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Periodically-driven quantum systems: Effective Hamiltonians and engineered gauge fields
N. Goldman, J. Dalibard
TL;DR
Periodically driven systems offer a route to nontrivial bands and synthetic gauge structures, but their dynamics includes effective, initial-phase, and micro-motion contributions. The paper develops a general N-step perturbative formalism to separate these effects and applies it to magnetic-field and spin-orbit-coupling engineering. The framework identifies driving schemes relevant to quantum simulation while highlighting convergence and launch-protocol considerations.
Problem
Periodic driving can generate nontrivial energy bands and synthetic gauge structures, motivating a framework that captures more than the effective Hamiltonian alone.
Source: §I · p. 1Method
The paper develops a general N-step perturbative formalism that separates the effective Hamiltonian, initial kick, and micro-motion.
Source: §I · p. 1, C. Conclusions · p. 14Results
The framework identifies driving schemes for synthetic magnetic fields and spin-orbit couplings, with applications including lattice and non-lattice settings.
Source: §I · p. 1, B. With optical lattices · p. 7, C. Conclusions · p. 14Takeaways & Limitations
Periodic modulation provides a route to configurations relevant for quantum simulation, including effective gauge structures and topological-order settings.
Source: §I · p. 1, C. Conclusions · p. 14
Abstract
from arXiv · showhide
Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating non-trivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micro-motion. This framework allows for the identification of driving schemes, based on general N-step modulations, which lead to configurations relevant for quantum simulation. In particular, we explore methods to generate synthetic spin-orbit couplings and magnetic fields in cold-atom setups.
I. INTRODUCTION
The paper motivates periodically driven quantum systems as a route to engineered gauge structures and develops a framework separating effective dynamics, micro-motion, and initial-phase effects. Simple traps, lattices, and two-step sequences show how driving reshapes slow motion while exposing limitations of first-order Trotter reasoning.
I. INTRODUCTION
- The framework targets synthetic gauge structures by tailoring periodic modulations so effective Hamiltonians reproduce useful static Hamiltonians.The motivation includes effective magnetic fields, spin-orbit couplings, and configurations inaccessible in static fabricated systems.Source: §I · p. 1
1. The Paul trap
- In the Paul trap, the slow dynamics is governed by an effective harmonic frequency, while the micro-motion contributes an equivalent average kinetic-energy scale.The example establishes micro-motion as an active ingredient rather than a negligible correction.Source: §1 · p. 2
2. The modulated optical lattice
- In a modulated optical lattice, the hopping rate is exactly renormalized by a Bessel function, an effect observed experimentally with cold atoms.Large quasimomentum micro-motion modifies tunneling substantially when its oscillation approaches the Brillouin-zone width.Source: §2 · p. 2
B. The two-step modulation and the ambiguity inherent to the Trotter approach
- For two-step driving, the first-order Trotter correction changes sign with the starting pulse but can be removed by a unitary transformation.Consequently, this term cannot generate nontrivial band structures or gauge fields; the first nontrivial terms arise at second order in T.Source: B. The two-step modulation and the… · pp. 2–3
C. Outline of the paper
- The paper proceeds from general time-dependent formalism to N-step modulations, magnetic-field and spin-orbit-coupling schemes, and convergence and adiabatic-launching discussions.The outline explicitly includes applications without and with optical lattices.Source: C. Outline of the paper · p. 3
II. THE NOTION OF EFFECTIVE HAMILTONIANS: USING A RELIABLE APPROACH
The formalism partitions driven evolution into a time-independent effective Hamiltonian, an initial kick, and periodic micro-motion, thereby separating phase-dependent launching effects from long-time dynamics. Applied to two-step and lattice examples, it recovers second-order effective terms, exact Bessel-renormalized hopping, and strong sensitivity to the initial phase.
