Acceleration measurement method and device based on centroid motion quantum state of light suspended microparticles

Abstract

The application discloses an acceleration measurement method and device based on optical suspended microparticle centroid motion quantum state, which utilizes optical tweezers to trap a four-level atom and a nano microparticle in an optical cavity simultaneously, the optical cavity is driven by driving laser LC, a stable cavity optical field, i.e., cavity mode, is excited and formed in the cavity; the cavity optical field is coupled with the internal energy level transition of the atom and the centroid motion of the nano microparticle simultaneously, and three control lasers LES, LER and LGR are used to drive three energy level transition processes of the atom; by controlling the three control lasers LES, LER and LGR, a rapid adiabatic process and an atomic state inversion process are realized; wherein, in the process of preparing the centroid motion of the nano microparticle to a target Fock state, the forward rapid adiabatic process and the atomic state inversion process are iteratively executed in sequence; in the acceleration measurement process, the reverse rapid adiabatic process, the atomic state inversion process and the population transfer process based on a pi pulse are comprehensively utilized. The application can further improve the measurement sensitivity.

Description

Technical Field

[0001] This invention relates to the field of quantum precision measurement, and in particular to an acceleration measurement method and apparatus based on the quantum state of motion of the center of mass of optically suspended particles. Background Technology

[0002] Optical tweezers, with their non-contact and high-precision manipulation capabilities, have become a key technology for manipulating micro- and nano-scale objects such as dielectric particles and Rydberg atoms, and are widely used in interdisciplinary fields such as biology, quantum information science, and physics and chemistry. In particular, the suspended optical tweezers system (i.e., the vacuum optical tweezers system), constructed by suspending nanoscale dielectric particles in a vacuum cavity, not only eliminates the vibration losses and noise introduced by mechanical support in traditional optical mechanics systems, but also greatly suppresses the thermal noise generated by the thermal motion of gas molecules by utilizing the high vacuum environment, giving the system extremely high mechanical detection sensitivity. Furthermore, this technology enables precise control of multiple degrees of freedom of motion of nanoparticles (such as translation, rotation, and vibration). Based on these advantages, the suspended optical tweezers system plays an important role in cutting-edge physics exploration (such as testing non-Newtonian gravity, matter-wave interference, classical-quantum boundary, and non-equilibrium thermodynamics) and high-precision engineering measurements (such as inertial acceleration, detection of extremely weak forces, and weak celestial signals).

[0003] In recent years, with the development of active feedback cooling and passive cavity cooling technologies, the center-of-mass motion of nanoparticles in levitated optical-mechanical systems has been successfully cooled to the quantum ground state, propelling the system into the realm of quantum research. In the field of quantum precision measurement, quantum levitated optical-mechanical systems have demonstrated enormous application potential in measuring physical quantities such as extremely weak forces, acceleration, and electric fields. However, current acceleration measurement schemes based on levitated optical-mechanical systems still rely on the traditional principle of measuring acceleration by extracting the perturbation caused by external interactions of the nanoparticle's center-of-mass motion, without utilizing the quantum states of the particle's center-of-mass motion, such as non-classical states—Fock states and superposition states. This limits further improvements in acceleration sensitivity. Utilizing quantum resources such as quantum states holds promise for further enhancing sensitivity and achieving ultrasensitive detection. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a method and apparatus for measuring acceleration based on the quantum state of the center-of-mass motion of optically suspended particles.

[0005] The objective of this invention is achieved through the following technical solution:

[0006] An acceleration measurement method based on the quantum state of the center-of-mass motion of optically suspended particles is proposed. This method utilizes optical tweezers to simultaneously confine a four-level atom and a nanoparticle within an optical cavity. The optical cavity is driven by a LC laser, which excites and forms a stable cavity optical field, i.e., a cavity mode. This cavity optical field simultaneously couples with the internal energy level transitions of the four-level atom and the center-of-mass motion of the nanoparticle. Three control lasers, LES, LER, and LGR, are used to drive the three energy level transitions of the four-level atom, respectively.

[0007] By manipulating three control lasers LES, LER and LGR, rapid adiabatic processes and atomic state reversal processes can be achieved;

[0008] In the process of preparing the center of mass of the nanoparticles to the initial target Fock state, a forward rapid adiabatic process and an atomic state flipping process are executed iteratively in sequence.

[0009] In the acceleration measurement process, the rapid adiabatic process in reverse, the atomic state reversal process, and the population transfer process based on π pulses are comprehensively utilized to achieve the measurement of acceleration.

