Method for enhancing microwave and light entanglement performance through mechanical auxiliary cavity electro-optical force converter

Abstract

A method for enhancing microwave and light entanglement performance by a mechanical auxiliary cavity electro-optical force converter can be applied to improving microwave-light entanglement and microwave quantum illumination performance, especially when an auxiliary mechanical oscillator and a microwave cavity are in parametric down-conversion type interaction. The microwave-optical reduction entanglement degree and the signal-to-noise ratio enhancement index are obviously improved, the system ratio stability is higher under the same condition, the driving power tunable range of the microwave cavity is enlarged, and the system availability is improved; compared with the prior art, the influence of light-auxiliary mechanical interaction on microwave-light entanglement is small, and the influence on the microwave quantum illumination performance is not obvious in a feasible parameter range.

Description

[Technical field]

[0001] The present invention relates to a method for enhancing microwave and light conversion entanglement performance, and in particular to a method for enhancing microwave and light entanglement performance by a mechanically assisted cavity electro-optical force converter. [Background technology]

[0002] With the continuous development of cavity electro-optical mechanics, microwave-optical conversion experiments have also made major breakthroughs based on this. In particular, Higginbotham et al. used silicon nitride film mechanical oscillators to achieve a photon conversion efficiency of 0.47 at a temperature of 35mK, with an effective bandwidth of about 12kHz. However, compared with the current highest efficiency of 0.8 achieved by the Rb atom system and the bandwidth of over 10MHz achieved by the electro-optical system, there is still a lot of room for improvement in microwave-optical conversion based on cavity electro-optical converters. Focusing on the application of quantum enhanced reception and processing of classical navigation signals, further improving the efficiency and bandwidth of microwave-optical conversion is very necessary to achieve efficient and distortion-free signal processing. However, the current experimentally feasible cavity electro-optical converters are difficult to meet this application requirement. The most critical problem is the limited working bandwidth, which is mainly determined by the maximum microwave (light)-mechanical rate that the cavity electro-optical converter can achieve. In addition, for the microwave quantum illumination navigation and ranging scheme, its signal-to-noise ratio enhancement performance is determined by the microwave-light reduced entanglement degree and the phase conjugation conversion efficiency, both of which are also limited by the experimentally feasible microwave (light)-mechanical rate. Although in practice the microwave (optical)-mechanical rate can be increased by increasing the driving power of the two cavities, the tunability is limited by the saturation range of the cavity, and the coupling matching conditions must be met as much as possible to achieve a higher photon conversion efficiency, which means that the one with a higher exchange rate with the mechanical mode in the two cavities needs to be suppressed to match the one with the worse performance, that is, sacrificing the effective bandwidth in exchange for higher efficiency.

[0003] In recent years, dissipative engineering has been widely used in various systems such as superconducting resonant cavities, Rydberg atoms, and trapped ions. By introducing new modes in quantum systems, the state density of the system modes can be effectively controlled, thereby suppressing or inducing the enhancement of certain interactions, which provides a new idea for improving the performance of cavity electro-optical converters. In the cavity optomechanical system, Zhang Yanlei, Shen Zhen and others have proved theoretically and experimentally that the interaction between the auxiliary mechanical mode and the optical cavity can modulate the dissipation of the cavity, thereby achieving enhancement and suppression effects in optomechanical cooling, opto-mechanical entanglement, etc., which provides a feasible method for adjusting the coupling matching of microwave-optical conversion and enhancing the performance of microwave-optical entanglement.

[0004] In view of this, the present invention provides a method for enhancing the microwave and optical entanglement performance of a mechanically assisted cavity electro-optical force converter. [Summary of the invention]

[0005] In order to solve the above problems, the present invention provides a method for enhancing the microwave and light entanglement performance of a mechanically assisted cavity electro-optical force converter.

[0006] The present invention is implemented by the following technical scheme, and provides a method for enhancing the microwave and light entanglement performance of a mechanically assisted cavity electro-optical force converter, comprising the following steps:

[0007] S1 analyzes the generation and mechanism of mechanically assisted photoacoustic interaction;

[0008] S2 constructs a mechanically assisted cavity electro-optical force converter model and derives the dynamic evolution equations under different interaction types.

[0009] In particular, the S1 is implemented in the following steps:

[0010] S11 is a three-mode optomechanical system based on mechanical-optical-mechanical coupling, with two independent mechanical modes and Simultaneous coupling to the optical mode The interaction Hamiltonian can be expressed as:

[0011]

[0012] In this model the light mode With mechanical mode Shows strong interaction, while assisting mechanical mode By using the light mode The weak interaction realizes the modulation of the optical mode, which further affects its interaction with the mechanical mode. Photoacoustic interaction;

[0013] S12 injects two frequencies of ω into the microwave cavity d,eb and ω d,em The driving field drives the microwave cavity mode and the coupling of the two mechanical modes respectively, then the total Hamiltonian of the system can be expressed as:

[0014]

[0015] In formula (1.2), the first line represents the free Hamiltonian of the microwave cavity and the two mechanical oscillators, the second line represents the interaction between the microwave cavity and the two mechanical oscillators, and the third line represents the Hamiltonian of the interaction between the two driving fields and the microwave cavity. Under the rotation representation, the system Hamiltonian shown in the above formula is unitarily transformed And ignoring the high-order oscillation terms, equation (1.2) becomes:

[0016]

[0017] In formula (1.3), Ω = ω d,em -ωd,eb ≠0, τ b(m) =ε eb(m) |(ε eb +ε em ), and again perform a unitary transformation on the Hamiltonian in the rotation representation And linearized, the interaction Hamiltonian is expressed as:

[0018]

[0019] In formula (1.4), G e,kl =g ek |ε el / (κ e / 2+iΔ el )|(k,l=m,b) represents the multiphoton photoacoustic coupling rate, is the quantum fluctuation of the microwave cavity mode. When the effective detuning of the two driving fields of the cavity is set to satisfy Δ eb =-ω m (ω m ), Δ em =-ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a parametric down-conversion manner, the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), then formula (1.4) can be simplified to:

