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Abstract
A collection of cold rubidium atoms in three-level configuration trapped in one dimensional (1D) optical lattice is revisited. The trapped atoms are considered in the Gaussian density distribution and study the realization of $\mathcal {PT}$-, non-$\mathcal {PT}$- and $\mathcal {PT}$ anti-symmetry in optical susceptibility in 1D atomic lattices in a periodic structure. Such a fascinating modulation is achieved by spatially modulating the intensity of the driving field. Interestingly, a nonreciprocal optical propagation phenomenon is investigated. In this system, we have introduced a microwave that couples to the two ground states, spatial modulation of the coupling field, and the atomic density with Gaussian distribution in practice. With a proper detuning and coupling field Rabi frequencies, we can find the condition of $\mathcal {PT}$-symmetry along with field propagation direction, and the novel properties of transmission and reflections have been discussed. The large difference of field reflections from the two ends of the atomic lattice medium shows strong evidence that the nonreciprocal behavior can be greatly enhanced by increasing the spatial modulation amplitude.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, meta-material has been paid great attention not only in fundamental researches but also in many applications. The non-Hermitian system having parity-time ($\mathcal {PT}$)-symmetric Hamiltonian [1,2] which is considered as a new-type meta-material extending the scope beyond Hermitian Hamiltonian has been extensively studied in various physical systems. One of the most surprising characteristics of $\mathcal {PT}$-symmetric system is the existence of the real eigen spectrum even in non-Hermitian Hamiltonian. On the other hand, the system undergoes a sudden phase transition for the system parameter above a critical threshold because of parity-time symmetry breaking. Besides the theoretical investigations, the characteristics of $\mathcal {PT}$-symmetry has been studied experimentally [3,4], and various topics have been discussed such as power oscillation [5–7], unidirectional invisibility [8,9], coherent perfect absorbers [10,11] and non-reciprocal light propagation [12–15]. With the controllable optical refractive index, optical systems have great advantages for studying the properties of $\mathcal {PT}$-symmetry, for example, in photonic structures [16,17], coupled waveguides [18,19], whispering-gallery micro-cavities [20,21], optomechanical systems [22,23] and transmission lines [24]. However, some schemes using interactions between light fields and atomic clouds or atomic vapors are proposed for the investigation of $\mathcal {PT}$-symmetry [25–31]. In the schemes mentioned above, multi-level atomic configuration has been used to realize some novel phenomena related to $\mathcal {PT}$ [29–31]. Unidirectional reflectionless atomic lattice with loss and no gain has been studied in $N$-type atomic system [29]. Similarly, in which $\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry in 1D atomic lattices was proposed by using modulations in driving field amplitude and frequency [30]. Recently, the realization of simultaneously $\mathcal {PT}$-symmetric and $\mathcal {PT}$-antisymmetric susceptibilities in 1D atomic system was studied theoretically. It shows that the well-controlled atomic coherence which arises from coherent interactions between fields and atomic transition levels plays an important role in realizing $\mathcal {PT}$-symmetric condition.
In this paper, we study the $\mathcal {PT}$-, non-$\mathcal {PT}$- and anti-$\mathcal {PT}$ symmetry in 1D atomic lattices. More interestingly, we explore the unidirectional scattering properties in atomic lattices under the condition of $\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry. The motivation comes from an earlier work [14], where the authors found that in a periodic structure such as Bragg grating the unidirectional reflection behavior directly attributes to the spontaneous $\mathcal {PT}$-symmetry breaking. In our investigation, we have proposed a scheme to coherently control the real and imaginary part of optical susceptibility by introducing a microwave, which is used to couple the two ground states and change the ground state atomic coherence [32]. By adjusting the modulation amplitude of the coupling field, we find that the reflections from the two ends of the atomic lattice medium gradually become different. It implies that we can enhance the nonreciprocal field reflection with the increment of amplitude modulation in the atomic lattice structure, which means one can properly manipulate the field reflections from reciprocal to nonreciprocal behavior.
