Probing, Quantifying and Freezing Coherence in a Thermal Ensemble of Atoms
Arif Warsi Laskar Niharika Singh Pratik Adhikary Arunabh Mukherjee Saikat Ghosh


Probing, Quantifying and Freezing Coherence in a Thermal  Ensemble of Atoms]Probing, Quantifying and Freezing Coherence in a Thermal  Ensemble of Atoms

[]gsaikat@iitk.ac.in
^1Department of Physics, Indian Institute of Technology, Kanpur-208016, India



	
	

Creating stable superposed states of matter is one of the most intriguing aspects of quantum physics, leading to a variety of counter-intuitive scenarios along with a possibility of restructuring the way we understand, process and communicate information. Accordingly, there has been a major research thrust in understanding and quantifying such coherent superposed states. Here we propose and experimentally explore a quantifier that captures effective coherent superposition of states in an atomic ensemble at room-temperature. The quantifier provides a direct measure of ground state coherence for electromagnetically induced transparency (EIT) along with distinct signature of transition from EIT to Autler-Townes splitting (ATS) regime in the ensemble. Using the quantifier as an indicator, we further demonstrate a mechanism to coherently control and freeze coherence by introducing an active decay compensation channel. In the growing pursuit of quantum systems at room-temperature, our results provide a unique way to phenomenologically quantify and coherently control coherence in atom-like systems.



		
				



[
    Arif Warsi Laskar^1, Niharika Singh^1, Pratik Adhikary^1, Arunabh Mukherjee^1, and Saikat Ghosh^1
    December 30, 2023
=====================================================================================================






Ability to generate, probe and control superposed states of physical systems provide distinct technological advantages in quantum protocols, when compared to their corresponding classical counterparts <cit.>. Even entanglement <cit.>, a critically important resource in quantum information, relies on superposed states of distinctly measurable channels. Over last few decades, there has therefore been a tremendous thrust in research, to better quantify such states theoretically <cit.>, and to generate and control them experimentally <cit.>. A widely used technique to generate superposed states in atom-like systems <cit.> is based on the phenomenon of electromagnetically induced transparency(EIT) <cit.>, where a strong control field is used to drive an  effective 
three level atomic system into a particular coherent superposition of ground-state sub-levels (|1⟩ and |2⟩, Fig. <ref>a), known as dark state, in  presence of a weak probe field. Such superposed dark states remain mostly decoupled from the lossy excited state (|3⟩) <cit.> leading to dramatic effects such as slow <cit.>, stopped <cit.> and stored <cit.> light, generation of entangled photons <cit.> and enhanced optical non-linearities at the level of single photons <cit.>. With such broad applicability <cit.>, harnessing superposed states to their complete potential requires a concrete means of quantifying the corresponding coherence.


Traditionally, superposition in EIT is characterized spectroscopically, through its signature transparency window in probe absorption profile (Fig. <ref>b). However, it is also well acknowledged that such a transparency is not necessarily a unique signature of superposed states, but can also appear due to the strong control field either optically pumping atoms out of the Λ system <cit.> or hybridizing the ground state with the excited state, leading to Autler-Townes splitting (ATS) in the absorption profile <cit.>. Early efforts to discern EIT from optical pumping effects relied on characterization of the transparency linewidth by fitting convolved absorption profiles <cit.>. Several recent works have explored the more subtle issue of discerning EIT from ATS using Akaike information criterion that provides a quantitative indicator based on Bayesian comparison of probe absorption profiles <cit.>.  Nevertheless, such techniques, based on post-processed spectroscopic data are cumbersome and do not directly quantify the useful ground state superposition. Experiments showing the storage and retrieval of light pulses provide an alternative characterization of ground state coherence via storage time and retrieval efficiency <cit.>. However, these parameters are also common to other mechanisms, including storage based on photon echo <cit.> or coherent population oscillation <cit.>. In particular, for comparing dynamically varying or coherently controlled superposed states, a direct experimental quantification of coherence, characterized by the off-diagonal matrix element ρ_12, can significantly boost applications in EIT.

 
Here we propose a phenomenological quantifier that accurately captures ground state coherence in EIT for an ensemble of rubidium (^85Rb) atoms at room temperature. The quantifier is based on single-shot time domain measurement of dynamical probe susceptibility and relies on the vastly differing classical and quantum time-scales in the system <cit.>. We experimentally demonstrate that with decreasing coherence, the quantifier decreases monotonically. Furthermore, with increasing control field strength, we observe emergence of a distinct splitting in the otherwise resonance peak of the quantifier, as a function of two-photon detuning. Such distinct spectroscopic signatures in the two regimes of EIT and ATS greatly simplify identification of the transition. Finally, using this quantifier as a tool for directly probing ground-state coherence, we propose and demonstrate a novel technique to phase coherently control and compensate its decay using an additional pair of fields. Our work complements growing theoretical initiatives in quantifying coherence in quantum systems <cit.> while the phase dependent control of coherence demonstrated here can be applied to improve the ability to store, process and retrieve quantum information in a variety of systems that use EIT to generate stable superposition <cit.>. 