[0020]

[0021] The first term represents the microwave-assisted mechanical parametric down-conversion interaction, under which the system produces microwave photon-phonon pairs, which leads to an increase in the microwave cavity mode density, which in turn modulates the mechanical mode. interaction, for which a detection field is input To analyze its dynamic process, the Langevin equation is expressed as:

[0022]

[0023] In formulas (1.6a) and (1.6b), δ e =ω p -ω e , ε p Represents frequency as ω p When the system is in a steady state, the steady-state field of the microwave mode can be obtained by solving the above equation:

[0024]

[0025] For ease of notation, the steady-state field is written as:

[0026]

[0027] Then κ e,eff and δ e,eff Can be expressed as:

[0028]

[0029] When |δ e |,|G em |,κ e <<γ ma When , the above formula can be simplified to:

[0030] κ e,eff ≈κ e -4|G e,mm | 2 / γ ma (1.10a)

[0031] δ e,eff ≈δ e (1.10b)

[0032] From formulas (1.7)-(1.10), it can be seen that the result of microwave-assisted mechanical interaction can be expressed as the modulation of the effective line width of the microwave cavity. The increase in cavity state density under parametric down-conversion interaction is the decrease in the equivalent line width of the microwave cavity, and this can be achieved by adjusting the driving field (ω d,em ) Power can achieve changes in equivalent line width;

[0033] Similarly, when Δ eb =-ω m (ω m ),Δ em =ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a beam splitter-type manner, ignoring the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), the Hamiltonian shown in equation (1.4) can be simplified to:

[0034]

[0035] Correspondingly, the density of states and effective linewidth of the microwave cavity mode are:

[0036]

[0037]

[0038] It can be seen that the result of microwave-assisted mechanical beam splitter type interaction is the decrease of the state density of microwave cavity mode, which can be equivalent to the increase of microwave cavity line width in mathematical expression, and increases with the increase of interaction strength; the state density of microwave cavity mode directly affects its multi-photon coupling rate with mechanical oscillator, so by adjusting the interaction between auxiliary mechanical oscillator and microwave cavity, the microwave cavity mode can be controlled. With mechanical mode Coupled modulation.

[0039] In particular, the S2 is implemented in the following steps:

[0040] S21 Construction of a quantum model of microwave-assisted mechanical interaction

[0041] From the three-mode microwave-mechanical system, it can be known that by connecting the two capacitors of the LC resonant circuit to the mechanical oscillator, an auxiliary mechanical-microwave-mechanical coupling system can be constructed. On this basis, a mechanical-assisted cavity electro-optical force converter model is constructed. The oscillator interacts with the microwave cavity through capacitive coupling. The capacitor plates of the microwave resonant circuit and the movable mirror of the FP cavity are connected to realize indirect microwave-light interaction to distinguish the mechanical modes. and model Refers to the mechanical mode, Refers to the auxiliary mechanical mode;

[0042] Two driving fields of different frequencies are applied to the microwave cavity to increase the interaction between the cavity and the two mechanical oscillators, and driving light is injected into the optical cavity to enhance the optical-mechanical coupling. The total Hamiltonian can be expressed as:

[0043]

[0044] Under the rotation representation, the unitary transformation of formula (2.1) is And ignoring the high-order oscillation terms, the transformed system Hamiltonian is expressed as:

[0045]

[0046] use Further rotation transformation and linearization of formula (2.2) yields:

[0047]

[0048] In formula (2.3), G e,kl =g ek |ε el / (κ e / 2+iΔ el)|(k,l=m,b),G o =g o |ε o / (κ o / 2+iΔ o )|;

[0049] S22 analyzes the process of mechanically assisted cavity electro-optical force converter to achieve microwave and optical conversion, and applies driving fields to the microwave cavity and optical cavity respectively, and the frequency detuning satisfies the relationship Δ eb =Δ o =ω m and Δ em =ω ma , at this time, microwave-mechanical beam splitter type interaction, optical-mechanical beam splitter type interaction and microwave-assisted mechanical beam splitter type interaction occur simultaneously in the system, then formula (2.3) can be simplified to:

[0050]

[0051] Formula (2.4) omits the frequency within ±2ω m ,±2ω ma and ±(ω m ±ω ma ) of the higher-order resonance term, when the frequencies of the two mechanical oscillators satisfy |ω ma -ω m |≥min{2ω ma ,2ω m}, the frequency is ±(ω m ±ω ma ) can be ignored. Since the slowly varying resonance term in formula (2.3) will cause the coupling rate between the two cavities and the mechanical oscillator to vary with the time period, this is not conducive to achieving long-term stable microwave-optical conversion. Therefore, the frequencies of the two mechanical oscillators must be strictly controlled to eliminate the frequency of ±(ω m ±ω ma ) resonant term. In the interaction Hamiltonian shown in equation (2.4), the contents of the first two brackets represent the beam splitter-type interaction between the two mechanical oscillators and the microwave cavity. The latter realizes microwave-light conversion through the bridge effect of the intermediate mechanical oscillator, and the former means that a dissipative channel is generated, which will reduce the number of cavity photons that undergo coherent conversion.

[0052] Correspondingly, when the auxiliary mechanical oscillator and the microwave cavity are considered to interact in a parametric down-conversion manner, the frequency detuning of the microwave cavity driving field satisfies Δ em =-ω ma , then the Hamiltonian shown in equation (2.4) becomes:

[0053]

[0054] This indicates that under microwave-assisted mechanical parametric down-conversion interaction, the microwave cavity photon number increases, and the cavity mode and mechanical mode are The coupling rate increases accordingly, which is more conducive to the conversion of microwave photons to mechanical phonons. However, the effect of the increase or decrease of the coupling rate on the microwave-light conversion performance also depends on the other parameters of the cavity electro-optical force converter. It is necessary to analyze the effect on the microwave-light conversion efficiency, effective working bandwidth, noise and other performance under the actual parameter settings.

