Quantum chaos of a mixed, open system of kicked cold atoms
Yevgeny Krivolapov, Shmuel Fishman, Edward Ott, Thomas M. Antonsen
QQuantum chaos of a mixed, open system of kicked cold atoms
Yevgeny Krivolapov and Shmuel Fishman
Physics Department, Technion - Israel Institute of Technology, Haifa 32000, Israel. ∗ Edward Ott and Thomas M. Antonsen
University of Maryland, College Park, Maryland 20742, USA
The quantum and classical dynamics of particles kicked by a gaussian attractive potential arestudied. Classically, it is an open mixed system (the motion in some parts of the phase space ischaotic, and in some parts it is regular). The fidelity (Loschmidt echo) is found to exhibit oscillationsthat can be determined from classical considerations but are sensitive to phase space structures thatare smaller than Planck’s constant. Families of quasi-energies are determined from classical phasespace structures. Substantial differences between the classical and quantum dynamics are foundfor time dependent scattering. It is argued that the system can be experimentally realized by coldatoms kicked by a gaussian light beam.
PACS numbers: 67.85.-d, 03.75.Lm, 03.75.Kk, 05.45.Pq, 05.45.Mt
I. INTRODUCTION
The quantum behavior of classically chaotic systems has been extensively studied both with time dependent and timeindependent Hamiltonians [1–7]. The main issue is that of determining fingerprints of classical chaos in the quantummechanical behavior. For example, the spectral statistics of closed classically integrable [8–10] and classically chaotic[11–13] quantum systems have been predicted to have clearly distinct properties. Many of the systems that are ofphysical interest are mixed, where some parts of the phase space are chaotic and some parts are regular. Spectralproperties of mixed systems with time independent Hamiltonians were studied by Berry and Robnik [14]. In thepresent paper we study the classical/quantum correspondence properties of a mixed, open, time dependent system.(Here by “open” we mean that both position and momentum are unbounded.)The system we study consists of a particle kicked by a Gaussian potential defined by the Hamiltonian, H = p m − K (cid:48) T e − x ∞ (cid:88) n = −∞ δ ( t − T n ) . (1)Models of this form were studied by Jensen who used it to investigate quantum effects on scattering in classicallychaotic [15] and mixed [16] systems. This system can be experimentally approximated by a Gaussian laser beamacting on a cloud of cold atoms, somewhat similar to the realization of the kicked rotor by Raizen and coworkers [17].As we will show, the study of Hamiltonian (1) is particularly suited to the investigation of generic behavior of kicked,open, mixed-phase-space systems. In particular, we will focus on issues of fidelity [18–21], decoherence [22, 23] andscattering [24–27]. Our main motivation in studying Hamiltonian (1) is that, with likely future technological advances(see Sec. VII for discussion), the phenomena we consider may soon become accessible to experimental investigation.Quantum mechanically, it is expected that classical phase space details on the scale of Planck’s constant are washedout [28, 29]. In contrast, one of our results will be that quantum dynamics can be sensitive to extremely fine structuresin phase space, and this sensitivity is stable in the presence of noise [22, 23]. Phase space tunneling has been studiedextensively [30–33]. For systems with many phase space structures complications arise due to transport between thesestructures. For our Hamiltonian (1) the motion is unbounded (i.e., the system is “open”), and therefore this system isideal for the exploration of tunneling out of phase space structures and, in particular, for study of resonance assistedtunneling, a current active field of research [31–33].The outline of our paper is as follows. Section II presents and discusses our model system. Section III considers thequasi-energies of quantum states localized to island chains. Section IV introduces the fidelity concept and applies itto study different regions of the phase space including the main, central KAM island (Sec. IV A), island chains (Sec.IV B), and chaotic regions (Sec. IV C). Experimentally there is always some noise present in such systems. Also, ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] O c t noise can be intentionally introduced. Section V considers this issue. Section VI presents a study of the scatteringproperties of the system. Conclusions and further discussion are given in Sec. VII. II. THE MODELA. Formulation
