Quantum memory based on phase matching control

Quantum memory based on phase matching control† Xi-Wen Zhang1 , A Kalachev2,3 , P Hemmer4 , M O Scully1,5,6 and O Kocharovskaya1 1

Department of Physics and Astronomy and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843-4242, USA 2 Zavoisky Physical-Technical Institute of the Russian Academy of Sciences, Sibirsky Trakt 10/7, Kazan, 420029, Russia 3 Kazan Federal University, Kremlevskaya 18, Kazan, 420008, Russia 4 Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas 77843-3128, USA 5 Princeton University, Princeton, New Jersey 08544, USA 6 Baylor University, Waco, Texas 76706, USA E-mail: [email protected] Abstract. We discuss a class of Quantum Memory (QM) schemes based on Phase Matching Control (PMC). A single-photon wave packet can be mapped into and retrieved on demand from the long-lived spin grating in the presence of the control field, forming along with signal field a Raman configuration, when the wave vector of the control field is continuously changed in time. Such mapping and retrieval takes place due to the phase matching condition and requires neither a variation of the amplitude of the control field nor inhomogeneous broadening of the medium. We discuss the general model of PMC QM and its specific implementation via (i) modulation of the refractive index, (ii) angular scanning of the control field, and (iii) its frequency chirp. We show that the performance of the PMC QM protocol may be as good as those realized in the Gradient Echo Memory (GEM) but achieved with less stringent requirements to the medium. We suggest the experimental realization of PMC QM in nitrogen vacancies (NV) and silicon vacancies (SiV) in diamond as well as in rare-earth doped crystals.

PACS numbers: 42.50.Ex, 42.50.Gy, 32.80.Qk

†We dedicate this paper to the memory of Professor Igor Yevseyev, the internationally renowned scientist, one of the pioneers of the field of quantum coherence effects. We are keeping warm memories of many fruitful and pleasant communications with Professor Yevseyev during the International Laser Physics Workshops, which he was so successfully organizing and hosting.

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1. Introduction Developing optical quantum memory [1, 2, 3] is considered to be one of the essential steps to build quantum repeaters that provide a way of implementing long distance fiber-based quantum communication. Quantum repeater protocols involve creating entangled photons, storing them in quantum memories, and swapping their entangled states [4]. Thus it is important to develop effective methods of storage and retrieval of single-photon wave packets. Several promising schemes for efficient quantum storage have been experimentally demonstrated, including electromagnetically induced transparency (EIT) [5, 6, 7, 8, 9, 10], Raman interaction [11, 12, 13], controlled reversible inhomogeneous broadening (CRIB) or gradient echo memory (GEM) in gases [14, 15, 16] and rare-earth-ion doped solids [17, 18], and atomic frequency comb (AFC) [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In EIT and Raman schemes, the storage is based on temporal variation of the control field amplitude. An optimal temporal shape of the control field and a control-signal fields synchronization are needed to efficiently store and recall the signal pulse. This means before the storage one needs to know the shape and arrival time of the signal field, which inevitably limits the real application of quantum memory. In GEM scheme, the storage of the signal relies on the creation of a resonance absorption frequency gradient by Stark or Zeeman effect. This confines the materials for implementation to those demonstrating these effects. In AFC scheme, a tailored inhomogeneous broadening (frequency comb) is created before storing the signal. In order to achieve good memory efficiency, a high quality frequency comb should be prepared via optical pumping. Such preparation requires a presence of the fine structure in the ground state and usually takes long time. Bearing in mind the above limitations, we study the possibility of a quantum storage scheme based on phase matching control (PMC), which does not require manipulation with the inhomogeneous broadening or amplitude variation in time of the control field and its synchronization with the signal field. Specifically we consider the PMC quantum memory protocol based on Raman interaction of the quantum signal field and classical control field with the medium, i.e. under two-photon resonance condition, when the frequency difference of these fields coincides with the spin transition frequency. The benefit of using the Raman interaction is the possibility to achieve wide-bandwidth, low-noise and inhomogeneous-broadening-insensitive storage. The pay-off is a relatively low coupling constant, which however can be overcome by using a strong transition, large driving power, and/or placing the medium into a cavity. Phase matching is a widely used concept in optics, especially in nonlinear optics. It stems from the conservation of momentum, meaning that the optical process is most efficient when the wave vectors match each other. In the context of quantum memory protocol, the phase matching condition in the case of Raman interaction means that the established spin coherence wave should have the wave vector matching the combination of the wave vectors of to the input fields. The idea of quantum storage scheme based on PMC is to map the different temporal parts of the signal field into the spin coherence

Quantum memory based on phase matching control

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waves with different wave vectors by exciting them selectively and subsequently via continues temporal variation of the wave vector of the control field. By the end of this mapping process the coherence grating is formed in the medium which stores the information about the signal field within the life time of the spin coherence. During the read out process, the control field wave vector is manipulated as a time reversal of the storage one. This allows the control field to pick up the wave-vector-matched spin wave in time ordered manner and to retrieve the signal field from the spin grating. Depending on in which way the control field wave vector is manipulated with time, there are three different schemes of the Raman PMC QM, namely, quantum storage based on (i) the refractive index control [29, 30, 31], (ii) the control field propagation direction angular scanning [32, 33], and (iii) the control field frequency chirp. In this paper, we develop the general model of the Raman PMC QM schemes, summarize the results and further develop the analysis of the first two schemes (proposed in our recent works [29, 30, 32, 33]), suggest and analyze the frequency chirp scheme, compare the basic conditions and requirements for all three PMC QM schemes and propose the experimental realization of PMC QM in color vacancy centers in diamond and in rare-earth doped crystals. The paper is organized as follows. In Sec. 2, the general model of the quantum memory based on Raman PMC is presented. In Sec. 3 and 4 the major requirements for realization of PMC QM, accordingly, via refractive index modulation and control field angular scanning are summarized. In Sec. 5, the Raman PMC QM via control-field frequency chirp is proposed and analyzed. In Sec. 6 the comparable analysis of all three Raman PMC schemes is presented and the experimental implementation issues of Raman PMC QM are discussed. The conclusions are given in Sec. 7. 2. The general model and basic equations for the Raman PMC QM In this section we introduce the general model of the quantum memory scheme based on PMC. We consider the storage and retrieval of a single-photon wave packet Es (r, t) through Raman interaction with a strong classical field Ec (r, t) in a three-level atomic medium with a Λ-type level structure, see Fig. 1(a). The atoms are assumed to be stationarily, uniformly distributed in a cylindrical geometry with medium length L. The coordinate system is originated at the center of the sample. The duration of storage and retrieval processes is equal to T . During the storage, t ∈ (−T, 0), the signal field interacts with the wave-vector-modulated control field, thus creates spin waves with different wave vectors, see Fig. 1(b, c). During retrieval, t ∈ (0, T ), the interaction between the spin waves and the control field with oppositely modulated wave vectors reconstructs the signal field. For convenience, we neglect the free decay of the spin wave between the end of the storage and the beginning of the retrieval in this paper. The signal field (single-photon wave packet) of average frequency ωs and wave vector

