Unidirectional and bidirectional photon transport blockade in driven atomic lattices of parity-time antisymmetry

We propose a scheme for realizing exact parity-time ( PT ) antisymmetry of complex susceptibility in a wide range of probe detuning by considering a one-dimensional lattice of cold atoms driven into the simplest three-level Λ configuration. This is attained by modulating the intensity of a standing-wave coupling field with a proper phase shift to counteract the product of a single-photon detuning and a two-photon detuning. Such a dynamically controlled PT -antisymmetric lattice supports the integration of a few nontrivial scattering behaviors including unidirectional light reflectionless, asymmetric perfect absorption, and directional signal quenching. These behaviors, depending in particular on atomic densities and lattice lengths, facilitate the on-demand realization of unidirectional or bidirectional photon transport blockade.

Compared to these optical platforms, a unique feature of cold or warm atomic samples is that their refractive indices or susceptibilities are easily controlled by external parameters such as intensities and frequencies of space-dependent driving fields.In this regard, it is more constructive to extend the optical PT symmetry studies by exploring near-resonant light-atom coherent interactions.In the first proposal by Hang et al [29], a PT -symmetric optical lattice is designed by considering a mixture of three-level Λ-type 85 Rb and 87 Rb atoms and making a control and a Stark fields to exhibit special distributions in space.In order to avoid using two species of atoms, the four-level N-type configuration is then proposed as an alternative scheme for realizing the optical PT symmetry in the standing-wave (SW) regime, but not the traveling-wave (TW) regime, of electromagnetically induced transparency (EIT) [30,31].Corresponding experiments have been implemented to demonstrate such a dynamically controlled PT -symmetric atomic lattice with balanced gain and loss distributions [32].The optical PT symmetry, if incorporated into atomic gratings, can lead to other novel phenomena like asymmetric light diffraction [33,34] and Talbot effects [35,36].
As a counterpart of PT symmetry, PT antisymmetry denoted by n(r) = −n * (−r) in the optical domain soon attracts equivalent attentions and has been investigated in diverse platforms, ranging from cold or warm atoms [37][38][39][40][41], magnonic systems [42,43], and electrical circuits [44] to cavity optomechanics [45], diffusive systems [46], microcavities [47][48][49], and optical fibers [50].We note in particular that the optical PT antisymmetry requires refractive index n(r) to be an odd function in its real part while an even function in its imaginary part.Consequently, in the presence of only a loss or gain distribution, the optical PT antisymmetry has been explored for displaying similar fascinating phenomena, like unidirectional reflectionless (URL) [37,38,41], asymmetric mode switching [49], energy-difference conservation [44,51] constant refraction [52], and asymmetric perfect absorption (APA) [53].Moreover, we note that it is viable to realize both optical PT symmetry and antisymmetry in a common atomic system [54].
In this paper, we propose a new scheme for realizing the optical PT antisymmetry and exploring relevant applications in asymmetric scattering manipulation.We consider, in particular, 133 Cs atoms driven into the three-level Λ configuration by a probe and a coupling fields and trapped in periodic dipole traps of a one-dimensional (1D) optical lattice.This scheme is easier to implement in experiment because it involves neither the complicated four-level N configuration [30][31][32][33][35][36][37][38] nor a mixture of two atomic isotopes [29,34].Our Λ configuration is also simpler than that in [41] needing an extra modulating field and that in [55] needing an extra microwave field.More specifically, a PT -antisymmetric susceptibility can be attained for a wide range of the probe detuning, but not at a specific probe detuning as in [41] and [55], when the atomic density exhibits a Gaussian distribution in each dipole trap and the coupling field bears a SW intensity with a proper phase shift.This PT -antisymmetric susceptibility is further explored to display a few asymmetric scattering effects, including URL as a probe beam is incident from one side of our atomic lattice and APA or directional signal quenching (DSQ) as two probe beams are incident from the opposite sides.The accomplishment of these nontrivial scattering effects in a single atomic system has not been reported before, and may be used to design versatile nonreciprocal devices with tunable scattering behaviors of unidirectional or bidirectional photon transport blockade.

