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Diagonalizing the Jaynes-Cummings Hamiltonian and Jaynes-Cummings coherent states

Stephen M. Barnett and Bryan J. Dalton

Open Access Open Access

Abstract

We determine the form of the unitary transformation that diagonalizes the Jaynes-Cummings Hamiltonian. This leads to operators the action of which has a simple interpretation in terms of the dressed states, the energy eigenstates. This suggests a set of coherent states and spin coherent states based on the dressed states.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. INTRODUCTION

The Jaynes-Cummings model (universally simplified to the JCM) is 60 years old [1]. Its simplicity and subtle behavior, as well as its links with experimentally realizable systems [2,3], have made it one of the most, perhaps the most, studied models in quantum optics [4–12]. Yet in all this work there seems not to be the unitary transformation that diagonalizes the Hamiltonian. In this paper we seek to correct this omission.

Finding the dressed states for the JCM is a straightforward exercise because the interaction between the atom and the field mode preserves the excitation number. This means that, in the Schrödinger picture, we only ever couple two states, an excited state with $n$ photons and a ground state with $n + 1$ photons. These two are the only states for which the total number of quanta, $N$, is $n + 1$. The dressed states are simply superpositions of such coupled bare states.

Addressing the dynamics in the Heisenberg picture is more of a challenge than the simpler Schrödinger picture treatment, but a complete solution was given by Ackerhalt [8,13]. His technique was to find superpositions of the photon annihilation and atom lowering operators that are eigenoperators of the Hamiltonian. The coefficients in this superposition are also operators, but are functions only of operator constants of the motion. We shall find that closely related ideas to this assist in our derivation of the unitary transformation that diagonalizes the JCM Hamiltonian.

We start with a short discussion of the JCM before, in Section 3, proceeding to derive the form of the unitary operator that diagonalizes the JCM Hamiltonian. In Section 4 we use the unitary transformation to derive the forms of the dressed operators for the coupled atom and cavity mode. Finally, in Section 5, we introduce a new set of coherent states, specifically related to the JCM, and show, in particular, that these tend to the familiar dressed states of the semi-classical model [8] in the appropriate limit.

2. JCM: A REMINDER

As a brief reminder, but also to set our notation, we give a very brief overview of the JCM Hamiltonian and of its energy eigenstates. The JCM Hamiltonian has the form

$$\hat H = \frac{{\hbar {\omega _0}}}{2}{\hat \sigma _3} + \hbar \omega {\hat a^\dagger}\hat a + \hbar \lambda ({\hat a^\dagger}{\hat \sigma _ -} + {\hat \sigma _ +}\hat a).$$
Here ${\hat \sigma _3} = |e\rangle \langle e| - |g\rangle \langle g|$ and ${\hat \sigma _ -} = |g\rangle \langle e| = \hat \sigma _ + ^\dagger$, where $|e\rangle$ and $|g\rangle$ are, respectively, the excited and ground states, and $\hat a$ and ${\hat a^\dagger}$ are the usual photon annihilation and creation operators for the single field mode. To proceed we transform this into an interaction picture and, for simplicity, impose resonance between the atom and the mode so that ${\omega _0} = \omega$. The resulting interaction picture Hamiltonian is
$${\hat H_I} = \hbar \lambda ({\hat a^\dagger}{\hat \sigma _ -} + {\hat \sigma _ +}\hat a).$$
The action of this simply exchanges a quantum of energy between the atom and the field.

It is straightforward to determine the energy eigenvalues and eigenstates (dressed states) for ${\hat H_I}$. There is a single, zero energy, ground state:

$${\hat H_I}|g\rangle \otimes |0\rangle = 0,$$
where $|0\rangle$ is the vacuum state for the field mode. The remaining eigenstates are simple superpositions of states with the same number of quanta:
$${\hat H_I}|n, \pm \rangle = \pm \hbar \lambda \sqrt n |n, \pm \rangle ,$$
where
$$|n, \pm \rangle = \frac{1}{{\sqrt 2}}(|e\rangle \otimes |n - 1\rangle \pm |g\rangle \otimes |n\rangle).$$

Here $|n\rangle$ are the photon number eigenstates.