[0055] The influence of the auxiliary mechanical oscillator on microwave-optical conversion lies in the adjustment of the effective line width and state density of the microwave cavity, which in turn affects the microwave-mechanical coupling rate. Therefore, the interaction Hamiltonian shown in equations (2.4) and (2.5) can be approximately equivalent to:

[0056]

[0057] In formulas (2.6a) and (2.6b), and are the annihilation operators of the equivalent microwave modes under microwave-assisted mechanical beam splitter type and parametric down-conversion type interactions, respectively. At this time, the auxiliary mechanical oscillator forms a new equivalent microwave cavity with an effective line width of κ e,eff ≈κ e ±4|G e,mm | 2 / γ ma Similarly, when considering the mechanically assisted cavity electro-optical force converter to prepare microwave-light entanglement, its interaction Hamiltonian is similar to equations (2.4)-(2.6). When considering the mechanically assisted cavity electro-optical force converter to prepare microwave-light entanglement, its interaction Hamiltonian is similar to equations (2.4)-(2.6), except that the last term becomes I won’t go into detail here;

[0058] S23 Quantum Model of Light-Assisted Mechanical Interaction

[0059] In the quantum model of the interaction between the auxiliary mechanical oscillator and the optical cavity, both mirrors of the FP cavity are movable and connected to the mechanical oscillator respectively. and in As an auxiliary mechanical oscillator to modulate the optical cavity, At the same time, the capacitor plates of the microwave resonant circuit are connected to realize indirect microwave-light interaction. The driving field is injected into the microwave cavity and the optical cavity at the same time. The optical cavity is driven by two lasers of different frequencies to increase the coupling between the cavity and the two mechanical oscillators. At this time, the total Hamiltonian of the system can be expressed as:

[0060]

[0061] From the analysis of mechanically assisted photoacoustic interaction, it can be seen that the optical cavity can be equivalent to a line width of κ under the interaction with the auxiliary mechanical oscillator. o,eff =κ o ±4|G o,mm | 2 / γ ma cavity, whereby the light-assisted mechanical beam splitter type interaction (Δ om =ω ma ) and parametric down-conversion interaction Δ om =-ω ma The system Hamiltonian of the cavity electro-optical force converter can be expressed as:

[0062]

[0063] When |G om |,κ o <<γ ma When , the Hamiltonian shown in the above formula can be approximately equivalent to:

[0064]

[0065] In formula (2.9), represents the annihilation operator of the equivalent optical mode. Although formula (2.6) and (2.9) are similar in form, the effects of mechanical oscillator-assisted microwave cavity or optical cavity on unidirectional microwave-optical conversion are different. The common point between the two is that the microwave / optical-mechanical rate is adjusted by changing the effective parameters of the cavity, thereby affecting the photon conversion efficiency and effective working bandwidth. The difference lies in the impact on noise introduction. The noise introduced in the microwave-optical up-conversion process mainly depends on the dissipation of the microwave cavity and the microwave-mechanical rate. The dissipation of the optical cavity and the optical-mechanical rate have a weaker effect on the noise, while the opposite is true in the optical-microwave down-conversion. Therefore, for the microwave-optical up-conversion process, microwave-assisted mechanical interaction can reduce the introduction of noise.

[0066] Compared with the existing design, the present invention provides a method for enhancing the microwave and light entanglement performance of a mechanically assisted cavity electro-optical force converter. The method can be applied to improve the performance of microwave-light entanglement and microwave quantum lighting, especially when the auxiliary mechanical oscillator interacts with the microwave cavity in a parametric down-conversion type, the microwave-light reduced entanglement degree and the signal-to-noise ratio enhancement index are significantly improved, and the system ratio stability under the same conditions is higher, the tunable range of the driving power of the microwave cavity is increased, and the availability of the system is improved; in comparison, the effect of light-assisted mechanical interaction on microwave-light entanglement is relatively small, and the effect on microwave quantum lighting performance is not obvious within the feasible parameter range.

Brief Description of the Drawings

[0067] Figure 1 Schematic diagram of multimode optical-mechanical coupling

[0068] Figure 2 A generalized three-mode opto-mechanical coupling system

[0069] Figure 3 It is a multi-mode cavity electro-optical force converter for mechanical vibrator-assisted microwave cavity;

[0070] Figure 4 A multi-mode cavity electro-optical force converter for mechanical oscillator-assisted optical cavity;

[0071] Figure 5 Microwave-optical entanglement and signal-to-noise ratio enhancement under microwave-assisted mechanical interaction;

[0072] Figure 6 Microwave-light entanglement and signal-to-noise ratio enhancement under light-assisted mechanical interaction. [Specific implementation method]

[0073] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings.

[0074] The present invention studies the microwave-light conversion and entanglement performance enhancement method around the mechanically assisted cavity electro-optical force converter. First, the generation and action mechanism of the mechanically assisted photoacoustic interaction are analyzed. On this basis, a mechanically assisted cavity electro-optical force converter model is constructed, and the dynamic evolution equations under different interaction types are derived. The feasibility of microwave-light conversion and entanglement performance enhancement is analyzed mathematically and physically. Afterwards, the influence of the auxiliary mechanical oscillator on the microwave-light coherent conversion performance is discussed in terms of photon conversion efficiency, effective conversion bandwidth and noise. The influence of the auxiliary mechanical oscillator on the microwave-light entanglement and signal-to-noise ratio enhancement performance is analyzed in the context of microwave quantum illumination navigation and ranging applications. The simulation results under experimentally feasible parameters verify the effectiveness of the mechanically assisted cavity electro-optical force converter. Specifically as follows:

[0075] 1. Mechanically Assisted Photoacoustic Interaction

[0076] Consider a three-mode optomechanical system with mechanical-optical-mechanical coupling, such as Figure 1 As shown, two independent mechanical modes and Simultaneous coupling to the optical mode The interaction Hamiltonian can be expressed as:

[0077]

[0078] In this model the light mode With mechanical mode Shows strong interaction, while assisting mechanical mode By using the light mode The weak interaction realizes the modulation of the optical mode, which further affects its interaction with the mechanical mode. The photoacoustic interaction can be realized in a variety of generalized three-mode optomechanical systems, such as the interaction between two mechanical films and optical modes in the FP cavity.

