Photon storage in Λ-type optically dense atomic media. IV. Optimal control using gradient ascent
Alexey V. Gorshkov,1 Tommaso Calarco,2 Mikhail D. Lukin,1 and Anders S. Sørensen3
Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA 2 Institut f¨r Quanteninformationsverarbeitung, Universit¨t Ulm, u a Albert-Einstein-Allee 11, D-89081 Ulm, Germany 3 QUANTOP, Danish National Research Foundation Center for Quantum Optics, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark (Dated: April 7, 2008) We use the numerical gradient ascent method from optimal control theory to extend eﬃcient photon storage in Λ-type media to previously inaccessible regimes and to provide simple intuitive explanations for our optimization techniques. In particular, by using gradient ascent to shape classical control pulses used to mediate photon storage, we open up the possibility of high eﬃciency photon storage in the non-adiabatic limit, in which analytical solutions to the equations of motion do not exist. This control shaping technique enables an order-of-magnitude increase in the bandwidth of the memory. We also demonstrate that the often discussed connection between time reversal and optimality in photon storage follows naturally from gradient ascent. Finally, we discuss the optimization of controlled reversible inhomogeneous broadening.
PACS numbers: 42.50.Gy, 03.67.-a, 32.80.Qk, 42.50.Fx
1

arXiv:0710.2698v2 [quant-ph] 7 Apr 2008

I.

INTRODUCTION

Faithful mapping between quantum states of light (ﬂying qubits) and quantum states of matter (storage and/or memory qubits) is an important outstanding goal in the ﬁeld of quantum information processing and is being pursued both theoretically and experimentally by a large number of research groups around the world. Photon storage in Λ-type atomic media is a promising avenue for achieving this goal. In a recent theoretical paper [1], we uniﬁed a wide range of protocols for photon storage in Λtype media, including the techniques based on Electromagnetically Induced Transparency (EIT), oﬀ-resonant Raman interactions, and photon-echo. In Ref. [1] we also demonstrated equivalence between all these protocols and suggested several eﬃciency optimization procedures, some of which have since been demonstrated experimentally [2]. In the three preceding papers of this series, Refs. [3, 4, 5], which we will refer to henceforth as papers I, II, and III, we presented some details and many extensions of the analysis of Ref. [1]. Most of the results in Ref. [1] and in papers I, II, and III were obtained based on physical arguments and on exact solutions available in certain limits. However, the optimization problems discussed there fall naturally into the framework of optimal control problems, for which powerful numerical optimization methods exist [6, 7]. Thus, in the present paper, we apply these optimal control methods to the problem of photon storage. As a result, we open up the possibility of eﬃcient photon storage in previously inaccessible regimes by increasing the bandwidth of the memory and provide simple intuitive understanding for the optimization methods underlying photon storage. We refer the reader to paper I for a comprehensive introduction to photon storage in Λ-type atomic media and for the full list of references. Here we summarize

only a few important points. In a typical photon storage protocol, an atomic ensemble with Λ-type level structure shown in Fig. 1 is assumed to start with all N atoms pumped into the metastable state |g . The incoming quantum light mode is coupled to the |g − |e transition √ with a collectively enhanced coupling constant g N and is mapped onto the collective coherence (called a spin wave) between the metastable states |s and |g using a classical two-photon-resonant control pulse with timedependent Rabi frequency Ω(t). Ideal mapping of the light mode onto the spin wave and back can be achieved in an ensemble that has inﬁnite resonant optical depth d on the |g −|e transition. However, despite the existence of proposals for achieving high values of d [8], in most current experiments d (or the cooperativity parameter C for ensembles enclosed in a cavity [9]) is limited to d ∼ 10 due to experimental imperfections such as competing four-

|e

(t )
|s

g N
|g

FIG. 1: (Color online) Λ-type medium coupled to a quantum ﬁeld (dashed) with a collectively enhanced coupling constant √ g N and a two-photon-resonant classical ﬁeld (solid) with time-dependent Rabi frequency Ω(t) .

2 wave mixing processes [10], spatially-varying light shifts [9], number of atoms in a trap [11, 12], or inhomogeneous broadening and short interaction lengths [13, 14]. As a result of the limited optical depth, the experimentally demonstrated eﬃciencies for the light-matter interface are low, which makes the optimization of photon storage protocols at ﬁnite values of d crucial. The optimization in Ref. [1], in papers I, II, and III, as well as in the present paper relies on the knowledge of the shape of the incoming photon mode. Note that such knowledge is not incompatible with storing unknown quantum states because the mode usually acts simply as a carrier while the information is stored in the quantum state of the harmonic oscillator corresponding to this mode [15]. A diﬀerent type of problem is the storage of an unknown mode or, equivalently, the storage of multiple photonic modes within an ensemble [16]. While we believe that the optimization procedures considered here will probably also be relevant to this situation, we shall not discuss it in more detail here. The main tool used in this paper is a numerical iterative optimization with respect to some set of control parameters, which are updated to yield higher photon storage eﬃciency at every iteration. Such iterative optimization methods are a standard tool in applied optimal control theory [6, 7]. These methods and their variations are already being used in a variety of applications including laser control of chemical reactions [17, 18, 19], design of NMR pulse sequences [20], loading of Bose-Einstein condensates into an optical lattice [21], atom transport in time-dependent superlattices [22], quantum control of the hyperﬁne spin of an atom [23], and design of quantum gates [24, 25]. Although advanced mechanisms for updating the control parameters from one iteration to the next exist and exhibit superior convergence characteristics [6, 21, 26, 27, 28], we will concentrate in the present paper on optimization via a simple gradient ascent method [6, 7, 20, 23], except for Sec. III where advanced updating mechanisms will also be used. Gradient ascent methods are often more eﬃcient than simple variations of the control parameters using, e.g., genetic algorithms. Moreover, we will show that gradient ascent optimization has the advantage that it can often be understood physically and can provide deeper intuition for the photon storage problem. In particular, in papers I, II, and III, we used involved physical arguments and exact analytical solutions available in certain limits to derive a time-reversal-based iterative optimization with respect to the shape of the incoming photon mode. In the present paper, we show that these time-reversal iterations and the general and often discussed connection between optimality and time reversal in photon storage [1, 29, 30, 33] naturally follow from the gradient ascent method. The results of papers I, II, and III are, however, still crucial since they show in certain cases that the solutions obtained via the local gradient ascent method represent global, rather than local, optima. In addition to considering optimization with respect to the shape of the input mode, we consider in the present paper optimization with respect to the storage control ﬁeld. In particular, we show that shaping the control ﬁeld via the gradient ascent method allows for eﬃcient storage of pulses that are an order of magnitude shorter than when the control ﬁeld is optimized in the adiabatic approximation discussed in Ref. [1] and in papers I and II. In other words, this new control shaping method increases the bandwidth of the memory. Finally, we discuss the performance of optimal control pulses in the context of photon storage via controlled reversible inhomogeneous broadening (CRIB) [30]. In particular, assuming one is interested in storing a single known incoming light mode and assuming one can shape control pulses with suﬃcient precision, we are not able to identify any advantages of CRIB-based photon storage compared to photon storage with optimal control pulses in homogeneously broadened media. The remainder of the paper is organized as follows. In Secs. II, III, and IV, we show how gradient ascent can be used to optimize with respect to the control ﬁeld, the input mode, and the inhomogeneous proﬁle, respectively. We summarize the discussion in Sec. V and, ﬁnally, present some details omitted in the main text in the Appendixes A-D.

II.

OPTIMIZATION WITH RESPECT TO THE STORAGE CONTROL FIELD

In principle, both the incoming light mode and the classical control pulse may be adjusted to maximize the light storage eﬃciency. However, it is often easier to vary the classical control pulse. In particular, the photonic state we wish to store may be some non-classical state generated by an experimental setup, where we cannot completely control the shape of the outgoing wave packet. This is, e.g., the case for single photons generated by parametric down conversion [16, 31] or by single nitrogen-vacancy centers in diamond [32], where the shape of the wave packet will be, respectively, set by the bandwidth of the setup and the exponential decay associated with spontaneous emission. Alternatively, the wave packet may also be distorted in an uncontrollable way by the quantum channel used for transmitting the photonic state [15]. In this section, we therefore discuss optimization with respect to the storage control ﬁeld in both the cavity model (Sec. II A) and the free space model (Sec. II B).

A.

