Wigner function for SU(1,1)

In spite of their potential usefulness, Wigner functions for systems with SU(1,1) dynamical symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples.


Introduction
Phase-space methods represent a self-standing alternative to the conventional Hilbertspace formalism of quantum theory. In this approach, observables are c-number functions instead of operators, with the same interpretation as their classical counterparts, although composed in novel algebraic ways. Quantum mechanics thus appears as a statistical theory on phase space, which makes the corresponding classical limit emerge in a natural and intuitive manner.
The realm of the method was established in the groundbreaking works of Weyl [1] and Wigner [2]. Later, Groenewold [3] and Moyal [4] established a solid formulation that has developed over time into a complete discipline that has been useful in many diverse fields [5][6][7][8] The main ingredient of this approach is a bona fide mapping that relates operators to functions defined on a smooth manifold, endowed with a very precise mathematical structure [9]. However, this mapping is not unique: actually, a whole family of functions can be consistently assigned to each operator. In particular, quasiprobability distributions are the functions connected with the density operator [10][11][12][13]. For continuous variables (such as Cartesian position and momentum), the quintessential example that fuelled the interest for this field, the most common choices are the P (Glauber-Sudarshan) [14,15], W (Wigner) [2], and Q (Husimi) [16] functions, respectively. This formalism has been applied to other different dynamical symmetries. Probably the most widespread case beyond the harmonic oscillator is that of SU (2), with the Bloch sphere as associated phase space [17][18][19]; this is of paramount importance in dealing with spinlike systems [20][21][22][23][24]. Suitable results have also been found for the Euclidean group E(2), this time with the cylinder as phase space [25][26][27][28]; this is of primary importance in treating the orbital angular momentum of twisted photons [29,30]. Moreover, the basic notions have been successfully extended to discrete qudits, where now the phase space is a finite grid [31][32][33][34][35][36][37].
Surprisingly, the phase-space description of systems having SU(1,1) symmetry has received comparatively little attention [38,39], in part because the representation theory of this group is not as familiar as SU (2) or even E(2). However, SU(1,1) plays a major role in connection with what can be called two-photon effects [40][41][42][43]. The topic is experiencing a revival in popularity due to the recent realization of a nonlinear SU(1,1) interferometer [44,45]. According to the pioneer proposal by Yurke et al. [46], this device would allow one to improve the phase measurement sensitivity in a stunning manner [47].
In spite of the importance of these systems, the mathematical complexity of the group SU(1,1) [48] has prevented a proper phase-space description. In this paper we approach this question using a physics-based approach. For SU (2), one can model the description in terms of a superposition of two harmonic modes. In technical terms, this corresponds to the Jordan-Schwinger bosonic realization of the algebra su(2) [49]. Here, we propose a similar way to deal with su(1, 1): starting with two orthonormal modes, and using the standard tools for continuous variables, we eliminate the irrelevant degrees of freedom and we get a description in the two-sheeted hyperboloid, which is the natural arena to represent the physics associated to these systems.
Our final upshot is that the Wigner function can be expressed as the average value of the displaced parity operator. This is reassuring, for it is also the case for continuous variables [50]. Moreover, as this property has been employed for the direct sampling of the Wigner function for a quantum field [51][52][53], our result opens the way for the experimental determination of the Wigner function for SU(1,1).
2 Phase-space representation of a single mode To keep the discussion as self-contained as possible, we first briefly summarize the essential ingredients of phase-space functions for a harmonic oscillator that we shall need for our purposes.
