Slow light, superadiabaticity and shape-preserving pulses

Slow light, superadiabaticity and shape-preserving pulses U. Leonhardt and L. C. D´avila Romero School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland We analyze the nonlinear dynamics of atomic dark states in Λ configuration that interact with light at exact resonance. We found a generalization of shape-preserving pulses [R. Grobe, F. T. Hioe, and J. H. Eberly, Phys. Rev. Lett. 73, 3183 (1994)] and show that the condition for adiabaticity of the atomic dynamics is never violated, as long as spontaneous emission is negligible.

arXiv:quant-ph/0506147v2 2 Sep 2005

PACS numbers: 42.50.Gy, 42.50.Md

Slow light in atomic media [1] relies on the adiabatic following of atomic dark states [2]. The atoms are brought into such states by optical pumping [3], a dissipative process based on spontaneous emission. Once they are in dark states, the atoms follow any changes in the applied light fields with remarkable ease [4, 5] and without further assistance by spontaneous emission, much beyond the expectations of adiabatic theory [6]. In this note, we develop a brief argument as to why this is the case. We consider the nonlinear dynamics of light and atoms in Λ configuration with two co-propagating light beams at exact resonance, see the figure. This case corresponds to slow and stopped light in Bose-Einstein condensates [4]. Our analysis extends the nonadiabatic approach by Matsko et al. [7] to a regime where the applied optical fields can be of almost arbitrary strength, whereas in their paper [7] one field dominates over the other. As an important ingredient, we construct a rather general approximate solution of the Maxwell-Schr¨odinger equations. Our solution generalizes the shape-preserving pulses, or adiabatons, previously studied by Grobe, Hioe, and Eberly [8] and Fleischhauer and Manka [9], to complex Rabi frequencies and complex time. The solution also shows that in the case of zero detuning not only slowlight solitons [10, 11, 12] represent stable structures that can be slowed down, stopped and accelerated depending on the total intensity. In this case, almost arbitrary optical polarization profiles can be stored and retrieved in atomic media, as long as the atoms are not significantly excited. Using the continuation of our solution to complex time, we show how the non-linear dynamics of atoms and light protects the atoms from violating the adiabaticity condition [13]. One may call such behavior superadiabaticity. Consider atoms in the Λ configuration illustrated in the figure. We denote the lower atomic levels by |±i and the excited state by |ei, and assume that the atoms are in the pure states |ψi = ψ+ |+i + ψ− |−i + ψe |ei. The probability amplitudes may differ for atoms located at different positions and they are time-dependent. Suppose that the atoms are illuminated by two light beams co-propagating in z direction with carrier frequencies that exactly match the atomic transitions from the |±i to the |ei level. We

e W-

W-

W+

W+

+ FIG. 1: Two co-propagating light beams interact with a medium composed of three-level atoms in Λ configuration at exact resonance. The electric field strengths of the light beams are characterized by the Rabi frequencies [14] Ω± . The atomic levels are denoted by |±i and |ei.

describe the field strengths of the two polarizations by the complex Rabi frequencies [14] Ω± . We assume that both the light and the atomic medium is uniform in x and y direction. We introduce the retarded time τ = t − z/c and the scaled z coordinate ζ = z/c that has the dimension of time. In such coordinates, the light wave obeys the Maxwell equations in the slowly varying envelope approximation [14] ∂ζ Ω± = ig ψ¯± ψe ,

¯ ± = −ig ψ± ψ¯e , ∂ζ Ω

(1)

where g(ζ) characterizes the atom-light interaction, assumed to be equal for the |±i states, that is proportional to the the atom-number density of the medium. At this stage we neglect any atomic relaxation, but we determine later the condition when this is justified. Without relaxation the atoms obey the Schr¨odinger equation i∂τ |ψi = H |ψi with the Hamiltonian in the interaction picture  X Ω ¯± Ω± H=− |±ihe| + |eih±| , 2 2 ±

(2)

(3)

