Abstract

Creating stable superposed states of matter is one of the most intriguing aspects of quantum physics, leading to a variety of counterintuitive 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 stable superposed states. Here, we propose and experimentally explore a quantifier that captures effective quantum coherence in an atomic ensemble at room temperature. The quantifier provides direct measure of ground-state coherence for electromagnetically induced transparency (EIT) along with a distinct signature of transition from EIT to Autler–Townes splitting 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 channel that compensates decay in the system. In the growing pursuit of quantum technologies at room temperature, our results provide a unique way to phenomenologically quantify and coherently control quantum coherence in atom-like systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Ability to generate, probe, and control superposed states of physical systems provides distinct technological advantages in quantum protocols, when compared to their corresponding classical counterparts [16]. Even entanglement [710], a critically important resource in quantum information, relies on superposed states of distinctly measurable channels. Over the last few decades, there has therefore been a tremendous thrust in research to better quantify such states theoretically [1113] and to generate and control them experimentally [1421]. A widely used technique to generate stable superposed states in atom-like systems [22,23] is electromagnetically induced transparency (EIT) [1417], where a strong control field is used to drive an effective three-level atomic system into a specific coherent superposition of ground states |1 and |2 [Fig. 1(a)]. These superposed dark states remain mostly decoupled from the lossy excited state (|3) [1417] leading to dramatic effects such as slow [24], stopped [25], and stored [2527] light, generation of entangled photons [710], and enhanced optical nonlinearities at the level of single photons [2830]. EIT-based technologies at room temperature have been actively pursued in a range of atom-like systems [22,23], leading to a need to characterize and quantify their effective, steady-state coherence.

 

Fig. 1. (a) Effective three-level atomic system where states |1 and |2 are coupled to |3 by a continuous probe and a pulsed control field of Rabi frequencies (detunings): Ωp(Δp) and Ωc(Δc), respectively. Level |4 depicts additional states of neighboring ground-state manifold. Solid, horizontal arrows indicate co-propagating fields. (b) Experimental trace (dots) of EIT resonance with two-photon detuning (δ=ΔpΔc). Here, Iin, Itrans are the initial and transmitted probe intensities, respectively. Solid line is the Lorentz fit that corresponds to a line width of 34 kHz. (c) Schematic of experimental setup. ECDL, external cavity diode laser; Rb, rubidium vapor cell; P, probe (purple); C, control (red); R, repumper (dark yellow); Ram, Raman beam (green); AOM, acousto-optic modulator; Q, quarter-wave plate; H, half-wave plate; M, mirror; BS, 50:50 beam splitter; PBS, polarizing beam splitter; GT, Glan–Thompson polarizing beam cube; PD, photo detector; DSO, digital storage oscilloscope.

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Traditionally, superposition in EIT is characterized spectroscopically through its signature transparency window in a probe absorption profile [Fig. 1(b)]. However, it is also well acknowledged that such transparency is not necessarily a unique signature of superposed states, but can also occur due to the strong control field either optically pumping atoms out of the Λ system [31] or hybridizing the ground state with the excited state, leading to Autler–Townes splitting (ATS) [32,33]. Early efforts to discern EIT from optical pumping effects relied on characterization of linewidths by fitting convolved absorption profiles [34]. Several recent works have explored the more subtle issue of discerning EIT from ATS using the Akaike information criterion, which provides a quantitative indicator based on Bayesian comparison of probe absorption profiles [23,33,35]. Nevertheless, such techniques, based on post-processed spectroscopic data are cumbersome and do not directly quantify useful ground-state superposition. Experiments showing storage and retrieval of light pulses provide an alternative characterization of ground-state coherence via storage time and retrieval efficiency [2527]. However, these parameters are also common to other mechanisms, including storage based on photon echo [36,37] or coherent population oscillation [38]. A direct experimental quantification of the off-diagonal atomic density matrix element ρ12 or coherence can significantly boost performance of devices that use EIT to delay, store, and switch classical fields at room temperature.

Here, we propose a quantifier that accurately captures ρ12, i.e., ground-state coherence in EIT for an ensemble of rubidium (Rb85) 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 of the system [6]. We experimentally demonstrate that with decreasing coherence, the quantifier decreases monotonically and satisfies the conditions to qualify as a quantifier of coherence (Supplement 1) [11]. Furthermore, with increasing control field strength, we observe emergence of a distinct splitting in the otherwise resonance peak of the quantifier. Such distinct spectroscopic signatures in EIT and ATS greatly simplify identification of transition between the two regimes. Finally, using this quantifier as a tool, we propose and demonstrate phase coherent control and compensate decay of ground-state coherence. Our work complements recent theoretical initiatives on quantifying coherence in quantum systems [1113]. Furthermore, the demonstrated phase-dependent control and freezing of coherence demonstrated here can improve the ability to store, process, and retrieve quantum information in a variety of systems that use EIT to generate stable ground-state superpositions in steady state [3943].