We note that the JCM has two natural constants of the motion, that is, Hermitian operators that commute with ${\hat H_I}$. The first is the interaction Hamiltonian itself and the second is

$$\hat N = {\hat \sigma _ +}{\hat \sigma _ -} + {\hat a^\dagger}\hat a,$$
which is the total number of excitations shared between the atom and the field mode. The dressed states are simple eigenstates of this quantity:
$$\begin{split}\hat N|g\rangle \otimes |0\rangle = 0, \\ \hat N|n, \pm \rangle = n|n, \pm \rangle .\end{split}$$

We shall find that this second constant of the motion plays a key role in our study.

3. UNITARY TRANSFORM TO DIAGONALIZE ${\hat H_I}$

We start by noting that the generator of the desired unitary transformation cannot be formed from a simple product of an annihilation or creation operator and a corresponding lowering or raising operator for the atom. It should also commute with the operator $\hat N$, corresponding to the total excitation number. To this end we introduce the (anti-Hermitian) operator

$$\hat {\cal O} = {\hat \sigma _ +}f({\hat a^\dagger}\hat a)\hat a - {\hat a^\dagger}f({\hat a^\dagger}\hat a){\hat \sigma _ -},$$
where $f$ is a real function, to be determined. The commutation relation between $\hat {\cal O}$ and ${\hat H_I}$ is
$$[\hat {\cal O},{\hat H_I}] = 2\hbar \lambda \!\left({{{\hat \sigma}_ +}{{\hat \sigma}_ -}\hat a{{\hat a}^\dagger}f({{\hat a}^\dagger}\hat a) - {{\hat \sigma}_ -}{{\hat \sigma}_ +}{{\hat a}^\dagger}\hat af({{\hat a}^\dagger}\hat a - 1)} \right),$$
where we have used the operator identity $f({\hat a^\dagger}\hat a)\hat a = \hat af({\hat a^\dagger}\hat a - 1)$ [14]. The fact that the eigenstates of ${\hat H_I}$ depend on the square root of the number of quanta suggests that we set $f({\hat a^\dagger}\hat a) = \frac{1}{2}{({\hat a^\dagger}\hat a + 1)^{- 1/2}} = \frac{1}{2}{(\hat a{\hat a^\dagger})^{- 1/2}}$ so that
$$[\hat {\cal O},{\hat H_I}] = \hbar \lambda \left({{{\hat \sigma}_ +}{{\hat \sigma}_ -}\sqrt {\hat a{{\hat a}^\dagger}} - {{\hat \sigma}_ -}{{\hat \sigma}_ +}\sqrt {{{\hat a}^\dagger}\hat a}} \right).$$
We note that the eigenvalues of this commutator are clearly ${\pm}\hbar \lambda \sqrt n$, as they must be, and that the eigenstates are the bare, or undressed, states:
$$\begin{split}[\hat {\cal O},{\hat H_I}]|g\rangle \otimes |n\rangle = - \hbar \lambda \sqrt n |g\rangle \otimes |n\rangle , \\ [\hat {\cal O},{\hat H_I}]|e\rangle \otimes |n - 1\rangle = \hbar \lambda \sqrt n |e\rangle \otimes |n - 1\rangle .\end{split}$$
These include the (trivial) ground state
$$[\hat {\cal O},{\hat H_I}]|g\rangle \otimes |0\rangle = 0.$$

It remains to construct the unitary transformation based on $\hat {\cal O}$. Before addressing this, it is interesting to note that we can write $\hat {\cal O}$ in a somewhat simpler form:

$$\begin{split}\hat {\cal O}& = {\hat \sigma _ +}\frac{1}{2}{({\hat a^\dagger}\hat a + 1)^{- 1/2}}\hat a - {\hat a^\dagger}{({\hat a^\dagger}\hat a + 1)^{- 1/2}}{\hat \sigma _ -} \\ &= \frac{1}{2}({\hat \sigma _ +}\hat E - {\hat E^\dagger}{\hat \sigma _ -}),\end{split}$$
where $\hat E$ and ${\hat E^\dagger}$ are the bare lowering and raising operators:
$$\begin{split}\hat E& = \sum\limits_{n^\prime = 0}^\infty {(n^\prime + 1)^{- 1/2}}|n^\prime \rangle \langle n^\prime |\sum\limits_{n = 0}^\infty {(n + 1)^{1/2}}|n\rangle \langle n| \\ &= \sum\limits_{n = 0}^\infty |n\rangle \langle n + 1|, \\ {\hat E^\dagger} &= \sum\limits_{n = 0}^\infty |n + 1\rangle \langle n|.\end{split}$$
These were introduced by Susskind and Glogower [15] to represent the exponentials of the phase of the field, although we now know that this is not correct [8,16–18].

