown in Fig. 4. The laser source is a single-frequency laser of 1550 nm. The half-wave plate HWP1 is employed to adjust the laser power reaching the modulator, and HWP2 is to adjust the polarization direction of the laser, perpendicular or horizontal to the modulator. This means that the direction is 45°from the main axis of the electro-optical crystal. The network analyzer sets the start and end frequencies of the test (starting from 1-100 MHz in the experiment, and then being refined according to the resonance frequency). The output signal is divided into two parts, one of which is loaded on the modulator for modulating the laser beam, and the other is returned as a reference signal. The second part is the measurement of the error signal, which adopts the electro-optical modulator (EOM). MC is closely related to the preparation of high level squeezed state. According to PDH technology, the anti-interference ability of locking is proportional to the peak-to-peak value of the error signal, and the larger peak-to-peak value will lead to stronger anti-interference ability. In addition to the incident laser power, the amplitude and SNR of the error signal also depend on the signal extraction capability of the photodetector, so the SNR of the error signal extracted by the photodetector determines the stability of the whole feedback loop. Therefore, the performance of the developed resonant detector is evaluated by the SNR and the stability of the error signal in MC cavity locking.Results and DiscussionsThis paper measures the transfer functions of commercial BPD (THORLABS PDA10D2) and RPD under the same conditions (Fig. 5). At the resonant frequency of 20 MHz, the gain of RPD is about 30 dB higher than that of BPD. The high gain helps to obtain a stable phase locking at lower power, thus improving the stability of the system in the squeezed state without reducing the quantum noise. Through external mixing and integrated circuit design, the 3 dB bandwidth of RPD is 0.285 MHz. The quality factor Q of RPD is 70 and can be calculated from the measurement results by Formula (5). The experimental results are shown in Figs. 6 and 7. The SNR improvement of the newly designed RPD is more obvious than that of BPD, and the SNR is defined as the ratio of the peak-to-peak value to the noise of the error signal. The error signal is a DC signal, which cannot be measured by a spectrum analyzer and can only be recorded by an oscilloscope. At the resonance frequency of 20 MHz, the peak-to-peak value of the RPD error signal is 560 mV, the peak-to-peak value of noise is 42 mV, and the SNR is about 22.5 dB. The peak-to-peak value of the BPD error signal is 35 mV, and the peak-to-peak value of noise is 20.8 mV. The SNR of the newly designed RPD is about 18 dB higher than that of BPD.ConclusionsBased on the theoretical analysis of the resonance circuit and cross-resistance amplifier circuit, the selection of low noise devices, and the optimized circuit layout, this paper develops a resonant detector with Q factor of 70 and SNR of 22.5 dB. Compared with the traditional broadband photodetector (BPD), the gain of RPD at 20 MHz is about 30 dB higher than that of BPD. By measuring the peak-to-peak value and SNR of error signals, the peak-to-peak value of RPD locking cavity error signals is 16 times that of BPD, and the SNR of RPD error signals is about 18 dB higher than that of BPD under the same condition. This RPD can provide a key device for photoelectric feedback control and the preparation of continuous variable nonclassical light fields. ObjectiveIn squeezing-enhanced system, the stability and squeezing level of the squeezed states directly affect the improvement of quantum enhancement sensitivity and signal-to-noise ratio (SNR). Squeezed states can be generated by an optical parametric oscillator (OPO) based on second-order nonlinearity. At present, Pound-Drever-Hall (PDH) is the most commonly employed method for locking the OPO cavity, and the photodetector plays a key role in extracting extremely weak signals. For PDH locking systems, the useful signals coupled to photodetectors are narrowband signals at the modulated frequency, while the traditional wideband photodetectors amplify signals and noise in the whole frequency band, which is not conducive to improving the SNR. It is worth noting that the seed light is employed in the active stable parametric cavity in the preparation of the bright squeezed state, and the increased seed power will lead to the coupling of the pump noise into the bright squeezed state, thereby resulting in the reduced squeezing level. However, the increased optical power of extracting signal can improve the error signal of the locked parameter cavity. Thus, it is important to design photodetectors with high gain and SNR. Photodiodes have certain junction capacitance, and combined with variable inductance, inductance and capacitance (LC) resonance circuit can be formed to enhance the resonance of specific frequency signals. The detector is named resonant photodetector (RPD). The LC resonance circuit can be equivalent to the parallel resonance circuit and is regarded as a bandpass filter, which only amplifies the required frequency band and suppresses the noise of unnecessary frequency bands. However, the quality factor Q directly characterizes the suppression effect on the external noise signal of the resonant frequency, and the SNR of the error signal directly affects the minimum jitter of the cavity length and phase after locking.MethodsTo evaluate the newly designed RPD, this paper builds a test platform to evaluate transfer functions and error signals, as shown in Fig. 4. The laser source is a single-frequency laser of 1550 nm. The half-wave plate HWP1 is employed to adjust the laser power reaching the modulator, and HWP2 is to adjust the polarization direction of the laser, perpendicular or horizontal to the modulator. This means that the direction is 45°from the main axis of the electro-optical crystal. The network analyzer sets the start and end frequencies of the test (starting from 1-100 MHz in the experiment, and then being refined according to the resonance frequency). The output signal is divided into two parts, one of which is loaded on the modulator for modulating the laser beam, and the other is returned as a reference signal. The second part is the measurement of the error signal, which adopts the electro-optical modulator (EOM). MC is closely related to the preparation of high level squeezed state. According to PDH technology, the anti-interference ability of locking is proportional to the peak-to-peak value of the error signal, and the larger peak-to-peak value will lead to stronger anti-interference