1. Introduction
Recently, much attention has been paid to impurity-doped Bose–Einstein condensate (BEC) systems due to their numerous applications in probing strongly correlated quantum many-body states [1–4], quantum state engineering [5–17], and quantum metrology [18, 19]. An impurity-doped BEC system [6–8, 20, 21] presents a totally new platform for studying micro-macro quantum effects where microscopic impurities meet macroscopic matter, the BEC. In particular, when Rydberg impurities are applied, one can tailor inter-impurity interaction by electric fields and microwave fields [22, 23]; while the BEC allows for an extremely precise control of interatomic interactions by manipulating the s-wave scattering length [24, 25], they provide a new tool for studying the many-body quantum physics of hybrid quantum systems [26] consisting of micro-macro systems. In the present paper, motivated by the recent progress of impurity-doped BEC physics theoretically and experimentally, we study the quantum dynamics of an impurity-doped BEC system. We show how to generate a macroscopic quantum superposition state (MQSS) of the BEC and control the nonclassicality of the MQSS by the use of impurity atoms. We will demonstrate the quantum collapse and revival (CR) phenomenon of the BEC matter-wave field and how to control micro-macro entanglement between the impurity atoms and the BEC in dynamic evolution.
An MQSS, or a superposition state of macroscopically distinct quantum states, was introduced in Schrödinger’s famous Gedanken experiment [27]. It was known as a Schrödinger’s cat state. Nowadays, the former two-component Schrödinger cat states have been extended to multi-component cat states which contain many macroscopically distinct quantum states which are being emulated in diverse physical systems [28]. MQSSs have significant applications in quantum metrology [29–32], quantum computation [33–38], and fundamental tests of quantum mechanics [39, 40]. Here we will show that multi-component MQSSs consisting of multi coherent-state components can be produced in the dynamic evolution of impurity-doped BEC systems.
A BEC can be regarded as the most classical matter wave, just as an optical laser emits the most classical form of an electromagnetic wave. Nevertheless, the BEC matter-wave field has a quantized structure owing to the granularity of the discrete underlying atoms. The quantum nature of the BEC matter-wave field can be revealed in the CR phenomenon in the dynamic evolution of the BEC matter-wave field [41], just as in the Jaynes–Cummings model [42, 43] in which the existence of the CR phenomenon of the Rabi oscillations provided direct evidence for the discreteness of the optical field excitation (photons) and hence for the truly quantum nature of the optical field. We will show here the impurity-induced CR phenomenon of the BEC matter-wave field which can reveal the quantum nature of the BEC matter-wave field.
As is well known, it is an open challenge for fundamental physics to reveal the quantum entanglement of a micro-macro system [44–57]. This endeavor could contribute to challenging the unobservability of quantum features at the macroscopic level, which is one of the very fascinating open problems in quantum physics. An impurity-doped BEC system presents a new platform for the study of micro-macro entanglement where microscopic impurities meet macroscopic BEC matter. We will investigate the quantum dynamics of a micro-macro entanglement in an impurity-doped BEC system and show how to control the micro-macro entanglement by manipulating the impurity atoms.
The remainder of this paper is organized as follows: in section 2 , we introduce the impurity-doped BEC model consisting of the BEC and N impurity atoms. We obtain an analytical solution of the impurity-doped BEC model. In section 3 , we show how to generate and manipulate the MQSSs of the BEC. In section 4 , we investigate the quantum dynamics of the BEC matter-wave field, and reveal the existence of the impurity-induced CR phenomenon of the BEC matter-wave field. In section 5 , we study micro-micro entanglement between impurity atoms and the BEC and find the physical mechanism that produces micro-micro entanglement. Finally, section 6 is devoted to some concluding remarks.
2. The impurity-doped BEC model
We consider an impurity-doped BEC system which consists of the BEC and N two-level impurity atoms. We assume that the N fixed impurities are immersed in the BEC, the impurities are independent of each other, and the number of the impurities is much smaller than the number of condensed atoms in the BEC. Under these conditions, the interactions between impurity atoms can be neglected. The Hamiltonian of the BEC in a trapping potential [58] $ \begin{eqnarray}{H}_{B}=\int {\rm{d}}{\boldsymbol{x}}{{\rm{\Psi }}}^{\dagger }({\boldsymbol{x}})\left[-\displaystyle \frac{1}{2m}{{\rm{\nabla }}}^{2}+V({\boldsymbol{x}})+\displaystyle \frac{U}{2}{{\rm{\Psi }}}^{\dagger }({\boldsymbol{x}}){\rm{\Psi }}({\boldsymbol{x}})\right]{\rm{\Psi }}({\boldsymbol{x}}),\end{eqnarray}$ where ${\rm{\Psi }}({\boldsymbol{x}})$ is the BEC field operator, $V({\boldsymbol{x}})$ is the external trapping potential, U is the self-interaction, and m is the mass of an atom. The condensate is assumed to be trapped in a deep potential such that the BEC can be well-described within a single-mode approximation ${\rm{\Psi }}({\boldsymbol{x}})\approx a\phi ({\boldsymbol{x}})$ . Here a and $\phi ({\boldsymbol{x}})$ are the the annihilator operator and the mode function of the condensate, respectively. Then, the Hamiltonian of the confined BEC can be written as a Kerr-type Hamiltonian $ \begin{eqnarray}{H}_{B}={\omega }_{b}{a}^{\dagger }a+\chi {a}^{\dagger }{a}^{\dagger }{aa},\end{eqnarray}$ where we have set the Planck constant ℏ = 1, and the mode frequency ωb and the coupling strength χ are defined as $ \begin{eqnarray}{\omega }_{b}=\int {\rm{d}}{\boldsymbol{x}}\left[-\displaystyle \frac{1}{2m}| {\rm{\nabla }}\phi ({\boldsymbol{x}}){| }^{2}+V({\boldsymbol{x}})| \phi ({\boldsymbol{x}}){| }^{2}\right],\end{eqnarray}$ $ \begin{eqnarray}\chi =\displaystyle \frac{U}{2}\int {\rm{d}}{\boldsymbol{x}}| \phi ({\boldsymbol{x}}){| }^{4}.\end{eqnarray}$
The impurities interact with the BEC via coherent collisions [2]. The impurity-BEC interaction Hamiltonian can be described as $ \begin{eqnarray}{H}_{I}=\displaystyle \frac{1}{2}\kappa \displaystyle \sum _{i=1}^{N}{\sigma }_{z}^{i}{a}^{\dagger }a,\end{eqnarray}$ where ${\sigma }_{z}^{i}$ is a Pauli operator of i-th impurity atom, and κ is the interaction strength between the impurities and the BEC.
