Can anyone give me the details of creating a cat state in Circuit QED?

In summary, the conversation is about the speaker's struggles with understanding circuit quantum electrodynamics and its various components such as resonators, qubit resonance frequencies, Hamiltonian, coupling strength, Hilbert-space cutoff, dissipation rate, and Wigner function of the cavity. They also discuss their attempts to learn about these concepts from various sources but still having difficulty comprehending them. The speaker then explains their proposed steps for manipulating the system, including applying a drive pulse to the qubit and cavity and using a displacement operator to generate a coherent state. They also question whether the resulting state is entangled and express confusion about the role of quantum back action in this situation. Overall, the speaker is seeking clarification and understanding on these complex topics.
  • #1
physicsclaus
20
5
TL;DR Summary
Hello everyone, I want to create a cat state by transforming a system of ground state to the system of quantum supposition of coherent states with the following steps. However, I do not know how to really achieve this experimentally and what I should consider.
I am so new to circuit quantum electrodynamics. As far as I know, there are few things I could manipulate, like resonator, qubit resonance frequencies, Hamiltonian, coupling strength, Hilbert-space cutoff, dissipation rate, but they do not make sense to me and I do not how they can relate to my desired way of transformation. I think I know the basic mathematics to transform the state in bracket notation, the physics concept is the most difficult part. I have to admit that I have quite a lot of misunderstanding on doing this transformation, I hope you could bear with me and answer my questions.

Besides, I do not understand Wigner function of cavity and why it is useful to Schrodinger Cat States.

I tried to read some wiki pages and some lecture notes, but I cannot comprehend them at this learning stage. I hope someone could me some digests.

Here is my reference: S. M. Girvin, Schrodinger Cat States in Circuit QED, arXiv:1710.03179 [quant-ph]
https://arxiv.org/abs/1710.03179v1

I mostly read section 1.4, other parts are quite hard for me to grasp the concept. I appreciate if someone can provide me more comprehensible learning materials.

\section{My First Step}
Initially, the system is in the ground state: $$\ket{\Psi_i} = \ket{0} \otimes \ket{g}$$
\section{My Second Step}
Then, I want to apply a drive pulse to the qubit to place it in an equal superposition of ground and excited states:

$$\ket{\psi} = \frac{1}{\sqrt{2}} \ket{0} \otimes [\ket{g}+\ket{e}]$$

However, I am not sure how to transform the initial state to superposition.

I propose that we can shine cohere electromagnetic radiation at frequency $\omega$ s.t. $E = \hbar\omega$ is the energy difference between the states, which causes Rabi oscillation between ground state ans excited state over time. But how the rabi frequency can be fixed? Is it by induced dipole moment and the intensity of the driving field? How can we induce the dipole moment then?

In quantum circuit, we simply apply a Hadamard gate to it. Thinking of Bloch Sphere, I think I need to apply a $\frac{\pi}{2}$ pulse to the qubit to make the state transform from $\ket{0}$ to be along with the x-axis.

\section{My Third Step}

After I have the superposition state, I want to apply a drive pulse to the cavity to displace it by an amount $2\alpha$, that I want to drive it into a coherent state $\ket{2\alpha}$, which is coupled to the ground state.

$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{2\alpha} \otimes \ket{g} + \ket{0} \otimes \ket{e}]$$

To achieve, I think I can apply a displacement operator to generate the coherent state.

I consider the context of quantum harmonic oscillator, I have the Hamiltonian characterized as follow:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega\hat{x}^2 = \hbar\omega(\hat{a}^\dag \hat{a}+\frac{1}{2}) = \hbar\omega(\hat{N}+\frac{1}{2})$$
, which can also be represented in terms of ladder operator or number operator.

