Scalable time reversal of Raman echo quantum memory and quantum waveform conversion of light pulse

In usual cases, the shape of IB G_2,μ(Δ_μ) is characterized by a smooth Gaussian or Lorentzian profile. It is worth noting additional possibilities for realization of the controlled IB besides those discussed in the introduction. The controlled IBs can be realized in solid state media for atoms (molecules) characterized by stochastically oriented frozen permanent dipole moments [22]. Here, the external electric fields can shift the resonant atomic (molecular) frequencies due to the interaction with the permanent dipole moments. The induced Stark shifts will be different for the atoms (molecules) depending on the spatial orientations of its permanent dipole moments. While changing the polarity of external electric field at $\tau = \tilde \tau$ will invert the atomic (molecular) spectral shifts providing the controlled inversion of IB. $\tilde \Delta _{2}(\tau \rangle \tilde \tau )\rightarrow -\tilde \Delta _1$. Such spectral inversion of IB will recover the excited atomic coherence leading to the echo signal emission, respectively.

Also one can use the atomic ensembles in optical waveguides [18] or on the interface of two media with different refractive indices [36]. Here, the controlled IBs can be a result of spatially inhomogeneous optical Stark shifts depending on the atomic coordinate in the waveguide cross section or on the atomic distance from the interface. While the opposite Stark shifts can be induced by changing the carrier frequencies of controlled intensive laser fields.

The time-reversibility of equations (9) and (10) is revealed by the following symmetry transformation [22] for the standard CRIB-protocol: (i) $\tilde \Delta _2\rightarrow -\tilde \Delta _1$, (ii) Z → − z, (iii) τ₂ → − τ₁ and (iv) $\skew3\hat{E_{2,0}}\rightarrow - \skew3\hat{E_{1,0}}$, where the conditions (i)–(iv) are met for the light–atoms equations (9) and (10) together with coupling condition Ω₂/Δ_0,2 = Ω₁/Δ_0,1. An opposite sign of the echo field $\skew3\hat{E}_2$ reflects probably the well-known difference of the relative phases between the light field and excited atomic dipole for the absorption or for the emission processes. This becomes clear if we take into account that equations (9) and (10) are held for a similar transformation but where the condition (iv) is replaced by (iv)': $\hat {M}_{12,2}\rightarrow - \hat {M}_{12,1}$. Inversion of the excited atomic dipole moments on the resonant transition ($|{1}\rangle \leftrightarrow |{2}\rangle$) can be performed by additional 2π-pulse on the adjacent atomic transition ($|{2}\rangle \leftrightarrow |{3}\rangle$). This well-known property of 4π symmetry of the two-level system has been realized experimentally in the QED cavity scheme [50].

Preserving the coupling condition by appropriate variation of the detuning Δ_0,2 and Rabi frequency Ω₂ provides the unitary wavelength conversion [32]. However, we can break the coupling condition transferring to more general symmetry transformation of equations (9) and (10) that makes possible a STR of echo field retrieval. In turn STR will provide an ideal c/d of the input light temporal duration. In order to find the STR conditions we note that the ideal quantum compression of the light pulse duration δt_e = δt₁/η is accompanied by appropriate enhancement of the light field amplitude given by the factor $\sqrt \eta$ [28] ($\langle \skew3\hat{E_2}\rangle =\sqrt \eta \langle \skew3\hat{E_1}\rangle$). By taking into account this requirement in equations (9) and (10) and assuming G_2,μ(Δ_μ) = G₂(Δ₁), we find five basic STR conditions for the echo emission (written in the first possible form):

• (i)  
  (c) $\tilde \Delta _2\rightarrow -\eta \tilde \Delta _1$,
• (ii)  
  (c) Z → − z,
• (iii)  
  (c) τ₂ → − τ₁/η,
• (iv)  
  (c) $\Omega _{2}(-\tau _2/\eta )/\Delta _{0,2}=\sqrt {\eta }\Omega _{1}(\tau _1)/\Delta _{0,1}$.
• (v)  
  (c) $\skew3\hat{E_{2,0}}\rightarrow - \sqrt {\eta }\skew3\hat{E_{1,o}}$.

Also we can find a second form of STR transformation. Here, by saving the first three conditions (i)(c), (iii)(c), we rewrite two others as follows: ${\mathrm { (iv)(c)^{\prime }}} \ \Omega _{2}(-\tau _2/\eta )/\Delta _{0,2}=-\sqrt {\eta }\Omega _{1}(\tau _1)/\Delta _{0,1}$ and ${\mathrm {(v)(c})}^{\prime } \skew3\hat{E_{2,0}}\rightarrow \sqrt {\eta }\skew3\hat{E_{1,0}}$. It is seen that STR indicates an increasing of the reading laser field amplitudes by the factor $\sqrt {\eta }$.