A. The formalism
- The formalism separates the initial phase, long-time evolution under a time-independent effective Hamiltonian, and final-time micro-motion.The kick operator is periodic with zero average, while the effective Hamiltonian is independent of the starting and final times.Source: A. The formalism · p. 3
- The perturbative construction provides formulas for effective Hamiltonians and kicks while omitting harmonic-mixing terms only when they do not contribute to the presented situations.It is intended to assess driving schemes for robust effects such as nonzero effective magnetic fields.Source: A. The formalism · p. 3
III. LAUNCHING THE DRIVING: ILLUSTRATION OF THE INITIAL KICK
- The perturbative expansion is not guaranteed to converge, and adiabatic launching is discussed as a way to reduce initial-kick effects.The formalism is therefore useful for identifying schemes but requires attention to convergence and launch protocol.Source: A. The formalism · p. 4, §III · p. 5
B. Illustration of the formalism: back to the two-step sequence
- For the two-step sequence, the effective Hamiltonian has no first-order contribution, while the initial kick and micro-motion reproduce the full evolution decomposition.The result is qualitatively equivalent to the smooth-driving examples.Source: B. Illustration of the formalism: back… · p. 4
- The modulated-lattice application recovers Bessel-renormalized hopping exactly and identifies a micro-motion kick with quasimomentum displacement Δk = κ/ω.When Δk ≈ π, the hopping changes dramatically because J0(π) ≈ −1/3 rather than its undriven value J0(0) = 1.Source: B. Illustration of the formalism: back… · p. 4
III. LAUNCHING THE DRIVING: ILLUSTRATION OF THE INITIAL KICK
- Abruptly launching the pulse sequence changes the initial velocity and can make long-time position strongly dependent on the modulation phase.For the alternating-force example, the kick is ±xFT/4 at two starting phases and vanishes at ti = ±T/4.Source: §III · p. 5
IV. MULTISTEP SEQUENCES
The general N-step framework separates effective Hamiltonian, micro-motion, and initial-phase effects, while showing that pulse ordering controls the initial kick but not the effective Hamiltonian. Four-step sequences provide distinct routes to first- and second-order effective terms relevant to synthetic magnetic fields and spin-orbit coupling.
IV. MULTISTEP SEQUENCES
- N-step driving uses repeated intervals of duration T/N with arbitrary operators V_m constrained by Σ_m V_m = 0, extending the two-step sequence.The sequence is represented as γ_N = {H_0 + V_1, ..., H_0 + V_N} and encoded through square functions and their Fourier series.Source: §IV · p. 5
- Figure 1 shows that reversing or shifting the starting pulse changes the long-time classical trajectory, while the particle retains a small micro-motion.The initial kick is inhibited for one shifted sequence and opposite for the two former cases.Source: Figure 1 · p. 5
- The initial kick K(0) depends on pulse ordering, whereas the effective Hamiltonian H_eff remains invariant under cyclic shifts of the sequence.Changing V_m to V_{m+p} alters K(0) but leaves H_eff unchanged; omitted harmonic-mixing terms do not contribute for the sequences considered.Source: §IV · pp. 5–6
- Sequences with N = 3 or N = 4 can generate non-trivial effects at first order in 1/ω, unlike the N = 2 case, motivating four-step applications to synthetic fields and spin-orbit couplings.The framework also identifies the Haldane construction as N = 3 and a topological Floquet-edge-state model as N = 5.Source: §IV · p. 6
A. The class of sequences α
- Class α alternates A and B with their negatives, producing both first- and second-order effective terms and encompassing a proposal for synthetic magnetic flux.Its pulsed effective Hamiltonian is approximately reproduced by the smooth drive V(t) = A cos(ωt) + B sin(ωt).Source: A. The class of sequences α · p. 6
B. The class of sequences β
- Class β orders A, −A, B, −B, eliminating first-order terms while adding mixed second-order terms such as [[A, H_0], B].Schemes for generating spin-orbit couplings in cold gases can be expressed in this sequence class.Source: B. The class of sequences β · p. 6
VI. PHYSICAL ILLUSTRATION: GENERATING SYNTHETIC MAGNETIC FIELDS IN COLD GASES
The paper applies its multi-step driving formalism to engineer synthetic magnetic fields and spin-orbit couplings in cold-atom systems, including lattice implementations. The examples show that effective dynamics, second-order corrections, initial-phase effects, and micro-motion are all relevant to faithful quantum simulation.