The eigenvalue equation of the Hamiltonian of the quantum harmonic oscillator
$$\hat{H} \ket{n} = E_n \ket{n}$$
The eigenvalue is quantized, so I have
$E_n = \hbar\omega(n+\frac{1}{2})$, for $n=0,1,2,3,\dots$
The associated eigenstate n can be constructed by acting n times of the raising operator on the ground state
$\ket{n} = \frac{(\hat{a}^\dag)^n}{\sqrt{n!}}\ket{0}$

I can have a lowering operator, which can move between eigenstates. So the lowering operator can lower the energy state by one quantum. Similarly, I can have the raising operator to have the following relation
$$\hat{a}\ket{n} = \sqrt{n}\ket{n-1}, \hat{a}^\dag\ket{n} = \sqrt{n+1}\ket{n+1}$$

Now I define my displacement operator as $\hat{D}$, displaced by an amount $\alpha$
$$\hat{D}(\alpha) = e^{\alpha \hat{a}^\dag - \alpha^*\hat{a}}$$

However, I only speculate and I am not sure if it is defined like this.

Assume it is unitary (I know I can prove it, but I don't want to make my post to lengthy)

I can express the unitary transformation of the displacement operator as:

$$\hat{D}^\dag(\alpha)\hat{a}\hat{D} (\alpha)= \hat{D}^\dag(\alpha)(\hat{D}(\alpha)\hat{a}+ \alpha\hat{D}(\alpha))$$

with the commutation relation

$[\hat{a}, \hat{D}(\alpha)] = \alpha\hat{D}(\alpha) \implies \hat{a} \hat{D}(\alpha) = \hat{D}(\alpha) \hat{a} + \alpha \hat{D}(\alpha)$

Then I can have

$$\hat{D}^\dag(\alpha)\hat{a}\hat{D} (\alpha) = \hat{D}^\dag(\alpha)\hat{D}(\alpha)\hat{a} + \hat{D}^\dag(\alpha)\hat{D}(\alpha)\alpha = \hat{a} + \alpha$$
Similarly, I have
$$\hat{D}^\dag(\alpha)\hat{a}^\dag\hat{D} (\alpha) = \hat{a}^\dag + \alpha^*$$

Considering the coherent state:

$$\hat{a}\ket{\alpha} = \alpha\ket{\alpha}$$

I have

$$\hat{a}(\hat{D}(\alpha)\ket{0}) = \alpha(\hat{D}(\alpha)\ket{0}) = \alpha \ket{0+\alpha} $$

, which confirms that $\hat{D}(\alpha)\ket{0}$ is the eigenstate of lowering operator with eigenvalue $\alpha$, so we can generate coherent state $\ket{\alpha}$ by the application of the displacement operator on the ground state. That is how I can have $\ket{0} \rightarrow \ket{2\alpha}$.
However, I do not know how the operation works in experimental setup.By the way, is this one an entangled state and why?
$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{2\alpha} \otimes \ket{g} + \ket{0} \otimes \ket{e}]$$

In my opinion, the cat's life is correlated to the existence of poison. But how can I sure if it is not a product state?

Should there be a phase and make it like this?:
$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{2\alpha} \otimes \ket{g} \pm \ket{0} \otimes \ket{e}]$$

I know there is a term called quantum back action, but I am not quite certain how this phenomenon related to this situation.

I think I have quite a lot of misunderstanding related to microwave engineering. For instance, let's say the qubit is in ground state why the cavity resonance frequency is $\omega_c$. When the qubit is in the excited state, what would be the cavity frequency? I do not know the terms like line-width, cavity response, susceptibility, strong-dispersive region, etc. And why does the cavity state can displace from $\ket{0}$ to $\ket{\alpha}$ if and only if the qubit is in ground state. Why the cavity remains in the vacuum state if the qubit is in the excited state? How can this relation correspond to the line-width? \section{My Fourth Step}
Next, I wan to have a state that the qubit can flip the excited state back to the ground state when the cavity is in the ground state by applying a drive pulse.

$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{2\alpha} + \ket{0}] \otimes \ket{g}$$

It seems like a CNOT gate in quantum circuit, but I do not know how it operate in physical experimentation.

