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Control of the Geometric Phase with Time-Dependent Fields Control of the geometric phase with time-dependent fields Xiaosong Zhu1;2, Peixiang Lu2,∗ and Manfred Lein1y 1Leibniz University Hannover, Institute of Theoretical Physics, D-30167 Hannover, Germany 2Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China (Dated: December 3, 2020) If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide, although the time evolution is adiabatic despite the degeneracy. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy-crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states. When a quantum-mechanical system undergoes a ing a degeneracy. closed circuit in parameter space, an adiabatically evolv- First, we investigate a Ne+ ion driven by periodic laser ing state may acquire a geometric phase, known as the fields, comparing two types of driving protocols: either Berry phase [1], in addition to its dynamical phase, where enclosing a degeneracy or passing through it. We con- the latter is essentially the time-integrated energy. The sider laser fields because their time-dependent electric- geometric phase is an important fundamental concept field vector follows a closed curve in the plane of polariza- and it appears in many fields including molecular physics tion, i.e., a circuit in parameter space. A circularly polar- [2{4], solid-state physics [5{8], ultracold atoms [9, 10], op- ized (CP) single-color field creates a circuit encircling the tics [11{15], and quantum computation [16{18]. Follow- degeneracy. Using additional colors allows for more com- ing Berry, the geometric phase can be evaluated from the plex paths. We employ bicircular fields, i.e., combining instantaneous eigenstates by integration of the Berry con- two circular fields of different colors, which have recently nection, without reference to time evolution. However, been used for fundamental studies and novel applications since the adiabatic approximation is not necessarily valid [23{32]. With equal strengths of both colors, the field [19, 20], one may resort to a nonadiabatic generalization crosses zero several times per cycle. Therefore, such a suggested by Aharonov and Anandan (AA) [21, 22]. To 1:1 field steers the system through a degeneracy if the calculate the geometric phase, both Berry and AA con- energy eigenstates are degenerate at zero field strength. sider the total phase acquired during the cyclic evolu- We demonstrate both analytically and numerically, by tion and they subtract the dynamical phase, but they introducing a numerical measure termed adiabaticity in- apply different definitions of the latter: Berry uses the dicator, that the evolution can be adiabatic despite the time-integrated instantaneous eigenenergy whereas AA degeneracy. However, we find a breakdown of the equiva- use the time-integrated energy expectation value. It is lence between AA phase and Berry phase in the adiabatic generally believed that the two geometric phases agree limit. We also show that the passage through the degen- in the adiabatic limit for unitary evolution [21, 22], but eracy causes a controlled π shift of the Berry phase. In we show a counter-example below. the end, we revisit AA's spin-1/2 example, finding that, due to a symmetry of time evolution, the AA phase for The vast majority of existing studies exclude circuits suitably chosen states is the same as the adiabatic Berry leading through a degeneracy (crossing or touching of phase, even in the nonadiabatic regime. the energy levels). It is worth asking what effects and applications will emerge in such a case. One example, Floquet theory and geometric phase.