The primary motivation for this study is based on the observation that in a three level atomic system [Fig. 1(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 (ρ13) that drives the probe transmission and takes a form (Supplement 1)

ρ13ss=[iΩc*ρ12ssiΩp*(ρ11ssρ33ss)]/Γ13,
in steady state (ss), where Γ13=iΔp+γex/2 with Δp being the probe detuning and γex the decay rate of excited state. While ρ12ss couples via the control field (with Rabi frequency Ωc), there is also a part that is set by steady-state populations ρ11ss and ρ33ss, modified from their thermal equilibrium values due to optical pumping and losses. These modified population terms lead to ambiguities in identifying useful ground state coherence ρ12ss from the probe spectrum. However, it can be observed that in absence of control, steady-state populations equilibrate (via thermal diffusion of atoms through a probe) in a time scale that is significantly long (10  μs for a probe beam diameter of 4 mm) compared to time scales associated with atomic dynamics (30  ns). By adiabatically turning off the control and taking the difference in response, one can thereby subtract out the population contribution. Accordingly, we define a phenomenological quantifier C, such that
C=|ρ13Ωcon,ssρ13Ωcoff,ss||ρ13Ωcoff,ss||Ωp||Ωc|,
where ρ13Ωcon,ss and ρ13Ωcoff,ss are the probe response in presence and absence of a control field. For an intermediate control turn-off time of 150  ns and in the limit of ρ11ss1, the quantifier C becomes proportional to ρ12ss. We set up an experiment to test the validity for the assertion of C as a quantifier of ρ12ss in different scenarios and compare observations with toy models and simulation results.

2. METHODS

We use a degenerate Λ system defined within D2 transition of Rb85 atoms as shown in Fig. 1(a), where |1|F=2,mF, |2|F=2,mF2, and |3|F=1,mF1. A schematic of the experimental setup is shown in Fig. 1(c). 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 locked at red detuning of 19 MHz with respect to |F=2|F=1. We experimentally simulate a quasi-closed three-level system with a counter-propagating continuous repumper field (resonant and locked at a transition F=3F=3) that cycles back atoms escaping out of F=2 manifold to F=3 [6]. A rubidium vapor cell of length 8 cm and diameter 2 cm is used as the atomic medium. The cell is shielded with three layers of μ-metal sheets along with magnetic coils to cancel any stray magnetic field. When a small magnetic field is scanned across resonance, a sharp two-photon resonance peak is observed in probe transmission, a typical signature of EIT [Fig. 1(b)]. Additionally two orthogonal circularly polarized Raman beams, counter-propagating to control and probe fields, are used to control the effective ground-state coherence, described in Section 5.

The cross-section diameter of all the beams is 4  mm. Laser frequencies are stabilized by a beat note offset frequency locking technique (Supplement 1). Laser pulses are controlled with acousto-optic modulators (AOMs) and field-programmable gate array (FPGA). The transmitted probe intensity is recorded in time, while the control and Raman fields are adiabatically turned on and off (in 150  ns1/γ3). In between, they are kept on for 10 μs, which is long enough for the system to reach steady state. Repetition time of the entire experiment is 50 μs. The probe beam is detected with a high-speed, low-noise, and amplified photo detector (PDB450A) after a Glan–Thompson polarizing beam splitter.

3. QUANTIFYING COHERENCE

A typical experimental trace for a closed Λ system is shown in Fig. 2(b) along with numerical simulations. When the control field is turned on, there is a sharp rise ab, followed by a slower decay bc. In a recent work [6], we have shown that this corresponds to an initial fast buildup of ground-state coherence ρ12, followed by optical pumping rearranging the populations ρ11 and ρ22.

