Detecting Fermi Surface Nesting Effect for Fermionic Dicke Transition by Trap Induced Localization

Introduction In recent decades, the developments in achieving strong coupling between atoms and light have led us to a new platform for studying non-equilibrium open quantum systemsEsslinger07 ; Colombe07 . With these accumulation, the Dicke model, a typical model of strong interactions between atoms and light is finally realizedEsslinger10 . It ends the era for Dicke transition being purely theoreticalDicke54 ; Lieb73 . The Dicke transition manifests itself through the emergence of the steady state superradiance together with a checkerboard density order in atomsEsslinger11 . The spontaneity of the self-organized crystalline is verified by the roton mode softening Roton12 and the critical behavior of the dynamical structure factorEsslinger13 ; Hemmerich14 ; Esslinger15 . Exotic phases like density-ordered Mott insulatorsHemmerich15ex ; Esslinger16 ; Esslinger18 ; Esslinger20 , as well as supersolid breaking U(1) symmetry and translation symmetry ContinousSS17 ; EsslingerSS17 ; EsslingerSS18 are observed successively experimentally. Excited topics with the combinations of quantum many-body systems and the traditional presentation of an open quantum system — cavity-QED (quantum electrodynamics) are enlightened following these new advancesReview21 .

Besides the developments in the superradiance of Bose gases, there are also many interesting statistical effects in fermionic superradiance. The most prominent signature for Dicke transitions of degenerate Fermi gases is its density dependence, namely, the Fermi surface nesting effect and the Pauli blocking effectSimons14 ; Piazza14 ; Yu14 . The Fermi surface nesting effect is due to the resonance between pairs of nested states on the Fermi surface whose momentum difference matches the cavity photon momentum. It results a sharp decreasing of critical pumping strength at specific fillings. The Pauli blocking effect, on the other hand, leads to a suppression of superradiance due to both of the momentum states connected by photon absorption being occupied. There are also further effects like liquid-gas like a transition for p-band fillingSimons14 and statistical crossover in interacting Fermi gasesYu15 , etc. For a long-time, these studies for statistical effects in fermionic superradiance are only theoretical. In a recent experiment, a steady state superradiance of fermions is realized in a cavity for the first time. A scaling law of critical pumping strength as $N^{-1/2}_{\rm at}$ against the particle number $N_{\rm at}$ at large $N_{\rm at}$ limit is verified in a three-dimensional trapped fermions systemWu21 . This is in sharp contrast with the bosonic Dicke transition whose scaling law is $N_{\rm at}^{-1}$. Pauli blocking effect in free space superradiance is also verified thereafterKetterle21 ; Deb21 ; YeJ21 . Unfortunately, contrary to the well-established Pauli blocking effect, Fermi surface nesting effect, which shows the enhancement of superradiance by Fermi statistics is still not verified in experiment. The main difficulty is the presence of the trap makes density ill-defined.

In this Letter, we offer a method to identify the Fermi surface nesting effect for two-dimensional Fermi gases within a trap. We find that, when trap and optical lattice are both presented, there are localized states on the trap edge. The mechanism is shown in Fig. 1(b). One can observe that on the trap edge, the onsite energy difference will outgo the hopping strength, which leads to localization. These localizations then contribute a lot of one-dimensional fermion tubes on the trap edge, where Fermi surface nesting effect is prominent. Phenomenologically, we find for different cavity detune $\Delta_{c}$, the critical pumping strength of Dicke transition as a function of particle number $N_{\rm at}$ falls into two universal functions up to a constant shift in log-log plot. One of the universal curves for small $\Delta_{c}$ shows a universal scaling $N_{\rm at}^{-1}$ for small $N_{\rm at}$ and $N_{\rm at}^{-1/3}$ at large $N_{\rm at}$ limit. There is no scaling faster than $N_{\rm at}^{-1}$ in this typical curve. Let us denote the scaling as $\varkappa$ for a critical pumping strength scaling law $N_{\rm at}^{-\varkappa}$ to simplify our description. For larger $\Delta_{c}$, however, we find a new scaling $\varkappa>1$ ($\varkappa=1.33$) in intermediate particle number region. We also find a crossover between these two typical critical pumping strength functions, which is sharp within a very small window of $\Delta_{c}$. In the same crossover $\Delta_{c}$ region, we find the eigenstate on the Fermi surface turns from an extended state to a localized state, which identifies the mechanism. Further, $\varkappa\rightarrow\infty$ in $\omega_{z}/E_{R}\rightarrow 0$ limit at a specific $N_{\rm at}$ is obtained as a result of Fermi surface nesting whose tendency is verified numerically. Therefore, the measurement of critical pumping strength as a function of particle number $N_{\rm at}$ for different cavity detune $\Delta_{c}$ and different trap frequencies can help us to identify Fermi surface nesting effect. We also stress that the trap induced localization is vital in detecting Fermi surface nesting effect, quite opposite from the original impression of trap being nothing but an obstacle.