A. Without a lattice
- A four-step sequence of directional motion and alternating quadrupolar fields generates a perpendicular, uniform synthetic magnetic field in two-dimensional atomic gases.The scheme also induces harmonic confinement through second-order corrections.Source: A. Without a lattice · p. 7
- Second-order corrections produce harmonic confinement that can dominate the centrifugal force, so truncating the expansion at first order can misrepresent the dynamics.The induced confinement is a special feature of the driving sequence.Source: A. Without a lattice · p. 7
B. With optical lattices
- In optical lattices, pulsed directional hoppings realize the sequence and yield a Hofstadter Hamiltonian in the low-flux regime Φ ≪1, with Φ = a^2κ/8ω.A partial resummation can extend the flux range to Φ ∼1.Source: B. With optical lattices · p. 7
- The lattice simulations show very good agreement between the pulsed dynamics and the effective Hamiltonian, producing spirographic and chiral center-of-mass motion.Changing the initial driving phase has little effect here, and real-space micro-motion is negligible.Source: B. With optical lattices · pp. 7–8, Figure 3 · p. 8
A. Spin-orbit coupling in 2D: general considerations
- The framework targets two-dimensional spin-orbit couplings, including Rashba, helical, and combined helical-Rashba forms for spin-1/2 atoms.The helical term is associated with intrinsic spin-orbit coupling, while the combined configuration parallels the Kane-Mele structure.Source: A. Spin-orbit coupling in 2D: general… · p. 8
- Perturbative analysis shows that Rashba and helical spin-orbit terms can become equivalent through a unitary transformation at a specific weight ratio, making second-order analysis essential for identifying non-trivial effects.The expansion in mλ_RL is distinct from the 1/ω expansion and becomes problematic near the Rashba-ring ground states.Source: A. Spin-orbit coupling in 2D: general… · p. 8
1. The helical-Rashba scheme
- A four-step driving sequence generates an effective helical-Rashba Hamiltonian containing both λ_R p·σ and Ω_SO L_zσ_z terms.Neither term can be eliminated by a unitary transformation when Ω_SO ≠ mλ_R^2.Source: §1 · pp. 8–9
- For the helical-Rashba scheme, second-order effective dynamics agrees well with the pulsed system, third-order corrections improve agreement, and the kick operator captures substantial spin-space micro-motion.The real-space micro-motion remains small, while strong spin-orbit coupling can make the initial kick important.Source: §1 · p. 9
2. The pure Dirac regime
The paper develops driving schemes that engineer effective spin-orbit couplings and related terms, including pure Dirac, topological-superfluid, Chern-insulator, and xy configurations. The β-based XA scheme highlights how a unitary transformation can recover Rashba spin-orbit coupling even when it is absent from the effective Hamiltonian's first-order terms.
2. The pure Dirac regime
- Controlling the static Hamiltonian can produce a pure Dirac system and tune the first-order effective spectrum while suppressing second-order spin-orbit contributions.The helical spin-orbit term arises from a nested commutator, and modifying the pulsed operators can access other regimes.Source: §2 · p. 10
3. Adding terms to the static Hamiltonian ˆH0: a route towards topological superfluids and topological insulators
- Adding a Zeeman term to the driving sequence preserves the first-order Rashba term and can support simulations of topological superfluids, while generating an additional second-order contribution.The resulting extra term corresponds to a spin-dependent harmonic potential, whose effect on the topological-superfluid phase remains open.Source: §3 · p. 10
- The same static-Hamiltonian strategy can target Chern insulators through spin-dependent hopping on square lattices or Rashba–Zeeman combinations on honeycomb lattices.These constructions are proposed as routes toward the quantum anomalous Hall effect.Source: §3 · p. 10
4. The xy scheme
- The xy scheme yields an effective Hamiltonian with Rashba coupling and a spin-dependent hyperbolic potential, while second-order terms do not contribute.The hyperbolic term can be problematic for strong spin-orbit coupling but may be treated almost exactly or inhibited by lattice parameter tuning.Source: §4 · p. 10
C. Generating spin-orbit couplings with the β sequence: the XA scheme
- The four-step XA scheme is qualitatively represented by the β sequence, whose first-order Rashba terms vanish but reappear through a unitary transformation.Its perturbative description involves both the expansion in ΩSO/ω and the expansion in mλRL.Source: C. Generating spin-orbit couplings… · p. 10
D. Quasi-exact treatments on special cases
The paper develops quasi-exact treatments for special driving sequences, recovering effective Rashba spin-orbit coupling while exposing strong dependence on initial phase and lattice conditions.