To construct the required unitary transformation we note that

$$\begin{split}{\hat {\cal O}^2} &= (- {\hat \sigma _ +}{\hat \sigma _ -}\hat E{\hat E^\dagger} - {\hat \sigma _ -}{\hat \sigma _ +}{\hat E^\dagger}\hat E)/4 \\ &= - \frac{1}{4}({\hat {\rm I}} \otimes {\hat {\rm I}} - |g\rangle \langle g| \otimes |0\rangle \langle 0|),\end{split}$$
which is a quarter of minus the projector onto all the states with the exception of the ground state. It is now straightforward to find the required unitary operator:
$$\begin{split}\hat U &= {e^{\frac{\pi}{2}\hat {\cal O}}} \\& = \frac{1}{{\sqrt 2}}({\hat {\rm I}} \otimes {\hat {\rm I}} - |g\rangle \langle g| \otimes |0\rangle \langle 0| + 2\hat {\cal O}) + |g\rangle \langle g| \otimes |0\rangle \langle 0|,\end{split}$$
where the first term couples the excited bare states and the last term leaves the ground state unchanged. We find, by direct calculation, that
$$\hat U{\hat H_I}{\hat U^\dagger} = \hbar \lambda ({\hat \sigma _ +}{\hat \sigma _ -}\sqrt {\hat a{{\hat a}^\dagger}} - {\hat \sigma _ -}{\hat \sigma _ +}\sqrt {{{\hat a}^\dagger}\hat a}).$$
We can take advantage of the Pauli operator algebra to write this in a yet more suggestive, diagonalized, form:
$${\hat H_{\rm{ID}}} = \hbar \lambda \sqrt {\hat N} {\hat \sigma _3}.$$
This is our main result. Henceforth we denote the transformed (dressed) operators by a subscript $D$.

It is clear that the eigenvalues of ${\hat H_{\rm{ID}}}$ are ${\pm}\hbar \lambda \sqrt n$ and that the bare atom-field states are its eigenstates:

$$\begin{split}{\hat H_{\rm{ID}}}|g\rangle \otimes |n\rangle &= - \hbar \lambda \sqrt n |g\rangle \otimes |n\rangle , \\ {\hat H_{\rm{ID}}}|e\rangle \otimes |n - 1\rangle& = \hbar \lambda \sqrt n |e\rangle \otimes |n - 1\rangle .\end{split}$$
We can also write the operator $\hat {\cal O}$ in a simpler form:
$$\hat {\cal O} = \frac{1}{{2\sqrt {\hat N}}}({\hat \sigma _ +}\hat a - {\hat a^\dagger}{\hat \sigma _ -}),$$
where we have used the fact that $\hat N$ commutes with ${\hat \sigma _ +}\hat a$ and with ${\hat a^\dagger}{\hat \sigma _ -}$.

4. DRESSED OPERATORS

To complete our analysis we should determine the forms of the atom and field operators in the transformed picture and provide their physical significance. The required dressed annihilation and lowering operators are

$$\begin{split}{\hat a_D} &= \hat U\hat a{\hat U^\dagger}, \\ {\hat \sigma _{- D}} &= \hat U{\hat \sigma _ -}{\hat U^\dagger}.\end{split}$$
We could determine the forms of these in terms of the original undressed operators, but it is both simpler and more informative to consider their action on the dressed states.

