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Lengthening of photon storage time has been an important issue in quantum memories for long distance quantum communications utilizing quantum repeaters. Atom population transfer into an auxiliary spin state has been adapted to increase photon storage time of photon echoes. In this population transfer process phase shift to the collective atoms is inevitable, where the phase recovery condition must be multiple of 2π to satisfy rephasing mechanism. Recent adaptation of the population transfer method to atomic frequency comb (AFC) echoes [Afzelius et al., Phys. Rev. Lett. 104, 040503 (2010)], where the population transfer method is originated in a controlled reversible inhomogeneous broadening technique [Moiseev and Kroll, Phys. Rev. Lett. 87, 173601 (2001)], however, shows contradictory phenomenon violating the phase recovery condition. This contradiction in AFC is reviewed as a general case of optical locking applied to a dilute medium for an optical depth-dependent coherence leakage resulting in partial retrieval efficiency.
©2010 Optical Society of America
1. Introduction
Photon storage time extension has been intensively studied recently for quantum memory applications to long-haul quantum communications such as quantum cryptography using quantum repeaters [1,2]. Optical deshelving has been successfully applied for extended photon storage in modified photon echo schemes [3–6], where optical coherence is converted into spin coherence through optical population transfer to an auxiliary spin state. Because spin states are much more robust than optical states, photon storage time can be easily extended [7–9]. However, the optical deshelving mechanism by a control π deshelving pulse accompanies a phase gain of π/2 to the transported individual atoms. Thus, the recovered individual atoms by a consecutive π−π control deshelving pulse pair, should gain a π phase shift. This π phase shift, however, exactly compensates the rephasing process in photon echoes resulting in no echo generation [10]. Therefore, an extra rf pulse needs to be added to compensate the π phase shift [3,11]. To avoid a complex photon storage system, deletion of the rf pulse has been studied [7,8,10], and experimentally demonstrated [4,5].
The idea of using identical control deshelving pulses for coherence conversion from an excited optical state to a ground spin state originates in a controlled reversible inhomogeneous broadening technique applied to an atomic Doppler broadened system [7]. In Ref. [7], however, phase reversal incurred by the identical π control deshelving pulses should exactly compensate the rephasing process occurred to the Doppler broadened medium under the counterpropagation scheme, resulting in no echoes. Although recent adaptation of the control deshelving technique into AFC echoes with identical control pulses demonstrates extended photon echoes, the contradictory phenomenon without satisfying phase recovery condition needs to be explained [4]. In this article, the function of control deshelving pulses is investigated to understand the contradictory phenomenon in the AFC echoes of Ref. [4], where retrieved photons should originate in undepleted (remnant) atoms. Because the requirement of retrieval efficiency of photon echoes is at least 67% to satisfy entanglement swapping in quantum repeaters, phase recovery condition in the control deshelving pulse technique plays a key role. This understanding sheds light on the usage of modified photon echoes in ultralong quantum memory applications.
2. Numerical analysis
Figure 1(a) shows an energy level diagram of a modified photon echo scheme using control deshelving pulses for the purpose of storage time extension utilizing stimulated photon echoes [12]. The reason of choosing stimulated photon echo scheme is to satisfy the origin of AFC echo: Detailed information of AFC is given in Ref. [13]. Adaptation of control deshelving technique into two-pulse photon echoes has already been studied for phase locked echoes [10], where the function of the control pulses is the same as that in AFC (will be explained later). The phase locked echo scheme utilizing rephasing halt in the two-pulse photon echoes can be easily implemented if R is combined with W preceding B1 [10].
Figure 1(b) shows an optical pulse sequence for Fig. 1(a). The pulses B1 and B2 are for atom deshelving control pulses. The pulses D (DATA), W (WRITE), and R (READ) represent a conventional stimulated photon echo pulse sequence [12], where each pulse area Φ is π/2: ; Ω is the Rabi frequency of each pulse. In this case inhomogeneously broadened individual atom phase evolutions triggered by D become frozen by W resulting in spectral grating [12]. When the first control pulse B1 is applied, the excited atoms on state |3> are transported into an auxiliary spin state |2>, resulting in storage time extension. For quantum memory applications, the pulse D must be a single photon or nonclassial light, but all descriptions are the same as many photon cases, because density matrix equations imply pure quantum mechanical approach [14].
