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 Lein1† 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 Schrödingerequation (TDSE) behaves adiabatically in our terminology) and that the i∂/∂t|Ψ(t)i = H(t)|Ψ(t)i with periodic Hamiltonian AA phase has a well-defined limit. Our analysis confirms H(t) = H(t + T ) has solutions |Ψ(t)i = e−iEF t|P (t)i both statements, which is important for quantum-state with quasienergies EF and time-periodic states |P (t + manipulation, but as we demonstrate, the second state- T )i = |P (t)i satisfying the Floquet equation (H − ment does not necessarily hold in other examples involv- i∂/∂t)|P (t)i = EF |P (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)|H(t)|Ψ(t)idt/T , we obtain the AA phase γA = (H¯ −EF )T . For practical calculations, the periodic state |P (t)i is Fourier expanded as P −inωt |P (t)i = n e |Fni with ω = 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 ¯ = nhFn|Fnin, 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)hm|li varies slowly and hFn|Fni is the weight of the contribution reached by compared to the exponential. It is therefore treated absorption of the photon energy nω. Hence the AA phase as constant. The integral is dominated by the re- equals the mean absorbed photon energy, in units of ω, 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) ∝ [ω(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 + ωEF + ω 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 ) ∝ ωN . 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) + αω (τ −t0) with a constant α. Sub- N/(N+1) that includes nonadiabaticity via EF (ω). The common stituting s = (τ −t0) ω , 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 = limω→0 γB = −2πEF . It is observable by ∞ 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)hm|lie e ω ds ∝ ω , (2) [43, 53]. −∞ ˙ Adiabaticity of time evolution.—We term an evolution where hm|li ∝ ω is used. Thus, at ω → 0, ∆am → 0 and adiabatic if the state |Ψ(t)i always remains close to an the evolution converges to adiabaticity despite the pas- instantaneous eigenstate |j(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) = |hj(t)|Ψ(t)i| 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 |Ψ(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. We treat a single active electron in an motion have excluded passages through degeneracies. effective potential [62] and we restrict the analysis to the Here, we provide a mathematical discussion of this 2p± states with magnetic quantum numbers ±1 and the case. The evolving state is expanded as |Ψ(t)i = 2s state [63–65]. The bicircular electric field composed R t P −i El(τ)dτ l al(t)e 0 |l(t)i with eigenstates |li chosen such of frequencies ω and 2ω is [31] ˙ that hl|li = 0 and energy eigenvalues El. The evolution E(t) = E1[cos(ωt)eˆx + sin(ωt)eˆy] is adiabatic if the change of the coefficient am, + E2[cos(2ωt)eˆx − sin(2ωt)eˆy]. (3) Z t X ˙ if(τ) ∆am(t) = − al(τ)hm|lie dτ, (1) A single-color CP field is a special case with E2 = 0. The l6=m 0 time-dependent Hamiltonian reads H(t) = H0 + r · E(t). In one optical cycle, the field follows a closed circuit in the R τ 0 0 remains small [55, 57], where f(τ) = 0 Eml(τ )dτ and parameter space given by the Ex-Ey-plane, see Fig.1(a), Eml = Em −El. The exponential oscillates rapidly except which shows also the adiabatic eigenenergies for the 2p 0 near the saddle points t0 satisfying f (t0) = Eml(t0) = 0 states as a function of the field. One of the orbitals tends 3