We start with the photon annihilation operator, the action of which on the bare states is

$$\begin{split}\hat a|g\rangle \otimes |n\rangle &= \sqrt n |g\rangle \otimes |n - 1\rangle , \\[-2pt] \hat a|e\rangle \otimes |n\rangle &= \sqrt n |e\rangle \otimes |n - 1\rangle \end{split}$$
for $n \ge 1$. It follows that
$$\begin{split}{\hat a_D}\hat U|g\rangle \otimes |n\rangle &= \hat U\hat a{\hat U^\dagger}\hat U|g\rangle \otimes |n\rangle = \sqrt n \hat U|g\rangle \otimes |n - 1\rangle , \\ {\hat a_D}\hat U|e\rangle \otimes |n\rangle &= \sqrt n \hat U|e\rangle \otimes |n - 1\rangle , \end{split}$$
with ${\hat a_D}\hat U|e\rangle \otimes |0\rangle = 0 = {\hat a_D}\hat U|g\rangle \otimes |0\rangle$. Hence we require only the forms of the transformed states $\hat U|g\rangle \otimes |n\rangle$ and $\hat U|e\rangle \otimes |n\rangle$:
$$\begin{split}\hat U|g\rangle \otimes |0\rangle& = |g\rangle \otimes |0\rangle , \\[-2pt] \hat U|g\rangle \otimes |n\rangle &= |n, + \rangle , \\[-2pt] \hat U|e\rangle \otimes |n\rangle &= |n + 1, - \rangle\end{split}$$
for $n \ge 1$. We can simplify these by defining the state
$$|0, + \rangle \equiv |g\rangle \otimes |0\rangle ,$$
so that the second line of Eq. (24) applies for all excitation numbers $n$. Note that there is no $|0, - \rangle$ state.

It now follows that the dressed annihilation operator acts, simply, to reduce the total number of excitations in both the symmetric and the antisymmetric dressed states:

$$\begin{split}{\hat a_D}|n, + \rangle &= \sqrt n |n - 1, + \rangle , \\[-2pt] {\hat a_D}|n, - \rangle &= \sqrt {n - 1} |n - 1, - \rangle .\end{split}$$
Hence the dressed annihilation operator is
$${\hat a_D} = \sum\limits_{n = 1}^\infty \sqrt n |n - 1, + \rangle \langle n, + | + \sum\limits_{n = 1}^\infty \sqrt {n - 1} |n - 1, - \rangle \langle n, - |.$$

It is straightforward to confirm that the commutation relation with the corresponding dressed creation operator has the form

$$[{\hat a_D},\hat a_D^\dagger] = {\hat {\rm I}} \otimes {\hat {\rm I}},$$
as must be the case as ${\hat a_D}$ and $\hat a_D^\dagger$ are obtained from $\hat a$ and ${\hat a^\dagger}$ via a unitary transformation.

Let us turn now to the atom lowering operator for which we have the actions

$$\begin{split}{\hat \sigma _ -}|g\rangle \otimes |n\rangle &= 0, \\ {\hat \sigma _ -}|e\rangle \otimes |n\rangle &= |g\rangle \otimes |n\rangle .\end{split}$$
It then follows that the dressed operator has the actions
$$\begin{split}{\hat \sigma _{- D}}\hat U|g\rangle \otimes |n\rangle &= \hat U{\hat \sigma _ -}{\hat U^\dagger}\hat U|g\rangle \otimes |n\rangle = 0, \\ {\hat \sigma _{- D}}\hat U|e\rangle \otimes |n\rangle &= \hat U|g\rangle \otimes |n\rangle \end{split}$$
for all $n$ and hence we can write ${\hat \sigma _{- D}}$ in the form
$${\hat \sigma _{- D}} = \sum\limits_{n = 0}^\infty |n, + \rangle \langle n + 1, - |,$$
which clearly satisfies the requirement that $\hat \sigma _{- D}^2 = 0$. It is simple to confirm that the dressed lowering operator and its conjugate satisfy the required anticommutation relation
$$\{{\hat \sigma _{- D}},{\hat \sigma _{+ D}}\} = {\hat {\rm I}} \otimes {\hat {\rm I}}.$$
We note also that $\hat N$ is a constant of the motion and commutes with $\hat {\cal O}$ and it follows that this quantity is invariant under the unitary transformation
$${\hat N_D} = {\hat \sigma _{+ D}}{\hat \sigma _{- D}} + \hat a_D^\dagger {\hat a_D} = \hat N.$$

