Schrodinger Cat States Prepared by Bloch Oscillation

8/12/2019 Schrodinger Cat States Prepared by Bloch Oscillation

1/5

. Letter .

SCIENCE CHINAPhysics, Mechanics & Astronomy

September 2013 Vol. 56 No. 9: 18101814doi: 10.1007 / s11433-013-5152-z

c Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com

Schr odinger cat states prepared by Bloch oscillationin a spin-dependent optical lattice

WU BaoJun & ZHANG JiangMin *

Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

Received March 27, 2012; accepted July 2, 2012; published online July 22, 2013

We propose to use Bloch oscillation of ultra-cold atoms in a spin-dependent optical lattice to prepare Schr odinger cat states.Depending on its internal state, an atom feels di ff erent periodic potentials and thus has di ff erent energy band structures for itscenter-of-mass motion. Consequently, under the same gravity force, the wave packets associated with di ff erent internal statesperform Bloch oscillation of di ff erent amplitudes in space and in particular they can be macroscopically displaced with respect toeach other. In this manner, a cat state can be prepared.

Bloch oscillation, Schr odinger cat, spin-dependent optical lattice

PACS number(s): 03.75.Lm, 03.65.Ud, 03.65.Sq

Citation: Wu B J, Zhang J M. Schrodinger cat states prepared by Bloch oscillation in a spin-dependent optical lattice. Sci China-Phys Mech Astron, 2013, 56:18101814, doi: 10.1007 / s11433-013-5152-z

Bloch oscillation is a peculiar response of a particle in a peri-odic potential to a weak external force [1,2]. Under the dragof a constant force F , the wave packet of the particle oscil-lates back and forth periodically in space without accelerat-ing catastrophically in the direction of the force. The strangephenomena is that a DC bias generates an alternating current.The reason underlying is because as long as the force is weakenough, interband transitions are prohibited by the gaps be-tween the energy bands. Con ned in a speci c energy band

by such a mechanism, the motion of the particle is to a goodextent captured by the semi-classical equations [3]

dr dt

= E n(q)

q,

dqdt

= F , (1)

where r and q are the center-of-mass and wave vector of theparticle, respectively, and E n(q) is the dispersion relation inthe n th band. In the k -space, the particle traverses the Bril-

*Corresponding author (email: [email protected])

louin zone and follows the topology of a circle, repeatedly ata constant rate. From these equations, one solves readily thedisplacement of the wave packet as

r (t ) =1F

E n q0 +Ft

E n(q0) , (2)

where q0 is the initial value of q. We see that r is a pe-riodic function of time with the period T = 2 / aF this

states the Bloch oscillation in the semi-classical theory. Herea is the period of the periodic potential and 2 / a is the sizeof the Brillouin zone. Note that T is independent of the de-tailed structure of the periodic potential but depends only onits period.

Suppose initially the particle is at the bottom of the low-est energy band ( n = 0) with q0 = 0. The maximum dis-placement r m is reached at t = T / 2, when the wave vectorq = / a arrives at the edge of the rst Brillouin zone. After-wards, the velocity of the particle d r / dt reverses. The value


2/5

Wu B J, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 1811

of r m is simply

r m = BF

, (3)

where B E 0( / a ) E 0(0) is the band width of the lowestband. Simple as it is, this equation is used as a reference pointin our work below.

So far, Bloch oscillation has been observed in a vari-ety of systems, such as semiconductor superlattices [4,5],cold atoms in optical lattices [611], and photonic lattices[1214]. For applications, it has found use in microwave gen-eration [4,5,15], precision force measurements [811], andcoherent transport of matter waves [1618].

In this paper, we propose that Bloch oscillation can alsobe used to prepare Schr odinger cat states [1923] in a spin-dependent optical lattice. The idea is actually very simple.In a spin-dependent optical lattice, atoms in di ff erent inter-nal states see di ff erent potentials (For example, of di ff erent

strengths). This non-trivial fact means that they also havediff erent energy band structures for their center-of-mass mo-tion, which in turn means they will have di ff erent Bloch os-cillation modes under the same force (eq. (2)). In particular,their maximum displacements will be di ff erent according toeq. (3) since the Bs may di ff er, and this implies that the wavepackets corresponding to di ff erent internal states will be dis-placed with respect to each other. That is, a cat state can beprepared.