In Fig. 2 , time dependent density matrix equations ( + decay terms) are numerically solved for Fig. 1 to present the phase recovery condition of the control pulses without considering reabsorption of photon echo signals [15]. In Fig. 2(a), the enhanced photon echo (blue curve) with a π−3π control pulse sequence (so-called optical locking) is compared with a corresponding conventional three-pulse photon echo (red curve). The enhancement is due to temporary stoppage of the optical population decay process via complete population transfer to the robust spin state |2> by control pulse B1. To understand the phase recovery condition for the control deshelving pulses, two symmetrically detuned atoms are taken out from Fig. 2(a) and examined for a π−7π control pulse pair of B1 and B2. According to the stimulated photon echoes [12], each atom phase evolution excited by D is stopped by W and reversed by R leading to photon echoes. In this case the inserted control pulses must be intact on the phase of individual atoms [see red dots in Fig. 2(b)]. Here it should be noted that absolute phase change on individual atoms by the control pulses does not influence on the rephasing process if there is no relative phase change among individual atoms. Therefore, the coherence matter of the control pulses with the excited atoms is not an issue. Details of individual atom’s phase evolution by B2 pulse are shown in Fig. 2(c), where the initial phase is retrieved at the pulse area of 3π (first arrow) and 7π (second arrow) of B2: see also Fig. 6 of Ref. [10]. This means that the 4π pulse area of B2 induces a 2π phase shift to the atoms due to a double round trip [see Fig. 2(d)]. Figures 2(c) and 2(d) present the π deshelving pulse mechanism, which swaps atom populations between two states with a π/2 phase gain: ρ33 ↔ ρ22. Thus, the phase recovery condition for B2 for a fixed B1 pulse (π) is:
where j is an integer. The phase recovery condition for the sum of B1 and B2 pulses is:where identical pulses of B1 and B2 used in Ref. [4] cannot be satisfied. Because density matrix approach is purely quantum mechanical, Eqs. (1) should be satisfied even for single photons of D. Considering packet quantum data composed of many photon qbits, the argument of spontaneous emission noise by a π rephasing pulse can be neglected. Moreover, in the scheme of modified stimulated photon echoes of Fig. 1, no spontaneous emission noise generates due to equal or less population on state |3> compared with that in state |1>.
Fig. 2 Optically locked echo using phase locked condition. (a) conventional three-pulse photon echo without B1 and B2 (red) versis optical locked echo with B1 and B2 (blue). Inset: pulse sequence. Pulse duration of D, W, and R: 100 ns corresponding to a π/2 pulse area. B1 and B2: 100 and 300 ns corresponding to π and 3π pulse area, respectively. TD = 5 μs; TW = 10 μs; TB1 = 10.1 μs; TB2 = 50 μs; TR = 51 μs; ρ11 (0) = 1; Δinh = 680 kHz; Γ31 = Γ32 = γ31 = γ32 = 2 kHz; Γ21 = γ21 = 0. (b). Individual atom phase evolution: δ = 30 kHz. (c) Expanded figure of (b). Pink shade: B2 pulse. (d) Corresponding figure to (c) for population on each state.
