Physics 566: Quantum Optics Quantization of the Electromagnetic Field

Transcription

1 Physics 566: Quantum Optics Quantization of the Eectromagnetic Fied Maxwe's Equations and Gauge invariance In ecture we earned how to quantize a one dimensiona scaar fied corresponding to vibrations on an eastic rod. The procedure invoved decomposing the fied into its norma modes of osciation and then quantizing each mode as a simpe harmonic osciator. We now want to appy this procedure to the eectromagnetic fied. One might thin of this a quantization of the vibrationa modes of the "ether" in the pre-reativity description of eectrodynamics. The dynamics of this fied are determined by Maxwe's equations, which in the absence of charges (and in CGS units) read E = 0, E = - 1 c B t, B = 0, B = 1 c E t. In contrast to our discussion of the quantization of the scaar fied, this probem is compicated by two factors. First we are deaing with a three dimensiona fied, and second it is a vector fied. A vector fied in three dimensions shoud be describabe in terms of three scaar fieds. However, at this point we have described our fied in terms of six fieds {E x, E y, E z, B x,b y, B z }. We can get a description of the dynamics with fewer fieds by using the vector a scaar potentias, which transforms ie the four vector in Minowsi space, A m = (f, A). The physica fieds which exert forces on the charges are the eectric an magnetic fieds which are defined in terms of the four-potentia as E = - f - 1 A c t, B = A. The equation of motion for f and A foow from Maxwe's equations 2 f = - 1 c t A 2 A c t 2 A = - Ê A + 1 f ˆ Ë c t At this point the four components of A m describe our fied. This set is actuay over compete. One of the fundamenta properties of Maxwe's equation is gauge invariance. Page 1

2 That is, the physica fieds E and B are unchanged if we redefine our four potentia by maing the transformation A m Æ A m + m c, or in terms of the components f Æ f + 1 c c t, A Æ A + c where c is an arbitrary scaar fied. Maxwe's Equations in the Cooumb Gauge The choice of gauge is competey arbitrary. For quantization of the fied the convenient choice is the Couomb gauge (a name we wi understand soon) which is defined by choosing the vector potentia such that A = 0. This is nown as the "transversaity" condition. If the vector potentia we start with does not satisfy this condition it is aways possibe to mae a gauge transformation so that A is transverse by choosing a gauge function that satisfies 2 c = 0. The equation for the scaar potentia is the Lapace's equation, 2 f = 0. Thus, f is no onger a dynamica variabe. It is just depends on the geometry of the probem, and is constant in time. The equation of motion for A^ (where ^ indicates that A satisfies the transversaity condition) is the wave equation, 2 A^ - 1 c 2 t 2 A^ = 0. The ony remaining dynamica fied which we must quantize is A^. The dynamica physica fieds are E^ = - 1 A ^ and B^ = A^. c t The name Couomb gauge becomes cear when we incude the sources: charge density r, and current density J. In the Couomb gauge, the equations are Page 2

3 2 f = 4pr 2 A^ c t 2 A^ = - 4p c J^ The first equation is Poisson's equation, now defined for the instantaneous charge density (which may be time varying), whose soution is f (x, t) = Ú d 3 x r( x,t) x - x. This is just the instantaneous Couomb potentia, and hence the name of the gauge. Note there is no retarded time in this integra. That is, the potentia at time t and a point x is instantaneousy reated to the charge density at a distant point x' the same time. One might worry if the causa aws of specia reativity are vioated by this fact. Of course this is not the case for the origina Maxwe equation were Lorentz invariant, and thus satisfy a the causa aws we now and ove. Remember that the physica fieds are E and B. These aways satisfy causa equations of motion, no matter what gauge we choose. The Couomb gauge is a bit bizarre in that f seems to be reated to the instantaneous vaues of the source. However, this effect is canceed by the fact that the vector potentia is not driven by the physica current, but ony by the "transverse" part of J, which itsef has an instantaneous part. In fact the transversaity condition is itsef dependent on the choice of reference frame (i.e. it is not manifesty covariant). However, we again emphasize that no aws of reativity are vioated. Norma Modes in the Couomb Gauge In order to quantize the fied by the procedure we deveoped in ecture notes on "quantum fied theory", we must introduce a norma mode expansion for the fied. Our dynamica variabe is the transverse deta function A^, which satisfies the wave equation. To arrive at a discrete set of modes, we wi introduce an artificia "quantization voume" in the same way that we too a finite ength of rod in our study of the vibrationa modes for a scaar fied. We wi tae periodic boundary conditions for compete generaity to aow for propagating soutions. The norma modes are then pane waves with a given poarization u (x) = r e e i x, Page 3

