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幺正算子对湮灭算子的作用

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幺正算子保持量子态的内积不变，描述量子系统的可逆演化。湮灭算子则用于描述量子场中粒子的湮灭过程。

位移算子

d^(α)=exp⁡(αa^†−α∗a^)\hat{d}(\alpha) = \exp\left(\alpha \hat{a}^\dagger - \alpha^* \hat{a}\right)

其中αα是复位移参数，α∗α^∗是其复共轭
幺正性

d^†(α)=d^(−α)hatd(α)d^†(α)=I^\hat{d}^\dagger(\alpha) = \hat{d}(-\alpha)\\hat{d}(\alpha)\hat{d}^\dagger(\alpha) = \hat{I}

对湮灭算子的作用

d^†(α)a^d^(α)=a^+αhatd(α)a^d^†(α)=a^−α\hat{d}^\dagger(\alpha)\hat{a}\hat{d}(\alpha) = \hat{a} + \alpha\\hat{d}(\alpha)\hat{a}\hat{d}^\dagger(\alpha) = \hat{a} - \alpha

乘积法则

d^(α)d^(β)=eiIm(αβ∗)d^(α+β)\hat{d}(\alpha)\hat{d}(\beta) = e^{i \text{Im}(\alpha\beta^*)}\hat{d}(\alpha+\beta)

相位平移算子

U^θ=eiθN^\hat{U}_\theta = e^{i\theta \hat{N}}

粒子数算子N^=a^†a^\hat{N} = \hat{a}^\dagger \hat{a}表示系统中的粒子数。
对湮灭算子的作用：

U^θ†a^U^θ=e−iθa^\hat{U}_\theta^\dagger \hat{a} \hat{U}_\theta = e^{-i\theta} \hat{a}

推导
计算对易子[N^,a^][\hat{N}, \hat{a}]
使用对易子的基本性质：[AB,C]=A[B,C]+[A,C]B[AB, C] = A[B, C] + [A, C]B

[N^,a^]=[a^†a^,a^]=a^†[a^,a^]+[a^†,a^]a^[\hat{N}, \hat{a}] = [\hat{a}^\dagger \hat{a}, \hat{a}] = \hat{a}^\dagger [\hat{a}, \hat{a}] + [\hat{a}^\dagger, \hat{a}] \hat{a}

简化对易关系
已知[a^,a^]=0[\hat{a}, \hat{a}] = 0和[a^†,a^]=−1[\hat{a}^\dagger, \hat{a}] = -1：

[N^,a^]=a^†⋅0+(−1)⋅a^=−a^[\hat{N}, \hat{a}] = \hat{a}^\dagger \cdot 0 + (-1) \cdot \hat{a} = -\hat{a}

结论

[N^,a^]=−a^ [N^,a^†]=a^†[\hat{N}, \hat{a}] = -\hat{a}\ [\hat{N},\hat{a}^\dagger]=\hat{a}^\dagger

Baker-Campbell-Hausdorff公式

BCH公式的一般形式，
对于任意两个算子X^\hat{X}和Y^\hat{Y}，BCH公式为：

eX^Y^e−X^=Y^+[X^,Y^]+12![X^,[X^,Y^]]+13![X^,[X^,[X^,Y^]]]+⋯e^{\hat{X}} \hat{Y} e^{-\hat{X}} = \hat{Y} + [\hat{X}, \hat{Y}] + \frac{1}{2!}[\hat{X}, [\hat{X}, \hat{Y}]] + \frac{1}{3!}[\hat{X}, [\hat{X}, [\hat{X}, \hat{Y}]]] + \cdots

对于相位平移算子

X^=iθN^,Y^=a^\hat{X} = i\theta \hat{N}, \quad \hat{Y} = \hat{a}

首先计算第一级对易子：

[X^,Y^]=iθ[N^,a^]=−iθa^[\hat{X}, \hat{Y}] = i\theta [\hat{N}, \hat{a}] = -i\theta \hat{a}

第二级对易子：

[X^,[X^,Y^]]=[iθN^,−iθa^]=(iθ)(−iθ)[N^,a^]=θ2(−a^)=−θ2a^[\hat{X}, [\hat{X}, \hat{Y}]] = [i\theta \hat{N}, -i\theta \hat{a}] = (i\theta)(-i\theta)[\hat{N}, \hat{a}] = \theta^2 (-\hat{a}) = -\theta^2 \hat{a}

第三级对易子：

[X^,[X^,[X^,Y^]]]=[iθN^,−θ2a^]=(iθ)(−θ2)[N^,a^]=−iθ3(−a^)=iθ3a^[\hat{X}, [\hat{X}, [\hat{X}, \hat{Y}]]] = [i\theta \hat{N}, -\theta^2 \hat{a}] = (i\theta)(-\theta^2)[\hat{N}, \hat{a}] = -i\theta^3 (-\hat{a}) = i\theta^3 \hat{a}

各级对易子

[X^,[X^,⋯[X^⏟n times,Y^]⋯ ]=(−iθ)na^[\hat{X}, \underbrace{[\hat{X}, \cdots [\hat{X}}_{n \text{ times}}, \hat{Y}] \cdots ] = (-i\theta)^n \hat{a}

