# (p.569) Appendix Quantum states in phase space

# (p.569) Appendix Quantum states in phase space

Phase space distributions are fundamental tools in classical statistical physics. For a particle undergoing one-dimensional motion, the phase space is a plane, with the position *x* and conjugate momentum *p* as coordinates. A point in this plane defines the mechanical state of the particle. In the case of a cavity mode, equivalent to a onedimensional oscillator, two orthogonal field quadratures play the role of position and momentum, and a point in the Fresnel plane defines a classical field state (Chapter 3). Statistical uncertainties or lack of knowledge about the system are accounted for by replacing the Dirac distribution associated to this point by a positive and normalized density of probability, *f*(*x, p*), which takes non-vanishing values in a limited region of phase space. The statistical average *ō* of any observable quantity, *o*(*x, p*), is given by:

The system's evolution due to external forces and to the coupling with the environment is described by a diffusion equation for *f*, relating its temporal and spatial derivatives.

The extension of this phase space representation to quantum states was discussed for the first time by Wigner (1932). A difficulty has immediately arisen because Heisenberg uncertainty relations forbid, even in the absence of any statistical indeterminacy, the precise and simultaneous determination of conjugate variables. In other words, a fundamental quantum blurring adds to the classical uncertainties. In spite of this problem, it remains possible to define a real phase space function for a quantum particle, which retains some of the essential characteristic features of the classical probability distribution. This quantum distribution is called the Wigner function *W*. Its definition and main properties are recalled in this appendix in the context of the physics of a harmonic oscillator or a field mode.

We will see that *W* is naturally defined from its Fourier transform, the system's symmetrical characteristic function *C _{s}*(

*λ*). Introducing

*C*will lead us to consider two simply related characteristic functions,

_{s}*C*(

_{n}*λ*) and

*C*(

_{an}*λ*), and we will see that their Fourier transforms define two phase space distributions different from

*W*, the Husimi-

*Q*function and the Glauber–Sudarshan

*P*distribution. The latter, which is highly singular, is of difficult mathematical use and we will not say much about it. The

*W*and

*Q*functions, which are ubiquitous tools in quantum optics, are used time and again in this book to give pictorial representations of oscillator and field states. Here, we will discuss their connexion, compare them and explain why

*W*is more useful than

*Q*for the analysis of the quantum features of a system. We will also study the (p.570) main properties of

*C*and

_{n}*C*, two functions which turn out to be very convenient to describe the dynamics of a quantum oscillator undergoing a relaxation process.

_{s}This appendix, while presenting the main results in a self-consistent way, is intended as a brief reminder.^{1} We start by introducing (Section A.1) the characteristic functions from which the *W*, *Q* and *P* functions are derived. We then describe the Wigner function (Section A.2), give its main properties and a few useful examples. The next section (A.3) is devoted to the *Q* distribution. The final section (A.4) deals with the evolution of the characteristic functions and *W* distribution under relaxation. We present a solution of the decoherence master equation for a Schrödinger cat state which is an instructive complement to the calculations presented in Chapter 7.

# A.1 Characteristic functions

The phase space density functions *W*, *Q* and *P* are the Fourier transforms of characteristic functions, linked to the quantum average of the displacement operator *D*(*λ*) = exp(*λa ^{†}* –

*λ*), in a state described by the density matrix ρ [see Section 3.1.3 for the main properties of

^{*}a*D*(

*λ*)]. When operators appear as arguments of a function, we must address the issue of their ordering. A very detailed discussion of this question is found in Cahill and Glauber (1969). Let us mention only here that three simple orderings can be used when expanding in power series a function of the annihilation and creation operators

*a*and

*a*. In the ‘normal’ order, all creation operators are placed to the left of the annihilation ones (the photon number operator

^{†}*N*=

*a*is in normal order). The ‘anti-normal’ order puts all the

^{†}a*a*'s on the left. Finally, the ‘symmetric’ order produces expressions in which all products of operators are symmetrized. For instance, the direct power series expansion of the displacement operator:

is naturally in symmetric order.