Cavity model

As discussed in papers I and II, the cavity model, in which the atomic ensemble is enclosed in a cavity, is theoretically simpler than the free space model because only one collective atomic mode can be excited. In addition, as shown in papers I and II, the cavity setup can yield

3 higher eﬃciencies in certain cases than the free space model due to the enhancement of the optical depth by the cavity ﬁnesse and due to (for certain spin wave modes) better scaling of the error with the optical depth d (1/d √ in the cavity vs. 1/ d in free space). We, therefore, start with the cavity model. As in paper I, to get the closest analogy to the free-space regime, we will discuss in the present paper only the so-called “bad cavity” limit, in which the cavity mode can be adiabatically eliminated. However, the method of gradient ascent can easily be applied outside of this limit, as well. To simplify the discussion, we ﬁrst consider the simplest example, in which one stores a given resonant input mode into a homogeneously broadened ensemble enclosed in a cavity and having negligible spin-wave decay rate. It is important to note that, because only one spin-wave mode is accessible in the cavity model, the retrieval efﬁciency is independent of how the storage is done (see paper I). This makes it meaningful to optimize storage separately from retrieval [the latter does not have to be optimized since its eﬃciency depends only on the cooperativity parameter (see paper I)]. We follow the derivation of paper I to adiabatically eliminate the cavity mode and to reduce the equations of motion to the following complex number equations on the time interval t ∈ [0, T ]: ˙ P (t) = −γ(1 + C)P (t) + iΩ(t)S(t) + i ˙ S(t) = iΩ(t)P (t). 2γCEin (t), (1) (2) ¯ ¯ tipliers P (t) and S(t) to ensure that the equations of motion are fulﬁlled, and turn the problem into an unconstrained maximization of [6, 7] J = S(T )S ∗ (T )
T

+
0

˙ ¯ dt P ∗ − P − γ(1 + C)P + iΩS +i 2γCEin + c.c.

T

+
0

˙ ¯ dt S ∗ −S + iΩP + c.c. ,

(3)

where c.c. stands for the complex conjugate [34]. The optimum requires that J is stationary with respect to any variation in P , S, and Ω. As shown in Appendix A, setting J to be stationary with respect to variations in P and S requires that the Lagrange multipliers (also ¯ ¯ referred to as the adjoint variables) P and S satisfy the equations of motion ˙ ¯ ¯ ¯ P = γ(1 + C)P + iΩS, ˙ ¯ ¯ S = iΩP , subject to boundary conditions at time t = T ¯ P (T ) = 0, ¯ S(T ) = S(T ). (6) (7) (4) (5)

Here the optical polarization P (t) on the |g − |e transition and the spin polarization S(t) on the |g − |s transition satisfy initial conditions P (0) = 0 and S(0) = 0, respectively, corresponding to the absence of atomic excitations at t = 0. In this example, the shape of the incoming mode Ein (t) is assumed to be speciﬁed, real, T 2 and normalized according to 0 dtEin (t) = 1. γ is the decay rate of the optical polarization and C is the collectively enhanced cooperativity parameter equal to the optical depth of the atomic ensemble times the cavity ﬁnesse. The goal is to ﬁnd the slowly varying control ﬁeld Rabi frequency envelope Ω(t) (assumed to be real) that maximizes the storage eﬃciency ηs = |S(T )|2 . [To avoid carrying around extra factors of 2, Ω(t) is deﬁned as half of what is usually called the Rabi frequency: it takes time π/(2Ω) to do a π pulse]. For the moment, we suppose that there is no constraint on the energy of the control pulse and return to the possibility of including such a constraint below. It is worth noting that due to their linearity, the equations of motion (and all the results of the present paper) apply equally well both to classical input ﬁelds with pulse shapes proportional to Ein (t) and to quantum ﬁelds whose excitations are conﬁned to the mode described by Ein (t). The eﬃciency η is thus the only parameter required to fully characterize the memory (see paper I). Since the optimization of ηs is constrained by the equations of motion (1) and (2), we introduce Lagrange mul-

These are the same equations as for S and P [Eqs. (1) and (2)] except that there is no input ﬁeld and that the decay with rate γ(1 + C) is replaced with growth, which will function as decay for backward evolution. This backward evolution, in fact, corresponds to retrieval with the time-reversed control ﬁeld and can be implemented experimentally as such (see papers I, II and Ref. [2]). It is satisfying to have obtained this purely mathematical and simple derivation of the often discussed connection between optimality and time reversal in photon storage [1, 2, 3, 4, 5, 29, 30, 33, 35]. As explained in the introduction to paper I and as shown in detail in paper II, this connection goes beyond the perfect reversibility of unitary evolution discussed in Refs. [29, 30, 35] by including systems with non-reversible dynamics, as exempliﬁed, for example, by the decay rate γ in the present model. Eqs. (4)-(7) ensure that J is stationary with respect to variations in P and S. To ﬁnd the optimum it remains to set to zero the functional derivative of J with respect to Ω. This functional derivative is given by δJ ¯ ¯ = −2 Im S ∗ P − P S ∗ , δΩ(t) (8)

where ”Im” denotes the imaginary part. In general, if one has a real function of several variables, one way to ﬁnd a local maximum is to pick a random point, compute partial derivatives at that point, move a small step up the gradient, and then iterate. The same procedure can be applied to our optimal control

4 problem [6, 7]. The gradient ascent procedure for ﬁnding the optimal storage control pulse Ω(t) is to take a trial Ω(t) and then iteratively update Ω(t) by moving up the gradient in Eq. (8) according to 1 ¯ ¯ Ω(t) → Ω(t) − Im S ∗ P − P S ∗ . λ (9)

Γ
4

a
3 2 1
in

where 1/λ regulates the step size. In order to compute the right hand side of Eq. (9), one has to evolve the system forward in time from t = 0 to t = T using Eqs. (1) and (2) to obtain S(t) and P (t). Then project the ﬁnal atomic state described by S(T ) and P (T ) onto S ¯ ¯ according to Eqs. (6) and (7) to obtain P (T ) and S(T ). ¯ and P backwards in time from t = T to ¯ Then evolve S t = 0 according to Eqs. (4) and (5). In general, as in any gradient ascent method, the step size 1/λ in Eq. (9) has to be chosen not too big (one should not go up the gradient so quickly as to miss the peak) but not too small (in order to approach the peak relatively quickly). To achieve faster convergence, one could use a diﬀerent step size 1/λ for each iteration; but for the problems considered in the present paper, convergence is usually suﬃciently fast that we do not need to do this (unless the initial guess is too far from the optimum, in which case changing λ a few times helps). Moreover, in some optimization problems [22], 1/λ has to be chosen such that it depends on the argument of the function we are trying to optimize, i.e., in this case the time t; this is not required for the present problems, and 1/λ is just taken to be a constant. For example, let us take C = 1, T γ = 10, and a Gaussian-like input mode √ 2 (10) Ein (t) = A(e−30(t/T −0.5) − e−7.5 )/ T , where A ≈ 2.09 is a normalization constant and where the mode is chosen to vanish at t = 0 and t = T for computational convenience. Starting with an initial guess Ω(t) = γ/T and using λ = 0.5, it takes about 45 iterations for the eﬃciency to converge to within 0.001 of the optimal eﬃciency of C/(1 + C) = 0.5 (see paper I for the derivation of this formula). If, however, λ is too small (e.g. λ = 0.1), then the step size is too large, and, instead of increasing with each iteration, the eﬃciency wildly varies and does not converge. We now compare the optimal control ﬁeld shaping to the adiabatic control ﬁeld shaping presented in paper I. We ﬁrst take C = 10 and consider the input mode in Eq. (10) with T = 50/γ. Following paper I, we calculate the storage control ﬁeld using the adiabatic equations [Eq. (26) in paper I], then numerically compute the storage eﬃciency with this control ﬁeld, and multiply it by the complete retrieval eﬃciency C/(1 + C) to obtain the total eﬃciency. Since we are in the adiabatic limit (T Cγ = 500 ≫ 1), the resulting total eﬃciency is equal to the maximum possible eﬃciency C 2 /(1 + C)2 = 0.83 (see paper I). Fig. 2(a) shows the input mode in Eq. (10) (dashed line) and the adiabatic storage control ﬁeld (dot-

0 25 20 15 10 5

Γ

10

20

30

40

50

tΓ

b

in

0.1

0.2

0.3

0.4

0.5

tΓ

FIG. 2: Adiabatic (dotted) and optimal (solid) control ﬁelds for the storage of a Gaussian-like input mode Ein (t) (dashed) in the cavity model with C = 10 and T = 50/γ (a) and T = 0.5/γ (b). The four diﬀerent optimal control pulses correspond to four diﬀerent initial guesses for the gradient ascent optimization. The adiabatic control ﬁeld agrees with the optimal one in the adiabatic limit (T Cγ ≫ 1) (a) and deviates from it otherwise (b).

