1 Introduction

For almost a century, observations of kinematics and mass profiles of galaxy clusters [76].

This paper aims to study the interaction between ALPs with neutral atoms moving in an SG interferometer using quantum field theory (QFT) techniques. Considering an initial superposition state for neutral atoms, we investigate the time evolution of the density matrix of the system by QBE to numerically calculate the relative phase shift induced by ALP-atom coupling inside the SG interferometer [77]. We consider both the forward scattering and the collision terms of the QBE, assuming the ALP scalar field exists in a coherent state. We find the exclusion regions provided by this process and compare them with other proposals for two types of neutral atoms, \(^{3}\)He and \(^{87}\)Rb.

In the rest of the paper, we use the natural unit set \(c=\hbar =1\), while we know that all subsequent values of velocity components are some multiples of c, where c is the speed of light in vacua. The paper is organized as follows. In Sect. 2, we introduce the effective Hamiltonian of the process and the density matrix of the system, and we describe our scheme. In Sect. 3, we study the QBE and calculate its forward scattering and collision terms related to the ALP scalar field to be in a coherent state. In Sect. 4, we report the results by plotting exclusion regions for the total forward scattering and collision terms. Finally, in Sect. 5, we end with conclusions.

2 Effective interaction Hamiltonian and the density matrix of the effective two-level system

In this section, we first write down the Hamiltonian density of the interacting neutral atoms and ALPs as DM in the SG interferometer configuration. To this end, we ignore the interaction between ALPs as DM and the magnetic field of the SG interferometer.

A magnetic field provides a spatial superposition for the spin particles. Their paths create a closed loop in such a way that finally recombines the different components (for more details about the full-loop SG interferometer, see Appendix A). We will effectively characterize the closed loop using time- and spin-dependent momenta. We assume that neutral atoms are effectively a two-level system and can initially be split into a superposition state [78,79,80,81]. During the movement of these neutral atoms in the interferometer, the ALPs interact with them. After recombination, the difference phase induced by this interaction contains information about the coupling and the mass of the ALPs which have occurred within the interferometer. We introduce an effective two-level system to describe the atoms in our interferometer. We assume that such a two-level system is effectively characterized by a Dirac spinor. Hence, the effective Hamiltonian density of the interaction between these neutral atoms and the ALPs as DM is expressed as

$$\begin{aligned} \mathcal {H}_{\textrm{int}}=g\bar{\psi }\gamma ^{\mu }\gamma ^{5}\psi \partial _{\mu }\phi \,, \end{aligned}$$
(1)

in which the coupling constant of the interaction is

$$\begin{aligned} g=\frac{g_{af}}{m_{f}}\,, \end{aligned}$$
(2)

where \(m_{f}\) is the mass of the two-level neutral atom system, \(g_{af}\) is the dimensionless coupling constant of ALP-atom interaction, \(\psi \) is the Dirac spinor with its adjoint \(\bar{\psi }=\psi ^{\dag }\gamma ^{0}\), and \(\phi (\textbf{x},t)\) is the ALP scalar field. We also assume that the ALP scalar field is homogeneous, and hence one can set \(\mu =0\) in the effective Hamiltonian density (1).

To express the corresponding S-matrix element describing this process, we write down the ALP field and the spinor field in terms of the creation and the annihilation operators. The Fourier transform of the ALP field is written in terms of the creation and the annihilation operators as

$$\begin{aligned} \phi (\textbf{x},t)=\phi ^{-}(\textbf{x},t)+\phi ^{+}(\textbf{x},t)\,, \end{aligned}$$
(3)

in which we have

$$\begin{aligned} \begin{aligned}&\phi ^{-}(\textbf{x},t)=\int \frac{d^{3}\textbf{q}}{(2\pi )^{3}\left( 2q^{0}\right) }\hat{d}^{\dag }(q) e^{\textrm{i}(q^{0}t-\textbf{q}\cdot \textbf{x})}\,,\\&\phi ^{+}(\textbf{x},t)=\int \frac{d^{3}\textbf{q}}{(2\pi )^{3}\left( 2q^{0}\right) }\hat{d}(q) e^{-\textrm{i}(q^{0}t-\textbf{q}\cdot \textbf{x})}\,, \end{aligned} \end{aligned}$$
(4)

where \(q^{0}\) is the 0-th component of four-momentum q of the ALP quantized scalar field, and \(\hat{d}(q)\) and \(\hat{d}^{\dag }(q)\) are the annihilation and creation operators of the ALPs, respectively, satisfying the following canonical commutation relation

$$\begin{aligned} {[}d(q),d^{\dag }(q')]=(2\pi )^{3}\left( 2q^{0}\right) \delta ^{3}(\textbf{q}-\textbf{q}')\,. \end{aligned}$$
(5)

In the non-relativistic regime, \(q^{0}\approx m_{a}\), where \(m_{a}\) is the mass of ALPs. As mentioned, the spinor field effectively describes the two-level atom. We can write the Fourier transform of the spinor field for neutral atoms as

$$\begin{aligned} \psi (\textbf{x},t)=\psi ^{-}(\textbf{x},t)+\psi ^{+}(\textbf{x},t)\,, \end{aligned}$$
(6)

in which we have

$$\begin{aligned} \begin{aligned}&\psi ^{-}(\textbf{x},t)=\int \frac{d^{3}\textbf{p}'}{(2\pi )^{3}}\sum _{r'}\bar{u}_{r'}(p') \hat{b}_{r'}^{\dag }(p')e^{\textrm{i}(Et-\textbf{p}'\cdot \textbf{x}')}\,,\\&\psi ^{+}(\textbf{x},t)=\int \frac{d^{3}\textbf{p}}{(2\pi )^{3}}\sum _{r}u_{r}(p)\hat{b}_{r}(p)e^{-\textrm{i} (Et-\textbf{p}\cdot \textbf{x})}\,, \end{aligned} \end{aligned}$$
(7)