The organization of this paper is given as follows. In Sec. II, we start from the fundamental field-atom interaction in the 1D atomic system, driving the optical susceptibility by considering the atomic distribution in the lattice system. In Sec. III, we will show that the $\mathcal {PT}$-, non-$\mathcal {PT}$- and $\mathcal {PT}$ anti-symmetry condition can be realized by adjusting the accessible physical quantities. Then, we discuss the scattering property in atomic lattice under $\mathcal {PT}$ symmetric and $\mathcal {PT}$-antisymmetry conditions in Sec. IV. Some applications are mentioned in Sec. V. Finally, we give a conclusion in Sec. VI.
2. Model
We consider 1D optical lattices having Gaussian distribution of clusters of cold three-level rubidium ($^{87}$Rb) atoms. Each cluster settled at the bottom of a dipole trap in the $x$-direction. The optical dipole trapping potential depends on the sign of field detuning. For the blue detuned case, the dipole interaction repels atoms out of the field so that the potential minimum would be the minimum intensity of the field. In this case, we can use a blue detuned trap to localize atoms in the lowest intensity region. Two optical fields are interacting with three-level atoms and coupled to the atomic transitions $|e\rangle \longleftrightarrow |b\rangle$ and $|e\rangle \longleftrightarrow |c\rangle$, see the energy level configuration in Fig. 1(a). A microwave field is also applied which coupled to the ground states as $|b\rangle \longleftrightarrow |c\rangle$. The interaction Hamiltonian under a rotating wave and dipole approximation is
Fig. 1. (a) Energy-level configuration for three-level atomic configuration. (b) In the $x$-direction, three-level atomic medium is tapped in 1D optical lattices in a Gaussian distribution.
The space dependent optical susceptibility can be calculated as
where $\wp _{eb}$ is the electric dipole transition from $\vert b\rangle$ to $\vert e\rangle$. $N_j(x)$ is considered in 1D lattices with constant density $N_0$ by a Gaussian distribution in the jth lattice as In Eq. (5), $x_j$ is the jth center of 1D atomic lattices of period $a$, whereas $\sigma$ stands for deviation of the Gaussian distribution from its peak. Further, we also consider that the coupling field $\Omega _2$ is periodically modulated in the form of a standing wave in the $x$-direction i.e., the coupling field is modulated in amplitude as where $\delta \Omega _2$ is known as the modulation amplitude.3. Results and discussion
In this section, we start our discussion by studying the real and imaginary parts of the optical susceptibility without considering any spatial modulation. We plot the real and imaginary parts of the optical susceptibility versus probe field detuning $\Delta$ for two values of microwave field i.e., $\Omega _{\mu }=0$ and $0.05\gamma$ , see Fig. 2. The other parameters are $\gamma =1$MHz, $\gamma _{eb}=1 \gamma$, $\gamma _{cb}=0.01\gamma$, $\Omega _p=0.01\gamma$, and $\Omega _2=5\gamma$. In the absence of microwave field, we plot the real and imaginary parts of the optical susceptibility versus probe field detuning. We get a typical electromagnetically induced transparency (EIT) spectrum as shown in Fig. 2 in black curves. In the typical EIT window, we have a normal dispersion and a transparency in between two absorption peaks. Further, the microwave field in our system plays a key role and can change the optical properties of the system. In the presence of microwave field, a steep positive dispersion appears in the real part of the optical susceptibility, whereas a gain appears in the imaginary part of the optical susceptibility, see the red curves in Fig. 2. It is also emphasized that by applying the microwave field, we get both loss and gain simultaneously at different frequencies, see the imaginary part (red curve) of the optical susceptibility in Fig. 2(b). Following the control of the optical susceptibility via microwave field, we expect to study the $\mathcal {PT}$-symmetry via microwave field using the modulation of Eq. (6).
Fig. 2. (a) Real and (b) imaginary parts of the optical susceptibility vs probe field detuning $\Delta$. The parameters are $\gamma =1$MHz, $\gamma _{eb}=1\gamma$, $\gamma _{cb}=0.01\gamma$, $\Omega _p=0.01\gamma$, $\Omega _2=5\gamma$ and $\phi =\pi /2$.
