Glauber P-representations for fermions

The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.


Introduction
The coherent states of light were first introduced in the context of the quantum theory of optical coherence [1][2][3][4][5][6]. These states are remarkable for representing the closest approximation, within quantum theory, to fully coherent classical light. They form, moreover, an overcomplete set of states and this property has led to their widespread application in quantum optics. Of particular interest to us is the representation of the state of a quantised field mode of the form 2 * ( ) | | () ò r a a a a a = ñ á d P , , 1 2 where |α〉 is the coherent state parametrised by the complex number α and the real function P(α) is a real quasiprobability distribution, and is a function of both α and α * [7]. This P-representation of the density operator has been widely employed in quantum optics, particularly in the evolution of open systems, and is usually introduced as an ansatz in the form of equation (1) [7][8][9][10][11][12][13][14][15]. It is possible, however, to derive this using a theorem due to Weyl [16] and presented in the context of quantum optics by Cahill and Glauber [17]. This derivation seems not to have not been widely appreciated and does not appear in most textbooks on quantum optics. We start, therefore, with a presentation of this derivation. Note, however, that not every density operator possess a P representation [18][19][20]. Before proceeding, we point out that there a number of ways of introducing phase space distributions, sometimes based on different choices of coherent states. These include working with coherent states based on particle-hole pairs [21], group theoretical methods [22] and spin or atomic coherent states [23]. We note also, that a fermionic Wigner function has been introduced [24] as has a phase space description for coupled boson-fermion systems using supercoherent states [25]. A comprehensive introduction to a wide variety of coherent states can be found in [26]. The coherent states for fermions are less familiar, certainly to those in the field of quantum optics, but are play an important role in the many-body physics and quantum field theory of fermions. In these states the complex c-numbers α for bosonic coherent states are replaced by anticommuting Grassmann variables [27][28][29][30][31][32][33][34][35].
Remarkably, however, two quite distinct forms have been reported for the corresponding P-representation of a fermionic state. When simplified to a single fermionic mode, these represent the density operator either as * ( ) | | () ò r f = ñ á d g g g g g , 2 2 or as * ( ) | | () ò r = ñ ád g P g g g g , , 3 2 where g and g * are Grassmann variables. The first of these is essentially that employed by Plimak et al in a study of Cooper-like pairing of fermions [36], and the latter was introduced by Cahill and Glauber [37]. We derive both of these and, in doing so, uncover the reason for the existence of two such representations, when only a single P representation appears for bosons. The two P functions, appearing in the above representations of the density operator, are quite distinct and this is the reason for writing the first as f(g, g * ).

P-representation for bosons
The P-representation of a density operator for a single boson mode is given in equation (1). Our task is to derive this expression and, in doing so, to arrive at the correct form of the function P(α, α * ). Note that we do not take the more familiar path of supposing this relation to be true and deriving the requisite properties of P(α, α * ) from this assumption. We follow closely the method of Cahill and Glauber [17] in deriving, first a completeness relation for the displacement operators, and then using this to derive equation (1). We start with just a single mode, but generalise to multi-mode states at the end of the section. For simplicity we consider only one type of boson.