where \(u_{r}\) is the free spinor solution of the Dirac equation, p is the four-momentum, \(\hat{b}_{r}^{\dag }(p)\) and \(\hat{b}_{r}(p)\) are the creation and annihilation operators associated with spin 1/2 systems, here neutral atoms, respectively, and the spin index \(r=1,2\) counts two spin states of neutral atoms. Also, \(\hat{b}_{r}^{\dag }(p)\) and \(\hat{b}_{r}(p)\) obey the following canonical anti-commutation relation

$$\begin{aligned} \{b_{r}(p),b^{\dag }_{r'}(p')\}=(2\pi )^{3}\delta ^{3}(\textbf{p}-\textbf{p}')\delta _{rr'}\,. \end{aligned}$$
(8)

Figure 1 shows the scattering process. Therefore, the effective interaction Hamiltonian describing this process can be written in the following form

$$\begin{aligned} \mathcal {H}\approx g\bar{\psi }^{-}\gamma ^{0}\gamma ^{5}\psi ^{+}\partial _{0}\phi ^{+}\,. \end{aligned}$$
(9)
Fig. 1
figure 1

The Feynman diagram for the neutral atom-ALP interaction

Full size image

Now, following the approach of Refs. [69,76], and using (3) and (6), the effective interaction Hamiltonian \(H^{0}_{\textrm{int}}(t)\) in Fourier space can be rewritten as follows

$$\begin{aligned} H^{0}_{\textrm{int}}(t)= & {} \int d^{3}\textbf{x}\int \textrm{D}\textbf{p}\int \textrm{D}\textbf{p}'\nonumber \\{} & {} \times \int \tilde{\textrm{D}}\textbf{q}\, e^{-\textrm{i}(p^{0}-p'^{0}+q^{0})t}e^{\textrm{i}(\textbf{p}-\textbf{p}'+\textbf{q}) \cdot \textbf{x}}\nonumber \\{} & {} \times (-\textrm{i}q^{0})\mathcal {M}(\textbf{p}\,r,\textbf{p}'\,r')\hat{b}_{r'}^{\dag }(p')\hat{b}_{r}(p)\hat{d}(q)\,, \end{aligned}$$
(10)

where the superscript 0 in \(H^{0}_{\textrm{int}}(t)\) denotes that the effective interaction Hamiltonian is a functional of the free fields, and \(\mathcal {M}(\textbf{p}\,r,\textbf{p}'\,r')=g\bar{u}_{r'}(\textbf{p}')\gamma ^{0}\gamma ^{5}u_{r} (\textbf{p})\) is the scattering matrix element, in which \(u_{r}\) is the free spinor describing the neutral atoms

$$\begin{aligned} u_{r}(\textbf{p})=\left( \begin{array}{c} \chi _{r} \\ \frac{\mathbf {\sigma }\cdot \textbf{p}}{2m_{f}}\chi _{r} \\ \end{array} \right) \,, \end{aligned}$$
(11)

where \(\chi _{r}\) is the bispinor of the two-level system with spin index r that generally can be read as

$$\begin{aligned} \chi _{r}=\frac{1}{2}\left( \begin{array}{c} 1-(-1)^{r} \\ 1+(-1)^{r} \\ \end{array} \right) \,. \end{aligned}$$
(12)

Moreover, in Eq. (10) we have used the following abbreviations

$$\begin{aligned} \frac{d^{3}\textbf{p}}{(2\pi )^{3}}\equiv \textrm{D}\textbf{p}\,,\qquad \frac{d^{3}\textbf{q}}{(2\pi )^{3}\left( 2q^{0}\right) }\equiv \tilde{\textrm{D}}\textbf{q}\,. \end{aligned}$$
(13)

As mentioned earlier, we provide a superposition state between spin states of neutral atoms as \((\left| {\uparrow } \right\rangle +\left| {\downarrow } \right\rangle )\), in which the initial phase is considered zero. Then, the superposition state of neutral atoms interacting with ALPs in the SG interferometer transforms into \((\left| {\uparrow } \right\rangle +f\left| {\downarrow } \right\rangle )\) state, where f is the decoherence factor. The superposition state will be preserved during the process. On the other hand, we take into account a spin-dependent momentum for neutral atoms. Hence, two paths of the SG interferometer are dependent on the spin indices of neutral atoms. Therefore, over the mesoscopic time scale \(t_{\textrm{mes}}\), the sign of the momentum of neutral atoms will change between the upper and the lower paths of the SG interferometer. Consequently, the momentum of the neutral atoms is time-dependent (as a similar approach, see Ref. [20]). We use the Heaviside step function \(\Theta \) to write the momentum of neutral atoms. Because the z-axis is considered in the down-to-up direction, only the sign of the z-component of the momentum of neutral atoms changes during the process. This situation is illustrated in Fig. 2. This figure represents the scheme of our setup over \(t_{\textrm{mes}}\) on which the evolution of the mesoscopic system occurs. Hence, we can express the momentum of neutral atoms in the form of

$$\begin{aligned} \mathbf {\textbf{k}_{r}}(t)=\left( k^{1},k^{2},(-1)^{r}\left( -1+2\,\Theta \left[ t-\frac{T}{2}\right] \right) k^{3}\right) \,, \end{aligned}$$
(14)

where T is the duration of the whole process. Moreover, we generally have \(k=(k^{0},\textbf{k})\), where in the non-relativistic regime we can write \(k^{0}\approx m_{f}\).

