Gaussian Operator Bases for Correlated Fermions

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Gaussian is an exponential of a quadratic form, with the basis set that includes mixed as well as pure quantum exponential deﬁned by its series representation. We fol- states. Second, the Gaussian operator basis states are low the standard convention that fermionic normal order- not orthogonal. Third, we also allow the basis set to ing includes a sign-change for every operator exchange, include operators which are not themselves density oper- † † so that : bkbj := bjbk. Normal ordering is utilized here ators. These additional degrees of freedom prove useful because it allows− us to most directly use Fermi coherent in obtaining the requisite completeness properties. We state methods[16]b b b involvingb Grassmann algebra[17][18], now give the mathematical form of these basis operators, which are described in greater detail in the Appendices. in unnormalised form; normalized forms ΛM , for which

We note that from particle-hole symmetry, it is also pos- T r ΛM = 1, will be introduced in Sec. V. sible to obtain an entirely analogous anti-normally or- b dered representation, which simply exchanges the roles b of particles and holes in the formalism. A. Number-conserving Gaussians The most general Gaussian form is a cumbersome object to manipulate, unless products of odd numbers of We first consider the special case obtained when the operators are excluded. Fortunately, restricting the set † of Gaussians to those containing only even products can quadratic is of form bi bj , so that it is number-conserving. be physically justified on the basis of superselection rules The most general, number-conserving Gaussian operator for fermions. Because it is constructed from pairs of oper- can be written as: b b ators, this type of Gaussian operator contains no Grass- mann variables[16][19]. Λ(u)(µ) = :exp b†µb : , In order to relate these quadratic expressions to the M − usual Dirac state-vector notation in the examples that M h i b b b † follow, we define number states as: = : exp b µij bj : − i i,j=1 Y h i ~n = n1 n2 ... nM M b b | i | i ⊗ | i ⊗ | i n1 n2 nM ˆ † † † † = : 1 bi µij bj : (2.5) = b1 b2 ... bM 0 M , (2.2) − | i i,j=1 Y h i b b where ~n = (n ,nb,...,nb ) is a vectorb of integer occu- where µ is a complex M M matrix. The last result 1 2 M × pations and where 0 M is the M-mode fermion vacuum follows because normally ordered products in which the state. | i same operator appears more than once are zero, from the We also recall some elementary identities which relate anti-commutation relations in Eq. (2.1). For the special quadratic forms in fermion operators to state projection case that µ is a hermitian matrix, the Gaussian operator operators. For individual operators, one has the well- is just the density operator of a thermal state. If the known identities: eigenvalues of µ are also either 0 or 1, then the Gaussian corresponds to a Slater determinant (i.e. a product of bj = 0 1 , single-mode number states). | ij h |j It should be noted here that we do not restrict the b† = 1 0 . (2.3) bj | ij h |j Gaussian operators of this type to be just thermal states. In general we would wish to consider density matrices Hence, the elementaryb identities for quadratic products are as follows, for i < j : that are linear combinations of Gaussian operators, and these can exist as hermitian, positive-definite operators † even when composed of Gaussians that have neither prop- b bi = 1 1 1ˆk i | ii h |i erty. k i Y6= b b† bib = 0 0 1ˆk i | ii h |i k i B. Non-number-conserving Gaussians Y6= b†b b bj = 1i, 0j 0i, 1j 1ˆk i | i h | k i,j If anomalous products of form bibj are included as well, Y6= b b then the most general non-number-conserving Gaussian bjbi = 0i, 0j 1i, 1j 1ˆk , (2.4) | i h | is a type of squeezed state: b b k i,j Y6= b b (u) + † 1 + † † ΛM (µ, ξ, ξ ) = :exp b µb bξ b + b ξb : where 1ˆk is the identity operator for the k th mode. − − 2 − Expanding quantum states in terms of an underlying † † † b = 1 bξijˆbbˆb : b 1b µbijˆb bˆbj : basis set is a widespread procedure in quantum mechan- − i j − i × i>j ij ics. However, using the general Gaussian operators in Y h i Y h i + this capacity is nonstandard in several respects. First, exp 1 ξ ˆbiˆbj . (2.6) × − ij one is expanding a quantum density operator over a i>j Y h i 3

+ Here ξ, ξ are complex antisymmetric M M matrices. If n is real, then Eq(3.4) shows directly that Λ1(n) is the For the special case that µ is a hermitian× matrix and density operator corresponding to a mixture of number ξ† = ξ+, then the Gaussian operator is just the density states in the one-mode case. The expectationb value of operator for a squeezed thermal state. Normalized forms the fermion occupation n b†b is just the variable n. ≡ ΛM , for which T r ΛM = 1, will be introduced later. b b b As before, we do not restrict the Gaussian operators to 1. Completeness and positivity beb just the squeezedb thermal states, even though this rep- resents a large and physically important class of fermionic density operators. By extending the definition to include As two special cases we obtain the number states: all exponentials of quadratic forms, we can obtain a more 0 0 = Λ (0) useful set, which is a complete basis with a positive- | ih | 1 definite representation for all density operators, as we 1 1 = Λ1(1), (3.5) | i h | show in Sec. V. b which implies that just these two single-mode Gaussians b form a complete basis set for a number-conserving subset of Hilbert space. Because super-selection rules pro- III. ONE AND TWO MODE GAUSSIAN OPERATORS hibit superpositions of states differing by odd numbers of fermions, this is the most general case possible. Further- more, we see from Eq (3.5) that the most general single- In this section we give examples of Gaussian operators mode density matrix can be expanded as a mixture of in elementary one and two-mode cases. These are suf- Gaussians with positive coefficients, since 0 n 1: ficient to illustrate the basic identities and ideas. More ≤ ≤ general results will be given in Sec. V. ρ = (1 n)Λ (0) + nΛ (1) . (3.6) − 1 1 Additionally in the one-mode case, all physical density b b b A. Single-mode Gaussian operators operators are also Gaussian operators ρ = Λ1(n), which is another proof of positivity and completeness. An unnormalised Gaussian operator for a single mode It is clear from these examples that theb Gaussianb oper- overcomplete has only one possible form: ators are : a physical density matrix may be represented by more than one positive distribution over the Gaussians. The most general single-mode Gaussian Λ(u)(µ) = :exp µb†b : 1 − operator, with n complex, provides an even larger over- ∞ h1 i k complete basis for physical density matrices, even though b : b µbb†b : , (3.1) such Gaussians are not always physical density matrices ≡ k! − k=0 themselves. For example, any uniform-phase distribution X b b over Gaussian states gives the zero-particle state: where the exponential is defined, as indicated, by its series representation. Just as in the general case of Eq (2.5), 0 0 = dφΛ (reiφ). (3.7) the anticommutivity of the fermionic operators means | ih | 1 that only the zeroth and first-order terms in the expan- Z From the usual hole-particle symmetryb of fermion states, sion contribute to the single mode Gaussian, giving the the single-particle state is similarly obtained from: simple result: iφ (u) † 1 1 = dφΛ (1 re ). (3.8) Λ (µ) = 1 µb b. (3.2) | ih | 1 − 1 − Z b The normalisationb of this Gaussianb b operator is B. Two-mode number-conserving Gaussian N TrΛ operators 1 ≡ 1 = 2 µ. (3.3) −b A straightforward extension of the generalized thermal Excluding the point µ = 2, we define the new parameter Gaussian form to two modes gives n = (1 µ)/(2 µ), which allows us to write the Gaussian − − 2 in normalised form as (u) † Λ (µ) = :exp µij b bj : 2 − i  i,j=1 X Λ (n) = (1 n) : exp (2+1/(n 1)) b†b : b 2 b b  1 − − − † † † = 1 µij bi bj + det µb1b2b2b1, † h † i − = (1 n)bb + nb b i,j=1 b − b b X = (1 n) 0 0 + n 1 1 . (3.4) b b b b b b (3.9) − b|b ih | b b | i h | 4 where the µij are the elements of the 2 2 matrix µ. The Because the matrix n is Hermitian for a Gaussian that last step follows by explicitly expanding× the general re- is a density operator, it can be diagonalised, correspond- sult in Eq (2.5), while taking account of the sign changes ing to a change of single-particle basis. In this diago- during normal ordering. Again, the series expansion con- nalised basis, since the coherences are zero, the density tains all possible normally ordered nonzero products of operator is a mixture of number states, totally charac- † the bi bj pairs. In terms of two-mode number-state pro- terised by average occupation numbers. In other words, jectors, the Gaussian operator is these Gaussian operators correspond to two-mode ther- b b mal states. Λ(u)(µ) = 00 00 + (1 µ ) 10 10 2 | i h | − 11 | i h | + (1 µ ) 01 01 − 22 | i h | 2. Completeness b + (1 µ µ + det µ) 11 11 − 11 − 22 | i h | µ12 10 01 µ21 01 10 . (3.10) We wish to show first that any number-conserving two- − | i h |− | i h | mode density matrix can be expanded in terms of Gaus- sian operators, and second that this can be done with 1. Normalisation and Moments positive expansion coefficients. The first result follows if we can represent all the number-state projectors be- Following from Eq. (3.10), the normalisation is tween states of the same total number using Gaussian operators. By inspection of Eq (3.12) above, we find N = 4 2µ 2µ + det µ. (3.11) that: 2 − 11 − 22 T −1 00 00 = Λ (0) Defining the matrix n = I 2I µ , where I is | i h | 2 − − 1 0 the 2 2 identity matrix, we can write the normalised 10 10 = Λ ( ) two-mode× Gaussian as | i h | b2 0 0 † T −1 b 0 0 Λ (n) = det [I n] : exp b 2I + n I b : 01 01 = Λ2( ) 2 | i h | 0 1 − − − = det [I n] 00 00h i 11 11 = Λb2(I) b − | i h |b b | i h | + (n11(1 n22)+ n12n21) 10 10 n1 0 n1 0 − | i h | n21 10 01 = Λ2( ) Λ2( ) + (n22(1 n11)+ n12n21) 01 01 | i h | b n21 n2 − 0 n2 − | i h | + det n 11 11 + n 10 01 + n 01 10 . 21 12 b n1 n12 b n1 0 | i h | | i h | | i h | n12 01 10 = Λ2( ) Λ2( ) (3.12) | i h | 0 n2 − 0 n2 . (3.15) If n is an Hermitian matrix, then the two-mode Gaus- b b sian corresponds to the density matrix of a mixture of Thus, the two-mode Gaussians form a complete operator states of different total number, with coherences n12 = basis for all number-conserving density matrices. ∗ n21 between states of the same total number. Normally ordered first-order correlations of the Gaus- sian density matrices correspond to elements of n: 3. Positivity

