User:物理极客铭/sandbox1

。

傑恩斯-卡明斯模型是一個量子光學的理论模型。它描述了一個雙態系統和量化光腔(optical cavity)的交互作用，這種交互作用和光子的存在與否無關(在电磁辐射能造成光子自發性的放射與吸收)。它主要被運用在原子物理學，量子光學，固態量子信息電路的理論與實驗上。

Formulation

描述整個系統的哈密頓量

{\hat {H}}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}+{\hat {H}}_{\text{int}}

包含了自由場的哈密頓量，原子激發態的哈密頓量，傑恩斯 - 卡明斯相互作用的哈密頓量組成：

{\begin{aligned}{\hat {H}}_{\text{field}}&=\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}\\{\hat {H}}_{\text{atom}}&=\hbar \omega _{a}{\frac {{\hat {\sigma }}_{z}}{2}}\\{\hat {H}}_{\text{int}}&={\frac {\hbar \Omega }{2}}{\hat {E}}{\hat {S}}.\end{aligned}}

為方便起見，真空場能量被設置為 $0$ .

For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single bosonic mode with the field operator ${\hat {E}}={\hat {a}}+{\hat {a}}^{\dagger }$ , where the operators ${\hat {a}}^{\dagger }$ and ${\hat {a}}$ are the bosonic creation and annihilation operators and $\omega _{c}$ is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-half whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom with spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic to a spin-half.) The atom is coupled to the field through its polarization operator ${\hat {S}}={\hat {\sigma }}_{+}+{\hat {\sigma }}_{-}$ . The operators ${\hat {\sigma }}_{+}=|e\rangle \langle g|$ and ${\hat {\sigma }}_{-}=|g\rangle \langle e|$ are the raising and lowering operators of the atom. The operator ${\hat {\sigma }}_{z}=|e\rangle \langle e|-|g\rangle \langle g|$ is the atomic inversion operator, and $\omega _{a}$ is the atomic transition frequency.

JCM Hamiltonian

Moving from the Schrödinger picture into the interaction picture (a.k.a. rotating frame) defined by the choice $H_{0}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}$ , we obtain

{\hat {H}}_{\text{int}}(t)={\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{-}e^{-i(\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{+}e^{i(\omega _{c}+\omega _{a})t}+{\hat {a}}{\hat {\sigma }}_{+}e^{i(-\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}e^{-i(-\omega _{c}+\omega _{a})t}\right).

This Hamiltonian contains both quickly $(\omega _{c}+\omega _{a})$ and slowly $(\omega _{c}-\omega _{a})$ oscillating components. To get a solvable model, when $|\omega _{c}-\omega _{a}|\ll \omega _{c}+\omega _{a}$ the quickly oscillating "counter-rotating" terms can be ignored. This is referred to as the rotating wave approximation. Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as

{\hat {H}}_{\text{JC}}=\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{a}{\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right).

Eigenstates

It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:

{\hat {H}}_{\text{JC}}={\hat {H}}_{I}+{\hat {H}}_{II},

where

{\begin{aligned}{\hat {H}}_{I}&=\hbar \omega _{c}\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {{\hat {\sigma }}_{z}}{2}}\right)\\{\hat {H}}_{II}&=\hbar \delta {\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)\end{aligned}}

with $\delta =\omega _{a}-\omega _{c}$ called the detuning (frequency) between the field and the two-level system.

The eigenstates of ${\hat {H}}_{I}$ , being of tensor product form, are easily solved and denoted by $|n,g\rangle ,|n,e\rangle$ , where $n\in \mathbb {N}$ denotes the number of radiation quanta in the mode.

As the states $|\psi _{1n}\rangle :=|n,e\rangle$ and $|\psi _{2n}\rangle :=|n+1,g\rangle$ are degenerate with respect to ${\hat {H}}_{I}$ for all $n$ , it is enough to diagonalize ${\hat {H}}_{\text{JC}}$ in the subspaces ${\text{span}}\{|\psi _{1n}\rangle ,|\psi _{2n}\rangle \}$ . The matrix elements of ${\hat {H}}_{\text{JC}}$ in this subspace, ${H}_{ij}^{(n)}:=\langle \psi _{in}|{\hat {H}}_{\text{JC}}|\psi _{jn}\rangle ,$ read

H^{(n)}=\hbar {\begin{pmatrix}n\omega _{c}+{\frac {\omega _{a}}{2}}&{\frac {\Omega }{2}}{\sqrt {n+1}}\\[8pt]{\frac {\Omega }{2}}{\sqrt {n+1}}&(n+1)\omega _{c}-{\frac {\omega _{a}}{2}}\end{pmatrix}}

