PT Symmetry via Electromagnetically Induced Transparency

We propose a scheme to realize parity-time (PT) symmetry via electromagnetically induced transparency (EIT). The system we consider is an ensemble of cold four-level atoms with an EIT core. We show that the cross-phase modulation contributed by an assisted field, the optical lattice potential provided by a far-detuned laser field, and the optical gain resulted from an incoherent pumping can be used to construct a PT-symmetric complex optical potential for probe field propagation in a controllable way. Comparing with previous study, the present scheme uses only a single atomic species and hence is easy for the physical realization of PT-symmetric Hamiltonian via atomic coherence.


Introduction
In recent years, a lot of efforts have been made on a class of non-Hermitian Hamiltonian with parity-time (PT) symmetry, which in a definite range of system parameters may have an entirely real spectrum [1,2]. PT symmetry requires that the real (imaginary) part of the complex potential in the Hamiltonian is an even (odd) function of space, i.e. V (r) = V * (−r). Even though the Hermiticity of quantum observables has been widely accepted, there is still great interest in PT symmetry because of the motivation for constructing a framework to extend or replace the Hermiticity of the Hamiltonian in ordinary quantum mechanics. The concept of PT symmetry has also stimulated many other studies, such as quantum field theory [3], non-Hermitian Anderson models [4], and open quantum systems [5], and so on.
Although a large amount of theoretical works exist, the experimental realization of PTsymmetric Hamiltonian in the fields mentioned above was never achieved. Recently, much attention has been paid to various optical systems where PT-symmetric Hamiltonians can be realized experimentally by balancing optical gain and loss [6][7][8][9]. In optics, PT symmetry is equivalent to demand a complex refractive index with the property n(r) = n * (−r). Such refractive index has been realized experimentally using two-wave mixing in an Fe-doped LiNbO 3 substrate [10]. The optical realization of PT symmetry has motivated various designs of PTsynthetic optical materials exhibiting many intriguing features, including non-reciprocal or unidirectional reflectionless wave propagation [10][11][12][13] In a recent work Hang et al.
[20] proposed a double Raman resonance scheme to realize PT symmetry by using a two-species atomic gas with Λ-type level configuration. This scheme is quite different from those based on solid systems mentioned above [6,[10][11][12][13][14][15][16][17][18][19], and possesses many attractive features. For instance, the PT-symmetric refractive index obtained in [20] is valid in the whole space; furthermore, the refractive index can be actively controlled and precisely manipulated by changing the system parameters in situ.
In the present article, we suggest a new scheme to realize the PT symmetry in a lifetime- Possible experimental arrangement. All the notation are defined in the text.
broadened atomic gas based on the mechanism of electromagnetically induced transparency (EIT), a typical and important quantum interference phenomenon widely occurring in coherent atomic systems [21]. Different from the two-species, double Raman resonance scheme proposed in [20], the scheme we suggest here is a single-species, EIT one. And due to the complexity of the susceptibility [20], it is difficult to design some PT potentials we wish, however, in our scheme, we can design many different periodic potentials and non-periodic potentials in light of our will, and the size of potential can also be adjusted conveniently. Especially, compared with the traditional idea that PT symmetric potential must be combined by the gain and loss parts, we utilize the atomic decay rate to design the imaginary part of PT potential, and use the giant cross-phase modulation (CPM) effect [21,22] of the resonant EIT system to realize the real part. We shall show that the cross-phase modulation contributed by the assisted field, the optical lattice potential provided by a far-detuned laser field, and the optical gain resulted from an incoherent pump can be used to construct a complex optical potential with PT symmetry for probe field propagation in a controllable way. The present scheme uses a single atomic species only and hence is simple for physical realization. The rest of the article is arranged as follows. In the next section, a description of our scheme and basic equations for the motion of atoms and light field are presented. In Sec. III, the envelope equation of the probe field and its realization of PT symmetry are derived and discussed. The final section is the summary of our main results.