ted line). The optimal control ﬁeld shaping using gradient ascent via Eq. (9) also yields the maximum possible eﬃciency C 2 /(1 + C)2 = 0.83 independent of the initial guess for Ω(t). The four solid lines in Fig. 2(a) show Ω(t) resulting from optimal control ﬁeld shaping for four diﬀerent initial guesses, Ω(t)/γ = 0.2, 1, 2, and 3. The four optimal control ﬁelds and the adiabatic control ﬁeld agree except at small times. The reason for the disagreement is that the dependence of storage eﬃciency on the front section of the control ﬁeld is very weak because this section aﬀects only the front part of the excitation, and a large part of this anyway leaks out at the back end of the atomic ensemble. In fact, the dependence is so weak that gradient ascent leaves the front part of the initial guesses almost unperturbed. It is worth noting that, in general, gradient ascent methods are not guaranteed to yield the global optimum, and the iterations may get trapped in a local maximum. However, for our photon storage problem, we know what the global optimum is in some cases. In particular, we

5 have shown in paper I (for the cavity model) and in paper II (for the free space model) that, in the adiabatic limit, adiabatic control ﬁeld shaping yields the global optimum. Since control shaping via gradient ascent agrees with the adiabatic shaping in this limit, we have a strong indication that gradient ascent always yields the global optimum also outside of the adiabatic limit. The global optimum is here the (unique) maximum possible eﬃciency, which, within the numerical error, is achievable for a variety of control ﬁelds due to the lack of sensitivity to the control ﬁeld for small times (see Fig. 2). We now repeat the same steps except that we use T = 0.5/γ. The resulting control ﬁelds are shown in Fig. 2(b). Again the four optimal control ﬁelds correspond to diﬀerent initial guesses [Ω(t)/γ = 2, 5, 8, and 11]. The adiabatic control ﬁeld now diﬀers from the optimal one on the entire time interval. The reason is that the adiabatic limit (T Cγ ≫ 1) is not satisﬁed to a suﬃcient degree (T Cγ = 5), and, as a result, the adiabatic approximation does not work well. Indeed, the eﬃciency yielded by the adiabatic control (0.49) is much smaller than that yielded by the optimal control (0.81). Physically, the breakdown of the adiabatic approximation means that the optical polarization P (t) no longer follows the spin wave S(t) adiabatically, but rather evolves dynamically according to the full diﬀerential equation (1). Since in this regime (T Cγ ∼ 1) the optimal control ﬁeld is turned on abruptly following a time period when it is oﬀ [see Fig. 2(b)], the optimal storage procedure acquires some characteristics of photon-echo type fast storage [1, 3, 4, 29]. In fast storage, the input pulse is ﬁrst absorbed on the |e − |g transition in the absence of the control ﬁeld, and is then mapped to the |s − |g coherence via a control π pulse. This connection is not surprising since fast storage is indeed optimal for certain input modes of duration T ∼ 1/(Cγ). Finally, we note that all the initial guesses for Ω that we tried yielded the same optimal control (up to the unimportant front part) and the same eﬃciency, which is a signature of the robustness of the optimization protocol and is another strong indication that, for this optimal control problem, gradient ascent yields the global, rather than local, optimum. Having performed the comparison of the control ﬁelds generated by adiabatic shaping and by gradient ascent, we turn to the investigation of the dependence on C and on T Cγ of the eﬃciency achieved by these two methods. In Fig. 3(a), we compare the eﬃciency of storage followed by retrieval of the input mode of Eq. (10) obtained using the adiabatic control ﬁeld (dotted lines) and using the control found via gradient ascent (solid lines). The eﬃciencies are plotted as a function of T Cγ for three indicated values of C (= 1, 10, 100). Dashed lines correspond to C 2 /(1+C)2, the maximum eﬃciency possible at any given C. We note that the dotted lines have already been shown in Fig. 2(a) of paper I. According to the arguments presented in papers II and III, we note that it is impossible to retrieve into a mode much shorter than 1/(γC), and hence, by time-reversal, it is impossible to

Η
1 0.75 0.5 0.25 0 0 1 0.75 0.5 0.25 0 0 2.5 5 7.5 5 10 15

100 10

a
1
20

TCΓ

Η

100 10

b
1
10

TCΓ

FIG. 3: (a) The total eﬃciency of storage followed by retrieval for the Gaussian-like input mode in Eq. (10) using adiabatic equations (dotted) and gradient ascent (solid) to shape the storage control ﬁeld. Results are shown as a function of T Cγ for the indicated values of C (= 1, 10, 100). The dashed lines are C 2 /(1+C)2 , the maximum eﬃciency possible at any given √ C. (b) Same for Ein (t) = 1/ T .

eﬃciently store such a short mode. Fig. 3(a) conﬁrms that indeed, when T Cγ ≪ 1, even optimal controls cannot give high eﬃciency. Using gradient ascent instead of adiabatic shaping, one can, however, eﬃciently store input modes that are about an order of magnitude shorter and, thus, an order of magnitude larger in bandwidth. It is worth repeating that although the method of gradient ascent is generally not guaranteed to yield the global maximum, the fact that it does give the known global maximum in the limit T Cγ ≫ 1 suggests that it probably yields the global maximum at all values of T Cγ. To conﬁrm the robustness and generality of the optimization procedure, we show in Fig. 3(b) the results of the same optimization as in Fig. 3(a) but for a square √ input mode Ein (t) = 1/ T instead of the Gaussian-like input mode of Eq. (10). As in Fig. 3(a), we see that gradient ascent control shaping improves the threshold in the value of T Cγ, where eﬃciency abruptly drops, by an order of magnitude. This can again be interpreted as an eﬀective increase in the bandwidth of the memory by an order of magnitude. The optimal storage eﬃciency for the square input pulse falls to half of the maximum at smaller T Cγ than for the Gaussian-like input pulse be-

6 cause the latter has a duration (half-width at half maximum, for example) signiﬁcantly shorter than T [see Eq. (10) or Fig. 2]. On the other hand, as T Cγ is increased, the maximum is approached slower for the square input mode than for the Gaussian-like mode. This is because the high frequency components contributed by the sharp edges of the square pulse are diﬃcult to store. Most experiments have features that go beyond the simple model we have just described. Therefore, in Appendix B, we generalize this model and the optimization procedure to include the possibility of complex control ﬁeld envelopes Ω(t) and input mode envelopes Ein (t), nonzero single-photon detuning ∆ and spin wave decay rate γs , and (possibly reversible [30]) inhomogeneous broadening. Our model of inhomogeneous broadening is applicable both to Doppler broadening in gases and to the broadening of optical transitions in solid state impurities caused by the diﬀerences in the environments of the impurities [36]. For the case of Doppler broadened gases, we also allow for the possibility of modeling velocity changing collisions with rate γc . Finally, in Appendix B, we also show how to take into account the possibility that the classical driving ﬁelds available in the laboratory are not suﬃciently strong to realize the optimal control ﬁelds, which may be the case for short input modes and/or large single-photon detuning ∆, both of which require control pulses with large intensities. Although a comprehensive study of optimization for ∆ = 0 is beyond the scope of the present paper, we will now prove that the maximum eﬃciency for ∆ = 0 is exactly equal to the maximum eﬃciency for ∆ = 0. Suppose we know the control ﬁeld Ω0 (t) that achieves the optimum for a given resonant input Ein (t). Then, for an input at ∆ = 0 with the same envelope Ein (t), we can construct the control ﬁeld Ω(t) as a sum of two parts [written in the two-photon-resonant rotating frame as in Eqs. (B1) and (B2)] Ω(t) = Ω2 e−i∆2 t + Ω0 (t)ei∆t . (11) mogeneous broadening at several diﬀerent values of T γ and C for the pulse shape in Eq. (10). Thus, the optimal oﬀ-resonant eﬃciency is greater than or equal to the optimal resonant eﬃciency for the same input envelope Ein (t). Carrying out the same argument backwards [i.e. using Ω2 (t) to shift |e out of resonance], we conclude that the optimal eﬃciency must be the same on and oﬀ resonance. When applying this idea in practice, one should, of course, realize that, in addition to a possible technical limit on the available control power, the three-level approximation and the rotating-wave approximation may start to break down for suﬃciently large values of ∆2 and Ω2 .
B. Free space model