   
  
The primary motivation for this study is based on the observation that in a three level atomic system (Fig. <ref>a), the probe transmission is driven by a part due to ground state (quantum) superposition along with another part due to (classical) population dynamics, both adding linearly.  This is particularly evident in the off-diagonal density matrix element (ρ_13) that drives the probe transmission and takes a form (see Supplementary Information)

    ρ^ss_13=[-i Ω^∗_cρ^ss_12-iΩ^∗_p(ρ^ss_11-ρ^ss_33)]/Γ_13.

in steady state (ss), where Γ_13 = iΔ_p +γ_ex/2 with Δ_p being the probe detuning and γ_ex includes decay channels out of the excited state. While the ground state quantum superposition ρ^ss_12 couples via the control field (with Rabi frequency Ω_c), there is also a part that is set by steady state populations ρ^ss_11 and ρ^ss_33, modified from their thermal equilibrium values due to optical pumping by the strong field and losses. This modified one-photon population term adds to probe transmission, leading to ambiguities in identifying useful ground state coherence ρ^ss_12 from probe spectrum. 
However, it can be observed that in absence of control, steady state populations equilibrate via thermal diffusion through probe in a time scale that is significantly long (∼ 10 μ s for a probe beam diameter of 4 mm) compared to atomic time-scales (∼ 30 ns). By adiabatically turning off the control in an intermediate time-scale and taking the difference in response, one can thereby subtract out this one-photon contribution. Accordingly, we define a phenomenological quantifier C, such that

    C=|ρ^Ω^on_c,ss_13-ρ^Ω^off_c,ss_13|/|ρ^Ω^off_c,ss_13||Ω_p|/|Ω_c|

where  ρ^Ω^on_c,ss_13 and  ρ^Ω^off_c,ss_13 are the probe response in presence and absence of control. For an intermediate control turn-off time scale of ∼ 150  ns and in the limit of ρ_11∼ 1, the quantifier C is then proportional to ρ_12. We set up an experiment to test the validity for the assertion of C as a quantifier of ρ_12 in different scenarios and compare observations with toy models and simulation results.

  
  
Experiment

We probe a thermal ensemble of ^85Rb atoms in a vapor cell, along with coils to apply a small magnetic field in the probe propagation direction (Fig. <ref>c). The hyperfine manifold, F=2 forms the two ground states (see Methods and Supplementary Information). The control (of 10 μ s pulse width) and probe (cw) fields of opposite circular polarization are derived from a single laser and are near resonant (Δ_p = Δ_c = 3.2γ_3) with the excited state manifold F'=1 (see Methods for details). When a small magnetic field is scanned across resonance, sharp two-photon Raman resonance peak is observed in probe transmission, a typical signature of EIT (Fig. <ref>b).


The transmitted probe intensity is recorded in time, while the control is adiabatically turned on and off (in ∼ 150 ns >> 1/γ_3) with acousto-optic modulator (AOM), keeping it on for long enough times (10 μ s) for the system to reach steady state. Furthermore, we experimentally simulate a quasi-closed three-level system with a counter-propagating continuous repumper field (resonant and locked at a transition F=3⇒ F'=3), that cycles back any atom escaping out of the F=2 manifold to F=3 <cit.>.

A typical experimental trace for the system (along with numerical simulations for a closed three-level system) is shown in Fig. <ref>b. With turn-on of control, there is a sharp rise a-b, followed by a slower decay b-c. In a recent work <cit.>, we have shown experimentally, that from the scaling of time scales with field strength, it can be inferred that there is an initial fast build up of ground state coherence ρ_12, followed by optical pumping, rearranging the populations ρ_11 and ρ_22. 

It is particularly intuitive to visualize this dynamics in an effective Bloch sphere corresponding to the ground state manifold spanned by |1⟩ and |2⟩. With an initial unpolarized ensemble (equally populated thermal states) corresponding to zero length, the Bloch vector grows in a convex path, first building up coherence along equatorial plane and then moving towards the pole due to optically pumped population imbalance (Fig. <ref>a). For an ideal EIT scenario with all the atoms in |1⟩, steady state vector points mostly down, with a slight tilt due to a small coherence, ρ_12∼ -Ω_p/Ω_c. It can also be noted that though ρ_12 depends on the relative phase between control and probe, setting the angle of the tilt in the equatorial plane, ρ_13 is independent of it (due to the additional excited state |3⟩). We use this visualization later for phase coherent control of ρ_12.

At control turn-off, there is a sharp fall in probe transparency going below the initial level (Fig. <ref>b), indicating a new steady state population difference ρ_11-ρ_22. We use the fall height from point c(h1) to d(h2), to estimate the quantifier

    C =h1-h2/h2|Ω_p|/|Ω_c|.


 

 In contrast, in absence of the repumper field, atoms escape out of the three level manifold and the system becomes open. This corresponds to traditional EIT experiments and experimentally, in absence of repumper, we observe a partial drop with a corresponding smaller C_ open (Fig. <ref>c). To test a scenario where the system is incoherent, we use a counter-propagating control field. Here one expects the Doppler averaging to wash out any built-up coherence in the system. Experimentally, we observe C_ incoherent = 0 (Fig. <ref>d). 
 
The long rise time in Fig. <ref>d corresponds to optical pumping, while the long time scales after control turn-off in all the three scenarios (Fig. <ref>b,c,d) correspond to thermal diffusion of atoms (∼ 10 μ s) which can be modelled using a simple rate equation picture <cit.>. 

