PHYSICAL REVIEW E 91 , 042901 (2015)
Chaos and band structure in a three-dimensional optical lattice Yingyue Boretz and L. E. Reichl The Center for Complex Quantum Systems, The University o f Texas at Austin, Austin, Texas 78712 (Received 22 December 2014; published 3 April 2015) Classical chaos is known to affect wave propagation because it signifies the presence of broken symmetries. The effect of chaos has been observed experimentally for matter waves, electromagnetic waves, and acoustic waves. When these three types of waves propagate through a spatially periodic medium, the allowed propagation energies form bands. For energies in the band gaps, no wave propagation is possible. We show that optical lattices provide a well-defined system that allows a study of the effect of chaos on band structure. We have determined the band structure of a body-centered-cubic optical lattice for all theoretically possible couplings, and we find that the band structure for those lattices realizable in the laboratory differs significantly from that expected for the bands in an “empty” body-centered-cubic crystal. However, as coupling is increased, the lattice becomes increasingly chaotic and it becomes possible to produce band structure that has behavior qualitatively similar to the “empty" body-centered-cubic band structure, although with fewer degeneracies. DOI: 10.1103/PhysRevE.91.042901
PACS number(s): 05.45.Ac, 42.50.Wk, 05.45.Gg
I. INTRODUCTION The effect o f classical chaos on wave m otion can be seen m ost directly when considering the behavior o f eigenvalues and eigenfunctions o f the Schrodinger equation for a particle in a potential well [1], W hen the particle has two degrees o f freedom , one can directly com pare Poincare surfaces o f section (PSS) o f the classical motion to quantum surfaces ol section of H usimi functions built from eigenstates o f the Schrodinger equation. The effect o f chaos on quantum system s w ith m ore than two degrees o f freedom is more difficult to determ ine in a system atic way because it is not easy to visualize either classical or quantum motion in system s with more than two degrees o f freedom . The effects o f chaos can also be seen in other types o f wave propagation. Electrom agnetic radiation exhibits the m anifestations o f chaos and this has been m easured experim entally in m icrowaves [2,3]. A coustic waves exhibit the effects o f chaos and this has special im portance in the study o f sound wave motion in the ocean [4], W hen waves propagate through a potential energy field that forms a periodic array, the allow ed energies or frequencies form bands o f values for w hich waves are allowed to propagate and gaps betw een the allow ed energy bands for w hich no wave propagation can occur. In crystalline solids, form ed from periodic arrays o f atom s or m olecules, electron m atter waves can propagate and the detailed structure o f electron energy bands and band gaps determ ines w hether the material is a conductor, sem iconductor, or an insulator o f electricity [5], Photonic crystals can be fabricated and consist o f regularly repeating regions o f low and high dielectric constant. Photonic crystals allow the propagation o f electrom agnetic waves whose w avelength is o f the order o f periodicity o f the dielectric lattice. Electrom agnetic waves have a scale invariance so that electrom agnetic waves o f any wavelength can propagate in a photonic crystal, provided the periodicity o f the crystal is the same order as the w avelength o f the radiation. E lectrom agnetic waves in photonic crystals have bands o f allow ed wavelengths that can propagate and band gaps in w hich no radiation can propagate [6,7], Elastic and acoustic waves through m aterials with spatially periodic arrays o f m aterial, with different elastic m oduli, have bands o f allowed wavelengths in w hich the
1539-3755/2015/91 (4)/042901(10)
