1. INTRODUCTION

In an inhomogeneously broadened two-level optical system, resonant optical fields can induce collective atomic coherence, where each atom’s coherence magnitude depends on the degree of population change resulting from the inhomogeneous detuning from the line center. The excitation of atomic coherence by a single optical pulse brings all interacting atoms into in phase, but they start to dephase as soon as the light pulse interacts due to different phase velocity determined by the inhomogeneous detuning. When a π optical pulse (or a Stark field) is applied to these coherently excited atoms after a complete dephasing process, as shown in conventional two-pulse photon echoes [1, 2] (or in gradient echoes [3]), all atoms’ phases are reversed, resulting in a collective coherence burst as a photon echo. In these cases, the photon echo storage time is limited by optical phase decay. To slow or even momentarily stop the atomic coherence decay, an independent control light pulse can be used to transfer excited atoms into an auxiliary ground spin state, where the spin state is robust against the optical phase decay processes. This on- demand atom population transfer, or an optical deshelving process, induces coherence conversion from an optically excited state to a robust spin state.

In an ultraslow light regime, atom population transfer-based coherence conversion has been experimentally demonstrated for coherence swing between optical and spin states [4]. In the coherence swing, optical phases of all atoms before and after the control pulse interaction must be in phase, so that nondegenerate four-wave mixing processes determine the outcome as a matched-pulse generation [5]. In the nondegenerate four-wave mixing or phase conjugate processes using the collective atomic coherence, the atomic phase must be within the coherence limit [5, 6]. However, this requirement of coherence between consecutive optical pulses can be relaxed if a rephasing process is involved, such as in photon echoes, where the rephasing is a time reversal of phase evolution to restore the initial collective coherence leading to photon echoes [1, 2, 7]. Thus, the rephasing-based coherence conversion process in photon echoes relaxes the strict in-phase condition, so that the deshelving control pulses do not need to be coherent with the initial excitation pulse(s). This benefit of rephasing in photon echoes makes it possible to extend photon storage time without relying on the group velocity control. Recently, such an optical deshelving-based coherence conversion has been experimentally demonstrated in atomic frequency comb (AFC) [8] and phase-locked echoes [9]. Because the transferred atoms from the optical state to the spin state and vice versa gain a $\pi /2$ phase shift, a phase recovery condition must be satisfied not to demolish the initial rephasing process [10].

In this paper, optical locking applied to the rephasing process is discussed for rephasing halt in both AFC and phase- locked echoes. The optical locking is a phase-conserved atomic coherence transfer between optical and spin states using a deshelving pulse pair satisfying the phase recovery condition. Decades ago, a simple population transfer technique originated in incoherent spontaneous emission process has been demonstrated [11] and has been adapted by AFC to solve the echo reabsorption dilemma by creating high-finesse spectral gratings with a low optical depth [12]. In contrast, conventional photon echoes need a high optical depth to fully absorb a data pulse, where photon echo reabsorption has been an inevitable problem. Because population transfer induces a phase gain to the atoms, the phase recovery condition must be carefully considered to resume the halted rephasing process [10]. Specifically here, optical locking in phase-locked echoes for the rephasing halt is analyzed as a photon storage extension protocol and compared with that in AFC echoes.

2. THEORY

Figure 1a shows an energy level diagram of an optical medium for the analysis of the optical locking applied to two-pulse photon echoes. Light P is used for input pulses in photon echoes or AFC preparation. Light pulses B1 and B2 represent an optical-locking pulse pair for coherence conversion via population transfer between optical and spin states. The light pulse B1 is to transfer atoms on state $|3〉$ into state $|2〉$, where quantum coherence ${\rho }_{13}$ excited by P is converted into spin coherence ${\rho }_{12}$ via population transfer only: ${\rho }_{ij}$ is a density matrix element between states $|i〉$ and $|j〉$. Light B2 is a match pulse to restore the coherence conversion process by returning both atoms’ phase and population into state $|3〉$ [10]. Figure 1b shows the interacting light pulse sequence, where the optical-locking pulses B1 and B2 function to extend photon storage time using longer coherence decay of ${\rho }_{12}$.

