Wave-Function Tomography of Topological Dimer Chains with Long-Range Couplings

May combine two claims.

The authors report generation of a time-reversal-breaking synthetic gauge field, which allows changing the band topology without modifying the strength of the couplings, a stringent prerequisite in the conventional SSH model.

“Finally, we report the generation of a time-reversal-breaking synthetic gauge field, which allows to change the band topology without modifying the strength of the couplings-a stringent prerequisite in the conventional SSH model.”

May combine two claims.

In the setup, the splitting between |m, ±⟩ is δ/2π = 3.43 MHz and the FSR is Ω/2π = 10.03 MHz.

“In our setup, the splitting between |m, ±⟩ is δ/2π = 3.43 MHz and the FSR is Ω/2π = 10.03MHz (Fig. 1 (e)).”

• Also stated in: § 1. Setup · plain

  “The cavity free spectral range (FSR) is 10.03 MHz.”

• Also stated in: § 1. Setup · plain

  “The hybridization of these modes lifts their degeneracy, giving rise to symmetric and anti-symmetric supermodes split by 3.43 MHz.”

May combine two claims.

The phase of the slow modulation is exactly the relative phase of the sublattice amplitudes in the Bloch modes, allowing experimental access to the phase of the wavefunction at every k point.

“Importantly, the phase of this modulation is exactly the relative phase of the sublattice amplitudes in the Bloch modes, allowing experimental access to the phase of the wavefunction in Eq. ( 2) at every k point.”

May combine two claims.

The quality factor is Q = 2.6 × 10^9 and the finesse is F = 133.7.

“The quality factor is therefore Q = 194.4THz/0.075MHz = 2.6 × 10 9 and the finesse is F = 133.7.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the setup, the different frequency components are f_a = 3.43 MHz, f_b = 6.60 MHz, f_c = 13.46 MHz, and f_d = 16.63 MHz.

“In our setup, the different frequency components are f a = ω a /2π = 3.43 MHz, f b = 6.60 MHz, f c = 13.46MHz, f d = 16.63MHz.”

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The authors observe an excellent agreement between measurements and tight-binding calculations, which validates their approach to unambiguously identify the winding number, including the full trajectory of g(k).

“We observe an excellent agreement between measurements and calculations, which validates our approach to unambiguously identify the winding number, including the full trajectory of g(k).”

May combine two claims.

The modulation gives rise to effective hopping terms a = iηVa and b* = iηVb*, where η is related to the electro-optical constant and has units of rad s^-1 V^-1.

“This modulation gives rise to effective hopping terms a = iηV a and b * = iηV * b , where η is related to the electro-optical constant and has units of rad s -1 V -1 [31].”

• cites [31]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The setup provides direct access to both the band dispersion and the geometry of the Bloch wavefunctions throughout the entire Brillouin zone, allowing extraction of the winding number for any possible configuration.

“This setup provides direct access to both the band dispersion and the geometry of the Bloch wavefunctions throughout the entire Brillouin zone allowing us to extract the winding number for any possible configuration.”

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Extracting the phase of the modulation at every k, the trajectory of g(k) throughout the entire Brillouin zone winds around the origin in the topological case but not in the trivial case.

“Extracting the phase of this modulation at every k, we can track the trajectory of g(k) throughout the entire Brillouin zone (Fig. 2 (c)-(d)), which winds around the origin in the topological case (d) but not in the trivial case (c).”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the experiment, δ = 3.43 × 10^6 MHz.

“In our experiment, δ = 3.43 × 10 6 MHz.”

Classified as probably not a substantial claim. May combine two claims.

Figure 3 shows measurements and numerical simulations of the band structure and g(k) trajectory for dimer chains with long-range hopping amplitudes, with each panel showing results for each possible winding number W = -1, 0, +1, +2.

“Measurements (top row) and numerical simulations (bottom row) of the band structure (left column) and of the trajectory of g(k) (central column) for dimer chains with long-range hopping amplitudes.Each panel shows results for each possible value of the winding number: W = -1 (a), W = 0 (b), W = +1 (c) and W = +2 (d).”

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The energy spectrum in the uncoupled basis consists of degenerate CW and CCW modes separated by a free spectral range Ω; in the coupled basis, symmetric and antisymmetric modes |m,±⟩ are separated by a frequency splitting δ and a free spectral range Ω.

“Energy spectrum of the loops in the uncoupled basis formed by a series of degenerate CW and CCW modes separated by a free spectral range Ω, and in the coupled basis of symmetric and anti-symmetric modes |m, ±⟩ separated by a frequency splitting δ and, again, a free spectral range Ω.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the experiment, δ/2π = 3.43 MHz.

“with δ = 4g proportional to the coupling strength (in our case δ/2π = 3.43 MHz)”

Classified as probably not a substantial claim.

Figure 4 shows measurements of the band structure and g(k) trajectory as the phase of the long-range hopping is tuned from θ = 0 to θ = π/2 to θ = π.

