PT Symmetry via Electromagnetically Induced Transparency
aa r X i v : . [ qu a n t - ph ] D ec PT symmetry via electromagneticallyinduced transparency
Hui-jun Li, , , ∗ Jian-peng Dou, and Guoxiang Huang , , Institute of Nonlinear Physics and Department of Physics, Zhejiang Normal University,Jinhua, 321004 Zhejiang, China State Key Laboratory of Precision Spectroscopy, East China Normal University, 200062Shanghai, China Department of Physics, East China Normal University, 200062 Shanghai, China [email protected] ∗ [email protected] Abstract:
We propose a scheme to realize parity-time (PT) symmetry viaelectromagnetically induced transparency (EIT). The system we consider isan ensemble of cold four-level atoms with an EIT core. We show that thecross-phase modulation contributed by an assisted field, the optical latticepotential provided by a far-detuned laser field, and the optical gain resultedfrom an incoherent pumping can be used to construct a PT-symmetriccomplex optical potential for probe field propagation in a controllableway. Comparing with previous study, the present scheme uses only asingle atomic species and hence is easy for the physical realization ofPT-symmetric Hamiltonian via atomic coherence. © 2018 Optical Society of America
OCIS codes: (270.0270) Quantum optics; (190.0190) Nonlinear optics.
References and links
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1. Introduction
In recent years, a lot of efforts have been made on a class of non-Hermitian Hamiltonian withparity-time (PT) symmetry, which in a definite range of system parameters may have an entirelyreal spectrum [1, 2]. PT symmetry requires that the real (imaginary) part of the complex poten-tial in the Hamiltonian is an even (odd) function of space, i.e. V ( r ) = V ∗ ( − r ) . Even though theHermiticity of quantum observables has been widely accepted, there is still great interest in PTsymmetry because of the motivation for constructing a framework to extend or replace the Her-miticity of the Hamiltonian in ordinary quantum mechanics. The concept of PT symmetry hasalso stimulated many other studies, such as quantum field theory [3], non-Hermitian Andersonmodels [4], and open quantum systems [5], and so on.Although a large amount of theoretical works exist, the experimental realization of PT-symmetric Hamiltonian in the fields mentioned above was never achieved. Recently, muchattention has been paid to various optical systems where PT-symmetric Hamiltonians can berealized experimentally by balancing optical gain and loss [6–9]. In optics, PT symmetry isequivalent to demand a complex refractive index with the property n ( r ) = n ∗ ( − r ) . Such refrac-tive index has been realized experimentally using two-wave mixing in an Fe-doped LiNbO substrate [10]. The optical realization of PT symmetry has motivated various designs of PT-synthetic optical materials exhibiting many intriguing features, including non-reciprocal or uni-directional reflectionless wave propagation [10–13], coherent perfect absorber [14, 15], giantwave amplification [16], etc. Experimental realization of PT symmetry using plasmonics [17],synthetic lattices [18], and LRC circuits [19] were also reported.In a recent work Hang et al. [20] proposed a double Raman resonance scheme to realize PTsymmetry by using a two-species atomic gas with L -type level configuration. This scheme isquite different from those based on solid systems mentioned above [6, 10–19], and possessesmany attractive features. For instance, the PT-symmetric refractive index obtained in [20] isvalid in the whole space; furthermore, the refractive index can be actively controlled and pre-cisely 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- ig. 1. (a) Energy-level diagram and excitation scheme used for obtaining a PT symmetric model. (b)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 coherentatomic systems [21]. Different from the two-species, double Raman resonance scheme pro-posed in [20], the scheme we suggest here is a single-species, EIT one. And due to the com-plexity 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 inlight of our will, and the size of potential can also be adjusted conveniently. Especially, com-pared with the traditional idea that PT symmetric potential must be combined by the gain andloss parts, we utilize the atomic decay rate to design the imaginary part of PT potential, and usethe giant cross-phase modulation (CPM) effect [21,22] of the resonant EIT system to realize thereal part. We shall show that the cross-phase modulation contributed by the assisted field, theoptical lattice potential provided by a far-detuned laser field, and the optical gain resulted froman incoherent pump can be used to construct a complex optical potential with PT symmetry forprobe field propagation in a controllable way. The present scheme uses a single atomic speciesonly and hence is simple for physical realization.The rest of the article is arranged as follows. In the next section, a description of our schemeand basic equations for the motion of atoms and light field are presented. In Sec. III, the enve-lope 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.
