User:Ninnatd/input output draft

The input-output formalism, is a formalism developed by M. J. Collett and C. W. Gardiner to model the behavior of systems coupled weakly to the environment in the Heisenberg picture. Therefore, it is appropriate for systems whose Heisenberg equations of motion are linear and thus can be solved easily.

In this model, the environment is treated as one or families of modes of excitation, each family being called a reservoir. An input field from a reservoir interacts with the system resulting in an output field. The reservoirs can be classified according to their uses into a noise reservoir and a signal reservoir.

• For a noise reservoir, which is a part of the environment we are not interested in, the goal is to eliminate the reservoir degree of freedoms: the input and output fields, from the description of the dissipative system. The input-output formalism gives a relation between quantum noise and the input fields.

• For a signal reservoir such as a classical driving field the goal is to find out about the system from the reservoir fields to be observed: the output. Historically, a system interacting with a reservoir such as a thermal reservoir or a classical driving field (coherent state reservoir) was studied using methods with little generality. The input-output formalism allows us to treat an arbitrary reservoir, for instance, a squeezed state reservoir, as an input.

The formalism can handle both types of reservoirs as long as the interaction is linear in the reservoir degree of freedom. This excludes, for example, a model for electrical resistance in metal, which arises from scattering of electrons by phonons and impurities and thus depends on the number of phonons.

Derivation of the formalism

In the standard input-output formalism, there are assumptions that mimic the Wigner-Weisskopf theory of spontaneous emission which captures a typical system-reservoir model in quantum optics.

• The spectrum of reservoir frequencies is continuous.

• The system has a discrete spectrum.

• The system is weakly coupled to a broad frequency range of the reservoir.

the Hamiltonian that describes the system $S$ and the reservoir $R$ is

$H=H_{S}+H_{R}+H_{int},$

$H_{R}=\hbar \int _{0}^{\infty }d\omega \omega b^{\dagger }(\omega )b(\omega ),$

$H_{SR}=i\hbar \int _{0}^{\infty }d\omega g(\omega )\left(a^{\dagger }b(\omega )-ab^{\dagger }(\omega )\right),$

where $a$ is the annihilation operator for the system, i.e. $\left[a,H_{S}\right]=\hbar \omega a,$ $b(\omega )$ is the annihilation operator for a reservoir mode with frequency $\omega$ that obeys the commutation relation

$\left[b(\omega ),b^{\dagger }(\omega ')\right]=\delta \left(\omega -\omega '\right),$

and $g(\omega )$ is a function involving the coupling strength and the density of the reservoir modes. Here, $g(\omega )$ is assumed to be real since any phase can be absorbed into the definition of $b(\omega )$. (The $i$ in $H_{SR}$ is for convenience.) For simplicity, we only consider a single reservoir. A generalization to multiple reservoirs is often necessary, even to accommodate a vacuum input, but straightforward by adding together the Hamiltonians of every reservoir and system-reservoir interaction.

The interaction Hamiltonian has a form similar to the Jaynes-Cummings interaction Hamiltonian after the rotating wave approximation. In the usual presentation, , the range of integration includes non-physical negative frequencies and is extended to $-\infty$ from the beginning. This can arise when we go into a rotating frame with angular frequency $\omega _{0}$, and the bandwidth of interest is very small compared to $\omega _{0}$. But we will follow Garrison and Chiao and keep the range of integration to only positive frequencies to make clear where and how we need to make further approximations.