2 in the case of exact resonance of the atomic transition with the carrier frequencies of the light. In the adiabatic dressed-state picture [9, 13], the atoms are assumed to be in instantaneous eigenstates of the Hamiltonian (3), in particular, in the atomic dark state [2] that depends on the applied optical fields. When the fields vary the atoms follow by remaining in the instantaneous eigenstate while acquiring a phase that corresponds to the instantaneous eigenvalue integrated over time. However, when the atomic levels coincide the atomic state is no longer well-defined and adiabaticity is clearly violated, because the atoms may easily go from one state to the other. In practice, most level crossings are avoided in real time, but they may occur at certain points of complex time for the analytically continued atomic dynamics. In fact, these level crossings in the complex time plane determine the Landau-Zener tunneling due to nonadiabaticity [13]. The exponential of the energy integrated from a complex level crossing to the real time axis gives the tunneling probability amplitude [13]. Consequemtly, in order to assess the adiabaticity of the nonlinear dynamics of light and atoms with Hamiltonian (3) we consider the analytic continuation of the MaxwellSchr¨odinger equations (1) and (2) indicated by a subtlety ¯ ± and ψ¯± , ψ¯e denotes the of the notation: the bar in Ω analytic continuation of the complex conjugate Ω∗± and ∗ ψ± , ψe∗ from real to complex retarded time τ [15]. Note that in general the analytic continuation of the complex conjugate does not coincide with the complex conjugate of the analytic continuation [15], and therefore this distinction must be made. The instantaneous eigenvalues of the Hamiltonian (3) are q 1 ¯ ¯ − Ω− . Ω+ Ω+ + Ω (4) E0 = 0 , E± = ± 2

First, we generalize a previously known approximate solution of the atomic dynamics [17] such that is is valid for complex time,   2N02 Ω− Ω− |+i + ¯ ∂τ |ei , |ψi = N |−i − Ω+ Ω+ Ω+  Z    ¯− ¯− 1 Ω Ω− Ω Ω− 2 N = N0 exp dτ , ∂τ ¯ − ¯ ∂τ N0 2 Ω+ Ω+ Ω+ Ω+  ¯ − Ω− −1/2 Ω . (7) N0 = 1 + ¯ Ω+ Ω+ ¯ ± are exWe note that N0 is invariant when Ω± and Ω changed, whereas  Z ¯ ¯   ¯ = N0 exp 1 N02 Ω− ∂τ Ω− − Ω− ∂τ Ω− dτ . N ¯+ ¯+ 2 Ω+ Ω+ Ω Ω (8) The state vector corresponds to an evolving dark state obeying the orthogonality condition (5). The expressions (7) exactly solve the Schr¨odinger equation (2) for the ground-state probability amplitudes ψ± , but not for the excited-state amplitude ψe . This is justified, as long as the population of the upper level is small in comparison with unity, a situation where the lower states enslave the dynamics at the top level. Given the solution (7) of the atomic dynamics, we solve the Maxwell equations (1) for the light that turn into 2N 4 ¯ Ω− ∂ζ Ω+ = g ¯ 20 Ω , − ∂τ Ω+ Ω+

2N 4 Ω− ∂ζ Ω− = −g ¯ 0 ∂τ Ω+ Ω+ (9) ¯ ± . We see that and the corresponding equations for Ω ¯ + Ω+ + Ω ¯ − Ω− ) vanishes, which implies ∂ζ (Ω ¯ + Ω+ + Ω ¯ − Ω− = Ω2 (τ ) , Ω where Ω is real for real time. We try the ansatz

The dark state is the eigenstate with the zero eigenvalue, a state that is orthogonal to the applied fields, Ω+ ψ+ + Ω− ψ− = 0 .

