Adiabatic condition for nonlinear systems

Topological invariant and anomalous edge modes of strongly nonlinear systems

Despite the extensive studies of topological states, their characterization in strongly nonlinear classical systems has been lacking. In this work, we identify the proper definition of Berry phase for nonlinear bulk waves and characterize topological phases in one-dimensional (1D) generalized nonlinear Schrödinger equations in the strongly nonlinear regime, where the general nonlinearities are beyond Kerr-like interactions. Without utilizing linear analysis, we develop an analytic strategy to demonstrate the quantization of nonlinear Berry phase due to reflection symmetry. Mode amplitude itself plays a key role in nonlinear modes and controls topological phase transitions. We then show bulk-boundary correspondence by identifying the associated nonlinear topological edge modes. Interestingly, anomalous topological modes decay away from lattice boundaries to plateaus governed by fixed points of nonlinearities. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.

Topological phases are challenging to identify in systems with general, strong nonlinearities. Here, the authors establish the analytic methodology that defines the topological invariant of nonlinear normal modes. Strongly nonlinear topological boundary modes are guaranteed by the nontrivial topological index.

Chern Number Governs Soliton Motion in Nonlinear Thouless Pumps

Nonlinear Thouless pumps for bosons exhibit quantized pumping via soliton motion, despite the lack of a meaningful notion of filled bands. However, the theoretical underpinning of this quantization, as well as its relationship to the Chern number, has thus far been lacking. Here we show that, for low-power solitons, transport is dictated by the Chern number of the band from which the soliton bifurcates. We do this by expanding the discrete nonlinear Schrödinger equation (equivalently, the Gross-Pitaevskii equation) in the basis of Wannier states, showing that a soliton's position is dictated by that of the Wannier state throughout the pump cycle. Furthermore, we describe soliton pumping in two dimensions.

Quantized nonlinear Thouless pumping

The topological protection of wave transport, originally observed in the context of the quantum Hall effect in two-dimensional electron gases¹, has been shown to apply broadly to a range of physical platforms, including photonics^2-5, ultracold atoms in optical lattices^6-8 and others^9-12. That said, the behaviour of such systems can be very different from the electronic case, particularly when interparticle interactions or nonlinearity play a major role^13-22. A Thouless pump²³ is a one-dimensional model that captures the topological quantization of transport in the quantum Hall effect using the notion of dimensional reduction: an adiabatically, time-varying potential mathematically maps onto a momentum coordinate in a conceptual second dimension^24-34. Importantly, quantization assumes uniformly filled electron bands below a Fermi energy, or an equivalent occupation for non-equilibrium bosonic systems. Here we theoretically propose and experimentally demonstrate quantized nonlinear Thouless pumping of photons with a band that is decidedly not uniformly occupied. In our system, nonlinearity acts to quantize transport via soliton formation and spontaneous symmetry-breaking bifurcations. Quantization follows from the fact that the instantaneous soliton solutions centred upon a given unit cell are identical after each pump cycle, up to translation invariance; this is an entirely different mechanism from traditional Thouless pumping. This result shows that nonlinearity and interparticle interactions can induce quantized transport and topological behaviour without a linear counterpart.

Cascaded frequency conversion under nonlinear stimulated Raman adiabatic passage

A comprehensive study of two simultaneous three wave mixing processes, a second harmonic generation followed by difference frequency generation, under nonlinear stimulated Raman adiabatic passage (STIRAP) is presented. An input pump is up-converted to its second harmonic, which then gets down-converted to a signal and idler pair with frequencies lying very close to the input pump in such a manner that complete conversion from the pump to the signal and idler takes place without exciting the second harmonic under counterintuitive adiabatic passage. This process involves nonlinear STIRAP with a nonlinear dark state similar to atomic population transfers, and we bring an analogy from atomic systems to our nonlinear dynamics to linearize the problem and analytically obtain the adiabaticity condition required for complete conversion. We also show that the nonlinear STIRAP mechanism results in a large bandwidth of about 380 nm with almost complete conversion of the pump to the signal and idler.

Quantum Hamiltonian daemons: Unitary analogs of combustion engines

Hamiltonian daemons have recently been defined classically as small, closed Hamiltonian systems which can exhibit secular energy transfer from high-frequency to low-frequency degrees of freedom (steady downconversion), analogous to the steady transfer of energy in a combustion engine from the high terahertz frequencies of molecular excitations to the low kilohertz frequencies of piston motion [L. Gilz, E. P. Thesing, and J. R. Anglin, Phys. Rev. E 94, 042127 (2016)2470-004510.1103/PhysRevE.94.042127]. Classical daemons achieve downconversion within a small, closed system by exploiting nonlinear resonances; the adiabatic theorem permits their operation but imposes nontrivial limitations on their efficiency. Here we investigate a simple example of a quantum mechanical daemon. In the correspondence regime it obeys similar efficiency limits to its classical counterparts, but in the strongly quantum mechanical regime the daemon operates in an entirely different manner. It maintains an engine-like behavior in a distinctly quantum mechanical form: a weight is lifted at a steady average speed through a long sequence of quantum jumps in momentum, at each of which a quantum of fuel is consumed. The quantum daemon can cease downconversion at any time through nonadiabatic Landau-Zener transitions, and continuing operation of the quantum daemon is associated with steadily growing entanglement between fast and slow degrees of freedom.