ability. In addition to the incident laser power, the amplitude and SNR of the error signal also depend on the signal extraction capability of the photodetector, so the SNR of the error signal extracted by the photodetector determines the stability of the whole feedback loop. Therefore, the performance of the developed resonant detector is evaluated by the SNR and the stability of the error signal in MC cavity locking.Results and DiscussionsThis paper measures the transfer functions of commercial BPD (THORLABS PDA10D2) and RPD under the same conditions (Fig. 5). At the resonant frequency of 20 MHz, the gain of RPD is about 30 dB higher than that of BPD. The high gain helps to obtain a stable phase locking at lower power, thus improving the stability of the system in the squeezed state without reducing the quantum noise. Through external mixing and integrated circuit design, the 3 dB bandwidth of RPD is 0.285 MHz. The quality factor Q of RPD is 70 and can be calculated from the measurement results by Formula (5). The experimental results are shown in Figs. 6 and 7. The SNR improvement of the newly designed RPD is more obvious than that of BPD, and the SNR is defined as the ratio of the peak-to-peak value to the noise of the error signal. The error signal is a DC signal, which cannot be measured by a spectrum analyzer and can only be recorded by an oscilloscope. At the resonance frequency of 20 MHz, the peak-to-peak value of the RPD error signal is 560 mV, the peak-to-peak value of noise is 42 mV, and the SNR is about 22.5 dB. The peak-to-peak value of the BPD error signal is 35 mV, and the peak-to-peak value of noise is 20.8 mV. The SNR of the newly designed RPD is about 18 dB higher than that of BPD.ConclusionsBased on the theoretical analysis of the resonance circuit and cross-resistance amplifier circuit, the selection of low noise devices, and the optimized circuit layout, this paper develops a resonant detector with Q factor of 70 and SNR of 22.5 dB. Compared with the traditional broadband photodetector (BPD), the gain of RPD at 20 MHz is about 30 dB higher than that of BPD. By measuring the peak-to-peak value and SNR of error signals, the peak-to-peak value of RPD locking cavity error signals is 16 times that of BPD, and the SNR of RPD error signals is about 18 dB higher than that of BPD under the same condition. This RPD can provide a key device for photoelectric feedback control and the preparation of continuous variable nonclassical light fields.
Acta Optica Sinica
- Publication Date: Apr. 10, 2023
- Vol. 43, Issue 7, 0727001 (2023)
E-Payment Protocols Based on Quantum Walk
Yefeng He, Mengmei Yang, Zhi Li, Yan Liu, and Zheming Tian
ObjectiveIn recent years, with the rapid development of e-commerce and computer, online shopping is more and more popular. In the meantime, the development of quantum algorithms makes the traditional e-payment protocols based on difficult mathematical problems more and more insecure, so the e-payment protocols based on quantum algorithms come into being. At present, most of the proposed quantum e-payment protocols use entangled states for quantum electronic signature protocols. However, the preparation and measurement of entangled states are very difficult, so in the case of ensuring the security of the protocols, using quantum states featuring more convenient preparation and measurement, instead of entangled states, has become a research direction of e-payment. Quantum walk is a technology that can produce the necessary entanglement resources spontaneously only by using the single-particle states without preparing entanglement resources in advance. This technology has been widely used in quantum computing and quantum simulation and is of certain practical value. At the same time, as people pay more attention to personal privacy, only users' shopping lists being confidential to banks have been unable to meet people's privacy needs. Therefore, in order to solve the above problems, we modify a classic e-payment agreement model and make the hidden users' identity information not affect the normal delivery and merchants. Furthermore, we combine the quantum walk with quantum e-payment and propose a quantum e-payment protocol to ensure that entanglement resources can be obtained without preparing entangled states in advance and guarantee that the buyers' bank accounts and real identity information can be kept confidential to merchants.MethodsQuantum walk is an extension of random walk in the quantum field. It takes the quantum as the carrier to simulate the chaotic nonlinear dynamic walk behavior. According to the characteristics of the quantum walk, the encrypted quantum communication channels are established accordingly. The quantum walk mainly contains the complex Hilbert space of two main quantum spaces, namely, coin space and position space. The protocol is based on the one-dimensional quantum walk teleportation, and through the two-step quantum walk, the shift operator can make the position space and coin space entangle with each other, so as to construct quantum channels for information transmission. The biggest advantage of quantum walk technology is that it can obtain the entanglement resources through the single photon operation. The measurement and preparation of the protocol are simpler compared with directly operating entangled states, and the randomness of the quantum walk makes the transmission more secure. In addition, by dividing the shopping information of buyers into the identity information accessible to the banks and making the shopping list open to the merchants, the banks and the merchants in the protocol will not know the information obtained by the other party so that the privacy of the buyers is greatly protected.Results and DiscussionsFirstly, in order to further protect the users' privacy, we modify a classic electronic payment agreement model (Fig. 1). In this model, we reduce the workload of buyers and give the processing and distribution of information to third-party platforms. While keeping the buyers' shopping lists confidential to the banks, the buyers' identity information is also unavailable to the merchants. So this protocol makes the merchants and the banks only have the information that they need and know nothing of the information obtained by the other party. Secondly, quantum walk technology is applied to various stages of the protocol including the trading purchase phase, trading payment phase, and verifying phase (Fig. 2). By applying quantum walk technology, the complexity of quantum resource preparation and measurement in the protocol is reduced. Finally, the security analysis of this protocol is conducted (Fig. 3), and the result shows that neither