acoustic waves can propagate and band gaps in w hich waves are not allow ed to propagate [8]. One o f the m ost recent additions to this collection of system s with wave propagation in periodic lattices is the propa gation o f atom ic m atter waves (alkali atom s such as rubidium, cesium , and sodium ) in optical lattices. O ptical lattices are form ed by superposing m ultiple pairs o f counterpropagating laser beam s in different patterns and have been realized in the laboratory by several groups [9-13], Such system s can be used to explore the quantum -classical correspondence o f the atoms interacting with the optical lattice, as was done in Ref. [14] for a 2D optical lattice and, by analogy, some properties o f electrons in crystalline solids [15,16], In subsequent sections, w e use a dynam ical model that, for a particular param eter range, describes the interaction of alkali atom s w ith a three-dim ensional (3D) optical lattice. However, the model more generally allows us to explore how the detailed structure o f the periodic lattice can affect band structure. As was shown in Ref. [14], the coupling between degrees o t freedom o f the lattice can be varied by varying the relative orientation o f the polarization o f the lasers. As a consequence, the nature o f the particle dynam ics can be varied from integrable behavior to alm ost fully chaotic behavior for a range ot energies. This transition from integrable to chaotic behavior occurs because resonances betw een the degrees o f freedom in the coupled system destroy local constants o f the m otion [ 1], thus allow ing the particles to access larger regions o f the lattice. In bounded system s, this transition is known to have a profound effect o f the dynam ics o f quantized system. Below we show how the transition to chaos can affect the energy band structure o f particles confined to the 3D BCC optical lattice. We begin in Sec. II with a description o f the H am iltonian and dynam ics o f atom s confined to a 3D optical lattice. In Sec. Ill, we describe the classical dynam ics o f noninteracting particles on the lattice and discuss properties o f invariant m anifolds em bedded in the 3D lattice. In Sec. IV. we describe the quantum dynam ics o f particles on the lattice and show the effect o f chaos on the energy band structure. Finally, in Sec. V, we make some concluding remarks.
042901-1
©2015 American Physical Society
PHYSICAL REVIEW E 9 1 , 042901 (2015)
YINGYUE BORETZ AND L. E. REICHL II. THREE-DIMENSIONAL OPTICAL LATTICE
We consider a gas of noninteracting particles of mass m interacting with a body-centered-cubic (BCC) lattice formed by the superposition of three pairs of mutually perpendicular laser beams (see Appendix). The laser beams are detuned from resonance with the atoms, so the interaction dynamics between the atoms and the optical lattice can be described by the Hamiltonian [17] d2£
dl
2m
2m
2m
between the laser beams occurs for a = b = c = \ with angles 9X — 9y = .£(:); an(j c = £(y).ito\ where e(x\ e(y\ and i
2 6°
«
ffiM = P x + pz + P im( v, z) = £) m .
(6)
where
2
^
Vim(>’. z) = u +
f
m[cos2(_v) +
cos2(z)]
+ 2an[cos( v) + cos(z) + cos(v)cos(z)]. (7)
11 ■)
2
4
4
4T 6
6
FIG. 2. The 3D potential energy surface for one unit cell of the 3D lattice for different energy intervals: (a) a = 0.1 and V(x,y,z) < 21.64d.u.; (b) a = 0.1 and 48 < V(x,y,z) < 72 d.u.; (c) a = 0.5 and V(x,y,z) < 15.33 d.u.; (d) a = 0.5 and 80 < V(x.y.z) < 120d.u.; (e) a = 1.0 and V(x,y,z) < 3.0d.u.; (f) a = 1.0 and 120 < V(x,y.z) < 180 d.u.. In subsequent sections, we consider the classical and quantum dynamics of noninteracting particles confined to this 3D optical lattice. III. CLASSICAL DYNAMICS ON THE THREE-DIMENSIONAL LATTICE
The first step in describing the classical dynamics of particles on this optical lattice is to write Hamilton’s equations of motion for the particles. They take the form Px = [2«cos(x) + 2aucos(v) + 2a«cos(z)]sin(x), Py — [2r, and evolve this new set of points for one time step r. This process is repeated N times, generating a sequence of distances dj — |d,-|, for j = The Lyapunov exponent is then defined as 1 ^
kN = — V In Nr ^
d2^ ( x , y , z ) + d2Vr(x„y^ ) + 8V (A ,y,z) + dy2 3z 2 3x2 2\z)
/= !
