Tavis–Cummings model

The Tavis–Cummings model assumes that for the purposes of electromagnetic interactions, atomic structures are dominated by their dipole, as they are for distant neutral atoms in the weak-field limit. Thus the only atomic quantity under consideration is its angular momentum, not its position nor fine electronic structure. Furthermore, the model asserts the atoms to be sufficiently distant that they don't interact with each-other, only with the electromagnetic field, modeled as a bosonic field (since photons are the gauge bosons of electromagnetism).

For two atomic-electronic states separated by a Bohr frequency $\backslash omega\_\{eg\}$, then transitions between the ground- and excited-states $|g\backslash rangle$ and $|e\backslash rangle$ are mediated by Pauli operators: $\backslash hat\{\backslash sigma\}\_z=|e\backslash rangle\backslash langle\; e|-|g\backslash rangle\backslash langle\; g|$, $\backslash hat\{\backslash sigma\}\_\{+\}=|e\backslash rangle\backslash langle\; g|$, and $\backslash hat\{\backslash sigma\}\_\{-\}=|g\backslash rangle\backslash langle\; e|$, and the Hamiltonian separating these energy states in the $j$th atom is $\backslash hat\{H\}\_A^\{(j)\}=\backslash frac\{\backslash hbar\; \backslash omega\_\{eg\}\}\{2\}\backslash hat\{\backslash sigma\}\_z^\{(j)\}$. With $N$ independent atoms each subject to this energy gap, the total atomic Hamiltonian is thus $\backslash hat\{H\}\_A=\backslash sum\_\{j=1\}^\{N\}\backslash hat\{H\}\_A^\{(j)\}=\backslash omega\_\{eg\}\backslash hat\{S\}\_z$ with total spin operators $\backslash hat\{S\}\_\{\backslash alpha\}=\backslash frac\{\backslash hbar\}\{2\}\backslash sum\_\{j=1\}^\{N\}\backslash hat\{\backslash sigma\}\_\{\backslash alpha\}^\{(j)\}$.

Similarly, in a free field with no modal restrictions, creation and annihilation operators dictate the presence of photons in each mode: $\backslash hat\{H\}\_\{F\}=\backslash sum\_\{\backslash vec\{k\},\backslash mu\}\backslash hbar\backslash omega\_\{k\}\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}^\backslash dagger\; \backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}$ with wave number $\backslash vec\{k\}$, polarization $\backslash mu$, and frequency $\backslash omega\_k=c^\{-1\}|\backslash vec\{k\}|$. If the dynamics occur within a sufficiently small cavity, only one mode (the cavity's resonant mode) will couple to the atom, thus the field Hamiltonian simplifies to $\backslash hat\{H\}\_\{F\}=\backslash hbar\backslash omega\_c\backslash hat\{a\}^\backslash dagger\backslash hat\{a\}$, just as in the Jaynes-Cummings and Dicke models.{{Not a typo (four dots on red and green levels) interacting symmetrically with a single mode photonic field (blue standing wave), isolated within a cavity. The atomic level separation is $\backslash omega\_\{eg\}$, the cavity's resonant frequency is $\backslash omega\_c$, and the coupling strength of atom-field interactions is $g\_c$.|400x400px]]

Finally, interactions between atoms and the field is determined by the atomic dipole, rendered quantumly as an operator $\backslash hat\{\backslash vec\{d\}\}$, and the similarly expressed electric field at the atoms' centers (assuming the field is the same at each atom's position) $\backslash hat\{\backslash vec\{E\}\}$, thus $\backslash hat\{H\}\_\{AF\}=-\backslash sum\_\{j=1\}^N\backslash hat\{\backslash vec\{d\}\}^\{(j)\}\backslash cdot\backslash hat\{\backslash vec\{E\}\}^\{(j)\}$ which acts on both the qubit and bosonic degrees of freedom. The dipole operator couples the excited and ground states of each atom $\backslash hat\{\backslash vec\{d\}\}^\{(j)\}=\backslash vec\{d\}\_\{eg\}\backslash hat\{\backslash sigma\}\_\{+\}^\{(j)\}+\backslash vec\{d\}\_\{eg\}^*\backslash hat\{\backslash sigma\}\_\{-\}^\{(j)\}$, while the electric free field solution is:

$\backslash hat\{\backslash vec\{E\}\}[\backslash vec\{R\},t]=\backslash sum\_\{\backslash vec\{k\},\backslash mu\}\; E\_\{k\}\; \backslash vec\{\backslash epsilon\}\_\{\backslash vec\{k\},\backslash mu\}\; e^\{i(\backslash vec\{k\}\backslash cdot\backslash vec\{R\}-\backslash omega\_\{k\}\; t)\}\; \backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}+\; E\_\{k\}\backslash vec\{\backslash epsilon\}\_\{\backslash vec\{k\},\backslash mu\}e^\{-i(\backslash vec\{k\}\backslash cdot\backslash vec\{R\}-\backslash omega\_\{k\}\; t)\}\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}^\backslash dagger$, which at a static point evaluates as:

$\backslash hat\{\backslash vec\{E\}\}=\backslash sum\_\{\backslash vec\{k\},\backslash mu\}\; \backslash sqrt\{2\backslash pi\backslash hbar\backslash omega\_k\}(e^\{-i\backslash omega\_k\; t\}\backslash vec\{u\}\_\{\backslash vec\{k\},\backslash mu\}\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}+\; e^\{i\backslash omega\_k\; t\}\; \backslash vec\{u\}\_\{\backslash vec\{k\},\backslash mu\}^*\; \backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}^\backslash dagger)$, thus the interaction Hamiltonian expands as

$\backslash hat\{H\}\_\{AF\}=\backslash sum\_\{\backslash vec\{k\},\backslash mu\}(e^\{-i\backslash omega\_k\; t\}g\_\{\backslash vec\{k\},\backslash mu\}\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}+e^\{i\backslash omega\_k\; t\}g\_\{\backslash vec\{k\},\backslash mu\}^*\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}\; ^\backslash dagger)(\backslash hat\{S\}\_\{+\}+\backslash hat\{S\}\_\{-\})$.

Here, $g\_\{\backslash vec\{k\},\backslash mu\}\backslash equiv-\backslash sqrt\{2\backslash pi\backslash hbar\backslash omega\_k\; N\; \}\backslash vec\{d\}\_\{eg\}\backslash cdot\backslash vec\{u\}\_\{\backslash vec\{k\},\backslash mu\}$ specifies the coupling strength of the total dipole to each electric field mode, and functioning as a Rabi frequency that scales with ensemble size $g\_c\backslash propto\backslash sqrt\{N\}$ due to the Pythagorean addition of single-atom dipoles. Then, in the rotating frame, $\backslash hat\{H\}\_\{AF\}=\backslash sum\_\{\backslash vec\{k\},\backslash mu\}(e^\{-i\backslash omega\_k\; t\}g\_\{\backslash vec\{k\},\backslash mu\}\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}+e^\{i\backslash omega\_k\; t\}g\_\{\backslash vec\{k\},\backslash mu\}^*\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}\; ^\backslash dagger)(e^\{i\backslash omega\_\{eg\}t\}\backslash hat\{S\}\_\{+\}+e^\{-i\backslash omega\_\{eg\}t\}\backslash hat\{S\}\_\{-\})$, which results in corotating terms $\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}\backslash hat\{S\}\_\{+\}$ (representing photon absorption causing atomic excitation), $\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}^\backslash dagger\; \backslash hat\{S\}\_\{-\}$ (representing spontaneous emission), and counterrotating terms $\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}\; \backslash hat\{S\}\_\{-\}$ and $\backslash hat\{a\}\_\{\backslash vec\{k\},\backslash mu\}^\backslash dagger\; \backslash hat\{S\}\_\{+\}$ (representing second-order effects like self-interaction and Lamb shifts). When $\backslash omega\_k\backslash approx\; \backslash omega\_\{eg\}$ and $g\_\{\backslash vec\{k\},\backslash mu\}\backslash ll\; \backslash omega\_k$ (close to resonance in a weak field), the corotating terms accumulate phase very slowly, while the counterrotating terms accumulate phase too fast to significantly affect time-ordered integrals, thus the rotating wave approximation allows counterrotating terms to drop in the rotating frame. The cavity permits only one field mode with energy sufficiently close to the Bohr energy, $\backslash omega\_k=\backslash omega\_c\backslash approx\; \backslash omega\_\{eg\}$, so the final form of the interaction Hamiltonian is $\backslash hat\{H\}\_\{AF\}=g\_c\; e^\{-i\backslash Delta\; t\}\backslash hat\{a\}\backslash hat\{S\}\_\{+\}+g\_c^*\; e^\{i\backslash Delta\; t\}\backslash hat\{a\}^\backslash dagger\; \backslash hat\{S\}\_\{-\}$ for dephasing $\backslash Delta=\backslash omega\_c-\backslash omega\_\{eg\}$.