4 where = n x 2p L e x + n y 2p L e y + n z 2p L e z, for periodic boundary conditions in a voume =L 3, for some integers {n x, n y,n z }. The vectors r e are the unit poarization vectors perpendicuar to wave vector, r e = 0. This insures the transversaity condition. The parameter (not to be confused with the wave ength) is an index with vaue 1 or 2 representing any two basis vectors in the pane perpendicuar to : The poarization vectors, r e, may be compex if they represent a genera eiptic poarization (for exampe right and eft circuary poarized fieds). The norma modes are orthogona Ú d 3 x u (x) u, (x) = d, d,. The competeness reation is a bit more compicated than for our scaar fied. Consider the (i) sum over a modes of the tensor u ( j) ( x ) u (x), (i) ( j) Â ( x ) u (x) = Â u e i ( x -x) (i ) e ( j) Â e. The poarization vectors r e span the pane perpendicuar to. Therefore (i) Âe e (j ) = dij - ˆ ˆ i j. That is, d ij - ˆ iˆ j is the tensor that projects any vector into the pane perpendicuar to. The competeness reation reads, Page 4

5  (i) u ( x ) u ( j) (x) = 1  d ij - ˆ ˆ i j e i ( x -x) = d^ij (x - x ). The tensor, d^ij (x - x ), is nown as the "transverse" deta function, defined by the property that it projects any vector fied (x) into the transverse component ^(x), Ú d 3 x i ( x ) d^ij (x - x ) = ^j (x), with zero divergence, ^ = 0. Since the vector potentia in the Couomb gauge is transverse, it can be expanded in the compete set of norma mode functions u, A^ (x, t) =  a (t)u (x) + a (t) u (x). w Here we have used the norma mode expansion for periodic boundary conditions, given on page 12 of ecture "Intro. to Quantum Fied Theory" in terms of the dimensioness compex ampitudes a (t) = a (0)e -iw t, where w = c. The normaization constant, for the norma modes we obtain, w, is the appropriate scae factor q 0,. Substituting in A^ (x, t) = w a (0) r e e i( x-w t) + a (0) r e e -i( x -w t ) Â. The transverse component of the eectric and magnetic fieds can be written using the definition at the bottom of page 2 E^(x, t) = i  B^ (x, t) = i  2phw a (0) r e e i( x -w t) - a (0) r e e -i( x- w t) a w (0) r e e i( x -w t ) - a (0) r e e -i( x- w t ) Page 5

6 The Quantized Eectromagnetic Fied Now that we've expanded the fied in terms of its norma modes, we can immediatey quantize it according the procedure in ecture. Thus, we associate the compex ampitudes with creation and annihiation operators a (0) Æ ˆ a, a (0) Æ ˆ a, are instate canonica commutation reations [ a ˆ, ˆ a, ] = d, d,. We have used the commutator, rather than the anti-commutator, impying that the quanta of the fied, otherwise nown as photons, are bosons. This foows from the fact that we have a vector fied with three components and thus must correspond to anguar momentum J=1 (2J+1=3 components). Athough the photon is a spin 1 partice, the transversaity condition requires that the poarization (which describes the anguar momentum of the ight) must rotate in the pane perpendicuar to the direction of propagation. Thus the spin vector can ony be parae, or anti-parae to the direction of propagation. Photons with the spin aong the direction of poarization are circuary poarized with their poarization rotating according to the right hand rue. These photons have positive heicity. the counter-rotating poarization describes negative heicity. J J Positive heicity Negative heicity Page 6

