Discerning quantum memories based on electromagnetically-induced-transparency and Autler-Townes-splitting protocols

Electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) are similar, but different quantum optical phenomena: EIT results from a Fano interference, whereas ATS is described by the ac Stark effect. Likewise, despite their close resemblance, light-storage techniques based on the EIT memory protocol and the recently proposed ATS memory protocol [Saglamyurek et al., Nat. Photon. 12, 774 (2018)] are distinct: The EIT protocol relies on adiabatic elimination of absorption, whereas the ATS protocol is based on absorption. In this article, we elaborate on the distinction between EIT and ATS memory protocols through numerical analysis and experimental demonstrations in a cold rubidium ensemble. We find that their storage characteristics manifest opposite limits of the light-matter interaction due to their inherent adiabatic versus nonadiabatic nature. Furthermore, we determine optimal memory conditions for each protocol and analyze ambiguous regimes in the case of broadband storage, where nonoptimal memory implementations can possess characteristics of both EIT and ATS protocols. We anticipate that this investigation will lead to deeper understanding and improved technical development of quantum memories, while clarifying distinctions between the EIT and ATS protocols.

Figure 1

(a) A three-level $\mathrm{\Lambda }$-type atom or atomlike system, coupled by a weak signal field $E(z,t)$ between $|g\rangle$ and $|e\rangle$ levels and a strong control field with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{C}}$ between $|s\rangle$ and $|e\rangle$ levels. The atomic coherences are indicated as shaded regions with polarization coherence $\widehat{P}\sim |g\rangle \langle e|+\text{H.c}.$ (orange) and the spin coherence $\widehat{S}\sim |s\rangle \langle g|+\text{H.c}$. (blue). (b) Forward retrieval scheme with copropagating write and read control pulses. The output photonic mode is emitted in the propagation direction of the input mode. (c) Backward retrieval scheme with counterpropagating write and read control pulses. The output photonic mode is emitted at the input side of the medium.

Figure 2

[(a)–(c)] Spectral domain representation of signal delay, for the parameters given in panels (d)–(f). The signal bandwidth (central peak, red) relative to the medium's absorption profile (outer peaks, blue) determines the dominant mechanism that would induce delay (Sec. 2b2). Absorption and input profiles are given in units of optical density and the associated dispersion curves (black) are shown in the upper panel. [(d)–(f)] Memory implementations via the EIT and ATS protocols for narrow- and broadband signals. A Gaussian input probe (dark red) can be subjected to a predetermined delay (light red and dashed curve) under a constant control field or can be stored and subsequently retrieved after a desired storage time (light red and solid curve) via an interrupted control field (black). In each case, the parameters are adjusted to achieve a memory efficiency of $\eta =90\%$, corresponding to the optimal efficiency at $d=100$. (d) Narrowband EIT protocol with $B=0.014\mathrm{\Gamma }/2\pi , d=100$, and ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{pk}}=0.5\mathrm{\Gamma }$. (e) Broadband EIT protocol with $B=14\mathrm{\Gamma }/2\pi , d=600$, and ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{pk}}=55\mathrm{\Gamma }$. (f) Broadband ATS protocol with $B=14\mathrm{\Gamma }/2\pi , d=100$, and ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{pk}}=14\mathrm{\Gamma }$. The animations depicting the coherence dynamics in the storage and on-demand recall processes for the above configurations are available [27].