Completeness of the displacement operators
The Glauber displacement operator has the form [7] which can be written in the normally or antinormally ordered forms: a  a  a  a  a  a  =  --=  -D  a  a  a  a  ,  exp  exp  exp  2  exp  exp  exp  2 . 5 2 2 This unitary operator acts to generate the coherent state |α, α * 〉 from the vacuum (or zero particle) state: , 0. 6 To condense the notation, we follow common practice and replace |α, α * 〉 and * ( ) a a D , by |α〉 andˆ( ) a D respectively. Our first task is to show that an operatorF can be written in the form corresponds to integrating the real and imaginary parts of over the whole of the complex ξ plane. Here the function f (ξ) is related to the operatorF by We can think of the expansion in equation (7) as the operator analogue of a Fourier expansion or transform, in which the operator is expanded in terms of a complete set of displacement operators in place of a complete set of functions of the form e ikx [16]. The representation of the operatorF in equation (7) is possible when f (ξ) is square-integrable or, equivalently, when the operatorF has finite Hilbert-Schmidt norm: To proceed with the proof, we follow Cahill and Glauber [17] by introducing a function of four complex variables It is straightforward to use the normally ordered form of the displacement operator, equation (5), and the overlap of two coherent states [7] to write this function in the form As the coherent states are complete (or more precisely overcomplete [7,38]), we have an identity between two operators if their coherent state matrix elements are identical. It follows, therefore, that we can write the outer product of operator |α〉〈β| in the form Any bounded operator,F, can be written in the form where we have used the resolution of the identity operator in terms of coherent states: It then follows thatˆ|ˆ( Finally, we note that the integral over β corresponds to the trace and so we find , 16 2 which is the expression in equation (7). For fermions we shall find and make use of a similar theorem, but in that case the final step leading to a trace operation does not hold.

Expressing the density operator in terms of the P-function
We can complete our derivation of the P-representation of the density operator by applying the general result in equation (7) to the density operator. This is allowable because the density operator will always have a finite Hilbert-Schmidt norm: ( )  r Tr 1 2 . Direct application of our theorem gives the form . 17 2 2 2 To obtain the P-representation form from this we first write the displacement operators in an ordered form, normal order for the first and antinormal order for the second: All that remains is to insert the identity in the form given in equation (14) between the final two operators: where the P function is Note that this function is the Fourier transform of the normally ordered characteristic function [7,18] T r 2 1 a a and the P-function will be well-behaved only if this Fourier transform exists. In the study of non-classical states of light this feature has often been used to distinguish between classical and intrinsically quantum properties of light [14]. We note that the expectation values of normally ordered functions of the creation and annihilation operators are readily expressed in terms of the P-function. In general we have We note, in particular, that this expression includes the normalisation of the P-function (for n = 0 = m): We shall encounter similar expressions for our fermionic P representations. It is often the case that we need to describe multimode field states and for this purpose we need a more general form for our P-representation. To this end we introduce a complex number, α i , for each of the modes. The completeness relation for the boson coherent states is where α is a shorthand for α 1 , α 2 , L α n , so that |α〉 is the multimode state |α 1 〉|α 2 〉 L |α n 〉 and d 2 α = ∏ i d 2 α i . The multimode P-function will then be a function of all of the c-numbers α i and * a i . The required form for an n mode state is simply The form of this multimode P-function is the natural analogue of the single-mode form given in equation (20): Tr exp exp exp , 26 with similar expressions for the other terms It is straightforward to verify that the multimode P-function is normalised: ∫d 2 αP(α, α * ) = 1. The normally ordered moments are given by the integral of P(α, α * ) multiplied by a product of the α i and * a j , with these variables replacing the annihilation and creation operatorsâ i andˆ † a j respectively.

P-representations for fermions
The title of this section deliberately employs the plural 'representations' as the construction we have employed above gives rise, in the fermionic case, to more than one expression for any given density operator. We shall find that the origin of this non-uniqueness lies in the existence of more than one integral representation of the identity in terms of fermionic coherent states. We follow closely the approach outlined in the preceding section for bosons, starting with a completeness relation for fermionic coherent states and following this with a derivation of the P-representations. As with the bosonic case, we begin with just a single mode before generalising to multi-mode states at the end of the section. Again, for simplicity, we consider only systems with one type of fermion.

