Driven Open Systems

A driven open system is a quantum system whose retained degrees of freedom are intentionally controlled while they also exchange information, energy, or coherence with unretained degrees of freedom. The unitary baseline and the distinction between external work and environmental exchange are developed in Driven Closed Quantum Systems. The simplest mental picture is:

Hamiltonian control changes what the system tries to do;
dissipation and noise change what actually survives.

This page explains the combined model. The closed-system time-ordering formalism belongs to Time-Dependent Hamiltonians. The standard Markovian generator belongs to Lindblad–GKSL Equation. This page is the canonical open-control bridge between them. Driven Many-Body Systems owns extensive heating, cycle energy balance, and many-body steady-regime evidence.

Basic Model

A common time-local Markovian description is

$$\dot\rho(t) = \mathcal L_t(\rho(t)) = - \frac{i}{\hbar} [H(t),\rho(t)] + \sum_\alpha \gamma_\alpha(t) \mathcal D[L_\alpha(t)]\rho(t),$$

where

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12 \{L^\dagger L,\rho\}.$$

The Hamiltonian is often split as

$$H(t) = H_0+\sum_j u_j(t)H_j,$$

with $u_j(t)$ controlled amplitudes and phases. The $L_\alpha(t)$ may be fixed natural noise channels, frame-transformed jump operators, or deliberately engineered channels. If all rates $\gamma_\alpha(t)$ are nonnegative and the equation is interpreted as a regular time-local Lindblad equation, the evolution is CP-divisible. It is not usually a time-homogeneous semigroup, because $\mathcal L_t$ changes with time.

The corresponding propagator is a time-ordered exponential of superoperators:

$$\Phi(t,t_0) = \mathcal T \exp \left[ \int_{t_0}^{t} ds\, \mathcal L_s \right].$$

This is the open-system analogue of the time-ordered unitary in closed dynamics. Since generators at different times need not commute, replacing this expression by an ordinary exponential is generally wrong.

What Counts as a Drive?

A drive is an externally controlled time dependence. It can enter as:

• a coherent field that changes $H(t)$;
• a modulation of detuning, tunneling, trapping frequency, or coupling strength;
• a shaped pulse sequence;
• a periodically driven Hamiltonian;
• a controlled dissipative channel or auxiliary loss process;
• a feedback signal computed from a measurement record.

This page focuses on unconditioned driven dynamics: the control waveform is specified independently of the measurement record for the current run. If the waveform depends on an observed record, the control entry point is Measurement-Based Feedback. If an unmeasured quantum field is routed back as a signal, see Coherent Feedback. If the environment is designed to stabilize a target autonomously, see Reservoir Engineering.

Rotating Frames

Driven systems are rarely analyzed only in the lab frame. Let

$$\tilde\rho(t) = R^\dagger(t)\rho(t)R(t)$$

for a time-dependent unitary frame transformation. If the lab-frame master equation has Hamiltonian $H(t)$ and jumps $L_\alpha(t)$, the transformed equation has

$$\tilde H(t) = R^\dagger H R - i\hbar R^\dagger\dot R, \qquad \tilde L_\alpha(t) = R^\dagger L_\alpha(t)R.$$

Thus

$$\dot{\tilde\rho} = - \frac{i}{\hbar} [\tilde H(t),\tilde\rho] + \sum_\alpha \gamma_\alpha(t) \mathcal D[\tilde L_\alpha(t)]\tilde\rho.$$

The extra Hamiltonian term $-i\hbar R^\dagger\dot R$ is the source of detunings in rotating-frame descriptions. Forgetting to transform the jump operators is a common error. A decay operator that is time independent in one frame may carry a phase in another; often that phase cancels inside a single dissipator, but not always in multi-channel or interference-sensitive settings.

Driven Two-Level Example

For a driven two-level system in a rotating frame, a common Hamiltonian is

$$H_{\mathrm{rot}} = \frac{\hbar}{2} \left( \Delta Z+\Omega_x(t)X+\Omega_y(t)Y \right),$$

where $\Delta$ is the detuning in the chosen sign convention and $\Omega_x,\Omega_y$ are controlled quadratures. Adding relaxation and pure dephasing gives a model of the form

$$\dot\rho = - \frac{i}{\hbar} [H_{\mathrm{rot}},\rho] + \Gamma\,\mathcal D[\sigma_-]\rho + \frac{\gamma_\phi}{2} \mathcal D[Z]\rho.$$

This is the conceptual skeleton behind Rabi oscillations with damping, Ramsey fringes with dephasing, driven qubit calibration, saturation spectroscopy, and many AMO control experiments. The detailed Bloch-vector equations and convention warnings are in Optical Bloch Equations.

