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arXiv:1801.08393

Three level Lambda system, Feynman amplitudes and relativistic invariance

Navin Khaneja

Published Jan 25, 2018• 0 citations
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Abstract

In this paper, we develop an analogy between the three level atomic system so called Lambda system and scattering processes in quantum electrodynamics (QED). In a Lambda system we have two ground state levels $|1 \rangle$ and $|3 \rangle$ at energy $E_1$ and excited level $|2 \rangle$ at energy $E_2$. The transition from $|1 \rangle$ to $|2 \rangle$ has strength $\Omega_1$ and transition from $|2 \rangle$ to $|3 \rangle$ has strength $\Omega_2$. When we adiabatically eliminate the excited state, i.e. go in the interaction frame of natural Hamiltonian of the system, we get a second order term connecting level $|1 \rangle$ to $|3 \rangle$ with strength $\frac{\Omega_1\Omega_2}{2(E_1-E_2)}$. This term creates an effective coupling between ground state levels and drives transition from $|1 \rangle$ to $| 3\rangle$. Scattering processes in QED can be modelled like this. Feynman amplitudes are calculation of second order term ${\cal M} = \frac{\Omega_1\Omega_2}{2(E_1-E_2)}$. In the rest frame $O'$ ({\bf CM} frame) where sum of momentum of incoming particles is $0$, if the Feynman amplitude is ${\cal M'}$, then in a frame $O$ in which $O'$ moves with velocity $v$ , the time gets dilated to $t = \frac{t'}{\sqrt{1-\frac{v^2}{c^2}}}$ and hence it must be true that ${\cal M} = {\cal M'} \sqrt{1-\frac{v^2}{c^2}}$. We show that to make this relativistic invariance of $\mathcal{M}$ to work, we have to modify the QED interaction Hamiltonian. Using this modification, we calculate the scattering amplitude for QED processes, Compton scattering and M{\o}ller Scattering.

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Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. 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In this paper, we develop an analogy between the three level atomic system so called Lambda system and scattering processes in quantum electrodynamics (QED). In a Lambda system we have two ground state levels ∣1⟩|1 \rangle∣1⟩ and ∣3⟩|3 \rangle∣3⟩ at energy E1E_1E1​ and excited level ∣2⟩|2 \rangle∣2⟩ at energy E2E_2E2​. The transition from ∣1⟩|1 \rangle∣1⟩ to
∣2⟩|2 \rangle∣2⟩ has strength Ω1\Omega_1Ω1​ and transition from ∣2⟩|2 \rangle∣2⟩ to
∣3⟩|3 \rangle∣3⟩ has strength Ω2\Omega_2Ω2​. When we adiabatically eliminate the excited state, i.e. go in the interaction frame of natural Hamiltonian of the system, we get a second order term connecting level ∣1⟩|1 \rangle∣1⟩ to ∣3⟩|3 \rangle∣3⟩ with strength Ω1Ω22(E1−E2)\frac{\Omega_1\Omega_2}{2(E_1-E_2)}2(E1​−E2​)Ω1​Ω2​​. This term creates an effective coupling between ground state levels and drives transition from ∣1⟩|1 \rangle∣1⟩ to ∣3⟩| 3\rangle∣3⟩. Scattering processes in QED can be modelled like this. Feynman amplitudes are calculation of second order term M=Ω1Ω22(E1−E2){\cal M} = \frac{\Omega_1\Omega_2}{2(E_1-E_2)}M=2(E1​−E2​)Ω1​Ω2​​. In the rest frame O′O'O′ ({\bf CM} frame) where sum of momentum of incoming particles is 000, if the Feynman amplitude is M′{\cal M'}M′, then in a frame OOO in which O′O'O′ moves with velocity vvv , the time gets dilated to t=t′1−v2c2t = \frac{t'}{\sqrt{1-\frac{v^2}{c^2}}}t=1−c2v2​​t′​ and hence it must be true that M=M′1−v2c2{\cal M} = {\cal M'} \sqrt{1-\frac{v^2}{c^2}}M=M′1−c2v2​​. We show that to make this relativistic invariance of M\mathcal{M}M to work, we have to modify the QED interaction Hamiltonian. Using this modification, we calculate the scattering amplitude for QED processes, Compton scattering and M{\o}ller Scattering.
A systematic description of a spin one-half system endowed with magnetic moment or any other two-level system (qubit) interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED (albeit in a greatly simplified form). The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two level system leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study two-level systems (qubits) can be extended to many-level systems (qudits). As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one SSS state and three degenerate PPP states.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). 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Abstract This book is unique in covering phenomena in photon- matter interactions in a unified way over a range of many orders in energy. The quantum field theoretic approach to the fully relativistic theory of quantum electrodynamics (QED) is presented together with the non- relativistic theory in both confined and unconfined geometries. The predictions of QED have been verified to a greater accuracy than any other physical theory. Moreover QED is a paradigm for other gauge theories and is presented in such a way that the generalisation to other gauge theories is natural. Gauge and Poincare symmetry properties and the non-existence of a photon wave function are thoroughly discussed. Starting from the Dirac equation the non-relativistic interaction of the electron with the electromagnetic field is derived as an effective Hamiltonian of multipole expansions. Much of quantum optics is based on the lowest order dipole approximation. From this point on the treatment of fully relativistic QED and quantum optics is done in parallel. Applications of perturbation theory such as Compton and Moller scattering and the theory photdetection are given. After the impressive successes of QED, the limitation of the theory and the necessity of electroweak theory and quantumchromodynamics are discussed. The remaining chapters are devoted to quantum optics inside cavaties. Various approaches to open systems such as master equations are discussed within the context of active systems (e.g. the laser) and passive systems. Semi- classical approximarions are shown to imply a rich non- linear dynamics including chaos for certain parameter regimes. The effect of fluctuations on such non-linear dynamics is also studied. The final chapter is devoted to highly non- classical states of the light field such as photon number, squeezed and two photon entangled states. The latter are studied for the important system of parametric down conversion and the localisation properties of photons are characterised in terms of asympotic tails in photodetection probabilities as a function of time delay. The range of the book has wider benefits. Workers in quantum optics will gain a deeper understanding of the foundations of their subject and field theorists will see concrete examples of open systems, which are beginning to impinge on fundamental theories.
The minimal-coupling Hamiltonian for a non-relativistic bound system of two charges in the dipole approximation is gauge transformed to a Hamiltonian of a variable gauge which mixes the usual p · A and q · E interactions, and re-scales the charges. Various photon transition processes—single photon absorption and emission; two-photon scattering and absorption; the Lamb shift—are investigated by perturbation theory and found to be gauge independent, providing the transitions occur on the energy shell. It is demonstrated how a particular choice of gauge removes the counter-rotating terms and diagonalises the Hamiltonian of a coupled oscillator and two-level atom.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′M'M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2M= \sqrt{P^2}M=P2​ M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0P^0P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′H'H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/MU= P / M U=P/M U = P / M but not with P, else there is no scattering. Even though H′H'H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a KlongK_{\text {long}}Klong​ K long are seen by an inversely shifted observer as a shifted KlongK_{\text {long}}Klong​ K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted KshortK_{\text {short}}Kshort​ K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.