To realize the $\mathcal {PT}$-symmetry, first we study the imaginary part of the optical susceptibility in one period of optical lattices, where the atoms are trapped in 1D and exhibiting the distribution as shown in Eq. (5) and the control field is periodically distributed as depicted in Eq. (6). Here, the probe field detuning plays a key role to attain gain and loss in the system simultaneously. Therefore, we show a spectrum of the imaginary part of optical susceptibility versus probe field detuning $\Delta$ and $x$-direction of the optical lattices, see Fig. 3. The plot clearly shows that there is gain and loss at positions $x/a=0.15$ and $x/a=-0.15$, respectively, versus probe field detuning. In our system, the gain and loss in the imaginary part of the optical susceptibility in 1D optical lattices are exactly equal at probe field detuning $\Delta =2.404\gamma$. Now, for the realization of $\mathcal {PT}$-symmetry, the balance of gain and loss in a system simultaneously plays an important role. To see the $\mathcal {PT}$-symmetry behavior of the optical susceptibility in 1D optical lattices, next we consider the real and imaginary parts of the optical susceptibility for only one cycle of the optical lattices. By considering $\Delta =2.404\gamma$, we plot the real and imaginary parts of the optical susceptibility in 1D lattices versus $x/a$, see Fig. 4. The plots in Fig. 4 shows that the real part of the optical susceptibility is an even function of space while the imaginary part is odd. Whereas the gain and loss of the imaginary part of the optical susceptibility in 1D lattices balance each other. This is a clear evidence for the investigation of $\mathcal {PT}$-symmetry in the form of optical susceptibility in 1D optical lattices. The imaginary part i.e., Im[$\chi$]$>\;0$ in one half period ($-0.25\;<\;x/a\;<\;0$) and Im[$\chi$]$<\;0$ for the another period ($0\;<\;x/a\;<\;0.25$) which clearly indicates a balanced gain and loss. Further, we study the $\mathcal {PT}$-symmetry in 1D optical lattices for different periods and plot the real and imaginary parts of the optical susceptibility versus $x/a$ for the probe field detuning $\Delta =2.404\gamma$, see Fig. 5. The real part of the optical susceptibility have different cycles and all are even function of space, similarly the imaginary part of the optical susceptibility have different cycles and all are odd function of space. Also all cycles in the imaginary part have equal gain and loss, then the condition for $\mathcal {PT}$-symmetry is satisfied. Hence, we have $\mathcal {PT}$-symmetric regime in 1D optical lattices in the presence of microwave field.
Fig. 3. Density plot of imaginary part of the optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a$. The parameters are $\gamma =1$MHz, $\gamma _{eb}=1\gamma$, $\gamma _{cb}=0.01\gamma$, $\Omega _p=0.01\gamma$, $\Omega _2=5\gamma$, $\phi =\pi /2$, $\Omega _{11}=5\gamma$, $\delta \Omega _2=0.3\gamma$, $\sigma =0.35 a$, $a= 1 \lambda$ and $\Omega _\mu =0.05\gamma$.
Fig. 4. (a) real and (b) imaginary parts of the optical susceptibility for a single cycle vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$, the other parameters remains the same as that in Fig. 3.
Fig. 5. (a) real and (b) imaginary parts of the optical susceptibility for many cycles vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$, the other parameters remains the same as that in Fig. 3.
As we discussed above that the microwave field switches the medium from absorptive to gain one i.e., in the absence of microwave field, we get a transparency window (EIT) having two absorption peaks. Whenever the microwave is switched on, we get gain in the imaginary part of the optical susceptibility. In the above study, we investigated the $\mathcal {PT}$-symmetry of the optical susceptibility in 1D optical lattices in the presence of the microwave field. Next, we study what will happen to the optical susceptibility in 1D optical lattices when there is no external microwave field? In the absence of the microwave field, we plot the real and imaginary parts of the optical susceptibility in 1D optical lattices as shown in Fig. 6. The real part of the optical susceptibility is not an even function of space whereas the imaginary part is not an odd function of space, also there is no gain in the system. This is clear evidence for the non-$\mathcal {PT}$-symmetry optical susceptibility in 1D optical lattices. We conclude that we have a controllable parameter i.e., microwave field that can switch a system from $\mathcal {PT}$-symmetry to non-$\mathcal {PT}$-symmetry and vice versa. The control of $\mathcal {PT}$-symmetry to non-$\mathcal {PT}$-symmetry in our system is not only dependent on the microwave field but also dependent on the relative phase of the optical fields and probe field frequency. It means that by manipulating the relative phase of the driving fields, we can switch the medium from $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry.
