Controlled multiple spectral hole burning via a tripod-type atomic medium

In limit of saturation spectroscopy, we theoretically study the spectral hole burning (SHB) in the absorption spectrum of a probe field through a tripod atomic system. The response function for the probe field is calculated in a Doppler-broadened medium. Burning of spectral holes is observed only for the counter propagation of either one or both the coupling fields in the medium. The SHB is not observed below some critical temperature which is a condition for the electromagnetically induced transparency (EIT) in the medium. The most interesting and significant feature is that the Doppler broadening acts as a decoherence effect in case of EIT, however, the Doppler broadening acts inversely in case of SHB and consequently the burning effect enhances. The SHB is further enhanced and controlled by classes of the average velocity of atoms. The classes of high average atomic velocity in the medium increase the number of spectral hole burns (HBs). The widths of HBs can be controlled by the intensity of the driving fields. A single HB can be switched to multiple HBs in a well-controlled manner using different classes of high average atomic velocity. The various switchable holes can be burned in a desired position of the absorption spectrum which in turn simultaneously slow down multiple probe fields. The phenomenon of SHB may be useful in the construction of multichannel optical switching and storage devices.


Introduction
The Fano type interference effect [1] induced by lasers is a quantum coherence phenomenon that turns an opaque absorbing medium into a transparent [2] is traditionally termed as electromagnetically induced transparency (EIT) [3].While spectral hole burning (SHB) is the phenomenon, where a saturated laser burns holes into the population distribution of an inhomogeneous Doppler-broadened medium [4], typically called the Bennet holes [5] or Lamb holes [6], within the absorption profile of a probe beam.The resonant absorption spectrum of a probe beam can be bleached, by a control field, at a definite frequency either by fabricating a hole or by generating an EIT window in the probe absorption profile.In an inhomogeneous medium with ground hyperfine states, a coupled laser beam can bleach atoms from any of the hyperfine states, where the population decay rate must be less than their transition.A group of depleted atoms thus appears as a spectral hole in the absorption profile.Note that, the storage capability in the holography is limited by cross talk due to the Bragg degeneracy [7].However, the SHB does not affect by this and it behaves as an unperturbed channel.Hence, the storage capacity can be amazingly enhanced with wave multiplexing [8].
In the last few decades, countless approaches have been adopted for slowing light [9][10][11][12][13][14] to the desired value, in which EIT is one of the dominant way [15][16][17][18], based on the population trapping in absorption-less dark state.An EIT window is generated in the homogeneously broadened absorption spectrum due to the destructive quantum interference between the indistinguishable transition paths to the excited state.The probe pulse transmitting through the EIT window causes a large delay [19].The width of the EIT window depends on the intensity of the Rabi frequency of driving field and the pulse delay is constrained by bandwidth of EIT window, which is of the order of MHz [20].The wide bandwidth of the EIT window can be obtained by the intense pulsed lasers [21,22], but with a limited lifetime.In addition, a number of techniques have been reported on the development of extremely coherent light sources.For example, the most widely used ones include the Pound-Drever-Hall method wherein the laser frequency is stabilized to an optical stable cavity to narrow the laser linewidth [23,24], the active optical clock technique [25][26][27][28][29] in which the stimulated emission from the atomic gain medium is used to generate narrow linewidth lasers [30,31], and the SHB phenomenon in cryogenically cooled crystals [32,33] which is the focus of interest of this paper.
The SHB, a high-resolution spectroscopic technique [34], is used in inhomogeneously broadened emitters, with applications from spectroscopy to slow light [35] and frequency combs [36].In this framework, an interesting theory of coherent hole burns (HBs) has proposed, where the EIT window is analyzed with four HBs at most when two laser beams, one saturating and other coupling field, drive a Doppler-broadened three-level atomic medium [37].Later on, the idea has expanded into a Λ-type Doppler-broadened atomic medium where coherent HBs are much distinct due to the resonant transition frequencies and thus more suitable for experimental realizations [38].Note that, SHB is mostly observed in solid state systems at room temperature [39].For instance, SHB in a three-level Λ-type ( 87 Rb) atomic system is studied both theoretically and experimentally, where different system parameters are used to control their widths, depths, and positions [4].Moreover, the phenomenon of SHB has validated through various experiments, which developed a Doppler free spectroscopy and settled the hyperfine structure of atomic and molecular lines [37,40,41].
