Geometric phases along quantum trajectories

Ludmila Viotti1,2, Ana Laura Gramajo2, Paula I. Villar3, Fernando C. Lombardo3, and Rosario Fazio2,4

1Departamento de Física Juan José Giambiagi, FCEyN UBA Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
2The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
3Departamento de Fí sica Juan José Giambiagi, FCEyN UBA and IFIBA CONICET-UBA, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
4Dipartimento di Fisica, Università di Napoli "Federico II'', Monte S. Angelo, I-80126 Napoli, Italy

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Abstract

A monitored quantum system undergoing a cyclic evolution of the parameters governing its Hamiltonian accumulates a geometric phase that depends on the quantum trajectory followed by the system on its evolution. The phase value will be determined both by the unitary dynamics and by the interaction of the system with the environment. Consequently, the geometric phase will acquire a stochastic character due to the occurrence of random quantum jumps. Here we study the distribution function of geometric phases in monitored quantum systems and discuss when/if different quantities, proposed to measure geometric phases in open quantum systems, are representative of the distribution. We also consider a monitored echo protocol and discuss in which cases the distribution of the interference pattern extracted in the experiment is linked to the geometric phase. Furthermore, we unveil, for the single trajectory exhibiting no quantum jumps, a topological transition in the phase acquired after a cycle and show how this critical behavior can be observed in an echo protocol. For the same parameters, the density matrix does not show any singularity. We illustrate all our main results by considering a paradigmatic case, a spin-1/2 immersed in time-varying a magnetic field in presence of an external environment. The major outcomes of our analysis are however quite general and do not depend, in their qualitative features, on the choice of the model studied.

The geometric phase (GP) accumulated by an isolated quantum system holds significant importance across various domains, ranging from the mathematical foundations of quantum mechanics to the explanation of physical phenomena and even practical applications. While several generalizations have been proposed to incorporate geometric phases in open quantum systems, where the state is described by a density operator undergoing non-unitary evolution, there exists an additional level of description for such systems.

This alternative description of open quantum systems is accessed, for example, when the state of the system is continuously monitored. In this case, the wave function becomes a stochastic variable that follows a different quantum trajectory on each realization of the evolution. The randomness in a given trajectory introduces stochastic characteristics in the GPs. Understanding the fluctuations induced in GPs through indirect monitoring remains largely unexplored. The goal of the present work is therefore to describe the properties of accumulated GP along quantum trajectories.

Our work presents a thorough study of the GPs distribution arising within this framework for the paradigmatic model of a spin-½ particle in a magnetic field, and whether, how, and when it is related to the corresponding distribution in the interference fringes in a spin-echo experiment. We also show that depending on the coupling to the external environment, the monitored quantum system will show a topological transition in the phase accumulated and we argue that this transition is visible in echo dynamics.

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[83] Note, a. Real implementations of the protocol require two extra steps. Preparing and measuring the system in the equal-superposition state |ψ(0)⟩ might be quite involved. Instead, the $\sigma_z$-goundstate |0⟩ is prepared and a pulse driving it to |ψ(0)⟩ is applied afterwards. Then, the protocol usually ends with a last spin rotation taking the final state back to the $\sigma_z$ basis, where the actually compute probability is that of being in |0⟩.

[84] Note, b. Different measurement schemes and physical situations can be described using symmetries of the Lindbland equation as a way of generating different unraveling. Given the invariance of Eq. (1) under some joint transformation $W_m\rightarrow W'_m$, $H \rightarrow H'$, the Lindblad evolution of the averaged density matrix $\rho(t)$ is consequently unchanged, while the different possible trajectories may undergo nontrivial changes, therefore describing different scenarios. Such a procedure can be followed to go from direct photodetection to discrete homodyne detection schemes, in which a beam-splitter mixes the output field with an additional coherent field.

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