Model We consider spinless fermions trapped inside a high-Q single mode cavity within a harmonic trap($\hbar=1$ throughout),

Here $\hat{a}$ is the annihilation operator of the cavity field, $\hat{\psi}$ is an annihilation operator of fermionic atoms. Here $V_{P}(\textbf{r})=V_{P}\cos^{2}(k_{0}x)$ is the optical lattice generated by the pumping field, $U_{0}(\textbf{r})=U_{0}\cos^{2}(k_{0}z)$ is the cavity field self-energy, $\eta(\textbf{r})=\eta_{0}\cos(k_{0}x)\cos(k_{0}z)$ is the interference between the pumping field and the cavity field. To be more specific, $V_{P}=\Omega^{2}_{p}/\Delta_{a}$, $U_{0}=g_{0}^{2}/\Delta_{a}$, and $\eta_{0}=g\Omega_{p}/\Delta_{a}$, where $\Delta_{a}=\omega_{a}-\omega_{p}$ is AC stark shift ($\omega_{a}$ is the excited state energy), $\Delta_{c}=\omega_{p}-\omega_{c}$ is the cavity detuning. Here the cavity in consideration has a photon decay rate of $\kappa$. $\Omega_{p}$ is the strength of the pumping lasers, $g_{0}$ is the single-photon Rabi frequency of the cavity field, and $k_{0}$ is the wave number of the pumping field which is close to the wave number of the cavity field. The wave number of the cavity field is approximated by $k_{0}$. The recoil energy is defined by $E_{R}\equiv k_{0}^{2}/2m$.

Mean Field Theory Since the cavity photon is lossy, the equation of cavity photon field should follow Lindblad equation,

where ${\cal L}_{\hat{a}}\hat{a}=\hat{a}^{\dagger}\hat{a}\hat{a}-\frac{1}{2}\{\hat{a}^{\dagger}\hat{a},\hat{a}\}$ is the Lindblad operator on cavity field operator $\hat{a}$. Assuming that the cavity field is in a coherent state $|\alpha\rangle$, such that $\langle\alpha|\hat{a}|\alpha\rangle=\alpha$. Then the above equation can be written as

where the effective detuning $\Delta_{c}^{\prime}(\alpha)=\Delta_{c}-\int d\textbf{r}U(\textbf{r})n(\textbf{r})$ and $\Theta(\alpha)=\int d\textbf{r}n(\textbf{r})\eta(\textbf{r})/\eta_{0}$. The fermionic density function is $n(r)\equiv\braket{\hat{\psi}^{{\dagger}}(\textbf{r})\hat{\psi}(\textbf{{r}})}={\rm Tr}(\hat{\psi}^{{\dagger}}(\textbf{r})\hat{\psi}(\textbf{{r}})\hat{\rho}(t))$. Here ${\rm Tr}$ is over the atom’s Hilbert space and a coherent state of cavity field is assumed. Here we assume the steady state density matrix of atoms is $\hat{\rho}_{\rm st}=e^{-\beta\hat{H}_{at}(\alpha)}/Z$ ($Z={\rm Tr}(e^{-\beta\hat{H}_{at}(\alpha)})$), which is justified by a full dynamical handling by Keldysh contour method. Here, $\hat{H}_{at}(\alpha)$ is defined by replace $\hat{a}$ in $\hat{H}_{at}$ with $\alpha$. In the steady state, $\partial\alpha/\partial t=0$, we have

As $\alpha$ is complex, the above steady state equation is indeed two equations. If we assume the Dicke transition is a second order transition, then ${\cal B}=\int d{\bf r}U({\bf r})n({\bf r})\approx U_{0}N/2$. The phase of $\alpha$ is locked by $\kappa$. The other equation can be understood as a minimization of free energy ${\cal F}_{\alpha}$,

One can check $\eta_{0}\Theta+\alpha\int d{\bf r}U({\bf r})n({\bf r})=\partial{\cal F}_{\alpha}/\partial\alpha^{*}$. Then the steady state equation become,

Further we can find in the second-order expansion of ${\cal F}_{\alpha}$, ${\cal F}_{\alpha}=-\eta^{2}_{0}\chi(\alpha+\alpha^{*})^{2}+{\cal B}\alpha^{*}\alpha$. $\chi=\int d{\bf r}d{\bf r^{\prime}}\langle\delta n({\bf r})\eta({\bf r})\delta n({\bf r^{\prime}})\eta({\bf r}^{\prime})\rangle/\eta^{2}_{0}$ is the static structure factor, which characterizes the density fluctuation at momentum $\pm k_{0}{\bf e}_{x}+\pm k_{0}{\bf e}_{z}$.