Nonlinear Stimulated Raman Exact Passage by Resonance-Locked Inverse Engineering

We derive an exact and robust stimulated Raman process for nonlinear quantum systems driven by pulsed external fields. The external fields are designed with closed-form expressions from the inverse engineering of a given efficient and stable dynamics. This technique allows one to induce a controlled population inversion which surpasses the usual nonlinear stimulated Raman adiabatic passage efficiency.

Dynamic stabilization of a coupled ultracold atom-molecule system

We numerically demonstrate the dynamic stabilization of a strongly interacting many-body bosonic system which can be realized by coupled ultracold atom-molecule gases. The system is initialized to an unstable equilibrium state corresponding to a saddle point in the classical phase space, where subsequent free evolution gives rise to atom-molecule conversion. To control and stabilize the system, periodic modulation is applied that suddenly shifts the relative phase between the atomic and the molecular modes and limits their further interconversion. The stability diagram for the range of modulation amplitudes and periods that stabilize the dynamics is given. The validity of the phase diagram obtained from the time-average calculation is discussed by using the orbit tracking method, and the difference in contrast with the maximum absolute deviation analysis is shown as well. A brief quantum analysis shows that quantum fluctuations can put serious limitations on the applicability of the mean-field results.

Experimental implementation of assisted quantum adiabatic passage in a single spin

Quantum adiabatic passages can be greatly accelerated by a suitable control field, called a counter-diabatic field, which varies during the scan through resonance. Here, we implement this technique on the electron spin of a single nitrogen-vacancy center in diamond. We demonstrate two versions of this scheme. The first follows closely the procedure originally proposed by Demirplak and Rice [J. Phys. Chem. A 107, 9937 (2003)]. In the second scheme, we use a control field whose amplitude is constant but whose phase varies with time. This version, which we call the rapid-scan approach, allows an even faster passage through resonance and therefore makes it applicable also for systems with shorter decoherence times.

Discrete chaotic states of a Bose-Einstein condensate

We examine spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For a low laser barrier, chaotic regions of the parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high-barrier case, bounded perturbed solutions that describe a set of discrete chaotic states are constructed for discrete barrier heights and magic numbers of condensed atoms. Chaotic density profiles are exhibited numerically for the lowest quantum number, and analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.

Effect of nonlinearity on adiabatic evolution of light

We investigate the effect of nonlinearity in a system described by an adiabatically evolving Hamiltonian. Experiments are conducted in a three-core waveguide structure that is adiabatically varying with distance, in analogy to the stimulated Raman adiabatic passage process in atomic physics. In the linear regime, the system exhibits an adiabatic power transfer between two waveguides which are not directly coupled, with negligible power recorded in the intermediate coupling waveguide. In the presence of nonlinearity the adiabatic light passage is found to critically depend on the excitation power. We show how this effect is related to the destruction of the dark state formed in this configuration.

Quantum noise in the collective abstraction reaction A + B2-->AB + B

We demonstrate theoretically that the collective abstraction reaction A + B2-->AB + B can be realized efficiently with degenerate bosonic or fermionic matter waves. We show that this is dominated by quantum fluctuations, which are critical in triggering its initial stages with the appearance of macroscopic nonclassical correlations of the atomic and molecular fields as a result. This study opens up a promising new regime of quantum-degenerate matter-wave chemistry.

Experimental study of the validity of quantitative conditions in the quantum adiabatic theorem

The quantum adiabatic theorem plays an important role in quantum mechanics. However, counter-examples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.

Atom-molecule dark state: the exact quantum solution

Developments in ultracold atomic physics have enabled the generation of an atom-molecule dark state in dilute quantum gases. Previous studies of this dark state were all carried out in the mean-field regime. Here we present an exact quantum many-body wave function for the atom-molecule dark state for an ideal system (in the absence of losses and two-body collisions). Using this exact quantum wave function, we are able to validate the mean-field solution to be a good approximation when the particle number N is large. For small N, unique quantum features will become important.

Integrability, stability, and adiabaticity in nonlinear stimulated Raman adiabatic passage

We study the dynamics of a two-color photoassociation of atoms into diatomic molecules via nonlinear stimulated Raman adiabatic passage process. The system has a famous counterpart in (linear) quantum mechanics, and has been discussed recently in the context of generalizing the quantum adiabatic theorem to nonlinear systems. Here we use another approach to study adiabaticity and stability in the system: we apply methods of classical Hamiltonian dynamics. We find nonlinear dynamical instabilities, cases of complete integrability, and improved conditions of adiabaticity.

Coherent atom-trimer conversion in a repulsive Bose-Einstein condensate

We show that the use of a generalized atom-molecule dark state permits the enhanced coherent creation of triatomic molecules in a repulsive atomic Bose-Einstein condensate, with further enhancement being possible in the case of heteronuclear trimers via the constructive interference between two chemical reaction channels.

Third-harmonic generation in quasi-phase-matched chi(2) media with missing second harmonic

What we believe to be a conceptually novel scheme for third-harmonic generation in engineered quasi-phase-matched chi((2)) optical structures is proposed in which the fundamental-frequency field is directly converted into the third-harmonic field without the intermediate generation of the second-harmonic field. This counterintuitive scheme, which exploits the concept of adiabatic passage and the existence of a nonlinear dark state, bears a close connection to the "stimulated Raman adiabatic passage" technique of population transfer in atomic physics.