In total, the Tavis–Cummings Hamiltonian includes the atomic and photonic self-energies and the atom-field interaction:

$\backslash hat\{H\}\_\{TC\}=\backslash hat\{H\}\_\{A\}+\backslash hat\{H\}\_\{F\}+\backslash hat\{H\}\_\{AF\}$,

$\backslash hat\{H\}\_\{A\}=\backslash omega\_\{eg\}\backslash hat\{S\}\_z$,

$\backslash hat\{H\}\_\{F\}=\backslash hbar\backslash omega\_c\backslash hat\{a\}^\backslash dagger\; \backslash hat\{a\}$,

$\backslash hat\{H\}\_\{AF\}=g\_c\; e^\{-i\backslash Delta\; t\}\backslash hat\{a\}\backslash hat\{S\}\_\{+\}+g\_c^*\; e^\{i\backslash Delta\; t\}\backslash hat\{a\}^\backslash dagger\; \backslash hat\{S\}\_\{-\}$.

The Tavis–Cummings model as described above exhibits two symmetries arising from the Hamiltonian's commutation{{Cite journal |last1=Tavis |first1=Michael |last2=Cummings |first2=Frederick W. |date=10 June 1968 |title=Exact Solution for an N-Molecule-Radiation-Field Hamiltonian |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.170.379 |journal=Physical Review |volume=170 |issue=2 |pages=379–384 |doi=10.1103/PhysRev.170.379 |bibcode=1968PhRv..170..379T |via=APS}} with excitation number $\backslash hat\{M\}=\backslash hat\{S\}\_z+\backslash hbar\; \backslash hat\{a\}^\backslash dagger\; \backslash hat\{a\}$ and angular momentum magnitude $\backslash hat\{S\}^2=\backslash hat\{S\}\_x^2+\backslash hat\{S\}\_y^2+\backslash hat\{S\}\_z^2$. Since $[\backslash hat\{M\},\backslash hat\{H\}\_\{TC\}]\; =0=\; [\backslash hat\{S\}^2,\backslash hat\{H\}\_\{TC\}]$, it is possible to find simultaneous eigenstates $|s,m,j\backslash rangle$ such that:

$\backslash hat\{H\}\_\{TC\}|s,m,j\backslash rangle=E\_\{s,m,j\}\; |s,m,j\backslash rangle$,

$\backslash hat\{S\}^2|s,m,j\backslash rangle=\backslash hbar^2\; s(s+1)|s,m,j\; \backslash rangle$,

$\backslash hat\{M\}|s,m,j\backslash rangle=\backslash hbar(m-s)|s,m,j\backslash rangle$.

The quantum number $s$ is bounded by $0\backslash leq\; s\backslash leq\backslash frac\{N\}\{2\}$, and $m\backslash geq\; 0$, but due to the infinity of Fock space, excitation number is unbounded above, unlike angular momentum projection quantum numbers. Just as the Jaynes-Cummings Hamiltonian block-diagonalizes into infinite $2\backslash times2$ blocks of constant excitation number, the Tavis–Cummings Hamiltonian block-diagonlizes into infinite blocks of size up to $2^\{N\}\backslash times2^\{N\}$ with constant $m-s$, and within these larger blocks, further block-diagonalizes into (usually degenerate) blocks of size $J+1\backslash times\; J+1$ where $J\backslash equiv\; \backslash min(2s,m)$, with constant cooperation number $s$. The size of each of these smallest blocks (irreps of SU(2)) determine the bounds of the final quantum number that specifies the eigenenergy: $0\backslash leq\; j\backslash leq\; J$, with $j=0$ signifying the ground state of each irrep, and $j=J$ the maximally excited state.