Two practical lessons already appear in this simple example.

First, the drive and dissipators need not commute as superoperators. The final behavior is not “unitary Rabi oscillation plus a separate exponential” unless the model actually reduces to that limit.

Second, the rates $\Gamma$ and $\gamma_\phi$ are not properties of the pulse alone. They summarize environmental channels over the bandwidth and frame relevant to the controlled system.

Driven Oscillator Example

A damped driven oscillator or cavity mode is often modeled by

$$\dot\rho = - \frac{i}{\hbar} [H(t),\rho] + \kappa\,\mathcal D[a]\rho,$$

with a rotating-frame Hamiltonian such as

$$H(t) = - \hbar\Delta\,a^\dagger a + i\hbar \left[ \epsilon(t)a^\dagger-\epsilon^*(t)a \right].$$

Here $\epsilon(t)$ is the input drive envelope, $\Delta$ is detuning, and $\kappa$ is the mode linewidth. This model underlies cavity ringdown, coherent-state preparation, linear response, dispersive readout, and many input–output calculations. If the drive is applied through a physical port, Input–Output Theory gives the relation between the intracavity mode and traveling fields.

Periodic Drives

If $H(t+T)=H(t)$, closed-system dynamics can often be organized using Floquet theory. For open systems, the same drive can change which transition frequencies the bath sees. In weak-coupling derivations, one may need a dressed-state or Floquet master equation rather than an undriven dissipator pasted onto a driven Hamiltonian.

The warning is simple: a dissipator derived for $H_0$ is not automatically valid for a strongly driven Hamiltonian. The system operators that couple to the bath should be decomposed in the relevant energy, quasienergy, or rotating-frame basis when the drive changes the important time scales.

For the closed periodic formalism, see Floquet Theorem in Quantum Mechanics. For near-resonant simplification, see Rotating-Wave Approximation.

Time-Dependent Rates

Time-dependent rates can mean different things.

Source of time dependence	Typical interpretation
controlled coupling to a reservoir	engineered dissipation or tunable loss
changing system Hamiltonian	bath sees time-dependent transition operators
phenomenological fit	compact model for observed nonexponential behavior
exact time-local representation	memory hidden in time-dependent coefficients

Nonnegative time-dependent rates in a Lindblad-form generator give a legitimate time-local Markovian model under the assumptions of that model. Temporarily negative rates can appear in exact time-local descriptions of non-Markovian dynamics; they should not be inserted casually into phenomenological equations. For the diagnostic distinction, see CP Divisibility and Time-Convolutionless Master Equations.

Validity Checklist

Before trusting a driven open-system equation, check:

• what frame the Hamiltonian, jump operators, and rates are written in;
• whether the drive is weak, strong, slow, fast, resonant, or periodic relative to bath correlation times;
• whether the dissipator was derived in the bare basis, dressed basis, rotating frame, or Floquet basis;
• whether the secular approximation remains valid under the drive;
• whether control noise, amplitude drift, phase noise, and leakage states are included when they dominate;
• whether the bath spectrum is approximately smooth over the frequencies sampled by the driven system;
• whether a time-independent steady state or a periodic limit cycle is the right asymptotic object;
• whether feedback conditioning is being ignored even though the control depends on a record.

The Approximation Checklist gives a broader validation workflow for open-system models.

Relation to Other Control Pages

Driven open systems provide the baseline model for the rest of the chapter.

Rabi and Ramsey Control specializes to the basic two-level calibration protocols built from this driven model. Dynamical Decoupling specializes to pulse sequences that reshape noise sensitivity. Reservoir Engineering specializes to designing dissipators or lossy auxiliaries. Feedback pages specialize to control laws that depend on measurement records. Optimal Control specializes to choosing $u_j(t)$ by optimizing an objective under constraints.

The unifying rule is to keep the model explicit. Say which controls are Hamiltonian, which channels are dissipative, which fields are measured, and which approximations justify the reduced equation.

