An optical microcavity assisted super-radiant phase transition implementation method and system

Abstract

The present disclosure relates to the technical field of quantum spin-based optical and matter interaction research models, and proposes an optical microcavity-assisted super-radiation phase transition implementation method and system. First, a time-oscillating magneto-optical trap is adopted to generate a simple harmonic potential field and bind a Bose cold atomic beam, and the bound Bose cold atomic beam is coupled to a high-precision optical microcavity to study the physical properties in the microcavity. An external magnetic field is applied to split the Bose-Einstein condensate ground state, and after coupling, the super-radiation quantum phase transition is obtained by adjusting the coupling strength. The super-radiation quantum phase transition can be used to study the self-organization phase transition and spin dynamics of the Bose-Einstein condensate in the microcavity, and also provides a reference for studying other spin-based Bose-Einstein condensate problems.

Description

Technical Field

[0001] This disclosure relates to the technical field of light-matter interaction research models based on quantum spin, specifically to a method and system for realizing superradiative phase transitions assisted by an optical microcavity. Background Technology

[0002] The statements in this section are merely background information relating to this disclosure and do not necessarily constitute prior art.

[0003] Since scientists successfully experimentally realized Bose-Einstein condensate (BEC) of a rarefied atomic gas in 1995, highly controllable ultracold atom physics has become a hot research area in physics, attracting increasing attention from scientists both experimentally and theoretically. For example, Feshbach resonances have been predicted and observed in ultracold atom systems, optical lattice constructions have been realized, and spin-orbit coupling has been achieved. With the continuous development of experimental techniques, quantum spin systems, the most crucial element in the field of ultracold atoms, not only possess rich physical properties but also have broad application prospects, especially in quantum information processing such as quantum computing, quantum storage, and quantum sensing. Therefore, studying the many-body effects of quantum spin systems has become a popular research direction in physics.

[0004] Compared with the traditional method of changing the spin degree of freedom by applying an external magnetic field, the spin-orbit coupling method, which couples the spin degree of freedom and the motion degree of freedom of atoms, is a new way to control spin. With the continuous realization of artificial spin-orbit coupling in cold atom systems in experiments, the study of many novel physical phenomena based on spin-orbit coupling has been widely promoted.

[0005] On the other hand, since the successful experimental realization of the superradiative quantum phase transition in 2010, systems coupling ultracold atomic gases with cavity quantum electrodynamics have become ideal platforms for exploring novel many-body physics, sparking a research boom among theoretical and experimental scientists. This coupling system couples ultracold atoms into a high-precision optical microcavity. Under specific electromagnetic boundary conditions, light interacts with the ultracold atoms, inducing novel many-body quantum properties. In this coupling system, researchers can not only explore the complex quantum behavior induced by infinite-range interactions between atoms mediated by cavity photons, but also understand the collective dynamics of cavity photons and ultracold atoms at the single-photon level. Furthermore, the optical microcavity, with its inherent drive and dissipation mechanisms, is a naturally non-equilibrium system, allowing for the study of non-equilibrium steady-state dynamics. With the continuous development of experimental techniques, artificial spin-orbit coupling has recently been achieved in an ultracold atom-optical microcavity coupling system.

[0006] The inventors discovered in their research that the time-dependent spin dynamics of cavity-assisted spin systems have not been considered experimentally or theoretically. This is because, on the one hand, the Schrödinger equation, which is explicitly time-dependent, is mathematically difficult to solve precisely; on the other hand, the physical processes expressed by the Schrödinger equation, which is explicitly time-dependent, involve complex energy changes, time evolution, and interaction problems, which further increases the difficulty of solving the problem. Summary of the Invention

[0007] To address the aforementioned issues, this disclosure proposes an optical microcavity-assisted superradiative phase transition realization method and system. By coupling the optical microcavity system with a Bose-Einstein condensate confined in a time-oscillating harmonic potential well, a new model is obtained. This model can be used to study the self-organized phase transition and spin dynamics characteristics of Bose-Einstein condensates within microcavities. It also provides a reference for studying other spin-based Bose-Einstein condensate problems and offers a research model for the field of optical fiber communication.

[0008] To achieve the above objectives, the present disclosure adopts the following technical solution:

[0009] One or more embodiments provide a method for realizing an optical microcavity-assisted superradiative phase transition, comprising the following steps:

[0010] Bose-Einstein condensates were prepared in a magneto-optical trap that oscillates over time, and the Bose atom gas, which is bound in an oscillating harmonic potential field, was coupled in an optical microcavity.