Quantum memory based on phase matching control (a)

4 (b)

Δ

|2\

r kc

r K

r ω c , kc (t )

r ω s , ks

r ks δ

|1\

|3\

(c)

Input signal field

Output signal field 0

-T

T

t

modulating refractive index n(t )

r Spin wave vectors K during retrieval

r Spin wave vectors K during stroage modulating propagation direction nˆ(t )

Figure 1. (a) Energy diagram of the Raman interaction in a three-level Λ system; (b) Phase matching between signal wave vector ks , control field wave vector kc and the spin wave vector K. (c) Schematic illustration of the principle of quantum storage based on off-resonance Raman phase matching control. By manipulating the control field wave vector as a function of time, the temporal profile of the signal field is mapped into (storage) or out from (retrieval) the spin waves with different wave vectors. For illustration purpose, discrete spin wave vectors are sketched.

ks propagating along the zˆ direction can be written as √ ~ωs Es (r, t) = i a(r, t) ei(ks z−ωs t) + H.c., (1) 2ε0 ns c where ns is the refractive index on the frequency of the signal field taking into account the contributions from the host material and the resonant atoms, and c is the speed of light in the vacuum, a(z, t) is the slowly varying annihilation operator. The classical control field contains wave vector modulation, which is described by the phase factor ϕ(r, t): ¯

Ec (r, t) = E0 ei[kc r−ωc t+ϕ(r,t)] + c.c. ,

(2)

¯ c are the average angular frequency and wave where E0 is a constant amplitude, ωc and k vector of the control field, respectively. The collective atomic operators are defined as the mean values of the single-atom operators 1 ∑ σmn (r, t) = |mj ⟩⟨nj | δ (3) (r − rj ) , (3) N j where N is the constant atomic number density, and |nj ⟩ is the nth state (n = 1, 2, 3) of jth atom with the energy ~ωn (set ω1 = 0). The collective spin coherence operator can be expressed into a slowly varying amplitude of coherence on the Raman transition s(z, t) multiplied by a fast oscillating factor: ¯

σ13 (r, t) = s(r, t) e−i[(ωs −ωc )t−(ks −kc )r] .

(4)

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We assume the medium has a large Fresnel number of the signal field to neglect the transverse diffraction. Consider all the atoms are initially in the ground state |1⟩ so that Langevin noise atomic operators are not present, then the off-resonance Raman interaction can be described by the following equations (see Appendix): ( ) ∂ 1 ∂ + a(r, t) = −g ∗ N s(r, t)eiϕ(r,t) , (5) ∂z vsg ∂t ∂ s(r, t) = (−γ + iδ)s(r, t) + ga(r, t)e−iϕ(r,t) , (6) ∂t where the decoherence rate between level |2⟩ and |1⟩ is neglected comparing with onephoton detuning ∆√= ωs − ω2 , vsg is the group velocity of the signal field inside the ∗ ωs medium, g = d21∆Ω is the coupling constant between the atoms and the weak 2~ε0 ns c quantized field, and Ω = d23 E0 /~ is the Rabi frequency of the classical control field, dij is the dipole moment of the transition between |i⟩ and |j⟩, γ is the rate of dephasing of the spin coherence, and δ = ωs − ωc − ω3 is the two-photon detuning. Equation (5) and (6) can be reduced to the equations describing the GEM scheme if the incoming signal pulse duration is much longer than the propagation time inside the medium and the phase factor ϕ(z, t) is linearly dependent on zt. Such equivalence can be understood by regarding the phase factor ϕ(z, t) as a result of either a time dependent wave vector or a spatial dependent frequency of the control field. This analogy provides a useful interpretation of the considered approach in frequency domain. There are three different ways to manipulate the control field wave vector and, accordingly, three different schemes to realize the Raman PMC QM: (i) by modulating the refractive index, (ii) the propagation direction, and (iii) the control field frequency. These schemes are discussed below. 3. Quantum storage via variation of refractive index Let the refractive index of the storage material be changed linearly in time during the interaction of atoms with the counter-propagating input and control fields. The counter propagation is necessary when refractive index is modulated in the same way for both fields. Otherwise, co-propagating geometry is also possible. Then the magnitude of the wave vector of a spin wave created via off-resonance Raman interaction also becomes a linear function of time so that during storage the single-photon wave packet is mapped into a superposition of spin waves with different magnitudes of the wave vector (see Fig. 1(c)). In doing so the amplitude of the signal pulse as a function of time, say Ein (t), is imprinted in the amplitude of the spin wave as a function of the wave vector, say S(K), where K = ks − kc , and ks and kc are wave vectors of the input/output and control fields, respectively. Retrieval is achieved by interaction of the atomic system with the control field as the same values of refractive index are scanned again. If they are scanned in the reversed order as during storage, the output pulse Eout (t) becomes a time-reversed replica of the input one, while in the case of scanning in the same order,

