La fase geométrica en mecánica cuántica:teoría y experimentos

J. C. Loredo O. Ortíz

A. Ballón S. Chávez

M. J. BustamanteA. 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

)cos(kx

)cos( kx

ikxe

( ) ( )i kx ikxiee e

Fases

Fase relativa entre dos ondas

Interferometría

ikxe

( )i kxe

En la mecánica cuántica se usa la “función de onda”, una función que toma valores complejos

( , )x t

La función de onda encierra toda la información que se tiene sobre un sistema físico.

( , )x t debe ser físicamente equivalente a ( , )i te x

Por ejemplo, la cantidad:*( , ) ( , )x t x t

es invariante bajo el cambio

* *

( , ) ( , )

( , ) ( , )

i

i

x t e x t

x t e x t

Ondas que pasan a través de una doble rendija

Experimento de la doble rendija

1*22

*1

2

2

2

1

2

21 I

12 ie

1 cos( )I

1

2

B 1 cos ( )BI

Efecto Aharonov-Bohm

1

2

Φ(B)

Berry’s phase

In 1984 Berry analyzed an adiabatic, unitary and cyclic evolution of a quantum system that obeys Schrödinger’s 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.

La función de onda que representa a un sistema físico evoluciona en el tiempo.

La evolución la rige la ecuación de Schrödinger:

opi Ht

Los estados cuánticos se pueden describir matemáticamente mediante “vectores” o “kets”:

Spin evolves following adiabatically a slowly changing magnetic field

B(T) = B(0)

ψS (T) = eiφ ψS (0)

Fast variation

Slow variation

H(t) = -μ.B(t)

Parallel transport

iv

fv

A vector vi parallel-transported along a closed path generally doesn’t return to its original value: vi ≠ vf

The difference vf – vi depends on the underlying space

One important field of interest is quantum computation

0 and 1

0 1

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.

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

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, 187902 (2003))

Pancharatnam’s phase

1956: Pancharatnam addressed polarization states and defined for them the notion of being „in phase“, thereby anticipating Berry‘s phase.

121 2 1 2( ) ( ) ( ) ( )ie

Consider two states that depend on some parameters ξ and write

The relative phase is naturally defined as φ12 and calculated as

12 1 2Im log ( ) ( )

1( )

2( )

3( )

4( )

Several states joined by a closed path

For several states joined by a closed path we define the total phase as

12 23 34 41

1 2 2 3 3 4 4 1Im log ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

This is an observable quantity, invariant under gauge transformations.

( )( ) ( )

( ) ( ) ( ) ( )

ie

For a continuous, closed path we define

( ) ( )

exp( ) ( )

i

( ) ( )i

, 11

NN

s ss

d

( ) ( )d i d

Generalization to nonadiabatic evolution and open trajectories

For ξ=ξ(s) we define the total phase as

1 2arg ( ) ( )tot s s

Defining

2

1

1 2arg ( ) Im ( ) (( ))geo to dyn

s

m t

s

s ss d s one can prove that Φgeom is gauge invariant:

( )( ) ( )i sA ses A

and also under parameter changes: s →u

Φgeom depends only on the curve traced back by |ψ(s)ψ(s)| in “ray space”, e.g., on the Poincaré sphere.

The total phase Φtot can be measured using, e.g., polarization states by

Polarimetry

(robust)

Interferometry (supposed to be unstable)

Measuring the total phase

As well known, polarization states can be represented through Jones vectors or through Stokes vectors.

cos

sinx

iy

EP

Ee

1ˆ

2P P I s

ˆ (cos 2 ,sin 2 cos ,sin 2 sin )s

The action of intensity-preserving optical elements is represented by matrices belonging to SU(2).

s lies on the Poincaré sphere and represents |P.

Different trajectories described by s on the Poincaré sphere

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

( , , ) exp exp exp2 2z y zU i i i

cosii f e

so that arg , mod( )tot i f

we have

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 σz/2):

/ 21

2 2

ii

z z z z z z

ei ie V

Now apply U(βγδ) and then the inverse transformation V-1. Then project back to |+z and measure the intensity:

2 2 2 2cos cos sin cosI

z zi

/ 2 / 2i i

z ze ei

/ 2 / 2i i

z zUe iUe

z

(2)SU

Phase-shift

State-splitting

2 2 221 2 coscos sin cos ( )I V UV

Analyzer

+

Pancharatnam’s phase can be extracted from I by measuring Imin and Imax :

2 min

max min

cos1

I

I I

Utot = 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):

( , , ) exp exp exp2 2 2y z yU i i i

3 5 2 9 2( ) 7

4 4 42 2 2 4 42 2Q Q Q H Q

Experimental arrangement for polarimetry

Polarimetric measurements of cos2(ΦP) as a function of one Euler angle. J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, Phys. Rev. A 80, 012113 (2009)

Interferometry

Consider two non-orthogonal polarization states and A B

Introduce a variable phase-shift φ on one state and measure the interference pattern

22 2 cos argiI e A B A B A B

Maximal intensity is obtained forarg A B

This is, by definition, Pancharatnam’s relative phase.

Interferometry (Wagh and Rakhecha)

2

11 cos cos

2 z z

i UeI

By interferometry we could in principle measure the relative shift of the pattern

with respect to a reference pattern corresponding to U = 1:

2

0

11 cos

2 z z

iI e

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 I0

Reference pattern

U(αβγ)

z zi

/ 2 / 2i i

z ze ei

/ 2 / 2i i

z ze ieU U

z

(2)SU

Phase-shift

Flipper (state-splitting)

2 2 221 2 coscos sin cos ( )I V UV

Analyzer

Let’s go back to polarimetry

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:

11 cos cos

2VI

11 cos cos

2HI

The relative shift between IV and IH is 2δ, i.e., twice Pancharatnam’s phase.

Mach-Zehnder like, robust interferometric arrangement

Interferograms and corresponding filtered patterns to measure relative phase shift

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

Interferometric measurements of cos2ΦP as a function of two Euler angles.

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.

2

1

1 2arg ( Im ( ) ( )) ( )tot

s

d n

s

g y A s A A s dA ss

( )( ) ( ) (

can always be achieved by choosing a

.

0 "gauge":

(In such a case ) ( ) "horizon, This is called a tal lift"

)

. .

dyn

g to

i

t

sA s

C

A s e s

C

A

tot

g dy

2

n

Alternatively, we can make

by choosing the phase

0,

of A(s ) .

Horizontal lift ( ): ( ) / 0 g totA s dA s ds

Being able to make Φg = Φtot 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

Φdyn(C) = 0

See, e.g., Y. Ota et al., Phys. Rev. A 80, 052311 (2009)

1

1ˆ( ) ( ) ( ) ( )

2ˆ( ) (cos ( )sin ,sin (

arg cos tan cos tan2 2

)sin ,cos )

g

s A s A s I n s

n s s s

( )( ) ( , )( ),z

tiA t e U t

0

cos( )

2( )t tc

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 Φg along non-geodesic paths

Interferometry

U{ Φdyn

Polarimetry

Array with seven retarders (another version uses five)

Φg by interferometry

Φg by polarimetry

Φg by interferometry

Φg by polarimetry

Φg by interferometry

Φg by polarimetry

Φg by interferometry

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.