The SHB has gained a great deal of research interest due to their potential applications.For example, the technique is used in atomic, molecular and solid state physics [42,43], which makes possible to build powerful devices for efficient optical information storage [44], high-resolution radio-frequency signal processing [45], and spectroscopy [46].The technique is also used for the optical tomography [47], multidimensional holography [48], programmable laser frequency stabilization [49,50], spectral spatial correlation [51], and photosynthesis [52].Furthermore, the remarkable research reveals that group velocity of light pulse depends on the parameters of the control fields [53][54][55].A usable HB can be generated by a coherent beam because of the coherent population oscillation [56].The suppression of decoherence through SHB has performed in hybrid quantum systems with simultaneously coupled spin states [57].To this end, the simultaneous use of coupling fields, allows us to observe multiple coherent HBs with controllable depths, widths, and positions in a Doppler-broadened absorption and dispersion spectrum.
In this connection, keeping in view the aforementioned practical applications of SHB, here we contribute by studying the switching from single to multiple HBs in a four level double Λ (tripod) system.The control over the depth, width, and frequency of HBs is thoroughly investigated, which are influenced by several parameters, for instance, the Doppler widths, average velocity of atoms in a cell, detunings, and the coupling field intensities.It is worth mentioning that here we consider a thermal medium wherein the SHB in the probe profile is subjected to the Doppler effect when a light buffer gas is also injected into the vapor cell of the medium under appropriate pressure.Usually noble gasses, such as, Xe or He, are added as the buffer gas to the atomic vapor cell.Following the experiment [58], we choose 2π × 20 MHz as the linewidth γ of the probe resonance as the unit of the relevant parameters of the system if Xe buffer gas is injected into vapor cell of the medium.By introducing a lighter buffer gas, coherence preserving velocity changing (CPVC) collisions could be produced between atoms of the buffer gas and atoms of the medium [59] under suitable pressure and temperature.At the probe resonance, this may create classes of high average atomic-velocity in the medium [60] upon the resonant interaction of the two strong coherent field with the medium, while dephasing rate in the atomic medium in vapor due to this effect could safely be neglected.In such a situation, as experimentally reported in [60], coherence phenomenon of the linewidth narrowing exhibits for a probe field absorption.In limit of saturation spectroscopy, we expect unprecedented behavior of controlled multiple SHBs when the medium of our system is investigated with classes of high average atomic-velocity.Note that, multiple SHBs may provide significant contributions in optical information and communication, multichannel optical switching, and optical information storage [59,60].
The remaining paper is organized as follows.In section 2, we present the model (tripod) system, its dynamical equations, and susceptibility and group index of the probe field.In section 3, we thoroughly discuss the quantum coherence phenomenon of EIT under two aspects of symmetric and asymmetric resonance of the tripod atomic system.Further, in section 4, the Doppler broadening and switching of multiple SHB under various considerations are analyzed in detail.Finally, the main results are summarized in section 5.