Fig. 2
figure 2

The scheme of the experimental setup over \(t_{\textrm{mes}}\), in which \(k_{3}\) is the third component of the momentum of neutral atoms. The blue-red circle illustrates the superposition of two neutral atom states at the beginning and end of the process. One of two states,i.e., the up state, proceeds in the upper loop with a blue line, and the other, the down state, moves in the lower path with a red line. Due to the effect of the magnetic field, at \(t_{\textrm{mes}}=0\), the superposition states of neutral atoms separate from each other so that each state (up and down) moves in one of the SG loops. Around \(t_{\textrm{mes}}=T/2\), the ALP DM interacts with neutral atoms. Then, at \(t_{\textrm{mes}}=T\), the inverse effect of the magnetic field leads to bringing the two states of neutral atoms back together (for more details, see Appendix A)

Full size image

One can read the density operator of a system of neutral atoms in the form of

$$\begin{aligned} \hat{\rho }=\int \textrm{D}\textbf{p}'\rho _{ij}(\textbf{p}')\hat{b}_{i}^{\dag }(\textbf{p}')\hat{b}_{j}(\textbf{p}')\,, \end{aligned}$$
(15)

in which the macroscopic properties of the neutral atoms in the SG interferometer are described by \(\rho _{ij}\) as the neutral atom polarization (density) matrix. Consequently, the expectation value of the neutral atom number operator \(\mathcal {\hat{D}}_{ij}(\textbf{k})=\hat{b}_{i}^{\dag }(\textbf{k})\hat{b}_{j}(\textbf{k})\) can be read as [70]

$$\begin{aligned} \left\langle \mathcal {\hat{D}}_{ij}(\textbf{k})\right\rangle =(2\pi )^{3}\delta ^{3}(0)\rho _{ji}(\textbf{k})\,. \end{aligned}$$
(16)

The density matrix \(\rho _{ij}(\textbf{k})\) of a system of neutral atoms can be parameterized using Bloch vector [75, 82]. Therefore, we can write

$$\begin{aligned} \rho _{ji}=\frac{1}{2}\left( \mathbb {I}+\mathbf {\sigma }\cdot \mathbf {\zeta }\right) =\frac{1}{2}\left( \begin{array}{cc} 1+\zeta _{3} &{} \zeta _{1}-\textrm{i}\zeta _{2} \\ \zeta _{1}+\textrm{i}\zeta _{2} &{} 1-\zeta _{3} \\ \end{array} \right) \,, \end{aligned}$$
(17)

where \(\mathbb {I}\) is the \(2\times 2\) identity matrix, \(\mathbf {\sigma }\) is the Pauli vector made by Pauli matrices, and \(\zeta _{i}\) are the components of the Bloch vector constructing the Bloch sphere. One can read \(\zeta _{1}-\textrm{i}\zeta _{2}=f\), in which the decoherence factor can be expressed as

$$\begin{aligned} f=\exp [-\epsilon +\textrm{i}\varphi ]\,, \end{aligned}$$
(18)

where \(\epsilon \) is a dimensionless factor to measure the decoherence, and \(\varphi \) is a phase measuring the path differences between two paths of the SG interferometer [76] gives the time evolution of the density matrix elements as

$$\begin{aligned}{} & {} (2\pi )^{3}\delta ^{3}(0)\frac{d}{dt_{\textrm{mes}}}\rho _{ji}(\textbf{k},t_{\textrm{mes}})\nonumber \\{} & {} =\textrm{i}\left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}), \mathcal {\hat{D}}_{ij}(\textbf{k},t_{\textrm{mes}})\right] \right\rangle \nonumber \\{} & {} \quad -\int _{0}^{t_{\textrm{mes}}}\left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}),\left[ {H^{0}}^{\dag }_{\textrm{int}}(t_{\textrm{mic}}),\right. \right. \right. \nonumber \\{} & {} \qquad \left. \left. \left. \mathcal {\hat{D}}_{ij}(\textbf{k},t_{\textrm{mes}} -t_{\textrm{mic}})\right] \right] \right\rangle dt_{\textrm{mic}}\,,\nonumber \\ \end{aligned}$$
(19)

where the interaction time scale of individual particles takes place on the microscopic time scale \(t_{\textrm{mic}}\). The first term on the right-hand side of Eq. (19) is known as the forward scattering term, while the second is the usual collision term. The forward scattering term vanishes if the ALP scalar field is in a vacuum state. However, assuming the scalar field to be in a coherent state, this term is non-vanishing. We display operator expectation values needed here using Wick’s theorem as follows [70]

$$\begin{aligned}{} & {} \left\langle \hat{b}_{r'}^{\dag }\hat{b}_{r}\right\rangle =(2\pi )^{3}\delta ^{3}(\textbf{p}'-\textbf{p}) \rho _{rr'}(\textbf{p})\,, \end{aligned}$$
(20)
$$\begin{aligned}{} & {} \left\langle \hat{d}^{\dag }(q_{2})\hat{d}(q_{1})\right\rangle =(2\pi )^{3}(2q^{0})\delta ^{3} (\textbf{q}_{2}-\textbf{q}_{1})\frac{1}{2}n_{a}(\textbf{q}_{2},\textbf{x})\,,\nonumber \\ \end{aligned}$$
(21)

where \(n_{a}(\textbf{q},\textbf{x})\) is the ALP number density of momentum \(\textbf{q}\) per unit volume. One can also express the number density in terms of the velocity distribution of the ALPs as DM [83]

$$\begin{aligned} n_{a}(\textbf{v}_{a},\textbf{x})=n_{a}(\textbf{x})\left( \frac{4\pi }{2\sigma _{v}^{2}}\right) ^{3/2}\exp \left( -\frac{\left( \textbf{v}_{a}+\textbf{V}^{E}\right) ^{2}}{2\sigma _{v}^{2}}\right) \,, \end{aligned}$$
(22)