PT -antisymmetric susceptibility
We start by considering a 1D optical lattice of period a along the z direction filled with cold 133 Cs atoms at the bottom of each dipole trap as depicted in figure 1(a).The lattice period a = λ 0 /(2 cos θ 0 ) is determined by the common wavelength λ 0 and a small angle θ 0 of two red-detuned laser beams, which form the dipole traps as propagating at angles ±θ 0 , respectively, with respect to the ±z directions.Atomic density in the jth dipole trap centered at z j is assumed to exhibit a Gaussian distribution N j (z) = N 0 exp[−(z − z j ) 2 /d 2 ] with d being the 1/e half width and N 0 the maximal density for a finite atomic lattice extending from j = −k to j = k periods.Under the rotating-wave and electric-dipole approximations, for each Λ-type atom, it is viable to write down the following interaction Hamiltonian, which can be brought into the master equation of density operator ρ to yield a set of dynamic equations for density matrix elements ρ µν with {µ, ν} ∈ {g, e, m}.In the limit of a weak probe field, it is appropriate to set ρ gg ≃ 1 and ρ ee ≃ ρ mm ≃ ρ me ≃ 0 so that these equations restricted by ρ µν = ρ * νµ and µ ρ µµ = 1 reduce to where γ µν denotes the dephasing rate on transition |µ⟩ ↔ |ν⟩.By setting ∂ t ρ µν = 0, we can solve equation (2) to attain the space-invariant atomic coherence on the probe transition.Typically, γ mg is three-four orders smaller than γ eg and other quantities, hence we will set γ mg = 0 for simplicity as we examine below the space-dependent probe susceptibility in the jth lattice period We recall that the out-of-phase spatial modulations of two independent quantities are requisite to pursue a PT -antisymmetric probe susceptibility whose real and imaginary parts are odd and even functions, respectively, along the spatial modulation direction [37,38,41].Then, a PT -antisymmetric probe susceptibility may be attained provided the coupling field strength ∝ Ω 2 c can be engineered to exhibit an odd function against lattice position z since atomic density N j (z) has already been an even function of lattice position z in the case of z 0 = 0.This can be realized, e.g. by taking a SW coupling field composed of two oppositely traveling beams ).Note that different maxima of the SW coupling field have been assumed to exhibit a π/4 phase shift relative to corresponding centers of the atomic lattice, and there exists a small angle θ c between the coupling beams and the z direction so that it is viable to attain k c = (2π/λ c ) cos θ c ≃ π/a by modulating this angle θ c or angle θ 0 of the lattice beams.With above considerations, we have which is generally non-Hermitian due to the interplay of a Gaussian modulation in the numerator and a Sine modulation in the denominator, but not PT -antisymmetric due to the presence of two constants ∆ p and For realizing a PT -antisymmetric probe susceptibility, we can try to eliminate the two constants by taking Ω 2 c = ∆ p (∆ p − ∆ c ) and ∆ p ̸ = 0 (probe off resonance).In this case, equation ( 4) will be largely simplified to result in,  and detuning ∆ a on a third transition |g⟩ ↔ |r⟩ restricted by |∆ a | ≫ Ω a as illustrated in figure 1(c).In this case, the probe detuning ∆ p should be replaced by ∆ eff p = ∆ p − Ω 2 a /∆ a so that it can take a small and even zero value because ) is required instead for attaining equation ( 5).Next, we examine via numerical calculations that χ pj (z) in equation ( 4) satisfies the exact PT antisymmetry for ∆ p (∆ p − ∆ c ) = Ω 2 c .To this purpose, we choose the D1 line of 133 Cs atoms by taking |g⟩ ≡ |6 2 S 1/2 , F = 3⟩, |e⟩ ≡ |6 2 P 1/2 , F = 3⟩ and |m⟩ ≡ |6 2 S 1/2 , F = 4⟩ as the three levels in figure 1(b), which then yields the dephasing rate γ eg /2π = 4.6 MHz.In addition, we assume that the atomic lattice exhibits a finite length L = 0.1 ∼ 0.6 mm, a maximal density N 0 = 10 11 ∼ 10 12 cm −3 , and a 1/e half width d = a/3 for the Gaussian distribution.It is then viable to plot in figure 2 imaginary and real parts of the probe susceptibility χ pj (z) in the jth lattice period as a function of lattice position z − z j by taking ∆ c = 0 and Ω 2 c = ∆ 2 p for simplicity without the loss of generality.We find that (i) Im[χ pj (z)] and Re[χ pj (z)] become, respectively, an even and an odd function of z − z j and (ii) Re[χ pj (z)] will flip over with respect to z = z j while Im[χ pj (z)] remains unchanged as ∆ p changes its sign.Moreover, with the increase of Ω c = ±∆ p , we can see that (iii) the spatial profiles of both Im[χ pj (z)] and Re[χ pj (z)] become thinner/sharper and (iv) the magnitude of Re[χ pj (z)] becomes higher while that of Im[χ pj (z)] remains unchanged.These findings are typical features of a PT -antisymmetric susceptibility and can be well understood by resorting to equation ( 5) attained directly from equation (4) restricted by Ω 2 c = ∆ p (∆ p − ∆ c ) and ∆ p ̸ = 0. Similar results can be attained in the more general case of ∆ c ̸ = 0 except it is impossible to attain the exact PT antisymmetry in the range of 0 < ∆ p < ∆ c for ∆ c > 0 and ∆ c < ∆ p < 0 for ∆ c < 0. Note also that the coupling field cannot be too weak and is restricted by Ω 2 c ≫ γ eg γ mg with γ mg /γ eg ≈ 10 −3 ∼ 10 −4 as required for attaining a good EIT effect of our interest.