As a first test, we therefore conclude that the defined quantifier behaves monotonically, with 

    C_ closed>C_ open> C_ incoherent


Transition from EIT to ATS 

As a second test, we probe C with increasing control intensity. At large control fields, it is well-known that the system hybridizes in a fragile superposition of ground (|2⟩) and excited (|3⟩) states, with corresponding Autler-Townes splitting <cit.>. In such hybridized basis (Fig. <ref>a), the corresponding low-field strength EIT regime is usually understood as a Fano resonance <cit.>, which vanishes monotonically with increasing field strength.

One therefore expects a corresponding signature of splitting in probe transmission <cit.>. However, due to power broadening, such a signature is not always discernible  <cit.>. The situation is particularly severe in an ensemble of hot atoms, where a large Doppler broadened one-photon background profile washes out any signature of ATS (Fig. <ref>b, inset).

However, the quantifier C precisely subtracts out this one-photon background (i.e. h2 at d in Fig. 2a), corresponding to the term ρ^ss_11-ρ^ss_33, which hardly evolves during control turn-off. C can thereby be viewed as distilling the information of coherent superposed states, ρ_12 in the system. Experimentally, it captures excellently a splitting in the corresponding two-photon resonance peak, a signature otherwise un-discernible in a steady state measurement (Fig. <ref>b and inset). The behavior matches well with a wave-function model (Fig. <ref>c), where C decreases monotonically with increasing control field strength (Fig. <ref>d).



Freezing coherence:

Along with magnitude, there is also an absolute phase of ρ_12 which remains undetected in EIT and can only be characterized with respect to a reference phase. An additional field, usually a radio-frequency field, coupling the ground states can provide such a reference <cit.>. Instead, here we use a a pair of far-detuned, two-photon resonant, counter-propagating Raman fields. While large detuning (Δ_R in Fig. <ref>a) ensures a build up of two-photon coherence with minimal population reshuffle, counter-propagation of the fields washes out higher-order multi-photon effects, simply adding a perturbative correction to the coherence, in the form (see Supplementary Information):

    ρ_12=-Ω_cΩ^*_p/|Ω_c|^2+i^*_RΓ_3/|Ω_c|^2.

where Γ_3=γ_3/2-iΔ_p. The resulting, modified ρ_12 is now sensitive to the phase difference Δϕ  between control-probe and the Raman fields  Ω^- and Ω^+ through the effective two-photon Rabi frequency Ω_R=Ω^+Ω^-e^iϕ_R/Δ_R <cit.> where ϕ_R is the phase difference between Raman fields (Fig. <ref>a).  

Experimentally, we use an additional counter-propagating laser for a closed system (with repumper), locked at a programmable detuning with respect to the probe (Methods and Supplementary Information). While  linear polarization generates both Ω^+ and Ω^-, rotation of the polarization with a half-wave plate changes their relative phase difference (ϕ_R). By changing ϕ_R, we experimentally observe sinusoidal variation of the total coherence (Fig. <ref>b).

Such a reference field controlled EIT coherence offers an intriguing consequence. For the closed system, we have observed (Fig. <ref>b) that an initial large build up in ρ_12 (segment a-b) eventually decays down (segment c-d) due to optical pumping by control field. With the reference Raman fields as an additional channel, one can now compensate for this decay.  Notably, at phase difference ϕ_R ∼120^o (ϕ_f), coherence C freezes to its initial maximum (Fig. <ref>c). In particular, the phase dependence (Fig. <ref>b) indicates that one can over compensate with phase, with the visibility (Δ C) encompassing the difference between initial and final steady state coherence (Δ C_o, corresponding dotted lines a-b and c-d, respectively). Furthermore, with increasing detuning Δ_R, the visibility decreases (Fig. <ref>d). These observations match well with simulation and a simple rate equation model accounting for far-detuned optical pumping effects of the Raman fields (Fig. <ref>d, inset and Supplementary Information).




Conclusions

To conclude here we have demonstrated a phenomenological quantifier for ground state coherence in an atomic ensemble for EIT at room-temperature. The quantifier is based on a single shot time-domain measurement of probe susceptibility and relies on the differing time-scales for classical and quantum dynamics in the system. A variety of platforms with such widely differing time-scales, including ensembles of cold atoms <cit.>, trapped ions <cit.>, defect centres in diamond <cit.>, synthesized or fabricated quantum dots <cit.> and rare-earth doped solid state materials <cit.> are currently being pursued for quantum devices. While, traditionally, steady state spectroscopic signatures have been usually used to claim coherence in them <cit.>, here we show that time domain measurements can be used to settle observational ambiguities. Accordingly, this work complements recent efforts of quantifying many-body entanglement through dynamical susceptibilities <cit.> and experimental efforts to store and retrieve light based on EIT <cit.>. While in the later case, the retrieval efficiency provides an indirect measure of the left-over coherence, here we provide its more direct quantification. We believe that along with the quantifier, the mechanism presented here to phase coherently control and freeze coherence can be used to enhance performance of quantum devices at room-temperature. 



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Acknowledgements


We thank H.M. Bharath, B. Deb, H. Wanare, K. Saha and G.S. Agarwal for insightful discussions and comments. We also thank Om Prakash for his numerous help in construction of the experimental setup. This work was supported under DST grant no: SERB/PHY/2015404. 