(5)

Adiabaticity is violated in the vicinity of a level crossing in complex time [13], unless the corresponding eigenstates are degenerate. In our case, level crossings occur when ¯ + Ω+ + Ω ¯ − Ω− = 0 . Ω

Ω+ = Ω e+iφ+ cos Θ , Ω− = Ω e+iφ− sin Θ , ¯ − = Ω e−iφ− sin Θ , ¯ + = Ω e−iφ+ cos Θ , Ω Ω

(11)

which gives N0 = cos Θ .

(12)

One verifies that Θ and φ± solve the Maxwell equations (9) when

(6)

For real times, this implies that the fields vanish. In this case, the Hamiltonian (3) vanishes altogether and hence the eigenstates are degenerate. No state mixing occurs. On the other hand, the analytically continued Ω± and ¯ ± may satisfy the crossing condition for non-zero Ω± Ω ¯ ± , and in general they will [16]. However, we show and Ω that the non-linear dynamics of light and atoms prevents this from happening.

(10)

Θ = Θ(ξ) ,

φ+ − φ− = φ(ξ)

(13)

Z

cos(2Θ) dφ

(14)

Z

(15)

and φ+ + φ− = with 1 ξ=ζ− 2g

Ω2 dτ .

3 Our solution generalizes a previous result by Grobe, Hioe, and Eberly [8] to complex Rabi frequencies and complex time, a prerequisite in assessing the adiabaticity of the ¯ + Ω+ + system. Equations (10) and (11) imply that Ω ¯ Ω− Ω− vanishes at the zeros of Ω± where the Hamiltonian (3) vanishes. Consequently, the levels are degenerate here and hence no adiabatic crossing occurs. Our solution describes general shape-preserving pulses [8]. The total envelope Ω(τ ) of the two matched light waves propagates through the medium with the speed of light in vacuum, whereas the polarization structure propagates with the characteristic speed of slow light [1] v=c

Ω2 g

for g ≫ Ω2 .

We thank J. H. Eberly for the correspondence that inspired this paper and we acknowledge the support of the Leverhulme Trust, COVAQIAL, and the Engineering and Physical Sciences Research Council.

(16)

The total envelope plays the role of the control beam that sets the speed of the polarization pulse. When Ω vanishes the structure is brought to a halt and the polarization profile is stored in the atomic medium, as long as it is not eroded by loss mechanisms. Illuminating the medium sets the structure in motion again. Therefore, in the case of zero detuning, arbitrary polarization profiles can be stored and retrieved, not only slow-light solitons [10], provided the profile does not significantly excite the atoms (7). To estimate the validity of our result, we calculate |ψe |2 for real retarded times. We obtain from Eqs. (7) and (11) and the shape-invariant solution (13) and (15) |ψe |2 =

the nonlinear dynamics of atoms and light protects the atoms from violating the adiabaticity condition, as long as spontaneous emission is negligible. Slow light is superadiabatic.

 Ω2 c2 Ω2  2 2 2 (2∂ Θ) + (∂ φ) sin (2Θ) ≈ 2 2 . (17) ξ ξ 4g 2 g a

Here a denotes the characteristic length scale over which the polarization profile varies. Clearly, |ψe |2 ≪ 1 when Ω2 c2 ≪ g 2 a2 , which limits the gradient of the polarization profile. If this condition is not satisfied our analysis is not applicable and the adiabaticity of the atomic dynamics may be violated. In practice, however, spontaneous emission from the excited state usually poses a much more severe limitation. Spontaneous emission amounts to losses and possibly the destruction of the Bose-Einstein condensate in which the slow-light polarization structure is contained. To estimate the propagation length l over which spontaneous emission is negligible, we follow the procedure of Ref. [10] and obtain the loss rate 32π lλ (18) ηL = nλ3 a2 where λ denotes the optical wavelength. For BoseEinstein condensates [4] nλ3 has been in the order of 1 or larger. Losses are thus negligible for pulses that are significantly longer than the geometric mean of wavelength and distance travelled. Summary. We derived analytic solutions for a general class of shape-preserving pulses interacting with threelevel atoms in Λ configuration. Our theory indicates that

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