[133] , optical micro-ring cavity mode coupling breathing mode and Brillouin mechanical mode, as well as microwave cavity and two aluminum film oscillator capacitive coupling, etc. Figure 2 Several feasible system designs are shown.

[0079] Figure 2 represents a feasible generalized three-mode opto-mechanical coupling system. To analyze the mechanically assisted opto-acoustic interaction, the present invention uses Figure 2 The dynamic process is analyzed by taking the model of two mechanical oscillators coupled to a microwave cavity as an example. Two frequencies of ω are injected into the microwave cavity. d,eb and ω d,em The driving field (driving the microwave cavity mode and the coupling of the two mechanical modes respectively), the total Hamiltonian of the system can be expressed as:

[0080]

[0081] The first row represents the free Hamiltonian of the microwave cavity and the two mechanical oscillators, the second row represents the interaction between the microwave cavity and the two mechanical oscillators, and the third row represents the Hamiltonian of the interaction between the two driving fields and the microwave cavity. And ignoring the high-order oscillation terms, equation (1.2) becomes:

[0082]

[0083] Where Ω=ω d,em -ω d,eb ≠0, τ b(m) =ε eb(m) / (ε eb +ε em ). Again, the Hamiltonian above is unitarily transformed in the rotation representation. And linearized, the interaction Hamiltonian is expressed as:

[0084]

[0085] Among them G e,kl =g ek |ε el / (κ e / 2+iΔ el )|(k,l=m,b) represents the multiphoton photoacoustic coupling rate, is the quantum fluctuation of the microwave cavity mode. When the effective detuning of the two driving fields of the cavity is set to satisfy Δeb =-ω m (ω m ), Δ em =-ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a parametric down-conversion manner, the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), then formula (1.4) can be simplified to:

[0086]

[0087] The first term represents the microwave-assisted mechanical parametric down-conversion interaction, under which the system produces microwave photon-phonon pairs, which leads to an increase in the microwave cavity mode density, which in turn modulates the mechanical mode. For this input a detection field To analyze its dynamic process, the Langevin equation can be expressed as:

[0088]

[0089] where δ e =ω p -ω e , ε p Represents frequency as ω p When the system is in steady state, by solving the above equation, the steady-state field of the microwave mode can be obtained as:

[0090]

[0091] For ease of notation, the steady-state field is written as:

[0092]

[0093] Then κ e,eff and δ e,eff Can be expressed as:

[0094]

[0095] When |δ e |,|G em |,κ e <<γ ma When , the above formula can be simplified to:

[0096] κ e,eff ≈κ e -4|G e,mm | 2 / γ ma (1.10a)

[0097] δ e,eff ≈δ e (1.10b)

[0098] From formulas (1.7)-(1.10), it can be seen that the result of microwave-assisted mechanical interaction can be expressed as the modulation of the effective line width of the microwave cavity. The increase in cavity state density under parametric down-conversion interaction is the decrease in the equivalent line width of the microwave cavity, and this can be achieved by adjusting the driving field (ω d,em ) power can achieve changes in the equivalent line width.

[0099] Similarly, when Δ eb =-ω m (ω m ),Δ em =ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a beam splitter-type manner, ignoring the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), the Hamiltonian shown in equation (1.4) can be simplified to:

[0100]

[0101] Correspondingly, the density of states and effective linewidth of the microwave cavity mode are:

[0102]

[0103] It can be seen that the result of microwave-assisted mechanical beam splitter type interaction is the decrease of the density of states of microwave cavity mode, which can be equivalent to the increase of microwave cavity line width in mathematical expression, and increases with the increase of interaction strength. According to the basic principle of cavity optomechanics, the density of states of microwave cavity mode directly affects its multi-photon coupling rate with mechanical oscillator. Therefore, by adjusting the interaction between auxiliary mechanical oscillator and microwave cavity, the microwave cavity mode can be controlled. With mechanical mode Coupled modulation.

[0104] 2. Quantum Model and Dynamical Equations of Mechanically Assisted Cavity Electro-Optical Force Converter

[0105] From the above analysis, it can be seen that the auxiliary mechanical oscillator realizes the modulation of the generalized photoacoustic interaction by changing the effective linewidth of the cavity. The following will further expand the mechanically assisted cavity optomechanical system, discuss the microwave-mechanical-optical interaction of the mechanically assisted cavity electro-optical force converter, and further explore the influence mechanism of the auxiliary mechanical oscillator on microwave-optical conversion and microwave-optical entanglement.

[0106] 2.1 Microwave-assisted mechanical interaction quantum model

[0107] Depend on Figure 2 From the three-mode microwave-mechanical system shown in (c), it can be seen that an auxiliary mechanical-microwave-mechanical coupling system can be constructed by connecting the two capacitors of the LC resonant circuit to the mechanical oscillator respectively. On this basis, a mechanically assisted cavity electro-optical force converter model is constructed, as shown in Figure 3 As shown, the mechanical oscillator The oscillator interacts with the microwave cavity through capacitive coupling. The capacitor plates of the microwave resonant circuit and the movable mirrors of the FP cavity are connected to achieve indirect microwave-light interaction. and Unless otherwise specified in the subsequent analysis, the mechanical mode refers to the mode The auxiliary mechanical mode indicates the mode

[0108] Two driving fields of different frequencies are applied to the microwave cavity to increase the interaction between the cavity and the two mechanical oscillators, and driving light is injected into the optical cavity to enhance the optical-mechanical coupling. The total Hamiltonian of the system can be expressed as:

[0109]

[0110] Under the rotation representation, perform a unitary transformation on the above formula And ignoring the high-order oscillation terms, the transformed system Hamiltonian is expressed as:

[0111]

[0112] Similarly, using Further rotation transformation and linearization of the above formula can be obtained:

[0113]

[0114] Among them G e,kl =g ek |ε el / (κ e / 2+iΔ el )|(k,l=m,b),G o =g o |ε o / (κ o / 2+iΔ o )|.