[0010] An acceleration measurement device based on the quantum state of motion of the center of mass of optically suspended particles is used to realize the acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles. The device includes a laser I, a laser II, an optical cavity, optical tweezers I and II, a polarization beamsplitter I and II, an optical modulator I, an optical modulator II and III, a reflector I, a reflector II, a reflector III and IV, a half-wave plate I, a half-wave plate II, a half-wave plate III, a half-wave plate IV, a half-wave plate V, a half-wave plate VI, a half-wave plate VII and a half-wave plate VIII. The optical axis of the laser is aligned with the optical axis of the optical cavity, which drives the optical cavity from one side to excite and form a stable cavity optical field. The optical tweezers are used to trap four-level atoms in the optical cavity and optical tweezers are used to trap nanoparticles in the optical cavity. The optical tweezers can freely adjust the positions of the four-level atoms and nanoparticles in the optical cavity to change the coupling strength of the interaction. The beam generated by laser 2 is split into control lasers LES, LER, and LGR to drive the three energy level transitions of the four-level atoms. Specifically, half-wave plate 1 and polarization beam splitter 1 are located sequentially in the optical path of laser 2. Polarization beam splitter 1 splits the beam generated by laser 2 into beam 1 and beam 2. Half-wave plate 3, optical modulator 1, half-wave plate 4, and mirror 2 are sequentially arranged in the optical path of beam 1, so that beam 1 illuminates the four-level atoms to form control laser LES. Half-wave plate 2 and polarization beam splitter 2 are sequentially arranged in the optical path of beam 2, so that beam 2 is split into beam 3 and beam 4 again. Half-wave plate 5, optical modulator 2, half-wave plate 6, and mirror 3 are sequentially arranged in the optical path of beam 3, so that beam 3 illuminates the four-level atoms to form control laser LER. Mirror 1, half-wave plate 7, optical modulator 3, half-wave plate 8, and mirror 4 are sequentially arranged in the optical path of beam 4, so that beam 4 illuminates the four-level atoms to form control laser LGR.

[0011] The beneficial effects of this invention are as follows:

[0012] Unlike traditional methods for detecting levitation optical acceleration, the acceleration measurement method proposed in this invention fully follows the theory of quantum sensing and quantum measurement. It obtains the acceleration information of the target signal by means of the quantum state after the interaction between the measurement system and the target signal, realizing acceleration detection based on the Fock state of the particle's center of mass motion. Using the non-classical state—the Fock state—for acceleration measurement can further improve the measurement sensitivity. Attached Figure Description

[0013] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.

[0014] The block diagrams shown in the accompanying drawings are merely functional entities and do not necessarily correspond to physically independent entities. That is, these functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0015] The flowcharts shown in the accompanying drawings are merely illustrative and do not necessarily include all content and operations / steps, nor do they necessarily have to be performed in the described order. For example, some operations / steps can be broken down, while others can be combined or partially combined; therefore, the actual execution order may change depending on the specific circumstances.

[0016] Figure 1 This is a schematic diagram of the acceleration measurement device based on the quantum state of motion of the center of mass of optically suspended particles according to an exemplary embodiment of the present invention, wherein, reference numeral 1 is laser one, 2 is laser two, 3 is optical cavity, 4 is optical tweezer one, 5 is optical tweezer two, 6 is four-level atom, 7 is nanoparticle, 8 is polarization beam splitter one, 9 is polarization beam splitter two, 10 is optical modulator one, 11 is optical modulator two, 12 is optical modulator three, 13 is mirror one, 14 is mirror two, 15 is mirror three, 16 is mirror four, 17 is half-wave plate one, 18 is half-wave plate two, 19 is half-wave plate three, 20 is half-wave plate four, 21 is half-wave plate five, 22 is half-wave plate six, 23 is half-wave plate seven, and 24 is half-wave plate eight.

[0017] Figure 2 The diagram illustrates the energy level structure and energy level transitions of the present invention according to an exemplary embodiment; wherein (a) is an atomic energy level structure, (b) is an energy level transition during a rapid adiabatic process, and (c) is an energy level transition during an atomic state flipping process.

[0018] Figure 3 This is a flowchart illustrating an acceleration measurement method based on the quantum state of motion of the center of mass of an optically suspended particle, according to an exemplary embodiment.

[0019] Figure 4 Numerical simulation results of a forward rapid adiabatic process illustrated according to an exemplary embodiment; wherein, (a) is the detuning amount of the control laser LES. The curves showing the change over time, (b) represent the intensity of the controlled laser LES. The curves showing the change over time, where (c) represents the quantum state. and The population change curve over time.

[0020] Figure 5 Numerical simulation results of the reverse rapid adiabatic process in the present invention, according to an exemplary embodiment, are shown; wherein, (a) is the detuning amount of the laser LES. The curves showing the change over time, (b) represent the intensity of the controlled laser LES. The curves showing the change over time, where (c) represents the quantum state. and The population change curve over time. Detailed Implementation

[0021] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments. The purpose and effects of the present invention will become clearer. It should be understood that the specific embodiments described herein are merely for explaining the present invention and are not intended to limit the present invention.

