Input–Output Theory

Input–output theory relates a localized quantum system to traveling fields incident on it and emitted from it. It is the language behind cavity reflection, transmission, homodyne measurement, photon counting, circuit-QED readout, and many Markovian quantum-optics experiments. When the ports and drives are chosen to stabilize useful states, the same formalism becomes a workhorse for Reservoir Engineering.

This page owns the general Markovian field boundary, arbitrary system coupling operator, stochastic description, and relation to continuous measurement. Input–Output Theory Overview owns the practical empty-cavity dictionary: partial linewidths, reflection and transmission, critical coupling, ringdown, and port-resolved fitting.

The simplest slogan is:

output field = input field + field radiated by the system

For a single Markovian port with coupling operator $L$, a common convention is

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t) + L(t).$$

For a cavity mode coupled to one port with $L=\sqrt{\kappa}\,a$,

$$b_{\mathrm{out}}(t) = b_{\mathrm{in}}(t) + \sqrt{\kappa}\,a(t).$$

Some references use the opposite sign for the radiated term. The sign depends on phase conventions for the port field and coupling Hamiltonian; measurable spectra and fluxes are unchanged when the convention is used consistently.

Field Normalization

The input field $b_{\mathrm{in}}(t)$ is a traveling-wave annihilation operator normalized so that

$$[b_{\mathrm{in}}(t),b_{\mathrm{in}}^\dagger(t')] = \delta(t-t').$$

The photon flux has units of inverse time:

$$\Phi_{\mathrm{in}}(t) = \langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t)\rangle.$$

For vacuum input,

$$\langle b_{\mathrm{in}}(t)b_{\mathrm{in}}^\dagger(t')\rangle = \delta(t-t'), \qquad \langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t')\rangle = 0.$$

For a thermal input with occupation $\bar n$,

$$\langle b_{\mathrm{in}}^\dagger(t)b_{\mathrm{in}}(t')\rangle = \bar n\,\delta(t-t'),$$

and

$$\langle b_{\mathrm{in}}(t)b_{\mathrm{in}}^\dagger(t')\rangle = (\bar n+1)\delta(t-t').$$

These delta correlations are the time-domain version of the broadband Markov approximation.

Langevin Equation

For one cavity mode with Hamiltonian $H=\hbar\omega_c a^\dagger a$ and coupling $L=\sqrt{\kappa}\,a$, the Heisenberg–Langevin equation in the same convention is

$$\dot a = i[H/\hbar,a] - \frac{\kappa}{2}a - \sqrt{\kappa}\,b_{\mathrm{in}}(t).$$

Since

$$i[H/\hbar,a] = -i\omega_c a,$$

one obtains

$$\dot a = - \left( i\omega_c+\frac{\kappa}{2} \right)a - \sqrt{\kappa}\,b_{\mathrm{in}}(t).$$

The noise term is not optional. Without the input field, damping would not preserve the correct commutation relations.

Master Equation Connection

If the output field is not measured or retained, tracing it out gives the Lindblad dissipator

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \mathcal D[L]\rho,$$

where

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

For a cavity port with $L=\sqrt{\kappa}a$,

$$\mathcal D[L]\rho = \kappa\mathcal D[a]\rho.$$

Thus the same coupling operator appears in three related descriptions:

unconditional master equation:  D[L] rho
input–output boundary:          b_out = b_in + L
trajectory measurement:         output record determined by L

This is why collapse operators must be interpreted with their physical ports and measurement schemes specified.

Multiple Ports

If several independent ports couple to the same system, each has its own input and output field:

$$b_{j,\mathrm{out}}(t) = b_{j,\mathrm{in}}(t) + L_j(t).$$

The unconditional master equation contains

$$\sum_j \mathcal D[L_j]\rho.$$

For a two-sided cavity,

$$L_1=\sqrt{\kappa_1}\,a, \qquad L_2=\sqrt{\kappa_2}\,a,$$

and internal loss can be represented by another unmonitored channel

$$L_{\mathrm{int}} = \sqrt{\kappa_{\mathrm{int}}}\,a.$$

The total energy-decay rate is

$$\kappa = \kappa_1+\kappa_2+\kappa_{\mathrm{int}},$$

but only monitored ports contribute to measured output records.