A particle kicked by a Gaussian beam is modeled by the Hamiltonian, Eq. (1), with the classical equations ofmotion, ˙ p = − ∂H∂x = − K (cid:48) T x ∆ e − x ∞ (cid:88) n = −∞ δ ( t − T n ) , (2) ˙ x = ∂H∂p = pm . We rewrite the equations of motion in dimensionless form by defining ¯ x = x/ ∆ , ¯ t = t/T . The dimensionless momentumis correspondingly defined as ¯ p = pT / ( m ∆) . Thus we obtain the dimensionless equations of motion, ˙¯ p = − K ¯ xe − ¯ x ∞ (cid:88) n = −∞ δ (¯ t − n ) , (3) ˙¯ x = ¯ p, where K = K (cid:48) T m ∆ . (4)Since in what follows we deal with the rescaled position, momentum and time we will drop the bar notation forconvenience. By integrating (2) and defining p n = p ( t = n − ) , x n = x ( t = n − ) , where t = n − is a time just beforethe n- th kick, we can rewrite the differential equations of the motion as a mapping, M , M : (cid:40) p n +1 = p n − Kx n e − x n ,x n +1 = x n + p n +1 . (5)The corresponding quantum dynamics in rescaled units is given by the Hamiltonian, H = p − Ke − x ∞ (cid:88) n = −∞ δ ( t − n ) , (6)where p = − iτ ∂ x , and τ = (cid:126) T / (cid:0) m ∆ (cid:1) is the rescaled (cid:126) , namely, [ x, p ] = iτ . The quantum evolution is given by iτ ∂ t ψ = − τ ∂ xx ψ − Ke − x ∞ (cid:88) n = −∞ δ ( t − n ) ψ, (7)or by the one kick propagator U = e − i p τ exp (cid:18) i Kτ e − x (cid:19) . (8) B. Properties of the Classical Map and the phase portrait
In this subsection the classical properties of the map Eq. (5) will be presented. The first property is reflectionsymmetry, ( x, p ) → ( − x, − p ) . Phase portraits such as these presented in Fig. 1 and Fig. 2 are clearly seen to satisfy Figure 1: (Color online) The phase space for K = 1 . Colors (shades) distinguish different orbits.Figure 2: (Color online) The phase space for K = 4 . . Colors (shades) distinguish different orbits. this property. Like the standard map, Eq. (5) can be written as a product of two involutions, M = J J , where J : ( p, x ) → ( − p, x + p ) (9) J : ( p, x ) → (cid:16) − p − Kxe − x , x (cid:17) . We will use this property for the calculation of the periodic orbits. From (5) we see that the only fixed point is ( x = 0 , p = 0) . Linearizing around this point, we find that the trace of the tangent map is − K . Therefore, thispoint is elliptic for < K < , and, for K = 1 , the phase portrait of Fig. 1 is found, while for K > this point ishyperbolic, leading to phase portraits like that of Fig. 2. Since the kicking as a function of x is bounded by K , forlarge initial momentum the particle is nearly not affected by the kicks, and continues to move in its initial direction.For < K < we find a large island around the elliptic point ( x, p ) = (0 , , and, for nearly all initial conditions near x = p = 0 , the motion is regular (i.e., lies on KAM surfaces). Further away from this fixed point, one finds islandchains embedded in a chaotic strip. And even further away, the motion is unbounded. III. QUASI-ENERGIES OF AN ISLAND CHAIN
In the semiclassical regime quasi-energies are related to classical structures. In this section we assume the existenceof quasi-energy eigenfunctions u n ( x ) , U u n ( x ) = e − iE n u n ( x ) , (10)such that u n ( x ) is strongly localized to an island chain of order r , and we attempt to calculate the quasi-energy E n .For this purpose we use the one-kick propagator U to generate successive jumps in the island chain, U ψ i = ψ i +1 , (11)where ψ i is a wavefunction which is localized in island number i within the island chain. Further, we assume thatthis wavefunction can be expanded using the quasi-eigenstates of the island chain, ψ i = (cid:88) n c in u n ( x ) . (12)Using this expansion we obtain a system of equations, U ψ i = (cid:88) n c in u n ( x ) e − iE n , (13)and U r ψ i = (cid:88) n c in u n ( x ) e − iE n r . (14)Classically the i − th island is transformed to itself by r successive applications of the map M . In particular, an ellipticfixed point of the map M r is located in the center of the island. In the semiclassical limit the eigenstates of U r aredetermined by M r and are close to the eigenstates of a harmonic oscillator centered on the fixed point of M r . Thefrequency of the oscillator, ν i , is such that the