A wavelength-convertible quantum memory: Controlled echo

Quantum coherence control is reinvestigated for a new physical insight in quantum nonlinear optics and applied for a wavelength-convertible quantum memory in a solid ensemble whose spin states are inhomogeneously broadened. Unlike typical atomic media whose spin decays are homogeneous, a spin inhomogeneously broadened solid ensemble requires a counter-intuitive quantum coherence control to avoid spontaneous emission-caused quantum noises. Such a quantum coherence control in a solid ensemble satisfying both near perfect retrieval efficiency and ultralong photon storage offers a solid framework to quantum repeaters, scalable qubit generations, quantum cryptography, and highly sensitive magnetometry. Here, the basic physics of the counter-intuitive quantum coherence control is presented not only for a fundamental understanding of collective ensemble phase control but also for a coherence conversion mechanism between optical and spin states involving Raman rephasing.

is also presented for a wavelength-convertible quantum memory satisfying both ultralong storage time and near perfect retrieval efficiency, where its retrieval mechanism is new with a succinct pulse scheme. Surprisingly the theoretical investigation of the present controlled echo-based quantum memory reveals that our conventional understanding on quantum coherence control is actually a special case, and thus it has misled quantum optics community working on solid media so far 24,27,33 . For this, a simple lambda-type three-level optical system is first investigated to derive the basic physics of quantum coherence control for the controlled echo in Figs 1 and 2. Then, a wavelength-convertible quantum memory is presented in a double-lambda-type four-level system in Fig. 3. Figure 1a shows the energy level diagram of a typical lambda-type three-level optical medium, but the access of the control pulse C is counter-intuitive for the present controlled echo-based quantum memory, where the data pulse A is to be stored. This counter-intuitive control access is the essence of the controlled echo, where its physics investigation is the key task of the present paper. Figure 1b is the pulse sequence for Fig. 1a, and the Raman rephasing pulse R is composed of a balanced Raman pulse for − − 1 3 2 transition for the control of spin coherent transients. The NDFWM signal e in Fig. 1c results from the light-matter interactions among optical pulses A, B, and C, where Raman rephasing by R play a key role. In spin homogeneous media such as alkali atoms 4,7,8,10,11,[15][16][17][18][19][20][21]32,40,42 , however, the control pulse C must be the same as B, if they are collinear, for the transition − 2 3 without R 7,8,10,11,21 , resulting in NDFWM signal e' in the same frequency as A (see Fig. 1e).  To describe the basic physics of the controlled echo, first, analytic discussions are performed and then numerical demonstrations follow to prove them. According to the CDR theory 22,23,29 , the function of CCC is for the coherence conversion between optical and spin states by the control pulse B acting on the data pulse-excited population ρ 33 on the excited state 3 : Here, B(π) stands for a π pulse area of B. The successive CCC by the second control pulse C results in coherence inversion: C A 13 13 where C is identical to B for the same transition. Equation (2) represents a correct understanding of conventional NDFWM process without rephrasing 7,8 . As a result, there is a π phase shift between the input and output [21][22][23]29,30 . Because the input pulse A must induce absorptive coherence ρ 13 (t A ) via absorption process, the π-phase shift in equation (2) to the output represents emissive coherence. Thus, the output coherence either from a rephasing system applied by an identical control pulse set 24 or from a doubly rephrased system 27,33 must be also absorptive 29 (see the Supplementary Information Fig. S1).