Model and dynamical equations
We consider a tripod four-level, double Λ-type, scheme with inhomogeneous broadening (see figure 1) which has various potential spectral characteristics.The transitions |1⟩-|4⟩ (frequency ω 14 ), |3⟩-|4⟩ (frequency ω 34 ), and |2⟩-|4⟩ (frequency ω 24 ) are respectively driving with coupling fields of Rabi frequencies and carrier frequencies ω 1 , ω 2 , and ω p .Here, E p and E 1,2 are the amplitudes of the probe and coupling laser fields, respectively, while µ 14 , µ 34 , and µ 24 are the dipole moments of the respective transitions.The atomic transitions may interact at off resonance with coupling fields showing the detunings ∆ 1 = ω 1 − ω 14 , ∆ 2 = ω 2 − ω 34 , and ∆ p = ω p − ω 24 .To derive the equations of motion of the tripod system, we start in the framework of semi-classical theory with the following interaction picture Hamiltonian in the dipole and rotating wave approximations Following the above equation with suitable transformation formalisms under the condition of slowly and fast varying amplitudes, the equations of motion for various density matrix elements are obtained as where γ 4 = γ 1 + γ 2 + γ 3 and γ i (i = 1, 2, 3) = γ are the population decay rates from the excited level |4⟩ to other levels.The atomic coherence decay rates between the upper level and ground levels γ 14 , γ 24 , γ 34 , are given by γ i4 = (γ 1 + γ 2 + γ 3 )/2.The dephasing decay rates γ 21 , γ 13 , γ 23 causes the decoherence in the warm atoms, which are usually much smaller than the population decay rates γ i .Such a tripod configuration can be found in many real atomic systems, such as alkali atoms.We consider Ω p up to the first order while Ω 1 and Ω 2 in all order of perturbation.We use the following relation for solving the above density matrix equations for ρ24 under the condition ρ11 + ρ22 + ρ33 + ρ44 = 1 and ρij = ρ * ji , and obtain the solution where The factor (ρ 44 − ρ22 ) is the population difference between levels |4⟩ and |2⟩.We further simplify equation ( 4) to with P = (∆ p + iγ 24 )(∆ p1 + iγ 21 )(∆ p2 + iγ 23 ).The population in the excited level is calculated as where

Susceptibility and group index
The response function due to an applied electric field is given by the susceptibility χ = χ R + iχ I of the medium.The real and imaginary parts of electric susceptibility for our cold atomic system are, respectively, written as where ℘ 24 is the dipole matrix element for the transition |2⟩ to |4⟩.For the Doppler broadening free set-up, we consider cold atomic medium.In most of Bose-Einstein condensate (BEC) experiments, quantum degeneracy for the atomic 23 Na prevails between 500 nK and 2 µK at the density between 10 14 and 10 15 [61].
In case of sodium D 1 line, the pre-factor β = N|℘24| 2 ϵ0ℏ , with N ∼ 1.3 × 10 14 cm −3 and ℘ 24 ≈ 2.492 × ea 0 , where a 0 is the Bohr radius, is about 2π × 9.89 MHz.In this system the pre-factor is associated with the decay rate γ.We consider it as natural unit for relevant quantities of our system when we the medium is free of the Doppler broadening effect.
A wave pulse is the superposition of many frequencies of waves, each with its own phase velocities (v = ω i /k i ).When a light pulse of frequency ω and bandwidth ∆ω incident on a dispersive medium of refractive index n g (ω), it propagates with group velocity v g = c/n g .The frequency dependent group index of a dispersive medium is where the last term shows the slope of the dispersion.The group velocity, however, can attain values of v g < 0 for normal dispersion with positive slope (∂n/∂ω > 0) and v g > 0 for anomalous dispersion with negative slope (∂n/∂ω < 0).

EIT and switching
In case of cold atomic medium with a BEC, Doppler-broadened effect is negligible as the atomic velocity of the medium is significantly small.The decoherence dephasing rates in this case are negligible hence the equation ( 10) is reduced to The positions of dark lines are shown in figure 2, which can be determined when absorption in equation ( 12) is set to zero.Thus two dark lines with three spectral components appear at positions of . In case of resonance (∆ 1 = ∆ 2 = 0) a single dark line is obtained at probe resonance (figure 2(a)), while in case of (∆ 1 = ∆ 2 = ∆ ̸ = 0) then the single dark line shifts to the position of ∆ (see inst of density plot in figure 2(a)).Likewise, by setting the denominator of equation ( 12) to zero, we obtain the positions of spectral components from the roots of the following expression The solution of this triplet equation is given in appendix.In a simple case of ∆ 1 = ∆ 2 = 0, the two roots appears at , while the third one collapse in the central dark line due to destructive interference.
The real and imaginary parts of the susceptibility are shown in the figure 2, where dark lines in the absorption spectrum are due to the quantum coherence phenomenon of EIT.The figure shows two aspects of the resonance of the tripod atomic system, which we call symmetric and asymmetric resonance.