The two dressed lowering operators, one for the field and one for the atom, have simple effects on the dressed states. The dressed annihilation operator, ${\hat a_D}$, reduces by one the number of quanta in a dressed state while retaining the symmetry of the state; a symmetric state (+) is transformed into a symmetric state and an antisymmetric one (−) into an antisymmetric one. The dressed atom lowering operator also reduces the total number of excitations by one but also changes an antisymmetric state into a symmetric one. Acting on a symmetric state it gives zero. This situation is reminiscent of Mollow’s treatment of resonance fluorescence [19] (see also [20]) in which a triplet of fluorescent spectral lines can be observed given sufficiently strong driving. The central line, at the natural transition frequency may be compared with the action of ${\hat a_D}$, while the observed lower frequency $(\omega - \lambda)$ may be associated with the action of ${\hat \sigma _{- D}}$. The remaining line (centered at frequency $\omega + \lambda$) may be associated with the more complicated action of $\hat a_D^2{\hat \sigma _{+ D}}$.

5. JCM COHERENT STATES

The coherent states were introduced by Glauber in connection with the coherence properties of light [21,22] and have since played a crucial role in contrasting quantum effects from those that have a (semi-)classical explanation [23]. Although intrinsically associated with electromagnetic field modes, with the associated harmonic oscillator algebra, it was quickly realized that similar states could be formed for other quantum systems [24,25]. In this section we consider the forms of coherent states associated with the properties of our dressed operators, ${\hat a_D}$ and ${\hat \sigma _{- D}}$. We find that there are clear similarities between these and those more commonly associated with a single field mode or a single atom. We should note that the JCM has been studied using a combination of regular coherent states for the field and Grassmann coherent states for the atom [26]. The coherent states introduced in this section are distinct from these, but both sets are aimed at introducing a useful description of the properties and dynamics of the JCM.

We start with the coherent states associated with the annihilation operator. Recall that for a single (bare) oscillator, we can define a coherent state by means of a unitary operator acting on the ground or vacuum state:

$$|\alpha \rangle = \exp (\alpha {\hat a^\dagger} - {\alpha ^*}\hat a)|0\rangle = {e^{- |\alpha {|^2}/2}}\sum\limits_{n = 0}^\infty \frac{{{\alpha ^n}}}{{\sqrt {n!}}}|n\rangle .$$
The quasi-classical behavior of the mode prepared in this state may be traced to the fact that the coherent states are right-eigenstates of the annihilation operator
$$\hat a|\alpha \rangle = \alpha |\alpha \rangle .$$
The operator ${\hat a_D}$ acts on both the atom and the field mode and there are two effective ground states. The coherent states are
$$\begin{split} |\alpha , + \rangle& = \exp (\alpha {\hat a^\dagger} - {\alpha ^*}\hat a)|0, + \rangle = {e^{- |\alpha {|^2}/2}}\sum\limits_{n = 0}^\infty \frac{{{\alpha ^n}}}{{\sqrt {n!}}}|n, + \rangle , \\ |\alpha , - \rangle &= \exp (\alpha {\hat a^\dagger} - {\alpha ^*}\hat a)|0, - \rangle = {e^{- |\alpha {|^2}/2}}\sum\limits_{n = 0}^\infty \frac{{{\alpha ^n}}}{{\sqrt {n!}}}|n + 1, - \rangle . \end{split}$$
Note that the state $|\alpha , + \rangle$ is similar to that for the familiar single-mode coherent state, Eq. (34), but that the state $|\alpha , - \rangle$ is a superposition of states starting with $|1, - \rangle$, which is the lowest energy antisymmetric state. These two coherent states reflect, naturally, the existence of two distinct coherent states, $|g\rangle \otimes |\alpha \rangle$ and $|e\rangle \otimes |\alpha \rangle$, for the bare atom and field mode. As with the bare coherent states, both $|\alpha , + \rangle$ and $|\alpha , - \rangle$ are right eigenstates of ${\hat a_D}$:
$$\begin{split}{\hat a_D}|\alpha , + \rangle& = \alpha |\alpha , + \rangle , \\ {\hat a_D}|\alpha , - \rangle& = \alpha |\alpha , - \rangle . \end{split}$$
The two sets of states are mutually orthogonal,
$$\langle \alpha , + |\alpha ^\prime , - \rangle = 0,$$
and together they form an overcomplete representation of the atom-field state space:
$$\int \frac{{{{\rm d}^2}\alpha}}{\pi}|\alpha , + \rangle \langle \alpha , + | + |\alpha , - \rangle \langle \alpha , - | = {\hat {\rm I}} \otimes {\hat {\rm I}}.$$