— 8aqcos(y)cos(z)is(x,y,z) = 0,
If r is chosen small enough, then the Lyapunov exponent kff will be independent of r and do. If X0.o is in a regular region of the phase space, then lim^^oc^v = 0. If X0.o is in a chaotic region of the phase space, then lim/v^oo^jv approaches a positive value [19,21]. In Fig. 5, we show the values of the dominant Lyapunov exponent for 1 ^ A ^ 1 0 5, t = 2d.u.,and Jo = 5 x l 0 -3 d.u., for initial conditions taken both on the InM and off the InM. The energies used to compute the Lyapunov exponents are the same as those in Figs. 4(c), 4(f), and 4(i). The initial conditions, for trajectories on and off the InM, are the same except
with e = E — y and q = to 32 t/f(x,y,z) 3x2
(12)
For the case a = 0 this reduces
32 tA(x,y,z) , d2f ( x , y , z ) H------- — -----+ ------ — 5----- + eiA(x,y,z) ' dy2 3z2 - 2Lt + [A(t)( * T ) + A (v)(y ,t) + A ‘zXzJ)] *ei(OL'. (A l)
102 N
10'
"TO4
FIG. 9. The Lyapunov exponents for trajectories in the 3D unit cell (diamonds) for coupling constants and energies: (a) a = 0.1 and £ = I3d.u.;(b)« = 0.5 and £ = 12d.u.;(c)a = 1.0 and £ = 5 d.u. All energies are above the high saddle so trajectories are free to move throughout the optical lattice, regardless of whether they are on the InM or off of it.
The particular form o f A (r ,t) will depend on the lasers. Let us assum e that the total electric field am plitude E(x,_y,r) is superposition o f the electric fields due to two counterpropagating lasers along the x direction, two counterpropagating lasers along the y direction, and two counterpropagating lasers along the z direction. The lasers propagating along the x, y, and z directions have polarizations e (x> = cos(0x )z + sin(0r )j>, i ,y> = cos(0v)z + sin(0v)x, a n d e (:) = cos(0-)x + sin(0-)y, re spectively. A ssum e that the lasers form standing waves along their respective directions. The standing wave along the x direction is Au )( x ,0 = — i (x)exp[+i(kLx +
Chaos and band structure in a three-dimensional optical lattice Yingyue Boretz and L. E. Reichl The Center for Complex Quantum Systems, The University o f Texas at Austin, Austin, Texas 78712 (Received 22 December 2014; published 3 April 2015) Classical chaos is known to affect wave propagation because it signifies the presence of broken symmetries. The effect of chaos has been observed experimentally for matter waves, electromagnetic waves, and acoustic waves. When these three types of waves propagate through a spatially periodic medium, the allowed propagation energies form bands. For energies in the band gaps, no wave propagation is possible. We show that optical lattices provide a well-defined system that allows a study of the effect of chaos on band structure. We have determined the band structure of a body-centered-cubic optical lattice for all theoretically possible couplings, and we find that the band structure for those lattices realizable in the laboratory differs significantly from that expected for the bands in an “empty” body-centered-cubic crystal. However, as coupling is increased, the lattice becomes increasingly chaotic and it becomes possible to produce band structure that has behavior qualitatively similar to the “empty" body-centered-cubic band structure, although with fewer degeneracies. DOI: 10.1103/PhysRevE.91.042901
PACS number(s): 05.45.Ac, 42.50.Wk, 05.45.Gg
I. INTRODUCTION The effect o f classical chaos on wave m otion can be seen m ost directly when considering the behavior o f eigenvalues and eigenfunctions o f the Schrodinger equation for a particle in a potential well [1], W hen the particle has two degrees o f freedom , one can directly com pare Poincare surfaces o f section (PSS) o f the classical motion to quantum surfaces ol section of H usimi functions built from eigenstates o f the Schrodinger equation. The effect o f chaos on quantum system s w ith m ore than two degrees o f freedom is more difficult to determ ine in a system atic way because it is not easy to visualize either classical or quantum motion in system s with more than two degrees o f freedom . The effects o f chaos can also be seen in other types o f wave propagation. Electrom agnetic radiation exhibits the m anifestations o f chaos and this has been m easured experim entally in m icrowaves [2,3]. A coustic waves exhibit the effects o f chaos and this has special im portance in the study o f sound wave motion in the ocean [4], W hen waves propagate through a potential