Grassmann variables and fermionic coherent states
Before proceeding with our derivation of the P-representation, we present a brief review of the principal properties of Grassmann variables and of the coherent states constructed with them. Fuller accounts can be found in [27,[35][36][37][39][40][41]. Grassmann variables are, in effect, anti-commuting 'numbers'. 3 If g and h are any two Grassmann variables then which implies that the square of any Grassmann variable is zero. It follows that expressions containing an even or an odd number of Grassmann variables behave rather differently: an even expression commutes with other Grassmann variables, but an odd one anti-commutes: g 1 g 2 g 3 = g 2 g 3 g 1 , but g 1 g 2 g 3 g 4 = − g 2 g 3 g 4 g 1 .
We can define complex conjugation for Grassmann variables by introducing g * as the complex conjugate of g. It is convenient to define this operation to reverse the order of the variables: which is reminiscent of the Hermitian conjugation operation for matrices and operators. We can define differentiation and integration for Grassmann variables [27,35]. We shall require only Grassmann integration, for which the rules are When there is more than one Grassmann variable involved we need to be careful with the order both of the variables in the integrand and also with the differentials: We denote the fermionic annihilation and creation operators byĉ andˆ † c , which satisfy the anticommutation relation The two possible fermionic number states are the vacuum, or no particle, state |0〉 and the one particle state |1〉: The properties of the annihilation and creation operators are complicated by the fact that they anticommute with the Grassmann variables in thatˆˆˆˆ( These features mean that we need to be careful with the ordering of both our Grassmann variables and our operators.
The fermionic coherent states are characterised by a Grassmann variable g rather than a c-number as in the bosonic case. Specifically, they are generated from the vacuum state by means of a unitary transformation generated by a fermionic displacement operator: 3 4 which can be expressed in normal or antinormal ordered forms as We note that these have same form as their bosonic counterparts. As with their bosonic counterparts, equation (6), the fermionic coherent states are given by Again we simplify the notation by using |g〉 andˆ( ) D g . It follows that our fermionic coherent states have the form The coherent states are right-eigenstates of the annihilation operator with eigenvalue g: and are left-eigenstates of the creation operator with eigenvalue g * : The overlap between two fermionic coherent states is We can write this in terms of exponential functions which has the same form as for the bosonic coherent states [7]. We may require that the vacuum state is an even Grassmann function so that h|0〉 = |0〉h. It follows that the one particle state, |ˆ| † ñ = ñ c 1 0, is odd, and it follows that our coherent states are even Grassmann functions and therefore commute with Grassmann variables: Finally, we can resolve the identity in terms an integral over the coherent states: which is the natural analogue of the bosonic expression given in equation (14). In contrast with the bosonic case, however, this is not the only way to resolve the identity in terms of the coherent states. In particular we can also express the identity as an integral over the operators |g〉〈 − g| in the form We shall find that it is this existence of a second resolution of the identity that leads to the existence of two P-representations for fermions. 4

Completeness of the displacement operators
We follow the derivation of the completeness of the bosonic displacement operators, given above, by introducing the Grassmann function Careful evaluation of the matrix elements and Grassmann integrals gives the same result as found for bosons: To proceed we can insert the identity operator either before or after the displacement operatorˆ( ) D h and use the fact that the overlap between two coherent states is an even Grassmann function to write where k and its complex conjugate k * are a further pair of Grassmann variables. We can extract a general relationship for the four independent single-mode operators, |0〉〈0|, |0〉〈1|, |1〉〈0| and |1〉〈1|, and hence for any operator. We can achieve this either by comparing the coefficients of the Grassmann variables or, more formally, by Grassmann integration. It follows that for a general single-mode operator,F we have the identity which has the same form as that given in equation (15) for a bosonic operator. Note, however, that unlike in the bosonic case, the k-integral does not, in general, correspond to the trace of the operator productˆˆ( ) FD h .

c h h c h c c h
At this point we can insert the identity operator, expressed in terms of coherent states, between the final two operators and hence obtain expressions for our two Grassmann functions f(g, g * ) and P(g, g * ). For f(g, g * ) we insert the identity as given in equation (43) so that Comparing this with the form ∫dg * dg f(g, g * )|g〉〈g| leads, directly, to an explicit form for the Grassmann function f(g, g * ): .