 

Fig. 2. (a) Evolution of the system in an effective Bloch-sphere corresponding to ground states |1 and |2 (see text). (b) Experimental trace (solid) of probe transmission, with a control field turned on (at t=0  μs) and off after 10 μs for an experimentally simulated, quasi-closed three-level system. Here, Δp=Δc=3.2γ3, Ωc=2.3×101γ3, and R=2.7×101γ3, where γ3=6.0(2π)  MHz is the radiative decay rate of |3. The level diagram in inset shows a counter-propagating repumper field (R) effectively closing the system. The dashed trace corresponds to numerical simulations. Frames (c) and (d) correspond to similar experimental and simulated traces for an open EIT system and an EIT configuration with counter-propagating fields. Ωc=2.3×101γ3 and 1.7×101γ3 for frames (c) and (d), respectively. Simulation parameters are chosen according to experiment.

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It is particularly intuitive to visualize this dynamics in an effective Bloch sphere corresponding to the ground-state manifold spanned by |1 and |2. An initial unpolarized ensemble with equally populated thermal states corresponds to a zero-length Bloch vector at origin. With turn on of control, the tip of the Bloch vector grows in a convex path, first building up coherence along the equatorial plane and then moving towards the pole due to optically pumped population imbalance [Fig. 2(a)]. For an ideal EIT scenario with all the atoms in |1, the steady-state Bloch vector points mostly down, with a slight tilt due to a small coherence, ρ12ssΩp/Ωc. It can also be noted that though ρ12ss depends on the relative phase between control and probe setting the angle of the tilt in the equatorial plane, ρ13ss is independent of it (due to the additional excited state |3). We use this visualization in Section 5 for phase coherent control of ρ12.

When control is turned off, there is a sharp fall in probe transmission going below the initial level [Fig. 2(b)], indicating a new steady-state population difference ρ11ssρ22ss. We use the fall height from point (c(h1) to d(h2), to estimate the quantifier

C=h1h2h2|Ωp||Ωc|.

In absence of the repumper field, atoms escape out of the three-level manifold, and the system becomes open. Experimentally, we observe a partial drop with a corresponding smaller Copen [Fig. 2(c)] as compared to Cclosed [Fig. 2(b)]. Furthermore, to construct a scenario where the system is incoherent, we use counter-propagating control and probe fields. Here, one expects Doppler averaging to wash out any coherence in the system, and we observe Cincoherent=0 [Fig. 2(d)]. The rise time in Fig. 2(d) corresponds to optical pumping, while the long time scales after control turn-off in all three scenarios [Figs. 2(b)2(d)] correspond to thermal diffusion of atoms (10  μs) [6].

As a first test, we therefore conclude that the defined quantifier decreases monotonically with decrease in coherence, i.e.,

Cclosed>Copen>Cincoherent.

4. 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 ATS in a probe absorption profile [1]. In such a hybridized basis [Fig. 3(a)], the corresponding low-field strength EIT regime is usually understood as a Fano resonance [23,33,35], which vanishes monotonically with increasing field strength. However, due to power broadening, such a signature of splitting in probe transmission is not always discernible [23,33,35]. The situation is particularly severe in an ensemble of hot atoms, where a large Doppler-broadened background profile washes out any signature of ATS [Fig. 3(b), inset].

 

Fig. 3. (a) EIT and Autler–Townes basis states in a Λ system. (b) Coherence quantifier C as a function of δ for varying control intensities. Near saturation intensity, the observed splitting in C is a signature of transition from EIT to ATS regime. Ωp=2.5×103γ3 and R=2.6×101γ3. Inset shows the steady-state probe transmission profile where any such signature is washed out due to power broadening. (c) Theoretically calculated coherence |ρ12ss| as a function of δ for varying control intensities. Here, Ωp=1.0×103γ3˜, 2Ωc=(I/2Isat)γ3˜, where γ3˜=1  MHz. (d) Experimentally measured C as a function of control intensity at δ=0. The red solid line is ΩpΩc/(Ωp2+Ωc2) fitting. Inset shows the variation of theoretically calculated |ρ12ss| with control intensity at δ=0. ID=IsatΓD/0.89γ3 is the saturation intensity for thermal atoms where ΓD=308  MHz (Supplement 1). Simulation parameters are chosen according to experiment.

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The quantifier C precisely subtracts out this one-photon background [i.e., h2 at d in Fig. 2(a)], corresponding to the term ρ11ssρ33ss, which hardly evolves during control turn-off. C can thereby be viewed as distilling the information of ρ12ss from probe transmission. Experimentally, it captures excellently a splitting in the corresponding two-photon resonance peak, a signature otherwise undiscernible in a steady-state measurement [Fig. 3(b) and inset]. The behavior is well understood in a wave-function framework [Fig. 3(c) and Supplement 1] where C decreases monotonically with increasing control field strength [Fig. 3(d)].