[^ Present address: The Institute of Optics, University of Rochester, New York-14627, USA]


Methods


We employ a degenerate Λ system formed within  D_2 transition of ^85Rb atoms as shown in Fig. <ref>a, where |1⟩≡|F=2, m_F⟩,  |2⟩≡|F=2,m_F-2⟩ and |3⟩≡|F^'=1, m_F-1⟩. Two orthogonal circularly polarized beams: a continuous σ_- probe and a pulsed σ_+ control beam are used to drive the transitions |1⟩→|3⟩ and |2⟩→|3⟩ respectively. These are derived from a single laser ECDL1, locked at red detuning of 19 MHz with respect to |F=2⟩→|F'=1⟩. In addition, we use an incoherent continuous repumper beam derived from ECDL2 locked at |F=3⟩→|F'=3⟩ to experimentally simulate a closed Λ system. A rubidium vapour cell at room temperature, of length 8 cm and diameter 2 cm is used as the atomic media. The cell is shielded with three layers of μ-metal sheets along with magnetic coils to cancel any stray magnetic field.

An additional phase dependent coherence is introduced in the medium to control the effective ground state coherence. This is realized by sending two orthogonal circularly polarized Raman beams through the medium, counter propagating to the control. These beams are derived from a single linearly polarized laser ECDL3. The phase difference between these beams ϕ_R is controlled by changing the polarization of the linearly polarized beam with a half-wave plate.  

 The beam diameter of all the lasers used is ∼4 mm.  Frequency of the beams is stabilized by beat note offset locking technique (see Supplementary Information for details). Laser pulses are controlled with field-programmable gate array (FPGA) card (Opal Kelly XEM-3001) and acousto optic modulators (AOMs). Control and Raman coherent beams are kept on for 10 μ s (with adiabatic turn on/off time of ∼150 ns). Repetition time of the entire experiment is 50 μ s (it takes about 30 μ s for the atomic population to relax back to thermal equilibrium after control turn-off). 


A schematic of the experimental setup is shown in Fig. <ref>c. The repumper and Raman beams are sent counter-propagating to the probe and control through the vapor cell.  A Glan-Thompson polarizing beam splitter is used to separate the control and probe beams after the cell. Probe beam is detected with a high speed, low noise and amplified photo detector (PDB450A). 



























    
    
[figure]format=default,justification=RaggedRight,labelfont=bf,labelformat=default,labelsep=none,name=Figure


 



Supplementary Information



		
				


 
 




S.I. Experimental techniques



 (a) Magnetic Field Shielding

Ground state coherence, responsible for electromagnetically induced transparency (EIT), is highly sensitive to external magnetic field. This is used as an advantage for high precision magnetometers. However the presence of stray magnetic field for e.g. earth's magnetic field shifts the system from an ideal Λ configuration (Fig. <ref>a) by lifting the degeneracy of energy levels involved, thereby changing the shape and broadening the EIT resonance. Large stray magnetic fields can destroy the resonance completely. Further noisy magnetic field introduces decoherence in the medium which also broadens EIT resonance. Therefore we shield the vapor cell with three layers of μ-metal sheets.


The blue curve in Fig. <ref>b shows a typical EIT signal obtained through  controlled scanning of an external magnetic field B in the absence of magnetic shielding. A solenoid with 47 turns is placed around the vapor cell for creating this field, and its amplitude is controlled by sending current through this coil. Current is regulated with a feedback control circuit. B introduces a shift of μ_B B in the atomic energy levels, where μ_B is the Bohr magneton. This shift is opposite for + and - magnetic sub-levels of ground state (see Fig. <ref>a). Therefore B is related to the two photon detuning as δ=Δ_p-Δ_c=2μ_B B. The line-width of EIT signal is 93 kHz. The blue curve in Fig. <ref>b shows the narrowing of EIT resonance to 30 kHz in the presence of shielding. Another evidence of decoherence introduced by stray magnetic field can be seen in the transient response of the cw probe by scanning it across a 10μs control pulse in Fig. <ref>c, where δ=0. When control is turned off we observe an oscillation of time period 2.45 μs in the probe transmission which corresponds to a magnetic field of 9.5μT. This indicates population oscillation between states |1⟩ and |2⟩. In the presence of magnetic shielding, this oscillation completely vanishes as seen in Fig. <ref>d.



 
 
 (b) Beat note laser locking


 Saturation absorption spectroscopy (SAS) is a widely established technique to stabilize the frequency of a laser with respect to an atomic transition. However locking several independent lasers requires independent SAS set-ups, which can be experimentally cumbersome. Another drawback of this technique is that the locked frequency can not be changed arbitrarily. Therefore as a better alternative, we employ a beat note locking technique <cit.> to stabilize the frequency difference of two independent lasers, one of which is frequency locked. Figure <ref>a shows the experimental set up for beat note lock. 
 
