{'fraction_non_alphanumeric': 0.02798687512063308, 'fraction_numerical': 0.011066074760342276, 'mean_word_length': 4.5893563466379, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 33, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Formation of field induced absorption in the probe response signal of a four-level V type atomic system a theoretical study

A density matrix based analytical model is developed to study the coherent probe field propagation through a four-level V type system in presence of a coherent control field. The model allows coupling of the probe field from the upper ground level to both of the excited levels keeping the control field locked to a particular transition. The addition of an extra ground level to a conventional three-level V type system creates extra decay paths to the ground levels for the upper level population. A set of sixteen density matrix based equations are formed and then solved analytically under rotating wave approximation to study the probe response under steady state condition. The simulated probe absorption spectra shows absorption dip at the centre of a transparency window only under Doppler broadened condition although the conventional EIT window appears under Doppler free condition. The dependence of the field induced absorption signal on the Rabi frequency of the control field, population transfer rate among the ground levels and temperature of the vapour medium has been studied in details.


Introduction
Electromagnetically induced transparency (EIT) [1][2][3][4][5], a prime example of quantum interference effect, has been the centre of attraction to a large section of researchers who are engaged in the field of quantum optics. When the frequency difference between the control and the probe beams is set equal to the separation between the ground levels or two excited levels of a multi-level system under consideration, EIT occurs [2] and an otherwise absorptive medium becomes transparent to a resonant (or detuned) low power coherent probe beam in presence of a coherent high-power control beam due to the formation of 'dark state'.
Alternatively, depending upon the choice of level schemes, this may also be explained on the basis of the splitting of the energy level, which is common to both the probe and the pump beams, as seen by the probe beam [2]. The reason behind the worldwide attraction of EIT among researchers is of course the potential application of EIT in developing future optical devices that are supposed to be used in optical logic gates, all optical switches [6,7], optical delay generators [8] etc. In addition to this, pure academic interest is also driving researchers to investigate this effect using various level schemes like inverted-Y [9,10], four-level cascade [10], M and N-type systems [11,12] etc. The effect of thermal velocity averaging also influences the EIT line shape in a way so as to reduce its width compared to the Doppler free condition. The Doppler broadening affects the dispersive properties of a medium substantially. The extremely low line width of the EIT window under Doppler broadened regime corresponds to a very steep dispersion that results in very low group velocity of a probe pulse passing through a medium in presence of high power control field. This phenomenon points towards a possibility to manipulate the probe pulse propagation through an atomic vapour medium. In fact, the theoretical prediction [13] has been well supported by the experimental demonstration of reduction of the group velocity of a probe pulse propagating through hot 87 Rb vapour [14]. The deceleration and storage of a light pulse in an atomic vapour and then its release on demand ultimately fulfilled the dream of stopping and storage of light. The observation of EIT is most easily realized in a three-level Λ type system [15] although in reality it is almost impossible to form a true three-level Λ type system even in the alkali atomic vapour medium. All most all alkali atoms have complex multilevel atomic structure. Many reports in this regard can be found in the literature. Vasant Natarajan et. al. [16] has demonstrated experimentally how the Doppler averaging in a rubidium vapour cell at room temperature reduces the EIT line width to sub-natural value for a Λ type system.
They have also presented a simple theoretical model in this connection. Very recently D.
Bhattacharyya et. al. [17] showed the formation of EIT with sub-natural line width in a sixlevel Λ type system in 87 Rb as well as 85 Rb vapour at room temperature. An analytical model has been presented there to explain the sub-natural line width of the observed EIT signal.
They have also stated there that the inclusion of all the hyperfine levels in the theoretical modelling has helped them to reproduce the experimentally observed spectra. But in threelevel 'V' and cascade type systems too EIT can be produced. D. J. Fulton et. al. [15] reported a comparative study of EIT in 'V', Λ and cascade (Ξ) type systems along with their experimental findings. They have explained why the Λ type system is the most well suited to study EIT on the basis of coherence dephasing rate between the dipole-forbidden transitions.
There has been very large number of reports on both the theoretical and experimental studies on EIT [15-17 and the references in 17] as well as electromagnetically induced absorption (EIA) [18,19] using Λ type level scheme. Although the three-level systems serve well in understanding the physical picture behind formation of EIT, these are hard to be found in reality. Formation of hyperfine levels in atoms almost everywhere makes the level scheme complicated enough to consider more than three energy levels in the theoretical models in order to explain the experimentally observed spectra. In this report, we shall present a theoretical investigation on the propagation of a resonant weak probe field through a fourlevel 'V' type atomic system in presence of a strong coupling field known as the pump field or control field. The treatment is completely analytic and the simulated probe absorption spectra exhibits formation of a field induced absorption dip on the background of an EIT peak under the Doppler broadened regime whereas under Doppler free condition a transparency window in the probe absorption profile is created. We shall also demonstrate how the different velocity groups of atoms contribute to the probe absorption profile and form the absorption dip. The dispersion property of the system will be studied too.