We now introduce three characteristic functions of the state of the system corresponding to these orders. The symmetric-order characteristic function, ${C}_{s}^{[\rho ]}(\text{\lambda})$, is the average of *D*(*λ*) in the state described by the density operator ρ:

The operator *D*(λ) being unitary, all its eigenvalues have a unit modulus. The modulus of its average is thus always bounded by one:

Note also the identity:

The upper bound of ${C}_{s}^{[\rho ]}$ is thus reached at the origin. In addition, ${C}_{s}^{[\rho ]}(\text{\lambda})$ obeys the conjugation relation:

Let us finally give the expression of the symmetric characteristic function for a pure state |Ψ>:

It is merely the overlap between |Ψ> and the state obtained by translating |Ψ> in phase space by *D*(λ). It is thus an autocorrelation function of the quantum state in phase space.

The normal- and anti-normal-order characteristic functions are likewise defined as:

and:

These functions can be related to each other with the help of the Glauber identity:

valid when both *A* and *B* commute with [*A, B*] (eqn. 3.39, on page 115). We obtain immediately:

Any couple of characteristic functions can thus be determined from the knowledge of the third one by a mere multiplication by ${e}^{\pm |{\lambda |}^{\text{2}}/2}$ or ${e}^{\pm |{\lambda |}^{\text{2}}}$.

As a simple example, let us determine the characteristic functions for a coherent state |α>. We have:

Recalling that:

we get:

a complex Gaussian function from which we derive:

and:

The characteristic functions of a Fock state |n> are obtained in a similar way, by expanding the normal-ordered displacement operator in powers of *a* ^{†} and *a*, retaining terms with the same number of creation and annihilation operators which are the only
(p.572)
ones to have non-vanishing expectation values in an |n> state. The final result for *C _{s}* is (Barnett and Radmore 1997):

where L_{n} is the *n*th Laguerre polynomial:

Let us finally mention the symmetric-order characteristic function for a thermal field, obtained by summing the Fock state result over a thermal distribution:

where *n* _{th} is the average number of thermal photons. We get a remarkably simple result:

# A.2 The Wigner distribution

The Wigner function is defined as the two-dimensional Fourier transform of the symmetric order characteristic function:^{2}

Using the notation λ′, λ″ for the real and imaginary parts of λ and setting *α* = α′ + *iα*″ = *x* + *ip* to emphasize the analogy between the field quadratures and the normalized dimensionless position and momentum of a particle, we can also write:

The Wigner function is real, as a direct consequence of the conjugate property of its Fourier transform (eqn. A.6). It is also normalized, which results from the identity:

where we have used an integral definition of the two-dimensional Dirac distribution *δ*(λ):

Before exploring further the properties of *W*, we will write it in two other equivalent and useful forms.

##
(p.573)
A.2.1 Two equivalent expressions of *W*

We first make explicit the trace operation in the definition of *C _{s}*(λ), by using the continuous eigenbasis {|

*x*>} of the quadrature operator

*X*

_{0}= (

*a*+

*a*

^{†})/2:

Using the Glauber identity, we can write:

where *P* _{0} = *i*(*a* ^{†} – *a*)/2 is the field quadrature conjugate of *X* _{0}. It follows that:

Within phase factors, the displacement operator translates the *X* _{0} quadrature eigenstates by an amount λ′. Hence:

The integral over λ″ is a one-dimensional Dirac function:

The integration over λ′ is then simple, leading to:

Setting finally *u* = 2(*x*′ –; *x*), we obtain a well-known form for the Wigner function:

which is the Fourier transform of non-diagonal density matrix elements in the position eigenstates basis. The function *W* is clearly sensitive to quantum coherence in the field state. The Fourier transform can be inverted yielding the matrix elements of *ρ* in terms of *W* as:

This equation shows that the knowledge of *W* is equivalent to that of the density operator, and hence that the Wigner function contains all information needed to compute the expectation value of any observable in the state of the system. We come back to this point below.

