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There is an elegant and beautiful way of deriving the form of the coupling between matter and radiation: to use the basic principle of gauge invariance. This principle says that the physics cannot depend on our choice of gauge for the electromagnetic potentials; that is, when we make the gauge change A'(r, t) = A(r, t)
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cj/(r,t) = cjJ(r,t) -
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where X is any given function of rand t, the observable quantities should not change. To get any further, we must make this physical requirement more precise by translating it into equations. Let us denote by unprimed quantities the quantum-mechanical description in the old gauge and, by primed quantities, the description in the new gauge. If the two descriptions are equivalent, they must be related by a unitary transformation 1'. This way, both descriptions will yield the same values for any observable quantity. Suppose that 6 is an observable; then its expectation value in the new gauge is given by (5.123) as 1't1' = 1. The unitarity of l' also allows us to write l' as
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where 6 is a Hermitian operator. Topnd out how the Hamiltonian in the new gauge iI' is related to the Hamiltonian in the old gauge iI, we write down the Schrodinger equation in the new gauge and use the fact that 11/1') = 1'11/1). Then, as
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i1i~ (1'11/1)) = i1iT~11/I) 8t
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(86) 1'11/1)
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mUltiplying the Schrooinger equation in the new gauge by 1't from the left, we find that
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i1i~11/I) = 1't (iI' + 8t 8t
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Comparing (5.126) with the Schrooinger equation in the old gauge, (5.127)
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we see that the Hamiltonian in the new gauge, gauge,H,by
H', must be related to that in the old
H' = fHft -
So the Hamiltonian in the new gauge will only be an observable when explicit function of time. Now consider the typical Hamiltonian for an atom or molecule:
ais not an
HA = L
+ LqacP(ra,t).
For instance, the Hamiltonian for the hydrogen atom that is studied in elementary quantum mechanics has this form. What happens when we make the general gauge change (5.122) Substituting cP by cP' given by (5.122) in the Hamiltonian (5.129), we find the following expression for the atomic Hamiltonian in the new gauge: (5.130) Comparing (5.128) and (5.130), we see that we must take (5.131) Unfortunately, this does not work, as the resulting t yields (5.132) which can be rewritten as
,, HA
'" qa {) 1, (\7 aa)2 THATt - L --{) x(ra,t)+ L -(\7 a G)'Pa- L (5.133) act a ma a 2ma
Looking at (5.130), we see that the last two terms in (5.133) should not have been present. The presence of these two terms shows that HA is not gauge invariant. But this is not very surprising, as HA does not include the radiation field. Any system of charged particles subject to acceleration would also involve electromagnetic radiation. Let us then complement HA adding the Hamiltonian of the radiation field, HF, which was derived in 2~ and a,yet u~known c:.ouplin~ ten!l ih. The complete Hamiltonian'is given by H = HA + HI + HF. As H'p = HF, HI must be such that when transformed, it generates terms that cancel the last two terms on the right-hand side of (5.133). In other words, writing HI as a functional of the electromagnetic potentials, HI [A, cPl, it must be such that (5.134)
As you can easily verify, the minimal-coupling interaction derived in Section 5.2 is the solution of this equation. Without the minimal-coupling interaction, the Hamiltonian is not gauge invariant. Historically, the form of the coupling between matter and radiation was known long before the discovery of the symmetry under gauge transformations of the quantummechanical system of charged particles interacting with the electromagnetic field [315]. This symmetry was discovered by Fock in 1926 [215]. But the idea of gauge invariance as a basic unifying principle predates quantum mechanics and can be traced back [413,414] to Weyl's [636] failed attempt to unify electromagnetism and gravitation in 1919 [315]. Much later, in 1928, Weyl wrote [637]: "Butl now believe that this gauge invariance does not tie together electricity and gravitation, but rather electricity and matter in the manner described above." The quantum nature of the electromagnetic field gives rise to many physical phenomena, some of which we mentioned earlier. In 6 we use the formalism we have presented here to study spontaneous emission. First, we examine spontaneous emission in free space and show how it is triggered by the zero-point fluctuations in the field. Then we describe how a cavity can alter this zero-point field and the consequences it has for spontaneous emission.
RECOMMENDED READING For a discussion of minimal coupling as a consequence of gauge invariance using a field description for matter as well as forthe field, see Sec. 3.5 of [116]. For a gauge-invariant formulation of perturbation theory in QED, see [366]. The minimal coupling can also be seen as a consequence of Lorentz invariance [406]. For a nonrelativistic particle, a rephrasing of the principle of restricted Galilean invariance also leads to the minimal coupling Hamiltonian [320, 321]. For a fully relativistic derivation of the interaction between electric and magnetic dipoles with an external electromagnetic field, see [20]. For a discussion of the Aharonov-Bohm phase in this context, see also [397,572]. See the book by Allen and Eberly [13] for a detailed discussion of the experimental requirements for the validity of the two-level atom approximation. Richard P. Feynman once derived the Lorentz force and two of Maxwell equations from the assumption that a nonrelativistic point particle's position and momentum satisfy the well-known quantum-mechanical commutation relation and that the particle obeys Newton's second law of motion [176]. This crazy derivation can be reinterpreted as a proof that the most general velocity-dependent force that can be described in a Lagrangian or Hamiltonian formalism is one that has the same general form of the Lorentz force [302]. This is related to the question of whether knowing the equations of motion of a physical sys-
tern is enough to allow us to construct a Lagrangian or Hamiltonian formalism describing that system. See [301,470, 646}.
Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.
Spontaneous emission: From irreversible decay to Rabi oscillations!
Humpty Dumpty sat on a wall, Humpty Dumpty had a great fall . All the Icing's horses, And all the Icing's men, Couldn't put Humpty together again! -Old nursery rhyme2
This chapter is divided into two sections. In the first section we use the formalism developed in 5 to discuss spontaneous emission in free space. We address the problem of atomic stability when the coupling with the dynamic electromagnetic field is included and the atom is allowed to radiate. We examine the roles of vacuum fluctuations and radiation reaction in the spontaneous emission process and the stability of the ground state of the atom [3, 129, 131,449,450,453,456,557]. Then
(This chapter is based on [165). 2The popular image of Humpty Dumpty today is that of the egglike creature invented by Lewis Carroll. In fifteenth-century England, however, "Humpty Dumpty" was a colloquial term describing an obese person. The "Humpty Dumpty" in this rhyme might have referred to a big fat cannon used by the Royalists during the English Civil War (1642-1649). It was mounted on top of the SI. Mary's at the Wall Church in Colchester defending the city against siege in the summer of 1648. A shot from a Parliamentary cannon damaged the wall beneath and "Humpty Dumpty had a great fall." The Royalists, "all the King's men," attempted to raise Humpty Dumpty onto another part of the wall. However, the cannon was so heavy that "All the King's horses and all the King's men couldn't put Humpty together again'"