Qutrit squeezing via semiclassical evolution

We introduce a concept of squeezing in collective qutrit systems through a geometrical picture connected to the deformation of the isotropic fluctuations of su(3) operators when evaluated in a coherent state. This kind of squeezing can be generated by Hamiltonians non-linear in the generators of su(3) algebra. A simple model of such a non-linear evolution is analyzed in terms of semiclassical evolution of the SU(3) Wigner function.


Introduction
The concept of squeezing in different systems has attracted significant attention due to its transparent physical meaning, related to the reduction of quantum fluctuations below some given threshold. Although most of applications of squeezing are related to the improvement of measurements precision, squeezing intrinsically reflects the existence of some particular correlations between parts of a quantum system. Since the squeezing parameters contains easily measurable first and second order moments of collective operators, this entails a successful application of squeezing criteria to detect quantum entanglement [1], [2], [3].
Historically, much attention has been paid to squeezing of the electromagnetic field modes or squeezing in SU (2) -or spin-like -systems. Recently, more complex experiments on quantum systems having higher symmetries have been proposed, particularly in relation to some possible applications to quantum information processes. Candidate qutrit systems described by the group SU (3) include Bose-Einstein condensates and three-level atomic ensembles interacting with quantized fields.
The definition of squeezing, while universal for harmonic oscillator-like systems, is otherwise far from unique. In spin-like systems there are several approaches used to define a squeezing parameters [4,1,5,6,7,8,3]. All parameters compare fluctuations of some suitably chosen observables with a certain threshold given by fluctuations in some reference state (or family of states). The coherent states of the corresponding quantum system are often taken as the family of reference states.
One of the crucial properties of coherent states is the invariance of the fluctuations of some observables under certain type of continuous transformations. This property allows the definition of the so-called Quantum Standard Limit [9].
In this article we use this property of coherent states to introduce the concept of squeezing for systems with SU (3) symmetry (the extension to systems with SU (N ) symmetry can also be done). The main idea consists in defining the full family K of collective operators (which in practice are some linear combinations of generators of the su(3) algebra) for which the fluctuations evaluated using SU (3) coherent states are invariant under the same group transformation that leaves invariant the fiducial state used to construct the set of coherent states.
We will show that, for a Hilbert space carrying an irreducible representation of SU (3) of the symmetric type, we can use 3 continuous parameters α 3 , β 3 , χ to label a generic element K(α 3 , β 3 , χ) ∈ K, but fluctuations of K(α 3 , β 3 , χ), when evaluated using a suitable SU (3) coherent state, are isotropic, i.e. do not depend on α 3 , β 3 , χ. Considering these (invariant) fluctuations as defining our threshold, we introduce squeezing as a reduction of fluctuations below the limit of these isotropic fluctuations in coherent states.
Since our objective is to show how SU (3) squeezing can emerge rather than propose a general criterion, we will focus on the deformations of probability distributions resulting from the Hamiltonian evolution of an initial coherent state. Geometrically, a group transformation obtained by exponentiating a linear combination of generators and acting on a state produces a simple rigid displacement of the associated probability distribution and is not associated with the introduction of correlations. A deformation of the probability density does mean that quantum correlations between parts of the system are generated; hence quantum correlations which generate the squeezing can only arise from non-linear interactions.
As the characteristic times needed to produce such correlations are inversely proportional to some power of the dimension of the system, correlations develop very rapidly and the analysis can be done using semi-classical methods. In this article we will use the SU (3) Wigner function method [10] to describe a non-linear evolution of a quantum system with the SU (3) symmetry group.
The article is organized as follows: in Section II we briefly recall general ideas on the coherent states for systems with SU (2) and SU (3) symmetries and construct the operators with isotropic fluctuations in the corresponding coherent states. In Section III we analyze squeezing generated by a simple non-linear SU (3) Hamiltonian. In Section IV the SU (3) Wigner function formalism is presented and applied to find the evolution of the squeezing parameter under the non-linear Hamiltonian.

Coherent states
Following the general construction [11,12] a coherent state for a system with a given symmetry group G acting irreducibly in a Hilbert space H is defined as a fiducial state displaced by a group transformation in G. We take this fiducial state to be the highest weight state of the irreducible representation carried by H. The highest weight state is invariant (up to a phase) under transformation from the subgroup H ⊂ G, so displacements of this state are labelled by points Ω on the coset G/H. The latter is known to be the classical phase space of the corresponding quantum system [13].
Below, we briefly review coherent states for the SU (2) and SU (3) groups, focusing only on the symmetric representations. In this case a coherent state can be considered as a composite state, occurring as a direct product of identical "single particle" states of systems with 2 or 3 energy levels, and invariant under permutation of the "particle" labels. In other words, coherent states can be conveniently thought of as symmetric (under permutation of particles) factorized states, thus displaying maximal classical correlations. Given any coherent state we can always find a operator written as linear combination of generators such that the fluctuations of this operator evaluated in the coherent state is invariant with respect to the transformations generated by the stationary subgroup H. Moreover, the fluctuations of this operator reach a value determined by the dimension of H.
|ϑ, ϕ is completely specified geometrically through the direction n = (n x , n y , n z ) of the mean spin vector S : A property of coherent states essential to us is the existence of a special tangent plane orthogonal to the direction n. If we define a direction vector n ⊥ (χ) as D(ϑ, ϕ)T (χ)x, we find n ⊥ (χ) · n = 0 for any χ.
The observablê independent of the angles ϑ, ϕ and χ when evaluated using |ϑ, ϕ . We will use the condition (10) to fix the threshold of quantum fluctuations and use this to define spin squeezing as was done by many authors [14] : a state of angular momentum j is squeezed if there is an orientation of n ⊥ (χ * ) in the tangent plane, defined for T (χ * ) ∈ H, for which 2.2. SU(3) coherent states for (λ, 0) irreps.