Common Mistakes

• Adding a dissipator derived for an undriven system to a strongly driven Hamiltonian without checking the dressed or rotating-frame basis.
• Forgetting the $-i\hbar R^\dagger\dot R$ term when changing frames.
• Transforming the Hamiltonian but not the jump operators.
• Treating a time-dependent Lindblad equation as a time-independent semigroup.
• Calling any time-dependent rate non-Markovian without checking CP divisibility or memory diagnostics.
• Assuming a steady state is time independent when the drive produces a periodic limit cycle.
• Ignoring drive noise while modeling environmental noise in detail.
• Comparing Rabi frequencies, detunings, or phases across papers without checking conventions.

Exercises

Trace preservation with a time-dependent dissipator

Show that $\mathcal D[L(t)]$ is trace preserving at each time.

Solution

Using cyclicity of the trace,

$$\operatorname{Tr} \left[ L\rho L^\dagger \right] = \operatorname{Tr} \left[ L^\dagger L\rho \right].$$

Therefore

$$\operatorname{Tr}\mathcal D[L]\rho = \operatorname{Tr}(L^\dagger L\rho) - \frac12 \operatorname{Tr}(L^\dagger L\rho) - \frac12 \operatorname{Tr}(\rho L^\dagger L) = 0.$$

This argument does not require $L$ to be time independent. It only proves trace preservation of the instantaneous dissipator, not the validity of a chosen physical model.

Covariance of a dissipator under a rotating frame

Let $\tilde\rho=R^\dagger\rho R$ and $\tilde L=R^\dagger L R$. Show that

$$R^\dagger\mathcal D[L]\rho\,R = \mathcal D[\tilde L]\tilde\rho.$$

Solution

For the jump term,

$$R^\dagger L\rho L^\dagger R = (R^\dagger L R) (R^\dagger\rho R) (R^\dagger L^\dagger R) = \tilde L\tilde\rho\tilde L^\dagger.$$

For the anticommutator,

$$R^\dagger L^\dagger L\rho R = \tilde L^\dagger\tilde L\tilde\rho, \qquad R^\dagger \rho L^\dagger L R = \tilde\rho\,\tilde L^\dagger\tilde L.$$

Combining the three terms gives the stated identity.

Driven qubit with pure dephasing

Consider

$$\dot\rho = - i\frac{\Omega}{2}[X,\rho] + \frac{\gamma_\phi}{2}\mathcal D[Z]\rho,$$

and write

$$\rho = \frac12 (I+xX+yY+zZ).$$

Find the Bloch equations.

Solution

The Hamiltonian rotates the Bloch vector around the $x$ axis:

$$\dot x\big|_H=0, \qquad \dot y\big|_H=-\Omega z, \qquad \dot z\big|_H=\Omega y.$$

The dephasing term satisfies

$$Z\rho Z-\rho = -xX-yY.$$

Since $\dot\rho=(\dot x\,X+\dot y\,Y+\dot z\,Z)/2$, the dissipator contributes

$$\dot x\big|_\phi=-\gamma_\phi x, \qquad \dot y\big|_\phi=-\gamma_\phi y, \qquad \dot z\big|_\phi=0.$$

Thus

$$\dot x=-\gamma_\phi x, \qquad \dot y=-\Omega z-\gamma_\phi y, \qquad \dot z=\Omega y.$$

The drive and dephasing together produce damped rotations, not a closed-system Rabi oscillation multiplied by an arbitrary decay factor.

Cross-Links

• Quantum Control and Feedback
• Rabi and Ramsey Control
• Optimal Control
• Time-Dependent Hamiltonians
• Lindblad–GKSL Equation
• Optical Bloch Equations
• Steady States and Relaxation
• Markov Approximation
• Secular Approximation
• Time-Convolutionless Master Equations
• CP Divisibility
• Noise Spectra
• Input–Output Theory
• Dynamical Decoupling
• Measurement-Based Feedback
• Coherent Feedback
• Reservoir Engineering
• Feedback from Measurement Records
• Floquet Theorem in Quantum Mechanics
• Rotating-Wave Approximation
• Approximation Checklist

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer (1999).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• D. D’Alessandro, Introduction to Quantum Control and Dynamics, Chapman and Hall/CRC (2007).
• C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present and future,” New Journal of Physics 12, 075008 (2010).
• S. Kohler, J. Lehmann, and P. Hänggi, “Driven quantum transport on the nanoscale,” Physics Reports 406, 379–443 (2005).
• M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Physics Reports 304, 229–354 (1998).