ky (π/a) kx (π/a)
4 2 0
1 0 ν1Energy (a.u.)
Φ Φ Φ Φ
δ δ
b a
1 0 0 1
ky (π/a) 0 0 kx (π/a)
1 1
ky (π/a) 0 0 kx (π/a)
1 1
ky (π/a) 0 0 kx (π/a)
1 1
Detuning δ (J)
x
y Topologically trivial Topologically non-trivial Hofstadter
model for Φ=π/2 B
D C
A
Figure 8.3: Energy spectra and topology of the Bloch bands as a function of the staggered detuningδ. aIllustration of the Hofstadter-like optical lattice with additional staggered detuning δalong thex- andy-direction. The unit cell of the lattice is depicted by the green shaded area and the four non-equivalent sites are labeled as A,B,C,D. b Energy spectra as a function of the detuning. For a detuning larger than 2J the bands are topologically trivial and the Chern numbers are zero. At δ = 2J there is a topological phase transition, where the gaps in the spectrum close and the system enters the topologically non-trivial regime. In this regime the Chern number of the lowest band is ν1 = +1 forΦ= π/2. For vanishing detuningδ = 0 the system realizes the Hofstadter model with fluxΦ= π/2. Note that the energy axis is rescaled for each spectrum. (Figure adapted from Ref. [73])
Hˆ → Hˆ =−2J
δ/(2J) isin(kxa) −sin(kya) 0
−isin(kxa) 0 0 cos(kya)
−sin(kya) 0 0 cos(kxa) 0 cos(kya) cos(kxa) −δ/(2J)
. (8.16)
During the loading sequence the detuning is decreased to zero,δ →0. The correspond-ing energy spectra are displayed in Fig. 8.3b. Since the unit cells are equivalent, the number of bands is preserved during the loading sequence. Forδ >2J the topology of the bands is trivial and all Chern numbers are zero. At δ=2J a topological phase tran-sition occurs and the gaps in the spectrum close. Forδ <2Jthe topologically non-trivial regime is reached, where the lowest band has a Chern number of ν1 = +1, and at the end of the sequence (δ =0) the Harper-Hofstadter Hamiltonian forΦ=π/2 is realized.
Note that the horizontal axes of the energy spectra in Fig. 8.3are different compared to the ones shown in Ref. [73]. Depending on the definition of the gauge and the detuning term given in Eq. (8.15) the dispersion relation might be shifted in momentum space but this has no impact on the general loading scheme.
8.3 Adiabatic loading into the Hofstadter bands 133
8.3.2 Experimental sequence
The experimental sequence is illustrated in Fig.8.4. It started by loading a Bose-Einstein condensate of 87Rb atoms within 150 ms into a two-dimensional optical superlattice of depths Vx = 6.0(2)Ers, Vxl = 5.25(16)Erl, Vy = 10(1)Ers and Vyl = 1.75(5)Erl. The phases between the short- and long-lattice standing waves, ϕxSL = ϕySL=π/2 (Sect.4.3), where chosen so as to create a staggered potential with energy offset∆+δx alongxand δy alongy. Along thez-direction the atoms were confined by a weak harmonic potential generated by a crossed optical dipole trap in the horizontal plane, ωz/(2π) ≈ 20 Hz.
Initially tunneling was inhibited along both directions due to the potential detuning (∆+δx) Jx and δy Jy and all atoms occupied the low-energy sites, denoted as A-sites (Fig. 8.3a). Then, the modulation was switched on off-resonant within 30 ms with V0 ' 1.6 Erl and ω/(2π) = ±∆/h ' ±2.7 kHz. The resonance condition was calibrated independently for the final lattice parameters along x, Vx = 6.0(2)Ers and Vxl=3.25(10)Erl, by performing spectroscopy measurements as discussed in Sect.4.3.2.
The detuning was chosen larger than the effective coupling strength on resonanceδx J such that tunneling remained suppressed along this direction. Additionally along the y-direction the conditionδy ∆was fulfilled in order to assure that the modulation with frequency ¯hω = ∆did not induce tunneling in the perpendicular direction (Fig. 8.4b).
In these limits tunneling was suppressed in both directions and atoms stayed inA-sites.
The values for δx andδy were optimized experimentally such that less than 10% of the atoms were transferred to higher bands after switching on the modulation.
The loading into the Hofstadter bands was achieved by ramping down the detunings to zero within 30 ms, by changing the long lattice depth along x to Vxl = 3.25(10)Erl and the long lattice alongy toVyl =0 Erl. For these values resonant tunneling occurred along both directions and the parameters were chosen such that the effective coupling strengths along both directions were the same, J = 75(3)Hz. This lattice configuration realizes a lattice with uniform fluxΦ= ±π/2 per plaquette, where the direction of the flux depends on the sign of the frequencyω.
8.3.3 Momentum distribution and initial band population
The Chern-number measurement as sketched in Sect.8.2is based on the assumption that the atoms in each Hofstadter band populate the corresponding band homogeneously in k-space. This assumption was verified experimentally by measuring the momentum distribution in the different bands.
For this purpose the loading sequence described above (Fig.8.3) was reversed. The se-quence started by ramping up the staggered detuningsδxandδywithin 30 ms in order to suppress tunneling along both directions. The final lattice depths wereVx = 6.0(2)Ers, Vxl = 5.25(16)Erl,Vy = 10(1)Ers andVyl =1.75(5)Erl. The number of energy bands is preserved during this ramp and the populations of the topological Hofstadter bands are
Transport measurement
150 30 30
Time (ms)
150 30 30
Detection
0 5 10
tBO
Time (ms)
Vy (Ers)
Vx (Ers)
Vyl (Erl) Vxl (Erl) V0 (Erl)
Lattice depth
b a
Detuning
δy δx
ħω Δ
Figure 8.4: Schematic drawing of the experimental sequence. a Lattice depths as a function of time. Note that the distances on the time-axis are not to scale. After loading the atoms into the Hofstadter bands as described in the main text the transport measurements were performed for a certain Bloch oscillation timetBObefore several detection techniques were applied: in-situ position of the cloud, band-population measurements, momentum distribution (Sect. 8.3.3). b Staggered energy offsets alongxandyas they evolve during the sequence (green). For compar-ison the modulation frequency ¯hωis shown in red.
mapped onto the topologically trivial Hofstadter-like bands.
Due to the detuning tunneling was suppressed and the modulation could be switched off instantaneously to map the populations of the explicitly time-dependent Hamilto-nian onto the ones of the static superlattice potential with staggered offsets ∆+δx and δy. The size of the Brillouin zone is unchanged during the whole mapping sequence, hence, the population of different k-states is preserved if scattering processes and heat-ing effects durheat-ing the ramp are neglected. Consequently the momentum distribution of the Hofstadter bands is reflected in the momentum distribution of the static two-dimensional superlattice potential.
All lattice potentials are subsequently ramped down adiabatically to map the momen-tum distribution in the lattice onto the real-space momenmomen-tum distribution. Then the atoms were released from the trap and detected via absorption imaging after 10 ms TOF (Fig. 8.5a). The connection between the different Brillouin zones and the corresponding Hofstadter bands is illustrated in Fig.8.5b and c. There are two informations we obtain from these images: (a) we achieve typically a population of about 60% in the lowest Hofstadter band; (b) the distribution is homogeneous in each of the individual bands.
This data is consistent with the assumption of homogeneous band populations.
In principle the band populations can be inferred by counting the atom numbers in the different Brillouin zones. However, the zones are connected, thus, to ease the counting of the occupations we apply a slightly different sequence that separates the different Brillouin zones from each other (Sect.8.4.2).