In a solid ensemble whose spin transition is inhomogeneous, however, the Raman rephasing by R is required as shown in Fig. 1b, otherwise the system is quickly dephased within spin T 2 *9, 24,30 . Here, I introduce new physics of controlled echo in Kerr nonlinear optics. By using equation (1) and the following Raman rephrasing by R [ 21 12 , the NDFWM signal e at t = t e in Fig. 1a is expressed by: e C A 23 21 13 where the control access of C must be counter-intuitive for the transition 1 3 − to work with the same transferred atom (ion) by B, satisfying no spontaneous emission-caused quantum noise. Here, equation (4) is the same as the conventional case in equation (2). Thus, the present controlled echo in Fig. 1 works for the quantum memory with emissive coherence under no population inversion via unitary transformation of Raman rephasing. By the way, if C is identical to B for the same transition 9 , the NDFWM signal e' at t = t e becomes: 12 13 where 33 22 ρ ρ , causing spontaneous emission noise (see Fig. 1e). The controlled echo in equation (4) is now discussed in more detail for the coherent transients. By the first control pulse B, the transferred coherence ρ 12 in equation (3) begins to evolve due to the spin inhomogeneity: where j δ is the detuning of the j th spin from the two-photon line center, and t k is the ending time of pulse k. Here, the pulses A and B must be consecutive to neglect optical dephasing regardless of the optical inhomogeneity. By the Raman rephasing pulse R at t = t R , the completely dephased spin coherence ρ 12 (t R ) starts to rephase: , the rephrased spin coherence reaches at maximum value (spin echo) with a π/2 phase shift: where ′ t C is the arrival time of the pulse C. If the second control pulse C follows the conventional NDFWM process as in equation (2), the output e' becomes: in rare earth doped solids 2,5,6,9,[24][25][26]30 . In equation (9), the output e' is obviously violates quantum memory due to population inversion (ρ ρ  33 11 ). If there is no Raman rephasing as in ref. 24 , then, the output e' still becomes absorptive due to the intrinsic optical rephrasing property 25,31 , which again violates the quantum memory (will be discussed elsewhere).
On the contrary, for the case of controlled echo in Fig. 1a, the final coherence for the NDFWM signal e becomes: Thus, the NDFWM signal e at t = t e is described as:  33 22 due to the Raman rephrasing by R at t = t R . In equations (9) and (10), both NDFWM signals, e' and e, represent the same emissive coherence, because all real components are zero at t = t e (see the Supplemental Information Fig. S2). Thus, the present controlled echo in Fig. 1a is now analytically proved for a quantum memory protocol in a spin inhomogeneously broadened solid ensemble.
Figure 1c-f represent numerical results for the controlled echo in Fig. 1a: see Methods for details. Figure 1c is the numerical result of the controlled echo, where the control pulse access is counter-intuitive with respect to conventional one (see Fig. 1e). In Fig. 1c, the data pulse-excited coherence ρ 13 (t A ) is transferred into spin coherence ρ 12 (t B ) by the first control pulse B as discussed in equation (6) (see the inset in Fig. 1e). The maximally rephased coherence (ρ 12 * ) by R is again coherently transferred by the second control pulse C, resulting in the emissive NDFWM signal e [Imρ 23 (t e )] (see the red curve) under no population inversion, as discussed in equation (11).
The magnitude of e Im t t ( ( )) e 23 ρ ≥ in Fig. 1c should decay down exponentially as a function of time by optical phase relaxation rate γ 23 (1/T 2 ). This decay is of course accelerated by optical inhomogeneous broadening. The excited state population ρ 33 (see the dotted line) for e at t = t e is exactly the same as that for A at t = t A (see the dotted line and the inset in Fig. 1d), resulting in no spontaneous emission-caused quantum noise. Figure 1d shows population evolutions for Fig. 1c, where the data A-excited population ρ 33 (dotted curve/line) is fully transferred into state 2 by B at t = 1.2. The function of R at t = t R is to swap the population between two ground states 1 and 2 , resulting in spin rephasing 5,6,9,48 (2 ) 22 and R 12 For the physics of 2π pulse area of R, refer to refs 6,48 .