The ﬁrst part is a far-detuned control (∆2 ≫ Ω2 , γ) that Stark-shifts level |e into resonance with the input (i.e. such that Ω2 /∆2 = ∆), while the second part is resonant 2 with the Stark-shifted |e − |s transition and has the envelope equal to the optimal resonant control. The reason why an extra detuning ∆ is needed to bring the second term in two-photon resonance is because Ω2 Stark-shifts both |e and |s by ∆. The resulting eﬃciency must be equal to the optimal resonant eﬃciency up to an error on the order of the small population mixing between |e and |s caused by Ω2 ; that is, ∼ (Ω2 /∆2 )2 = ∆/∆2 . To verify mathematically that the control in Eq. (11) works, one can write P and S as a sum of a slowly varying piece and a rapidly oscillating piece, extract separate equations for the rapidly and slowly oscillating variables, and ﬁnally adiabatically eliminate the rapidly oscillating variables. We have also numerically veriﬁed the performance of the control in Eq. (11) and the scaling of the error (∼ ∆/∆2 ) by integrating the equations of motion for the case of ho-

Although the cavity model discussed in Sec. II A is theoretically simpler and results, in certain cases, in higher eﬃciencies than the free space model, the latter is easier to set up experimentally. Moreover, because of the accessibility of a large number of spin wave modes, the free space model can provide higher eﬃciencies in some other cases (see paper II) and can, in principle, function, as a multi-mode memory. Therefore, we turn in the present section to the analysis of the free space model. To demonstrate how optimization with respect to the control ﬁeld works in the free space model, we again begin with a simple example of resonant photon storage in a homogeneously broadened atomic ensemble with negligible spin-wave decay. It is important to note that, in contrast to the cavity model, the free space model gives access to many diﬀerent spin-wave modes, which makes the retrieval eﬃciency dependent on how storage is carried out (see paper II). Therefore, optimization of storage alone is not a priori very practical. However, as shown in paper II, the optimization of storage alone is indeed useful because, in many cases, it also optimizes storage followed by backward retrieval. In order to have slightly simpler mathematical expressions, we work in the co-moving frame (see paper II), although the same argument can be carried out using the original time variable, as well. The complex number equations of motion on the interval t ∈ [0, T ] are then (see Ref. [1] and paper II) √ ∂z E(˜, t) = i dP (˜, t), z ˜ (12) ˜ z ˜ √ ˜ ˜ z ˜ ˜) = −P (˜, t) + i dE(˜, t) + iΩ(t)S(˜, t), (13) ˜ ˜ ∂t P (˜, t z z z ˜ ˜ ˜ z ˜ ∂t S(˜, t) = iΩ(t)P (˜, t), ˜ z ˜ (14) with initial and boundary conditions ˜ ˜ E(0, t) = Ein (t), P (˜, 0) = 0, z S(˜, 0) = 0. z (15) (16) (17)

These equations are written using dimensionless variables, in which (co-moving) time and Rabi frequency are

7 ˜ ˜ rescaled by γ (t = tγ and Ω = Ω/γ) and the position is rescaled by the length L of the ensemble (˜ = z/L). z E(˜, t) describes the slowly varying electric ﬁeld envez ˜ ˜ lope, the input mode Ein (t) satisﬁes the normalization ˜ 2 T ˜ ˜ constraint 0 Ein (t) dt = 1, d is the resonant optical ˜ z ) and Ein (t) are for now assumed to be ˜ depth, and Ω(˜ real. [To avoid carrying around extra factors of 2, d is deﬁned as half of what is often referred as the optical depth: the steady-state solution with Ω = 0 gives probe intensity 2 2 attenuation |E(˜ = 1)| = e−2d |E(˜ = 0)| .] The goal is z z to maximize the storage eﬃciency
1

Γ
1.5

a
1

0.5
in

ηs =
0

d˜ S(˜, T ) z z ˜

2

(18)

0 20

Γ

10

20

30

40

50

tΓ

˜ ˜ with respect to Ω(t). A procedure analogous to that used in the cavity model in Sec. II A yields equations of motion (also referred to as the adjoint equations) for the ¯z ˜ ¯ z ˜ ¯z ˜ Lagrange multipliers E(˜, t), P (˜, t), and S(˜, t): √ ¯ ¯ (19) ∂z E = i dP , ˜ √ ¯ ¯ ¯ ˜¯ ∂˜P = P + i dE + iΩS, (20)
t

b
15 10 5
in

¯ ˜¯ ∂t S = iΩP , ˜

(21)

with initial and boundary conditions ¯ ˜ E(1, t) = 0, ¯ z ˜ P (˜, T ) = 0, ¯z ˜ S(˜, T ) = S(˜, T ). z ˜ (22) (23) (24)

0.1

0.2

0.3

0.4

0.5

tΓ

As in the cavity discussion in Sec. II A, these equations describe backward retrieval and provide a simple mathematical connection between optimality and time-reversal. ˜ In order to move up the gradient, one should update Ω(t) according to 1 ˜ ˜ ˜ ˜ Ω(t) → Ω(t) − λ
1 0

FIG. 4: Adiabatic (dotted) and optimal (solid) control ﬁelds for the storage followed by backward retrieval of a Gaussianlike input mode Ein (t) (dashed) in the free space model with d = 10 and T = 50/γ (a) and T = 0.5/γ (b). Four optimal control pulses were obtained using four diﬀerent initial guesses for the gradient ascent procedure. The adiabatic control ﬁeld agrees with the optimal one in the adiabatic limit (T dγ ≫ 1) (a) and deviates from it otherwise (b).

¯ z ˜ z ˜ d˜ Im S ∗ (˜, t)P (˜, t) z (25)

¯ z ˜ −P (˜, t)S ∗ (˜, t) . z ˜

We showed in Ref. [1] and in paper II that, in the adiabatic limit (T dγ ≫ 1) and for a certain class of input modes of duration T ∼ 1/(dγ), one can achieve a universally optimal (for a ﬁxed d) storage eﬃciency that cannot be exceeded even if one chooses a diﬀerent input mode. We showed that in that case the obtained control ﬁeld will also maximize the total eﬃciency of storage followed by backward retrieval. However, this would not necessarily be the case for a general input mode in the nonadiabatic limit (T dγ 1), which is precisely the limit, in which gradient ascent optimization becomes most useful. Moreover, for the case of forward retrieval, the control ﬁeld that maximizes the storage eﬃciency does not maximize the total eﬃciency of storage followed by retrieval even in the adiabatic limit. Thus, in Appendix C, we describe how to use gradient ascent to maximize (still with

respect to the storage control ﬁeld) the total eﬃciency of storage followed by retrieval. As in the cavity model in Sec. II A, we now compare adiabatic shaping of the storage control ﬁeld (see Ref. [1] and paper II) to the optimal shaping via gradient ascent. To compare with the results of paper II, we maximize the total eﬃciency of storage followed by backward retrieval rather than the storage eﬃciency alone. We assume that d = 10 and that Ein (t) is the Gaussian-like input mode in Eq. (10), shown as a dashed line in Figs. 4(a) and 4(b). We ﬁrst consider the case T = 50/γ and shape the storage control using adiabatic shaping (Sec. VI B of paper II). Then we numerically compute the total eﬃciency of storage followed by complete backward retrieval using this storage control ﬁeld (the total eﬃciency is independent of the retrieval control ﬁeld provided no excitations are left in the atoms). The adiabatic storage control is shown as a dotted line in Fig. 4(a). Since for this input mode the adiabatic limit is satisﬁed (T dγ = 500 ≫ 1), the adiabatic storage control yields an eﬃciency of 0.66,

8

Η
1 0.75 0.5 0.25 0 0 10 20 30 40

100 10

1
50

TdΓ

FIG. 5: The total eﬃciency of storage followed by backward retrieval for the Gaussian-like input mode in Eq. (10) using adiabatic equations (dotted) and gradient ascent (solid) to shape the storage control ﬁeld. The results are shown for the indicated values of d (= 1, 10, 100), as a function of T dγ. The dashed lines represent the maximum eﬃciency possible at the given d [1].

