Emergent time crystals from phase-space noncommutative quantum mechanics

It has been argued that the existence of time crystals requires a spontaneous breakdown of the continuous time translation symmetry so to account for the unexpected non-stationary behavior of quantum observables in the ground state. Our point is that such effects do emerge from position ($\hat{q}_i$) and/or momentum ($\hat{p}_i$) noncommutativity, i.e., from $[\hat{q}_i,\,\hat{q}_j]\neq 0$ and/or $[\hat{p}_i,\,\hat{p}_j]\neq 0$ (for $i\neq j$). In such a context, a predictive analysis is carried out for the $2$-dim noncommutative quantum harmonic oscillator through a procedure supported by the Weyl-Wigner-Groenewold-Moyal framework. This allows for the understanding of how the phase-space noncommutativity drives the amplitude of periodic oscillations identified as time crystals. A natural extension of our analysis also shows how the spontaneous formation of time quasi-crystals can arise.

Time crystals are time-periodic self-organized structures that are supposed to emerge in the time domain due to the spontaneous breaking of time translation symmetry [1,2]. They are analogous to spatial crystal lattices that are formed when the spontaneous breaking of space translation symmetry takes place [3]. Their features have been quantum mechanically probed in ultra-cold atoms [4,5] and spin-based solid state [6][7][8][9][10][11][12] systems, through which it has been claimed that periodically driven systems exhibit a discrete time symmetry which corresponds to a time translation led by the period of an external driver [13,14]. Such plethora of follow up experiments [4][5][6][7][8][12][13][14] suggests the emergence of novel phases of matter [9][10][11] that exhibit a discrete time translation symmetry, mostly described as arising from the breakdown of the continuous time translation symmetry,T H ≡ e −iĤt .
In fact, if a time-independent system driven by a time-independent Hamiltonian H is prepared in an eigenstate |ψ n , such that H|ψ n = E n |ψ n , for the energy eigenvalue E n , quantum mechanics (QM) implies that the probability density at a fixed position in the configuration space is also time-independent. Nevertheless, according to the above arguments and experiments, the existence of time crystals would admit that [Ĥ, ρ n ] ≡ [T H , ρ n ] = 0 for ρ n = |ψ n ψ n |, which would correspond to a spontaneous breakdown of time translation symmetry followed by a non-stationary behavior of the eigensystem solutions.
In this letter, we argue that such a non-stationary behavior, and its straightforward connection with time crystal properties, both emerge from position and/or momentum noncommutativity in the phase-space.
As its inception, noncommutativity was firstly considered in the space coordinate domain as a way to regularize quantum field theories [15]. Subsequently it appeared in the formulation of string theories [16][17][18][19]. As a natural extension, the phase-space noncommutative (NC) QM considered here [20][21][22][23][24][25] was formulated in terms of the Weyl-Wigner-Groenewold-Moyal (WWGM) framework [26][27][28], which is supported by a 2n-dim phase-space that satisfies a deformed Heisenberg-Weyl algebr, where position and momentum operators,q i andp j , obey the following commutation relations, with i, j = 1, ..., d, and η ij and θ ij identified as entries of invertible antisymmetric real constant (d × d) matrices, Θ and N, such that an equally invertible matrix, Σ, with Σ ij ≡ where A ij , B ij , C ij and D ij are real entries of constant matrices, A, B, C and D. In this case, one recovers the algebra of ordinary QM, from which it is straightforward to obtain the following matrix equation constraints [22], The algebra, Eq. (1), in the context of the WWGM framework [22,23], allows for examining striking features which include putative violations of the Robertson-Schrödinger uncertainty relation [29,30], quantum correlations and information collapse in gaussian quantum systems [31][32][33][34][35], and regularizing features in minisuperspace quantum cosmology models [29,36] and in black-hole physics [37,38]. The generalized WWGM star-product, the extended Moyal bracket and the NC Wigner function framework ensure that observables are independent of any particular choice of the SW map [23].
To make our proposal more concrete, the 2-dim harmonic oscillator in the NC phase-space [39] will be evaluated in order to show that time crystal patterns on quantum observables and quantum states naturally emerge from the NC QM. Hence, let us consider the quantum on the NC "x − y" plane, with position and momentum satisfying the NC algebra, Eq. (1), now with i, j = 1, 2, θ ij = θ ij and η ij = η ij , where ij is the 2-dim Levi-Civita tensor. The map to commutative operators is given bŷ in terms of the SW map, which is invertible for θη = 2 , and the parameters λ and µ satisfying the condition The Hamiltonian in terms of the so-called commutative variables,Q i andΠ i , reads [39] H where , and γ ≡ mω 2 θ/(2 )+ η/(2m ), from which one obtains the following set of coupled equations of motion, with Q i ≡ Q i and Π i ≡ Π i . In this case, Q = (Q 1 , Q 2 ) and Π = (Π 1 , Π 2 ) may be interpreted as the dynamical variables within the WWGM formalism for which the solutions are given by [39] Q 1 (t) = x cos(Ωt) cos(γt) + y cos(Ωt) sin(γt) + β α [π y sin(Ωt) sin(γt) + π x sin(Ωt) cos(γt)] , where x, y, π x , and π y are arbitrary parameters, and with λ and µ being eliminated by the constraint Eq. (7). In particular, by setting θ = η = 0, and therefore γ = 0, one recovers the solutions for the 2-dim harmonic oscillator with uncoupled x − y coordinates and Ω = ω. For θ, η = 0, the above results lead to two The meaning of the modifications introduced by the NC variables can be evinced by setting π x = π y = α /2β, and x = y = β /2α, so that the associated x and y translational energy contributions can be shown to evolve as with i = 1, 2, from which a typical low frequency γ-dependent beating behavior is found [39].
where W (Q, Π) is the eigenstate associated Wigner function, from which one has [22], where L 0 n are the associated Laguerre polynomials, n 1 and n 2 are non-negative integers, and such that the energy spectrum is given by E n1,n2 = [2αβ(n 1 + n 2 + 1) + γ(n 1 − n 2 )].
by Eq. (11), one obtains an unexpected non-stationary behavior for each of the energy contributions, from which the time crystal non-stationary behavior driven by Ω and a beating behavior driven by γ both emerge. As depicted in Fig. 1, if either θ or η vanishes, one has Ω 2 = ω 2 +γ 2 For the arbitrary choice of γ/Ω = 0.002, in the smaller window of Fig implying that at first order in γ. In this case, a time crystal periodic behavior with a measurable energy time derivative oscillation amplitude, γΩ, driven by both the NC parameter, γ, and the external oscillation frequency Ω ∼ ω.
In fact, QM describes quantum superpositions of normalized solutions of uncoupled pairs of two 1-dim harmonic oscillators in terms of decoupled stationary functions, W n 1 (ξ 1 (t)) and W n 2 (ξ 2 (t)). However, under the NC QM perspective, given the time dependent behavior of ξ 1(2) (t), Fig. 3 depicts the analogous non-stationary behavior in terms of, ∂W n 1 (ξ 1 (t))/∂t = 0, for n 1 = 0, 1 and 2, from which the highly oscillatory behavior is also evinced. Of course, for too small values of γ/Ω (say γ/Ω 0.002), time crystal and NC beating patterns would be hard to measure. Plots are for ground state (n 1 = 0, black solid lines), first (n 1 = 1, red dotted lines) and second (n 1 = 2, green dashed lines) excited states. Similar results can be obtained for ∂W n 2 (ξ 2 (t))/∂t.