Symmetric resonance.A symmetric resonance is produced due to the destructive interference effect in the absorption spectrum using the resonant coupling fields (∆ 1 = ∆ 2 = 0).At the resonance interactions, the EIT with single dark line in the absorption spectrum appears to be symmetrically doublet, like a typical Λ-type atomic system, as shown in figures 2(a) and (b).
Asymmetric resonance.Asymmetric resonance is developed in the spectrum when the couplings with the medium are set forth off the resonance.The off resonance interactions with different values of the detunings of the coupling fields switches to the behavior of double EIT (double Λ or triplet) system with two dark lines and three asymmetric spectral components (see figures 2(c) and (d)).The EIT1 window is always at probe resonance, while EIT2 window is at positive (negative) side when the frequency of the coupling field is larger (less) than the transition frequency.Furthermore, in case of equal values of the detunings other than zero, the system switches to the asymmetric single EIT, which can be seen in the density plot at the crossing point of the dark lines (see inset in figure 2(a)).This is the typical example of the Fano resonance.
Traditionally, Fano resonance in presence of detuned control field, provides the probe pulse with three indistinguishable paths for the excitation, which consequences the asymmetric profile of a single or double EIT in tripod medium.As a response, two dark lines develop in the spectrum due to destructive quantum interference effect among the excitation probability amplitudes.The degree of asymmetry in the present system depends mainly on the detunings of the coupling fields.The larger the detunings of the coupling fields with large difference from each other, the more enhanced is the degree of the asymmetry in form of a From this discussion, we depict that, with appropriate choices of the detunings, our tripod system switches, from symmetric doublet (single EIT of Λ-type) to asymmetric doublet and asymmetric triplet (Fano-like) resonance and vice versa [62,64].Thus the intensity and detunings of the coupling fields control the width of transparency window (spectral component), which is responsible for the steepness of the slope (∂n/∂ω) of normal (anomalous) dispersion which consequences time delay (advancement), see figures 2(c) and (d) for comparison.Furthermore, the increase in the radiative decay rates γ i =1,2,3 shows the decrease in height of the spectral components, and hence minimization of the EIT (see figures 2(a) and (b)).

Doppler broadening and SHB
The phenomenon of SHB depends on the velocity component of the thermal atomic medium, which induces Doppler effect.Hence, no HBs are observed in the cold atomic medium without Doppler effect, as can be seen in figure 2. When certain group of atoms with certain velocities v collectively interact with the probe and coupling fields by resonant absorption, coherent HBs are probable to generate in the probe absorption line due to population depletion.The higher the temperature, the greater is the chance of the Doppler effect, and hence it will be the evidence of HBs at higher rate due to depletion in the population.To account the Doppler effect in the system, we replace the detuning parameters ∆ 1 , ∆ 2 , and ∆ p in the susceptibility given in equations ( 9) and ( 10) by , and ∆ p + α p k p v, respectively.Here, α i =1,2,p = 1(−1) indicates co-propagation (counter-propagation) direction of coupling fields to the probe field, while k 1 , k 2 , and k p are, respectively, the wavevectors of the two coherent coupling fields and probe field, given by k i = ω i /c.After introducing the Doppler-broadened parameters, the susceptibility becomes where the expressions of D 1k , D 2k , and other variables can be calculated form equation ( 5) by replacing ∆ i by with and In contrast to homogeneous broadening, Doppler broadening is characterized by an atomic velocity parameter v that varies over the observed ensemble.In a thermal assembly of a gas at temperature T, the velocity distribution is obtained by the Doppler-broadened susceptibility χ(kv), which is the average over the Maxwell-Boltzmann velocity distribution, given by where V D is the Doppler width with K B is the Boltzmann constant and v p = √ 2K B T/M represents the most probable atomic velocity.Here, we consider k 1 = k 2 = k p = k for simplicity.