where \(\textbf{v}_{a}\) is the velocity of the ALPs, and we have \(\sigma _{v}=\upsilon _{0}/\sqrt{2}\) in which \(\upsilon _{0}=220\,\mathrm {km/s}\) is the circular rotation speed of the standard halo model; \(\left| \textbf{V}^{E}\right| =230\,\mathrm {km/s}\) is the velocity of the Earth with respect to the galactic frame, and we also have \(n_{a}(\textbf{x})=\rho ^{DM}/m_{a}\), with \(\rho ^{DM}=0.3\,\mathrm {GeV/cm^{3}}\) as the energy density of the DM near the Earth [83]. Also, we have

$$\begin{aligned} \left\langle \hat{b}_{i}^{\dag }\hat{b}_{j}\hat{b}_{k}^{\dag }\hat{b}_{l}\right\rangle \simeq (2\pi )^{6}\delta ^{3}(\textbf{p}_{j}-\textbf{p}_{k})\delta ^{3}(\textbf{p}_{i}-\textbf{p}_{l}) \delta _{jk}\rho _{li}(\textbf{p}_{l})\,. \end{aligned}$$
(23)

Therefore, we can find the following expectation value

$$\begin{aligned} \left\langle \hat{b}_{i}^{\dag }\hat{b}_{j}\hat{b}_{k}^{\dag }\hat{b}_{l}\hat{b}_{m}^{\dag } \hat{b}_{n}\right\rangle\simeq & {} (2\pi )^{9}\delta ^{3}(\textbf{p}_{l}-\textbf{p}_{m}) \delta ^{3}\nonumber \\{} & {} (\textbf{p}_{i}-\textbf{p}_{n})\delta ^{3}(\textbf{p}_{j}-\textbf{p}_{k}) \delta _{lm}\delta _{jk}\rho _{ni}(\textbf{p}_{n})\,.\nonumber \\ \end{aligned}$$
(24)

For the case where the ALP scalar field is in a coherent state, the expectation value of the annihilation operator of the ALP scalar field in the non-relativistic limit yields [26, 84]

$$\begin{aligned} \left\langle \hat{d}(q)\right\rangle =(2\pi )^{3/2}m_{a}A\,\delta ^{3}(0)\,, \end{aligned}$$
(25)

where the amplitude of the coherently oscillating ALP scalar field is [26, 84]

$$\begin{aligned} A=2\times 10^{-6}\sqrt{\frac{\rho ^{DM}}{0.3\,\mathrm {GeV/cm^{3}}}}\left( \frac{10^{-6}\,\textrm{eV}}{m_{a}}\right) \, \textrm{GeV}\,. \end{aligned}$$
(26)

Now, from Eqs. (17) and (19), we can find the time derivatives of Bloch vector components as the building blocks of the density matrix of neutral atoms in the following form

$$\begin{aligned} \begin{aligned}&\dot{\zeta }_{1}(\textbf{v},t_{\textrm{mes}})=\dot{\rho }_{12}(\textbf{v},t_{\textrm{mes}})+\dot{\rho }_{21} (\textbf{v},t_{\textrm{mes}})\,,\\&\dot{\zeta }_{2}(\textbf{v},t_{\textrm{mes}})=-\textrm{i}\left[ \dot{\rho }_{21}(\textbf{v},t_{\textrm{mes}}) -\dot{\rho }_{12}(\textbf{v},t_{\textrm{mes}})\right] \,,\\&\dot{\zeta }_{3}(\textbf{v},t_{\textrm{mes}})=\dot{\rho }_{11}(\textbf{v},t_{\textrm{mes}})-\dot{\rho }_{22} (\textbf{v},t_{\textrm{mes}})\,. \end{aligned} \end{aligned}$$
(27)

where we use \(\textbf{k}=m_{f}\textbf{v}\) in the non-relativistic regime, in which \(\textbf{v}=(v_{1},v_{2},v_{3})\) is the velocity of neutral atoms. Equation (27) results in a set of coupled differential equations for the Bloch vector components. Solving these equations, one finds the relative phase shift arising from the interaction.

3.1 Forward scattering term

The forward scattering term on the right-hand side of Eq. (19) can be approximately expressed as

$$\begin{aligned} \textrm{i}\left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}),\mathcal {\hat{D}}_{ij}(\textbf{k},t_{\textrm{mes}})\right] \right\rangle{} & {} \simeq \left\langle \hat{d}(q)\right\rangle \left[ \left\langle \hat{b}_{r'}^{\dag } \hat{b}_{r}\hat{b}_{i}^{\dag }\hat{b}_{j}\right\rangle \right. \nonumber \\{} & {} \quad \left. -\left\langle \hat{b}_{i}^{\dag }\hat{b}_{j} \hat{b}_{r'}^{\dag }\hat{b}_{r}\right\rangle \right] \,. \end{aligned}$$
(28)

Therefore, using Eqs. (23) and (25) we get

$$\begin{aligned} \begin{aligned}&(2\pi )^{3}\delta ^{3}(0)\frac{d}{dt_{\textrm{mes}}}\rho _{ji}(\textbf{k},t_{\textrm{mes}})\\ {}&= \textrm{i}\left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}), \mathcal {\hat{D}}_{ij}(\textbf{k},t_{\textrm{mes}})\right] \right\rangle \\&\simeq \textrm{i}\int d^{3}\textbf{x}\int \textrm{D}\textbf{p}\int \textrm{D}\textbf{p}'\\&\quad \times \int \tilde{\textrm{D}}\textbf{q}\,e^{-\textrm{i}(p^{0}-p'^{0}+q^{0})t_{\textrm{mes}}} e^{\textrm{i}(\textbf{p}-\textbf{p}'+\textbf{q})\cdot \textbf{x}}\\&\quad \times \mathcal {M}(\textbf{p}\,r,\textbf{p}'\,r',t_{\textrm{mes}})\\&\quad \times (-\textrm{i}q^{0})(2\pi )^{3/2}m_{a}A\,\delta ^{3}(0)(2\pi )^{6}\\&\quad \times \Big (\delta ^{3}(\textbf{p}_{r}-\textbf{k}) \delta ^{3}(\textbf{p}'_{r'}-\textbf{k})\delta _{ri}\rho _{jr'}(\textbf{k})\\&\quad -\delta ^{3}(\textbf{k}-\textbf{p}'_{r'})\delta ^{3}(\textbf{k}-\textbf{p}_{r})\delta _{jr'}\rho _{ri}(\textbf{p})\Big ). \end{aligned} \end{aligned}$$
(29)