(p.574)
An even simpler expression for *W* can be derived from eqn. (A.31). We start from the translation relation:

[which follows directly from eqn. (A.27) in which we set *x*′ = –*u*/2 and λ = *x* + *ip*] and from the conjugate relation, in which we change *u* into –*u*:

We then replace |*x* – *u*/2> and <*x* + *u*/2| by their expressions (A.33) and (A.34) in eqn. (A.31) and we introduce the hermitian parity operator ƿ which performs a symmetry around the phase space origin according to:

We finally get:

The Wigner function is thus the average value of 2ƿ/*π* in the state obtained by displacing the oscillator in phase space by the amount −*α*, transforming its density operator according to *ρ* → *D*(−*α*)*ρD*(*α*). It is easy to show that the ƿ operator is the photon number parity observable introduced in Chapters 6 and 7. The |*n*> Fock state wave functions in the |*x*> and |*p*> representations is a product of an even parity Gaussian multiplied by a Hermite polynomial which has the parity of *n* (eqn. 3.13, on page 108). Reversing the sign of *x* or *p* in these functions thus amounts to a multiplication by (−1)^{n}, so that wecan write:

Important properties of this operator are summarized in Section 7.1.2.

Being the expectation value of an observable, *W* is a directly measurable quantity. We present in Section 6.5 a determination of *W* for the vacuum and the single-photon Fock state based on the definition given by eqn. (A.36). The eigenvalues of the parity operator being +1 and −1, its expectation lies between these values. It follows that the Wigner function is bounded:

We now use the equivalent definitions of *W* to establish its main properties.

## A.2.2 Main properties

### Average operator values

As noted above, the average value of any observable can be obtained from the knowledge of the Wigner distribution. There is in fact a very simple recipe to express this average by a simple integral, provided the observable, considered as a function of *a*
(p.575)
and *a* ^{†}, has been cast in the symmetric order. The relation (A.21) defining *W* can be written:

in which the integral is reminiscent of the two-dimensional Dirac distribution (see eqn. A.24). We can then formally write:

The Wigner function is thus the average value of the operator-Dirac distribution of *α* − *a*. Obviously, such a formal derivation has to be taken with care. We do not enter here in the mathematical justification of this expression [see for instance Cahill and Glauber (1969)], which is valid provided all quantum operators are put in the symmetric order.

Consider now an observable *O* of the field mode. It can be expanded as a power series of *a* and *a* ^{†} and cast in the symmetric order by repeated commutations of these operators. Let us note *O _{s}*(

*a*,

*a*

^{†}) the resulting expression. Formally, we can write:

where *o _{s}*(

*α*,

*α*

^{*}) is the complex function of

*α*obtained by replacing

*a*by

*α*and

*a*

^{†}by

*α*

^{*}in

*O*(

_{s}*a*,

*a*

^{†}) [the precise rules for this substitution can be found in Scully and Zubairy (1997)]. The average value of

*O*is then:

_{s}This simple expression reminds us of eqn. (A.1) giving the average value of any classical observable as an integral over the probability density in phase space. The real and normalized Wigner function plays in this respect the role of *ƒ*(*x*, *p*) in classical physics. We will see however that, contrary to *ƒ*, *W* can take negative values which are a signature of the oscillator's quantum behaviour.

### Marginal distributions

The analogy between *ƒ*(*x*, *p*)and *W*(*x*, *p*) can be pushed further by considering the marginal distributions of *x* and *p*, obtained by integrating the distribution over the conjugate variable. Setting *u* = 0 in eqn. (A.32), we get immediately the probability density of finding the value *x* of the oscillator's position (or field quadrature *X* _{0})as:

A simple calculation involving changes between the |*x*> and |*p*> bases shows that the integration of *W* over *x* yields likewise the probability density of finding the value *p* of the oscillator's momentum (or field quadrature *X* _{π/2}):

(p.576)
More generally, the Wigner distribution can be expressed as a function of a any couple of orthogonal field quadratures *x* _{ϕ} and *p* _{ϕ} defined by:

and corresponding to axes rotated in the phase plane by an angle *ϕ* with respect to the *x* and *p* coordinate directions. The marginal distributions properties apply to any such couple of conjugate quadratures:

It can be shown that *W* is the only quantum phase space distribution obeying this interesting property (Bertrand and Bertrand 1987), on which the ‘quantum tomographic’ methods for the reconstruction of *W* are based (Section 3.2.4).