[0022] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.

[0023] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. The singular forms “a,” “the,” and “the” used in this application and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0024] It should be understood that although the terms first, second, third, etc., may be used in this application to describe various information, such information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, without departing from the scope of this application, first information may also be referred to as second information, and similarly, second information may also be referred to as first information. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to determination."

[0025] like Figure 1 As shown, the acceleration measurement device based on the quantum state of the center of mass motion of optically suspended particles in this embodiment includes a laser 1, a laser 2, an optical cavity 3, optical tweezers 1, optical tweezers 2, a polarization beam splitter 1, a polarization beam splitter 2, an optical modulator 10, an optical modulator 2, an optical modulator 3, a reflector 13, a reflector 2, a reflector 3, a reflector 4, a half-wave plate 17, a half-wave plate 2, a half-wave plate 3, a half-wave plate 4, a half-wave plate 5, a half-wave plate 6, a half-wave plate 7, and a half-wave plate 8, and a half-wave plate 8, and a half-wave plate 9.

[0026] In this system, the four-level atom 6 and the nanoparticle 7 are located within the optical cavity 3. The optical axis of laser 1 coincides with the optical axis of the optical cavity 3, driving the optical cavity 3 from one side to excite and form a stable cavity optical field. Optical tweezers 1 4 is used to confine the four-level atom 6 within the optical cavity 3, and optical tweezers 2 5 is used to confine the nanoparticle 7 within the optical cavity 3. Optical tweezers 1 4 and 2 5 can freely adjust the positions of the four-level atom 6 and the nanoparticle 7 within the optical cavity 3 to change the coupling strength of their interaction.

[0027] The beam generated by laser 2 is split into control lasers LES, LER, and LGR to drive the three energy level transitions of the four-level atom 6. Specifically, half-wave plate 17 and polarization beam splitter 8 are located sequentially in the optical path of laser 2. Polarization beam splitter 8 splits the beam generated by laser 2 into two beams, beam one and beam two. Half-wave plate 3 19, optical modulator 10, half-wave plate 4 20, and reflector 2 14 are sequentially arranged in the optical path of beam one, so that the beam illuminates the four-level atom 6, forming the control laser LES; the beam splitter 8 splits the beam into beam two. In the optical path of beam two, half-wave plate two 18 and polarization beam splitter two 9 are arranged in sequence. The polarization beam splitter two 9 splits beam two into beam three and beam four. In the optical path of beam three, half-wave plate five 21, optical modulator two 11, half-wave plate six 22 and reflector three 15 are arranged in sequence, so that beam three illuminates the four-level atom 6 to form the control laser LER. In the optical path of beam four, reflector one 13, half-wave plate seven 23, optical modulator three 12, half-wave plate eight 24 and reflector four 16 are arranged in sequence, so that the beam illuminates the four-level atom 6 to form the control laser LGR.

[0028] Optical modulator 10, optical modulator 21, and optical modulator 32 are all used to adjust the driving frequency and driving intensity of the output laser beam. They can be electro-optic modulators or acousto-optic modulators.

[0029] The optical cavity 3 is preferably a high-precision cavity with a precision greater than 100,000 and a cavity mirror reflectivity greater than or equal to 99.998%.

[0030] The fourth-level atom 6 has two excited states. and and two ground states and Among them, controlling the laser-driven energy level transition process of LES. Controlling the laser LER-driven energy level transition process Controlling the laser LGR-driven energy level transition process The cavity mode excited by the driving laser LC within the optical cavity is simultaneously with the energy level transition process. The motion of the center of mass of nanoparticle 7 is coupled.

[0031] Figure 2 This is a schematic diagram of the energy level structure and energy level transitions according to an embodiment of the present invention. The specific physical operation process is as follows:

[0032] When the control lasers LER and LGR are turned off and the control laser LES is turned on, the center-of-mass motion of the four-level atom 6 is coupled with that of the nanoparticle 7, forming an equivalent atom-phonon interaction to perform a rapid adiabatic process.

[0033] When the control laser LES is turned off and the control lasers LER and LGR are turned on, the center-of-mass motion of the four-level atom 6 is decoupled from that of the nanoparticle 7 to perform the atomic state flipping process.

[0034] In addition, auxiliary atomic states are used to perform population transfer processes based on π pulses.

[0035] like Figure 3 As shown, the acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles in this embodiment includes the following steps:

[0036] Step 1: Cool the center of mass motion of nanoparticles 7 to the ground state through active feedback cooling. The four-level atom is initialized in the ground state. Above; at this point, the joint quantum state of atoms and nanoparticles is .

[0037] Step 2: Prepare the center of mass motion of nanoparticle 7 to the initial target Fock state.

