Efficient adiabatic tracking of driven quantum nonlinear systems

PHYSICAL REVIEW A 88, 063622 (2013)

Efficient adiabatic tracking of driven quantum nonlinear systems S. Gu´erin,1,* M. Gevorgyan,1,2 C. Leroy,1 H. R. Jauslin,1 and A. Ishkhanyan2 1

Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS UMR 6303, Universit´e de Bourgogne, BP 47870, 21078 Dijon, France 2 Institute for Physical Research NAS of Armenia - 0203 Ashtarak-2, Armenia (Received 13 June 2013; published 11 December 2013) We derive a technique of robust and efficient adiabatic passage for a driven nonlinear quantum system, describing the transfer to a molecular Bose-Einstein condensate from an atomic one by external fields. The pulse ingredients are obtained by tracking the dynamics derived from a Hamiltonian formulation, in the adiabatic limit. This leads to a nonsymmetric and nonmonotonic chirp. The efficiency of the method is demonstrated in terms of classical phase space, more specifically with the underlying fixed points and separatrices. We also prove the crucial property that this nonlinear system does not have any solution leading exactly to a complete transfer. It can only be reached asymptotically for an infinite pulse area. DOI: 10.1103/PhysRevA.88.063622

PACS number(s): 03.75.Nt, 02.30.Hq, 05.30.Jp, 32.80.Qk

I. INTRODUCTION

Nonlinear quantum systems are at the heart of modern applications such as Bose-Einstein condensation (BEC) and nonlinear optics. It has been recognized that the formation of molecules from ultracold atom gases by external fields, and more generally the coherent oscillations between the atomic and molecular BECs, are well described by a semiclassical mean-field Gross-Pitaevskii theory [1,2]. More specifically, a driven two-level model features already a good approximation accounting for the one-color photoassociation or Feshbach resonance [3–5]. This corresponds to the set of nonlinear equations including third-order nonlinearities, t

i a˙1 = U e−i (s)ds a¯ 1 a2 + (11 |a1 |2 + 12 |a2 |2 )a1 , U t i a˙2 = ei (s)ds a1 a1 + (21 |a1 |2 + 22 |a2 |2 )a2 , 2

(1a) (1b)

where a1 (t) and a2 (t) are the atomic- and molecular-state probability amplitudes, respectively, satisfying |a1 |2 + 2|a2 |2 = 1 (corresponding to a 1:2 resonance), U ≡ U (t) is the (real) Rabi frequency associated to the external field, and the atom-atom, atom-molecule, and molecule-molecule elastic interactions are described by the terms 11 , 12 = 21 , and 22 , respectively, proportional to the scattering lengths. The Rabi frequency is proportional to the amplitude of the photoassociating laser field and  ≡ (t) the detuning between the frequency of the two-atom-molecule transition and the chirped driving laser frequency. In the case of a Feshbach resonance, the Rabi frequency is proportional to the square root of the magnetic field width of the resonance, and the detuning is proportional to the external magnetic field. The language of photoassociation theory is adopted in this paper, however the derived results are general for nonlinear two-state problems that are described by (1), arising, for instance, in nonlinear optics [6]. Such equations have been widely studied for given models of the parameters U (t) and (t) [7,8], taken from the ones usually studied for the corresponding linear problem, namely, Rabi model and crossing models such as Landau-Zener and Allen-Eberly models, leading to an adiabatic evolution in

*

[email protected]

1050-2947/2013/88(6)/063622(5)