Quantum memory based on phase matching control

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the output pulse reconstructs the input signal [29]. Such retrieval without reversal is possible in a cavity with an optically thin medium. For a single pulse to be stored or retrieved, the total change of refractive index needs to be on the order of λc /L, where L is the sample length and λc is the control field wavelength [29], which gives ∆nmin ∼ 10−5 under typical experimental conditions. The ratio between the total accessible range ∆n and this minimum value determines the number of pulses that can be stored in a series. One of the problems on the way to experimental implementation of the suggested technique is that in general it is rather difficult to realize a modulation of refractive index without modulations of atomic levels. If a doped nonlinear crystal is used as a storage medium, we can take advantage of the linear electro-optic effect. However, using an external electric field for refractive index control in doped nonlinear crystals meets significant constraints due to the linear Stark effect. It is necessary to select a specific class of impurity ions having a definite orientation of the difference between the permanent electric dipole moments of the ground and excited states connected by the working resonant transition. A possible candidate is LiNbO3 crystal doped by rare-earth ions, which allows one to achieve the maximum value of index change on the order of 10−3 [30]. Nonlinear crystals doped by rear-earth-metal ions are considered as promising materials for quantum storage applications [34]. It is worth to note that PMC QM in general and, it’s particular scheme based on modulation of the refractive index, does not necessary imply the usage of the control field and may be realized in a two level system. Such two-level version of PMC QM based on refractive index change was developed in LiNbO3 doped by Tm3+ ions [31], where the existence of the ion sites with appropriate local symmetry was shown. 4. Quantum storage via angular scanning of the control field Quantum storage via refractive index control may be considered as a particular onedimantional case of the more general three-dimensional approach, wherein an input pulse is mapped to a superposition of collective atomic states corresponding to orthogonal modes of a finite interaction volume. By continuously changing refractive index or frequency of the control field we project the input pulse shape onto a subsystem of modes which differ in length of their wave vectors. In a similar manner, by changing the direction of propagation of the control field we project the input pulse on the modes with different directions of the wave vectors (Fig. 1(c)). Let the control field propagate at a non-vanishing angle with respect to the signal field, and this angle be continuously changed with time. As a result, the projection of the spin wave vectors along the signal field propagation direction changes its length as a function of time (Fig. 1(c)), thereby mapping the signal pulse shape at different moment of time into different longitudinal modes of the spin wave. This scheme can be implemented both in free space [32] and in a cavity [33]. In the case of transverse control field the developed scheme proves to be equivalent to

Quantum memory based on phase matching control

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GEM scheme in free space. But unlike GEM, this scheme does not require a direct control of atomic levels (which potentially reduces decoherence in the system) and can be implemented in resonant media which demonstrate neither linear Stark or Zeeman effects nor electro-optic effects. It has the advantage of GEM in that high efficiency can be achieved without backward retrieval. Good storage efficiency can be achieved under the following condition [32] λc 2∆θ < < λc |g|2 N (7) ∆tL T for transverse excitation regime, where λc is the control field wavelength, ∆t is the signal pulse duration, 2∆θ is the total angle of rotation of the control beam during the storage (−T, 0) or retrieval (0, T ) process, N is the atomic number density, and L is the medium length. For a single pulse to be stored or recalled in the free space model, the minimum increment |∆kc /kc | (or the rotation angle) of the control field wave vector kc should be on the order of 10λc /L. In other words, the ratio between the minimum angle of rotation and the beam diffraction angle (resolvable spot number) should be equal to ∼ 10. The necessary angular rate of rotation proves to be of the order of 103 rad/s for L = 1 cm [32], which can be achieved by commercial equipment. The optimal configuration is to rotate the angle near the transverse direction with respect to the signal field, because this case corresponds to the maximum change of the longitudinal component of the control field wave vector and hence the spin wave at the given variation of the angle. Enclosing an atomic ensemble in a cavity makes it possible to achieve high efficiency of quantum storage with optically thin materials, which is typical for Raman transitions, thereby making it possible to realize the scheme even in the case of relatively small dipole moments of optical transitions. We develop a three-dimensional theory, which allows us to consider storage and retrieval of spatially multimode states containing information not only in the pulse envelope but also in the transverse profile of the field [33]. As in the case of the free-space storage, retrieval is almost perfect in the case of transverse control field. The spatial multimode storage is crucially important for multiplexing in quantum repeaters, which can significantly increase the rate of quantum communication in possession of short-time quantum storage, and for holographic quantum computers. It was experimentally demonstrated using EIT [35, 36, 37, 38] and GEM [39, 40], but only in free space. As shown in [33], if we take confocal cavities with Fresnel number equal to 10, the total number of accessible transverse modes T EMmn approaches 1000. Storage of such a multimode transverse field can be combined with that of the multimode longitudinal profile, i.e., with storage of a complicated pulse shape or a sequence of pulses. The minimum angle of rotation per pulse proves to be ∼ 5λc /L, while accessible angular scanning range around transverse direction is about ≈ 20◦ for typical values of experimental parameters (see Ref. [33] for details). Therefore, one can store about 100 of the spatially multimode pulses in a sequence. Thus we can predict a large storage capacity for the proposed memory scheme. The experimental realization of this scheme is discussed in Sec. 6.

Quantum memory based on phase matching control

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5. Quantum storage based on control field frequency chirp In this section we suggest and analyze the third and the last possible scheme of PMC QM. It is based on frequency chirp of the control field. Adding a frequency chirp to the control field provides one more way to achieve the modulation of wave vector, different from those discussed above. Indeed, the propagation of the frequency chirped control beam produces time dependent part of the wave vector, which will sweep the phase matching condition temporally during the storage. Such scheme illustrates the principle of Raman PMC, offers additional degree of freedom to achieve quantum storage, and completes the discussion of Raman PMC QM schemes. However, as is shown below, changing the frequency as a function of time introduces additional two-photon detuning that needs to be compensated in this scheme. Two figures of merit for the quantum memory need to be considered: total efficiency η and fidelity F . We adopt similar definitions as in [32], namely, the total efficiency is defined as Nout η= , (8) Nin ∫0 ∫∞ where Nin = −∞ dt ⟨a†in (t)ain (t)⟩ and Nout = 0 dt⟨a†out t)aout (t)⟩. Here ain (t) = a(z = −L/2, t < 0) and aout (t) = a(z = L/2, t > 0). The fidelity can be defined as ∫ ∞ 2 1 † ′ ¯ − t)aout (t)⟩ , F = dt ⟨a ( t (9) in Nin Nout 0 where t¯ − t takes into account the time reversal and possible delay of the output pulse.