Assume that $H_{S}$ has a characteristic frequency $\omega _{0}$ much larger than a characteristic response frequency of the interaction since $H_{SR}$ is weak compared to $H_{S}$. So to separate the fast and slow evolution, let us define the slowly-varying operators ${\bar {a}}(t)=a(t)e^{i\omega _{0}t}$ and ${\bar {b}}(\omega ,t)=b(\omega ,t)e^{i\omega _{0}t}$ whose Heisenberg equations of motion are

${\begin{aligned}{\frac {d}{dt}}{\bar {a}}\left(\omega ,t\right)&=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]+\int _{0}^{\infty }d\omega g(\omega ){\bar {b}}(\omega ,t),\\{\frac {d}{dt}}{\bar {b}}\left(\omega ,t\right)&=-i\left(\omega -\omega _{0}\right){\bar {b}}(\omega ,t)-g(\omega ){\bar {a}}(t).\end{aligned}}$

Now we can choose to calculate everything w.r.t. to a time before the interaction $t_{0}<t$ or a time after the interaction $t_{1}>t$.

In-fields

For $t_{0}<t$,

${\bar {b}}\left(\omega ,t\right)={\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}-g(\omega )\int _{t_{0}}^{t}dt'{\bar {a}}(t')e^{-i\left(\omega -\omega _{0}\right)\left(t-t'\right)}$

So

${\frac {d}{dt}}{\bar {a}}\left(\omega ,t\right)=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]+\int _{0}^{\infty }d\omega g(\omega ){\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}-\int _{-\omega _{0}}^{\infty }d\Omega \left|g\left(\Omega +\omega _{0}\right)\right|^{2}\int _{t_{0}}^{t}dt'{\bar {a}}(t')e^{-i\Omega \left(t-t'\right)},$

where $\Omega =\omega -\omega _{0}$. Since ${\bar {a}}(t')$ is slowly varying compared to the oscillating exponential function, it can be taken out of the $t'$ integral. Moreover, $\int _{t_{0}}^{t}dt'e^{-i\Omega t'}$ is sharply peaked at $\Omega =0$. Therefore, the lower limit of the $\omega$ integral can be extended to $-\infty$ with negligible error, yielding

${\frac {d}{dt}}{\bar {a}}\left(\omega ,t\right)=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]+\int _{0}^{\infty }d\omega g(\omega ){\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}-\int _{-\infty }^{\infty }d\Omega \left|g\left(\Omega +\omega _{0}\right)\right|^{2}{\bar {a}}(t)\int _{t_{0}}^{t}dt'e^{-i\Omega \left(t-t'\right)}$

The Markov approximation dictates that the system coupling equally to broad spectrum of the reservoir $2\pi \left|g(\omega )\right|^{2}\simeq 2\pi \left|g(\omega _{0})\right|^{2}=\kappa$.

Defining $b_{\mbox{in}}(t)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{\infty }d\omega {\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}$,

${\begin{aligned}{\frac {d}{dt}}{\bar {a}}\left(\omega ,t\right)&=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]+{\sqrt {\frac {\kappa }{2\pi }}}\int _{0}^{\infty }d\omega {\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}-{\frac {\kappa }{2\pi }}{\bar {a}}(t)\int _{t_{0}}^{t}dt'\int _{-\infty }^{\infty }d\Omega e^{-i\Omega \left(t-t'\right)}\\&=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]-{\frac {\kappa }{2}}{\bar {a}}(t)+{\sqrt {\kappa }}b_{\mbox{in}}(t),\end{aligned}}$

where $\int _{t_{0}}^{t}dt'\delta \left(t-t'\right)={\frac {1}{2}}$. This is a retarded quantum Langevin equation with the noise operator $\xi (t)={\sqrt {\kappa }}b_{\mbox{in}}(t)$.

