Dynamicallymodulated structures from reciprocitybreaking to synthetic dimensions Shanhui

Dynamically-modulated structures: from reciprocitybreaking to synthetic dimensions Shanhui Fan Department of Electrical Engineering, and Ginzton Laboratory Stanford University Stanford, CA 94305 Email: shanhui@stanford. edu http: //www. stanford. edu/~shanhui/

Outline • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic dimension

Outline • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic dimension

Maxwell’s equations In this talk, we consider the physics of a time-dependent dielectric function, i. e.

Experimental mechanism for modulating dielectric constant Injecting carriers into Si allows modulation of refractive index as a function of time Q. Xu et al, Nature 435, 325 (2005)

Typical strength of index modulation Modulation Phase Dielectric constant of Si: Modulation strength Modulation Frequency Typical optical frequency

Outline • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic dimension

Optical isolator Output signal Device Parasitic reflection Output signal Device Isolator Parasitic reflection An optical isolator suppresses the reflection of arbitrary parasitic reflected wave.

Towards large-scale on-chip information network Large-scale communication network Large-scale on-chip network

Achieving optical isolation on chip How does one achieve optical isolation on a standard optoelectronic platform? Silicon Photonics Platform The discussion here applies to most other standard optoelectronic material platform as well.

Lorentz Reciprocity Theorem The theorem applies to any electromagnetic system that is: • linear, • time-independent, • has a symmetric permittivity and permeability tensor, including medium that has gain or loss. H. Lorentz (1896); H. A. Haus, Waves and Fields in Optoelectronics (1984) It applies independent of structural complexity, e. g. Metal (Al, Cu, …) Dielectric (Si, Si. O 2, Ga. As, Ge, …. ) If the optical properties are entirely described by

Reciprocal system has a symmetric scattering matrix a 1 a 2 a 3 b 2 b 3 Device b 1 Input-output is defined by the scattering matrix (S-matrix) Reciprocity theorem implies that e. g. Reciprocity relates two pathways that are related by time-reversal. Reciprocity therefore is closely related to time-reversal symmetry.

Conventional optical isolators 5 cm Images from www. ofr. com Use magneto-optical materials

Magneto-optical effect is non-reciprocal e. g. YIG M z Dielectric tensor • Asymmetric • Non-reciprocal • Hermitian • Energy conserving

Break time-reversal symmetry and reciprocity in dynamically modulated system It is relatively straightforward to imagine modulated system in which:

Using dynamic modulation to replace magneto-optical effects Underlying physical mechanism: Photonic transition by dynamic refractive index modulation. Modulated Si waveguide

Silicon Waveguides Air Silicon

Modulations Air Silicon z

Modulation creates interband transition Air Silicon Modulated Area, Length Lc z

Time-Reversal Symmetry Breaking Air Silicon z The modulation does not phase match any transition in the backward direction Modulation is not invariant for or

Non-reciprocal frequency and modal conversion

Device Considerations • Converted light can be removed using modal filters. • Modulation frequency far smaller than signal bandwidth. • Broad-band operation results from transition between parallel bands. Linear with respect to the signal light. Z. Yu and S. Fan, Nature Photonics, vol. 3, pp. 91 -94 (2009).

Implementation with electro-optic modulator Optical Input

Modulation through carrier injection V 1 V 2 V 3 V 4 pn junction array modulation period

Contrast Ratio Forward/Backward Electrically-driven non-reciprocity H. Lira, Z. Yu, S. Fan, and M. Lipson, Physical Review Letters 109, 033901 (2012).

Outline • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic space

Static magnetic field breaks time-reversal symmetry for electrons B B Can we create an effective magnetic field for photons?

Si Metal electrode: applying a time-dependent voltage gauge potential for photons K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012).

Magnetic field for electrons in quantum mechanics • Electron couples to the vector gauge potential

Gauge potential results in a direction-dependent phase Propagation phase 1 2 Propagation phase

Direct transition Uniform modulation along z-direction Air Silicon z

Direct transition independent of the modulation phase

Modulation phase provides a gauge transformation of the photon wavefunction Gauge potential that couples to the photon

Downward and upper-ward transition acquires a phase difference An interferometer that detects such a gauge degree of freedom is an Aharonov-Bohm (AB) interferometer.

A direction-dependent phase shifter in real space Phase accumulated

A Photonic Aharonov-Bohm Interferometer as an Optical Isolator silicon air K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012)

AB Interferometer from Photon-Phonon Interaction He-Ne Laser (633 nm) Local oscillator (50 MHz) AOM (Acoustic. Optic Modulator) 13 d. B forward-backward contrast E. Li, B. Eggleton, K. Fang and S. Fan, Nature Communications 5, 3225 (2014).

AB interferometer on a silicon platform L. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, Nature Photonics 8, 701 (2014).

Electron on a lattice Electron hopping on a tight-binding lattice Single unit cell Magnetic field manifests in terms of a non-reciprocal round-trip phase as an electron hops along the edge of a unit cell.

