Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures

For wavepackets near a Dirac point, straining the system is mathematically equivalent to introducing magnetic fields.

“It has been shown for graphene that for wavepackets that lie near a Dirac point, straining the system is mathematically equivalent to introducing magnetic fields 3 .”

• cites [3] Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011). sourceCould not verify

  The 2D Photonic System and Quantum Spin Hall Hamiltonian section (pp. 1–3) describes synthetic magnetic fields produced by phases in coupled optical resonators, not by straining graphene. Searches for graphene, strain, and Dirac points found no passage addressing the claimed equivalence for wavepackets near a Dirac point. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

In honeycomb photonic lattices, the pseudomagnetic field separates the spectrum into Landau levels with β = ±ω√N, where ω = 3Bc0a/2.

“€ β = ±ω N € N = 0,1,2,... € ω = 3 Bc 0 a /2”

The gaps between the Landau levels give rise to transverse confinement of the optical modes.

“We show experimentally and numerically that the gaps between these Landau levels give rise to transverse confinement of the optical modes.”

The present work demonstrates the existence of pseudomagnetism in photonic lattices by virtue of the inhomogeneous strain.

“In the present work, we demonstrate the existence of pseudomagnetism in such photonic lattices by virtue of the inhomogeneous strain.”

By properly structuring a dielectric lattice, it is possible to induce a pseudomagnetic field at optical frequencies in a photonic lattice, where the propagation dynamics is equivalent to the evolution of an electronic wavepacket in graphene.

“Here, we show experimentally and theoretically that, by properly structuring a dielectric lattice, it is possible to induce a pseudomagnetic field at optical frequencies in a photonic lattice, where the propagation dynamics is equivalent to the evolution of an electronic wavepacket in graphene.”

The numerically calculated positions of the Landau levels coincide exactly with the analytically predicted values.

“Note that the numerically calculated positions of the Landau levels coincide exactly with the analytically predicted values (which assume that the states at the vicinity of the Dirac point obey the Dirac equation).”

• AssumptionThe authors state this as an assumption: The states at the vicinity of the Dirac point obey the Dirac equation.

The induced field gives rise to multiple photonic Landau levels, which are singularities in the density of states, separated by band gaps.

“The induced field gives rise to multiple photonic Landau levels (singularities in the density of states) separated by band gaps.”

The strain introduces edge states that are normally associated with the zig-zag and bearded edges at β/c0=0.

“Furthermore, the strain introduces edge states that are normally associated with the zig-zag and bearded edges 31 at .”

• cites [31] Kohmoto, M. & Hasegawa, Y. Zero modes and edge states of the honeycomb lattice. Phys. Rev. B 76, 205402 (2007). sourceNot checked — no full text of the source

Some of the Landau level eigenstates have significant power residing on the armchair edges of the sample.

“As a result, some of the Landau level eigenstates have significant power residing on the armchair edges of the sample.”

Launching light in a single waveguide on the armchair edge of the strained photonic lattice excites only modes within the Dirac region.

“In the paper, we claim that by launching light in a single waveguide on the armchair edge of the strained photonic lattice, we only excite modes within the Dirac region.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

Next-nearest-neighbor hopping terms are negligible compared to nearest-neighbor terms.

“Furthermore, we have confirmed that next-nearest-neighbor hopping terms are negligible compared to nearest-neighbor terms.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

In the unstrained system, the total fraction of power that resides in eigenstates in the Dirac region is 9%.

“In the unstrained system, the total fraction of power that resides in eigenstates in the Dirac region is 9%.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

For increasing strain, a monotonically increasing fraction of the projected power resides in the Dirac region.

“We find that for increasing strain, a monotonically increasing fraction of the projected power resides in the Dirac region.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

In a pseudomagnetically strained honeycomb lattice, a defect waveguide with a refractive index slightly higher or slightly lower than the rest of the waveguides in the array leads to spreading into the bulk.

“As we show in that figure, if the defect waveguide has a refractive index that is either slightly higher or slightly lower than the rest of the waveguides in the array, this leads to spreading into the bulk.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

At very strong defect (! !/! != 4), the light becomes localized again.

“(g) Once the defect mode is very strong (! !/! != 4) it becomes a defect mode outside of the band and therefore in another band gap.As a result, it becomes localized again.”

The Landau levels reside at the center of the spectrum, where the Dirac cones lie in the unstrained lattice.

“The Landau levels reside at the center of the spectrum, where the Dirac cones lie in the unstrained lattice.”

Once the defect mode is very strong (! !/! != 4), it becomes a defect mode outside of the band and therefore in another band gap.

“(g) Once the defect mode is very strong (! !/! != 4) it becomes a defect mode outside of the band and therefore in another band gap.”

For a slightly negative defect (! !/! != -0.5), some spreading of light occurs.

“As a result, coupling with the bulk bands causes some spreading.”

In the strained system with strain c=0.0015a -1 , 55% of the power resides in the Dirac region.

