Multiple self-organized phases and spatial solitons in cold atoms mediated by optical feedback
Giuseppe Baio, Gordon R. M. Robb, Alison M. Yao, Gian-Luca Oppo, Thorsten Ackemann
MMultiple self-organized phases and spatial solitons in cold atoms mediated by opticalfeedback
Giuseppe Baio, ∗ Gordon R. M. Robb, Alison M. Yao, Gian-Luca Oppo, and Thorsten Ackemann
SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom (Dated: February 4, 2021)We study the transverse self-structuring of a cloud of cold atoms with effective atomic inter-actions mediated by a coherent driving beam retro-reflected by means of a single mirror. Theresulting self-structuring due to optomechanical forces is much richer than that of an effective-Kerrmedium, displaying hexagonal, stripe and honeycomb phases depending on the interaction strengthparametrized by the linear susceptibility. Phase domains are described by real Ginzburg-Landauamplitude equations. In the stripe phase the system recovers inversion symmetry. Moreover, thesubcritical character of the honeycomb phase allows for light-density feedback solitons functioningas self-sustained dark atomic traps with motion controlled by phase gradients in the driving beam.
Spontaneous self-organization phenomena are ubiqui-tous in out-of-equilibrium classical and quantum dynam-ics [1, 2]. In recent years, cold and ultracold gaseshave provided useful platforms for probing light-atomself-structuring by means of density modes or internalstates, resulting in crystalline (density) or magnetic or-der respectively [3–7]. In the first case, the emerging dy-namical potential for the atoms induces a density grat-ing which, in turn, scatters photons into the side-bandmodes creating the potential and leading to optome-chanical self-structuring [8]. Several experimental real-izations of such phenomena in cold atoms setups haveoffered groundbreaking insight into different quantummany-body physics aspects such as quantum phase tran-sitions [9], supersolidity [10–13], topological defects [14],and structural phase transitions [15].A key aspect, analogous to soft-matter realizations[16], is that the collective bunching of the scatterers givesrise to a self-focusing Kerr-like optomechanical nonlinear-ity [17]. Hence, Ashkin et al. coined the term ‘artificialKerr medium’ [18]. Transverse optical pattern formationin effective-Kerr media (and beyond) has been the sub-ject of wide theoretical and experimental efforts since the1990’s, in both cavity and single-feedback-mirror (SFM)configurations [19–22]. Among the major advantagesof cold atoms is the possibility to significantly reducethreshold intensities when the atoms are laser cooled tohundreds of µ K [3, 23, 24].In this Letter, we show that, despite some similari-ties between Kerr media and mobile dielectric scatterers,the latter is source of a much richer structural transi-tion behaviour characterized by three light-atom crys-talline phases, i.e., hexagonal, stripe and honeycomb.We explore phase stability for an SFM setup in termsof a weakly nonlinear analysis, leading to the amplitudeequations (AEs) and relative free energy functional inthe universal Ginzburg-Landau form [25]. This providesan accurate description of the selection mechanism andspatial soliton formation in a cloud of atoms undergoingoptomechanical self-structuring. Our results can be ap- E +,0 ( r ) Λ L , b , T E + E − Mirror
Atoms ( Rb ) FIG. 1.
Optomechanical SFM scheme. A far detuned input beamof amplitude E + , and wavenumber k illuminates a cloud of Ru-bidium atoms of thickness L , optical density b and temperature T . The reflected E − provides feedback by means of the dipole po-tential, leading to self-structuring with critical wavelength Λ [3]. plied to other configurations of interest, e.g, in free-spaceor longitudinally pumped cavities [26, 27], and can shednew light on the ongoing discussion of potential phasesin the rapidly developing field of dipolar supersolids [28–30]. Indeed, although current experimental realizationsare limited to quasi-1D cases, 2D structural transitionsare predicted to occur in dipolar condensates [31]. Basedon a close correspondence between the condensate en-ergy functional and the Lyapunov functional discussedbelow, we conjecture that our analysis will motivate fur-ther studies in the potential of a new stripe supersolidphase in between the hexagonal and honeycomb phasesalready predicted in [31].We consider a thermal cloud of two-level atoms attemperature T , where atomic motion is overdamped bymeans of optical molasses [23]. In this regime, the trans-verse dynamics is described by density modulations only,i.e., n ( r , t ) = 1 + δn ( r , t ), where the atom density n ( r , t )obeys a Smoluchowski drift-diffusion equation [8, 27]: ∂ t n ( r , t ) = − βD ∇ ⊥ · [ n ( r , t ) f dip ( r , t )]+ D ∇ ⊥ n ( r , t ) , (1)where D is the cloud diffusivity, and β = 1 /k B T , with k B being the Boltzmann constant. The dipole force reads: f dip ( r , t ) = − (cid:126) Γ∆4 ∇ ⊥ s ( r , t ) , (2) a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b where s ( r , t ) is the total light intensity (saturation pa-rameter), and ∆ corresponds to the light-atom detuningin units of half the linewidth Γ. The SFM setup, rep-resented in Fig. 1, is a paradigmatic scheme for Talbot-based optical pattern formation [19, 21, 32]. For a diffrac-tively thin cloud, the field equations are: ∂ z E ± ( r , t ) = ± i χL n ( r , t ) E ± ( r , t ) , (3)where the + sign relates to E + and vice versa. As typ-ical for the dispersive regime, we assume large detun-ing and low saturation, so that scattering forces are ne-glected and the susceptibility of the cloud is real andreads χ = b ∆ / ) (See Fig. 1) [3]. The feedbackloop is closed by considering propagation to the mirror(at distance d from the cloud) and back [33]. Let us in-troduce a constant σ = (cid:126) Γ∆ / k B T , representing compe-tition between the dipole potential and spatial diffusion.We first study the linear stability of spatial modulations[34]. By parametrizing δn ( q , t ) = a exp( i q · r + νt ) + c.c.,one obtains the following growth rate: ν ( q ) = − D | q | (cid:20) − σR |E + , | b ∆ sin Θ(1 + ∆ ) (cid:21) , (4)where Θ = d | q | /k is the total diffractive phase slip-page, R is the mirror reflectivity and q is the transversewavevector. Imposing ν ( q ) = 0, we arrive at the thresh-old condition: I = |E + , | = 1 + ∆ σR b ∆ sin Θ ≥ σR b ∆ = I , (5)where I represents the minimum threshold, i.e., atthe critical wavenumber q c = k π/ d (purely disper-sive case). We explore the coexistence of self-structuredphases by means of numerical and analytical observa-tions. Unlike the molasses-free case considered in [35],the dissipative dynamics of Eq. (1) admits a (quasi) sta-tionary state given by the Gibbs distribution [27]: n eq ( r , t ) = exp[ − σs ( r , t )] (cid:82) Ω d r exp[ − σs ( r , t )] , (6)where Ω is the integration domain and s ( r , t ) = |E + ( r , t ) | + |E − ( r , t ) | . The feedback loop is integratedaccording to the following scheme: first, we propagate theincident field through the cloud, i.e., E + ( z = L, r , t ) = E + , exp { iχn ( r , t ) } . We then propagate in free-spaceover 2 d to determine E − ( z = L, r , t ), and update theatom density according to n eq ( r , t ) in Eq. (6). By ex-panding Eq. (6) to first order, one shows that, regardlessof the sign of χ , the total refractive index of the cloud isof self-focusing Kerr type and, thus, the atom density isexpected to choose an hexagonal (honeycomb) geometryabove threshold for ∆ < >
0) [3, 21]. However, forthe optomechanical interaction we numerically observethe formation of three self-structured phases shown in
FIG. 2. Optomechanical self-structured phases obtained atfixed b = 110 and T = 300 µ K. (a),(d) H − phase at ∆ = 25.(b),(e) S phase at ∆ = 55. (b),(e) H + phase at ∆ = 90. Fig. 2 for different values of ∆ at fixed b , i.e., hexagons( H + ), stripes ( S ) and honeycombs ( H − ), where the la-bels identify the atom-density states. To characterizetransitions between such phases we span the two dimen-sional space (∆ , b ) within the experimentally achievableranges of ∆ = [10 , b = [50 , S state and iterating theloop long enough to let the stucture stabilize. A simplediscriminant between phases is the number of peaks inthe resonant circle of the far field. In Fig. 3, we report astability domain of S states (in grey) for I/I = 1 .
2, sand-wiched between two disjoint H ± regions (in yellow/cyan)and separated by lines of constant χ .A weakly nonlinear analysis based on the AEs repre-sents the canonical approach to describe pattern selectionprocesses above threshold [36]. The first step is to for-mally integrate Eq. (3) for the backwards field with ahomogenous pump, namely: E − ( r , t ) = √ RI ˆ L e iχn ( r ,t ) , (7)where we defined the the differential operator ˆ L = e − id ∇ ⊥ /k . Thus, we are left with one equation for thedensity perturbation δn ( r , t ) only:( −∇ ⊥ + ∂ t ) χδn ( r , t ) = RσIχ ∇ ⊥ · (cid:104) (1 + δn ( r , t )) ∇ ⊥ | ˆ L e iχδn ( r ,t ) | (cid:105) . (8)A similar approach was used to derive a closed equa-tion capturing the features of the long-range interactionsmediated by feedback in a SFM scheme [37]. We now ex-pand | ˆ L e iχδn ( r ,t ) | up to O (cid:2) ( χδn ) (cid:3) and introduce slowspatial scales up to third order [38]. Furthermore, wederive the solvability conditions for our model and sub- FIG. 3. Numerically observed stability domains of the S , H ± phases at fixed I/I . The observed boundaries match the val-ues of the susceptibility χ from Eqs. (15)-(16). The S phase(grey) is absolutely stable on a domain sandwiched betweenthe lines corresponding to the χ S , points (red), around χ = 1(black). Moreover, the S phase coexists with H ± and mini-mizes the free energy in the region between the χ H , (dashed-black) and χ ∗ , points (dashed-green). H ± phases are stablewithin the yellow and cyan domains and absolutely stableoutside χ H , . stitute a hexagonal ansatz for the resonant terms n : n = 12 (cid:34) (cid:88) i =1 A i exp ( i q i · r ) + c.c. (cid:35) , (9)where q + q + q = 0 and | q i | = q c . After lengthyalgebra [34], we obtain the AEs in the real Ginzburg-Landau form, namely: ∂ t A i = µA i + λA ∗ j A ∗ k − γ (cid:88) j (cid:54) = i | A j | A i − γ | A i | A i . (10)where i, j, k = 1 , , i (cid:54) = j (cid:54) = k . In many circum-stances, pattern stability close to threshold is universallydescribed in terms of the AEs critical points, dependingon the coefficients in Eq. (10) [21, 32]. Defining p = I/I ,and for a generic critical shift Θ c , the linear growth andthree-mode mixing coefficients read: µ ( p ) = 2 RI σ ( p − χ sin Θ c , (11) λ ( p, χ ) = RI σpχ c + χ (cos Θ c − . (12)Already at this level, a number of interesting consider-ations arise. Indeed, in sharp contrast with the Kerrmodel, the coefficient λ changes sign around the point FIG. 4. (a) Lyapunov functionals for the three phases at p =1 . F H + ( χ ) (blue), F H − ( χ ) (orange), F S ( χ ) (green). Theresulting minimum determines the observed self-structuredphase while χ ∗ and χ ∗ identify the boundaries in Fig. 3. Notethat F H + = F H − for χ = 1 (dotted line). (b) Critical µ > S and µ > H ± (dashed-black/red lines) and phase boundaries (dashed-green) as functions of χ . Intersections with µ (blue) determinethe size of the S / H competition region. χ = cot(Θ c /
2) ( χ = 1 with Θ c = π/ H + for λ > χ >
0, i.e., for blue-detuning (∆ >
0) while, instead,no phase other than H − is expected at threshold for red-detuning. As in the Hamiltonian case, phase selectionprocesses, such as the one in Fig. 3, are described interms of Lyapunov or free energy functionals associatedwith the AEs in Eq. (10) [36]. To this aim, we computethe self and cross-cubic coefficients as follows: γ ( p, χ ) = RI σpχ (cid:20) χ sin Θ c + 2 −
12 (cos 3Θ c + cos Θ c ) (cid:21) (13) γ ( p, χ ) = RI σpχ (cid:20) χ (cid:18) sin Θ c − sin 3Θ c + 23 (cid:19) +2(1 − cos 4Θ c )] , (14)The Lyapunov functional assumes the following quarticform, as in the weak crystallization scenario [39, 40]: F [ { A i } ] = − µ (cid:88) i =1 | A i | − λ ( A ∗ A ∗ A ∗ + c.c.) + γ (cid:88) i,j =1 | A i | | A j | + γ (cid:88) i =1 | A i | , (15)where i = 1 , , i (cid:54) = j . Non-zero λ implies that self-structuring is a first-order phase transition. We obtainthe Lyapunov functional for the three phases F H ± and F S , and compute the corresponding minimum as a func-tion of χ , shown in Fig. 4(a) [34]. In addition we havethe critical points [41]: µ > S = λ γ ( γ − γ ) , µ < H ± = λ (2 γ + γ )( γ − γ ) , (16)representing the lower S and the higher H ± stabilitylimits respectively. We overall single out six values of χ , as shown in Fig. 3. A first couple χ ∗ , arises fromthe intersections F H + ( χ ∗ ) = F S ( χ ∗ ) and F H − ( χ ∗ ) = F S ( χ ∗ ) (Fig. 4(a)) and provide phase boundaries in goodagreement with the observed ones in Fig. 3 (dashed-green lines). The extremal points in Eq. (16), shownin Fig. 4(b), yield two other pairs of intersections χ S , and χ H , , delimiting the S / H competition regions around χ = 1 (dashed-black/solid-red lines). Furthermore, atthe same point, the system (displaying S states) recovers inversion symmetry (IS) whereas the H + and H − statesbreak IS (but are inversion-symmetric to each other).This phenomenon is known for dissipative pattern for-mation [41–43]. The highly interesting feature here isthat such a recovery results from a self-tuning depend-ing on the interaction strength χ , while, otherwise, ittypically results from different boundary conditions (e.g.in Maragoni compared to Rayleigh-B´enard convection[42]), symmetry-breaking external fields [44] or polar-ization imbalances [32, 45], and strong changes in thehomogeneous solution [46–49].A second intriguing consequence of the optomechanicalnonlinearity, elucidated by the AEs, is the possibility ofexciting light-density spatial solitons when λ (cid:54) = 0 [50, 51].Indeed, as a universal feature of the AEs (10), the H ± branches display subcriticality, i.e. they are stable in anegative range µ SN < µ <
0, originating in a saddle-nodebifurcation at: µ SN = − λ γ + 2 γ ) . (17)This is shown for ∆ > H + branch A = A = A = A (blue line), computedanalytically from the AEs coefficients above, is in goodagreement with the numerical amplitude (max( n eq ) − min( n eq )) / p ∈ [0 . , .
2] [34]. The stability of H + for χ < FIG. 5. Optomechanical solitons for blue detuning. (a) Am-plitude of the H ± , S branches as functions of p for χ ≈ . b = 50 , ∆ = 80 , σ ≈ . p = 0 . × atoms witha driving beam possessing OAM (index l = 1) [34]. (a) Den-sity evolution n (¯ x, ¯ t ), numerically reconstructed from particletrajectories. (b) Phase space distribution at ¯ t = 120. characterized by a dark intensity profile |E − ( r ) | , whichserves as a self-sustained trap for a bright density peak, asdisplayed in Fig. 5(b)-(c). Controlling soliton motion viaexternal phase gradients enables atomic transport appli-cations [52, 53]. We address that by means of 1D particledynamics simulations, where parameters are tuned in or-der to match those in Fig. 5 in the thermodynamic limit[34]. Assuming periodic boundary conditions, the atomsare effectively confined in an annular trap and, thus, alinear phase on the input field corresponds to the 1Dangular equivalent of orbital angular momentum (OAM)[54]. The density profile is shown in Fig. 6(a) where, af-ter a transient behaviour, the atoms initially prepared ina density peak reach steady state angular drift, inducedby OAM. This is illustrated by the phase space distribu-tion in Fig. 6(b), where the non-zero momentum of thetrapped region is visible.In summary, we have demonstrated transverse optome-chanical self-structuring to hexagonal, stripe and honey-comb phases in cold atomic clouds subject to optical feed-back. Focusing on a simple model of overdamped motion,we pointed out that the Kerr picture of the optomechan-ical nonlinearity fails to capture structural transitionsamong hexagons, stripes, and honeycombs, depending onthe coupling strength. Indeed, in that case, only the sec-ond addend in Eq. (12) arises as the resulting nonlinearityinvolves the intensity alone, resulting in a pure quadraticdependence on the susceptibility [21]. By contrast, theoptomechanical nonlinearity involves the transport gen-erating product n ( r , t ) ∇ ⊥ s ( r , t ) [27], so that the mixingof linear terms from both factors gives rise to a shiftedquadratic dependence on χ , becoming effective-Kerr for χ > χ = 1, the system is inversion symmetric,undergoing a structural transition to a stripe state.Structural phase transitions received recent attentionin the context of dipolar supersolids [31], and drivenBose-Einstein condensates [55]. To our knowledge, thereis no experimental confirmation of a stripe-like inver-sion symmetric supersolid phase, as current experimentsaddress only quasi-1D configurations [56, 57]. In lightof the similarity between the free energy discussed hereand the energy functional for a dipolar condensate, weconjecture the existence of an intermediate, inversion-symmetric, supersolid phase (stripe or square-like). Theproposed scheme provides relative ease of experimentalimplementation of 2D symmetry-breaking phenomena incold atoms [3], and quantum degenerate gases [31, 58].The overdamped limit under scrutiny here simplifies theanalytical treatment but the present phase selection oc-curs in the Hamiltonian case, i.e., without optical mo-lasses [34]. Finally, the existence of optomechanical feed-back solitons motivates us to explore analogues in quan-tum degenerate gases [58], in connection with the conceptof quantum droplets [59, 60].All authors acknowledge financial support from the Eu-ropean Training Network ColOpt, which is funded by theEuropean Union (EU) Horizon 2020 program under theMarie Sk(cid:32)lodowska-Curie Action, Grant Agreement No.721465. 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