Finally we write STR symmetry in a third form. By saving the first four conditions (i)(c)–(iv)(c) we add two new conditions: (v)(c)'' $\skew3\hat{E}_{2,0}\rightarrow =\sqrt {\eta }\skew3\hat{E}_{1,0}$ and (vi)(c)'' $\hat {M}_{12,2}\rightarrow - \hat {M}_{12,1}$. Concerning the new STR conditions (i)(c)–(v)(c), we note that the fourth condition (iv)(c) arises for preserving the interaction strength of the echo field with the atomic system characterized by new IB bandwidth on this stage. It is worth noting the used initial relation between the bandwidths of IB Δ_1,in and input light signal δω₁ is held for the atomic and light parameters Δ_2,in and δω₂ at the echo emission stage. For example, the scaling spectral factor η > 1 will decrease the interaction strength of light with atoms due to larger IB bandwidth Δ_2,in = ηΔ_1,in. However increasing the Rabi frequency in accordance with (iv)(c) is possible to compensate this negative effect for time reversibility. We change the Rabi frequency of the reading laser pulse by the factor $\sqrt \eta$. In this case we preserve the same absorption coefficient for both stages. Thus preserving the condition (iv)(c) (or its similar form (iv)(c)') via appropriately changing the Rabi frequency of the control laser field Ω₂ or optical detuning Δ_0,2, we can realize the unitary c/d with wavelength conversion of the signal light pulse. Similar consideration is valid for the second and third forms of STR transformations based on the additional conditions: (iv)(c)'–(v)(c)' or (v)(c)''–(vi)(c)''.

The discussed new conditions clearly demonstrate considerable opportunities for the realization of STR dynamics in the studied scheme of PEQM. The question arises about the possibility of such STR symmetry for other realizations of the Raman echo QM scheme.

Following [28, 29] in this case we can switch the initial IB $G_{2,1}=\delta (\tilde \Delta _{1} - \chi _{1} Z)$ to the new shape with the inverted and scaled spatial gradient $G_{2,2}=\delta (\tilde \Delta _{1}+\eta \chi _{1} Z)$ with compression factor η (χ₂ = −ηχ₁ so the light pulse compression occurs for η > 1 and decompression is for η < 1). By using IBs G_2,1 and G_2,2 in equations (9) and (10) we find that all the four basic considerations of STR (i)(c)–(v)(c) with two supplement conditions are met for this type of IB if the echo field is irradiated in the backward direction (see also equation (34)). Thus, the four conditions constitute the generalized time reversal of CRIB based techniques and open a perfect method for quantum c/d of the single photon light fields.

It is worth noting a possibility of QM realization using Raman atomic interactions in the media with time controlled refractive index [51]. So one can also realize the STR of signal pulse in this scheme by combining the time modulation of refractive index with appropriate intensity variation of the control laser field determined by the conditions (i)(c)–(v)(c). For efficient realization of the described schemes, we have to evaluate the optimal physical parameters where the ideal system of equations (9) and (10) can be used for real atomic systems.

The discussed ideal scheme of generalized STR assumes a negligibly small IB width of the atomic transition $1\leftrightarrow 3$ and sufficiently large optical spectral detuning Δ_0,1. Also it is important to optimize the switching rate of control laser fields in order to provide an accessible quantum efficiency. Below we perform such analysis following the original system of equations (3)–(5) and conditions (i)(c)–(v)(c) of the ideal pulse compression.

By using the initial condition $\langle \hat R_{12}(-\infty )\rangle =\langle R _{13}(-\infty )\rangle =0$ in equations (3)–(5) with constant control laser field Ω₁(τ) = Ω_1,0 and applying Fourier transformation $\tilde {B}(\nu ,Z) = \int ^{+\infty }_{-\infty } B(\tau _1,Z)\, \mathrm {e}^{\mathrm {i} \nu \tau _1}\,\mathrm {d}\tau _1$ (where $\hat B(\tau ,Z)$ is atomic or field operator and $\tilde {B}(\nu ,Z)=\langle \hat {\tilde {B}}(\nu ,Z)\rangle$) we get the solution

$$\begin{equation}\fl A_{1,0}(\tau_1,Z) = \frac{1}{2 \pi} \int\nolimits_{- \infty}^{\infty}\mathrm{d} \nu \skew3\tilde{A}_{1,0}(\nu,0) \exp\!\left( -\mathrm{i} \nu \tau_1 - \frac{1}{2}\int_{0}^{Z} \alpha_1 (\nu,Z)\,\mathrm{d} Z\right), \end{equation} \tag{ 11 }$$

$$\begin{equation}\fl R_{12}(\delta_1,\Delta_1;\tau_1,Z) = \int\nolimits_{-\infty}^{\infty} \mathrm{d} \nu \frac {g \Omega_{1,0} \skew3\tilde{A}_{1,0}(\nu ,0) \exp ( -\mathrm{i} \nu \tau_1 - \frac{1}{2}\int_{0}^{Z} \alpha_1 (\nu,Z)\,\mathrm{d} Z) } { 2 \pi\{ ( \Delta_{0,1} +\delta_1-\nu - \mathrm{i}\gamma_{31} )(\Delta_1-\nu-\mathrm{i}\gamma_{21})-\Omega^{2}_{1,0} \}}, \end{equation} \tag{ 12 }$$

