Coherent model of the AC Stark shift

Coherent model of the AC Stark shift

The AC Stark shift is a shift in the energy levels of an atom, in the presence of an electromagnetic field of frequency $\omega$, which is typically detuned far from the atomic resonance frequency $\omega _{0}$. For a two-level atom with dipole moment $d$, in an electric field of amplitude $E$, this "light shift" is characterized as a shift in the ground and excited state energies,

$\Delta E=\pm {\frac {(dE)^{2}}{4(\omega _{0}-\omega )}}\,,$

where the $+$ and $-$ signs refer to the shifts given to the excited and ground states, respectively.

This result is typically obtained through perturbation theory, but can also be obtained through analysis of the Bloch equations. In terms of the Bloch equations, the AC Stark shift arises from the fact that a far off-resonant field causes a rotation of the two-state "qubit" transition, primarily about its ${\hat {z}}$ axis.

No population change is typically associated with the AC Stark shift. As the detuning $\delta =\omega _{0}-\omega$ becomes infinitely large, no change of population between the excited and ground states is expected, because the excitation is so off-resonance.

In reality, however, for finite detunings, some population change does occur. Traditionally, this is neglected, since at worst it leads to a small amount of off-resonance scattering. This is then a source of decoherence, which will adversely impact coherent control experiments, such as in spin squeezing, where the effective Rabi frequency $dE$ can be high. Thus, it is important to known when the dynamics of off-resonant excitation can, and cannot be well approximated by the simple AC Stark shift model.

This writeup provides a non-perturbative derivation of the AC Stark shift, with which we can quantify the degree to which a simple AC Stark shift model is appropriate. We use the model to quantify the amount of population change as a function of detuning, and we also present a pulse scheme to reduce undesired population changes, while retaining a strong light shift.

Exact treatment off-resonant excitation of an atom

The Hamiltonian for a two-level atom (a "qubit") interacting with a classical field may be written in general as

$H={\frac {\omega _{0}}{2}}Z+g(X\cos \omega t+Y\sin \omega t)\,,$

where $g$ parameterizes the field strength, $\omega _{0}$ is the atomic transition frequency, and $X,Y,Z$ are the Pauli matrices as usual. Define $|\phi (t)\rangle =e^{i\omega tZ/2}|\chi (t){\rangle }$, such that the Schrödinger equation

$i\partial _{t}|\chi (t)\rangle =H|\chi (t){\rangle }$

can be re-expressed as

$i\partial _{t}|\phi (t){\rangle }=\left[{e^{i\omega Zt/2}He^{-i\omega Zt/2}-{\frac {\omega }{2}}Z}\right]|\phi (t){\rangle }\,.$

Since

$e^{i\omega Zt/2}Xe^{-i\omega Zt/2}=(X\cos \omega t-Y\sin \omega t)\,,$

Eq.(\ref{eq:nmr:schrB}) simplifies to become

$i\partial _{t}|\phi (t){\rangle }=\left[{{\frac {\omega _{0}-\omega }{2}}Z+gX}\right]|\phi (t){\rangle }\,,$

where the terms on the right multiplying the state can be identified as the effective `rotating frame' Hamiltonian. The solution to this equation is

$|\phi (t)\rangle =e^{i\left[{{\frac {\omega _{0}-\omega }{2}}Z+gX}\right]t}|\phi (0){\rangle }\,.$

It arises from the rotating frame Hamiltonian

$H_{rot}=-\left[{{\frac {\omega _{0}-\omega }{2}}Z+gX}\right]\,.$

The concept of resonance arises from the behavior of this time evolution, which can be understood as being a single qubit rotation about the axis

${\hat {n}}={\frac {{\hat {z}}+{\frac {2g}{\omega _{0}-\omega }}\,{\hat {x}}}{\sqrt {1+\left({\frac {2g}{\omega _{0}-\omega }}\right)^{2}}}}$

by an angle

$|{\vec {n}}|=t{\sqrt {\left({\frac {\omega _{0}-\omega }{2}}\right)^{2}+g^{2}}}\,.$

When $\omega$ is far from $\omega _{0}$, the qubit is negligibly affected by the laser field; the axis of its rotation is nearly parallel with ${\hat {z}}$, and its time evolution is nearly exactly that of the free atom Hamiltonian. On the other hand, when $\omega _{0}\approx \omega$, the free atom contribution becomes negligible, and a small laser field can cause large changes in the state, corresponding to rotations about the ${\hat {x}}$ axis.