Making use of equations (2 ) and (5 ), we can obtain the Hamiltonian of the total system in the interaction picture as $ \begin{eqnarray}H=\lambda {\left({a}^{\dagger }a\right)}^{2}+\kappa \hat{{J}_{z}}{a}^{\dagger }a,\end{eqnarray}$ where we have introduced collective operators for the N two-level impurities $ \begin{eqnarray}{J}_{x}=\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}{\sigma }_{x}^{i},\hspace{0.5cm}{J}_{y}=\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}{\sigma }_{y}^{i},\hspace{0.5cm}{J}_{z}=\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}{\sigma }_{z}^{i},\end{eqnarray}$ which obey the following commutation relations of the su(2) algebra $ \begin{eqnarray}\begin{array}{rcl}\left[{J}_{x},{J}_{y}\right] & = & {\rm{i}}{J}_{z},\hspace{0.5cm}[{J}_{y},{J}_{z}]={\rm{i}}{J}_{x},\hspace{0.5cm}[{J}_{z},{J}_{x}]={\rm{i}}{J}_{y},\\ \left[{J}_{+},{J}_{-}\right] & = & 2{J}_{z},\hspace{0.5cm}[{J}_{z},{J}_{\pm }]=\pm {J}_{\pm },\end{array}\end{eqnarray}$ where ${J}_{\pm }={J}_{x}\pm {\rm{i}}{J}_{y}$ is the ladder operator of the angular momentum.
The Hamiltonian (6 ) has the following eigenvalues and eigenstates $ \begin{eqnarray}{E}_{{nm}}=\lambda {n}^{2}+\kappa {nm},\hspace{0.5cm}| {\rm{\Psi }}{\rangle }_{{nm}}=| n\rangle \otimes | j,m\rangle ,\end{eqnarray}$ where $| n\rangle $ is a Fock state of the BEC with $n=0,1,2,\cdots ,\infty $ , and $| j,m\rangle $ is a common eigenstate of the operators J2 and Jz with j = N/2, $m=-j,-j+1,-j+2,\,...j$ .
In what follows we will use the Hamiltonian (6) to study the quantum dynamics of the impurity-doped BEC system. We will show how to generate the MQSS of the BEC, a micro-macro entanglement between the impurity atoms and the BEC, and demonstrate the quantum CR phenomenon of the BEC matter-wave field.
3. Generation and manipulation of the MQSS
In this section we show that the BEC MQSS can be produced by the use of the impurity-BEC interaction and projective measurements of the impurity atoms, and the nonclassicality of the MQSS can be manipulated by the use of the impurity atoms and their interaction with the BEC. We consider such a situation in which the impurity atoms and the BEC are initially in a Greenberger–Horne–Zeilinger (GHZ) state and a coherent state $| \alpha \rangle $ , respectively. The GHZ state of N impurity atoms can be represented in terms of the basis of the angular momentum space as $ \begin{eqnarray}\begin{array}{rcl}| {\rm{GHZ}}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}[| 00\ldots 0\rangle +| 11\ldots 1\rangle ]\\ & = & \displaystyle \frac{1}{\sqrt{2}}\left[\left|N/2,-N/2\right\rangle +\left|N/2,N/2\right\rangle \right],\end{array}\end{eqnarray}$ where $\left|N/2,N/2\right\rangle $ and $\left|N/2,-N/2\right\rangle $ are the highest-weight and lowest-weight states in the angular momentum representation of the su(2) algebra with j = N/2, respectively.
The initial state of the impurity-doped BEC system is then given by $ \begin{eqnarray}| {\rm{\Psi }}(0)\rangle =\displaystyle \frac{1}{\sqrt{2}}\left[| j,-j\rangle +| j,j\rangle \right]\otimes | \alpha \rangle ,\end{eqnarray}$ where j = N/2, and $| \alpha \rangle =D(\alpha )| 0\rangle $ with the displacement operator being given by $D(\alpha )=\exp (\alpha {a}^{\dagger }-{\alpha }^{* }a)$ .