First of all we need a spin-dependent optical lattice. Asproposed in refs. [24,25] and realized in ref. [26], such an

optical lattice can be constructed by interfering two counter-propagating laser beams linearly polarized but with an an-gle between the polarization vectors. The resulting stand-ing light eld can be decomposed into a + and a po-larized one with intensities I + = I m cos2(kx + / 2) and I = I m cos2(kx / 2), respectively. Here k = 2 / is the wavevector of the laser beams. Such a decomposition is helpfulsince the dipole potential for an atom in a state |F , mF (thequantization axis of the atom is along the optical lattice) issimply the sum of the contributions of the two components.Below we will use the same system as in ref. [26]. That is,we choose 87Rb as the atom, and |1 |F = 2, mF = 2and |0 |F = 1, mF = 1 as the two atomic internal states.The spin dependence of the dipole potential is realized bychoosing the laser frequency to resolve the ne structure of the Rubidium D line. Speci cally, as in ref. [26], by tuningthe wave length of the optical lattice laser to = 785 nm,the dipole potentials for an atom in the |1 and |0 states, arerespectively,

V 1( x ; ) = V m cos2 kx 2

,

V 0( x ; ) =34

V m cos2 kx + 2

+14

V m cos2 kx 2

,

where V m I m. The point is that if 0 or , V 1( x ) and

V 0( x ) are shifted relative to each other, and more importantly,have di ff erent amplitudes. The latter e ff ect results in di ff erentband structures as we see in Figure 1. There the potentialsV 1,0 and the corresponding energy bands for the two internalstates are depicted. The parameters chosen are V m/ E r = 5

and =

/ 2, where E r = 2

k 2

/ 2m = 2

3.72 kHz is therecoil energy of the atom. Note that the width of the lowestband for the |1 state is 2 0.983 kHz, while that for the|0 state is 2 1.925 kHz. The two di ff er almost by a factorof 2.

Our scheme to generate a Schr odinger cat is according tothe following. Suppose initially the angle = 0 (for thisvalue of , V 1 = V 0) and the atom is in the |1 state. As for itsexternal state, it is assumed to be

i + / a

/ adq f (q)0(q). (4)

Here 0(q) is the Bloch state in the lowest band with wavevector q. The weight function f (q) is localized around q = 0but otherwise unspeci ed. This condition is easily satis edas long as the spatial size of the wave packet i is muchlarger than the lattice constant a = / 2. Actually, the con-densate wave function should satisfy this condition if thecondensate is loaded adiabatically from a magnetic trap intothe lattice as is usually done in cold atom experiments. Itcan be con rmed that the results presented in the followingare barely a ff ected with di ff erent choices of f (q), as long asthe localization condition is satis ed. This fact is consis-tent with the semi-classical theory in which the details of thewave packets are irrelevant. Speci cally, in the simulationsto be presented, f (q) is of the form f (q) exp( q2 / w2) withwa / = 0.1 1. This value of w corresponds to a wavepacket with a size on the order of 10 a .

Then at some moment, by using a microwave pulse we canprepare the internal state of the atom into an arbitrary super-position of the |0 and |1 states. Let it be |0 + |1 with| |2 + | |2 = 1. Subsequently, is adjusted to / 2 (suddenlyor smoothly, it does not matter; but in our simulation we sup-pose the sudden scenario), and the lattice is tilted by with

respect to the horizontal plane. The Bloch oscillation thenbegins. At a later time t , the wave function of the atom is of the form (t ) = |0 0( x , t ) + |1 1( x , t ). The evolution of the external wave function j( x , t ) is given by ( j = 0, 1)

i t

j = 2

2m2

x 2 + V j( x ) Fx j, (5)

with the force F = mg sin and the initial condition j(t = 0) = i .