Coherent Drives

A coherent incident field can be written as a mean amplitude plus quantum noise:

$$b_{\mathrm{in}}(t) = \beta_{\mathrm{in}}(t) + \xi_{\mathrm{in}}(t),$$

where $\xi_{\mathrm{in}}$ has vacuum or thermal correlations. For a cavity, the mean drive enters the equation of motion as

$$\dot a = - \left( i\omega_c+\frac{\kappa}{2} \right)a - \sqrt{\kappa}\,\beta_{\mathrm{in}}(t) - \sqrt{\kappa}\,\xi_{\mathrm{in}}(t).$$

Equivalently, the coherent drive may be moved into an effective Hamiltonian,

$$H_{\mathrm{drive}} = i\hbar \left( \sqrt{\kappa}\,\beta_{\mathrm{in}}^*a - \sqrt{\kappa}\,\beta_{\mathrm{in}}a^\dagger \right),$$

with the remaining input treated as noise. Different phase conventions move minus signs between $\beta_{\mathrm{in}}$, $H_{\mathrm{drive}}$, and the output relation.

Empty-Cavity Reflection

For a single-sided empty cavity probed at angular frequency $\omega$, define detuning

$$\Delta=\omega-\omega_c.$$

In steady state,

$$a = - \frac{\sqrt{\kappa}} {\kappa/2-i\Delta} b_{\mathrm{in}}.$$

Using $b_{\mathrm{out}}=b_{\mathrm{in}}+\sqrt{\kappa}a$ gives the reflection amplitude

$$r(\omega) = \frac{b_{\mathrm{out}}}{b_{\mathrm{in}}} = 1 - \frac{\kappa}{\kappa/2-i\Delta}.$$

For a lossless single-sided cavity, $|r(\omega)|=1$: the cavity changes the phase of the reflected field. On resonance, this convention gives $r=-1$.

With internal loss or a second output port, some incident power can be absorbed or transmitted, so the measured reflection is no longer a pure phase.

Output Flux

The output photon flux is

$$\Phi_{\mathrm{out}}(t) = \langle b_{\mathrm{out}}^\dagger(t)b_{\mathrm{out}}(t)\rangle.$$

Using $b_{\mathrm{out}}=b_{\mathrm{in}}+L$ gives

$$\begin{aligned} \Phi_{\mathrm{out}} =& \langle b_{\mathrm{in}}^\dagger b_{\mathrm{in}}\rangle + \langle L^\dagger L\rangle \\ &+ \langle b_{\mathrm{in}}^\dagger L\rangle + \langle L^\dagger b_{\mathrm{in}}\rangle. \end{aligned}$$

For vacuum input with no coherent amplitude, the interference terms vanish in the usual Markov treatment, and the emitted flux is

$$\Phi_{\mathrm{emit}} = \langle L^\dagger L\rangle.$$

For a driven cavity, the interference terms are essential. Reflection and transmission are field-amplitude phenomena, not just photon-number addition.

Homodyne and Heterodyne Records

Input–output theory supplies the field that is measured by homodyne or heterodyne detection. For a monitored port with vacuum input and coupling $L$, a homodyne quadrature record has the schematic form

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}L+e^{i\phi}L^\dagger\rangle_c\,dt + dW_t,$$

after subtracting known coherent input amplitudes. This is the same structure used in Diffusive Trajectories, with the detector model discussed in Homodyne Detection.

Photon counting instead samples the output intensity. For vacuum input and ideal detection, the count rate is

$$\mathbb E[dN_t\mid\rho_c] = \eta\, \langle L^\dagger L\rangle_c\,dt.$$

This is the field-theory origin of Photon Counting and Quantum Jump Trajectories. For two-quadrature diffusive monitoring of the same output field, see Heterodyne Detection.

Cascaded and Feedback Systems

Input–output theory also describes cascaded systems, where the output of one component becomes the input of another. In the simplest Markovian limit, propagation delay is neglected and the connection is one-way. The control-oriented entry point for such routed quantum signals is Coherent Feedback.

If delays, reflections, or finite-bandwidth propagation matter, the dynamics may become non-Markovian. Then the simple boundary relation remains locally meaningful, but the reduced system may require delay equations, explicit waveguide modes, or memory kernels.

Assumptions

The basic Markovian input–output formula assumes:

• a continuum of traveling modes near the system frequency;
• approximately frequency-independent coupling over the relevant bandwidth;
• rotating-wave coupling between system and field;
• short propagation time across the coupling region;
• initially specified input field states;
• no unresolved delayed feedback;
• ports that can be treated as independent channels.