Both of the JCM coherent states are entangled states of the atom and of the field. As the magnitude of $\alpha$ increases, however, the degree of entanglement reduces and tends, in the limit, to analogues of the semiclassical dressed states, with the atom driven by a classical (c-number) field: ${2^{- 1/2}}(|e\rangle \pm |g\rangle)|\alpha \rangle$.

The spin, or atomic, coherent states [8,27,28], like their oscillator counterparts, can be parameterized by a single complex number. For our two-level atom, we can write this in the form

$$|\zeta \rangle = (1 + |\zeta {|^2}{)^{- 1/2}}(|g\rangle + \zeta |e\rangle).$$
We note that all pure states of the atom can be written in this form. The spin coherent state is not an eigenstate of the lowering operator, ${\hat \sigma _ -}$, but it does have the non-zero expectation value
$$\langle \zeta |{\hat \sigma _ -}|\zeta \rangle = \frac{\zeta}{{1 + |\zeta {|^2}}}.$$
It is straightforward to obtain the analogous coherent states for the dressed operators ${\hat \sigma _{- D}}$. As with the coherent states for ${\hat a_D}$, there will be more than one of these; there are, in fact, infinitely many with one associated with each pair of dressed states $|n, + \rangle$ and $|n + 1, - \rangle$. For these two energy eigenstates we have the spin coherent state
$$|\zeta ,n\rangle = (1 + |\zeta {|^2}{)^{- 1/2}}(|n, + \rangle + \zeta |n + 1, - \rangle).$$
This is not an eigenstate of ${\hat \sigma _{- D}}$ but has the expectation value
$$\langle \zeta ,n|{\hat \sigma _ -}|\zeta ,n\rangle = \frac{\zeta}{{1 + |\zeta {|^2}}}.$$

The oscillator and spin coherent states are a natural consequence of the unitary connection between our bare and dressed states and operators. In particular we have

$$\begin{split} |\alpha , + \rangle &= \hat U|g\rangle \otimes |\alpha \rangle , \\ |\alpha , - \rangle &= \hat U|e\rangle \otimes |\alpha \rangle , \\ |\zeta ,n\rangle &= \hat U|\zeta \rangle \otimes |n\rangle . \end{split}$$
This means that the well-known properties and applications of the coherent states can readily be extended to the JCM dressed states.

6. CONCLUSION

Part of the appeal of the JCM, certainly to theorists, is the comparative simplicity with which it can be analyzed, coupled with the richness of the associated dynamics. There is scarcely a feature of quantum optics that has not been explored using the JCM, with important examples including collapses and revivals [29], the generation of squeezed light [30,31], and the generation of Schrödinger cat states [32].

We have derived the unitary transformation that diagonalizes the Jaynes-Cummings Hamiltonian. The full Hamiltonian in its diagonalized form is

$${\hat H_D} = \hbar \omega \hat N + \hbar \lambda \sqrt {\hat N} {\hat \sigma _3}.$$
It is immediately clear that the spectrum of the energy eigenvalues is $\hbar \omega n \pm \hbar \lambda \sqrt n$, which corresponds to the single zero energy ground state and then pairs of near-degenerate states split by the interaction term.

Having the unitary transformation means that we can transform the atom and field operators into operators that act directly on the basis of the dressed states. We have constructed the annihilation and atom lowering operators for the dressed Hamiltonian, ${\hat a_D}$ and ${\hat \sigma _{- D}}$, and explored some of their properties. It is to be hoped that these may find further application in the next 60 years of the JCM.

Funding

Royal Society (RP150122).

Acknowledgment

SMB is grateful to the Royal Society for the award of a Research Professorship. We are grateful to Sarah Croke and James Cresser for helpful comments.

Disclosures

The authors confirm that there is no conflict of interest.