Figure 3

Temporal evolution of normalized coherences in the photonic ${\left|E(z,t)\right|}^2$ (red, upper), polarization ${\left|P(z,t)\right|}^2$ (orange, middle), and spin ${\left|S(z,t)\right|}^2$ (blue, lower) modes for the EIT protocol in the narrowband [(a), (d)] and broadband [(b), (e)] signal regimes, and for the ATS protocol in the broadband regime [(c), (f)]. The polarization coherence is normalized with respect to the maximum value of spin. The lightest shading shows coherence at the entrance to the medium ($z=0$), with progressively darker shading toward later sections at $L/5,L/2$, and $3L/4$. The coherence dynamics are analyzed for delay [(a)–(c)] and on-demand memory [(d)–(f)] operations in the forward retrieval scheme, such that the output photonic mode is emitted at $z=L$. The dashed line indicates the normalized ${\mathrm{\Omega }}_{\mathrm{C}}$ profile. The parameters $\{\tau ,B,d,{\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{pk}}\}$ are [(a), (d)] $\{100/\gamma ,0.014\mathrm{\Gamma }/2\pi , 40, 0.33\mathrm{\Gamma }\}$; [(b), (e)] $\{0.1/\gamma ,14\mathrm{\Gamma }/2\pi ,800,55\mathrm{\Gamma }\}$; and [(c), (f)] $\{0.1/\gamma ,14\mathrm{\Gamma }/2\pi ,40,14\mathrm{\Gamma }\}$. For the ATS protocol, the periodic exchange of coherence between $P$ and $S$ modes occurs at the control field Rabi frequency.

Figure 4

Normalized memory-character factor $\tilde{\mathcal{C}}$ as a function of optical depth for various signal bandwidths, extending from the narrow ($B=0.0007\mathrm{\Gamma }/2\pi$) to the broadband regime ($B=136\mathrm{\Gamma }/2\pi$) for the EIT protocol. Pure EIT and ATS storage correspond to $\tilde{\mathcal{C}}\lesssim 0.1$ (zone III) and $\tilde{\mathcal{C}}\gtrsim 1$ (zone I), respectively. The progression from light to dark shading represents transition from EIT to ATS storage. For narrow bandwidths (red, orange, yellow lowest curves), the values of $\tilde{\mathcal{C}}$ stay well below the threshold. For improperly optimized EIT protocols, the memory operation lies within zone II, indicating mixed storage due to the presence of both EIT and ATS mechanisms. Optimal ATS operation is shown by $\tilde{\mathcal{C}}=1$ (dashed line).

Figure 5

Optical-depth dependence of the delay characteristics in (a) EIT and (b) ATS protocols for a broadband signal with $B=7\mathrm{\Gamma }/2\pi$ under forward retrieval. Color bar indicates the normalized intensity of delayed pulses. (a) A constant control with ${\mathrm{\Omega }}_{\mathrm{C}}=28\mathrm{\Gamma }$ results in a slow-light-mediated group delay which monotonically increases with the optical depth. Here, the signal delay is smooth and continuous unlike the discretized delays in the ATS protocol. (b) A constant control with ${\mathrm{\Omega }}_{\mathrm{C}}=2\pi B=7\mathrm{\Gamma }$ generates a fixed delay of one Rabi period ${\tau }_{\mathrm{D}}=2\pi /{\mathrm{\Omega }}_{\mathrm{C}}\approx 24\text{ns}$ for all $d\lesssim 3F$. Large optical depths result in the suppression of first-order “echo” (reabsorption) followed by higher order emissions occurring at integer multiples of the Rabi period.

Figure 6

Mixed EIT-ATS delay in the intermediate signal regime by varying the control strength in the range $B\le {\mathrm{\Omega }}_{\mathrm{C}}/2\pi <2.5B$, where $B=14\mathrm{\Gamma }/2\pi$ and optical depth $d=30$. (a) The ATS delay (blue, front curve), using the forward retrieval scheme, is shown as a reference. Upon increasing ${\mathrm{\Omega }}_{\mathrm{C}}$, the distinct transmitted and delayed peaks tend to merge toward each other, indicating a mixed storage character due to both EIT- and ATS-based delay mechanisms. For ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi \gtrsim 2B$, the two peaks are no longer separate and EIT delay starts to dominate. (b) The associated delay character values. Transition from ATS- to EIT-based delay is shown by the decrease in $\widetilde{{\mathcal{C}}_{\mathrm{D}}}$ values.