In contrast to Fig. 1c, Fig. 1e is for the conventional control pulse access to the transition 2 3 − , resulting in an emissive e' output (Imρ 13 ) at the same frequency as A for the transition − 1 3 7-10 . The emissive NDFWM signal e' at ≥ t t e is, however, under population inversion as shown in Fig. 1e: ρ 33 ≫ ρ 11 . Figure 1f shows the relation between equations (9) and (11). Thus, the present controlled echo theory is numerically proved for quantum memories. Ω Ω . Except for the Raman data D, all others are the same as in Fig. 1b. As shown in Fig. 2a, the controlled echo still works for the resonant Raman data for quantum memories. However, the excited population by the data pulse A is half-shelved in the excited state 3 , where ρ 22 = ρ 33 (see Fig. 2b). Because the coherence is induced by the population difference, e.g., 12 22 11 ρ ρ ρ ∝ − , the spin coherence reduction in the resonant Raman case is ~67% (see Fig. 2c and d). Although the reduced coherence itself by the atom shelving has nothing to with quantum fidelity, the shelved population on 3 may deteriorate the read-out conversion efficiency by C. To avoid this mat- ter, a double lambda-type four-level system may be used as in Fig. 3a, so that C pulse can circumvent the shelved population. The optical locking technique also gives a solution to avoid the shelving problem even for ultralong quantum memories 48 . In ref. 48 , quantum decoherence by spontaneous emission decay is negligible due to  T T spin optical 1 1 , where the photon storage time is extended up to spin population decay time. The Raman gradient-echo protocol based on off-resonant interactions may be free from the population shelving or population decay-caused quantum noises discussed in Fig. 2 32 . Instead of the direct Raman rephasing, an oppositely polarized gradient electric field applied to the C transition functions the Raman rephrasing without population control 32 . For the case of multiple data pulses (As), however, the mandatory use of a long, single control pulse C induces at least 50% coherence loss due to unwanted coherence read-out for 'OFF' data pulse timing. Moreover, such a Raman gradient technique can never be applied to a solid ensemble due to the intrinsic spin inhomogeneity. Figure 3 is an application of the present controlled echo for a wavelength-convertible quantum memory in a double lambda-type four-level solid medium whose spin transition is inhomogeneouly broadened. For this, an extra state 4 is simply added in Fig. 1a. Unlike Fig. 1, the resonant Raman rephasing pulse R composed of balanced C and D is applied for the transition − − 1 4 2, so that any potential defect by the shelved atoms on state 3 can be removed on the coherence recovery process. The frequency difference between A and e is the origin of the frequency up-or down-conversion. The control pulse (Cn) access is for the transition − 1 4 and results in the photon signal e, resonant between states 2 and 4 . Here it should be noted that the control pulse Cn must be resonant to maximize the coherence conversion process as shown in most quantum memory cases [7][8][9][10][15][16][17][18][19][20][21][22][23][28][29][30] , where preliminary NDFWM generation in a double-Λ solid medium of Fig. 3 has been observed in a cw scheme 49 . The Rabi frequency Ω A of the data pulse A in Fig. 3 is decreased by a factor of 2 for the purpose of comparison with Fig. 1. As a direct result, the coherence excitation by A is also decreased by 2 as shown in Fig. 3c (see also the inset in Fig. 3d). Either for 3 or 4 , a resonant Raman pulse deals only with two-photon coherence between the two ground states 1 and 2 , resulting in the same spin rephasing result. The retrieved photon signal e must satisfy Kerr nonlinear optics among pulses A, B, and Cn, resulting in up-or down-conversion depending on the relative energy level of 2 .
In contrast to Fig. 3c, the control pulse (Cn) access is applied for the transition − 2 4 in Fig. 3d, which is conventional as in Fig. 1e. The resultant coherence of the NDFWM signal e' is, however, under population inversion (ρ 44 ~ 1 at t = t e′ ). Even if such an inverted echo signal may be useful for classical applications of associative memories 50 , the potential quantum noise from ρ 44 should prevent Fig. 3d from quantum memories. The relation between equations (9) and (11) is also numerically demonstrated in Fig. 3e and f for the signals e' and e in Figs. 3c and d, respectively. This proves again a clear distinction between rephasing-based quantum memories and the present controlled echo: ρ * vs. −ρ. Thus, the present wavelength-convertible quantum memory protocol in a resonant four-level solid system can be used for multimode quantum wavelength conversion, which is essential for future spectral division multiplexing in quantum networks.