 In our experiment scheme (Fig. 1c of the main text), control and probe are derived from an external cavity diode laser (ECDL1), which is locked to  |F=2⟩→|F'=2,3⟩ crossover by SAS locking technique, and later down-shifted by 80 MHz with an AOM. This implies that control and probe are 19 MHz red detuned with respect to  |F=2⟩→|F'=1⟩ transition. Repumper (ECDL2) and Raman beams (ECDL3) are locked with respect to ECDL1 via beat note lock. In this technique a part of two beams from ECDL1 and ECDL2 (or ECDL3) are mixed on a 50:50 beam splitter, and the beat note signal is recorded on a fast MSM detector (Hamamatsu G4176-03) with a rise time of 30 ps. The beat signal is amplified and then mixed with a reference signal from a VCO (Mini-Circuits ZX95-3360+, ν_VCO≃ 3 GHz). Voltage across VCO is computer controlled using a DAQ card (NI PXIe-6738). The mixer (Mini-Circuits ZX05-C42+) output is split into two equal parts with a power splitter, and a fixed delay is introduced in one part via a longer BNC cable. The two parts are recombined on a phase detector (Mini-Circuits ZAD-1-1+), and the output at their sum frequency is blocked by a low pass filter. Fig. <ref>b shows the output signal while scanning free running laser. Here the time axis corresponds to frequency of the laser. The frequency of free running laser can then be locked with respect to reference laser (ECDL1) using a PID circuit. An added advantage of this robust locking technique is that the frequency of a locked laser can be changed arbitrarily by controlling the VCO frequency. Using this technique repumper is locked at |F=3⟩⟶|F'=3⟩, while the Raman coherent beams are kept highly red detuned with respect to |F=2⟩⟶|F'=1⟩ transition, where the detuning is controlled by the VCO voltage. 
 
  
 
  
  (c) Saturation intensity in a Doppler broadened medium

Rabi frequency Ω is related to saturation intensity I_sat and natural line-width γ_3 as

    I/I_sat=2(2Ω/γ_3)^2 ⇒ I_sat=cϵ_0γ_3^2ħ^2/4|ϵ̂.d|^2

where ϵ̂ is the unit polarization vector, d is the atomic dipole moment, Ω=d.ε_o/2ħ is the resonant Rabi frequency and ε_o is the electric field amplitude <cit.>. The scattering cross-section in a homogeneously broadened medium is <cit.>

    σ(Δ)=σ_0γ_3^2/4/γ_3^2/4+Δ^2,σ_0=ħωγ_3/2I_sat

where σ_0 is the on-resonance scattering cross-section. For a Doppler broadened medium scattering cross-section is Gaussian instead of a Lorentzian,

    σ'(Δ)=σ_De^-Δ^2/Γ_D^2

where σ_D, Γ_D are the modified on-resonant scattering cross-section and line width due to Doppler effect. The corresponding change in saturation intensity can be obtained by 
integrating equations (S.1) and (S.2), which gives
 
    σ=∫_-∞^∞σ(Δ)dΔ=σ_0πγ_3,σ'=∫_-∞^∞σ'(Δ)dΔ=σ_D√(π/2)Γ_D

Since σ= σ', using equation (S.1) we have
   
    σ_D≈0.89γ_3/Γ_Dσ_0, σ_D=ħωγ_3/2I_D

 which finally gives
  
    I_D=I_satΓ_D/0.89γ_3
  
 where I_D is the saturation intensity for thermal atoms. γ_3, Γ_D and I_sat for ^85Rb atoms are 6 MHz, 308 MHz and 1.66 mw/cm^2 respectively <cit.>, which yields I_D≈ 95.75 mW/cm^2.
 
   
  

 
 

 S.II. Theoretical modelling
 

  
 
(a) Density matrix picture
 
 
The scheme of Fig. 4a in the main text can be simulated by considering a four level atomic system where the fourth level corresponds to a virtual level which is highly red detuned from the excited state |3⟩. For analytical simplicity the the fourth level can be adiabatically eliminated assuming an effective Raman coupling  Ω_R=Ω^+Ω^-e^iϕ_R/Δ_R between the two ground states. 

 

  
  The effective Hamiltonian of the three level system, under dipole and rotating wave approximation, can be expressed as
  
    Ĥ =-(Δ_p-Δ_c)|2⟩⟨2|-Δ_p|3⟩⟨3|-Ω_p(z,t)|3⟩⟨1|-Ω^∗_p(z,t)|1⟩⟨3|
    
      	-Ω_c(z,t)|3⟩⟨2|-Ω^∗_c(z,t)|2⟩⟨3|+Ω_R(z,t)|2⟩⟨1|+Ω^∗_R(z,t)|1⟩⟨2|
 
Here control and Raman field Rabi frequency are defined as Ω_c,±(z,t)Ω_c,±(t)=Ω_c,±(0)e^-(t-t_on)^2/2τ^2 for t≤ t_on, Ω_c,±(0) for t_on<t<t_off and Ω_c,±(0)e^-(t-t_off)^2/2τ^2 for t≥ t_off, where τ = 150 ns is the ramp time. The turn-on and turn-off times are t_on= 0 and t_off= 10 μ s respectively. The Rabi frequencies are defined as  Ω_c(0)=d_cε_c(z)/2ħ, Ω_p(z,0)=d_pε_p(z)/2ħ,  Ω_±(0)=d_±ε_±(z)/2ħ, with d_c, d_p and d_± being the respective transition dipole moments. ε_c,p,±(z) are the electric field amplitudes of the corresponding fields. The detunings of these lasers from the corresponding atomic transitions are Δ_c=ω_32-ω_c, Δ_p=ω_31-ω_p and Δ_R=ω_31-ω_-=ω_32-ω_+ where ω_p, ω_c and ω_±  correspond to the respective carrier frequencies.
  
