Spatial Kramers-Kronig relation and controlled unidirectional reflection in cold atoms

We propose a model for realizing frequency-dependent spatial variations of the probe susceptibility in a cold atomic sample. It is found that the usual Kramers-Kronig (KK) relation between real and imaginary parts of the probe susceptibility in the frequency domain can be mapped into the space domain as a far detuned control field of intensity linearly varied in space is used. This non-Hermitian medium exhibits then a unidirectional reflectionless frequency band for probe photons incident from either the left or the right sample end. It is of special interest that we can tune the frequency band as well as choose the direction corresponding to the vanishing reflectivity by changing, respectively, the control field intensity and frequency. The nonzero reflectivity from the other direction is typically small for realistic atomic densities, but can be largely enhanced by incorporating the Bragg scattering into the spatial KK relation so as to achieve a high reflectivity contrast.


I. INTRODUCTION
Asymmetric-reflection and unidirectional reflectionless control of the flow of photons, a key technique for realizing photonic quantum manipulation and communication, has attracted intense research efforts because of its imminent applications in developing novel photonic circuits and devices [1][2][3][4][5][6][7][8].The reflection control of light signals is usually reciprocal and static (i.e., determined by growth design) as achieved, e.g., via fixed band gaps of photonic crystals possessing certain periodic structures of the real refractive index [9,10].A tunable photonic band gap has been proved to be viable by establishing controlled periodic structures of the complex susceptibility in the regime of electromagnetically induced transparency (EIT) [11,12], with standing-wave coupling fields to dress homogeneous atomic clouds [13][14][15][16][17] or travelingwave coupling fields to dress periodic atomic lattices [18][19][20][21][22][23][24][25][26][27][28][29].Generally speaking, it is hard to achieve asymmetric light transport in the familiar linear optical processes [30][31][32], though significant progress has been made in the recent years by considering moving atomic lattices [1,2,28] and fabricating materials of parity-time (PT) symmetry or asymmetry [3,4,33,34].Experimental implementations of these schemes, however, are rather challenging due to the needs of complicated atom-light coupling configurations, precise spatial field arrangement, and balanced gain and loss in a single period.
Alternatively, asymmetric and unidirectional reflection can be realized in an inhomogeneous continuous medium as the usual Kramers-Kronig (KK) relation, satisfied by its complex optical response function, is mapped from the frequency domain to the space domain [5][6][7][8][35][36][37][38][39][40][41][42].This intriguing idea is first proposed by Horsley et al. [5], who * Electronic address: jhwu@nenu.edu.cnfound that a non-Hermitian medium would not reflect radiation from one side for all incident angles if its complex permittivity shows a space instead of a frequency dependence.The spatial KK relation is also found to promise the realization of omnidirectional perfect absorber [7] and transmissionless media [43].When extended into discrete lattices, complex potentials exhibiting the spatial KK relation may further become invisible to support a bidirectional reflectionless behavior [41,42].These works not only deepen our understanding of light propagation, but also provide a new platform for realizing multi-functional optical elements, especially those requiring perfect antireflection.In particular, the original idea has been experimentally demonstrated via a suitable design of different inhomogeneous media [7,36].Once again, these schemes have the disadvantage of lacking dynamic tunability, being based on fixed spatial structures of complex refractive index, susceptibility, or permittivity.
Here we propose an efficient scheme for mapping the KK relation of a probe susceptibility from the frequency domain into the space domain in a cold atomic sample.The essence is to generate a position-dependent ground level shift with a far detuned control field of intensity linearly varied in space.Depending on the probe frequency, the sample is found to exhibit the unbroken, transitional, or broken regime in regard of the spatial KK relation.The unbroken regime with a well-satisfied spatial KK relation is of particular interest because it allows the reflectionless manipulation of probe photons incident from one sample end in a tunable spectral range.The transitional regime with a partially-destroyed spatial KK relation is also of interest because it may be explored to realize the reflectionless manipulation from both sample ends, albeit at different frequencies.More importantly, we can swap the direction of vanishing reflectivity and that of nonzero reflectivity by changing the sign of ground level shift, and enhance the nonzero reflectivity while retaining the vanishing reflectivity by increasing the magnitude of ground level shift.Last but not least, the Bragg scattering in an atomic lattice can be incorporated into the spatial KK relation to yield a high forward-backward reflectivity contrast, corresponding to an enhanced nonzero reflectivity and an invariant vanishing reflectivity.

