Concurrence of Two Identical Atoms in a Rectangular Waveguide: Linear Approximation with Single Excitation

TL;DR

This paper investigates how two identical two-level atoms interact within a rectangular waveguide, focusing on their quantum entanglement dynamics using a linear approximation with a single excitation.

Contribution

It introduces a simplified model for analyzing entanglement evolution of two TLSs in a waveguide near cutoff frequencies, considering inter-atom distance effects.

Findings

Concurrence depends on inter-TLS distance and waveguide mode structure.

Delay-differential equations describe the entanglement dynamics.

Energy separation far from cutoff simplifies the interaction model.

Abstract

We study two two-level systems (TLSs) interacting with a reservoir of guided modes confined in a rectangular waveguide. For the energy separation of the identical TLSs far away from the cutoff frequencies of transverse modes, the delay-differential equations are obtained with single excitation initial in the TLSs. The effects of the inter-TLS distance on the time evolution of the concurrence of the TLSs are examined.

Equations

$$H_a=\sum _{l=1}^2\hslash {\omega }_A{\sigma }_l^{+}{\sigma }_l^{-}$$

$$H_f=\sum _j\int dk\hslash {\omega }_{jk}{\hat{a}}_{jk}^{†}{\hat{a}}_{jk}$$

$$H_{int}=\sum _{l=1}^2\sum _j\int dk\hslash \frac{g_{jl}}{\sqrt{{\omega }_{jk}}}{e}^{ik{z}_l}{S}_l^{-}{\hat{a}}_k^{†}+h.c.$$

$$g_j=\frac{{\Omega }_j\mu \sin \left(\frac{m\pi }{2}\right)\sin \left(\frac{n\pi }{2}\right)}{\sqrt{\hslash A\pi {ϵ}_0}},$$

$$H=H_f+H_a+H_{int}$$

$$|\psi (t)⟩=b_1|20⟩+b_2|30⟩+\sum _j\int dk{b}_{jk}{a}_{jk}^{†}|10⟩$$

$${\dot{b}}_1$$

$${\dot{b}}_2$$

$${\dot{b}}_{jk}$$

$$b_1(t)$$

$$b_2(t)$$

$$b_{jk}(t)$$

$${\omega }_{jk}={\omega }_A+v_j\left(k-k_{j0}\right),$$

$$v_j\equiv \frac{d{\omega }_{jk}}{dk}{|}_{k=k_{j0}}=\frac{c\sqrt{{\omega }_A^2-{\Omega }_j^2}}{{\omega }_A}$$

$$\left({\partial }_t+\gamma \right)B_1(t)$$

$$\left({\partial }_t+\gamma \right)B_2(t)$$

$$\left({\partial }_t+\gamma \right)C_s(t)$$

$$\left({\partial }_t+\gamma \right)C_a(t)$$

$$C(t)=\max (0,2\left|B_1(t)B_2^{\ast }(t)\right|)$$

$$C_s(t)$$

$$C_a(t)$$

$$C_s(t)$$

$$C_a(t)$$

$$C_s(t)=C_{s0}\exp \left[-t{\gamma }_1(1+\cos {\phi }_1)-it{\gamma }_1\sin {\phi }_1\right]$$

$$C_s(t)\propto \frac{{\left(-{\gamma }_1{e}^{i{\phi }_1}\right)}^n}{n!}{t}_n^n{e}^{-{\gamma }_1{t}_n}$$

$$\left({\partial }_t+\gamma \right)C_s(t)$$

$$\left({\partial }_t+\gamma \right)C_a(t)$$

$$C_s(t)$$

$$C_a(t)$$

$$C_s=C_{s0}\sum _{n=0}^{\infty }\frac{{\left(-{\alpha }_2\right)}^n}{n!}{e}^{-\left(\gamma +{\alpha }_1\right)\left(t-n{\tau }_2\right)}{\left(t-n{\tau }_2\right)}^n$$