[0011] A tunable external magnetic field is applied to the Bose-Einstein condensate coupled in the optical microcavity so that the ground state of the Bose-Einstein condensate in the optical microcavity is split by Zeeman to obtain two hyperfine ground states.

[0012] The two hyperfine ground states of Bose-Einstein condensation are coupled through two independent Raman channels via pump light and cavity mode light field;

[0013] The coupling strength between the hyperfine ground state of an atom and the cavity field within the optical microcavity is controlled until the coupling strength reaches a critical value, at which point a superradiative quantum phase transition occurs.

[0014] One or more embodiments provide an optical microcavity-assisted superradiative phase transition realization system, including: a time-oscillating magneto-optical trap, an optical microcavity, an external magnetic field application device, a driving light supply device, a pump, a coupling module, and a control device;

[0015] A magneto-optical trap that oscillates over time is used to prepare Bose-Einstein condensates;

[0016] Optical microcavities are used to couple Bose atom gas, which is bound in an oscillating harmonic potential field, into optical microcavities.

[0017] An external magnetic field application device is used to apply a controllable external magnetic field to a Bose-Einstein condensate coupled to an optical microcavity and confined to an oscillating harmonic potential field, so that the ground state of the Bose-Einstein condensate in the cavity is split by Zeeman to obtain two hyperfine ground states.

[0018] Based on the driving light providing device, it is used to inject driving light along the cavity axis of the optical microcavity to drive the cavity mode optical field;

[0019] Pumping is used to inject two traveling wave pump lights in a direction perpendicular to the cavity axis of the optical microcavity to pump Bose-Einstein condensates.

[0020] The coupling module is used to couple the Bose-Einstein condensate with the cavity mode light field and the pump light. The two hyperfine states of the Bose-Einstein condensate are coupled by the pump light and the cavity mode light field through two independent Raman channels.

[0021] The control device is used to control the Rabi frequency of the coupling between the pump light and the cavity mode light field and the Bose-Einstein condensate, thereby controlling the coupling strength between the hyperfine ground state of the atom and the cavity field. When the coupling strength is greater than a certain critical value, the system undergoes a superradiative quantum phase transition.

[0022] Compared with the prior art, the beneficial effects of this disclosure are as follows:

[0023] In this embodiment, a time-oscillating magneto-optical trap is first used to generate a harmonic potential field and confine the Bose cold atom gas within it. The confined Bose cold atom gas is then coupled into a high-precision optical microcavity, where its physical properties can be studied. An external magnetic field is applied to split the ground state of the Bose-Einstein condensate. After coupling, the superradiative quantum phase transition is obtained by adjusting the coupling strength. The superradiative quantum phase transition can be used to study the self-organized phase transition and spin dynamics of the Bose-Einstein condensate within the microcavity, and also provides a reference for studying other spin-based Bose-Einstein condensate problems.

[0024] The advantages of this disclosure, as well as its additional advantages, will be described in detail in the following specific embodiments. Attached Figure Description

[0025] The accompanying drawings, which form part of this disclosure, are used to provide a further understanding of this disclosure. The illustrative embodiments of this disclosure and their descriptions are used to explain this disclosure and do not constitute a limitation thereof.

[0026] Figure 1 This is a schematic diagram of the structure of an optical microcavity-assisted superradiative phase transition realization system according to Embodiment 2 of this disclosure;

[0027] Figure 2 This is an atomic energy level diagram containing four internal states, according to Embodiment 1 of this disclosure;

[0028] Figure 3 This is a schematic diagram of the change in self-organized order density of Bose-Einstein condensate before and after the superradiative phase transition in Embodiment 1 of this disclosure;

[0029] Figure 4(a) is the first planar ground state phase diagram of Embodiment 1 of this disclosure;

[0030] Figure 4(b) is the second planar ground state phase diagram of Embodiment 1 of this disclosure;

[0031] Figure 5(a) shows the curves of the average value of the Pauli operator over time under different coupling intensities when the vibration intensity is fixed in Embodiment 1 of this disclosure.

[0032] Figure 5(b) shows the curves of the average value of the Pauli operator over time under different vibration intensities when the coupling strength is fixed in Embodiment 1 of this disclosure.

[0033] Figure 6 This is a flowchart of the superradiative phase transition implementation method of Embodiment 1 of this disclosure. Detailed Implementation

[0034] The present disclosure will be further described below with reference to the accompanying drawings and embodiments.

[0035] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of this disclosure. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure pertains.