2. Model and equations of motion
The system under consideration is a cold, lifetime-broadened Rb atomic gas with N-type levelconfiguration; see Figure 1. The levels of the system are taken from the D line of Rb atoms,with | i = | S / , F = i , | i = | S / , F = i , | i = | P / , F = i , and | i = | P / , F = i .A weak probe field E p = e x E p ( z , t ) exp [ i ( k p z − w p t )] + c . c . and a strong control field E c = e x E c exp [ i ( − k c y − w c t )] + c . c . interact resonantly with levels | i → | i and | i → | i , respec-tively. Here e j and k j ( E j ) are respectively the polarization unit vector in the j th direction andthe wave number (envelope) of the j th field. The levels | l i ( l = , ,
3) together with E p and E c constitute a well-known L -type EIT core.Furthermore, we assume an assisted filed E a = e y E a ( x ) exp [ i ( − k a z − w a t )] + c . c . (1)is coupled to the levels | i → | i , where E a ( x ) is field-distribution function in transverse direc-ion. The assisted filed E a , when assumed to be weak (satisfying E p ≤ E a ≪ E c ), will contributea CPM effect to the probe field E p . Note that the levels | l i ( l = , , ,
4) together with E p , E c , and E a form a N-type system, which was considered firstly by Schmidt Imamoˇglu [22] forobtaining giant CPM via EIT.In addition, we assume there is another far-detuned (Stark) optical lattice field E Stark = e y √ E s ( x ) cos ( w L t ) (2)is applied to the system, where E s ( x ) and w L are respectively the field-distribution function andangular frequency. Due to the existence of E Stark , a small and x -dependent Stark shift of level E j to the state | j i occurs, i.e., E j → E j + D E j with D E j = − a j (cid:10) E (cid:11) t = − a j | E s ( x ) | , here a j is the scalar polarizability of the level | j i , and h· · · i t denotes the time average in an oscillatingcycle. The explicit forms of E a ( x ) and E s ( x ) in (1) and (2) will be chosen later on according tothe 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 provideperiodic complex refractive index to the evolution of probe-filed envelope. However, they arestill not enough to obtain a refractive index with PT symmetry since a gain to the probe fieldis needed. Therefore, we introduce an incoherent optical pumping which can pump atoms fromthe ground-state level | i to the excited-state level | i with the pumping rate G [see equa-tions (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 microwavedischarge lamps [23].In Fig. 1(a), G , G , and G are spontaneous emission rates denoting the population decaysrespectively from | i to | i , | i to | i , and | i to | i ; W p = ( e x · p ) E p / ¯ h , W c = ( e x · p ) E c / ¯ h ,and W a = ( e y · p ) E a / ¯ h are respectively the half Rabi frequencies of the probe, control, andassisted fields, here p i j signifies the electric dipole matrix element of the transition from state | i i to | j i , D , D , and D are respectively one-, two-, and three-photon detunings in relevanttransitions. Fig. 1(b) shows a possible experimental arrangement. Under electric-dipole and rotating-wave approximations, the Hamiltonian of the system in inter-action picture reads ˆ H int = − ¯ h (cid:229) j = D ′ j | j ih j | − ¯ h ( W p | ih | + W c | ih | + W a | ih | + h . c . ) , whereh . c . denotes Hermitian conjugate, and D ′ j = D j + a j h | E s ( x ) | . (3)The motion of atoms interacting with the light fields is described by the Bloch equation ¶s¶ t = − i ¯ h (cid:2) ˆ H int , s (cid:3) − G s , (4)where s jl is the density-matrix elements in the interaction picture, G is a 4 × G ) from the level | i to the level | i is introduced [see equations (18a)and (18c)].Under a slowly varying envelope approximation, Maxwell equation of the probe field isreduced to i (cid:18) ¶¶ z + c ¶¶ t (cid:19) W p + c w p ¶ W p ¶ x + k s = , (5)here k = N w p | e x · p | / ( e ¯ hc ) with N being the atomic concentration. Note that, for sim-plicity, we have assumed W p is independent on y , which is valid only for the probe beam havinga large width in the y -direction so that the diffraction term ¶ W p / ¶ y can be neglected; in addi-tion, we have also assumed that the dynamics of W a is negligible during probe-field evolution,which is a reasonable approximation because the assisted field couples to the levels | i and | i that have always vanishing population due to the EIT effect induced by the strong control field.