  
  Time dynamics of the system is governed by master equation  ρ̇(̇ṫ)̇= -i[Ĥ,ρ] +L̂(ρ). Here the first term on the right accounts for coherent interactions while the second term  L̂ represents  irreversible incoherent processes in the system. The evolution of atomic states is therefore given by the following set of equations:

    ∂ρ_11/∂ t=-i Ω_pρ_13+iΩ^∗_pρ_31+i Ω_Rρ_12-iΩ^∗_Rρ_21+γ_31ρ_33-Γ_th(ρ_11-ρ^eq_11)
    ∂ρ_22/∂ t=-i Ω_cρ_23+iΩ^∗_cρ_32-i Ω_Rρ_12+iΩ^∗_Rρ_21+γ_32ρ_33-Γ_th(ρ_22-ρ^eq_22) 
    ∂ρ_33/∂ t=i Ω_cρ_23-iΩ^∗_cρ_32+i Ω_pρ_13-iΩ^∗_pρ_31-γ_exρ_33-Γ_th(ρ_33-ρ^eq_33)
    ∂ρ_12/∂ t=-Γ_12ρ_12+i Ω^∗_pρ_32-i Ω_cρ_13+iΩ^∗_R(ρ_11-ρ_22)
    ∂ρ_13/∂ t=-Γ_13ρ_13-i Ω^∗_cρ_12-i Ω^∗_Rρ_23-iΩ^∗_p(ρ_11-ρ_33)
    ∂ρ_23/∂ t=-Γ_23ρ_23-i Ω^∗_pρ_21-i Ω_Rρ_13-iΩ^∗_c(ρ_22-ρ_33)

 where
  
    Γ_12=Γ_decoh+Γ_th+i(Δ_p-Δ_c)
    Γ_13=Γ_decoh+γ_ex/2+Γ_th+iΔ_p
    Γ_23=Γ_decoh+γ_ex/2+Γ_th+iΔ_c
    γ_3=γ_31+γ_32, γ_ex=γ_3+γ_out

Here γ_out, Γ_decoh and Γ_th are the radiative decay from excited state |3⟩ out of the closed Λ system, decoherence rate and transit time decay  respectively. The steady state solutions for ρ_12 and ρ_13 are
  
    ρ_12=[iΩ^∗_pρ_32-i Ω_cρ_13+iΩ^∗_R(ρ_11-ρ_22)]/Γ_12
    ρ_13=[-i Ω^∗_cρ_12-i Ω^*_Rρ_23-iΩ^∗_p(ρ_11-ρ_33)]/Γ_13
 
For weak probe field Ω_p<<Ω_c,Ω_R, steady state analytical solutions for ρ_ij can be obtained using the perturbation approach as ρ_ij=ρ_ij^(0)+ρ_ij^(1).
The zeroth order solutions are
   
    ρ_11^(0)≃1,ρ_22^(0)=ρ_33^(0)≃0
    ρ_12^(0)=iΩ^*_R/Γ_12+|Ω_c|^2/Γ_13(1+|Ω_R|^2/Γ^2_13)
    ρ_13^(0)=Ω^*_cΩ^*_R/Γ_12Γ_13(1+|Ω_R|^2/Γ^2_13)+|Ω_c|^2

   where we have assumed that Γ_13=Γ_23. The above equations represent the contribution of Raman fields to the three level EIT system. First order solutions are
    
    ρ_12^(1)=-Ω_cΩ^*_p[Γ_13(Γ^*_13|Ω_c|^2+Γ^*_12Γ^*2_13+Γ^*_12|Ω_R|^2)+Ω_R^2(Γ^2_13-Γ^*2_13)]/(Γ^*_13|Ω_c|^2+Γ^*_12Γ^*2_13+Γ^*_12|Ω_R|^2)(Γ_13|Ω_c|^2+Γ_12Γ^2_13+Γ_12|Ω_R|^2)
    ρ_13^(1)=-iΩ^*_p[Γ_13|Ω_c|^2(Γ_12Γ^*_13-Ω_R^2)+Γ_12(Γ^*2_13+|Ω_R|^2)(Γ^*_12Γ_13-Ω_R^2)/(Γ^*_13|Ω_c|^2+Γ^*_12Γ^*2_13+Γ^*_12|Ω_R|^2)(Γ_13|Ω_c|^2+Γ_12Γ^2_13+Γ_12|Ω_R|^2)

The above equations comprises of the EIT terms and higher order corrections of Raman fields. When Γ_2,Γ_3 are real, the above equations assume a very simple form:
     
    ρ_12^(1)=-Ω_cΩ^*_pΓ_13/Γ_13|Ω_c|^2+Γ_12Γ^2_13+Γ_12|Ω_R|^2
    ρ_13^(1)=-iΩ^*_p(Γ_12Γ_13-Ω_R^2)/Γ_13|Ω_c|^2+Γ_12Γ^2_13+Γ_12|Ω_R|^2

 which are the usual EIT terms in absence of Ω_R.

The time dependent response of the medium is obtained by numerically integrating the density matrix equations along with Maxwell wave propagation equation 

    1/c∂Ω_p/∂ t+∂Ω_p/∂ z=-i μ_a ρ_13(z,t)
 
which is integrated self-consistently. Here μ_a=Nd_p^2ω_p/ħϵ_0 where c and ϵ_0 correspond to speed of light and dielectric susceptibility in vacuum  respectively.  The transmitted probe pulse is Ω_p(t+τ)=Ω_p(t)+∫_0^Lα Im(ρ_13(z,t))dz, where L is the propagation length in the cell and α is a constant <cit.>.