$$C_a=C_{a0}\sum _{n=0}^{\infty }\frac{{\alpha }_2^n}{n!}{e}^{-\left(\gamma -{\alpha }_1\right)\left(t-n{\tau }_2\right)}{\left(t-n{\tau }_2\right)}^n$$

$$C_s$$

$$C_a$$

Full text

I Introduction

Quantum entanglement is a nonlocal correlation of multipartite quantum systems, which distinguishes the quantum world from the classical world. Due to its important role in quantum computation and communication, it is a physical resource which quantum technologies are based on. However, the inevitable interaction of quantum systems with their surrounding environments induces decoherence of quantum systems, which degrades the entanglement of quantum systems. Understanding the dynamics of the entanglement is desirable to be able to manipulate entanglement states in a practical way as well as the question of emergent classicality from quantum theory. Entanglement dynamics is studied under local decoherence (two particles in an entangled state are coupled to its own environment individually), a peculiar dynamical feature of entangled state is that complete disentanglement is achieved in finite time although complete decoherence takes an infinite time, which is termed “entanglement sudden death” ESD1 ; ESD2 . The assumption of local decoherence requires that two two-level systems (TLSs), e.g. atoms, are sufficiently separated. It is well known that the radiation field emitted by an atom may influence the dynamics of its closely spaced atoms Dicke ; Lehmberg70 ; Feng41 ; Ordon70 ; BermanPRA76 . The entanglement can be generated in a two-atom system after a finite time by their cooperative spontaneous emission, or the destroyed entanglement may reappear suddenly after its death, which is known as sudden birth of entanglement ESB .

In quantum network, stationary qubits generate, store, and process quantum information at quantum nodes, and flying qubit transmit quantum information between the nodes through quantum channels. A distributed quantum network requires coherently transferring quantum information among stationary qubits, flying qubits, and between stationary qubits and flying qubits. With the development of techniques in quantum information, an alternative waveguide-based quantum electrodynamics (QED) system has emerged as a promising candidate for achieving quantum network Tien ; HamRMP82 ; Caruso . In this system, atoms are located at quantum nodes and photons propagating along the network are confined in a waveguide. Inside a one-dimensional (1D) waveguide, the electromagnetic field is confined spatially in two dimensions and propagates along the remaining one, which is called guided modes. The spectrum of the guided modes is continuous. The coupling of the electromagnetic field to a TLS can be increased by reducing the transverse size of the guided modes. Therefore the study of entanglement dynamics in systems embedded in waveguides is of importance. A waveguide with a cross section has many guided modes TETMmode , e.g. transverse-magnetic (TM) modes or transverse-electric (TE) ones. However, most work only consider one guided mode of the waveguide Fans ; ZLPRL08 ; ZLQrouter ; Zheng ; LawPRA78 ; TShiSun ; PRA14Red ; Ordonez . In this paper, we consider the dynamic behavior of bipartite entanglement involving two identical TLSs which is implanted into the 1D rectangular hollow metallic waveguide. Since local addressing is difficult, we assume that there is no direct interaction between the TLSs, the TLSs and the field share initially a single excitation. By considering the energy separation of the TLSs is far away from the cutoff frequencies of the transverse modes, the delay differential equations are obtained for two TLSs’ amplitudes with the field initially in vacuum, where multiple guided modes are included. The spatial separation of the two TLSs introduces the position-dependent phase factor and the time delay (finite time required for light to travel from one TLS to the other) in each transverse mode. The phase factors and the time delays are different in different transverse modes. The effect of the phase factors and the time delays on the entanglement dynamics of the TLSs are studied in details by considering the TLSs interacting with single transverse mode and double transverse modes.

This paper is organized as follows. In Sec. II, we introduce the model and establish the notation. In Sec. III, we derive the relevant equations describing the dynamics of the system for the TLSs being initially excited and the waveguide mode in the vacuum state, and investigate the effect of spatial separation on the dynamics of entanglement between two identical TLSs, which is characterized by concurrence. We make a conclusion in Sec. IV.