Asymmetric scattering effects
It is known that symmetric reflection usually happens in traditional (Hermitian) optical structures because the real and imaginary parts of their susceptibilities keep in phase in spatial variation.But, high-contrast asymmetric reflection and even URL are expected to occur in PT -antisymmetric optical structures in which the real and imaginary parts of their susceptibilities exhibit out-of-phase spatial modulations [37,38,41,55].This motivates us to examine how our PT -antisymmetric atomic lattice can be explored to manipulate asymmetric reflection by incorporating the probe susceptibility χ pj (z) into the transfer matrix formula [56] (see the appendix for details).In brief, as shown in figure 1(a), (E + L , E − L ) and (E + R , E − R ) are electric-field amplitudes of a probe field on the left and right lattice sides, respectively.They are connected by a standard transfer matrix M restricted by detM = 1 via from which it is straightforward to calculate the following (reciprocal) transmissivity and (asymmetric) reflectivities with M (ij) being an element of matrix M on the ith row and the jth column.
With reflection coefficients r L,R and transmission coefficient t in hand, one can further relate the outgoing field amplitudes (E − L , E + R ) to the incoming field amplitudes (E − R , E + L ) via a scattering matrix S, i.e.
In general, matrix S is non-Hermitian because its eigenvalues λ ± s = t ± √ r L r R are complex and its eigenvectors (± r L /r R , 1) T for r R ̸ = 0 or (1, ± r R /r L ) T for r L ̸ = 0 are not orthogonal.The two eigenvalues would merge into one another with λ + s = λ − s as the two eigenvectors coalesce into a single one (0, 1) T or (1, 0) T in the case of r L = 0 or r R = 0 to generate a non-Hermitian degeneracy.It is clear that such a degeneracy is associated with the URL behavior (e.g.E − L = 0) of a probe field incident from one side (e.g.
As two probe fields (E + L ̸ = 0 and E − R ̸ = 0) are simultaneously incident from the opposite sides of our atomic lattice, one may realize another scattering behavior: APA indicating zero outgoing fields (E − L = E + R = 0), which could be attained if we have λ + s = 0 or λ − s = 0 with t 2 = r L r R and r L ̸ = r R .In both cases of λ ± s = 0, the APA eigenvector of scattering matrix S reads (−t/r R , 1) T for r R ̸ = 0 or (1, −t/r L ) T for r L ̸ = 0, which tells us how two incoming probe fields (E − R ̸ = E + L ) should be applied so as to realize APA via bidirectional perfect destructive interference.One may also realize a third scattering behavior: DSQ with With appropriate parameters, now we numerically examine all three asymmetric scattering behaviors mentioned above.First, we plot in figure 3  MHz, respectively, indicating a non-Hermitian degeneracy where a probe field incident from the left or right side will be reflectionless and suffer a π or −π phase shift.A comparison between the dilute and dense atomic lattices confirms that this degeneracy is determined just by the ratio of Re[χ pj (z)] and Im[χ pj (z)] while irrelevant to their magnitudes proportional to N 0 .However, an increase of N 0 or L can be used to improve the magnitude of |r L | and |r R | on different sides of this degeneracy, though same results would be attained when the product N 0 L remains fixed.Moreover, we have plotted transmission amplitude |t| and phase ϕ t = arg(t) as a reference, which are respectively sensitive to and independent of {∆ p , N 0 , L}.This finding indicates that it is possible to realize APA in a suitable range of ∆ p by modulating N 0 and L, as examined below.
Then, we switch to the second case where two probe fields are simultaneously incident form the opposite sides of an atomic lattice by plotting in figure 4 real and imaginary parts of scattering eigenvalues λ ± s as a function of probe detuning ∆ p for different values of N 0 and L. It is easy to see that λ ± s become degenerate due to r L = 0 or r R = 0 at two points ∆ p /2π = ±5.2MHz, remain to be real in the unbroken regime of PT antisymmetry referring to |∆ p /2π| < 5.2 MHz, but are always complex in the broken regime of PT antisymmetry referring to |∆ p /2π| > 5.2 MHz.That means, we can realize a phase transition from complex λ ± s to real λ ± s at the left degeneracy point while a phase transition from real λ ± s to complex λ ± s at the right degeneracy point by gradually increasing ∆ p .We further note from figure 4(a) that the two degeneracy points also refer to λ ± s = 0, indicating the overlap of APA and URL for a dense and long enough atomic lattice.Figure 4(b) shows instead that the points of APA move away from the points of URL into the unbroken regime of PT antisymmetry for a diluter and/or shorter atomic lattice.When the atomic lattice becomes dilute and/or short enough, APA cannot be observed anywhere as we can see from figures      Based on the observations in figures 5 and 6, we conclude that a weaker signal coming from one side can be tuned in amplitude and phase to control a stronger signal coming from the other side for attaining either APA or DSQ at a desired probe detuning.This is relevant to the non-Hermitian degeneracy characterized by r L = 0 or r R = 0 of a PT -antisymmetric atomic lattice, and may find applications in developing nonreciprocal optical devices exhibiting scattering behaviors of unidirectional or bidirectional photon transport blockade.It is also worth noting that we always have figures 5 and 6 as required by energy conservation based on the fact that our atomic lattice is lossy everywhere.