eigenvalues of the tangent map of M r , which transforms the i − th islandto itself, are e ± iν i . This tangent map can be written in terms of the product of the tangent maps of M ( i → i + 1) , which transform the i − th island to the ( i + 1) − th island. Consequently, since the eigenvalues are determined bythe trace of the product of the tangent maps, they are independent of i (due to the invariance of the trace to cyclicpermutations). In what follows we therefore drop the index i from ν i .Choosing ψ i as the eigenstate of U r , means that U r ψ i = e i ¯ β ψ i , (15)where ¯ β = τ ν/ and we have taken ψ i to be the ground state of the harmonic oscillator. Therefore, ψ i = (cid:88) n c in u n ( x ) e − i ( E n r + ¯ β ) . (16)Using the orthogonality of the u n ( x ) , Eqs. (16) and (12) yield e − i ( E n r + ¯ β ) = 1 . (17)The quasi-energies, obtained from (17) are, therefore, E n = 2 πr n + β, ≤ n ≤ r, (18)where β = − ¯ β/r . Approximations to the quasi-energies can be calculated numerically by launching a wavepacket intoone island in the island chain and propagating it in time, which gives ψ ( x, N ) = (cid:88) n c in u n ( x ) e − iE n N . (19)Taking a Fourier transform with respect to N gives the quasi-energies. We have found that for K = 1 the chains with r = 8 and r = 16 accurately satisfy (18). IV. FIDELITY
The concept of quantum fidelity was introduced by Peres [18] as a fingerprint of classical chaos in quantum dynamics.It has subsequently been extensively utilized in theoretical [19, 20, 34, 35] and experimental studies [35–38], for areview see [21]. Most of this research has focused on the difference between chaotic and regular systems. Here wediscuss fidelity for a mixed system. We have calculated the fidelity, S ( t ) = (cid:12)(cid:12)(cid:12) (cid:104) φ | e iH t/τ e − iH t/τ | φ (cid:105) (cid:12)(cid:12)(cid:12) , (20)where H , are Hamiltonians of the form (6) with with slightly different kicking strengths, K , , and φ is the initialwavefunction. We note that the fidelity S ( t ) can be experimentally measured by the Ramsey method, as used in Ref.[37]. The fidelity is related to an integral over Wigner functions, S ( t ) = 2 πτ ˆ dxdp P φ ( x, p ) P φ ( x, p ) , (21)where P φ , are the Wigner functions of φ , = e − iH , t/τ φ , respectively.We study separately the fidelity in the central island, in the island chain, and in the chaotic region (i.e., Eq.(20)with φ localized to these regions). A. Fidelity of a wavepacket in the central island
First we prepare the initial wavefunction φ as a Gaussian wavepacket with a minimal uncertainty, namely, ∆ x =∆ p = ( τ / / , φ ( x ) = 1 (cid:16) π (∆ x ) (cid:17) / e − ip x/τ exp (cid:34) − ( x − x ) x ) (cid:35) . (22)We place φ in the center of the island, namely, x = p = 0 . Since the center of the wavepacket is initially at thefixed point, for ∆ x and ∆ p classically small, its dynamics are approximately determined by the tangent map of thefixed point. For this purpose we linearize the classical map (5) around the point x = p = 0 . This gives the equationfor the deviations, (cid:18) δx n +1 δp n +1 (cid:19) = (cid:18) (1 − K ) 1 − K (cid:19) (cid:18) δx n δp n (cid:19) . (23)The eigenvalues of this equation are, α , = (2 − K )2 ± i (cid:112) K (4 − K )2 ≡ e ± iω , (24)with ω = arctan (cid:112) K (4 − K )(2 − K ) , (25)which is the angular velocity of the points around the origin. In the vicinity of the fixed point, the system behaveslike a harmonic oscillator with a frequency ω . Classically, the motion of the trajectories, starting near the ellipticfixed point, x = p = 0 , stays there because the region is bounded by KAM curves that surround this point. Forsmall effective Planck’s constant, τ , the quantum behavior is expected to mimic the classical behavior for a long time.Inspired by the relation between the fidelity and the Wigner function (see (21)), we have defined a classical fidelity, S c ( t ) , as the overlap between coarse-grained Liouville densities of H and H (this is similar to the classical fidelitydefined in [39]). To do this we first randomly generate a large number of initial classical positions using the initialdistribution function, f ( x, p ) = 12 π ∆ x ∆ p exp (cid:40) − (cid:34)(cid:18) x − x ∆ x (cid:19) + (cid:18) p − p ∆ p (cid:19) (cid:35)(cid:41) , (26) Figure 3: (Color online) Quantum fidelity, S ( t ) , (dashed red) and classical fidelity, S