Driven Closed Quantum Systems

A driven closed quantum system has a Hamiltonian that depends explicitly on time while the retained quantum state still evolves unitarily:

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

The word closed refers to the absence of unmodeled environmental degrees of freedom in this evolution. It does not mean that the retained system’s energy is constant. An external drive can perform work on the system, and the prescribed source is not included in the system’s energy bookkeeping.

This distinction is the foundation for coherent spectroscopy, magnetic resonance, laser and microwave control, trap modulation, periodically driven matter, and quantum gates. It is also the boundary between this page and Driven Open Systems, where relaxation, dephasing, noise, and dissipative generators are included. Driven Many-Body Systems owns extensive power, energy-density absorption, periodic heating, and thermodynamic evidence standards.

Time-Dependent Hamiltonian Model

A useful decomposition is

$$H(t) = H_0+V(t),$$

where $H_0$ is a reference Hamiltonian and $V(t)$ is the drive. More generally, a control-affine model has

$$H(t) = H_0 + \sum_{a=1}^{m} u_a(t)H_a.$$

The functions $u_a(t)$ are prescribed control waveforms. The operators $H_a$ specify how each available control couples to the quantum system.

Examples include:

• an electric field coupled to a dipole operator,

  $$V(t)=-\mathbf d\cdot\mathbf E(t);$$

• a magnetic field coupled to a magnetic moment,

  $$V(t)=-\boldsymbol{\mu}\cdot\mathbf B(t);$$

• a force driving an oscillator coordinate,

  $$V(t)=-F(t)x;$$

• a gate voltage or flux waveform changing a qubit Hamiltonian;

• a modulated lattice depth, tunneling amplitude, or trapping frequency.

The drive is often treated semiclassically: the quantum system is dynamical, while the applied field or control waveform is a specified classical function. This can be accurate when source depletion, quantum fluctuations, and backaction are negligible. If those effects matter, the source must be quantized or incorporated into a larger open-system model.

Unitary Propagation

The evolution operator satisfies

$$i\hbar \frac{\partial}{\partial t} U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

If $H(t)$ is self-adjoint with suitable domain control, then

$$U(t,t_0)^*U(t,t_0)=I.$$

The state and density operator evolve as

$$\lvert\psi(t)\rangle = U(t,t_0)\lvert\psi(t_0)\rangle,$$

and

$$\rho(t) = U(t,t_0)\rho(t_0)U(t,t_0)^*.$$

When Hamiltonians at different times fail to commute,

$$[H(t_1),H(t_2)]\neq0,$$

the propagator is not generally

$$\exp \left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right].$$

The correct formal expression is time ordered:

$$U(t,t_0) = \mathcal T \exp \left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right].$$

Time-Dependent Hamiltonians owns the general propagation problem, and Time Ordering owns the ordering operation and Dyson expansion. Here the propagator is the engine that turns a drive waveform into state motion and transition amplitudes.

External Driving

An external parameter $\lambda(t)$ may enter through

$$H(t)=H(\lambda(t)).$$

The protocol is the path $\lambda(t)$ together with its timing. Two protocols can trace the same path at different speeds and produce different states unless an adiabatic, sudden, or other controlled approximation applies.

Several time scales can matter:

• intrinsic Bohr periods set by energy gaps;
• the drive period or pulse duration;
• the envelope turn-on and turn-off times;
• detuning and Rabi time scales;
• coherence and relaxation times when the closed approximation is relaxed.

Calling a drive “slow” or “fast” without identifying the comparison scale is incomplete. Slow relative to a carrier period can still be fast relative to a small avoided-crossing gap.

The instantaneous eigenvectors of $H(t)$ are not by themselves solutions of the Schrödinger equation. Their time dependence introduces nonadiabatic couplings. Exact following requires special structure; approximate following requires the hypotheses of an adiabatic theorem.

