# Understanding of Collective Atom Phase Control in Modified Photon Echoes for a Near-Perfect Storage Time-Extended Quantum Memory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Theory: Maxwell–Bloch Equations

#### 1.1.1. A. Conventional Two-Pulse Photon Echo

_{1}) and the second rephasing (R

_{2}) pulses were resonant to the transition of $|1\rangle -|2\rangle $ to satisfy the requirements of the DR photon echo scheme [10,11,12,13,14,15,16,17]. The control pulses, C

_{1}and C

_{2}, were resonant between states $|2\rangle $ and $|3\rangle $ to enable CCC [14,15,16,17,18,19]. Initially, all atoms of the medium were in the ground state, $|1\rangle $. For an ideal system, all decay rates were neglected, unless specified otherwise. All light pulses were collinear (or near collinear) in the z-axis. To make the first echo (E

_{1}) silent to avoid affecting the final echo (E

_{2}), both rephasing pulse propagation directions were set to be opposite (backward) to that of the D pulse [10]. To satisfy the backward photon echo condition, the control pulses were set to counter-propagate each other [19]: ${\overrightarrow{k}}_{{\mathrm{E}}_{2}}={\overrightarrow{k}}_{{\mathrm{C}}_{1}}+{\overrightarrow{k}}_{{\mathrm{C}}_{2}}-{\overrightarrow{k}}_{\mathrm{D}}$; each pulse j was characterized by a wave vector ${\overrightarrow{k}}_{\mathrm{j}}$. Unlike phase mismatching in silent echo (E

_{1}) formation [10], which is determined by D and R

_{1}, the final echo (E

_{2}) propagation direction (${\overrightarrow{k}}_{{\mathrm{E}}_{2}}$) is determined by the control pulses [14]. The MB equations for the atomic coherence and the D pulse were respectively denoted, as follows:

_{D}is the data pulse, α is the optical depth parameter, also known as the attenuation coefficient with unit m

^{−1}, and $\Delta $ is the detuning of the atom (see Supplementary Materials). The ensemble was inhomogeneously broadened by ${\mathsf{\Delta}}^{\prime}={\sum}_{j}{\mathsf{\Delta}}_{j}$. Here, we considered the case of a detuned atom for simplicity. Thus, the solution for atomic coherence is given by:

_{1}) arrives at $t={t}_{{\mathrm{R}}_{1}}.$ The R

_{1}pulse has a π pulse area, and its propagation direction is opposite to that of the D pulse (${\overrightarrow{k}}_{{\mathrm{R}}_{1}}=-{\overrightarrow{k}}_{\mathrm{D}}$). Here, the function of the R

_{1}pulse is to swap the populations between states $|1\rangle $ and $|2\rangle $: ${\sigma}_{11}(z,{t}_{\mathrm{D}},\Delta )\stackrel{{\mathrm{R}}_{1}}{\leftrightarrow}{\sigma}_{22}(z,{t}_{{\mathrm{R}}_{1}},\Delta )$. With the weak D pulse, the swapping result is ${\sigma}_{22}(z,t,\Delta )=1\mathrm{and}{\sigma}_{11}(z,t,\Delta )=0$. Thus, the atomic coherence arising from the application of the R

_{1}pulse becomes:

_{1}at $t={t}_{{\mathrm{E}}_{1}}=2{t}_{{\mathrm{R}}_{1}}-{t}_{\mathrm{D}}$. The propagation direction of the first echo, E

_{1}, determined by the first rephasing pulse, R

_{1}, was ${\overrightarrow{k}}_{{\mathrm{E}}_{1}}=2{\overrightarrow{k}}_{{\mathrm{R}}_{1}}-{\overrightarrow{k}}_{\mathrm{D}}=-3{\overrightarrow{k}}_{\mathrm{D}}$. Due to the phase mismatch between D and E

_{1}, the echo signal (macroscopic coherence) could not be generated due to complete out of phase [10]: silent echo.