We also introduce a quantity F ′′ to describe the conservation of the pulse amplitude of the retrieved field: (∫ ∞ )2 1 † ¯ ′′ dt ⟨ain (t − t)aout (t)⟩ . (10) F = Nin Nout 0

Specifically, we consider a single-photon signal field Es (z, t) and a strong classical control field Ec (z, t) propagate collinearly along z direction with orthogonal polarizations. The two fields interact with each other through off-resonance Raman interaction (Fig. 1(a)). The control field has a frequency chirp, meaning that its frequency changes with time. Such a frequency chirp leads to a time dependent wave vector, which is used to achieve phase matching control in the process of quantum storage. It is worth noting that frequency chirped control field has been considered in an AFC scheme [41], where the chirp as an aide helps to increase the control filed bandwidth and thus reduce the requirement of the intensities. In present proposal, the chirp plays a principal role, leading to a time dependent excited spin wave vector, which records the temporal shape of the single-photon wave packet. Our model is outlined in Fig. 2. During the storage, the Raman interaction between the signal and control fields creates spin coherence, and the temporal information of the signal field is recorded into the spin wave with difference wave vectors, see Fig. 2(a). During retrieval, the frequency chirp of the control field is reversed, resulting in an

Quantum memory based on phase matching control

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oppositely modulated control wave vector. Such a control field will read the spin wave into the signal field in a time-ordered manner, see Fig. 2(b).

Storage

R -L/2 Interaction volume

rt kr s kc

L/2

z

Spin wave

x

r r ks - kc

Atomic medium

(b) Spin wave

Control field

x

r kc

z

0

r r ks - kc

Forward Retrieval

Output signal

Input signal

Control field

(a)

t r ks

Figure 2. Model of quantum storage based on control field frequency chirp in a offresonance Raman interaction. (a) During storage, the temporal profile of the signal field is mapped into the spin wave distributed over different spin wave vectors; (b) During forward retrieval, the spin wave profile is mapped back into the output signal field.

Let the sample be of a cylindrical geometry with the length L and a large Fresnel number. For such a pencil shape sample, we adopt a one dimensional model to describe the system. The signal field is given by Eq. (1) with plain wave approximation, and the classical control field contains a frequency chirp: Ec (z, t) = E0 ei[kc0 z−ωc0 t+ϕ(z,t)] + c.c. ,

(11)

in which ωc0 is the angular frequency of the control field at t = 0, and kc0 = ωc0 /vc , where vc is the phase velocity of the control field in the medium. We consider only the case of the linear chirp, so that the phase factor reads 1 2 α 1 (z − z0 )2 ϕ(z, t) = − αt + (z − z0 )t − α , 2 2 vcg 2 vcg

(12)

where α is the chirp parameter and vcg is the group velocity of the control field. The first term on the right hand side of Eq. (12) is a time dependent phase from the frequency chirp. The second term is a time and space dependent phase coming from the chirp-leaded wave vector modulation. z = z0 is a phase stationary plane in which wavevector-modulation part of the phase remains constant during storage or retrieval. In the long pulse regime (propagation time of the signal and control field inside the medium tp is much shorter than the signal pulse duration ∆t), which is considered below, the 2 0) may be neglected in (12). Indeed, as it follows from higher spatial order term − 21 α (z−z 2 vcg the left hand side of condition (18) (derived below), with a proper choice of the chirp 2 p cg , the second order phase is at most on the order of 12 α vL2 ∼ πt , parameter α ∼ 2πv ∆tL ∆t cg which is ≪ π in the long pulse regime.

Quantum memory based on phase matching control

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Defining the slowly varying collective spin coherence operator as s(z, t) = σ13 (z, t)e

[ ] i (ωs −ωc0 )t− 21 αt2 − vα z0 t−(ks −kc0 )z cg

,

the quantum storage process can be described by the following equations: ) ( ∂ 1 ∂ i α zt + a(z, t) = −g ∗ N s(z, t)e vcg , ∂z vsg ∂t ∂ −i α zt s(z, t) = [−γ + iδ(t)] s(z, t) + ga(z, t)e vcg . ∂t

(13)

(14) (15)

Denote δ0 = ωs − ωc0 − ω3 − |Ω| − vαcg z0 (where ω3 is the Raman transition frequency), ∆ then δ(t) = δ0 −αt is the time dependent two-photon detuning. The time dependent part originates from the changing of the control field frequency. Namely, the frequency chirp of the control field provides the phase matching control phase factor, but simultaneously introduces a resonance frequency shift. Equations (14) and (15) are similar to the equations describing GEM scheme except for a presence of an inherent time dependent two-photon detuning. The term i vαcg zt in Eqs. (14) and (15) is responsible for the creation and erasure of the spin grating (leading to quantum storage), while iδ(t), on the other hand, destroys the quantum storage by detuning the system away from two-photon resonance. In order to store a single photon wave packet, this time-dependent two-photon detuning has to be compensated. i α zt Assuming it is compensated and defining new variable S(z, t)=−g ∗ N s(z, t)e vcg , in long pulse regime equations (14) and (15) in a co-moving frame (τ = t − z/vsg ) read: ∂ a(z, τ ) = S(z, τ ), (16) ∂z ( ∂ α ) S(z, τ ) = − γ + i z S(z, τ ) − |g|2 N a(z, τ ) , (17) ∂τ vcg 2

These equations are equivalent to those describing other PMC QM schemes (namely, via refractive index modulation and angular scanning) which, in its turn, are equivalent to the equations describing the GEM scheme [42, 32]. Indeed, the time dependent wave vector of the spin wave in all PMC QM schemes (produced by the control field wave vector changing) can be equivalently viewed as a spatial dependent spin transition absorption frequency corresponding to GEM. The quantum memory performance of GEM scheme and other schemes of PMC QM has been well studied; see Refs. [43, 17, 14, 42, 44, 32] for details. It is convenient to introduce the “chirp depth” ∆d = αT , which shows the amount of chirp during the storage or retrieval time T . High efficiency and field amplitude correlation can be achieved when vcg T ∆d . . vcg |g|2 N T . (18) L ∆t 2π Let us consider now the possibilities for compensation of the time-dependent twophoton detuning. In order to do that, one needs to modulate the atomic energy level |3⟩ simultaneously with the chirp in a way that two-photon detuning δ(t) = 0, namely, ω3 (t) = ωs − ωc0 −

|Ω|2 α − z0 − αt . ∆ vcg

(19)

