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Geometric phases in superconducting nanocircuits
GEOMETRIC
PHASES
IN S U P E R C O N D U C T I N G
NANOCIRCUITS
Rosario Fazio
NEST- INFM & Scuola Normale Superiore, piazza dei Cavalieri 7, 1-56126 Pisa, Italy
When a quantum mechanical system undergoes an adiabatic cyclic evolution, it acquires a geometrical phase factor in addition to the dynamical one. This remarkable effect, found by Berry, has been demonstrated in a variety of systems [ 1]. Since recent years all the attention in detecting geometric phases have been confined to microscopic systems. More recently, however, due the advances in nanofabrication the possibility to observe geometric phases in mesoscopic systems has become an interesting possibility. Quantum dynamics has been already observed in superconducting nanocircuits [2], therefore these systems have been indicated as promising candidates to measure geometric interference. The simplest configuration in which to observe the Berry phase consists of a superconducting electron box formed by an asymmetric SQUID, pierced by a magnetic flux ~ and with an applied gate voltage Vx [3]. See also the course by D. Est~ve in this volume. The Hamiltonian is H -- Ech(n - nx) 2 - E j ( O ) cos(0 - or)
(1)
where E J1
-
E J2 tan
tanol -- E j1 -+- E J2
7c -~0
'
and E j ( ~ ) is the effective Josephson coupling of the loop and ~0 = h / 2 e is the (superconducting) quantum of flux. The phase difference across the junction 0 and the number of Cooper pairs n are canonically conjugate variables [0, n] = i. Both external parameters of the Hamiltonian can be controlled. The offset 589
590
R. Fazio
charge 2enx can be tuned by changing Vx and the coupling E j ( ~ ) depends on 9 . The device operates in the regime where the Josephson couplings EJI(2~ of the junctions are much smaller than the charging energy Ech. At temperatures much lower than Ech, if nx varies around the value 1/2, only two charge states, n = 0, 1, are important. The effective Hamiltonian is obtained by projecting Eq.(1) on the computational Hilbert space, and reads H8 = - ( 1 / 2 ) B 9 ~, where we have defined the fictitious field B ---- ( E j ( ~ ) cos or, - E j ( ~ ) sin or, E c h ( 1 - 2 n x ) ). Charging couples the system to Bz whereas the Josephson term determines the projection in the xy plane. By changing Vx and 9 the qubit Hamiltonian HB describes a cylindroid in the parameter space {B }. By means of a more complicated circuit it is also possible to obtain a degenerate subspace and hence to observe non-Abelian holonomies [4]. Besides the interest in itself, the detection of geometric phases in superconducting nanocircuits may have an impact in the area of solid state quantum computation. Quantum computers are usually analysed in terms of qubits and gates. In most of the implementations proposed quantum gates are realized by varying in time in a controlled way the Hamiltonian of the individual qubits as well as their mutual coupling. An alternative design [5, 6] makes use of quantum geometric phases, obtained by adiabatically varying the qubits' Hamiltonian in such a way to describe a suitably chosen closed loop in its parameter space. It is believed that geoemtric quantum computation may be more robust to certain type of errors. It has already been proposed that geometric computation can be implemented with Josephson nanocircuits. As a final remark it is worth to mention that the detection of geometric phases is intimately related to coherent pumping of Cooper pairs [7]. Additional informations concerning work done on geometric phases in superconducting nanocircuits can be found in the reference list of the papers quoted in this abstract. An important topic related to the role of the environment on geometric interferometry is discussed by Yu. Makhlin in this Volume.
References [1] Geometricphases in physics, Shapere A. and Wilczek E, Eds. World Scientific, (Singapore, 1989). [2] Yu. Makhlin, G. Sch6n, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). [3] G. Falci, R. Fazio, G.M. Palma, J. Siewert, and V. Vedral, Nature 407, 355 (2000). [4] L. Faoro, J. Seiwert, and R. Fazio, Phys. Rev. Lett. 90, 028301 (2003). [5] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [6] J. Jones, V. Vedral, A. K. Ekert and G. Castagnoli, Nature 403, 869 (2000). [7] J. Pekola et al., Phys. Rev. B 60, R9931 (1999).