Under the simplifications of real $g\_c$ and quasistatically small $\backslash Delta$, the Hamiltonian becomes $\backslash hat\{H\}\_\{TC\}=\backslash omega\_\{eg\}\backslash hat\{S\}\_z+\backslash hbar\backslash omega\_c\backslash hat\{a\}^\backslash dagger\backslash hat\{a\}+g\_c(\backslash hat\{a\}^\backslash dagger\; \backslash hat\{S\}\_\{-\}+\backslash hat\{a\}\backslash hat\{S\}\_\{+\})$, whose matrix elements one can express in a joint Dicke and Fock basis $|s,s\_z\backslash rangle\backslash otimes\; |n\backslash rangle$ such that $\backslash hat\{S\}\_\{z\}|s,s\_z\; \backslash rangle=\backslash hbar\; s\_z|s,s\_z\; \backslash rangle$ and $\backslash hat\{a\}^\backslash dagger\; \backslash hat\{a\}|n\; \backslash rangle\; =n|n\; \backslash rangle$. Necessarily, $m=s+s\_z+n$, so the matrix elements are as follows:

{{bi|left=1.6|$\backslash langle\; s,s\_z,n|\backslash hat\{H\}|s,s\_z,n\backslash rangle=\; \backslash hbar\; (\backslash omega\_\{eg\}\; s\_\{z\}\; +\; \backslash omega\_\{c\}\; n)=\backslash hbar(\backslash omega\_c(m-s)-\backslash Delta\; s\_z)$}}

{{bi|left=1.6|$\backslash begin\{align\}$

\langle s,s_z,n|\hat{H}_{TC}|s,s_z+1,n-1 \rangle&=\langle s,s_z+1,n-1|\hat{H}_{TC}|s,s_z,n\rangle\\

&=\hbar g_c \sqrt{n(s+s_z+1)(s-s_z)}\\

\end{align}}}

{{bi|left=1.6|$\backslash langle\; s,s\_z,n|\backslash hat\{H\}\_\{TC\}|s\text{'},s\_z\text{'},n\text{'}\backslash rangle=0$ if $s\backslash neq\; s\text{'}$, $s\_z+n\backslash neq\; s\_z\text{'}+n\text{'}$, or $|s\_z-s\_z\text{'}|>1$.}}

From these elements, one can express Schrödinger equations of motion to demonstrate the photon field's ability to mediate entanglement formation between atoms without atom-atom interactions:{{Cite journal |last1=Tessier |first1=T.E. |last2=Deutsch |first2=I.H. |last3=Delgado |first3=A. |last4=Fuentes-Guridi |first4=I. |date=18 December 2003 |title=Entanglement sharing in the two-atom Tavis–Cummings model |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.68.062316 |journal=Physical Review A |volume=68 |issue=6 |page=062316 |doi=10.1103/PhysRevA.68.062316 |arxiv=quant-ph/0306015 |bibcode=2003PhRvA..68f2316T |via=APS}}

{{bi|left=1.6|$\backslash begin\{align\}$

{d\over dt}|s,s_z,n\rangle=&{}-i(\omega_c(m-s)-\Delta s_z)|s,s_z,n\rangle\\

&{}-ig_c\sqrt{n(s+s_z +1)(s-s_z)}|s,s_z+1,n-1\rangle\\

&{}-ig_c\sqrt{(n+1)(s+s_z)(s-s_{z}+1)}|s,s_z-1,n+1\rangle,\end{align}}}

for which the fine-tuned, multivariate dependence on quantum numbers demonstrates the difficulty of solving the Tavis-Cumming model's eigensystem. Here, a few approximate methods, and an exact solution involving Stark shifts and Kerr nonlinearities follow.

In 1969, Tavis and Cummings found approximate eigenenergies and eigenstates for a nondimensionalized Hamiltonian in three different regimes of approximation:{{Cite journal |last1=Tavis |first1=Michael |last2=Cummings |first2=Frederick W. |date=10 December 1969 |title=Approximate Solutions for an N-Molecule-Radiation-Field Hamiltonian |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.188.692 |journal=Physical Review |volume=188 |issue=2 |pages=692–695 |doi=10.1103/PhysRev.188.692 |bibcode=1969PhRv..188..692T |via=APS}} first, for $j\backslash ll\; J$ near the ground-state of each irrep; second, for $m\; \backslash gtrsim\; 2s$ and $m-s\backslash gg\; 1$ when each atom sees a highly saturated "averaged" field; third, for $m\backslash ll\; s$ with sparse excitations. In all solutions, eigenstates are related to Dicke-Fock joint states by $|s,m,j\backslash rangle=\backslash sum\_\{n=\backslash max(0,m-2s)\}^\{m\}\; A\_\{s,m,j\}^\{(n)\}|s,s\_z=m-s-n\backslash rangle\backslash otimes\; |n\backslash rangle$, for coefficients $A\_\{s,m,j\}^\{(n)\}$ that are solved from the Hamiltonian spectrum.