Figure 7

Resource scaling with respect to the bandwidth for EIT (red, dash-dotted) and ATS (blue, dashed) protocols along with the narrow-band EIT scaling (green, dotted) applied in the broadband regime. (a) Control power scaling, (b) character ratio scaling, (c) optical depth scaling, and (d) efficiency scaling. For each bandwidth, ${\mathrm{\Omega }}_{\mathrm{C}}$ and $d$ are adapted to maintain a memory efficiency $\eta =90\%$. For narrowband signals, the EIT protocol gives the optimal efficiency for $d=100$ and ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{EIT}}\propto \sqrt{B}$. Maintaining the same ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{EIT}}$ scaling (with fixed $d$) for broadband signals results in a nonoptimal ATS-based storage as shown by the dramatic rise in $\tilde{\mathcal{C}}$ and drop in the efficiency values (green). The shaded region, where $B\sim \mathrm{\Gamma }/2\pi$, indicates the crossover between EIT and ATS phenomena, as well as between the protocols. For true broadband EIT memory, we set ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{EIT}}/2\pi =4B$ and scale $d$ such that ${\tau }_{\mathrm{D}}^{\mathrm{EIT}}/\tau \approx 2$, thereby maintaining $\tilde{\mathcal{C}}$ at its threshold limit. For the broadband ATS protocol, ${\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{ATS}}/2\pi =B$ and the effective optical depth is fixed such that $d=6F$.

Figure 8

Transition from the EIT to ATS storage mechanism by variation of the temporal profile of the control field in the broadband regime. Here, $B=7\mathrm{\Gamma }/2\pi , {\mathrm{\Omega }}_{\mathrm{C}}^{\mathrm{pk}}=28\mathrm{\Gamma }$, and $d=60$. (a) Writing stage control field profiles, characterized by different slopes during the switch-off (dashed). Also shown are the Gaussian input (shaded gray) and the leaked (for EIT-dominated storage, as in configurations 1–3) or transmitted (for ATS-dominated storage, as in configurations 4–8) pulses. “Configuration 1” (top) corresponds to true-EIT storage with $\tilde{\mathcal{C}}=0.1$ and $\eta =20\%$. Steeper switch-off leads to coherent absorption and hence a mixed-character memory, as seen by an increase in both $\tilde{\mathcal{C}}$ and $\eta$, moving from top to bottom. (b) $\tilde{\mathcal{C}}$ values (diamonds) and memory efficiencies (circles) for different control profiles in panel (a).

Figure 9

Adiabatic vs nonadiabatic character in optimal EIT and ATS memories. (a) Efficiency at fixed bandwidth [$B=5\mathrm{\Gamma }/2\pi$ (dashed) and $10\mathrm{\Gamma }/2\pi$ (dash-dotted)] vs optical depth using the EIT (pink, light) and ATS (blue, dark) protocols, along with the maximum achievable efficiency (solid gray curve) [26] at each $d$. The ATS intersection (also shown in the inset) and EIT merger points are indicated by arrows. (b) Optimality at fixed optical depth values [$d=50$ (dashed) and $250$ (dash-dotted)] vs bandwidth. Efficiency is normalized with respect to the optimal efficiency for each optical depth. For moderate optical depths, the EIT curve (pink, light) starts to deviate from optimality around $B\approx \mathrm{\Gamma }/2\pi$, with a rapid drop in efficiency thereafter. In the ATS protocol, each optical depth is related to a unique bandwidth mode (in the broadband regime) for optimal storage. (c) Normalized efficiency (indicated by color or shading) for varying bandwidths and optical depth in the EIT protocol depicting the full topology for optimal or nonoptimal operations. (d) As in panel (c), for the ATS protocol.