Making use of equations (6 ) and (9 ), we can obtain the wave function of the impurity-doped BEC system at a time t $ \begin{eqnarray}| {\rm{\Psi }}(t)\rangle =\displaystyle \frac{1}{\sqrt{2}}\left[| {\rm{\Psi }}{\left(t\right)}_{-j,\alpha }\rangle \otimes | j,-j\rangle +| {\rm{\Psi }}{\left(t\right)}_{+j,\alpha }\rangle \otimes | j,j\rangle \right],\end{eqnarray}$ where we have introduced an impurity-dependent BEC state $ \begin{eqnarray}| {\rm{\Psi }}{\left(t\right)}_{j,\alpha }\rangle ={{\rm{e}}}^{-{\rm{i}}t(\lambda {\hat{n}}^{2}+\kappa j\hat{n})}| \alpha \rangle ,\end{eqnarray}$ which can be expressed as a form of a generalized coherent state [59–63] $ \begin{eqnarray}| {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }\rangle ={{\rm{e}}}^{-\tfrac{1}{2}| \alpha {| }^{2}}\displaystyle \sum _{n=0}^{\infty }{{\rm{e}}}^{{\rm{i}}\tau {\theta }_{{Nn}}}\displaystyle \frac{{\alpha }^{n}}{\sqrt{n!}}\otimes | n\rangle ,\end{eqnarray}$ where the scaled time is given by τ = λt, and the running frequency ${\theta }_{{Nn}}$ is defined by $ \begin{eqnarray}{\theta }_{{Nn}}={NKn}-{n}^{2},\hspace{0.5cm}K=\kappa /(2\lambda ),\end{eqnarray}$ where K is the coupling ratio of the impurity-BEC interaction strength and the interatomic self-interaction strength in the BEC.
For a given number of impurity atoms N, we have $ \begin{eqnarray}\begin{array}{rcl}\langle {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }| {\rm{\Psi }}{\left(\tau \right)}_{-N/2,\alpha }\rangle & = & \exp \left[-| \alpha {| }^{2}(1-{{\rm{e}}}^{2{\rm{i}}{NK}\tau })\right],\\ \langle {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }| {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }\rangle & = & 1.\end{array}\end{eqnarray}$
It is noted that the generalized coherent state given by equation (14 ) can be expressed as a continuous superposition of coherent states in the following form $ \begin{eqnarray}| {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }\rangle =\displaystyle \sum _{n=0}^{\infty }\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}{\phi }_{N}f({\phi }_{j},{\theta }_{N,n})| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{N}}\rangle ,\end{eqnarray}$ where the time-dependent phase function is defined by $ \begin{eqnarray}f({\phi }_{N},{\theta }_{N,n})=\exp \left[{\rm{i}}(\tau {\theta }_{N,n}-n{\phi }_{N})\right].\end{eqnarray}$
From equations (14 )–(18 ), we can see that the values of the interaction ratio K may substantially affect the form of the states given by equation (7 ). Of particular interest is the situation where K takes the value of a nonzero integer. In this case, ${\theta }_{{Nn}}$ also takes integer values. Making use of equations (14 ) and (15 ) from equations (17 ) and (18 ) we can see that $| {\rm{\Psi }}{(\tau +2\pi )}_{N/2,\alpha }\rangle =| {\rm{\Psi }}{\left(\tau \right)}_{N/2,\alpha }\rangle $ , which implies that the time evolution of the joint wave function given by equation (17 ) is a periodic evolution with respect to scaled time τ with a period of 2π, when the scaled interaction time τ takes its values in the following manner: $ \begin{eqnarray}\tau =\displaystyle \frac{M}{S}2\pi ,\end{eqnarray}$ where M and S are mutually prime integers, we can find that the running phase (18 ) is also a periodic function with respect to the parameter n $ \begin{eqnarray}f({\phi }_{N},{\theta }_{N,n+S})=f({\phi }_{N},{\theta }_{N,n}).\end{eqnarray}$
If the scaled time takes its values according to equation (19 ), as a fraction of the period, then the state given by equation (17 ) becomes a discrete MQSS of coherent states, which can be expressed as follows $ \begin{eqnarray}| {\rm{\Psi }}{\left(\tau =2\pi M/S\right)}_{N/2,\alpha }\rangle =\displaystyle \sum _{r=1}^{S}{c}_{r}| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{r}}\rangle ,\end{eqnarray}$ where the phase φr is defined by the following expression $ \begin{eqnarray}{\phi }_{r}=2\pi \displaystyle \frac{r}{S}\hspace{0.5cm}(r=1,2,\,...,N),\end{eqnarray}$ and the superposition coefficients are given by $ \begin{eqnarray}{c}_{r}=\displaystyle \frac{1}{S}\displaystyle \sum _{n=1}^{S}\exp \left\{-\displaystyle \frac{2\pi {\rm{i}}}{S}({nr}-M{\theta }_{N,n})\right\},\end{eqnarray}$ which is determined by the number of impurity atoms N = 2j and the coupling ratio K.
The MQSS given by equation (21 ) is a discrete superposition of S equivalent-amplitude coherent states. The quantum structure and properties of the MQSS (21) depend on the superposition coefficients cr and the running phase φr. From equations (21 )–(23 ), we can see that the MQSS (21) can be controlled by manipulating the number of impurity atoms N and the coupling ratio K.
In order to observe the capability of the impurity manipulation on the MQSS, as a specific example, we consider the case of M = one and S = three. In this case, from equation (21 ) we can get a three-component MQSS as follows $ \begin{eqnarray}| {\rm{\Psi }}{\left(\tau =2\pi /3\right)}_{N/2,\alpha }\rangle ={c}_{1}| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{1}}\rangle +{c}_{2}| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{2}}\rangle +{c}_{2}| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{3}}\rangle ,\end{eqnarray}$ where φi are given by equation (22 ) and from equation (23 ) we can obtain the superposition coefficients $ \begin{eqnarray}\begin{array}{rcl}{c}_{1} & = & \displaystyle \frac{1}{3}\left[{{\rm{e}}}^{-\tfrac{2\pi {\rm{i}}}{3}(2-{NK})}+{{\rm{e}}}^{\tfrac{4\pi {\rm{i}}}{3}\cdot {NK}}+1\right],\\ {c}_{2} & = & \displaystyle \frac{1}{3}\left[{{\rm{e}}}^{\tfrac{2\pi {\rm{i}}}{3}\cdot {NK}}+{{\rm{e}}}^{\tfrac{4\pi {\rm{i}}}{3}(1-{NK})}+1\right],\\ {c}_{3} & = & \displaystyle \frac{1}{3}\left[{{\rm{e}}}^{-\tfrac{2\pi {\rm{i}}}{3}(1-{NK})}+{{\rm{e}}}^{-\tfrac{2\pi {\rm{i}}}{3}(1-2{NK})}+1\right],\end{array}\end{eqnarray}$ which are explicitly determined by the number of impurity atoms N and the coupling ratio K.