† † bi bj Trbi bj Λ2 = nij , (3.13) Not only is the two-mode Gaussian a complete repre- Λ ≡ D E sentation, but it is also a positive one: any two-mode and higher-orderb correlationsb b b reduceb b to products of first- number-conserving density operator can be written as order averages, for example, a positive distribution over Gaussian operators. To see this, note that while the expression for the projection † † operators, Eq (3.15), includes terms with negative coeffi- b1b1b2b2 = n11n22 n12n21. (3.14) Λ − cients, the projectors involved are the off-diagonal ones. D E Since density matrices are positive-definite, off-diagonal This kind ofb factorisationb b b b of higher-order correlations projectors can only occur in combination with diagonal could be taken as the defining characteristic of a Gaus- projectors. We take this into account in what follows. sian state, and more generally, Gaussian operators, Any two-mode density operator can be expanded into rather than the more formal operator definition given number-state projector operators as follows: by Eq. (3.9). In other words, Gaussian operators have both a Gaussian form and generate Gaussian statistics. ′ ρ = ρ~n,~n′ ~n ~n , (3.16) The connection between these two defining features can | i h | ~n ~n′ be made more explicit by use of a moment-generating X X function, or characteristic function, which is considered where ~n and ~n′ areb vectors of integer occupation numbers: ′ ′ ′ in Appendix B. ~n = (n ,n ), ~n = (n ,n ). Here ρ~n,~n′ = 0 if nj = 1 2 1 2 j 6 P 5

′ j nj , because of total-number conservation. Using the C. Two-mode squeezed Gaussian operators relations in Eq. (3.15), we can write the density operator P as Equation (3.9) does not represent the most general

1 n1 2ρ(01),(10) two-mode Gaussian form, as the quadratic form does ρ = ρ~n,~n Λ2( ) 2 0 n2 not yet include terms such as b1b2. Incorporating such X~n anomalous products, we can write the most general Gaus- b bn1 0 sian operator in normalised form as: +Λ2( ) . (3.17) b b 2ρ(10),(01) n2 2 Since the diagonalb coefficients ρ~n,~n are positive and sum (u) + † + † † Λ (µ,ξ,ξ ) = :exp µij b bj ξ b b ξb b : to one, the Gaussian operators form the basis of a proba- 2 − i − 1 2 − 2 1 i,j=1 bilistic representation of any two-mode density operator. X While any two-mode number-conserving density ma- b 2 b b b b b b  † † † + = 1 µij b bj ξb b ξ b b trix can be expanded in the form in Eq. (3.17), there − i − 2 1 − 1 2 i,j=1 are often more direct representations. For example, X as Eq. (3.20) below shows, the entangled state φ = b +b † b† b b b + det µ + ξξ b1b2b2b1, (3.21) α 10 +β 01 can be represented by just one term, rather| i than| i the four| i terms that result from Eq. (3.17). where ξ and ξ+ are independent complex numbers. The b b b b † † two additional operator terms in the expansion, b2b1 and b1b2, are projectors between states of different total num- 4. Correlation and entanglement ber: b b b b We first note that any uncorrelated product of number b1b2 = 00 11 , − | i h | state mixtures can be represented exactly: b†b† = 11 00 , (3.22) b2b1 − | i h | ρn1 ρn2 (1 n1) 0 0 + n1 1 1 ⊗ ≡ { − | ih | | ih |} which are the kinds of coherences that appear in the den- (1 n ) 0 0 + n 1 1 b b ⊗{ − 2 | ih | 2 | ih |} sity matrices of two-mode squeezed states. b b n1 0 = Λ2( ). (3.18) 0 n2 1. Normalisation and Moments As well as these uncorrelatedb mixtures, the Gaussian basis can also be used to represent a mixture with correlations between the modes, this time as a sum (with posi- The normalisation of the squeezed Gaussian is tive weights) of two terms: N = 4 2µ 2µ + det µ + ξξ+. (3.23) 2 − 11 − 22 A 00 00 + B 11 11 = AΛ (0)+ BΛ (I). | i h | | i h | 2 2 To incorporate this into a normalised Gaussian, we rede- (3.19) fine the n matrix to be b b Importantly, superpositions of number states, corre- N2 −1 sponding to entangled states, can also be represented, n = I 2I µT , (3.24) − N ξξ+ − subject to total-number conservation. For example, the 2 − density matrix corresponding to the state φ = α 10 + and introduce rescaled squeezing parameters m = | i | i β 01 is ξ/N , m+ = ξ+/N . The normalised form is then | i − 2 − 2 2 ∗ φ φ = α 10 10 + αβ 10 01 n + I n + | i h | | | | i h | | i h | Λ2( ,m,m ) = det [ ]+ mm +α∗β 01 10 + β 2 01 01 − × | i h | | | | i h | † T −1 b 2 ∗ b : exp b b I σ /2 :, α α β − b†T = Λ2( | |∗ 2 ). (3.20) αβ β b | | b b (3.25) Thus the number-conservingb Gaussian operators even in- b where the 4 4 matrices I and σ are defined to be clude entangled density matrices, making them a power- × ful tool for representing highly correlated states. 1 0 00 This property of being able to represent such Bell-like − entangled states with the hermitian subset of Gaussian 0 1 0 0 I =  −  , operators, is different to the case of Gaussian expan- 0 0 10 sions for bosons. A typical example of this type of non-  0 0 01  classical representation is the positive P-representation[7,  n 1 n  0 m 11 − 21 8], which must use a non-hermitian basis to obtain a pos- n12 n22 1 m 0 σ = −+ − .(3.26) itive distribution that can represent all two-mode density  0 m 1 n11 n12  matrices, such as those that violate a Bell inequality[20]. + − − −  m 0 n21 1 n22   − −  6