the internal nor external attackers of this protocol can obtain the secret information in the protocol, and this protocol can resist both internal and external attacks.ConclusionsThis protocol, compared with the existing quantum e-payment protocols, not only retains the third-party platforms and the inter-bank payment function but also combines the quantum walk and electronic payment. It makes the participants of the protocol in the initial stage free from preparing particles in entangled states and makes them only prepare the single-particle states which can get the required entanglement resources. This move reduces the complexity of quantum state preparation and measurement. At the same time, the shopping information of the buyers is divided, with the information of the purchased goods confidential to the banks and the buyers' private information unavailable to the merchants. In addition, when the merchants really need this part of the information of the buyers, they can apply to third-party platforms for the information. Being reviewed by the platforms and approved by the buyers, the merchants can know the information they want, so as to complete the corresponding operation. Finally, security analysis shows that this protocol can resist internal and external attacks and is safe and feasible under current technology. ObjectiveIn recent years, with the rapid development of e-commerce and computer, online shopping is more and more popular. In the meantime, the development of quantum algorithms makes the traditional e-payment protocols based on difficult mathematical problems more and more insecure, so the e-payment protocols based on quantum algorithms come into being. At present, most of the proposed quantum e-payment protocols use entangled states for quantum electronic signature protocols. However, the preparation and measurement of entangled states are very difficult, so in the case of ensuring the security of the protocols, using quantum states featuring more convenient preparation and measurement, instead of entangled states, has become a research direction of e-payment. Quantum walk is a technology that can produce the necessary entanglement resources spontaneously only by using the single-particle states without preparing entanglement resources in advance. This technology has been widely used in quantum computing and quantum simulation and is of certain practical value. At the same time, as people pay more attention to personal privacy, only users' shopping lists being confidential to banks have been unable to meet people's privacy needs. Therefore, in order to solve the above problems, we modify a classic e-payment agreement model and make the hidden users' identity information not affect the normal delivery and merchants. Furthermore, we combine the quantum walk with quantum e-payment and propose a quantum e-payment protocol to ensure that entanglement resources can be obtained without preparing entangled states in advance and guarantee that the buyers' bank accounts and real identity information can be kept confidential to merchants.MethodsQuantum walk is an extension of random walk in the quantum field. It takes the quantum as the carrier to simulate the chaotic nonlinear dynamic walk behavior. According to the characteristics of the quantum walk, the encrypted quantum communication channels are established accordingly. The quantum walk mainly contains the complex Hilbert space of two main quantum spaces, namely, coin space and position space. The protocol is based on the one-dimensional quantum walk teleportation, and through the two-step quantum walk, the shift operator can make the position space and coin space entangle with each other, so as to construct quantum channels for information transmission. The biggest advantage of quantum walk technology is that it can obtain the entanglement resources through the single photon operation. The measurement and preparation of the protocol are simpler compared with directly operating entangled states, and the randomness of the quantum walk makes the transmission more secure. In addition, by dividing the shopping information of buyers into the identity information accessible to the banks and making the shopping list open to the merchants, the banks and the merchants in the protocol will not know the information obtained by the other party so that the privacy of the buyers is greatly protected.Results and DiscussionsFirstly, in order to further protect the users' privacy, we modify a classic electronic payment agreement model (Fig. 1). In this model, we reduce the workload of buyers and give the processing and distribution of information to third-party platforms. While keeping the buyers' shopping lists confidential to the banks, the buyers' identity information is also unavailable to the merchants. So this protocol makes the merchants and the banks only have the information that they need and know nothing of the information obtained by the other party. Secondly, quantum walk technology is applied to various stages of the protocol including the trading purchase phase, trading payment phase, and verifying phase (Fig. 2). By applying quantum walk technology, the complexity of quantum resource preparation and measurement in the protocol is reduced. Finally, the security analysis of this protocol is conducted (Fig. 3), and the result shows that neither the internal nor external attackers of this protocol can obtain the secret information in the protocol, and this protocol can resist both internal and external attacks.ConclusionsThis protocol, compared with the existing quantum e-payment protocols, not only retains the third-party platforms and the inter-bank payment function but also combines the quantum walk and electronic payment. It makes the participants of the protocol in the initial stage free from preparing particles in entangled states and makes them only prepare the single-particle states which can get the required entanglement resources. This move reduces the complexity of quantum state preparation and measurement. At the same time, the shopping information of the buyers is divided, with the information of the purchased goods confidential to the banks and the buyers' private information unavailable to the merchants. In addition, when the merchants really need this part of the information of the buyers, they can apply to third-party platforms for the information. Being reviewed by the platforms and approved by the buyers, the merchants can know the information they want, so as to complete the corresponding operation. Finally, security analysis shows that this protocol can resist internal and external attacks and is safe and feasible under current technology.