c ( t ) (solid blue). K = 1 , K = 1 . , τ = 0 . , x = − . and p = 0 .Figure 4: (Color online) Classical density, which was initially placed at x = − . and p = 0 after kicks. Blue (dark)dots are for K = 1 and green (light) dots are for K = 1 . . corresponding to our initial φ given by (22). The coarse grained densities for H and H are then computed by firstintegrating these initial conditions and then coarse graining to a grid of squares in phase space of area τ [40]. Themotivation for this procedure is to check if structures in phase space of size smaller than τ are of importance to thefidelity. A comparison between S ( t ) and S c ( t ) for x = − . , p = 0 , and τ = 0 . is presented in Fig. 3. Theinitial wavepacket is smeared on a ring in the phase space due to the twist property of the map. Since the probabilitydensity is preserved, the “whorl” which is formed contains very dense and thin tendrils. In Fig. 4 two such “whorls”are presented for H with K = 1 and H with K = 1 . . When the two “whorls” coincide a fidelity revival is formed.Coarse graining the densities to a boxes of size τ averages the differences between the two “whorls”, obtained by H and H . This explains why the classical fidelity approaches as the number of kicks becomes large. On the otherhand, the quantum fidelity shows strong revivals which suggests that it feels the difference in trajectories between thetwo Hamiltonians. To understand the period of the revivals, we calculate, δω , the frequency difference between the P e r i od i n no . k i cks Figure 5: (Color online) A numerical ˙(solid blue) and an analytical (dashed green) computation of the period of the fidelityrevival as a function of K , δK = 0 . , x = p = 0 , τ = 0 . . two Hamiltonians, H , . Expanding ω around K gives ω ( K ) = ω ( K ) + K − K (cid:112) K (4 − K ) + O (cid:16) ( K − K ) (cid:17) . (27)Therefore, the difference in angular velocity between two orbits of Hamiltonians, H and H is given by δω = ω ( K ) − ω ( K ) = δK (cid:112) K (4 − K ) , (28)for δK = K − K . This suggests that the fidelity, S ( t ) will be periodic, with the period T = π/δω . Note that wepredict T = π/δω , rather than T = 2 π/δω . This is because of the symmetry of the initial condition. Each point of H is chasing a point of H which is its reflection through the origin of the phase space and, therefore, is found first at anangle of π and not π . To check this, we have calculated the period of the revivals numerically for < K < . First,fidelity was computed and Fourier transformed, then the second most significant value was taken as the period. InFig. 5 we present a comparison of the analytic calculation of the period of the fidelity and the numerical computation.The correspondence is good through the whole range of the stochasticity parameter K but degrades near K = 4 ,where the elliptic point at the origin becomes unstable. Also, near K = 2 , resonance chains appear near the fixedpoint x = p = 0 , which results in poor agreement with the theoretical prediction, see Fig. 7. Very often it is assumedthat quantum mechanical behavior is insensitive to phase space structures with areas smaller than Planck’s constant,which results in an effective averaging on this scale [28, 29]. While this assumption is often correct [32], sometimes itis not [41–46]. The difference between S ( t ) and S c ( t ) demonstrated in Fig. 3 shows that fidelity may be sensitive toextremely small details in the classical phase space. In particular, a “whorl” [28, 29] affects the quantum dynamics.The small decay of the quantum fidelity seen in Fig. 3 is a result of tunneling.We stress that to observe the oscillations which appear on Fig. 3 requires sensitivity to the structure of the “whorl”of Fig. 4. In our quantum calculation the effective Planck’s constant is τ = 0 . and it is obvious that the “whorl” ofFig. 