Photons on a dynamic lattice • Two sub-lattices of resonators • Coupling constant between nearest neighbor resonators dynamically modulated. K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012). See also M. Hafezi et al, Nature Physics 7, 907 (2011); R. O. Umucallar and I. Carusotto, Physical Review A 84, 043804 (2011).

Uniform effective magnetic field

Quantum Hall Effect 2 D electron gas at low temperature exhibits one-way edge mode.

Dynamically induced one-way edge mode resonator lattice One-way propagation source K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012). By-pass defect

Photons on a dynamic lattice • Two sub-lattices of resonators • Coupling constant between nearest neighbor resonators dynamically modulated. K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012). See also M. Hafezi et al, Nature Physics 7, 907 (2011); R. O. Umucallar and I. Carusotto, Physical Review A 84, 043804 (2011).

The effect of a constant gauge potential For electrons In general, a constant gauge potential shifts the wavevector

A constant gauge potential shifts the constant frequency contour A n 1 A

Gauge field induced negative refraction A n 1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

Gauge field induced total internal reflection A n 1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

A single-interface four-port circulator A n 1 • Both regions have zero effect B-field. • A B-field sheet at the interface. K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

Outline • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic dimension

Explore higher dimensional physics with lower dimensional physical structures For example, can we simulate three dimensional topological physics with two-dimensional physical structure? Previous work on synthetic dimension: Tsomokos et al, PRA 82, 052311 (2010); Boada et al, PRL 108, 133001 (2012); Jukic and Buljan, PRA 87, 013814 (2013).

A single ring resonator In the absence of group velocity dispersion (GVD) in the ring waveguide, the ring supports a set of resonances with equally spaced resonant frequencies. where T is the round trip time

Electro-optic phase modulator: • Modulation generates side bands • Phase of modulation imprinted in the side band • Upward and downward conversions have different phases associated with it

On resonance case Modulation frequency Mode spacing Modulation resonantly coupled different modes together

Near resonance case Modulation frequency Mode spacing Tight-binding model under a constant electric field

Bloch oscillation in the frequency domain k L. Yuan and S. Fan, Optica 3, 1014 (2016). See also similar concept in laser systems: S. Longhi, Optics Letters 30, 786 (2005).

One-way frequency translation Periodic switching of modulation frequency above and below the mode spacing leads to one-way frequency conversion L. Yuan and S. Fan, Optica 3, 1014 (2016).

One Dimensional Physics in a Zero-Dimensional Structure Modulation frequency Mode spacing Modulation resonantly coupled different modes together

Exploring 2 D physics in 1 D system Array of coupled resonator A two-dimensional space, having a real-space axis and a frequency axis with applied gauge field in the synthetic space.

A boundary on the frequency axis Group velocity dispersion (GVD) in the waveguide introduce a boundary in the frequency axis Around the zero-GVD point, modes are equally spaced, modulators induce onresonance coupling between the modes Away from the zero-GVD point, modes are no longer equally spaced and hence modulation no longer induce on-resonance coupling

Tight-binding lattice in synthetic space Array of coupled resonator A two-dimensional space, having a real-space axis and a frequency axis with applied gauge field in the synthetic space.

One-way edge mode in the synthetic space Input L. Yuan, Y. Shi and S. Fan, Optics Letters 41, 741 (2016) See also, Ozawa et al, PRA 93, 043827 (2016).

Three Dimensional Physics: Weyl Hamiltonian E E E Weyl point

Topological Robustness of Weyl Points • Weyl point describes a magnetic monopole in the momentum space. • Weyl point is topologically robust, in the sense that it can not be destroyed by any perturbation that preserves translational symmetry form a complete basis for 2 x 2 Hermitian matrices

Weyl point: an active area of contemporary physics Theoretical Proposal with Sr 2 Ti. O 4 X. Wang et al, PRB 83, 205101 (2011) S. –Y. Xu et al, Science 349, 613– 617 (2015) B. Q. Lv et al, PRX 5, 031013 (2015) L. Lu et al. Nature Photon. 7, 294 (2013) M. Xiao et al. Nature Phys. 11, 920 (2015) L. Lu et al. Science 349, 622 (2015)

Exploring 3 D physics in 2 D system

2 D Dirac point in honeycomb lattice The “spin” refers to the A and B sites B site A site Castro Neto et. al. Rev. Mod. Phys. 81, 109 (2009).

3 D Weyl point M. Xiao, W. -J. Chen, W. -Y. He & C. T. Chan. Nature Phys. 11, 920 (2015).

Implementation in 2 D lattice of ring resonators Q. Lin, X. Meng, L. Yuan and S. Fan. Nature Communications (2016, in press)

Inversion symmetry breaking

Time-reversal symmetry breaking

One-way surface states Q. Lin, X. Meng, L. Yuan and S. Fan. Nature Communications (2016, in press)

Summary • Brief introduction to dynamically modulated photonic structures • Non-reciprocity • Gauge field for photons • Synthetic dimension There is rich fundamental physics and device applications associated with dynamically modulated structure