3 c h h c h g g h
The second possibility is to insert the identity given in equation (44): If we compare this with the form ∫dg * dg P(g, g * )|g〉〈 − g| then we obtain an explicit form of the Grassmann function P(g, g * ): which clearly differs from the function f(g, g * ). The two P functions are simply related to each other, however: That this is satisfactory follows directly from the condition that (−2gg * − 1)(2gg * − 1) = 1.

Properties of the two P representations
We have two quantities, f(g, g * ) and P(g, g * ), that are the analogues for fermions of the bosonic quasiprobability distribution P(α, α * ). Like P(α, α * ), we cannot expect either of them to be a true probability distribution. This is, perhaps, yet clearer for our fermionic functions as they depend on Grassmann variables, which have no interpretation as physical properties of the fermions. Despite this, we can treat both f(g, g * ) and P(g, g * ) in a manner analogous to probability distributions; in particular we can use them to evaluate occupation probabilities and, in multimode situations, also correlation functions. We examine here these features of f(g, g * ) and P(g, g * ).

Normalisation?
The integral of P(α, α * ) over the whole of the complex plane gives unity: 2 and although P(α, α * ) can take negative values, this normalisation is indicative of its role as a quasiprobability distribution [7]. We can use our explicit expressions for f(g, g * ) and P(g, g * ) to determine whether either of these is normalised under Grassmann integration. Let us start with f(g, g * ): which is not the trace of the density operatorr and so is not generally equal to unity. Moreover, if the single particle probability exceeds the vacuum probability then this integral will be negative. In order to fulfil the role of a quasiprobability distribution, we need to introduce a weight function to include on the normalisation and moment integrals. It is straightforward to show that this weight function is so that the product w(g, g * )f(g, g * ) is normalised: The necessity of introducing a weight function appears, also, in the bosonic case, where the identity is given by | |ò a a ñá = a p I d 2 but the normalisation condition requires the weight function π: For the fermions we have *| |ò ñá = dgdg g g I but ∫dgdg * w(g, g * )f(g, g * ) = 1. For P(g, g * ) we find The function P(g, g * ) has the normalisation attribute of a quasiprobability distribution, while f(g, g * ) requires the addition of the weight function w(g, g * ) to produce a normalised Grassmann function.

Evaluating moments
For our single mode problem there are only two possible distinct moments. These are the zeroth moment,á ñ I , and the first moment,ˆ † á ñ c c , of the particle number operator. Superpositions of zero and one fermion are not allowed and soˆ † á ñ c andá ñ c are always zero. We have two coherent-state representations of the density operator for our single-mode state, and we can use either of these to evaluate moments.
We consider, first, the zeroth momentá ñ I , which must equal unity. For the representation in terms of f(g, g * ) we have Note that the weight function, w(g, g * ) = 2gg * + 1 appears naturally in this expression. Here we have used the fact that f(g, g * ) must be an even Grassmann function and so be formed only of products of even numbers of Grassmann variables. Were this not the case, then we would have non-zero expectation values ofĉ and/orˆ † c . For our Grassmann integral to equal unity (as it must) f(g, g * ) must have the highly restricted form where u is a constant. For the representation in terms of P(g, g * ) we have ò ò á ñ = á ñá-ñ + á ñá-ñ = dg dg P g g g g g g dg dg P g g which is the normalisation condition derived above. As with f(g, g * ), this condition restricts P(g, g * ) to a simple form where v is a constant. The simple forms of f(g) and P(g) arise because the allowed states of a single fermionic mode are limited to statistical mixtures of the vacuum and one particle states:ˆ( )| | | | r = -ñá + ñá p p 1 0 0 1 1. These forms follow also directly from equations (53) and (55) for a density operator given by this statistical mixture. We see that both P(g, g * ) and f(g, g * ) are even Grassmann functions as a consequence of this superselection rule.
The first order momentˆ † á ñ c c is the probability, p, that a single fermion is present in the mode. For f(g, g * ) we find c c dg dg w g g g g g g c c dg dg w g g g g c g g c dg dg gg g g gg dg dg g g gg For this to equal the single fermion probability we require u = p so that f(g, g * ) = p + (2p − 1)gg * . If we use the representation in terms of P(g, g * ) then we find * * * * * *ˆ( c c dg dg P g g g c c dg dg P g g c g g c dg dg P g g g g Tr Tr , , . For this expression to be the single fermion occupation probability we require v = − p so that P(g, g * ) = − p + gg * . We note that, in this single mode case, the forms of the moments for our two suitably weighted quasiprobability distributions, w(g, g * )f(g, g * ) and P(g, g * ), differ only by the ordering of the Grassmann variables g and g * . For the bosonic case, α is a c-number and so there is no distinction between α * α and αα * .