Interestingly, 2-dim (or 3-dim) Bose-Einstein condensates with time crystal behavior
have, in fact, been detected through a resonance between two (or three) oscillating atom mirrors through a configuration where the localized wave-packets are products of 1-dim wave-packets [40,41]. The results discussed above suggest a natural explanation for such effect, in this case, with quasi-periodic eigenstates driven by ξ 1 , ξ 2 , and γ replacing the In order to exemplify how both time crystal and time quasi-crystal patterns emerges simultaneously arise from NC QM, one should turn back to the choice of x, y, π x , and π y arbitrary parameters and constrain them by π x = π y = k α /2β, and x = y = s β /2α, with k and s corresponding to dimensionless phase-space coordinates; thus one can observe the NC effects on the s − k plane. Fig. 4, depicts the phase-space s − k Wigner function described in terms of the difference between the NC Wigner function, π 2 W n 1 (ξ 1 (t)) W n 2 (ξ 2 (t)), and the standard QM Wigner function, π 2 W n 1 ((s 2 + k 2 )/2) W n 1 ((s 2 + k 2 )/2), at Ωt = π/8 ( th -row), with from 1 to 8. The short time-scale pattern depicts the time crystal periodic behavior identified by an external Ω-frequency driven periodic rotation in the s − k plane. A larger time-scale pattern (last row) shows that the time crystal periodic behavior is turned into a quasi periodic behavior identified by NC γ-dependent corrections, which suppresses the Wigner function amplitude. Hence, at a large time-scale, depending on the magnitude of the NC parameters, amplitude suppression is converted into amplitude revival, due to the NC γ-driven quantum beating.