Acta Optica Sinica
- Publication Date: Mar. 10, 2023
- Vol. 43, Issue 5, 0527001 (2023)
Navigation and Ranging Scheme Based on Hybrid Entangled Quantum Signals
Zhaoheng Ren, Qiang Miao, Dewei Wu, and Tianli Wei
Results and Discussions Combined with the principle and scheme of quantum lighting, by modeling the ranging task as a multi-target hypothesis testing problem, the limitation of binary hypothesis testing in quantum lighting is broken (Fig. 2). The asymptotic performance of the classical ranging scheme and the entangled ranging scheme is discussed and analyzed, and simulations are carried out. The entanglement upper limit (dotted line) proves the advantages of entangled ranging, as well as the scale advantage in the error index, while the progressive performance (dotted solid line) further demonstrates the advantages of entangled ranging. The error index of the QCB simulation results ( dotted solid line in the entangled case and dotted dashed line in the classical case) shows a 6 dB (4-fold) advantage brought by entanglement, which is consistent with the theoretical analysis (Fig. 3). The relationship between the error probability measurement performance of different ranging schemes and the distance accuracy and target reflectivity is simulated and analyzed. When the target reflectivity is constant, there is an inverse relationship between the error probability performance of the entangled ranging scheme and the ranging accuracy. When the error probability is high, the ranging accuracy is low, and when the error probability is low, the ranging accuracy is high. Besides, the solution proposed in this paper is better than the classical ranging solution in terms of ranging accuracy.ObjectiveSince the birth of navigation, it has had a huge impact on human life. At present, various navigation and positioning methods based on physical foundations such as sound, light, electricity, magnetism, and force have emerged one after another. Among them, the systems based on radio navigation technology for navigation and detection are the most common, radio navigation is still the main means used in the field of military and civil aviation navigation. Navigation is the process of guiding the safe navigation of the operating body. The main tasks include ranging, angle measurement and positioning, etc. These tasks are premised on obtaining the required navigation parameters. The navigation parameters are mainly divided into four types, namely position, angle, distance and speed. The position is a space-time parameter, including the time and space information of the running body. With the progress of the times and the growth of human needs, although the traditional navigation detection method is still the mainstream application in related fields, the detection accuracy is limited. Besides, it is easy to be interfered, the long-distance weak signal detection ability is not strong, and the safety performance cannot be effectively guaranteed. It has become increasingly prominent that traditional radio navigation methods will gradually fail to meet human needs for navigation. It is of great significance to study new navigation ranging solutions combined with quantum lighting to solve the above problems. Meanwhile, quantum illumination can only interrogate the presence of a target in one polarization-azimuth-elevation-range-Doppler-resolution area array at a time. It sends a signal to a narrow area and judges the presence of a target moving at a fixed speed at a fixed time, while practical navigation systems usually estimate the target's polarization properties, azimuth and elevation, distance and velocity (via Doppler effect). Therefore, it is of great significance to study new solutions to solve this problem and give full play to its advantages.MethodsIn this paper, combined with the method and principle of quantum lighting, the cavity electro-optical force converter is used to solve the problem of signal detection, and the energy and quantum state transfer between light waves and microwaves are realized, so that the advantages of the two complement each other. The model is transformed into a multi-objective hypothesis testing problem, and the advantages of the proposed scheme are verified by simulating the target identification error probability index and ranging accuracy of the classical ranging scheme and the proposed scheme.ConclusionsIn this paper, by modeling the ranging task as a multi-target hypothesis testing problem, a quantum entanglement-based navigation ranging scheme is proposed. The principle and model of the ranging scheme are expounded, and the progressive performance of the classical ranging scheme and entanglement measuring scheme is analyzed. On this basis, a quantitative analysis of the common parameters of different ranging schemes is carried out, the ranging performance is compared, and the relationship between ranging accuracy and ranging error probability is analyzed. The proposed scheme outperforms the classical scheme, providing a 6 dB advantage in determining the error exponent for any number of possible ranges. In addition, this ranging scheme can also be used to realize entanglement-assisted communication of pulse position modulation, and provides an application direction for quantum ranging radar with quantum advantages. Results and Discussions Combined with the principle and scheme of quantum lighting, by modeling the ranging task as a multi-target hypothesis testing problem, the limitation of binary hypothesis testing in quantum lighting is broken (Fig. 2). The asymptotic performance of the classical ranging scheme and the entangled ranging scheme is discussed and analyzed, and simulations are carried out. The entanglement upper limit (dotted line) proves the advantages of entangled ranging, as well as the scale advantage in the error index, while the progressive performance (dotted solid line) further demonstrates the advantages of entangled ranging. The error index of the QCB simulation results ( dotted solid line in the entangled case and dotted dashed line in the classical case) shows a 6 dB (4-fold) advantage brought by entanglement, which is consistent with the theoretical analysis (Fig. 3). The relationship between the error probability measurement performance of different ranging schemes and the distance accuracy and target reflectivity is simulated and analyzed. When the target reflectivity is constant, there is an inverse relationship between the error probability performance of the entangled ranging scheme and the ranging accuracy. When the error probability is high, the ranging accuracy is low, and when the error probability is low, the ranging accuracy is high. Besides, the solution proposed in this paper