[0038] Step two includes the following sub-steps:

[0039] S2.1: Calculate and set the laser control parameters for the positive rapid adiabatic process;

[0040] S2.2: Iterate through the forward rapid adiabatic process and the atomic state reversal process sequentially until the center of mass of nanoparticle 7 is driven to the initial target Fock state. superior.

[0041] Step 3: Turn off the control lasers LES, LER, and LGR, and the driving laser LC, allowing the center of mass of nanoparticle 7 to freely evolve under the action of the acceleration to be measured for a period of time. .

[0042] Step 4: Extract the projection probability of nanoparticle 7 onto the initial target Fock state. Specifically, it includes:

[0043] When the initial target Fock state n=0, the laser control parameters for the reverse rapid adiabatic process are calculated and set, and the reverse rapid adiabatic process is executed for the atomic state. Fluorescence detection was performed to obtain the projection probability of the nanoparticles onto the initial target Fock state. ;

[0044] When the initial target Fock state n=1, the laser control parameters for the reverse rapid adiabatic process are calculated and set. The reverse rapid adiabatic process, the π-pulse-based population transfer process, the atomic state flipping process, and the reverse rapid adiabatic process are executed sequentially. Fluorescence detection was performed to obtain the projection probability P1 = P of the nanoparticles onto the initial target Fock state. g ;

[0045] When the initial target Fock state n≥2, the laser control parameters for the reverse rapid adiabatic process are calculated and set, and the reverse rapid adiabatic process, the population transfer process based on π pulses, and the atomic state flipping process are executed iteratively until the target projection probability amplitude c is reached. g,n Appears in dark state Up; then perform the reverse rapid adiabatic process, followed by the atomic state. Fluorescence detection was performed to obtain the projection probability P of the nanoparticle onto the initial target Fock state. n (n≥2)= P g .

[0046] By following the steps above, the evolution time can be accurately obtained. Projection probability of the lower particle onto the initial target Fock state .

[0047] Based on projection probability The acceleration a to be measured is obtained by solving the following system of equations:

[0048]

[0049] Among them, P n ξ is the projection probability of the particle's final state after the measured acceleration onto the initial target Fock state; ξ is an intermediate variable. The evolution time of the particle under the action of the measured acceleration is the time during which it freely evolves. It is an nth-order Laguerre polynomial; m is the mass of the nanoparticle; a is the acceleration to be measured; is the reduced Planck constant.

[0050] The positive rapid adiabatic process is achieved in the following way:

[0051] By shutting down the control lasers LER and LGR and turning on the control laser LES, the center-of-mass motion of the four-level atom 6 is coupled with that of the nanoparticle 7, forming an equivalent atom-phonon interaction. The detuning amount and intensity of the control laser LES are then controlled, causing the equivalent detuning amount to linearly change from a positive value to a negative value, and the change of the equivalent Rabi frequency intensity over time follows a Gaussian function, thereby achieving a forward rapid adiabatic process. .

[0052] The reverse rapid adiabatic process is achieved in the following way:

[0053] The control lasers LER and LGR are turned off, and the control laser LES is turned on, so that the four-level atoms are coupled with the center-of-mass motion of the nanoparticles, forming an equivalent atom-phonon interaction. The detuning amount and intensity of the control laser LES are adjusted so that the equivalent detuning amount changes linearly from a negative value to a positive value, and the change of the equivalent Rabi frequency with time satisfies a Gaussian function, thereby realizing a negative rapid adiabatic process.

[0054] The atomic state flipping process is achieved in the following way:

[0055] Turn off the control laser LES and turn on the control lasers LER and LGR to decouple the center-of-mass motion of the atoms from that of the nanoparticles; regulate the control lasers LER and LGR to generate a π pulse to achieve the atomic state flipping process.

[0056] During atomic state flips, the parameters of the π pulse depend only on the two ground states of the atom. and With excited state The detuning between the levels and the Rabi frequency; after determining the atomic level structure, the π pulse parameters used each time remain the same.

[0057] The population transfer process based on π pulses is achieved in the following way:

[0058] Apply a ground state to a four-level atom With auxiliary state The π pulse drives the transition process , dark state The population on the upper state is transferred to the auxiliary state, so that the population is hidden in the auxiliary state.

[0059] The core theoretical principles and physical mechanisms involved in this invention are explained in detail below, including:

[0060] (A) The principle of rapid adiabatic process;

[0061] (B) The principle of atomic state reversal process;

[0062] (C) The principle of population transfer process;

[0063] (D) Principle of acceleration measurement;

[0064] (E) Relationship between acceleration and projection probability.