“Measurements of the band structure (top left) and g(k) trajectory (top right) as the phase of the long-range hopping is tuned from θ = 0 (c), to θ = π/2 (d), and θ = π (e).”

Probably not a substantial claim: may not pass S3 (load-bearing).

In synthetic photonic lattices, a wide variety of hopping terms can be engineered through suitable modulation of the relevant degrees of freedom.

“In the specific case of synthetic photonic lattices, a wide variety of hopping terms can be engineered through suitable modulation of the relevant degrees of freedom [19]”

• cites [19]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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Driving the EOMs with the same electrical signal amplitude but with a π phase shift maximizes coupling between states of opposite parity while suppressing all other couplings, thereby enforcing chiral symmetry by suppressing hopping between sites of the same sublattice.

“In particular, by driving the EOMs with the same electrical signal amplitude but with a π phase shift, V cw (t) = -V ccw (t), we maximize the coupling between states of opposite parity while suppressing all other couplings [30].This choice enforces chiral symmetry by suppressing hopping processes between sites belonging to the same sublattice.”

• Also stated in: § 1. Setup · plain

  “The two EOMs are driven with the same electrical signal, but with a π phase shift that ensures that symmetric supermodes only couple to anti-symmetric ones and vice-versa.”

• cites [30]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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When only nearest-neighbor couplings are present, the lattice realizes the SSH model, which displays two topologically distinct phases associated with integer winding number W = 0, 1.

“When only nearest-neighbor couplings are present, our lattice realizes the well-known Su-Schrieffer-Heeger (SSH) model [26], which displays two topologically distinct phases associated with the integer-valued winding number being W = 0, 1.”

• cites [26]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

The experimental implementation and control of sizable hopping terms extending beyond nearest neighbors is typically not straightforward.

“The experimental implementation and control of sizable hopping terms extending beyond nearest neighbors is typically not a straightforward task [6].”

• cites [6]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim. May combine two claims.

The SSH model hosts winding number W = 0 and 1 for |a| > |b| and |a| < |b| respectively.

“This model hosts a winding number W = 0 and 1 for |a| > |b| and |a| < |b| respectively (Fig. 1 (b)).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The averaging process ensures that the band structures do not exhibit fringes associated with the beating modulation of the Dirac comb.

“This averaging process ensures that the band structures do not exhibit fringes associated to the beating modulation of the Dirac comb discussed in the main text.”

Classified as probably not a substantial claim.

In usual realizations of lattice models based on condensed matter or ultracold atomic systems, hopping typically occurs via tunneling mediated by spatial overlap of wavefunctions and is dominated by short-range processes.

“In usual realizations of lattice models based on condensed matter or ultracold atomic systems, hopping typically occurs via tunneling processes mediated by the spatial overlap of wavefunctions at different sites and is therefore dominated by short-range processes [7,8].”

• cites [7]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [8]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The approximation used to obtain the field amplitude is only accurate when the gap ω_g = |ω_{k,+}-ω_{k,-}| is larger than the linewidth γ; when the gap is comparable to the linewidth, deviations appear in the extracted phase and a small kink in the trajectory of g(k), as observed in Fig. 4d.

“Note that this approximation is only accurate when the gap ω g = |ω k,+ -ω k,-| is larger than the linewidth γ.When the gap is comparable to the linewidth, deviations appear in the extracted phase of the field and a small kink in the trajectory of g(k) (as observed in Fig. 4d of the main text).”

• AssumptionRests on an assumption: The linewidths of the laser and of the bare modes are much smaller than Ω and δ.

Classified as probably not a substantial claim.

The modulator Hamiltonian is given by Eq. (14).

“The Hamiltonian describing the effect of the modulators is given by:”

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The eigenstates of the beam-splitter Hamiltonian are symmetric and antisymmetric combinations of CCW and CW modes, with eigenfrequencies ω_{m,±}=mΩ ∓ δ/2, where δ=4g.

“The eigenstates are symmetric (+) and antisymmetric (-) linear combinations of the CCW and CW modes:”

• Also stated in: § 1. Derivation of the dimer chains' Hamiltonian · plain

  “With this choice for g, the eigenfrequencies are ω m,± = mΩ ∓ δ 2 , with δ = 4g proportional to the coupling strength”

Probably not a substantial claim: may not pass S3 (load-bearing).

The beam-splitter Hamiltonian in matrix form for the mth pair of modes is Eq. (11).

“Hence, we can write H BS in a matrix form for the m th pair of modes as:”

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Under the rotating-wave approximation, H_EOM becomes Eq. (19), which shows a π phase shift between nearest-neighbor and long-range couplings.

