1 Introduction
The quantum Rabi model (QRM) describes the simplest interaction between a two-level atom (qubit) and a light field (cavity). It has inspired exciting developments in quantum optics, quantum information science, and condensed matter physics for a long time.[1] The Hamiltonian reads
$ H_{R}=\frac{\Delta }{2}\sigma _{z}+\omega_0 a^{\dagger }a+g( a^{\dagger }+a) \sigma _{x} , $
where $\Delta $ and $\omega_0 $ are frequencies of two-level system and cavity, $\sigma _{x,z}$ is usual Pauli matrices for the two-level system, $ a~(a^{\dagger })$ is the annihilation (creation) bosonic operator of the cavity mode, and $g$ is the coupling strength. In the conventional quantum optics,[2] the coupling strength between the atom and the field is quite weak, $g/\omega \thicksim 10^{-6}$. It can be described by the well-known Jaynes-Cummings model[3] where the rotating-wave approximation (RWA) is made. Many physical phenomena can be described in this framework, such as collapse and revival of quantum state populations, entanglement, Schröding cat states.[4]
In recent years, the QRM can be implemented in the ultra-strong coupling regime $% ( g/\omega \thicksim 0.1) $ in some solid-state device, such as superconducting circuits.[5-7] More recently, even the deep-strong coupling $( g/\omega >1) $ regime[8] can be realized in the Josephson junctions.[9-10] The breakdown of the rotating-wave approximation has been shown at ultra-strong coupling[5] in the qubit-oscillator system experimentally. Many theoretical approaches in the ultra-strong coupling regime thus have been developed[11-15] where the effect of the counter-rotating wave terms are taken into account.
On the other hand, quantum simulations attract many researchers to design and realize ideal quantum models which have interesting physical phenomena. For example, trapped-ion technologies, can bring about many important phenomena with controllable parameters and coherence, such as system fidelity, dynamics and quantum phase transitions.[16] For the QRM with tunable parameters, it has been realized in quantum simulations based on Raman transitions in an optical cavity QED settings.[17] In this proposed scheme, a new nonlinear coupling item between the atom and cavity
$ H_{\rm NL}=\frac{U}{2}\sigma _{z}a^{\dagger }a , $
is added to the linear dipole coupling, where the coupling strength $U$ is determined by the dispersive energy shift. This generalized model was first proposed by Grimsmo and Parkins, which can be called quantum Rabi-Stark model because the new added item is associated with the dynamical Stark shift discussed in the quantum optics.[18] This emergent Stark-like nonlinear interaction has no parallel in the conventional cavity QED, which needs careful study. The spectrum of the total Hamiltonian $H_{0}=H_{R}+H_{\rm NL}$ has been studied mathematically in the Bargmann space[19] but their derivations are very complicated and the expressions are very lengthy.
The solutions to the linear QRM has been searched for a long time. Many analytical works have been proposed for decades.[20-31] The analytically exact solution was found by Braak[32] using the Bargmann representations. It was shown that Braak's solution can be constructed in terms of the mathematically well-defined Heun confluent function.[33] By Bogoliubov operators,[34] Braak's solution was reproduced straightforwardly in a more physical way. Before Braak's solution, Chen {\it et al}.[29] proposed an exact solution by using tunable coherent states. This approach is much simpler and has no pole structures, and therefore facilitates the calculations of the eigensolutions. Moreover, it can be easily extended to other related models which are the qubit and bosons coupling. Recently, the tunable coherent states approach has been generalized to cavity femtochemistry[35] by Mukamel's group.
In this work, we will study the quantum Rabi-Stark model by the tunable coherent states approach, and then explore some exotic physical phenomena. The paper is structured as follows: In Sec. 2, a concise equation is derived for this model by using tunable coherent states. In Sec. 3, we use the first-order approximation to give analytical solutions of the energy and mean photon numbers in the ground states. In Sec. 4, we explore the spectral collapse of this model when $U\rightarrow \pm 2\omega _{0}$. Section~5 contains some concluding remarks and future directions.