3. Realization of PT symmetric potential
The Maxwell equation (5) governs the propagation of the probe field. To solve it one must know s , which is controlled by the Bloch equation (4) and hence coupled to W p . For simplicity,we assume W p has a large time duration t so that G t >>
1. In this case a continuous-wave 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 thediffraction effect of the probe field in x direction is considered. In addition, because the probefield is weak, a perturbation expansion can be used for solving coupled equations (4) and (5)analytically [24, 25].We take the expansion s i j = s ( ) i j + es ( ) i j + e s ( ) i j + e s ( ) i j + · · · , W p = e W ( ) p + e W ( ) p + · · · . Here e is a small parameter characterizing the typical amplitude of the probe field (i.e W p , max / W c ). Substituting such expansion to equations (4) and (5), we obtain a series of linearbut inhomogeneous equations for s ( l ) i j and W ( l ) p ( l = , , , ... ) that can be solved order by order.To get a divergence-free perturbation expansion, s ( l ) i j and W ( l ) p are considered as functions ofthe multiple scale variables z l = e l z ( l = ,
2) and x = e x [24, 25]. In addition, we assume W a = e W ( ) a ( x ) , E s = e E ( ) s ( x ) . Thus we have d i j = d ( ) i j + e d ( ) i j with d ( ) i j = D i − D j + i g i j and d ( ) i j = [( a i − a j ) / ( h )] | E ( ) s | .At O ( ) -order, we obtain non-zero density-matrix elements s ( ) = − ( − X ) X , s ( ) =( − X ) X , s ( ) = X , s ( ) = [ W ∗ c / ( d ( ) ) ∗ ] X X , with X = G / [ ( | W c / d ( ) | )] and X = G / [ G + G ( − X )] . It is the base state solution of the MB equations (i.e., the solution for W p = W a = G = | i , | i , and | i . Because G takes the order of MHz in ourmodel, the populations in | i and | i are small. In particular, s ( ) = s ( ) = s ( ) = G = O ( e ) -order, the solution is given by W ( ) p = F e iKz , (6a) s ( ) = W ∗ c ( s ( ) − s ( ) ) − d ( ) s ( ) D Fe iKz ≡ a ( ) Fe iKz , (6b) s ( ) = K k Fe iKz ≡ a ( ) Fe iKz , (6c) s ( ) = d ( ) s ( ) + W c s ( ) D W ( ) a ≡ a ( ) W ( ) a , (6d) s ( ) = W ∗ c s ( ) + d ( ) s ( ) D W ( ) a ≡ a ( ) W ( ) a , (6e)with other s ( ) jl =