[0065] (A) Principle of rapid adiabatic process

[0066] like Figure 2 As shown in (b), when the control lasers LER and LGR are turned off and the control laser LES is turned on, the auxiliary atom excited state Decoupled from the system, the Hamiltonian of the entire hybrid suspended optical-mechanical system is ( )

[0067]

[0068] The first line of the formula represents the cavity optical field, i.e., the free Hamiltonian of the cavity mode, nanoparticles, and atoms. and These represent the annihilation operators for cavity modes and nanoparticles, respectively. The first row represents the frequencies of the cavity mode, nanoparticles, and atomic energy levels, respectively; the second row represents the interaction terms between atoms and the cavity mode. This indicates the coupling strength between the atom and the cavity mode. and To control the intensity and frequency of the laser LES; the third row is the interaction term between the nanoparticles and the cavity mode. The fourth row represents the coupling strength between the center of mass motion of the nanoparticles and the cavity mode; the fifth row represents the driving term of the cavity mode. and ω l This indicates the intensity and frequency of the driving laser LC.

[0069] Under strong driving conditions, cavity mode and nanoparticle operators can be rewritten as the sum of classical average values ​​and quantum fluctuation operators, i.e., a linearization process. and Rotate the entire system to the frequency that drives the laser LC. ω l and atomic ground state frequency The linear Hamiltonian is obtained by approximating the rotating wave using the standard linearization process.

[0070]

[0071] in, This represents the cavity mode detuning, which can be adjusted by driving the laser LC; the atom-dependent detuning is defined as follows: , , ; The equivalent coupling strength between the nanoparticles and the cavity mode.

[0072] Under conditions of large detuning, atomic excited states are eliminated sequentially through adiabatic processes. After combining the cavity mode, the equivalent atom-phonon interaction Hamiltonian is obtained.

[0073]

[0074] The relevant parameters are defined as follows:

[0075]

[0076]

[0077]

[0078] .

[0079] In the joint state space of atoms and phonons In this process, the diagonalization of the atom-phonon interaction Hamiltonian yields two eigenstates of the system:

[0080]

[0081]

[0082] The probability amplitude is defined as follows:

[0083]

[0084]

[0085] The relevant parameters are defined as follows:

[0086]

[0087]

[0088] .

[0089] Based on the standard Rapid Adiabatic Passage (RAP) strategy, the method for implementing the rapid adiabatic process is designed as follows:

[0090] Positive rapid adiabatic process: Selecting to use eigenstates The evolution time is Initially, the frequency (detuning) and intensity of the control laser LES are adjusted to... ,at this time, , The system is in In terms of attitude; subsequently, in Within a given time period, the detuning of the laser LES is adjusted and controlled to achieve an equivalent detuning. linearly from Change to Simultaneously, the intensity of the laser LES is adjusted to change the equivalent Rabi frequency. So that its variation with time satisfies a Gaussian function, i.e. ,in, The width of the Gaussian function. Finally, we arrived. At time t, make the probability amplitude , The system is in In terms of attitude.

[0091] Reverse rapid adiabatic process: Selecting to use eigenstates The evolution time is Initially, the frequency (detuning) and intensity of the control laser LES are adjusted to... ,at this time, , The system is in In terms of attitude; subsequently, in Within a given time period, the detuning of the laser LES is adjusted and controlled to achieve an equivalent detuning. linearly from Change to Simultaneously, the intensity of the laser LES is adjusted to change the equivalent Rabi frequency. So that its variation with time satisfies a Gaussian function, i.e. ,in, The width of the Gaussian function. Finally, we arrived. At time t, make the probability amplitude , The system is in In terms of attitude.

[0092] The equivalent mistuning amount is calculated based on the above process. and equivalent Rabi frequency Then, by using their relationship with the detuning amount and intensity of the control laser LES, the relationship between the detuning amount and intensity of the control laser LES and time is solved in reverse, such as... Figure 4 and Figure 5 As shown.

[0093] Utilizing the system's dark state The reverse of the aforementioned rapid adiabatic process can separate the phonon population in the atomic internal states, that is, when the reverse rapid adiabatic process acts on the superposition state... When it is above, the following process occurs:

[0094]

[0095] During rapid adiabatic processes, the strength of the equivalent atom-phonon interaction increases with the phonon number n, and the required adiabatic evolution time... It decreases as the phonon number n increases. This is achieved by uniformly setting the adiabatic evolution time. A reverse rapid adiabatic process is performed to achieve the overall adiabatic transformation of the Fock superposition state, i.e. .

[0096] In addition, during rapid adiabatic processes, the cavity mode detuning is positive and approximately equal to the resonant frequency of the nanoparticles, meaning that the frequency driving the laser LC is less than the cavity mode frequency.

[0097] This implementation example uses numerical simulations to verify the forward and reverse rapid adiabatic processes. Relevant results can be found in... Figure 4 (Positive) and Figure 5 (Reverse). For ease of calculation and analysis, all physical parameters involved in the system are expressed in terms of nanoparticle frequencies. ω m Normalization was performed on the units. Figure 4 and Figure 5 (a) represents the amount of laser LES detuning. The curves showing the change over time, (b) represent the control of the laser LES intensity. The curves showing the change over time, where (c) represents the quantum state. and The population change curve over time.