is better than the classical ranging solution in terms of ranging accuracy.ObjectiveSince the birth of navigation, it has had a huge impact on human life. At present, various navigation and positioning methods based on physical foundations such as sound, light, electricity, magnetism, and force have emerged one after another. Among them, the systems based on radio navigation technology for navigation and detection are the most common, radio navigation is still the main means used in the field of military and civil aviation navigation. Navigation is the process of guiding the safe navigation of the operating body. The main tasks include ranging, angle measurement and positioning, etc. These tasks are premised on obtaining the required navigation parameters. The navigation parameters are mainly divided into four types, namely position, angle, distance and speed. The position is a space-time parameter, including the time and space information of the running body. With the progress of the times and the growth of human needs, although the traditional navigation detection method is still the mainstream application in related fields, the detection accuracy is limited. Besides, it is easy to be interfered, the long-distance weak signal detection ability is not strong, and the safety performance cannot be effectively guaranteed. It has become increasingly prominent that traditional radio navigation methods will gradually fail to meet human needs for navigation. It is of great significance to study new navigation ranging solutions combined with quantum lighting to solve the above problems. Meanwhile, quantum illumination can only interrogate the presence of a target in one polarization-azimuth-elevation-range-Doppler-resolution area array at a time. It sends a signal to a narrow area and judges the presence of a target moving at a fixed speed at a fixed time, while practical navigation systems usually estimate the target's polarization properties, azimuth and elevation, distance and velocity (via Doppler effect). Therefore, it is of great significance to study new solutions to solve this problem and give full play to its advantages.MethodsIn this paper, combined with the method and principle of quantum lighting, the cavity electro-optical force converter is used to solve the problem of signal detection, and the energy and quantum state transfer between light waves and microwaves are realized, so that the advantages of the two complement each other. The model is transformed into a multi-objective hypothesis testing problem, and the advantages of the proposed scheme are verified by simulating the target identification error probability index and ranging accuracy of the classical ranging scheme and the proposed scheme.ConclusionsIn this paper, by modeling the ranging task as a multi-target hypothesis testing problem, a quantum entanglement-based navigation ranging scheme is proposed. The principle and model of the ranging scheme are expounded, and the progressive performance of the classical ranging scheme and entanglement measuring scheme is analyzed. On this basis, a quantitative analysis of the common parameters of different ranging schemes is carried out, the ranging performance is compared, and the relationship between ranging accuracy and ranging error probability is analyzed. The proposed scheme outperforms the classical scheme, providing a 6 dB advantage in determining the error exponent for any number of possible ranges. In addition, this ranging scheme can also be used to realize entanglement-assisted communication of pulse position modulation, and provides an application direction for quantum ranging radar with quantum advantages.
Acta Optica Sinica
- Publication Date: Feb. 25, 2023
- Vol. 43, Issue 4, 0427002 (2023)
Quantum-Enhancement Microscopy with Six-Photon Twin-Fock State
Xiaoju Ren, Huili Zheng, ZeZhun Shi, and GuangRi Jin
Results and Discussions With a large enough repeated binary-outcome photon counting measurement, it is shown that the likelihood function can be well approximated by a Gaussian function [Figs. 2 (c) and 2 (d)], where its peak determines the MLE. To confirm it, we analytically derive the approximate results of the likelihood function and the MLE [Eqs. (12)-(19)], which shows that the MLE can saturate the CRB asymptotically. The above results also hold for a combination of two binary-outcome measurements with and without an offset phase shift [Figs. 4 (b)-(e) and Eqs. (23)-(32)]. For the six-photon twin-Fock state, the divergence of the phase sensitivity at a certain phase shift can be removed by comparing Fig. 3(a) and Fig. 4(f). Therefore, the microscopy imaging with a combination of two binary-outcome measurements can avoid the imaging speckles [Fig. 3 (c) and Fig. 5 (a)]. The overall quality of the imaging in Fig. 5 (a), quantified by the root-mean-square error of the MLE, outperforms that of classical light illumination by a factor of 1.82, approaching to its theoretical prediction.ObjectiveQuantum multiphoton microscopy utilizes quantum correlation effects of photons to improve the imaging quality of biological samples at low light illumination. Based on a N-photon NOON state, the microscopy imaging has been successfully demonstrated in recent two experiments, which shows the imaging quality better than that of coherent light illumination by a factor of N (N=2, 3). However, the NOON states are difficult to prepare and are easily subject to the loss-induced decoherence. Furthermore, the microscopy imaging shows speckles within a local region, due to the divergence of the phase sensitivity. The twin-Fock states of light are believed to be more robust to the decoherence. For a binary-outcome photon counting measurement, it has been shown that a better phase sensitivity can be obtained in a comparison with that of the NOON states. Therefore, it is interesting to investigate the super-sensitive microscopy using the N-photon twin-Fock states of light. Recently, it is shown that the visibility of the six-photon count rate can reach 94%, which is significantly better than that of the five-photon NOON state (42%). Here, we investigate quantum-enhanced microscopy illuminated by the twin-Fock state of the light. With a combination of two binary-outcome measurements with and without an offset phase shift, it is shown that the divergence of the phase sensitivity at certain phase shifts can be removed, which avoids the imaging speckles. We hope our observations can be helpful on the quantum-enhancement microscopy with the large-N twin-Fock states.MethodsA binary-outcome photon counting measurement is employed in present work, where the detection event with equal number of photons is a measurement outcome. All the other