$$\begin{equation}\fl R_{13}(\delta_1,\Delta_1;\tau_1,Z) =\int\nolimits_{- \infty}^{\infty}\mathrm{d} \nu \frac {g (\Delta_1- \nu -\mathrm{i}\gamma_{21} ) \skew3\tilde{A}_{1,0}(\nu ,0) \exp( -\mathrm{i} \nu \tau_1 - \frac{1}{2}\int_{0}^{Z} \alpha_1(\nu,Z)\,\mathrm{d} Z )} { 2 \pi \{( \Delta_{0,1} +\delta_1-\nu - \mathrm{i}\gamma_{31} )(\Delta_1-\nu-\mathrm{i}\gamma_{21}) -\Omega^{2}_{1,0}\}}, \end{equation} \tag{ 13 }$$

where for completeness we have added the phenomenological decay constants γ₂₁ and γ₃₁ for the atomic coherences R₁₂, R₁₃ and effective absorption coefficient on the Raman transition,

$$\begin{equation}\fl \alpha_1( \nu,Z ) =-\mathrm{i}\beta \int\int \nolimits^{\infty}_{-\infty}\frac{G(\delta_1,\Delta_1;Z) (\Delta_1 -\nu -\mathrm{i} \gamma_{21})\, \mathrm{d} \delta_1 \,\mathrm{d}\Delta_1 }{( \Delta_{0,1}+ \delta_1 -\nu -\mathrm{i}\gamma_{31}) (\Delta_1-\nu -\mathrm{i}\gamma_{21}) - \Omega^{2}_{1,0} } . \end{equation} \tag{ 14 }$$

Excited atomic coherences are given by general solutions (12), (13) for time τ₁ ≫ δt_s after complete disappearance of the signal light pulse (A_1,0(τ₁ > τ₀,Z) = 0). General analysis of the atomic coherences can be performed on equations (11)–(13) and on similar equations for the echo emission stage. Here, we take into account that the highest quantum efficiency is possible only for the sufficiently weak decoherence of Raman transition γ₂₁ ≪ Δ_1,in. We also assume a sufficiently large spectral detuning on the optical transition Δ_0,1 > δω₁, Ω₁, δ_1,in, Δ_1,in in equations (12) and (13). After such calculation we get the following relations for atomic coherences:

$$\begin{eqnarray*} &&R_{12} (\tau_1)\cong \mathrm{i} \zeta_{12}g \tilde A_{1,0}(\nu_1,0) \,\mathrm{e}^{-\mathrm{i}\nu_1 \tau_1} \exp \left( - \frac{1}{2} \int_{0}^{Z} \alpha_1(\nu_1, Z)\, \mathrm{d}Z\right) \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&R_{13}(\tau_1)\cong\zeta_{13} R_{12}(\tau_1), \end{eqnarray*}$$

where $\zeta _{12}= \Omega _{1,0} / (\Delta _{0,1} + \delta _{1} - \Delta _{1} + 2\frac { \Omega ^{2}_{1,0} }{\Delta _{0,1} + \delta _{1} - \Delta _{1}} )$ and ζ₁₃ = Ω_1,0/(Δ_0,1 + δ₁ − Δ₁) in accordance with equation (6), $\nu _1 \cong \Delta _1 -\frac {\Omega ^{2}_{1,0}}{\Delta _{0,1} + \delta _1 - \Delta _1} - \mathrm {i} \gamma _{\mathrm { eff}}$, $\gamma _{\mathrm { eff}}=\gamma _{21}+\gamma _{31}(\frac {\Omega _{1,0}^2}{\Delta _{0,1}})$. More detailed analysis of ζ₁₂(δ₁) and ζ₁₃(δ₁) shows that influences of these factors on δ₁ will not lead to essential effects for sufficiently large optical detuning Δ_0,1. This property of the Raman photon echo quantum storage reveals a nonadiabatic feature of the light–atom interaction in contrast to the EIT based QM. In this case (Δ_μ,0 ≫ δ_1,in,Δ_1,in), with large accuracy we can take ζ₁₂≅ζ₁₃≅Ω_1,0/Δ_0,1 and to simplify the complex absorption coefficient α₁(ν,Z) as follows:

$$\begin{equation} \alpha_1(\nu,Z) \cong 2\beta \frac {\Omega_{1,0}^2}{\Delta_{0,1}^2} \int_{-\infty}^{\infty}\mathrm{d} \Delta_1 \frac {G_2 (\Delta_1,Z)}{(\Delta_1-\nu-\mathrm{i}\gamma_{\rm eff})}, \end{equation} \tag{ 15 }$$

where both the real α_abs,1(ν,Z) and the imaginary α_dis,1(ν,Z) parts of the complex absorption coefficient can play a significant role in the efficient quantum storage of broadband signal light fields [48].