[0036] It should be noted that the terminology used herein is for descriptive purposes only and is not intended to limit the exemplary embodiments according to this disclosure. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof. It should be noted that, without conflict, the various embodiments and features described in this disclosure can be combined with each other.

[0037] This disclosure couples an optical microcavity system with a Bose-Einstein condensate confined in a time-oscillating harmonic potential well, further realizing and exploring the factors influencing the superradiative quantum phase transition and the spin dynamics of the time-oscillation within this oscillating system. The novel model proposed in this invention has significant research value and provides a feasible scheme for in-depth research on cavity quantum manipulation. The embodiments will be described in detail below with reference to the accompanying drawings.

[0038] Example 1

[0039] In one or more of the technical solutions disclosed in the embodiments, such as Figures 1 to 6As shown, a method for realizing superradiative phase transition assisted by an optical microcavity includes the following steps:

[0040] Step 1: Prepare Bose-Einstein condensate in a magneto-optical trap that oscillates over time, and couple the Bose atom gas confined in the oscillating harmonic potential field into an optical microcavity;

[0041] Step 2: Apply a tunable external magnetic field to the Bose-Einstein condensate coupled in the optical microcavity so that the ground state of the Bose-Einstein condensate in the optical microcavity is split by Zeeman to obtain two hyperfine ground states.

[0042] Step 3: Couple the two hyperfine ground states of Bose-Einstein condensation through two independent Raman channels using pump light and cavity mode light field;

[0043] Step 4: Adjust the coupling strength between the hyperfine ground state of the atom and the cavity field inside the optical microcavity until the coupling strength reaches a critical value, at which point a superradiative quantum phase transition occurs.

[0044] In this embodiment, a time-oscillating magneto-optical trap is first used to generate a harmonic potential field and confine Bose cold atom gas within it. The confined Bose cold atom gas is then coupled into a high-precision optical microcavity, where its physical properties can be studied. An external magnetic field is applied to split the ground state of the Bose-Einstein condensate. After coupling, a superradiative quantum phase transition is obtained by adjusting the coupling strength. This superradiative quantum phase transition can be used to study the self-organized phase transition and spin dynamics of Bose-Einstein condensate within the microcavity, and also provides a reference for studying other spin-based Bose-Einstein condensate problems.

[0045] In step 1, based on current experimental techniques, Bose-Einstein condensate is first prepared in a magneto-optical trap that oscillates over time, and then the Bose atom gas confined in the oscillating harmonic potential field is coupled in a high-precision optical microcavity.

[0046] Among them, the magneto-optical trap is used to provide a harmonic potential field to trap Bose-Einstein condensates.

[0047] In some embodiments, Bose-Einstein condensates are prepared using a magneto-optical trap, such as... Figure 1 As shown, Bose-Einstein condensates can be confined within a time-oscillating harmonic potential field confined in the yz plane and along the x-direction, and the corresponding harmonic potential field is:

[0048] V(x,t)=mω 2 [x-x0(t)] 2 / 2

[0049] Where m is the atomic mass, ω is the frequency of the bound potential field in the x direction, and the center x0(t) of the harmonic potential field changes with time.

[0050] The process of the atom starting to undergo simple harmonic periodic oscillation along the x-direction at time t0 is completely determined by the expression for x0(t). Without loss of generality, x0(t) can be adopted as a Gaussian distribution, with the following specific form:

[0051]

[0052] Where ξ0 is the peak amplitude, also known as the vibration intensity; σ t t represents the time width, t represents the time, and t0 represents the initial time.

[0053] In this embodiment, the Bose-Einstein condensate confined within the oscillating harmonic potential field is coupled to a high-precision optical microcavity, thus realizing a one-dimensional coupling system. The Bose-Einstein condensate moves only along the x-direction, the cavity mode is driven by a linearly polarized laser along the cavity axis x, and the condensate is pumped by two transverse laser beams along the y-direction. Furthermore, this embodiment considers the use of a time-oscillating magneto-optical trap, which can generate a harmonic potential field and confine the Bose cold atom gas within the trap.

[0054] In step 2, an adjustable external magnetic field is applied to the Bose-Einstein condensate coupled to the optical microcavity and confined in the oscillating harmonic potential field, so that the ground state of the Bose-Einstein condensate in the cavity is split by Zeeman to obtain two hyperfine ground states.

[0055] In step 3, optionally, driving light can be injected along the cavity axis to drive the cavity mode light field, and two traveling wave pump light beams can be injected in the direction perpendicular to the cavity axis to pump the Bose-Einstein condenser. The Bose-Einstein condenser is coupled to the cavity mode light field and the pump light respectively, and the two hyperfine ground states of the Bose-Einstein condenser are coupled together by the pump light and the cavity mode light field through two independent Raman channels.