detection events are treated as another outcome. Starting from general principle of quantum metrology, we first calculate the Fisher information and the Cramer-Rao lower bound (CRB) of the phase sensitivity, which determine the enhancement factor of the imaging quality for the N-photon twin-Fock states. Then, we derive the phase distribution (the likelihood function) and the maximum likelihood estimator (MLE) by considering the binary-outcome measurements. Using Monte Carlo method, we simulate the measurement probabilities of the six-photon twin-Fock state and the single-photon state, where the experimental imperfection is added artificially. The microscopy imaging is reconstructed using numerical result of the MLE. Finally, we derive the likelihood function and show the microscopy imaging for a combination of two binary-outcome measurements with and without an offset phase shift.ConclusionsRegardless of the specific model, we first prove analytically that the likelihood functions of single and two groups of binary-outcome photon counting measurements can approximate a Gaussian function, the maximum likelihood estimator is asymptotically unbiased which can saturate the lower limit of phase measurement of the above two measurement schemes. Based on the six-photon twin-Fock state, this paper studies the maximum likelihood estimator and phase sensitivity of the binary-outcome photon counting measurements, and reconstructs the two-dimensional microscopy imaging of the birefringent sample with the MLE. Our results show that a combination of binary-outcome photon counting measurements can avoid the divergence of phase sensitivity at dark spots, thus overcoming the speckle problem of microscopy imaging. The maximum likelihood estimator at each pixel in the reconstructed image is close to the optimal phase working point, and the overall quality factor of the image is measured by the root-mean-square error of the estimator. Results and Discussions With a large enough repeated binary-outcome photon counting measurement, it is shown that the likelihood function can be well approximated by a Gaussian function [Figs. 2 (c) and 2 (d)], where its peak determines the MLE. To confirm it, we analytically derive the approximate results of the likelihood function and the MLE [Eqs. (12)-(19)], which shows that the MLE can saturate the CRB asymptotically. The above results also hold for a combination of two binary-outcome measurements with and without an offset phase shift [Figs. 4 (b)-(e) and Eqs. (23)-(32)]. For the six-photon twin-Fock state, the divergence of the phase sensitivity at a certain phase shift can be removed by comparing Fig. 3(a) and Fig. 4(f). Therefore, the microscopy imaging with a combination of two binary-outcome measurements can avoid the imaging speckles [Fig. 3 (c) and Fig. 5 (a)]. The overall quality of the imaging in Fig. 5 (a), quantified by the root-mean-square error of the MLE, outperforms that of classical light illumination by a factor of 1.82, approaching to its theoretical prediction.ObjectiveQuantum multiphoton microscopy utilizes quantum correlation effects of photons to improve the imaging quality of biological samples at low light illumination. Based on a N-photon NOON state, the microscopy imaging has been successfully demonstrated in recent two experiments, which shows the imaging quality better than that of coherent light illumination by a factor of N (N=2, 3). However, the NOON states are difficult to prepare and are easily subject to the loss-induced decoherence. Furthermore, the microscopy imaging shows speckles within a local region, due to the divergence of the phase sensitivity. The twin-Fock states of light are believed to be more robust to the decoherence. For a binary-outcome photon counting measurement, it has been shown that a better phase sensitivity can be obtained in a comparison with that of the NOON states. Therefore, it is interesting to investigate the super-sensitive microscopy using the N-photon twin-Fock states of light. Recently, it is shown that the visibility of the six-photon count rate can reach 94%, which is significantly better than that of the five-photon NOON state (42%). Here, we investigate quantum-enhanced microscopy illuminated by the twin-Fock state of the light. With a combination of two binary-outcome measurements with and without an offset phase shift, it is shown that the divergence of the phase sensitivity at certain phase shifts can be removed, which avoids the imaging speckles. We hope our observations can be helpful on the quantum-enhancement microscopy with the large-N twin-Fock states.MethodsA binary-outcome photon counting measurement is employed in present work, where the detection event with equal number of photons is a measurement outcome. All the other detection events are treated as another outcome. Starting from general principle of quantum metrology, we first calculate the Fisher information and the Cramer-Rao lower bound (CRB) of the phase sensitivity, which determine the enhancement factor of the imaging quality for the N-photon twin-Fock states. Then, we derive the phase distribution (the likelihood function) and the maximum likelihood estimator (MLE) by considering the binary-outcome measurements. Using Monte Carlo method, we simulate the measurement probabilities of the six-photon twin-Fock state and the single-photon state, where the experimental imperfection is added artificially. The microscopy imaging is reconstructed using numerical result of the MLE. Finally, we derive the likelihood function and show the microscopy imaging for a combination of two binary-outcome measurements with and without an offset phase shift.ConclusionsRegardless of the specific model, we first prove analytically that the likelihood functions of single and two groups of binary-outcome photon counting measurements can approximate a Gaussian function, the maximum likelihood estimator is asymptotically unbiased which can saturate the lower limit of phase measurement of the above two measurement schemes. Based on the six-photon twin-Fock state, this paper studies the maximum likelihood estimator and phase sensitivity of the binary-outcome photon counting measurements, and reconstructs the two-dimensional microscopy imaging of the birefringent sample with the MLE. Our results show that a combination of binary-outcome photon counting measurements can avoid the divergence of phase sensitivity at dark spots, thus overcoming the speckle problem of microscopy imaging. The maximum likelihood estimator at each pixel in the reconstructed image is close to the optimal phase working point, and the overall quality factor of the image is measured by the root-mean-square error of the estimator.