Data availability

No data were generated in this work.

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Equations (45)

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H ^ = ℏ ω 0 2 σ ^ 3 + ℏ ω a ^ † a ^ + ℏ λ ( a ^ † σ ^ − + σ ^ + a ^ ) .
H ^ I = ℏ λ ( a ^ † σ ^ − + σ ^ + a ^ ) .
H ^ I | g ⟩ ⊗ | 0 ⟩ = 0 ,
H ^ I | n , ± ⟩ = ± ℏ λ n | n , ± ⟩ ,
| n , ± ⟩ = 1 2 ( | e ⟩ ⊗ | n − 1 ⟩ ± | g ⟩ ⊗ | n ⟩ ) .
N ^ = σ ^ + σ ^ − + a ^ † a ^ ,
N ^ | g ⟩ ⊗ | 0 ⟩ = 0 , N ^ | n , ± ⟩ = n | n , ± ⟩ .
O ^ = σ ^ + f ( a ^ † a ^ ) a ^ − a ^ † f ( a ^ † a ^ ) σ ^ − ,
[ O ^ , H ^ I ] = 2 ℏ λ ( σ ^ + σ ^ − a ^ a ^ † f ( a ^ † a ^ ) − σ ^ − σ ^ + a ^ † a ^ f ( a ^ † a ^ − 1 ) ) ,
[ O ^ , H ^ I ] = ℏ λ ( σ ^ + σ ^ − a ^ a ^ † − σ ^ − σ ^ + a ^ † a ^ ) .
[ O ^ , H ^ I ] | g ⟩ ⊗ | n ⟩ = − ℏ λ n | g ⟩ ⊗ | n ⟩ , [ O ^ , H ^ I ] | e ⟩ ⊗ | n − 1 ⟩ = ℏ λ n | e ⟩ ⊗ | n − 1 ⟩ .
[ O ^ , H ^ I ] | g ⟩ ⊗ | 0 ⟩ = 0.
O ^ = σ ^ + 1 2 ( a ^ † a ^ + 1 ) − 1 / 2 a ^ − a ^ † ( a ^ † a ^ + 1 ) − 1 / 2 σ ^ − = 1 2 ( σ ^ + E ^ − E ^ † σ ^ − ) ,
E ^ = ∑ n ′ = 0 ∞ ( n ′ + 1 ) − 1 / 2 | n ′ ⟩ ⟨ n ′ | ∑ n = 0 ∞ ( n + 1 ) 1 / 2 | n ⟩ ⟨ n | = ∑ n = 0 ∞ | n ⟩ ⟨ n + 1 | , E ^ † = ∑ n = 0 ∞ | n + 1 ⟩ ⟨ n | .
O ^ 2 = ( − σ ^ + σ ^ − E ^ E ^ † − σ ^ − σ ^ + E ^ † E ^ ) / 4 = − 1 4 ( I ^ ⊗ I ^ − | g ⟩ ⟨ g | ⊗ | 0 ⟩ ⟨ 0 | ) ,
U ^ = e π 2 O ^ = 1 2 ( I ^ ⊗ I ^ − | g ⟩ ⟨ g | ⊗ | 0 ⟩ ⟨ 0 | + 2 O ^ ) + | g ⟩ ⟨ g | ⊗ | 0 ⟩ ⟨ 0 | ,
U ^ H ^ I U ^ † = ℏ λ ( σ ^ + σ ^ − a ^ a ^ † − σ ^ − σ ^ + a ^ † a ^ ) .
H ^ I D = ℏ λ N ^ σ ^ 3 .
H ^ I D | g ⟩ ⊗ | n ⟩ = − ℏ λ n | g ⟩ ⊗ | n ⟩ , H ^ I D | e ⟩ ⊗ | n − 1 ⟩ = ℏ λ n | e ⟩ ⊗ | n − 1 ⟩ .
O ^ = 1 2 N ^ ( σ ^ + a ^ − a ^ † σ ^ − ) ,
a ^ D = U ^ a ^ U ^ † , σ ^ − D = U ^ σ ^ − U ^ † .