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At the retrieval stage the sign of the frequency chirp parameter should be changed (so that the equivalent “inhomogeneous broadening” would be reversed), while the modulation of the atomic level should be switched simultaneously to compensate the two-photon detuning introduced by this chirp and to ensure keeping of the two-photon resonance. One way to achieve linearly shift of the ground state |3⟩ is to use dc Stark or Zeeman effect. The shift of the latter is proportional to µB /2π~ = 14 GHz/T, where µB is the Bohr magneton. In the presence of polarization selection rules, the control field acts only on the transition of |2⟩-|3⟩ levels, which are essentially almost empty. So the medium is dispersionless for the control field. As a result, its group velocity vcg is equal to phase velocity vc , which is typically on the order of the speed of light in the vacuum. Then according to (18), for medium length L = 2 cm and signal field T /∆t = 20, the chirp depth ∆d /2π should be on the order of 286 GHz. Numerical calculation shows that a chirp depth ∆d /2π = 80 GHz can store and recall such a signal pulse with high efficiency and fidelity in a medium with |g|2 N T ∼ 2000 /m. A linear time dependent magnetic field up to ∼ 1 T is able to compensate a chirp depth ∆d /2π ∼ 30 GHz. Such a magnetic field can be modulated as fast as with 10 µs [45] time period. If the g-factors for the ground and excited state are the same, the levels |1⟩ and |2⟩ can be shifted roughly by the same amount, while the level |3⟩ can be shifted substantially by different amount. As an example, in Fig. 3, we show a single photon wave packet of Gaussian temporal shape ∆t/T = 1/20 is stored and retrieved with a chirp depth ∆d /2π = 95 GHz in a 2 cm medium with |g|2 N T ∼ 2650 m−1 with high efficiency and fidelity. The lifetime of the spin wave is assumed to be much longer than the signal pulse duration. 8

7

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|aout| (a.u.)

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ain (a.u.)

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Figure 3. Incoming signal field ain = a(−L/2, τ < 0) and forward retrieved field aout = a(L/2, τ > 0) via Raman PMC QM based on control field frequency chirp. The corresponding parameters are: λs ≈ λc = 700 nm, |g|2 N T = 2650 m−1 , ∆t/T = 1/20, L = 2 cm, ∆d = 6 × 1011 rad/s. The efficiency η and fidelity F ′ are 96%. The quantity F ′′ , which describes the preservation of the pulse amplitude without taking into account possible phase modulation, is 98.6%.

Alternatively, another way to shift the ground state |3⟩ is to take advantage of acStark effect, which potentially allows the compensation of the time dependent frequency shift caused by the chirp on a much shorter time scale. Since the two-photon detuning

Quantum memory based on phase matching control

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includes the ac Stark shift, at the first sight, it provides an attractive possibility to compensate the time-dependent term originating from chirp via modulation of its own amplitude of the control field. Unfortunately, it is not the case. Indeed, the Rabi frequency of the control field is the function of the running time: t − z/vcg . Thus, when ac-Stark effect compensates two-photon detuning the propagation effect produces a term αz/vcg which exactly cancels the time dependent wave vector in the phase factor ϕ(z, t), since the envelope of the control field propagates at the same group velocity as the frequency chirp. In principal, a second coherent external control field (Stark field) with time-varying amplitude, propagating in the transverse direction with respect to the signal and the first control fields, could be used for compensation of the time-dependent frequency shift via ac-Stark effect. However, its scattering at spin coherence would introduce a noise decreasing the fidelity. Its frequency should be carefully selected to provide, on one hand a large detuning to other levels to minimize a scattering rate, on the other hand small enough detuning sufficient for compensation of time-dependent twophoton detuning. The ac Stark gradient echo quantum memory scheme was suggested in [46], where a MHz shift can be obtained with 10W Stark laser power (with 1 cm by 10 µm medium) in rubidium 87. It can be seen in both situations a smaller chirp depth is favorable. In principle the requirement of the chirp depth can be reduced by a slowly propagating control field. This is possible if both optical transitions are of the same polarization in Raman interaction. The missing of the polarization selection rules is very common in solid state systems. In such a case, the control field becomes a slow light due to its interaction with |1⟩-|2⟩ transition dipole moment. The detuning of the control field on |1⟩-|2⟩ transition should be small enough to produce substantial reduced group velocity while large enough to suppress the noise due to inelastic scattering into the spin wave. The slowly propagating control field will enhance the sweeping speed of the phase matching condition, or equivalently enlarge the frequency gradient in GEM point of view. This releases the requirement of chirp parameter α (or chirp depth ∆d for the same T). For example, if the control field group velocity is reduced by 1000 times (which is expected under experimental conditions discussed in Sec. 6), the compensation of the chirp depth through linear Zeeman or ac Stark effect becomes possible. The next question is how to generate a long chirped control beam. Experimentally, sub-nanosecond chirp pulses at joule energy level can be produced using the ultra-fast optics technology [47, 48]. But first, the spectrum width (and thus chirp depth) of such a pulse is too large for the compensation of the frequency shift induced by the chirp. Second, it becomes difficult to achieve a longer than nanosecond chirp pulse, which is required, in order to provide a compensation of frequency shift via linear Stark or Zeeman effect, as it was discussed above. If a slow control field allows to reduce chirp depth ∆d , a long chirp pulse with narrow bandwidth is needed. By optical means such a chirp pulse would require an extraordinarily dispersive system which is very difficult to be achieved in conventional grating or prism pair. One promising solution to unravel the above difficulty is to stay in the narrow