“However, in the strained system (with a strain given by c=0.0015a -1 , the same as that considered in Fig. 2), 55% of the power resides in the Dirac region.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

Larger defect strengths (! !/! != -2) cause more spreading of light into the photonic lattice.

“(d) Larger defect strengths (! !/! != -2) causes more spreading of the light into the photonic lattice.”

Without strain, the power is spread quite evenly over all eigenmodes.

“Indeed, without the strain, the power is spread quite evenly over all eigenmodes, and the relatively low amount of power in the Dirac region is a result of the relatively low density-of-states there.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

In the unstrained system (q=0.0a-1), light launched into an armchair-edge waveguide is not confined to the edge and diffracts into the bulk.

“no light is confined to the edge and it diffracts into the bulk.”

Light incident on the armchair edge excites modes in the zeroth Landau level.

“Furthermore, the light that is incident on the armchair edge excites modes that are within gaps between Landau levels, and those in the zeroth level.”

• Also stated in: § Section 3: Coupled-mode theory analysis: exciting single waveguides on armchair edge excites modes entirely within the Dirac region · hedged

  “It is clear from Fig. S2 that the power projected on the eigenmodes resides in modes in between Landau levels and to some extent in the zeroth Landau level, as described in the paper.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

A mildly positive defect (! !/! != 0.5) also causes some spreading, as in the slightly negative defect case.

“(e) A mildly positive defect (! !/! != 0.5) also causes some spreading, as in (c).”

The fact that the negative and positive defects both lead to spreading proves conclusively that the gap, in between the strain-induced Landau levels, lies at the center of the band.

“Taken together, the fact that the negative and positive defects both lead to spreading proves conclusively that the gap, in between the straininduced Landau levels, lies at the center of the band.”

Upon straining the system as described in the text, the Dirac cone splits into Landau levels with energies given by E_N = ±√N, where N = 0, 1, 2, ...

“(b) The Dirac cone splits into Landau levels, with !!!= ±√!where N=0,1,2,…, upon straining the system as described in the text.”

With increasing strain, light becomes highly confined on the edge.

“Light becomes highly confined on the edge with increasing strain.”

Light incident on the armchair edge excites modes that are within gaps between Landau levels.

“Furthermore, the light that is incident on the armchair edge excites modes that are within gaps between Landau levels, and those in the zeroth level.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The plane-wave expansion method applied to the Schrödinger equation yields the guided mode and the propagation constant of a single waveguide.

“This yields both the guided mode of the waveguide (in the present work the waveguides are designed to be singlemoded), as well as the propagation constant of the mode (which is the eigenvalue of the operator defined on the left-hand side of Eq. ( S2)).”

• Also stated in: § Section 2: Calculating guided eigenmodes of femtosecond-written waveguides · plain

  “In Fig. S1 we show the structure of an eigenmode of a single waveguide, as calculated using the plane-wave expansion method.”

In a properly strained honeycomb photonic lattice, when a single waveguide on the armchair edge is excited, the light is confined.

“The light is confined as a result of the gaps between the Landau levels, as described in the text and as shown in Fig. 3.”

In the presence of the pseudomagnetic field, light escapes from the defect waveguide.

“here -in the presence of the pseudomanetic field -the light escapes from the defect waveguide.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The band structure of the propagation constant is similar to that of electrons in graphene.

“This band structure of the propagation constant is similar to that of electrons in graphene.”

The strained honeycomb photonic lattice exhibits pseudomagnetic effects as a result of the inhomogeneous strain applied to it, which make it aperiodic.

“It exhibits pseudomagnetic effects as a result of the inhomogeneous strain applied to it, which make it aperiodic.”

The introduction of Landau levels provides a new mechanism, only achievable in aperiodic systems, to achieve very high degeneracies.

“The introduction of Landau levels provides a new mechanism, only achievable in aperiodic systems, to achieve very high degeneracies (as opposed to the van Hove mechanism typical of photonic crystals, and associated with band edges 32 ).”

• cites [32] Lodahl, P. et al. Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals. Nature 430, 654–657 (2004). sourceNot checked — no full text of the source

Equation (1) is mathematically equivalent to the Schrodinger equation for graphene in the tight-binding limit, where propagation distance z replaces time t.

“Equation ( 1) is mathematically equivalent to the Schrodinger equation for graphene in the tight-binding limit, where the propagation distance, z replaces time, t.”

• AssumptionThe authors state this as an assumption: In Eq. (1), the waveguides are sufficiently distant from one another that only nearest-neighbor coupling needs to be incorporated.
• AssumptionThe authors state this as an assumption: Each of the waveguides supports only a single mode.

The input waveguide resides on the armchair edge of the honeycomb lattice, which has no localized states associated with it.

“Importantly, this input waveguide resides on the armchair edge of the honeycomb lattice, which has no localized states associated with it 31 .”

• cites [31] Kohmoto, M. & Hasegawa, Y. Zero modes and edge states of the honeycomb lattice. Phys. Rev. B 76, 205402 (2007). sourceNot checked — no full text of the source

The expressions for the pseudomagnetic field in graphene and a honeycomb photonic lattice are identical, if units are chosen such that h/e=1.