Though the semi-classical theory above gives us an overallidea of the motion of the wave packets j( x , t ), here we shall

solve eq. (5) numerically. Snapshots of j( x , t ) are shown in


3/5

1812 Wu B J, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9

2 1 0 1 20

1

2

3

4

5

2 x/

V 1 , 0

( x ) / E

r

(a)

0.5 0 0.52

3

4

5

6

7

8

q/ 4

E n ( q ) / E

r

(b)

Figure 1 (Color online) (a) Dipole potentials V 1 ( x ) (solid line) and V 0( x ) (dashed line) for an atom in the internal states |1 and |0 , respectively. Theparameters are V m/ E r = 5 and = / 2. Here E r = 2k 2 / 2m is the recoil energy of the atom. (b) The three lowest energy bands for an atom in the |1 (solidlines) and |0 (dashed lines) internal states.

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(a)

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(b)

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(c)

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(d)

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(e)

0 50 100 1500

0.5

1.0

1.5

2.0

2.5

3.0

2 x/

| 1

, 0 ( x ) | 2

( 1 / )

(f)

Figure 2 (Color online) Time evolution of the wave packets 1( x , t ) and 0( x , t ) in a Bloch oscillation cycle. (a) t / T = 0; (b) t / T = 1/ 6; (c) t / T = 1/ 3; (d)t / T = 1/ 2; (e) t / T = 3/ 4; (f) t / T = 1. The parameters are the same as in Figure 1 and Fa / E r = 0.005. Initially, 1,0 = i as de ned in (4). Note that the wavepackets are not normalized to unity.

in Figure 2. As expected, in the interval 0 t T / 2, thetwo wave packets move rightward and gradually depart. Atthe turning point t = T / 2 (Figure 2(d)), the distance betweenthe two reaches the maximum. Remarkably, at this point,the two wave packets are well separatedthe separation be-tween them is about 50 a , which is much larger than their sizes( 10a ). Note that the wave packets themselves are largeenough to deserve the name macroscopic. Therefore we ob-

tain a desired Schrodinger cat state

(t = T / 2) = |1 1(t = T / 2) + |0 0(t = T / 2), (6)

for which the internal and external states of the atom areentangled. The critical condition is that the latter is in twomacroscopically displaced macroscopic states, which corre-spond to a live and a dead cat, respectively. The coher-

ence between the two packets can be checked by applying a


4/5

Wu B J, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 1813

microwave pulse to achieve a rotation in the {|1 , |0 } space,and then turning o ff the lattice and observing the momen-tum distribution of the atom in the |1 state by the absorptionimaging method [26]. Note that 1,0(T / 2) are both peakedaround q = / a in momentum space and thus the interfer-

ence pattern will primarily consist of two peaks whose am-plitudes depend on the rotation as well as and .Afterwards, the two wave packets move backwards and

return approximately 1) to their original states at t = T (Fig-ure 2(f)). That is, 1,0(T ) i up to some global phases [27].The global wave function is then (T ) ( |0 + ei |1 ) i,where is the diff erence of the phases 1,0 accumulated in acycle. Thus the atom completes a Bloch oscillation cycle bygetting its internal state rotated somehow. From the point of view of the cat state, a cycle of Bloch oscillation is a cycle of birth-growth-death. The cycle can be interrupted by puttingF = 0 at an appropriate time, e.g., at t = T / 2, when the cat isin its largest size.

We have followed the center-of-mass motion of the wavepackets in time. The results are shown in Figure 3. We seethat the semi-classical theory is consistent quantitatively. Thenumerically exact results deviate signi cantly from the semi-classical predictions only in the vicinities of t = 0 and t = T .The reason is that due to the sudden change of , both wavepackets are partially excited and some packets (not visible inthe snapshots) belonging to higher bands are emitted aroundt = 0, which may perform Bloch oscillation also and returnaround t = T .