(b) Wave function picture 

     
    
The evolution of probability amplitudes corresponding to Hamiltonian in equation (S.6) can be written as
  
    Ċ_1=iΩ^*_pC_3-iΩ^*_RC_2-iΩ^2_-/Δ_R^2γ_3C_1
    Ċ_2=-(γ_2/2-i(δ+Ω^2_+/Δ_R^2γ_3))C_2+iΩ^*_cC_3-iΩ_RC_1
    Ċ_3=-(γ_3/2-iΔ_p)C_3+iΩ_pC_1+iΩ_cC_2

where γ_2 is decay rate of |2⟩. For C_1≃1, the steady state solutions are
   
    C_2Γ_2=iΩ^*_cC_3-iΩ_R
    
      C_3Γ_3=iΩ_p+iΩ_cC_2

where Γ_2=γ_2/2-i(δ+Ω^2_+/Δ_R^2γ_3) and Γ_3=γ_3/2-iΔ_p. Substituting the value of C_2 in equation(S.31) we get
  
    C_3=iΩ_p/Γ_3+iΩ_c/Γ_3(iΩ^*_c/Γ_2C_3-iΩ_R/Γ_2)

which implies

    C_3=Γ_2/Γ_2Γ_3+|Ω_c|^2[iΩ_p+Ω_RΩ_c/Γ_2]

Substituting C_3 in equation(S.30) we have

    C_2=-Ω^*_cΩ_p/Γ_2Γ_3+|Ω_c|^2-iΩ_R/Γ_2[1-|Ω_c|^2/Γ_2Γ_3+|Ω_c|^2]

which means

    ρ_21=-Ω^*_cΩ_p/Γ_2Γ_3+|Ω_c|^2-iΩ_R/Γ_2[1-|Ω_c|^2/Γ_2Γ_3+|Ω_c|^2]
    ρ_31=iΩ_pΓ_2/Γ_2Γ_3+|Ω_c|^2+Ω_RΩ_c/Γ_2Γ_3+|Ω_c|^2

For Ω_c>>Ω_p,Γ_3,Γ_2

    ρ_21=-Ω^*_cΩ_p/|Ω_c|^2-iΩ_RΓ_3/|Ω_c|^2=-|Ω_p|/Ω_ce^iϕ_0-|Ω_R|/|Ω_c|^2|Γ_3| e^i(ϕ_R+ϕ_3+π/2)
    
    =e^iϕ_0[A+Be^i(ϕ_R+ϕ_3+π/2-ϕ_0)]

where ϕ_0=ϕ_p-ϕ_c is the relative phase between control and probe fields, ϕ_R is the phase between Raman beams, Γ_3=|Γ_3| e^iϕ_3 and A=-|Ω_p|/|Ω_c| and B=-|Ω_R|/|Ω_c|^2|Γ_3|. The absolute value of ρ_21 is therefore given as

    |ρ_21|=√(A^2+B^2+2ABcos(ϕ_R-ϕ_0+ϕ_3+π/2))
	
and

    ρ_31=iΩ_pΓ_2/|Ω_c|^2+Ω_RΩ_c/|Ω_c|^2
	
Since susceptibility χρ_31/Ω_p, in the limit of γ_2=0 we have

    ρ_31/Ω_p=δ'/|Ω_c|^2+|Ω_R||Ω_c|/|Ω_c|^2|Ω_p|e^i(ϕ_R-_(ϕ_p-ϕ_c))
    
    =δ'/|Ω_c|^2+|Ω_R|/|Ω_c||Ω_p|e^i(ϕ_R-ϕ_0)

where δ'=δ+Ω^2_+/Δ^2γ_3. Figure <ref> shows the dependence of phase ϕ_R on ρ_12 and ρ_13. From equation(S.39) we have

    Re(ρ_31/Ω_p)=δ'/|Ω_c|^2+|Ω_R|/|Ω_c||Ω_p|cos(ϕ_R-ϕ_0)
    
    Im(ρ_31/Ω_p)=|Ω_R|/|Ω_c||Ω_p|sin(ϕ_R-ϕ_0)

which implies 

    tan(ϕ_eff)=|Ω_R|/|Ω_c||Ω_p|sin(ϕ_R-ϕ_0)/δ'/|Ω_c|^2+|Ω_R|/|Ω_c||Ω_p|cos(ϕ_R-ϕ_0)


   (c) Numerical simulations for the complete Zeeman manifold 


Though the numerical simulation of a simple three level model agrees well with our experiments, in reality there are other levels in the Zeeman manifold which makes the system very complicated. The coherent dynamics in such systems is essentially dominated by the formation of a dark state. It was first observed by Parkins et.al. <cit.> that even for a multilevel atom, with a ground state Zeeman structure, there is always formation of similar dark states in presence of circularly polarized control and probe fields. Subsequently, such dark states, formed effectively between the stretched states for multilevel systems have been used explicitly for various schemes <cit.>. Eventually, experiments on such multilevel atoms also started using and effective three-level model to simulate observations <cit.>. We follow this approach, relying on the fact that the simplicity and power of a three-level model can elucidate most of the essential features of a coherent dark state dynamics.