Energy Nonconservation and Power

Let

$$E(t) = \langle H(t)\rangle = \operatorname{Tr} \left[ \rho(t)H(t) \right].$$

Closed unitary dynamics obeys

$$\dot\rho = -\frac{i}{\hbar}[H(t),\rho].$$

Differentiate the energy:

$$\begin{aligned} \frac{dE}{dt} &= \operatorname{Tr}(\dot\rho H) + \operatorname{Tr}(\rho\dot H)\\ &= -\frac{i}{\hbar} \operatorname{Tr} \left( [H,\rho]H \right) + \left\langle \frac{\partial H}{\partial t} \right\rangle. \end{aligned}$$

Cyclicity of the trace makes the commutator term vanish, so

$$\frac{d}{dt} \langle H(t)\rangle = \left\langle \frac{\partial H(t)}{\partial t} \right\rangle.$$

This is the exact energy-balance law for a closed driven system. Explicit time dependence supplies or extracts energy even though the state evolution is unitary.

For a parameter protocol,

$$\frac{\partial H}{\partial t} = \dot\lambda \frac{\partial H}{\partial\lambda},$$

so the instantaneous power delivered to the system is

$$P(t) = \dot\lambda(t) \left\langle \frac{\partial H}{\partial\lambda} \right\rangle.$$

For several controls,

$$P(t) = \sum_a \dot u_a(t) \langle H_a\rangle$$

when the control operators are time independent.

This formula does not by itself settle every definition of quantum work. Work distributions, two-time measurements, strong coupling, and open-system heat are separate topics. Its role here is simpler: a time-dependent Hamiltonian permits energy exchange without violating unitarity.

Transition Amplitudes

If the system begins in $\lvert i\rangle$ at $t_0$, the amplitude to find $\lvert f\rangle$ at time $t$ is

$$\mathcal A_{fi}(t,t_0) = \langle f\vert U(t,t_0) \vert i\rangle.$$

The corresponding probability is

$$P_{i\to f} = \left| \mathcal A_{fi} \right|^2.$$

The exact amplitude contains the full time ordering, pulse shape, phase, detuning, and interference between alternative transition histories.

For weak driving, the interaction picture isolates the drive. Let

$$H_0\lvert n\rangle = E_n\lvert n\rangle, \qquad \omega_{fi} = \frac{E_f-E_i}{\hbar}.$$

To first order,

$$c_f^{(1)}(t) = -\frac{i}{\hbar} \int_{t_0}^{t} dt'\, e^{i\omega_{fi}t'} V_{fi}(t'),$$

where

$$V_{fi}(t) = \langle f\vert V(t)\vert i\rangle.$$

This formula shows why the frequency content of the drive matters. For

$$V(t) = \lambda A \cos(\omega t+\phi),$$

the integrand contains phases with frequencies

$$\omega_{fi}-\omega \qquad \text{and} \qquad \omega_{fi}+\omega.$$

A long weak pulse can accumulate a large amplitude near resonance, while a short pulse has broad spectral bandwidth. Selection rules enter through the matrix element $\langle f\vert A\vert i\rangle$.

The perturbative derivation and validity conditions belong to First-Order Transition Probability. The general meaning and composition of amplitudes belong to Transition Amplitudes.

Driven Two-Level System

A standard model is

$$H(t) = \frac{\hbar\omega_0}{2}\sigma_z + \frac{\hbar\Omega(t)}{2} \cos \left( \omega_dt+\phi \right) \sigma_x.$$

The first term sets the undriven splitting. The second is a coherent drive with carrier frequency $\omega_d$, envelope $\Omega(t)$, and phase $\phi$.

This compact Hamiltonian already contains several regimes:

• near-resonant population transfer;
• off-resonant dispersive shifts;
• short pulses and broad bandwidth;
• periodic continuous driving;
• composite pulse sequences;
• breakdown of the rotating-wave approximation at strong drive.

The Rabi formula and Bloch-sphere interpretation belong to Rabi Oscillations: First Encounter. Calibrated pulse protocols and decoherence effects belong to Rabi and Ramsey Control.

Rotating Frames

The exact transformation of states, observables, density operators, and propagators is developed in Rotating Frames. The summary here records only the basic Hamiltonian formula.

A time-dependent unitary change of frame can remove rapid known motion. Let

$$\lvert\psi_R(t)\rangle = R(t)^* \lvert\psi(t)\rangle.$$

Differentiation gives the transformed Hamiltonian

$$H_R(t) = R(t)^*H(t)R(t) - i\hbar R(t)^*\dot R(t).$$

The second term is essential. It records the motion of the frame itself.