We consider the standard harmonic oscillator described by annihilation and creation operatorsâ andâ † , which obey the bosonic commutation relation They are the generators of the Heisenberg-Weyl algebra, which has become the hallmark of noncommutativity in quantum theory [54]. The classical phase space is here the complex plane C. These complex amplitudes are expressed in terms of the quadrature operatorsx andp asâ and the commutation relation (1) reduces then to the canonical form [x,p] = i1. A central role in what follows will be played by the unitary operator which is called the displacement operator for it displaces a state localized in phase space at α 0 to the point α 0 + α. The Fourier transform of the displacement is the Cahill-Glauber kernel [55]ŵ which is an instance of a Wigner-Weyl quantizer [56]. The operatorsŵ(α) constitute a complete trace-orthonormal set that transforms properly under displacementsŵ is the parity operator. IfÂ is an arbitrary (trace-class) operator acting on the Hilbert space of the system, the Wigner-Weyl quantizer allows one to associate toÂ a function WÂ(α) representing the action of the corresponding dynamical variable in phase space: The function WÂ(α) is the symbol of the operatorÂ. Such a map is one-to-one, so we can invert it to get the operator from its symbol througĥ We focus on what follows on the Wigner function, although everything can be immediately extended to any other case. Actually, the Wigner function is nothing but the symbol of the density matrixˆ . Consequently, The Wˆ (α) defined in (9) fulfills the basic properties required for any good probabilistic description [57]. First, due to the Hermiticity ofŵ(α), it is real for Hermitian operators. Second, the probability distributions for the canonical variables can be obtained as the corresponding marginals. Third, Wˆ (α) is translationally covariant, which means that for the displaced stateˆ =D(α )ˆ D † (α ), one has so that it follows displacements rigidly without changing its form, reflecting the fact that physics should not depend on a certain choice of the origin. Finally, the overlap of two density operators is proportional to the integral of the associated Wigner functions: This property (known as traciality) offers practical advantages, since it allows one to predict the statistics of any outcome, once the Wigner function of the measured state is known.
For what follows it is important to stress that (5) and (6) together imply the following alternative form for the Wigner function: that is, the displaced parity [50].
To conclude, we mention that the displacements constitute also a basic ingredient in the concept of coherent states. If we choose a fixed normalized reference state |Ψ 0 , we have [58] |α =D(α) |Ψ 0 , so they are parametrized by phase-space points. These states have a number of remarkable properties inherited from those ofD(α). The standard choice for the fiducial vector |Ψ 0 is the vacuum |0 .
3 Phase-space representation of two modes Next, we consider the superposition of two modes in two orthogonal directions, say x and y, with momenta p x and p y , respectively. Since they are kinematically independent, the complex amplitudes of these modes (we denote byâ andb) commute ([â,b] = 0) and the total Wigner-Weyl quantizer can be expressed as the product of the corresponding ones for each mode; that is,ŵ Using the form in (12) and disentangling the exponentials, we get As we can see, this kernel depends on the four real variables α = (x, p x ) and β = (y, p y ).
As a consequence, the resulting Wigner function W (α, β) contains all the information, but it is hard to grasp any physical flavor from it: in particular, it cannot be plotted, which is always a big advantage in depicting complex phenomena. To avoid this drawback we use the parametrization where the radial variable r 2 = |α| 2 −|β| 2 represents the difference in intensities between the two modes, and the parameters χ and τ can be interpreted as azimuthal and "polar" angles on a two-sheeted hyperboloid H 2 [59]. A similar parametrization as in (16), wherein the hyperbolic functions are replaced with trigonometric ones, maps two complex modes into the Bloch sphere S 2 . This is an instance of a Hopf fibration [60]. Therefore, the hyperboloid H 2 can be properly called the Bloch hyperboloid and the map (16) is a noncompact Hopf fibration [61]. After some lengthy algebra, the kernel can be recast in the form whereN =â †â +b †b is the total number and we have introduced the two-mode squeeze operatorŜ which is defined in terms of the two-mode realization of the su(1, 1) algebrâ with commutation relations Using the Baker-Hausdorff formula, one can check that and i.e., the transformationŜ(ζ) depends only on the sum of the phases. We proceed to integrate over the physically irrelevant phase ϕ to get Finally, we integrate over r (with a measure 2rdr): If we note that we arrive at our central resultŵ Since (−1)K 0 is the SU(1,1) parity andŜ(ζ) is a displacement operator, this shows that the Wigner function for SU(1,1) can be understood much in the same way as for the harmonic oscillator: just a displaced parity. It is worth mentioning that this parity operator has been recently proposed as a scheme to beat the Heisenberg limit in SU(1,1) interferometry [62][63][64][65]. Bear in mind though that the SU(1,1) parity is not, in general, the parity of the photon numbers [66].