1. The XA scheme
- The XA sequence admits a quasi-exact treatment that partially resums the infinite series in the effective Hamiltonian and removes dependence on mλ_RL from the evolution operator.Source: §1 · p. 11
- The lattice XA scheme recovers the continuum Rashba result for weak driving, but its Rashba coupling is maximized at λ_R=(π/2)aJ.Source: §1 · p. 11
- Changing the initial phase from +Â to −B̂ gives a y-directed kick that ejects the momentum distribution from the Rashba ring and changes isotropic Dirac spreading into anisotropic dynamics.Source: §1 · p. 11
2. The xy scheme
- The xy scheme is almost exactly tractable only on the lattice at κT=4π/a, equivalently λ_R=(π/2)aJ, where it is gauge-equivalent to XA and the hyperbolic potential disappears.Source: §2 · p. 12
- At λ_R≈(π/2)aJ, the effective Hamiltonian reproduces the pulsed-lattice spin and center-of-mass dynamics, although micro-motion remains absent from the effective description.Source: §2 · p. 12, Figure 6 · p. 12
A. Convergence of the (1/ω) expansion
- The perturbative expansion requires case-specific convergence checks because higher-order terms can remain comparable to lower orders for relevant states.Source: A. Convergence of the (1/ω) expansion · p. 12, §1 · p. 13
2. The perturbative approach revisited
The paper revisits perturbative convergence by accounting for driving operators that may scale with frequency, and examines adiabatic launching as a way to reduce initial-phase effects.
2. The perturbative approach revisited
- An alternative perturbative method partially resums the expansion when driving operators depend on the modulation frequency, yielding a controlled parameter Ω/ω.Source: §2 · p. 13
- The proposed convergence guideline computes the effective Hamiltonian, analyzes its relevant states or dispersion, and tests the next perturbative order before truncation.Source: §2 · p. 13
B. Adiabatic launching
- Adiabatic launching suppresses the initial kick and can erase the system’s memory of abrupt modulation onset, with a ramp of order 1/J in the Fig. 5 situation.Source: B. Adiabatic launching · pp. 13–14
- Micro-motion can exchange target states or produce oscillations around the effective-Hamiltonian ground state, especially when that state is degenerate or momentum and spin stability matter.Source: B. Adiabatic launching · p. 14
C. Conclusions
- The framework is formulated for off-resonant driving, while resonant schemes require a generalization of the effective-Hamiltonian method.Source: C. Conclusions · p. 14
- Micro-motion is typically large in momentum and spin space, making detection and collision-induced heating potentially important concerns in cold-matter applications.Source: C. Conclusions · p. 14
- Additional probe potentials can alter the effective Hamiltonian and band structure, so measuring topological order may itself destroy the targeted order.Source: C. Conclusions · pp. 14–15
Appendix A: The Paul trap
The appendices derive effective dynamics for the Paul trap and modulated optical lattices, showing how micro-motion generates slow trapping and how lattice hopping is renormalized exactly.
Appendix A: The Paul trap
- A periodically driven Paul trap produces an effective harmonic potential whose nontrivial contribution appears at second order in the period T.Source: Appendix A · pp. 17–18
- The effective harmonic potential governing slow motion equals the average kinetic energy associated with the particle’s micro-motion.Source: Appendix A · p. 17
- The Paul-trap ground-state momentum dispersion is comparable to the momentum generated by average micro-motion, illustrating its dynamical significance.Source: Appendix A · p. 17
Appendix B: Renormalization of the hopping in modulated optical lattices
- For a modulated optical lattice, the effective Hamiltonian yields an exact Bessel-function renormalization of the hopping rate.Source: Appendix B · p. 18
Appendix C: General expression for the effective Hamiltonian: Derivation of Eqs. (16)-(17)
The formalism separates periodically driven evolution into effective slow dynamics and kick operators, then derives perturbative expressions for general and two-step drives.
