Semiclassical Treatment of Interference Phenomena in Bosonic Quantum Many-Body Systems

A major goal of the semiclassical theory is to understand and treat a quantum system in the regime of large quantum numbers using information from its classical limit. In this framework, quantum phenomena are addressed through a path integral perspective, i.e. the weighted interference of phases accumulated along classical trajectories. While this formalism has been successful in explaining interference phenomena, for instance, in mesoscopic single-particle quantum systems, the recent extension of the semiclassical theory allows now also the treatment of interacting quantum many-body systems. This thesis aims to contribute to these ongoing efforts to apply semiclassical techniques to quantum many-body systems. It is focused on bosonic quantum many-body systems, where the thermodynamic limit of a large number of particles represents an alternative version of a semiclassical limit. Within the derivation of the corresponding semiclassical theory, mean-field equations, i.e. effective nonlinear wave equations describing the matter wave, re-emerge as Hamilton's equations of motion of an abstract Hamilton function, defining the above thermodynamic limit as the classical limit. Within this thesis, two different topics are studied. The first one deals with the so-called “Out-of-Time-Order Correlator” (OTOC), the expectation value of the squared commutator of two local operators at different times. The OTOC provides a direct probe for the presence of chaos in the classical limit (if any) of a quantum many-body system. This is based on that for short times the OTOC directly relates to the stability of classical solutions upon changes in their initial conditions. One of the defining properties of chaos is that this stability displays an exponential growth, with the rate called the classical Lyapunov exponent. Immediately, this implies an exponential growth of the OTOC for short times, from which the Lyapunov exponent can be extracted. For later times, one observes a saturation of the OTOC as a consequence of unitary quantum time evolution. The Ehrenfest time, the time scale which marks the onset of quantum interference, separates these different dynamical behaviors. In this thesis a thorough understanding of the concept of chaos in the classical limit of quantum many-body systems is provided, and with that the underlying interference mechanisms involved in the early exponential growth and the later saturation of OTOCs are identified. It is found that the pre-Ehrenfest time exponential growth is given through the interference of multiple contributions, all of them essentially following only a single solution of mean-field equations. Conversely the post-Ehrenfest time behavior stems from the contributions of fundamentally different mean-field solutions, which display correlated dynamics only for a limited amount of time, of the order of Ehrenfest time. The second topic covered in this thesis deals with the coherent transport of cold bosonic atoms through an Aharonov-Bohm ring structure. This setup consists of two semi-infinite wave guides attached on the opposite sides of a wave guide ring which is pierced by a synthetic gauge field. Within the ring, the atoms are further subject to both a weak disorder potential and particle-particle interaction. In the non-interacting case, the disorder-averaged transmission probability as a function of the encircled flux displays the well-known Al'tshuler-Aronov-Spivak (AAS) oscillations, i.e. oscillations with a frequency twice as large as Aharonov-Bohm oscillations. This thesis provides further insight into the influence of a weak particle-particle interaction on AAS oscillations, for which numerical results predict an interaction-based inversion of peaks. Here, using semiclassical techniques to solve the corresponding mean-field problem, this inversion is successfully reproduced, and a first indicator of the relevant mechanism leading to the inversion is found through contributions of self-averaging scattering path constellations

Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators

Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τE in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τE=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τE, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems

Al'tshuler-Aronov-Spivak oscillations of bosonic matter-wave beams in the presence of interaction

We study theoretically the propagation of a guided atom laser across an Aharonov-Bohm ring which is exposed to a synthetic gauge field. The presence of disorder within the ring gives rise to Al'tshuler-Aronov-Spivak oscillations, seen in the disorder average of the transmission as a function of the effective gauge flux that is contained within the ring. Those oscillations are induced by coherent backscattering and represent a manifestation of weak localization. Through analytical and numerical calculations that are based on the mean-field Gross-Pitaevskii approximation for the propagating Bose-Einstein condensate, we show that the presence of a very weak atom-atom interaction within the ring leads to an inversion of the Al'tshuler-Aronov-Spivak oscillations, in a very similar manner as for the coherent backscattering of Bose-Einstein condensates within two-dimensional disorder potentials. Numerical simulations based on the truncated Wigner method reveal that this signature of weak antilocalization becomes washed out if the interaction strength is increased

Altshuler-Aronov-Spivak oscillations of coherent bosonic matter-wave beams in the presence of interaction

We theoretically study the propagation of a guided atom laser across an Aharonov-Bohm ring which is exposed to a synthetic gauge field. The presence of disorder within the ring gives rise to Al’tshuler-Aronov-Spivak oscillations [1], seen in the disorder-averaged of the transmission as a function of the effective gauge flux that is contained within the ring. Those oscillations are induced by coherent backscattering and represent a manifestation of weak localisation. Through analytical and numerical calculations that are based on the mean-field Gross-Pitaevskii approximation for the propagating Bose-Einstein condensate, we show that the presence of a weak atom-atom interaction within the ring leads to an inversion of the Al’tshuler-Aronov-Spivak oscillations, in a very similar manner as for the coherent backscattering of Bose-Einstein condensates within two-dimensional disorder potentials [2]. Numerical simulations based on the Truncated Wigner method reveal that this signature of weak antilocalisation becomes washed out if the interaction strength is increased, which is in qualitative agreement with the findings of the diagrammatic study undertaken in Ref. [3]. [1] B. L. Al’tshuler, A. G. Aronov and B. Z. Spivak, JETP Lett. 33, 94 (1981). [2] M. Hartung, T. Wellens, C. A. Müller, K. Richter, and P. Schlagheck, Phys. Rev. Lett. 101, 020603 (2008). [3] T. Geiger, A. Buchleitner and T. Wellens, New J. Phys. 15, 115015 (2013)

Altshuler-Aronov-Spivak oscillations of coherent bosonic matter-wave beams in the presence of interaction

We theoretically study the propagation of a guided atom laser across an Aharonov-Bohm ring which is exposed to a synthetic gauge field. The presence of disorder within the ring gives rise to Al’tshuler-Aronov-Spivak oscillations [1], seen in the disorder-averaged of the transmission as a function of the effective gauge flux that is contained within the ring. Those oscillations are induced by coherent backscattering and represent a manifestation of weak localisation. Through analytical and numerical calculations that are based on the mean-field Gross-Pitaevskii approximation for the propagating Bose-Einstein condensate, we show that the presence of a weak atom-atom interaction within the ring leads to an inversion of the Al’tshuler-Aronov-Spivak oscillations, in a very similar manner as for the coherent backscattering of Bose-Einstein condensates within two-dimensional disorder potentials [2]. Numerical simulations based on the Truncated Wigner method reveal that this signature of weak antilocalisation becomes washed out if the interaction strength is increased, which is in qualitative agreement with the findings of the diagrammatic study undertaken in Ref. [3]. [1] B. L. Al’tshuler, A. G. Aronov and B. Z. Spivak, JETP Lett. 33, 94 (1981). [2] M. Hartung, T. Wellens, C. A. Müller, K. Richter, and P. Schlagheck, Phys. Rev. Lett. 101, 020603 (2008). [3] T. Geiger, A. Buchleitner and T. Wellens, New J. Phys. 15, 115015 (2013)