We now numerically study the properties of the MQSS and the influence of impurity atoms, by means of the Wigner function [64] defined by $ \begin{eqnarray}W(z)=\displaystyle \frac{2}{\pi }\mathrm{Tr}\left[{\hat{D}}^{\dagger }(z)\rho \hat{D}(z)\hat{P}\right],\end{eqnarray}$ where ρ is the density operator of the BEC, $\hat{D}(z)\,=\exp (z{\hat{a}}^{\dagger }-{z}^{* }\hat{a})$ and $\hat{P}={(-1)}^{{\hat{a}}^{\dagger }\hat{a}}$ are the displacement operator and the parity operator, respectively. The Wigner function has turned out to be remarkably useful in quantum physics, particularly in the characterization and visualization of nonclassical fields. The negativity of the Wigner function of a quantum state represents its nonclassicality.
In figure 1 we have plotted the Wigner function of the three-component MQSS (24 ). The related parameters are taken as the initial-state parameter α = 14, the impurity number N = one, and the coupling ratio K = one. From figure 1 we can see that there are three main peaks numbered P1, P2, and P3, and three different quantum interference regimes in the phase space for the Wigner function of the three-component MQSS. One quantum interference regime exists between each of the two main peaks. The three-component MQSS exhibits nonclasicality in each quantum interference regime described by the negativity of the Wigner function. In the following discussion, we show that the impurity atoms significantly affect the quantum interference effect of the MQSS in the phase space.
Figure 1. The Wigner function of the three-component MQSS. The related parameters are taken as the initial-state parameters of the BEC α = 14, the impurity number N = one, and the coupling ratio K = one. |
In order to observe the effect of impurity atoms on the quantum interference effect in the phase space, in figure 2 we have plotted the quantum interference patterns of the Wigner function of the three-component MQSS (24 ), in the quantum interference regime between the first main peak and the second main peak when the impurity numbers are N = 1, 2, 3, and 4, which correspond to figures 2(a)–(d), respectively. From figure 2 we can see that the displacement of the quantum interference patterns takes place with an increasing number of impurities from the left-hand side to the right-hand side. After increasing one impurity, each peak of the Wigner function as a whole moves one step towards the right-hand side. The greater the number of impurities, the greater the steps by which the peaks move. The number of impurities determines the number of steps that each peak moves by. Hence, the quantum interference patterns are very sensitive to the number of impurity atoms. In this sense, the quantum interference patterns in the phase space provide a possible way to probe the existence of impurity atoms in the BEC.
Figure 2. Quantum interference patterns for the Wigner function of the three-component MQSS in the phase space. The related parameters are the number of condensed atoms n0 = 105, the coupling ratio K = one, and the impurity numbers N = 1, 2, 3, and 4, which correspond to figures 1(a)–(c), respectively. |
In order to further understand the influence of impurity atoms on the Wigner function of the BEC, we calculate the Wigner function of the MQSS (24 ), which can be expressed as the sum of a classical term and a quantum interference term $ \begin{eqnarray}W(z)={W}_{C}(z)+{W}_{Q}(z),\end{eqnarray}$ where the classical term is the summation of the independent Wigner functions of three coherent states which build the MQSS (24 ) $ \begin{eqnarray}{W}_{C}(z,\alpha )=\displaystyle \frac{1}{3}[{W}_{1}(z,\alpha )+{W}_{2}(z,\alpha )+{W}_{3}(z,\alpha )],\end{eqnarray}$ where Wn(z, α) is the Wigner function of the coherent state $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{n}}\rangle $ in equation (24 ) given by $ \begin{eqnarray}\begin{array}{rcl}{W}_{1}(z,\alpha ) & = & \displaystyle \frac{2}{\pi }\exp \left(-2{\left|z+\alpha {{\rm{e}}}^{-{\rm{i}}\tfrac{\pi }{3}}\right|}^{2}\right),\\ {W}_{2}(z,\alpha ) & = & \displaystyle \frac{2}{\pi }\exp \left(-2{\left|z+\alpha {{\rm{e}}}^{{\rm{i}}\tfrac{\pi }{3}}\right|}^{2}\right),\\ {W}_{3}(z,\alpha ) & = & \displaystyle \frac{2}{\pi }\exp \left(-2{\left|z-\alpha \right|}^{2}\right).\end{array}\end{eqnarray}$ From equations (28 ) and (29 )we can see that the classical part of the Wigner function consists of three Gaussian peaks at z = α and $z=\alpha {{\rm{e}}}^{\pm {\rm{i}}\tfrac{\pi }{3}}$ , which are clearly indicated in figure 1.