However, for the sake of completeness we consider the complete Zeeman manifold of ^85Rb D_2 |F=2⟩→|F^'=1⟩ transition as shown in Fig. <ref>a. This system comprises of three mutually coupled Λ system and its evolution is governed by 36 coupled equations. We solve the density matrix equations and propagate the field with Maxwell equation (S.26) similar to the case of three level system.  Figure <ref>b shows the simulated probe transmission for closed eight level scheme which is similar to the response obtained by simulating a three level Λ system, thereby validating the fact that a simplified three-level scheme captures most of the essential physics for a complicated multi-level system.


   

 S.III. Some comments on experimental results
 

 
(a)  Coherence quantification vs measure, in an experimentally simulated closed system 
     
     
The presence of neighbouring levels make it impossible to experimentally realize a closed three level system as shown in Fig. <ref>a. To counter the leakage of atoms outside the Λ manifold, we use an incoherent repumping field which pumps the scattered atoms back to the system, thereby effectively closing the system. By tuning the repumper intensity we can change the system from closed to open. This can be verified in Fig. <ref>a, where it is seen that as the repumper intensity decreases, the fall height Δ h=h1-h2 decreases making the system more and more incoherent(open). 
    
    We have defined coherence quantifier C in terms of Δ h. For a quantity to qualify as a measure of coherence, it needs to satisfy the following conditions <cit.>: (1) The quantifier should be positive. (2) It should not increase under incoherent operation. (3) It should be monotonic. (4) It should be an additive quantity. We find that C indeed satisfies all these conditions. As C is proportional to Δ h , the condition Δ h_closed>Δ h_open>Δ h_incoherent implies
     
    C_closed>C_open>C_incoherent

      which also proves that C cannot be negative because the minimum bound of C is given by Δ h_incoherent =0. 
      Further equation(S.43) implies that C decreases under incoherent operation. Monotonicity of C can be verified from Figure <ref>b, where it is observed that as the repumper intensity increases, i.e. as the system gets more and more closed (coherent), C also increases. Numerical results show an excellent agreement with the experiment as shown by solid line in Fig. <ref>b. Additivity of C can be established from the linear region of Fig. <ref>b (R≃ 0 - 0.07), where it is clear that C is linearly proportional to number of atoms in the closed  Λ system.  
     
   
       
           
  
(b) Signature of EIT to ATS transition in  ρ_21
       
       
The EIT to ATS transition shown in Fig. 3 of the main text is not captured in the probe transmission, but is clearly visible in  C i.e |ρ_21|. This can be explained in terms of steady state solution of ρ_21 as given by equation (S.34) (for Ω_R=0)
         
    ρ_21=-Ω^*_cΩ_p/Γ_2Γ_3+|Ω_c|^2

where Γ_2=γ_2/2-iδ and Γ_3=γ_3/2-iΔ_p. For Ω_c<<Γ_2Γ_3, we can see that there is one pole of |ρ_21| at Δ_p=0, while for larger control strength, there are two poles at ±√(2)Ω_c, which indicates splitting in |ρ_21| at large control strengths and agrees with the observations in Fig. 3c of the main text.
    
  (c) Offset in C with respect to C^EIT 

                   
                  
  For the phase coherent decay compensation scheme as shown in Fig. 4a of the main text, large detuning Δ_R ensures a build up of two-photon coherence with minimal population reshuffle, simply adding a perturbative correction to the coherence. However the sinusoidal variation of C with ϕ_R is not symmetric with respect to C^EIT as may be seen in Fig. <ref>a (also in Fig. 4b of the main text). This is indicative of limitation of the perturbative analysis, with Raman fields competing with control in redistributing the populations. Figure <ref>b shows the variation of offset in C with respect to C^EIT as a function of |Ω_-|^2/|Δ_R|^2 where we have considered that Ω_-=Ω_+.
        
     For a three level EIT scheme the steady state population in the ground states varies as ρ_22/ρ_11=|C_2|^2/|C_1|^2≃|Ω_p|^2/|Ω_c|^2. The Raman beams Ω_+ and Ω_- cause population transfer within the ground states via optical pumping thereby departing the system from ideal steady state EIT behavior. The ground state populations in steady state can be written as 
     
    |C_2|^2/|C_1|^2≃|Ω_p|^2/|Ω_c|^2+(Δ_c^2+γ_3^2)/(Δ_R^2+γ_3^2)|Ω_-|^2/|Ω_c|^2+(Δ_R^2+γ_3^2)/(Δ_p^2+γ_3^2)|Ω_p|^2/|Ω_+|^2

    Here the last two terms are the correction terms due to optical pumping by the Raman beams, out of which the second term plays the dominant role. Therefore the population ratio ρ_22/ρ_11∝|Ω_-|^2/|Δ_R|^2 as can be seen in Fig. <ref>c. The offset is decided by the last two terms of equation (S.45), where we have considered that Ω_+=Ω_-. Initially for small |Ω_-|^2/|Δ_R|^2=|Ω_+|^2/|Δ_R|^2 the last term dominates and the offset is proportional to |Δ_R|^2/|Ω_+|^2 and when the second term dominates, offset is proportional to |Ω_-|^2/|Δ_R|^2, before reaching saturation. 
      
 




[^ Present address: The Institute of Optics, University of Rochester, New York-14627, USA]


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