Shanxi University Atomic Physics Chapter 7 The interaction of atoms with radiation Atomic Physics

Shanxi University Atomic Physics 7.1 Setting up the equations 7.2 The Einstein B coefficients 7.3 Interaction with monochromatic radiation 7.4 Ramsey fringes 7.5 Radiative damping 7.6 The optical absorption cross-section 7.7 The a.c. effect or light shift 7.8 Comment on semiclassical theory

Shanxi University Atomic Physics 7.1 Setting up the equations We start from the time-dependent Schrōdinger equation (7.1) The Hamiltonian has two parts, (7.2)

Shanxi University Atomic Physics Time dependent QM of a two-level atom interacting with an EM-field |1> |2> Starting point - time dependent Schrodinger Equation These are stationary states and the spatial functions obey the TISE Solutions look like

Shanxi University Atomic Physics In what follows we assume  n (r) are known. Postulate a wavefunction that looks like

Shanxi University Atomic Physics Inserting this wavefunction into the full TDSE, including the perturbation gives Premultiplying by  i * (where i=1,2) and integrating gives Where we have defined I ij as

Shanxi University Atomic Physics The interaction of the field with the atom is given by the electric dipole Hamiltonian. With this Hamiltonian it can be shown that So that only I 12 is required. It is given by

Shanxi University Atomic Physics We then define the Rabi frequency as For allowed transitions  R is real. Thus, for this choice of Interaction Hamiltonian, the coupled equations take the following form

Shanxi University Atomic Physics Rotating wave approximation Recipe for solving these for the special case of  =  0. 1.Write the cos  t term in exponential form. This yields a pair of Esq. each containing a pair of terms in exp[i(  -  0 )t] and exp[i(  +  0 )t]. 2.Formally, solving the equations involves an integration. The exp[i(  +  0 )t] term in the integrand oscillates rapidly and gives a very small contribution to the integral. We omit these terms. This is the Rotating Wave Approximation. Solving for the case of    0 is more tedious but can be done.

Shanxi University Atomic Physics Rabi Oscillations With the initial conditions c 1 (0)=1 and c 2 (0)=0 the solutions are We identify |c 2 (t)| 2 as the population of the excited state. This is given by

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7.2 The Einstein B coefficient To relate this to Einstein’s treatment of the interaction with broadband radiation we consider what happens with radiation of energy density p(w) in the frequency interval w to w+dw.

Shanxi University Atomic Physics We make assumption of long interaction to find the steady-state excitation rate for broad-band radiation. The probability of transition from level 1 to 2 increases linearly with time corresponding to a transition rate of :

Shanxi University Atomic Physics B 12 in Einstein ’ s treatment of radiation Where |X 12 | 2 → |D 12 | 2 /3, and D 12 is the magnitude of the vector

Shanxi University Atomic Physics 7.3 Interaction with monochromatic radiation We shall now find a solution without assuming a weak field. The term with w+w 0 t oscillates very fast and therefore averages to zero over any reasonable interaction time– this is the rotating-wave approximation.

Shanxi University Atomic Physics Given the probability of being in the upper state as: At resonance so and, so

Shanxi University Atomic Physics The concepts of -pulses and -pulses A pulse of resonant radiation that has a duration of is called a More precisely, a sways states in a superposition: Interferometry experiments also use that have half the duration of a (for the same Rabi frequency ).

Shanxi University Atomic Physics The Bloch vector and Bloch sphere We find the electric dipole moment induced on a atom by radiation, and introduce a very powerful way of describing the behaviors of two-level systems by the Bloch vector. Assume that the electric field is along Here

Shanxi University Atomic Physics To calculate this dipole moment induced by the applied field we need to know the bilinear quantities and. These are some of the elements of the density matrix Off-diagonal elements of the density matrix are called coherences and they represent the response of the system at the driving frequency

Shanxi University Atomic Physics The diagonal elements and are the population. We define the new variables whereis the detuning of the radiation from the resonance In term of this coherence the dipole moment become: (7.34) (7.35) (7.37)

Shanxi University Atomic Physics To find expressions for, and hence and, we start by writing eqns for and in terms of as follows: We find that the time derivatives (7.42) (7.41)

Shanxi University Atomic Physics In term of u and v in eqns 7.37 these equations become: (7.44) We can write the population difference as (7.45) So that finally we get the following compact set of equations:, (7.46)

Shanxi University Atomic Physics These eqns7.46 can be written in vector notation as: (7.47) by take,and as the components of the Bloch vector and define the vector (7.49) hat has magnitude The constant is unity so that :

Shanxi University Atomic Physics The Bloch vector corresponds to the position vector of points on the surface of a sphere with unit radius: Bloch sphere

Shanxi University Atomic Physics 7.4 Ramsey fringes In this section, we shall now consider what happens when an atom interacts with two separate pulses of radiation, from time t=0 to and again from t=T to T ＋. Integration of eqn7.10 with the initial condition at t=0 yields: When there are two pulses the amplitudes in 2the excited state interfere giving (7.51) (7.52)