In the next step we evaluate the quantum storage for the conditions which are close to the case described by equations (9) and (10). Here, we identify the influence of IB on the optical transition to the excited atomic coherences via the frequency $Re(\nu _{1})\cong \Delta _1(1-\frac {|\Omega _{1,0}|^2}{\Delta _{0,1}^2})-\frac {|\Omega _{1,0}|^2}{\Delta _{0,1}}+\delta _{1,R}$, where δ_1,R = δ₁(Ω_1,0/Δ_0,1)² (see for comparison equation (7)). One can see that additional spectral detuning δ_1,R in Re(ν₁) will lead to additional irreversible dephasing of the macroscopic atomic coherences R₁₂(τ) and R₁₃(τ) due to averaging over the exponential factor $\langle \exp \{-\mathrm {i}\delta _{1,R}\tau \}\rangle$. Thus, taking into account δ_1,R in the macroscopic atomic coherence we find that the echo field will decay proportionally to the factors

$$\begin{equation} \Gamma_{\mathrm{G}} = \exp \left( -\frac{1}{4}(\Omega_{1,0}/\Delta_{0,1})^4 (1+\eta^2)(\delta_{1,{\rm in}}(\tau_{\rm echo}(\eta)-\tau_{\rm st}))^2\right), \end{equation} \tag{ 16 }$$

$$\begin{equation} \Gamma\mathrm{_{L}} = \exp \left( - \frac{1}{2} (\Omega_{1,0}/\Delta_{0,1})^2(1+\eta) \delta_{1,{\rm in}}(\tau_{\rm echo}(\eta)-\tau_{\rm st})\right) \end{equation} \tag{ 17 }$$

for the Gaussian (G) and Lorentzian (L) shapes of the IB line on transition $1\leftrightarrow 3$, where we have assumed $(\Omega _{2,0}/\Delta _{0,2})=\sqrt {\eta }(\Omega _{1,0}/\Delta _{0,1})$, $\tau _{{\mathrm { echo}}}(\eta )=\frac {1+1/\eta }{2}\tau _{\mathrm { echo}}(1)$ is the time moment of echo pulse irradiation [28] and τ_st is the storage time on the long-lived atomic levels 1 and 2 (see figure 1).

Then we deal with the control laser field switching to transfer of the excited atomic coherence on the long-lived levels 1 and 2. This problem has not been discussed in the previous studies of resonant Raman echo QM, while we show that it considerably influences the quantum efficiency. It is worth noting here that a similar problem has been studied early in the QM based on the electromagnetically induced transparency [53].

We assume sufficiently large times of the interaction when the input light field (τ₀ ≫ δt_s) and excited macroscopic atomic coherence disappear completely in the medium ($\langle \skew3\hat{E}_1(\tau _0,Z) \rangle =0$). At the same time the resonant atoms are excited to the state characterized by nonzero atomic coherences R₁₂ and R₁₃. Obviously, the ideal quantum storage will occur for complete adiabatic transfer of the optical coherence R₁₃ on the long-lived levels 1 and 2. Below we evaluate the realistic conditions of the perfect transfer of the optical coherence during the both stages.

Taking into account the vanished input light field in equations (3) and (4), we evaluate the influence of switching rate with the exponential decay of the control laser field Ω₁(τ₁) = Ω_1,0 e^{−k(τ₁−τ₀)} (where τ₀ is a moment of time when the switching off procedure starts, k is a switching rate). Here, we have the following atomic equations:

$$\begin{equation} \frac{\mathrm{d} R_{12}}{\mathrm{d}\tau_1} = -\mathrm{i}(\Delta_{1}-\mathrm{i} \gamma_{21}) R_{12} + \mathrm{i}\Omega_{1,0} \,\mathrm{e}^{-k(\tau_1 -\tau_0)} R_{13}, \end{equation} \tag{ 18 }$$

$$\begin{equation} \frac{\mathrm{d} R_{13}}{\mathrm{d}\tau_1} = -\mathrm{i}(\Delta_{0,1} +\delta_{1} -\mathrm{i} \gamma_{31})R_{13} + \mathrm{i}\Omega_{1,0} \,\mathrm{e}^{-k(\tau_1 -\tau_0)} R_{12}. \end{equation} \tag{ 19 }$$

The atomic evolution of equations (18) and (19) will not be accompanied by macroscopic atomic coherence and coherent irradiation of the stored light field, respectively, due to the strong dephasing of excited atoms during the switching procedure.

After the substitutions $R_{12}= \tilde {R}_{12} \,\mathrm {e}^{-(\mathrm {i}\Delta _1+ \gamma _{21})(\tau _1 - \tau _0) }$, $R_{13}= \tilde {R}_{13}\, \mathrm {e}^{-(\mathrm {i}\Delta _1+ \gamma _{21}) ( \tau _1 - \tau _0 )}$, we transfer to the new atomic variables $\tilde {R}_{12} = \chi ^{(1+\mathrm {i}\tilde \alpha )/2} Y(\chi )$, $\tilde {R}_{13} =\chi ^{(1+\mathrm {i}\tilde \alpha )/2} \tilde {Y}(\chi )$ with appropriate timescale χ = e^{−k(τ₁−τ₀)}Ω_1,0/k (where $\tilde \alpha \equiv \left (\Delta _{0,1}+\delta _{1}-\Delta _{1}-\mathrm {i}(\gamma _{31}-\gamma _{21}) \right )/k$) that leads equations (18) and (19) to the form of Bessel equations

$$\begin{equation} \frac{\mathrm{d}^2 Y}{\mathrm{d}\chi^2} +\frac{1}{\chi} \frac{\mathrm{d} Y }{\mathrm{d}\chi} + \left(1- \frac{(1+\mathrm{i} \tilde\alpha)^2}{4 \chi^2}\right)Y =0, \end{equation} \tag{ 20 }$$

$$\begin{equation} \frac{\mathrm{d}^2 \tilde{Y}}{\mathrm{d}\chi^2} +\frac{1}{\chi} \frac{\mathrm{d} \tilde{Y} }{\mathrm{d}\chi} + \left( 1- \frac{(-1+\mathrm{i} \tilde\alpha)^2}{4 \chi^2} \right)\tilde{Y} =0. \end{equation} \tag{ 21 }$$