Figure 10

Experimental demonstration of EIT and ATS memories in ${}^{87}\mathrm{Rb}$ atoms for narrowband [$B=0.25\mathrm{\Gamma }/2\pi$, (a) and (d)] and broadband [$B=2.5\mathrm{\Gamma }/2\pi$, (b), (c), (e), and (f)] signals. [(a)–(c)] Measured spectra of the absorption profile of the atomic medium (blue, outer peaks) and the input signal (red, inner peaks) for (a) ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =7.5\mathrm{MHz}$, (b) ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =45\mathrm{MHz}$, and (c) ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =15\mathrm{MHz}$. [(d)–(f)] The time domain picture of signal delay and storage-retrieval processes associated with the top panels; coherence dynamics in the middle panels; and input (gray), delayed (light red, dashed), transmitted or leaked (dark red, solid), and on-demand recall (dark red, solid) signals in the bottom panels. The coherences are shown at a given slice of the medium: $z=L/2$ for EIT memories and $z=0$ for ATS memory. The control power and Rabi frequency are related as ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =\alpha \sqrt{P}$, where $\alpha =5.3\mathrm{MHz}/\sqrt{\mathrm{mW}}$.

Figure 11

Delay characterization for the narrowband EIT protocol. The Gaussian input (shaded) has a bandwidth $B_{\mathrm{narrow}}=1.62\mathrm{MHz}\approx 0.25\mathrm{\Gamma }/2\pi$ and the delay is measured as the difference between the peak positions of input and delayed pulses. (a) Delayed pulses for varying control powers (dark to light is low to high power). (b) Measured delay times vs ${\mathrm{\Omega }}_{\mathrm{C}}$ (circles) and calculated ${\tau }_{\mathrm{D}}^{\mathrm{EIT}}$ (curve, assuming $d=9$).

Figure 12

Delay characterization for EIT, mixed, and ATS protocols in the broadband signal regime. (a) EIT delay for control Rabi frequencies in the range $2.5B<{\mathrm{\Omega }}_{\mathrm{C}}/2\pi <4B$ (here and in all parts, ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =5.3\mathrm{MHz}\sqrt{P\left[\mathrm{mW}\right]}$, where $P$ is indicated in legends.) (b) Mixed EIT-ATS delay for control Rabi frequencies $1.2B<{\mathrm{\Omega }}_{\mathrm{C}}/2\pi <2.2B$. (c) ATS delay for control Rabi frequencies $0.7B<{\mathrm{\Omega }}_{\mathrm{C}}/2\pi <1.3B$, where the ATS peaks' separation is close to the signal bandwidth. In panels (a)–(c), the power of the curves is indicated by color (online) and is found such that the higher power curves always have higher peak signals. (d) Complilation of data in panels (a), (b), and (c) (same color coding), with the peak locations indicated in the lower plane: dark-filled circles indicate ATS peaks and light-filled are the EIT maxima. (e) Signal delay vs control Rabi frequency, with diamonds corresponding to EIT delays in panel (a), circles to mixed character in panel (b), and squares to ATS delays in panel (c). Inset shows delay definitions: ATS delay is the time between the transmitted and delayed peaks, and EIT delays are defined between the input signal peak and the output peak, with an estimate for the peak “envelope” made in the cases where ATS character is significant.

Figure 13

Experimental demonstration of the transition from EIT- to ATS-based broadband storage by varying the control temporal profile. Here $B=16.3\mathrm{MHz}$ and ${\mathrm{\Omega }}_{\mathrm{C}}/2\pi =44\mathrm{MHz}$. (a) Writing stage showing control fields at different switch-off (ramp down) positions. The leaked and transmitted signals correspond to the EIT- and ATS-dominated storage. (b) $\tilde{\mathcal{C}}$ values and the memory efficiencies $\eta$ for different control profiles. “Configuration 3” bears the maximum EIT character with minimal value of $\tilde{\mathcal{C}}=0.19$ corresponding to $\eta =10\%$. While “configuration 6” gives the highest efficiency of $19\%$, it benefits from the ATS storage ($\tilde{\mathcal{C}}=0.35$), and the efficiency remains below the measured efficiency of $23\%$ (blue right-pointing arrow) for optimal ATS operation.