For $j$ close to zero, a differential approach provides the eigenvalues: $E\_\{s,m,j\}\; \backslash approx\; -\backslash sqrt\{\backslash alpha\}\; (j+\backslash frac\{1\}\{2\})+\backslash sqrt\{(j+\backslash frac\{1\}\{2\})^2\backslash alpha+n\_\{0\}\; C\_\{m-s-n\_\{0\}\}^2\}$ with $\backslash alpha=\backslash sqrt\{3s(s+1)+(m-s)(m-s+1)+1\}$, average photon number $n\_0=\backslash frac\{1\}\{3\}(2m-2s+\backslash alpha+1)$, and differential coefficient $C\_\{x\}=\backslash sqrt\{s(s+1)-x(x+1)\}$.

For an averaged photon field when $m\backslash gtrsim\; 2s$ in a large photon-rich system, the off-diagonal matrix elements in $\backslash hat\{H\}\_\{TC\}$ (above) replace $n$ with $n\_0$, and each two-level atom interacts independently with a photon field that conveys no information about the other atoms. For each of these atoms, there are two dressed eigenstates coupled to "pseudophoton" number states: $|\backslash pm\backslash rangle=\backslash frac\{1\}\{\backslash sqrt\{2\}\}(|\backslash uparrow\backslash rangle|\backslash otimes|\backslash frac\{m-s\}\{N\}-\backslash frac\{1\}\{2\}\backslash rangle\backslash pm\; |\backslash downarrow\backslash rangle|\backslash otimes|\backslash frac\{m-s\}\{N\}+\backslash frac\{1\}\{2\}\backslash rangle)$, with single-atom eigenenergies $\backslash lambda\_\{\backslash pm\}\backslash approx\backslash frac\{m-s\}\{N\}\backslash pm\; g\_c\backslash sqrt\{n\_0\}$. Superpositions of these single-atom dressed states construct the full eigenstates $|s,m,j\backslash rangle$ according to addition of angular momenta, mediated by Clebsch-Gordan coefficients. The full eigenvalues are approximately $E\_\{s,m,j\}\backslash approx\; m-s-2(s-j)g\_c\backslash sqrt\{n\_0\}$.

For

In 1996, Nikolay Bogoliubov (son of the 1992 Dirac Medalist of the same name), Robin Bullough, and Jussi Timonen found that adding quadratic excitation-dependent terms to the Tavis–Cummings Hamiltonian allowed for an exact analytic eigensystem.{{Cite journal |last1=Bogoliubov |first1=N.M. |last2=Bullough |first2=R.K. |last3=Timonen |first3=J. |date=2 May 1996 |title=Exact solution of generalized Tavis - Cummings models in quantum optics |url=https://iopscience.iop.org/article/10.1088/0305-4470/29/19/015 |journal=Journal of Physics A: Mathematical and General |volume=29 |issue=19 |pages=6305–6312 |doi=10.1088/0305-4470/29/19/015 |bibcode=1996JPhA...29.6305B |via=IOP Science}} In the limit where these Kerr and Stark shifts vanish, this solution can recover the eigensystem of the unmodified Tavis–Cummings system.{{Cite book |last=Eckle |first=Hans-Peter |title=Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz |publisher=Oxford University Press |year=2019 |isbn=9780199678839 |edition=1 |pages=474–488 |chapter=Quantum Tavis–Cummings Model}}