2 Exact Solutions with Tunable Coherent States
To facilitate the study, the Hamiltonian $H_{0}=H_{R}+H_{\rm NL}$ is rotated around the $y$-axis by an angle $\pi /2$,
$ H_{0}=-\frac{1}{2}( \Delta +Ua^{\dagger }a) \sigma _{x}+\omega _{0}a^{\dagger }a+g( a^{\dagger }+a) \sigma _{z} . $
In terms of two eigenstates of $\sigma _{z}$, the above Hamiltonian takes the following matrix form in unit of $ \omega _{0}=1 $,
$ H=\left(\begin{matrix} a^{\dagger }a+g( a^{\dagger }+a) & \quad -\frac{1}{2}( \Delta +Ua^{\dagger }a) \\ -\frac{1}{2}( \Delta +Ua^{\dagger }a) & \quad a^{\dagger }a-g(a^{\dagger }+a)\end{matrix} \right). $
This Hamiltonian enjoys a discrete $Z_{2}$ symmetry, meaning the total excitation number $\hat{N}=( 1-\sigma _{x}) /2+a^{\dagger }a$ is conserved. The parity operator defined as $\Pi =exp ( i\pi \hat{N}) $ commutes with the Hamiltonian $H$. Here $\Pi$ has two eigenvalues $\pm 1$ depending on whether $\hat{N}$ is even or odd. Thus the whole Hilbert space can be divided into two isolated subspaces (even or odd) via the parity operator.
Based on the method originally proposed in Ref. [29], the wave function is supposed as
$ | \Psi \rangle = \left( \begin{matrix} \sum_{n=0}^{\infty}c_{n}( a^{\dagger }) ^{n}exp ( \alpha a^{\dagger }) | 0\rangle \\ \Pi \sum_{n=0}^{\infty}c_{n}( -a^{\dagger }) ^{n}exp ( -\alpha a^{\dagger }) | 0\rangle \end{matrix}\right), $
where the coefficients $c_{n}$ and coherent parameter $\alpha $ are to be determined by the following calculations.
Fig. 1 (Color online) The lowest ten solutions of the polynomial equation for $\Delta =0.5$ with different parameters $U$ and $g$. The left panel is for even parity and the right panel is for odd parity. The black circles are numerical solutions. |
Substituting Eq. (5) into Eq. (4), the Schrödinger equation gives
$ Ec_{n}( a^{\dagger }) ^{n}exp ( \alpha a^{\dagger }) | 0\rangle \\ = c_{n}[ a^{\dagger }a + g( a^{\dagger }+a) ] ( a^{\dagger }) ^{n}exp ( \alpha a^{\dagger }) | 0\rangle \\ -\frac{\Pi }{2}( \Delta +Ua^{\dagger }a) \sum_{n}c_{n}(-a^{\dagger }) ^{n}exp ( -\alpha a^{\dagger }) | 0\rangle .$
Each terms in both sides of the above equation can be expanded in terms of $(a^{\dagger})^{n}exp ( \alpha a^{\dagger }) | 0\rangle $. Equating the coefficients of these terms in $n$-th order yields
$ Ec_{n} = ( n+\alpha g) c_{n}+( \alpha +g) c_{n-1}+g(n+1) c_{n+1} \\ -\frac{\Pi }{2}( -1) ^{n}\sum_{j=0}^{n}\frac{( 2\alpha) ^{j}}{j!} \\ \times \{ [ \Delta +U( n-j) ] c_{n-j}+U\alpha c_{n-j-1}\} .$
Similar to Ref. [29], looking at the wavefunction (5), one can note that $c_0$ is flexible in the Schrödinger equation where the normalization for the eigenfunction is not necessary, so we select $c_0=1$. The linear term in $a^{\dagger}$ in the Fock space can be also determined by the displacement $\alpha$ in the pure coherent state so we do not need $c_1$, which can be set to zero. When putting $n=0$, we can obtain the energy function for the tunable parameter $\alpha$,
$ E=\alpha g-\frac{\Pi }{2}\Delta . $
Inserting the energy into Eq. (7) leads to the following recursive relation for $c_{n}$,
$ c_{n+1}=\frac{( n+({\Pi }/{2})\Delta ) c_{n}+( \alpha +g) c_{n-1}}{-g( n+1) } \\ +\frac{\Pi ( -1) ^{n}}{2g( n+1) }\sum_{j=0}^{n}\frac{( 2\alpha ) ^{j}}{j!} \\ \times \{ [ \Delta +U( n-j) ] c_{n-j}+U\alpha c_{n-j-1}\} .$
Now all the coefficients can be obtained by this $\alpha $-dependent polynomial equation. The zero locations of Eq. (9) converge reasonably well with increasing truncated number in the eigenstates (5). In principle, we can set infinite truncated number, but it is impossible to implement practically in the numerical calculations. Thus by setting $c_{N_{tr}+1}=0$, we have a closed polynomial equation about $\alpha $, which can be solved exactly. We actually can determine the solutions of the model with any desired accuracy by increasing truncated number $N_{\rm tr}$. In numerical sense, it can be called exact solutions.