Multimode multiparticle states
The Pauli exclusion principle means that single-mode states are limited to the presence or the absence of a single particle. Most physically interesting states are therefore multimodal. Cold atom states, for example, often span a very large number of modes. To describe such states we require multimode generalisations of the expressions derived above. The two important completeness relations, equations (43) and (44) where f (g, g * ) is which is clearly an even Grassmann function, and The multimode coherent states are generated from the vacuum (or zero particle) state by the displacement operator * ( ) D g g , : Here the vector quantities g andĉ denote {g 1 , g 2 , L ,g i , L } and {ˆˆˆ}   c c c , , , , respectively. To condense the notation, |g, g * 〉 and * ( ) D g g , will be replaced by |g〉 andˆ( ) D g . It follows that our multimode coherent states and displacement operators are  We note that the super-selection rule ensures that P(g, g * ) is also an even Grassmann function. It follows that our two multimode quasiprobability distributions are related by the consistency of which follows from the fact that f (g, g * )f (g, − g * ) = 1.
As with the single-mode case, the function P(g, g * ) is normalised but f(g, g * ), in general, is not: where F(g, g * ) is a general Grassmann function, which we may split into the sum of an even and an odd Grassmann function. These contain products of only even or odd numbers of Grassmann variables respectively: , , e o 5 The range of possible matrix elements depends very much on the physical system being modelled. This is particularly true if we consider states of more than one type of fermion. For cold atoms, the number of atoms is a conserved quantity and so the only non-vanishing matrix elements will be those between states with the same number of atoms. In relativistic quantum theory, an intense electric field can create an electron-positron pair and thereby increase the number of fermions present by two. Similarly, in models of solids, an external electromagnetic field may act to create an electron-hole pair, again increasing the number of effective particles present by two. If this is a coherent process, then the state can evolve to one in which there is a superposition of the vacuum state and a two particle state.
As we have seen, the coherent states are even and so commute with Grassmann variables and it follows that . 8 9 2 To proceed, it is simplest and clearest to specialise to a single mode, before generalising to the full multimode problem. For a single-mode operator,F 1 , we have where we have used the fact that there are two Grassmann integrations and so the integral of any odd Grassmann function, such as (1 + gg * )F 1,o (g, g * ) and gF 1,o (g, g * )g * , must be zero. For the multimode case, we can write the trace ofF, in equation (89), as For the present case (see equations (85) and (87)) F(g, g * ) is given by * ( ) ( )  g g P g g , m m q 1 (( ) ( ))  -g g l l p 1 . The physical constraint of the super-selection rule on fermion number requires P(g, g * ) to be an even Grassmann function. It follows, therefore, that F(g, g * ) is an even function if p and q differ by an even number and F(g, g * ) is an odd function if they differ by an odd number. Hence * * * ( ) ( ) ( ) ( )(( ( ))   = = --F F g g P g g g g g g g g , , , if p − q is even and F e (g, g * ) = 0 if p − q is odd. Returning to our correlation function we therefore find