Now let us turn to the problem of the feasibility of thescheme in experiment. In our simulation, Fa / E r = 0.005.It is essential to make sure that this ratio is much smallerthan unity. There are two reasons. First, the potential dropFa between two neighboring sites should be much smallerthan the gap between the zeroth (lowest) and the rst bands(Figure 1(b)) so as to suppress Zener tunneling [2]. Or equiv-alently, the Brillouin zone, especially its boundary, shouldbe traversed slowly so that transition into excited bands canbe neglected. Second, according to eq. (3), the distance be-tween the two wave packets is inversely proportional to F ,thus smaller F means larger separation or larger cat. Of course, there should be an optimal value of F since the pe-riod T is also inversely proportional to F . For 87 Rb and alattice constant a = / 2 = 392 .5 nm, the ratio above corre-sponds to a tilt angle = 4 and a period T = 53 ms. On thecontrary, under the chosen detuning and strength of the opti-cal lattice, the spontaneous radiation rate eff of the atom isabout 0 .2 s 1 . Thus the atom is long lived enough to oscillateseveral circles before incoherent processes set in.

0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

120

140

t/T

2 x /

|1 exact|0 exact|1 semi-classical|0 semi-classical

Figure 3 (Color online) Evolution of the center-of-mass of the wave pack-ets 1,0 ( x , t ) in a Bloch oscillation cycle. The parameters are the same as inFigures 1 and 2. The results obtained by solving eq. (5) numerically / exactlyagree very well with those by the semi-classical theory (eq. (2)).

In conclusion, we have proposed that the spin-dependentoptical lattice may o ff er an opportunity to create Schr odingercat states by using Bloch oscillation. Our scheme has severalinteresting advantages. First, the cat state experiences birth-growth-death cycles repeatedly. It would be worthy to studyexperimentally how this process is damped in a real opticallattice, which is believed to be well isolated from the envi-ronment. Second, if we start from a Bose-Einstein conden-sate and minimize the atom-atom interaction which is delete-rious to the Bloch oscillation, it might be possible to createa collection of atoms condensed in a cat state. We note thatsome generalizations are also possible. For example, though

here we focused on the one dimensional case, the schemecan be directly extended to higher dimensions [18] sincetwo-dimensional spin-dependentoptical lattices have alreadybeen demonstrated experimentally [28]. Furthermore, in con-trast to the static force considered here, periodically modula-tions [16,17] are worth consideration also since they may helpto increase the size of Schr odingers cat.

We are grateful to DUAN L M, PAN Y, and ZHOU D L for stimulating discus-

sions and valuable suggestions. ZHANG J M was supported by the National

Natural Science Foundation of China (Grant No. 11091240226).

1 Bloch F. Uber die quantenmechanik der elektronen in kristallgittern. ZPhys, 1929, 52: 555600

2 Zener C. A theory of the electrical breakdown of solid dielectrics. ProcR Soc Lond A, 1934, 145: 523529

3 Kittel C. Introduction to Solid State Physics. 7th ed. New York: JohnWiley & Sons, 1996

4 Esaki L, Tsu R. Superlattice and negative di ff erential conductivity insemiconductors. IBM J Res Dev, 1970, 14: 6165

1) The recurrence of i is imperfect since on the one hand both 1,0 are partially excited into higher bands at the beginning due to the sudden change of , and on the other hand the subsequent evolution is not strictly adiabatic and higher band components still mix in ref. [2]. This feature is not captured by thesemi-classical theory


5/5

1814 Wu B J, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9

5 Waschke C, Roskos H G, Schwedler R, et al. Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice.Phys Rev Lett, 1993, 70: 33193322

6 Dahan M B, Peik E, Reichel J, et al. Bloch Oscillations of atoms in anoptical potential. Phys Rev Lett, 1996, 76: 45084511

7 Salger T, Ritt G, Geckeler C, et al. Bloch oscillations of a Bose-

Einstein condensate in a subwavelength optical lattice. Phys Rev A,2009, 79: 011605