2 1 to spatially align along the field (labeled pk), exhibiting a (a) -1.503 (b) parabolic energy surface, whereas the other one tends to 0.9 1 align perpendicularly (labeled p⊥) with flat energy sur- 0.8 -1.504 face [63]. The degeneracy at E = 0 is reminiscent of a 0 Renner-Teller level touching, which shows no adiabatic 0.7 -1 Berry phase when encircled by an adiabatic path [66]. 0.6 P -1.505 In the Floquet calculation, we expand |Fni = j cnj|φji -2 0.5 where |φji are the field-free states (j = p−, s, p+). Field- 0 10 20 30 0 10 20 30 induced transitions obey angular-momentum conserva- tion rules. The possible photon channels from 2p and − FIG. 2. Results for a CP field with intensity 5 × 1014 W/cm2. 2s as initial states are illustrated in Fig.1(b). A CP field (a) AA phases γA and adiabaticity indicator F for p-type permits only few channels, indicated by the thick arrows. states. (b) Energy curves. |p−, 0i denotes a solution for |P (t)i We first consider a CP driving field. The numerically that resembles the p− state at the smallest considered N , calculated AA phases γA for the Floquet states are pre- while |p−, ni denotes a solution with quasienergy differing by sented in Fig.2(a) as a function of the dimensionless −nω. 2 2 2 2 2 parameter N = 2D E0 /(ω∆E), where E0 = E1 + E2 , ∆E = 0.8509 a.u. is the gap between the field-free 2p As Fig.2(b) shows, the quasienergies EF of the p states and 2s energies, and D = hφp± |x|φsi = −0.3513 a.u. is the transition dipole. The AA phases for the two 2p converge to the time-averaged instantaneous eigenenergy states have opposite signs and vary continuously from E¯ at N → ∞. A fit of the asymptotic behavior reveals ¯ 2 zero at small N to ±2π at N → ∞ [67]. It is intuitive that E −EF ∝ ω (mod ω). This means that the Berry ¯ that γA vanishes at large frequencies as there is no time phase γB = (E−EF )T vanishes in the low-frequency limit ¯ (T → 0) to accumulate phase. To understand the low- (up to integer multiples of 2π). Similarly, H−EF behaves frequency limit, we plot the adiabaticity indicator F for as ω2 (mod ω), implying that the AA phase approaches p-type states in Fig.2(a). It quantifies the similarity of zero (mod 2π). the Floquet state to the instantaneous energy eigenstates, A bicircular field permits more photon channels (Fig.1(b)) and multiphoton resonances appear. Figure3 iωt p 2 2 2 2iωt e ( ∆E + 8D E0 − ∆E) T shows results for E :E = 1:1. Three times per cycle, this |pk(t)i ∝ (e , , 1) (4) 1 2 2DE0 bicircular field (yellow curve in Fig.1(a)) passes through 2iωt T zero and thus drives the system through the degener- and |p⊥(t)i ∝ (−e , 0, 1) , where the three compo- nents refer to the basis given by the field-free states acy. As inferred from the saddle-point method, the evolution can still be adiabatic despite the degeneracy. The p−, s, p+. The adiabaticity indicator converges to 1 at low frequency (N → ∞), indicating that the evolution adiabaticity indicator F in Fig.3(a) shows sharp drops becomes adiabatic. Next, we derive the value of the at the multiphoton resonances, implying highly nonadi- AA phase assuming F = 1. From Eq. (4), we see abatic evolution. Nevertheless, at off-resonance low fre- 2 2 2 quencies, F approaches unity. For one such case, the |hφp− |pki| = |hφp+ |pki| = (1 − |hφs|pki| )/2. Hence, time-dependent fidelity is shown in Fig.3(b). It stays when the field-free p− state is turned into the state |pki by the presence of the external field (see upper panel close to unity despite noticeable oscillations. of Fig.1(b)), the mean number of absorbed quanta ω In the following, we focus on the off-resonance geomet- for the transitions to s and p+ isn ¯ = 1. Inserting into ric phases of p states. (Those for the s state are zero). γA = 2πn¯ gives γA = 2π, in agreement with the nu- We choose representative off-resonance points and plot merical finding at low frequency. In analogy, one finds the corresponding phases γA and γB, shifted by integer −2π for the other p-type state. At intermediate N , the multiples of 2π into the interval [−π, π], see Fig.3(c). system deviates from adiabaticity, resulting in nontrivial Here, an astounding anomalous behavior is found. Un- values of γA. For the 2s state, the transitions from s like the result in Fig.2(a), the AA phase does not have to p± via absorbing/emitting photons are symmetric, so an adiabatic limit, although the time evolution becomes thatn ¯ = 0 and γA = 0. Apparently, symmetry breaking adiabatic. We have further confirmed this in calculations of the photon channels for the ring-current initial states up to N = 900 (not shown). causes nonzero geometric phases. A similar conclusion Integration of the Berry connection for the instanta- −iωt T drawn for pseudorotating molecules [68] was that the nu- neous eigenstate |p⊥(t)i ∝ (−e , 0, 1) yields π (mod clear ring currents allowed by a degeneracy play a role 2π), in agreement with γB tending to π in Fig.3(c). in the nonzero geometric phase. The CP field (red curve Fitting the asymptotic behavior of the energies at ω →0 0.797 in Fig.1(a)) drives the system around the degeneracy. yields E−E¯ F ≈0.5ω and H¯−EF ∝ω . Indeed, the time- With this driving, the adiabatic limit of the AA phase averaged expectation value H¯ and the time-averaged equals the adiabatic Berry phase obtained by integrating eigenvalue E¯ converge to the same limit EF , but with the Berry connection. different speeds. When computing the geometric phases 4

1.0 (a) 1 (b) (a) 2.0 (b) 1.0

0.8 1 0.8 1.5 0.99 0.6 0.5 0.6 1.0 0.4 0 0.4 0.98 -0.5 0.5 0.2 0.2 -1 0.0 0.97 0.0 0.0 0 10 20 30 0 0.5 1 0 10 20

1.0 (c) (d)

phase FIG. 4. (a) Illustration of the spin-1/2 state during one ro- 0.5 tation of the magnetic ﬁeld in the limit ω → 0. Thick arrows indicate the spin direction. The colors represent the dynam- 0.0 ical phase. See also the supplementary movie. (b) AA phase γA and Berry phase γB for the superpositions; adiabaticity -0.5 indicator F for Floquet states and superpositions.