7 The cassica fieds now become vector operators, A ˆ ^ (x, t) = w a ˆ (0) r e e i( x -w t ) + a ˆ (0) r e e -i( x-w t) Â, E ˆ 2phw ^(x, t) = i ˆ a (0) r e e i( x- w t ) - a ˆ (0) r  e e -i( x -w t), ) B ^ (x, t) = i  w a ˆ (0) r e e i( x- w t ) - a ˆ (0) r e e -i( x-w t). The Hibert space describing the quantized fied is a Foc space. A state with n photons in the mode with wave vector and poarization is given by acting with the creation operator ˆ a n times on the vacuum state, and normaizing. n = ( ˆ a ) n n! 0. Any state wi be a superposition of such states for a modes. At times one wi see the fied operator decomposed as E ˆ ^(x, t) = E ˆ ^(+) (x, t) + E ˆ ^(-) (x, t) where E ˆ 2phw ^(+) (x, t) = i  a ˆ (0) r e e i( x-w t) is nown as the positive frequency component of the eectric fied, and E ˆ ^(-) (x, t) = ( E ˆ ^(+) (x, t) ) is nown as the negative frequency component. The positive frequency component contains ony annihiation operators, and is thus responsibe for absorption, whie the negative frequency components are responsibe for emission of photons. From this picture, the "rotating wave" approximation, introduced in ecture for cassica fieds becomes cear. The dipoe interaction Hamitonian then has then form = - e ˆ d g ( e g + g e ) E ˆ ^(+) (x, t) + E ˆ ^(-) (x, t) ( e g E ˆ ^(+) (x, t) + g e E ˆ ^(-) (x, t ) ). H ˆ int = -ˆ d E ˆ ^(+) (x, t) + E ˆ ^(-) (x, t) ª - e ˆ d g Page 7

8 That is, the annihiation of the photon is aways accompanied by the transition of the atom from the ground to excited state, and the creation of a photon is accompanied by the transition from the excited to ground state. The Hamitonian of the fied is given by the integra of the energy density over the quantization voume H ˆ = 1 8p Ú Ê 2 2 d3 x E ˆ ^ + B ˆ ˆ Ë ^. If we write the fied operators in terms of their norma mode expansions, E ˆ ^(x, t) = iâ 2phw a ˆ e -iw t u (x) - a ˆ e iw t u (x), B ˆ ^ (x, t) = iâ a ˆ e -iw t u (x) - a ˆ e iw t u (x), w where is a composite index for the mode (), and u (x) is the norma mode vector function, then, Ú d 3 x E ˆ ^ 2 = 2ph w w m (-ˆ a a ˆ m e -i( w t +w m  t) Ú d 3 xu (x) u m (x),m + a ˆ ˆ a m e -i(w t -w m t ) d 3 Ú x u (x) u m (x) + H.c) where H.c. stands for the Hermitian conjugate of the terms in parentheses. Using the competeness reations on the norma mode functions we have Ú d 3 x u (x) u m (x) = d m, Ú d 3 xu (x) u m (x) = d -m so that, d 3 x E ˆ 2 Ú ^ = 2phw ( a ˆ ˆ a + a ˆ a ˆ - a ˆ a ˆ - e -2iw t  - ˆ a ˆ a - e 2iw t ). Simiary for the magnetic fied we have, Page 8