which is the maximum eﬃciency possible at this d [1]. For the same reason, the adiabatic control agrees with the control ﬁeld computed via gradient ascent (solid line), which also yields an eﬃciency of 0.66. Fig. 4(a) shows four solid lines (optimal control ﬁelds) corresponding to four initial guesses Ω(t)/γ = 0.2, 0.5, 1, and 1.5. As in the cavity model discussion in Sec. II A, the diﬀerence between the four optimal controls and the adiabatic control is inconsequential. Repeating the calculation for T = 0.5/γ, we obtain Fig. 4(b). Since the adiabatic limit (T dγ ≫ 1) is no longer satisﬁed (T dγ = 5), the adiabatic approximation does not work and the adiabatic control diﬀers from the optimal control and gives a lower eﬃciency: 0.24 vs. 0.58. As in Fig. 4(a), the four optimal control ﬁelds plotted correspond to diﬀerent initial guesses Ω(t)/γ = 1, 3, 5, and 7. As in the cavity discussion, Fig. 4(b) indicates that, in the regime T dγ ∼ 1, where the adiabatic approximation no longer holds, the optimal control ﬁeld acquires characteristics of the control ﬁeld used in fast storage. As in the analysis of the cavity model in Sec. II A, we now analyze the dependence on d and T dγ of the eﬃciency yielded by the adiabatic control shaping and the optimal control shaping. In Fig. 5, we compare the eﬃciency of storage followed by complete backward retrieval of the input mode in Eq. (10) obtained using the control ﬁeld shaped using the adiabatic equations (dotted lines) and using gradient ascent (solid lines). The eﬃciencies are plotted as a function of T dγ for three indicated values of d (= 1, 10, 100). Horizontal dashed lines represent the maximum eﬃciency possible at the given d [1]. The dotted lines are the same as in Fig. 6(a) of paper II. Similar to the corresponding discussion of the cavity model in Sec. II A, Fig. 5 conﬁrms the predictions of papers II and III that eﬃcient photon storage is not possible for

T dγ 1. It also illustrates that optimal control ﬁelds open up the possibility of eﬃcient storage of input modes with a bandwidth that is an order of magnitude larger than the bandwidth allowed by the adiabatic storage. In addition, the same reasoning as in the cavity discussion leads to the conclusion that for this problem, gradient ascent most likely yields the global, rather than local, maximum at all values of T dγ. Various generalizations of the presented procedure can be made. First, the generalization to limited control pulse energy, (possibly reversible [30]) inhomogeneous broadening, complex Ω and Ein , and nonzero ∆, γs , and γc can be carried out exactly as in the cavity case (Appendix B). Second, in the case of backward retrieval, if the two metastable states are nondegenerate and have a frequency diﬀerence ωsg , one should incorporate an appropriate position-dependent phase shift of the spin wave ˜z ˜ of the form exp(−2i∆k˜), where ∆k = Lωsg /c (see Sec. VIII of paper II). Finally, another extension can be made for the cases when the total eﬃciency depends on the retrieval control ﬁeld (e.g. if γs and/or γc are nonzero). In those cases, one can simultaneously optimize with respect to both the storage and the retrieval control ﬁelds. However, one may then need to put a limit on the energy in the retrieval control pulse since, for the case of γs = 0, for example, the faster one retrieves, the higher is the eﬃciency, and the optimal retrieval control ﬁeld may, in principle, end up having unlimited power (e.g. an inﬁnitely short π pulse).

III.

OPTIMIZATION WITH RESPECT TO THE INPUT FIELD

Although it is usually easier to optimize with respect to the control ﬁeld, optimization with respect to the input mode can also be carried out in certain systems. For both classical and quantum light, the mode shape can often be controlled by varying the parameters used during the generation of the mode. For example, if the photon mode is created by releasing some generated collective atomic excitation, one can, under certain assumptions, generate any desired mode shape [1]. For the case of classical light, one can also shape the input light pulse simply using an acousto-optical modulator. An important advantage of optimizing with respect to the input mode is that the iterations can be carried out experimentally [2, 4]. In this section, we consider the maximization of light storage eﬃciency with respect to the shape of the input mode. The gradient ascent method, used in Sec. II to optimize with respect to the control ﬁeld, can be easily applied to the optimization with respect to the input mode shape both in the cavity model and in the free space model. Since one is interested in ﬁnding the optimal input mode shape, the optimization has to be carried out subject to T 2 the normalization condition 0 dt |Ein (t)| = 1. This condition can be included by adding an extra term with a Lagrange multiplier to the functional J to be optimized.

9 The iterations are then done as follows: one ﬁrst integrates the storage equations for a trial input mode; then integrates the adjoint equations corresponding to backward retrieval (as in Secs. II A and II B); then updates the trial input mode by adding to it a small correction ¯ proportional to the output of backward retrieval [−iP (t) ¯ ˜ in the cavity model or E(0, t) in the free space model]; and ﬁnally renormalizes the new input mode to satisfy the normalization condition. An important feature that distinguishes the optimization with respect to the input mode from the optimization with respect to the control ﬁeld is the possibility of making ﬁnite (not inﬁnitesimal) steps. Standard gradient-ascent improvement [such as via Eqs. (9) and (25)] is, in principle, inﬁnitesimal due to its reliance on the small parameter 1/λ. Several decades ago, Krotov introduced and developed an important powerful and rapidly converging global improvement method [6, 21, 26, 27, 28] that is not characterized by a small parameter. Largely thanks to the presence of the normalization condition on the input mode, this method can be applied to derive non-inﬁnitesimal quickly converging updates for the problem of optimization of light storage eﬃciency with respect to the input mode. For the cavity model of Sec. II A, this update is given by ¯ Ein (t) → −iP (t), (26) geneous broadening and (for the case of Doppler broadened gases) the presence of velocity changing collisions. One can show that, even with these features, the iterative optimization procedure still works in exactly the same way by updating the input mode with the output of time-reversed retrieval.

IV.

OPTIMIZATION WITH RESPECT TO THE INHOMOGENEOUS PROFILE

followed by a renormalization of Ein (t), while for the freespace model of Sec. II B, the update is given by ¯ ˜ ˜ Ein (t) → E(0, t), (27)

Having discussed optimization with respect to the control ﬁeld and the input mode, we now turn to the optimization with respect to the shape of the inhomogeneous proﬁle. This optimization is most relevant in the context of controlled reversible inhomogeneous broadening (CRIB) [30]. The main idea of CRIB is that by introducing inhomogeneous broadening into a homogeneously broadened medium (via Stark or Zeeman shifts, for example) and by optimizing the shape and width of this inhomogeneous proﬁle, one can better match the absorption proﬁle of the medium to the spectrum of the incoming photon mode and, thus, increase the storage efﬁciency [30]. At the same time, one can minimize the losses caused by dephasing of diﬀerent frequency classes with respect to each other by using an echo-like process triggered by a reversal of the inhomogeneous proﬁle between the processes of storage and retrieval [29, 30]. We refer the reader to papers I and III for a full list of references, and to Ref. [37] and paper III for examples of recent theoretical studies.

followed by renormalization. These updates precisely correspond to the time-reversal-based iterations suggested in Ref. [1] and explained in more detail in papers I and II. In these iterations, optimization of light storage with respect to the input ﬁeld is done by carrying out storage of a trial input mode followed by backward retrieval, and then using the normalized output of backward retrieval as the input mode in the next iteration. The beauty of this update procedure is the possibility of carrying it out experimentally. In fact, the extension of this procedure to the optimization of storage followed by forward retrieval, suggested in Ref. [1] and in paper II, has already been demonstrated experimentally [2]. In the language of gradient ascent, one can still think of Eqs. (26) and (27) as steps along the gradient. These steps are, however, ﬁnite, not inﬁnitesimal. This allows one to think of time-reversal-based optimization with respect to the input mode as simple intuitive walk up the gradient. As shown in paper I, the fact that only one collective atomic mode can be excited in the cavity model makes the iterations of Eq. (26) converge to the optimum in a single step. Using the terminology of gradient ascent, the optimization with respect to the input ﬁeld in the cavity model can, surprisingly, be achieved with a single large step up the gradient. We note that the optimization procedure discussed in this section can be easily generalized to include inhomo-

A.

Cavity model

As in Sec. II, we begin the discussion with the theoretically simpler cavity model. Although one can, of course, optimize with respect to the inhomogeneous proﬁle in the problem of storage alone (i.e. not followed by retrieval), in the context of CRIB it is more relevant to consider the problem of storage followed by retrieval with the reversed inhomogeneous proﬁle [30]. Moreover, although the approach can be extended to nonzero single-photon detuning and arbitrary control ﬁelds, we suppose for simplicity that the input mode Ein (t) is resonant and that the storage and retrieval control pulses are π pulses. Following the convention of Ref. [1] and papers I-III, we refer to this use of π-pulse control ﬁelds as fast storage and fast retrieval. We leave most of the mathematical details of the problem to Appendix D. Here we only note that we describe the inhomogeneous proﬁle by a discrete number of frequency classes. The index j labels the frequency class with detuning ∆j from the center of the line containing a fraction pj of atoms ( j pj = 1). In Appendix D, we show how to carry out optimization with respect to pj and/or ∆j .