5. FREEZING COHERENCE

Along with magnitude, there is also an absolute phase of ρ12, which remains undetected in EIT and can be characterized only with respect to a reference phase. An additional radio-frequency/Raman fields coupling the ground states or an auxiliary atomic level configuration can provide such a reference. Such phase-dependent probe transmission (Im(ρ13)) has been studied extensively [4451]. Here, we directly probe the generated coherence as a function of the reference phase of a pair of far-detuned and counter-propagating Raman fields. While large detuning [ΔR in Fig. 4(a)] ensures a buildup of two-photon coherence with minimal population reshuffle, counter-propagating fields wash out higher-order multi-photon effects, simply adding a perturbative correction to the coherence, in the form (Supplement 1)

ρ12ss=ΩcΩp*|Ωc|2+iΩR*Γ3|Ωc|2,
where Γ3=γ3/2iΔp. The resulting, modified ρ12ss 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 ΩR=Ω+ΩeiϕR/ΔR [1], where ϕR is the phase difference between Raman fields [Fig. 4(a)].

 

Fig. 4. (a) Energy-level scheme for controlling coherence. Here, Ω and Ω+ are highly detuned Raman coherent fields with ϕR being the phase difference between them. Figure on the right shows the projection of Bloch vector in ρ12 plane for EIT and Raman field controlled coherence schemes, where ρ12EIT=ΩcΩp*/|Ωc|2 and ρ12R=iΩR*Γ3/|Ωc|2. Δϕ=(ϕcϕp)ϕR, where ϕp(c) is the phase of probe (control) field. (b) Sinusoidal variation of C as a function of ϕR. Circles and solid line show experimental and simulation results, respectively. Dashed (Cab) and dotted (Ccd) black lines indicate C at initial peak (region ab) and steady state (region cd) for EIT case. ΔC is the visibility of coherence, and ΔCo=CabCcd. ϕf corresponds to ϕR=120° (100° in simulation), at which C freezes to its initial maxima. (c) Experimental (solid) and simulated (dashed) time trace of probe transmission for EIT (red) and Raman field controlled coherence schemes at ϕf (blue). Here Ωc=2.5×101γ3, Ωp=3.5×103γ3, |Ω+|=|Ω|=3.7×101γ3, and ΔR=1.5ΓD. (d) ΔC as a function of ΔR with the dashed line showing 1/ΔR fit. Red dashed line indicates ΔCo for EIT system. Inset shows the simulations where Δρ12ss is the visibility in terms of ground-state coherence. Here, γex=γ3+γout as defined in Supplement 1.

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Experimentally, we use an additional counter-propagating linearly polarized laser beam for the Raman fields Ω+ and Ω. The relative phase difference (ϕR) between them is controlled by rotating the linear polarization with a half-wave plate. We experimentally observe sinusoidal variation of the total coherence as a function of ϕR [Fig. 4(b)].

There is an intriguing consequence of this phase dependence for our closed system. We have observed [Figs. 2(b) and 4(c)] that only in such a scenario, an initial large buildup of ρ12 corresponding to Cab eventually decays down due to optical pumping to Ccd [dashed and dotted lines in Fig. 4(b), respectively] [6]. Since the system is closed, this corresponds to a decrease in total number of superposed atoms. One can now compensate for this loss with the Raman fields, which thereby act as a freezing channel, e.g., at ϕR120°(ϕf), the modified steady-state coherence C freezes to the transient maximum Cab [Fig. 4(c)]. In particular, this phase dependence [Fig. 4(b)] indicates that one can overcompensate, with the visibility (ΔC) getting larger than the difference between initial and final steady-state coherence (ΔCo=CabCcd). Furthermore, with increasing detuning ΔR, this visibility decreases [Fig. 4(d)]. These observations match well with simulation and simple rate equation models accounting for far-detuned optical pumping effects of the Raman fields [Fig. 4(d), inset and Supplement 1]. It may be noted that the freezing channel restores the initial number of superposed atoms in the closed system; however, the coherence lifetime of individual atoms remains unchanged.