-1.0 0 10 20 30 adiabatic Berry phase of Nπ. Spin-1/2 particle in a magnetic field.—Since AA and FIG. 3. Results for a 1:1 bicircular field with total inten- Berry phases may differ in the adiabatic limit for a cir- sity 5 × 1014 W/cm2. (a) Adiabaticity indicator for p-type cuit that crosses a degeneracy, it is important to re- 2 states. (b) Time-dependent fidelity Fp⊥ (t) = |hp⊥(t)|Ψ(t)i| examine AA’s example [21] of a spin-1/2 particle in a for N = 19.6 (see red cross in (a)). (c) AA phase γA magnetic field with a zero crossing. AA assumed a well- and Berry phase γB of p states at off-resonance frequencies. defined adiabatic limit for their phase. The Hamiltonian Gray curves serve as guide to the eye. (d) Evolution of the is H(t) = B · σ/2 with the Pauli vector σ. The time- pk-like Floquet orbital at N = 1.7. The colors represent the position-dependent phase after subtracting the dynam- varying magnetic field reads ical phase − R t E(τ)dτ. The change of this phase over one 0 B(t) = B eˆ + B [cos(ω t)eˆ + sin(ω t)eˆ ]. (5) period gives the Berry phase γB . 0 z 0 B z B x An aligned spin following the field direction cannot suddenly reverse when the field passes zero (degeneracy of ¯ ¯ as γA = (H −EF )T and γB = (E −EF )T , the energies are the eigenstates), see Fig.4(a). Thus, the system is ex- multiplied with a diverging factor T =2π/ω → ∞. Thus, pected to return to the initial state only after two ro- ¯ γB converges to the finite value π, while |H −EF | does tations of the field [21]. We therefore choose the Flo- not decrease fast enough to let γA converge. The striking quet frequency ω =ωB/2. For both instantaneous eigen- conclusion is that AA phase and Berry phase differ in the ωt ωt T ωt ωt T states (cos 2 , sin 2 ) and (− sin 2 , cos 2 ) , integra- adiabatic limit. tion of the Berry connection yields π. A bicircular field with E1:E2 6= 1:1 creates multiphoton The adiabaticity indicator F for the Floquet states, resonances, too, but the field does not cross zero. In presented as a function of NB = B0/ωB in Fig.4(b), this case, integration of the Berry connection gives the is below 0.5, i.e., the Floquet states differ strongly from adiabatic Berry phase2 π and the AA phase varies from 0 the adiabatic states. Yet, we find that at certain dis- to (integer multiple of) ±2π when selecting off-resonance crete frequencies, the two Floquet states have the same frequencies, similar as in Fig.2(a). quasienergy and one can superpose them with equal The adiabatic Berry phase π for the 1:1 bicircular field weights such that the superpositions have F ≈ 1, i.e., can be understood as follows. When the field passes zero, their time evolution is nearly adiabatic (Fig.4(b)). For its direction suddenly reverses, while the aligned orbital the superpositions, we obtain the geometric phases γA = must evolve continuously. After one cycle with three γB = π, surprisingly without any frequency dependence, zero-crossings, the field returns to its initial direction, see Fig.4(b). This confirms AA’s conclusion. Moreover, whereas, relative to the field, the antisymmetric p or- we can explain that γA = γB arises in this special ex- bital has reversed three times and thus gained the phase ample because the two energies used for the dynamical 3π. Hence, passage through the degeneracy offers control phase by Berry and AA are equal: E¯= H¯ = 0 due to the of the geometric phase. Figure3(d) illustrates this for an antisymmetry of the spin dynamics about the degener- example (see also the supplementary movie). Here, the p acy. The accumulated dynamical phase after one period orbital gains a Berry phase of about 0.8π within one pe- is always zero, see Fig.4(a), which is significant for the riod. Because of nonadiabaticity, this value is not exactly implementation of geometric quantum gates. π. More generally, a 1:1 bicircular field with frequencies Conclusion.—Our evaluation of geometric phases for ω and (N −1)ω has N zero-crossings [28], implying an cyclic states passing through a degeneracy reveals both 5 surprises and benefits. Although the Aharonov-Anandan [12] H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and phase often agrees with the adiabatic Berry phase at low D. A. Reis, “High-harmonic generation from an atomi- frequencies, we have presented a counter-example where cally thin semiconductor,” Nat. Phys. 13, 262 (2017). they differ although the time evolution is evidently adia- [13] A. Chacón,D. Kim, W. Zhu, S. P. Kelly, A. Dauphin, E. Pisanty, A. S. Maxwell, A. Picón,M. F. Ciappina, batic. 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