Using the well-known solutions [52] of these equations,we find the atomic coherences

$$\begin{eqnarray} &&\fl R_{12}(\tau_1) = \mathrm{e}^{ - \left( \mathrm{i}\Delta_1+\gamma_{21} \right)(\tau_1 -\tau_0)} \frac{(2k/\Omega_{1,0})^{ \frac{1 + \mathrm{i} \tilde\alpha}{2}}}{ \Gamma\left(\frac{1 - \mathrm{i}\tilde\alpha}{2}\right) M} \left( J_{\frac{-1 +\mathrm{ i} \tilde\alpha}{2}}\left(\frac{\Omega_{1,0}}{k}\right) R_{12}(\tau_0)+\mathrm{i}J_{\frac{1 + \mathrm{i} \tilde\alpha}{2}}\left(\frac{\Omega_{1,0}}{k}\right) R_{13}(\tau_0) \right),\nonumber\\ \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} &&\fl R_{13}(\tau_1) = \mathrm{i}\, \mathrm{e}^{ - ( \mathrm{i}(\Delta_{0,1}+\delta_1)+\gamma_{31})(\tau_1 -\tau_0)}\frac{(2k/\Omega_{1,0})^{\frac{1 - \mathrm{i} \tilde\alpha}{2}}}{ \Gamma\left(\frac{1 +\mathrm{ i}\tilde\alpha}{2}\right) M} \left(\! J_{\frac{1 - \mathrm{i} \tilde\alpha}{2}}\left(\frac{\Omega_{1,0}}{k}\right)\!R_{12}(\tau_0)-\mathrm{i}J_{\frac{-1 -\mathrm{ i} \tilde\alpha}{2}}\left(\frac{\Omega_{1,0}}{k}\right)\!R_{13}(\tau_0)\! \right)\!,\nonumber\\ \end{eqnarray} \tag{ 23 }$$

where the complete switching off the control laser field occurs for $\tau _1>\tilde \tau _1$, $(\tilde \tau _1 -\tau _0)\gg 1/k$, $J_{\frac {1 + \mathrm {i} \tilde \alpha }{2}}(\frac {\Omega _{1,0}}{k})$ and $\Gamma \left (\frac {1 + \mathrm {i} \tilde \alpha }{2}\right )$ are the Bessel and Gamma functions [52], $M = J_{\frac {1 + \mathrm {i} \tilde \alpha }{2}}(\frac {\Omega _{1,0}}{k})J_{\frac {1 - \mathrm {i} \tilde \alpha }{2}}(\frac {\Omega _{1,0}}{k})+J_{\frac {-1 + \mathrm {i} \tilde \alpha }{2}}(\frac {\Omega _{1,0}}{k}) J_{\frac {-1 - \mathrm {i}\tilde \alpha }{2}}(\frac {\Omega _{1,0}}{k})$.

Figure 2 demonstrates a decrease of the optical coherence $R_{13}(\tau _1=\tilde \tau _1)$ with enhancement of switching rate k for various atomic detunings Δ_0,1 (for weak atomic decoherence $\gamma _{21}(\tilde \tau _1-\tau _0)\ll 1$, $\gamma _{31}(\tilde \tau _1-\tau _0)\ll 1$ and negligibly small IBs δ_1,in,Δ_1,in ≪ Δ_0,1) after complete switching of the control laser field ($\tilde \tau _1-\tau _0\gg 1/k$). Effective depopulation of $R_{13}(\tilde \tau _1)$ occurs for large enough detuning Δ_0,1 where almost adiabatic transfer of the optical coherence occurs to the long-lived levels. The transfer efficiency _t is given by $\epsilon _{\mathrm {t}}= \vert R _{12}(\tilde \tau _1) \vert ^2 / \left ( \vert R_{12}(\tau _0) \vert ^2+\vert R_{13}(\tau _0) \vert ^2 \right )$ together with equation (6) and equations (20), (21) for sufficiently large optical detuning. Remnant coherence R₁₃(τ) will not contribute to the echo signal emitting in the backward direction due to the mis-phasematching. While it is still possible to think about using the remnant optical coherence in the case of fast control laser field manipulations for the forward echo emission. Thus, the control field switching can lead to irreversible losses in quantum transfer to the long-lived atomic levels.