In terms of number-state projectors, the normalised over Gaussian operators: Gaussian is a generalisation of the number-conserving ′ case, but with the additional non-number-conserving pro- ρ = ρ~n,~n′ ~n ~n jectors: ′ | i h | X~n X~n b 1 I n + = ρ~n,~n Λ2(σ) = det [ ]+ mm 00 00 2 − | i h | ~n + X + n11(1 n22)+ n12n21 mm 10 10 b − − | i h | n1 2ρ(01),(10) + Λ2( , 2ρ(11,00), 0) + n22(1 n11)+ n12n21 mm 01 01 × 0 n2 − − − | i h | n + + det + mm 11 11 b n1 0 | i h | +Λ2( , 0, 2ρ(00,11)) (3.31). +n 10 01 + n 01 10 2ρ(10),(01) n2 − 21 12 | i h | + | i h | m 11 00 m 00 11 . (3.27) b − | i h |− | i h | Thus the Gaussians form the basis of a probabilistic representation of any physical two-mode density operator. In addition to the normal fluctuations of Eq. (3.13), the The ability to represent physical density matrices with squeezed Gaussians also contain anomalous fluctuations, + the Gaussian basis, either as a single element or by a posi- which are just equal to the new variables m and m : tive distribution over basis elements, is important for cal- culating dynamical simulations of quantum systems via b1b2 = m, probabilistic methods. Λ D † †E + b2b1 b = m , (3.28) Λ D E 4. Entanglement b b b + which implies that for Λ2(n,m,m ) to be a density matrix, m and m+ must be complex-conjugate. The second- Again, these types of Gaussians can directly corre- order correlation generalisesb to spond to entangled density matrices, without having to consider expansions over several elements. But now † † + superpositions of different total number can be repre- b1b1b2b2 = n11n22 n12n21 + mm , (3.29) Λ − sented. As an example of these additional kinds of physi- D E cal states that the squeezed Gaussians can represent, con- which againb b correspondsb b b to the decorrelation that occurs sider the entangled non-number-conserving superposition in a state with Gaussian statistics. state: ψ = [ 00 + 11 ] /√2. This can be represented by a single| i Gaussian:| i | i 1 2. Completeness ψ ψ = [ 00 00 + 00 11 + 11 00 + 11 11 ] | i h | 2 | i h | | i h | | i h | | i h | 1 1 1 From Eq. (3.27), it follows that the squeezed Gaussians = Λ ( I, , ). (3.32) 2 2 2 2 provide a complete two-mode fermionic basis, not only for − − the number conserving subspace, but also for all states Hence, the moreb general type of Gaussian basis element containing superpositions of states whose difference in considered here is considerably more powerful for repre- total number is even. To see this, note that the projectors senting entangled and correlated states than the number- between number the 00 and 11 number states can be conserving basis set. written explicitly in terms| i of the| i Gaussian operators as

m 11 00 = Λ (n, m, 0) Λ (n, 0, 0), IV. MULTIMODE DECOMPOSITION OF | i h | 2 − − 2 + + FERMI SYSTEMS m 00 11 = Λ2(n, 0, m ) Λ2(n, 0, 0), | i h | b − −b (3.30) b b So far we have considered only one- and two-mode systems, to illustrate the basic physical properties of Gaus- for any n. These projectors, together with those of sian operators. As we saw, the Gaussian operators could Eq. (3.15), span the complete Hilbert space of density easily be written in terms of number-state projectors. matrices in question. The power of the Gaussian operators as a basis in its own right becomes apparent for multimode systems, for which the number-state basis becomes unusable. 3. Positivity Before defining the general Gaussian basis for a system with many degrees of freedom, we define some mathemat- Just as for the number-conserving case, we can write ical notation and conventions that will be of subsequent any physical density operator as a positive distribution use. As before, we define b as a column vector of the M

b 7 annihilation operators, and b† as a row vector of the cor- where I is the constant matrix responding creation operators. We also introduce an extended column vector of all 2bM operators: b = (bT , b†)T , I 0 † I − , (4.5) † T ≡ 0 I with an adjoint row vector defined as b = (b , b ). Writ- ing these out in full, we get: b b b in which 0 and I are the M M zero and identity ma- b b b trices, respectively. × b1 . When ordering products that contain a Gaussian oper-  .  b ator Λ (and later the density operator), we do not change  bM  † † † the ordering of Λ itself; the other operators are merely b =  †  , b = b1, ..., bM , b1, ..., bM . (4.1)  b1  b † †   reordered around it. Thus : Λ bi : bj = bj bi Λ. The  b.  b − b  .  b b b b b different possible quadraticn orderings containingo a Gaus-  b†   b  sian operator can thus be writtenb b inb matrixb formb b as  M    Throughoutb the paper, we print length-M vectors and M M matrices in bold type, and index them where T † b†T ΛbT ΛbbT necessary× with Latin indices: j = 1, ..., M. Length-2M : b b Λ: = − ,  †T † †T T  vectors we denote with an underline and 2M 2M matri- b b Λ b Λb × b bb bbb ces we denote with a double underline. These extended bb b   b b† bbT † Λb b Λb b vectors and matrices are indexed where necessary with b b Λ = b Tb , Greek indices: µ = 1, ..., 2M. Note that an object such  Λb†T b† bΛb†  † n o bbb − bb b as b b is a 2M 2M matrix: bb b  † T × † bb bbb Λ bbb bΛb T T † bb† bbT b : b Λ: = †T † † , bb b b = , (4.2)  b Λb Λbb  b†T b† b†T bT n o − bb b − bbb b b b  T  bb bb † T bb † Λbbb bb bΛb bbb † † b b † b b : Λb : b = − . (4.6) whereas b b = k bk bk + bkbk(= M) is a scalar. More  T  b†T b† bb† general kinds of vectors are denoted with an arrow nota- n o bbΛb bbbΛ P  −  tion: −→µ .b b b b b b bb b   For products of operators, we make use of normal and b bb bb b antinormal ordering concepts. Normal ordering, denoted V. NORMALIZED GAUSSIAN OPERATORS by : : , is defined as in the bosonic case, with all annihilation··· operators to the right of the creation operators, A. Definition except that each pairwise reordering involved induces a † † sign change, e. g. : bibj := bj bi . The sign changes are necessary so that the anticommuting− natures of the We define a normalized Gaussian operator Λ to be Fermi operators canb beb accommodatedb b without ambigu- the most general Gaussian form of fermionic annihila- ity. We define antinormal ordering similarly, and de- tion and creation operators, with zero displacementb and † † unit trace. Using the extended-vector notation, we first note it via curly braces: bjbi = bibj . More generally, we can define nested{ orderings,} − in which the outer write the normally-ordered Gaussian in an unnormalized form as ordering does not reorderb theb innerb one.b For example, † † : Ob : bi = bib : O : , where O is some operator. { j } − j For example, the different orderings of pairs are, in † Λ(u)(µ, ξ, ξ+) = :exp b I σ−1/2 b : , blockbb matrixb form,b b b b − h i (5.1) b b b † † † bjbi bibj : bµ b := : b bµ : = − , where we have introduced a new extended 2M 2M co- ν ν † † † + × − " bi bj bi bj # variance matrix σ defined in terms of µ, ξ, ξ so that: b b b b b b b b b b† b b b b† = b† b = bi bj bi bj . (4.3) µ ξ µ ν ν µ † † † σ−1 =2I + . (5.2) − " bi bj bjbi # ξ+ µT n o n o b b −b b − b b b b Note that this convention meansb thatb theb b relation be- The introduction of the generalized covariance σ allows tween the two orderings is the matrix to be written in a normalised form, using the † † results of Appendix B, together with an explicit complex : b b : = I + b b , (4.4) amplitude factor Ω: n o bb bb 8