#### 1.1.2. B. DR Echo

_{2}) π pulse was followed by E

_{1}to satisfy the requirements of the DR photon echo scheme. In the DR scheme, the excited state population after the final echo, E

_{2}, was the same as that after the D pulse excitation. This means that all excited-state atoms in the DR scheme should contribute to the echo signal without contributing to quantum noises. The MB equation for DR was similar to that for D, and the optical coherence solution for the final echo, E

_{2}, is:

_{2}, was formed at $t={t}_{{\mathrm{E}}_{2}}$ as the rephasing result of E

_{1}by R

_{2}, and its propagation direction was forward if ${\overrightarrow{k}}_{{\mathrm{R}}_{2}}={\overrightarrow{k}}_{{\mathrm{R}}_{1}}$: ${\overrightarrow{k}}_{{\mathrm{E}}_{2}}=2{\overrightarrow{k}}_{{\mathrm{R}}_{2}}-{\overrightarrow{k}}_{{\mathrm{E}}_{1}}=2{\overrightarrow{k}}_{{\mathrm{R}}_{2}}-(2{\overrightarrow{k}}_{{\mathrm{R}}_{1}}-{\overrightarrow{k}}_{\mathrm{D}})={\overrightarrow{k}}_{\mathrm{D}}$. However, the retrieved signal, E

_{2}, was absorbed by the medium due to absorptive coherence, as shown in Equation (8). In other words, echo E

_{2}could not be radiated from the medium. This fact has already been explored numerically [14,15,16] and analytically [17]. From now on, we discuss and correct critical mistakes in previous analyses [8,9,10,11,12,13]. The observation of DR echoes [10,11,12,13] has been understood as a coherence leakage due to an imperfect rephasing process by Gaussian-distributed light in a transverse mode, resulting in the leakage-caused maximum retrieval efficiency at 26% [22].

## 2. Discussion

#### 2.1. A. CDR Echo

_{1}and C

_{2}were added, as shown in Figure 1b [14]. As discussed in references [14,15,16,17,20], the function of the control pulses is not only to convert the coherence between optical and spin states via population transfer, but also to induce a collective phase shift. Unlike the DR scheme of Equation (8), the propagation vector of E

_{2}can be controlled to be backward if ${\overrightarrow{k}}_{{\mathrm{C}}_{1}}=-{\overrightarrow{k}}_{{\mathrm{C}}_{2}}$: ${\overrightarrow{k}}_{{\mathrm{E}}_{2}}={\overrightarrow{k}}_{{\mathrm{C}}_{1}}+{\overrightarrow{k}}_{{\mathrm{C}}_{2}}-{\overrightarrow{k}}_{\mathrm{D}}=-{\overrightarrow{k}}_{\mathrm{D}}$ [14,21]. Here, the rephasing pulses had nothing to do with the phase matching for ${\overrightarrow{k}}_{{\mathrm{E}}_{2}}$. Unlike the forward echo in the conventional two-pulse photon echo scheme, which suffers from reabsorption by noninteracting atoms, according to Beer’s law, the backward echo E

_{2}is free from reabsorption, resulting in near-perfect echo efficiency [1,2,21]. To eliminate any potential two-photon coherence between states $|1\rangle \mathrm{and}|3\rangle $, the C

_{1}pulse was delayed by δT from R

_{1}. Here, it should be noted that the macroscopic two-photon coherence was sustained only within the overall optical coherence time determined by the inverse of the atom broadening ${\mathsf{\Delta}}^{\prime}$. However, individual atom coherence was sustained regardless of atom broadening for the optical phase decay time T

_{2}, where ${\mathrm{T}}_{2}\gg \mathsf{\Delta}\mathrm{T}$ [20]. Here, the D pulse duration was practically comparable to (or a bit longer than) $1/{\mathsf{\Delta}}^{\prime}$. Thus, by simply neglecting $\delta \mathrm{T}$, the atomic coherence at ${t}_{{\mathrm{C}}_{1}}$ can be expressed as:

_{1}, is silent, and the second echo, E

_{2}, is not emitted, owing to its absorptive coherence, we can remove the last two terms of Equation (8). With a π C

_{1}pulse, optical and spin coherence, respectively, satisfy the following relations: ${\sigma}_{12}(z,t,\Delta )=\mathrm{cos}(\frac{\pi}{2}){\sigma}_{12}(z,{t}_{{\mathrm{C}}_{1}},\Delta )=0$; ${\sigma}_{13}(z,t,\Delta )={e}^{i\pi /2}{\sigma}_{12}(z,{t}_{{\mathrm{C}}_{1}},\Delta )$ (see references. [14,15,16,17,20]). The C

_{1}pulse induced a π/2 phase shift and locked the optical coherence until C

_{2}arrived. Here, the novel property of the π/2 phase shift by the π pulse area of C

_{1}(or C

_{2}) originated in the Raman (two-photon) coherence, where a 2π Raman pulse in a three-level system actually played as a π pulse does in a two-level system: see Figures 3 and 4 of reference [25] and Figure 4 of reference [26]. There is no way to expect this novel property from the MB approach. Without a priori understanding of the coherence behavior in a three-level system, the same mistake has been repeated in the controlled AFC echoes [8,9]; this will be discussed further in section B.