Theoretical Model:
We shall now describe the four-level system in details. There are two ground levels and two close upper levels. We have set the separation between the two excited states |3> and |4> to be equal to 266.6 MHz and that between the ground levels as 6.8 GHz following the energy level of 87 Rb [20]. The frequency of the probe beam is scanned from the upper ground state |2> to the excited states |3> and |4>. Hence it couples both the upper states |3> and |4> to the ground level |2>. The frequency of the pump beam is kept locked between the levels |2> and |3>. The coupling of the probe beam to both the upper levels makes the formulation different from the conventional 'V' type system where the probe field couples one of the upper states from the ground level and the control (pump) field acts between the ground level and the other upper level. The spontaneous decay of the population from the uppermost level |4> to the ground level |2> is dipole-allowed whereas that of population from the uppermost level |4> to the lowest ground level |1> may or may not be dipole-allowed. We assume that the population of the level |3> can always decay spontaneously to both the ground levels |1> and |2>. The provision of population transfer between the two ground levels has been kept in the theoretical formulation so that the collision induced transfer of population among the ground levels may be taken into consideration if so desired.
Here the control and probe Rabi frequencies are defined by = μ ij. E c /ℏ and = μ ij. E p /ℏ.
, stands for the atomic decay rates. The atomic transition operators are given by + and and they are complex conjugate of each other [21,22]. The population and the coherence terms are (diagonal and off-diagonal elements of the density matrix respectively) obtained by solving the following equations and their complex conjugates analytically following rotating wave approximation [21,22]: We have considered, subject to the boundary condition + + + = 1. The probe absorption and dispersion are determined by using the imaginary and real parts of the coherence terms induced by the probe between the levels |2> and |3> ( ), |2> and |4> ( ) [17,21,22]. The analytical expression of the and can be derived by solving sixteen optical Bloch equations (OBE) under steady state condition.
The nature of probe absorption and probe dispersion can be obtained by simulating ( + ) and then plotting its imaginary and real parts separately as functions of probe detuning ( ).