After integrating over \(\textbf{x},\, \textbf{p},\, \textbf{p}',\) and \(\textbf{q}\), the following result is achieved

$$\begin{aligned} \begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t_{\textrm{mes}}}\rho _{ji}(\textbf{k},t_{\textrm{mes}})&\simeq \frac{1}{2}\sqrt{2\pi }g m_{a}A\,e^{-\textrm{i}m_{a}t_{\textrm{mes}}}\\&\quad \times \left[ u_{r'}^{\dag }(\textbf{k},t_{\textrm{mes}})\gamma ^{5}u_{i}(\textbf{k},t_{\textrm{mes}})\rho _{jr'}(\textbf{k})\right. \\&\quad \left. -u_{j}^{\dag }(\textbf{k},t_{\textrm{mes}})\gamma ^{5}u_{r}(\textbf{k},t_{\textrm{mes}})\rho _{ri}(\textbf{k})\right] \,. \end{aligned} \end{aligned}$$
(30)

3.2 Collision term

The integrand in the collision term on the right-hand side of Eq. (19) can be approximately expressed as follows

$$\begin{aligned}{} & {} \left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}),\left[ {H^{0}}^{\dag }_{\textrm{int}}(t_{\textrm{mic}}),\mathcal {\hat{D}}_{ij}^{(f)} (\textbf{k},t_{\textrm{mes}}-t_{\textrm{mic}})\right] \right] \right\rangle \nonumber \\{} & {} \quad \simeq \left\langle \hat{d}_{1}(t_{\textrm{mes}})\hat{d}^{\dag }_{2}(t_{\textrm{mic}})\right\rangle \nonumber \\{} & {} \qquad \times \left[ \left\langle \hat{b}_{r'_{1}}^{\dag }\hat{b}_{r_{1}}\hat{b}_{r_{2}}^{\dag }\hat{b}_{r'_{2}} \hat{b}_{i}^{\dag }\hat{b}_{j}\right\rangle -\left\langle \hat{b}_{r'_{1}}^{\dag }\hat{b}_{r_{1}} \hat{b}_{i}^{\dag }\hat{b}_{j}\hat{b}_{r_{2}}^{\dag }\hat{b}_{r'_{2}}\right\rangle \right] \nonumber \\{} & {} \qquad +\left\langle \hat{d}^{\dag }_{2}(t_{mic})\hat{d}_{1}(t_{\textrm{mes}})\right\rangle \left[ \left\langle \hat{b}_{i}^{\dag }\hat{b}_{j}\hat{b}_{r_{2}}^{\dag }\hat{b}_{r'_{2}}\hat{b}_{r'_{1}}^{\dag } \hat{b}_{r_{1}}\right\rangle \right. \nonumber \\ {}{} & {} \qquad \left. -\left\langle \hat{b}_{r_{2}}^{\dag }\hat{b}_{r'_{2}} \hat{b}_{i}^{\dag }\hat{b}_{j}\hat{b}_{r'_{1}}^{\dag }\hat{b}_{r_{1}}\right\rangle \right] \end{aligned}$$
(31)

It should be noted that operators in Eq. (31) with index “1” associated with \(H^{0}_{\textrm{int}}(t_{\textrm{mes}})\) are functions of \(t_{\textrm{mes}}\), and operators with index “2” associated with \(H^{0\dagger }_{\textrm{int}}(t_{\textrm{mic}})\) are functions of \(t_{\textrm{mic}}\).

Thus, using Eqs. (21) and (24), one can express the QBE (19) associated with the collision term as