|A systematic a spin-1/2 particle driven by a slowly varying periodic way [33{35] to obtain the AA phase [36] follows from magnetic field that passes through zero, was suggested Floquet theory, the tool to describe periodically driven by AA [21]. Without rigorous calculation, they argued quantum matter [37{43]. According to the Floquet the- that the system remains in an energy eigenstate (i.e., it orem, the time-dependent Schr¨odingerequation (TDSE) behaves adiabatically in our terminology) and that the i@=@tjΨ(t)i = H(t)jΨ(t)i with periodic Hamiltonian AA phase has a well-defined limit. Our analysis confirms H(t) = H(t + T ) has solutions jΨ(t)i = e−iEF tjP (t)i both statements, which is important for quantum-state with quasienergies EF and time-periodic states jP (t + manipulation, but as we demonstrate, the second state- T )i = jP (t)i satisfying the Floquet equation (H − ment does not necessarily hold in other examples involv- i@=@t)jP (t)i = EF jP (t)i [44]. These solutions return 2 exactly to the initial state, up to an overall phase de- (a) termined by the quasienergy: The total phase accumula- tion in one period is −EF T . Subtracting the dynamical phase −HT¯ based on the time-averaged energy expec- ¯ R T tation value H = 0 hΨ(t)jH(t)jΨ(t)idt=T , we obtain the AA phase γA = (H¯ −EF )T . For practical calcula- tions, the periodic state jP (t)i is Fourier expanded as P −in!t jP (t)i = n e jFni with ! = 2π=T so that the Flo- quet states are determined by a matrix eigenvalue equa- + tion [37, 38]. The AA phase takes the form γA = 2πn¯ FIG. 1. (a) Energy surfaces of the 2p states for Ne in an elec- P tric field and paths of the electric field in the parameter space withn ¯ = nhFnjFnin, see also [34, 45]. We interpret this result using the photon-channel perspective [24, 46]: for CP (red) and 1:1 bicircular (yellow) fields. (b) Illustration An eigenstate of the field-free Hamiltonian is a Floquet of the possible transitions in the bicircular field. state with only n=0 populated; if the system remains in a single Floquet state while turning on the field, popu- [58, 59]. These are the instants when the system crosses lations are redistributed by photon absorption/emission _ the degeneracy. The prefactor al(τ)hmjli varies slowly and hFnjFni is the weight of the contribution reached by compared to the exponential. It is therefore treated absorption of the photon energy n!. Hence the AA phase as constant. The integral is dominated by the re- equals the mean absorbed photon energy, in units of !, gions around the saddle points, so we expand f(τ) = multiplied by 2π. 0 f(t0)+f (t0)(τ −t0)+··· . Close to t0, we also exploit An alternative view on geometric phases is provided N Eml(t) / [!(t−t0)] with positive integer N. (In the ex- by low-frequency Floquet theory, where the quasienergy amples below, we have N =2 in Ne+ due to the quadratic ¯ (1) 2 (2) is expanded as EF = E + !EF + ! EF + ::: [45, 47{ Stark shift and N = 1 for the spin-1/2 particle.) Hence, ¯ R T the first nonzero derivative of f is f (N+1)(t ) / !N . 52] with E = 0 E(t)dt=T . Here, E(t) is an eigenvalue 0 of H(t). Using the dynamical phase −E¯T as in [1], we Keeping the two lowest contributing orders, we have N N+1 introduce a nonadiabatic Berry phase γB = (E¯ −EF )T f(τ) = f(t0) + α! (τ −t0) with a constant α. Sub- N=(N+1) that includes nonadiabaticity via EF (!). The common stituting s = (τ −t0) ! , the contribution of one adiabatic Berry phase is then equal to the low-frequency saddle point to the integral in Eq. (1) is (0) (1) limit γB = lim!!0 γB = −2πEF . It is observable by 1 measuring the quasienergy, for example from positions of Z N+1 N 1 _ if(t0) iαs − N+1 N+1 peaks in electron spectra from light-induced ionization al(t0)hmjlie e ! ds / ! ; (2) [43, 53]. −∞ _ Adiabaticity of time evolution.|We term an evolution where hmjli / ! is used. Thus, at ! ! 0, ∆am ! 0 and adiabatic if the state jΨ(t)i always remains close to an the evolution converges to adiabaticity despite the pas- instantaneous eigenstate jj(t)i of the Hamiltonian H(t), sage through the degeneracy and irrespective of the value 2 i.e., the time-dependent fidelity Fj(t) = jhj(t)jΨ(t)ij of N. However, our analysis indicates that adiabaticity satisfies mint Fj(t) ≈ 1. To quantify the adiabaticity of is approached slower for larger N. any numerical wavefunction, we introduce the adiabatic- Ne+ ion in a light field.|This set of examples is mo- ity indicator F = maxj mint Fj(t), where the maximum tivated by the experimental feasibility of preparing rare- over j selects the closest eigenstate. Values F ≈ 1 indi- gas ions in one of their degenerate ring-current states cate that jΨ(t)i is always close to this eigenstate. [60, 61] and by the possibility to lift this degeneracy by Previous discussions [54{57] of criteria for adiabatic an external field.
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