[0115] First, the process of mechanically assisted cavity electro-optical force converter to achieve microwave-optical conversion is analyzed. The driving field is applied to the microwave cavity and the optical cavity respectively, and the frequency detuning satisfies the relationship Δ eb =Δ o =ω m and Δ em =ωma , at this time, microwave-mechanical beam splitter type interaction, optical-mechanical beam splitter type interaction and microwave-assisted mechanical beam splitter type interaction occur simultaneously in the system, then formula (2.3) can be simplified to:

[0116]

[0117] The above formula omits the frequency within ±2ω m ,±2ω ma and ±(ω m ±ω ma ) of the higher-order resonance term. However, it should be pointed out that only when the frequencies of the two mechanical oscillators satisfy |ω ma -ω m |≥min{2ω ma ,2ω m}, the frequency is ±(ω m ±ω ma ) can be ignored. Since the slowly varying resonance term in formula (2.3) will cause the coupling rate between the two cavities and the mechanical oscillator to vary with the time period, which is not conducive to achieving long-term stable microwave-optical conversion, the frequencies of the two mechanical oscillators must be strictly controlled in the experiment to eliminate the frequency of ±(ω m ±ω ma ). In the interaction Hamiltonian shown in equation (2.4), the contents of the first two brackets represent the beam splitter-type interaction between the two mechanical oscillators and the microwave cavity. The latter realizes microwave-light conversion through the bridge effect of the intermediate mechanical oscillator, and the former means that a dissipative channel is generated, which will reduce the number of cavity photons that undergo coherent conversion.

[0118] Correspondingly, when the auxiliary mechanical oscillator and the microwave cavity are considered to interact in a parametric down-conversion manner (the frequency detuning of the microwave cavity driving field satisfies Δ em =-ω ma ), then the Hamiltonian shown in equation (2.4) becomes:

[0119]

[0120] This indicates that under microwave-assisted mechanical parametric down-conversion interaction, the microwave cavity photon number increases, and the cavity mode and mechanical mode are The coupling rate increases accordingly, which is theoretically more conducive to the conversion of microwave photons to mechanical phonons. However, the effect of the increase or decrease in the coupling rate on the microwave-light conversion performance also depends on the other parameters of the cavity electro-optical force converter. It is necessary to analyze the effect on the microwave-light conversion efficiency, effective working bandwidth, noise and other performance under the actual parameter settings.

[0121] Specifically, the influence of the auxiliary mechanical oscillator on microwave-optical conversion lies in the regulation of the effective line width and state density of the microwave cavity, which in turn affects the microwave-mechanical coupling rate. Therefore, the interaction Hamiltonian shown in equations (2.4) and (2.5) can be approximately equivalent to:

[0122]

[0123] in and are the annihilation operators of the equivalent microwave modes under microwave-assisted mechanical beam splitter type and parametric down-conversion type interactions. At this time, the auxiliary mechanical oscillator forms a new equivalent microwave cavity with an effective line width of κ e,eff ≈κ e ±4|G e,mm | 2 / γ ma Similarly, when considering the mechanically assisted cavity electro-optical force converter to prepare microwave-light entanglement, its interaction Hamiltonian is similar to equations (2.4)-(2.6), with the only difference being the last term becomes I won’t go into details here.

[0124] 2.2 Quantum model of light-assisted mechanical interaction

[0125] Similar to the cavity electro-optical force converter under microwave-assisted mechanical interaction, the following will further discuss the interaction between the auxiliary mechanical oscillator and the optical cavity. Its quantum model is as follows Figure 4 As shown. The two mirrors of the FP cavity are movable and are connected to mechanical oscillators. and in As an auxiliary mechanical oscillator to modulate the optical cavity, At the same time, the capacitor plates of the microwave resonant circuit are connected to realize indirect microwave-optical interaction. The driving field is injected into the microwave cavity and the optical cavity at the same time, and the optical cavity is driven by two lasers of different frequencies to increase the coupling between the cavity and the two mechanical oscillators. At this time, the total Hamiltonian of the system can be expressed as:

[0126]

[0127] From the analysis of mechanically assisted photoacoustic interaction, it can be seen that the optical cavity can be equivalent to a line width of κ under the interaction with the auxiliary mechanical oscillator. o,eff =κ o ±4|G o,mm | 2 / γ ma The light-assisted mechanical beam splitter type interaction (Δ om =ω ma ) and parametric down-conversion interaction Δ om =-ω maThe system Hamiltonian of the cavity electro-optical force converter can be expressed as:

[0128]

[0129] When |G om |,κ o <<γ ma When , the Hamiltonian shown in the above formula can be approximately equivalent to:

[0130]

[0131] in represents the annihilation operator of the equivalent optical mode.

[0132] It should be noted that although formula (2.6) and (2.9) are similar in form, the effects of mechanical oscillator-assisted microwave cavity or optical cavity on unidirectional microwave-optical conversion are different. The common point between the two is that they both adjust the microwave / optical-mechanical rate by changing the effective parameters of the cavity, thereby affecting the photon conversion efficiency and effective working bandwidth. The difference lies in the impact on noise introduction.

[0133] Due to the non-ideality of the cavity electro-optical force converter, a certain amount of noise is inevitably introduced during the conversion process, which mainly includes four parts: the inherent dissipation of the microwave cavity, the vibration thermal effect of the mechanical oscillator, the inherent dissipation of the optical cavity, and the reflection noise of the microwave cavity (corresponding to the down-conversion process) or the optical cavity (corresponding to the up-conversion process). Since the thermal noise of the optical mode (~200THz) at room temperature is extremely low, it can be approximated as Therefore, only the noise introduced by microwave and mechanical modes needs to be considered in the calculation. For the microwave-optical upconversion process, the noise part is Then the number of photons introduced into the noise corresponding to a single conversion photon can be expressed as:

[0134]

[0135] In formula (2.91), and They represent the temperature noise of the microwave cavity and the mechanical oscillator respectively, specifically: k B and T E are the Boltzmann constant and the cavity electro-optical converter temperature respectively. Accordingly, the average noise photon number of the down-conversion process can be expressed as:

[0136]

[0137] When the inherent dissipation of the two cavities is small enough to be ignored, that is, Formulas (2.91) and (2.92) can be simplified to:

[0138]

[0139] At this time, the noise introduced in the bidirectional microwave-optical conversion process can be considered to be mainly caused by mechanical vibration thermal noise. From formulas (2.91)-(2.94), it can be seen that the noise introduced in the microwave-optical up-conversion process mainly depends on the dissipation of the microwave cavity and the microwave-mechanical rate. The dissipation of the optical cavity and the optical-mechanical rate have a weaker effect on the noise, while the opposite is true in optical-microwave down-conversion.