$$\begin{aligned}{} & {} (2\pi )^{3}\delta ^{3}(0)\frac{\mathrm{{d}}}{\mathrm{{d}}t_{\textrm{mes}}}\rho _{ji}(\textbf{k},t_{\textrm{mes}})\nonumber \\{} & {} \quad =-\int _{0}^{t_{\textrm{mes}}} \left\langle \left[ H^{0}_{\textrm{int}}(t_{\textrm{mes}}),\left[ {H^{0}}^{\dag }_{\textrm{int}}(t_{\textrm{mic}}),\right. \right. \right. \left. \left. \left. \mathcal {\hat{D}}_{ij} (\textbf{k},t_{\textrm{mes}}-t_{\textrm{mic}})\right] \right] \right\rangle \mathrm{{d}}t_{\textrm{mic}}\nonumber \\{} & {} \quad \simeq -\int _{0}^{t_{\textrm{mes}}}\mathrm{{d}}t_{\textrm{mic}}\int \mathrm{{d}}^{3}\textbf{x}\int \mathrm{{d}}^{3}\textbf{x}'\int \textrm{D}\textbf{p}_{1}\int \textrm{D}\textbf{p}'_{1}\int \textrm{D}\textbf{p}_{2}\nonumber \\{} & {} \qquad \times \int \textrm{D}\textbf{p}'_{2}\int \tilde{\textrm{D}}\textbf{q}_{1}\int \tilde{\textrm{D}}\textbf{q}_{2}\,(q^{0}_{1})(q^{0}_{2})\nonumber \\{} & {} \qquad \times e^{-\textrm{i}(p^{0}_{1}-p'^{0}_{1}+q^{0}_{1})t_{\textrm{mes}}}e^{\textrm{i}(\textbf{p}_{1}-\textbf{p}'_{1} +\textbf{q}_{1})\cdot \textbf{x}}e^{\textrm{i}(p^{0}_{2}-p'^{0}_{2}+q^{0}_{2})(t_{\textrm{mic}})}\nonumber \\{} & {} \qquad \times e^{-\textrm{i}(\textbf{p}_{2}-\textbf{p}'_{2}+\textbf{q}_{2})\cdot \textbf{x}'} \mathcal {M}_{1}(\textbf{p}_{1}\,r_{1},\textbf{p}'_{1}\,r'_{1},\textbf{q}_{1},t_{\textrm{mes}})\nonumber \\{} & {} \qquad \times \mathcal {M}_{2} (\textbf{p}_{2}\,r_{2},\textbf{p}'_{2}\,r'_{2},\textbf{q}_{2},t_{\textrm{mic}})\nonumber \\{} & {} \qquad \times \Big \{(2\pi )^{3}\left( 2q^{0}_{1}\right) \delta ^{3}(\textbf{q}_{1}-\textbf{q}_{2})\frac{1}{2} n(\textbf{q}_{1},\textbf{x})\nonumber \\{} & {} \qquad \times \Big [(2\pi )^{9}\delta ^{3}(\textbf{p}'_{2}-\textbf{k})\delta ^{3} (\textbf{p}'_{1}-\textbf{k})\delta ^{3}(\textbf{p}_{1}-\textbf{p}_{2}) \delta _{r'_{2}i}\delta _{r_{1}r_{2}}\rho _{jr'_{1}}(\textbf{k})\nonumber \\{} & {} \qquad -(2\pi )^{9}\delta ^{3}(\textbf{p}_{1}-\textbf{k})\delta ^{3}(\textbf{p}'_{1}-\textbf{p}'_{2}) \delta ^{3}(\textbf{k}-\textbf{p}_{2})\delta _{r_{1}i}\delta _{jr_{2}}\rho _{r'_{2}r'_{1}} (\textbf{p}'_{2})\Big ]\nonumber \\{} & {} \qquad +(2\pi )^{3}\left( 2q^{0}_{2}\right) \delta ^{3}(\textbf{q}_{2}-\textbf{q}_{1}) \frac{1}{2}n(\textbf{q}_{2},\textbf{x})\nonumber \\{} & {} \qquad \times \Big [(2\pi )^{9}\delta ^{3}(\textbf{p}'_{2}-\textbf{p}'_{1})\delta ^{3}(\textbf{k}-\textbf{p}_{1}) \delta ^{3}(\textbf{k}-\textbf{p}_{2})\delta _{jr_{2}}\delta _{r'_{2}r'_{1}}\rho _{r_{1}i}(\textbf{p}_{1})\nonumber \\{} & {} \qquad -(2\pi )^{9}\delta ^{3}(\textbf{p}_{2}-\textbf{p}_{1})\delta ^{3}(\textbf{p}'_{2}-\textbf{k})\delta ^{3}\nonumber \\{} & {} \qquad (\textbf{k}-\textbf{p}'_{1})\delta _{jr'_{1}}\delta _{r'_{2}i}\rho _{r_{1}r_{2}}(\textbf{p}_{1})\Big ]\Big \} \end{aligned}$$
(32)

After integrating over \(\textbf{x},\, \textbf{x}',\, \textbf{p}_{1},\, \textbf{p}_{2},\, \textbf{p}'_{1},\, \textbf{p}'_{2},\) and \(\textbf{q}_{1}\) and using Eqs. (11)–(14), one can read Eq. (32) in the following form in terms of velocities of neutral atoms and ALPs