Shanxi University Atomic Physics Ramsey fringes

Shanxi University Atomic Physics 7.5 Radiative damping The damping of a classical dipole For a harmonic oscillator of natural frequency, the equation of motion (7.55) To solve this we look for a solution of the form (7.56) (7.57) The amplitudes U and V change in time as the amplitude of the force changes, but we assume that these changes occur slowly compared to the fast oscillation at : slowly-varying envelope approximation

Shanxi University Atomic Physics By setting, We find the form of the solution that is a good approximation when the amplitudes and the force change slowly compared to the damping time of the system from eqn7.57 for and we find that (7.57 ) (7.60) power as the force times the velocity (7.62) only the cosine term contributes to the cycle- averaged power

Shanxi University Atomic Physics When any changes in the driving force occur slowly eqn7.60 has the following quasi-steady-state solution: (7.65) This shows that the energy of the classical oscillator increases linearly with the strength of the driving force, whereas in a two-level system the energy has an upper limit when all the atoms have been excited to the upper level

Shanxi University Atomic Physics The optical Bloch equations A two-level atom has an energy proportional to the excited-state population In this analogy, between the quantum system and a classical oscillator corresponds to. The coherences and have a damping factor of optical Bloch equations eqns7.46 (7.67) These optical Bloch equations describe the excitation of a two-level atom by radiation close to resonance for a transition that decays by spontaneous emission

Shanxi University Atomic Physics we shall concentrate on the steady-state solution that is established at times which are long compared to the lifetime of the upper level ( ), namely These show that a strong driving field ( ) tends to equalise the populations. Equivalently, the upper has a steady-state population of (7.69)

Shanxi University Atomic Physics 7.6 The optical absorption cross-section The probability of absorption equals the fraction of intensity lost, so the attenuation of the beam is described by (7.70) Consider a beam of particles (in this case photons) passing through a medium with N atoms per unit volume is the absorption coefficient at the angular frequency Integration gives an exponential decrease of the intensity with distance, namely (7.71) of the incident photons

Shanxi University Atomic Physics Atoms in level 2 undergo stimulated emission and this process leads to a gain in intensity (amplification) that offsets some of the absorption. Equation7.70 must be modified to (7.72) In the steady state conservation of energy per unit volume of the absorber requires that (7.73) (7.75) (7.76) (7.77) line shape function

Shanxi University Atomic Physics a real atom with degenerate levels has a cross-section of (7.79) Cross-section for pure radiative broadening The peak absorption cross-section given by eqn7.76, when (7.80) The saturation intensity [ saturation intensity]

Shanxi University Atomic Physics we find that the absorption coefficient depends on intensity as follows: (7.84) The minimum value of occurs on resonance where the cross-section is largest (7.85) Power broadening The expression for the absorption coefficienthas a Lorentzian line shape Full Width at Half Maximum (FWHM)

Shanxi University Atomic Physics 7.7 The a.c. Stark effect or light shift The perturbing radiation also changes the energy of the levels and we calculate this light shift in this section Write eqns 7.41 for and in matrix form as = is The equation for the eigenvalues (7.89)

Shanxi University Atomic Physics Normally light shifts are most important at large frequency detuning where the effect of absorption is negligible; in this case and the eigenvalues are The states are shifted from their unperturbed eigenfrequencies by the light shift This system of atom plus photon is called a ‘dressed atom ’

Shanxi University Atomic Physics 考虑封闭的三能级 Λ 型系统， 系统哈密顿量为： 其中 Complementarity: Quantum coherence effect EIT and EIA １、 EIT 的理论

Shanxi University Atomic Physics 由光学 Bloch 方程： 得介质对探测场的极化率为 在满足双光子共振且 时 ，可以看出正比于 。 其中

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２、电磁感应吸收的理论 图 3 N- 型四能级原子结构. 其中 EIA 的原子密度方程如下式 系统哈密顿量为： b

Shanxi University Atomic Physics 获得探测场的密度矩阵元 需要强调的是：分支比率 (branching ratio) b(0≤b≤1) 是电子由激发态回到基态的几率， 它是系统开放性的一个量度。 b=1 形成循环 跃迁。

Shanxi University Atomic Physics (a)(b) (c) (d) 图 4 数值计算结果 参数如下 : (a) 、 (c)b=1 (b) 、 (d)b=0 A 2 =B 2 =0.5 Ω 1 =0.1Γ

Shanxi University Atomic Physics 图 8 激光器频率分别调谐于 Cs 原子 D 2 线 6S 1/2 (F=3)→6P 3/2 和 6S 1/2 (F=4)→6P 3/2 时的 EIT 和 EIA 光谱图

Shanxi University Atomic Physics group velocity

Shanxi University Atomic Physics 图 14 强耦合场获得的慢光速脉冲实验结果 激光共振于 EIA 产生的分裂光谱 在强耦合光作用下的慢光脉冲传输

Shanxi University Atomic Physics 图 20 Cs 原子蒸气中的光脉冲存储 (a)50μs(b)75μs

Shanxi University Atomic Physics For more information to see reference Nature, 397, (1999) Phys. Rev. Lett., 82, (1999) Nature, 413, (2001)

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