Conclusions
In summary, we have investigated a convenient realization and relevant scattering applications of the PT -antisymmetric probe susceptibility by considering a 1D optical lattice of 133 Cs atoms.Our strategy is relating to the spatially correlated modulations of atomic lattice density and coupling field intensity, the former of which exhibits a Gaussian distribution in each dipole trap while the latter with a proper phase shift contains a constant and a Sine components.As compared to previous schemes involving the four-level N configuration or a mixture of two isotopes, our scheme is simpler and hence easier to be implemented in experiment since it is based on a single isotope driven into the three-level Λ configuration needing no extra fields except a probe and a coupling fields.Analytical results show that the probe susceptibility will exhibit an exact PT antisymmetry in a wide frequency range provided the constant coupling component is adjusted to cancel the product of a single-photon and a two-photon detunings.This PT -antisymmetric susceptibility promises a flexible manipulation of diverse nontrivial scattering effects and has been examined via numerical calculations to realize URL, APA, and DSQ for a single probe beam incident upon either side or two probe beams incident upon different sides of our atomic lattice.where t l jξ and t r jξ (r l jξ and r r jξ ) are the transmission (reflection) coefficients on the left and right sides of this layer, respectively.Rearranging equation (A.1), it is easy to attain describing the transfer relation between right-side two fields (E r+ jξ , E r− jξ ) and left-side two fields (E l+ jξ , E l− jξ ).Hence, m jξ is the transfer matrix for the ξth layer in the jth period with its four elements determined by the following transmission and reflection coefficients (according to Fresnel's formula) [56]  These coefficients allow us to further relate two outgoing fields (E − L , E + R ) to two incoming fields (E − R , E + L ) as a result of non-Hermitian scattering as follows, where S refers to the scattering matrix of our atomic lattice.

Figure 1 (
b) further shows that all trapped atoms are driven into the three-level Λ configuration by a weak probe field of frequency (amplitude) ω p (E p ) and a strong coupling field of frequency (amplitude) ω c (E c ) acting, respectively, upon transitions |g⟩ ↔ |e⟩ and |m⟩ ↔ |e⟩.Corresponding detunings (Rabi frequencies) are defined as ∆ p = ω p − ω eg and ∆ c = ω c − ω em (Ω p = E p d ge /2h and Ω c = E c d me /2h) with ω µν being transition frequencies and d µν electric-dipole moments.