the linear model [9–11]. The main known results in terms of transfer from the atomic condensate [|a1 (ti )| = 1 with ti the initial time] to the molecular BEC can be summarized as follows: (i) The Rabi model for  = 0 and any pulse shape leads to a complete transfer [|a2 (tf )|2 = 1/2 with tf the final time] only asymptotically, in the limit of an infinite pulse area. (ii) A crossing model [12] leads also to a complete transfer in the limit of infinite pulse area (adiabatic limit), like the corresponding linear problem. However, unlike the corresponding linear problem, near the end of the process, the transfer becomes oscillatory due to the crossing of a separatrix which strongly reduces the expected efficiency [13–15]: a much larger pulse area is needed to reach the same efficiency as the corresponding linear model. Typically, the exponential efficiency of the linear model (for analytic parameters), that is an infidelity of the form e−CT with C a constant and T the characteristic time of interaction, becomes of the form C/T n with n  1 for the nonlinear model. In this paper, we first show the important result of the nonexistence of a solution leading exactly to a complete transfer in the finite pulse area, whatever the parameters (adiabatic or not), unlike the linear model. On the other hand, since the simple Rabi solution is strongly nonrobust with respect to the detuning or to the presence of third-order nonlinearity, it is desirable to derive alternative robust solutions. We construct an improved adiabatic procedure featuring an infidelity of the form C/T n , n  2, i.e., improving it by more than one order of magnitude with respect to standard procedures. The pulse ingredients are obtained by tracking the corresponding classical dynamics derived from a Hamiltonian formulation, in the adiabatic limit. This leads to a nonsymmetric and nonmonotonic chirp: the pulse parameters start, as in the linear case, out of resonance, but end at the resonance (the latter in the case of no third-order nonlinearity). The efficiency of the adiabatic method is demonstrated in the phase space: the fixed points do not cross the separatrix, except at the final asymptotic time when the amplitude of the coupling goes to zero. The resulting robustness makes this adiabatic passage very attractive for a practical implementation. The paper is organized as follows: We first establish the Hamiltonian formulation of the quantum problem. We next prove several general results which motivate the need of an

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efficient and robust adiabatic passage technique. It is then derived and discussed within the Hamiltonian formulation. We finally conclude. II. HAMILTONIAN FORMULATION

One first recast of Eqs. (1) removing the phase and redefining the coupling is     2 2 i c˙1 = − + 11 |c1 | + 12 |c2 | c1 + √ c¯1 c2 , (2a) 3 2     2 2 i c˙2 = √ c1 c1 + (2b) + 21 |c1 | + 22 |c2 | c2 , 3 2 2 with  |c1 |2 + 2|c2 |2 = 1  using the transformation a1 = √ c1 e−i (s)ds/3 , a2 = c2 ei (s)ds/3 , and (t) = 2U (t). For comparison purposes, the corresponding linear√model for the variables b1 ,b2 is defined by b1 = c1 , b2 = 2c2 , |b1 |2 + |b2 |2 = 1 with ij = 0 (for all i and j ) and c¯1 (c1 ) dropped from Eq. (2a) [Eq. (2b)] in the coupling term: i b˙1 = − 3 b1 +  b , i b˙2 = 2 b1 + 3 b2 . 2 2 It is well known that an adiabatic treatment of a nonlinear model, which will allow the extension of the standard adiabatic passage of linear models, can be accomplished by a classical Hamiltonian formulation of the problem. We derive, first without considering the third-order nonlinearities for simplicity, the Hamiltonian      h = − I1 + I2 + √ c12 c¯2 + c¯12 c2 , Ij ≡ |cj |2 , (3) 3 3 2 2 whose corresponding solutions cj , j = 1,2, of the complex Hamilton equations i c˙j = ∂h/∂ c¯j [from √ the definition of the standard coordinates cj = (qj + ipj )/ 2], coincide with the solution of the original quantum  problem (2). The variables Ij and ϕj , defined from cj = Ij e−iϕj , are canonically conjugate: {Ij ,Ik } = 0, {ϕj ,ϕk } = 0, {I j ,ϕj } = 1, {Ij ,ϕk=j } = 0 with the Poisson bracket {α,β} = i j =1,2 (∂α/∂cj ∂β/∂ c¯j − ∂β/∂cj ∂α/∂ c¯j ). Since J = |c1 |2 + 2|c2 |2 = 1 (action variable) is a conserved quantity, i.e., {h,J } = 0, it is relevant to use it explicitly, accompanied with its conjugate angle variable γ ≡ ϕ1 = i ln(c1 /|c1 |), indeed satisfying {J,γ } = 1. The solution of Eqs. (2) can be thus parameterized in the most general way as

√

√ J − 2I I1 e−iϕ1 c1 (4) = √ −iϕ = √ e−iγ , c2 I2 e 2 I e−i(α+γ ) where γ is the global phase of the wave function, α + γ is its internal relative phase, and I ≡ I2 with p ≡ 2I corresponds thus to the probability of formation of a molecular BEC. The pair of variables (γ ,α) can be determined from the original angles (ϕ1 ,ϕ2 ) by the following canonical transformation:





1 0 γ ϕ1 , T = , (5) =T ϕ2 −2 1 α with their corresponding conjugate variables: [J I ] = [I1 I2 ]T −1 . With these variables, the Hamiltonian reads √   (6) h = − J + I + √ (J − 2I ) I cos α, 3 2

with the corresponding equations of motion: J˙ = −∂h/∂γ = 0, γ˙ = ∂h/∂J , I˙ = −∂h/∂α, and α˙ = ∂h/∂I , i.e., using J = 1: √  (7a) I˙ = √ (1 − 2I ) I sin α, 2  1 − 6I cos α, (7b) α˙ =  + √ √ I 2 2   √ γ˙ = − + √ I cos α. (7c) 3 2 This definition (4) of α and γ has thus led to a Hamiltonian independent of γ . This allows one to define a reduced phase space of only two dimensions (I,α), and γ is determined from (7c). We notice that the angle α is not well defined for I = 0, nor for I = 1/2. This is similar to the linear two-state problem formulated on the Bloch sphere, on which the angle at the poles is not well defined. III. GENERAL RESULTS

The first statements of this paper, which motivate the further development of an efficient and robust adiabatic passage in the next section, are as follows: (i) In order to have a complete transfer from an initial purely atomic system I (ti ) = 0 (at time ti ) to a final (at time t = tf ) molecular BEC, I (tf ) = 1/2, one needs an infinite pulse area whatever the parameters  and . (ii) The Rabi model (for  = 0) gives the highest fidelity molecular BEC transfer for a given pulse. (iii) The Rabi model is strongly nonrobust. The result (i) is proved by solving Eq. (7a) exactly [for any (t) and α(t), the latter being still unknown]:  p(t) ≡ 2I (t) = tanh

t

2 ti

 (s) sin α(s)ds . 2

(8)

It shows directly the result that the targeted molecular state can be reached only asymptotically, i.e., for BEC tf ti (s) sin[α(s)]ds → +∞, which is possible only for an infinite pulse area, and with 0 < α(t) < π for large times. For  = 0, the solution of (7b) is given by α = π/2. This also proves statement (ii),since α = π/2 is the value which t gives the largest integral ti (s) sin[α(s)]ds. The condition for a high-fidelity transfer of the Rabi model can be simply t rewritten, from Eq. (8): exp[ ti f (s)ds]  1. Because of the exponential dependence, this can be well satisfied even for a relatively modest area. One can remark, however, that the π -pulse area, which corresponds to the smallest pulse area leading to a complete transfer in the corresponding linear case [16], gives here at the end of the pulse only p = 2|c2 |2 ≈ 0.84. Concerning the statement (iii), the presence of a nonzero detuning or of a third-order nonlinearity induces, in general, oscillations in the integral of (8) due to sin α, which ruins the fidelity of the Rabi model. This is shown in Fig. 1 displaying the transfer profile as a function of a constant detuning 0 for a Rabi model. We emphasize that the excitation profile oscillates more and more for a larger pulse area.

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1 0.8

p

0.6 0.4 0.2 0 0

0.2

0.4

Δ0 T

0.6

0.8

1

FIG. 1. (Color online) Robustness of the final transfer profile p = 2|c2 (tf )|2 = 2I (tf ) (dimensionless) as a function of the static detuning 0 (in units of 1/T ) for (t) = 0 sech(t/T ) with 0 T = 3, ij = 0 and the detuning (t) + 0 with (i) (t) ≡ 0 (Rabi model, solid line) and (ii) (t) [(9) with α = 0], p(t) (11b) (Allen-Eberly nonlinear tracking model, dashed line). IV. ADIABATIC PASSAGE FOR NONLINEAR SYSTEMS AND ADIABATIC TRACKING A. Adiabatic passage for nonlinear systems: Formulation and conditions

Adiabatic passage can be defined as follows for the system (7) (see, for instance, [13–15]): For a sufficiently slow evolution of the parameters  and , featured by the quantity 1/T (where T is their characteristic duration), i.e., satisfying 1/T ωd with ωd a typical frequency of the dynamics, the solution follows the instantaneous fixed points of the reduced phase space, given by I˙ = 0 and α˙ = 0 at the considered (constant) values  and . The obstacles to the adiabatic following come from (i) regions surrounding the instantaneous separatrices involving arbitrary small frequencies for the dynamics and (ii) the crossing of fixed points. The conditions of validity of the adiabatic theorem are thus met if the adiabatically followed fixed point does not cross any separatrix or other fixed points, in the same manner as the crossing of eigenvalues have to be avoided in linear models. We determine below the fixed points and the separatrices, and show that any adiabatic solution I (t) which satisfies the appropriate initial and final conditions, i.e., I (ti ) = 0 and I (tf ) = 1/2, can be obtained by a choice of (t) and (t) satisfying the conditions of the adiabatic theorem. This is referred to as an adiabatic tracking. For given values  and , the fixed points, corresponding to (I˙ = 0,α˙ = 0), are given by  1 − 3p iα α = 0 or π, and α = − √ e , 2 p

