I have come across terms such as "dispersive regime" and "dispersive measurement" often.

To understand this, I have been reading this, this, and this.

In the first paper, the gist I got is the following:

• in the dispersive regime, the qubit is detuned from the resonator by an amount $\Delta$ and thus induces a shift $g^2/\Delta$ in the resonance frequency (I believe the shift is happening to the resonator's resonance frequency).

In the second paper, what I got out of is that dispersive regime is a perturbative approximation of the Jaynes-Cummings Hamiltonian. Also, I got the following:

• In the limit that detuning between the cavity and the qubit is large, no energy is exchanged. In this situation, the interaction is said to be dispersive.

To improve my understanding, the third paper was useful with Fig 5 in the paper, which shows the resonator's frequency shift in either direction, dependent on the state of the qubit.

My question centers around the equation in the second paper (shown below).

It seems you can understand the effect of $2g \lambda[a^+ a +\frac{1}{2}] \frac{\sigma_z}{2}$ in two ways, as stated below the equation. For this though, are we ignoring $\frac{1}{2}$ term after $a^+ a$ term?

Additionally, I would appreciate it if someone could provide a more holistic picture of the so-called dispersive regime. I feel that I haven't internalized it yet.

to second order in $\lambda$, it is simple to obtain the effective Hamiltonian describing the dispersive regime \begin{align} H_\mathrm{eff} & = \mathbf D_\mathrm{Linear}^\dagger H_s\mathbf D_\mathrm{Linear} \\ & = \hbar \omega_r a^\dagger a + \hbar \left( \omega_a + 2g\lambda \left[a^\dagger a+\frac12\right]\right) + \mathcal O(\lambda^2). \tag{3.3} \end{align}

The qubit transition frequency is shifted by a quantity proportional to the photon population $2g\lambda\langle a^\dagger a\rangle$. Alternatively, this shift can be seen as a qubit dependent pull of the resonator frequency $\omega_r \to \omega_r \pm g\lambda$. As a result, shinning microwaves at the input port of the resonator at a frequency close to $\omega_r$ and measuring the transmitted signal using standard homodyne

I have come across terms such as "dispersive regime" and "dispersive measurement" often.

To understand this, I have been reading this, this, and this.

In the first paper, the gist I got is the following:

• In the dispersive regime, the qubit is detuned from the resonator by an amount $\Delta$ and thus induces a shift $g^2/\Delta$ in the resonance frequency (I believe the shift is happening to the resonator's resonance frequency).

In the second paper, what I got out of is that dispersive regime is a perturbative approximation of the Jaynes-Cummings Hamiltonian. Also, I got the following:

• In the limit that detuning between the cavity and the qubit is large, no energy is exchanged. In this situation, the interaction is said to be dispersive.

To improve my understanding, Fig 5 in the third paper was useful as it shows the resonator's frequency shift in either direction, dependent on the state of the qubit.

My question centers around the equation in the second paper (shown below). It seems you can understand the effect of $$2g \lambda \left( a^+ a +\frac{1}{2} \right) \frac{\sigma_z}{2}$$ in two ways, as stated below the equation. For this though, are we ignoring $\frac{1}{2}$ term after $a^+ a$ term?

Additionally, I would appreciate it if someone could provide a more holistic picture of the so-called dispersive regime. I feel that I haven't internalized it yet.

to second order in $\lambda$, it is simple to obtain the effective Hamiltonian describing the dispersive regime \begin{align} H_\mathrm{eff} & = \mathbf D_\mathrm{Linear}^\dagger H_s\mathbf D_\mathrm{Linear} \\ & = \hbar \omega_r a^\dagger a + \hbar \left( \omega_a + 2g\lambda \left[a^\dagger a+\frac12\right]\right) + \mathcal O(\lambda^2). \tag{3.3} \end{align}

The qubit transition frequency is shifted by a quantity proportional to the photon population $2g\lambda\langle a^\dagger a\rangle$. Alternatively, this shift can be seen as a qubit dependent pull of the resonator frequency $\omega_r \to \omega_r \pm g\lambda$. As a result, shinning microwaves at the input port of the resonator at a frequency close to $\omega_r$ and measuring the transmitted signal using standard homodyne

I have come across terms such as "dispersive regime" and "dispersive measurement" often.

To understand this, I have been reading this, this, and this.

In the first paper, the gist I got is the following: "in the dispersive regime, the qubit is detuned from the resonator by an amount $\Delta$ and thus induces a shift $g^2/\Delta$ in the resonance frequency (I believe the shift is happening to the resonator's resonance frequency).

In the second paper, what I got out of is that dispersive regime is a perturbative approximation of the Jaynes-Cummings Hamiltonian. Also, I got the following: "In the limit that detunning between the cavity and the qubit is large, no energy is exchange. In this situation, the interaction is said to be dispersive."

To improve my understanding, the third paper was useful with Fig 5 in the paper, which shows the resonator's frequency shift in either direction, dependent on the state of the qubit.

My question centers around the equation in the second paper (shown below).

It seems you can understand the effect of $2g \lambda[a^+ a +\frac{1}{2}] \frac{\sigma_z}{2}$ in two ways, as stated below the equation. For this though, are we ignoring $\frac{1}{2}$ term after $a^+ a$ term?

Additionally, I would appreciate it if someone could provide a more holistic picture of the so-called dispersive regime. I feel that I haven't internalized it yet.

I have come across terms such as "dispersive regime" and "dispersive measurement" often.

To understand this, I have been reading this, this, and this.

In the first paper, the gist I got is the following:

• in the dispersive regime, the qubit is detuned from the resonator by an amount $\Delta$ and thus induces a shift $g^2/\Delta$ in the resonance frequency (I believe the shift is happening to the resonator's resonance frequency).

In the second paper, what I got out of is that dispersive regime is a perturbative approximation of the Jaynes-Cummings Hamiltonian. Also, I got the following:

• In the limit that detuning between the cavity and the qubit is large, no energy is exchanged. In this situation, the interaction is said to be dispersive.

To improve my understanding, the third paper was useful with Fig 5 in the paper, which shows the resonator's frequency shift in either direction, dependent on the state of the qubit.

My question centers around the equation in the second paper (shown below).

It seems you can understand the effect of $2g \lambda[a^+ a +\frac{1}{2}] \frac{\sigma_z}{2}$ in two ways, as stated below the equation. For this though, are we ignoring $\frac{1}{2}$ term after $a^+ a$ term?

Additionally, I would appreciate it if someone could provide a more holistic picture of the so-called dispersive regime. I feel that I haven't internalized it yet.

to second order in $\lambda$, it is simple to obtain the effective Hamiltonian describing the dispersive regime \begin{align} H_\mathrm{eff} & = \mathbf D_\mathrm{Linear}^\dagger H_s\mathbf D_\mathrm{Linear} \\ & = \hbar \omega_r a^\dagger a + \hbar \left( \omega_a + 2g\lambda \left[a^\dagger a+\frac12\right]\right) + \mathcal O(\lambda^2). \tag{3.3} \end{align}

The qubit transition frequency is shifted by a quantity proportional to the photon population $2g\lambda\langle a^\dagger a\rangle$. Alternatively, this shift can be seen as a qubit dependent pull of the resonator frequency $\omega_r \to \omega_r \pm g\lambda$. As a result, shinning microwaves at the input port of the resonator at a frequency close to $\omega_r$ and measuring the transmitted signal using standard homodyne