For numerical simulations, time-dependent density matrix equations for Fig. 1 are derived for the Liouville equation:

3. ANALYSES AND DISCUSSION

In Fig. 2, the AFC echo is calculated and compared with a conventional three-pulse photon echo to analyze its storage mechanism. Figure 2a shows the AFC pulse sequence of P without use of control pulses, B1 and B2. For AFC preparation, a light pulse train composed of consecutive five two-pulse pairs is applied long before the last single pulse, INPUT. The population decay time is set to as short as $\sim 10\mathrm{\mu s}$ (${\mathrm{\Gamma }}_{31}={\mathrm{\Gamma }}_{32}=20\mathrm{kHz}$), so that there is no atom left on state $|3〉$ when the INPUT pulse is applied.

Figure 2b shows the result of Fig. 2a for absorption, $\mathrm{Im}{\rho }_{13}$. Each light pulse area Φ is set to be small enough: $\mathrm{\Phi }=\pi /5$. The AFC echo amplitude (dotted circle) is proportional to the magnitude of INPUT power in a weak field limit, as shown in Fig. 2c. For the INPUT pulse Rabi frequency reduced by 80% (red curve), the echo amplitude reduction also shows the same ratio against the blue curve. This is because information is stored in the spectral gratings (or ground state atom population redistribution) made by the preparation pulse sets (see [11]), and the INPUT pulse scatters off the grating to generate an echo signal (explained in Fig. 2d).

Figure 2d shows the ground state atom population ${\rho }_{11}$ resulting from each preparation pulse set in Fig. 2a as a function of spectral detuning of interacted atoms from the line center of inhomogeneous broadening. The spectral spacing Δ is determined by the temporal spacing τ of the preparation pulse set: $\mathrm{\Delta }=1/\tau$. Because all pulse sets have the same temporal separation τ, the spectral gratings are superposed and sharpened as the pulse sets are repeated. Thus, the AFC (or spectral grating) contains the information (optical phase and time delay τ) of the input pulse pair. Eventually, with many pulse sets, the finesse (the ratio of Δ to each line width) of the spectral grating formed on the ground state increases. Here, having high finesse is important to reduce the echo reabsorption and decoherence. However, at the same time, high finesse causes low retrieval efficiency due to decreased optical depth even to single-photon INPUT.

Figure 2e shows a conventional three-pulse (stimulated) photon echo to compare with the AFC echo in Fig. 2b. For this, only the first preparation pulse set in Fig. 2a is used with a $\pi /2$ pulse area to maximize echo efficiency [2]. To satisfy conventional three-pulse (stimulated) photon echoes, the last pulse INPUT in Fig. 2a is used as a READ pulse. The magnitude of the stimulated photon echo (“2”) decreases due to population decay rate ${\mathrm{\Gamma }}_{3j}(j=1,2)$ as storage time increases. The mark “1” is a reference of two-pulse photon echo. The rephasing process resumes when the third pulse (INPUT) arrives, resulting in the photon echo.

Figure 2f shows the corresponding spectral gratings of Fig. 2e: the red curve is for $t=136\mathrm{\mu s}$ [red dotted line of Fig. 2e], while the blue curve is for $t=16\mathrm{\mu s}$, immediately following the second pulse in Fig. 2e. The modulation depth decreases due to population decay, ${\mathrm{\Gamma }}_{31}$, where modulation has a 50% duty cycle. Compared with AFC in Fig. 2d, finesse has nothing to do with the retrieval efficiency if a phase conjugate scheme is used to avoid the echo reabsorption suggested in [13]. Thus, Fig. 2 proves that the AFC [12] belongs to the conventional stimulated photon echoes, where the storage mechanism is in the spectral grating. It should be noted that, unlike the explanation in [8], the AFC echo is a retrieval of the spectral grating as a delayed coherence response, where INPUT is not the origin of the echo but plays as a READ pulse as in the conventional three-pulse photon echoes shown in Figs. 2d, 2f.