(9) with α = 0 (α = π ) is initially connected. The detuning has to change its sign during the dynamics to lead to a final molecular BEC p(tf ) = 1, as in the linear model. An important result, which significantly differs from the linear model, is that the molecular BEC p(tf ) = 1 is adiabatically finally reached only if (tf ) = ±(tf ) (the sign depends on the initial connection), with the detuning ending up at resonance while the pulse is switched off. The crucial property of the underlying solution given by the fixed points (9) is that it always satisfies the adiabatic conditions of noncrossing of the separatrix. We prove it for the fixed point (9) with α = 0. We define as representation the nonlinear generalized Bloch sphere [17] embedded in R3 of coordi√ nates [p, 2 = 2(c12 c¯2 + c¯12 c2 ) = √42 (1 − p) p cos α, 3 = √ −2i(c12 c¯2 − c¯12 c2 ) = √42 (1 − p) p sin α] (the first coordinate p is taken for convenience instead of the usual 1 = |c1 |2 − 2|c2 |2 = 1 − 2p2 ). The coordinates of the fixed points (α0 = √ 0,p0 ) are [p0 , 2,0 = √42 (1 − p0 ) p0 , 3,0 = 0]. Separatrices are curves ps ,αs in the reduced phase space of energy hs passing through the hyperbolic fixed points, i.e., p = 1. From Eq. (6), we get hs = /6, which gives the equation for the √ separatrix (1 − ps )( −  ps cos αs ) = 0, i.e., 1  √ = (3p0 − 1)/(2 p0 ). cos αs = μ √ , μ ≡ ps 

(10)

It exists only when |μ|  1 with μ2  ps  1. Note that μ is an increasing (decreasing) function of p0 with μ  1 (μ  −1) for (9) with α = 0 (α = π ). For μ > 0, the coordinates of the separatrix curve on the nonlinear Bloch  sphere read [ps , 2,s = √42 (1 − ps )μ, 3,s = √42 (1 − ps ) ps − μ2 ]. Since the fixed point lies on a curve satisfying 3,0 = 0, the only point of the separatrix which could intersect the fixed point is for 3,s = 0. This implies ps = μ2 = (3p0 − 1)2 /(4p0 ) (ps = 1 is excluded since this point cannot be reached strictly) and 2,s = √42 (1 − μ2 )μ. We finally compare ps and p0 : Solutions of ps = p0 are p0 = 1 and p0 = 1/5, but the latter is excluded since it gives μ < 0. We thus get ps  p0 where the equality arises only for p = 1. For μ < 0, 2,s is negative, which excludes the intersection of the fixed point with the separatrix. Figure 2 shows the fixed point (9) and the corresponding separatrix curve (10) on the nonlinear Bloch sphere for various values of p0 . They both contract to a single point (the north pole of the sphere) for p0 = 1. B. Adiabatic tracking

(9)

and also (iii) I = 0,1/2 at points where α is not defined. I = 1/2 is a hyperbolic (unstable) fixed point. If one excludes the latter singular fixed points, Eq. (9) shows the relation between the detuning (t) and the coupling (t) for two possible adiabatic approximations p(t) for the probability of formation of a molecular BEC. We consider without loss of generality a positive pulse (t) > 0 such that (ti ) = (tf ) = 0. For an initial atomic system p(ti ) = 0, and if the initial detuning is negative (positive), (ti ) < 0 [(ti ) > 0], then the fixed point

The adiabatic dynamics can be concretely designed by the tracking of a desired given solution I (t). This can be achieved by additionally choosing, for instance, the pulse shape (t). The dynamics is then realized if one assumes (t) given by one of the two possibilities (9) [depending on the initial sign of (ti )], and if the above conditions of the adiabatic theorem are satisfied. A possible (however arbitrary) choice consists in choosing p(t) such that the transfer tracks the solution given by the linear case, in the adiabatic limit. This is achieved if one takes the probability p(t) and (t) from the corresponding linear model and determines the nonlinear (t) from Eq. (9).

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Populations

p

p0=0.4 0.5

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0

0

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p0=0.96

Detuning

p

p0=0.8 0.5

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0 −1

−1 0 Π

1

−1

0 Π

3

2

1

−5

0 Π

2

1

−1

0

1

Π

3

FIG. 2. (Color online) Representation of the fixed point (9) with α = 0 for p = p0 = 0.4, 0.6, 0.8, 0.96 and the corresponding separatrix curve (10) on the nonlinear generalized Bloch sphere (all the quantities are dimensionless).