The retarded Langevin equation for ${\bar {c}}(t)=c(t)e^{i\omega _{0}t}$, where $c(t)$ is an arbitrary system operator, is

${\begin{aligned}{\frac {d}{dt}}{\bar {c}}(t)&=-{\frac {i}{\hbar }}\left[{\bar {c}},H_{S}\right]+\int _{0}^{\infty }d\omega {\sqrt {\frac {\kappa }{2\pi }}}\left(\left[{\bar {c}},a^{\dagger }\right]b(\omega ,t)-\left[{\bar {c}},a\right]b^{\dagger }(\omega ,t)\right)\\&=-{\frac {i}{\hbar }}\left[{\bar {c}},H_{S}\right]+{\sqrt {\frac {\kappa }{2\pi }}}\int _{0}^{\infty }d\omega \left(\left[{\bar {c}},{\bar {a}}^{\dagger }\right]{\bar {b}}\left(\omega ,t_{0}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}-\left[{\bar {c}},{\bar {a}}\right]{\bar {b}}^{\dagger }\left(\omega ,t_{0}\right)e^{i\left(\omega -\omega _{0}\right)\left(t-t_{0}\right)}\right)\\&-{\frac {\kappa }{2\pi }}\int _{0}^{\infty }d\omega \left(\left[{\bar {c}},{\bar {a}}^{\dagger }\right]\int _{t_{0}}^{t}dt'{\bar {a}}(t')e^{-i\left(\omega -\omega _{0}\right)\left(t-t'\right)}-\left[{\bar {c}},{\bar {a}}\right]\int _{t_{0}}^{t}dt'{\bar {a}}^{\dagger }(t')e^{i\left(\omega -\omega _{0}\right)\left(t-t'\right)}\right)\\&=-{\frac {i}{\hbar }}\left[{\bar {c}},H_{S}\right]-\left\{\left[{\bar {c}},{\bar {a}}^{\dagger }\right]\left({\frac {\kappa }{2}}{\bar {a}}(t)-{\sqrt {\kappa }}{\bar {b}}_{\mbox{in}}(t)\right)-\left({\frac {\kappa }{2}}{\bar {a}}^{\dagger }(t)-{\sqrt {\kappa }}{\bar {b}}_{\mbox{in}}^{\dagger }(t)\right)\left[{\bar {c}},{\bar {a}}\right]\right\}.\end{aligned}}$

The unequal-time commutation relation for the in-field can be calculated using the equal time commutation relation to be

$\left[b_{\mbox{in}}(t),b_{\mbox{in}}^{\dagger }(t')\right]=\int _{-\omega _{0}}^{\infty }{\frac {d\Omega }{2\pi }}e^{-i\Omega \left(t-t'\right)}.$

Integrating this againts a test function $f(t)$ gives

$\int dt'\int _{-\omega _{0}}^{\infty }{\frac {d\Omega }{2\pi }}e^{-i\Omega \left(t-t'\right)}f(t')=\int _{-\omega _{0}}^{\infty }{\frac {d\Omega }{2\pi }}e^{-i\Omega t}f(\Omega ).$

Since we are only interested in a slowly-varying function $f(t)$ in this approximation, $f(\Omega )$ is peaked at $\Omega =0$ and the same argument to extend the lower limit of integration to $-\infty$ is applied again, giving

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

There is a few things to note:

• $b_{\mbox{in}}(t)$ is defined in terms of $b(t_{0},\omega )$ at time $t_{0}<t$. It is not evolving as a Heisenberg operator.

• The term $-{\frac {\kappa }{2}}{\bar {a}}(t)$ gives the damping effect regardless of the reservoir quantum state, whereas the noise statistics does depend on the reservoir state. The damping is Markovian, having no memory, because the damping term only depends on the system operator at the current time.

• The in-field is required to preserve the commutation relation $\left[a,a^{\dagger }\right]$. There is always a vacuum input in an "empty" input port.