Figure 3 represents a storage time-extended photon echo using optical locking pulses, B1 and B2. Temporal separation τ of the consecutive pulses in each AFC preparation pulse set is $5\mathrm{\mu s}$. Figures 3a, 3b, 3c show the pulse sequence for Figs. 3d, 3e, 3f, respectively, for different conditions. In Figs. 3a, 3d, two different cases for B1 delay (blue and red) are intentionally overlapped: the longer the delay of B1, the shorter the delay of the echo. The sum of the B1 delay from the INPUT and the echo delay from B2 is always the same as τ [10]. If the B1 delay $\mathrm{\Delta }T$ from INPUT is longer than τ, however, the echo disappears, as shown in Fig. 3e (see the dotted red circle). This implies that the mechanism of the photon storage in AFC is the same as the conventional three-pulse photon echoes.

Figures 3c, 3f represent optical deshelving with an identical pulse set B1 and B2 applied to AFC echoes (see [8]), where usage of identical deshelving control pulses violates the phase recovery condition resulting in no echo signal [see the red dotted circle for negative echo, Figs. 3e, 3f] [10]. The contradictory observation of AFC echoes in [8] has been explained due to imperfect population transfer-based coherence leakage in a dilute medium [14].

Figures 3g, 3h represent optical locking applied to conventional two-pulse photon echoes; the so-called phase locked echoes to compare with the AFC echoes in Fig. 3d. For this, Fig. 2e is modified: the second pulse satisfies π in pulse area; right after the second pulse, B1 is added, and the third pulse is replaced by B2. To satisfy the phase recovery condition, ${\mathrm{\Phi }}_{\mathrm{B}1}+{\mathrm{\Phi }}_{\mathrm{B}2}=4\pi$ [10], the pulse area of B1 and B2 is set to π and $3\pi$, respectively. In Fig. 3g, the blue curve (smaller curve) denotes the stimulated photon echo in Fig. 2e for $\tau =10\mathrm{\mu s}$ as a reference. The phase-locked echo magnitude (red curve [bigger curve]) is much greater than the stimulated photon echo (blue curve). This is due to population transfer by B1 to freeze population decay-caused coherence loss. Here, the two-pulse echo signal (third peak from the left) in Fig. 3g is shown as a reference, where its magnitude is 50% reduced due to half pulse area ($\pi /2$) of the second pulse for the calculations of the three-pulse photon echo without applying the phase-locking condition (see the blue curve). The B1 delay-dependent echo position (red and blue) in Fig. 3h for the phase-locked echo in Fig. 3g shows the same pattern as in Fig. 3d for AFC echoes, implying that the storage time extension mechanism of both AFC and phase-locked echoes is based on the same physics of rephasing halt. The observed storage time extension in both AFC [8] and phase-locked echoes [9] is about $10\mathrm{\mu s}$ in ${\mathrm{Pr}}^{3+}$-doped ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ (Pr:YSO), which is limited by spin inhomogeneous broadening ($30\mathrm{kHz}$). Compared with optical phase decay time of $110\mathrm{\mu s}$ in Pr:YSO, the storage time extension in both AFC and phase-locked echoes is too short. However, using magnetic field gradient the spin dephasing time can be increased [15].

Figure 3i represents the rephasing mechanism of Fig. 3g for two-pulse photon echoes. In the phase evolution process of two-pulse photon echoes, the phase of symmetrically δ detuned atoms is swapped (or rephased) by the followed π rephasing pulse. The mirrorlike behavior of the atom phase before B1 and after B2 reveals a π phase shift as shown in the same color. This means that B1 and B2 simply function for rephasing halt without affecting the phase change as discussed above. As denoted by the green curve (see open circles), the atom population in the excited state has no change. In summary, the AFC echo uses spectral gratings for quantum optical storage, while the storage extension mechanism by the optical-locking pulse pair is to halt the rephasing process as in the phase-locked echoes. The function of optical locking pulses in both AFC and phase-locked echoes is to hold the rephasing process by converting optical coherence into long-lived spin coherence.