In the phenomenon of SHB, only the absorbing atoms have negligible velocity component along the axis of counter propagating beams, and hence temporarily depleted.This technique is generally used for laser frequency stabilization, e.g. with the iodine locked He-Ne laser.A specific atom with velocity v in the direction of probe field with wavevector k experiences the Doppler shifted frequency ω i − kv relative to a stationary atom, where the effective detuning ∆ eff = ∆ i + kv replaces the ∆ i .The atoms in the lower population (ρ 22 ), distributed according to equation (21), are now depleted from the velocity class around v = (ω p − ω 24 )/k.When the coupling field frequencies ω i are tuned, the hole sweeps over the atom's velocity distribution.The atomic response will saturate at the velocity group of the hole, which indicates the generation of HBs.The absorption spectrum is obtained by averaging the single atom response over the velocity distribution of equation (21).Since the equation of the population in the excited state ρ44 is identical to the atomic coherence ρ24 , thus the depleted atoms in the population show HB in the absorption spectrum.

Multiple SHB and switching
Here, we analyze and interpret the phenomenon of SHB in the dispersion and absorption probability spectra, given in equations ( 20) and ( 21) respectively, of a probe field during its propagation through a thermal medium of the tripod atomic system.In the limit of the saturation spectroscopy, we recall that a light buffer gas, for example, atomic Xe is injected into the vapor cell of the medium under appropriate pressure and temperature to the control the classes of average atomic velocity in the medium.The CPVC collisions may initiate between atoms of the Xe buffer gas and atoms of our medium [59,60].In such a situation, the probe field resonance, according to the experimental results, appears as γ = 2π × 20 MHz [58].In what follows, we will use γ = 2π × 20 MHz as the unit of the relevant parameters of the thermal medium.
In order to eliminate the Doppler broadening, for example, from Λ (Ξ) -type [2] or multi-level atomic medium [63,64], one can take the coupling fields counter (co-linearly) propagate to the direction of the probe field, the scheme then behaves as typical EIT.It is well-known that EIT allows us to obtain a steeper slope of the normal dispersion with large group index, however, it is usually necessary to cool the medium to very low temperature.The Doppler broadening is a temperature dependent incoherent effect which minimizes the EIT.We need a technique, in which it becomes possible to slow down the group velocity at room temperature.Here, we propose an alternate option, using coherent SHB technique, to create slow light with multiple channels even at room temperature.In a specific case, it behaves as an inverse Doppler effect, which can instead be used to enhance the coherence in the form of HBs and spectrally switch the slow light propagation.Because of the inverse Doppler shift, the transition frequency is different for atoms with different velocities along a given axis.
Resonance.At resonance of the coupling fields (∆ 1 = ∆ 2 = 0), two HBs appear with central EIT window in the absorption spectrum by introducing the Doppler effect provided that at least one of the coupling field counter propagates to the probe field, see figure 3.Under this condition, a coherent beam Ω 1 with detuning ∆ 1 − ω 41 v/c depletes a group of atoms in the population ρ44 with velocity v = γ/k 1 , such that, the phase of the velocity components are opposite to the probe field.At the same time, this group of atoms burns two holes (one each side) in the Doppler-broadened absorption profile of probe beam, which can be further understood as, there were two Autler-Townes absorption components [65,66] for that group of atoms.
Figure 3 further illustrates that the two HBs become deeper and deeper up to some constant value with increase in Doppler width V D at some velocity.The solid blue line shows a minor hole which increases gradually toward the dashed black line with Doppler width.Moreover, the widths of the HBs can be controlled by intensities of the coupling fields.A HB will be wider at relatively high intensity of the coupling fields, as shown in figures 3(c) and (d).The two HBs are asymmetric, with corresponding steep normal dispersion as can be seen in figure 3. The position of these HBs for a group of atoms with velocity v are given by ±(∆ i + (k p + k i )v), where i stands for the counter-propagating field.Consider k p = k 1 = k = 1 at resonance (∆ i = 0), for atoms with velocity v = γ/k i in (m s −1 ), the two HBs are observed at position of ∆ p = ±2γ (see figure 3).Thus, the positions of HBs depend on the detunings of the coupling fields and the atoms velocity in medium.We also note that the positions of these HBs on the frequency axis are static with respect to the increase in the Doppler width or the intensity of the coupling fields.
Off-resonance.The condition shows that a HB will be displaces by ∆ i toward positive or negative sides only in case of positive or negative detuning of the counter-propagating field, respectively.The detuning of the co-propagating field does not effect the position of HBs, it only causes the depths compensation of the EIT1 and the neighboring HB.This condition is illustrated in figures 4 and 5.