6. CONCLUSION

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 time scales, including ensembles of cold atoms [52], trapped ions [53], defect centers in diamond [41], synthesized or fabricated quantum dots [40], and rare-earth doped solid-state materials [42] are currently being pursued as quantum-enabled devices for delays, filters, and memories. While traditionally steady-state spectroscopic signatures have been used to establish coherence [1417], here, we show that in such dirty systems with a variety of classical lossy channels, time-domain measurements can be used to settle observational ambiguities. Accordingly, this work complements efforts of quantifying many-body entanglement through dynamical susceptibilities [13] and experimental efforts to store and retrieve light based on EIT [25,26]. While in the latter case, the retrieval efficiency of the stored excitation provides an indirect measure of the leftover 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 devices that use EIT as a mechanism, at room temperature.

Funding

Department of Science and Technology, Ministry of Science and Technology (DST) (SERB/PHY/2015404).

Acknowledgment

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 generous help in construction of the experimental setup.

 

See Supplement 1 for supporting content.

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44. L. Yang, L. Zhang, X. Li, L. Han, G. Fu, N. B. Manson, D. Suter, and C. Wei, “Autler-Townes effect in a strongly driven electromagnetically induced transparency resonance,” Phys. Rev. A 72, 053801 (2005). [CrossRef]  

45. C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A 65, 043805 (2002). [CrossRef]  

46. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]  

47. G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001). [CrossRef]  

48. H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009). [CrossRef]  

49. M. Ghosh, A. Karigowda, A. Jayaraman, F. Bretenaker, B. C. Sanders, and A. Narayanan, “Demonstration of a high-contrast optical switching in an atomic delta system,” J. Phys. B 50, 165502 (2017). [CrossRef]  

50. M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: theory,” Phys. Rev. A 74, 013812 (2006). [CrossRef]  

51. M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity,” Phys. Rev. A 76, 033835 (2007). [CrossRef]  

52. R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nat. Phys. 5, 100–104 (2009). [CrossRef]  

53. H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, “Deterministic single-photon source from a single ion,” New J. Phys. 11, 103004 (2009). [CrossRef]  

References

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  6. A. W. Laskar, N. Singh, A. Mukherjee, and S. Ghosh, “Interplay of classical and quantum dynamics in a thermal ensemble of atoms,” New J. Phys. 18, 053022 (2016).
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  7. M. D. Lukin, S. F. Yelin, and M. Fleischhauer, “Entanglement of atomic ensembles by trapping correlated photon states,” Phys. Rev. Lett. 84, 4232–4235 (2000).
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  8. D. N. Matsukevich, T. Chanelière, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of remote atomic qubits,” Phys. Rev. Lett. 96, 030405 (2006).
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  9. J. Simon, H. Tanji, S. Ghosh, and V. Vuletic, “Single-photon bus connecting spin-wave quantum memories,” Nat. Phys. 3, 765–769 (2007).
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  17. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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  18. D. Suter and G. A. Álvarez, “Colloquium: protecting quantum information against environmental noise,” Rev. Mod. Phys. 88, 041001 (2016).
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  21. Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.-Y. Xu, C.-F. Li, and G.-C. Guo, “Directly measuring the degree of quantum coherence using interference fringes,” Phys. Rev. Lett. 118, 020403 (2017).
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  24. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
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  25. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
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  26. D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
    [Crossref]
  27. Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87, 031801 (2013).
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    [Crossref]
  29. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995).
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    [Crossref]
  31. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [Crossref]
  32. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
    [Crossref]
  33. P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning Autler-Townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107, 163604 (2011).
    [Crossref]
  34. A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
    [Crossref]
  35. L. Giner, L. Veissier, B. Sparkes, A. S. Sheremet, A. Nicolas, O. S. Mishina, M. Scherman, S. Burks, I. Shomroni, D. V. Kupriyanov, P. K. Lam, E. Giacobino, and J. Laurat, “Experimental investigation of the transition between Autler-Townes splitting and electromagnetically-induced-transparency models,” Phys. Rev. A 87, 013823 (2013).
    [Crossref]
  36. N. Gisin, S. A. Moiseev, and C. Simon, “Storage and retrieval of time-bin qubits with photon-echo-based quantum memories,” Phys. Rev. A 76, 014302 (2007).
    [Crossref]
  37. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
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  38. P. Neveu, M.-A. Maynard, R. Bouchez, J. Lugani, R. Ghosh, F. Bretenaker, F. Goldfarb, and E. Brion, “Coherent population oscillation-based light storage,” Phys. Rev. Lett. 118, 073605 (2017).
    [Crossref]
  39. S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005).
    [Crossref]
  40. D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vuckovic, “Controlling cavity reflectivity with a single quantum dot,” Nature 450, 857–861 (2007).
    [Crossref]
  41. M. K. Bhaskar, D. D. Sukachev, A. Sipahigil, R. E. Evans, M. J. Burek, C. T. Nguyen, L. J. Rogers, P. Siyushev, M. H. Metsch, H. Park, F. Jelezko, M. Lončar, and M. D. Lukin, “Quantum nonlinear optics with a germanium-vacancy color center in a nanoscale diamond waveguide,” Phys. Rev. Lett. 118, 223603 (2017).
    [Crossref]
  42. T. Zhong, J. M. Kindem, E. Miyazono, and A. Faraon, “Nanophotonic coherent light-matter interfaces based on rare-earth-doped crystals,” Nat. Commun. 6, 8206 (2015).
    [Crossref]
  43. J. Hansom, C. H. H. Schulte, C. Le Gall, C. Matthiesen, E. Clarke, M. Hugues, J. M. Taylor, and M. Atatüre, “Environment-assisted quantum control of a solid-state spin via coherent dark states,” Nat. Phys. 10, 725–730 (2014).
    [Crossref]
  44. L. Yang, L. Zhang, X. Li, L. Han, G. Fu, N. B. Manson, D. Suter, and C. Wei, “Autler-Townes effect in a strongly driven electromagnetically induced transparency resonance,” Phys. Rev. A 72, 053801 (2005).
    [Crossref]
  45. C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A 65, 043805 (2002).
    [Crossref]
  46. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
    [Crossref]
  47. G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
    [Crossref]
  48. H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
    [Crossref]
  49. M. Ghosh, A. Karigowda, A. Jayaraman, F. Bretenaker, B. C. Sanders, and A. Narayanan, “Demonstration of a high-contrast optical switching in an atomic delta system,” J. Phys. B 50, 165502 (2017).
    [Crossref]
  50. M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: theory,” Phys. Rev. A 74, 013812 (2006).
    [Crossref]
  51. M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity,” Phys. Rev. A 76, 033835 (2007).
    [Crossref]
  52. R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nat. Phys. 5, 100–104 (2009).
    [Crossref]
  53. H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, “Deterministic single-photon source from a single ion,” New J. Phys. 11, 103004 (2009).
    [Crossref]