II Two TLSs in a rectangular waveguide

We consider a rectangular hollow metallic waveguide with the area $A=ab$ ($a=2b$) of its cross section, as shown in Fig. 1. The axes of the waveguide parallel to the $z$ axis, and the waveguide is infinite along the $z$ axis. Since the translation invariance is maintained along the $z$ axis, all the components of the electromagnetic field describing the guided mode depend on the coordinate z as $e^{ikz}$. The guided mode can be characterized by three wave numbers $\{k_{x},k_{y},k_{z}\}$. The spatial confinement of the electromagnetic field along the $xy$ plane makes the appearance of the two non-negative integers $m$ and $n$, which related to wave numbers along the $x$ and $y$ directions by $k_{x}=m\pi/a$ and $k_{y}=n\pi/b$. In this waveguide, there are two types of guiding modes TETMmode ; HuangPRA ; Shahmoon ; liqongPRA ; zhouCTP69 :the transverse-magnetic modes $TM_{mn}$ ($H_{z}=0$) and the transverse-electric modes $TE_{mn}$ ($E_{z}=0$). Two idential TLSs, named TLS $1$ and TLS $2$, are separately located inside the waveguide at positions $\vec{r}_{1}=(a/2,b/2,z_{1})$ and $\vec{r}_{2}=(a/2,b/2,z_{2})$, the distance between the TLSs is denoted by $d=z_{2}-z_{1}$. The free Hamiltonian of the TLSs read

[TABLE]

where $\omega_{A}$ are the energy difference between the excited state $|e\rangle$ and the ground state $|g\rangle$, and $\sigma_{l}^{+}\equiv\left|e_{l}\right\rangle\left\langle g_{l}\right|$ ($\sigma_{l}^{-}\equiv\left|g_{l}\right\rangle\left\langle e_{l}\right|$ ) is the rising (lowing) atomic operator of the $l$-$th$ TLS. We assume the dipoles of TLSs are along the $z$ axis. In this case, only the $TM_{mn}$ guided modes are interacted with the TLSs. The free Hamiltonian of the field reads

[TABLE]

where $\hat{a}_{jk}^{\dagger}$ ($\hat{a}_{jk}$) is the creation (annihilation) operator of the $TM_{mn}$ modes. Here,we have replaced $(m,n)$ with the sequence number $j$, i.e., $j=1,2,3...$ denoting $TM_{11},TM_{31},TM_{51}\cdots$, respectively. For each guided mode, the dispersion relation is given by $\omega_{jk}=\sqrt{\Omega_{j}^{2}+c^{2}k^{2}}$, where $\Omega_{mn}=c\sqrt{(m\pi/a)^{2}+(n\pi/b)^{2}}$ is the cutoff frequency. No electromagnetic field can be guided if their frequency is smaller than the cutoff frequency $\Omega_{1}$. The interaction between the TLSs and the the electromagnetic field is written as

[TABLE]

in the electric dipole and rotating wave approximations, where $g_{jl}=\Omega_{j}\mu_{l}/\sqrt{A\pi\epsilon_{0}}$ and $\mu_{l}$ the magnitude of the dipole of the $l$-th TLS. We assume that $\mu_{1}=\mu_{2}=\mu$ is real. Then the parameter $g_{jl}$ becomes

[TABLE]

where $\epsilon_{0}$ is the permittivity of free space. The TLS’s position is presented in the exponential function in Eq.(3).

The total system, the two TLSs and the photons in quantum electromagnetic field, is described by the Hamiltonian

[TABLE]

The total system is a closed system. However, each subsystem is an open system. When we are only interested in the dynamics of TLSs, the quantum electromagnetic field can be regarded as an environment.