energy field that forms a periodic array, the allow ed energies or frequencies form bands o f values for w hich waves are allowed to propagate and gaps betw een the allow ed energy bands for w hich no wave propagation can occur. In crystalline solids, form ed from periodic arrays o f atom s or m olecules, electron m atter waves can propagate and the detailed structure o f electron energy bands and band gaps determ ines w hether the material is a conductor, sem iconductor, or an insulator o f electricity [5], Photonic crystals can be fabricated and consist o f regularly repeating regions o f low and high dielectric constant. Photonic crystals allow the propagation o f electrom agnetic waves whose w avelength is o f the order o f periodicity o f the dielectric lattice. Electrom agnetic waves have a scale invariance so that electrom agnetic waves o f any wavelength can propagate in a photonic crystal, provided the periodicity o f the crystal is the same order as the w avelength o f the radiation. E lectrom agnetic waves in photonic crystals have bands o f allow ed wavelengths that can propagate and band gaps in w hich no radiation can propagate [6,7], Elastic and acoustic waves through m aterials with spatially periodic arrays o f m aterial, with different elastic m oduli, have bands o f allowed wavelengths in w hich the
1539-3755/2015/91 (4)/042901(10)
acoustic waves can propagate and band gaps in w hich waves are not allow ed to propagate [8]. One o f the m ost recent additions to this collection of system s with wave propagation in periodic lattices is the propa gation o f atom ic m atter waves (alkali atom s such as rubidium, cesium , and sodium ) in optical lattices. O ptical lattices are form ed by superposing m ultiple pairs o f counterpropagating laser beam s in different patterns and have been realized in the laboratory by several groups [9-13], Such system s can be used to explore the quantum -classical correspondence o f the atoms interacting with the optical lattice, as was done in Ref. [14] for a 2D optical lattice and, by analogy, some properties o f electrons in crystalline solids [15,16], In subsequent sections, w e use a dynam ical model that, for a particular param eter range, describes the interaction of alkali atom s w ith a three-dim ensional (3D) optical lattice. However, the model more generally allows us to explore how the detailed structure o f the periodic lattice can affect band structure. As was shown in Ref. [14], the coupling between degrees o t freedom o f the lattice can be varied by varying the relative orientation o f the polarization o f the lasers. As a consequence, the nature o f the particle dynam ics can be varied from integrable behavior to alm ost fully chaotic behavior for a range ot energies. This transition from integrable to chaotic behavior occurs because resonances betw een the degrees o f freedom in the coupled system destroy local constants o f the m otion [ 1], thus allow ing the particles to access larger regions o f the lattice. In bounded system s, this transition is known to have a profound effect o f the dynam ics o f quantized system. Below we show how the transition to chaos can affect the energy band structure o f particles confined to the 3D BCC optical lattice. We begin in Sec. II with a description o f the H am iltonian and dynam ics o f atom s confined to a 3D optical lattice. In Sec. Ill, we describe the classical dynam ics o f noninteracting particles on the lattice and discuss properties o f invariant m anifolds em bedded in the 3D lattice. In Sec. IV. we describe the quantum dynam ics o f particles on the lattice and show the effect o f chaos on the energy band structure. Finally, in Sec. V, we make some concluding remarks.
042901-1
©2015 American Physical Society
PHYSICAL REVIEW E 9 1 , 042901 (2015)
YINGYUE BORETZ AND L. E. REICHL II. THREE-DIMENSIONAL OPTICAL LATTICE
We consider a gas of noninteracting particles of mass m interacting with a body-centered-cubic (BCC) lattice formed by the superposition of three pairs of mutually perpendicular laser beams (see Appendix). The laser beams are detuned from resonance with the atoms, so the interaction dynamics between the atoms and the optical lattice can be described by the Hamiltonian [17] d2£
dl
2m
2m
2m
between the laser beams occurs for a = b = c = \ with angles 9X — 9y = .£(:); an(j c = £(y).ito\ where e(x\ e(y\ and i
2 6°
«
ffiM = P x + pz + P im( v, z) = £) m .