[0140] 3. Microwave-optical entanglement performance analysis

[0141] The following will further discuss the impact of microwave / light-assisted mechanical interaction on the performance of microwave-light entanglement using microwave quantum illumination navigation and ranging as the application background, and specifically analyze the performance of microwave-light entanglement and signal-to-noise ratio enhancement.

[0142] First, the effect of microwave-assisted mechanical interaction on microwave-optical entanglement and signal-to-noise ratio enhancement is considered. The system is driven by microwave cavity and optical cavity respectively, and the frequency detuning satisfies Δ eb =-Δ o =ω m , Δ em = ±ω ma (representing microwave-assisted mechanical beam splitter type and parametric down-conversion type interactions respectively). The microwave cavity driving field power is adjusted to make the microwave-assisted mechanical coupling rate be a certain G e,mm / 2π=5kHz>γ ma In this paper, the microwave-optical entanglement preparation based on the cavity electro-optical force converter is more concerned with the microwave-optical entanglement degree, and the actual frequency requirement of the microwave mode is lower, so the small signal frequency deviation caused by the microwave-assisted mechanical interaction can be ignored. Under this condition, the reduced entanglement degree under two microwave-assisted mechanical interactions is calculated respectively. Signal-to-Noise Enhancement Index The results are as follows Figure 5 As shown, Figure 5 (a), (b) and (c) represent the reduced entanglement under the interaction without auxiliary mechanical oscillator, microwave-assisted mechanical beam splitter type interaction and microwave-assisted mechanical parameter down-conversion type interaction respectively. (d), (e) and (f) are the corresponding signal-to-noise ratio enhancement indices.

[0143] By comparison, when the microwave cavity and the auxiliary mechanical oscillator interact in a beam splitter type, the reduced entanglement degree under the same driving power is SNR are smaller than those without mechanical assistance, that is, the microwave-assisted mechanical beam splitter type interaction will weaken the microwave-light entanglement and quantum lighting performance; and when the microwave cavity and the auxiliary mechanical oscillator are in a parametric down-conversion interaction, the reduced entanglement degree under the same conditions is SNR The increase in microwave-light entanglement and quantum lighting performance indicates that the system stability boundary line in the comparison diagram shows that the microwave-assisted mechanical beam splitter type interaction expands the unstable domain of the system. Under the same optical cavity driving power, the microwave cavity driving power required for the system to reach the stability boundary is greater, which reduces the tunable range of the microwave cavity driving field power to a certain extent; while the parametric down-conversion type interaction expands the stable domain of the system and reduces the requirements of the system stability on the microwave cavity driving power.

[0144] Correspondingly, the microwave-light-reduced entanglement degree under light-assisted mechanical interaction Signal-to-Noise Enhancement Index Changes such as Figure 6 where the light-assisted mechanical coupling rate is G o,mm / 2π=5kHz, other parameters are set the same Figure 5 , Figure 6 (a), (b), and (c) are the reduced entanglement under the interaction without auxiliary mechanical oscillator, light-assisted mechanical beam splitter type interaction, and light-assisted mechanical parameter down-conversion type interaction, respectively. (d), (e), and (f) are the corresponding signal-to-noise ratio enhancement indices.

[0145] By comparison, it can be seen that the interaction between the auxiliary mechanical oscillator and the optical cavity has an effect on the microwave-light reduced entanglement degree. SNR There is no obvious change. This is because within the tunable range of the microwave cavity and optical cavity driving field power (within the system stability domain), the sensitivity of the microwave-light entanglement to the optical cavity driving power is lower than that of the microwave cavity driving power (corresponding to the sensitivity of the photomechanical rate and the microwave-mechanical rate, respectively). Therefore, the change in the photomechanical rate caused by the light-assisted mechanical interaction cannot significantly change the microwave-light entanglement and the signal-to-noise ratio enhancement index. That is, the cavity electro-optical force converter under the light-assisted mechanical interaction in practice is not beneficial to improving the performance of microwave quantum lighting.

[0146] Based on the above analysis, it can be seen that the mechanically assisted cavity electro-optical force converter can be used to improve the performance of microwave-light entanglement and microwave quantum lighting. In particular, when the auxiliary mechanical oscillator interacts with the microwave cavity in a parametric down-conversion type, the microwave-light reduced entanglement degree and the signal-to-noise ratio enhancement index are significantly improved, and the system ratio stability under the same conditions is higher, the microwave cavity drive power tunable range becomes larger, and the system availability is improved; in comparison, the light-assisted mechanical interaction has a smaller effect on microwave-light entanglement, and has no obvious effect on microwave quantum lighting performance within the feasible parameter range.

[0147] In this invention, the performance limitations of the current cavity electro-optical force converter in terms of microwave-light entanglement degree and other aspects are studied around the mechanically assisted cavity electro-optical force converter to enhance the microwave-light entanglement performance. First, a mechanically assisted opto-mechanical coupling model is constructed, and the influence mechanism of the weak coupling between the auxiliary mechanical oscillator and the cavity on the photoacoustic interaction is analyzed. On this basis, a mechanically assisted cavity electro-optical force converter model is constructed, and the microwave-light conversion and entanglement realization are discussed in detail and simulated and verified in four forms of microwave / light-assisted mechanical beam splitter type and parametric down-conversion type interaction. The research content and main conclusions are as follows:

[0148] (1) The mechanism of the influence of the auxiliary mechanical oscillator on the photoacoustic interaction was studied. First, a quantum model of the mechanically assisted generalized cavity optomechanical system was constructed, and its dynamic evolution process was derived in detail. When the auxiliary mechanical oscillator is weakly coupled with the generalized optical cavity, the interaction result can be manifested as a modulation of the cavity linewidth, and the beam splitter-type interaction and parametric down-conversion-type interaction correspond to the increase and decrease of the cavity linewidth, respectively, and the corresponding cavity mode density is manifested as a decrease and increase, respectively, thereby further realizing the modulation of the opto-mechanical rate, which is the key to the subsequent enhancement or suppression of microwave-optical entanglement performance.