In order to use the Gaussian operator basis, we need to make use of a number of basic results. The proof of many † −1 Λ(−→λ ) = ΩPf σA : exp b I σ /2 b : . − of these can be established, as we show in Appendix B, with Fermi coherent states and Grassmann algebra, the h i h i (5.3) b b b basics of which are given in Appendix A. However the The normalisation is one obvious difference with the con- final results do not contain any Grassmann variables. ventional complex-number or bosonic Gaussian forms[9]. Chosen to ensure that Tr Λ = Ω , it contains the Pfaf- fian of an antisymmetric form σA of the covariance. The B. Trace Properties choice of anti-symmetrisationb is given in Appendix B; other choices will lead, in general, to additional sign fac- Some basic traces are tors. Now the square of the Pfaffian of an antisymmetric matrix is equal to its determinant, and the determinant of σA differs from that of σ by a constant sign (see Ap- Tr Λ = Ω , pendix B). Thus Pf σA = det σ , and as we shall h i | | | | Tr b Λb = 0 , see, the relative phase between the two does not appear h i q †h i in later identities. The additional variable Ω plays the Tr : b b Λ:b = Ωσ . (5.8) role of a weighting factor in the expansion that allows h i us to represent unnormalised density operators and to The first of these is theb normalisation,b b proved as theorem introduce ‘stochastic gauges’ in the drift[21, 22]. 1 in Appendix B. That the second is zero follows from The covariance has a type of generalized Hermitian the fact that the Gaussians are constructed from pairs of antisymmetry, which can be written as σ = σ+, with ladder operators and thus cannot correspond to a super- − the definition that: position of states whose total fermion numbers differ by one. The same result holds for the trace with any odd a b + d c T product. The third trace, proved as theorem 2 in Ap- . (5.4) pendix B, allows us to calculate first-order moments. In c d ≡ b a terms of the M M submatrices, these become: × It is this generalised antisymmetry that allows the covari- † ance to be transformed into an explicitly antisymmetric Tr bi bj Λ = Ωnij , matrix. The covariance can also be broken down into the + h i physically significant M M submatrices n , m and m : Tr bbibj Λb = Ωmij , × nT I m h † † i + Tr bibj Λb = Ωmij . (5.9) σ = m−+ I n , (5.5) − h i Here n is a complex matrix, which corresponds to the nor- These results implyb thatb b for Λ itself to correspond to a mal Green’s function in many-body terminology, while m physical density matrix, n must be a Hermitian matrix, ∗ and m+ are two independent antisymmetric complex ma- † † mb+ since bi bj = bj bi , and must be the Hermitian trices that correspond to anomalous Green’s functions, as ∗ D E D mE † † we will show in the next section. conjugateb b matrixb ofb , since bibj = bj bi . Physi- Thus the total parametrization of a general Gaussian cally, n gives the number, or normal,D E correlations,D E and m operator is and m+ give the squeezing, or anomalous,b b b correlations.b Another restriction on Λ being a physical density ma- −→λ = (Ω, n, m, m+) , (5.6) trix that follows from Eq. (5.9) is that the eigenvalues consisting of 1 + p =1+ M(2M 1) parameters in all. of the matrix n lie in theb interval [0, 1], because of the However, for many systems, only− a subset of all Gaus- Pauli exclusion principle for fermions. Furthermore, the sian operators is required for a complete representation variance in the number correlations is of the density operator. One important subset is the set + † † † of generalised thermal states, for which m = m = 0. var b bj = b bj 1 b bj , (5.10) i i − i In this case, from Appendix B, and using the result that nD Eo D E D E [2I µ] = [I n]−1, the normalization factor is det [I n]. which impliesb thatb if all theb b eigenvaluesb b of n are 0 or − − − The normalized thermal Gaussians therefore can be writ- 1, then Λ is a number state, as the variance in number ten: vanishes. If the eigenvalues are not limited to 0 or 1, then thebΛ corresponds to a mixture of number states in T −1 the eigenbasis and is thus a (possibly squeezed) thermal Λ(−→λ ) = Ω det [I n] : exp b† 2I + n I b : . − − − state, characterisedb by average occupation numbers nj = h (5.7)i eig (n), and squeezing matrix m. b b b j 9

A general Gaussian operator will not necessarily ful- and positive: ρ~n~n > 0. It is sufficient for completeness fill the Hermiticity condition and thus will not neces- to prove that any elementary fermionic operator of form sarily correspond to a physical state. However the set ρˆ~n~m = ρ~n~m ~n ~m corresponds to a normalised Gaussian | i h | of operators that do correspond to physical states is an Λ(−→λ ), apart from a positive scaling factor. The demon- important subclass, because the expansion allows these stration proceeds by constructing limiting cases of Gaus- states to be represented with exact precision. The inclu- siansb that correspond to each of the elementary compo- sion of nonphysical operators in the expansion, on the nents of the density matrix. As demonstrated in the one other hand, makes the Gaussian basis an (over)complete and two mode cases, such expansions are not unique, and basis in which to expand a physical density operator of generally one can obtain other more compact representa- an arbitrary state (This is proved below for the general tions by combining diagonal and off-diagonal elements. case). The overcompleteness of the Gaussian operators To prove the elementary result, we proceed in three as a basis set has important implications for representing steps: arbitrary states with a positive distribution function, a fact that we discuss in detail elsewhere[15]. 1. Diagonal operators

C. Completeness and Positivity First, the generic diagonal operator in the number basis is: We next wish to show that the previous results on completeness and positivity obtained for one and two mode ρˆ~n~n = ρ~n~n ~n ~n density matrix representations can be generalized to the | i h | = ρ~n~n ni ni , (5.13) multi-mode case. That is, we will prove that: | ii h |i i Y For any physical density matrixρ ˆ, a positive set of • with a total occupation number coeﬃcients Pj exists such that