0. Here F is yet to be determined envelope function, D = | W c | − d ( ) d ( ) , I m K ( c m - ) Fig. 2.
The imaginary part Im K of K as a function of D / g for D = D . Solid (red), dashed (green),and dashed-dotted (blue) lines correspond to ( W c , G ) = ( , ) , (5 × Hz, 0), and (5 × Hz,0 . g ), respectively. For illustration, the value of dashed-dotted (green) line has been amplified 7.8times. D = | W c | − d ( ) d ( ) , and K = k d ( ) ( s ( ) − s ( ) ) + W c s ( ) D . (7)Obviously, in the linear case W p (cid:181) e iKz , and K is complex. Thus K , particularly its imaginarypart, controls the behavior of the probe-field propagating along z .Figure 2 shows the imaginary part Im K of K as a function of D / g for D = D . The sys-tem parameters used are [26] g = D = , g = × Hz , G = g =
36 MHz , k = . × cm − Hz. Solid (red), dashed (green), and dashed-dotted (blue) lines correspond to ( W c , G ) = ( , ) , (5 × Hz, 0), and (5 × Hz, 0 . g ), respectively.From the solid line of Figure 2, we see that in the absence of the control field and incoher-ent pumping (i.e., W c = G = W c takes the value of 5 × Hz, a transparency win-dow is opened (as shown by the dashed line). This is well-known EIT quantum interferencephenomenon induced by the control field [21]. However, there is still a small absorption (i.e.,Im K >
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 ( G = . g )is introduced. One sees that a gain (i.e., negative Im K in the region near D =
0) occurs. Suchgain is necessary to get a PT-symmetric optical potential for the probe-field propagation, asshown below.At O ( e ) -order of the perturbation expansion, we obtain the closed equation for F , whichcan be converted to the equation for W p : i ¶ W p ¶ z + c w p ¶ W p ¶ x + ˜ V ( x ) W p = V ( x ) = a | e y · p | ¯ h | E a ( x ) | + a | E s ( x ) | + K , (9)here W p = e F exp ( iKz ) , the coefficients a and a are given in Appendix.We now make some remarks about the potential ˜ V ( x ) given by Eq. (9):(1). The coefficients a and a are complex . We stress that the occurrence of a complex po-tential for the evolution of probe-field envelope is a general feature in the system with resonantinteractions. The reason is that, due to the resonance, the finite lifetime of atomic energy statesmust be taken into account. As a result, the variation of the probe-field wavevector resulted bythe external light laser fields (here the Stark and the assisted fields) are complex. It is just thispoint that provides us the possibility to realize a PT symmetric potential in our system by usingthe periodic external laser fields.(2). If the incoherent pumping is absent, the probe field has only absorption but no gain andhence 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 regionnear D =
0, which can be used to suppress an absorption constant (i.e. the term not dependenton x ) appearing in the previous two terms of ˜ V ( 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 thejoint action between the Stark field, the assisted field, and the incoherent pumping can givePT-symmetric potentials in the system.The susceptibility of the probe field is given by c ( x ) = c ˜ V ( x ) / w p . Because the potential(9) is a complex function of x , which is equivalent to a space-dependent complex refractiveindex n ( x ) = p + c ( x ) ≈ + c ˜ V ( x ) / w p for the probe-field propagation. PT symmetry requires˜ V ∗ ( − x ) = ˜ V ( x ) , which is equivalent to the condition n ∗ ( − x ) = n ( x ) . Equation (8) is a linear Schr¨odinger equation with the “external” potential (9). To realize aPT-symmetric model we assume the field-distribution functions in (1) and (2) taking the forms E a ( x ) = E a [ cos ( x / R ) + sin ( x / R )] , (10) E s ( x ) = E s cos ( x / R ) , (11)with E a and E s being typical amplitudes and R − being typical “optical lattice” parameter.For convenience of later discussion, we write Eq. (8) into the following dimensionless form i ¶ u ¶ s + ¶ u ¶x + V ( x ) u = , (12)with V ( x ) = ( g + g sin 2 x ) + g cos x + K , (13)where u = W p / U , s = z / L diff , x = x / R , g = a | e y · p | E a L diff / ¯ h , g = a E s L diff , and K = KL diff . Here, L diff ≡ w p R / c is the typical diffraction length and U denotes the