Including a Kerr term of the form $\backslash hat\{a\}^\{\backslash dagger\}\; \backslash hat\{a\}^\{\backslash dagger\}\; \backslash hat\{a\}\; \backslash hat\{a\}$, or a Stark term $\backslash hat\{S\}\_z^2$ equivalently results in a new Hamiltonian: $\backslash hat\{H\}\_\{TC2\}=\backslash hat\{H\}\_\{TC\}+\backslash hbar\backslash gamma\; \backslash hat\{a\}^\{\backslash dagger\}\; \backslash hat\{a\}^\{\backslash dagger\}\; \backslash hat\{a\}\; \backslash hat\{a\}+\; \backslash frac\{\backslash gamma\}\{\backslash hbar\}\; \backslash hat\{S\}\_\{z\}^\{2\}$, which obeys the same operator symmetries (above) as does the unmodified Tavis–Cummings Hamiltonian, and which reduces to Tavis–Cummings in the limit $\backslash gamma\backslash to\; 0$. Thus the transformation $\backslash hat\{H\}\_0=\backslash frac\{1\}\{g\_c\}(\backslash hat\{H\}\_\{TC2\}+(\backslash gamma-\backslash omega\_c)\backslash hat\{M\}+\backslash frac\{\backslash gamma\}\{\backslash hbar\}\backslash hat\{M\}^2)$ preserves the dynamics and shares joint eigenvectors with the untransformed Hamiltonian. The transformed Hamiltonian, explicitly, is $\backslash hat\{H\}\_0=d\backslash hat\{S\}\_z+\backslash hat\{a\}^\backslash dagger\; \backslash hat\{S\}\_\{-\}+\backslash hat\{a\}\backslash hat\{S\}\_\{+\}+c\; \backslash hat\{a\}^\backslash dagger\backslash hat\{a\}\backslash hat\{S\}\_z$ for new parameters $c=-2\; \backslash frac\{\backslash gamma\}\{g\_c\}$ and $d=\backslash frac\{\backslash gamma-\backslash Delta\}\{g\_c\}$, which is integrable using quantum inverse methods. Separating the dynamics into two $2\backslash times2$ complex-parametrized operator matrices (that is, matrices whose elements are operators), one acting on the bosonic degrees of freedom and the other on the spin degrees of freedom produces a monodromy matrix whose determinant is directly proportional to $\backslash hat\{S\}^2$, whose trace is proportional to $\backslash hat\{H\}\_0$, and whose trace's parametric derivative is proportional to $\backslash hat\{M\}$. Manipulating the monodromy matrix allows its spectral parameter to determine the Hamiltonian eigenstates and eigenenergies as the complex roots $\backslash lambda\_\{p,j\}$ of a Bethe ansatz. Every $\backslash lambda\_\{p,j\}$ for $p\backslash in\; \backslash \{1,2,...\; m\backslash \}$ must satisfy the following Bethe equations:

$(1+cd-c\backslash lambda\_\{p,j\})\backslash frac\{\backslash lambda\_\{p,j\}+cs\}\{\backslash lambda\_\{p,j\}-cs\}=\backslash prod\_\{q\backslash neq\; p\}^\{m\}\backslash frac\{\backslash lambda\_\{p,j\}-\backslash lambda\_\{q,j\}+c\}\{\backslash lambda\_\{p,j\}-\backslash lambda\_\{q,j\}\; -c\}$,

then the Hamiltonian eigenvalues arise from the roots as:

$E\_\{s,m,j\}^\{(TC2)\}=(\backslash omega\_c-\backslash gamma)(m-s)+\backslash gamma(m-s)^2+\backslash frac\{sg\_c\}\{c\}\backslash prod\_\{p=1\}^\{m\}\; (1-\backslash frac\{c\}\{\backslash lambda\_\{p,j\}\})-g\_c(sd+\backslash frac\{s\}\{c\})\backslash prod\_\{p=1\}^\{m\}\; (1+\backslash frac\{c\}\{\backslash lambda\_\{p,j\}\})$.

In the limit where $\backslash gamma\backslash to\; 0$ (and thus $c\backslash to\; 0$), the above Bethe equations simplify to $\backslash frac\{2s\}\{\backslash lambda\_\{p,j\}\}-\backslash lambda\_\{p,j\}-\backslash frac\{\backslash Delta\}\{g\_c\}=\backslash sum\_\{q\backslash neq\; p\}^\{m\}\backslash frac\{2\}\{\backslash lambda\_\{p,j\}-\backslash lambda\_\{q,j\}\}$, and the eigenenergies to $E\_\{s,m,j\}=\backslash omega\_\{eg\}(m-s)-g\_c\backslash sum\_\{p=1\}^\{m\}\; \backslash lambda\_\{p,j\}$. Eigenstates follow similarly.