The quantum term in the Wigner function of the MQSS consists of three cross terms in the Wigner function, and can be expressed as $ \begin{eqnarray}{W}_{q}(z,\alpha )=\displaystyle \frac{2}{3}[{W}_{32}(z,\alpha )+{W}_{31}(z,\alpha )+{W}_{21}(z,\alpha )].\end{eqnarray}$
Substituting equation (24 ) into equation (26 ), we can obtain three cross terms in the Wigner function $ \begin{eqnarray}\begin{array}{l}{W}_{32}(z,\alpha )=\displaystyle \frac{2}{\pi }\exp \left(-2{\left|z-\displaystyle \frac{1}{2}\alpha {{\rm{e}}}^{-{\rm{i}}\tfrac{\pi }{3}}\right|}^{2}\right)\\ \quad \times \,\cos \left[\displaystyle \frac{2\pi }{3}({NK}-1)+\sqrt{3}\alpha \left(x-\sqrt{3}y-\displaystyle \frac{1}{2}\alpha \right)\right],\\ {W}_{21}(z,\alpha )=\displaystyle \frac{2}{\pi }\exp \left(-2{\left|z+\displaystyle \frac{\alpha }{2}\right|}^{2}\right)\\ \quad \times \,\cos \left[\displaystyle \frac{2\pi }{3}{NK}+\sqrt{3}\alpha \left(2x+\displaystyle \frac{1}{2}\alpha \right)\right],\\ {W}_{31}(z,\alpha )=\displaystyle \frac{2}{\pi }\exp \left(-2{\left|z-\displaystyle \frac{1}{2}\alpha {{\rm{e}}}^{{\rm{i}}\tfrac{\pi }{3}}\right|}^{2}\right)\\ \quad \times \,\cos \left[\displaystyle \frac{2}{3}\pi +\sqrt{3}\alpha \left(x+\sqrt{3}y-\displaystyle \frac{1}{2}\alpha \right)\right],\end{array}\end{eqnarray}$ where we define $x={\rm{Re}}z$ and $y={\rm{Im}}z$ . The cross terms above depend on the phase of the superposition of three coherent states $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{1}}\rangle $ , $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{2}}\rangle $ , and $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{3}}\rangle $ in equation (24 ). They are not positive definite. Their negative values indicate the nonclassicality of the MQSS.
From the expressions of the cross terms we can see that the function WQ(z, α) describes the quantum interference effects between the states $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{1}}\rangle $ , $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{2}}\rangle $ , and $| \alpha {{\rm{e}}}^{{\rm{i}}{\phi }_{3}}\rangle $ in the phase space due to the appearance of the cosine and sine terms. Since W31(z, α) does not depend on the number of impurity atoms, the impurities are not the cause of the quantum interference effect in the quantum interference regime between the first and third main peaks. The quantum interference term W21(z, α) is responsible for the appearance of the quantum interference patterns in figure 2. The impurity-induced shift of the quantum interference patterns in figure 2 is a result of the fact that the number of impurity atoms directly affects the phase of the cosine term of W21(z, α). In fact, the quantum interference term W21(z, α) can be rewritten as $ \begin{eqnarray}\begin{array}{rcl}{W}_{21}(z,\alpha ) & = & \displaystyle \frac{2}{\pi }\exp \left(-2{\left|z+\displaystyle \frac{\alpha }{2}\right|}^{2}\right)\\ & & \times \cos \left[2\sqrt{3}\alpha \left(x+{x}_{D}N\right)-\displaystyle \frac{\sqrt{3}}{2}{\alpha }^{2}\right],\end{array}\end{eqnarray}$ where the shift amount of the quantum interference pattern induced by one single impurity xD is given by $ \begin{eqnarray}{x}_{D}=\displaystyle \frac{\sqrt{3}\pi K}{9\alpha },\end{eqnarray}$ which indicates that the smaller the initial state parameter αof the BEC and/or the larger the coupling ratio K, the larger the shift amount of the quantum interference pattern induced by one single impurity xD.
4. Quantum CR phenomenon of the BEC
In this section we investigate the quantum dynamics of the BEC matter-wave field. We show that impurity atoms immersed in the BEC can induce the quantum CR phenomenon of dynamic evolution of the BEC matter-wave field. This CR phenomenon is a consequence of the impurity-BEC interaction and the interatomic nonlinear self-interaction in the BEC. It can reflect the quantized structure of the matter-wave field.
We suppose that the BEC and the impurity atoms are initially in the following separable state $ \begin{eqnarray}| {\rm{\Psi }}(0)\rangle =[\cos (\theta )| j,-j\rangle +\sin (\theta )| j,j\rangle ]\otimes | \alpha \rangle ,\end{eqnarray}$ which indicates that the BEC is initially in the coherent state $| \alpha \rangle $ while the initial state of the impurity atoms is a GHZ-like state $\cos (\theta )| j,-j\rangle +\sin (\theta )| j,j\rangle $ .
From equations (6 ) and (34 ) we can obtain the wave function of the impurity-doped BEC system at a time t $ \begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}(t)\rangle & = & \cos (\theta )| {\rm{\Psi }}{\left(t\right)}_{-j,\alpha }\rangle \otimes | j,-j\rangle \\ & & +\sin (\theta )| {\rm{\Psi }}{\left(t\right)}_{+j,\alpha }\rangle \otimes | j,j\rangle ,\end{array}\end{eqnarray}$ where the impurity-dependent BEC state $| {\rm{\Psi }}{\left(t\right)}_{\pm j,\alpha }\rangle $ is given by equation (13 ).