Figure 1 shows the lowest ten zeroes of the polynomial equation for even (blue lines) and odd (red lines) parity for different $U$ and $g$. The numerical solutions by exact diagonalzations denoted by black circles are also collected. It is found that two results agree excellently with each other. This is to say, the present method can yield the exact solution by searching for the zero of the derived simple polynomial equation.
3 Results for the Ground State
In this section, we present some results for the ground states based on the tunable coherent state approach. The ground-state wavefunction can be obtained exactly by solving polynomial equation (9), then we can calculate many physical observables. Here we calculate the entanglement entropy[36] between two-level system and cavity, which is defined as
$ S=-Tr( \rho \log _{2}\rho ) , $
where $\rho =Tr_{\rm cavity}( | \Psi \rangle \langle \Psi | ) $ is the reduced density matrix by tracing out the cavity degrees of freedom. We plot the ground-state entanglement entropy with black solid lines for $\Delta =0.5$, $U=1.9$ in Fig. 2(a). Surprisingly, one can observe that the entropy jumps at a critical coupling for these model parameters. For the one-mode cavity case,[37] the entanglement entropy in the ground-state usually increases smoothly with the coupling strength. To account for this unexpected findings, we calculate the energy spectra in low energy regime, as demonstrated in Fig. 2(b). We find that the parity of the ground state energy changes from even to odd at the level crossing. It is just this level crossings of the two lowest energy level that cause the abrupt jump of the entanglement entropy in the ground state. Both the discontinuous parity and entanglement entropy imply the existence of the first-order phase transition in the Rabi-Stark model, which has never been found in the isotropic linear QRM.
Fig. 2 (Color online) (a) The entanglement entropy as a function of the coupling strength for the even parity (blue line) and odd parity (red line). The ground state is denoted by the black solid line, corresponding to the lowest energy level. (b) Some low energy spectra with even (blue) and odd (red) parity obtained by the polynomial equation. $\Delta =0.5$, $U=1.9$. |
Next, we will present the analytical closed-form expressions for the ground-states up to ultra-strong coupling (i.e. $ g/\omega \thicksim 0.1 $). According to the polynomial equation (9) above, we have
$ c_{2}=\frac{1}{-2g}\Big[ \alpha + g +\Pi \alpha \Big( \Delta +\frac{U}{2}\Big) \Big] .$
With wave function (5) the energy and the mean photon number of the ground state can be determined analytically by performing the first-order approximation via $c_{3}( \alpha ) =0$, which leads to a quadratic equation
$ g( U+\Delta ) \alpha ^{2}+\Big( \Pi -\frac{U}{2}\Big) \Big[1+\Pi \Big( \Delta +\frac{U}{2}\Big) \Big] \alpha \\ + \Big( \Pi -\frac{U}{2}\Big) g = 0 .$
At the weak and ultra-strong coupling, the ground-state is usually in the even parity $\Pi=1$. The lowest energy is achieved accordingly if
$ \alpha _{0} = -\frac{( 1-\frac{U}{2}) ( 1+\Delta +\frac{U}{2}) }{2g( U+\Delta ) } \\ +\frac{\sqrt{( 1-\frac{U}{2}) ^{2}( 1+ \Delta + \frac{U}{2} ) ^{2} -2g^{2}( 2- U) ( U+ \Delta ) }}{2g( U+\Delta ) }, $
$ E_{0} = \alpha _{0}g-\frac{\Delta }{2} .$
For the real physical solution, the coupling strength should be confined to
$ g<\frac{( 2-U) ( 2+2\Delta +U) }{4\sqrt{2( 2-U) ( U+\Delta ) }} . $
Therefore the analytical results (13) and (14) cannot be applied to the strong coupling regime. Fortunately, we will demonstrate below that the analytical results work very well up to the ultra-strong coupling regime.
