The Specific Case of P-function

Historically, the starting point for construction of a phase-space representation for quantum mechanics is that of Wigner [18]. However, we prefer to introduce the subject using a more advanced approach. As we have seen in Eq. (1.51), the set S_coh forms a basis for the Hilbert space of a single-mode bosonic system H_bos and thus any state vector |ψi can be expanded in terms of optical coherent states as

There is, however, a more surprising fact regarding projections onto the optical-coherent basis

|αihα|, and that is, the set of such projections S_pcoh = {|αihα|}_α∈C forms a basis for linear operators L(H_bos) acting on the Hilbert space H_bos. That is, any operator ˆA acting on H_boson can be expanded in terms of optical-coherent projections as

A =ˆ

Z d²α

π PAˆ(α)|αihα|. (1.56)

We are not going to prove this statement here, and refer the interested reader to Ref. [17] for a comprehensive discussion. The expansion coefficient PAˆ(α) is called the Glauber-Sudarshan P-function, or shortly the P-function, corresponding to the operator ˆA. Considering a harmonic oscillator prepared in the optical-coherent state |αi, its density operator can be written as

ˆ

% = |αihα| =

Z d²β

π δ⁽²⁾(β − α)|βihβ|. (1.57)

Let us now consider the free evolution of our oscillator given by the unitary evolution

Uˆ_free(t) = e^−i~tHˆ ≈ e^−itωa†a. (1.58)

The action of ˆU_free(t) on the coherent states using the expansion in Eq. (1.50) is simply given by

Uˆ_free(t)|αi = e^−itωa†a|αi = |e^−itωαi, (1.59) known as the phase rotation operation. Hence, for the density operator, we find that

ˆ

We see that, in the P-function picture, the coherent state is represented by a two-dimensional Dirac delta function which rotates around the center of the complex plane α with the mechanical

angular frequency ω. Interestingly, interpreting the complex plain α as a quantum mechanical phase-space and the P-function of the optical-coherent state of the oscillator as a distribution over this phase space, we recover exactly the description of a classical harmonic oscillator evolving in a classical phase-space in classical statistical mechanics. Based on this analogy, one can also interpret the optical-coherent state |αihα| as the best quantum approximation to a classical harmonic oscillator in a pure state, commonly called “classical states” within the context of quantum optics [19].

1.3.3 Displacements

Despite the nice similarity of the optical-coherent states to the pure classical states of a harmonic oscillator, not all the expectations from a classical phase-space distribution is satisfied via the use of P-function representation. For instance, the marginal distributions of P-functions do not faithfully correspond to marginal position and momentum distributions as in classical mechanics. In general, it is now known that (i) the quantum phase-space picture is not unique, and (ii) none of the possible space representations faithfully resemble the classical phase-space. Consequently, we are mainly concerned about how to generate various phase-space representations.

The main building block in constructing a phase-space representation is the structure of transformations over the desired phase-space plane. A very important class of such transfor-mations is translations of possible distributions. We now briefly study the emergence of the family of phase-space representations in bosonic systems from the group of translations. Con-sider the optical-coherent state |α = 0i. Using the expansion in Fock basis as in Eq. (1.50), we find that

|α = 0i = |0i, (1.61)

that is, the zero optical-coherent state coincides with the energy ground state of a harmonic oscillator. Taking this state as a natural reference point, the phase-space displacement operator is defined as

D(α) = eˆ ^{αa†−α∗a} (1.62)

= e^−|α|22 e^αa†e^α∗a (1.63)

= e^|α|22 e^α∗ae^αa†. (1.64) Both forms in Eqs. (1.63) and (1.64) have been obtained from Eq. (1.62) using the well-known Baker-Campbell-Hausdorff relation. In particular, while the definition of Eq. (1.62) is known as the displacement operator in standard order, the disentangled form of Eq. (1.63) is called the displacement operator in normal order in which all the creation operators are placed on the left-hand-side of the annihilation operators. In contrast, in Eq. (1.64) the displacement operator has been disentangled into the antinormal order where all the creation operators are on the right-hand-side of the annihilation operators. The three ordered forms above can be

unified using a single parameter s ∈ C,

D(α; s) = ˆˆ D(α)e^s2|α|2, (1.65)

such that for s = −1, 0, 1 one obtains the normal, symmetric, and antinormal orderings of the exponentials e^αa† and e^α∗a, respectively.