PHASES
IN S U P E R C O N D U C T I N G
NANOCIRCUITS
Rosario Fazio
NEST- INFM & Scuola Normale Superiore, piazza dei Cavalieri 7, 1-56126 Pisa, Italy
When a quantum mechanical system undergoes an adiabatic cyclic evolution, it acquires a geometrical phase factor in addition to the dynamical one. This remarkable effect, found by Berry, has been demonstrated in a variety of systems [ 1]. Since recent years all the attention in detecting geometric phases have been confined to microscopic systems. More recently, however, due the advances in nanofabrication the possibility to observe geometric phases in mesoscopic systems has become an interesting possibility. Quantum dynamics has been already observed in superconducting nanocircuits [2], therefore these systems have been indicated as promising candidates to measure geometric interference. The simplest configuration in which to observe the Berry phase consists of a superconducting electron box formed by an asymmetric SQUID, pierced by a magnetic flux ~ and with an applied gate voltage Vx [3]. See also the course by D. Est~ve in this volume. The Hamiltonian is H -- Ech(n - nx) 2 - E j ( O ) cos(0 - or)
(1)
where E J1
-
E J2 tan
tanol -- E j1 -+- E J2
7c -~0
'
and E j ( ~ ) is the effective Josephson coupling of the loop and ~0 = h / 2 e is the (superconducting) quantum of flux. The phase difference across the junction 0 and the number of Cooper pairs n are canonically conjugate variables [0, n] = i. Both external parameters of the Hamiltonian can be controlled. The offset 589
590
R. Fazio
charge 2enx can be tuned by changing Vx and the coupling E j ( ~ ) depends on 9 . The device operates in the regime where the Josephson couplings EJI(2~ of the junctions are much smaller than the charging energy Ech. At temperatures much lower than Ech, if nx varies around the value 1/2, only two charge states, n = 0, 1, are important. The effective Hamiltonian is obtained by projecting Eq.(1) on the computational Hilbert space, and reads H8 = - ( 1 / 2 ) B 9 ~, where we have defined the fictitious field B ---- ( E j ( ~ ) cos or, - E j ( ~ ) sin or, E c h ( 1 - 2 n x ) ). Charging couples the system to Bz whereas the Josephson term determines the projection in the xy plane. By changing Vx and 9 the qubit Hamiltonian HB describes a cylindroid in the parameter space {B }. By means of a more complicated circuit it is also possible to obtain a degenerate subspace and hence to observe non-Abelian holonomies [4]. Besides the interest in itself, the detection of geometric phases in superconducting nanocircuits may have an impact in the area of solid state quantum computation. Quantum computers are usually analysed in terms of qubits and gates. In most of the implementations proposed quantum gates are realized by varying in time in a controlled way the Hamiltonian of the individual qubits as well as their mutual coupling. An alternative design [5, 6] makes use of quantum geometric phases, obtained by adiabatically varying the qubits' Hamiltonian in such a way to describe a suitably chosen closed loop in its parameter space. It is believed that geoemtric quantum computation may be more robust to certain type of errors. It has already been proposed that geometric computation can be implemented with Josephson nanocircuits. As a final remark it is worth to mention that the detection of geometric phases is intimately related to coherent pumping of Cooper pairs [7]. Additional informations concerning work done on geometric phases in superconducting nanocircuits can be found in the reference list of the papers quoted in this abstract. An important topic related to the role of the environment on geometric interferometry is discussed by Yu. Makhlin in this Volume.
References [1] Geometricphases in physics, Shapere A. and Wilczek E, Eds. World Scientific, (Singapore, 1989). [2] Yu. Makhlin, G. Sch6n, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). [3] G. Falci, R. Fazio, G.M. Palma, J. Siewert, and V. Vedral, Nature 407, 355 (2000). [4] L. Faoro, J. Seiwert, and R. Fazio, Phys. Rev. Lett. 90, 028301 (2003). [5] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [6] J. Jones, V. Vedral, A. K. Ekert and G. Castagnoli, Nature 403, 869 (2000). [7] J. Pekola et al., Phys. Rev. B 60, R9931 (1999).