4 exhibits structures on smaller scale, for example, in a square with sides of length . in phase space (of Fig.4) one finds several stripes of the “whorl”. Indeed, averaging over such a square leads to the classical fidelity thatdoes not exhibit oscillations as the quantum fidelity does. We conclude that the structures on the scale smaller thanthe effective Planck’s constant, τ , are crucial for the oscillations in the quantum fidelity. Hence, structures of scalessmaller than Planck’s constant may dominate fidelity, which is a quantum quantity.For a wavepacket started around an initial point ( x , p ) (cid:54) = (0 , the behavior is similar, but with a slightlydifferent period due to a decrease in the angular velocity for points far from the fixed point. Similarly to the case of ( x , p ) = (0 , , we have calculated numerically the revival period for different values of K ; this is shown in Fig. 6 .For K > . resonances appear near the launching point which introduce additional periods into the fidelity, makingthe analysis more complicated. P e r i od i n no . k i cks Figure 6: A numerical (blue circles) and an analytical (solid blue line) computation of the period of the fidelity revival as afunction of K , δK = 0 . , x = − . , p = 0 , τ = 2 × − .Figure 7: (Color online) The phase space for K = 2 . . Colors (shades) distinguish different orbits. B. Fidelity for a wavepacket in an island chain
We consider two different island chains occurring for different values of K . For K = 2 . we have examined achain of order r = 4 (see Fig. 7) , and for K = 1 we have studied a chain of order r = 8 (see Fig. 1). The initialwavepacket was launched inside one of the islands of the chain, and the we numerically computed the fidelity. In Figs.8 ( K = 2 . , K = 2 . ) and 9 ( K = 1 . , K = 1 . ) we show the results of these computations.It is notable that there are three timescales in the graph of the fidelity. The shortest timescale is visible only inthe inset of Fig. 9 and may be understood taking into account the symmetry of the equations of motion, x → − x,p → − p . This symmetry implies that each island has a “twin” which is found by reflection through the origin, x = p = 0 . Therefore, the overlap between the islands of H and H is a periodic function with a period of r / ,where r is the number of islands in the chain. Consequently, for the island chains used to obtain Fig. 8 and 9, thefidelity has periods of and , respectively, on its shortest timescale. The intermediate timescale is due to a rotation
500 1000 1500 20000.10.20.30.40.50.60.70.80.91 No. kicks F i de li t y Figure 8: Fidelity of packet started inside an island chain of order . K = 2 . , K = 2 . , τ = 2 × − , and the center ofthe packet is started at x = 0 . , p = 0 , in the center of one of the islands of the chain.
500 1000 1500 2000 2500 30000.10.20.30.40.50.60.70.80.9 500 1000 1500 2000 2500 30000.10.20.30.40.50.60.70.80.9 No. kicks F i de li t y
50 100 1500.20.40.60.8
Figure 9: Fidelity of packet started inside an island chain of order . K = 1 , K = 1 . , τ = 2 × − , and the center of thepacket is started at x = 1 . , p = 0 , in the center of one of the islands of the chain. The inset is a zoom on the graph. of the wavepacket around the elliptic points of the island where it is initially launched. The central point in theisland is a fixed point of M r . In r iterations, points in the island rotate with an angular velocity ω and ω for H and H , respectively. The angular velocities can be calculated numerically by linearization of the tangent map of M r around the fixed point of the map M r . We find the fixed point by reducing M to a product of involutions (9),which allows us to reduce the search for the fixed points to the line p = 0 in the phase space since any point onthis line is a fixed point of J [47, 48]. For K = 1 and K = 1 . , the angular velocities are found to be ω = 1 . and ω = 1 . . For K = 2 . and K = 2 . , the angular velocities are found to be ω = 0 . and ω = 0 . .Therefore, the time it takes for a packet to accomplish a full revolution around the fixed points of M r is πr/ ¯ ω , where ¯ ω = ( ω + ω ) ≈ ω ≈ ω (see Table I). The longest timescale of the fidelity is the timescale when the differencebetween the angular velocities is resolved T = 2 πr/δω . In Table I we compare those periods deduced directly fromFig. 8 and Fig. 9 and the periods calculated by finding ω , from the tangent map. We see that the agreement isexcellent.0 r = 4 r = 8 Fig.8 Tangent map Fig.9 Tangent mapshortest period 2 2 4 4medium period 62 . . longest period 651 . . Table I: This table compares two ways of calculating the periods of revivals for the resonance chains. In one way we havededuced them from the Figures 8,9, and in the other way we have calculated them using the tangent map. This is done for twodifferent resonances: r = 4 , for K = 2 . , K = 2 . ; and r = 8 for K = 1 , K = 1 . . For both cases τ = 2 × − .