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We evaluate the transfer efficiency _t as a function of switching rate k (see figure 3) for various spectral detunings Δ_0,1. Here we assume a negligibly weak influence of relatively small IBs (Δ_2,in,δ_1,in ≪ Δ_0,1) on _t. This figure demonstrates a negative influence of fast switching procedure to the efficiency _t that is considerably suppressed for large optical spectral detuning Δ_0,1 ≫ Ω_1,0. However, the transfer efficiency _t ≈ 0.96 can occur for relatively small optical detuning Δ_0,1/Ω_1,0 ≈ 5 and fast switching rate. The switching rate k ⩾ 20 leads to an apparently negative impact for intermediate optical detuning Δ_0,1/Ω_1,0 ⩽ 10 where _t ⩽ 0.99. As also seen in figure 3, retardation of the switching off procedure increases the transfer efficiency. However, using the low-speed switching rate k is limited by the relaxation processes of the optical atomic coherence. Moreover, the optical spectral detuning is limited since we have provided a large enough optical depth of the atomic medium in accordance with $\alpha _1\sim \beta \left ( \frac {\Omega _{1,0}}{\Delta _{0,1}} \right )^2$. Thus, we have to use an optimal exchange for choice of the optimal switching rate in order to get the maximum quantum efficiency for really used parameters of the light–atom interaction and IBs.

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Let us assume the control reading laser field is switched exponentially $\Omega _2(\tau _2) = \Omega _{2,0} \,\mathrm {e}^{k_{\mathrm {r}} (\tau _2-\tilde \tau _2)}$ with the switching rate k_r. Evolution of atomic coherences are described by the atomic equations which are quite similar to equations (16)–(19) where k is replaced by ' − k_r' and Δ₁ by ' − ηΔ₁'. By taking into account the initial coherence on the long-lived level $R_{12}(\tilde \tau _1)$, we find the solution for atomic equations after the reading control field is switched on at the moment of time $\tau _2=\tilde \tau _2$

$$\begin{equation} R_{12} (\tilde\tau_2,\delta_1,\Delta_1,Z) = C_{1,2} R_{12}(\tilde\tau_1,\delta_1,\Delta_1,Z), \end{equation} \tag{ 24 }$$

$$\begin{equation} R_{13} (\tilde\tau_2,\delta_1,\Delta_1,Z) = \mathrm{i} C_{1,3} R_{12}(\tilde\tau_1,\delta_1,\Delta_1,Z), \end{equation} \tag{ 25 }$$

where the functions C_1,2 and C_1,3

$$\begin{equation} C_{1,2} = \left( \frac{ \sqrt{\eta} \Omega_{1,0}}{2 k_r} \right)^{\frac{1-\mathrm{i}\Delta_{0,2}/k_{\mathrm{r}}}{2}} \Gamma\left( \frac{1+\mathrm{i}\Delta_{0,2}/k_{\mathrm{r}}}{2} \right) J_{\frac{\mathrm{i}\Delta_{0,2}/k-1}{2}} \left( \frac{ \sqrt{\eta} \Omega_{1,0}}{k_{\mathrm{r}}} \right), \end{equation} \tag{ 26 }$$

$$\begin{equation} C_{1,3} = \mathrm{i} \left( \frac{ \sqrt{\eta} \Omega_{1,0}}{2 k_{\mathrm{r}}} \right)^{\frac{1-\mathrm{i}\Delta_{0,2}/k_{\mathrm{r}}}{2}} \Gamma\left( \frac{1+\mathrm{i}\Delta_{0,2}/k_{\mathrm{r}}}{2} \right) J_{\frac{i\Delta_{0,2}/k_{\mathrm{r}}}{2}} \left( \frac{ \sqrt{\eta} \Omega_{1,0}}{k_{\mathrm{r}}} \right), \end{equation} \tag{ 27 }$$

are given with accuracy of the same phase factor e^iϕ and it was assumed $\Omega _{2,0}=\sqrt {\eta } \Omega _{1,0}$. Here it is worth noting an important property for the functions: |C_1,2|² + |C_1,3|² = 1. Further evolution of atoms and light is accompanied by the rephasing of the macroscopic coherences of the echo field emission. The optimal switching on procedure of the reading laser pulse can be clarified from general analysis of the echo emission.

Here we use the coupled system of light–atom equations with new atomic detuning Δ₂ = −ηΔ₁ on the long-lived transition $1\leftrightarrow 2$ and echo field emission in the backward direction

$$\begin{equation} \frac{ \partial R_{13}}{ \partial \tau_2} = - \mathrm{i} \left( \Delta_{0,2} + \delta_1 \right) R_{13} + \mathrm{i}g A_{2,0} + \mathrm{i} \Omega_{2,0} R_{12}, \end{equation} \tag{ 28 }$$

$$\begin{equation} \frac{ \partial R_{12}}{ \partial \tau_2} = \mathrm{i} \eta \Delta_{1} R_{12} +\mathrm{ i} \Omega_{2,0} R_{13}, \end{equation} \tag{ 29 }$$

$$\begin{equation} \frac{\partial A_{2,0}}{ \partial Z} = -\mathrm{i} \frac{ \pi n g S}{ v_{\mathrm{g}}} \int \nolimits^{\infty}_{-\infty} \mathrm{d} \delta_1 \,\mathrm{d} \Delta_{1} G( \delta_1, \Delta_1, Z ) R_{13}, \end{equation} \tag{ 30 }$$

where the initial condition for atomic coherences and irradiated field are determined by $R_{12} (\tilde \tau _2)$ and $R_{13} (\tilde \tau _2)$ and $A_{2,0}(\tilde \tau _2,Z)=0$ at time $\tau _2=\tilde \tau _2$.

