The tight-binding Hamiltonian describ-
ing our system is
H¼
Xm;s½wma†m;sam;sþXm′Jmm′ðtÞa†m;sam′;sXmKa†m;↑am;↓eimf^0 þH:c:
ð 1 Þwheream;sanda†m;sare the annihilation and
creation operators for themth longitudinal
cavity mode with frequencywm¼mWRand
with pseudospins∈f↓;↑g, and H.c. is the
Hermitian conjugate.Jmm^0 ðtÞis the coupling
along the synthetic frequency dimension
( 6 , 13 , 15 , 16 ), produced by the electro-optic
modulation ( 7 ). Because a small portion of
the ring is modulated, this coupling can be
simplified asJmm 0 ðtÞ¼JcosWRt;thatis,the
modemcan couple to all other modes of
the system, and the coupling strength is
independent of the mode indices ( 7 ). Here,
WRis the free spectral range (FSR), corre-
sponding to the separation between the lon-
gitudinal modes.Kin Eq. 1 is the strength
of the coupling between the two legs of the
ladder, created by the 8-shaped coupler com-
prising two-directional couplers with splitting
amplitudeffiffiffiffi
Kp. This coupling has a frequency-
and direction-dependent phaseTmf 0 (Fig.
1B), withf 0 ∼pDL=L 0 ( 17 ), whereDLis the
length difference between the two connect-
ing waveguides, andL 0 is the length of the
ring. To explain how this phaseTmf 0 is in-
troduced, we note that the connecting wave-
guide depicted by the blue solid line in Fig. 1A
couples exclusively from the CW to the CCW
mode, whereas the connecting waveguide de-
picted by the dashed line couples only from
the CCW to the CW mode. The phase dif-
ference between the coupling in the two di-
rections is thereforeDfðwÞ¼f↓→↑f↑→↓¼
bðwÞDL,wherebðwÞis the propagation con-
stantatfrequencywfor a mode in the connect-
ing waveguides. Assuming that the connecting
waveguides are the same as the waveguide of
the ring, and becausebðwmÞ¼ 2 pm=L 0 ,the
phase differenceDfincreases linearly with
m:DfðwmÞ¼ 2 pmDL=L 0 ¼ 2 mf 0.
To transform Eq. 1 into a time-independent
Hamiltonian, we definebm;↑¼am;↑eimðWRtþf^0 =^2 Þ
andbm;↓¼am;↓eimðWRtf^0 =^2 Þand use the
rotating-wave approximation to getH¼J
2Xmb†mþ 1 ;↓bm;↓eif^0 =^2 þb†mþ 1 ;↑bm;↑eif^0 =^2
KXmb†m;↑bm;↓þH:c:ð 2 ÞThis Hamiltonian describes a two-legged lad-
der pierced by a uniform magnetic field (a Hall
ladder) ( 18 ), as each plaquette is threaded by
an effective magnetic fluxf 0 (Fig. 1, B and C).
Thus, by choosing a nonzeroDL, our structure
in Fig. 1A naturally implements an effective
magnetic field. Large magnetic fluxes span-
ning the entire range in½p;pare achievableDuttet al.,Science 367 ,59–64 (2020) 3 January 2020 2of5
A BCωin
D EFG HICWCCWCW–1–200001 –1 –1 02–2020.90.10.90.1–202111φ 0φ 0φ 0Fig. 2. Chiral band structure and spin-momentum locking in the synthetic
Hall ladder.(A) Projected band structure of a 2D quantum Hall insulator infinite
along the vertical direction and finite along the horizontal direction (as shown
in the inset), showing topological chiral edge states (CES) highlighted in blue and
red between the bulk band gaps.f 0 ¼ 2 p=3. (B) Band structure of the two-
legged synthetic Hall ladder from the tight-binding HamiltonianHðkÞ(Eqs. 2 and
3) forJ/K= 2. The bulk bands disappear but signatures of chiral edge states
are preserved ( 19 ). (C) Schematic setup to directly measure band structure by
coupling an input-output waveguide to the ring in Fig. 1A. By varyingwinand
detecting the time-resolved transmission through the ring, the band structure
can be directly read out in experiments. The CW (CCW) spin-resolved band
structure can be detected by exciting the waveguide from the left (right) and
recording its transmission. (DandG) Theoretical band structures, with color-
coded pseudospin projectionsn↑andn↓for corresponding eigenstates. For the
lower band,k> 0 states have predominantly CW pseudospin character, signifying
spin-momentum locking. The dashed lines are band structures for the same
Jbut forK=0.(EandH) Experimental time-resolved transmission through the
ring for CW excitation (E) and CCW excitation (H).Dwis the detuning of the input
frequencywinfrom the resonance frequency of the uncoupled CW and CCW
modes. (FandI) Theoretical time-resolved transmission based on Floquet
analysis ( 17 ). Experimental parameters:J/2p= 1.95 MHz andK/2p= 0.97 MHz.
f 0 ≈ 3 p=4. Cavity linewidthg= 2 p¼480kHz.RESEARCH | REPORT