The spontaneous emission noise induced by a rephasing pulse in conventional photon echoes has been thought to be a major problem for single-photon-based quantum memory applications. Although the phase-locked echo based on a conjugate scheme with a pencil-like geometry can contribute to near-perfect retrieval efficiency [9, 13], it actually does not reduce the spontaneous emission noise. However, in a rare earth ${\mathrm{Pr}}^{3+}$-doped ($0.05\text{\hspace{0.17em} at.}\%$; ${\mathrm{Pr}}^{3+}$ ion number: $4.7\times {10}^{18}/{\mathrm{cm}}^3$) ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ using a $0.1\mathrm{ns}$ data pulse focused by a $10\mathrm{cm}$ focal length lens to excite $4\mathrm{GHz}$ inhomogeneously broadened atoms, the spontaneous emission noise contributing to the echo signal is less than 0.03 photons, which is nearly negligible. If multi-photon entangled photons or squeezed light is used, the spontaneous emission problem becomes out of question. The actual problem in photon echoes is the population inversion-caused echo gain, which violates the no-cloning theorem in quantum information processing. To solve this problem, a double rephasing technique with deshelving has been proposed for a spontaneous emission-free scheme [16].

4. CONCLUSION

In conclusion, the atomic coherence conversion process between an optically excited state and a robust spin state using optical locking was analyzed for AFC and phase-locked echoes. From this analysis, it has been discussed that the storage mechanism of AFC has the same origin as the conventional three-pulse (or stimulated) photon echoes in a form of spectral grating. The optical locking applied to AFC echoes is for rephasing halt, which is the same as in the phase-locked echoes. For quantum optical memory applications, the phase-locked echo has no limit for the data pulse intensity ranging from single photons to many photons of squeezed light.

ACKNOWLEDGMENTS

This work was supported by the Creative Research Initiative program (grant 2010-0000690) of the Korean Ministry of Education, Science, and Technology via the National Research Foundation.

 

Fig. 1 (a) Energy level diagram and (b) pulse sequence.

Download Full Size | PDF

 

Fig. 2 (a) AFC echo pulse sequence. Each pulse turns on at 5, 15, 35, 45, 65, 75, 95, 105, 125, 135, and $175\mathrm{\mu s}$. Each pulse duration is $100\mathrm{ns}$. (b), (c) Absorption versus time for (a). For the red curve in (c), the INPUT pulse in (a) is reduced to one fifth. The dotted circle indicates the AFC echo. (d) Population ${\rho }_{11}$ versus atom detuning. $t=16$ (red), 46 (green), 76 (magenta), 106 (cyan), and $136\mathrm{\mu s}$ (black). (top to bottom) Dotted curve represents initial spectral distribution at $t=0$. (e) Conventional three-pulse (stimulated) photon echo (see mark “2”). Mark “1” is for a reference as a two-pulse photon echo. (f) Population ${\rho }_{11}$ versus atom detuning. Blue, $t=16\mathrm{\mu s}$; red, $t=136\mathrm{\mu s}$. For all, ${\mathrm{\Gamma }}_{31}={\mathrm{\Gamma }}_{32}=20\mathrm{kHz}$ (population decay rate), ${\gamma }_{31}={\gamma }_{32}=30\mathrm{kH}$ (phase decay rate), ${\mathrm{\Gamma }}_{21}={\gamma }_{21}=0$. Inhomogeneously broadened atoms are Gaussian distributed: $\mathrm{FWHM}=680\mathrm{kHz}$. Initial population is ${\rho }_{11}=1$, ${\rho }_{22}={\rho }_{33}=0$.

Download Full Size | PDF

 