Acta Optica Sinica
- Publication Date: Feb. 25, 2023
- Vol. 43, Issue 4, 0427001 (2023)
Multi-User Switching Strategy of Evolutionary Game-Based Low-Orbit Quantum Satellite Under Snowfall Disturbance
Changchun Xue, Min Nie, Guang Yang, Meiling Zhang, Aijing Sun, and Changxing Pei
ObjectiveLow-orbit quantum satellites are part of building a global secure communication network. However, as single quantum satellites move fast relative to ground terminals with limited service time and the satellite-ground quantum link is susceptible to atmospheric conditions (e.g. rain, snow, haze, etc.), ground end-users need to switch to other satellites available for service in time to meet the sustainable communication demands. In the common coverage area, if the user only chooses the currently proposed single-attribute decision strategy, such as the minimum communication elevation angle, the optimal entanglement degree or the minimum link attenuation, the optimal single attribute can be achieved with losing the advantage of other attributes. This will easily result in load imbalance and uneven resource allocation of quantum satellites, and even communications may be interrupted in serious cases. To this end, we consider the attenuation interference of snowfall on the satellite-ground link and the process of end-user-associated switching of quantum satellites, and build a multi-attribute evolutionary game switching model to achieve Nash equilibrium in the allocation of quantum satellite resources. Meanwhile, satellite resource allocation can be maximized to enhance the switching success rate of users in the low earth orbit (LEO) quantum satellite communication network.MethodsEvolutionary game theory combines biological evolutionary properties with game theory to make the system stable through constant comparison and imitation in multiple choices. In actual quantum satellite switching, the users switching satellites at the same time do not know each other's state information in a completely rational way, which satisfies the non-rational conditions of evolutionary games. We analyze the quantum channel attenuation characteristics under snowfall and atmospheric turbulence according to the Gamma spectral distribution function of snow and obtain the variation of channel attenuation with transmission distance. The bandwidth that can be allocated to users by the quantum satellite corresponds to the current satellite load state, which means that more users result in fewer quantum satellite bandwidth resources available to each user. The longer remaining service time of the quantum satellite indicates that the selection of this strategy prolongs the service cycle of the user and reduces the user switching number. The communication elevation angle reflects the channel condition of the satellite-ground link, and the larger communication elevation angle leads to shorter communication distance, lower link attenuation, and better channel conditions. As the elevation angle is difficult to measure in real time with the terrain environment obstruction, the measured elevation angle cannot reflect the channel conditions, and thus the communication elevation angle is converted into the link attenuation characteristics which can directly represent the channel conditions. Therefore, we define a utility function based on the user's bandwidth, remaining satellite service time, and link attenuation, and define an overhead function based on the inter-satellite transmission delay and channel entanglement to obtain the user's payoff function. Finally, we derive the dynamic replication equation of the satellite to build an evolutionary game switching model.Results and DiscussionsFirstly, the effect of snowfall on the satellite-ground link attenuation is analyzed. Under certain snowfall intensity, the total attenuation suffered by the link increases as the light quantum propagation distance rises. In the case of a certain propagation distance of the light quantum signal, as the snowfall intensity increases, the total communication link attenuation grows due to the scattering and absorption effect between the light quantum signal and snow particles in the atmosphere. It results in the subsequent increase in the total communication link attenuation, and the atmospheric snowfall environment will exert a significant influence on quantum satellite communications (Fig. 1). The number of users to be switched simultaneously is 1000 and the number of available strategies is 2. In the six switching experiments, the number of users selected for Quantum LEO1 varies with the iteration number (Fig. 3), and the number of users plateaus with the increasing iteration number, which demonstrates that the proposed quantum satellite multi-user switching strategy has sound convergence stability. As the game proceeds, the gains gradually converge to the average gain of all users and reach equilibrium by the sixth iteration. Then the users revenue do not increase and stabilize to ensure the multi-user fairness during the quantum satellite switching (Fig. 4). The single-attribute judgment switching method converges faster than the multi-attribute switching decision, and the number of users connected to Quantum LEO1 increases when equilibrium is finally reached, but this is at the expense of the other attributes (Fig. 5). Experimental results show that the switching strategy based on the evolutionary game improves the switching success rate by 1.2% over the switching strategy based on the lowest link attenuation (Fig. 6) under the switching user number of 660. When the minimum entanglement threshold is set to 0.8, the switching success rate of the evolutionary game-based switching strategy improves by 1.5% over the switching strategy based on the optimal entanglement (Fig. 7) under the switching user number of 700.ConclusionsAn evolutionary game-based quantum satellite switching strategy is proposed for the multi-user switching scenario of quantum satellites under a snowfall environment. Various attributes affecting the quantum satellite switching decision are analyzed and combined with the attenuation characteristics between quantum star-ground links to obtain the effect function, the overhead function, and then the average gain function of users. An evolutionary game model is built by considering the transmission among users and between users and quantum satellites, and the performance of the switching strategy using this model is simulated. The results show that the proposed strategy not only has sound stability with the influence among multiple attributes considered but also can make the quantum satellite load relatively balanced. Finally, compared with the single-attribute quantum satellite switching strategy based on minimum link attenuation and optimal entanglement, the proposed strategy can also improve the success rate of user switching more effectively, providing references for future multi-user dynamic switching design of low-orbit