_{2}pulse is:

_{2}) into Equation (S4) of the Supplementary Information. The solution of Equation (10) is:

_{2}pulse also brings a π/2 phase shift, while swapping the spin and optical coherence:

_{2}is emissive due to the π phase shift induced by the control pulse pair, which has already been extensively explored numerically [14,15,16] and analytically [17]. Thus, the echo E

_{2}propagates backward without reabsorption and is radiated out of the medium with near-perfect retrieval efficiency; this will be discussed further in Figure 2.

#### 2.2. A. Single Rephased Photon Echo

_{2}= 0 in Figure 1 [20,21]. This scheme itself cannot be used for quantum memory because the population inversion has not been solved yet. However, this scheme itself implies the AFC echo scheme, whereby all excited-state atoms freely decay into a dump state. The AFC scheme can be easily obtained by swapping the pulse sequence of D and R

_{1}and substituting R

_{1}with a repeated weak two-pulse train (see Supplementary Information). Therefore, by neglecting the population constraint issue without R

_{2}, the final optical coherence could be obtained by applying the control pulses to Equation (6), as follows (see Section III of the Supplementary information):

_{2}pulse area was increased to 3π, i.e., ${\sigma}_{12}(z,t,\Delta )\stackrel{{\mathrm{C}}_{1}(\pi )}{\to}{e}^{-(\frac{\pi}{2})i}{\sigma}_{13}(z,t,\Delta )\stackrel{{\mathrm{C}}_{2}(\pi )}{\to}{e}^{\pi i}{\sigma}_{12}(z,t,\Delta )\stackrel{{\mathrm{C}}_{2}(\pi )}{\to}{e}^{-(\frac{3\pi}{2})i}{\sigma}_{13}(z,t,\Delta )\stackrel{{\mathrm{C}}_{2}(\pi )}{\to}{e}^{2\pi i}{\sigma}_{12}(z,t,\Delta )$, and thus ${\sigma}_{12}(z,t,\Delta )$ was negative. This means that the π–π control pulse sequence in references [8,9] must be corrected to be a π–3π control pulse sequence. The π–π control pulse sequence in reference [1] uses a Doppler effect–caused π-phase shift, resulting in an emissive echo. This is not the case for a solid medium [2,14,20]. However, the observation of controlled AFC echoes with a π–π control pulse set [8,9] is due to the imperfect rephasing by commercial light sources with Gaussian spatial distributions [22]. The coherence leakage in a DR scheme induced by the Gaussian pulse limits the echo efficiency to ~10% on average, regardless of the rephasing pulse area, whereas its maximum efficiency reaches 26% for a π/2 pulse area in a DR scheme [22].

#### 2.3. A. Near Perfect Retrieval Efficiency in CDR

_{2}was emitted in the backward direction, the atomic coherence and optical field in the backward direction could be respectively represented, as follows [27]:

_{b}(L, ω) = 0 because there is no field at z = L. Assuming an ideal case of complete absorption at z = L, solves Equations (19) and (16) yields:

_{2}is given by:

_{2}can be obtained by setting ${\overrightarrow{k}}_{{\mathrm{C}}_{1}}={\overrightarrow{k}}_{{\mathrm{C}}_{2}}$, where ${\overrightarrow{k}}_{{\mathrm{C}}_{1}}={\overrightarrow{k}}_{\mathrm{D}}$: ${\overrightarrow{k}}_{{\mathrm{E}}_{2}}={\overrightarrow{k}}_{{\mathrm{C}}_{1}}+{\overrightarrow{k}}_{{\mathrm{C}}_{2}}-{\overrightarrow{k}}_{\mathrm{D}}=2{\overrightarrow{k}}_{{\mathrm{C}}_{1}}-{\overrightarrow{k}}_{\mathrm{D}}={\overrightarrow{k}}_{\mathrm{D}}$. Here, the difference frequency between C

_{1}and D is just ~10 MHz, which is about 10

^{−8}times the frequency of D in a rare-earth doped solid. The atomic coherence and the forward field satisfy the following relations:

_{2}is (αL)

^{2}e

^{−αL}(see the dotted curve in Figure 2). The forward echo E

_{2}suffers from reabsorption.