Model
The system under consideration is a cold, lifetime-broadened 87 Rb atomic gas with N-type level configuration; see Figure 1. The levels of the system are taken from the D 1 line of 87 Rb atoms, c. interact resonantly with levels |1 → |3 and |2 → |3 , respectively. Here e j and k j (E j ) are respectively the polarization unit vector in the jth direction and the wave number (envelope) of the jth field. The levels |l (l = 1, 2, 3) together with E p and E c constitute a well-known Λ-type EIT core.
Furthermore, we assume an assisted filed is coupled to the levels |2 → |4 , where E a (x) is field-distribution function in transverse direc-tion. The assisted filed E a , when assumed to be weak (satisfying E p ≤ E a ≪ E c ), will contribute a CPM effect to the probe field E p . Note that the levels |l (l = 1, 2, 3, 4) together with E p , E c , and E a form a N-type system, which was considered firstly by Schmidt Imamoǧlu [22] for obtaining giant CPM via EIT. In addition, we assume there is another far-detuned (Stark) optical lattice field is applied to the system, where E s (x) and ω L are respectively the field-distribution function and angular frequency. Due to the existence of E Stark , a small and x-dependent Stark shift of level E j to the state | j occurs, i.e., here α j is the scalar polarizability of the level | j , and · · · t denotes the time average in an oscillating cycle. The explicit forms of E a (x) and E s (x) in (1) and (2) will be chosen later on according to the requirement of PT symmetry (see Sec. 3.2).
As will be shown below, the CPM effect contributed by the assisted field E a given by (1) and the Stark shift contributed by the far-detuned Stark field E Stark given by (2) will provide periodic complex refractive index to the evolution of probe-filed envelope. However, they are still not enough to obtain a refractive index with PT symmetry since a gain to the probe field is needed. Therefore, we introduce an incoherent optical pumping which can pump atoms from the ground-state level |1 to the excited-state level |3 with the pumping rate Γ 31 [see equations (18a) and (18c) in Appendix]. Such optical pumping can be realized by many techniques, such as intense atomic resonance lines emitted from hollow-cathode lamps or from microwave discharge lamps [23].
In Fig. 1(a), Γ 13 , Γ 23 , and Γ 24 are spontaneous emission rates denoting the population decays respectively from |3 to |1 , |3 to |2 , and |4 to |2 ; and Ω a = (e y · p 24 )E a /h are respectively the half Rabi frequencies of the probe, control, and assisted fields, here p i j signifies the electric dipole matrix element of the transition from state |i to | j , ∆ 3 , ∆ 2 , and ∆ 4 are respectively one-, two-, and three-photon detunings in relevant transitions. Fig. 1(b) shows a possible experimental arrangement.

Maxwell-Bloch equations
Under electric-dipole and rotating-wave approximations, the Hamiltonian of the system in inter- , where h.c. denotes Hermitian conjugate, and The motion of atoms interacting with the light fields is described by the Bloch equation where σ jl is the density-matrix elements in the interaction picture, Γ is a 4 × 4 relaxation matrix.
where κ 13 = Nω p |e x · p 13 | 2 /(2ε 0h c) with N being the atomic concentration. Note that, for simplicity, we have assumed Ω p is independent on y, which is valid only for the probe beam having a large width in the y-direction so that the diffraction term ∂ 2 Ω p /∂ y 2 can be neglected; in addition, we have also assumed that the dynamics of Ω a is negligible during probe-field evolution, which is a reasonable approximation because the assisted field couples to the levels |2 and |4 that have always vanishing population due to the EIT effect induced by the strong control field.