(j) N = ni . (5.14) ρˆ = Pj Λ(σ ) . (5.11) i j X X b Each diagonal multi-mode projector ~n ~n is simply an This central result does not rely on utilising the com- outer-product of single-mode density| matrices.i h | From the plex amplitudes Ω, which are part of the most general single-mode example in Section III A, each single-mode Gaussian operator. If these were used, then positivity density matrix corresponds to a Gaussian as in Eq(3.6). of the coefficients would be trivial, since these additional Thus, we see that a diagonal projector is exactly equal to amplitudes could be used to absorb any phase or sign fac- a normalized Gaussian (we suppress the trivial arguments tors arising in the expansion. Instead, we wish to prove for simplicity): a much stronger result, that a positive expansion exists without any additional amplitude factors. This result is ~n ~n = i Λ1 [ni] analogous to a similar result known for the positive-P | i h | = ΛM [n] , (5.15) bosonic representation[7, 8]. Q b From the number state basis of Eq (2.2), the full set where the matrix n is defined as n = n δ . of possible fermionic many-body number states is the set b ij i ij Because the ρ are real and positive, Eq. (5.13) shows ~n where ~n is varied over all 2M possible permuta- ~n~n that a positive Gaussian expansion exists for all diagonal tions.{| i} This defines a complete basis of 22M number-state density matrices: projectors for the set of all fermionic density operators. While not all the number-state projectors are Hermitian, ρ = ρ Λ [n] . (5.16) it is no restriction to use this larger set of operators as a ~n~n ~n~n M basis for the density matricesρ ˆ. In summary, a diagonal multi-mode projector ~n ~n is b b Next, we expand: simply an outer-product of single-mode density matrices,| i h | and hence corresponds exactly to a multi-mode normal- ρˆ = ~n ~n ρˆ ~m ~m | i h | | i h | ized Gaussian. X~n X~m = ρ~n~m ~n ~m | i h | 2. Generalized thermal operators X~n X~m = ρˆ~n~m (5.12) ~n ~m Next, consider off-diagonal projectors in the number- X X state expansion that conserve total number, i.e. The positive definiteness of the density operator means in particular that diagonal density matrix elements are real ρ~n~m = ρ~n~m ~n ~m , (5.17) | i h | b 10 for which 3. Squeezed operators

ni = N = mi . (5.18) i i Finally, we consider the remaining elements of the den- X X sity operator expansion, i.e. those squeezed projectors We show that any such component can be written as the ρ~n~m for which limiting form of a number-conserving Gaussian, up to a positive scaling factor. b N = ni = mi +2S, (5.24) Let i i ′ X X n = δij min ni,mi , (5.19) ij { } where S is an integer denoting the change in the number and define an off-diagonal Gaussian in terms of the diag- of fermion pairs. We suppose that S > 0, since the case onal normalized Gaussian: of S < 0 is obtained trivially by hermitian conjugation. Let n be obtained from n by removing 2S occupied (o) ′ † ′ i i Λ (µ, n ) = : 1 µijˆb ˆbj Λ[n ]: . (5.20) − i sites, labeled as successive pairs i, j belonging to a set σ, i6=j Y h i so thate N = i ni = i mi. The occupation numbers b b ′ Now for every distinct mode i with δni = ni n =1 ni,mi now define a generalised thermal density matrixe − i P P one can define a corresponding distinct index j(i) with componente as previously,e and hence equate to a limiting ′ case of a Gaussian operator from Eq (5.23) above. δmj = mj mj = 1. It follows from the minimum con- e dition, Eq(5.19)− that j(i) = i, and from the number- Now define a squeezed off-diagonal Gaussian in terms conservation equation, Eq (5.18),6 we have of the thermal case, which we have already proved has a positive representation: δni = δmi . (5.21) (s) ′ † † ′ i i Λ (µ(ε), ξ(ε), n )= 1 ξijˆb ˆb Λ[µ(ε), n ] , X X − i j i>j Similarly, for every distinct pairs of indices i,i’ with Y h i ′ δni = δni′ = 1, it follows that j(i ) = i, since other- b b (5.25) 6 wise δni = 0. The mapping therefore generates distinct where: pairs so that j(i) = j(i′) i = i′. Next, we note that 6 ⇐⇒ 6 this mapping is not a permutation of the set of modes i δii′ δjj′ ′ with δni = 0, since if it were the condition that j(i ) = i ξij (ε)= . (5.26) ′ 6 ′ ′ ε would be violated for some i . Similarly, the mapping is {i X,j }∈σ − not a permutation of any subset of these modes. This Then consider the limit of ε e 0 as before, so that to means that the only non-vanishing terms in the normal- → ization factor det[2I µ] are the diagonal terms, which leading order, are already normalized.− Proceed by defining the resulting set of δM M pairs † † ′ ≤ ρ~n~m = : aˆ aˆ ~n ~m : as σ = i, j , and let: i j | i h | { } {i,j}∈σ˜ h i ′ ′ Y µδii δjj S+δM (s) ′ µij (ε, ~n)= b = lim ε Λ [µ(ε), ξ(ε), n ] (5.27) ′ ′ ε ε→0 {i X,j }∈σ − A positive expansion parameterb is obtained here as where µδM = ρ , so that: ~n~m well, thus completing the proof.

′ † ′ Λ[µ(ε), n ] = : 1 µijˆb ˆbj Λ[n ]: − i i j D. Differential Properties Y6= h i b µˆb†ˆb b = : 1+ i j ~n′ ~n′ : In order to use the Gaussian basis in a time-evolution ε | i h | {i,j}∈σ " # problem, we need to be able to map the evolution of the Y density operator onto an evolution of the expansion coef- (5.22) ficients Pj . To achieve this, one must be able to write the Then consider the limit of ε 0, so that to leading order, action of ladder operators on a Gaussian basis element → in differential form. δM † ′ ′ We can differentiate the Gaussian operators with re- ρ~n~m = µ : ˆb ˆbj ~n ~n : i | i h | spect to their parameters to get {i,jY}∈σ h i δM d 1 b = lim ε Λ[µ(ε), n′] (5.23) Λ = Λ , ε→0 dΩ Ω d † Again, we see that a positiveb expansion parameter is Λb = σ−b1Λ σ−1 : b b Λ: σ−1 , (5.28) obtained. dσ − b b bb b 11 where the matrix derivative is defined as special cases, the set of Gaussian operators include the density operators for thermal states and squeezed states, ∂ ∂ = , (5.29) and thus the physics of the noninteracting Fermi gas and ∂σ ! ∂σνµ the BCS state is incorporated into the basis itself. Fur- µ,ν thermore, the basis also includes more general operators, i. e. involving a transpose. These expressions for the which ensure an overcompleteness that makes it possible derivative can be inverted to obtain the important iden- to express any physical density operator as a probabilis- tities: tic distribution over the Gaussian operators, without the need to use Grassmann algebra. ∂ We have calculated the normalisation and moments, Λ = Ω Λ , and proved completeness and positivity for the most ∂Ω general basis, and also for specific subsets, such as the † ∂Λ : b b Λ:b = σΛ bσ σ , (5.30) number-conserving thermal basis. These results mean − ∂σ that the phase-space representation defined by the Gaus- b sian operators can be used for first-principles simulations bb b b and thus we can write the normally ordered action of any of many-body fermionic systems. The mapping from the pair of operators on a Gaussian as a first-order deriva- quantum operator to the probabilistic c-number descrip- tive. As theorems 5 and 6 in Appendix B show, there are tion is enabled by the Gaussian differential identities that analogous identities for antinormally ordered and mixed have been derived here. The application of these identi- pairs: ties will be dealt with elsewhere, when we consider the phase-space representation in more detail.