Simulation:
The analytical expressions for ( + ) given in Eq. (12) and Eq. (13)   between the control and probe fields equals the frequency difference between the upper levels |3> and |4> (Fig.3). This is the usual EIT signal.   A transparency like peak at = − 266.6 MHz is generated in the Doppler broadened probe response spectra (Fig.2) but the Doppler free condition shows only an absorption dip at this value of the probe detuning. At this condition, both the pump and the control fields are onresonant with the |2> → |3> transition. The probe absorption decreases here since the control field creates a hole in the population of the zero velocity groups of atoms, hence the probe field would see less population to interact with. This is similar to saturation absorption spectroscopy and formations of Lamb dip [23]. The width of this EIT like peak created under Doppler broadened condition is seen to increase with increase in the control Rabi frequency (Fig.2).The formation of an absorption dip on the background of a transparency window in the probe absorption spectra and the corresponding shift with increase in the control Rabi frequency are present only under Doppler broadened regime (please compare Fig. 2 and Fig.   3 around = 0). These features cannot be observed in the probe absorption spectra for a simple three-level 'V' type system [24]. The presence of an extra ground level (|1> in this case) provides an extra decay channel to the population pumped to the upper levels. To get an idea of the contribution of the non-zero velocity groups of atoms towards the probe response we have plotted imaginary part of ( + ) vs. probe detuning ( ) for specific velocity groups ( Fig.4). At zero probe field detuning, the contribution towards the probe absorption by different velocity groups adds up to create the observed absorption dip in Fig.2. If we vary the temperature of the ensemble, we observe variation of the width of the field induced absorption (Fig.6). We have also shown the zoomed absorption dip at different temperature in the inset of Fig.6 below.   A linear shift of the field induced absorption dip with increase in the value of the Rabi frequency of the control field is observed (Fig.8 below) although the fitted width of this dip does not show any linear dependence on the control Rabi frequency (Fig.9 below). The probe Rabi frequency is kept fixed at 1 MHz. The temperature of the system is kept at 300 K. Fig.8 just quantifies the observation made in fig.2 in this regard. The shift of the absorption dip with increase in the control Rabi frequency is attributed to the AC Stark shift [25]. We have not given any analytical expression for the width and shift of the peak since the mathematical expression for the probe response ( + ) is very complicated (Eq.12 and Eq.13). It is very difficult to derive any simple expression for these parameters from the algebraic expressions written in Eq.12 and Eq.13 without any approximation. We have tried to avoid the use of approximation throughout the treatment as far as possible and relied on the graphical representation of the analytically obtained probe response.

Control Rabi frequency ( c ) in MHz
Data points Fit to data points  It is evident from the above figure that with increase in the Rabi frequency of the control field the probe dispersion will get flatter and flatter. Under Doppler broadened regime the steepness of the dispersion curve however decreases with increase in the Rabi frequency of the control field, a feature just similar to that of Doppler free condition (please compare Fig.10 and Fig.11). A kink like structure is appearing in the dispersion curve around the zero probe field detuning ( = 0, fig. 11 below). This kink becomes prominent at higher intensities of the control field. This has been shown in the inset of Fig.11 below. This kink modulates the probe dispersion around zero probe detuning. This may have useful application in studying the probe pulse propagation through such an atomic vapour system. This will be quantified in a future correspondence. In the inset of the above figure we have shown the zoomed probe response around zero probe field detuning. If this is compared (Fig.13 below) with the zoomed part of Fig.2 we can see that the absorption dip starts appearing when the Rabi frequency of the control field is increased to 30 MHz and beyond but the absorption dips are not as prominent as in Fig.2 We have shown in the Fig.13   If we investigate the effect of the variation in the population transfer rate between the two ground levels |1> and |2> on the probe response signal, we can have a better understanding of the physical phenomenon responsible in creating the absorption dip. We have already shown that the formation of this absorption dip is related to Doppler broadening (please compare Fig.2 and Fig.3, Fig.3 does not show any enhancement in the probe absorption in the EIT background at zero probe detuning whereas we observe absorption dip in the transparency background at zero probe detuning when Doppler broadening is taken into account in the simulation). If we still keep = = = 6 MHz and = 0 and vary the population transfer rates ( and ) between the ground levels (|1> and |2>) we can clearly see that the dip appearing at the zero probe detuning is becoming more and more prominent with increase in . This is shown in the fig.14 below along with the statistics of fits in table-1.
For each of the graphs in Fig.14    The availability of more atoms in the level |2> means these atoms would interact with the coherent fields and thereby reduce the transparency at the zero probe detuning (under Doppler broadened regime) when the population transfer rate from |1> to |2>is increased. In

Conclusion
We has also been investigated in details. In fact, to the best of our knowledge, formation of an absorption dip on the transparency background in a four-level 'V' type system is shown here for the first time through a complete analytical treatment. The approach has been to incorporate the experimental conditions to the closest possible way in the simulation. A scheme is also presented on the basis of the outcome of this theoretical study for producing a dispersive medium where rapid manipulation of the group velocity of the probe pulse propagation can be achieved within a very small range of probe frequency detuning. The model can also be used to study the effect of dephasing and other collision related population transfer phenomena among the ground levels on the probe response. We believe that this simple analytical model will be useful in probing the 'V' type system in further details in the future and exciting new results can be obtained.