Fig. 3 (a), (b) AFC echo pulse sequence with control pulses B1 and B2 for $\mathrm{\Delta }T<\tau$ and $\mathrm{\Delta }T<\tau$, respectively. Pulse area of B1 and B2 satisfies π and $3\pi$. (c) For identical control pulses B1 and B2, as shown in [8], whose pulse area is π, resulting in no AFC echo: $\mathrm{\Delta }T<\tau$. (d)–(f) Numerical simulations of (a)–(c), respectively. Each pulse is turned on at 5, 10, 30, 35, 55, 60, 120, 120.1, and $160\mathrm{\mu s}$. Each pulse duration is $100\mathrm{ns}$. $\tau =5\mathrm{\mu s}$, $\delta =3\mathrm{\mu s}$. (g), (h) Phase-locked echo corresponding to (a) and (d). Each pulse is turned on at 5, 15, and $175\mathrm{\mu s}$ for the blue line (three-pulse echo) with identical pulse durations of $100\mathrm{ns}$ to satisfy $\pi /2$ pulse area. For the red curve (phase-locked echo), each pulse turns on at 5, 15, 15.1, and $175\mathrm{\mu s}$, with pulse areas $\pi /2$, π, π, and $3\pi$, respectively. The last two pulses are optical-locking pulses, B1 and B2. Two different cases of B1 delay from the rephasing pulse at $t=15\mathrm{\mu s}$ are overlapped. The delay difference is $\mathrm{\Delta }{T}^{\prime }=5\mathrm{\mu s}$, which is shorter than the two-pulse delay ${\tau }^{\prime }=10\mathrm{\mu s}$. (i) Individual atom phase evolution of (g) as a function of time: $\delta =40\mathrm{kHz}$. Green curve is for ${\rho }_{33}$. For all, ${\mathrm{\Gamma }}_{31}={\mathrm{\Gamma }}_{32}=20\mathrm{kHz}$ (population decay rate), ${\gamma }_{31}={\gamma }_{32}=30\mathrm{kH}$ (phase decay rate), ${\mathrm{\Gamma }}_{21}={\gamma }_{21}=0$. Inhomogeneously broadened atoms are Gaussian distributed: $\mathrm{FWHM}=680\mathrm{kHz}$. Initial population is ${\rho }_{11}=1$, ${\rho }_{22}={\rho }_{33}=0$.

Download Full Size | PDF

1. N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a photon echo,” Phys. Rev. Lett. 13, 567–570 (1964). [CrossRef]  

2. T. W. Mossberg, “Time-domain frequency-selective optical data storage,” Opt. Lett. 7, 77–79 (1982). [CrossRef]  

3. M. Hosseini, B. M. Sparkes, G. Hetet, J. J. Longdell, P. K. Lam, and B. C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009). [CrossRef]  

4. B. S. Ham and J. Hahn, “Atomic coherence swing in a double L-type system using ultraslow light,” Opt. Lett. 34, 776–778 (2009). [CrossRef]  

5. V. Boyer, D. F. McCormick, E. Arimondo, and P. D. Lett, “Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor,” Phys. Rev. Lett. 99, 143601 (2007). [CrossRef]  

6. B. S. Ham, P. R. Hemmer, and M. S. Shahriar, “Efficient phase conjugation via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583–R2586 (1999). [CrossRef]  

7. N. W. Carlson, W. R. Babbitt, and T. W. Mossberg, “Storage and phase conjugation of light pulses using stimulated photon echoes,” Opt. Lett. 8, 623–625 (1983). [CrossRef]  

8. M. Afzelius, I. Usmani, A. Amari, B. Lauritzen, A. Walther, C. Simon, N. Sangouard, J. Minar, H. de Riedmatten, N. Gisin, and S. Kröll, “Demonstration of atomic frequency comb memory for light with spin-wave storage,” Phys. Rev. Lett. 104, 040503 (2010). [CrossRef]  

9. B. S. Ham and J. Hahn, “Phase locked photon echoes for near perfect retrieval efficiency and extended storage time,” arXiv:0911.3869.

10. B. S. Ham, “Control of photon storage time using phase locking,” Opt. Express 18, 1704–1713 (2010). [CrossRef]  

11. M. Mitsunaga and N. Uesugi, “$248\mathrm{bit}$ optical storage in ${\mathrm{Eu}}^{3+}\text{:}{\mathrm{YAlO}}_3$ by accumulated photon echoes,” Opt. Lett. 15, 195–197 (1990). [CrossRef]  

12. H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. A. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008). [CrossRef]  

13. S. A. Moiseev and S. Kroll, “Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a Doppler-broadened transition,” Phys. Rev. Lett. 87, 173601 (2001). [CrossRef]  

14. B. S. Ham, “A contradictory phenomenon of deshelving pulses in a dilute medium used for lengthened photon storage time,” Opt. Express 18, 17749–17755 (2010). [CrossRef]  

15. E. Fraval, M. J. Sellars, and J. J. Longdell, “Method of extending hyperfine coherence times in ${\mathrm{Pr}}^{3+}\text{:}{\mathrm{Y}}_2{\mathrm{SiO}}_5$,” Phys. Rev. Lett. 92, 077601 (2004). [CrossRef]  

16. B. S. Ham, “Atom phase controlled noise-free photon echo,” arXiv:1101.5480.