The AC Stark shift

The usual expression for the AC Stark shift can be derived from the above treatment of two-level resonance by recognizing that the energy level shifts are given by the eigenvalues $\lambda _{\pm }$ of the rotating frame Hamiltonian $H_{rot}$. These are

$\lambda _{\pm }=\pm {\sqrt {g^{2}+{\frac {(\omega -\omega _{0})^{2}}{4}}}}\,.$

When $\omega$ is large, or equivalently, for this purpose, $g$ is small, these eigenvalues can be expanded using the fact that ${\sqrt {\epsilon ^{2}+b^{2}}}\approx b+\epsilon ^{2}/2b+O(\epsilon ^{3})$, giving

$\lambda _{\pm }\approx \pm \left[{{\frac {\omega -\omega _{0}}{2}}+{\frac {g^{2}}{\omega -\omega _{0}}}}\right]\,.$

These give energies in the laboratory frame corresponding to shifts of

$\Delta E\approx \pm {\frac {g^{2}}{\omega -\omega _{0}}}\,,$

which is in agreement with the usual result, recognizing that $g=\Omega /2$ is defined as being half the Rabi frequency $\Omega =dE$.

Phase and population change from off-resonant excitation

The AC Stark shift is often interpreted as an extra phase shift associated with the energy levels of the two-level atom, induced by an off-resonant field. However, that is imprecise, particularly at non-inifinite detunings. While the energy levels do shift by $\lambda _{\pm }$ as derived above, the energy eigenbasis of $H_{rot}$ is different from that of the bare two-level system Hamiltonian. More specifically, we have seen above that the rotation of the atomic state is actually about a tilted axis, with both ${\hat {z}}$ and ${\hat {x}}$ components. How much of the rotation is a phase shift (rotation about ${\hat {z}}$) and how much is a population change (rotation about ${\hat {x}}$), at a given detuning?

Bound on the maximum population change

Let us quantify the maximum population change possible by considering the largest effect possible on a state which is initially along $+{\hat {z}}$, that is, the excited state $|0{\rangle }$. Specifically, we are interested in calculating the probability of ending up in the state $-{\hat {z}}$, the ground state $|1{\rangle }$, due to off-resonant excitation.

We work in the rotating frame, and define the rotation operator induced by $H_{rot}$ acting for time $t$ as

$R(t)=e^{-iH_{rot}t}\,.$

Define

$F_{z}(t)=|\langle 1|R(t)|0{\rangle }|^{2}$

as being the probability of a z-axis flip from the rotation $R$, and for convenience, we let

$\theta _{g\omega }={\sqrt {g^{2}+\left({\frac {\omega -\omega _{0}}{2}}\right)^{2}}}\,.$

For the z-axis flip probability, we find

$F_{z}(t)=\left|g{\frac {\sin(t\theta _{g\omega })}{\theta _{g\omega }}}\right|^{2}\,.$

This is largest for time $t=t_{\pi }=\pi /2/\theta _{g\omega }$, corresponding to a $\pi$ rotation of the Bloch sphere about ${\hat {n}}$, giving

$F_{z}(t_{\pi })=\left|{\frac {g}{\theta _{g\omega }}}\right|^{2}\approx {\frac {4g^{2}}{(\omega -\omega _{0})^{2}}}\,.$

This measure gives an upper bound on how much the population can change due to off-resonant excitation of a two-level system.

Bound on the maximum phase shift

Similarly, we may define a "x-axis" flip probability

$F_{x}(t)=|{\langle }-|R(t)|+{\rangle }|^{2}\,,$

where $|\pm \rangle =(|0{\rangle }\pm |1\rangle )/{\sqrt {2}}$ correspond to the $\pm {\hat {x}}$ axes of the Bloch sphere. $F_{x}(t)$ thus quantifies how much phase shift occurs. For the x-axis flip probability, we find

$F_{x}(t)=\left|{\frac {\omega -\omega _{0}}{2}}{\frac {\sin(t\theta _{g\omega })}{\theta _{g\omega }}}\right|^{2}\,.$

At time $t_{\pi }$, this is

$F_{x}(t_{\pi })=\left|{\frac {\omega -\omega _{0}}{2\theta _{g\omega }}}\right|^{2}\approx 1-{\frac {4g^{2}}{(\omega -\omega _{0})^{2}}}\,.$

This measure gives an upper bound on the phase shift due to off-resonant excitation of a two-level system.