For many quantum states, the probability distributions *P*(*x*) or *P*(*p*) present nodes. This is, for instance, the case for the *x* distribution of the excited Fock states (Fig. 3.2, on page 107). When *P*(*x* _{0}) = 0, it follows that:

Hence, *W* cannot be everywhere positive. When it takes negative values, it cannot be assimilated to a genuine probability distribution. In fact, as we now show, negativities in *W* are clear-cut indicators of the non-classical nature of a field state.

## A.2.3 Examples of Wigner functions

We give in this paragraph the explicit expressions of *W* for a few quantum states classified in two categories: those whose Wigner function is everywhere positive, which we call ‘quasi-classical’ and those who present negativities, called ‘non-classical’.

### Quasi-classical states

Let us consider first a coherent state |*β*> (including obviously the vacuum state). Using the symmetric-order characteristic function (eqn. A.14, in which we replace *α* by *β*), the corresponding Wigner function, *W* ^{[|β> <β|]}(*α*)is:

It is a Gaussian, with a width $1/\sqrt{2}$, centred on the classical amplitude *β* and taking for *α* = *β* its maximum allowed value, 2/*π*. Figure A.1(a) and (b) present the Wigner functions of the vacuum (*β* = 0) and of a coherent state with $\beta =\sqrt{5}$ ($n\xaf=5$ photons on the average).

The Wigner function of a thermal field (Fig. A.1c), with *n* _{th} photons on the average is derived also from the corresponding Gaussian characteristic function (eqn. A.20):

It is again a Gaussian, centred at the origin and represented in Fig. A.1(c) for *n* _{th} = 1. The width is now $\sqrt{{n}_{th}+1/2}$ and the peak value is reduced to 1/*π*(*n* _{th} + 1/2).

Let us finally examine the case of a squeezed vacuum state (Kimble 1992), often considered in quantum optics as ‘non-classical’. It is obtained by the action on the vacuum of the ‘squeezing’ operator (Vogel *et al*. 2001):

where we assume for the sake of this simple discussion that ξ is real. The squeezed states are minimal uncertainty states satisfying the Heisenberg uncertainty relation Δ*X* _{0}Δ*P* _{0} = 1/4. For a coherent state, the variances of *X* _{0} and *P* _{0} are equal. For a squeezed state the standard deviation of the *X* _{0} quadrature is reduced to:

and the fluctuations of *P* _{0} accordingly increased:

Note that the limit of infinite squeezing corresponds to the position eigenstate |*x* = 0>, with no position fluctuations and infinite momentum uncertainty. The Wigner function of a squeezed vacuum is a non-degenerate Gaussian:

represented in Fig. A.1(d) for ξ = 0.5. A general squeezed state corresponds to a complex ξ parameter and to a displacement towards a non-vanishing classical amplitude in phase space. Its Wigner function is also a non-degenerate Gaussian, centred on the classical amplitude, whose principal axis (minimum and maximum fluctuations) are tilted with respect to *x* and *p*.

For all states considered so far, including the squeezed ones, the Wigner function is definite and positive. It has all the properties of a classical probability distribution in phase space. For the computation of any observable eigenvalue, these states can be considered as classical fields with a stochastic complex amplitude whose probability distribution over phase space is given by *W*. This is not the case for the states we consider below.

### Non-classical states

The Fock state Wigner function can be derived from the corresponding characteristic function (eqn. A.17):

Note that *W* ^{[|n> <n|]} does not depend upon the phase of *α*, which reflects the complete phase indeterminacy in Fock states. Since ℒ_{n}(0) = 1:

The Wigner function of odd photon number states takes the minimal value −2/*π* at the origin of phase space. It cannot thus be interpreted as a probability distribution of classical field fluctuations, as was the case for the states considered above. In fact, all photon number states, but |0>, present negativities in some region of phase space. The single-photon Wigner function:

has a Mexican-hat shape. It is shown in Fig. 6.22(b), on page 349. As another example, we present in Fig. A.2 the five-photon state Wigner function. It presents circular rims around the origin, with alternating negative and positive values. Note that this representation does not coincide with the intuitive picture of a Fock state (a well-defined |*α*| value, corresponding to a single circular rim). We show below that the Husimi-*Q* function is closer to this simple picture. The inner rims in *W* ^{[|n> >n|]} correspond to quantum interference features, revealing the non-classical nature of the Fock states.