This procedure allows one, from a given linear crossing model, to define the corresponding nonlinear tracking models whose dynamics is expected to closely follow the corresponding linear dynamics in the adiabatic limit. This procedure is numerically shown in Fig. 3 for a specific example: We consider an Allen-Eberly model for the corresponding linear model with a symmetric detuning: (t) = 0 sech(t/T ), lin (t) = B tanh(t/T ), (11a)  t  2 sech(s/T )ds/2T . (11b) p(t) = sin ti

The simple form of p(t) is obtained for the choice B = 30 /2 and 0 T  1 (adiabatic limit) (changing B, but keeping it of the order of 0 , does not lead to a significant difference for the transfer). For the numerics, we use the above (t), and the modified detuning (t) is determined from Eq. (9) (with α = 0), which defines the corresponding nonlinear Allen-Eberly adiabatic tracking (AEAT) model. We compare it with the dynamics having the symmetric detuning lin (t) for the same pulse area. Figure 3 shows the remarkable efficiency of the derived AEAT model compared to the standard Allen-Eberly detuning which fails to achieve a complete transfer. The traditional Allen-Eberly detuning leads to a typical dynamics when symmetric chirped detuning is considered: The transfer at some point becomes oscillatory and inefficient. The failure is due to the crossing of the fixed point with the separatrix (10) [13,14]. On the other hand, the derived AllenEberly nonlinear tracking model, by construction, satisfies the boundary limit at the initial and final times, and thus satisfies the conditions of the adiabatic following. Figure 3 shows the detuning falling to zero near the end of the dynamics, when the pulse is switched off. We have numerically determined

0

5

Time (in units of T) FIG. 3. (Color online) Numerical dynamics for (i) the AEAT model (solid line): (t) = 0 sech(t/T ) with 0 T = 6 and (t) determined from (9) with the tracked solution p(t) (11b), and for (ii) the standard nonlinear Allen-Eberly model (dashed line), i.e., with (t) = B tanh(t/T ), BT = 9 (both with ij = 0). Upper graph: Populations p(t) (dimensionless). Lower graph: chirped detuning (in units of 1/T ). The superiority of the asymmetric AEAT detuning is shown.

that the infidelity decreases asymptotically as C/(0 T )2 for this model, which surpasses the traditional Allen-Eberly model of more than one order of magnitude. The derived adiabatic passage features robustness with respect to the pulse area and a static detuning. This is shown in Fig. 1: The transfer profile first decreases (with a smaller decrease for a larger pulse area) before featuring a slowly decreasing plateau.

V. CONCLUSION

In conclusion, we have derived an efficient and robust adiabatic passage technique based on the tracking of a desired solution rather than imposing the parameters. The proposed method is versatile and can be applied for other types of nonlinearities, such as the ones described in [13] or in nonlinear optics [6]. One can also treat more complicated problems, such as  systems with stimulated Raman processes. For instance, the presence of third-order nonlinearities [see Eqs. (1) and (2)] can be treated explicitly: The classical Hamiltonian has to be changed accordingly, which results in a new separatrix and fixed points, imposing another form for the instantaneous detuning. The crossing of the separatrix by the fixed points in this condition is under investigation. The resulting nonmonotonic chirps can be nowadays well implemented in any time scale regime, even in the picosecond and subpicosecond regimes where the pulse has to be shaped in the spectral domain [18].

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The robustness of the derived technique, with respect to the pulse area (as traditional adiabatic techniques for linear problems) and with respect to a static detuning (see Fig. 1), is anticipated to be a decisive property for practical implementations. Robustness for a particular implementation could necessitate optimizing the tracking specifically, as recently proposed for the linear case [19,20]. The achievement of an ultrahigh fidelity with an optimal exponential efficiency (see, for instance, [21]) is an open question.

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ACKNOWLEDGMENTS

This research has been conducted in the scope of the International Associated Laboratory (CNRS-France and SCSArmenia) IRMAS. We acknowledge additional support from the European Union Seventh Framework Programme through the International Cooperation ERA WIDE No. GA-INCO295025-IPERA. M.G. thanks the Cooperation and Cultural Action Department (SCAC) of the French Embassy in Armenia for a grant. A.I. acknowledges the support from the Armenian State Committee of Science (SCS Grant No. 11RB-026).

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