10

Η
1

a
0.1

0.01
200 100

0.1
I

1

10

100

TCΓ

b

20 10 1 10 100

TCΓ

FIG. 6: Comparison of the eﬃciency for storage followed by retrieval in the cavity model with and without controlled reversible inhomogeneous broadening (CRIB). We consider storage of the Gaussian-like input mode of duration T [Eq. (10)] in a cavity with C = 50. (a) The ﬁgure shows the eﬃciency of fast storage followed by fast retrieval with a homogeneous line (dash-dotted), fast storage followed by fast retrieval with a reversible optimized inhomogeneous proﬁle, i.e. CRIB (circles), optimal storage and retrieval with a homogeneous line as in Fig. 3(a) (solid), and the asymptotic value C 2 /(C + 1)2 (dashed). (b) The optimal inhomogeneous width ∆I for CRIB.

We now present the results of gradient ascent optimization with respect to the inhomogeneous proﬁle for a particular example. We suppose that the input pulse is the Gaussian-like mode in Eq. (10) and that C = 50. The total eﬃciency of storage followed by retrieval, as a function of T Cγ, is shown in Fig. 6(a) for various storage protocols. The dash-dotted line gives the eﬃciency of fast storage (i.e. storage obtained by applying a control π pulse on the |g − |e transition at the end of the input mode at time T ) followed by fast retrieval using a homogeneous line. As discussed in papers I and II, a homogeneous ensemble enclosed in a cavity has only one accessible spin-wave mode and can, therefore, fast-store only one input mode, which has duration T ∼ 1/(Cγ). As a result, the decay at T Cγ ≫ 1 of the eﬃciency represented by the dash-dotted line is dominated by leakage of the input mode into the output mode and not by polarization decay. We now consider intoducing reversible inhomogeneous broadening and iteratively optimizing with respect to its shape (using Eq. (D6) or Eq. (D8)). As ex-

pected, the eﬃciency grows with each iteration independently of the choice of the number of frequency classes, the choice of ∆j , and the initial guess for pj . The landscape in the control space, however, depends on the number of frequency classes and on ∆j . This landscape is also not as simple as in Secs. II and III, i.e. there exist local maxima. We did not perform an exhaustive search, but out of all the initial conﬁgurations, number of frequency classes, and ∆j distributions that we tried, the highest eﬃciencies were obtained for the cases when gradient ascent converged to only two nonempty frequency classes with opposite detunings (we have not been able to come up with a simple physical reason for this). We therefore focus on the case of only two frequency classes with detunings ±∆I and optimize with respect to ∆I [using Eq. (D9)]. The optimized eﬃciency is shown with circles in Fig. 6(a). For T Cγ less than about 0.75, it is optimal to have ∆I = 0. For larger T Cγ, the optimal ∆I is shown in Fig. 6(b): at small T Cγ, it scales approximately as ∝ (T Cγ)−1 and then slower. The presence of two frequency classes and hence two accessible spin wave modes instead of one allows us to reduce the leakage error, so that the eﬃciency [circles in Fig. 6(a)] is now limited by polarization decay. Finally, we would like to compare the broadeningoptimized eﬃciency to the homogeneous controloptimized eﬃciency. Repeating the optimization procedure of Sec. II A for C = 50, we obtain the solid line in Fig. 6(a). The maximum eﬃciency possible at this C is C 2 /(C +1)2 and is shown as the dashed line. The dashed line and the solid line are the same as in Fig. 3(a), except that now C = 50. The fact that the solid line in Fig. 6(a) lies above the circles indicates that we have not been able to identify any advantage of fast storage with CRIB compared to optimal storage in the homogeneous medium. Moreover, all inhomogeneous broadening conﬁgurations we tried to introduce into the optimized homogeneous protocol converged back to the homogeneous proﬁle. These results suggest that if one wants to store a single mode of known shape using a homogeneously broadened ensemble of Λ-type systems enclosed in a cavity and can shape and time the control ﬁeld with suﬃcient precision, it may be better to use optimal homogeneous storage and not to use CRIB. It is, however, worth noting that we have only carried out the simplest optimization of fast storage with CRIB. In particular, the performance of fast storage with CRIB may be further enhanced by optimizing with respect to the time, at which the storage π pulse is applied. Such optimization represents an optimal control problem with a free terminal time [6] and is beyond the scope of the present paper (although it can be carried out in a straightforward manner by repeating the optimization above systematically for diﬀerent times of the π-pulse application). It is also important to note that the use of CRIB in the cavity model may allow for implementing a multimode memory [16] in the cavity setup. Unlike the free space

11

Η
1 0.75 0.5 0.25 0 0

G L

10

20

30

40

50

TdΓ

FIG. 7: Comparison of optimized homogeneous-line storage with storage based on CRIB. For d = 100, the plot shows the eﬃciency of storage followed by backward retrieval of the Gaussian-like input mode of duration T [Eq. (10)]. The curves show results for fast storage and retrieval with a homogeneous line (dash-dotted), fast storage and retrieval with an optimized reversible Gaussian (G) or Lorentzian (L) inhomogeneous proﬁle, i.e. CRIB (solid lines labeled G and L), storage and retrieval with a homogeneous line using adiabatic (dotted) or optimal (unlabeled solid line) control ﬁeld shaping (same as in Fig. 5), and the asymptotic value (dashed).

model, which allows for the storage of multiple temporal input modes using, e.g., Raman- or EIT-based protocols [1, 4, 38, 39], the homogeneously broadened cavity model only has a single accessible spin-wave mode. Therefore, if we do not use CRIB or some other inhomogeneous broadening mechanism, it can only store a single input mode

followed by fast backward retrieval using a homogeneous line. The two solid lines labeled G and L are obtained using fast storage with optimal-width reversible inhomogeneous broadening with Gaussian proﬁle and Lorentzian proﬁle, respectively. Although the optimization with respect to the inhomogeneous width can be done eﬃciently via gradient ascent [using Eq. (D11)], we have already performed this optimization in paper III by sampling a suﬃciently large set of inhomogeneous widths. The remaining third solid line and the dotted line (both taken from Fig. 5) correspond to homogeneous storage with optimal storage controls (solid) and with adiabatic controls (dotted). The dashed line (also from Fig. 5) is the maximum possible eﬃciency at this d. The plot shows that while adiabatic control ﬁeld shaping (dotted) makes homogeneous storage less eﬃcient for some values of T dγ than fast storage with CRIB (solid lines labeled G and L), optimal control ﬁeld shaping (unlabeled solid line) may enable homogeneous storage to be more eﬃcient than fast storage with CRIB at all values of T dγ. As in Sec. IV A, we note, however, that we have presented only the simplest optimization of CRIB and that the full investigation of the advantages of CRIB is beyond the scope of the present paper. In particular, the CRIB eﬃciency may be enhanced by optimizing with respect to the time, at which the storage π-pulse is applied. Moreover, CRIB might be useful in circumstances such as when a homogeneously broadened three-level system is not available, when more complicated inputs (such as time-bin qubits) are used, or when precise shaping and timing of the control pulse is harder to achieve than controlled reversible broadening. Finally, CRIB-based memories may even be implemented without any optical control ﬁelds [37].

B.

Free space model V. SUMMARY

Having discussed the optimization with respect to the inhomogeneous proﬁle in the cavity model, we note that the same procedure can be carried out for the free space model in an analogous manner. The appropriate update equations are listed at the end of Appendix D. In paper III, we compared storage using adiabatic control shaping in a homogeneous ensemble to fast storage with CRIB. We found that fast storage with CRIB can indeed do better than adiabatic homogeneous storage for T dγ ∼ 1. We show now that this result was mainly due to imperfect control ﬁeld optimization outside of the adiabatic limit and that, in the present work, we have not been able to identify any advantages of fast storage with CRIB compared to optimal homogeneous storage. We consider storage of the resonant Gaussian-like input mode in Eq. (10) in a free space atomic ensemble with d = 100 followed by backward retrieval. The total eﬃciency for various storage protocols is shown in Fig. 7 as a function of T dγ. The dash-dotted line and the two solid lines labeled G and L are taken from Fig. 8(a) of paper III. The dash-dotted line is the eﬃciency of fast storage

In conclusion, we have shown that the powerful numerical optimal control method of gradient ascent allows one to obtain simple intuitive understanding and to achieve a signiﬁcantly improved eﬃciency and a higher bandwidth in the problem of photon storage in Λ-type atomic ensembles. First, we showed how to apply gradient ascent to numerically compute optimal control ﬁelds even outside of the adiabatic limit both with and without a constraint on the energy in the control pulse. In particular, this opens up the possibility of eﬃcient storage of input modes that are an order of magnitude shorter (and hence an order of magnitude larger in bandwidth) than the shortest modes that can be eﬃciently stored using adiabatic control ﬁeld shaping. Second, we showed that gradient ascent provides an alternative justiﬁcation for the often discussed connection between optimality and time-reversal in photon storage, as well as for the iterative time-reversal-based optimization procedure with respect to the input ﬁeld suggested in Ref. [1], discussed in detail in papers I, II, and III, and demonstrated experi-