Fig. 6. (a) real and (b) imaginary parts of the optical susceptibility for many cycles vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$ and $\Omega _{\mu }=0$, the other parameters remains the same as that in Fig. 3.
In order to study the $\mathcal {PT}$ anti-symmetric optical susceptibility, next, we change the relative phase of the optical fields from $\pi /2$ to $\pi$ in the presence of microwave field. The $\mathcal {PT}$ anti-symmetry realization must be achieved at a different detuning i.e., $\Delta =2.502\gamma$. Therefore, we consider $\Delta =2.502\gamma$ and plot the real and imaginary parts of the optical susceptibility in 1D lattices versus $x/a$, see Fig. 7. The plot shows that the real part of the susceptibility is an odd function of space where the imaginary part is an even. This is an evidence of the $\mathcal {PT}$ anti-symmetric optical susceptibility of the optical lattices.
Fig. 7. (a) real and (b) imaginary parts of the optical susceptibility for one cycle vs lattice position $x/a$ with probe field detuning $\Delta =2.502\gamma$ and $\phi =\pi$, the other parameters remains the same as that in Fig. 3.
4. Nonreciprocal reflection in 1D system
In Sec.III, we have proposed a scheme to generate $\mathcal {PT}$ symmetric condition by using atomic lattice. Such a periodic structure with $\mathcal {PT}$ symmetric property provides us a good platform to study scattering problems. In this section, we are going to study the behaviors of reflection and transmission of the incident field, which have nonreciprocal properties under $\mathcal {PT}$ symmetric environment along the propagation direction. In Fig. 8, we have shown how the field scattered in one-dimension under the condition of $\mathcal{PT}$-symmetry. The forward fields at the two ends of the lattices are $E_f(-L/2)$ (green) and $E_f(+L/2)$ (red), and backward fields are $E_b(-L/2)$ (yellow) and $E_b(+L/2)$ (blue). In order to study the field propagation dynamics, we start from the Helmholtz equation in the 1D system.
where $n(x) = \sqrt {1+\chi (x)} \simeq 1 + \chi (x)/2$, is the refractive index of the medium along the field propagation direction and $\chi (x) = \chi _R(x) + i\chi _I(x)$. Since the optical susceptibility $\chi (x)$ is spatially modulated as shown in Fig. 5, under $\mathcal {PT}$ symmetric condition the real and imaginary parts of optical susceptibility can be approximated by $\chi _R(x) = \bar {\chi } + \chi _1\cos (2\beta x)$ and $\chi _I(x)= \chi _2 \sin (2\beta x)$, respectively. Here, $\bar {\chi }$ is the constant term of $\chi _R$, $\beta = \pi /a$ is the spatial frequency of atomic lattice whereas $\chi _1$ and $\chi _2$ are the amplitudes of modulation in real and imaginary part of $\chi$. The refractive index $n(x)$, thus take the form as $n(x) = n_0 + n_1 \cos (2\beta x) + in_2\sin (2\beta x)$, where $n_0 = \sqrt {1+\bar {\chi }}$, $n_1 = \chi _1/2n_0$, and $n_2 = \chi _2/2n_0$. In general, the optical susceptibility $\chi (x)$ in our case is not a perfect sine and cosine function so that we can express the modulated refraction index with Fourier series, in which the real and imaginary parts of $\chi$ are expanded with even and odd functions, respectively. Thus, the refractive index take the form as $n(x) = n_0 + \eta _1(x) + i\eta _2(x)$, where where $a_0$, $a_n$ and $b_n$ are constants which can be determined with the physical parameters given by $\Omega _p$, $\Omega _2$, $\Omega _\mu$, and $\Delta$. Under $\mathcal {PT}$-symmetric condition, $\eta _1$ and $\eta _2$ are plotted in Fig. 9 in which the real and imiginary parts are even and odd function of space.Next, we are dealing with the field part. As shown in Fig. 8, we have forward and backward fields, $E_f$ and $E_b$, at the two ends of atomic lattice, respectively. Thus we can assume that the field solution is given by
Fig. 8. Field scattering in one-dimension under the condition of $\mathcal {PT}$-symmetry. The forward fields at the two ends of lattice are $E_f(-L/2)$ (green) and $E_f(+L/2)$ (red), and backward fields are $E_b(-L/2)$ (yellow) and $E_b(+L/2)$ (blue).