II. MODEL AND EQUATIONS
We consider in Fig. 1(a) a cold atomic sample extending from x = 0 to x = L, driven by a weak probe field of amplitude (frequency) E p (ω p ) and a strong control field of amplitude (frequency) E c (ω c ).The control field is assumed to illuminate the sample along the −y direction while the probe field can travel through the sample along either x or −x direction.All atoms are driven into the three-level V configuration, as shown in Fig. 1 Most importantly, we will assume that the control field is linearly varied in intensity along the x direction, e.g., by a neutral density filter (NDF).In this case, With the electric-dipole and rotating-wave approximations, working in the weak probe limit, we can solve density matrix equations for the three-level V configuration to attain the steady-state probe susceptibility where N 0 denotes the homogeneous atomic density while It is worth noting that this level shift can also be induced by a magnetic field of linearly varied magnitude in space, which would however be less amenable to dynamic remote control than an optical driving.Then the probe susceptibility can be cast into a more compact form as given by whose real (χ ′ 2 ) and imaginary (χ ′′ 2 ) parts govern the local dispersive and absorption properties around the probe resonance, respectively.The possibility to control the effective detuning ∆ ef f p (x) = ∆ p + δ(x) by changing the ground level shift δ(x) amounts to a direct mapping from the spatially linear variation of the control intensity to that of the probe detuning.The real and imaginary parts of χ 2 are well known to be related via the KK relation in the frequency domain based on the causality principle and Cauchy's theorem if we set δ(x) ≡ 0 [44].Thus, for appropriate probe detunings and sufficiently long samples such that the spatial variation of the effective probe detuning induced by δ(x) is fully developed, as shown by the red solid lines in Figs.2(a) and 2(b), the KK relation holds in the space domain and can be defined by the following Cauchy's principal value integral over spatial coordinate s along the x direction.
A medium described by χ 2 in Eq. ( 2) is expected to exhibit asymmetric light transport features, which can be examined via the standard transfer-matrix method [45][46][47] as sketched below.First, the atomic sample is partitioned along the x direction into a large number (J) of slices labeled by j ∈ [1, J], which exhibit slightly different susceptibilities χj (∆ p ) = χ 2 (∆ p , jl) but identical length l = L/J = 10 nm.Second, a 2 × 2 unimodular transfer matrix M j (∆ p , l) characterized by l and χj (∆ p ) can be established to describe the propagation of an incident probe field of wavelength λ p through the jth slice by where E + p and E − p denote the forward and backward components of the probe field, respectively.Third, the total transfer matrix of the atomic sample turns out to be as a multiplication of the individual transfer matrices of all atomic slices.Finally, we can write the reflectivities (R l = R r ) and the transmissivities (T = T l = T r ) as in terms of the four matrix elements of M (∆ p , L).Here the subscripts 'l' and 'r' have been used to denote that the weak probe field is incident from the left and right sides along the x and −x directions, respectively.

