Observation of Geometric Phases for Mixed States using NMR Interferometry

Abstract

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well understood, noiseless, pure-state case.PACS numbers: 03.65. Bz, 42.50.Dv, A quantum system can retain a memory of its motion when it undergoes a cyclic evolution, e.g its quantum state may acquire a geometric phase factor in addition to the dynamical one [1,2]. For pure quantum states this effect is well understood and it has been demonstrated in a wide variety of physical systems [3]. Its potential application to perform the fault-tolerant quantum computation has been the subject of more recent investigations [4,5,6]. In contrast, relatively little is known about geometric phases, and more generally, about quantum holonomies of mixed or entangled quantum states. Here we report an NMR experiment which constitutes the first experimental study of quantum holonomies for mixed quantum states. We observed and measured the geometric phase of a mixed state of a spin half nuclei. Our experimental data are in a good agreement with the recent theoretical predictions by Sjöqvist et al [7].The geometric phase of pure states is an intriguing property of quantum systems undergoing parallel cyclic evolutions. The parallel transport of a particular vector |Ψ implies no change in phase when |Ψ(t) evolves into |Ψ(t + dt) , for some infinitesimal change of the parameter t. Although locally there is no phase change, the system may acquire a non-trivial phase after completing a closed loop parameterized by t. The origin of this phase can be traced to an underlying curvature of the parameter space, depending only on the geometry of the path and is resilient to certain dynamical perturbations of the evolution, e.g. it is independent of the speed of the evolution. Therefore, it is a potential method for performing intrinsically fault-tolerant quantum logic gates, a very desirable feature for practical implementations of quantum computation. However, quantum systems that interact with other systems, be it components in a quantum computer or otherwise, become entangled and cannot be described by a state vector |Ψ . In this context the notion of parallel transport and geometric phases must be extended to mixed quantum states.Mathematically, Uhlmann was the first to address the issue of a mixed state holonomy [8]. In his approach a system in a mixed state is embedded, as a subsystem, in a larger system that is in a pure state. Given a mixed state of the subsystem there are infinitely man...

Use of non-adiabatic geometric phase for quantum computing by NMR

“…The indirect spin-spin coupling (the J-coupling) between the two qubits is 210 Hz. Starting from equilibrium, the |00 pseudopure state was prepared by spatial averaging method using the pulse sequence [19],…”

“…A two level subsystem was made to undergo a cyclic evolution in the Hilbert space by applying a time-dependent magnetic field, while geometric phase was observed in the modulation of the coherence of the other two-level subsystem [18]. Recently, non-adiabatic geometric phase has also been observed for mixed states by NMR using evolution in tilted Hamiltonian frame [19]. In the work reported here, we have adopted a scheme similar to that of Pines et.al.…”

Experimental measurement of mixed state geometric phase by quantum interferometry using NMR

“…The geometric phase can be directly measured from the shift of the interference pattern. Mixed state geometric phase has been experimentally demonstrated by Du et al [12] using NMR and by Ericsson et al [13] using single photon interferometry. Du et al have experimentally demonstrated the mixed state geometric phase by measuring the relative phase change of an auxiliary spin.…”

Kinematic approach to off-diagonal geometric phases of nondegenerate and degenerate mixed states

“…[11,16] by identifying ǫ as the length of the Bloch vector. For N ≥ 3 a necessary condition for a nodal point is Ω/2 = (2k + 1)π, k integer, and η = 0.…”

“…[13,14,15] and experimental test of the phase concept in Ref. [11] has been carried out using nuclear magnetic * Electronic address: eriks@kvac.uu.se † Electronic address: sfilipp@ati.ac.at ‡ Electronic address: phyohch@nus.edu.sg resonance [16].…”