Another example of non-classical state is given by the ‘phase cat states’, discussed in Chapter 7, which are superposition of two coherent fields with opposite phases:

(p.579) where:

is a normalization factor resulting from the finite overlap of |*β*> and |−*β*>. The density matrix is ${\rho}_{\text{cat}}^{\pm}=|{\Psi}_{\text{cat}}^{\pm}>\text{}{\Psi}_{\text{cat}}^{\pm}|$ and the Wigner function:

The first line in this equation corresponds to the weighted sum of the Wigner functions of the coherent states |*β*> and |−*β*>. The second line corresponds to non-diagonal terms involving the quantum coherence between the two components. These terms are absent in the case of a statistical mixture (|*β*> <*β*| + |*β*> <*β*|)/2. Assuming, for the sake of simplicity, that *β* is real we get:

The coherence terms also correspond to Gaussian integrations which can be performed explicitly, leading finally to:

These functions, represented in Fig. 7.4, on page 362, exhibit large negative values near the phase-space origin, a clear signature of non-classical behaviour which is analysed in detail in Chapter 7.

# A.3 The Husimi-*Q* distribution

We have focused so far on the symmetrically ordered function *C _{s}* and its Fourier transform

*W*. Let us consider now the anti-normal-order characteristic function

*C*and the related Husimi-

_{an}*Q*function. As we will see, the features of the

*Q*function are very different from the ones of

*W*. Being always positive,

*Q*is in some cases more convenient to use for simple representations of quantum states such as Fock states. It is however less well-adapted than the Wigner function to exhibit explicitly the non-classical features of Schrödinger cats.

## A.3.1 Definition and main properties

The *Q* distribution is the Fourier transform of the anti-normal-order characteristic function:

This expression looks formidable, but in fact, *Q* ^{[ρ]}(*α*) is simply related to the expectation value of the density operator *ρ* in state |*α*>. To show this, let us write *Q* as:

(p.580)
where we have used the definition (A.9) of *C _{an}* and the closure relation on coherent states:

The coherent state |*β*> being an eigenstate of exp(−λ^{*} *a*), *Q* reduces to:

and the integration over λ leads to a Dirac distribution *δ*(*α* − *β*) (eqn. A.24). We thus finally get:

which can also be written as:

The *Q* function is thus the average of the projector onto the vacuum state,|0> <0|, in the field displaced in phase space by −*α*. Being the expectation value of an observable, it is thus – as *W* – a directly measurable quantity. The *Q* function reconstruction method presented in Section 7.3.2 is based on this expression.

The *Q* distribution is positive, bounded by 1/*π* and normalized $\left(\int {d}^{2}\alpha \text{}Q(\alpha )=1\right)$, this last property resulting directly from eqn. (A.64). Using eqn. (A.66), it is easy to compute the *Q* functions of our favourite classical and non-classical states.

## A.3.2 Examples of *Q* functions

The *Q* distribution of a coherent state |*β*> is:

It is plotted in Fig. A.3(a) for $\beta =\sqrt{5}$. It is a Gaussian, with a width 1, reaching for *α* = *β* the maximum allowed value 1/*π*. Besides their widths, the *Q* and *W* functions are thus very similar for a coherent state. Note that the pictorial representation of coherent states as an uncertainty disk superposed on a classical amplitude, used repeatedly from Chapter 3 onwards, corresponds to a cut at 1/*eπ* in the *Q* ^{[|β> <β|]}(*α*) distribution.

Expanding |*α*> on the Fock state basis, it is easy to find the *Q* function for an n-photon Fock state:

Represented in Fig. A.3(b) for *n* = 5, it appears as a single circular rim around the origin, which peaks for |*α*|^{2} = *n*, an intuitive result. For Fock states, the *Q* function seems thus more ‘reasonable’ than the Wigner distribution.