Quantum memory based on phase matching control

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bandwidth regime and generate long chirp control field by electronic means. In this case, frequency chirp is added on top of a long control pulse by electronic modulation. Such technique is used, e.g., in high resolution lidar systems. The chirp depth ∆d is determined by the RF-signal frequency tuning range, which can be as high as GHz. Using different techniques, 1-10 GHz chirp depth with pulse duration 0.10.5 µs [49, 50, 51] were demonstrated. In our application, for instance, let us take a control field of duration 1 µs with chirp depth ∆d /2π ∼ 0.2 GHz [52]. Such a chirp depth permits the storage of the signal field of duration ∆t ∼ 50 ns by a slow control field with group velocity vcg = c/1000 in a 1 cm medium. Since 20-30 mT magnetic field can be switched within 10 ns [53, 54], the compensation of this chirp depth operating at microsecond time duration can be achieved with either linear Zeeman effect, or ac Stark effect via large Stark laser power discussed above. It is worth noting that another type of QM scheme, which is closely related to the frequency chirp scheme, has been suggested recently[55]. It explores a spatial chirp, i.e. z dependence of the frequency of the control field propagating in the transverse direction to mimic inhomogeneous broadening produced by external gradient field. In such a case, there is no two-photon-detuning drift problem as in the frequency chirp Raman PMC scheme, and variation of any parameters in time is not required. 6. Experimental implementation of Raman PMC As it was shown above, the Raman PMC QM scheme has essentially the same performance in terms of efficiency and fidelity as GEM QM. Thus it has the same advantages over traditional Raman and EIT QM schemes (where the storage is based on modulation of the control field amplitude) as GEM does. Namely, synchronization and optimization of Ec (t) with respect to the signal field is not required and hence a preliminary knowledge about arriving time and shape of the signal field is not needed. Second, no tailoring of the inhomogeneous absorption profile into the well separated narrow peaks (as in AFC QM) via optical pumping (which implies, in turn, a presence of proper sublevel structure in the ground state and the proper selection rules) is required for PMC QM. Finally, as the major potential advantage of the Raman PMC QM schemes with respect to GEM scheme, they have less stringent requirements for the material system. This leads to wider class of potentially appropriate materials and to a simpler practical implementation of Raman QM. As it follows from the above analysis of three different PMC QM schemes these advantages can be fully realized only in the scheme based on angular scanning of the control field propagation. Therefore, we focus below on the experimental realization of this scheme. A realization of the proposed PMC approach relies on an ensemble of stationary atoms such as impurities embedded in solid lattice or cold atoms in an optical lattice in order to avoid washing out of the spin waves. The estimates presented below show that nitrogen vacancy centers in diamond (NVD) is a very promising material for realization of Raman PMC. This is due to the fact that it has much stronger (compared

Quantum memory based on phase matching control

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to rare-earth ions) electro-dipole Raman transitions. It is worth noting that quantum memories were not demonstrated yet in NVD. Raman PMC has important advantage with respect to other methods for realization of quantum memory in NVD. Indeed, EIT technique requires rather high EIT contrast, while GEM and AFC based on optical transition require long life time of the excited state. Both are hardly achievable in NVD. EIT contrast in NVD is only 6% [56]. Three level AFC scheme is limited by the small inhomogeneous broadening of the spin transition, and three level GEM is difficult because the external electric or magnetic field strongly affects the excited spin state, which in turn affects the required Λ level structure. So Raman scheme combined with PMC approach for quantum memory application is naturally suitable for NVD. At the same time the rare-earth doped crystals placed into a cavity also can be used for demonstration of PMC QM. 6.1. NV centers in diamond (NVD) in free space Nitrogen vacancy centers in diamond (NVD) [57] consists of a carbon-vacancy in the diamond lattice with one of the neighboring carbon atoms substituted by a nitrogen atom. This vacancy can be thought of as a carbon atom with zero nuclear charge. The effective energy level scheme of electronic transitions of NVD can be found in, for example, Refs. [58, 59]. It has a number of unique features such as strong coupling of light with an individual vacancy and long-lived electron spin coherence (∼ 1 ms at the room temperature [60]). Recently some important experimental works have been done in ensembles of NVD, including EIT [56], spin-superconducting resonator coupling [61], quantum correlation between phonon mediated Stokes/anti-Stokes pulses [62], etc.. These results clearly show the potential of bulk NVD for realization of quantum memory. Raman phase-matched quantum memory requires the realization of the Λ scheme, i.e., the presence of the allowed spin-flip Raman transitions. Such a Λ scheme can be realized in NVD between the spin triplet ground states 3 A2 and excited state 3 E at zero phonon line 637 nm. The spin transition in the ground states between spin sublevels ms = 0 and ms = ±1 is approximately 2.88 GHz. The spin coherence time up to 0.6 s is demonstrated using decoupling techniques at temperatures below 100 K [63]. A Λ system in the case of arbitrary strain was used in pioneering demonstration of EIT in NV-diamond by the group of Philip Hemmer [64] through ground state spin level anticrossing under magnetic field. A Λ system in the case of moderate transverse strain was used in very recent demonstration of EIT by the group of Dmitry Budker [56] by using ∼ 20 ppb NVD with ∼ 30 GHz optical inhomogeneous broadening and > 1 µs spin coherence time. Quite recently a Λ system under external electric field and low magnetic field due to mixing of the ground states is suggested for Raman scheme quantum storage [65]. Potentially, any of the described above Raman transitions can be used for realization of PMC storage. For the implementation of Raman PMC the scheme with transverse control field in Ref [32] could be used. In this case, the main condition for high efficiency and fidelity of

Quantum memory based on phase matching control

15

quantum storage in free space is given by Eq. (7). Taking into account that commercial electro-optical beam deflectors can provide ∆θ/T ∼ 103 rad/s for beam diameters about 10 mm, from the left hand side of the condition we conclude that in samples of length L = 1 cm it is possible to store and recall pulses of duration as small as ∼ 30 ns with refractive index of diamond n = 2.4. The right hand side of Condition (7) allows us to estimate necessary intensity of the control field. Let us assume both transitions forming the Λ-structure have the same dipole moment corresponding to the oscillator strength 0.1DW , where DW = 0.035 is the Debye-Waller factor for NVD. Taking N = 5 × 1013 cm−3 and inhomogeneous broadening of 1 GHz, and one-photon detuning ∆/2π = 2 GHz, we obtain the required intensity of the control field ∼ 4.57 kW/cm2 , which gives |g|2 N = 1.5 × 1010 s−1 m−1 . The intensity up to 280 W/cm2 was used in EIT experiments in NVD ensemble studies [64] and much higher intensities are routinely used in single NV experiments making use of microscope objectives. In order to reduce the required power, a planar dielectric waveguide for the signal field can be considered for implementation. Optical waveguide tightly confines the signal pulse and allows us to consider beam diameters about 10 µm. The transverse control field is able to penetrate the waveguide channel. If we take advantage of a cylindrical lens and assume, e.g., the control beam cross section of 1 cm by 10 µm, then the total required power is 4.57 W. 6.2. Rare-earth-ion-doped crystals and NV diamond in a cavity Cavity-assisted quantum storage via control-field angular scanning can be implemented in crystals doped by rare-earth ions as was shown in recent works [32, 33]. One promising candidate is a low-strain crystal YLiF4 doped by 167 Er3+ ions (see [33] for details). The uniqueness of such a material is extremely narrow inhomogeneous linewidth (as low as 15 MHz [66] for optical transitions of Er3+ ions, which is much smaller than frequencies of the hyperfine transitions. This makes preparing of narrow absorption lines within spectral pits unnecessary. Another promising variant is YLiF4 doped by Nd3+ ions. In this case, the inhomogeneous broadening of optical transitions may be also very small (about 45 MHz [67]), while the oscillator strength is usually much larger than for other rare-earth ions. Let us consider the feasibility of implementing Raman PMC in the second crystal. First, it is necessary to fulfill the impedance matching condition, which in terms of the angular scanning approach may be written as ( )2 Ω 1 ε 0 ~ λc n s = , (20) ∆ F d221 N δt 2w0 where F is the cavity finesse (2π/F is the losses per round trip), w0 is the beam radius of the control field in the center of the sample, and δt is the switching time between excitation of two orthogonal spin modes during the angular scanning. The latter takes the form δt = (T /∆θ)(λc /L) in the case of the transverse control field [33]. Eq. (20) is valid for dispersive cavities, where signal field propagates with a group velocity vsg = c/ng (ng is the group refractive index). Second, in order to achieve high