The operatorŜ(ζ) displaces by a complex number. However, there is a one-to-one correspondence between points in the complex plane and the upper sheet of the hyperboloid H 2 , established via stereographic projection from the south pole: where n is a unit vector on H 2 , with the metric n 2 = n 2 0 − n 2 1 − n 2 2 . This construction provides a complex structure on the upper sheet of the hyperboloid H 2 , which can be treated as a noncompact complex manifold.

Explicit form of the Wigner function for SU(1,1)
To gain further insights into this formalism, we will work out the structure of the Wigner function for SU(1,1) with more detail.
Before going ahead we recall that the irreducible representations (irreps) of SU(1,1) are labeled by the eigenvalues of the Casimir operator whereK ± = ±i(K 1 ± iK 2 ). The irrep k is carried by a Hilbert space spanned by the common eigenstates ofK 2 andK 0 : {|k, µ : µ = k, k +1, . . .}. The action of the generators {K 0 ,K ± } therein isK All unitary irreps are infinite dimensional. There are several different series of irreps for SU(1,1) fixed by the domains of the eigenvalues k [67]. In this paper we are interested in the positive discrete series, where k is a nonnegative integer or half integer. The carrier space is denoted by D + k . If the number of excitations in modes a and b are n a and n b , respectively, then k and µ satisfy k = 1 2 (|n a − n b | + 1) , µ = 1 2 (n a + n b + 1) .
As n b > n a can be obtained from n a > n b by just a relabelling of modes, with no physical consequences, we consider ±(n a − n b ) to be equivalent irreps. The total Hilbert space of the two oscillators decomposes then as This decomposition allows us to expand any (pure) state in an SU(1,1)-invariant way; viz, where Ψ kµ = k, µ|Ψ . The Wigner function reads where d (k) µµ (τ ) are the d-functions for SU (1,1), which are the hyperbolic counterparts of the Wigner d functions for SU(2) [68]; that is, They can be expressed as [69,70] d (k) Equation (33) is a closed expression for the SU(1,1) Wigner function we were looking for. Alternatively, one can rewrite the Wigner kernelŵ(ζ) in the form which can be disentangled aŝ with γ ± = e ±iχ tanh(τ /2) , γ 0 = 1/ cosh 2 (τ /2) .
The action of this operator in the basis {|k, µ } can be easily found by expanding the exponentials. After a lengthy calculation the final result coincides with (35).
A word of caution is in order here. Strictly speaking, Wigner functions can be properly defined for a single irrep, where the concept of phase space is uniquely defined. However, since the Hilbert space of our original two-mode problem splits as in (31), our Wigner function appears as a sum over all the irreps in the discrete positive series.
Let us consider the relevant example of SU(1,1) coherent states, defined as [58] |ξ, k =Ŝ(ζ)|k, k , where ξ = tanh(τ /2)e iχ and ζ = 1 2 τ e iχ . These states live in the irrep k and for k = 1/2 they are nothing but two-mode squeezed vacuum states, which in the photon-number basis read Using this form in our general formula, we get an involved expression. The result appears in Fig. 1. As we can appreciate, the squeezing appears here as a displacement (and not merely as a deformation, as in the case of continuous variables). The limit of infinite squeezing corresponds to displacing the state to the infinity in the upper sheet, which means that the function tends to the border of the unit disk. Note that the metric in the unit disk is not Euclidean, but hyperbolic. This means, that as we approach the boundary, big squeezing translates in small displacements.
As a second example, we consider the factorized state where |α a is a coherent state in mode a and |ξ b a single-mode squeezed state in mode b. The decomposition (33) now involves a sum over all irreps. This sum can be split into integer and half-integer values, which translates in the presence of two peaks in the corresponding Wigner function. The displacement from the origin of these peaks is related to the squeezing, as before, and can be appreciated in Fig. 1.

Concluding remarks
Quantum phenomena must be depicted in the proper phase space. This is unanimously recognized for continuous variables (with the complex plane as phase space), for spinlike systems (with the Bloch sphere as the underlying manifold) and for orbital angular momentum (represented in the cylinder). Surprisingly, the physics related to the SU(1,1) symmetry is not displayed on the hyperboloid, the natural arena for these phenomena.
What we expect to have accomplished here is to provide a practical framework to represent SU(1,1) states in an appropriate way. Apart from the intrinsic beauty of the formalism, our compelling arguments should convince the community of the benefits that arise using the proper phase-space tools to deal with these systems.