III Entanglement Dynamics

Any state of the two TLSs are linear superposition of the basis of the separable product states $\left|1\right\rangle=\left|g_{1}g_{2}\right\rangle$, $\left|2\right\rangle=\left|e_{1}g_{2}\right\rangle$, $\left|3\right\rangle=\left|g_{1}e_{2}\right\rangle$, and $\left|4\right\rangle=\left|e_{1}e_{2}\right\rangle$. Since the number of quanta is conserved in this system, the wavefunction of the total system can be written as:

[TABLE]

in single excitation subspace, where $\left|0\right\rangle$ is the vacuum state of the quantum field. The first term in Eq.(6)presents TLS $1$ in the excited state with no excitations in the field, $b_{1}\left(t\right)$ is the corresponding amplitude, the second term in Eq.(6) presents TLS $2$ in the excited state with photons in the vacuum, whereas the third term in Eq.(6) describes all TLSs in the ground state with a photon emitted at a mode $k$ of the TMj guided mode, $b_{jk}\left(t\right)$ is the corresponding amplitude. The initial state of the system is denoted by the amplitudes $b_{1}\left(0\right),b_{2}\left(0\right)$, $b_{jk}\left(0\right)=0$. The Schrödinger equation results in the following coupled equation of the amplitudes

[TABLE]

[TABLE]

[TABLE]

where the wavelength of the emitted radiation $k_{j0}=\sqrt{\omega_{A}^{2}-\Omega_{j}^{2}}/c$ is determined by $\omega_{jk_{0}}=\omega_{A}$, and the group velocity

[TABLE]

is different for different TMj guided modes. Integrating over all wave vectors $k$ gives rise to a linear combination of $\delta\left(t-\tau-\tau_{j}\right)$ and $\delta\left(t-\tau_{j}\right)$, where $\tau_{j}=d/v_{j}$ is the time delay taking by a photon traveling from one TLS to the other TLS in the given transverse mode $j$. The differential equations governing the dynamics of two TLSs read

[TABLE]

[TABLE]

To measure the amount of the entanglement, we use concurrence as the quantifier Wootters . By taking a partial trace over the waveguide degrees of freedom, the initial density matrix of the two TLSs is of an X-form in the two-qubit standard basis $\{\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle,\left|4\right\rangle\}$. The concurrence for this type of state can be calculated easily as

[TABLE]

which can also expressed as the function of the amplitudes of the Dicke states by the relation

[TABLE]

III.1 Single transverse mode

In the frequency band between $\Omega_{11}$ and $\Omega_{31}$, the waveguide is said to be single-moded. The TLSs with the transition frequency $\omega_{A}\in\left(\Omega_{11},\Omega_{31}\right)$ only emit photons into the TM11 ($j=1$) guided mode. In this case, the time behavior of Dicke states reads

[TABLE]

From Eq.(15), one can observe that there is a $\pi$ phase difference between the amplitudes $C_{s}(t)$ and $C_{a}(t)$. Hence, we assume that the two TLSs are initially prepared in the symmetric state $C_{s0}=1$ to study the effect of the inter-TLS distance on the the dynamics of entanglement between the TLSs. In this case, the concurrence takes the maximum value between [math] and $|C_{s}(t)|^{2}$. In Fig. 2, we have numerically plotted the concurrence as a function of time $t$ in units of $\tau_{1}$ with $\gamma_{1}\lambda_{1}/\tau_{1}=0.05$, where the wavelength $\lambda_{1}k_{10}=2\pi$. In the time interval $t\in\left[0,\tau_{1}\right]$, two TLSs radiate spontaneously, so the concurrence decays exponentially with time. As time goes on, the inter-TLS distance have influenced the entanglement dynamics via phase $\varphi_{1}$ and delay time $\tau_{1}$. When the two TLSs are close together, two TLSs act collectively, the system dynamics is independent of the finite propagating time of the light, which has been shown in Fig. 2(a) with $\gamma_{1}\tau_{1}\ll 1$. There is stationary two-TLS entanglement when the inter-TLS distance equals an odd integer number of $\lambda_{1}/2$, (i.e., $\varphi_{1}=2n\pi+\pi$). And a small deviation of the special position leads to the entanglement decaying asymptotically to zero. The entanglement loses fast when the inter-TLS distance equals an integer number of $\lambda_{1}$ corresponding to $\varphi_{1}=2n\pi$. For $\gamma_{1}\tau_{1}\ll 1$, The dependence of the entanglement on phase in Fig. 2(a) can be understood by letting $\tau_{1}\rightarrow 0$. In this case, the amplitude of state $\left|s\right\rangle$ becomes