代入BCH公式

eiθN^a^e−iθN^=∑n=0∞(−iθ)nn!a^=a^∑n=0∞(−iθ)nn!e^{i\theta \hat{N}} \hat{a} e^{-i\theta \hat{N}} = \sum_{n=0}^{\infty} \frac{(-i\theta)^n}{n!} \hat{a} = \hat{a} \sum_{n=0}^{\infty} \frac{(-i\theta)^n}{n!}

求和项正是指数函数的泰勒展开：

eiθN^a^e−iθN^=a^e−iθU^θa^U^θ†=e−iθa^U^θ†a^U^θ=eiθa^e^{i\theta \hat{N}} \hat{a} e^{-i\theta \hat{N}} = \hat{a} e^{-i\theta}\\\hat{U}_\theta \hat{a} \hat{U}_\theta^\dagger = e^{-i\theta} \hat{a}\\\hat{U}_\theta^\dagger \hat{a} \hat{U}_\theta = e^{i\theta} \hat{a}

对于位移算子
令X^=αa^†−α∗a^\hat{X} = \alpha \hat{a}^\dagger - \alpha^* \hat{a}，Y^=a^\hat{Y} = \hat{a}：

d^†(α)a^d^(α)=e−X^a^eX^=a^+[−X^,a^]+12![−X^,[−X^,a^]]+⋯\hat{d}^\dagger(\alpha)\hat{a}\hat{d}(\alpha) = e^{-\hat{X}} \hat{a} e^{\hat{X}} = \hat{a} + [-\hat{X}, \hat{a}] + \frac{1}{2!}[-\hat{X}, [-\hat{X}, \hat{a}]] + \cdots

计算对易子

[X^,a^]=[αa^†−α∗a^,a^]=α[a^†,a^]−α∗[a^,a^]=−α [X^,[X^,a^]]=[X^,−α]=0[\hat{X}, \hat{a}] = [\alpha \hat{a}^\dagger - \alpha^* \hat{a}, \hat{a}] = \alpha [\hat{a}^\dagger, \hat{a}] - \alpha^* [\hat{a}, \hat{a}] = -\alpha\ [\hat{X}, [\hat{X}, \hat{a}]] = [\hat{X}, -\alpha] = 0

所有更高阶的对易子都为零

d^†(α)a^d^(α)=a^+αd^(α)a^d^†(α)=a^−α\hat{d}^\dagger(\alpha)\hat{a}\hat{d}(\alpha) = \hat{a} + \alpha\\\hat{d}(\alpha)\hat{a}\hat{d}^\dagger(\alpha) = \hat{a} - \alpha

相位平移算子的性质

性质1：幺正性

U^θ†=e−iθN^=U^−θU^θU^θ†=eiθN^e−iθN^=eiθN^−iθN^=e0=I^\hat{U}_\theta^\dagger = e^{-i\theta \hat{N}} = \hat{U}{-\theta}\\\hat{U}_\theta \hat{U}_\theta^\dagger = e^{i\theta \hat{N}} e^{-i\theta \hat{N}} = e^{i\theta \hat{N} - i\theta \hat{N}} = e^0 = \hat{I}

性质2：乘积法则

U^θU^ϕ=eiθN^eiϕN^=ei(θ+ϕ)N^=U^θ+ϕ\hat{U}_\theta \hat{U}_\phi = e^{i\theta \hat{N}} e^{i\phi \hat{N}} = e^{i(\theta+\phi)\hat{N}} = \hat{U}_{\theta+\phi}

因为N^\hat{N}与自身对易，指数可以直接相加
性质3：周期性

U^θ+2π=ei(θ+2π)N^=eiθN^ei2πN^ei2πN^∣n⟩=ei2πn∣n⟩=∣n⟩U^θ+2π=U^θ\hat{U}_{\theta + 2\pi} = e^{i(\theta+2\pi)\hat{N}} = e^{i\theta \hat{N}} e^{i2\pi \hat{N}}\\e^{i2\pi \hat{N}} |n\rangle = e^{i2\pi n} |n\rangle = |n\rangle\\\hat{U}_{\theta + 2\pi} = \hat{U}_\theta

例

对相干态的作用
相干态∣α⟩=d^(α)∣0⟩|\alpha\rangle = \hat{d}(\alpha)|0\rangle，计算相位平移后的状态：

U^θ∣α⟩=U^θd^(α)∣0⟩=d^(eiθα)U^θ∣0⟩=d^(eiθα)∣0⟩=∣eiθα⟩\hat{U}_\theta |\alpha\rangle = \hat{U}_\theta \hat{d}(\alpha)|0\rangle\\ =\hat{d}(e^{i\theta}\alpha) \hat{U}_\theta |0\rangle = \hat{d}(e^{i\theta}\alpha)|0\rangle = |e^{i\theta}\alpha\rangle

对粒子数态的作用

U^θ∣n⟩=eiθN^∣n⟩=eiθn∣n⟩⟨m∣U^θ∣n⟩=eiθnδmn\hat{U}_\theta |n\rangle = e^{i\theta \hat{N}} |n\rangle = e^{i\theta n} |n\rangle\\\langle m | \hat{U}_\theta |n\rangle = e^{i\theta n} \delta_{mn}