Let us now turn to the phase-cat states $|{\Psi}_{\text{cat}}^{\pm}>$. Assuming again *β* real, and expanding coherent states scalar products, we get:

As for the Wigner function, the cat's Husimi distributions are made up of two Gaussians [first two terms in the right-hand side of eqn. (A.70)], corresponding to the (p.581)

*α*= 0, is exponentially small for large

*β*values. The

*Q*function of the $|{\Psi}_{\text{cat}}^{\pm}>$ cat state, plotted in Fig. A.3(c) for $n\xaf=5$, is thus almost identical to that of a statistical mixture, represented in Fig. A.3(d). The

*Q*function is not able to display the quantum coherence of such a state superposition. Note that this exponential suppression of interference makes it experimentally difficult to reconstruct the field density matrix from

*Q*(we would need to perform a measurement with an unrealistic precision).

The *W* and *Q* functions are connected in reciprocal space by the relation (A.11) between their Fourier transforms, *C _{s}(λ)* being obtained from

*C*by a multiplication by ${e}^{|{\lambda |}^{\text{2}}/2}$. This relation explains why

_{an}(λ)*W*is more sensitive than

*Q*to a cat's state coherence. We have seen that this coherence is described by vanishingly small terms in

*Q*around the phase space origin. The value of

*Q*at

*α*= 0 is equal to the integral over the λ plane of its Fourier transform

*C*. This integral is thus very small for a cat. The same integral, performed on the Fourier transform of

_{an}(λ)*W*, yields a much larger result since

*C*(λ)/

_{s}*C*diverges for |λ| → ∞, entailing

_{an}(λ)*W*(0) ≫

*Q*(0). The features of the phase space distribution around

*α*= 0, critical to the description of the coherence, are thus greatly amplified by the ${e}^{|{\lambda |}^{\text{2}}/2}$ multiplication transforming

*Q*into

*W*in reciprocal space.

We have defined *W* and *Q* as the Fourier transforms of *C _{s}* and

*C*. What about

_{an}*C*? Its Fourier transform is yet another phase space density function, the Glauber– Sudarshan-

_{n}*P*distribution (Vogel

*et al*. 2001). We do not use it here, since it turns out to be highly singular. For instance, the

*P*distribution of the coherent state |

*β*> is a

*δ*-Dirac distribution centred at

*β*and the

*P*distribution of an |

*n*> Fock state involves the n first derivatives of

*δ*. In spite of their mathematical interest, these distributions (p.582) do not provide a very practical representation of quantum states.

^{3}

# A.4 Phase-space representations of relaxation

The phase space description of quantum states is well-suited to the description of relaxation mechanisms. The Lindblad equation introduced in Section 4.3.2 can be expressed as a differential equation for characteristic functions or phase space distributions. The solution of these equations provides an insightful view on relaxation, particularly in the case of Schrödinger cat states.

## A.4.1 Relaxation of the normally-ordered characteristic function

We derive here first the differential equation for *C _{n}*, which takes a particularly simple form. The relations between characteristic functions (eqn. A.11) can then be used to obtain the corresponding equations for

*C*or

_{s}*C*. For the sake of simplicity, we examine only the

_{an}*T*= 0 K case and we adopt the interaction representation with respect to the oscillator's free Hamiltonian. The Lindblad master equation is then simply (eqn. 7.112):

Combining it with eqn. (A.8), we get the evolution equation for ${C}_{n}^{[\rho ]}$:

This equation involves, in the trace operation, a sum of products of *ρ* with *a* and *a* ^{†}. In order to deal with these products, it is convenient to extend the definition of *C _{n}* to these operator combinations. We start by generalizing eqn. (A.8) to an arbitrary operator

*U*and define the normal order characteristic function of this operator as:

We next consider the action on ${C}_{n}^{[U]}$ of the differential operators (∂/∂λ) and (∂/∂λ^{*}), where λ and λ^{*} are formally taken as independent quantities when evaluating the derivatives. We get for instance:

Similar relations can be established for other products of *U* with *a* and *a* ^{†} and we easily obtain simple correspondence rules linking partial derivatives of ${C}_{n}^{[U]}$ to the characteristic functions of *Ua* ^{†}, *Ua*, *a* ^{†} *U* and *aU*:

Iterating these rules, all the terms in the right-hand side of eqn. (A.72) can be put in the form of products of differential operators acting on ${C}_{n}^{[\rho ]}$ and we finally get an ordinary differential equation describing the evolution of this quantity:

This equation admits for solution the simple ansatz (Barnett and Radmore 1997):

as can be checked by plugging it in eqn. (A.76). The relaxation of ${C}_{n}^{[\rho ]}$ thus takes a simple form which can be used to derive the explicit expressions versus time of various quantum states.