Evaluating the BEC field operator $\hat{a}$ for a quantum state of the BEC then yields the macroscopic matter-wave field, which has an intriguing dynamic evolution. We let $ \begin{eqnarray}E(t)=\langle {\rm{\Psi }}(t)| \hat{a}| {\rm{\Psi }}(t)\rangle ,\end{eqnarray}$ which can be expressed as $ \begin{eqnarray}E(t)={\cos }^{2}(\theta ){E}_{-}(t)+{\sin }^{2}(\theta ){E}_{+}(t),\end{eqnarray}$ where we have introduced two functions $ \begin{eqnarray}{E}_{\pm }(t)=\langle {{\rm{\Psi }}}_{\pm j\alpha }| \hat{a}| {{\rm{\Psi }}}_{\pm j\alpha }\rangle .\end{eqnarray}$
Substituting equation (13 ) into the above equation, we can obtain $ \begin{eqnarray}\begin{array}{rcl}{E}_{\pm }(t) & = & \langle \alpha | {{\rm{e}}}^{{\rm{i}}t(\lambda {\hat{n}}^{2}\pm j\kappa \hat{n})}\hat{a}{{\rm{e}}}^{-{\rm{i}}t(\lambda {\hat{n}}^{2}\pm j\kappa \hat{n})}| \alpha \rangle \\ & = & \alpha \exp [-2| \alpha {| }^{2}{\sin }^{2}(\lambda t)]{{\rm{e}}}^{-{\rm{i}}{\phi }_{\alpha \lambda }^{\pm }(t)},\end{array}\end{eqnarray}$ where we have introduce the following two phase functions $ \begin{eqnarray}{\phi }_{\alpha \lambda }^{\pm }(t)=\displaystyle \frac{1}{2}(2\lambda \pm N\kappa )t+| \alpha {| }^{2}\sin (2\lambda t).\end{eqnarray}$
Without loss of generality, we consider the case of α being a real number; from equations (37 ), (39 ) and (40 ) we can get $ \begin{eqnarray}\begin{array}{rcl}E(t) & = & \sqrt{{n}_{0}}\exp [-2{n}_{0}{\sin }^{2}(\lambda t)][{\cos }^{2}(\theta ){{\rm{e}}}^{-{\rm{i}}{\phi }_{\alpha \lambda }^{-}(t)}\\ & & +{\sin }^{2}(\theta ){{\rm{e}}}^{-{\rm{i}}{\phi }_{\alpha \lambda }^{+}(t)}],\end{array}\end{eqnarray}$ where ${n}_{0}=| \alpha {| }^{2}$ is the number of the condensed atoms in the initial state.
In what follows, we study the quantum CR phenomenon of the BEC matter-wave field for two cases: (1) without the initial entanglement of the impurities, in this case we have θ = zero or θ = π/2. (2) The impurities are initially in a maximally entangled state with θ = π/4.
In the absence of the initial entanglement with the impurities, from equation (41 ) we can obtain the following expression of the amplitude of the BEC matter-wave field $ \begin{eqnarray}\begin{array}{rcl}A(t) & = & | {E}_{\theta =0}(t){| }^{2}=| {E}_{\theta =\pi /2}(t){| }^{2}\\ & = & {n}_{0}\exp \left[-4{n}_{0}{\sin }^{2}(\lambda t)\right],\end{array}\end{eqnarray}$ which indicates that the dynamic evolution of the amplitude of the BEC matter-wave field is a simple periodic oscillation which only depends upon the two self-parameters of the BEC, the number of the condensed atoms and the interatomic self-interaction strength in the BEC. The amplitude oscillation is determined by the number of condensed atoms while the period of the oscillation is determined by the interatomic self-interaction strength in the BEC.
We now take into account the case of the maximally entangled state of the impurities (θ = π/4). In this situation, the BEC matter-wave field is given by $ \begin{eqnarray}\begin{array}{rcl}{E}_{\theta =\pi /4}(t) & = & \sqrt{{n}_{0}}\cos \left(\displaystyle \frac{N\kappa t}{2}\right)\exp \left[-2{n}_{0}{\sin }^{2}(\lambda t)\right]\\ & & \times {{\rm{e}}}^{-{\rm{i}}[\lambda t+{n}_{0}\sin (2\lambda t)]},\end{array}\end{eqnarray}$ which leads to the amplitude of the BEC matter-wave field given by $ \begin{eqnarray}A(t)=| {E}_{\theta =\pi /4}(t){| }^{2}={n}_{0}{\cos }^{2}\left(\displaystyle \frac{N\kappa t}{2}\right)\exp \left[-4{n}_{0}{\sin }^{2}(\lambda t)\right],\end{eqnarray}$ which can be expressed in terms of the scaled time τ = λt and the coupling ratio K = κ/λ as $ \begin{eqnarray}A(\tau )={n}_{0}{\cos }^{2}\left(\displaystyle \frac{{NK}\tau }{2}\right)\exp \left[-4{n}_{0}{\sin }^{2}(\tau )\right].\end{eqnarray}$
From equation (45 ) we can see that the amplitude of the BEC matter-wave field is factorized as the product of the BEC part ${n}_{0}\exp \left[-4{n}_{0}{\sin }^{2}(\tau )\right]$ and the impurity part ${\cos }^{2}\left(\tfrac{{NK}\tau }{2}\right)$ . It is straightforward to observe the CR phenomenon of the BEC matter-wave field when detuning exists between the oscillation of the BEC part and the oscillation of the impurity part. The CR of the BEC matter-wave field depends on the number of impurities and condensed atoms, and the coupling ratio between the impurity-BEC coupling and the interatomic self-interaction strength in the BEC. Comparing equation (45 ) with equation (42 ), we find that it is the initial entanglement among impurities that leads to the CR phenomenon of the BEC matter-wave field. In this sense we say that the quantum CR phenomenon of the BEC matter-wave field is a kind of impurity-induced CR effect. This nature of the Wigner function could also be used as evidence that the dispersive impurity-BEC interaction is valid.
