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La fase geométrica en mecánica cuántica: teoría y experimentos J. C. Loredo O. Ortíz A. Ballón S. Chávez M. J. Bustamante A.P. Galarreta C. Sihuincha F. De Zela Departamento de Ciencias – Sección Física – Grupo de Óptica Cuántica Pontificia Universidad Católica del Perú Coloquios de la Sección Física, 19 de mayo 2011

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Fases

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Fase relativa entre dos ondas

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Interferometría

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En la mecánica cuántica se usa la función de onda, una función que toma valores complejos La función de onda encierra toda la información que se tiene sobre un sistema físico. debe ser físicamente equivalente a Por ejemplo, la cantidad: es invariante bajo el cambio

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Ondas que pasan a través de una doble rendija

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Experimento de la doble rendija

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B Efecto Aharonov-Bohm Φ(B)

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Berrys phase In 1984 Berry analyzed an adiabatic, unitary and cyclic evolution of a quantum system that obeys Schrödingers equation. He discovered that the quantum state acquires a geometric phase besides the (expected) dynamical phase. Later on, it was shown that geometric phases appear even in non-adiabatic, non-unitary and non-cyclic situations of a general kind. Geometric phases are an interesting subject for many fields: differential geometry and topology, classical dynamics, relativity, quantum dynamics, classical and quantum optics, quantum computation, etc.

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La función de onda que representa a un sistema físico evoluciona en el tiempo. La evolución la rige la ecuación de Schrödinger: Los estados cuánticos se pueden describir matemáticamente mediante vectores o kets:

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Spin evolves following adiabatically a slowly changing magnetic field B (T) = B (0) ψ S (T) = e iφ ψ S (0) Fast variation Slow variation H(t) = -μ.B(t)

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Parallel transport A vector v i parallel- transported along a closed path generally doesnt return to its original value: v i v f The difference v f – v i depends on the underlying space

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One important field of interest is quantum computation Classical computation requires bits: Quantum computation requires qubits: Qubits in a register (memory) must be submitted to (unitary) transformations. Having a universal set of elementary transformations one may perform any computational task.

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A key task in quantum computation: to cop with decoherence. A possible solution: all-geometric quantum computation, which is robust against decoherence. Different scenarios: NMR, Josephson junctions, quantum dots, ion traps, polarized states. Using NMR Jones et al. (Nature 403, 869 (2000)) removed the dynamical phase, leaving a geometric phase alone. They used a setup that adiabatically changed the spin-state

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Motivation Goals for (nonadiabatic) geometric quantum computation: To find paths, along which the dynamical phase vanishes. To implement robust one- and two-qubit phase-gates. D. Leibfried et al. Nature 422, 412 (2003): two-qubit phase gates (using beryllium ions) in which the geometric phase is proportional to the dynamical phase (Zhu and Wang, PRL 91, (2003))

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Pancharatnams phase 1956: Pancharatnam addressed polarization states and defined for them the notion of being in phase, thereby anticipating Berrys phase. Consider two states that depend on some parameters ξ and write The relative phase is naturally defined as φ 12 and calculated as

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Several states joined by a closed path

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For several states joined by a closed path we define the total phase as This is an observable quantity, invariant under gauge transformations.

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For a continuous, closed path we define

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Generalization to nonadiabatic evolution and open trajectories For ξ=ξ(s) we define the total phase as Defining one can prove that Φ geom is gauge invariant: and also under parameter changes: s u Φ geom depends only on the curve traced back by |ψ(s) ψ(s)| in ray space, e.g., on the Poincaré sphere.

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The total phase Φ tot can be measured using, e.g., polarization states by Polarimetry (robust) Interferometry (supposed to be unstable) Measuring the total phase

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As well known, polarization states can be represented through Jones vectors or through Stokes vectors. The action of intensity-preserving optical elements is represented by matrices belonging to SU(2). s lies on the Poincaré sphere and represents |P.

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Different trajectories described by s on the Poincaré sphere

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Measuring the total phase Initial state |i evolves under U(β,γ,δ) SU(2) to |f = U |i. U can be realized using three retarders: two λ/4 and one λ/2. For the ZYZ parametrization of U so that we have

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Polarimetry (Wagh and Rakhecha) Take initial state |+ z Subject it to /2-rotation: |+ z (|+ z - i | z )/ 2 Phase-shift the state by applying exp(- i φ σ z /2): Now apply U(βγδ) and then the inverse transformation V -1. Then project back to |+ z and measure the intensity:

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+

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Pancharatnams phase can be extracted from I by measuring I min and I max : U tot = V U can be implemented with 5 retarders: 1 Half-wave and 4 Quarter-wave plates: The angles ξ, η, ζ refer to a YZY form of U SU(2):

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Experimental arrangement for polarimetry

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Polarimetric measurements of cos 2 (Φ P ) as a function of one Euler angle. J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, Phys. Rev. A 80, (2009)

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Interferometry Consider two non-orthogonal polarization states Introduce a variable phase-shift φ on one state and measure the interference pattern Maximal intensity is obtained for This is, by definition, Pancharatnams relative phase.

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Interferometry (Wagh and Rakhecha) By interferometry we could in principle measure the relative shift of the pattern with respect to a reference pattern corresponding to U = 1: One can use a Mach-Zehnder, a Sagnac, or a Michelson array. But the array should be robust against mechanical and thermal disturbances, to allow capture of reference pattern I 0

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Reference pattern

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U(αβγ)

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Lets go back to polarimetry

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Take a laser beam and let it pass through two polarizers, so that half the beam is vertically and the other half is horizontally polarized. Let both halves go through the same optical components of an interferometer. We have two patterns now: The relative shift between I V and I H is 2δ, i.e., twice Pancharatnams phase.

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Mach-Zehnder like, robust interferometric arrangement

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Interferograms and corresponding filtered patterns to measure relative phase shift

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Interferometric measurement of visibility v (θ1,θ2,θ3).

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Interferometric measurements of cos 2 Φ P as a function of two Euler angles.

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Using the same techniques we can measure the geometric phase. Generally, we can choose the gauge so as to make zero either the total or else the dynamical phase.

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Being able to make Φ g = Φ tot for any curve we avoid the restriction of using only those special curves C for which the dynamical phase automatically cancels. C = loop 1 + loop 2 Φ dyn (C) = 0 See, e.g., Y. Ota et al., Phys. Rev. A 80, (2009)

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Consider the state obtained by applying U(γβχ) SU(2): The phase-shift necessary to make the dynamical phase zero is We can thus measure Φ g along non-geodesic paths

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Interferometry U { Φ dyn

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Polarimetry Array with seven retarders (another version uses five)

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Φ g by interferometry

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Φ g by polarimetry Φ g by interferometry

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Φ g by polarimetry Φ g by interferometry

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Φ g by polarimetry Φ g by interferometry

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Conclusions Polarimetric and interferometric methods could be applied in an all-optical setup that allowed us to generate geometric phases with great versatility. Our interferometric arrangement is robust against mechanical and thermal disturbances. We expect to upgrade our approach to deal with single-photons, in order to implement one and two- qubit gates, testing them against decoherence.

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