III. RESULTS AND DISCUSSION
In this section, we show via numerical calculations how to implement the spatial KK relation in a narrow spectral range by tailoring the complex probe susceptibility, and how to implement the unidirectional reflection of a high reflectivity contrast by utilizing the spatial KK relation.All numerical calculations will be done with realistic parameters corresponding to the three states of 87 Rb atoms chosen above, though our driving configuration can also be realized, e.g., in other alkali metal atoms.
First, we plot in Figs.2(a) and 2(b), respectively, the real and imaginary parts of χ 2 against position x for two sets of parameters making the choice ∆ p −δ 0 /2, which allows the effective probe detuning ∆ ef f p (x) to be in the range of {−δ 0 /2, δ 0 /2}.It is clear that χ ′ 2 and χ ′′ 2 show an odd profile and an even profile, respectively, centered at z = L/2 and practically fully contained by the atomic sample, so they should satisfy the spatial KK relation described by Eq. (3).We also can see that smaller (larger) values of δ 0 /γ 12 , a key dimensionless parameter in our reduced two-level system, will result in broader (narrower) spatial profiles of χ ′ 2 and χ ′′ 2 yet without changing their peak amplitudes.To reveal the frequency-dependent feature, we plot in Figs.2(c) and 2(d), respectively, the real and imaginary parts of χ 2 against both position x and detuning ∆ p instead.The profiles of χ ′ 2 and χ ′′ 2 are found to move simultaneously toward the left (right) sample end with the increasing (decreasing) of ∆ p .Accordingly, the spatial KK relation will be gradually destroyed because Eq. (3) becomes less and less satisfied.
In order to assess the extent to which the spatial KK relation is satisfied for different probe detunings in a finite atomic sample, here we propose the following integral the spatial KK relation is fully satisfied while D kk = ±1 denote the broken regime where the spatial KK relation is fully destroyed.According to Eq. ( 2) and Fig. 2, the spatial profiles of χ ′ 2 and χ ′′ 2 are well contained within our finite atomic sample only for a small range of ∆ p , so the validity of the spatial KK relation is expected to increasingly deteriorate (D kk = 0 → D kk = ±1) as ∆ p is gradually modulated out of this range.This is different from all previous works on spatial KK relations [5][6][7][8][35][36][37][38][39][40][41][42], where the susceptibility or permittivity has been assumed to be fixed by design, i.e. not tunable.
Then we plot in Fig. 3(a) the reflection and transmission spectra for the parameters used in Figs.2(a) and 2(b) based on Eq. ( 5).It is easy to see that these spectra can be divided into three regions: (I) where we have R l = R r → 0 and T → 1; (II) where R l = R r and T are sensitive to ∆ p ; (III) where T ≃ 0.05, R l → 0, but R r oscillates around 0.2.The generation of three different regions can be understood by examining in Fig. 3(b) the figure of merit D kk against probe detuning ∆ p , which clearly shows, as compared to Fig. 3(a), that D kk governs the relations between R l , R r , and T .The symmetric (I), asymmetric (II), and unidirectional (III) reflection regions correspond, respectively, to the broken (D kk = ±1), transitional (0 < |D kk | < 1), and unbroken (D kk = 0) regimes, and to the cases when the absorption (χ ′′ 2 ) and dispersion (χ ′ 2 ) profiles move out of the sample, lie at the sample boundaries, and are well contained by the sample.It is worth noting that in region (III) a left (right) incident probe beam is reflectionless (partially reflected) because it first sees the negative (positive) peak of χ ′ 2 [5], and the resonant absorption (χ ′′ 2 ) is already strong enough to yield T → 0.0 for forward photons while the dispersion profile (χ ′ 2 ) is not too sharp to yield R r → 1.0 for backward photons.One way for further reducing T and simultaneously increasing R r is to produce enhanced absorption profiles and sharper dispersion profiles in denser atomic samples.The restricted range of densities of cold atoms available in experiment, however, places a constraint on this approach.
Fig. 4(a) further shows the different regimes on a diagram with D kk plotted against δ 0 and ∆ p , in which the green region (D kk = −1) and the red region (D kk = 1) refer to the broken regime (I); the four narrow blue regions (0 < |D kk | < 1) refer to the transitional regime (II); the two triangular yellow regions (D kk = 0) refer to the unbroken regime (III).It is also clear that the widths of two yellow regions depend critically on the magnitude of δ 0 ; a broken regime may be converted into an unbroken regime and vice versa for a given ∆ p by changing the sign of δ 0 .Accordingly, it is viable to enlarge or reduce the reflectionless frequency band by varying the magnitude of δ 0 and convert the sample from left reflectionless to right reflectionless or vice versa by changing the sign of δ 0 .This potentially dynamic controllability, a chief feature of our proposal, is well demonstrated in Figs.4(b) and 4(c) in terms of reflectivities R l and R r .
It is also interesting to examine what could happen for reflectivities R l and R r when sample length L is multi-FIG.5: Reflectivities (a) R l and (b) Rr against detuning ∆p for L = 10 µm (black-squares), L = 15 µm (red-circles), and L = 20 µm (blue-triangles).Other parameters used in calculations are the same as in Fig. 2(a,b).Black (red) curves are shown with a vertical offset 0.1 (0.05) in both insets.
plied while atomic density N 0 remains invariant.In this case, we can see from δ(x) = xδ 0 /L that the linear variation occurs in a much larger range while its magnitude δ 0 is unchanged.Then, as shown in Fig. 5, the spatially wider/smoother dispersion (χ ′ 2 ) and absorption (χ ′′ 2 ), of L-independent maxima and minima, together result in a notable reduction of R r while R l remains vanishing in the unbroken regime.The peak of R l (R r ) accompanied by R r → 0 (R l → 0) in the transitional regime has a L-independent position because it only appears as the main profiles of χ ′ 2 (x) and χ ′′ 2 (x) approach and even partially leave the left (right) sample end.It is clearly not a result of the spatial KK relation and allows two probe beams of different frequencies to be simultaneously reflected or not when they are incident upon the opposite sample ends.The damped oscillations of R r against ∆ p in the unbroken regime can be understood as a multiple interference effect due to the discontinuities of the probe susceptibility at the right sample end and at the resonant position inside the sample.It is clear that stronger (weaker) oscillations occur at larger (smaller) values of |∆ p | because the resonant position of χ ′ 2 (x) and χ ′′ 2 (x) is close to (far from) the right sample end, yielding thus stronger (weaker) discontinuities.The oscillation period can be roughly estimated as d∆ p ≃ δ 0 /L•λ p /2 by considering that the interval d∆ p of two adjacent maxima corresponds to a 2π phase shift (λ p /2 spatial shift) gained by the reflected photons (χ ′ 2 (x) and χ ′′ 2 (x)).Finally, we note that an experiment may typically have a lower atomic density and a larger sample length than in simulations presented so far.To overcome this difficulty, we need to find an alternative way to enhance the nonzero reflectivity in the unbroken regime.This can be done by loading cold atoms into a 1D optical lattice to create a spatially periodic density N j (x) as described in the caption of Fig. 6, yielding thus Bragg scattering incorporated into the spatial KK relation.As shown in Fig. 6(a) and Fig. 6(b), both dispersion χ ′ 2 (x) and absorption χ ′′ 2 (x) of one-order lower values now exhibit the comb-like spatial profiles while satisfying to a less extent the spatial KK relation.In this case, we can find from Fig. 6(c) that R l and R r are strongly asymmetric in a much smaller frequency range, e.g., with R r exhibiting a maximal value up to 0.54 while R l 0.01 for −25.5 ∆ p /γ 21 −11.0.Fig. 6(d) further shows that the reflectivity contrast C = (R r −R l )/(R r +R l ), an important figure of merit on the asymmetric reflection, could be up to 0.97 and is over 0.90 for −27.0 ∆ p /γ 21 −4.5.It is noticeable that the incorporation of Bragg scattering, typically yielding symmetric reflectivities, has negligible effects on the vanishing reflectivity but largely enhances the nonzero reflectivity and the reflectivity contrast.That means, replacing a constant density N 0 with a periodic density N j (x) does not hamper the implementation of spatial KK relation, which is essential for developing nonreciprocal optical devices requiring a high reflectivity contrast.
It has been shown that atoms could be trapped and guided using nanofabricated wires and surfaces to form atom chips [48].These chips provide a versatile experimental platform with cold atoms and constitute the basis for wide and robust applications ranging from atom optics to quantum optics.They have been used, for instance, in diverse experiments involving quantum simulation, metrology, and information processing [49][50][51].We then believe that our proposal is well poised to atom-chip implementations in integrated optical devices.

