The Q Function

In classical mechanics, the state of a point particle is completely speciﬁed by giving its position and momentum at any given time. For one-dimensional motion, the pair of values x and p denotes a point in a two-dimensional phase space.

In quantum mechanics, it is impossible to specify both x and p simultaneously because of the uncertainty principle, which asserts that the “uncertainty” (as given by the standard deviation of a series of measurements on identically-prepared systems) in x and p must satisfy

ˉh Δx Δp ≥ -- 2

(1)

This means that it is impossible in practice, and probably meaningless in theory, to “locate” the particle’s state within a region of phase space of area smaller than about $ˉh$ .

For a quantum free particle, or a harmonic oscillator, minimum uncertainty states where the relation (1) is satisﬁed with the equal sign are given by Gaussian wavefunctions. For the harmonic oscillator, a special kind of Gaussian state is the so-called coherent state, for which not only $Δx Δp = ˉh∕2$ , but also Δx and Δp remain constant throughout the evolution. A coherent state is usually denoted by $∣α 〉$ , where the complex number α is related to the expectation values $〈x 〉$ and $〈p〉$ of the position and momentum by

∘ -mω- 〈p 〉 α = ----〈x〉+ i√------- 2ˉh 2m ωˉh

(2)

(where m is the mass of the particle, and ω the oscillation frequency). The variance of x in a coherent state is $2 (Δx) = ˉh∕m ω$ , and that of p is $2 (Δp) = m ωˉh$ , so the dimensionless phase-space coordinates $′ ∘ ------- x = x∕ 2ˉh∕m ω$ and $′ √------ y = p∕ 2m ω ˉh$ have variances (Δx′)² = (Δy′)² = 1∕4. By Eq. (2), we also have $′ ′ α = 〈x 〉+ i〈y 〉$ .

The Q function makes use of the coherent states to represent the state of a quantum-mechanical oscillator in the (x′,y′) phase space in the following way. Instead of trying to locate the particle precisely at a speciﬁc point (x′,y′) we ask about the overlap of the particle’s state $∣ψ〉$ with the coherent state $′ ′ ∣α〉 = ∣x + iy 〉$ which is centered at that point. That is, we ask for the probability to ﬁnd the particle not necessarily precisely at x and p but within a “minimum uncertainty region,” of area $~ ˉh$ , around that point:

Q(α) = Q(x ′,y ′) = 1∣〈α ∣ψ 〉∣2 π

(3)

where the 1∕π is included for normalization purposes, so that $∫ ∫ ∞-∞ ∞-∞ Q(x ′,y′)dx′dy′ = 1$ . Q can then be viewed as a sort of joint probability distribution for the position-like variable x′ and the momentum-like variable y′.

The electromagnetic ﬁeld, of course, is not a mechanical oscillator, but the Maxwell equations in free space—essentially, the wave equation—can be written as the equations of motion for an inﬁnite set of harmonic oscillators, one for each mode of the ﬁeld. Then the Q function can be used to visualize the evolution of the quantized modes. Naturally, in this case x′ and y′ are not position and momentum, but rather something called the ﬁeld’s “quadratures.” Basically, $-------- ∘ ′2 ′2 x + y$ gives the ﬁeld’s amplitude, and tan^-1(y′∕x′) gives the ﬁeld phase. Classical optics often makes use of the “phasor” representation, where an optical ﬁeld is represented by a vector in a two-dimensional space (the complex plane); the Q function may be regarded as a (quasi-)probability distribution for the tip of such a phasor, over the complex plane, as a function of time.

Our applets use x and y, rather than x′ and y′, to label the axes in the Q-function plots. The scale of the plot can be gauged from the initial conditions: the initial state is always a coherent state, and in that case the Q function is a Gaussian which peaks at the point representing the complex amplitude α, with a width of about 1 unit, in the dimensionless units used for x′ and y′. Note also that, in these units, x′² + y′² gives the number of photons in the ﬁeld mode.

For the exactly integrable Jaynes-Cummings model, we have made sure that the plot is always wide enough to display all the relevant parts of the Q function, but for the much more complex spin-boson Hamiltonian it is easy to come up with initial conditions for which signiﬁcant parts of the Q function eventually leave the plot area.

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