CC BY
28
УДК 53.01
Вестник СПбГУ. Сер. 4. 2013. Вып. 4
D. K. Efimov, M. Bruvelis, N. N. Bezuglov, K. Miculis, A. Ekers
FORMATION AND CONTROL OF DARK STATES IN HYPERFINE LEVELS OF Na ATOMS*
Study of dark state formation in different quantum systems, such as optically excited atoms and molecules [1], artificial atoms in superconducting circuits [2] and excitons [3], is one of developing parts of modern quantum optics. Dark (decoupled) [1] states do not interact with other states in multilevel quantum system thus preserving its population for the time of the dark state's existence. This can be extremely useful for almost a complete population transfer between energy levels during Stimulated Raman Adiabatic Passage (STIRAP) [4], Electromagnetically Induced Transparency [5] and applications of the latter.
Creation of multiple dark states is possible in a system with partially degenerate discrete energy spectrum. A number of examples were studied by Morris and Shore for laser-coupled two-level systems manifesting degenerate splitting of one or both states [6]. If state A is splitted into Na sublevels and state B is splitted into NB sublevels then the system can be reduced to a set of N< = min{NA,NB} two-level systems (pairs of bright states) and \Na — NB \ decoupled (dark) states.
We have shown that formation of dark states is also possible in systems with nondegen-erate splitting. In the system of 3si/2 — 3pk/2 (k = {1, 3}) Na levels coupled by a laser field we take into account its Hyperfine splitting. In a process of laser Rabi frequency enlarging we observe formation of "grey" states described by diminishing coupling with other levels. At large values of Rabi frequency 1000 MHz) these "grey" states evolute into the dark states. It turns out that one can control this evolution via varying a shape and an amplitude of the laser pulse.
Here we report on a study of the dark state evolution in a three-level ladder scheme. The system under consideration consists of two lower states coupled by a strong pump field EP. The third level (usually, 4d3/2) is coupled with the system by a weak probe laser field ES with variable detuning. The excitation spectrum, or the dependence of a fluorescence intensity registered from this level on S-laser detuning is the main experimental information. To clarify what occurs under such situation we performed simulations of the ladder level scheme for the typical Doppler-free arrangement of an experiment in atomic/molecular spectroscopy [7, 8] when a collimated supersonic beam of particles with z-axis crosses two counterpropagating laser beams. The beams are focused using a cylindrical lens such that the long axis of the focus is perpendicular to the molecular beam axis. The chosen here typical in experiments
(P) (S)
waists 2ш0 of the P (pump) (ш0 ' = 41 ^m) and S (probe) laser (ш0 ' = 140 ^m) beams
Dmitry Kirillovich Efimov — student, Saint Petersburg State University; e-mail: dmitry.eflmov@de29866.spb.edu
Martins Bruvelis — PhD student, University of Latvia; e-mail: martins.bruvelis@gmail.com
Nikolai Nikolaevich Bezuglov — Professor, Saint Petersburg State University; e-mail: bezuglov@nb16672.spb.edu
Kaspars Miculis — PhD, University of Latvia; e-mail: michulis@latnet.lv
Aigars Ekers — PhD, University of Latvia; e-mail: aigars.ekers@lu.lv
* По материалам международного семинара «Collisional processes in plasmas and gas laser media», 22—24 апреля 2013 г., физический факультет СПбГУ.
Семинар был проведён при софинансировании фондом «Династия».
This work is supported by European programs EU FP7 Centre of Excellence FOTONIKA-LV-FP7-REGPOT-CT-2011-285912 and EU FP7 IRSES Project COLIMA.