IV. CONCLUSIONS
In summary, we have investigated the spatial KK relation and relevant reflection features in a short and dense sample of cold 87 Rb atoms.This nontrivial relation in regard of the probe susceptibility is enabled by generating a position-dependent ground level shift δ(x) with a far detuned control field of intensity linearly varied along the x direction.We find, in particular, that the figure of merit D kk characterizing the spatial KK relation may switch from the unbroken regime of unidirectional reflection, via the transitional regime of asymmetric reflection, to the broken regime of symmetric reflection, or vice versa.This is attained by increasing the maximal level shift δ 0 from a negative value to a positive value or considering an inverse process, depending on the sign of probe detuning ∆ p .A swapping between the nonzero reflectivity and the vanishing reflectivity at opposite sample ends is also vi-able by changing the sign of maximal level shift δ 0 .It is of more interest that the nonzero reflectivity can be well enhanced to result in a high reflectivity contrast for lower densities and larger lengths in a cold atomic lattice, indicating that Bragg scattering does not hamper or spoil the main effects of spatial KK relation.

FIG. 1 :
FIG. 1: (a) A cold atomic sample illuminated by a control beam Ec(x) along the −y direction exhibits a strongly asymmetric reflection for a probe beam Ep incident in the ±x directions.(b) A three-level atomic system driven by a probe field of Rabi frequency Ωp (detuning ∆p) and a control field of Rabi frequency Ωc(x) (detuning ∆c) into the V configuration.(c) A two-level atomic system with a dynamic shift δ(x) of level |1 upon the adiabatic elimination of level |3 .

FIG. 3 :
FIG. 3: (a) Reflectivity R l , reflectivity Rr, and half transmissivity T ; (b) figure of merit D kk against detuning ∆p.Other parameters used in calculations are the same as in Fig. 2(a,b).

FIG. 4 :
FIG. 4: (a) Figure of merit D kk , (b) reflectivity R l , and (c) reflectivity Rr (c) against shift δ0 and detuning ∆p.Other parameters used in calculations are the same as in Fig. 2(a,b).