With the energy at hand, we can obtain the corresponding eigenstate as
$ | \Psi _{0}\rangle = \frac{1}{\sqrt{2+2c_{2}^{2}}}\left(\begin{matrix} exp ( \alpha _{0}a^{\dagger }) | 0\rangle +c_{2}( a^{\dagger }) ^{2}exp ( \alpha _{0}a^{\dagger }) | 0\rangle \\ exp (-\alpha _{0}a^{\dagger }) | 0\rangle +c_{2}( -a^{\dagger }) ^{2}exp ( -\alpha _{0}a^{\dagger }) | 0\rangle \end{matrix}\right). $
Transforming back to the original representation, the mean photon number is derived as
$ \langle a^{\dagger }a\rangle = \frac{\alpha _{0}^{2}+c_{2}^{2}( 4+14\alpha _{0}^{2}+8\alpha _{0}^{4}+\alpha _{0}^{6}) +2c_{2}( \alpha _{0}^{4}+2\alpha _{0}^{2}) }{1+c_{2}^{2}}e^{\alpha _{0}^{2}} .$
Fig. 3 (Color online) The ground state energy $E_{0}$ (left panel) and mean photon number $\langle a^{\dagger }a\rangle $ (right panel) for $g=0.25$, $U=0.5$ (upper panel) and $\Delta=0.5$ (low panel). |
The analytical results for the ground state energy and the mean photon number for $g=0.25$ at different values of $U$ and $\Delta$ are plotted in Fig. 3. The exact solutions are also presented. Very interestingly, one can note that the analytical results agree well with the exact ones for a wide range of $U$ and $\Delta$ at ultra-strong coupling, such as $g=0.25$. We have checked that the situation become even better with the decrease of the coupling strength. So in practise, the present analytical results for the ground-state can be applied to the recent experiments which is usually performed at from weak to ultra-strong coupling regime.
4 Spectral Collapse
The solution to the Rabi-Stark model at $U=\pm 2$ is extremely difficult to achieve in both the Bargmann space[19] and the present tunable coherent state approach due to the extremely slow convergence with the expansion number. Here we propose a new analytical method for this special case. We mainly discuss the $U=2$ case. The results of $U=-2$ can be obtained from those of $U=2$ just by replacing $\Delta$ by $-\Delta$.
In terms of the basis of $\sigma_z$, Hamiltonian (3) in the matrix form can be written as
$ H_{0}=\left(\begin{matrix} 2a^{\dagger }a+{\Delta }/{2} & \quad g( a^{\dagger }+a) \\ g( a^{\dagger }+a) & \quad -{\Delta }/{2} \end{matrix} \right).$
In principle, the wave function can be expanded in the Fock space as
$ | \Psi \rangle =\binom{| \Psi _{1}\rangle }{| \Psi _{2}\rangle }=\left(\begin{matrix} \sum_{n}^{\infty}e_{n}| n\rangle \\ \sum_{n}^{\infty}f_{n}| n\rangle \end{matrix} \right) . $
The Schrödinger equation then gives
$ \Big( 2a^{\dagger }a+\frac{\Delta }{2}\Big) \sum_{n}^{\infty}e_{n}| n\rangle + g( a^{\dagger }+a) \sum_{n}^{\infty}f_{n}| n\rangle \\ = E\sum_{n}^{\infty}e_{n}| n\rangle , $
$ g( a^{\dagger }+a) \sum_{n}^{\infty}e_{n}| n\rangle = \Big(\frac{\Delta }{2}+E\Big) \sum_{n}^{\infty}f_{n}| n\rangle . $
By inspection of Eq. (21), we obtain
$ \sum_{n}^{\infty}f_{n}| n\rangle =\frac{g( a^{\dagger }+a) }{{\Delta }/{2}+E}\sum_{n}^{\infty}e_{n}| n\rangle .$
Inserting it into Eq. (20) leads to the effective Hamiltonian for the bosonic wavefunction in the upper level $| \Psi _{1}\rangle$
$ H^{\prime }=2( 1+\chi ) a^{\dagger }a+\chi ( a^{\dagger 2}+a^{2}) +\chi +\frac{\Delta }{2} , $
where $\chi={g^{2}}/({{\Delta }/{2}+E})$ is dependent on the energy.