We solve the system equations in spectral Fourier representation of the atomic coherences $R_{12(13)}(\nu ,\ldots ,Z)=\int _{-\infty }^{\infty } \mathrm {d} \tau _2 R_{12(13)}(\tau _2,\ldots ,Z)\,\mathrm {e}^{\mathrm {i}\nu \tau _2}$ and field amplitude $\skew3\tilde{A}_{2,0}(\nu ,Z)=\int _{-\infty }^{\infty } \mathrm {d} \tau _2 A_{2,0}(\tilde \tau _2,Z)\, \mathrm {e}^{\mathrm {i}\nu \tau _2}$. After performing a number of standard calculations [20, 28] we find following equation for the Fourier component of echo field amplitude $\skew3\tilde{A}_{2,0}(\nu ,Z)$:

$$\begin{eqnarray}&&\fl \frac{\partial \skew3\tilde{A}_{2,0}(\nu,Z)}{ \partial Z} = -\frac{\beta}{2g} \int\nolimits^{\infty}_{-\infty} \mathrm{d} \delta_1\, \mathrm{d} \Delta_{1} G( \delta_1,\Delta_1, Z ) \nonumber \\ &&\times \left( \frac{ R_{13} ( \tilde\tau_2,\delta_1,\Delta_1,Z)\left( \nu + \eta \Delta_1 \right) + \Omega_{2,0} R_{12} (\tilde\tau_2,\delta_1,\Delta_1,Z )}{ \left( \Delta_{0,2} + \delta_1 -\nu \right) \left( \nu - \eta \Delta_1\right) + \Omega^{2}_{2,0} }\right. \nonumber \\ &&\left.+\frac{ \mathrm{i}g (\nu + \eta \Delta_{1} ) \skew3\tilde{A}_{2,0}(\nu,Z)}{ \left( \Delta_{0,2} + \delta_1 -\nu \right) (\nu + \eta \Delta_{1}) + \Omega^{2}_{2,0} } \right). \end{eqnarray} \tag{ 31 }$$

In evaluation of this equation we use the early discussed approximations determined by sufficiently large optical detuning Δ_0,1. Then similarly to the calculation of integrals in equations (11)–(13), we find the following equation for the echo field amplitude:

$$\begin{eqnarray} &&\fl \frac{\partial \tilde{E}_{2,0}(\nu,Z)}{ \partial Z}=\frac{\eta^{\prime}}{2\eta}\left(2\sqrt{\tilde\epsilon}\frac{\alpha_{{\rm abs},1}(\nu/\eta,Z)}{\sqrt{\eta^{\prime}}} \tilde{E}_{1,o}(-\nu/\eta,0) \,\mathrm{e}^{\mathrm{i}\nu \tau_{\rm echo}-\frac{1}{2}\int_{0}^{Z} \mathrm{d}z \alpha_1(-\nu/\eta,z)}\right. \nonumber \\ &&\left. + \alpha_1(\nu/\eta,Z) \tilde{E}_{2,0}(\nu,Z) \right), \end{eqnarray} \tag{ 32 }$$

where optical Stark shifts have been taken into account in the new field amplitude $E_{2,0}=A_{2,0} \exp {\{-\mathrm {i}(\beta Z/2\Delta _{0,2})\}}$ and $\Omega _{2,0}/\Delta _{0,2}=\sqrt {\eta ^{\prime }}\Omega _{1,0}/\Delta _{0,1}$, $\alpha _{{\mathrm { abs}},1}(\Delta _1/\eta ,z)=2\pi \beta \frac {\Omega _{1,0}^2}{\Delta _{0,1}^2}G_2 (\Delta _1/\eta ,z)$ is an absorption coefficient, $\sqrt {\tilde \epsilon }=\sqrt {\epsilon _{\mathrm {t}}} (C_{1,2} + \mathrm {i} \frac {\sqrt {\eta \prime }\Omega _{1,0}}{\Delta _{0,1}} C_{1,3})\Gamma _{\mathrm {G},\mathrm {L}}$.

The solution of equation (32) can be written for the arbitrary ratio η'/η. However, we discuss only the case η' = η. Taking into account

$$\begin{eqnarray*} &&\alpha_1(-\nu/\eta,z)+ \alpha_1(\nu/\eta,z)= 2\alpha_{{\rm abs},1}(\nu/\eta,z), \end{eqnarray*}$$

in equation (32), we get the solution

$$\begin{eqnarray} &&\fl \tilde{E}_{2,0}(\nu,Z)=\sqrt{\tilde \epsilon}\frac{\tilde{E}_{1,0}(-\nu/\eta,0)}{\sqrt{\eta}} \mathrm{e}^{\mathrm{i}\nu\tau_{\rm echo} +\frac{1}{2}\int_{0}^{Z} \mathrm{d}z \alpha_1(-\nu/\eta,z)}\nonumber \\ &&\times( \mathrm{e}^{-\int_{0}^{Z} \mathrm{d}z^\prime \alpha_{{\rm abs},1}(-\nu/\eta,z^\prime)} -\mathrm{e}^{-\int_{0}^{L} \mathrm{d}z^\prime \alpha_{{\rm abs},1}(-\nu/\eta,z)}). \end{eqnarray} \tag{ 33 }$$