† ∂Λ b : b Λ: = σΛ+ σ I σ , − − ∂σ n o b Acknowledgments b b b† b ∂Λ b b Λ = σ I Λ σ I σ I . − − − ∂σ − n o b (5.31) Funding for this research was provided by an Aus- bb b b tralian Research Council Centre of Excellence grant. For the subset of Gaussian operators that correspond to (generalised) thermal states, i. e. m+ = m = 0, we obtain a simpler set of differential identities: Appendix A: GRASSMANN ALGEBRA ∂Λ b†T bT Λ = nΛ + (I n) n , − ∂n This appendix introduces the basic concepts of non- ∂Λ b commuting algebra and lists some results pertaining to bΛb†bT bTb = nΛ+b n (I n) , ∂n − Grassmann calculus and Fermi coherent states. These results are used in Appendix B to establish important b ∂Λ b†bT ΛbT = (Ib n) Λ + (I n) (I n) , properties of the Gaussian operators. Proofs and fur- − − ∂n − ther discussion of these results can be found in the T ∂Λ b bΛbb† b = nΛ n b n . (5.32) literature[16, 17, 18]. − ∂n Let α be a vector of M anticommuting (Grassmann) b Theb actionbb of fourb ladder operators on a Gaussian op- numbers, i.e. erator can be obtained by applying the previous identities twice. Thus in a Gaussian expansion, any two-body [αi, αj ]+ = 0 . (A1) interaction term can be written as a second-order differential operator. As we show in detail elsewhere[15], this 2 Since αj = 0, any function of Grassmann numbers can central result allows the evolving density operator to be be at most linear in any one of its arguments. Thus, for mapped to a Fokker-Planck equation for the expansion example, the single mode exponential is coefficients, thus enabling a Monte-Carlo sampling of the many-body quantum state. exp(αj ) = 1+ αj , (A2)

VI. CONCLUSION and a multimode exponential, e. g. exp ( αj ) will be an ordered product of such single-mode exponentials. P In summary, we have introduced here a generalised The Grassmann numbers anticommute with all Fermi Gaussian operator basis, as a means of deﬁning a phase- annihilation and creation operators, but commute with space representation for correlated fermionic states. As c-numbers and bosonic operators. 12

1. Grassmann Calculus 2. Grassmann coherent states

Differentiation of a single Grassmann variable is de- For each Grassmann number αj we can associate an- fined as other Grassmann number, denoted αj , to play the role ∂ of a complex conjugate. This is formally regarded as an αj = δij , (A3) independent variable for calculus purposes. The conju- ∂αi gates αj anticommute with all other Grassmann vari- with the derivative of products obtained by the Grass- ables. By use of such a complex Grassmann algebra, we mann chain rule: can define a fermionic coherent state, which is formally an eigenstate of the annihilation operator: ∂f g + f ∂g , f even ∂ ∂αj ∂αj 1 [f(α)g(α)] = ∂f ∂g ,(A4) − 2 ∂αj g f , f odd αj = (1+ αj αj ) ( 0 + 1 αj ) ( ∂αj − ∂αj | i | i | i † 1 i. e. the derivative operator also anticommutes. = exp b αj αj αj 0 , (A10) j − 2 | i Grassmann integration is defined to be the same as 1 − 2 differentiation, but written in a different way: αj = (1+ αbj αj ) ( 0 + αj 1 ) h | h | h | 1 = exp αj bj αj αj 0 . (A11) dαj = 0 − 2 h | Z Like the bosonic coherent state,b the fermionic coherent αj dαj = 1 . (A5) state can be written as (Grassmann) displacement from Z the vacuum: Multivariate integrals are ordered sequences of single- variable integrations, which can be written without am- † αj = exp b αj αj bj 0 . (A12) biguity if the integration measure (as for a derivative) | i j − | i is also taken to be an anticommuting number. For an Multimode coherent statesb are productsb of the single- integral over all variables in a vector, we define the in- mode states: tegration measure to be ordered in increasing numerical M order: dα dα1...dαN . ≡ α = αj Note that integrating a total derivative gives zero: | i | i j=1 Y ∂ M dαj f = 0 . (A6) † ∂αj = exp b αj αj bj 0 Z  j −  | i j=1 This fact, coupled with the product rule, gives a result X for partial integration:  b b 

∂g dαj f , f even The inner product of two states is: ∂f ∂αj dαj g = − ∂g . (A7) ∂αj dαj f , f odd ( R ∂αj αi αj = exp(αiαj αiαi/2 αj αj /2) , (A13) Z h | i − − One very useful result isR the multivariate Gaussian in- and thus as two special cases: tegral: αi αi = 1 T h | i exp α Aα/2 dα = Pf [A] (A8) αi αi = exp( 2αiαi) . (A14) − h− | i − Z The usefulness of the coherent states lies in the fact for an antisymmetric matrix A of complex numbers. The 2 that they form a complete set: Pfaffian is related to the determinant (Pf [A]) = det A , and thus, apart from a sign change, has many of the same properties. For example, Pf AT = ( 1)N Pf [A] dαj dαj αj αj = 0 0 + 1 j 1 = I1 , (A15) − | ih | | ih | | i h | and Pf A−1 = 1/Pf [A] . Another useful type of Z Gaussian| integral| is:| | or, for the multimode case, M dα α α = I , (A16) T M exp β Bα (dβj dαj ) = det [B] , (A9) | ih | − Z Z j=1 Y where we have defined a 2M-variate integration measure where B is a square matrix of complex numbers and β as and α are two independent Grassmann vectors. This M second integral is in fact a special case of the first, except dα (dαj dαj ) . (A17) ≡ with 2M total Grassmann variables. j=1 Y 13

Finally, we can express the trace of an arbitrary operator where the constant matrix X is defined as O as 0 I X I 0 . (B4) b Tr O = dα α O α . (A18) ≡ h− | | i Z h i When applied to the left of a matrix, X swaps the upper b b and lower halves; when applied to the right, it swaps Appendix B: GAUSSIAN FERMION OPERATORS the left and right halves. This structure means that the matrices σ and µ can be transformed into explic- We prove some useful results concerning the most itly antisymmetric forms by certain permutations of rows general multi-mode Gaussian operator constructed from and columns. For example, it follows immediately from fermionic ladder operators. The proofs make use of the Eq. (B3) that interchanging left and right halves, or up- properties of Grassmann coherent states and anticom- per and lower halves, will generate an antisymmetric ma- muting algebra, which are summarized in Appendix A. trix. Alternatively, inserting each row in the lower half However the final results do not contain any Grassmann after the corresponding row in the upper half, and insert- variables. The results establish the trace and differential ing each column in the right half before the corresponding properties of the Gaussian operators that are discussed row in the left half generates the antisymmetric form in Sec. V. b a b M a M 11 11 ··· 1 1 d11 c11 d1M c1M 1. General Gaussian Operator a b  ···  ...... (B5) c d ≡ . . . . . A    bM1 aM1 bMM aMM  In this appendix, we use the vector and ordering nota-  ···   dM1 cM1 dMM cMM  tions introduced in Sec. IV. We find it convenient to use  ···  an unnormalised Gaussian form of Fermi operators: With the covariance matrix antisymmetrized in this way, the Gaussian operator becomes † Λ(u)(µ) = :exp b µ b/2 : , T − (u) −1 ∞ h i Λ (σ ) = :exp bA I σ /2 bA : , (B6) 1 † n A A − A b b: b b µ b : ≡ 2nn! − h i n=0 whereb the vector of operatorsb is now b X 2M b b 1 † b = : 1 bµ µµν bν : , (B1) 1 − 2 † µ,ν=1 b Y  1  b b b = b. . (B7) where µ is a 2M 2M matrix of parameters, given by A  .  ×  b   bM  b  †  µ ξ  bM  µ = + T .   ξ µ  b  − The generalised antisymmetryb of σ and µ has impli- In terms of the covariance matrix σ, the unnormalised cations for matrix derivatives. Because each element Gaussian is: of the matrix appears twice, we have ∂σµν /∂σθφ = δµθδνφ XµφXνθ, where δµν is the Kronecker delta func- − † tion. The extra terms here appear in the derivative of an Λ(u)(σ) = :exp b I σ−1/2 b : , (B2) − inverse: h i −1 where theb relation betweenb the two parametrizationsb is ∂σµν −1 −1 −1 −1 −1 = σµθ σφν + σµγ XγφXθǫσǫν , (B8) σ = [µ +2I] and where the diagonal matrix I is as ∂σθφ − defined in Eq. (4.5). Because of the anticommuting property of fermions, and they also give an additional factor of two in the both σ and µ possess a generalized antisymmetry: σ = derivative of a determinant, thus: σ+ ,µ = µ+, or more in block matrix form, d − − det σ = det σ σ−1, (B9) dσ q q a b a b T X X where we define the matrix derivative as c d ≡ − c d d d dT bT = , (B10) = T T , (B3) dσ dσ − c a ! νµ µν 14 i. e. involving a transpose. The result (B8) allows us to is determined by the determinant of the covariance: relate the derivatives with respect to σ and µ : 2 1 ( 1)M 1 Tr Λ(u) = = − = , df df df = = σ σ . (B11) det σ det σ det iσ dµ dσ−1 − dσ h i A b h i (B16)

and thus we may write the Gaussian operator in nor- 2. Normalisation malised form as

† Λ = det iσ : exp b I σ−1/2 b : , (B17) Theorem 1: The trace of the unnormalised Gaussian ± − operator is equal to the Pfaffian of the inverse of the q h i b b b antisymmetrized covariance, i. e. where the choice of plus/minus sign is determined by −1 det σA Pf σA . This extra sign, or phase, which r N Tr Λ(u)(σ) = Pf σ−1 , (B12) does noth i appearh ini the normalizations of the familiar ≡ A complex-number or bosonic Gaussians, fortunately does h i h i not appear in any of the identities needed to make use where Pf stands for Pfaffian.b of the Gaussian operators as a basis for a phase-space representation. Proof: Because the Gaussian is in normally ordered A specific case where the determinant appears is for form, it is straightforward to evaluate the trace with mul- the generalised thermal Gaussian without squeezing pa- timode Grassmann coherent states α , using the Grass- rameters, so that m = m+ = 0. In this case, the nor- mann trace result of Eq. (A18): | i malization follows directly from the second Grassmann M Gaussian integral identity, Eq. (A9). Following the same (u) (u) procedure as before, we find that: Tr Λ = α Λ α [dαj dαj ] h− | | i Z j h i Y (u) b b M Tr Λ = det [2I µ] . (B18) † −1 − = exp α σ α/2 [ dαj dαj ] , h i − − Z j b Y (B13) 3. First-order moments where we have made use of the fact that, from the Grass- mann inner product result of Eq. (A13), α α = h− | i Theorem 2: The fermionic Gaussian operator is com- exp( 2αα) . We have also changed variables: αj − → pletely characterised by its first-order moments. In par- αj , and introduced 2M-vectors of Grassmann variables ticular, the covariance matrix corresponds to the first- − T † T α = (α, (α) ) and α = (α, α ) . Changing to the anti- order moments in normally ordered trace form, i. e. symmetric form of the covariance σ , and swapping the A order of the pairs in the integration measure, we obtain † M Tr : b b Λ: = σ , (B19) (u) T −1 Tr Λ = exp αA σA αA/2 [dαj dαj ] , h i − † j b b b h i Z h i Y where b b is a matrix multiplication of two vectors, re- b (B14) sulting in the 2M 2M matrix of Eq. (4.2). × Proof:bbWe proceed as in the proof of Theorem 1, by tak- where the reordered Grassmann vector is αA = ing the trace of the unnormalised form using Grassmann T (α1, α1, ..., αM , αM ) . Noting that the arrangement of el- coherent states and then changing variables: αj αj : ements in αA matches the order of integration, we can →− apply the Gaussian integral result [Eq. A8)]: M † (u) † (u) Tr : b b Λ : = α : b b Λ : α [dαj dαj ] h− | | i Tr Λ(u) = Pf σ−1 . (B15) j A h i Z Y bb b bb b M h i h i † † −1 QED. b = α α exp α σ α/2 [dαj dαj ] . − Z j Y Corollaries: (B20) Now the square of the Pfaffian of a matrix is equal to its determinant. Thus, to within a sign, the normalisation Next we put the integral into the form Eq. (A8): ± 15

† Tr : b b Λ(u) : = 4. Higher-order moments h i bb b M One can calculate higher-order moments along similar †T T T † −1 α α exp α σ α/2 [dαj dαj ] lines, i. e. by expanding the trace as a Grassmann integral − − Z j then converting this into a higher-order derivative of a Y M determinant. The results of this procedure in the general d † −1 case can be written in terms of a moment generating = exp α σ α/2 [dαj dαj ] , (B21) dσ−1 − function. Z j Y where in taking the derivative with respect to σ−1 we Theorem 3: Any even moment of a Gaussian opera- have taken account of the fact that each element of it tor, in normally ordered trace form, can be calculated by appears twice, owing to its generalized antisymmetry. means of the moment generating function Evaluating the Grassmann integral, and employing the M(τ) det I2 σ τ (B27) determinant result , we get ≡ − q † as follows (u) d −1 Tr : b b Λ : = −1 ( ) det iσ dσ ± † † ∂ h i q Tr : bµ1 bµ2 bµr−1 bµr Λ: = ··· ∂τ 1 2 ··· bb −1 µ ,µ b = det iσ σ . (B22) h i ± ∂ q b b b b b M(τ) . ··· ∂τ −1 Finally, dividing through by the normalisation in µr ,µr τ=0

Eq. (B12) gives the normalised result . QED. (B28)

Corollaries: We can put Eq. (B19) into a more familiar form by Proof: We start by writing the normally ordered oper- using the cyclic property of trace: ator product as a derivative of a normally ordered Gaus- sian: T † b†T bT bbT ∂ ∂ Λ Λ : b† b b† b : = Tr : b b Λ: = Tr − µ1 µ2 µr−1 µr ··· ∂τ 1 2 ··· ∂τ −1  b†T b†Λ b†T ΛbT  µ ,µ µr ,µr h i + b bb bbb b b b b : exp b τ b/2 : ,(B29) b b b  bb† bbT  b b b b τ=0 = Tr − b T Λb h i  b†T b† bb†  b b bb bb where τ is a 2M 2M matrix of complex numbers with × + nT I m  b the generalised antisymmetry τ = τ . Using this result b b bb − = m−+ I n , (B23) in Eq. (B28) shows that the moment generating function − M(τ) must satisfy where we have defined the matrix n for the number, or + normal, moments, and the matrices m and m+ for the M(τ) = Tr : exp b τ b/2 Λ: , (B30) squeezing, or anomalous, moments, as follows: h i which can be evaluated by writingb theb traceb as a Grass- mann integral n = b†T bT , Λ M m DbbT E 1 † −1 = b b b, M(τ) = exp α σ τ α/2 [dαj dαj ] . Λ N − − Z j m+ Db†T bE† Y = bb b , (B24) Λ (B31) D E b b b Using the Gaussian integral result (A8), we get where O Tr O Λ . Thus we can write covariance Λ ≡ matrix in terms of the moments as −1 −1 D E h i M(τ) = Pf σA τ A /Pf σA . (B32) b b b b − T h i h i n I m Because of the relationship between Pfaffians and deter- σ = m−+ I n , (B25) − minants, we can rewrite this as or inverting the relationship, M(τ) = det σ−1 τ / det σ−1 A − A A † r h i r h i : b b : = I I σ I . (B26) 2 Λ − = det I σ τ , (B33) D E − bb b q 16 where the sign of the square root is chosen to give a Proof: The proof can be established easily without positive result when τ = 0. QED Grassmann algebra. We first take the derivative of the Examples: unnormalised Gaussian operator: To calculate the derivatives of the moment generating ∂ (u) † (u) function, one makes use of the results for the derivative Λ (µ) = : bµ Λ (µ) bν : . (B39) of a determinant Eq. (B9) and of an inverse Eq. (B8). A ∂µµν − general second-order moment is of the form We write this asb a derivative withb b respectb to the covari-