For a near-resonantly driven two-level system, a rotation at the carrier frequency can turn rapid laboratory-frame oscillations into slow detuning and coupling terms. A rotating-wave approximation may then discard counter-rotating contributions, but only after a scale comparison.

Rotating frames are exact changes of representation. The rotating-wave approximation is an additional approximation. Conflating those steps can hide the origin of errors such as the Bloch–Siegert shift.

Periodic and Stroboscopic Drives

The periodic setup, one-period covariance, and basic examples are developed in Periodic Hamiltonians. If

$$H(t+T)=H(t),$$

the drive is periodic. Continuous evolution is then organized by the one-period propagator

$$U(T,0).$$

Repeated periods give a discrete sequence of stroboscopic states. Floquet theory expresses solutions using periodic modes and quasienergies; see Floquet Theorem in Quantum Mechanics.

Periodic driving is a special case of driven closed dynamics, not a separate postulate. Its additional time-translation symmetry makes one-period evolution more informative than a generic pulse.

Links to Quantum Control and AMO Physics

In atomic, molecular, and optical physics, coherent driving is used to:

• prepare superpositions and transfer populations;
• perform spectroscopy and estimate level splittings;
• manipulate spins, atoms, ions, molecules, and superconducting circuits;
• engineer effective interactions and synthetic gauge structures;
• suppress unwanted evolution through pulse sequences;
• realize periodic Floquet Hamiltonians.

Quantum control asks which target operations are reachable, how to design controls under constraints, and how robustly a protocol survives calibration errors and noise. Those questions go beyond writing $H(t)$.

Links to Quantum Control turns the propagator language developed here into control Hamiltonians, pulse and gate objectives, rotating-frame design, average-Hamiltonian theory, and constrained optimization.

Pulse Sequences owns composite protocols, Optimal Control owns optimization methods, and Control Limits and Noise owns performance boundaries. This page supplies the closed-system dynamical substrate.

Closed Versus Open Description

A time-dependent Hamiltonian model is a closed description when

$$\rho(t) = U(t,t_0)\rho(t_0)U(t,t_0)^*$$

is adequate for the retained degrees of freedom.

An open-system description is needed when unretained degrees of freedom produce effects such as:

• relaxation or dephasing;
• stochastic amplitude or phase noise;
• leakage followed by irreversible loss;
• measurement backaction;
• source backaction or depletion;
• uncontrolled entanglement with a bath.

A laboratory device can be externally driven and approximately closed on one time scale, yet clearly open on a longer one. The classification is a modeling statement, not an intrinsic label attached to the apparatus.

Validity Checklist

Before trusting a driven Hamiltonian, identify:

1. which degrees of freedom are retained;
2. which controls are prescribed classically;
3. the domain or finite-level truncation of each coupling operator;
4. whether Hamiltonians at different times commute;
5. the relevant drive frequencies, envelopes, and bandwidths;
6. the approximation behind the coupling, such as dipole or rotating-wave form;
7. leakage levels omitted from the model;
8. the time scale on which dissipation and noise can be neglected;
9. whether instantaneous eigenstate, perturbative, or Floquet reasoning is actually justified.

Unitary numerics can be exact for an approximate Hamiltonian. Numerical precision does not validate the physical truncation or drive model.

Common Mistakes

• Assuming a closed system must conserve $\langle H(t)\rangle$ when $H$ depends explicitly on time.
• Writing an ordinary exponential of $\int H(t)\,dt$ without checking time commutators.
• Treating instantaneous eigenvectors as exact dynamical solutions.
• Confusing a transition amplitude with its probability.
• Calling a short pulse monochromatic or a long pulse spectrally broad without checking its Fourier width.
• Dropping the frame term $-i\hbar R^*\dot R$.
• Treating a rotating-frame transformation and the rotating-wave approximation as the same step.
• Ignoring the energy and backaction of a drive while claiming the combined laboratory is closed.
• Applying a two-level truncation at drive strengths that populate leakage levels.
• Moving directly from coherent drive equations to pulse design without control constraints or robustness analysis.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Volume Two, Wiley, 1977.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley, 1992.
• B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, 1990.
• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.
• N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Reviews of Modern Physics 89, 015006, 2017.