Equation of the probe-field envelope
The Maxwell equation (5) governs the propagation of the probe field. To solve it one must know σ 31 , which is controlled by the Bloch equation (4) and hence coupled to Ω p . For simplicity, we assume Ω p has a large time duration τ 0 so that Γ 31 τ 0 >> 1. In this case a continuouswave approximation can be taken. As a result, the time derivatives in the Maxwell-Bloch (MB) equations (4) and (5) (i.e. the dispersion effect of the probe field) can be neglected, and only the diffraction effect of the probe field in x direction is considered. In addition, because the probe field is weak, a perturbation expansion can be used for solving coupled equations (4) and (5) analytically [24,25]. We take the expansion σ i j = σ p + · · · . Here ε is a small parameter characterizing the typical amplitude of the probe field (i.e Ω p,max /Ω c ). Substituting such expansion to equations (4) and (5), we obtain a series of linear but inhomogeneous equations for σ At O(1)-order, we obtain non-zero density-matrix elements σ (0) At O(ε)-order, the solution is given by with other σ
From the solid line of Figure 2, we see that in the absence of the control field and incoherent pumping (i.e., Ω c = Γ 31 = 0), the probe field has a very large absorption; however, when the incoherent pumping still absent but Ω c takes the value of 5 × 10 7 Hz, a transparency window is opened (as shown by the dashed line). This is well-known EIT quantum interference phenomenon induced by the control field [21]. However, there is still a small absorption (i.e., ImK > 0, which can not be seen clearly due to the resolution of the figure). That is to say, although EIT can suppress largely the absorption, it can not make the absorption become zero.
The dashed-dotted line in Fig. 2 is the situation when the incoherent pumping (Γ 31 = 0.7γ 3 ) is introduced. One sees that a gain (i.e., negative ImK in the region near ∆ 3 = 0) occurs. Such gain is necessary to get a PT-symmetric optical potential for the probe-field propagation, as shown below.
At O(ε 3 )-order of the perturbation expansion, we obtain the closed equation for F, which can be converted to the equation for Ω p : after returning to original variables, with where Ω p = εF exp(iKz), the coefficients α 12 and α 13 are given in Appendix. We now make some remarks about the potentialṼ (x) given by Eq. (9): (1). The coefficients α 12 and α 13 are complex. We stress that the occurrence of a complex potential for the evolution of probe-field envelope is a general feature in the system with resonant interactions. The reason is that, due to the resonance, the finite lifetime of atomic energy states must be taken into account. As a result, the variation of the probe-field wavevector resulted by the external light laser fields (here the Stark and the assisted fields) are complex. It is just this point that provides us the possibility to realize a PT symmetric potential in our system by using the periodic external laser fields.
(2). If the incoherent pumping is absent, the probe field has only absorption but no gain and hence not possible to realize PT symmetry. With the incoherent pumping present, the parameter K [given by the Eq. (7)] in the Eq. (9) is complex and has negative imaginary part in the region near ∆ 3 = 0, which can be used to suppress an absorption constant (i.e. the term not dependent on x) appearing in the previous two terms ofṼ (x).
(3). It is easy to show that if only a single external laser field (the Stark or the assisted field) is applied, it is impossible to realize a PT symmetry. That is why the two separated light fields (i.e. both the Stark and the assisted fields) have been adopted. We shall show below that the joint action between the Stark field, the assisted field, and the incoherent pumping can give PT-symmetric potentials in the system.
The susceptibility of the probe field is given by χ(x) = 2cṼ (x)/ω p . Because the potential (9) is a complex function of x, which is equivalent to a space-dependent complex refractive index n(x) = 1 + χ(x) ≈ 1+cṼ (x)/ω p for the probe-field propagation. PT symmetry requires V * (−x) =Ṽ (x), which is equivalent to the condition n * (−x) = n(x).

The design of PT symmetric potential
Equation (8) is a linear Schrödinger equation with the "external" potential (9). To realize a PT-symmetric model we assume the field-distribution functions in (1) and (2) taking the forms with E a0 and E s0 being typical amplitudes and R −1 being typical "optical lattice" parameter. For convenience of later discussion, we write Eq. (8) into the following dimensionless form with V (ξ ) = (g 12 + g 12 sin 2ξ ) + g 13 cos 2 ξ + K 0 , Here, L diff ≡ 2ω p R 2 /c is the typical diffraction length and U 0 denotes the typical Rabi frequency of the probe field.
PT symmetry of Eq. (12) requires V * (−ξ ) = V (ξ ). In general, such requirement is difficult to be satisfied because resonant atomic systems have very significant absorption. However, in the system suggested here the absorption can be largely suppressed by the EIT effect induced by the control field. The remainder small absorption that can not be eliminated by the EIT effect may be further suppressed by the introduction of the incoherent optical pumping. If the optical pumping is large enough, the system can acquire a gain. This point can be understood from Fig. 2 for the case of (Ω c , Γ 31 ) = (5 × 10 7 Hz, 0.7γ 3 ) where near the EIT transparency window ImK is negative, which means that the probe field acquires a gain contributed by the optical pumping. Such gain can be used to suppress the imaginary parts of g 12 and g 13 through choosing suitable system parameters, and hence one can realize a PT symmetry of the system.
Equation (17) satisfies the PT-symmetry requirement V * (−ξ ) = V (ξ ) when exact to the accuracy O(10 −2 ). The constant term −11.7 in V (ξ ) can be removed by using a phase transformation u → u exp(−i11.7s). Equation (17) is a kind of PT-symmetric periodic potential. In fact, one can design many different periodic potentials or non-periodic potentials with PT symmetry in our system by using different assisted and far-detuned laser fields. Consequently, our system has obvious advantages for actively designing different PT-symmetric optical potentials and manipulating them in a controllable way.

Conclusion
We have proposed a scheme to realize PT symmetry via EIT. The system we considered is an ensemble of cold four-level atoms with an EIT core. We have shown that the cross-phase modulation contributed by an assisted field, the optical lattice potential provided by a far-detuned laser field, and the optical gain coming from an incoherent pumping can be used to construct a PT-symmetric complex optical potential for probe field propagation in a controllable way. Comparing with previous study in [20], our scheme has the following advantages: (i) Our scheme uses only one atomic species, which is much simpler than that in [20]. (ii) The mechanism of realizing the PT-symmetric potential is based on EIT, which is different from that in [20] where a double Raman resonance was used. (iii) One can design many different PT-symmetric potentials at will in our scheme in a simple way.