[0056] In step 4, the coupling strength can be adjusted by controlling the Rabi frequency of the coupling between the pump light and the Bose-Einstein condensate, and by controlling the Rabi frequency of the coupling between the cavity mode light field and the Bose-Einstein condensate. That is, the pump light is coupled with the Einstein condensate, and the cavity mode light field is coupled with the Bose-Einstein condensate.

[0057] By separately controlling the Rabi frequency of the pump light, cavity mode light field, and Bose-Einstein condensate coupling, the coupling strength between the hyperfine ground state of the atom and the cavity field can be controlled. When the coupling strength is greater than a certain critical value, the system will undergo a superradiative quantum phase transition.

[0058] Furthermore, by adjusting the applied magnetic field, the critical value for the phase transition of the superradiative quantum changes with the change in magnetic field strength, thereby adjusting the critical value for the phase transition of the superradiative quantum.

[0059] In this embodiment, adjusting the phase transition influencing factor is achieved by indirectly adjusting the resonant frequency of the atomic ground state by adjusting the strength of the external magnetic field, thereby causing a change in the phase transition point of Bose-Einstein condensation.

[0060] Furthermore, by controlling the oscillation intensity and oscillation time of the oscillating harmonic potential field, the spin dynamics characteristics assisted by the microcavity are obtained.

[0061] The following is a detailed explanation.

[0062] like Figure 2 As shown, consider the atom as a Bose-Einstein condensate with four internal energy levels, namely two hyperfine ground states (|↑> and |↓>) and two excited states (|1> and |2>).

[0063] and The transition between them is caused by a quantized cavity field with a coupling strength of g. and The transition is caused by a transverse pump light with a Rabi frequency of Ω.

[0064] Bose-Einstein condensation interacts with both the cavity mode light field and the pump light, generating two independent Raman processes that couple the two hyperfine ground states |↑ and |↓>. The Rabi frequency of the atom with the pump light is Ω, the coupling strength between the atom and the cavity mode is g, and the cavity mode frequency is ω. c With the frequency ω of the pump light p They are close, but both are far off-harmonic to the transition frequency ω of the Bose atom. a That is, the detuning frequency is Δ A =ω p -ω a And satisfy |Δ A |>>g, Ω, is the condition for large detuning;

[0065] Under the aforementioned large detuning conditions, the excited states of Bose-Einstein condensates can be adiabatically removed, taking into account a deep harmonic potential well (the optical potential well of the cavity field is negligible) and a second quantization of the atomic degrees of freedom. Figure 1 The Hamiltonian of the system can be expressed as

[0066]

[0067] Where, ξ σ =ξ ↑ =-ξ ↓ =1,2δ=ω ↑ -ω ↓ =m z It is the ultrafine ground state |↑> and ↓> energy splitting, ω ↑ and ω ↓These are the characteristic frequencies of the two ground states, respectively. For the photon generation operator, For the annihilation operator of photons, η = Ωg / Δ a For the coupling strength of the cavity-assisted Raman process, Δ C =ω p -ω c This represents cavity field mistuning; k0 is the pump's reaction impulse. Let x be the momentum operator of the atom in the x-direction. For the boson field operator corresponding to the spin state, and Follows the boson commutation relation;

[0068] It can be seen from the Hamiltonian (1) that the interaction between light and matter couples the internal spin degree of freedom of the atom with the external motion degree of freedom, and achieves spin flipping.

[0069] Optionally, the average field approximation method is applied to the field operators of the cavity field and atoms, and the time-dependent system is processed to obtain the superradiative phase transition of the system, thereby identifying the factors affecting the superradiative quantum phase transition, including the following steps:

[0070] (1) The Heisenberg equations of motion for the cavity field operator and the Bose field operator are given, and the cavity field degrees of freedom and the atomic field operator are treated with mean field methods, i.e. (The field operator is approximately expanded into a condensed-state wave function), and the following coupled mean-field equation can be obtained from the Hamiltonian (1):

[0071] Rate, N is the number of atoms, α is the optical field order parameter, ψ σ For the wave function corresponding to the spin state, For ψ σ The complex conjugate of i, where i is an imaginary number;

[0072] (2) The coupled mean-field equations are simplified; and the Hamiltonian of the Bose-Einstein condensate system is transformed by a unitary transformation to obtain the Hamiltonian that evolves with time.