2017 (5)

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.-Y. Xu, C.-F. Li, and G.-C. Guo, “Directly measuring the degree of quantum coherence using interference fringes,” Phys. Rev. Lett. 118, 020403 (2017).
[Crossref]

P. Neveu, M.-A. Maynard, R. Bouchez, J. Lugani, R. Ghosh, F. Bretenaker, F. Goldfarb, and E. Brion, “Coherent population oscillation-based light storage,” Phys. Rev. Lett. 118, 073605 (2017).
[Crossref]

M. K. Bhaskar, D. D. Sukachev, A. Sipahigil, R. E. Evans, M. J. Burek, C. T. Nguyen, L. J. Rogers, P. Siyushev, M. H. Metsch, H. Park, F. Jelezko, M. Lončar, and M. D. Lukin, “Quantum nonlinear optics with a germanium-vacancy color center in a nanoscale diamond waveguide,” Phys. Rev. Lett. 118, 223603 (2017).
[Crossref]

M. Ghosh, A. Karigowda, A. Jayaraman, F. Bretenaker, B. C. Sanders, and A. Narayanan, “Demonstration of a high-contrast optical switching in an atomic delta system,” J. Phys. B 50, 165502 (2017).
[Crossref]

2016 (4)

X. Gu, S.-N. Huai, F. Nori, and Y.-X. Liu, “Polariton states in circuit QED for electromagnetically induced transparency,” Phys. Rev. A 93, 063827 (2016).
[Crossref]

P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, “Measuring multipartite entanglement through dynamic susceptibilities,” Nat. Phys. 12, 778–782 (2016).
[Crossref]

D. Suter and G. A. Álvarez, “Colloquium: protecting quantum information against environmental noise,” Rev. Mod. Phys. 88, 041001 (2016).
[Crossref]

A. W. Laskar, N. Singh, A. Mukherjee, and S. Ghosh, “Interplay of classical and quantum dynamics in a thermal ensemble of atoms,” New J. Phys. 18, 053022 (2016).
[Crossref]

2015 (2)

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen quantum coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref]

T. Zhong, J. M. Kindem, E. Miyazono, and A. Faraon, “Nanophotonic coherent light-matter interfaces based on rare-earth-doped crystals,” Nat. Commun. 6, 8206 (2015).
[Crossref]

2014 (4)