Note that Hamiltonian (23) can be easily solved by a simple Bogoliubov transformation. Applying the following squeezed operator transformation
$S=exp \Big[ \frac{\lambda }{2}( a^{2}-a^{\dagger 2}) \Big] , \quad \lambda =\frac{1}{4}\ln \Big( \frac{1}{1+2\chi }\Big) $
to the Hamiltonian $H^{\prime}$, we have
$ SH^{\prime }S^{\dagger }=\sqrt{1+2\chi }( 2a^{\dagger }a+1) -1+\frac{ \Delta }{2} , $
which is corresponding to an exactly solvable free particle system. Then we can easily get a univariate cubic equation for the energy eigenvalue $E$
$ \frac{g^{2}+\chi ( 1-\Delta ) }{\chi \sqrt{1+2\chi }}=2n+1,n=0,1,2,\ldots ,$
which requires $E>{\Delta }/{2}-1$ or $E<E_{c}^{+}=-{\Delta }/{2}-2g^{2}$, $g<g_{c}^{+}=\sqrt{({1-\Delta })/{2}}$. This method using squeezed states recovers the results in Ref. [38] with Bargmann space and our previous work in Ref. [39] with harmonic oscillator. For $g>g_{c}^{+}$, the low energy levels can only collapse to the same point $E=E_{c}^{+}$, because no real solution can be found in the energy interval [$E_{c}^{+}, {\Delta }/{2}-1$] by Eq. (25).
To this end, the exact solutions of the Rabi-Stark spectrum can be obtained in the whole range of $| U| \le 2$. So we can study effect of the additional nonlinear coupling term $({U}/{2})a^{\dagger }a\sigma _{z}$ on the energy spectra. Figure~4 displays the lowest five energy levels with both even and odd parity as a function of $U$ for $\Delta =0.5$, $g=0.25$ and $g=0.75$. One can see that many discrete energy levels tend to a narrow region around or collapse to $E_{c}^{+}=-{\Delta }/{2}-2g^{2}$ in the $U\rightarrow \pm 2$ limit. From the inset of Fig. 4(b), we observe that all the spectral collapse to $E_{c}^{+}$, and no discrete energy levels lie below this energy. But the insets of other figures demonstrate many discrete levels below $E_{c}^{\pm}=\mp(\Delta/2)-2g^2$ in the $U\rightarrow \pm 2$ limit.
Note that the energy spectral collapse has been observed and discussed extensively in the two-photon QRM when the normalized coupling approaches the half cavity frequency.[40-42] Very interestingly, the similar behavior is also observed in this Rabi-Stark model, which adds a new member to the list of quantum Rabi models with the spectral collapse.
Fig. 4 (Color online) Energy spectrum as a function of $U$ for $\Delta=0.5$, $g=0.25$ (upper panel) and $g=0.75$ (low panel). Blue lines denote even parity and red lines odd parity. Insets show the corresponding enlarged views around the accumulation area. |
5 Conclusion
In this work, we use tunable coherent states approach to derive a one variable polynomial equation for the quantum Rabi-Stark model. The solutions to the polynomial equation determine the spectrum exactly. Based on the exact solution, we fist study the entanglement entropy in the ground state. Interestingly, we find that the entropy could jump abruptly with the coupling strength, in sharp contrast to the previous two-level system coupling with one mode bosons. This unexpected feature is attributed to the first-order phase transition in the ground state, which can be detected by the level crossing of the ground state and the first excited state. We also derive closed-form analytical results of the energy and mean photon numbers in the ground states by a first-order approximation in the tunable coherent states scheme. These analytical results agree well with the exact ones in a wide range of detuning $\Delta$ and energy shift $U$. The present model at critical value $U=\pm 2$ is studied in a new way based on a squeezed states transformation. We finally analyze the energy spectral collapse in the $U\rightarrow \pm 2$ limit. It turns out that the spectral collapse not only occurs in the two-photon QRM, but also in the Rabi-Stark model. One common feature for these two model is the non-linear coupling between the atom and cavity.
The spectral collapse and the discrete levels observed at the critical points might be qualitatively understood in the polaron picture by the tunneling induced potential well.[43] Due to the parity symmetry, the second-order phase transition in the present model should also occur, like that in the linear QRM.[44-46] The applications of the present approach to these studies are in progressing.