If my previous step is an entangled state, how can I disentangle the qubit from the cavity? Why we need to flip the qubit if and only if the cavity is in the vacuum state, I mean why it works like that? Why we need to have a large amplitude of $\alpha$?

What is strong-dispersive coupling? Why do we apply a $\pi$ pulse to the qubit can cause a quantized light shift to the frequency and how this transition can make a product state?

\section{My Fifth Step}
I want to transform the state by displacing an amount of $-\alpha$ by driving a pulse to the cavity.
$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{\alpha}+ \ket{-\alpha}] \otimes \ket{g}$$

This is actually a cat state by definition. I think it is not an entangled state but a product state, because the ground state can be separated.

$$\ket{\psi} = \frac{1}{\sqrt{2}}[\ket{\alpha} \otimes \ket{g} \pm \ket{-\alpha} \otimes \ket{g}]$$

So the cavity is in a quantum superposition of two different coherent states. For the $\pm$ sign, is there anything to deal with the photon number parity of the state or the spatial parity symmetry? Why the photon number parity reverses the position and momentum of the oscillator? Why we need to consider even and odd parity cat states?

I do not understand how this displacement relate to the phase space to produce the cat state. I know there few concepts to look at, like quantum state tomography, quantum jump spectroscopy, photon number distribution, even or odd cats, Wigner function fringes. But I lacks these concepts for discussion.

\section{other way}
I know there is another way to produce the cat states with non-deterministic parity. How the measurement back action create a cat state?

$$\ket{\alpha} - \frac{1}{\sqrt{2}}[\ket{\Psi_+}+ \ket{\Psi_-}]$$

Can someone provide me anything I need to consider to get into my final state?
Is it that I need to adjust the frequency of the driving pulse so to transform every step of the state? Is there any other thing I need to consider?
 
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  • #2
I try to visualize how the states look like by using Qutip, still, I am not sure if it is correct. I think these graphs can help me check the results, but I have be certain the way I input the states are all correct. I hope someone can check it for me, thanks a lot!The installation is very easy, but putting the following codes into Jupyter Notebook.

Installation:
pip install qutip

from qutip import *
import numpy as np
import matplotlib.pyplot as plt

A plot function generating 2D and 3D to see how the result looks like.

Gallery of Wigner Function:
def plot_wigner_2d_3d(psi):
    #fig, axes = plt.subplots(1, 2, subplot_kw={'projection': '3d'}, figsize=(12, 6))
    fig = plt.figure(figsize=(17, 8))
    
    ax = fig.add_subplot(1, 2, 1)
    plot_wigner(psi, fig=fig, ax=ax, alpha_max=6);

    ax = fig.add_subplot(1, 2, 2, projection='3d')
    plot_wigner(psi, fig=fig, ax=ax, projection='3d', alpha_max=6);
    
    plt.close(fig)
    return fig
The first step
$$ \vert \Psi_i \rangle = \vert 0 \rangle \otimes \vert g \rangle$$

System in Ground State:
psi1 = tensor(basis(2,0), basis(2,0))
plot_wigner_2d_3d(psi1)

psi1.png

The second step

$$\vert \Psi \rangle = \frac{1}{\sqrt{2}}\vert 0 \rangle \otimes [\vert g \rangle + \vert e \rangle ] $$

Superposition of Ground and Excited State by applying a drive pulse:
psi2 = tensor(basis(2,0), (basis(2, 0) + basis(2, 1)).unit())
plot_wigner_2d_3d(psi2)

psi2.png


The third step

$$\vert \Psi \rangle = \frac{1}{\sqrt{2}} [ \vert 2\alpha \rangle \otimes \vert g \rangle + \vert 0 \rangle \otimes \vert e \rangle] $$

