## 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 [1–6]. Even entanglement [7–10], 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 [11–13] and to generate and control them experimentally [14–21]. A widely used technique to generate stable superposed states in atom-like systems [22,23] is electromagnetically induced transparency (EIT) [14–17], where a strong control field is used to drive an effective three-level atomic system into a specific coherent superposition of ground states $|1\u27e9$ and $|2\u27e9$ [Fig. 1(a)]. These superposed *dark states* remain mostly decoupled from the lossy excited state ($|3\u27e9$) [14–17] leading to dramatic effects such as slow [24], stopped [25], and stored [25–27] light, generation of entangled photons [7–10], and enhanced optical nonlinearities at the level of single photons [28–30]. 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.

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 $\mathrm{\Lambda}$ 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 [25–27]. 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 ${\rho}_{12}$ 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 ${\rho}_{12}$, i.e., ground-state coherence in EIT for an ensemble of rubidium (${}^{85}\mathrm{Rb}$) 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 [11–13]. 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 [39–43].

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 (${\rho}_{13}$) that drives the probe transmission and takes a form (Supplement 1)

## 2. METHODS

We use a degenerate $\mathrm{\Lambda}$ system defined within ${D}_{2}$ transition of ${}^{85}\mathrm{Rb}$ atoms as shown in Fig. 1(a), where $|1\u27e9\equiv |F=2,{m}_{F}\u27e9$, $|2\u27e9\equiv |F=2,{m}_{F}-2\u27e9$, and $|3\u27e9\equiv |{F}^{\prime}=1,{m}_{F}-1\u27e9$. A schematic of the experimental setup is shown in Fig. 1(c). Two orthogonal circularly polarized beams—a continuous ${\sigma}_{-}$ probe and a pulsed ${\sigma}_{+}$ control beam—are used to drive the transitions $|1\u27e9\to |3\u27e9$ and $|2\u27e9\to |3\u27e9$, respectively. These are derived from a single laser locked at red detuning of 19 MHz with respect to $|F=2\u27e9\to |{F}^{\prime}=1\u27e9$. We experimentally simulate a quasi-*closed* three-level system with a counter-propagating continuous *repumper* field (resonant and locked at a transition $F=3\Rightarrow {F}^{\prime}=3$) 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 $\mu $-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 $\sim 4\text{\hspace{0.17em}}\mathrm{mm}$. 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 $\sim 150\text{\hspace{0.17em}}\mathrm{ns}\gg 1/{\gamma}_{3}$). 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 $\mathrm{\Lambda}$ system is shown in Fig. 2(b) along with numerical simulations. When the control field is turned on, there is a sharp rise $a-b$, followed by a slower decay $b-c$. In a recent work [6], we have shown that this corresponds to an initial fast buildup of ground-state coherence ${\rho}_{12}$, followed by optical pumping rearranging the populations ${\rho}_{11}$ and ${\rho}_{22}$.

It is particularly intuitive to visualize this dynamics in an effective Bloch sphere corresponding to the ground-state manifold spanned by $|1\u27e9$ and $|2\u27e9$. 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 $|1\u27e9$, the steady-state Bloch vector points mostly down, with a slight tilt due to a small coherence, ${\rho}_{12}^{ss}\sim -{\mathrm{\Omega}}_{p}/{\mathrm{\Omega}}_{c}$. It can also be noted that though ${\rho}_{12}^{ss}$ depends on the relative phase between control and probe setting the angle of the tilt in the equatorial plane, ${\rho}_{13}^{ss}$ is independent of it (due to the additional excited state $|3\u27e9$). We use this visualization in Section 5 for phase coherent control of ${\rho}_{12}$.

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 ${\rho}_{11}^{ss}-{\rho}_{22}^{ss}$. We use the fall height from point ($c(h1)$ to $d(h2)$, to estimate the quantifier

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 ${C}_{\mathrm{open}}$ [Fig. 2(c)] as compared to ${C}_{\mathrm{closed}}$ [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 ${C}_{\mathrm{incoherent}}=0$ [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 ($\sim 10\text{\hspace{0.17em}}\mathrm{\mu s}$) [6].

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

## 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 ($|2\u27e9$) and excited ($|3\u27e9$) 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].

The quantifier $C$ precisely subtracts out this one-photon background [i.e., $h2$ at $d$ in Fig. 2(a)], corresponding to the term ${\rho}_{11}^{ss}-{\rho}_{33}^{ss}$, which hardly evolves during control turn-off. $C$ can thereby be viewed as distilling the information of ${\rho}_{12}^{ss}$ 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 ${\rho}_{12}$, 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 $(\mathrm{Im}({\rho}_{13}))$ has been studied extensively [44–51]. 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 [${\mathrm{\Delta}}_{R}$ 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)

Experimentally, we use an additional counter-propagating linearly polarized laser beam for the Raman fields ${\mathrm{\Omega}}^{+}$ and ${\mathrm{\Omega}}^{-}$. The relative phase difference (${\varphi}_{R}$) 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 ${\varphi}_{R}$ [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 ${\rho}_{12}$ corresponding to ${C}_{a-b}$ eventually decays down due to optical pumping to ${C}_{c-d}$ [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 ${\varphi}_{R}\sim 120\xb0({\varphi}_{f})$, the modified steady-state coherence $C$ freezes to the transient maximum ${C}_{a-b}$ [Fig. 4(c)]. In particular, this phase dependence [Fig. 4(b)] indicates that one can overcompensate, with the visibility ($\mathrm{\Delta}C$) getting larger than the difference between initial and final steady-state coherence ($\mathrm{\Delta}{C}_{o}={C}_{a-b}-{C}_{c-d}$). Furthermore, with increasing detuning ${\mathrm{\Delta}}_{R}$, 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 [14–17], 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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