J. Hansom, C. H. H. Schulte, C. Le Gall, C. Matthiesen, E. Clarke, M. Hugues, J. M. Taylor, and M. Atatüre, “Environment-assisted quantum control of a solid-state spin via coherent dark states,” Nat. Phys. 10, 725–730 (2014).
[Crossref]

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
[Crossref]

B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref]

2013 (3)

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref]

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87, 031801 (2013).
[Crossref]

L. Giner, L. Veissier, B. Sparkes, A. S. Sheremet, A. Nicolas, O. S. Mishina, M. Scherman, S. Burks, I. Shomroni, D. V. Kupriyanov, P. K. Lam, E. Giacobino, and J. Laurat, “Experimental investigation of the transition between Autler-Townes splitting and electromagnetically-induced-transparency models,” Phys. Rev. A 87, 013823 (2013).
[Crossref]

2012 (2)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

F. De Martini and F. Sciarrino, “Colloquium: multiparticle quantum superpositions and the quantum-to-classical transition,” Rev. Mod. Phys. 84, 1765–1789 (2012).
[Crossref]

2011 (2)

A. Noguchi, Y. Eto, M. Ueda, and M. Kozuma, “Quantum-state tomography of a single nuclear spin qubit of an optically manipulated ytterbium atom,” Phys. Rev. A 84, 030301 (2011).
[Crossref]

P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning Autler-Townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107, 163604 (2011).
[Crossref]

2010 (2)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

2009 (5)

H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuletić, “Heralded single-magnon quantum memory for photon polarization states,” Phys. Rev. Lett. 103, 043601 (2009).
[Crossref]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
[Crossref]

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nat. Phys. 5, 100–104 (2009).
[Crossref]

H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, “Deterministic single-photon source from a single ion,” New J. Phys. 11, 103004 (2009).
[Crossref]

2008 (1)

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

2007 (4)

J. Simon, H. Tanji, S. Ghosh, and V. Vuletic, “Single-photon bus connecting spin-wave quantum memories,” Nat. Phys. 3, 765–769 (2007).
[Crossref]

N. Gisin, S. A. Moiseev, and C. Simon, “Storage and retrieval of time-bin qubits with photon-echo-based quantum memories,” Phys. Rev. A 76, 014302 (2007).
[Crossref]

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity,” Phys. Rev. A 76, 033835 (2007).
[Crossref]

D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vuckovic, “Controlling cavity reflectivity with a single quantum dot,” Nature 450, 857–861 (2007).
[Crossref]

2006 (2)

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: theory,” Phys. Rev. A 74, 013812 (2006).
[Crossref]

D. N. Matsukevich, T. Chanelière, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of remote atomic qubits,” Phys. Rev. Lett. 96, 030405 (2006).
[Crossref]

2005 (3)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005).
[Crossref]

L. Yang, L. Zhang, X. Li, L. Han, G. Fu, N. B. Manson, D. Suter, and C. Wei, “Autler-Townes effect in a strongly driven electromagnetically induced transparency resonance,” Phys. Rev. A 72, 053801 (2005).
[Crossref]

2002 (2)

C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A 65, 043805 (2002).
[Crossref]

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[Crossref]

2001 (2)

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
[Crossref]

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

2000 (2)

M. D. Lukin, S. F. Yelin, and M. Fleischhauer, “Entanglement of atomic ensembles by trapping correlated photon states,” Phys. Rev. Lett. 84, 4232–4235 (2000).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

1995 (2)

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995).
[Crossref]

Y.-Q. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51, R2703–R2706 (1995).
[Crossref]

1991 (1)

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

1972 (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[Crossref]

Adesso, G.

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen quantum coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Álvarez, G. A.

D. Suter and G. A. Álvarez, “Colloquium: protecting quantum information against environmental noise,” Rev. Mod. Phys. 88, 041001 (2016).
[Crossref]

Anisimov, P. M.