Semiclassical squeezing
Squeezing related to a given algebra of observables is understood to reflect correlations (commonly called quantum correlations) between components of a basis of an irrep. As mentioned before group transformations, obtained by exponentiating linear combinations of elements from the algebra, produce rigid displacements of the basis states. Correlations between basis states cannot as a matter of definition be induced by such group transformations. Rather, correlations can be either constructed through a special preparation, or obtained as a result of non-linear (in terms of the algebra of observables) transformations (usually from non-linear Hamiltonian evolution) applied to initially uncorrelated systems.
In the case of large systems, it is convenient to analyze the evolution using the phase-space approach. The reasons are twofold: we can not only represent the initial state as a real-valued function and "draw" it (for some appropriately chosen cuts) in the form a distribution "covering" some slices of the phase-space, but more importantly also deduce many qualitative features of the time-evolution of this distribution. For a wide class of quantum systems with a symmetry group G, the phase-space functions are defined through an invertible map [16], so that we associate to an operatorX a phase-space symbol where the quantization kernelŵ(Ω) is a Hermitian operator defined on the classical manifold G/H and Ω denotes the phase-space coordinates. A feature of this mapping is that the commutator of two elementsX andŶ of the Lie algebra g corresponding to the group G is mapped to the Poisson brackets of the respective symbols: The commutator of two generic operators is in general mapped to the so-called Moyal bracket. For SU (3) irreps of the type (λ, 0) and λ 1, and for sufficiently localized initial states in a class dubbed " semiclassical" [19], [17], the short time dynamics can be well described by the Liouville-type equation for the evolution of the Wigner function: where W ρ (Ω) is the Wigner function, i.e. the symbol of the density matrixρ of the system, W H (Ω) is the symbol of the Hamiltonian, and ε is the so-called semiclassical parameter. The Poisson bracket is in fact, the leading term in an expansion of the Moyal bracket in inverse powers of the square root of eigenvalue of one of the Casimir operators in the SU (3) irrep (λ, 0); we found that, for the mapping defined in [10] on SU (3)/U (2) the semiclassical parameter ε is The solution of (26) can be written in general form as where Ω(t) denotes classical trajectories on SU (3)/U (2). The approximation of dropping in Eqn.(26) higher order terms in ε describes well the initial stage of the nonlinear dynamics, when self-interference is negligible. In physical applications, semiclassical states often have the form of localized states (e.g. coherent states) and their "classicality" depends on non-invariance under the transformations induced by symmetry subgroups of the (nonlinear) Hamiltonian ("classicality" is a subtle and delicate question not addressed here) [20], [21]. The method of the Wigner functions allows us to calculate average values of the observables giving drastically better results than the "naive" solution of the Heisenberg equations of motion with decoupled correlators. On the other hands, the quantum phenomena which are due to self-interference (like Schrödinger cats) are beyond the scope of this semiclassical approximation.

Phase space considerations
From the parametrization of the coherent state of Eqn.(15), we deduce a Poisson bracket on S 4 , given by where f and g are any two functions on SU (3)/U (2). Following the prescription of [10], we associate to an operatorX a phase-space symbol W X (Ω) according to Eq.(24). This map is linear onX so we only need to consider the phase space symbols of a basis set constructed from su(3) tensors T (σ,σ) (ν1ν2ν3)I , which transforms under conjugation by g ∈ G as the state |(σ, σ)ν 1 ν 2 ν 3 I in irrep (σ, σ) transforms under g Notational details can be found in [10]. For irreps of the type (σ, σ), some weights occur multiple times and the label I, which specifies transformation properties of the states under SU (2) transformations generated by R 23 , is required to fully distinguish states with the same weights. The tensors T where D is an SU (3) group function defined in the usual way as the overlap The Wigner function corresponding to |λ00 λ00| is given by (σσ)(σσσ)0 n1n2n3;n1n2n3 2(σ + 1) (λ + 1)(λ + 2) with P a Legendre polynomial of order . For λ 1, we have found, with much similarity to the SU (2) case [22], that W |λ00 λ00| (β 2 ) is well approximated by where A = 4λ 2 (λ+1)(λ+2) is a constant obtained so that the normalization condition is satisfied. The approximate expression (35) does not describes very well the tail of the Wigner function, but for our purposes this is not essential. For the coherent state |ω = R(ω)|λ00 , the density operatorρ ω = |ω ω| is mapped to the Wigner function W ρω (Ω)