The Tavis–Cummings model has seen numerous experimental implementations verifying its phenomena, including several since 2009 virtually realizing the model on quantum computational platforms like superconducting qubits and circuit QED. Such experiments utilize the Tavis–Cummings Hamiltonian's ability to generate superradiance wherein the artificial atoms emit and absorb light from the field coherently, as though they were a single atom with a large total angular momentum. Superradiance, scaling dipole-interaction strength $g\_c\backslash propto\backslash sqrt\{N\}$, and other features allow Tavis–Cummings-type dynamics to manifest quantum computationally and metrologically desirable states, such as Dicke states (joint eigenstates of $\backslash hat\{S\}^2$ and $\backslash hat\{S\}\_z$) through global interactions, as was explored in the 2003 paper by Tessier et al.

One realization by Tuchman et al., in 2006, used a stream of ultracold Rubidium-87 atoms ($N\backslash approx\; 2\backslash times\; 10^5$), and observed cooperation number $s\backslash approx\; 1.2\; \backslash times10^4$, or 12% its maximum possible value,{{Cite journal |last1=Tuchman |first1=A.K. |last2=Long. |first2=R. |last3=Vrijsen |first3=G. |last4=Boudet |first4=J. |last5=Lee |first5=J. |last6=Kasevich |first6=M.A. |date=27 November 2006 |title=Normal-mode splitting with large collective cooperativity |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.74.053821 |journal=Physical Review A |volume=74 |issue=5 |page=053821 |doi=10.1103/PhysRevA.74.053821 |bibcode=2006PhRvA..74e3821T |via=APS}} indicating very high interatomic coherence relative to experimental capabilities of the time. This experiment also confirmed the scaling of the dipole-interaction; at the level of single atoms, dipole interactions are much weaker than monopolar interactions, so the ability of Tavis–Cummings dynamics to counteract the weakness of $\backslash lVert\; \backslash vec\{d\}\_\{eg\}\backslash rVert$ with quadrature addition of dipoles makes neutral atom control more feasible.

A seminal result from Fink et al. in 2009 involved 3 transmons as virtual "atoms"{{Cite journal |last1=Fink |first1=J.M. |last2=Bianchetti |first2=R. |last3=Baur |first3=M. |last4=Göppl |first4=M. |last5=Steffen |first5=L. |last6=Filipp |first6=S. |last7=Leek |first7=P.J |last8=Blais |first8=A. |last9=Wallraff |first9=A. |date=17 August 2009 |title=Dressed Collective Qubit States and the Tavis–Cummings Model in Circuit QED |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.083601 |journal=Physical Review Letters |volume=103 |issue=8 |page=083601 |doi=10.1103/PhysRevLett.103.083601 |pmid=19792728 |arxiv=0812.2651 |bibcode=2009PhRvL.103h3601F |via=APS}} with qubit-dependent Bohr frequencies $\backslash hbar\; \backslash omega\_\{eg\}^\{(j)\}=\backslash sqrt\{8\; E\_C^\{(j)\}E\_J^\{(j)\}\}\; -\; E\_C^\{(j)\}$ for controllable Josephson energy $E\_J^\{(j)\}$ and experimentally determined single electron charging energy $E\_C^\{(j)\}$, inside a microwave waveguide resonator which supplies a standing electric field at $\backslash frac\{\backslash omega\_c\}\{2\backslash pi\}\; =\; 6.729$GHz. To ensure symmetric coupling of the qubits to the field, each transmon was placed at an antinode of the standing wave, and to best conserve excitations by minimizing photon leakage, the resonator was kept ultracold (20mK) which ensured a high quality factor. Manipulating each qubit's Bohr frequency so only one qubit resonated with the field, the team measured each single-qubit coupling strength $g\_\{c\}^\{(j)\}\; (1)$, then reintroduced the other qubits to compare the total coupling strength $g\_c(N)$ with the average strength of the resonating qubits, $\backslash overline\{g\_c\; (1)\}$, confirming that $g\_c(N)=\backslash sqrt\{N\}\backslash overline\{g\_c\; (1)\}$. In addition, the team observed bright and dark states characterized by high emission rates and zero emissions respectively, for 2 and 3 active qubits, with the 3-qubit bright and dark states each being degenerate.

In addition to superconducting qubits, semiconducting qubits have also been a platform for Tavis–Cummings dynamics, such as in a 2018 investigation by van Woerkom et al. at ETH Zürich, in which two qubits constructed of double quantum dots (DQDs) coupled to a SQUID resonator, with the two DQDs separated by a distance of 42μm. The micrometer regime is a far greater distance than that over which semiconducting qubits had previously achieved entanglement, and the difficulty of long-range interactions in semiconducting qubits was at the time a major weakness compared to other quantum computing platforms, for which the Tavis–Cummings model's ability to form entanglement through global atom-field interactions is one solution. By observing the reflection amplitude of field waves between the SQUID array and the DQDs, the team isolated the photon number states as they smoothly coupled to the first qubit to form superpositional Jaynes-Cummings eigenstates when the first qubit tuned to the resonator. Similarly, they observed these hybrid states shift into a pair of bright states and a dark state (which did not interact with the light, and thus did not cause a dip in the reflection amplitude) when the second qubit was tuned to resonance. In addition to physical photons mediating the long-range entanglement at $\backslash Delta=0$, the team found similar energy shifts at $\backslash Delta\backslash gg\; g\_c$ signalling qubits interacting with "virtual" photons, measured by the phase shift of the field rather than the reflection amplitude.

Recent investigations by Johnson, Blaha, et al., have verified and explained two major regimes where the Tavis–Cummings model fails to predict physical reality,{{Cite journal |last1=Blaha |first1=Martin |last2=Johnson |first2=Aisling |last3=Rauschenbeutel |first3=Arno |last4=Volz |first4=Jürgen |date=28 January 2022 |title=Beyond the Tavis–Cummings model: Revisiting cavity QED with ensembles of quantum emitters |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.105.013719 |journal=Physical Review A |volume=105 |issue=1 |page=013719 |doi=10.1103/PhysRevA.105.013719 |arxiv=2107.04583 |bibcode=2022PhRvA.105a3719B |via=APS}} both following from systemic parameters approaching or exceeding the free spectral range $\backslash nu\_\{FSR\}=\; \backslash frac\{c\}\{L\}$ based on the waveguide length $L$. The violating quantities are the coupling strength $g\_c$, and the rate of photon-loss from atomic emissions into non-cavity modes, $\backslash frac\{g\_c^2\}\{\backslash gamma\_l\}$, where $\backslash gamma\_l$ is the single-atom spontaneous-emission rate into all modes. When $\backslash nu\_\{FSR\}\backslash gg\; g\_c$ and $\backslash nu\_\{FSR\}\backslash gg\; \backslash frac\{g\_c^2\}\{\backslash gamma\_l\}$, the Tavis–Cummings model well-describes the system, since the atom-light interactions are suppressed for all but one mode, and the field intensity is not significantly attenuated due to atomic emissions into other modes. However, when $\backslash nu\_\{FSR\}\backslash lesssim\; g\_c$ then the coupling enters the so-called "superstrong" regime and atom-light interactions must consider multiple field-modes. More severely, when $\backslash nu\_\{FSR\}\; \backslash lesssim\; \backslash frac\{g\_c^2\}\{\backslash gamma\_l\}$ then the atomic ensemble becomes optically thick, and the model must consider time-ordered interactions between the field and each atom, as atoms at the front of the ensemble will experience a more intense photonic wavefront than those at the back, due to the frontal atoms' absorptions and non-cavity mode emissions. This has the effect of interactions and correlations cascading successively across multiple atoms. As photons cross the waveguide and interact sequentially with the atoms in the ensemble, they accumulate phase at phenomenon-dependent rates. The total phase accumulated by electromagnetic waves in one round-trip of the waveguide may manifest resonances causing high transmission rates under specific dephasings $\backslash Delta$ and emission rates $\backslash frac\{g\_\{c\}^2\}\{\backslash gamma\_\{l\}\}$, and the locations of these resonances differs between the standard Tavis–Cummings model and the team's proposed "cascade" model. Using a fluid of ultracool Cesium atoms surrounding a nanofiber-section of a 30m fiber-ring resonator, the team coupled the atoms to the light passing through the nanofiber via an evanescent field, measuring the light's transmission for variable $\backslash Delta$ and $g\_c$, into the superstrong coupling and cascade regimes. The data from the nanofiber-Cesium experiment agreed better with the cascade model's predictions than with the Tavis–Cummings', specifically in the parametrically violating regimes above.

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Category:Quantum models

Category:Quantum optics