Short time behavior

Many experiments involve weak AC Stark shifts, corresponding to interaction times which give a rotation angle much less than $\pi$. The relevant expressions for such regimes are thus short time expansions of $F_{z}(t)$ and $F_{x}(t)$. For small $t$, $R\sim 1-iH_{rot}t$, such that

$F_{z}(t)=g^{2}t^{2}+O(t^{4})$

and

$F_{x}(t)=\left({\frac {\omega -\omega _{0}}{2}}\right)^{2}t^{2}+O(t^{4})\,.$

The $F_{z}(t)$ expansion may appear somewhat strange, as it does not apparently depend on the detuning, but there is a dependency: the expression is valid only as long as the $O(t^{4})$ term is not too large, meaning that

$t\ll {\frac {\sqrt {3}}{\theta _{g\omega }}}\,.$

These expressions describe the short time behavior of the phase shift and population change due to off-resonant excitation, and emphasize the fact that the two effects go hand-in-hand. There is no AC Stark shift without some degree of population change, and the change can be small or large, depending on the amount of detuning.

Pulse scheme to reduce population change

Often, it is desirable to utilize off-resonant excitation as a way to obtain a phase shift, with no population change, and this is usually accomplished by increasing the detuning. The downside of having to use large detunings is either longer times or higher Rabi frequencies are needed to get the desired phase shift, and both will cause more population change as well.

Another way to reduce the amount of population change $F_{x}(t)$, while leaving $F_{z}(t)$ relatively unchanged, is to apply not a simple continuous off-resonant excitation, but rather, a sequence of pulsed excitations, or a shaped excitation.

Specifically, consider a sequence in which the phase of the incident laser is shifted between $0$ and $\pi$. The interaction Hamiltonian in the laboratory frame, with a laser of phase $\phi$, is a slight modification of the expression we used above, namely

$H={\frac {\omega _{0}}{2}}Z+g(X\cos \omega t+e^{i\phi }Y\sin \omega t)\,.$

When $\pi =\pi$, then the rotating frame Hamiltonian changes from the $\pi =0$ expression,

$H_{rot,0}=-\left[{{\frac {\omega _{0}-\omega }{2}}Z+gX}\right]$

to become

$H_{rot,\pi }=-\left[{{\frac {\omega _{0}-\omega }{2}}Z-gX}\right]\,.$

By using a sequence of $\phi =0$ and $\phi =pi$ pulses, the amount of rotation about ${\hat {x}}$ can be reduced, as long as each pulse is applied for small time $t$.

The improvement offered by this method can be quantified by defining the phase and population change metrics

$F'_{x}(t)=|{\langle }-|R_{\pi }(t/2)R_{0}(t/2)|+{\rangle }|^{2}$

and

$F'_{z}(t)=|\langle 1|R_{\pi }(t/2)R_{0}(t/2)|0{\rangle }|^{2}\,,$

where $R_{\phi }(t)=\exp(-iH_{rot,\phi }t)$ is the rotation induced by an incident laser with phase $\phi$ applied for time $t$. We find, for small time $t$,

$F'_{x}(t)\approx \left({\frac {\omega -\omega _{0}}{2}}\right)^{2}t^{2}\,,$

and

$F'_{z}(t)\approx \left({\frac {g(\omega -\omega _{0})}{4}}\right)^{2}t^{4}\,,$

so the population change induced now goes as $O(t^{4})$ instead of $O(t^{2})$, which is a dramatic improvement, while the phase shift induced is of the same order as previously.

Note that instead of using a pulsed excitation scheme, equivalently the laser's phase could be modulated continuously between $0$ and $\pi$ to achieve essentially the same effect. The main criterion is that the modulation rate must be fast compared with higher order terms in the time expansion of the rotation operator, else higher order terms appear.