Coherent State Coupled to the Ground State by applying a drive pulse:
psi3a = tensor(coherent(2, 2.0), basis(2,0))
psi3b = tensor(basis(2, 0), basis(2, 1))
psi3 = psi3a + psi3b
plot_wigner_2d_3d(psi3)

psi3.png


The fourth step

$$ \vert \Psi \rangle = \frac{1}{\sqrt{2}} [ \vert 2\alpha \rangle + \vert 0 \rangle ] \otimes \vert g \rangle$$

The qubit flips the excited state back to the ground state when the cavity is in the ground state by applying a drive pulse:
psi4 = tensor(coherent(2, 2.0) + basis(2,0), basis(2,0))
plot_wigner_2d_3d(psi4)

psi4.png
The fifth step

$$\vert \Psi \rangle = \frac{1}{\sqrt{2}} [ \vert \alpha \rangle + \vert -\alpha \rangle ] \otimes \vert g \rangle $$

transform the state by displacing an amount of −α by driving a pulse to the cavity:
psi5 = tensor(coherent(2, 1.0) + coherent(2, -1.0), basis(2,0))
plot_wigner_2d_3d(psi5)

psi5.png
 
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  • #3
I think I have made a mistake in my above plots
https://www.physicsforums.com/threa...cat-state-in-circuit-qed.1048821/post-6840193

since I forgot to include 1/√ 2 to the states for normalisation.

As the microwave resonator is coupled to the two levels of superconducting qubit, I can set N = 20. Assume that ground state is 1 in basis and excited state is 0 in basis, I think it is a common practice. Here I set α as 1, I am not sure if it is appropriate.

Step 1
Code:
psi1 = tensor(basis(N,0), basis(2,1))
plot_wigner_2d_3d(psi1)

psi1.png


Step 2
Code:
psi2 = tensor(basis(N,0), ((basis(2, 1) + basis(2, 0)) / np.sqrt(2) ))
plot_wigner_2d_3d(psi2)

psi2.png


Step 3
Code:
psi3a = tensor(coherent(N, 2.0), basis(2,1))
psi3b = tensor(basis(N, 0), basis(2, 0))
psi3 = (psi3a + psi3b) / np.sqrt(2)
plot_wigner_2d_3d(psi3)

psi3.png


Step 4
Code:
psi4 = tensor((coherent(N, 2.0) + basis(N,0))/ np.sqrt(2), basis(2,1))
plot_wigner_2d_3d(psi4)

psi4.png


Step 5
Code:
psi5 = tensor((coherent(N, 1.0) + coherent(N, -1.0)) / np.sqrt(2) , basis(2,1))
plot_wigner_2d_3d(psi5).savefig('psi5.png')
psi5.png


Although I know Wigner function represents the probability distribution, I cannot interpret those graphs.

I hope someone can shed me some lights, or even ask me some questions for clarity. Thanks a lot!
 
  • #4
physicsclaus said:
I hope someone can shed me some lights, or even ask me some questions for clarity. Thanks a lot!
Let me be honest: your question is "too long", and has "too many" typos and markup mistakes:
  • Better ask small self-contained questions for each topic that is unclear to you. For example, your line: "Although I know Wigner function represents the probability distribution, I cannot interpret those graphs." indicates that you might want to ask specific questions about the Wigner function representation.
  • Inline latex on PF is not indicated by "$...$", but by "##...##". The latex command for dagger on PF is not \dag but \dagger (##\dagger##). If you want, you can ask a "Mentor" to fix your initial post.

physicsclaus said:
Assume it is unitary (I know I can prove it, but I don't want to make my post to lengthy)
This is a typical example of your typos, but there are many more. But let us better ignore those for the moment. If you write shorter posts, then you also have a better chance to avoid such typos yourself.
 
