Optimal coherence via adiabatic following Svetlana A. Malinovskaya Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, United States article info Article history: Received 2 January 2009 Received in revised form 25 May 2009 Accepted 27 May 2009 abstract We propose a new method for creating an optimal coherence in a two-level system (TLS) based on adi- abatic following. Optimal coherence is achieved by choosing the time-dependent phase of the pulse that provides vanishing field-TLS detuning dðtÞ at central time when the pulse amplitude reaches maximum. For example, dðtÞ¼ d 0 ð1  expððt  t 0 Þ=s 0 ÞÞ 4 for t 6 t 0 and d ¼ 0 for t > t 0 . The pulse envelope used in cal- culations has Gaussian form such that the Rabi frequency is X R ðtÞ¼ X R expððt  t 0 Þ 2 =s 2 0 ÞÞ, where s 0 is the pulse duration. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Creation of an optimal coherence in a TLS is an important prob- lem in quantum optics. Optimal value of coherence is called as such because it provides optimal properties of coherence-depen- dent physical quantities. Cutting edge research areas that make use of this fundamental concept are nonlinear optical spectros- copy, e.g., [1], and quantum information and quantum computa- tion, e.g., [2]. In a vast majority of cases an optimal coherence corresponds to a maximum value of coherence, which is known to be 1/2 [3] owing to equal population distribution between two states of interest. For example, a maximum coherence between the ground and excited vibrational states in a molecule optimizes the magnitude of the Raman fields generated upon propagation of an incident light through a molecular medium. Well established solutions in strong fields are based on Rabi oscillations or adiabatic passage, see, e.g., [4]. Recent research into this problem has brought several new realizations of these approaches. Maximal coherence has resulted from crossing between the dynamic Stark sublevels in a three-level atom interacting with a strong field, as studied in [5]. With the fractional STIRAP, a maximum coherence was created between the initial and final states according to [6]. The fractional STIRAP was implemented, e.g., to get entangled mul- tiple Fock states from a single photon [7]. A maximally coherent dark state was formed in the electromagnetically induced trans- parency (EIT) scheme [8]. A coherent state superposition was cre- ated using two time-delayed pulses, one shaped to reproduce the field-frequency detuning, vanishing at early and late times [9]. Cre- ation of a maximum coherence using Rabi oscillations and adia- batic passage methods with femtosecond chirped pulses in the Raman configuration was described in [10]. In this paper we pro- pose a robust way to induce a maximum coherence in an atomic or molecular system via a one photon transition by making use of a single ultrafast pulse possessing a specifically shaped phase. Solution of this problem opens the door for variety of applications including the signal enhancement in atomic and molecular spec- troscopy and microscopy and quantum state information processing. 2. Theoretical analysis We propose a robust way to generate a maximum value of coherence in a two-level system (TLS), having transition frequency x, via adiabatic following achieved by a pulse having a time- dependent phase /ðtÞ. The time-dependent phase causes a varia- tion in the field-TLS detuning dðtÞ with time, for example, in such a way that dðtÞ¼ d 0 ð1  expððt  t 0 Þ=s 0 ÞÞ 4 for times before the cen- tral time t 0 , and d ¼ 0 after that. The central time t 0 determines time when pulse amplitude reaches maximum, and s 0 is the pulse duration such that 1=s 0 ¼ x. This choice of the s 0 was made for simplicity of presentation. (Generally, it may be multiples of x 1 providing a picosecond to femtosecond region for the pulse dura- tion.) Under these conditions, the detuning dðtÞ smoothly vanishes when the Rabi frequency X R ðtÞ¼ X R expððt  t 0 Þ 2 =s 2 0 ÞÞ is at the peak value X R . The choice of s 0 –x would change the rate at which the time-dependent phase dðtÞ arrives at zero value, preserving the mechanism of creating a maximum coherence. The interaction Hamiltonian in the basis of the bare states j 1ðtÞi; j 2ðtÞi written in the field-interaction representation [11], reads as H int ¼ dðtÞ X R ðtÞ X R ðtÞ dðtÞ   : ð1Þ Here X R ðtÞ¼ X R expððt  t 0 Þ 2 =s 2 0 ÞÞ is the Rabi frequency with Gaussian form of the pulse envelope. This choice is most natural and convenient for the description of the pulse envelope. Alterna- tively, a hyperbolic secant function can be used without any restrictions and modifications to the proposed method. In the pres- 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.066 E-mail address: smalinov@stevens.edu Optics Communications 282 (2009) 3527–3529 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom