WWigner functional theory for quantum optics
Filippus S. Roux and Nicolas Fabre National Metrology Institute of South Africa, Meiring Naudé Road, Brummeria 0040, Pretoria, South Africa Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université de Paris, CNRS UMR 7162,75013 Paris, France
Using the quadrature bases that incorporate the spatiotemporal degrees offreedom, we develop a Wigner functional theory for quantum optics, as an ex-tension of the Moyal formalism. Since the spatiotemporal quadrature basesspan the complete Hilbert space of all quantum optical states, it does not re-quire factorization as a tensor product of discrete Hilbert spaces. The Wignerfunctions associated with such a space become functionals and operations areexpressed by functional integrals — the functional version of the star product.The resulting formalism enables tractable calculations for scenarios where bothspatiotemporal degrees of freedom and particle-number degrees of freedom arerelevant. To demonstrate the approach, we compute examples of Wigner func-tionals for a few well-known states and operators.
Quantum information technology promises to provide secure communication [1], more accu-rate measurements [2] and more efficient computations [3], among other benefits. However,quantum states are often fragile. The purity and coherence of such states, for instance, areeasily lost when such states interact with the environment [4].To increase the information capacity of quantum systems [5–7] and to improve thesecurity in quantum cryptography [8–10], the states are often prepared in higher dimen-sional Hilbert spaces. An example is the spatial modes of photons, such as orbital angularmomentum (OAM) modes [11, 12]. They represent an infinite dimensional Hilbert space.Applications that use such higher dimensional Hilbert spaces are usually implemented interms of individual photons encoded in terms of their spatial degrees of freedom. Lossesand stray photons tend to reduce the fidelity, purity or signal-to-noise ratio in quantumstates, slowing down the rate at which such systems can operate [13].A way to overcome the losses and noise issues is to prepare multiphoton states thatalso incorporate different spatial modes. Such quantum systems are defined in terms ofboth their spatiotemporal degrees of freedom and particle-number degrees of freedom [14].They are often rather complex and difficult to analyze. One approach is to duplicate theoperator formalism for a single-mode multi-particle system several times to handle severaldiscrete modes [15, 16]. The result is best applied in cases of Gaussian states that can berepresented in terms of a few discrete spatial modes [17].In a different development, started during the Second World War, it was independentlyshown by Groenewold [18] and Moyal [19] that quantum mechanics can be successfully
Filippus S. Roux: [email protected] Fabre: [email protected] a r X i v : . [ qu a n t - ph ] J un ormulated without operators. This formulation of quantum mechanics in phase space [20]represents the states and operators by functions of phase space variables (analogues toposition and momentum for the harmonic oscillator). Examples of such functions are thequasi-probability distributions that include the Glauber-Sudarshan P -distribution [21, 22],the Husimi Q -distribution [23] and the Wigner distribution [24]. Products of operators arerepresented by so-called star products of the phase space functions. The Moyal formulationwas shown to reproduce all the uncertainty relations associated with quantum mechanics.One of the challenges initially encountered with the Moyal formulation was how toincorporate other degrees of freedom (apart from the particle-number degrees of freedom)into the formulation. In the case of spin (and other internal symmetries), the problem wasovercome with the aid of the Stratonovich-Weyl correspondence [25–27]. For the spatialdegrees of freedom, one can use a similar approach [28] or other approaches (see for example[29]), but these again lead to a finite set of discrete spatial modes (and often tend to returnto an operator-based approach).Recently, the spatiotemporal degrees of freedom and the particle-number degrees offreedom were combined into one comprehensive Hilbert space for all quantum opticalstates [30, 31], by introducing spatiotemporal quadrature bases that are generalizationsof the quadrature bases associated with only the particle-number degrees of freedom. Thisapproach was subsequently used to investigate the evolution of arbitrary multiphoton statespropagating through turbulence [32] and to investigate the amount of entanglement in aparametric down-converted state when all the degrees of freedom are included [33]. How-ever, a full description of the approach has not been published yet.The purpose of this article is to provide such a description. Guided by the Moyalformalism, we use the spatiotemporal quadrature bases to develop a formalism that in-corporates both the spatiotemporal degrees of freedom and the particle-number degreesof freedom. For this purpose, we choose the Wigner distribution, since it is naturally re-lated to the quadrature bases. (However, the resulting formalism can be used with anyof the other quasi-probability distributions.) In this approach, these quadrature bases areused to generalize the standard Wigner distributions to become Wigner functionals . Thedevelopment parallels the normal theory of Wigner functions (and of the Moyal formal-ism), showing that most properties can be carried over to the functional formalism, exceptthat analyses now tend to involve functional integrals. Though the expressions may appearfamiliar, the resemblance is deceptive in that it now incorporates all the spatiotemporal de-grees of freedom. While the current development is done in the context of quantum optics,similar developments have been done in the context of quantum field theory [34–36].The involvement of functional integrals in the new formalism may create the impressionthat any analysis that is done with this formalism would be severely complex and oftenintractable. It is true that, apart from some special cases, one can evaluate such functionalintegrals only when the integrand is in the form of a Gaussian functional. However, withthe aid of auxiliary variables, source terms and generating functionals, it is often possibleto represent the quantum states and operations in terms of Gaussian functionals, even ifthe original expressions are not of that form.To demonstrate its usefulness, we use the formalism to compute the Wigner functionalsfor a few well-known states and operators. For example, we compute a generating func-tional for the Wigner functionals of fixed-spectrum Fock states. The term fixed-spectrum indicates that the spatiotemporal degrees of freedom of all the photons in the state arerepresented by the same spectrum of plane waves. Although the Wigner functionals ofFock states are not in Gaussian form, their generating functional is in Gaussian form andcan therefore be used in calculations involving functional integrals. he functional phase space introduced here can be considered as the “mother” of allphase spaces for quantum optics in that it allows one to recover all those that are studiedin quantum optics. For example, by restricting the spatiotemporal degrees of freedom toa single spectral function, the functional phase space reduces to the phase space that onlyrepresents the particle-number degrees of freedom.The paper is organized as follows. In Sec. 2, we review the spatiotemporal quadraturebases, together with some background on other aspects that we need in the rest of thepaper. The definition of the Wigner functionals and related quantities are discussed inSec. 3. Some examples of Wigner functionals are computed in Sec. 4. We provide adiscussion in Sec. 5 and end with conclusions in Sec. 6. The quadrature bases in terms of which the Wigner functionals for quantum optics isformulated, are obtained as eigenstates of the wave vector dependent quadrature operators: ˆ q s ( k ) | q i = | q i q s ( k ) , ˆ p s ( k ) | p i = | p i p s ( k ) . (1) Here, k represents the three-dimensional wave vector and the subscript s is the spin index.These quadrature operators are directly defined in terms of the standard ladder operators ˆ a † s ( k ) and ˆ a s ( k ) used for the quantization of the electromagnetic field: ˆ q s ( k ) = 1 √ h ˆ a s ( k ) + ˆ a † s ( k ) i , ˆ p s ( k ) = − i √ h ˆ a s ( k ) − ˆ a † s ( k ) i . (2) The ladder operators obey a Lorentz covariant commutation relation, given by h ˆ a s ( k ) , ˆ a † r ( k ) i = (2 π ) ω δ s,r δ ( k − k ) , (3) where ω = c | k | is the angular frequency, given in terms of the vacuum dispersion relation, δ s,r is the Kronecker delta for the spin indices and δ ( k − k ) is a three-dimensional Dirac δ function for the wave vectors. The equivalent Lorentz covariant commutation relationfor the quadrature operators reads [ˆ q s ( k ) , ˆ p r ( k )] = i (2 π ) ω δ s,r δ ( k − k ) . (4) Although the two quadrature operators ˆ q s ( k ) and ˆ p s ( k ) in (1) are unique operator-valuedfunctions of the wave vector, the real-valued eigenvalue functions q s ( k ) and p s ( k ) are notunique — there are an infinite number of them. Indeed, q s , p s ∈ L { R } . Moreover, foreach eigenvalue function, q s ( k ) or p s ( k ) , there is a unique eigenstate, | q i or | p i , whichis associated with the function as a whole and not with particular function values of theeigenvalue function. For that reason, an eigenstate does not explicitly depend on the valueof the wave vector.To simplify notation, we shall neglect the spin degrees of freedom and not display thespin indices in the remainder of this paper. It is nevertheless straight-forward to reintroducethem if necessary. he eigenstates in (1) can be created from the vacuum | q i = ˆ a † q | vac i , | p i = ˆ a † p | vac i , (5) with operators given by ˆ a † q = π − Ω / exp (cid:16) − k q k + ˆ a † Q − ˆ a † R (cid:17) , ˆ a † p = 2 Ω / π Ω / exp (cid:16) − k p k + i ˆ a † P + ˆ a † R (cid:17) , (6) where ˆ a † Q = √ Z ˆ a † ( k ) q ( k ) d ¯ k, ˆ a † P = √ Z ˆ a † ( k ) p ( k ) d ¯ k, ˆ a † R = 12 Z ˆ a † ( k )ˆ a † ( k ) d ¯ k, (7) and, k f k ≡ Z | f ( k ) | d ¯ k, (8) for f : R → C . The quantity Ω in (6) is defined as Ω ≡ Z δ (0) d k, (9) and (within the current context) it represents the cardinality of a countable infinite set Ω = ℵ . The integration measures in (6) – (8) and below are given in terms of a simplifiednotation: d ¯ k ≡ d k (2 π ) ω . (10) Note that all the wave vector dependencies are integrated out in (6) , so that the elementsof the quadrature bases do not explicitly depend on the wave vector.
The quadrature bases obey orthogonality conditions [30, 31], expressed in terms of Dirac δ functionals (cid:10) q | q (cid:11) = δ [ q − q ] , (cid:10) p | p (cid:11) = (2 π ) Ω δ [ p − p ] . (11) The square brackets indicate that the quantity is a functional (a function of functions),where q and p represent functions. It depends on the entire functions and not on particularfunction values of those functions. For that reason, the quantity does not explicitly dependon k and we do not show the arguments of the functions inside the square brackets.The completeness conditions for the spatiotemporal quadrature bases [30, 31] are rep-resented as functional integrals Z | q i h q | D [ q ] = , Z | p i h p | D ◦ [ p ] = , (12) here is the identity operator for the entire Hilbert space of all quantum optical states.The functional measures in (12) run over all finite-energy real-valued functions. The mea-sure for the integral over p incorporates a normalization constant: D ◦ [ p ] ≡ D (cid:20) p π (cid:21) = 1(2 π ) Ω D [ p ] . (13) Since these functional integrals can, apart from some special cases, only be evaluated whentheir integrands are in Gaussian form, one can simplify the notation. The Gaussian formimplies an exponential function with an argument consisting of integrals over some degreesof freedom, typically the three-dimensional wave vectors. The integrands of these integralsare products of functions of the wave vectors. There may be multiple distinct wave vectorsthat are being integrated. Usually, a given wave vector would appear exactly twice asarguments of functions in each term, thus connecting a pair of functions in the term. Wedenote such a connection by a binary operator (cid:5) and we call it a (cid:5) - contraction .As an example, we introduce the following notation for the inner-product between twofunctions h f, g i ≡ Z f ∗ ( k ) g ( k ) d ¯ k ≡ f ∗ (cid:5) g. (14) If there is a kernel function involved, we have f ∗ (cid:5) B (cid:5) g ≡ Z f ∗ ( k ) B ( k , k ) g ( k ) d ¯ k d ¯ k . (15) Note that it is not equivalent to f ∗ (cid:5) T (cid:5) g = Z f ∗ ( k ) T ( k ) g ( k ) d ¯ k. (16) An important quantity is the overlap h q | p i , which reads h q | p i = exp (cid:20) i Z q ( k ) p ( k ) d ¯ k (cid:21) ≡ exp( iq (cid:5) p ) . (17) It appears when expressions are converted from one quadrature basis into another mutuallyunbiased quadrature basis and thus represents the kernel of a functional Fourier transform.These Fourier transforms indicate that the quadrature bases are related by functionalFourier transforms. As a result, one can express one in terms of the other as | p i = Z | q i exp( iq (cid:5) p ) D [ q ] , | q i = Z | p i exp( − iq (cid:5) p ) D ◦ [ p ] . (18) Using the expressions of the eigenvalue equations in (1) , the Fourier relationships allowthe quadrature operators to be represented in their dual bases by functional derivatives ˆ p ( k ) = Z | q i (cid:20) − i δδq ( k ) (cid:21) h q | D [ q ] , ˆ q ( k ) = Z | p i (cid:20) i δδp ( k ) (cid:21) h p | D ◦ [ p ] . (19) he operation of a functional derivative is defined by δδf ( k ) f ( k ) = (2 π ) ωδ ( k − k ) ≡ . (20) One can also use the definitions of the quadrature bases in terms of their dual bases,given in (18) , to define shift operators | q − q i = exp( iq (cid:5) ˆ p ) | q i , | p − p i = exp( − i ˆ q (cid:5) p ) | p i . (21) These shift operators will come in handy later. δ functionals Using (11) , (12) , and (17) , one can show that Z exp( iq (cid:5) p − iq (cid:5) p ) D ◦ [ p ] = δ [ q − q ] , Z exp( − iq (cid:5) p + iq (cid:5) p ) D [ q ] = (2 π ) Ω δ [ p − p ] . (22) Combining these integrals and converting the real-valued field variables into complex-valued field variables, given by α ( k ) = 1 √ q ( k ) + ip ( k )] , (23) one obtains Z exp ( α ∗ (cid:5) α − α ∗ (cid:5) α ) D ◦ [ α ] = (2 π ) Ω δ [ α ] , (24) where D ◦ [ α ] ≡ D [ q ] D ◦ [ p ] , (25) and δ [ α ] ≡ δ [ q ] δ [ p ] . (26) The complex-valued function α ( k ) , defined in (23) , can serve different purposes. Itcan be regarded as an independent field variable in the context of functional expressionsand thus can become the integration variable in functional integrals. It can also serve as a parameter function , representing for instance the spectral function in fixed-spectrum coher-ent states, considered below. Such parameter functions can also be turned into integration(fields) variables for functional integrals. The generic functional integral with an integrand in isotropic Gaussian form can be eval-uated to give Z exp ( − α ∗ (cid:5) K (cid:5) α − α ∗ (cid:5) ξ − ζ ∗ (cid:5) α ) D ◦ [ α ] = exp (cid:0) ζ ∗ (cid:5) K − (cid:5) ξ (cid:1) det { K } , (27) where K is an invertible kernel, and ξ and ζ are arbitrary complex functions. Invertibilityimplies that the kernel must have an inverse K − , such that K (cid:5) K − = Z K ( k , k ) K − ( k , k ) d ¯ k = δ ( k − k ) . (28) he functional determinant det { K } can be expressed as det { K } = exp[ tr { ln (cid:5) ( K ) } ] , (29) where ln (cid:5) ( · ) is defined as the inverse of exp (cid:5) ( H ) ≡ + ∞ X n =1 n ! ( H ) (cid:5) n , (30) for an arbitrary kernel function H ( k , k ) , and the trace of such a kernel is given bytr { H } ≡ Z H ( k , k ) d ¯ k. (31) The fixed-spectrum
Fock states are defined as | n F i = 1 √ n ! (cid:16) ˆ a † F (cid:17) n | vac i , (32) in terms of fixed-spectrum creation operators, defined by ˆ a † F ≡ Z ˆ a † ( k ) F ( k ) d ¯ k. (33) The angular spectrum (or Fourier domain wave function) F ( k ) that parameterizes a fixed-spectrum Fock state is normalized: k F k = Z | F ( k ) | d ¯ k = 1 . (34) It ensures that the fixed-spectrum ladder operators obey a simple commutation relation [ˆ a F , ˆ a † F ] = 1 and that each fixed-spectrum Fock state is normalized h n F | n F i = 1 . Theinner-product between Fock states with different spectra reads h m F | n G i = δ mn ( h F, G i ) n , (35) where h F, G i is defined in (14) . When the annihilation operator in the momentum basis isapplied to the fixed-spectrum Fock states, we obtain ˆ a ( k ) | n F i = | ( n − F i F ( k ) √ n. (36) The fixed-spectrum Fock states are eigenstates of the number operator ˆ n ≡ Z ˆ a † ( k )ˆ a ( k ) d ¯ k. (37) Using (36) , one can show that ˆ n | n F i = | n F i n. (38) .8 Fixed-spectrum coherent states The fixed-spectrum coherent states are defined as eigenstates of the standard (wave vectordependent) annihilation operator ˆ a ( k ) | α F i = | α F i α ( k ) , (39) where the eigenvalue function α ( k ) is an arbitrary finite-energy complex-valued spectralfunction α : R → C . The fixed-spectrum coherent states | α F i do not explicitly depend on k . There exists a unique fixed-spectrum coherent state for every spectral function α ( k ) .The concept of a fixed-spectrum coherent states is not new [37]. They have been used invarious contexts, for instance in condensed matter theory to calculate Green functions [38].The subscript F in | n F i , | α F i and ˆ a † F is a reminder that the state or operator isparameterized in terms of a fixed spectrum and should not necessarily be seen as a labelfor the associated complex-valued function. The latter is thus represented as α ( k ) and not α F ( k ) . Later, where we use different fixed-spectrum coherent states in the same expression,we will use the parameter functions to label the coherent states, instead of α F .The fixed-spectrum coherent states can be expressed in terms of functional displacementoperators given by ˆ D [ α F ] ≡ exp (cid:16) α (cid:5) ˆ a † − α ∗ (cid:5) ˆ a (cid:17) , (40) which are generalizations of the multimode displacement operator [39]. The inner-productbetween different fixed-spectrum coherent states can be derived from their displacementoperators and reads h α F | β G i = exp (cid:16) − k α k − k β k + h α, β i (cid:17) . (41) As a consequence, it follows that the inner-product between a fixed-spectrum coherentstate and the vacuum state is h α F | vac i = exp (cid:16) − k α k (cid:17) . (42) Although not orthogonal, the fixed-spectrum coherent states resolve the identity oper-ator. The completeness condition for the fixed-spectrum coherent states reads [31] = Z | α i h α | D ◦ [ α ] . (43) To expand fixed-spectrum coherent states in terms of the spatiotemporal quadrature bases,we use the operators defined in (7) and employ the eigenstate property of the coherentstates in (39) . As a result, we obtain ˆ a Q | α F i = | α F i √ q (cid:5) α , ˆ a R | α F i = | α F i α (cid:5) α , (44) where α represents the complex-valued parameter function of the fixed-spectrum coherentstate. Therefore, h q | α F i = π − Ω / h vac | exp (cid:16) − k q k + ˆ a Q − ˆ a R (cid:17) | α F i = π − Ω / exp (cid:16) − k q k − k α k + √ q (cid:5) α − α (cid:5) α (cid:17) . (45) If we express α ( k ) in terms of its real and imaginary parts, as in (23) , we obtain h q | α F i = π − Ω / exp h ip (cid:5) (cid:16) q − q (cid:17) − k q − q k i . (46) Wigner functional theory
Here, we develop the formalism for Wigner functionals in quantum optics. To avoid clutter-ing the notations, we proceed to neglect the spin indices. However, we emphasize that onecan incorporate spin into the formalism when necessary. Therefore, the resulting formalismrepresents all the degrees of freedom of quantum optics.The expressions that are obtained resemble those in the standard formalism that onlyinvolves the particle-number degrees of freedom. The expressions we obtain here often lookthe same. However, we emphasize that due to the functional nature of these expressions,they also incorporate all the spatiotemporal degrees of freedom.
The generic definition of a
Wigner functional is W [ q, p ] ≡ Z D q + x (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) q − x E exp( − ip (cid:5) x ) D [ x ] , (47) where ˆ A is an operator on the Hilbert space of all quantum optical states, incorporatingboth particle-number degrees of freedom and spatiotemporal degrees of freedom. Thesquare brackets in W [ q, p ] indicate that the quantity is a functional. While q and p representtwo real-valued functions, we’ll eventually combine them into a single complex-valuedfunction α and represent the Wigner functional as W [ α ] . As in the case of Wigner functions,which only represent the particle-number degrees of freedom, Wigner functionals, whichrepresent both the particle-number degrees of freedom and the spatiotemporal degrees offreedom, can be negative and thus need to be interpreted as quasi-probability distributions.If the operator is a density operator ˆ ρ , the resulting Wigner functional represents aquantum state. The density operator can also be represented as a density ‘matrix’, whichwe refer to as a density functional ρ h q + x, q − x i ≡ D q + x (cid:12)(cid:12)(cid:12) ˆ ρ (cid:12)(cid:12)(cid:12) q − x E . (48) In the case of a pure state, the density functional becomes a product of a wave functional ψ [ q ] = h q | ψ i and its complex conjugate ρ h q + x, q − x i = ψ h q + x i ψ ∗ h q − x i . (49) One can convert the calculation of the Wigner functional into a purely operator-basedcalculation. For this purpose, we use the shift operators defined in (21) and pull theFourier kernel through the quadrature basis elements to convert it into operators. Whenthese operators on either side of the density operator are combined with the aid of theBaker-Campbell-Hausdorff formula the result reads W [ q, p ] = 2 Ω tr n ˆ ρ ˆ D [ q, p ] ˆΠ ˆ D † [ q, p ] o , (50) where ˆ D [ q, p ] is the displacement operator, defined in (40) , and the parity operator for theentire Hilbert space is given by ˆΠ ≡ Z (cid:12)(cid:12) − q (cid:11) (cid:10) q (cid:12)(cid:12) D [ q ] . (51) It thus follows that one can compute the Wigner functional of an operator by computingits trace with the displaced parity operator, similar to the way it is done without thespatiotemporal degrees of freedom [40]. .2 Functional Weyl transformation The inverse process whereby a density functional in either of the quadrature bases is repro-duced from its Wigner functional is given by a generalization of the Weyl transformation.For the q -basis, we have ρ [ q, q ] = Z W h ( q + q ) , p i exp[ ip (cid:5) ( q − q )] D ◦ [ p ] . (52) A similar expression applies for the p -basis. The functional Weyl transformation can alsobe used to reproduce the density operator: ˆ ρ = Z (cid:12)(cid:12)(cid:12) q + x E W [ q, p ] exp( ip (cid:5) x ) D q − x (cid:12)(cid:12)(cid:12) D ◦ [ p ] D [ q, x ] . (53) It then follows that the trace of the density operator is represented by the functionalintegral of the associated Wigner functionaltr { ˆ ρ } = Z W [ α ] D ◦ [ α ] = 1 , (54) where we used (23) and (25) to express it in terms of α ’s, instead of q ’s and p ’s.The expectation value of an observable, which is given by the trace of the product ofdensity operator of the state and the operator for the observable, is represented by thefunctional integral of the product of their Wigner functionalstr { ˆ ρ ˆ O } = Z W ˆ ρ [ α ] W ˆ O [ α ] D ◦ [ α ] , (55) in the same way it is done with Wigner functions without the spatiotemporal degrees offreedom. The Wigner functional for the observable can be computed in the same way thatthe Wigner functional is computed for the density operator. The characteristic functional is the functional Fourier transform of the Wigner functional χ [ ξ, ζ ] = Z exp( ip (cid:5) ζ − iξ (cid:5) q ) W [ q, p ] D [ q ] D ◦ [ p ] . (56) The symplectic form of the functional Fourier kernel is due to the simultaneous Fouriertransformation of q and p , which are already related by a Fourier relationship (18) . TheWigner functional is obtained from the characteristic functional via the inverse functionalFourier transform W [ q, p ] = Z exp( iξ (cid:5) q − ip (cid:5) ζ ) χ [ ξ, ζ ] D [ ζ ] D ◦ [ ξ ] . (57) The characteristic functional of an operator is directly given by χ [ ξ, ζ ] = Z D q + ζ (cid:12)(cid:12)(cid:12) ˆ ρ (cid:12)(cid:12)(cid:12) q − ζ E exp( − iξ (cid:5) q ) D [ q ] . (58) The calculation of a characteristic functional can be converted into a purely operator-based calculation, similar to the way it is done for Wigner functionals, by using the shiftoperators defined in (21) and the Baker-Campbell-Hausdorff. The result χ [ ξ, ζ ] = tr n ˆ ρ ˆ D † [ ζ, ξ ] o , (59) hows that the characteristic functional of an operator is given by its trace with the adjointof the displacement operator. Note the interchange in the roles of ξ and ζ .The characteristic functional serves as a generating functional for the moments of theWigner functional, in the same way it does for Wigner functions without the spatiotemporaldegrees of freedom. However, here the derivatives are replaced by functional derivatives.The ( m, n ) -th moment of the Wigner functional is obtain by the ( m, n ) -th functionalderivatives with respect to the respective functional variables Z q m p n W [ q, p ] D [ q ] D ◦ [ p ] = ( i ) m ( − i ) n δ mξ δ nζ χ [ ξ, ζ ] (cid:12)(cid:12)(cid:12) ξ = ζ =0 . (60) Here, q m p n is the Wigner functional of the operator with the equivalent powers of quadra-ture operators in symmetrized order. The functional expression in (60) was previouslyshown in the context of quantum field theory for the spatial degrees of freedom only [34]. The Wigner functional for the product of two operators can be obtained by expressingthese operators in terms of Weyl transformations (53) . The result is a functional integralover the Wigner functionals of these operators: W ˆ A ˆ B [ q, p ] =2 Z W ˆ A [ q − q , p − p ] W ˆ B [ q − q , p − p ] × exp[ i q (cid:5) p − q (cid:5) p )] D [ q , q ] D ◦ [ p , p ] . (61) It represents the
Moyal product or star product for Wigner functionals [20]. An alternativeway to express the star product is in terms of functional derivatives W ˆ A ˆ B [ q, p ] = W ˆ A [ q, p ] exp (cid:20) i (cid:16) ←− δ q −→ δ p − −→ δ q ←− δ p (cid:17)(cid:21) W ˆ B [ q, p ] , (62) where δ q and δ p represent functional derivatives with respect to q and p , respectively, andthe arrows indicate to which side the derivatives are applied. One can use the expressionsof the Wigner functionals in terms of their characteristic functionals, given in (57) , to showthat (62) leads to (61) .For the product of three operators, the functional integral expression is W ˆ A ˆ B ˆ C [ α ] = Z exp[( α ∗ − α ∗ a ) (cid:5) α b − α ∗ b (cid:5) ( α − α a )] W ˆ A h ( α a + α b + α ) i × W ˆ B [ α a ] W ˆ C h ( α a − α b + α ) i D ◦ [ α a , α b ] . (63) Here, we used (23) and (25) to render the expression in terms of α ’s. As a quasi-probability distribution over the functional phase space, the Wigner functionalof a state does not qualify as a true probability density. However, one can compute aprobability density from it by integrating over either p or q (or any linear combination of p and q ). Integrating the Wigner functional over p , we obtain Z W [ q, p ] D ◦ [ p ] = Z D q + q (cid:12)(cid:12)(cid:12) ˆ ρ (cid:12)(cid:12)(cid:12) q − q E δ [ q ] D [ q ]= h q | ˆ ρ | q i = ρ [ q, q ] . (64) ence, we recover the diagonal of the density functional, which represents the probabilities.To perform the integration over q , we first need to insert identities resolved in the p -basis Z W [ q, p ] D [ q ] = Z D q + q (cid:12)(cid:12)(cid:12) p E h p | ˆ ρ | p i D p (cid:12)(cid:12)(cid:12) q − q E exp( − ip (cid:5) q ) D ◦ [ p , p ] D [ q, q ]= h p | ˆ ρ | p i = ρ [ p, p ] . (65) If we integrate these probability distributions over the remaining variables, we obtain 1,thanks to the normalization.
Although the spatiotemporal quadrature bases naturally lead to a Wigner functional for-mulation, other types of quasi-probability distributions can also be used in the formal-ism. The functional versions of the Glauber-Sudarshan P -distribution and the Husimi Q -distribution, which are of significant interest in quantum optics, are respectively definedby the following expressions, incorporating fixed-spectrum coherent states: ˆ ρ = Z | α i P [ α ] h α | D ◦ [ α ] , (66) and Q [ α ] = h α | ˆ ρ | α i . (67) To show how these functional quasi-probability distributions are related to the Wignerfunctional formulation, we use the Baker-Campbell-Hausdorff formula in (88) to definedifferent versions of the displacement operator, based on their operator ordering: ˆ D s [ η ] ≡ exp (cid:16) s k η k (cid:17) ˆ D [ η ] = exp (cid:16) ˆ a † (cid:5) η (cid:17) exp ( − η ∗ (cid:5) ˆ a ) for s = 1exp (cid:16) ˆ a † (cid:5) η − η ∗ (cid:5) ˆ a (cid:17) for s = 0exp ( − η ∗ (cid:5) ˆ a ) exp (cid:16) ˆ a † (cid:5) η (cid:17) for s = − . (68) So, for s = 1 , the operator is normal ordered and for s = − it is anti-normal ordered.The usual displacement operator in symmetrical order is obtained for s = 0 .The different versions of the displacement operator can now be used, in analogy with (59) , to compute the associated characteristic functional for a density operator ˆ ρ : χ s [ ξ, ζ ] = tr n ˆ ρ ˆ D † s [ η ] o , (69) where η = ( ζ + iξ ) / √ . The functional Fourier transform of the characteristic functional(assuming it is well-defined), as given in (57) , then leads to the associated functionalquasi-probability distribution W s [ α ] . Hence, W s [ α ] = tr n ˆ ρ ˆ T s [ α ] o , (70) where ˆ T s [ α ] ≡ Z ˆ D † s [ η ] exp( α ∗ (cid:5) η − η ∗ (cid:5) α ) D ◦ [ η ] . (71) One can now use (70) and (71) , together with either (66) or (67) to show that W [ α ] = P [ α ] and W − [ α ] = Q [ α ] . In the latter case, one can insert an identity resolved in terms of thecoherent states (43) to aid the calculation. The association between the operator orderingand the type of functional quasi-probability distributions is the same as in the case withoutthe spatiotemporal degrees of freedom [41]. Examples of Wigner functionals
We substitute ˆ ρ → | α F i h α F | into (47) to obtain the Wigner functional for a fixed-spectrumcoherent state, W [ q, p ] = Z D q + x (cid:12)(cid:12)(cid:12) α F E D α F (cid:12)(cid:12)(cid:12) q − x E exp( − ip (cid:5) x ) D [ x ] . (72) The expressions for the two overlaps are obtained from (46) . After substituting them into (72) and evaluating the functional integral over x , we obtain W [ α ] = N exp (cid:16) − k α − α k (cid:17) , (73) where N is a normalization constant, and α ( k ) is the parameter function of the fixed-spectrum coherent state. We emphasize that, while the expression looks familiar, one mustremember that it represents a functional — the argument contains an integral of functionsover all wave vectors.The normalization constant N can be obtained either by keeping track of the constantsduring the calculation or by imposing the requirement that the state is normalized, as in (54) . Both ways lead to N = 2 Ω . (74) It is often convenient to employ coherent states in the computation of the Wigner func-tionals. Inserting identity operators, resolved in terms of coherent states (43) , into (47) ,we obtain W ˆ A [ q, p ] = Z D q + x (cid:12)(cid:12)(cid:12) α E h α | ˆ A | α i D α (cid:12)(cid:12)(cid:12) q − x E exp( − ip (cid:5) x ) D [ x ] D ◦ [ α , α ] , (75) where ˆ A is an arbitrary operator, and α and α are the parameter functions for with thefixed-spectrum coherent states, serving as integration variables. Next, we substitute (46) into the result and evaluate the functional integral over x to obtain W ˆ A [ α ] = N Z exp (cid:16) − k α k + 2 α ∗ (cid:5) α + 2 α ∗ (cid:5) α − k α k − k α k − α ∗ (cid:5) α (cid:17) × h α | ˆ A | α i D ◦ [ α , α ] , (76) where N is given in (74) . It now remains to evaluate the overlap of the operator ˆ A by thetwo coherent states and to perform the functional integrations over α and α to obtainthe Wigner functional for ˆ A . Next, we use the coherent state assisted approach to compute the Wigner functionals forthe fixed-spectrum Fock states, defined in (32) . The overlap between such a Fock stateand two arbitrary fixed-spectrum coherent states gives h α | n F i h n F | α i = exp (cid:16) − k α k − k α k (cid:17) n ! ( h α , F ih F, α i ) n , (77) here we used (32) , (33) , (39) and (42) . One can simplify the calculation by representingthe above result as a generating functional K ≡ exp (cid:16) − k α k − k α k + η h α , F i + η h F, α i (cid:17) , (78) where η and η are auxiliary parameters, such that h α | n F i h n F | α i = 1 n ! ∂ n ∂η n ∂ n ∂η n K (cid:12)(cid:12)(cid:12)(cid:12) η = η =0 . (79) Substituting h α | ˆ A | α i → K into (76) , we obtain a generating functional for the Wignerfunctionals of the Fock states, expressed as a functional integral W ( η ) = X n η n W | n i [ α ]= N Z exp (cid:16) − k α k + 2 α ∗ (cid:5) α + 2 α ∗ (cid:5) α − k α k − k α k − α ∗ (cid:5) α + η α ∗ (cid:5) F + η F ∗ (cid:5) α ) D ◦ [ α , α ] . (80) The functional integration over α and α produces W ( η , η ) = N exp (cid:16) − k α k + 2 η α ∗ (cid:5) F + 2 η F ∗ (cid:5) α − η η (cid:17) . (81) One can show that the Wigner functionals for the individual fixed-spectrum Fock statesare given by W | n i [ α ] = ( − n N L n (cid:16) |h F, α i| (cid:17) exp (cid:16) − | α | (cid:17) , (82) where L n is the n -th order Laguerre polynomial. Although the Wigner functionals for theindividual fixed-spectrum Fock states are not in Gaussian form, the generating functionalfor the Wigner functionals of these Fock states, given in (81) , is in Gaussian form and cantherefore be used in functional integrals. Wigner functionals are not only associated with quantum states — they can also representarbitrary operators. We now use the coherent state assisted approach to obtain a Wignerfunctional for the number operator defined in (37) . For this purpose, we compute h α | ˆ n | α i = h α , α i exp (cid:16) − k α k − k α k + h α , α i (cid:17) , (83) where we used (41) . We again use a generating functional to simplify calculations. Thegenerating functional G = exp (cid:16) − k α k − k α | + J h α , α i (cid:17) , (84) reproduces the overlap through h α | ˆ n | α i = ∂ J G| J =1 . (85) We substitute h α | ˆ A | α i → G into (76) and evaluate the functional integrals to obtain W ( J ) = N Z exp (cid:16) − k α k + 2 α ∗ (cid:5) α − k α k − k α k +2 α ∗ (cid:5) α − α ∗ (cid:5) α + J α ∗ (cid:5) α ) D ◦ [ α , α ]= N (1 + J ) Ω exp (cid:20) − (cid:18) − J J (cid:19) α ∗ (cid:5) α (cid:21) . (86) inally, we evaluate the derivative with respect to J and set J = 1 to get the Wignerfunctional for the number operator: W ˆ n [ α ] = α ∗ (cid:5) α − Ω . (87) Next, we consider the displacement operator given in (40) . Using the Baker-Campbell-Hausdorff formula exp (cid:16) ˆ X + ˆ Y (cid:17) = exp (cid:16) − [ ˆ X, ˆ Y ] (cid:17) exp( ˆ X ) exp( ˆ Y ) , (88) which assumes [ ˆ X, [ ˆ X, ˆ Y ]] = [ ˆ Y , [ ˆ X, ˆ Y ]] = 0 , to separate the displacement operator into aproduct of exponential operators, we obtain ˆ D [ α ] = exp (cid:16) − k α k (cid:17) exp (cid:16) α (cid:5) ˆ a † (cid:17) exp ( − α ∗ (cid:5) ˆ a ) . (89) It allows us to compute the overlap h α | ˆ D [ α ] | α i = exp (cid:16) − k α k + α ∗ (cid:5) α − k α k − k α k + α ∗ (cid:5) α − α ∗ (cid:5) α (cid:17) . (90) for the coherent state assisted approach. After substituting it into (76) , we obtain W ˆ D = N Z exp (cid:16) − k α k + 2 α ∗ (cid:5) α + 2 α ∗ (cid:5) α − k α k − k α k − α ∗ (cid:5) α + α ∗ (cid:5) α + α ∗ (cid:5) α − α ∗ (cid:5) α − k α k (cid:17) D ◦ [ α , α ] . (91) Finally, we evaluate the functional integrals over α and α and obtain a familiar form: W ˆ D [ α ; α ] = exp ( α ∗ (cid:5) α − α ∗ (cid:5) α ) . (92) One can use the Wigner functional expression for the product of three operators in (63) to obtain a general expression for the Wigner functional of an arbitrary state afterdisplacement operators are applied to it: W ˆ D ˆ ρ ˆ D † [ α ] = Z exp ( α ∗ b (cid:5) α − α ∗ (cid:5) α b ) exp[( α ∗ − α ∗ a ) (cid:5) α b − α ∗ b (cid:5) ( α − α a )] × W ˆ ρ [ α a ] D ◦ [ α a , α b ] . (93) There are no quadratic terms for α b in the exponent. Hence, the functional integrationover α b produces a Dirac δ functional, as in (24) , which leads to W ˆ D ˆ ρ ˆ D † [ α ] = (2 π ) Ω Z W ˆ ρ [ α a ] δ [ α a − α + α ] D ◦ [ α a ] = W ˆ ρ [ α − α ] . (94) As expected, the effect of the displacement operation on an arbitrary Wigner functional isa shift in its argument. Discussion
The notation for the formalism that we present here leads to expressions that are almostidentical to those that exclude the spatiotemporal degrees of freedom. While the notationmay alleviate the complexities in its use, this resemblance could be misleading — causingone to confuse the more general expressions for those of the simpler case. Therefore, it isnecessary to emphasize the difference in meaning.Consider for example the Wigner function of the coherent state incorporating only theparticle-number degrees of freedom, given by W ( α ; α ) = 2 exp (cid:16) − | α − α | (cid:17) , (95) and compare it to the Wigner functional of the fixed-spectrum coherent state, which in-corporates both the spatiotemporal and particle-number degrees of freedom W [ α ; α ] = 2 Ω exp (cid:16) − k α − α k (cid:17) . (96) While the two expressions may seem very similar, they represent different content. In (95) , α and α respectively represent a complex variable and a complex parameter. Onthe other hand, in (96) they represent functions . That is why the former is a Wignerfunction, while the latter is a Wigner functional. The argument of the former contains themodulus square of the difference between the complex values. In contrast, the argumentof the latter contains an integral over the space of wave vectors that computes the squaredmagnitude of the difference between the complex functions. Due to the integral, the wavevector dependencies of the complex functions are removed (integrated out). As a result,the Wigner functional does not explicitly depend on the wave vector. However, it doesdepend on the complex function as a whole.Another difference between the expressions in (95) and (96) is the normalization con-stant. In the former case, the constant is a finite number. In the latter case, it becomes adivergent constant that one can associate with the cardinality of the space. If Ω is associ-ated with the cardinality of countable infinity, then Ω would represent the cardinality ofthe continuum. It is inevitable that functional integrals would produce such divergent con-stants. However, when these functional integrals are employed to compute the predictedresults of measurements, one expects to obtain finite quantitative results. Therefore, thedivergent constants must cancel. According to cardinal arithmetic [42], all divergent con-stants of the same cardinality are formally equal. However, unless one can keep carefultrack of these constants, their cancellation may hide finite (ordinal) numbers that are im-portant for the correct quantitative predictions. For this reason, we retain the precise formof the divergent constants, be it π Ω / , Ω , (2 π ) Ω , or whatever else, even though all theseconstants are formally equal.One can also compare the current formalism with the continuous variable formalism[17]. The latter represents the spatiotemporal degrees of freedom as a finite number ofdiscrete modes. The implication is that the complete Hilbert space of quantum opticalstates is stratified into a tensor product of discrete Hilbert spaces, each representing onemode. Such a stratification cannot completely separate these Hilbert spaces because theyshare the same vacuum state. Practical calculations require a truncation to get a finitenumber of such Hilbert spaces. A coherent state in this formalism is expressed as W [ Q ; Q ] = 2 M exp h − Q − Q ) T J ( Q − Q ) i , (97) where Q is a vector consisting of M pairs of quadrature variables, one pair for each of the M Hilbert spaces; Q is a vector of the associated parameters and J is a symplectic matrix hat maintains the correct multiplications among the quadrature variables. Although thereare many applications where the latter formalism has been used successfully, such as repre-senting the spectral components in a frequency comb [43], these cases invariably truncatethe set of discrete modes to a finite number. Therefore, such cases do not represent thecomplete Hilbert space of all quantum optical states. The existence of a complete orthogonal basis for the full Hilbert space of quantum opti-cal states, which incorporates all the degrees of freedom associated with photonic states,allows one to formulate powerful tools to analyze quantum optical systems. Since thecomplete orthogonal basis is a quadrature basis, the natural choice of such a formalism isa generalization of the well-known Wigner distribution.Here, the Wigner functional formalism is presented, based on the spatiotemporal quadra-ture bases. The result demonstrates a clear analogy between the functional formalism andthe well-known Wigner function, characteristic function and Weyl transform. Even thestar product is reproduced in a similar form.We use the functional formalism to compute some examples of Wigner functionals.These examples include: the Wigner functional for fixed-spectrum coherent states, a gen-erating functional for the Wigner functionals of fixed-spectrum Fock states and the Wignerfunctionals for the number operator and the displacement operator.Functional Wigner distributions were previously introduced in the context of quantumfield theory to represent bosonic [34] and fermionic [35] quantum fields. (See also [36].)Here, our interest is quantum optics. While we only focus on the (bosonic) optical field,one could also investigate the extension of the current formalism to fermionic fields, whichcould be applied in solid state physics for instance.
Acknowledgement
We gratefully acknowledge fruitful discussions with Thomas Konrad. This work was sup-ported in part by funding from the National Research Foundation of South Africa (GrantNumbers: 118532).
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