Dynamic behavior at a Feshbach resonance

arealsovisibleintheuncertaintyoftheatoms’positions. Foranincreasedharmonic perturbationnorepulsivelyboundstateisinresonancewiththeunboundstate. Hence, asshownintherightgraphofFig.5.6,theatomsoscillatepredominant lybetweende-localizedstatesandstateslocalizedatthecentrallatticesite. Sincetheatomsrepel eachother,theoscillationsareexactlyopposingeachother. Theoff-resonantcoupling totheboundstateleadstosmallandfastoscillationsofthe meandistance r_x² between0.8dand1.0d.

Figure5.6:Time-dependentbehaviorforthesuddenturn-onofanadditionalharmonic confinement.Left:For ˆW=5.3J1(ˆx²₁+ˆx²₂)/d²=0.034 ω1(ˆx²₁+ˆx²₂)/d² os-cillationsbetweenboundandunboundstatesappear.Right: Forstronger confinementˆW=10.5J1(ˆx²₁+ˆx²₂)/d²=0.067 ω1(ˆx²₁+ˆx²₂)/d² thebound-stateoccupationis muchweaker,howevertheparticlestunnelalternating betweenthecentralandouterwells.LegendasinFig.5.4.

5 .5 . Dynam icbehav iorataFeshbachresonance

ByreplacingtheBorn-Oppenheimerinteractionpotentialbythesquare-wellpotential definedinSec.4.4,thedynamicalbehaviorinthepresenceofan MFRcanberealist i-callynumericallysimulated. ThisallowsforstudyingthevalidityoftheBHMnotonly forastationarybutalsoforadynamicallyperturbedOL.ForM identicalatomsany time-dependentperturbation

W(t,r1,···,rM)=f(t)^M

i=1

Vpert(ri) (5.36) actsoneachatominthesameway.

Normally,anyexternalperturbationVpert(ri)isapproximatelyconstantonthelength scaleoftheboundstate. Hence,theperturbationcannotcoupletheorthogonalclosed andopen-channelstatesatan MFR.The matrixelementsoftheperturbationofthe

5. Time propagation of two atoms in an optical lattice

Hence, in second quantization the perturbation is expressed as W(t) =ˆ f(t)

As usual, only next-neighbor coupling and on-site coupling are considered and the basis is restricted to the firstN Bloch bands.

In the following the case of a linear perturbation in x direction with increasing strength is considered. That is,V_pert(~r_i) =x_i and f(t) =λt.

Of course, like the eigenenergies in Chap. 4 also the dynamical behavior depends crucially on the value of the resonance energy E_res, i.e. the energy of the bound state in the BHM. For the dynamical studies a resonance energy is chosen such that an inclination leads to the resonant next-neighbor coupling of two separated atoms in the ground state to a bound state in the first and second Bloch band. The corresponding dynamical behavior is sketched in Fig. 5.7. As one can see, the COM movement of the system upon accelerating the lattice depends crucially on the energy of the bound states.

Depending on the bound state and its COM excitation that comes into resonance, the system can move against the direction or in direction of the acceleration. A precise representation of the system is thus necessary to predict the mobility behavior of two atoms at an MFR.

Fig. 5.8 shows the projections | hn|Ψ(t)i |² of the time-dependent wave functions

|Ψ(t)i onto the eigenstates |ni of the unperturbed system for a slow inclination with λ= 0.0003^Er

~

~ω

d. If the perturbation would be suddenly switched off, the projections give the probability of finding the system in the corresponding eigenstate. For the same three coupling energies as shown in Fig. 4.9 the qualitative agreement between the result of the non-perturbative approach (upper row) and the dressed BHM (middle row) is very good. As described in Fig. 5.7 initially the bound state in the second Bloch band is slowly occupied. Aftert≈1300~/Er this bound state gets into resonance with the bound state in the first Bloch band, which is then occupied. Aftert≈1500~/Erthe main occupation moves back to the initial state. Additionally to the behavior described in Fig. 5.7 the inclination leads to a strong coupling of the bound states in the first and second Bloch bands. Due to the large energy separation of these states this coupling

82

5.5. Dynamic behavior at a Feshbach resonance

leads to fast oscillations of the population of the eigenstates. These oscillations appear, once the excited bound state in the left well comes into resonance with the lowest bound state in the right well att≈1300~/E_r.

In order to examine the quantitative agreement between the non-perturbative and BHM results, the time-dependent COM motion of the system ^DΨ(t)Rˆ_xΨ(t)^E is re-garded. As one can see in the lower row in Fig. 5.7, the overall quantitative agreement between the non-perturbative calculations and the dressed BHM is good. Especially for the smallest coupling energyχ= 0.41~ω the dressed BHM accurately recovers the correct dynamical behavior. For the larger coupling energies the fast oscillations ap-pearing aftert ≈ 1300~/E_r are less accurately reproduced by the dressed BHM. The phase shift and altered frequency of the oscillations is mainly due to a small underesti-mation of the coupling strength between the stationary eigenstates within the dressed BHM by about 1% . In contrast to the dressed BHM, the undressed BHM leads even for small coupling energies to a dynamical behavior significantly disagreeing from the one of the non-perturbative calculations.

5. Time propagation of two atoms in an optical lattice

0 d

xPosition

Energy

a

b

c

d

Figure 5.7: Sketch of the dynamical behavior while accelerating (inclining) the double-well. (a) The initial state consists of separated atoms (red disks) in the ground state of the left and right well. The four molecular states in the COM ground state (blue double disk below red disks) and in the first excited COM state (blue double disk above red disks) are not in resonance. (b) Upon inclining the potential the energy of an excited molecular state in the left well (dark blue) comes in resonance with the energy of the separated atoms. The molecular state is occupied and the COM of the system moves to the left. (c)After a further inclination the energy of the excited molecule in the left well comes into resonance with the ground-state molecule in the right well. By occupying this state the COM of the system moves to the right. (d)Finally, the molecule on the right well comes into resonance with the initial state of two separated atoms and the COM of the system moves again to the left.

84

5.5. Dynamic behavior at a Feshbach resonance

0 500 1000 1500 2000

0.0

0 500 1000 1500 2000

0.001

0 500 1000 1500 2000

0.0

0 500 1000 1500 2000

0.001

0 500 1000 1500 2000

0.001

0 500 1000 1500 2000

0.0

Figure 5.8: Dynamic behavior of two initially separated atoms in the ground state of the double-well potential during an inclination of the lattice for different cou-pling energies χand resonance energiesEres. Att=t_end= 2000~/Er each atom experiences a perturbation of ˆW = 0.7~ωx/d, which suffice to bringˆ both the first and second bound state into resonance (see Fig. 5.7). The projection of the time-dependent wave function|Ψ(t)i onto the eigenstates

|ni of the unperturbed system is shown in the first row (non-perturbative results) and the second row (results of the dressed BHM) using the same color coding as in Figs. 4.9 and 4.10. In the lowest row the mean COM po-sition ^DΨ(t)Rˆ_xΨ(t)^E is shown for the non-perturbative calculations and the time propagation of the dressed and undressed BHM. The insets show a magnified region of the beginning of the fast oscillations between 1200~/E_r and 1400~/Er.

6. Quantum computation with ultracold