For the near perfect retrieval efficiency in the present controlled echo quantum memory, the control pulse propagation directions must be opposite each other 21,30 , so that a backward echo signal e can be generated by the following phase matching conditions: where j ω (k j ) is the angular frequency (wave vector) of pulse j, and the subscript 'e' stands for the controlled echo. Although a perfect collinear scheme between A and e cannot be satisfied due to + ≠ k k 0 B C n 30 , the wavelength deviation among them is negligibly small at 10 −8 for most rare-earth doped solids 46 . Thus, the refractive index-dependent phase walk-off is also negligibly small at far less than π as experimentally demonstrated 5,6,9,30 . This flexible NDFWM offers a great advantage in echo-based quantum memories for spatial multiplexing. According to the theory 21 and experimental observations 30 , a near perfect retrieval efficiency η e can be achieved for the backward controlled echo scheme even for an optically thick ensemble 45 Here, the higher optical depth (αl) is actually necessary for the single write-and read-out process, otherwise an optical cavity is needed sacrificing bandwidth. The above phase matching conditions can also be expanded for light polarizations.
In conclusion, a wavelength-convertible quantum memory protocol based on controlled echo was introduced, analyzed, and discussed for a spin inhomongeneously broadened double lambda-type four-level optical ensemble. Unlike alkali atoms whose spin transitions are homogeneous, the control pulse access in a spin inhomogeneous solid ensemble must be counter-intuitive to avoid spontaneous emission-caused quantum noises. In the present study, quantum coherence control in a spin inhomogeneously broadened solid ensemble was explicitly analyzed and discussed to elucidate the basic but novel physics of ensemble phase control for both quantum coherence conversion and Raman rephasing, resulting in near perfect, ultralong, and emissive photon echoes without quantum noises. With a backward control pulse set and balanced Raman rephasing, the retrieval efficiency can be near perfect due to the absence of echo reabsorption, and the photon storage time can be extended up to the spin homogeneous decay time. Here, the spin homogeneous decay time can be as long as minutes or even hours in rare-earth doped solids under external magnetic fields 41 . Moreover, the present controlled echo scheme is simple in configuration and applicable to spectral and spatial multiplexing in all-optical quantum information processing in the future quantum networks. In magnetometry, sensing ability increases by a factor of T 2 /T 2 * 44 , where T 2 can be extended by several orders of magnitude in the present controlled echo. The present research sheds light on potential quantum memory applications in various quantum information areas such as scalable qubit generations, recursive operations, sensing, and quantum repeaters for long-distance quantum communications.

Methods
For the numerical calculations, total sixteen time-dependent density matrix equations are solved for a four-level ensemble medium in an interacting Heisenberg picture under rotating wave approximations 51 : , where ρ is a density matrix element, H is Hamiltonian, and γ is a decay parameter. In the calculations, 99.55% of the Gaussian distribution is taken for total 201 distributed spin groups at 2 kHz spacing, where the spectral spin inhomogeneous width (FWHM) is set at 170 kHz. Here, the exaggerated spin bandwidth (x30) is only to save computer calculation time. Those 201 spectral groups are calculated for the time domain and summed up for all spectral groups. For the optical transition, optical inhomogeneity is neglected because it does not violate physics of Raman coherence nor affect the result, unless optical pulse delay between A and B is given. The following equations are for the coherence terms of  ρ ij in a four-level system interacting with three resonant optical fields: where the interaction Hamiltonian matrix H is given by: here Ω 1 (Ω 3 ) is the Rabi frequency of the optical field between the ground state 1 and the excited state 3 ( 4 ), and Ω 2 (Ω 4 ) is the Rabi frequency of the optical field between the ground state 2 and the excited state 3 ( 4 ). The δ 1 , δ 2 , and δ 3 are the atom detuning from the resonance frequency for Ω 1 , Ω 2 , and Ω 3 (Ω 4 ), respectively. For visualization purpose and simplification, all decay terms are neglected except for the optical phase relaxation rates γ ij . The optical pulse duration is set at 0.1 μs, otherwise specified. The time increment in the calculations is 0.01 μs. Initially all atoms are in the ground state 1 ( (0) 1 11 ρ = ), and thus all initial coherence is ρ = (0) 0 ij , where i (j) = 1, 2, 3, 4. The program used for the numerical calculations is time-interval independent, so that there is no accumulated error depending on the time interval settings. The corresponding Rabi frequency is set at 100 2 MHz for the pulse duration of 0.01 μs to satisfy a 2π pulse area.