[TABLE]

It can be observed from Eq. (16) that $|C_{s}(t)|$ exponentially decays with time, it decays fast when $\varphi_{1}=2n\pi$ and keeps its initial value when $\varphi_{1}=2n\pi+\pi$. Although one can explain the relation of entanglement with phase by Eq. (16), the probability of finding the TLSs in the initial state is less than unity in Fig. 2(a) when $\varphi_{1}=2n\pi+\pi$. Hence, the stationary two-TLS entanglement indicates a superposition of the symmetry state in the absence of photons and the ground TLS state in the presence of a photon. As the inter-TLS separation increases a little bit to meet $\gamma_{1}\tau_{1}\sim 1$ in Fig. 2(b), both the decay time and the phase play important roles due to the interference. The interference produced by multiple reemissions and reabsorptions of photon results in an oscillatory entanglement. Panel (c) of Fig. 2 illustrates the dynamics of entanglement for a larger inter-TLS distance with $\gamma_{1}\tau_{1}\gg 1$. It can be observed that the phase does not make any sense. At early time, each initially excited TLS emits light to the waveguide, the entanglement begins to decrease abruptly from one. Then the concurrence keeps small and approximates to zero until the time $t=\tau_{1}$, at which the excitation get absorbed by each TLS. As soon as the emitted photon returns to the TLSs, the entanglement is created. Then, the decrease of entanglement begins. The entanglement of the TLSs exhibits peaks due to the iteration of the process where a photon emitted by one of the atoms is reabsorbed by another atom, but its periodic maxima are reduced in magnitude as $t$ increases because the energy is carried away from TLSs by the forward-going waves emitted by TLS 1 and the backward-going waves emitted by TLS 2. In this case, the amplitudes are approximately described by

[TABLE]

in each time interval $\left[n\tau_{1},\left(n+1\right)\tau_{1}\right]$.

For TLSs initially in the antisymmetry state, the concurrence exhibits the similar behavior to the symmetry state with a $\pi$ phase difference. However, when the inter-TLS spacing $d=0$, the antisymmetry state is a dark state which does not interact with the electromagnetic field, it means the the probability of finding the TLSs in the initial state is unity at any time, so the concurrence is unchanged and remains its initial value one (see the solid black line in Fig. 3). We also plot the concurrence as a function of the dimensionless time $t/\tau_{1}$ for phase $\varphi_{1}=2n\pi$ with $n=10$ (the green dot-dashed line) and $n=200$ (red dashed line) in Fig. 3, where all the position-dependent phase factors $e^{i\varphi_{1}}$ are equal. It can be seen that the cooperative effect become lower and lower as the inter-TLS distance increases, so does the maximum of concurrence.

III.2 two transverse modes

A TLS in its upper state radiates waves into the continua resonant with itself. As the transition frequency of the TLSs increases, there are additional guided modes taking part in the interaction with the TLSs. Due to their different wave numbers, the inter-TLS distance introduces different flight time $\tau_{j}$ of light between the TLSs as well as different phases $\varphi_{j}$. From the definition of $\tau_{j}$ and $\varphi_{j}$, we know that $\tau_{j}<\tau_{j+1}$ and $\varphi_{j}<\varphi_{j+1}$ for the given distance $d$. In this section, we assume that the transition frequency of the TLSs is smaller than the cutoff frequency $\Omega_{51}$ and larger than the cutoff frequency $\Omega_{31}$, this means that the emit photons will propagate in guided modes TM11 and TM51. The delay-differential equation Eq.( 12) reduces to