To drive a simplified analytical solution of the deshelving process by the control pulses B1 and B2 in Fig. 1, we set an absorption parameter η to determine the amount of population transfer. The absorption parameter η is represented by exp(−d), where d is an optical depth (d = αl, where α is an absorption coefficient in Beer’s law, and l is the length of an optical medium along the light pulse propagation direction). If initial population on the excited state |3> for the B1 control pulse is A, where A is an entity of optical coherence for photon echoes, then the resulting populations on the excited (|3>; ρ33) and ground (|2>; ρ22) states by the π B1 control pulse are:
Because the pulse separation T between B1 and B2 is much shorter than the optical population decay time, T1 opt, (as in modified photon echo protocols including gradient echoes [6,16], AFC echoes [4,17], and phase locked echoes [5,10]), the exponential terms in Eqs. (2) can now be conveniently dropped. Thus, the population ρ33 on the excited state |3> by the second control pulse B2 for the first π pulse area is simply given by as:where the first term (1−η)2 represents no phase shift to the untransferred atoms, while the second term η2 is for round trip atoms via state |2> with an accumulated π phase shift. This explains why the second control pulse area of B2 should need another 2π pulse area for another round trip as discussed in Fig. 2(c), which is three times longer (or stronger) than B1. Thus, for the case of identical deshelving control pulse usage in Ref. [4], the effective term to photon echoes is (1-η)2, where echo signals show quadratic decrease as a function of absorption η. The auxiliary spin state population ρ22 on state |2> by the second control pulse B2 for the corresponding first π pulse area isFor the second π pulse area of B2 (total 2π pulse area), the population ρ33 and ρ22 are calculated from Eqs. (3): showing no echo contribution due to no phase recovery except the zeroth order, (1−η)3. This is obvious because no atoms exists on the excited state |3> if η = 1. For the third π of B2 (total 3π pulse area), the populations ρ33 and ρ22 are obtained from Eqs. (4): where the first, second, and third terms of ρ33 imply zero, π, and 2π phase shift, respectively. Here, the only effective term for the lengthened photon echo generation is the last term with satisfaction of the phase recovery condition. Unlike phase locked echoes or AFC echoes, the first term in Eq. (5-1), however, which adds to the echo generation, cannot be separated due to spatiotemporal overlap: This subject will be discussed elsewhere.Owing to the recursive relation between ρ22 and ρ33 as a function of the B2 pulse area, Table 1 shows the following: , where Bnm = Bn(m-1) + B(n-1)(m-1), Bjj = 1; m≥n, and ii = 33 (22) for even (odd) n’s. Here “n” stands for the order of η, while “m” stands for the total pulse area mπ of the control pulses. The zeroth, 4th, and 8th terms of n imply 0, 2π, and 4π phase shifts, respectively, and contribute to the photon echoes according to Eq. (1-2).
Table 1. Coefficient Bmn of nth η for deshelving pulse area mπ for .
3. Discussions
In Table 1, for m≥4, all pulse areas include the fourth order of η, which is contrary to the phase recovery condition in Fig. 2. This is complete contradiction to the phase recovery condition obtained in Eqs. (1). Before proceeding, the AFC echo is briefly reviewed. Unlike using a single pair of D and W in Figs. 1 and 2, many sets of weak pulses (say Ds with the same pulse separation as in D and W) forming a long pulse train are used in AFC to prepare a long lasting spectral grating on the ground state |1> through complete population transfer into state |2> by a spontaneous decay process [13]. Later, another weak pulse (input mode in Ref. 4) playing as the READ pulse R in conventional three-pulse echoes is applied, scatters off the grating, and generates a photon echo (output mode). Thus, AFC is an extreme case of the conventional three-pulse photon echoes, where usage of sharp spectral gratings is an advantage [13]. The “beauty” of AFC is to use single photon or very weak optical pulse as R, so that the spectral gratings accumulated with many weak pulse pairs in advance can be read out several times. The disadvantage of AFC, however, is in the low retrieval efficiency due to decreased optical depth resulting from the comb creation process. Here it should be mentioned that the control pulse B1 in AFC echoes comes last. Unlike Figs. 1 and 2, the function of the control pulse B1 in AFC is to transfer the retrieved (or rephased) atoms into state |2>, where the AFC echo position is strongly dependant on the delay of B1 from R, but independent of the delay between B1 and B2 as demonstrated in Refs. 5 and 10. Moreover, the storage time or longest delay time of B2 from B1 is limited by spin inhomogeneous decay, which is much shorter than T1 opt [4,5].