Exercises

1. Derive the exact energy-balance law for a closed driven density operator.

Solution

Start from

$$E(t) = \operatorname{Tr}(\rho H)$$

and

$$\dot\rho = -\frac{i}{\hbar}[H,\rho].$$

Then

$$\begin{aligned} \dot E &= \operatorname{Tr}(\dot\rho H) + \operatorname{Tr}(\rho\dot H)\\ &= -\frac{i}{\hbar} \operatorname{Tr} \left( [H,\rho]H \right) + \operatorname{Tr}(\rho\dot H). \end{aligned}$$

Cyclicity gives

$$\operatorname{Tr}(H\rho H) = \operatorname{Tr}(\rho H^2),$$

so the commutator term vanishes. Therefore

$$\dot E = \operatorname{Tr}(\rho\dot H) = \left\langle \frac{\partial H}{\partial t} \right\rangle.$$

2. Suppose

$$H(t)=f(t)H_0$$

with fixed self-adjoint $H_0$. Find the propagator and determine whether the drive creates transitions between $H_0$ eigenstates.

Solution

All Hamiltonians commute:

$$[H(t_1),H(t_2)] = f(t_1)f(t_2)[H_0,H_0] =0.$$

Hence

$$U(t,t_0) = \exp \left[ -\frac{i}{\hbar} H_0 \int_{t_0}^{t} f(s)\,ds \right].$$

If

$$H_0\lvert n\rangle = E_n\lvert n\rangle,$$

then

$$U(t,t_0)\lvert n\rangle = \exp \left[ -\frac{iE_n}{\hbar} \int_{t_0}^{t}f(s)\,ds \right] \lvert n\rangle.$$

The drive changes phases but does not mix the eigenstates of $H_0$.

3. Identify the resonant term in first-order perturbation theory for

$$V(t) = \lambda A\cos(\omega t+\phi).$$

Solution

The first-order integrand contains

$$e^{i\omega_{fi}t} \cos(\omega t+\phi).$$

Using

$$\cos(\omega t+\phi) = \frac{1}{2} \left( e^{i(\omega t+\phi)} + e^{-i(\omega t+\phi)} \right),$$

the phases are

$$\frac{1}{2} e^{i(\omega_{fi}+\omega)t+i\phi} + \frac{1}{2} e^{i(\omega_{fi}-\omega)t-i\phi}.$$

For a positive transition frequency $\omega_{fi}$, the second term varies slowly when

$$\omega\approx\omega_{fi}.$$

It can then accumulate coherently over a long pulse. The transition is absent at this order if

$$\langle f\vert A\vert i\rangle=0.$$

4. Derive the Hamiltonian in a time-dependent rotating frame.

Solution

Let

$$\lvert\psi_R\rangle = R^*\lvert\psi\rangle.$$

Differentiate:

$$\lvert\dot\psi_R\rangle = \dot R^*R\lvert\psi_R\rangle + R^*\lvert\dot\psi\rangle.$$

Using

$$\lvert\dot\psi\rangle = -\frac{i}{\hbar}H\lvert\psi\rangle$$

and $\dot R^*R=-R^*\dot R$ gives

$$i\hbar \lvert\dot\psi_R\rangle = \left( R^*HR - i\hbar R^*\dot R \right) \lvert\psi_R\rangle.$$

Thus

$$H_R = R^*HR - i\hbar R^*\dot R.$$

5. Classify each description as closed or open: a noiseless prescribed microwave drive on a qubit, the same drive with measured dephasing, and a quantized cavity mode retained together with the qubit.

Solution

A noiseless prescribed microwave waveform driving a qubit can be modeled as a closed time-dependent Hamiltonian for the qubit, provided leakage and environmental effects are negligible on the time scale of interest.

If dephasing is included phenomenologically or through a bath, the qubit description is open. Unitary evolution under $H(t)$ alone no longer gives the retained state.

If the cavity mode is quantized and retained together with the qubit, the enlarged qubit–cavity composite can be treated as closed when its own losses are negligible. Energy exchanged between qubit and cavity is internal to that enlarged model.