[0098] The relevant parameter settings in the numerical simulation are shown in the table below:

[0099]

[0100] (B) Principle of atomic state reversal process

[0101] like Figure 2 As shown in (c), when the control laser LES is turned off and LER and LGR are turned on, the auxiliary atom excited state Simultaneously with two ground states and They are coupled together. (Transition process) Driven by the control laser LER; transition process Driven by the control laser LGR; and the excited state Due to the cavity mold and Large mistuning between states is not related to A transition occurs between states, causing the atoms and the center of mass of the particles to decouple. The Hamiltonian of the entire system is:

[0102]

[0103] in, It is an auxiliary excited state The frequency; It controls the driving intensity and frequency of the laser LER; This controls the driving intensity and frequency of the laser LGR. Under large detuning conditions, the atomic excited states and cavity modes are successively adiabatically eliminated, yielding the equivalent Hamiltonian of the system as:

[0104]

[0105] in, It is the equivalent resonant frequency of the nanoparticles; It is the equivalent detuning quantity; It is the equivalent Rabi frequency.

[0106] This shows that the center-of-mass motion of nanoparticles is completely decoupled from the atoms, meaning that manipulating the atoms does not affect the center-of-mass motion of the nanoparticles. Ignoring the nanoparticle term in the equivalent Hamiltonian, the Hamiltonian for the atoms represents the standard classical light field-two-level atom interaction model in quantum optics—the atoms interact at Rabi frequencies. It oscillates periodically between the upper and lower energy levels. Therefore, an atomic state flip can be achieved using a π pulse, i.e. and .

[0107] (C) Principle of Population Transfer Process

[0108] Similar to the principle of atomic state flipping, the population transfer process is also modeled in quantum optics as the standard classical light field-two-level atom interaction model, which uses a laser beam to drive the ground state. With auxiliary state Energy level transition process Their interaction Hamiltonian is:

[0109]

[0110] in, Rabi frequency (coupling strength).

[0111] Therefore, an atomic state transition process can be achieved using a single π pulse. This allows the dark-state population to be transferred to the auxiliary energy level. It will no longer participate in subsequent measurement processes.

[0112] (D) Principle of Acceleration Measurement

[0113] Initially, the center of mass motion of the nanoparticles is cooled to the ground state through active feedback cooling. The four-level atom is initialized in the ground state. Above, the joint quantum state of four-level atoms and nanoparticles is .

[0114] (I) Preparation:

[0115] After one rapid adiabatic process, the quantum state of the system evolves as follows:

[0116]

[0117] Subsequently, an atomic state flipping process is performed, and the system's quantum state evolves as follows:

[0118]

[0119] Next, the above two steps are executed iteratively in sequence. (until the particle's center of mass is driven to the target Fock state) superior.

[0120] (II) Evolution:

[0121] With the control lasers LES, LER, and LGR shut off, and the driving laser LC turned off, atoms and nanoparticles are completely decoupled and evolve independently. The center-of-mass motion of the nanoparticles evolves for a period of time under the action of the measured acceleration. Atoms, because they are not subject to interaction, remain in a state of flux. Above. After the evolution is complete, the center of mass of the nanoparticles is in a Fock superposition state. Above. The acceleration to be measured is encoded in the projection probability of the particle's center of mass motion onto the initial target Fock state. For a detailed analysis, see (E) Relationship between acceleration and projection probability.

[0122] (III) Measurement:

[0123] After the evolutionary steps are completed, the joint quantum state of atoms and nanoparticles is: The principle for measuring projective probability is as follows:

[0124] Based on the target projection probability to be obtained Select and perform the following corresponding operation steps:

[0125] (i) When the initial target Fock state n=0, obtain the projection probability P0;

[0126] After performing a reverse rapid adiabatic process, the quantum state of the system evolves as follows:

[0127]

[0128] Utilizing dark states By leveraging the non-evolutionary nature of the system under the influence of the Hamiltonian, the rapid adiabatic process enables the population distribution of phonon numbers within the atomic internal states. Effective separation. Subsequently, through the analysis of atomic states. By performing fluorescence detection, the corresponding probability can be obtained. .

[0129] (ii) When the initial target Fock state n=1, obtain the projection probability P1;

[0130] If a probability P1 is needed to be obtained further, fluorescence detection of the atomic states will not be performed for the time being based on the evolution in the previous step. First, a population transfer process based on π pulses will be executed to transfer the dark state... The population is transferred from the upper energy level to the auxiliary energy level, so that it no longer participates in the subsequent measurement process. At this time, the system is in a state of... .