quantum satellite networks under snowfall interference environment. ObjectiveLow-orbit quantum satellites are part of building a global secure communication network. However, as single quantum satellites move fast relative to ground terminals with limited service time and the satellite-ground quantum link is susceptible to atmospheric conditions (e.g. rain, snow, haze, etc.), ground end-users need to switch to other satellites available for service in time to meet the sustainable communication demands. In the common coverage area, if the user only chooses the currently proposed single-attribute decision strategy, such as the minimum communication elevation angle, the optimal entanglement degree or the minimum link attenuation, the optimal single attribute can be achieved with losing the advantage of other attributes. This will easily result in load imbalance and uneven resource allocation of quantum satellites, and even communications may be interrupted in serious cases. To this end, we consider the attenuation interference of snowfall on the satellite-ground link and the process of end-user-associated switching of quantum satellites, and build a multi-attribute evolutionary game switching model to achieve Nash equilibrium in the allocation of quantum satellite resources. Meanwhile, satellite resource allocation can be maximized to enhance the switching success rate of users in the low earth orbit (LEO) quantum satellite communication network.MethodsEvolutionary game theory combines biological evolutionary properties with game theory to make the system stable through constant comparison and imitation in multiple choices. In actual quantum satellite switching, the users switching satellites at the same time do not know each other's state information in a completely rational way, which satisfies the non-rational conditions of evolutionary games. We analyze the quantum channel attenuation characteristics under snowfall and atmospheric turbulence according to the Gamma spectral distribution function of snow and obtain the variation of channel attenuation with transmission distance. The bandwidth that can be allocated to users by the quantum satellite corresponds to the current satellite load state, which means that more users result in fewer quantum satellite bandwidth resources available to each user. The longer remaining service time of the quantum satellite indicates that the selection of this strategy prolongs the service cycle of the user and reduces the user switching number. The communication elevation angle reflects the channel condition of the satellite-ground link, and the larger communication elevation angle leads to shorter communication distance, lower link attenuation, and better channel conditions. As the elevation angle is difficult to measure in real time with the terrain environment obstruction, the measured elevation angle cannot reflect the channel conditions, and thus the communication elevation angle is converted into the link attenuation characteristics which can directly represent the channel conditions. Therefore, we define a utility function based on the user's bandwidth, remaining satellite service time, and link attenuation, and define an overhead function based on the inter-satellite transmission delay and channel entanglement to obtain the user's payoff function. Finally, we derive the dynamic replication equation of the satellite to build an evolutionary game switching model.Results and DiscussionsFirstly, the effect of snowfall on the satellite-ground link attenuation is analyzed. Under certain snowfall intensity, the total attenuation suffered by the link increases as the light quantum propagation distance rises. In the case of a certain propagation distance of the light quantum signal, as the snowfall intensity increases, the total communication link attenuation grows due to the scattering and absorption effect between the light quantum signal and snow particles in the atmosphere. It results in the subsequent increase in the total communication link attenuation, and the atmospheric snowfall environment will exert a significant influence on quantum satellite communications (Fig. 1). The number of users to be switched simultaneously is 1000 and the number of available strategies is 2. In the six switching experiments, the number of users selected for Quantum LEO1 varies with the iteration number (Fig. 3), and the number of users plateaus with the increasing iteration number, which demonstrates that the proposed quantum satellite multi-user switching strategy has sound convergence stability. As the game proceeds, the gains gradually converge to the average gain of all users and reach equilibrium by the sixth iteration. Then the users revenue do not increase and stabilize to ensure the multi-user fairness during the quantum satellite switching (Fig. 4). The single-attribute judgment switching method converges faster than the multi-attribute switching decision, and the number of users connected to Quantum LEO1 increases when equilibrium is finally reached, but this is at the expense of the other attributes (Fig. 5). Experimental results show that the switching strategy based on the evolutionary game improves the switching success rate by 1.2% over the switching strategy based on the lowest link attenuation (Fig. 6) under the switching user number of 660. When the minimum entanglement threshold is set to 0.8, the switching success rate of the evolutionary game-based switching strategy improves by 1.5% over the switching strategy based on the optimal entanglement (Fig. 7) under the switching user number of 700.ConclusionsAn evolutionary game-based quantum satellite switching strategy is proposed for the multi-user switching scenario of quantum satellites under a snowfall environment. Various attributes affecting the quantum satellite switching decision are analyzed and combined with the attenuation characteristics between quantum star-ground links to obtain the effect function, the overhead function, and then the average gain function of users. An evolutionary game model is built by considering the transmission among users and between users and quantum satellites, and the performance of the switching strategy using this model is simulated. The results show that the proposed strategy not only has sound stability with the influence among multiple attributes considered but also can make the quantum satellite load relatively balanced. Finally, compared with the single-attribute quantum satellite switching strategy based on minimum link attenuation and optimal entanglement, the proposed strategy can also improve the success rate of user switching more effectively, providing references for future multi-user dynamic switching design of low-orbit quantum satellite networks under snowfall interference environment.
Acta Optica Sinica
- Publication Date: Dec. 25, 2023
- Vol. 43, Issue 24, 2427001 (2023)
Topics
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