The relaxation of the coherent state |*α*> provides a simple application of this general solution. Its initial normal order characteristic function given by eqn. (A.15) evolves at time *t* into:

which corresponds to the coherent state |*α* exp(−*κt*/2)>. We recover the simple result that a coherent state remains coherent in the relaxation process, with an exponentially damped amplitude. We give in Sections 3.2.5 and 4.4.4 detailed physical interpretations of this important result.

This simple formalism is also well-adapted to take into account a unitary evolution. Let us consider, for instance, the evolution of a field mode under the combined effects of its coupling to a classical source current and to an environment producing a relaxation described by eqn. (A.71). The additional term in the master equation due to the classical current is, using the notation of Section 3.1.3 [see in particular eqn. (3.45), on page 116]:

where *S* = *ε* _{0} *J* _{1}/*ħω _{c}* is the normalized current amplitude (assumed to be real without loss of generality). This commutator source term corresponds to an additional contribution to the right-hand side of eqn. (A.76), which is obtained by using again the correspondence rules (A.75). The final differential equation:

admits as an explicit solution (Barnett and Radmore 1997):

The steady state, obtained for *t* ≫ 1/*κ* is:

corresponding to a coherent state with an amplitude:

This simple expression coincides with the steady-state solution of the classical differential equation for the field amplitude *ε*:

## A.4.2 Relaxation of the Wigner function

A similar approach can be used to express the Lindblad master equation as a differential equation for the Wigner distribution *W*. We start by defining, as above, the symmetric characteristic function ${C}_{n}^{[U]}$ associated to an arbitrary operator *U* and, by Fourier transform, the corresponding operator *W*[^{U}]. With this generalized notation, the time derivative of W^{[ρ]} is:

To express this equation in terms of the Wigner function W^{[ρ]} alone, we need the algebraic correspondence rules between W^{[U]} and the four quantities ${W}^{[U{\alpha}^{\u2020}]}$, ${W}^{[U\alpha ]}$, ${W}^{[{\alpha}^{\u2020}U]}$ and ${W}^{[\alpha U]}$. These rules are:

where *α* and *α* ^{*} are to be considered as independent variables for the calculation of the derivatives. To obtain these equations, we establish first the correspondence rules linking ${C}_{s}^{[U{\alpha}^{\u2020}]}$, ${C}_{s}^{[U\alpha ]}$, ${C}_{s}^{[\alpha \u2020U]}$, and ${C}_{s}^{[\alpha U]}$ to ${C}_{s}^{[U]}$ (with the help of eqns. A.75 and A.11). These relations involve four differential operators (∂/∂λ + λ^{*}/2), (∂/∂λ^{*} − λ/2), (∂/∂λ − λ^{*}/2) and (∂/∂λ^{*} + λ/2) acting on ${C}_{s}^{[U]}$. In the Fourier correspondence between the *C _{s}*’s and the

*W*'s, these differential operators are changed into the operators appearing in (A.86).

Iterating the rules (A.86), we transform eqn. (A.85) into a differential equation of the Fokker–Planck type for *W* ^{[ρ]}(*α*), considered as a function of the two independent variables *α* and *α* ^{*}:

This equation describes the relaxation of the Wigner function of an oscillator coupled to a *T* = 0 K bath. Its right-hand side is the sum of two contributions. The firstorder derivative terms produce a global drift of the distribution towards the origin
(p.585)
and account for energy dissipation. The second-order derivative contribution represents a diffusion process which tends to broaden the distribution and prevents it from collapsing into a Dirac distribution at the origin, which would violate the Heisenberg uncertainty relations. This diffusion process blurs rapidly any small scale feature in the Wigner distribution (Habib *et al*. 1998). The evolution of a Schrödinger cat state illustrates this blurring in a striking way, as shown by the snapshots of *W* in Fig. 7.25, on page 406, where we observe the fast decay of the Wigner function interference feature. This decoherence phenomenon is analysed in detail in Chapter 7 and in the next section.