“Injecting this expression of V (t) in H EOM and neglecting non-resonant terms in the rotating-wave approximation (RWA), we obtain:”

• Also stated in: § 1. Derivation of the dimer chains' Hamiltonian · boosted

  “In particular, note how in Eq.( 19) one clearly sees a π phase shift between the nearest-neighbor and long-range couplings that needs to be considered when defining V (t).”

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These features are used in this work to extract experimentally the Bloch wavefunction for every k point and then the winding number.

“These features are is used in this work for extracting experimentally the Bloch wavefunction for every k point and, then, the winding number.”

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To emulate Hamiltonians with chiral symmetry, it is necessary to couple eigenmodes of opposite parity, which requires modulating CW and CCW modes independently; identical in-phase signals only couple eigenmodes of similar parity.

“In order to emulate Hamiltonians with chiral symmetry, it is necessary to couple eigenmodes of opposite parity, i.e. symmetric with anti-symmetric and vice-versa.This requires modulating the CW and CCW modes independently; if they are driven with identical, in-phase electrical signals only eigenmodes of similar parity will couple.”

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Direct comparison with the extended Hamiltonian yields the relationships a = iηV_a, b* = iηV_b*, c = -iηV_c, d* = iηV_d*.

“Direct comparison with the function g(k) in the extended Hamiltonian presented in Eq. ( 3) of the main text, yields the following relationships between the hopping coefficients (a, b, c, d) and EOM voltages (V a,b,c,d ):”

• Also stated in: § 1. Derivation of the dimer chains' Hamiltonian · plain

  “a = +iηV a b * = +iηV * b c = -iηV c d * = -iηV * d .”

Classified as probably not a substantial claim. The quote only partly states this claim.

The beam-splitter Hamiltonian for the CW/CCW coupler is given by Eq. (8).

“The Hamiltonian describing the effect of the beam-splitter is given by:”

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Imposing V_ccw(t) = -V_cw(t) = V(t) minimizes coupling between states of the same parity and maximizes coupling between states of opposite parity.

“We impose V ccw (t) = -V cw (t) = V (t) to minimize (maximize) coupling between states of the same (opposite) parity.”

May combine two claims.

The shape of each pulse is exactly a δ-function in the limit of small cavity linewidth γ; finite γ leads to broadening of the pulses in time as visible in Fig. 2(a,b).

“Note also that the shape of each pulse is exactly a δ-function in the limit of a small cavity linewidth γ.In the general case, the finite γ leads to a broadening of the pulses in time as visible in Fig. 2(a,b).”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

For a simple 1D lattice, the band structure must follow a cosine function, and a small deviation of the driving frequency from the FSR results in an effective electric field and distorted bands associated with Bloch oscillations.

“For such a simple 1D lattice, the band structure must follow a cosine function and a small deviation of the driving frequency from the FSR results in an effective electric field and distorted bands (associated to Bloch oscillations).”

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Measuring the position of the Dirac comb within each period at every laser detuning allows extraction of the band structure, while the phase of the slow modulation provides information on the relative phase of the two sublattice components of the Bloch mode wavefunctions.

“Measuring the position of the Dirac comb within each single period at every laser detuning allows to extract the band structure, whereas the phase of its slow modulation across the different periods provides direct information on the relative phase of the two sublattice components of the Bloch mode wavefunctions.”

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In reciprocal space, H_EOM becomes Eq. (22), with g(k) = iη(V_a + V_b^* e^{ikΩ} - V_c e^{-ikΩ} - V_d^* e^{2ikΩ}).

“In this k-space, H EOM becomes:”

• Also stated in: § 1. Derivation of the dimer chains' Hamiltonian · plain

  “g(k) = |g(k)|e iϕ(k) = iη V a + V * b e ikΩ -V c e -ikΩ -V * d e 2ikΩ .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A semiconductor optical amplifier is used to compensate losses and optimize the quality factor of the loop.

“To optimize the quality factor of the loop, we use a semiconductor optical amplifier (SOA) to compensate the losses.”

Classified as probably not a substantial claim. May combine two claims.

The field at the photodiode is given by the input-output relation Eq. (39), and the intensity is Eq. (44).

“Putting all together, this gives the final expression for the field amplitude:”

• Also stated in: § 3. Input-output relations · plain

  “Hence the intensity measured on the photodiode for a mode |k, ±⟩ is given by:”

• AssumptionThe authors state this as an assumption: The laser is in the vicinity of the mth symmetric mode so that only a single term in C1 is retained.
• AssumptionThe authors state this as an assumption: The laser is resonant with the ± Bloch band, so contributions from the other band can be neglected.
• AssumptionThe authors state this as an assumption: The laser resonantly excites a single state in this band.

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Expressing H_EOM with symmetric and antisymmetric mode operators gives Eq. (17), and the dependence on Δ vanishes.

“We now express the H EOM with the symmetric and anti-symmetric mode operators (c m,± ):”

• Also stated in: § 1. Derivation of the dimer chains' Hamiltonian · plain

  “Note that the dependence on ∆ has vanished.”