Appendix C: General expression for the effective Hamiltonian: Derivation of Eqs. (16)-(17)
- The evolution operator factors into an initial kick, time-independent effective-Hamiltonian evolution, and final kick.Source: Appendix C · p. 19
- The effective Hamiltonian and kick operator are constructed perturbatively when no general analytical expression is available.Source: Appendix C · p. 19
Appendix D: The two-step sequence: Derivation of Eqs. (18)-(19)
- The two-step effective Hamiltonian is obtained by inserting the drive’s Fourier components into the general expansion through second order in 1/ω.Source: Appendix D · p. 19, Appendix F · p. 21
Appendix E: The modulated optical lattice revisited
The appendix revisits the modulated optical lattice using a unitary-transformation framework that isolates a time-independent effective Hamiltonian and its kick operator. The non-perturbative treatment recovers Bessel-function hopping renormalization, while the modulation phase changes the initial kick without changing the effective Hamiltonian.
Appendix E: The modulated optical lattice revisited
- The non-perturbative analysis recovers the well-known Bessel-function renormalization of the hopping amplitude.This reproduces the result obtained for the modulated lattice in Eq. (B9).Source: Appendix E · p. 20
- The method determines a time-independent effective Hamiltonian through a unitary transformation with periodic functions α(t), β(t), and γ(t).The transformed Hamiltonian is constrained to remove explicit time dependence.Source: Appendix E · p. 20
- The zero-mean condition on γ(t) fixes γ1 = 0 and yields the familiar hopping renormalization.The constraint determines the integration constants needed for the kick operator.Source: Appendix E · p. 20
- The modulation phase φ changes the initial kick K(0) = ξ0 sin φ V but has no consequence for the effective Hamiltonian.The phase therefore affects the initial micromotion rather than the long-time effective dynamics.Source: Appendix E · p. 20
Appendix F: N-step sequence: Derivation of the general expression in Eq. (31)
- For general N-step pulse sequences, the appendix derives first- and selected second-order contributions to the effective Hamiltonian and initial-kick operator.The derivation uses Fourier components, commutator expansions, and a unitary transformation; harmonic-mixing terms do not contribute for the sequences treated.Source: Appendix F · pp. 20–21
Appendix G: Effective Hamiltonians for general sequences with N = 3, 4 different steps
This appendix applies the general N-step expansion to three- and four-step sequences. Proper operator choices can generate non-trivial effects at first order in 1/ω, while the four-step treatment includes higher-order corrections and a stated scheme-specific simplification.
1. The case N = 3
- Three-step sequences can produce non-trivial first-order effects in 1/ω through an appropriate choice of H0 and the pulse operators.The text relates this possibility to pulsed-hopping proposals for realizing the Haldane model.Source: §1 · p. 21
2. The case N = 4
- Four-step sequences are principally explored and can likewise generate non-trivial effects at first order in 1/ω.Their effective Hamiltonian contains specific commutator combinations, including nearest-neighbor sequence terms.Source: §2 · pp. 21–22
- For the α sequence, third-order corrections can be evaluated using either a smooth single-harmonic approximation or an alternative perturbative method for the square wave.The smooth driving is written as V(t) = A cos(ωt) + B sin(ωt).Source: §2 · p. 22
- The four-step expression omits selected terms at this order, but those terms vanish identically for the SOC scheme analyzed in Section VII B 1.The associated third-order effects are illustrated for that specific scheme in Fig. 4.Source: §2 · p. 22
Appendix H: Tight-binding operators in 2D lattice systems and useful commutators
The appendices establish a tight-binding operator dictionary and commutator toolkit for one- and two-component two-dimensional lattices. They then use exact subsequence treatments to derive the lattice-free Rashba spin-orbit-coupling Hamiltonian for the XA sequence.