Out-fields

For $t_{1}>t$,

${\bar {b}}\left(\omega ,t\right)={\bar {b}}\left(\omega ,t_{1}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{1}\right)}+{\sqrt {\kappa }}\int _{t}^{t_{1}}dt'{\bar {a}}(t')e^{-i\left(\omega -\omega _{0}\right)\left(t-t'\right)}.$

Defining

${\bar {b}}_{\mbox{out}}\left(t\right)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{\infty }d\omega {\bar {b}}\left(\omega ,t_{1}\right)e^{-i\left(\omega -\omega _{0}\right)\left(t-t_{1}\right)},$

apparently with the same commutation relation as ${\bar {b}}_{\mbox{in}}(t)$, we obtain an advanced quantum Langevin equation

${\frac {d}{dt}}{\bar {a}}(t)=-{\frac {i}{\hbar }}\left[{\bar {a}}(t),H_{S}(t)\right]+{\frac {\kappa }{2}}{\bar {a}}(t)+{\sqrt {\kappa }}{\bar {b}}_{\mbox{out}}$

Note again that $b_{\mbox{out}}(t)$ is defined in terms of $b(t_{1},\omega )$ at time $t_{1}>t$. It is not evolving as a Heisenberg operator.

The advanced Langevin equation for an arbitrary system operator can similarly be derived by the substitution

${\begin{aligned}b_{\mbox{in}}(t)&\mapsto b_{\mbox{out}}(t),\\{\sqrt {\kappa }}&\mapsto {\sqrt {\kappa }},\\{\frac {\kappa }{2}}&\mapsto -{\frac {\kappa }{2}}:\end{aligned}}$

${\frac {d}{dt}}{\bar {c}}(t)=-{\frac {i}{\hbar }}\left[{\bar {c}},H_{S}\right]-\left\{\left[{\bar {c}},{\bar {a}}^{\dagger }\right]\left(-{\frac {\kappa }{2}}{\bar {a}}(t)-{\sqrt {\kappa }}{\bar {b}}_{\mbox{in}}(t)\right)-\left(-{\frac {\kappa }{2}}{\bar {a}}^{\dagger }(t)-{\sqrt {\kappa }}{\bar {b}}_{\mbox{in}}^{\dagger }(t)\right)\left[{\bar {c}},{\bar {a}}\right]\right\}$

Inputs and outputs

The input-output relation can be found by equating the two Langevin equations:

${\bar {b}}_{\mbox{out}}(t)={\bar {b}}_{\mbox{in}}(t)-{\sqrt {\kappa }}{\bar {a}}(t).$

This allows us to obtain the unequal time commutation relation between an arbitrary system operator and the fields in terms of the system operators alone. First, because the definition of $b_{\mbox{in}}$ and $b_{\mbox{out}}$, the retarded and advanced Langevin equations respect causality:

${\begin{aligned}\left[{\bar {c}}(t),{\bar {b}}_{\mbox{in}}(t')\right]&=0,\ t'>t,\\\left[{\bar {c}}(t),{\bar {b}}_{\mbox{out}}(t')\right]&=0,\ t'<t,\end{aligned}}$

Therefore, using the input-output relation,

${\begin{aligned}\left[{\bar {c}}(t),{\bar {b}}_{\mbox{in}}(t')\right]&=\left[{\bar {c}}(t),{\bar {b}}_{\mbox{out}}(t')+{\sqrt {\kappa }}{\bar {a}}(t')\right]=\Theta \left(t-t'\right){\sqrt {\kappa }}\left[{\bar {c}}(t),{\bar {a}}(t')\right],\\\left[{\bar {c}}(t),{\bar {b}}_{\mbox{out}}(t')\right]&=\left[{\bar {c}}(t),{\bar {b}}_{\mbox{in}}(t')-{\sqrt {\kappa }}{\bar {a}}(t')\right]=-\Theta \left(t'-t\right){\sqrt {\kappa }}\left[{\bar {c}}(t),{\bar {a}}(t')\right],\end{aligned}}$

where $\Theta \left(t-t'\right)$ is the Heaviside step function.

Our derivation is now completed. For a noise reservoir, the evolution of any system operator can be calculated without referring to the reservoir. For a signal reservoir, the output can be computed from the input and the system operator ${\bar {a}}(t)$.

The field interpretation

Because of the input-output relation and causality of the Langevin equations, $b_{\mbox{in}}(t)$ and $b_{\mbox{out}}(t)$ can be interpreted as inputs and outputs to the system, which acts as a boundary condition.