[0149] (2) The microwave-light entanglement under microwave / light-assisted mechanical interaction was studied, and the microwave-light reduced entanglement degree and signal-to-noise ratio enhancement index and other performance were analyzed based on the application background of microwave quantum illumination navigation and ranging. The results show that within the feasible cavity driving power range of the system, the microwave-assisted mechanical parameter down-conversion type interaction can improve the microwave-light reduced entanglement degree and signal-to-noise ratio enhancement index, and effectively expand the stability domain of the system, increase the tunable range of microwave cavity driving power, and improve the availability of the system. In comparison, the microwave-assisted mechanical beam splitter type interaction and the two light-assisted mechanical interactions did not show a significant improvement in the microwave quantum illumination performance. In practice, the microwave-assisted mechanical parameter down-conversion type interaction should be selected to improve the microwave quantum illumination navigation and ranging performance.

Claims

1. A method for enhancing microwave and optical entanglement performance by a mechanically assisted cavity electro-optical force converter, characterized in that: The following steps are involved: S1 analyzes the generation and mechanism of mechanically assisted photoacoustic interaction; S2 constructs a mechanically assisted cavity electro-optical force converter model and derives the dynamic evolution equations under different interaction types.

2. The method for enhancing microwave and optical entanglement performance of a mechanically assisted cavity electro-optical force converter according to claim 1, characterized in that: The S1 is specifically implemented according to the following steps: S11 is a three-mode optomechanical system based on mechanical-optical-mechanical coupling, with two independent mechanical modes and Simultaneous coupling to the optical mode The interaction Hamiltonian can be expressed as: In this model the light mode With mechanical mode Shows strong interaction, while assisting mechanical mode By using the light mode The weak interaction realizes the modulation of the optical mode, which further affects its interaction with the mechanical mode. Photoacoustic interaction; S12 injects two frequencies of ω into the microwave cavity d,eb and ω d,em The driving field drives the microwave cavity mode and the coupling of the two mechanical modes respectively, then the total Hamiltonian of the system can be expressed as: In formula (1.2), the first line represents the free Hamiltonian of the microwave cavity and the two mechanical oscillators, the second line represents the interaction between the microwave cavity and the two mechanical oscillators, and the third line represents the Hamiltonian of the interaction between the two driving fields and the microwave cavity. Under the rotation representation, the system Hamiltonian shown in the above formula is unitarily transformed And ignoring the high-order oscillation terms, equation (1.2) becomes: In formula (1.3), Ω = ω d,em -ω d,eb ≠0, τ b(m) =ε eb(m) / (ε eb +ε em ), and again perform a unitary transformation on the Hamiltonian in the rotation representation And linearized, the interaction Hamiltonian is expressed as: In formula (1.4), G e,kl =g ek |ε el / (κ e / 2+iΔ el )|(k,l=m,b) represents the multiphoton photoacoustic coupling rate, is the quantum fluctuation of the microwave cavity mode. When the effective detuning of the two driving fields of the cavity is set to satisfy Δ eb =-ω m (ω m ), Δ em =-ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a parametric down-conversion manner, the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), then formula (1.4) can be simplified to: The first term represents the microwave-assisted mechanical parametric down-conversion interaction, under which the system produces microwave photon-phonon pairs, which leads to an increase in the microwave cavity mode density, which in turn modulates the mechanical mode. interaction, for which a detection field is input To analyze its dynamic process, the Langevin equation is expressed as: In formulas (1.6a) and (1.6b), δ e =ω p -ω e , ε p Represents frequency as ω p When the system is in a steady state, the steady-state field of the microwave mode can be obtained by solving the above equation: For ease of notation, the steady-state field is written as: Then κ e,eff and δ e,eff Can be expressed as: When |δ e | , |G em |,κ e <<γ ma When , the above formula can be simplified to: k e,eff ≈k e -4|G e,mm | 2 / c ma (1.10a)d e,eff ≈d e (1.10b) From formulas (1.7)-(1.10), it can be seen that the result of microwave-assisted mechanical interaction can be expressed as the modulation of the effective line width of the microwave cavity. The increase in cavity state density under parametric down-conversion interaction is the decrease in the equivalent line width of the microwave cavity, and this can be achieved by adjusting the driving field (ω d,em ) Power can achieve changes in equivalent line width; Similarly, when Δ eb =-ω m (ω m ),Δ em =ω ma When the auxiliary mechanical oscillator interacts with the microwave cavity in a beam splitter-type manner, ignoring the ±2ω m ,±2ω ma and ±(ω m ±ω ma ), the Hamiltonian shown in equation (1.4) can be simplified to: Correspondingly, the density of states and effective linewidth of the microwave cavity mode are: It can be seen that the result of microwave-assisted mechanical beam splitter type interaction is the decrease of the state density of microwave cavity mode, which can be equivalent to the increase of microwave cavity line width in mathematical expression, and increases with the increase of interaction strength; the state density of microwave cavity mode directly affects its multi-photon coupling rate with mechanical oscillator, so by adjusting the interaction between auxiliary mechanical oscillator and microwave cavity, the microwave cavity mode can be controlled. With mechanical mode Coupled modulation.