Appendix H: Tight-binding operators in 2D lattice systems and useful commutators
- The tight-binding framework represents hopping on a two-dimensional lattice and connects its continuum limit to an effective-mass kinetic energy.The operator dictionary translates bulk position and momentum notation into lattice analogues.Source: Appendix H · p. 22
1. One-component lattices
- Tables I–IV and the associated commutator relations provide the operator definitions needed for one- and two-component lattice calculations.The two-component operators obey cyclic commutation conditions used repeatedly in the effective-Hamiltonian derivations.Source: §1 · p. 22, §2 · p. 23
Appendix I: Exact treatment of the XA scheme
- The XA sequence is treated exactly over each half-period before applying the lowest-order Trotter expansion.The half-period evolution results remain valid for arbitrary ω or λR proportional to κ/ω.Source: Appendix I · p. 23, §1 · p. 23
1. Without the lattice
- The lowest-order Trotter expansion yields the Rashba spin-orbit-coupling Hamiltonian for the lattice-free XA sequence.The derivation identifies the resulting effective Hamiltonian after combining the exactly treated subsequences.Source: §1 · pp. 23–24
2. With the lattice
For the lattice implementation, exact half-period calculations provide lattice analogues of the XA sequence and recover the continuum Rashba result in the weak-driving limit. The lattice also imposes a maximum effective Rashba coupling, while the xy scheme admits an exact factorization under specific operator relations.
2. With the lattice
- The lattice XA sequence is analyzed with the same unitary transformation used in the lattice-free case, producing exact half-period evolution operators.The resulting expressions are direct lattice analogues of the continuum formulas.Source: §2 · p. 24
- In the weak-driving regime and continuum limit, the lattice calculation exactly recovers the lattice-free XA result.The effective Rashba strength is λR = πκ/(8m*ω), with m* = 1/(2Ja^2).Source: §2 · p. 24
- The lattice bounds the effective Rashba strength, whose maximum occurs at λR = π/(4am*) = aJ(π/2).This limit distinguishes the lattice implementation from the continuum treatment.Source: §2 · p. 24
Appendix J: Exact treatment of the xy scheme
- The xy scheme is treated by factorizing its four primitive evolution operators with the Zassenhaus formula under cyclic commutation relations.The relations are [X,Y] = Z, [X,Z] = aY, and [Y,Z] = 0.Source: Appendix J · p. 24
- Applying the factorized operators and the Trotter expansion yields the effective Rashba Hamiltonian for the xy sequence.The lattice-specific simplification uses exp(iπmσx) = (−1)^m in the special regime λR = (π/2)aJ.Source: Appendix J · p. 24
Appendix K: An alternative perturbative approach
Appendix K develops an alternative perturbative approach for general time-dependent Hamiltonians, modifying the expansion when one driving operator scales with frequency. It derives effective Hamiltonians for pulse-sequence and smooth harmonic drives, while identifying omitted terms and conditions under which the approach can improve convergence.
General perturbative construction
- The approach modifies the perturbative expansion for Hamiltonians containing a term proportional to the driving frequency ω, with a zero-mean kick operator.The construction identifies frequency-proportional terms first and imposes a zero time average on the kick operator over one period.Source: Appendix K · p. 25
- The resulting time-averaged expression provides the effective Hamiltonian to lowest order in 1/ω.The derivation organizes higher-order contributions through the operator R(B, K(1)), which absorbs zero-average periodic terms.Source: Appendix K · p. 25
α pulse sequence
- For the α four-step pulse sequence, substituting the sequence-specific square-wave functions recovers the announced third-order commutator term.The sequence alternates H0+A, H0+B, H0−A, and H0−B, and produces the third-order contribution proportional to 1/ω^3.Source: Appendix K · p. 25
Smooth harmonic driving
- For smooth single-harmonic driving, the method recovers the result only partially because it disregards additional commutator terms.The drive uses A cos(ωt)+B sin(ωt), and the omitted terms are explicitly identified in the expansion.Source: Appendix K · p. 25
Scope and limitations
- The omitted terms arise because the method assigns different perturbative orders to H0 and A versus B, with B scaling as ω.This ordering is built into the modified expansion and limits which terms appear at the computed order.Source: Appendix K · p. 25
- In the lowest-Landau-level example, retained terms are all of order ∼Ω while neglected terms are ∼Ω3/ω2 ≪Ω, potentially partially resumming the formal series and supporting convergence.The example assumes H0 ∼ A ∼ Ω ≪ ω and B ∼ 1 in the rescaled formulation.Source: Appendix K · p. 25