## 3. Conclusions

## Supplementary Materials

_{j}is the arrival time of pulse j.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Moiseev, S.A.; Kröll, S. Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a Doppler-broadened transition. Phys. Rev. Lett.
**2001**, 87, 173601. [Google Scholar] [CrossRef] [PubMed] - Moiseev, S.A.; Tarasov, V.F.; Ham, B.S. Quantum memory photon echo-like techniques in solids. J. Opt. B Quantum Semiclass Opt.
**2003**, 5, S497–S502. [Google Scholar] [CrossRef] - Kraus, B.; Tittel, W.; Gisin, N.; Nilsson, M.; Kröll, S.; Cirac, J.I. Quantum memory for nonstationary light fields based on controlled reversible inhomogeneous broadening. Phys. Rev. A
**2006**, 73, 020302. [Google Scholar] [CrossRef][Green Version] - Alexander, A.L.; Longdell, J.J.; Sellars, M.J.; Manson, N.B. Photon echoes produced by switching electric fields. Phys. Rev. Lett.
**2006**, 96, 043602. [Google Scholar] [CrossRef][Green Version] - Hedges, M.P.; Longdell, J.J.; Li, Y.; Sellars, M.J. Efficient quantum memory for light. Nature
**2010**, 465, 1052–1056. [Google Scholar] [CrossRef] - De Riedmatten, H.; Afzelius, M.; Staudt, M.U.; Simon, C.; Gisin, N. A solid-state light-matter interface at the single-photon level. Nature
**2008**, 456, 773–777. [Google Scholar] [CrossRef] - Saglamyurek, E.; Sinclair, N.; Jin, J.; Slater, J.A.; Oblak, D.; Bussieres, F.; George, M.; Ricken, R.; Sohler, W.; Tittel, W. Broadband waveguide quantum memory for entangled photons. Nature
**2011**, 469, 512–515. [Google Scholar] [CrossRef][Green Version] - Afzelius, M.; Usmani, I.; Amari, A.; Lauritzen, B.; Walther, A.; Simon, C.; Sangouard, N.; Minář, J.; de Riedmatten, H.; Gisin, N.; et al. Demonstration of atomic frequency comb memory for light with spin-wave storage. Phys. Rev. Lett.
**2010**, 104, 040503. [Google Scholar] [CrossRef][Green Version] - Gündoğan, M.; Mazzera, M.; Ledingham, P.M.; Cristiani, M.; de Riedmatten, H. Coherent storage of temporally multimode light using a spin-wave atomic frequency comb memory. New J. Phys.
**2013**, 15, 045012. [Google Scholar] [CrossRef][Green Version] - Damon, V.; Bonarota, M.; Louchet-Chauvet, A.; Chanelière, T.; Le Gouët, J.-L. Revival of silenced echo and quantum memory for light. New. J. Phys.
**2011**, 13, 093031. [Google Scholar] [CrossRef] - Dajczgewand, J.; Ahlefeldt, R.; Böttger, T.; Louchet-Chauvet, A.; Le Gouët, J.-L.; Chanelière, T. Optical memory bandwidth and multiplexing capacity in the erbium telecommunication window. New J. Phys.
**2015**, 17, 023031. [Google Scholar] [CrossRef][Green Version] - Arcangeli, A.; Ferrier, A.; Goldner, P. Stark echo modulation for quantum memories. Phys. Rev. A
**2016**, 93, 062303. [Google Scholar] [CrossRef][Green Version] - McAuslan, D.L.; Ledingham, P.M.; Naylor, W.R.; Beavan, S.E.; Hedges, M.P.; Sellars, M.J.; Longdell, J.J. Photon-echo quantum memories in inhomogeneously broadened two-level atoms. Phys. Rev. A
**2011**, 84, 022309. [Google Scholar] [CrossRef][Green Version] - Ham, B.S. Atom phase controlled noise-free photon echoes. arXiv
**2011**, arXiv:1101.5480v2. [Google Scholar] - Ham, B.S. Coherent control of collective atom phase for ultralong, inversion-free photon echoes. Phys. Rev. A
**2012**, 85, 031402(R). [Google Scholar] [CrossRef][Green Version] - Ham, B.S. A controlled ac Stark echo for quantum memories. Sci. Rep.
**2017**, 7, 7655. [Google Scholar] [CrossRef][Green Version] - Ullah, R.; Ham, B.S. Analysis of Controlled Coherence Conversion in a Double Rephasing Scheme of Photon Echoes for Quantum Memories. arXiv
**2016**, arXiv:1612.02167v2. [Google Scholar] - Kalachev, A.A.; Karimullin, K.; Samartsev, V.; Zuikov, V. Optical echo-spectroscopy of highly doped Tm:YAG. Laser Phys. Lett.
**2008**, 5, 882. [Google Scholar] [CrossRef] - Samartsev, V.V.; Shegeda, A.M.; Shkalikov, A.V.; Karimullin, K.R.; Mitrofanova, T.G.; A Zuikov, V. Incoherent backward photon echo in ruby upon excitation through an optical fiber. Laser Phys. Lett.
**2007**, 4, 534. [Google Scholar] [CrossRef] - Ham, B.S. Control of photon storage time using phase locking. Opt. Exp.
**2010**, 18, 1704. [Google Scholar] [CrossRef] - Hahn, J.; Ham, B.S. Rephasing halted photon echoes using controlled optical deshelving. New J. Phys.
**2011**, 13, 093011. [Google Scholar] [CrossRef] - Ham, B.S. Gaussian beam profile effectiveness on double rephasing photon echoes. arXiv
**2017**, Entropy (to be published). arXiv:1701.04291. [Google Scholar] - Steane, A.M. Efficient fault-tolerant quantum computing. Nature
**1999**, 399, 124. [Google Scholar] [CrossRef] - Lo, H.-K.; Curty, M.; Qi, B. Measurement-Device-Independent Quantum Key Distribution. Phys. Rev. Lett.
**2012**, 108, 130503. [Google Scholar] [CrossRef][Green Version] - Ham, B.S. Collective atom phase controls in photon echoes for quantum memory applications I: Population inversion removal. Adv. Appl. Sci. Res.
**2018**, 9, 32–46. [Google Scholar] - Ham, B.S.; Shahriar, M.S.; Kim, M.K.; Hemmer, P.R. Spin coherence excitation and rephasing with optically shelved atoms. Phys. Rev. B
**1998**, 58, R11825. [Google Scholar] [CrossRef][Green Version] - Sangouard, N.; Simon, C.; Afzelius, M.; Gisin, N. Analysis of a quantum memory for photons based on controlled reversible inhomogeneous broadening. Phys. Rev. A
**2007**, 75, 032327. [Google Scholar] [CrossRef][Green Version] - Zhong, M.; Hedges, M.; Ahlefeldt, R.; Bartholomew, J.G.; Beavan, S.E.; Wittig, S.M.; Longdell, J.J.; Sellars, M.J. Optical addressable nuclear spin in a solid with a six-hour coherence time. Nature
**2015**, 517, 177. [Google Scholar] [CrossRef]