3. The method for enhancing microwave and optical entanglement performance of a mechanically assisted cavity electro-optical force converter according to claim 2, characterized in that: The S2 is specifically implemented according to the following steps: S21 Construction of a quantum model of microwave-assisted mechanical interaction From the three-mode microwave-mechanical system, it can be known that by connecting the two capacitors of the LC resonant circuit to the mechanical oscillator, an auxiliary mechanical-microwave-mechanical coupling system can be constructed. On this basis, a mechanical-assisted cavity electro-optical force converter model is constructed. The oscillator interacts with the microwave cavity through capacitive coupling. The capacitor plates of the microwave resonant circuit and the movable mirror of the FP cavity are connected to realize indirect microwave-light interaction to distinguish the mechanical modes. and model Refers to the mechanical mode, Refers to the auxiliary mechanical mode; Two driving fields of different frequencies are applied to the microwave cavity to increase the interaction between the cavity and the two mechanical oscillators, and driving light is injected into the optical cavity to enhance the optical-mechanical coupling. The total Hamiltonian can be expressed as: Under the rotation representation, the formula (2.1) is unitary transformed And ignoring the high-order oscillation terms, the transformed system Hamiltonian is expressed as: use Further rotation transformation and linearization of formula (2.2) yields: In formula (2.3), G e,kl = g ek |ε el / (κ e / 2 + iΔ el )| (k, l = m, b), G o = g o |ε o / (κ o / 2 + iΔ o )|; S22 analyzes the process of mechanically assisted cavity electro-optical force converter to achieve microwave and optical conversion, and applies driving fields to the microwave cavity and optical cavity respectively, and the frequency detuning satisfies the relationship Δ eb =Δ o =ω m and Δ em =ω ma , at this time, microwave-mechanical beam splitter type interaction, optical-mechanical beam splitter type interaction and microwave-assisted mechanical beam splitter type interaction occur simultaneously in the system, then formula (2.3) can be simplified to: Formula (2.4) omits the frequency within ±2ω m ,±2ω ma and ±(ω m ±ω ma ) of the higher-order resonance term, when the frequencies of the two mechanical oscillators satisfy |ω ma -ω m |≥min{2ω ma ,2ω m }, the frequency is ±(ω m ±ω ma ) can be ignored. Since the slowly varying resonance term in formula (2.3) will cause the coupling rate between the two cavities and the mechanical oscillator to vary with the time period, this is not conducive to achieving long-term stable microwave-optical conversion. Therefore, the frequencies of the two mechanical oscillators must be strictly controlled to eliminate the frequency of ±(ω m ±ω ma ) resonant term. In the interaction Hamiltonian shown in equation (2.4), the contents of the first two brackets represent the beam splitter-type interaction between the two mechanical oscillators and the microwave cavity. The latter realizes microwave-light conversion through the bridge effect of the intermediate mechanical oscillator, and the former means that a dissipative channel is generated, which will reduce the number of cavity photons that undergo coherent conversion. Correspondingly, when the auxiliary mechanical oscillator and the microwave cavity are considered to interact in a parametric down-conversion manner, the frequency detuning of the microwave cavity driving field satisfies Δ em =-ω ma , then the Hamiltonian shown in equation (2.4) becomes: This indicates that under microwave-assisted mechanical parametric down-conversion interaction, the microwave cavity photon number increases, and the cavity mode and mechanical mode are The coupling rate increases accordingly, which is more conducive to the conversion of microwave photons to mechanical phonons. However, the effect of the increase or decrease of the coupling rate on the microwave-light conversion performance also depends on the other parameters of the cavity electro-optical force converter. It is necessary to analyze the effect on the microwave-light conversion efficiency, effective working bandwidth, noise and other performance under the actual parameter settings. The influence of the auxiliary mechanical oscillator on microwave-optical conversion lies in the adjustment of the effective line width and state density of the microwave cavity, which in turn affects the microwave-mechanical coupling rate. Therefore, the interaction Hamiltonian shown in equations (2.4) and (2.5) can be approximately equivalent to: In formulas (2.6a) and (2.6b), and are the annihilation operators of the equivalent microwave modes under microwave-assisted mechanical beam splitter type and parametric down-conversion type interactions, respectively. At this time, the auxiliary mechanical oscillator forms a new equivalent microwave cavity with the microwave cavity, and its effective line width is κ e,eff ≈κ e ±4|G e,mm / 2 / γ ma Similarly, when considering the mechanically assisted cavity electro-optical force converter to prepare microwave-light entanglement, its interaction Hamiltonian is similar to equations (2.4)-(2.6), with the only difference being the last term becomes S23 Quantum Model of Light-Assisted Mechanical Interaction In the quantum model of the interaction between the auxiliary mechanical oscillator and the optical cavity, both mirrors of the FP cavity are movable and connected to the mechanical oscillator respectively. and in As an auxiliary mechanical oscillator to modulate the optical cavity, At the same time, the capacitor plates of the microwave resonant circuit are connected to realize indirect microwave-light interaction. The driving field is injected into the microwave cavity and the optical cavity at the same time. The optical cavity is driven by two lasers of different frequencies to increase the coupling between the cavity and the two mechanical oscillators. At this time, the total Hamiltonian of the system can be expressed as: From the analysis of mechanically assisted photoacoustic interaction, it can be seen that the optical cavity can be equivalent to a line width of κ under the interaction with the auxiliary mechanical oscillator. o,eff =κ o ±4 / G o,mm / 2 / γ ma cavity, whereby the light-assisted mechanical beam splitter type interaction (Δ om =ω ma ) and parametric down-conversion interaction Δ om =-ω ma The system Hamiltonian of the cavity electro-optical force converter can be expressed as: When |G om |,κ o <<γ ma When , the Hamiltonian shown in the above formula can be approximately equivalent to: In formula (2.9), represents the annihilation operator of the equivalent optical mode. Although formulas (2.6) and (2.9) are similar in form, the effects of mechanical oscillator-assisted microwave cavity or optical cavity on unidirectional microwave-optical conversion are different. The common point between the two is that the microwave / optical-mechanical rate is adjusted by changing the effective parameters of the cavity, thereby affecting the photon conversion efficiency and effective working bandwidth. The difference lies in the impact on noise introduction.

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