Quantum memory based on phase matching control

16

efficiency and fidelity for different pulse shapes, it is necessary to fulfill bad-cavity limit which reads 2κδt ≫ 1,

(21)

where 2κ = (2π/F )(c/2Lc ng ) is the cavity decay rate, and Lc is the cavity length. In particular, for Gaussian pulses of duration (FWHM) ∆t = δt the maximum efficiency, 0.99, is achieved when 2κδt ≈ 9. Now, taking the concentration of impurities N = 7 × 1016 cm−3 , which corresponds to 5 ppm, and assuming that d12 = 0.0386 Debye (f = 8 × 10−6 ), and ∆/2π = 900 MHz, we obtain ng ≈ 1000. Such a group index significantly reduces cavity linewidth [28], thereby limiting the minimum value of input pulse duration. For Lc = 2w0 = 0.2 mm, δt = 300 ns, ns = 1.5, and F/2π = 25, we obtain 2κδt ≈ 9 and Ω/∆ = 0.0105. The latter corresponds to the control field power about 400 mW, provided that d13 ≈ d12 . In the case of a Gaussian beam of the control field, inhomogeneous broadening of the Raman transition due to its spatial profile proves to be about 100 kHz. Such broadening may be completely eliminated by using top-hat spatial profiles, which can be prepared by diffraction beam shapers. Similar estimations can be performed for other promising materials, in particular for NVD. The possibility of implementing off-resonance Raman memory scheme in this material has been recently considered in [65]. Following this proposal, we can take d12 = 1 Debye, N = 5 × 1013 cm−3 , and obtain ng ≈ 1000 for ∆/2π = 2 GHz. Then 300 ns pulses can be stored and recalled under similar conditions (Lc = 2w0 = 0.2 mm and F/2π = 25) but only with 7 mW power of the control field. In the case of larger inhomogeneous broadening such as 10 GHz and concentration N = 8 × 1015 cm−3 , we can take ∆/2π = 30 GHz, F/2π = 10, and realize storage of 100 ns-pulses under the same conditions with control field power about 76 mW. To store and recall shorter pulses we can take advantage of microcavities as suggested in [65]. Beside the materials discussed above, we would like to point out another candidate: negatively charged silicon vacancy (SiV) in diamond. SiV caused a lot of attention in the field of quantum information due to some of its remarkable features. It has similar oscillator strength as NV center [68], but more than 70% of single SiV center fluorescence falls in zero phonon line at 738 nm [69], comparing with a small fraction of 3% in NV center [70]. This makes SiV a very promising candidate for room temperature bright single-photon source. Each of the ground 2 Eg and excited 2 Eu state has a twofold orbital and a twofold spin degeneracy. The orbital degeneracy is lifted by spin-orbit coupling and Jahn-Teller effect into double fine structure with ∆2 Eg = 50 GHz and ∆2 Eu = 260 GHz [71, 68]. This gives rise to 4 lines in the fluorescence spectrum with polarization selection rules [72], and in total makes a Λ system for Ramanbased quantum memory schemes. Optical access to electronic spin shows a spin purity approaching unity in the excited state [73]. Small inhomogeneous broadening can be achieved to bring the linewidth of individual fine structure component of SiV ensemble to 9 GHz [74]. However, spin coherence lifetime is yet to be studied in this crystal.

Quantum memory based on phase matching control

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7. Conclusion A general method to realize quantum memory for a weak signal pulse (single photon wave packet) based on phase matching control (PMC) is discussed. Based on different PMC methods, there are three different implementations. Namely, quantum memory (QM) via (i) refractive index modulation, (ii) control-field angular scanning and (iii) controlfield frequency chirp. We analyze and compare these three PMC quantum memory schemes. Specifically, we illustrate in detail PMC QM based on control-field frequency chirp in a Raman configuration. The frequency chirp is used to create a continuous set of spin waves with different lengths of wave vectors, which will record the temporal profile of the signal pulse. Later on, the same information can be read out by applying a strong control field with the opposite chirp. In order to stay on two-photon resonance, the atomic level should be modulated through a dc or ac external field to compensate the two-photon detuning. In a dispersive medium the control-field frequency chirp scheme involves also the refractive index modulation so that we actually have a combination of both approaches. This enhances the control field wave vector scanning rate and makes it easier for the sake of chirp-leaded two-photon detuning compensation. If the two-photon detuning is compensated, the proposed quantum memory scheme has a mathematical analogy with the gradient echo memory scheme. High efficiency and fidelity can be achieved with forward retrieval. This is true for the whole class of free space Raman PMC quantum memory schemes. Among the three PMC quantum memory schemes, the first and second ones are much easier to be implemented since there is no two-photon detuning shift. While the second one seems to be the easiest in the sense that no direct manipulation of the material is involved. So we discuss the experimental implementation of the control-field angular scanning scheme. It turns out that in free space, among the considered materials nitrogen vacancy centers in diamond is the most promising one for demonstrating the proposed Raman PMC QM scheme. While in a cavity-assisted set up, a series of rareearth-doped crystals can also be used. The phase matching control quantum memory demonstrates high efficiency and fidelity with the same performance as gradient echo memory. However the former (in cases of refractive index modulation and control-field angular scanning schemes) does not require Stark or Zeeman effect, and no synchronization between signal and control fields is needed. Acknowledgments The authors thank A. Sokolov, A. Zheltikov and C. O’Brien for very useful discussions. This work was supported by NSF under Grant No. 0855688, and RFBR under Grant No. 12-02-00651.