Violating these assumptions does not make input–output language useless, but it does mean the single-line formula may no longer produce a valid Markovian master equation.

Common Mistakes

• Confusing the intracavity operator $a$ with the traveling output field $b_{\mathrm{out}}(t)$.
• Dropping the input noise term while keeping damping.
• Forgetting that $b_{\mathrm{in}}$ has units of square root of photon flux, while $a$ is dimensionless.
• Mixing sign conventions for $b_{\mathrm{out}}=b_{\mathrm{in}}\pm L$.
• Treating internal loss as if it were a monitored output port.
• Adding output intensities while ignoring coherent interference terms.
• Confusing cavity energy-decay rate $\kappa$ with field-amplitude decay rate $\kappa/2$.
• Applying Markovian input–output theory to long-delay feedback without keeping the delay.

Exercises

Output Flux from Vacuum Input

Assume $b_{\mathrm{out}}=b_{\mathrm{in}}+L$ and vacuum input with no coherent amplitude. Show why the emitted flux is $\langle L^\dagger L\rangle$.

Solution

The output flux is

$$\Phi_{\mathrm{out}} = \langle b_{\mathrm{out}}^\dagger b_{\mathrm{out}}\rangle.$$

Substitute the input–output relation:

$$\Phi_{\mathrm{out}} = \langle b_{\mathrm{in}}^\dagger b_{\mathrm{in}}\rangle + \langle L^\dagger L\rangle + \langle b_{\mathrm{in}}^\dagger L\rangle + \langle L^\dagger b_{\mathrm{in}}\rangle.$$

For vacuum input with no coherent amplitude, the normally ordered input flux and the interference terms vanish in the Markov treatment. Therefore

$$\Phi_{\mathrm{emit}} = \langle L^\dagger L\rangle.$$

Cavity Loss Channel

For $L=\sqrt{\kappa}a$, show that the unconditional dissipator is $\kappa\mathcal D[a]\rho$.

Solution

By definition,

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

With $L=\sqrt{\kappa}a$,

$$L\rho L^\dagger = \kappa a\rho a^\dagger, \qquad L^\dagger L = \kappa a^\dagger a.$$

Thus

$$\mathcal D[L]\rho = \kappa \left( a\rho a^\dagger - \frac12\{a^\dagger a,\rho\} \right) = \kappa\mathcal D[a]\rho.$$

Empty-Cavity Reflection

Using

$$r(\omega) = 1 - \frac{\kappa}{\kappa/2-i\Delta},$$

show that a lossless single-sided cavity has $|r(\omega)|=1$.

Solution

Rewrite

$$r(\omega) = \frac{\kappa/2-i\Delta-\kappa} {\kappa/2-i\Delta} = \frac{-\kappa/2-i\Delta} {\kappa/2-i\Delta}.$$

The numerator and denominator have the same magnitude:

$$\left|-\frac{\kappa}{2}-i\Delta\right|^2 = \left|\frac{\kappa}{2}-i\Delta\right|^2 = \frac{\kappa^2}{4}+\Delta^2.$$

Therefore $|r(\omega)|=1$.

Homodyne Signal

For a monitored port with $L=\sqrt{\kappa}a$, write the expected homodyne signal at phase $\phi$ after subtracting known coherent input.

Solution

The record has the form

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}L+e^{i\phi}L^\dagger\rangle_c\,dt + dW_t.$$

Substituting $L=\sqrt{\kappa}a$ gives

$$\mathbb E[dY_t\mid\rho_c] = \sqrt{\eta\kappa}\, \langle e^{-i\phi}a+e^{i\phi}a^\dagger\rangle_c\,dt.$$

Cross-Links

• Quantum Optical Master Equation
• Lindblad Operators
• Quantum Langevin Equations
• Thermal and Vacuum Noise
• Stochastic Master Equations
• Diffusive Trajectories
• Quantum Jump Trajectories
• Coherent Feedback
• Reservoir Engineering
• Noise Spectra
• Fluctuation–Dissipation Relation
• Approximation Checklist
• Jaynes–Cummings Model

References

• C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Physical Review A 31, 3761–3774 (1985).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer (2004).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• D. F. Walls and G. J. Milburn, Quantum Optics, Springer (2008).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010).