10 20 30 400.20.40.60.81 No. kicks F i de li t y Figure 10: Fidelity of packet started inside a chaotic layer. K = 1 , K = 1 . , τ = 2 × − and the center of the packet isstarted at x = − , p = 0 . C. Fidelity of the wavepacket in the chaotic strip
For the fidelity of a packet started inside the chaotic strip (see Fig. 10), we notice a strong revival after 6 kickswhich is dependent on K . This is half a period in this chain/strip. After this revival the fidelity decays to zero,which is a characteristic of chaotic regions. Detailed exploration of this region is left for further studies. V. DEPHASING
We now investigate the effect of dephasing by adding temporal noise to the time between the kicks. The classicalequations of motion with the dephasing are given by p n +1 = p n − Kx n e − x n , (29) x n +1 = x n + (1 + δt n ) · p n +1 , and the quantum one kick propagator is U = e − i p τ (1+ δt n ) exp (cid:18) i Kτ e − x (cid:19) , (30)where δt n is a random variable which is normally distributed with zero mean and a standard deviation σ t . Thestandard deviation of the δt n , corresponds to the strength of the noise. We find that the noise results in an escape1 F i de li t y Figure 11: (Color online) Quantum fidelity, S ( t ) , (dashed red) and classical fidelity, S c ( t ) (solid blue) for a dephasing noise ofstrength σ t = 0 . , K = 1 , K = 1 . , τ = 0 . , x = − . and p = 0 . F i de li t y Figure 12: (Color online) Quantum fidelity, S ( t ) , (dashed red) and classical fidelity, S c ( t ) (solid blue) for a dephasing noise ofstrength σ t = 0 . , K = 1 , K = 1 . , τ = 0 . , x = − . and p = 0 . outside of the island, which yields additional decay in the fidelity. Since we are interested in the difference betweenthe two wavefunctions only inside the main island, for each kick we normalize the wavefunctions of H and H suchthat their norm is equal to inside a region of | x | ≤ x b = 3 . This gives the following expression for the fidelity S ( t ) = ´ x b − x b (cid:0) e − iH t/τ φ ( x (cid:48) ) (cid:1) (cid:0) e − iH t/τ φ ( x (cid:48) ) (cid:1) dx (cid:48) (cid:16) ´ x b − x b (cid:12)(cid:12) e − iH t/τ φ ( x (cid:48) ) (cid:12)(cid:12) dx (cid:48) (cid:17) / (cid:16) ´ x b − x b (cid:12)(cid:12) e − iH t/τ φ ( x (cid:48) ) (cid:12)(cid:12) dx (cid:48) (cid:17) / , with he classical fidelity S c ( t ) defined in a similar way. We have numerically calculated the fidelity for the samesituation as in Fig. 3 with added relative noise of σ t = 0 . (Fig. 11) and σ t = 0 . (Fig. 12). We notice that thenoise introduces additional decay in the quantum fidelity.2 F i de li t y Figure 13: (Color online) Quantum fidelity, S ( t ) , (dashed light red) and classical fidelity, S c ( t ) (solid dark blue) for twodifferent realizations of a dephasing noise of strength σ t = 0 . , K = K = 1 , τ = 0 . , x = − . and p = 0 .Figure 14: (Color online) Classical density, which was initially placed at x = − . and p = 0 after × kicks, K = K = 1 .Colors (shades) correspond to two different realizations of a dephasing noise of strength σ t = 0 . . To isolate the effect of noise from the decay in the fidelity due to the difference between K and K we set K = K and use two different noise realizations with the same strength σ t . From Fig. 13 we notice that classical fidelityinitially decays very fast due to the noise and than slowly recovers approaching a value of . . This is due to thecoarse graining to the scale of τ . To illustrate this we plot in Fig. 14 the classical densities after × kicks for apacket initially launched at x = − . . We notice that the densities for the two Hamiltonians highly overlap, whichexplains the high fidelity. In Fig. 15 we observe the corresponding quantum wavepackets. Contrary to the classicalfidelity, the quantum fidelity decays rather slowly with the noise, suggesting that it is more robust to noise than theclassical fidelity.3 −2 −1 0 1 200.20.40.60.8 x | ψ | Figure 15: (Color online) Wavepackets, which were initially placed at x = − . and p = 0 after × kicks, K = K = 1 .Colors (shades) correspond to two different realizations of a dephasing noise of strength σ t = 0 . . VI. SCATTERING