In order to observe how the impurity atoms modulate the CR phenomenon of the BEC matter-wave field, in figure 3 we have plotted the quantum evolution of the amplitude of the BEC matter-wave field with respect to the scaled time τ when the BEC part is rapidly changing while the impurity part is slowly changing for the two cases of the absence and the presence of impurities, respectively. Figure 3(a) corresponds to the case of the absence of impurities, we take the number of the BEC n0 = 105 and the coupling ratio κ = zero. Figure 3(b) corresponds to the case of the presence of impurities, we take the number of the BEC n0 = 105, the number of impurities N = 20, and the coupling ratio κ = 0.005. From figure 3(a) we can see that the time evolution of the BEC matter-wave field is a simple equivalent-amplitude and periodic oscillation in the absence of the impurity atoms. Figure 3(b) indicates that the complete CR phenomenon of the BEC matter-wave field takes place in the quantum evolution of the BEC matter-wave field.
Figure 3. The quantum evolution of the amplitude of the BEC matter-wave field versus the scaled time τ; the BEC part is the rapidly changing part. (a) The case of the absence of impurities with the number of the BEC n0 = 105 and the coupling ratio κ = zero. (b) The case of the presence of impurities with the number of the BEC n0 = 105, the number of impurities N = 20, and the coupling ratio κ = 0.005. |
In figure 4 we have plotted the quantum evolution of the amplitude of the BEC matter-wave field with respect to the scaled time τ when the impurity part is rapidly changing while the BEC part is slowly changing. Figure 4(a) corresponds to the case of the number of the BEC n0 = 105, the number of impurities N = ten, and the coupling ratio κ = 500. Figure 4(b) corresponds to the case of the number of the BEC n0 = 105, the number of impurities N = 20, and the coupling ratio κ = 500. The dashed line in figures 4(a) and (b) corresponds to the situation of the absence of impurities with κ = zero. From figures 4(a) and (b) we can see that the number of impurities can manipulate the internal structure of the CR patterns, but it does not change the envelope of the CR patterns, the collapse time and the revival time.
Figure 4. The quantum evolution of the amplitude of the BEC matter-wave field versus the scaled time τ when the impurity part is rapidly changing, for different values of the number of impurities. Figures 4(a) and (b) correspond to the number of impurities N = ten and N = 20, respectively. The other related parameters are the number of the BEC n0 = 105 and the coupling ratio κ = 500. The dashed line in figures 4(a) and (b) corresponds to the situation of the absence of the impurities with κ = zero. |
From figures 3 and 4 we can conclude that the CR phenomenon of the BEC matter-wave field originates from the impurity-BEC interaction. The quantum CR phenomenon of the BEC matter-wave field can be understood as an impurity-induced quantum effect of the BEC matter-wave field. This quantum CR effect reveals the quantum nature of the BEC matter-wave field. This is just like the quantum CR effect of the Rabi oscillations in the Jaynes–Cummings model [42, 43] which describes the interaction between a two-level atom and a single-mode optical field, which reflects the quantum nature of the optical field.
5. Quantum entanglement between impurities and a BEC
In this section we study quantum entanglement between impurity atoms and a BEC in an impurity-doped BEC system during dynamic evolution. From the discussion in the second section we can see that the wave function of an impurity-doped BEC system is a micro-macro entangled state given by equation (12 ) when the BEC is initially in the coherent state $| \alpha \rangle $ and the initial state of the impurity atoms is a GHZ-like state given by equation (10 ). In order to study the quantum dynamics of the micro-macro entanglement, we need to calculate the micro-macro entanglement between the impurities and the BEC. Note that the micro-macro entangled state (12 ) is a two-component entangled state. Its entanglement amount can be measured by quantum concurrence through changing it to an equivalent two-qubit state. The concurrence of an arbitrary quantum state of two qubits with a density operator ρR(t) [65] is given by $ \begin{eqnarray}{ \mathcal C }=\max \{0,{\lambda }_{1}-{\lambda }_{2}-{\lambda }_{3}-{\lambda }_{4}\},\end{eqnarray}$ where the λi (i = 1, 2, 3, 4) are the square roots of the eigenvalues in descending order of the operator $R={\rho }_{R}(t)({\sigma }_{y}^{1}\otimes {\sigma }_{y}^{2}){\rho }_{R}^{* }(t)({\sigma }_{y}^{1}\otimes {\sigma }_{y}^{2})$ with Σy being the Pauli operator in the computational basis. It ranges from ${ \mathcal C }=0$ for a separable state to ${ \mathcal C }=1$ for a maximally entangled state.
