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. Hamiltonian control and dissipation generally act at the same time, and their effects need not commute. Combining them in one dynamical equation explains damped pulses, driven steady states, and periodic responses.

Required background. Time-Dependent Hamiltonians introduces time ordering; Lindblad–GKSL Equation defines the dissipative generator. For heating and energy balance in extended systems, see Driven Many-Body Systems.

Hamiltonian Control with Dissipation

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; a scalar phase always cancels in $\mathcal D[e^{i\phi}L]=\mathcal D[L]$, including each term of a sum of independent dissipators. Relative phases between contributions to one coherently combined jump operator, or in measured output amplitudes, can still matter.

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 $\epsilon(t)$ has units of angular frequency, $\Delta=\omega_{\mathrm d}-\omega_c$, and $\kappa$ is the energy-decay linewidth. In this chapter’s convention $b_{\mathrm{out}}=b_{\mathrm{in}}+\sqrt\kappa a$, driving through that sole port gives $\epsilon=-\sqrt\kappa\,\beta_{\mathrm{in}}$. The steady coherent amplitude for constant drive is $\langle a\rangle=\epsilon/(\kappa/2-i\Delta)$. 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 canonical rates signal failure of CP divisibility where an invertible time-local generator exists. Rates in an arbitrary redundant jump representation need not be canonical. An exact nondivisible evolution can still be CPTP from its initial time; arbitrary negative rates need not have that property. For the diagnostic distinction, see CP Divisibility and Time-Convolutionless Master Equations.

When the Drive Changes the Noise Model

The bare dissipators can remain accurate when the drive is slow or weak on the bath’s relevant scales and the spectrum is smooth across the sampled transition frequencies. Strong driving can split transitions into dressed-state or Floquet sidebands; using the undriven rates then misses the noise at those frequencies. Rapid control can also invalidate the coarse graining used to derive the equation.

Transform the Hamiltonian and jumps in the same frame, check that any secular frequency separations exceed dissipative linewidths, and include leakage states if the pulse bandwidth reaches them. For a periodic generator, the natural asymptotic state is often a periodic cycle. Convergence to a unique cycle still requires mixing; conserved sectors can retain initial-state dependence. Approximation Checklist gives a compact comparison of these assumptions.

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.

References

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