I Displacement operation acting on individual quadrature components

waadles
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Hi all,

I have a naive understanding of how operators work and wondered if someone could help me. I have tried to understand this myself, but alas, I think my knowledge is too premature to understand what I am reading online. Is someone able to explain?

I want to perform the operation ##D(\alpha)^\dagger X D(\alpha)##, where ##X## is one of the quadrature components ##X = \frac{1}{\sqrt{2}} (a + a^\dagger)##. I know the following:

$$
D(\alpha)^\dagger a D(\alpha) = a + \alpha
$$

Does this mean ##D(\alpha)^\dagger a^\dagger D(\alpha) = a^\dagger + \alpha^*##?

If so, then:

$$
D(\alpha)^\dagger X D(\alpha) = \frac{1}{\sqrt{2}} D(\alpha)^\dagger (a + a^\dagger) D(\alpha) = \frac{1}{\sqrt{2}} (a + a^\dagger + 2|\alpha|)
$$

This is not correct. I would have expected something like ##\frac{1}{\sqrt{2}} (a + a^\dagger + |\alpha|)##.

I am struggling even more to work out ##P##, where ##P = \frac{i}{\sqrt{2}} (a - a^\dagger)##. I get:

$$
D(\alpha)^\dagger P D(\alpha) = \frac{1}{\sqrt{2}} D(\alpha)^\dagger (a - a^\dagger) D(\alpha) = \frac{1}{\sqrt{2}} (a - a^\dagger)
$$

and it remains the same, which is not correct. I would also expect ##\frac{1}{\sqrt{2}} (a - a^\dagger + |\alpha|)##.

Can someone tell me where I am going wrong?

I appreciate any help you can provide.
 
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Since operator a is not Hermitian, I am not sure and accoustomed to translational operation on it. Is it an exercise on textbook ?
 
waadles said:
Hi all,

I have a naive understanding of how operators work and wondered if someone could help me. I have tried to understand this myself, but alas, I think my knowledge is too premature to understand what I am reading online. Is someone able to explain?

I want to perform the operation ##D(\alpha)^\dagger X D(\alpha)##, where ##X## is one of the quadrature components ##X = \frac{1}{\sqrt{2}} (a + a^\dagger)##. I know the following:

$$
D(\alpha)^\dagger a D(\alpha) = a + \alpha
$$

Does this mean ##D(\alpha)^\dagger a^\dagger D(\alpha) = a^\dagger + \alpha^*##?
That follows by taking the Hermitian conjugate of the equation.
waadles said:
If so, then:

$$
D(\alpha)^\dagger X D(\alpha) = \frac{1}{\sqrt{2}} D(\alpha)^\dagger (a + a^\dagger) D(\alpha) = \frac{1}{\sqrt{2}} (a + a^\dagger + 2|\alpha|)
$$
Note that ##\alpha + \alpha^* = 2Re(\alpha)##.
waadles said:
This is not correct. I would have expected something like ##\frac{1}{\sqrt{2}} (a + a^\dagger + |\alpha|)##.
Why did you expect that?
waadles said:
I am struggling even more to work out ##P##, where ##P = \frac{i}{\sqrt{2}} (a - a^\dagger)##. I get:

$$
D(\alpha)^\dagger P D(\alpha) = \frac{1}{\sqrt{2}} D(\alpha)^\dagger (a - a^\dagger) D(\alpha) = \frac{1}{\sqrt{2}} (a - a^\dagger)
$$
In this case you should use ##\alpha - \alpha^* = 2iIm(\alpha)##.
 
PeroK said:
In this case you should use α−α∗=2iIm(α).
May we say about meaning of translation of a as
\alpha=\frac{d_x}{\sqrt{2}}-i\frac{d_p}{\sqrt{2}}
where d_x and d_p are displacement in coordinate and momentum space ?
If so it holds for the translated Hamiltoian, i.e.
(a^{\dagger} +\alpha^*)(a+\alpha)+\frac{1}{2}? It seems not obvious to me.
 
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