P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning Autler-Townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107, 163604 (2011).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Supplements

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Figures (4)

Fig. 1.
Fig. 1. (a) Effective three-level atomic system where states |1 and |2 are coupled to |3 by a continuous probe and a pulsed control field of Rabi frequencies (detunings): Ωp(Δp) and Ωc(Δc), respectively. Level |4 depicts additional states of neighboring ground-state manifold. Solid, horizontal arrows indicate co-propagating fields. (b) Experimental trace (dots) of EIT resonance with two-photon detuning (δ=ΔpΔc). Here, Iin, Itrans are the initial and transmitted probe intensities, respectively. Solid line is the Lorentz fit that corresponds to a line width of 34 kHz. (c) Schematic of experimental setup. ECDL, external cavity diode laser; Rb, rubidium vapor cell; P, probe (purple); C, control (red); R, repumper (dark yellow); Ram, Raman beam (green); AOM, acousto-optic modulator; Q, quarter-wave plate; H, half-wave plate; M, mirror; BS, 50:50 beam splitter; PBS, polarizing beam splitter; GT, Glan–Thompson polarizing beam cube; PD, photo detector; DSO, digital storage oscilloscope.
Fig. 2.
Fig. 2. (a) Evolution of the system in an effective Bloch-sphere corresponding to ground states |1 and |2 (see text). (b) Experimental trace (solid) of probe transmission, with a control field turned on (at t=0  μs) and off after 10 μs for an experimentally simulated, quasi-closed three-level system. Here, Δp=Δc=3.2γ3, Ωc=2.3×101γ3, and R=2.7×101γ3, where γ3=6.0(2π)  MHz is the radiative decay rate of |3. The level diagram in inset shows a counter-propagating repumper field (R) effectively closing the system. The dashed trace corresponds to numerical simulations. Frames (c) and (d) correspond to similar experimental and simulated traces for an open EIT system and an EIT configuration with counter-propagating fields. Ωc=2.3×101γ3 and 1.7×101γ3 for frames (c) and (d), respectively. Simulation parameters are chosen according to experiment.
Fig. 3.
Fig. 3. (a) EIT and Autler–Townes basis states in a Λ system. (b) Coherence quantifier C as a function of δ for varying control intensities. Near saturation intensity, the observed splitting in C is a signature of transition from EIT to ATS regime. Ωp=2.5×103γ3 and R=2.6×101γ3. Inset shows the steady-state probe transmission profile where any such signature is washed out due to power broadening. (c) Theoretically calculated coherence |ρ12ss| as a function of δ for varying control intensities. Here, Ωp=1.0×103γ3˜, 2Ωc=(I/2Isat)γ3˜, where γ3˜=1  MHz. (d) Experimentally measured C as a function of control intensity at δ=0. The red solid line is ΩpΩc/(Ωp2+Ωc2) fitting. Inset shows the variation of theoretically calculated |ρ12ss| with control intensity at δ=0. ID=IsatΓD/0.89γ3 is the saturation intensity for thermal atoms where ΓD=308  MHz (Supplement 1). Simulation parameters are chosen according to experiment.
Fig. 4.
Fig. 4. (a) Energy-level scheme for controlling coherence. Here, Ω and Ω+ are highly detuned Raman coherent fields with ϕR being the phase difference between them. Figure on the right shows the projection of Bloch vector in ρ12 plane for EIT and Raman field controlled coherence schemes, where ρ12EIT=ΩcΩp*/|Ωc|2 and ρ12R=iΩR*Γ3/|Ωc|2. Δϕ=(ϕcϕp)ϕR, where ϕp(c) is the phase of probe (control) field. (b) Sinusoidal variation of C as a function of ϕR. Circles and solid line show experimental and simulation results, respectively. Dashed (Cab) and dotted (Ccd) black lines indicate C at initial peak (region ab) and steady state (region cd) for EIT case. ΔC is the visibility of coherence, and ΔCo=CabCcd. ϕf corresponds to ϕR=120° (100° in simulation), at which C freezes to its initial maxima. (c) Experimental (solid) and simulated (dashed) time trace of probe transmission for EIT (red) and Raman field controlled coherence schemes at ϕf (blue). Here Ωc=2.5×101γ3, Ωp=3.5×103γ3, |Ω+|=|Ω|=3.7×101γ3, and ΔR=1.5ΓD. (d) ΔC as a function of ΔR with the dashed line showing 1/ΔR fit. Red dashed line indicates ΔCo for EIT system. Inset shows the simulations where Δρ12ss is the visibility in terms of ground-state coherence. Here, γex=γ3+γout as defined in Supplement 1.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ρ13ss=[iΩc*ρ12ssiΩp*(ρ11ssρ33ss)]/Γ13,
C=|ρ13Ωcon,ssρ13Ωcoff,ss||ρ13Ωcoff,ss||Ωp||Ωc|,
C=h1h2h2|Ωp||Ωc|.
Cclosed>Copen>Cincoherent.
ρ12ss=ΩcΩp*|Ωc|2+iΩR*Γ3|Ωc|2,

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