We use the slowly varying envelope approximation and express the forward and backward amplitudes as $E_f(x) = \mathcal {E}_f(x) e^{+i\delta x}$ and $E_b(x) = \mathcal {E}_b(x) e^{-i\delta x}$, in which $\delta = \beta -k$ is the spatial detuning between field wave number and spatial frequency of lattice. Here, we let $\delta \ll 1$ such that we can only focus on the main contribution from spatial frequency component, i.e. $k\simeq \beta$. Substituting Eq. (10) into Eq. (7) and ignoring the second derivative terms of slowly envelope fields, we can obtain coupled equations between forward and backward given as
in which $B$ and $D$ are coupling coefficients connected between $\mathcal {E}_f(x)$ and $\mathcal {E}_b(x)$. Their values are determined from Eqs. (8) and (9) as where $I_1 \equiv a_0a_1 + a_1a_2 + a_2a_3 -b_1b_2 - b_2b_3$, and $I_2 \equiv a_0b_1 - a_2b_1 + a_1b_2 -a_3b_2 + a_2b_3$. The quantities of $B$ and $D$ shown in Eqs. (13) and (14) can be obtained from Fourier series analysis under the case of $k$ being closed to $\delta$. Numerically, solving Eqs. (11) and (12) with the coefficients given by Eqs. (13) and (14), we can obtain the matrix elements of transfer matrix defined byFig. 9. Spatial refractive index. The blue and red lines represent $\eta _1(x)$ and $\eta _2(x)$, respectively. The parameters are given as $\Omega _{11} = 5\gamma$, $\delta \Omega _2 = 0.3\gamma$, $\Omega _{\mu } = 0.05\gamma$, $\Omega _p = 0.01\gamma$, $\sigma = 0.35a$, and $\phi = \pi /2$.
Fig. 10. Transmission and reflections versus spatial detuning under $\delta \Omega _2 = 0.3\gamma$ (figure a) and $\delta \Omega _2 = 1.0\gamma$ (figure b). The red curve represents the field transmission $T$, and the reflections $R_R$ and $R_L$ are plotted by blue and green curves. The other parameters we’ve used are given by $\Omega _{11} = 5\gamma$, $\Omega _{\mu } = 0.05\gamma$, $\Omega _p = 0.01\gamma$, $\sigma = 0.35a$, $k = 500$ and $\phi = \pi /2$.
Fig. 11. Transmission and reflections v.s. modulation amplitude $\delta \Omega _2$. The transmission is plotted by the red curve, and the two reflections, $R_R$ and $R_L$, are represented by blue and green curves, respectively. Other parameters are the same as that of in Fig. 10.
The field transmission and reflections can be calculated from the transfer matrix elements, which can be analytically obtained by solving Eqs. (11) and (12). Here, we show the transmission and reflections as
Fig. 12. The values of $B^2$ and $D^2$ v.s. $\delta \Omega _2$. Blue curve represents square of $B$, and blue curve represents square of $D$. We’ve set the same parameters used in Fig. 11.
Fig. 13. The modulation refraction index with two different $\delta \Omega _2$’s (a and c), and the corresponding reflections and transmissions (b and d) under $\mathcal {PT}$-antisymmetry condition. Blue lines and red lines in (a) and (c) represent $\eta _1$ and $\eta _2$, respectively. In (a) and (b), $\delta \Omega _2 = 0.3\gamma$, and $\delta \Omega _2 = 0.7\gamma$ for (c) and (d). Other parameters are given by $\Omega _{11} = 5\gamma$, $\Omega _{\mu } = 0.05\gamma$, $\Omega _p = 0.01\gamma$, $\sigma = 0.35a$, and $\phi = \pi$.