It is interesting to note that the entanglement degree of an arbitrary two-component bipartite state can be measured by quantum concurrence through transformation to an equivalent two-qubit state [62, 66]. In fact, if we consider the following bipartite state $ \begin{eqnarray}| {\rm{\Psi }}\rangle =\mu | \eta \rangle \otimes | \gamma \rangle +\nu | \xi \rangle \otimes | \delta \rangle ,\end{eqnarray}$ where $| \eta \rangle $ and $| \xi \rangle $ are two normalized states of the first subsystem while $| \gamma \rangle $ and $| \delta \rangle $ are two normalized states of the second subsystem. After normalization the wave function becomes $ \begin{eqnarray}| {\rm{\Psi }}\rangle =\displaystyle \frac{1}{M}[\mu | \eta \rangle \otimes | \gamma \rangle +\nu | \xi \rangle \otimes | \delta \rangle ],\end{eqnarray}$ where the normalization coefficient is given by $ \begin{eqnarray}{M}^{2}=| \mu {| }^{2}+| \nu {| }^{2}+2\mathrm{Re}({\mu }^{* }\nu {p}_{1}{p}_{2}^{* }),\end{eqnarray}$ where p1 and p2 are overlapping functions between two quantum states for each subsystem $ \begin{eqnarray}{p}_{1}=\langle \eta | \xi \rangle ,{p}_{2}=\langle \delta | \gamma \rangle .\end{eqnarray}$
After transformation to an equivalent two-qubit state, the quantum concurrence of the quantum state (66) is given by $ \begin{eqnarray}C=\displaystyle \frac{2| \mu | | \nu | }{{M}^{2}}\sqrt{(1-| {p}_{1}{| }^{2})(1-| {p}_{2}{| }^{2})}.\end{eqnarray}$
For the micro-macro entangled state of the impurity-doped BEC system given by equation (12 ), which is a normalized state with M = one, we have $ \begin{eqnarray}\begin{array}{rcl}{p}_{1} & = & \langle {{\rm{\Psi }}}_{-j\alpha }(t)| {{\rm{\Psi }}}_{+j\alpha }(t)\rangle =\exp [-| \alpha {| }^{2}(1-{{\rm{e}}}^{-{\rm{i}}2j\kappa t})],\\ {p}_{2} & = & \langle j,j| j,-j\rangle =0,\end{array}\end{eqnarray}$ and the two-component superposition coefficients given by $ \begin{eqnarray}\mu =\cos (\theta ),\nu =\sin (\theta ).\end{eqnarray}$
Substituting equations (52 ) and (53 ) into equation (51 ), we can obtain the quantum concurrence of the micro-macro entangled state (12 ) $ \begin{eqnarray}C=\sqrt{2{C}_{0}\left[1-{{\rm{e}}}^{-4{n}_{0}{\sin }^{2}(N\kappa t/2)}\right]},\end{eqnarray}$ where ${n}_{0}=| \alpha {| }^{2}$ is the number of condensed atoms, ${C}_{0}=| \sin (2\theta ){| }^{2}/2$ is the initial quantum coherence of N impurities and ${n}_{0}=| \alpha {| }^{2}$ is the number of condensed atoms with the maximal quantum coherence being C0 = 0.5.
From equation (54 ) we can see that micro-macro entanglement dynamics in the impurity-doped BEC system depend upon four parameters: the number of impurity atoms (N), the initial quantum coherence of the impurities (C0), the impurity-BEC interaction (κ), and the number of condensed atoms (n0).
It should be noted that micro-macro entanglement is a kind of quantum entanglement induced by the initial quantum coherence of impurity atoms. From equation (50 ) we can see that the impurity-BEC entanglement dynamic vanishes without the initial quantum coherence of the impurity atoms. In this sense the micro-macro entanglement between the impurity atoms and the BEC is a transferring result of the initial quantum coherence of the impurities to the impurity-BEC system. In fact, the initial quantum coherence of the impurity atoms C0 determines the maximal entanglement amount between the impurities and the BEC in the process of dynamic evolution. On the other hand, equation (50 ) indicates that the impurity-BEC entanglement dynamic is a simple periodic oscillation with an evolution period $T=4\pi /(N\kappa )$ , which is determined by the number of impurity atoms and the impurity-BEC coupling strength κ. The stronger the impurity-BEC coupling strength, the faster the entanglement oscillation in the dynamic evolution. The larger the number of impurity atoms, the faster the entanglement oscillation.
Equation (50 ) indicates that one can control the micro-micro entanglement dynamics between the impurities and the BEC by changing the amount of initial quantum coherence of the impurities, the number of the impurities and the BEC, and the impurity-BEC interaction strength. The impurity-BEC entanglement vanishes at times ${t}_{n}=2n\pi /(M\kappa )$ ( $n=0,1,2,\cdots ,\infty $ ) with the concurrence C = zero. The impurity-doped BEC system reaches the maximally entangled state at times ${t}_{n}=(2n+1)\pi /(M\kappa )$ ( $n=0,1,2,\cdots ,\infty $ ) with the concurrence $C=\sqrt{2{C}_{0}(1-{{\rm{e}}}^{-4{n}_{0}})}$ , which indicates that the maximal amount of micro-macro entanglement can be manipulated through enhancing the initial quantum coherence of the impurity atoms and/or increasing the number of the condensed atoms.
6. Summary and conclusion
We have studied the quantum dynamics of an impurity-doped BEC system consisting of a BEC and N two-level immersed impurities. We have found that the presence of impurities in the BEC substantially affects the quantum dynamics of the BEC system. Although the number of impurity atoms is much less than that of the condensed atoms, the impurities manifest their effects when they interact with the BEC. We have shown how to generate an MQSS of a BEC by the use of projective measurements on impurity atoms. It has been indicated that one can control the nonclassicality of the MQSS described by the Wigner function in the phase space through varying the number of impurities. In particular, it was found that the impurities can induce the shift of the quantum interference patterns of the MQSS in the phase space. The very sensitivity of the quantum interference patterns of the MQSS to the impurity influence provides an effective way to detect the existence of the impurities in the BEC. It has been shown that the BEC matter-wave field exhibits quantum CR phenomenon in its temporal evolution. The quantum CR patterns depend on the initial-state parameters of the impurities and the BEC and impurity-BEC interaction. This quantum CR effect reveals the quantum nature of the BEC matter-wave field. It is similar to the quantum CR effect of the Rabi oscillations in the Jaynes–Cummings model that describe the interaction between a two-level atom and a single-mode optical field, which reflects the quantum nature of the optical field. We have investigated the quantum dynamics of the micro-macro entanglement between impurities and the BEC, and found enhancement of the impurity-BEC entanglement induced by the initial quantum coherence of the impurity atoms. Our results cast a new light on impurity-doped BEC physics. It would be interesting to experimentally observe the quantum effects predicted in the present paper, for instance the MQSSs and their quantum interference effects in phase space, the CR phenomenon of the BEC matter-wave field, and the micro-macro entanglement in an impurity-doped BEC system.