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

A near-perfect storage time-extended photon echo-based quantum memory protocol has been analyzed by solving the Maxwell–Bloch equations for a backward scheme in a three-level system. The backward photon echo scheme is combined with a controlled coherence conversion process via controlled Rabi flopping to a third state, where the control Rabi flopping collectively shifts the phase of the ensemble coherence. The propagation direction of photon echoes is coherently determined by the phase-matching condition between the data (quantum) and the control (classical) pulses. Herein, we discuss the classical controllability of a quantum state for both phase and propagation direction by manipulating the control pulses in both single and double rephasing photon echo schemes of a three-level system. Compared with the well-understood uses of two-level photon echoes, the Maxwell–Bloch equations for a three-level system have a critical limitation regarding the phase change when interacting with an arbitrary control pulse area.


Supplementary Information for "Understanding of collective atom phase control in modified photon echoes for a near perfect, storage time extended quantum memory" by
where, l= D or R1 and j= C1 or C2. The ε ( ) is the optical field, is the optical depth parameter and = 12 − = 23 − is the detuning of the atom. For the controlled single rephasing photon echo scheme without R2, the controlled coherence conversion is studied below as discussed in numerically in ref.

I.
D-pulse A weak D-pulse propagates through the medium along z-direction. We assume that D-pulse is much weaker than the -pulse and neglect population change by D: 11 ( , , ) = 1. The resultant Maxwell-Bloch equations by D are obtained by putting ε = ε D and ε = 0 in equations (S4) and (S7) The equation (S8) is the first order linear differential equation. By applying integrating factor − ∆ , the solution is obtained as: Here, the coherence is positive (absorptive), so does the echo. It should be noted that 12 ( , , ) = − 12 ( , , ), where 12 ( , , ) is the density matrix element. Taking the Fourier transform of equation (S11) and substituting into the Fourier version of (S9), we obtain: The solution of equation (S12) in time domain is: The D-pulse exponentially decays as it propagates through the medium.

II. R1-pulse
To retrieve D, we apply a R1-pulse in delay time T after the D-pulse. By the R1-pulse the atoms are excited ( 22 ( , , ) = 1), and the corresponding equations of motion are: (S15) The equations (S14) and (S15) are obtained by substituting ε = R 1 and ε = 0 into equations (S4) and (S7). The solution of equation (S14) yields: (S16) The R1-pulse results in a phase conjugate of the D-excited coherence, so the coherence at = R 1 is equal to the conjugate of equation (S11): (S17) The negative sign in equation (S17) shows the emissive coherence of the photon echo. With some mathematical calculations, we obtain the following equation in a frequency domain: By taking the Fourier transform of equation (S15) and substituting equation (S18), we get: The echo is emitted at = 2 R 1 − D . The efficiency of the echo is 4sinh 2 � 2 �, which is greater than unity for a large optical depth due to the stimulated emission in the inverted medium [1,2].

III. C1 and C2 -pulses
The function of C1-pulse is to convert the optical coherence 12 ( , , ) into spin coherence 13 ( , , ) . The coherence at C 1 is given by: (S20) Here, we consider the first term of equation (S17) related to the evolution of the coherences excited by the D-pulse.
The optical coherence is transferred to spin coherence via relation; 12 ( , , ) = cos � The solution of (S21) is: The C2 -pulse re-swaps the coherences with another /2 phase shift.

(S24)
Here the final atomic coherence is positive. Thus, the echo becomes absorptive and cannot be radiated from the medium [18]. If the area of the C2 is 2 , it brings the atomic coherence halted ( 12 ( , ) = 0) again transferring it to the spin state |3⟩. For 3 -C2, the atoms are returned in the excited state |2⟩, and 12 � , C 2 C 2 , � = − 12 � , C 2 C 2 , � in equation (S24), where the echo becomes emissive. Therefore, -3 control pulse sequence is valid for a single rephasing scheme as shown in ref. [18]. Thus, the controlled AFC echoes with the -control pulse sequence [8,9] results in an absorptive echo [3]. The observation of the controlled AFC is due to coherence leakage by spatial Gaussian distribution of laser light [20]. _____________________________________________________________________________________________