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Appendix In this Appendix, we provide the derivation of the equations of motion (5), (6). Let us start with the one-dimensional Maxwell equation for the signal field ( 2 ) ∂ 1 ∂2 ∂ 2 P (r, t) − E (r, t) = µ , (A.1) s 0 ∂z 2 c2 ∂t2 ∂t2 and define slow varying amplitudes of the electric field, E(r, t), and atomic polarization density, p(r, t), via relations Es (r, t) = E(r, t) eiks z−iωs t + c.c.,

(A.2)

P (r, t) = N d12 p(r, t) eiks z−iωs t + c.c.

(A.3)

Then inserting (A.2), (A.3) into (A.1), and neglecting second derivatives of the slowly varying amplitudes, we obtain ( ) ∂ ωs ∂ ωs2 /c2 − ks2 iω 2 N d12 ωs N d12 ∂p(r, t) + 2 + E(r, t) = s 2 p(r, t) − . (A.4) ∂z c ks ∂t 2iks 2ε0 c ks ε 0 c 2 ks ∂t If we take ks = ωs /c and neglect the first derivative of polarization, we come to the standard propagation equation, where dispersive effects are not present explicitly. To reveal them we take into account the equations of motion for the slowly varying amplitudes of polarization and spin coherence densities, which describe interaction of three-level atoms with the weak signal and strong control fields: d id21 p(r, t) = (−Γ + i∆)p(r, t) + E(r, t) + iΩs(r, t) eiϕ(r,t) , (A.5) dt ~ d s(r, t) = (−γ + iδ)s(r, t) + iΩ∗ p(r, t) e−iϕ(r,t) . (A.6) dt where ∆ = ωs − ω2 , δ = ωs − ωc − ω3 , Γ and γ are dephasing rates for optical and spin transitions, respectively, Ω = d23 E0 /~ is the Rabi frequency of the control field, and ϕ(r, t) is the phase shift due to the angular or frequency manipulation with the control field. When considering off-resonance Raman interaction (|∆| ≫ Γ) and doing adiabatic elimination of the polarization, we usually set time derivative p(z, ˙ t) equal to zero and then express the polarization amplitude through the field one. But this procedure does not take into account dispersion. To do this we can take advantage of the frequency domain: id21 −iωp(r, ω) = (−Γ + i∆)p(r, ω) + E(r, ω) + iΩF (r, ω), (A.7) ~ where F (r, ω) stands for the Fourier transform of s(r, t) eiϕ(r,t) . Then iΩF (r, ω) 1 id21 E(r, ω) + (A.8) p(r, ω) = (Γ − i∆ − iω) ~ (Γ − i∆ − iω) ≡ ε0 χ(ω ¯ s + ω)E(r, ω) + η(ωs + ω)F (r, ω). (A.9) Now we expand χ(ω ¯ s + ω) and η(ωs + ω) in a series around the point ωs : ∂ χ¯ χ(ω ¯ s + ω) = χ(ω ¯ s) + ω + ..., ∂ω ωs ∂η η(ωs + ω) = η(ωs ) + ω + ..., ∂ω ωs

(A.10) (A.11)

Quantum memory based on phase matching control and return to the time domain: p(r, t) = ε0 χ(ω ¯ s )E(r, t) + iε0

19

∂ χ¯ ∂E(r, t) + η(ωs )s(r, t) eiϕ(r,t) + . . .(A.12) ∂ω ωs ∂t

Then from (A.4) and (A.6) we have ( ) ∂ 1 ∂ iωs N d12 + E(r, t) = η(ωs )s(r, t) eiϕ(r,t) , (A.13) ∂z vgs ∂t 2ε0 cns [ ] d s(r, t) = (−γ + iδ)s(r, t) + iΩ∗ ε0 χ(ω ¯ s )E(r, t) e−iϕ(r,t) + η(ωs )s(r, t) , (A.14) dt √ provided that ks = ns ωs /c, where ns = 1 + χ(ωs ) is the refractive index and χ = N d12 χ¯ is the linear susceptibility of the atomic system, and ns 1 ωs ∂χ = + (A.15) vgs c 2cns ∂ω ωs is the reciprocal of group velocity. Taking into account that id21 iΩ ε0 χ(ω ¯ s) = , η(ωs ) = , ~(Γ − i∆) Γ − i∆ ∂ χ¯ d21 ε0 = − , ∂ω ωs ~(Γ − i∆)2 and considering the limit |∆| ≫ Γ, we obtain ( ) ∂ 1 ∂ i~ωs N d12 Ω + E(r, t) = − s(r, t) eiϕ(r,t) , ∂z vgs ∂t 2ε0 cns ~∆ ( [ ]) 2 d |Ω| Γ |Ω|2 d21 Ω∗ s(r, t) = −γ − + i δ − s(r, t) − i E(r, t) e−iϕ(r,t) , dt ∆2 ∆ ~∆

(A.16) (A.17)

(A.18) (A.19)

where 1 ns ωs N |d|2 = + . vgs c 2ε0 ~cns ∆2 Finally, in terms of the slowly varying annihilation operator we have √ ~ωs E(r, t) = i a(r, t), 2ε0 cns which gives ( ) ∂ 1 ∂ + a(r, t) = −g ∗ N s(r, t) eiϕ(r,t) , ∂z vgs ∂t ∂ s(r, t) = (−γ + iδ)s(r, t) + g a(r, t) e−iϕ(r,t) , ∂t where √ d21 Ω∗ ωs g= , ∆ 2ε0 ~cns

(A.20)

(A.21)

(A.22) (A.23)

(A.24)

the dephasing rate γ is redefined so that it includes |Ω|2 Γ/∆2 , while excitation induced frequency shift |Ω|2 /∆ is compensated by tuning the coupling field frequency. The annihilation operator is normalized so that a† (r, t)a(r, t) corresponds to the photon flux density in the dispersive medium.

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