© D. K. Efimov, M. Bruvelis, N.N. Bezuglov, K. Miculis, A. Ekers, 2013
allows safely ignoring the effects of lasers wave-front curvatures [8]. Along z-axis the lasers spatial intensity profiles are Gaussian with characteristic radii Rj = CDgJ-)/v/2:
Ij (z) = exp(-z2/R2); j = P,S. (1)
The mean flow velocity vf of atoms was measured to be 1160 m/s [9], so that the transit times Tj = 2u>q')/vf [8] of atoms through the laser beams are tP = 35 ns and ts = 121 ns for P- and S-lasers respectively. The intensity distributions (1) result in the time varying reduced Rabi frequencies
in the moving frame of an atom (t = z/vf); definition for Rabi frequencies is
Hp = Ep|(3s || D || 3p)|; ils = Es\(4d || D || 3p)\, (2)
where Ej is an amplitude of the laser field j, and D signifies an operator of the electric dipole transition. Since the both transit times exceed the natural lifetime tnat = 16.23 ns for the 3pi/2 state, one can treat the dynamics of the excitation process adiabatically from the concept of dressed-states. Each dressed state is an eigenfunction of the Hamiltonian H written in Rotating Wave Approximation (RWA):
tt % i ( o Am
tlM = A H--n— t
2 \Am 0
The first diagonal matrix consists of the bare states energies: A\n\n = AP (AP is the pump laser detuning); Ayy = -190 MHz; A22/ = 0. The second matrix is determined by the Rabi frequencies (2) within the HF components F" = 1" for 3sy/2; F' = 1', F' = 2' for 3pi/2 and with the fixed Zeeman quantum number M. The corresponding vector lines A ={Ai»i/, Ai"2/} turnout to be A M=o = {0, 0.408} and AM=i = {-0.204,0.354}. The dark state is a dressed state decoupled from the laser interaction with slightly varied eigenenergy eD. Other dressed states with eigenenergies e± strongly dependent on are usually called "bright".
Probing the space varying dressed states energies e±, eD at the second step of the ladder scheme by the weak S-probe laser (3p1/2(F',M) ^ 4d3/2(F, M)) has interesting consequences. The probe signal If from the final 4d3/2 state is proportional to its integral population nf over four degenerate HF sublevels F = 0,1, 2, 3. The energy of the final state f depends on the probe laser detuning As and, because of small ^S^ values 4d3/2 states could not be noticeably perturbed by the coupling. The adiabatic states \f), consequently, are identical to 4d3/2 HF bare components with non-varying energies £f.
Due to an unfamiliar behaviour of e± and eD, the bright and dark states contribute into the formation of signal If as a function of the probe As detuning in quite a different way. Since the laser beam intensity has Gaussian profile, the space profiles of bright states energies e±(z) are bell-like. At the same time the dark state has a horizontal profile eD (z) because it's energy does not depend on the strong light intensity. Such a configuration of energy profiles allows one to translate the entire population dynamics into the excitation spectra [7, 9], thus revealing the evolution of initially visible bare states into the dark states. The typical spectrogram is shown in Figure; central intensive peak corresponds to dark
1.0-,
0.8-
0.6-
0.4-
«
0.0
0.2-
300 -200 -100 0 100 200 300 400 500 A„ MHz
Dependence of fluorescence signal If from 4d3/2 level of Na on detuning of probe laser As: pump laser detuning Ap = —51 MHz; the Rabi frequencies are Qp = 1200 MHz, Qs = 7 MHz; the mean flow velocity Vf = 1160 m/s; solid line corresponds to numerical calculations, black circle line shows the experimental data
state, it appears as a result of long duration of interaction between parallel 4d3/2 level and £D in the central area of laser beam. Under the increase of pump laser intensity, position of the peak is weakly affected, that is also consequence of weak dark state dependence on Qp. In contrast, bright states energies £±(0) increasingly shift from bare state energies (at zero coupling) with enlarging of the laser power.
The appearance of wide wings in If (AS) with an interference structure was shown in [7]. The physical motivation of such phenomenon is based on the dynamics of the excitation. The population resides initially in ground (3si/2) state which is associated with the adiabatic state \ —). If the evolution is adiabatic, the state vector remains in the bright state \ —). The latter becomes a superposition of states 3s1/2 and 3p1/2 due to the increasing coupling of the atomic states by the P laser and can be excited to state \f) by the S laser. The excitation of \—) is located mainly at the vicinities of two Landau—Zener points z1j2 where the curves £-(z) and £f (z) cross. Each point z1j2 contributes into probability amplitude Cf with an individual phase, so that the total amplitude Cf is a subject of the constructive or destructive interference between both contributions depending on the details of evolution, i. e. on the parameters Hp, Ap, AS.
To illustrate the above predictions, we calculate the excitation spectrum If (AS) using numerical solutions of Optical Bloch Equations for the density matrix p (see Figure) in the system under consideration. The spectrum peculiarities explanation is done via examining two independent models. The first one gives an approximate law for the evolution of dressed states population p = A(t)p, p = {pn, p22, p33}; A having been determined by parameters of the laser/atom beams and atomic constants. The second model provides calculations of transition probabilities between the dressed states and \f) and hence gives If (AS) provided the evolution law p(t) is known. Using this two models one can relatively easily predict the dependence of the both spectrum peculiarities and the dressed states population evolution on the experiment initial parameters.
References
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Статья поступила в редакцию 22 апреля 2013 г.