We now investigate the difference between quantum and classical scattering behavior by studying the evolution ofa wavepacket initialized outside of the main island of the phase space, Eq. (22) with x = − , p = 0 , ∆ x = ∆ p =( τ / / . In the classical case both the classical chaos, as well as the numerous small island structures introduce,an erratic behavior for the transmission and reflection coefficients as a function of the initial launching position andenergy [15, 16]. Due to effective phase space smoothing of areas much smaller than our effective Planck’s constant, τ , we expect that fine scale fractal-like features in the classically erratic scattering dependence will be averaged out.To quantify this behavior, we measure the transmission and reflection coefficients for a wavepacket defined as thetransfered or reflected probability mass, either quantum or classical. Classically, it is the fraction of initial trajectories(generated using (26)) reflected or transmitted by the main island for a given time, while quantum mechanically, wemeasure the total escaped probability up to time t from the island area, | x | ≤ x b , L ( t ) = ˆ t dt (cid:48) ˆ − x b −∞ | ψ ( x, t (cid:48) ) | dx (31) R ( t ) = ˆ t dt (cid:48) ˆ ∞ x b | ψ ( x, t (cid:48) ) | dx, where x b is the margin of the main island (we choose x b = 4 ), L ( t ) and R ( t ) are probabilities to be scattered tothe left or the right of the island till time t , correspondingly. To determine those probabilities, we use the continuityequation for the probability, ∂ t (cid:32) ˆ ba | ψ | dx (cid:33) = τ Im [( ψ∂ x ψ ∗ ) | x = b − ( ψ∂ x ψ ∗ ) | x = a ] , (32)so that, L ( t ) = τ i ˆ t dt (cid:48) ˆ t (cid:48) dt (cid:48)(cid:48) ( ψ∂ x ψ ∗ − ψ ∗ ∂ x ψ ) | x = − x b , (33) R ( t ) = − τ i ˆ t dt (cid:48) ( ψ∂ x ψ ∗ − ψ ∗ ∂ x ψ ) | x = x b . In Figs. 16-21 we compare the quantum and classical scattering of a wavepacket launched from the left of the4
200 400 600 800 100000.10.20.30.4 No. kicks L Figure 16: (Color online) Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the left of theisland ( x < − x b ) as a function of the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 .
200 400 600 800 100000.10.20.30.4 No. kicks R Figure 17: (Color online) Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the right ofthe island ( x > x b ) as a function of the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 . main island. We notice that there is a substantial difference, which decreases when we decrease the effective Planck’sconstant, τ . Figures 16-18 and Figs. 19-21 differ in the initial launching position of the wavepacket ( x = − , forFigs. 16-18 and x = − for Figs. 19-21). We notice that the scattering is sensitive to x . Different aspects of chaoticscattering for this problem were explored in [16], and in particular, the effect of small (cid:126) on washing out rainbowsingularities of the classical scattering function.5
200 400 600 800 10000.20.40.60.81 No. kicks P r obab ili t y i n t he i s l and Figure 18: Total quantum (solid blue line) and classical (blue dots) probabilities to stay in the island ( | x | ≤ x b ) as a functionof the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 .
200 400 600 800 100000.10.20.3 No. kicks L Figure 19: Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the left of the island ( x < − x b ) as a function of the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 . VII. DISCUSSION AND CONCLUSIONSA. Discussion of experimental realizability
In the present work the classical and quantum dynamics of a system with a mixed phase space were studied. It isproposed to realize this system by injecting cold atoms into a coherent, pulsed, gaussian light beam. The phase spacestructures, which can be seen on Figs. 1,2 and 7 are controlled by the parameters of the beam via the parameter K .Since it is relatively straightforward to control the parameters of gaussian beams, the proposed system is ideal forthe exploration of dynamics of mixed systems. In what follows limitations on experimental realizations are discussed.First we consider the realizability of an approximately one dimensional situation necessary for the validity of our6
200 400 600 800 100000.20.40.6 No. kicks R Figure 20: Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the right of the island ( x > x b ) as a function of the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 .