[0073] It can be seen that equation (3) is the nonlinear Schrödinger equation for the condensed matter wave function assisted by the microcavity, which is the Gross-Pitaevskii (GP) equation; from equations (2) and (3), it can be seen that using Replacing the field amplitude in the equation allows for parameter scaling without affecting the final calculation result. Keep it constant. That is, in the mean-field model, the number of atoms can be incorporated into the system parameters and field amplitude variables; therefore, during calculation, it can be... It is considered as a whole.

[0074] Hamiltonian (1) and GP equation (3) are time-dependent equations. To facilitate solving, the Hamiltonian (1) is subjected to the following unitary transformation to obtain the Hamiltonian that evolves with time:

[0075] U = exp[-imω] 2 F(t)] (4)

[0076]

[0077]

[0078] Where erf(x) is the error function;

[0079] Then the Hamiltonian (1) can be written as:

[0080]

[0081] Assume that the time width and vibration intensity satisfy σ t →0 and ξ o →∞, and It can remain unchanged. Under this condition, the graph of x0(t) with respect to time can be regarded as a Delta pulse, which can be expressed as F(t)=F0Θ(t-t0), where Θ(t-t0) is a step function. The Hamiltonian (5) that evolves with time can be rewritten in the following form:

[0082]

[0083] (3) Based on the transformed Hamiltonian, the time-independent GP equation is obtained, and the order parameter characterizing the superradiative phase transition is calculated under the corresponding steady-state cavity field.

[0084] Starting from the Hamiltonian (6), we can re-obtain the time-independent GP equation as follows:

[0085]

[0086] Here, Re{} refers to solving for the real number part within the parentheses, and is a mathematical symbol.

[0087] Based on the understanding of the fundamental properties of the microcavity-ultracold atom coupled system, it is first necessary to use ω as the energy normalization unit. -1 For time normalization units, and The equations (2) and GP (7) are simplified to length normalization units. Next, the many-body ground state properties of the system need to be analyzed. Due to the dissipation of the cavity field in the experiment, when the cavity field dissipation is large, 1 / κ is much smaller than the atomic degrees of freedom. At this point, the cavity field can be adiabatically removed, i.e. The specific form of the corresponding steady-state cavity field can be obtained from equation (2), and the order parameter α0 is as follows:

[0088]

[0089] (4) According to the order parameter and the time-independent GP equation, the condensed matter wave function is obtained by a numerical calculation method, and then the magnitudes of various factors in the condensed matter wave function formula are adjusted to obtain the critical phase transition point at which superradiance phase transition can occur.

[0090] Substituting equation (8) into GP equation (7), the condensed matter wave function is obtained by a numerical calculation method. Specifically, the imaginary-time evolution method is used to solve the GP equation (7), that is τ=it. Conversely, substituting the obtained condensed matter wave function into equation (8) can obtain the evolution of the order parameter α0 that characterizes the occurrence of the superradiance phase transition with the coupling strength. When the coupling strength reaches a certain critical value, collective excitation of the cavity field occurs, that is, the superradiance phase transition occurs in the system, as shown in Figure 3 It is worth noting that, for the two different cases of t<t0 and t>t0, comparative analysis shows that changing the two cases changes the critical phase transition point for the occurrence of the superradiance phase transition, but the trend is the same. Therefore, only the ground state properties when t>t0 are given in this embodiment.

[0091] In addition to the coupling strength of light-matter interaction, the effective magnetic field m experienced by atoms z The magnitude of will also affect its phase transition. The presence or absence of a magnetic field will correspondingly change the symmetry of the system. By adjusting m z the steady-state phase diagram of the system is obtained, as shown in Figure 4(a). There are two phases in the system: the normal phase (N) and the superradiant phase (SR). It can be seen that the critical point of the superradiance phase transition increases monotonically with the increase of the magnetic field strength m z

[0092] The vibration intensity ξ0 of the harmonic potential well also affects the occurrence of the phase transition. Since the vibration intensity is coupled with the atomic motion, changing the vibration intensity will correspondingly change the motion of the atoms. Therefore, under different vibration intensity conditions, the critical point of the superradiance phase transition will change accordingly, as shown in Figure 4(b).