**Figure 1.**Schematics of controlled double rephasing (CDR) echoes. (

**a**) Energy level diagram for the CDR echo. The short black arrows indicate the pulse propagation direction. (

**b**) Pulse sequence for (

**a**), where t

_{j}is the arrival time of pulse j.

**Figure 2.**Efficiency of CDR echoes versus optical depth αL. The solid curve is for the backward final echo (E

_{2}): see Equation (22). The dotted curve is for the forward echo E

_{2}: (αL)

^{2}e

^{−αL}.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ullah, R.; Ham, B.S.
Understanding of Collective Atom Phase Control in Modified Photon Echoes for a Near-Perfect Storage Time-Extended Quantum Memory. *Entropy* **2020**, *22*, 900.
https://doi.org/10.3390/e22080900

**AMA Style**

Ullah R, Ham BS.
Understanding of Collective Atom Phase Control in Modified Photon Echoes for a Near-Perfect Storage Time-Extended Quantum Memory. *Entropy*. 2020; 22(8):900.
https://doi.org/10.3390/e22080900

**Chicago/Turabian Style**

Ullah, Rahmat, and Byoung S. Ham.
2020. "Understanding of Collective Atom Phase Control in Modified Photon Echoes for a Near-Perfect Storage Time-Extended Quantum Memory" *Entropy* 22, no. 8: 900.
https://doi.org/10.3390/e22080900