200 400 600 800 10000.20.40.60.81 No. kicks P r obab ili t y i n t he i s l and Figure 21: Total quantum (solid blue line) and classical (blue dots) probabilities to stay in the island ( | x | ≤ x b ) as a functionof the number of kicks. K = 1 , τ = 0 . , x = − , p = 0 . theoretical results. Let us assume that the gaussian beam propagates in the z direction. Its profile in the xy plane is e − x − y y . (34)Assuming that the extent of the light beam is much smaller than the Rayleigh length, z R = π ∆ /λ where λ isthe wavelength, and the z dependence of the potential can be ignored. The potential of Eq. (34) can be wellapproximated by exp (cid:0) − x / (cid:0) (cid:1)(cid:1) in (1), for sufficiently small values of y / ∆ y , and, to facilitate this, it is appropriateto consider ∆ y (cid:29) ∆ , i.e., a quasi-sheet-like beam. Such beams are experimentally realizable via routine methods.To analyze this situation, the normalized map M of (5) should be replaced by one with exp (cid:0) − x / (cid:1) replaced by7 exp (cid:16) − (cid:104) x / y (∆ / ∆ y ) (cid:105)(cid:17) . In addition, there are equations for y n and its conjugate momentum p y,n , which indimensionless units with y and p y,n rescaled by ∆ and T / ( m ∆) , respectively, take the form, p y,n +1 = p y,n − K y y n e − x n − (cid:16) ∆∆ y (cid:17) y n ,y n +1 = y n + p y,n +1 , (35)where K y = K (cid:18) ∆∆ y (cid:19) . (36)Since K ≈ and ∆ / ∆ y (cid:28) , it can be assumed that K y (cid:28) . Therefore, the motion in the y direction is slow relativeto the motion in the x direction. Thus exp (cid:0) − x n / (cid:1) can be approximated by its time average (cid:10) exp (cid:0) − x / (cid:0) (cid:1)(cid:1)(cid:11) ≡ ρ (which is of order unity), and, for sufficiently small y the y − motion (35) can be described by a Harmonic oscillatorwith a force constant K y ρ (cid:28) . Conservation of energy E y implies that the maximal value of y satisfies E y = 2 K y ρy . (37)The energy E y is determined by the initial preparation. Let us assume that initially the atoms form a Bose-EinsteinCondensate (BEC) and are in a harmonic trap that is anisotropic where the frequency in the y direction is ν (cid:48) y inexperimental units, and ν y = T ν (cid:48) y in our rescaled units. We assume that the center of this trap y satisfies y (cid:28) y max .The experiment starts when the trap is turned off. Assuming the atoms are in the ground state, their energy in ourrescales units is (cid:126) ν (cid:48) y (cid:18) T m ∆ (cid:19) = 12 ν y τ ≤ E y . (38)We desire the effect of the motion in the y direction on the motion in the x direction (Eq.(1) with exp (cid:0) − x / (cid:0) (cid:1)(cid:1) replaced by V of (34)) to be negligible. Thus it is required that η ≡ y (cid:18) ∆∆ y (cid:19) (cid:28) . (39)In this case the y motion corresponds to a variation in K of the order ∆ K ∼ Kη . Using (37) and (38), condition (39)reduces to ν y τK ≤ E y K = η (cid:28) , (40)where, since we are interested only in crude estimates, we have replaced ρ by one. The initial spread in y is given bythe ground state of the harmonic oscillator, where (cid:10) y (cid:11) = τ / (2 ν y ) , and we require that the expectation value of y satisfies (cid:10) y (cid:11) (cid:28) y , resulting in (cid:18) ∆∆ y (cid:19) τ ν y (cid:28) η. (41)For both inequalities (40) and (41) to be satisfied it is required that (cid:18) ∆∆ y (cid:19) τ η (cid:28) ν y ≤ Kτ η. (42)The resulting fundamental lower bound on η is (cid:18) ∆∆ y (cid:19) τ K (cid:28) η. (43)Reasonable experimental values are ∆ / ∆ y ≈ − and ν y ≈ . . For τ = 10 − and K ≈ the lower bound on η is − leaving a wide range for ‘engineering’ of BEC traps so that the ν y is in the range (42). For ν y ≈ . and τ = 10 − and K ≈ we can make η (cid:46) − . Since this value of η is small compared to the value of ∆ K = K − K ,used in our fidelity calculations (Figs 3, 5, 11, 12), those calculations are expected to be uneffected by y motion forour assumed parameters. It is also encouraging to see that noise of a higher level does not destroy fidelity oscillations(see Fig. 12). One should note, however, that the variation of the effective K of the motion in the x direction isslow, with effective frequency ∆ / ∆ y that for ∆ / ∆ y ≈ − is of order − . For these reasons, we expect that, themodel that we have explored theoretically in the present work should be realizable for a wide range of experimentalparameters.8 B. Conclusions
The main result of this paper is that the quantum fidelity is sensitive to the phase space details that are finer thanPlanck’s constant, contrary to expectations of Refs. [28, 29]. In particular, the fidelity was studied and predicted tooscillate with frequencies that can be predicted from classical considerations. This behavior is characteristic of regularregions. Fidelity exhibits a periodic sequence of peaks. For wavepackets in the main island, it was checked that the peakstructure is stable in the presence of external noise but the amplitude decays with time. For wavepackets initializedin a chain of regular islands, it was found that the fidelity exhibits several time scales that can be predicted fromclassical considerations. For wavepackets initialized in the chaotic region, the fidelity is found to decay exponentiallyas expected. It was shown how quasi-energies are related to classical structures in phase space. Substantial deviationbetween quantum and classical scattering was found. These quantum mechanical effects can be measured with kickedgaussian beams as demonstrated in the present work.
Acknowledgments
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