[0093] Further, on the basis of obtaining the order parameter α0 by the numerical self-consistent method, in order to describe the non-trivial spin dynamics induced by the effective light-atom interaction, it is necessary to select a physical quantity that can describe this property. Based on the light-matter interaction, the two internal states with opposite boson spins are coupled, and the two orbital states with opposite spins are non-orthogonal. The Pauli operator σ along the x-direction is selected x The average value of (t) <σ x (t)> is used to analyze the spin dynamics properties, as shown in Figure 5.​

[0094] Where, σ x (t) represents the mixture of two spin internal states;

[0095] This embodiment studies the nontrivial spin dynamics by qualitatively analyzing the average value of the Pauli operator, specifically the spin dynamics of the time-varying Bose-Einstein condensate within a microcavity. Specifically, it describes the spin dynamics of the oscillations over time as follows: when the binding potential vibration intensity is constant, and when superradiation does not occur (coupling strength is small), i.e., when the hyperfine ground state is not coupled, <σ x (t)> oscillates symmetrically at zero time; however, when the system undergoes superradiation (high coupling strength), <σ x The oscillation at zero (t) over time remains symmetrical but not smooth; while keeping all parameters except vibration intensity constant, changing the value of vibration intensity and ensuring superradiation occurs, <σ x The oscillations over time are asymmetric; the violation of symmetry means that the spin resonance effect is affected by the vibrational potential of the bound atoms.

[0096] To illustrate the above process, a specific experiment was conducted, as follows:

[0097] The selected high-precision optical microcavity is a Fabry-Perot resonator with a wavelength of 780 nm. This resonator consists of two curved mirrors with a radius of curvature of approximately 1 cm. Ignoring the basic TEM... 00 For all cavity modes other than the standard mode, the waist radius of the microcavity is 35 μm, and the free spectral range is approximately 15 GHz, therefore the cavity accuracy is approximately 5.5 × 10⁻⁶. 4 The frequency of the bound potential field is approximately ω≈2π×40Hz. By applying a gradient magnetic field to the bound resonant potential, a bound potential that vibrates continuously can be realized, with a vibration time span of σ. t =0.05 / ω≈0.2ms, and the values ​​of other parameters are approximately κ≈50kHz, Δ C ≈-130kHz, and m z ≈25kHz. The system design diagram is attached. Figure 1 The Bose-Einstein condensate selected in this embodiment is 4.1(3)×10⁻⁶. 5 indivual 87 Rb atom, 87 The two hyperfine internal states of the Rb atom are |F,m F >=|1,-1>≡|↓> and |F,m F >=|2,-2>≡|↑>, such as Figure 2 As shown.

[0098] When both photons and atoms are collectively excited, the system undergoes a corresponding superradiative phase transition. Therefore, the superradiative phase transition can be manifested through the collective excitation of photons. The number of photons (corresponding to the change in light intensity leaking from the cavity) can be detected outside the optical microcavity using a calibrated single-photon counting module. By adjusting the Rabi frequency and the single-photon coupling strength, the single-photon counting module can monitor the light intensity change in real time, thus obtaining a graph showing the change of the order parameter with the coupling strength. When the vibration intensity is adjusted to... The relevant parameter value is Δ C / ω=-400, κ / ω=200 and m z When ω = 0.1, the critical point of the phase transition is approximately... like Figure 3 The figure shows the change in self-organized order density θ of Bose-Einstein condensate before and after the superradiative phase transition.

[0099] In this embodiment, The phase boundary in the plane can be determined by time-flight imaging of Bose-Einstein condensates and changes in light intensity in the leaking cavity. Experimentally, by adjusting the vibrational intensity and coupling strength of the harmonic potential well, and simultaneously detecting changes in light intensity and atomic changes in quasi-momentum space, the transition from the normal phase to the superradiative phase can be further obtained, yielding the corresponding phase diagram. Other relevant parameters are taken as follows: Δ C / ω=-400, κ / ω=200 and m z / ω=0.1, as shown in Figure 4(a) and Figure 4(b).

[0100] Figure 4(a) is a diagram. Planar ground-state phase diagram, effective magnetic field m on the atom z The magnitude of the magnetic field affects its phase transition (the presence or absence of the magnetic field will correspondingly change the symmetry of the system), by adjusting m. z The magnitude of the magnetic field m is used to determine the steady-state phase diagram of the system, which exhibits two phases: a normal phase (N) and a superradiative phase (SR). It is found that the critical point for the superradiative phase transition increases with the magnetic field strength m. z The increase is monotonically enhanced. Figure 4(b) shows... Planar ground-state phase diagram. It can be seen that, under different vibration intensities, adjusting the coupling strength of the system can also achieve a jump from the normal (N) phase to the superradiative (SR) phase. Specifically, when... At that time, the critical point for the superradiative phase transition decreases with increasing vibration intensity. At this point, the critical point increases with increasing vibration intensity. Other relevant parameters in the figure take the following values: Δ C / ω=-400, κ / ω=200, and in Figure 4(a) And m in Figure 4(b) z / ω=0.1.