a ^ | g ⟩ ⊗ | n ⟩ = n | g ⟩ ⊗ | n − 1 ⟩ , a ^ | e ⟩ ⊗ | n ⟩ = n | e ⟩ ⊗ | n − 1 ⟩
a ^ D U ^ | g ⟩ ⊗ | n ⟩ = U ^ a ^ U ^ † U ^ | g ⟩ ⊗ | n ⟩ = n U ^ | g ⟩ ⊗ | n − 1 ⟩ , a ^ D U ^ | e ⟩ ⊗ | n ⟩ = n U ^ | e ⟩ ⊗ | n − 1 ⟩ ,
U ^ | g ⟩ ⊗ | 0 ⟩ = | g ⟩ ⊗ | 0 ⟩ , U ^ | g ⟩ ⊗ | n ⟩ = | n , + ⟩ , U ^ | e ⟩ ⊗ | n ⟩ = | n + 1 , − ⟩
| 0 , + ⟩ ≡ | g ⟩ ⊗ | 0 ⟩ ,
a ^ D | n , + ⟩ = n | n − 1 , + ⟩ , a ^ D | n , − ⟩ = n − 1 | n − 1 , − ⟩ .
a ^ D = ∑ n = 1 ∞ n | n − 1 , + ⟩ ⟨ n , + | + ∑ n = 1 ∞ n − 1 | n − 1 , − ⟩ ⟨ n , − | .
[ a ^ D , a ^ D † ] = I ^ ⊗ I ^ ,
σ ^ − | g ⟩ ⊗ | n ⟩ = 0 , σ ^ − | e ⟩ ⊗ | n ⟩ = | g ⟩ ⊗ | n ⟩ .
σ ^ − D U ^ | g ⟩ ⊗ | n ⟩ = U ^ σ ^ − U ^ † U ^ | g ⟩ ⊗ | n ⟩ = 0 , σ ^ − D U ^ | e ⟩ ⊗ | n ⟩ = U ^ | g ⟩ ⊗ | n ⟩
σ ^ − D = ∑ n = 0 ∞ | n , + ⟩ ⟨ n + 1 , − | ,
{ σ ^ − D , σ ^ + D } = I ^ ⊗ I ^ .
N ^ D = σ ^ + D σ ^ − D + a ^ D † a ^ D = N ^ .
| α ⟩ = exp ⁡ ( α a ^ † − α ∗ a ^ ) | 0 ⟩ = e − | α | 2 / 2 ∑ n = 0 ∞ α n n ! | n ⟩ .
a ^ | α ⟩ = α | α ⟩ .
| α , + ⟩ = exp ⁡ ( α a ^ † − α ∗ a ^ ) | 0 , + ⟩ = e − | α | 2 / 2 ∑ n = 0 ∞ α n n ! | n , + ⟩ , | α , − ⟩ = exp ⁡ ( α a ^ † − α ∗ a ^ ) | 0 , − ⟩ = e − | α | 2 / 2 ∑ n = 0 ∞ α n n ! | n + 1 , − ⟩ .
a ^ D | α , + ⟩ = α | α , + ⟩ , a ^ D | α , − ⟩ = α | α , − ⟩ .
⟨ α , + | α ′ , − ⟩ = 0 ,
∫ d 2 α π | α , + ⟩ ⟨ α , + | + | α , − ⟩ ⟨ α , − | = I ^ ⊗ I ^ .
| ζ ⟩ = ( 1 + | ζ | 2 ) − 1 / 2 ( | g ⟩ + ζ | e ⟩ ) .
⟨ ζ | σ ^ − | ζ ⟩ = ζ 1 + | ζ | 2 .
| ζ , n ⟩ = ( 1 + | ζ | 2 ) − 1 / 2 ( | n , + ⟩ + ζ | n + 1 , − ⟩ ) .
⟨ ζ , n | σ ^ − | ζ , n ⟩ = ζ 1 + | ζ | 2 .
| α , + ⟩ = U ^ | g ⟩ ⊗ | α ⟩ , | α , − ⟩ = U ^ | e ⟩ ⊗ | α ⟩ , | ζ , n ⟩ = U ^ | ζ ⟩ ⊗ | n ⟩ .
H ^ D = ℏ ω N ^ + ℏ λ N ^ σ ^ 3 .
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