Semiclassical evolution
A simple Hamiltonian that leads to squeezing iŝ where the factor 2λ+3 5 is chosen so that no terms in T (1,1) (ν1ν2ν3)I appear in the expansion of H; this guarantees that no rigid motion on the S 4 sphere is produced. This choice of H is motivated on the following physical grounds. The operatorĥ 1 is invariant under the same U (2) transformations that leave the highest weight invariant. Squeezing resulting from its evolution is thus a pure SU (3) effect, distinct from SU (2) correlations that are present in the individual U (2) subspaces contained in (λ, 0). Pure SU (2) correlations generated by non-linear Hamiltonians have been analyzed elsewhere [23]. The symbol for this Hamiltonian is (up to a constant factor) We choose as initial state a coherent state with coordinates ω = (A 1 , B 1 , A 2 , B 2 ) so it "sits" above the minimum of H in (40), i.e. is located at A 1 = B 1 = A 2 = 0 and B 2 = arccos(−1/5). If we write the coset representative of ω −1 Ω as (ᾱ 1 ,β 1 ,ᾱ 2 ,β 2 ), we find for the coherent state and its symbol respectively: with W |λ00 λ00| given in Eqn.(34). Typical squeezing times scale as t ∼ λ −p , p > 0 and are much shorter than selfinterference times. Hence, using |ω as initial state, we can use Eqn.(26) to obtain the approximate evolution as dW ρω dt = 9 5 (λ − 1)(λ + 4)(1 + 5 cos β 2 ) ∂W ρω ∂α 2 .
This in turn implies that the angle α 2 evolves in time according to

Semiclassical squeezing
On Fig.1 we present as a 3D plot and as a contour plot the Wigner function for the initial state (41), time-evolved using the exact quantum mechanical evolution equation. The slices are taken at α 1 = β 1 = 0 and at specific values of t = 0, 0.008 and 0.015 as indicated. (The value of t = 0.015 is the time at which the fluctuation of (∆K ⊥ (Ω; α * 3 , β * 3 , χ * ) (t)) 2 reaches a minimum, as seen on Fig.3.) One observes that initial coherent state is rapidly deformed from its nearly Gaussian shape in S 4 , spreads and leaves the tangent hyperplane. In particular, small negative regions are generated in the vicinity of the main peak.   (18) characterizing |ω , we can use the same observableŝ K ⊥ (Ω; α 3 , β 3 , χ) to detect squeezing. Operationally this means the fluctuations of K ⊥ (Ω; α 3 , β 3 , χ) will now depend on the parameters α 3 , β 3 , χ = 6γ 1 + γ 2 of the transformation T of Eqn. (19) through the combinations of Eqn. (21), in such a way that there may exist "directions" parametrized by α * 3 , β * 3 , χ * in the tangent hyperplane where the fluctuations are smaller than in the coherent state |ω . It remains to select from those directions the one along which the fluctuations are smallest to complete our definition of squeezing.

Conclusion
We have shown that the reduction of fluctuations in the systems with SU (3) symmetry can be achieved in a manner similar to the reduction in spin-like systems: by correlating initially factorized coherent states via an evolution generated by a Hamiltonian non-linear on the generators of the su(3) algebra. We constructed the Hamiltonian in a such way that it does not produce a rigid motion of the initial state, so we can use as observables those having uniform fluctuations in a coherent state as a reference to detect squeezing. Although we have not established a general criteria for SU (3) squeezing, we have shown how quantum correlations (in the sense described above) can lead to a reduction of fluctuations, which is reflected through a specific deformation ("squeezing") of the initial coherent state. It must be emphasized that, in quantum systems with higher symmetries, different types of squeezing can be identified, and these types can be conceptually different from the so-called one and two axis squeezing typically found in spin-like systems. Here we used the Hamiltonian invariant under U (2) transformations and thus producing " true" , (i.e. not reducible to the U (2)-type interactions) SU (3) correlations.
It should be also observed that in, contrast to spin-like systems, the exact quantum mechanical calculations for physical models with SU (3) symmetries can be extremely cumbersome. Thus, application of the phase-space methods are extremely helpful not only for the geometrical interpretation and state visualization, but also for estimating the evolution of systems in the limit of large dimensions through the use of semiclassical calculations. In particular, important physical effects such as squeezing, which originate from non-trivial evolutions of collective qutrit fluctuations, can be described in terms of semiclassical evolution of initial Wigner distribution for suitable initial states. This is ultimately possible because the approximate solutions (35) and (46) describe well the dynamics of initial semiclassical states for times of order t ∼ 1, while the major squeezing effect is achieved for times t ∼ λ −p , where p > 0.
The work of ABK is partially supported by the Grant 106525 of CONACyT (Mexico). The work of HDG is partially supported by NSERC of Canada. HTD would like to acknowledge the financial support from Lakehead University.