“Note that the expressions for the pseudomagnetic field in graphene and a honeycomb photonic lattice are identical, if we choose units such that .”

This work is the first realization of a pseudomagnetic field in optics.

“To our knowledge, this is the first realization of a pseudomagnetic field in optics.”

If the waveguides in the honeycomb photonic lattice are transversely displaced in the specified way, then a constant magnetic field is achieved.

“In this vein, if the waveguides in our honeycomb photonic lattice are transversely displaced from their original positions in the following way, then a constant magnetic field is achieved:”

Engineering a certain inhomogeneous strain corresponding to a constant magnetic field gives rise to Landau levels separated by gaps.

“In particular, engineering a certain inhomogeneous strain corresponding to a constant magnetic field gives rise to Landau levels separated by gaps 10 .”

• cites [10] Bahat-Treidel, O., Peleg, O. & Segev, M. Symmetry breaking in honeycomb photonic lattices. Opt. Lett. 33, 2251–2253 (2008). sourceNot checked — no full text of the source

The transverse wavefunction of the light that emerges from the lattice is equivalent to a finite-time evolution of an electron wavepacket in graphene.

“The transverse wavefunction of the light that emerges from the lattice is therefore equivalent to a finite-time evolution of an electron wavepacket in graphene.”

Exciting the armchair edge for large values of strain entirely confines light to being in the Dirac region of the spectrum.

“Thus, exciting the armchair edge for large values of the strain entirely confines light to being in the Dirac region of the spectrum.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The unstrained lattice has no edge states associated with it.

“The unstrained lattice consists of only armchair edges, and therefore has no edge states associated with it whatsoever.”

Taken together, the results situate the excitation of the defect-free case (! !/! != 0) within the Landau level gaps.

“These results, taken together, situate the excitation of the defect-free case (! !/! != 0) within the Landau level gaps.”

The lack of light confinement on the armchair edge in the unstrained system is due to the armchair edge having no edge states.

“Due to the lack of edge states on the armchair edge, no light is confined to the edge and it diffracts into the bulk.”

• AssumptionRests on an assumption: The armchair edge has no edge states.

Since the unstrained honeycomb lattice does not have edge states on the armchair edge, light starting on that edge immediately spreads away from it into the bulk.

“Since the unstrained honeycomb lattice does not have edge states on the armchair edge, light starting on that edge immediately spreads away from it into the bulk.”

The spreading for a slightly negative defect is caused by coupling with the bulk bands.

“As a result, coupling with the bulk bands causes some spreading.”

The greater spreading for a stronger positive defect is due to strong coupling to the bulk bands.

“(f) A stronger positive defect (! !/! != 2) causes greater spreading (as in (d)) as a result of strong coupling to the bulk bands.”

The features of modes in the zeroth Landau level resemble those of the edge states of the zig-zag or bearded edges.

“Notably, the modes in the zeroth Landau level result from distortions of the armchair edge, and their features resemble those of the edge states of the zig-zag or bearded edges.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The confinement in the properly strained lattice is due to the gaps between the Landau levels.

“The light is confined as a result of the gaps between the Landau levels, as described in the text and as shown in Fig. 3.”

The relatively low amount of power in the Dirac region in the unstrained system is a result of the relatively low density-of-states there.

“Indeed, without the strain, the power is spread quite evenly over all eigenmodes, and the relatively low amount of power in the Dirac region is a result of the relatively low density-of-states there.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The localization at very strong defect (! !/! != 4) results from the mode being in another band gap.

“(g) Once the defect mode is very strong (! !/! != 4) it becomes a defect mode outside of the band and therefore in another band gap.As a result, it becomes localized again.”

Strained structures exhibit greater field-enhancement and slow-light effects than periodic structures via the high density-of-states at Landau levels.

“Figure: greater field-enhancement and slow-light effects than periodic structures via the high density-of-states at Landau levels.”

Modes in the zeroth Landau level result from distortions of the armchair edge.

“Notably, the modes in the zeroth Landau level result from distortions of the armchair edge, and their features resemble those of the edge states of the zig-zag or bearded edges.”

• AssumptionThe authors state this as an assumption: In coupled-mode theory, the continuum wavefunction ψ(x,y,z) can be written as the sum of waveguide modes centered on each waveguide.

The observed confinement is strong evidence of the existence of the band gaps that lie in between the Landau levels.

“This is strong evidence of the existence of the band gaps that lie in between the Landau levels.”

Since the honeycomb edge states are necessarily degenerate, a wavepacket made up of a superposition of such states will not spread.

“Since the honeycomb edge states are necessarily degenerate 35 , i.e., they all possess the same propagation constant, a wavepacket (beam) made up of a superposition of such states will not spread.”

• cites [35] Akhmerov, A. R. & Beenakker, C. W. J. Boundary conditions for Dirac fermions on a terminated honeycomb lattice. Phys. Rev. B 77, 085423 (2008). sourceNot checked — no full text of the source