[TABLE]

To analyze the influence of the number of the transverse modes on the entanglement which interact with TLSs, we fix the phase $\varphi_{1}=2n\pi$ with $n=4$ in Fig. 4(a), $n=10$ in Fig. 4(b), $n=30$ in Fig. 4(c), $n=3000$ in Fig. 4(d). We plotted the time behavior of the concurrence between TLS by only considering the TM11 mode in Eq.(18), which is shown in blue dashed lines in Fig. 4. The red solid lines in Fig. 4 present the time behavior of the concurrence by considering both TM11 and TM31 modes in Eq. (18). It can be found that increasing the number of the transverse modes which interact with TLSs leads to exponentially decay with a rate $\gamma_{1}+\gamma_{2}$ up to time $t=\tau_{1}$. After this, the phase $\varphi_{1}$ begins to have an effect until time $t=\tau_{2}$. After time $\tau_{2}$, the dynamics can be dramatically affected by the phases $\varphi_{j}$ and delay times $\tau_{j}$ induced by the inter-TLS distance. The blue lines of panels (a) and (b) in Fig. 4 show that the concurrence remains constant in time after the time $t=\tau_{1}$ in TM11 mode, which indicate that the delay time $\tau_{1}$ is negligibly small. As the phase factor $e^{i\varphi_{1}}$ is fixed, the behavior of the red solid lines is completely determined by the phase $\varphi_{2}$ and delay time $\tau_{2}$ in TM31 mode. We see from panel (a) that the entanglement decays almost exponentially in time after delay time $\tau_{2}$, which means that a part of the emitted energy from one TLS is transferred directly to another TLS, so there is no delay in the absorption of the energy by another TLS in both TM11 and TM31 modes. In this case, we can solve Eq.(18) by letting $\tau_{1},\tau_{2}\rightarrow 0$

[TABLE]

[TABLE]

[TABLE]

IV conclusion

We have studied the effects of the inter-TLS distance on the entanglement properties of two identical TLSs located inside a rectangular hollow metallic waveguide of transverse dimensions $a$ and $b$. When the energy separation of the TLS is far away from the cutoff frequencies of the transverse modes and there is single excitation in the system, the Schrodinger equation for the wave function with single excitation initial in the TLSs is reduced to the delay differential equations for the amplitudes of two TLSs, where phase factors and delay times are induced by the finite distance between the TLSs. The delay differential equations are solved exactly for the TLSs interacting with either single transverse mode or double transverse modes of the waveguide, which directly reveals the retarded character of multiple reemissions and reabsorptions of photons between the TLSs. For the inter-TLS distance $d=0$, there exists an anti-symmetry state decoupled with the field modes, so the entanglement can be generated if the TLSs are initial in a separate state, later trapped in the anti-symmetry state. As the TLSs are close together such that the time delay $\max\{\tau_{j}\}$ is much smaller than the TLS decay time $\gamma^{-1}$, the excitation emitted by one TLS into the field is absorbed immediately by the other. The dynamic of the entanglement are dramatically affected by phases, leading to an enhanced and inhibited exponential decay of the concurrence when only one transverse mode are considered, and an exponential decay when more transverse modes are involved. As the inter-TLS distance increases, both phases and delay times affect the concurrence. There is a proper delay of reabsorption after reemission of photons but interference is possible if the amplitudes of TLSs do not decay appreciably over time $\tau_{nk}$. As $d$ is large enough so that $\min_{p,q}{\gamma|p\tau_{2}-q\tau_{1}|}\gg 1$ with non-negative integers $p$ and $q$ which are not zero at the same time,the phase factor plays no role. There appear collapses and revivals of the entanglement of the TLSs. We note that our studies focus on the the dependence of the concurrence on the inter-TLS distance but it is easy to study the dependence of the concurrence on the initial state of the system with the exact solution.

Acknowledgements.

This work was supported by NSFC Grants No. 11434011, No. 11575058.