12 mentally in Ref. [2]. In particular, we conﬁrmed that the iterative procedure works even in the presence of inhomogeneous broadening and (for the case of Doppler broadened gases) in the presence of velocity changing collisions. Finally, we showed how to use gradient ascent to optimize with respect to inhomogeneous broadening and demonstrated how this can signiﬁcantly increase the eﬃciency of fast storage followed by fast backward retrieval in the presence of controlled reversible inhomogeneous broadening (CRIB) [30]. Provided one is interested in storing a single input photon mode of known shape and provided the control pulses can be generated with suﬃcient precision, we have not, however, been able to identify any advantages of CRIB-based photon storage compared to photon storage with optimal control pulses in homogeneously broadened media. In general, gradient ascent methods do not guarantee the attainment of the global maxima. The global maximum is, however, indeed attained for our problem in the regimes where this maximum is known. This strongly suggests that, for the optimization with respect to the input mode and with respect to the storage control, gradient ascent may indeed be yielding the global optimum. We also note that one can optimize simultaneously with respect to various combinations of the control parameters simply by simultaneously updating each of them along the corresponding gradient. One can also include other possible control parameters that are available in a given experimental setup but have not been discussed in the present paper. For example, for the case of photon storage in solid-state systems, one can consider optimizing with respect to the number of atoms put back into the antihole [30, 36] or with respect to a timedependent reversible inhomogeneous proﬁle. Other light storage systems, such as photonic crystals [40] or cavity models where the cavity ﬁeld cannot be eliminated, are also susceptible to gradient ascent optimization. Therefore, we expect the optimization procedures described in the present paper to allow for increased eﬃciencies and increased bandwidths in many current experiments on quantum memories for light, many of which are narrowband and suﬀer from low eﬃciencies. Such improvements would facilitate advances in ﬁelds such as quantum communication and quantum computation.
APPENDIX A: DERIVATION OF THE ADJOINT EQUATIONS OF MOTION IN THE CAVITY MODEL

In Sec. II A, we omitted the derivations of the adjoint equations of motion (4) and (5) and of the corresponding boundary conditions (6) and (7). We provide these derivations in this appendix. Varying J given in Eq. (3) with respect to S, S ∗ , P , and P ∗ , we obtain δJ = S(T )δS ∗ (T )
T

+
0 T

˙ ¯ dtP ∗ − δ P − γ(1 + C)δP + iΩδS ˙ ¯ dtS ∗ −δ S + iΩδP (A1)

+
0

+ c.c.,

where ”c.c.” means complex conjugate taken of the whole expression after the equal sign. Integrating by parts the terms containing time derivatives, we obtain δJ = S(T )δS ∗ (T ) ¯ −P ∗ (T )δP (T ) +
T 0 T 0

˙ ¯ dtP ∗ δP +

¯ dtP ∗ iΩδS

−γ(1 + C)δP ¯ −S ∗ (T )δS(T ) + + c.c..
T 0

˙ ¯ dtS ∗ δS +
0

T

¯ dtS ∗ (iΩδP ) (A2)

Since the initial conditions are ﬁxed, we have here used δS(0) = δP (0) = δS ∗ (0) = δP ∗ (0) = 0 to simplify the expression. The optimum requires that δJ = 0 for any variations δP and δS. Hence we collect the terms multiplying, e.g., δP (T ) and set the result to zero. Carrying out this procedure for δP (T ), δS(T ), and their conjugates, we obtain the boundary conditions (6) and (7). Collecting terms proportional to δP , δS, and their conjugates, we obtain adjoint equations of motion (4) and (5).
APPENDIX B: CONTROL FIELD OPTIMIZATION IN THE CAVITY MODEL: GENERALIZATION

VI.

ACKNOWLEDGEMENTS

We thank N. Khaneja, L. Jiang, I. Novikova, V.F. Krotov, and M. Shapiro for fruitful discussions. This work was supported by the National Science Foundation, Danish Natural Science Research Council, DARPA, the Packard Foundation, and the EC-funded projects SCALA and QOQIP. This work has also been partially supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.

In Sec. II A, we showed how to perform control ﬁeld optimization in the simplest possible version of the cavity model: a resonant input mode with a real envelope was stored using a control pulse with a real envelope and unlimited power into a homogeneously broadened ensemble with inﬁnite spin-wave lifetime. In this appendix, we show how to optimize the control ﬁeld in a more general model that includes the possibility of complex control ﬁeld envelopes Ω(t) and input mode envelopes Ein (t), nonzero single-photon detuning ∆ and spin wave

13 decay rate γs , and (possibly reversible [30]) inhomogeneous broadening such as Doppler broadening in gases or the broadening of optical transitions in solid state impurities [36]. For the case of Doppler broadened gases, we also include velocity changing collisions with rate γc . We also show how to take into account possible experimental restrictions on the strength of the classical control ﬁelds. Using the notation of paper III, the complex number equations describing the generalized model are √ ˙ Pj = − [γ + i(∆ + ∆j )] Pj − γC pj P + iΩSj √ √ (B1) +i 2γC pj Ein + γc ( pj P − Pj ), √ ∗ ˙ Sj = −γs Sj + iΩ Pj + γc ( pj S − Sj ), (B2) where j labels the frequency class with detuning ∆j from the center of the line containing a fraction pj of atoms ( j pj = 1) and where the total optical and spin polar√ √ izations are P = k pk Pk and S = k pk Sk , respectively. The terms proportional to γc describe completely rethermalizing collisions with rate γc [41]. One can, of course, also take γc to be diﬀerent for P and S. For example, if γc ≪ γ, which is often the case, one can drop the terms proportional to γc in Eq. (B1) [42]. In addition to moving atoms from one frequency class to the other, collisions also result in line broadening, which can be taken into account by increasing γ [43]. We assume that the goal is to maximize the eﬃciency ηs = |S(T )|2 of storage into the symmetric mode S(T ) with respect to the control pulse Ω(t) for a given input mode shape Ein (t) T satisfying the normalization condition 0 dt |Ein (t)|2 = 1. A procedure very similar to that described in Sec. II A and in Appendix A yields the following equations of motion for the adjoint variables: √ ¯ ˙ ¯ ¯ ¯ Pj = [γ − i(∆ + ∆j )] Pj + γC pj P + iΩSj √ ¯ ¯ (B3) −γc ( pj P − Pj ), ˙ = γ S + iΩ∗ P − γ (√p S − S ), ¯ ¯j ¯ ¯j ¯ (B4) Sj j c s j simpler optimization problem with a limit on the control T pulse energy 0 |Ω(t)|2 dt ≤ E for some E [44]. In order to carry out the optimization subject to this constraint, one should ﬁrst carry out the optimization without the constraint and see whether the optimal control satisﬁes the constraint or not. If it does not satisfy the constraint, T 2 one has to add a term µ′ (E − 0 |Ω(t)| dt) to J, so that the update becomes   1 ¯∗ ¯ ∗ Ω(t) → Ω(t)+ Sj Pj − Pj Sj − µ′ Ω(t) , (B8) i λ j where µ′ is adjusted to satisfy the constraint. By redeﬁning µ′ and λ, this update can be simpliﬁed back to Eq. (B7) followed by a renormalization to satisfy the constraint. Depending on how severe the energy constraint is, one can then sometimes (but not always) further simplify the update by completely replacing Ω(t) with the gradient [i.e. set λ = µ′ in Eq. (B8)] followed by a renormalization of Ω(t), as is done, for example, in Ref. [17] for the problem of laser control of chemical reactions. We note that these optimization protocols can be trivially extended to the full process of storage followed by retrieval, which, in the presence of inhomogeneous broadening, one might not be able to optimize by optimizing storage and retrieval separately. Similarly, one may include the possibility of reversing the inhomogeneous proﬁle between the processes of storage and retrieval [30].
APPENDIX C: CONTROL FIELD OPTIMIZATION IN THE FREE-SPACE MODEL: GENERALIZATION TO STORAGE FOLLOWED BY RETRIEVAL

√ ¯ √ ¯ ¯ ¯ where P = pk Pk and S = pk Sk . The cork k responding initial conditions for backward propagation are ¯ Pj (T ) = 0, √ ¯ Sj (T ) = pj S(T ). (B5) (B6)