Figure 1 .
Figure 1.(a) A 1D lattice of cold atoms loaded into 2k + 1 dipole traps of period a along the z direction and exhibiting a Gaussian distribution in each dipole trap.This atomic lattice is illuminated by a standing-wave coupling field of amplitude Ec with a π/4 phase shift.Two oppositely incoming probe fields of amplitudes (E + L , E − R ) will be scattered by this atomic lattice into two oppositely outgoing probe fields of amplitudes (E − L , E + R ).(b) All trapped atoms are driven into the three-level Λ configuration by a weak probe field of Rabi frequency (detuning) Ωp (∆p) on the |g⟩ ↔ |e⟩ transition and a strong coupling field of Rabi frequency (detuning) Ωc (∆c) on the |m⟩ ↔ |e⟩ transition.(c) A four-level N configuration where the ground state |g⟩ exhibits a tunable Stark shift (not shown) induced by an auxiliary field of Rabi frequency (detuning) Ωa (∆a) on the |g⟩ ↔ |r⟩ transition with Ωa ≪ |∆a|.
0 via unidirectional perfect destructive interference.Both APA and DSQ are close to a non-Hermitian degeneracy, hence accessible with an imbalanced input control due to |t| ̸ = |r L | ̸ = |r R |.
reflection amplitudes |r L | and |r R | as well as phases ϕ L = arg(r L ) and ϕ R = arg(r R ) as a function of probe detuning ∆ p in the case of ∆ c = 0 and Ω c = |∆ p | for a dilute and a dense atomic lattices.It is easy to find that both atomic lattices exhibit asymmetric reflections characterized by |r L (∆ p )| = |r R (−∆ p )|, whose spectral profiles and maximal magnitudes depend clearly on atomic density N 0 .Note also that |r L | and |r R | exhibit a vanishing minimum at ∆ p /2π = −5.2MHz and ∆ p /2π = 5.2 4(c) and (d), showing that λ + s and λ − s (unlike |r L | and |r R |) may be very different for the same product of N 0 L.

Figure 5 (
a) shows in particular that, with a largely imbalanced input ratio |E − R /E + L | = 0.1 and ϕ in = arg(E − R /E + L ) = 0, it is viable to realize DSQ by quenching the left output field with|E − L | = |tE − R + r L E + L | = 0 viaperfect destructive interference at two points ∆ p ≃ −23.9 MHz and ∆ p ≃ −6.6 MHz while the right output field |E + R | = |tE + L + r R E − R | exhibits a roughly symmetric spectral profile with a nonzero dip at ∆ p ≃ 0. As the input ratio gradually increases, we further find that |E − L | is quenched at only one point ∆ p ≃ −10.2 MHz in figure 5(b) but exhibits a larger and larger nonzero minimum in figures 5(c) and (d) such that DSQ cannot be realized again.Note also that the spectral profile of |E + R | looks more asymmetric, indicating different interference results between tE + L and r R E − R for positive and negative probe detunings.To observe DSQ for the right output field |E + R | at positive probe detunings, we need to choose appropriate reversed input ratios with |E − R /E + L | > 1 (not reported here).Another important scenario of the input-output relation is examined in figure 6 with ϕ in = π instead.For a largely imbalanced input ratio |E − R /E + L | = 0.1, we observe once again from figure 6(a) that only the left output field |E − L | can be quenched at one point ∆ p ≃ −4.5 MHz while the right output field |E + R | remains nonzero.Slightly increasing the input ratio, we find from figure 6(b) that both left and right output fields can be quenched to realize DSQ via perfect destructive interference, though at points ∆ p ≃ −3.6 MHz and ∆ p ≃ −2.2 MHz, respectively.It is more interesting that |E − L | and |E + R | can be quenched at a common point ∆ p ≃ −3.5 MHz to realize the usually more inaccessible APA with a proper input ratio |E − R /E + L | = 0.328 in figure 6(c).As the input ratio increases to |E − R /E + L | = 0.5, figure 6(d) shows that |E − L | is quenched at a second point ∆ p ≃ 39.6 MHz far from the probe resonance; |E + R | exhibits also two vanishing points close to but deviating evidently from the original vanishing point ∆ p ≃ −3.0 MHz of |E − L |.

Figure A1 .
Figure A1.Schematic of the ξth layer centered at z = z jξ , as a homogeneous medium, of thickness δz and refractive index n pj (z jξ ) in the jth period of our atomic lattice, where two incoming fields (E l+ jξ , E r− jξ ) are scattered into outgoing fields (E r+ jξ , E l− jξ ).