$$\begin{aligned}{} & {} \frac{\mathrm{{d}}}{\mathrm{{d}}t_{\textrm{mes}}}\rho _{ji}(\textbf{v},t_{\textrm{mes}})\approx g^{2}\int _{0}^{t_{\textrm{mes}}}\mathrm{{d}}t_{\textrm{mic}}\nonumber \\{} & {} \quad \times \int \textrm{D}\textbf{v}_{a}\frac{m_{a}n_{a}(\textbf{x})}{16m_{f}^{2}}\left( \frac{4\pi }{2\sigma _{v}^{2}}\right) ^{3/2}\nonumber \\{} & {} \quad \times \exp \left( -\frac{\left( \textbf{v}_{a}+\textbf{V}^{E}\right) ^{2}}{2\sigma _{v}^{2}}\right) \nonumber \\{} & {} \quad \times \Big \{-\left[ \bar{\chi }_{r'}\mathbf {\sigma }\cdot (m_{f}\textbf{v}_{r'}(t_{\textrm{mes}})-\textbf{q})\chi _{r}\right. \nonumber \\{} & {} \quad \left. +\bar{\chi }_{r'}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r}(t_{\textrm{mes}})\chi _{r} \right] \nonumber \\{} & {} \quad \times \left[ \bar{\chi }_{r}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r}(t_{\textrm{mic}})\chi _{i}+\bar{\chi }_{r} \mathbf {\sigma }\cdot (m_{f}\textbf{v}_{i}(t_{\textrm{mic}})-\textbf{q})\chi _{i}\right] \nonumber \\{} & {} \quad \times \rho _{jr'}(\textbf{v})e^{-\textrm{i}(P^{0}-k^{0}+q^{0})(t_{\textrm{mes}}-t_{\textrm{mic}})}\nonumber \\{} & {} \quad +\left[ \bar{\chi }_{j}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{j}(t_{\textrm{mes}})\chi _{r}+\bar{\chi }_{j} \mathbf {\sigma }\cdot (m_{f}\textbf{v}_{r}(t_{\textrm{mes}})+\textbf{q})\chi _{r} \right] \nonumber \\{} & {} \quad \times \left[ \bar{\chi }_{r'}\mathbf {\sigma }\cdot (m_{f}\textbf{v}_{r'}(t_{\textrm{mic}})+\textbf{q})\chi _{i} +\bar{\chi }_{r'}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{i}(t_{\textrm{mic}})\chi _{i}\right] \nonumber \\{} & {} \quad \times \rho _{rr'}(\textbf{v})e^{-\textrm{i}(k^{0}-P'^{0}+q^{0})(t_{\textrm{mes}}-t_{\textrm{mic}})}\nonumber \\{} & {} \quad -\left[ \bar{\chi }_{r'}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r'}(t_{\textrm{mes}})\chi _{r}+\bar{\chi }_{r'} \mathbf {\sigma }\cdot (m_{f}\textbf{v}_{r}(t_{\textrm{mes}})+\textbf{q})\chi _{r} \right] \nonumber \\{} & {} \quad \times \left[ \bar{\chi }_{j}\mathbf {\sigma }\cdot (m_{f}\textbf{v}_{j}(t_{\textrm{mic}})+\textbf{q})\chi _{r'} +\bar{\chi }_{j}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r'}(t_{\textrm{mic}})\chi _{r'}\right] \nonumber \\{} & {} \quad \times \rho _{ri}(\textbf{v})e^{-\textrm{i}(k^{0}-P'^{0}+q^{0})(t_{\textrm{mes}}-t_{\textrm{mic}})}\nonumber \\{} & {} \quad +\left[ \bar{\chi }_{j}\mathbf {\sigma }\cdot (m_{f}\textbf{v}_{j}(t_{\textrm{mes}})-\textbf{q})\chi _{r} +\bar{\chi }_{j}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r}(t_{\textrm{mes}})\chi _{r} \right] \nonumber \\{} & {} \quad \times \left[ \bar{\chi }_{r'}\mathbf {\sigma }\cdot m_{f}\textbf{v}_{r'}(t_{\textrm{mic}})\chi _{i}+ \bar{\chi }_{r'}\mathbf {\sigma }\cdot (m_{f}\textbf{v}_{i}(t_{\textrm{mic}})-\textbf{q})\chi _{i}\right] \nonumber \\{} & {} \quad \times \rho _{rr'}(\textbf{v})e^{-\textrm{i}(P^{0}-k^{0}+q^{0})(t_{\textrm{mes}}-t_{\textrm{mic}})}\Big \}\,, \end{aligned}$$
(33)

where \(P^{0}=E(\textbf{k}-\textbf{q})\) and \(P'^{0}=E(\textbf{k}+\textbf{q})\), and we employed the approximation \(\rho (\textbf{k}-\textbf{q})\approx \rho (\textbf{k}+\textbf{q})\approx \rho (\textbf{k})\) as well as \(q^{0}\approx m_{a}\). Also, we utilized the following abbreviation

$$\begin{aligned} \frac{d^{3}\textbf{v}_{a}}{(2\pi )^{3}}\equiv \textrm{D}\textbf{v}_{a}\,. \end{aligned}$$
(34)

Now, for the right-hand side of Eq. (33), one should first take the time integral over \(t_{\textrm{mic}}\). Then, by converting the exponential functions gained to the sinc functions and assuming energy conservation, one can calculate the integration over \(\textbf{v}_{a}\) in Eq. (33).

4 Estimate of the induced phase

We numerically solve the QBE for both the forward and the collision terms. The forward scattering term induces only the relative phase shift, while the collision term is responsible for both the relative phase shift and decoherence. Our numerical investigation shows that the effect of the collision term, however, is dominated by the forward scattering term. Accordingly, the induced decoherence is suppressed because of the smallness of the collision term effect. In the following, we will discuss these results in more detail.

Using Eqs. (27), (30), and (33), the following final results for the time evolution of the Bloch vector components associated with both the forward scattering and collision terms are obtained as the following three coupled differential equations