Equation (33) is valid for the arbitrary type of IB including the transverse and longitudinal IBs. In the case of almost constant optical depth ($\int _{0}^{L} \mathrm {d}z^\prime \alpha _{{\mathrm { abs}},1}(-\nu /\eta ,z^\prime )\cong \tilde \kappa$) within the input light pulse spectrum, we get for the irradiated echo field ($E_{2,0}(\tau _2,Z=0)=\frac {1}{2\pi }\int _{-\infty }^{\infty } \mathrm {d} \nu \tilde {E}_{2,0}(\nu ,0) \,\mathrm {e}^{-\mathrm {i}\nu \tau _2}$):

$$\begin{equation} E_{2,0}(\tau_2,0)=\sqrt{\tilde \epsilon \eta} E_{1,0}\left(-\frac{(\tau_2-\tau_{\rm echo})}{\delta t_{\mathrm{e}}},0\right) \mathrm{e}^{-\gamma_{21}\tau_{\rm echo}} (1-\mathrm{e}^{-\tilde\kappa}), \end{equation} \tag{ 34 }$$

where δt_e = δt₁/η is the temporal duration of the echo pulse, the exponential factor e^{−γ₂₁τ_echo} is determined by the relaxation of long-lived atomic coherence during storage time. The result confirms the general consideration of section 3 based on the formal using of STR symmetry. Using equation (34) we obtain the quantum efficiency:

$$\begin{equation} \epsilon= \epsilon_{\mathrm{t}} \epsilon_{\mathrm{r}}\Gamma_{\mathrm{G},\mathrm{L}}^2 \,\mathrm{e}^{-2\gamma_{21}\tau_{\rm echo}}|1-\exp(-\tilde \kappa)|^2, \end{equation} \tag{ 35 }$$

where the latter two factor in (phase relaxation of the atomic transition $1\leftrightarrow 2$ and finite optical depth $\tilde \kappa$) are quite typical for CRIB in particular for the Raman scheme [32, 33, 35]. The first three factors (result of this work) describe an influence of switching procedure and IB on the optical transition. Here, the quantum efficiency of switching on is determined by

$$\begin{equation} \epsilon_{\mathrm{r}} = \vert C_{1,2} \vert^2 + \vert \frac{\sqrt{\eta}\Omega_{1,0}}{\Delta_{0,1}} C_{1,3} \vert^2. \end{equation} \tag{ 36 }$$

Quantity _r demonstrates maximum efficiency for fast switching on of the control laser field in contrast to the switching off procedure.

We can conclude from equation (36), that only a small part (${\sim}\vert \frac {\sqrt {\eta }\Omega _{1,0}}{\Delta _{0,1}} C_{1,3}\vert ^2{\ll}\vert C_{1,3}\vert ^2$) of the excited optical coherence evolves adiabatically and rephases with long-lived atomic coherence. Such behavior is accompanied by frozen dephasing of this optical atomic coherence. While another leaving part of the atomic coherence will not participate in the echo emission that decreases the echo field retrieval. Particular properties of the switching on efficiency are demonstrated in figures 4–6. Here it is worth noting that the quantum efficiency _r is independent of scaling factor η for a sufficiently fast switching rate. In figures 7 and 8, we evaluate the overall efficiency only for the fast rate of switching on.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Thus in the case of highly perfect STR, the main influence to the quantum efficiency can be determined by the switching off procedure in accordance with the factor ∼_tr. All other decoherent processes are almost repeated for the echo emission stage. In particular as seen in equations (16) and (17), it is realized for the influence of extra detuning δ_1,R with weak modification due to the scaling factor η.

Efficiency of the quantum waveform conversion (35) is depicted in figure 7 using quite typical atomic parameters for experiments with condensed and gaseous atomic systems. Figure 7 demonstrates possible optimal values of optical detuning Δ_0,1 and interaction time τ_echo − τ_st in the presence of the writing and reading control fields. As seen in the figure, maximum efficiency takes place for a relatively small interaction time. For a larger interaction time, one can find a local maximum at the frequency detuning Δ_0,1 ≈ 6.5Ω_1,0. This maximum arises due to suppression of the atomic dephasing caused by IB on $1\leftrightarrow 3$, however further increasing of Δ_0,1 reduces considerably the effective optical depth. For highly efficient waveform conversion (realized at small interaction time), one can see the large influence of the switching rate in figure 8 which provides almost perfect c/d and wavelength conversion of the signal light pulse.

Leaving further discussion of STR symmetry we note that the quantum efficiency characterizes general features of perfect transformation for weak quantum light pulses and can be used for studying the delicate interaction of single and multi-pulse light fields with resonant media. We note that the relatively slow rate of the switching off procedure is preferable for the storage stage since we need to transfer the excited coherence to the long-lived levels. However the read-out stage is needed in a fast switching rate. Even the control laser field is turned off without the presence of the signal light field, the perfect storage requires an optimal switching of the writing laser field. The optimization problem of the control field shapes practically reminds the optical QM based on the electromagnetically induced transparency [53].