To further clarify the formalism and the nature of these fields, let us consider the implementation of the formalism as a one-dimensional semi-infinite transmission line. Suppose that the system is localized at position, say, $x=0$. There is a field $A\left(x,t\right)$ propagates from the right, interacts with the system, and propagates back to the right.

The one-dimensional semi-infinite transmission line model

The Lagrangian that describes the system $S$ and the (massless free scalar) field is

$L=L_{S}+{\frac {1}{2}}\int _{0}^{\infty }dx\left\{{\dot {A}}\left(x,t\right)^{2}-c^{2}\left[\partial _{x}A\left(x,t\right)\right]^{2}\right\}+c\int _{0}^{\infty }dxg(x){\dot {A}}\left(x,t\right),$

where $g(x)$ is the coupling function and $c$ is a system operator.

The field has a canonical momentum

$\pi \left(x,t\right)={\dot {A}}\left(x,t\right)+g(x)q.$

If $g(x)$ is strictly a Dirac delta function, then this Lagrangian cannot be turned into a Hamiltonian because $\pi ^{2}\left(x,t\right)$ contains the mathematically undefined square of Dirac delta function. Nevertheless, this is a continuum analog of a system interacting with an oscillator reservoir. It exhibits damping and, moreover, the input and output are actual quantum fields.

The reason we are considering this model is that several assumptions made in the derivation of the formalism above is rather natural in this context.

• The Markovian assumption that $g(\omega )$ is flat amounts to the assumption that the interaction is local: $g(x)\simeq \delta (x)$.

• The damping is caused by radiation damping.

• Once we define Fourier transform operators

${\begin{aligned}A\left(x,t\right)&={\sqrt {\frac {2}{\pi c}}}\int _{0}^{\infty }d\omega q\left(\omega ,t\right)\cos \left(\omega x/c\right),\\\pi \left(x,t\right)&={\sqrt {\frac {2}{\pi c}}}\int _{0}^{\infty }d\omega p\left(\omega ,t\right)\cos \left(\omega x/c\right),\\a(\omega ,t)&={\frac {\omega q\left(\omega ,t\right)+ip\left(\omega ,t\right)}{\sqrt {2\hbar \omega }}},\end{aligned}}$

and

${\begin{aligned}A_{\mbox{in}}(t)&={\frac {1}{2}}\int _{0}^{\infty }d\omega {\sqrt {\frac {\hbar }{\pi \omega c}}}\left[a\left(\omega ,t_{0}\right)e^{-i\omega \left(t-t_{0}\right)}+a^{\dagger }\left(\omega ,t_{0}\right)e^{i\omega \left(t-t_{0}\right)}\right],\\A_{\mbox{out}}(t)&={\frac {1}{2}}\int _{0}^{\infty }d\omega {\sqrt {\frac {\hbar }{\pi \omega c}}}\left[a\left(\omega ,t_{1}\right)e^{-i\omega \left(t-t_{1}\right)}+a^{\dagger }\left(\omega ,t_{1}\right)e^{i\omega \left(t-t_{1}\right)}\right].\end{aligned}}$

Then it can be shown that the in- and out- field satisfy the input-output relation

$A_{\mbox{out}}(t)=A_{\mbox{in}}(t)-{\frac {1}{2}}\int _{-\infty }^{\infty }d\tau g(c\tau )c\left(t-\tau \right),$

and that outside the range of interaction,

$A\left(x,t\right)=A_{\mbox{in}}\left(t+x/c\right)+A_{\mbox{out}}\left(t-x/c\right).$

That is, the total field is simply the sum of the in- and out-fields, and the out-field is the sum of a reflected in-field and the radiated field ${\frac {1}{2}}\int _{-\infty }^{\infty }d\tau g(c\tau )c\left(t-\tau \right)$ from the system.