In $\mathcal {PT}$-symmetric systems, it is worth to discuss the phenomenon of unbroken-$\mathcal {PT}$ and broken-$\mathcal {PT}$-symmetry, which is the phase transition from the system with entire real eigenvalues to complex eigenvalues when the system parameter exceeds a certain value called exceptional point. In our system, we can study the phase transition from unbroken-$\mathcal {PT}$ and broken-$\mathcal {PT}$ by changing the modulation amplitude $\delta \Omega _2$. The system is at exceptional point when $\delta \Omega _2 \simeq 1.3\gamma$, and the spatial modulation amplitudes in real part and imaginary part of refractive index are almost the same, i.e. $\vert a_1\vert \simeq \vert b_1\vert$. When $\delta \Omega _2\;>\;1.3\gamma$, the system enters into broken-$\mathcal {PT}$ region, and the corresponding eigenvalues become complex. As we can see from Fig. 11 and Fig. 12, $R_R$ approaches to zero, and $R_L$ becomes very large when $\delta \Omega _2$ reach the exceptional point. Meanwhile, the coefficients $B \gg D$, making the strong non-reciprocal reflection between $R_R$ and $R_L$. In our system, we restrict our discussion for the reflections and transmission in unbroken-$\mathcal {PT}$ region because $\eta _1$ and $\eta _2$ would suffer severe distortion when $\delta \Omega _2 \gg 1.3\gamma$ so that and the condition of $\mathcal {PT}$-symmetry can not be exact.
Besides the realization of $\mathcal {PT}$-symmetry, $\mathcal {PT}$-antisymmetry can also be realized in our system by using the correct detuning $\Delta = 2.502\gamma$, and the relative phase $\phi = \pi$, as shown in Fig. 7. Under $\mathcal {PT}$-antisymmetric condition, the real part of refraction index modulation becomes odd function, i.e. $\eta _1(x) = -\eta _1(-x)$, while the imaginary part becomes even function, i.e. $\eta _2(x) = \eta _2(-x)$. In Fig. 13, we have shown $\eta _1$ and $\eta _2$ with two different $\delta \Omega _2$’s in (a) and (c). With the increment of the magnitude of $\delta \Omega _2$, the value of $\eta _1$ is increasing. In contrast to the case of $\mathcal {PT}$ symmetry, $R_R$ is strongly enhanced, while $R_L$ is greatly suppressed.
4.1 Possible applications
The atomic lattices system with nonreciprocal reflections which can be controlled by amplitude modulation provides a great platform to design optical devices, having great potential in the application of optical computing. The nonreciprocal reflection using the atomic lattice with spatial modulation of the driving field provides a promising element to study the light dynamics in the metamaterial system. With the strong suppression of reflection from one side, one can have high transmission without any scattering, which can be applied to the one-way optical device. On the other hand, the enhanced reflection from the other end of the atomic lattice can be used in the amplification of weak non-classical light sources such as a squeezed state of light. Besides, one can verify the relative phase among fields by using the significant different properties of light scattering in $\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry conditions. It is worth to note that both $\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry can be realized in the same system, which makes the system promising in the application of design of special optical waveguide devices. The parameters in our system are tunable and accessible in experiment so that it is robust to some applications in modern optical devices.
5. Conclusion
In conclusion, we have studied the $\mathcal {PT}$, non-$\mathcal {PT}$ and $\mathcal {PT}$ anti-symmetric properties of the optical susceptibility by considering 1D atomic lattices. The atomic lattices consist of cold three-level configuration in a Gaussian density distribution. In the presence of the microwave $\Omega _\mu$, we can generate a $\mathcal {PT}$ and anti $\mathcal {PT}$-symmetric atomic lattice by introducing the modulation on $\Omega _2$. In the scenario, we study the scattering problem in the 1D atomic lattice under $\mathcal {PT}$-symmetric condition. Our results show that the nonreciprocal behavior between two reflections can be realized in $\mathcal {PT}$-symmetric environment, and can be greatly enhanced by increasing the modulation amplitude $\delta \Omega _2$.
Disclosures
The authors declare no conflicts of interest.
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