It is now easy to verify using Eq. (1.63) that

D(α)|0i = |αi.ˆ (1.66)

Displacement operators in symmetric order also possess the group properties, most importantly, D(α) ˆˆ D(β) = e^{αβ∗−α∗β}D(α + β),ˆ (1.67) Dˆ⁻¹(α) = ˆD^†(α) = ˆD(−α). (1.68)

Furthermore, the bosonic creation and annihilation operators transform under the action of displacement operator as

Dˆ^†(α)a ˆD(α) = a + α, Dˆ^†(α)a^†D(α) = aˆ ^†+ α^∗. (1.69)

To see the appropriateness of using the name “displacement” for the operator ˆD(α), we note that by the application of displacement operation, the P-function corresponding to the quantum state of a bosonic system ˆ% evolves into

ˆ

%(α) = ˆD(α) ˆ% ˆD^†(α) =

Z d²β

π P_%ˆ(β) ˆD(α)|βihβ| ˆD^†(α)

=

Z d²β

π P_%ˆ(β)|β + αihβ + α|

=

Z d²β

π P_%ˆ(β − α)|βihβ|.

(1.70)

Equation (1.70) shows that the displacement operator correctly translates the P-function over the complex phase-space.

The most important feature of displacement operators, however, is that they form a complete basis for (bounded) linear operators ˆA ∈ L(H_bos). In particular, we have

Tr ˆD(α) ˆD^†(β) = πδ⁽²⁾(α − β), (1.71) which can be seen as a duality relation for displacement operators. We refer the interested reader to Ref. [17] for a rigorous proof of Eq. (1.71) and the completeness relation of displacement operators. As a result, one may expand the operator ˆA ∈ L(H_bos) as

A =ˆ

Z d²α

π χAˆ(α) ˆD^†(α), (1.72)

in which the function χAˆ(α) is called the Wigner characteristic function corresponding to the operator ˆA and, using Eq. (1.71), it is given by

χAˆ(α) = Tr ˆA ˆD(α). (1.73)

Within the Hilbert space picture of quantum mechanics, indeed one requires the functions χAˆ(α) to be square integrable which is equivalent to ˆA being bounded in the sense that

Aˆ = Tr ˆA^†A < ∞ [17]. However, this is not necessary as one uses a rigged Hilbert space formalism [1]ˆ in which unbounded operators such as projections onto position eigenspaces are well-defined in a functional sense.

The duality relation (1.71) can simply be extended to any s-ordered form of the displacement operators in Eq. (1.65) as

Tr ˆD(α; s) ˆD^†(β; −s) = πδ⁽²⁾(α − β), (1.74) Using similar arguments as above, one can show that the set of s-ordered displacement operators given by Eq. (1.65) also form a basis for L(H_bos) and thus, any operator ˆA ∈ L(H_bos) can be formally expanded as

A =ˆ

Z d²α

π χ_Aˆ(α; s) ˆD^†(α; −s). (1.75) Here, the function χAˆ(α; s) is the s-parameterized characteristic function that corresponds to the operator ˆA and using the duality relation Eq. (1.74) is given by

χAˆ(α; s) = Tr ˆA ˆD(α; s). (1.76) As we will see shortly, the relations (1.75) and (1.76) form the core of the phase-space description of quantum mechanics for CV systems.

As a final remark, it is useful to investigate the effect of displacements on the characteristic function of operators in Eq. (1.75). We first combine Eqs. (1.65), (1.67), and (1.68) to obtain

D(β) ˆˆ D(α; s) ˆD^†(β) = e^{βα∗−β∗α}D(α; s).ˆ (1.77)

Unitarily displacing the operator ˆA in Eq. (1.75) and employing Eq. (1.77), we find

A(β) = ˆˆ D(β) ˆA ˆD^†(β) ←→ χA(β)ˆ (α; s) = e^{αβ∗−α∗β}χAˆ(α; s), (1.78)

representing the fact that the effect of displacements on the characteristic functions is mani-fested as a modulating phase-factor, rather than proper phase-space translations.