8 Ferrari G, Poli N, Sorrentino F, et al. Long-lived Bloch oscillationswith Bosonic Sr atoms and application to gravity measurement at themicrometer scale. Phys Rev Lett, 2006, 97: 060402

9 Poli N, Wang F Y, Tarallo M G, et al. Precision measurement of grav-ity with cold atoms in an optical lattice and comparison with a classicalgravimeter. Phys Rev Lett, 2011, 106: 038501

10 Gustavsson M, Haller E, Mark M J, et al. Control of interaction-induced dephasing of Bloch oscillations. Phys Rev Lett, 2008, 100:080404

11 Fattori M, DErrico C, Roati G, et al. Atom interferometry with aweakly interacting Bose-Einstein condensate. Phys Rev Lett, 2008,100: 080405

12 Pertsch T, Dannberg P, El ein W, et al. Optical Bloch oscillations intemperature tuned waveguide arrays. Phys Rev Lett, 1999, 83: 47524755; Morandotti R, Peschel U, Aitchison J S. Experimental observa-tion of linear and nonlinear optical Bloch oscillations. Phys Rev Lett,1999, 83: 47564759; Sapienza R, Costantino P, Wiersma D. Opticalanalogue of electronic Bloch oscillations. Phys Rev Lett, 2003, 91:263902

13 Trompeter H, Krolikowski W, Neshev D N, et al. Bloch oscillationsand Zener tunneling in two-dimensional photonic lattices. Phys RevLett, 2006, 96: 053903

14 Longhi S. Quantum-optical analogies using photonic structures. Laser

Photon Rev, 2009, 3: 24326115 Grenzer J, Ignatov A A, Schomburg E, et al. Microwave oscillator

based on Bloch oscillations of electrons in a semiconductor superlat-

tice. Ann Phys-Berlin, 1995, 3: 184190

16 Alberti A, Ivanov V V, Tino G M, et al. Engineering the quantum trans-port of atomic wavefunctions over macroscopic distances. Nat Phys,2009, 5: 547550

17 Haller E, Hart R, Mark M J, et al. Inducing transport in a dissipation-free lattice with super Bloch oscillations. Phys Rev Lett, 2010, 104:

20040318 Zhang J M, Liu W M. Directed coherent transport due to the Bloch

oscillation in two dimensions. Phys Rev A, 2010, 82: 025602

19 Wheeler J A, Zurek W H. Quantum Theory of Measurement. Prince-ton: Princeton University Press, 1983

20 Schr odinger E. Die gegenwartige situation in der quantenme chanik.Naturwissenschaften, 1935, 23: 807812, 823828, 844849. Re-printed in English in ref. [19]

21 Monroe C, Meekhof D M, King B E, et al. A Schr odinger cat super-position state of an atom. Science, 1996, 272: 11311136

22 Brune M, Hagley E, Dreyer J, et al. Observing the progressive deco-herence of the meter in a quantum measurement. Phys Rev Lett, 1996,77: 48874890

23 Friedman J R, Patel V, Chen W, et al. Quantum superposition ofdistinctmacroscopic states. Nature, 2000, 406: 4346

24 Brennen G K, Caves C M, Jessen P S, et al. Quantum logic gates inoptical lattices. Phys Rev Lett, 1999, 82: 10601063

25 Jaksch D, Briegel H J, Cirac J I, et al. Entanglement of atoms via coldcontrolled collisions. Phys Rev Lett, 1999, 82: 19751978

26 Mandel O, Greiner M, Widera A, et al. Coherent transport of neutralatoms in spin-dependent optical lattice potentials. Phys Rev Lett, 2003,91: 010407

27 Zak J. Berrys phase for energy bands in solids. Phys Rev Lett, 1989,62: 27472750

28 Soltan-Panahi P, Struck J, Hauke P, et al. Multi-component quantum

gases in spin-dependent hexagonal lattices. Nat Phys, 2011, 7: 434440

Documents

Schrodinger Cat States Prepared by Bloch Oscillation