The action of an additional Hamiltonian interaction can be easily incorporated to the right-hand side of eqn. (A.87). For instance, in the presence of a classical source, it becomes:

with:

Compared to eqn. (A.87), only the drift term has changed. The Wigner distribution maximum moves now towards the point *β* in phase space and the steady state of the process is the coherent state |*β*>.

## A.4.3 Relaxation of a Schrödinger cat state

We conclude this appendix by deriving the exact expression of the density matrix of a relaxing Schrödinger cat state. This result is useful for the analysis of cat state decoherence in Chapter 7. We consider initially a superposition of two coherent states with different phases:

(where we assume that *α* is real) with:

This state generalizes the even and odd phase cats (eqn. A.57), which we recover for *ϕ* = *π*/2 and Ψ = 0 or *π*. The distance between the two coherent components amplitude in phase space is:

Setting *β* = *αe ^{iϕ}*, the initial density matrix is:

in which the first two terms describes the coherent components and the last ones their mutual coherence. We then deduce the initial normal-order characteristic function *C _{n}*:

The first two terms in the right-had side of this equation correspond to the characteristic functions of the two coherent components. The remaining complex terms correspond to the coherence in the initial field's density matrix.

For a relaxation in a *T* = 0 K bath, the characteristic function becomes at time *t*:

with:

It is easy to recognize that this characteristic function corresponds to the density operator:

The first line describes two coherent components relaxing towards the origin. The damping time constant for the amplitude is 2/*κ* = 2*T _{c}*. By comparing eqns. (A.93) and (A.97), we see that the non-diagonal coherence terms (last line in eqn. A.97) are strongly affected by relaxation. On the one hand, the quantum phase of the superposition rotates. Equal to Ψ at

*t*= 0, it becomes Ψ + $n\xaf$ sin(2

*ϕ*)(1 −

*ε*

^{2}) attime

*t*. On the other hand, the amplitude of the coherence terms decays rapidly, since it is multiplied by the factor:

Let us consider times much shorter than *T _{c}*. The exponentials in the exponent of ξ can be expanded to first order, and we get:

The decoherence time *T _{D}* is thus:

reducing to *T _{c}*/(2$\text{n}\xaf$) in the case of a

*π*-phase cat with

*ϕ*=

*π*/2. This time becomes very short for $\text{n}\xaf$ large. The experiment presented in Section 7.5.3 has revealed the dynamics of the coherence damping and the quantum phase rotation of cat states corresponding to $\text{n}\xaf$ ∼ 3–5 and

*ϕ*≈ 50 degrees.

The Wigner function of the decaying cat at time t can be derived fom eqn. (A.97). We find, in the simple case of the even and odd cats (*ϕ* = *π*/2 and Ψ = 0 or *π*):

This equation reduces obviously to eqn. (A.61) for *t* = 0, i.e. *ε* = 1.

## Notes:

(1) More detailed discussions can be found in quantum optics textbooks: Walls and Milburn (1995); Barnett and Radmore (1997); Scully and Zubairy (1997); Schleich (2001); and Vogel et al. (2001).

(2)
To simplify the notation, we omit for the time being, since no confusion is possible, the indication of the state in the expression of the *C _{s}* function and its Fourier transform. We write here

*C*(λ) instead of ${C}_{s}^{[\rho ]}(\text{\lambda})$ and

_{s}*W*(

*α*) instead of

*W*

^{[ρ]}(

*α*). We come back later to the complete notation.

(3)
One can in fact consider a wide range of quasi-probability distributions, by defining the *s*-order of operators (*s* being a continuous parameter), whose symmetric, normal and anti-normal orders are limiting cases. None of these ‘exotic’ distributions is as useful as *W* or *Q*.