[0101] In this embodiment, it is necessary to observe the effects of light-matter coupling strength and vibrational intensity on spin dynamics characteristics. This is achieved by fixing the coupling strength and changing the vibrational intensity (or fixing the vibrational intensity and changing the coupling strength), and using time-of-flight absorption imaging technology of atoms to detect <σ. x The free flight time of atoms varies with the time of atomic flight during the detection process, ranging from 0 to 100 s, as shown in Figures 5(a) and 5(b), thus further revealing the spin dynamics characteristics induced by this coupled system.

[0102] Figure 5(a) shows the vibration intensity. When fixed, the dashed and solid lines represent the coupling strength. The graphs show the relationship curves for values ​​of 16.1 and 26.1, respectively; Figure 5(b) shows... When fixed, the dashed and solid lines represent vibration intensity. The graphs show the relationship curves for values ​​of 25 and 30, respectively. Other relevant parameters in the graphs have the following values: Δ C / ω=-400, κ / ω=200 and m z / ω=0.1.

[0103] Example 2

[0104] Based on Embodiment 1, this embodiment provides an optical microcavity-assisted superradiative phase transition realization system, including: a magneto-optical trap that oscillates with time, an optical microcavity, an external magnetic field application device, a driving light supply device, a pump, and a single-photon counting module disposed outside the optical microcavity;

[0105] A magneto-optical trap that oscillates over time is used to prepare Bose-Einstein condensates;

[0106] Optical microcavities are used to couple Bose atom gas, which is bound in an oscillating harmonic potential field, into optical microcavities.

[0107] An external magnetic field application device is used to apply a controllable external magnetic field to a Bose-Einstein condensate coupled to an optical microcavity and confined to an oscillating harmonic potential field, so that the ground state of the Bose-Einstein condensate in the cavity is split by Zeeman to obtain two hyperfine ground states.

[0108] Based on the driving light providing device, it is used to inject driving light along the cavity axis of the optical microcavity to drive the cavity mode optical field;

[0109] Pumping is used to inject two traveling wave pump lights in a direction perpendicular to the cavity axis of the optical microcavity to pump Bose-Einstein condensates.

[0110] The coupling module is used to couple the Bose-Einstein condensate with the cavity mode optical field and the pump light. The two hyperfine states of the Bose-Einstein condensate are coupled by the pump light and the cavity mode optical field through two independent Raman channels.

[0111] The control device is used to control the Rabi frequency of the coupling between the pump light and the cavity mode light field and the Bose-Einstein condensate, thereby controlling the coupling strength between the hyperfine ground state of the atom and the cavity field. When the coupling strength is greater than a certain critical value, the system undergoes a superradiative quantum phase transition.

[0112] The single-photon counting module is used to monitor changes in light intensity in real time, thereby obtaining a graph showing the change of order parameters with coupling strength.

[0113] It may also include a time-flight absorption imaging device for detecting the Pauli operator σ. x The average value of (t) varies with the free flight time of atoms.

[0114] The above description is merely a preferred embodiment of this disclosure and is not intended to limit this disclosure. Various modifications and variations can be made to this disclosure by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this disclosure should be included within the scope of protection of this disclosure.

[0115] While the specific embodiments of this disclosure have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of this disclosure. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of this disclosure are still within the scope of protection of this disclosure.

Claims

1. A method for realizing superradiative phase transition assisted by an optical microcavity, characterized in that, Includes the following steps: Bose-Einstein condensates were prepared in a magneto-optical trap that oscillates over time, and the Bose atom gas, which is bound in an oscillating harmonic potential field, was coupled in an optical microcavity. A tunable external magnetic field is applied to the Bose-Einstein condensate coupled in the optical microcavity so that the ground state of the Bose-Einstein condensate in the optical microcavity is split by Zeeman to obtain two hyperfine ground states. The two hyperfine ground states of Bose-Einstein condensation are coupled through two independent Raman channels via pump light and cavity mode light field; The coupling strength between the hyperfine ground state of an atom and the cavity field within the optical microcavity is controlled until the coupling strength reaches a critical value, at which point a superradiative quantum phase transition occurs.

2. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: Bose-Einstein condensates are influenced by a time-oscillating harmonic potential field, which is expressed as: V(x,t)=mω 2 [x-x0(t)] 2 / 2, Where ξ0 is the vibration intensity, σ t ω is the time width, m is the atomic mass, ω is the frequency of the bound potential field in the x-direction, t is the time, and t0 is the initial time.

3. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: A driving light is injected along the cavity axis to drive the cavity mode light field. At the same time, two traveling wave pump light beams are injected in the direction perpendicular to the cavity axis to pump the Bose-Einstein condensate. The Bose-Einstein condensate is coupled to the cavity mode light field and the pump light, respectively, and the two hyperfine ground states of the Bose-Einstein condensate are coupled together by the pump light and the cavity mode light field through two independent Raman channels.

4. The method for achieving superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: The coupling strength is adjusted by regulating the Rabi frequency of the coupling between the pump light and the Bose-Einstein condensate, as well as the Rabi frequency of the coupling between the cavity mode light field and the Bose-Einstein condensate.

5. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: By adjusting the applied magnetic field, the critical value for the phase transition of superradiative quantum particles can be changed with the change in magnetic field strength, thereby adjusting the critical value for the phase transition of superradiative quantum particles.

6. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: The superradiative phase transition of the system is obtained by applying the mean-field approximation method to the field operators of the cavity field and atoms and processing the time-dependent system. The factors affecting the superradiative quantum phase transition are obtained, including the following steps: The Heisenberg equations of motion for cavity field operators and Bose field operators are given, and the cavity field degrees of freedom and atomic field operators are averaged to obtain the coupled average field equations. The coupled mean-field equations are simplified; Furthermore, a unitary transformation is performed on the Hamiltonian of the Bose-Einstein condensate system to obtain the Hamiltonian that evolves over time. Based on the transformed Hamiltonian, the time-independent GP equation is obtained, and the order parameter characterizing the superradiative phase transition is calculated under the corresponding steady-state cavity field. Based on the order parameter and the time-independent GP equation, the condensed matter wave function is obtained through numerical calculation. Then, by adjusting the magnitudes of various factors in the condensed matter wave function formula, the critical phase transition point that can undergo superradiative phase transition is obtained.

7. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 1, characterized in that: Microcavity-assisted spin dynamics characteristics are obtained by controlling the oscillation intensity and oscillation time of the oscillating harmonic potential field.

8. The method for realizing superradiative phase transition assisted by an optical microcavity as described in claim 7, characterized in that: The spin dynamics are analyzed by averaging the Pauli operators along the x-direction of the bound potential field.

9. The method for achieving superradiative phase transition assisted by an optical microcavity as described in claim 8, characterized in that, The spin dynamics of time-varying Bose-Einstein condensates within microcavities are specifically described as follows: When the intensity of the bound potential vibration is constant, and when superradiation does not occur, i.e., when the hyperfine ground state is not coupled, the average value of the Pauli operator oscillates symmetrically at zero with time. When superradiation occurs, the average value of the Pauli operator oscillates at zero with time in a symmetrical but not smooth manner. Keeping all parameters except vibration intensity constant, when the value of vibration intensity is changed and superradiation is ensured, the average value of the Pauli operator oscillates asymmetrically over time; it is determined that the spin resonance effect is affected by the vibrational potential of the bound atoms.

10. A system for realizing superradiative phase transitions assisted by an optical microcavity, characterized in that, include: A magneto-optical trap that oscillates over time, comprising an optical microcavity, an external magnetic field application device, a driving light supply device, a pump, a coupling module, and a control device; A magneto-optical trap that oscillates over time is used to prepare Bose-Einstein condensates; Optical microcavities are used to couple Bose atom gas, which is bound in an oscillating harmonic potential field, into optical microcavities. An external magnetic field application device is used to apply a controllable external magnetic field to a Bose-Einstein condensate coupled to an optical microcavity and confined to an oscillating harmonic potential field, so that the ground state of the Bose-Einstein condensate in the cavity is split by Zeeman to obtain two hyperfine ground states. Based on the driving light providing device, it is used to inject driving light along the cavity axis of the optical microcavity to drive the cavity mode optical field; Pumping is used to inject two traveling wave pump lights in a direction perpendicular to the cavity axis of the optical microcavity to pump Bose-Einstein condensates. The coupling module is used to couple the Bose-Einstein condensate with the cavity mode optical field and the pump light. The two hyperfine states of the Bose-Einstein condensate are coupled by the pump light and the cavity mode optical field through two independent Raman channels. The control device is used to control the Rabi frequency of the coupling between the pump light and the cavity mode light field and the Bose-Einstein condensate, thereby controlling the coupling strength between the hyperfine ground state of the atom and the cavity field. When the coupling strength is greater than a certain critical value, the system undergoes a superradiative quantum phase transition.

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