9 Ú d 3 x B ˆ ^ 2 = (-a ˆ w w a ˆ m e -i(w t+ w m  t) Ú d 3 x( u ) ( m u m ) m,m + a ˆ ˆ a m e -i(w t -w m t ) d 3 Ú x( u ) ( m u m ) + H.c) Using vector identities, and the fact that u = 0 ( u ) ( m u m ) = ( m )(u u m ). Therefore, the orthogonaity reation yieds, Ú d 3 x ( u ) ( m u m ) = - 2 d -,m, Ú d 3 x ( u ) ( m u m ) = 2 d,m. The integra of the norm of the magnetic fied operator is thus, Ú d 3 x B ˆ ^ 2 =  2phw ( a ˆ a ˆ + a ˆ a ˆ + a ˆ a ˆ - e -2iw t + a ˆ ˆ e 2iw t ) a - Finay we arrive at the expression for the Hamitonian in terms of the creation and annihiation operators, H ˆ = 1 2  hw ( ˆ a a ˆ + a ˆ ˆ a ) =  hw ( ˆ a 1 a ˆ + 2 ), where we have reinstated the fu indexing of the modes, and used the commutator to move a the creation operators to the right. We recognize this Hamitonian as the sum of simpe harmonic osciator Hamitonians for the modes. acuum Fuctuations In ecture we discussed that even in a state with no quanta, i.e. the vacuum 0, the quantum uncertainties ead to fuctuations in the fied. The average vaue of the eectric fied operator vanishes in the vacuum, Page 9

10 E ˆ = 0 E ) 0 = 0 E )(+) E ) (-) 0 = 0, vac since E )(+) 0 = i  2phw = 0. u (x)e -iw t a ˆ 0 = 0, and 0 E )(-) = E )(+) 0 However, the fuctuation of the fied, given by the statistica variance, does not vanish DE ˆ 2 = 0 E ) E ) 0 2 vac = 0 E ) (+) E )(+) E ) (-) E ) (-) E ) (-) E )(+) E ) (+) E ) (-) 0 From the reation above 0 E )(+) E ) (+) 0 = 0 E ) (-) E ) (-) 0 = 0 E )(-) E ) (+) 0 = 0. Thus,. DE ˆ 2 = 0 E ) (+) E ) (-) 0 = vac Â2ph w w m u (x) um (x)e -i(w -w m )t,m 0 ˆ a m ˆ a 0, 0 a ˆ m a ˆ 0 = 0 a ˆ a ˆ m [ a ˆ m, a ˆ ] = d m 0 DE ˆ 2 = 0 E ) (+) E ) (-) 0 =  2phw vac u (x) 2 = d m  2phw. Therefore, the vacuum fuctuation of the eectric fied is on the order of hw / per mode. The expectation vaue of the Hamitonian in the vacuum is, H ˆ = 0 H ˆ 0 = vac  hw ( 0 ˆ a 1 a ˆ ) = hw  2 =! That is, each mode possesses a zero point energy hw, and for an infinite number of 2 modes in the vacuum the tota vacuum energy goes to infinity! Such divergences pague the mathematics of quantum fied theory. A proper treatment requires the sophisticated theory of renormaization. For our purposes we wi use the physica argument that the true energy of the system must be measured with respect to the ground state energy, i.e. the energy of the vacuum. In other words, we wi set the vacuum energy to zero. This is not to say the vacuum energy has no observabe effects. In the next ecture we wi show that the vacuum fuctuations in the eectromagnetic fied are responsibe for the decay of an atom initiay in the excited state to its ground sate. If we were abe in some way to experimentay effect the environment of modes which interact with the atom, then we Page 10

11 may be abe to manipuate the vacuum fuctuations which interact with the atom and thereby modify the spontaneous emission rate with respect to the free space vaue. This may be done by pacing the atom in a highy refecting cavity which aters the spectrum of modes that can be supported inside. Such experiments have been performed. This a research fied nown as "cavity QED", which again and again demonstrates the power of quantum mechanics to describe the interactions of eectromagnetic fieds and atoms. Page 11