typicalRabi frequency of the probe field.PT symmetry of Eq. (12) requires V ∗ ( − x ) = V ( x ) . In general, such requirement is difficultto be satisfied because resonant atomic systems have very significant absorption. However, inthe system suggested here the absorption can be largely suppressed by the EIT effect inducedby the control field. The remainder small absorption that can not be eliminated by the EITeffect may be further suppressed by the introduction of the incoherent optical pumping. If theoptical pumping is large enough, the system can acquire a gain. This point can be understoodfrom Fig. 2 for the case of ( W c , G ) = ( × Hz, 0 . g ) where near the EIT transparencywindow Im K is negative, which means that the probe field acquires a gain contributed by theptical pumping. Such gain can be used to suppress the imaginary parts of g and g throughchoosing suitable system parameters, and hence one can realize a PT symmetry of the system.For a practical example, we select the D line of Rb atoms, with the energy levels in-dicated in the beginning of Sec. 2.1. The system parameters are given by 2 g = × Hz, G , = g , =
36 MHz, | p | = . × C cm, w p = . × s − . Other (adjustable)parameters are taken as k = . × cm − s − , R = . × − cm, W c = . × s − , D = − . × s − , D = . × s − , and D =
0. Then we have L diff = . E a ( x ) = . ( cos x + sin x ) V / cm , (14) E s ( x ) = . × cos x V / cm , (15) G = . × Hz . (16)Based on these data and the assisted laser field (14), the far-detuned laser field (15) and theoptical pumping (16), we have g = . + . i , g = . + . i , and K = − . − . i .Here, the imaginary parts of g and K can be alone controlled by E a ( x ) and k , respectively.As a result, we obtain V ( x ) = − . + cos x + . i sin 2 x + O ( − ) . (17)Equation (17) satisfies the PT-symmetry requirement V ∗ ( − x ) = V ( x ) when exact to the accu-racy O ( − ) . The constant term − . V ( x ) can be removed by using a phase transforma-tion u → u exp ( − i . s ) . 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 symmetryin our system by using different assisted and far-detuned laser fields. Consequently, our systemhas obvious advantages for actively designing different PT-symmetric optical potentials andmanipulating them in a controllable way.
4. Conclusion
We have proposed a scheme to realize PT symmetry via EIT. The system we considered is anensemble of cold four-level atoms with an EIT core. We have shown that the cross-phase mod-ulation contributed by an assisted field, the optical lattice potential provided by a far-detunedlaser field, and the optical gain coming from an incoherent pumping can be used to construct aPT-symmetric complex optical potential for probe field propagation in a controllable way. Com-paring with previous study in [20], our scheme has the following advantages: (i) Our schemeuses only one atomic species, which is much simpler than that in [20]. (ii) The mechanismof 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-symmetricpotentials at will in our scheme in a simple way. ppendix
Explicit expression of Eq. (4)
Equations of motion for s i j are given by i ¶¶ t s + i G s − i G s + W ∗ p s − W p s ∗ = , (18a) i ¶¶ t s − i G s − i G s + W ∗ c s − W c s ∗ + W ∗ a s − W a s ∗ = , (18b) i (cid:18) ¶¶ t + G (cid:19) s − i G s − W ∗ p s + W p s ∗ − W ∗ c s + W c s ∗ = , (18c) i (cid:18) ¶¶ t + G (cid:19) s − W ∗ a s + W a s ∗ = , (18d) (cid:18) i ¶¶ t + d (cid:19) s + W ∗ c s + W ∗ a s − W p s ∗ = , (18e) (cid:18) i ¶¶ t + d (cid:19) s + W p ( s − s ) + W c s = , (18f) (cid:18) i ¶¶ t + d (cid:19) s + W a s − W p s = , (18g) (cid:18) i ¶¶ t + d (cid:19) s + W c ( s − s ) + W p s ∗ − W a s ∗ = , (18h) (cid:18) i ¶¶ t + d (cid:19) s + W a ( s − s ) − W c s = , (18i) (cid:18) i ¶¶ t + d (cid:19) s + W a s ∗ − W ∗ p s − W ∗ c s = , (18j)with d jl = D ′ j − D ′ l + i g jl ( D ′ j is given by Eq. (3) ), g jl = ( G j + G l ) / + g dph jl ( j = , l = g =( G + G ) / + g dph31 , and G l = (cid:229) E j < E l G jl . Here g dph jl denotes the dipole dephasing rates causedby atomic collisions; G jl is the rate at which population decays from | l i to | j i . Especially, G is the incoherent pumping rate from | i to | i . Perturbation expansion of the MB equations