In the presence of the detunings ∆ 1 = 0.7γ and ∆ 2 = 0, the same behavior of the SHB appears with two EIT windows EIT1 at position of probe resonance and EIT2 displaces from centre by ∆ 1 = 0.7γ.According to the condition ±(∆ 1 + (k p + k 1 )v), the two group of atoms (with velocities v = 1γ/k, 2γ/k(m s −1 )) produce four HBs at positions of ±2γ, ±4γ, respectively, with a shift of 0.7γ toward the right side (figures 4(a) and (b)).The depth of the second window (EIT2) attenuates with detuning ∆ 1 , which is compensated by the depths of the neighboring HBs.The figures 4(c) and (d) also shows that the EIT1 window is coherently evolved into EIT2 window at equal but non-zero vales of the detunings.While in case, if both the detunings are set to zero then EIT2 window merged into the EIT1 window, as shown already in  the figure 3. Further, if we alternate the detunings, i.e. ∆ 1 = 0 and ∆ 2 = 0.7γ then the EIT1 window starts attenuation with compensation in its neighboring HBs (see figures 5(c) and (d)).Since the the detuning ∆ 2 only displaces the EIT2 window but does not change the position of HBs, so the increase in ∆ 2 causes to merge the EIT2 at the respective value of the HB (see figures 5(e) and (f)).Thus the two EIT windows can be switched coherently, with the help of the detunings of the coupling fields, as a consequence, the position and depths of the HBs can be controlled.
Effect of atom's velocity.The injection of Xe buffer gas into the vapor cell of the medium under appropriate pressure and temperature initiate the process of CPVC in the vapor cell.Therefore, the average atomic velocity for the different classes of the atoms in the vapor cell can be varied using appropriate pressure.These different classes of atoms increases the probability of depletion in the population difference.Since the SHB phenomenon is directly influenced by the velocity of a class of atoms in the population, thus multiple HBs appear at both sides for large velocities of classes of atoms (see black-dashed line in figure 5).
Further, for more clarity, we add some more results at probe resonance of the coupling fields.We show that by subjecting the medium to have large velocity classes of atoms v = 0.5, 1, 2, 5γ/k in (m s −1 ), the multiple HBs appear at positions of ∆ p = ±(1, 2, 4, 10γ), respectively, as shown in figures 6(a)-(d).Hence, in the absorption spectrum, we can switch from super-luminal to sub-luminal by creating a HB in the spectral component.We can also switch over from one channel to another channel with the help of multiple HBs.We note that, at relatively high Doppler width and large group velocity, large number of HBs are generated, where deeper HBs are closer to the central dark line, see black-dashed line in figures 6(c) and (d).The appearances of the multiple HBs are the consequences of the increase in the inhomogeneity of the atomic medium due to large random motions of the atoms with different classes of atomic velocities, which can be controlled with CPVC in the atomic medium with changing pressure or temperature of the medium.
Further control on the velocity of atoms can be achieved by introducing the buffer gas into the vapor cell at different density.Collisions with the buffer gas atoms, changes the velocity and slow the diffusion rate of the subject atoms through the cell.This may be intended to reduce the frequency of collisions with the cell walls.Usually noble gasses, such as, He, Ne, or Xe are added as buffer gas to atomic vapor cell containing the subject species.
Effect of Doppler width.From figures 3 and 4, we depict some interesting results with respect to Doppler width v D .On one hand, the increase in Doppler width suppresses the coherence phenomenon in the shape of minimization in EIT and broadening in the absorption spectrum as expected.On the other hand, it dramatically enhances the depths of HBs which is the cancellation of the absorption in the spectral region.Thus in the present atomic scheme, the Doppler broadening plays an opposite role, the ultra-narrow HBs feature takes place just in the presence of the inverse Doppler shift when the average atomic velocity for the different classes of the atoms in the vapor cell increase.The absorption and dispersion properties in this scheme are strongly dependent on the propagation directions of the applied fields.