† † ance matrix, using Eq. (B11), and swap the pair of oper- Tr : bµ1 b bµ3 b Λ: = σµ1µ2 σµ3 µ4 σµ1µ4 σµ3µ2 µ2 µ4 − ators, to give, h i + (σX)µ1µ3 (Xσ)µ4µ2 . (B34) (u) b b b b b dΛ (σ) † σ σ = : ˆb ˆb Λ(u)(σ): . (B40) Thus the normally ordered number-number correlations dσ − are b Next, we take the derivative of theb normalisation, using + Tr : ninj : Λ = niinjj nij nji mij m , (B35) Eq. (B9): − − ij 1 h i d d − 2 i. e. containing theb three terms expected for a state with N = ( ) det iσ b b dσ dσ ± Gaussian correlations. −1 Similarly, a third-order moment is of the form = Nσ . (B41) − Combining both of these results, we ﬁnd that the deriva- † † † Tr : bµ1 bµ2 bµ3 bµ4 bµ5 bµ6 Λ: = tive of the normalised Gaussian is h i σµ1µ2 σµ3µ6 σµ5µ4 + σµ1 µ4 σµ3µ2 σµ5µ6 + σµ1 µ6 σµ3 µ4 σµ5µ2 d dN d b b b b b b b Λ = N −2Λ(u) + N −1 Λ(u) σµ1 µ2 σµ3µ4 σµ5µ6 σµ1µ4 σµ3 µ6 σµ5 µ2 σµ1µ6 σµ3µ2 σµ5µ4 dσ − dσ dσ − − − † σµ1 µ2 (σX) (Xσ) σµ3µ4 (σX) (Xσ) −1 −1 −1 − µ3µ5 µ6µ4 − µ1µ5 µ6µ2 b = σ Λ b σ : b b Λ: σ b, (B42) − σµ5 µ6 (σX) (Xσ) + σµ1µ4 (σX) (Xσ) − µ1µ3 µ4µ2 µ3µ5 µ6µ2 whose inverse is the requiredb result.bb b QED. σµ1 µ6 (σX) (Xσ) + σµ3µ2 (σX) (Xσ) − µ3µ5 µ4µ2 µ1µ5 µ6µ4 This result can also be proved by use of Grass- mann coherent-state expansions, in similar manner to the + σµ3 µ6 (σX)µ1µ5 (Xσ)µ4µ2 σµ5µ2 (σX)µ1µ3 (Xσ)µ6µ4 − proofs below for the products of diﬀerent ordering. + σµ5µ4 (σX)µ1µ3 (Xσ)µ6µ2 . (B36) Thus the normally ordered triple correlations are: 6. Mixed products

+ Tr : ninj nk : Λ = niinjj nkk nii njknkj + mjkm − jk + h i + nij njknki njj niknki + mikm Theorem 5: A product of mixed order of a pair of ladder b − ik b b b + operators and a Gaussian is equivalent to a first-order + njinkj nik nkk nij nji + mij m − ij differential operator on the Gaussian, as follows: + + + nij mikmjk + njimjkmik + + + njkmjimki + nkj mkimji † ∂Λ + + b : b Λ: = σΛ+ σ I σ . (B43) + nkimkj mij + nikmkj mij , (B37) − − ∂σ n o b again as expected for a state with Gaussian correlations. b b b b Proof: We first make use of the Fermi coherent-state completeness identity to replace all ladder operators by 5. Normally ordered products Grassmann integrals over coherent projection operators: † b : b Λ: = n o Theorem 4: A normally ordered product of a pair of † b bdγbdβdαdǫ γ γ b : β β b Λ α α : ǫ ǫ ladder operators and a Gaussian is equivalent to a first- | ih | | ih | | ih | | ih | Z order differential operator on the Gaussian, as follows: 1 n o α = dγdβdαdǫ γ ǫbexp βb, αb µ + I /2 N | ih | − β Z † ∂Λ β 1 1 : b b Λ: = σΛ σ σ . (B38) β, α exp γβ + αǫ αα ββ γγ ǫǫ − ∂σ × α − − − 2 − 2 b (B44) bb b b 17 where we have used the result that the inner product of 7. Antinormal products two coherent states is, from Eq. (A13) β α = exp βα ββ/2 αα/2 . (B45) h | i − − Next, we employ integration by parts [Eq. (A7)] to Theorem 6: An antinormally ordered product of a pair β of ladder operators and a Gaussian is equivalent to a first- replace by variables that appear in the Gaussian α order differential operator on the Gaussian, as follows: form: α exp β, α µ + I /2 β − † ∂Λ β b b Λ = σ I Λ σ I σ I .(B49) exp αα ββ β, α − − − ∂σ − × α − − n o b bb b b α = exp β, α µ + I /2 − β Proof: The proof initially proceeds in the same man- ∂ − β exp αα ββ β, α ner as for products of mixed ordering. We first insert × ∂α − − the coherent state identity to convert the action of the exp αα ββ operators into integrals over coherent-state projectors: → − − † ∂ α b b Λ = β exp β, α µ + I /2 β, α × ∂α − β n o − α bb b = exp αα ββ I µ + I β, α I † − − β − dγdβdαdǫ γ γ b b β β Λ α α ǫ ǫ | ih | | ih | | ih | | ih | α Z exp β, α µ + I /2 . 1 n o α × − β = dγdβdαdǫ γ ǫbexpb βb, α µ + I /2 N | ih | − β (B46) Z β 1 1 (α, β)exp γβ + αǫ αα ββ γγ ǫǫ We can now express the result as a derivative of the × α − − − 2 − 2 unnormalised Gaussian operator with respect to µ : (B50)

† 1 b : b Λ: = dγdβdαdǫ γ ǫ N | ih | This time, however, we integrate by parts twice: Z n o 1 1 expb b bαα ββ γγ ǫǫ + γβ + αǫ × − − − 2 − 2 α exp β, α µ + I /2 d 1 α − β I µ + I I exp β, α µ + I × dµ − −2 β β " # (α, β)exp αα ββ × α − − 1 d = I µ + I I Λ(u) . (B47) α N dµ − =exp β, α µ + I /2 " # − β b −1 Finally, we can change variables to σ = [µ +2I] and ∂β − ∂α, ∂β exp αα ββ use the result for the derivative of the normalisation: × ∂α − − − ∂ † (u) β 1 −1 dΛ (u) exp αα ββ − ∂α, ∂β b : b Λ: = I σ I σ σ IΛ → − − ∂α − N − − dσ − " # n o b α b b b b exp β, α µ + I /2 −1 dΛ Λ dN × − β = I σ I σ + σ IΛ − − dσ N dσ ! − α b b =exp αα ββ I + I µ + I β, α b − − − β dΛ −1 = σ I Λσ σ IΛ α − dσ − ! − µ + I I exp β, α µ + I /2 (B51) b × − β dbΛ b = σΛ+ σ I σ . (B48) − − dσ b which is now in a form that we can again express as a QED. b derivative of the unnormalised Gaussian operator with 18 respect to µ : Finally, we change variables to σ = [µ+2I]−1and use the result for the derivative of the normalisation: † 1 b b Λ = − dγdβdαdǫ γ ǫ N | ih | Z n o 1 1 bexpb b αα ββ γγ ǫǫ + γβ + αǫ (u) × − − 2 − − 2 † 1 (u) −1 dΛ −1 b b Λ: = IΛ I I σ σ σ I σ I N " − − dσ # − d 1 α n o I I µ + I exp β, α µ + I b × − dµ −2 β bb b b dΛ " # ! = σ I Λ σ I σ I . (B53) − − − dσ − µ + I I b × b 1 dΛ(u) = IΛ(u) + I µ + I µ + I I . (B52) N "− dµ # QED. b b

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