To study and visualize the leaked population contribution to the storage time-extended echoes by identical deshelving pulses, we consider a 3π−3π deshelving pulse sequence of B1 and B2 to analyze the echo observation in Ref. [4]. For AFC using identical deshelving pulses on a short time scale (T<<T1 opt) we start with a 3π B1 pulse (m = 3 in Table 1). Through recursive relation, the final status after a 3π B2 pulse is (m = 6):
where only the third term 15η4(1-η)2 is effective for photon echoes. Here even ordered η is for ρ33, while odd ordered η is for ρ22. Even with a higher order m (wider pulse area of B2), the fourth order of η still exists and increases: see m = 10 (Bnm = 210) for B1 = B2 = 5π in Table 1. This is due to the leaked population determined by η (discussed in Fig. 3 ). Thus, an optically shallow medium alleviates the phase recovery condition of Eqs. (1), e.g., π−3π deshelving pulse sequence [10].Fig. 3 Excited state population versus absorption. Deshelving pulse areas of B1 and B2: 3π−π (blue); 3π−2π (red); 3π−3π (dotted); 3π−4π (magenta); 3π−5π (green).
Figure 3 shows all extended B2 pulse areas for the fixed 3π B1 pulse: Eq. (6) is denoted as a dotted curve to analyze Ref. [4]. Especially with the 70% absorptive medium (η~0.7; d~1), the photon echo efficiencies are nearly the same for any deshelving pulse area combination, supporting the dilute medium used in Ref. [4] otherwise no echoes are expected. Although echo efficiency (ρ33) is low, the contradictory phenomenon of photon echo observation in Ref. [4]. with an identical control pulse pair is successfully explained as an effect of remnant population in a dilute sample. Here, the 3π−5π deshelving pulse sequence shown as green curve satisfying the phase recovery condition [see Eq. (1-2)], shows similar pattern to the reference of Eq. (5-1) shown as a blue curve: For the blue curve, the zeroth order in Eq. (5-1) is excluded to satisfy the condition of Ref. [4] as discussed above. The opposite order of π−3π deshelving pulse sequence gives identical result. For the case of π−π deshelving pulse sequence, however, absorption dependent echo efficiency shows an opposite phenomenon to the blue curve according to Eq. (3-1), [by (1-η)2 term], where only remnant population contributes to the echo signal. This means that echo intensity sharply drops down as the absorption increases. The numerical calculations of Eq. (3-1) shows that echo signal is negligible if η≥0.8 (not shown).
To discuss potential applications of ultralong photon storage, as in the optically locked echoes, we now lengthen the deshelving control pulse separation T between B1 and B2: . In the photon echo theory, decayed atoms do not contribute photon echoes. Thus, we simply discard ρ33 (Eq. (2-1) and use ρ22 (Eq. (2-2) for the recursive relation to obtain ρ33 satisfying the phase recovery condition (3π−π):
where effective term for photon echoes is the same as Eq. (5-1). For a higher order of B2 (m > 4), however, the Bnm of η4 decreases by half, explaining why conventional three-pulse photon echo efficiency is always less than 50%. Like short time scale in Table 1, any pulse length of B2 for does generate photon echoes due to population leakage in a dilute medium (η<1). In any case regardless of η, the echo efficiency, without satisfying the phase recovery condition, is always less than 50%, which cannot be used for quantum memories. For an optically dense medium (η~1), however, the phase recovery condition of Eqs. (1) is still satisfied in both cases, because all even-ordered terms of η for ρ33 contain (1−η)m-n except for the highest order given by m. Hence, using a deshelving pulse pair satisfying the phase recovery condition in an optically dense medium (η~1), and using a phase conjugate scheme [5,7], can give nearly 100% echo efficiency as well as ultralong photon storage time [18].4. Conclusion
In conclusion, we investigated the function of deshelving control pulses used for photon storage time extension. Due to incomplete population transfer in an optically shallow medium even with a π optical pulse, the phase recovery condition of a deshelving pulse set is alleviated to allow coherence leakage caused by remnant population in the excited state. Although observed photon echoes in Ref. [4] seem to contradict to the theory, the present analysis clearly supports it due to optical leakage in an optically dilute sample. However, the optical leakage-based photon echoes using identical control deshelving pulses in Refs. [4] and [7] may not be used for quantum memory applications in long haul quantum communications due to less than 50% echo efficiency. To maximize the echo efficiency, the phase recovery condition of the π−3π deshelving pulse sequence in an optically dense medium must be satisfied.
Acknowledgment
This work was supported by the CRI program (Center for Photon Information Processing) of the Korean Government (MEST) via National Research Foundation.
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