$$\begin{aligned} \dot{\zeta }_{1}(\textbf{v},t_{\textrm{mes}}){} & {} =Agm_{a}\sqrt{2\pi }e^{-\textrm{i}m_{a}t_{\textrm{mes}}} \Bigg \{v_{3}\zeta _{1}(t_{\textrm{mes}})\nonumber \\{} & {} \quad \times \left. \left( 1-2 \Theta \left[ t_{\textrm{mes}}-\frac{T}{2}\right] \right) -\textrm{i}v_{2}\zeta _{3}(t_{\textrm{mes}})+v_{1}\right\} \nonumber \\{} & {} \quad -\frac{g^{2}\pi \rho }{16m_{f}^{2}}\Bigg \{t_{\textrm{mes}}\left( m_{a}^{2} W_{1}-2m_{a}m_{f}W_{2}\right. \nonumber \\{} & {} \quad \left. \left. +4m_{f}^{2}W_{3}\right) -2m_{f}v_{3}W_{4}\Theta \left[ t_{\textrm{mes}}-\frac{T}{2}\right] \right\} \,,\nonumber \\ \dot{\zeta }_{2}(\textbf{v},t_{\textrm{mes}}){} & {} = Agm_{a}\sqrt{2\pi }e^{-\textrm{i}m_{a}t_{\textrm{mes}}} \Bigg \{v_{3}\zeta _{2}(t_{\textrm{mes}})\nonumber \\{} & {} \quad \times \left. \left( 1-2 \Theta \left[ t_{\textrm{mes}}-\frac{T}{2}\right] \right) +\textrm{i}v_{1}\zeta _{3}(t_{\textrm{mes}})+v_{2}\right\} \nonumber \\{} & {} \quad +\frac{g^{2}\pi \rho }{16m_{f}^{2}}\Bigg \{t_{\textrm{mes}}\left( m_{a}^{2} Z_{1}-2m_{a}m_{f}Z_{2}\right. \nonumber \\{} & {} \quad \left. +4m_{f}^{2}Z_{3}\right) +2m_{f}v_{3}Z_{4}\Theta \left[ t_{\textrm{mes}}-\frac{T}{2}\right] \Bigg \}\,,\nonumber \\ \dot{\zeta }_{3}(\textbf{v},t_{\textrm{mes}}){} & {} = Agm_{a}\sqrt{2\pi }e^{-\textrm{i}m_{a}t_{\textrm{mes}}} \Bigg \{v_{3}\zeta _{3}(t_{\textrm{mes}})\nonumber \\{} & {} \quad \times \left. \left( 1-2 \Theta \Bigg [t_{\textrm{mes}}-\frac{T}{2}\right] \right) \nonumber \\{} & {} \qquad -\textrm{i}v_{1}\,\zeta _{2}(t_{\textrm{mes}}) +\textrm{i}v_{2}\zeta _{1}(t_{\textrm{mes}})\Bigg \}\nonumber \\{} & {} \quad -\frac{g^{2}\pi \rho }{16m_{f}^{2}}\Bigg \{t_{\textrm{mes}}\left( m_{a}^{2} Y_{1}-2m_{a}m_{f}Y_{2}\right. \nonumber \\{} & {} \quad \left. +4m_{f}^{2}Y_{3}\right) +2m_{f}v_{3}Y_{4}\Theta \left[ t_{\textrm{mes}}-\frac{T}{2}\right] \Bigg \}\,, \end{aligned}$$
(35)

where the explicit forms of coefficients \(W_{i}, Z_{i}\), and \(Y_{i}\) are given in Appendix B.

We provide the system such that the initial phase is \(\varphi _{0}=0\). Then, from Eq. (18), one can obtain the relative phase shift as

$$\begin{aligned} \Delta \varphi =\varphi -\varphi _{0}=\left| \tan ^{-1}\left[ \frac{\Im (\zeta _{1}(t_{\textrm{mes}}))-\Re (\zeta _{2} (t_{\textrm{mes}}))}{\Im (\zeta _{2}(t_{\textrm{mes}}))+\Re (\zeta _{1}(t_{\textrm{mes}}))}\right] \right| \,. \end{aligned}$$
(36)

This equation gives us the time evolution of phase measure within the decoherence factor (18) during the process. One can numerically solve the set of Eqs. (35) to calculate the relative phase shift (36) due to the interaction of neutral atoms with ALPs. To this end, we consider two types of neutral atoms, \(^{3}\)He and \(^{87}\)Rb, moving in the SG interferometer.

Next, we turn our attention to the phase sensitivity of the system and the time duration of the process. The phase sensitivity is defined as the smallest measurable phase shift, which is determined by the number of nucleons, typical contrast, and other factors. On the other hand, the time duration of the process is the total time during which the effective spin 1/2 atom are within the SG interferometer. Among the available values chosen by the proposed experiments, we assume that the minimum phase (phase sensitivity) provided by the setup is of the order of \(\Delta \varphi _{\min }=10^{-10}\,\textrm{Rad}\) and the time duration of the process is of the order of \(T=2.6\times 10^{-4}\,\textrm{s}\) [15].

Fig. 3
figure 3

The exclusion areas in the plane of ALP mass and the coupling constant of ALP–electron interaction for \(^{\textit{3}}\)He and \(^{\textit{87}}\)Rb compared with other proposals [92]

Full size image

We plot the regions that our scheme can exclude for these particles. We compare the data from underground detectors PandaX-II [85], SuperCDMS [86], and XENON1T (ALP DM search single electron) [87]; data from haloscope QUAX [88, 89]; and Astro bound Red giant branch [90] and Solar \(\nu \) [91], with total outcome of \(^{3}\)He and \(^{87}\)Rb, in Fig. 3. In this figure, \(g_{ae}\) (equivalent to \(g_{af}\) in Eq. (2)) is the dimensionless coupling constant of the ALP–electron interaction, which is based on the assumption that ALPs interact with the electrons of the atom that we have considered as a two-level system. As Fig. 3 illustrates, our scheme has excluded the area between \(10^{-10}\le m_{a}\le 10^{2}\,\textrm{eV}\) and \(10^{-13}\le g_{ae}\le 10^{0}\). Again, it is worth noting that the effect of the collision term is tiny, so its corresponding induced decoherence is negligible.

5 Conclusion and discussion

This paper presented a new scheme for the detection of ALPs as DM by the SG interferometer. ALPs interact with neutral atoms moving in the SG interferometer. The neutral atoms are considered to be in a superposition state, and the time evolution of their corresponding density matrix is derived by considering the forward scattering and the collision terms of the QBE. The interaction results in a relative phase shift between the initial and final superposition states of the effectively two-level neutral atoms. We considered two neutral atoms, \(^{3}\)He and \(^{87}\)Rb, moving in the SG interferometer. Taking a zero initial phase into account, we calculated the induced relative phase shift. This way, we plot the exclusion areas for each neutral atom compared to other proposals. Consequently, we found that our scheme has excluded the area between \(10^{-10}\le m_{a}\le 10^{2}\,\textrm{eV}\) and \(10^{-13}\le g_{ae}\le 10^{0}\). We also propose considering other schemes that use lighter neutral particles such as neutrons with the same minimum phase. Providing such conditions can exclude a wider region compared with other experiments.