Similarly to Sec. II A, after taking an initial guess for ¯ ¯ Ω(t) and solving for Pj , Sj , Pj , and Sj , one updates Ω(t) by moving up the gradient Ω(t) → Ω(t) + 1 i λ ¯∗ ¯ ∗ Sj Pj − Pj Sj . (B7)

j

Short input modes and/or large single-photon detuning ∆ require control pulses with large intensities that might not be available in the laboratory. There exist ways to include a bound on the control ﬁeld amplitude [6]. Alternatively, one may want to consider a slightly

In Sec. II B, we showed how to use gradient accent to ﬁnd the control ﬁeld that maximizes the storage eﬃciency. However, the obtained storage control ﬁeld does not always maximize the total eﬃciency of storage followed by retrieval. Therefore, in this appendix, we consider the maximization of the total eﬃciency of storage followed by retrieval with respect to the storage control ﬁeld. While we demonstrate the procedure only for the case of forward retrieval, the treatment of backward retrieval is analogous. We suppose that the control ﬁeld Ω(t) consists of a storage control pulse on t ∈ [0, T ] and a retrieval control pulse on t ∈ [Tr , Tf ]. We want to optimize with respect to the former given the latter and the input mode (note that the total eﬃciency is independent of the retrieval control for suﬃciently strong retrieval control pulses, and it is therefore often less important to optimize with respect to the retrieval control pulse). Here 0 < T ≤ Tr < Tf , and the subscripts in Tr and Tf stand for ”retrieval” and ”ﬁnal”. The time interval [T, Tr ] corresponds to the waiting (i.e. storage) time between the processes of storage (which ends at t = T ) and retrieval (which begins at t = Tr ).

14 We suppose that storage is described by Eqs. (12)-(17) on t ∈ [0, T ]. Then forward retrieval that follows after the storage time interval [T, Tr ] is described by the same equations (12)-(14) but on the time interval t ∈ [Tr , Tf ] with initial and boundary conditions ˜ E(0, t) = 0, P (˜, Tr ) = 0, z ˜ S(˜, Tr ) = S(˜, T ), z ˜ z ˜ (C1) (C2) (C3) broadening without providing the mathematical details. In this appendix, we present these details. We ﬁrst consider the cavity model, but turn brieﬂy to the free-space model at the end of this appendix. We suppose for simplicity that the input mode Ein (t) is resonant and that the storage and retrieval control pulses are π pulses at t = T and t = Tr , respectively. In order √ to simplify notation, we deﬁne xj = pj , satisfying the 2 normalization j xj = 1. The storage equation (B1) on the interval t ∈ [0, T ] then becomes ˙ Pj = −(γ + i∆j )Pj − γCxj P + i 2γCxj Ein ,(D1)

˜ ˜ where Tr = Tr γ (similarly, Tf = Tf γ). Eq. (C2) assumes that the polarization has suﬃcient time to decay before retrieval starts, while Eq. (C3) assumes that spin-wave decay is negligible during the storage time. The goal is to maximize the total eﬃciency of storage followed by retrieval,
˜ Tf

ηtot =

˜ Tr

˜ ˜ dt E(1, t) ,

2

(C4)

with P = k xk Pk and with the initial condition Pj (0) = 0. A π-pulse at t = T mapping P onto S followed by another π-pulse at t = Tr mapping S back onto P result in an overall 2π pulse, so that Pj (Tr ) = −Pj (T ). Assuming the broadening is reversed at some time between T and Tr , the equations for retrieval on the interval t ∈ [Tr , Tf ] are ˙ Pj = −(γ − i∆j )Pj − γCxj P.
Tf Tf

with respect to the storage control ﬁeld. Constructing J and taking appropriate variations, we obtain initial and boundary conditions for backward propagation: ¯ ˜ ˜ ˜ ˜ ˜ E(1, t) = E(1, t) for t ∈ [Tr , Tf ], ¯ z ˜ P (˜, Tf ) = 0, ¯z ˜ S(˜, Tf ) = 0, and ¯ ˜ ˜ ˜ E(1, t) = 0 for t ∈ [0, T ], ¯ z ˜ P (˜, T ) = 0, ¯z ˜ ¯z ˜ S(˜, T ) = S(˜, Tr ). (C8) (C9) (C10) (C5) (C6) (C7)

(D2)

The total eﬃciency of storage followed by retrieval is then ηtot =
Tr

dt |Eout (t)| =

2

dt i
Tr

2γCP (t) . (D3)

2

One can show that the equations of motion for the ¯ adjoint variables (i.e. the Lagrange multipliers) Pj are ˙ ¯ ¯ ¯ Pj = (γ + i∆j )Pj + γCxj P − 2γCxj P ¯ for t ∈ [Tr , Tf ] with Pj (Tf ) = 0 and ˙ ¯ ¯ ¯ Pj = (γ − i∆j )Pj + γCxj P (D5) (D4)

By taking the variational derivative of J with respect to ˜ ˜ Ω(t) on the storage interval, we ﬁnd that the update is exactly the same as for the optimization of storage alone and can be done via Eq. (25). We note that if the retrieval control pulse leaves no atomic excitations, one can obtain the same optimization equations by solving the storage optimization problem in Sec. II B but changing the function to be maximized from 1 z z ˜ z ˜ the number of spin-wave excitations 0 d˜S(˜, T )S ∗ (˜, T ) ˜) [Eq. (6) to the complete retrieval eﬃciency from S(˜, T z of Ref. [1]]. It is also worth noting that the derivation presented here can trivially be extended to apply to backward (instead of forward) retrieval and to include complex Ω and Ein , (possibly reversible [30]) inhomogeneous broadening, and nonzero ∆, γs , and γc .
APPENDIX D: OPTIMIZATION WITH RESPECT TO THE INHOMOGENEOUS PROFILE: MATHEMATICAL DETAILS

¯ ¯ for t ∈ [0, T ] with Pj (T ) = −Pj (Tr ), where we deﬁned ¯k . The last term in Eq. (D4) describes an ¯ = P k xk P incoming ﬁeld that is the time-reverse of the retrieved ﬁeld. Assuming we are optimizing with respect to xj , the update is xj → xj + 1 Aj , λ (D6)

followed by a rescaling of all xj by a common factor to ensure the normalization j x2 = 1. Here Aj is given by j
T Tf

Aj = −γC Re

dt +
0 T Tr

dt

¯ ¯ Pj∗ P + P ∗ Pj

− 2γC Im +2γC Re
Tr

0 Tf

¯ dtEin Pj∗ (D7)

dtPj∗ P,

In Sec. IV, we presented the results on the optimization of photon storage with respect to the inhomogeneous

where Re denotes the real part. Numerics show that the update can usually be simpliﬁed in a way that avoids

15 the search for convenient values of λ and does not lose convergence. Speciﬁcally, taking λ → 0 in Eq. (D6), we obtain xj → Aj , (D8) By adjusting fj and xj , one can choose a particular inhomogeneous proﬁle shape (e.g. Lorentzian, Gaussian, or a square) and optimize with respect to its width. Having discussed the cavity case, we now list the corresponding free-space results. In free space, the update of xj via Eq. (D8) would use

followed by renormalization. By deﬁning a particular functional form for the dependence of xj on ∆j , one could also consider optimization with respect to only a few parameters, such as, for example, the width ∆I and the degree of localization n of the inhomogeneous proﬁle of 2 the form pj = xj ∝ 1/ [1 + (∆j /∆I )n ]. Equivalently, instead of optimizing with respect to xj , one can optimize with respect to ∆j . To illustrate this procedure, we consider a simple optimization procedure with respect to a single parameter, the inhomogeneous width ∆I . We write ∆j = ∆I fj for some ﬁxed dimensionless parameters fj and consider maximizing the eﬃciency with respect to ∆I for ﬁxed xj and fj . The equations of ¯ motion and the initial conditions for both Pj and Pj stay the same as in the optimization with respect to xj while the update becomes ∆I → ∆I + 1 Im λ
T j 0 Tf

√ Aj = − d Im

1

˜ T

d˜ z
0 0

˜ dt +

˜ Tf ˜ Tr

˜ dt

¯ ¯ Pj∗ E + E ∗ Pj . (D10)

˜ Similarly, the update of ∆I = ∆I /γ would be

1 ˜ ˜ ∆I → ∆I + Im λ

1

˜ T

d˜ z
j 0 0

˜ dt −

˜ Tf ˜ Tr

˜ ¯ dt Pj∗ fj Pj . (D11)

dt −

Tr

¯ dt Pj∗ fj Pj . (D9)

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