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  • #5
gentzen said:
Let me be honest: your question is "too long", and has "too many" typos and markup mistakes:
  • Better ask small self-contained questions for each topic that is unclear to you. For example, your line: "Although I know Wigner function represents the probability distribution, I cannot interpret those graphs." indicates that you might want to ask specific questions about the Wigner function representation.
  • Inline latex on PF is not indicated by "$...$", but by "##...##". The latex command for dagger on PF is not \dag but \dagger (##\dagger##). If you want, you can ask a "Mentor" to fix your initial post.
This is a typical example of your typos, but there are many more. But let us better ignore those for the moment. If you write shorter posts, then you also have a better chance to avoid such typos yourself.
Thank you for your suggestion. I will make another post soon.
 
  • #6
Based on your questions I suspect you would benefit from reading a bit more about the basic concepts of circuit-QED. Girvin's papers are not easy to understand and I would definitely start by reading a review paper.

This is a recent one with Girvin as a co-author
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.93.025005

Blais' "original" circuit-QED paper from 2004 might also be worth a look for some of the basic theory.

If you want to understand how the experiments are done I would recommend this review

https://arxiv.org/pdf/1904.06560.pdf
 
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  • #7
f95toli said:
Based on your questions I suspect you would benefit from reading a bit more about the basic concepts of circuit-QED. Girvin's papers are not easy to understand and I would definitely start by reading a review paper.

This is a recent one with Girvin as a co-author
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.93.025005

Blais' "original" circuit-QED paper from 2004 might also be worth a look for some of the basic theory.

If you want to understand how the experiments are done I would recommend this review

https://arxiv.org/pdf/1904.06560.pdf
Thank you so much for your information, however, my institution cannot access to this aps journal. The below review is quite comprehensive. I believe it's a good read, thanks a lot!
 
  • #8
Pretty much all papers in this field are also available from the arXiv. If you don't have access to a journal it is usually possible to find the paper there

https://arxiv.org/abs/2005.12667

I am pretty sure Blais' 2004 paper is also on the arXIv
 
  • #9
physicsclaus said:
Thank you so much for your information, however, my institution cannot access to this aps journal. The below review is quite comprehensive. I believe it's a good read, thanks a lot!
The RMP article by Blais is also on arXiv:

https://arxiv.org/abs/2005.12667
 
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  • #10
  • #11
Hello everyone, I have posted a new thread about closely related to this, please check it out!
 
  • #12
https://www.physicsforums.com/threads/sending-drive-pulse-in-cqed-and-visualize-the-state-by-qutip.1049103/

And I see that you got inline latex and \dagger correct now. It might still be a bit long, but I have no idea what is normal for Advanced Physics Homework Help.
 
  • #13
gentzen said:
https://www.physicsforums.com/threads/sending-drive-pulse-in-cqed-and-visualize-the-state-by-qutip.1049103/

And I see that you got inline latex and \dagger correct now. It might still be a bit long, but I have no idea what is normal for Advanced Physics Homework Help.
Yes, I know how to type latex here now. Do you think I should post somewhere else?
 
  • #14
physicsclaus said:
Do you think I should post somewhere else?
No, not at all. I just mean that I am unfamiliar with the "expectations" for that subforum.
 
  • #15
physicsclaus said:
Hello everyone, I have posted a new thread about closely related to this, please check it out!
And where did the thread go (o_o)?
 
  • #16
yucheng said:
And where did the thread go (o_o)?
The admin deleted my thread, they think I duplicate it. Actually, I am asking different question.
 
  • #17
physicsclaus said:
Hello everyone, I have posted a new thread about closely related to this, please check it out!
yucheng said:
And where did the thread go (o_o)?
physicsclaus said:
The admin deleted my thread, they think I duplicate it. Actually, I am asking different question.
The new thread appeared to be a re-post of this thread, and multiple posting threads is not allowed. After a PM conversation with the OP, I will close this thread and undelete the new thread with a note in that thread.

Update -- New thread is here: https://www.physicsforums.com/threads/sending-drive-pulse-in-cqed-and-visualize-the-state-by-qutip.1049103/

Update2 -- This original thread will now be closed. Thanks to all who tried to help the OP with these questions.
 
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