(6)
where
2
^
Vim(>’. z) = u +
f
m[cos2(_v) +
cos2(z)]
+ 2an[cos( v) + cos(z) + cos(v)cos(z)]. (7)
11 ■)
2
4
4
4T 6
6
FIG. 2. The 3D potential energy surface for one unit cell of the 3D lattice for different energy intervals: (a) a = 0.1 and V(x,y,z) < 21.64d.u.; (b) a = 0.1 and 48 < V(x,y,z) < 72 d.u.; (c) a = 0.5 and V(x,y,z) < 15.33 d.u.; (d) a = 0.5 and 80 < V(x.y.z) < 120d.u.; (e) a = 1.0 and V(x,y,z) < 3.0d.u.; (f) a = 1.0 and 120 < V(x,y.z) < 180 d.u.. In subsequent sections, we consider the classical and quantum dynamics of noninteracting particles confined to this 3D optical lattice. III. CLASSICAL DYNAMICS ON THE THREE-DIMENSIONAL LATTICE
The first step in describing the classical dynamics of particles on this optical lattice is to write Hamilton’s equations of motion for the particles. They take the form Px = [2«cos(x) + 2aucos(v) + 2a«cos(z)]sin(x), Py — [2r, and evolve this new set of points for one time step r. This process is repeated N times, generating a sequence of distances dj — |d,-|, for j = The Lyapunov exponent is then defined as 1 ^
kN = — V In Nr ^
d2^ ( x , y , z ) + d2Vr(x„y^ ) + 8V (A ,y,z) + dy2 3z 2 3x2 2\z)
/= !
— 8aqcos(y)cos(z)is(x,y,z) = 0,
If r is chosen small enough, then the Lyapunov exponent kff will be independent of r and do. If X0.o is in a regular region of the phase space, then lim^^oc^v = 0. If X0.o is in a chaotic region of the phase space, then lim/v^oo^jv approaches a positive value [19,21]. In Fig. 5, we show the values of the dominant Lyapunov exponent for 1 ^ A ^ 1 0 5, t = 2d.u.,and Jo = 5 x l 0 -3 d.u., for initial conditions taken both on the InM and off the InM. The energies used to compute the Lyapunov exponents are the same as those in Figs. 4(c), 4(f), and 4(i). The initial conditions, for trajectories on and off the InM, are the same except
with e = E — y and q = to 32 t/f(x,y,z) 3x2
(12)
For the case a = 0 this reduces
32 tA(x,y,z) , d2f ( x , y , z ) H------- — -----+ ------ — 5----- + eiA(x,y,z) ' dy2 3z2 - 2Lt + [A(t)( * T ) + A (v)(y ,t) + A ‘zXzJ)] *ei(OL'. (A l)
102 N
10'
"TO4
FIG. 9. The Lyapunov exponents for trajectories in the 3D unit cell (diamonds) for coupling constants and energies: (a) a = 0.1 and £ = I3d.u.;(b)« = 0.5 and £ = 12d.u.;(c)a = 1.0 and £ = 5 d.u. All energies are above the high saddle so trajectories are free to move throughout the optical lattice, regardless of whether they are on the InM or off of it.
The particular form o f A (r ,t) will depend on the lasers. Let us assum e that the total electric field am plitude E(x,_y,r) is superposition o f the electric fields due to two counterpropagating lasers along the x direction, two counterpropagating lasers along the y direction, and two counterpropagating lasers along the z direction. The lasers propagating along the x, y, and z directions have polarizations e (x> = cos(0x )z + sin(0r )j>, i ,y> = cos(0v)z + sin(0v)x, a n d e (:) = cos(0-)x + sin(0-)y, re spectively. A ssum e that the lasers form standing waves along their respective directions. The standing wave along the x direction is Au )( x ,0 = — i (x)exp[+i(kLx +