Moreover, we study the effect of the dephasing decay rates in the thermal environment on the behavior of the HBs.In addition to the Doppler effect, we observe that the dephasing decay rates which is a de-coherence effect and broaden the HBs in the absorption spectrum and hence suppression of the coherence phenomenon as expected.When the dephasing decay rates becomes too larger, the destructive interference is converted into the constructive interference, as a consequence, the HBs with EIT washed out in whole spectrum of the absorption which shows the behavior of Lorentzian, see figures 6(e) and (f).
Further we analyze the effect of the radiative decay rates on the behavior of the HBs in the presence of the inverse Doppler effect.We observe that the radiative decay rates being a de-coherence effect broaden the absorption spectrum as expected, while in contrast they narrower the spectral HB regions with more steeped dispersions, see figure 7. The narrowing of widths of the HBs with the decay are also because of the inverse Doppler effect.
Finally, we check the appearance and disappearance of the HBs in the absorption and dispersion profile for the direction of propagation of the coupling fields.We note that the HBs are possible only when either one or both of the coupling fields counter propagates to the probe field, which causes the destructive interference at some positions where holes are burned at these positions.In figure 8, both the coupling fields co-propagate, with α 1,2 = 1, to the probe field and hence HBs disappear in the absorption and dispersion profiles.

Conclusion
In the limit of saturation spectroscopy, SHB is another vivid phenomenon which provides with an alternative way to pass a probe light through a medium with numerous advantages and minimal probability loss.In this paper, we have studied SHB in the four level-tripod atomic medium, and investigate their positions, width, depths and numbers in a well controlled manner in the absorption and dispersion spectrum of light pulse propagation through the medium.
We have shown that, the HBs are observed only for the counter propagation of either one or both the coupling fields in a Doppler-broadened medium.Below some critical temperature, the HBs are not observed which is the condition for EIT.An abrupt depletion in the absorption spectrum is observed at a selective probe detuning with one spectral hole at each side of the central dark line.This is because of an inverse Doppler broadening effect in the population difference, which causes the depletion of atoms in the population at the positions of HBs.The HBs dip (the lamb dip) increases up to some constant value with the relative increase in the Doppler width, which causes the increase in the inhomogeneity of the atomic medium.The increase in the Doppler width enhances the HBs phenomenon, which is further enhanced and controlled by the velocity of atoms.The velocity of atoms increases the numbers and position of HBs.The widths can be controlled by the intensity and detunings of the coupling fields.Therefore, single pair HBs can be switched to multiple pairs of HBs in a well controlled manner.The most interesting and significant result is that in contrast to EIT, the SHB phenomenon enhances when the process of CPVC is initiated for different classes of the atoms average velocity with injection of a light buffer gas under suitable temperature or pressure.We also noticed that the radiative decay rates can also affect the shape of HBs.Further, the two EIT windows appeared in the presence of different values of detuning of the coupling fields.The slope of dispersion is observed steeply anomalous in the spectral HB regions.Hence, the SHB enhances sub-luminality (super-luminality) in the HB regions (between the two HB regions).
Our current study on the generation and control of coherent spectral HBs may provide potential applications in various fields of optical technologies.On one hand, the delayed propagation pulses have many useful applications in high resolution spectroscopy and frequency stabilization.On the other hand, multiple holes can be burnt in various regions of the inhomogeneous absorption spectrum which can slow down simultaneously many pulses with frequencies coinciding with the frequency of these holes positions.Such a coherence controlled SHB could be useful in on-chips functional photonics circuits, multichannel optical switching, and optical information storage devices.

Figure 2 .
Figure 2. χI and χR vs ∆p/γ (γ = 2π × 10 MHz of the probe linewidth of the atomic 23 Na D1 line in the absence of Doppler broadening effect), in the absence of Doppler effect and without detuning in (a), (b) and with detuning in (c), (d).The inset of (a) shows the density plot of χI vs ∆1 and ∆p/γ, with ∆2 = 1.5γ.

Figure 6 .
Figure 6.χI and χR ∆p/γ, for different atom's velocity with dephasing rates shown inside the figures.Other parameters are same as used in figure 3(c).

Figure 8 .
Figure 8. χI and χR vs ∆p/γ, here both the coupling fields are co-propagate to the probe field.Other parameters are same as used in figure 3(c).