Tipo y diseño de investigación

λ

λ = Tr ˆF (3.81)

which implies that|m| ˆF|n| has an upper bound. This being the case, it follows that the function F (β^∗, α) is an entire function in both β^∗andα.

The diagonal elements of an operator ˆF in a coherent state basis completely determine the operator. From Eqs. (3.76) and (3.77) we have

α| ˆF |α e^α∗α=

n

m

α^∗mαⁿ

√m!n!m| ˆF |n . (3.82)

Treatingα and α^∗as independent variables it is apparent that

√1 m!n!

#∂^n+m(α| ˆF|αe^α∗α)

∂α^∗m∂αⁿ

$

α=0

α^∗=0 = m| ˆF |n . (3.83) Thus from “diagonal” coherent-state matrix elements of ˆF we can obtain all the matrix elements of the operator in the number basis.

3.6 Phase-space pictures of coherent states

It is well known that the concept of phase space in quantum mechanics is problematic owing to the fact that the canonical variables ˆx and ˆp are incom-patible, i.e. they do not commute. Thus the state of a system is not well localized as a point in phase space as it is in classical mechanics. Neverthe-less, we have shown that the coherent states minimize the uncertainty relation for the two orthogonal quadrature operators and that the uncertainties of the two quadratures are equal. Recall that ˆX1= (ˆa + ˆa^†)/2 and ˆX2= (ˆa − ˆa^†)/2i.

These operators are dimensionless scaled position and momentum operators, respectively. Their coherent-state expectation values are ˆX1α= ¹₂(α + α^∗)= Reα,  ˆX2_α= _2i¹(α − α^∗)= Imα. Thus the complex α-plane plays the role of phase space where, up to scale factors, the real and imaginary parts ofα are posi-tion and momentum variables respectively. A coherent state|αwith α = |α|e^iθ may be represented pictorially then as in Fig.3.5, where the shaded circle repre-sents “area of uncertainty” of the coherent state, the fluctuations being equal in all directions of phase space, the center of the circle located at distance

|α| = ˆn^1/2 from the origin and at angle θ above the position axis. Further,

θ, in a qualitative sense, represents the phase uncertainty of the coherent state and it should be clear thatθ diminishes for increasing |α|, the fluctuations in X1 and X2 being independent ofα and identical to those of the vacuum. For the vacuum,|α| = 0, the phase-space representation is given in Fig.3.6, where it is evident that uncertainty in the phase is as large as possible, i.e.θ = 2π.

Fig. 3.5. Phase-space portrait of a coherent state of amplitude|α| and phase angleθ. Note the error circle is the same for all coherent states. Note that as|α| increases, the phase uncertaintyθ decreases, as would be expected in the “classical limit”.

Fig. 3.6. Phase-space portrait of the quantum vacuum state.

Fig. 3.7. Phase-space portrait of the number state|n. The uncertainty in the photon number is

n = 0 while the phase is entirely random.

A number state|n can be represented in phase space as a circle of radius n, the uncertainty in n being zero and the uncertainty in phase again being 2π, as in Fig.3.7. It must be understood that these pictures are qualitative in nature but are useful as a graphical way of visualizing the distribution of noise in various quantum states of the field. As most quantum states of the field have no classical

Fig. 3.8. The error circle of a coherent state (the black dot) revolves about the origin of phase space at the oscillator angular frequencyω and the expectation value of the electric field is the projection onto an axis parallel with X1.

analog, the corresponding phase-space portraits should not be taken too literally.

Yet these representations will be quite useful when we discuss the nature of the squeezed states of light in Chapter7.

Finally in this section, we make one more use of the phase-space diagrams, namely to illustrate the time evolution of quantum states for a non-interacting field. We have seen that, for a noninteracting field, a coherent state|α evolves to the coherent state|αe^−iωt. This can be pictured as a clockwise rotation of the error circle in phase sinceαe^−iωt| ˆX1|αe^−iωt = α cos ωt, αe^−iωt| ˆX2|αe^−iωt =

−α sin ωt, assuming α real. Because in the Schr¨odinger picture the electric field operator is given, from Eq. (2.15) and (2.52), as

Eˆx= 2E0sin(kz) ˆX1, (3.84)

the expectation value for the coherent state|αe^−iωt is

αe^−iωt| ˆEx|αe^−iωt = 2E0sin(kz)α cos ωt. (3.85)

Thus apart from the scale factor 2E0sin(kz), the time evolution of the electric field and its fluctuations, is given by the projection of the ˆX1 axis as a function of time as indicated in Fig.3.8.

The evolution of points within the error circle are shown indicating the uncer-tainty of the electric field – the “quantum flesh” on the “classical bones”, so to speak. Note that the greater the excitation of the field, i.e.α, the more classical the field appears since the fluctuations are independent ofα. But the coherent state is the most classical of all the quantum states so it is apparent that for the field in something other than a coherent state, the expectation value of the field may in no way appear classical-like. A number state is a very nonclassical state and, by using representative points from its phase-space portrait, it is easy to see, from Fig.3.7, that the expectation value of the field is zero. But it is possible to imagine other kinds of states having no vanishing expectation values of the field but where fluctuations may be less than those of a coherent state in one part of the field. These are the squeezed states to be taken up in Chapter7.

3.7 Density operators and phase-space probability