For a given $n$ , the energy eigenvalues of $H^{(n)}$ are

E_{\pm }(n)=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)\pm {\frac {1}{2}}\hbar \Omega _{n}(\delta ),

where $\Omega _{n}(\delta )={\sqrt {\delta ^{2}+\Omega ^{2}(n+1)}}$ is the Rabi frequency for the specific detuning parameter. The eigenstates $|n,\pm \rangle ~$ associated with the energy eigenvalues are given by

|n,+\rangle =\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle

|n,-\rangle =-\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle

where the angle $\alpha _{n}$ is defined through $\alpha _{n}:=\tan ^{-1}\left({\frac {\Omega {\sqrt {n+1}}}{\delta }}\right)$

Schrödinger picture dynamics

It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates. We consider a superposition of number states as the initial state for the field, $~|\psi _{\text{field}}(0)\rangle =\sum _{n}{C_{n}|n\rangle }~$ , and assume an atom in the excited state is injected into the field. The initial state of the system is

|\psi _{\text{tot}}(0)\rangle =\sum _{n}C_{n}\left[\cos \left({\frac {\alpha _{n}}{2}}\right)|n,+\rangle -\sin \left({\frac {\alpha _{n}}{2}}\right)|n,-\rangle \right].

Since the $~|n,\pm \rangle ~$ are stationary states of the field-atom system, then the state vector for times $~t>0~$ is just given by

|\psi _{\text{tot}}(t)\rangle =e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|\psi _{\text{tot}}(0)\rangle =\sum _{n}C_{n}\left[\cos \left({\frac {\alpha _{n}}{2}}\right)|n,+\rangle e^{-iE_{+}(n)t/\hbar }-\sin \left({\frac {\alpha _{n}}{2}}\right)|n,-\rangle e^{-iE_{-}(n)t/\hbar }\right].

The Rabi oscillations can readily be seen in the sin and cos functions in the state vector. Different periods occur for different number states of photons. What is observed in experiment is the sum of many periodic functions that can be very widely oscillating and destructively sum to zero at some moment of time, but will be non-zero again at later moments. Finiteness of this moment results just from discreteness of the periodicity arguments. If the field amplitude were continuous, the revival would have never happened at finite time.

Heisenberg picture dynamics

It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian:^[1]

{\begin{matrix}{\begin{aligned}{\hat {U}}(t)&=e^{-i{\hat {H}}_{\text{JC}}t/\hbar }\\&={\begin{pmatrix}e^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}})}\left(\cos t{\sqrt {{\hat {\varphi }}+g^{2}}}-i\delta /2{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\right)&-ige^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}})}{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\,{\hat {a}}\\-ige^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}})}{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}{\hat {a}}^{\dagger }&e^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}})}\left(\cos t{\sqrt {\hat {\varphi }}}+i\delta /2{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}\right)\end{pmatrix}}\end{aligned}}\end{matrix}}

where the operator $~{\hat {\varphi }}~$ is defined as

{\hat {\varphi }}=g^{2}{\hat {a}}^{\dagger }{\hat {a}}+\delta ^{2}/4

The unitarity of $~{\hat {U}}~$ is guaranteed by the identities

{\frac {\sin t\,{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\;{\hat {a}}={\hat {a}}\;{\frac {\sin t\,{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}},

\cos t\,{\sqrt {{\hat {\varphi }}+g^{2}}}\;{\hat {a}}={\hat {a}}\;\cos t{\sqrt {\hat {\varphi }}},

and their Hermitian conjugates.

By the unitary evolution operator one can calculate the time evolution of the state of the system described by its density matrix $~{\hat {\rho }}(t)~$ , and from there the expectation value of any observable, given the initial state:

{\hat {\rho }}(t)={\hat {U}}^{\dagger }(t){\hat {\rho }}(0){\hat {U}}(t)

\langle {\hat {\Theta }}\rangle _{t}={\text{Tr}}[{\hat {\rho }}(t){\hat {\Theta }}]

The initial state of the system is denoted by $~{\hat {\rho }}(0)~$ and $~{\hat {\Theta }}~$ is an operator denoting the observable.

Collapses and revivals of quantum oscillations

The plot of quantum oscillations of atomic inversion (for quadratic scaled detuning parameter $a=(\delta /(2g))^{2}=40$ , where $\delta$ is the detuning parameter) was built on the basis of formulas obtained^[2] by A.A. Karatsuba and E.A. Karatsuba.

References

^ S. Stenholm, "Quantum theory of electromagnetic fields interacting with atoms and molecules", Physics Reports, 6(1), 1-121 (1973).
^ A. A. Karatsuba, E. A. Karatsuba (2009). "A resummation formula for collapse and revival in the Jaynes–Cummings model". J. Phys. A: Math. Theor. (42): 195304, 16. Bibcode:2009JPhA...42s5304K. doi:10.1088/1751-8113/42/19/195304.