[0131] Subsequently, an atomic state flipping process is performed, and the system's quantum state evolves as follows:

[0132] ;

[0133] Next, the rapid adiabatic process is executed again, and the system's quantum state further evolves as follows:

[0134]

[0135] At this time, the measured probability amplitude c g,1 Successfully transferred to the dark state Above. Finally, regarding the atomic state. By performing fluorescence detection, the probability P1 = P g .

[0136] (iii) When the initial target Fock state n≥2, the projection probability P is obtained. n (n≥2):

[0137] And so on, if a higher order probability P is needed... n (n≥2), fluorescence detection is temporarily suspended. Instead, based on the previous step, the reverse rapid adiabatic process, the π-pulse-based population transfer process, and the atomic state flipping process are executed iteratively in sequence. Through multiple iterations, the probability amplitude c to be measured is reached. g,n Ultimately appears in the dark state Above; finally, a reverse rapid adiabatic process is performed to separate the state and to the atomic state. Perform fluorescence detection to obtain probability. .

[0138] By following the steps above, the evolution time can be accurately obtained. The projected probability under the following conditions .

[0139] (E) Relationship between acceleration and projection probability

[0140] In step (II) of the evolution, the control lasers LES, LER, and LGR, and the driving laser LC are turned off, completely decoupling the atoms and nanoparticles, allowing them to evolve independently. The center-of-mass motion of the nanoparticles evolves for a period of time under the action of the measured acceleration. Atoms, because they are not subject to interaction, remain in a state of flux. Above. The Hamiltonian of the nanoparticles is:

[0141]

[0142] Where f=ma is the external force introduced to the acceleration to be measured, m is the mass of the nanoparticle, and a is the acceleration to be measured. It is an introduced coupling parameter. This is achieved by introducing an operator. ,in, For the displacement operator, the Hamiltonian is diagonalized as follows:

[0143]

[0144] Therefore, the time evolution operator of the system can be obtained as follows:

[0145]

[0146] From the properties of the displacement operator, we know that:

[0147]

[0148]

[0149] Substituting this into the expression of the system's time evolution operator yields...

[0150]

[0151] Initially, the center-of-mass motion of the nanoparticles was prepared in In terms of state, the wave function at any given time can be obtained from the time evolution operator as follows:

[0152]

[0153] The projection probability of the wave function onto the initial Fock state at any given time is (to be measured)

[0154]

[0155] Using the relation of the shift operator in quantum optics:

[0156]

[0157] in, It is an nth-order Laguerre polynomial, and the projected probability can be written as

[0158]

[0159] In step (II) of the evolution, the time for the system to evolve freely is... Therefore, the relationship between the measured acceleration and the projected probability is as follows:

[0160]

[0161] It will be understood by those skilled in the art that the above descriptions are merely preferred examples of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent substitutions for some of the technical features. All modifications and equivalent substitutions made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. An acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles, characterized in that, A four-level atom and a nanoparticle are simultaneously confined in an optical cavity using optical tweezers. The optical cavity is driven by a driving laser LC, which excites and forms a stable cavity optical field, i.e., a cavity mode. The cavity optical field is simultaneously coupled with the internal energy level transitions of the four-level atom and the center-of-mass motion of the nanoparticle. Three control lasers, LES, LER, and LGR, are used to drive the three energy level transition processes of the four-level atom, respectively. The acceleration measurement method includes the following steps: Step 1: Cool the center-of-mass motion of the nanoparticles to the ground state through active feedback cooling. The four-level atom is initialized in the ground state. Above; at this point, the joint quantum state of atoms and nanoparticles is ; Step 2: Calculate and set the laser control parameters for the forward rapid adiabatic process. By manipulating the three control lasers LES, LER and LGR, the forward rapid adiabatic process and atomic state reversal process are executed iteratively until the center of mass of the nanoparticles is moved to the initial target Fock state. Step 3: Turn off the control lasers LES, LER, and LGR, and the driving laser LC, allowing the center of mass of the nanoparticles to freely evolve under the action of the acceleration to be measured for a period of time. ; Step 4: Extract the projection probability of the nanoparticles in the initial target Fock state, and obtain the acceleration to be measured based on the projection probability; the extraction of projection probability specifically includes: When the initial target Fock state n=0, the laser control parameters for the reverse rapid adiabatic process are calculated and set, and the reverse rapid adiabatic process is executed for the atomic state. Fluorescence detection was performed to obtain the projection probability P0 of the nanoparticles onto the initial target Fock state; When the initial target Fock state n=1, the laser control parameters for the reverse rapid adiabatic process are calculated and set. The reverse rapid adiabatic process, the π-pulse-based population transfer process, the atomic state flipping process, and the reverse rapid adiabatic process are executed sequentially. Fluorescence detection was performed to obtain the projection probability P1 of the nanoparticles onto the initial target Fock state; When the initial target Fock state n≥2, the laser control parameters for the reverse rapid adiabatic process are calculated and set, and the reverse rapid adiabatic process, the population transfer process based on π pulses, and the atomic state flipping process are executed iteratively until the target projection probability amplitude c is reached. g,n Appears in dark state Up; then perform the reverse rapid adiabatic process, followed by the atomic state. Fluorescence detection was performed to obtain the projection probability P of the nanoparticle onto the initial target Fock state. n .

2. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The four-level atom has two excited states. and and two ground states and ; Among them, controlling the laser LES-driven energy level transition process Controlling the laser LER-driven energy level transition process Controlling the laser LGR-driven energy level transition process ; The cavity mode excited by the driving laser LC within the optical cavity is simultaneously with the energy level transition process. The motion of the center of mass of the nanoparticles is coupled.

3. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The positive rapid adiabatic process is achieved in the following way: The control lasers LER and LGR are turned off, and the control laser LES is turned on, so that the four-level atoms and the center of mass motion of the nanoparticles are coupled to form an equivalent atom-phonon interaction. The detuning amount and intensity of the control laser LES are adjusted so that the equivalent detuning amount changes linearly from a positive value to a negative value, and the change of the equivalent Rabi frequency with time satisfies a Gaussian function, thereby realizing a positive rapid adiabatic process.

4. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The reverse rapid adiabatic process is achieved in the following way: The control lasers LER and LGR are turned off, and the control laser LES is turned on, so that the four-level atoms and the center of mass motion of the nanoparticles are coupled to form an equivalent atom-phonon interaction. The detuning amount and intensity of the control laser LES are adjusted so that the equivalent detuning amount changes linearly from a negative value to a positive value, and the change of the equivalent Rabi frequency with time satisfies a Gaussian function, thereby realizing a reverse rapid adiabatic process.

5. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The acceleration to be measured is obtained by solving the following system of equations: ； Among them, P n ξ is the projection probability of the particle's final state after the measured acceleration onto the initial target Fock state; ξ is an intermediate variable. The evolution time of the particle under the action of the measured acceleration is the time during which it freely evolves. It is an nth-order Laguerre polynomial; m is the mass of the nanoparticle; a is the acceleration to be measured; is the reduced Planck constant.

6. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The atomic state flipping process is achieved in the following way: Turn off the control laser LES and turn on the control lasers LER and LGR to decouple the center-of-mass motion of the four-level atoms from that of the nanoparticles; regulate the control lasers LER and LGR to generate a π pulse to realize the atomic state flipping process.

7. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The population transfer process based on π pulses is achieved in the following way: Apply a ground state to a four-level atom With auxiliary state The π pulse drives the transition process , will dark state The population on the state is transferred to the auxiliary state, so that the population is hidden in the auxiliary state.

8. The acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles according to claim 1, characterized in that, The frequency of the driving laser LC is lower than the frequency of the cavity mode.

9. An acceleration measurement device based on the quantum state of motion of the center of mass of optically suspended particles, characterized in that, Used to implement the acceleration measurement method based on the quantum state of motion of the center of mass of optically suspended particles as described in any one of claims 1 to 8; The optical axis of laser one coincides with the optical axis of the optical cavity, which is used to drive the optical cavity from one side, excite and form a stable cavity optical field within the cavity; optical tweezers one is used to trap four-level atoms in the optical cavity, and optical tweezers two is used to trap nanoparticles in the optical cavity; optical tweezers one and optical tweezers two can freely adjust the positions of four-level atoms and nanoparticles in the optical cavity to change the coupling strength of the interaction. The beam generated by laser 2 is split into control lasers LES, LER, and LGR to drive the three energy level transitions of the four-level atoms. Specifically: half-wave plate 1 and polarization beam splitter 1 are located sequentially in the optical path of laser 2. Polarization beam splitter 1 splits the beam generated by laser 2 into beam 1 and beam 2. Half-wave plate 3, optical modulator 1, half-wave plate 4, and mirror 2 are sequentially arranged in the optical path of beam 1, so that beam 1 illuminates the four-level atoms to form control laser LES. Half-wave plate 2 and polarization beam splitter 2 are sequentially arranged in the optical path of beam 2, so that beam 2 is split into beam 3 and beam 4 again. Half-wave plate 5, optical modulator 2, half-wave plate 6, and mirror 3 are sequentially arranged in the optical path of beam 3, so that beam 3 illuminates the four-level atoms to form control laser LER. Mirror 1, half-wave plate 7, optical modulator 3, half-wave plate 8, and mirror 4 are sequentially arranged in the optical path of beam 4, so that beam 4 illuminates the four-level atoms to form control laser LGR.

Images