Here $\phi_{n}({\bf r})$ is the eigenstate of $\hat{H}_{\rm 0}$, $E_{n}$ is the corresponding eigen energy of state $|n\rangle$, $n_{F}$ the Fermi distribution function. If we assume the Dicke transition is second order transition, the critical pumping strength is obtained on condition that $\Delta^{\prime}_{c}+4\eta^{2}_{0}\chi\Delta^{\prime 2}_{c}/(\Delta^{\prime 2}_{c}+\kappa^{2})=0$ for the second order coefficient of $\alpha$ in ${\cal F}_{\alpha}$ being zero. The critical pumping strength is

In this Letter, we will focus on the scaling of $V_{\rm P}^{\rm cr}$ over $N_{\rm at}$ to identify the statistical effect in fermionic superradiance transition in zero temperature.

Localization on Trap Edge In previous calculation for fermions in a trap, we have assumed that the fermions states are more close to plane wave. However, on the edge of the trap, the interplay between the pumping lattice and the trap can lead to localization. Here we show the mechanism in a two-dimensional non-interacting fermions in a two-dimensional trap.

The eigenstate $|n\rangle$ can be factorized as $\phi_{n_{x}}(x)\phi_{n_{z}}(z)$. $\phi_{n_{z}}(z)$ is the eigenstate of Harmonic trap, and its eigen-energy is $n_{z}\omega_{0}$. $\hbar=1$. Further, we solve the x direction hamiltonian with the trap, with $u_{1,k_{x}}(x)$ being the lowest band Bloch wave function. Then we can construct a Wannier basis by

where $j$ is the site index, $d=\frac{\pi}{k_{0}}$. In the Wannier basis, the hamiltonian in x direction can be rewritten as

where, $t=\int dxw_{i}(x)(-\frac{\partial^{2}_{x}}{2m}+V_{p}\cos^{2}(k_{0}x))w^{*}_{i+1}(x)$, $\mu_{i}=\int dy|w_{i}(x)|^{2}\frac{1}{2}\omega^{2}x^{2}$. Next-to-nearest neighbor hopping can be also included, which is not explicitly shown in above equation.

One can check that when $|\mu_{j+1}-\mu_{j}|>t$, the eigen-state is localized because the resonance condition between sites is broken down. Here we employ the wave packet width to quantitatively characterize the localization degree of these eigenstates for different eigen-energy. The width of a state is defined as:

where $\langle x^{2}\rangle_{n}=\int dxx^{2}|\phi_{n}(x)|^{2}$, $\langle x\rangle_{n}=\int dxx|\phi_{n}(x)|^{2}$. In Fig.2, we show $\Delta(E_{n})$ as a function of excitation energy $E_{n}$ for different pumping lattice strengths for fixed trap frequency. A clear sign for mobility edge is shown, and the high energy states can not be approximated as extended states. In the following, we will explore the physical consequences by the trap edge localization.

Scaling of the Critical Pumping Strength — Following Eq. 9 and Eq. 10, together with the calculation of eigen-state $|n\rangle=|n_{x},n_{z}\rangle$, we can obtain the numerical result for $V_{P}^{\rm cr}$ for different $\Delta_{c}$ and particle number $N_{\rm at}$. Here we have fixed $\omega_{x}=\omega_{z}=E_{R}/50$. One can find when $\Delta^{\prime}_{c}$ is larger, the critical pumping strength is larger, thus the localization effect at the trap edge is larger. The numerical result for critical pumping strength of particle number $N_{\rm at}$ for different $\Delta^{\prime}_{c}$ is given in Fig. 3(a). One can find for smaller $\Delta^{\prime}_{c}$, the log-log plot of $V_{P}^{\rm cr}(N_{\rm at})$ up to a multiplier constant factor falls into one universal curve. The initial scaling of $V_{P}^{\rm cr}$ against $N_{\rm at}$ is $-1$, then it crosses over to $-0.66$ and finally it crosses over to $-0.35$ in large $N_{\rm at}$ limit. The scaling reduction is due to Pauli blocking effect. One can also find there are no scalings larger than $1$, which means we only observe the suppression of superradiance due to Fermi statistics. The situation is suddenly changed when $\Delta^{\prime}_{c}>0.5$MHz. We find due to localization on the trap edge, the critical pumping strength function $V_{P}^{\rm cr}(N_{\rm at})$ fall into a new universal curve. In this new universal curve, the initial small $N_{\rm at}$ scaling and the large particle number scaling are the same as the previous universal curve. What is beyond, a new scaling being larger than $1$ emerges around the middle range $N_{\rm at}$. This middle range $N_{\rm at}$ matches the fermion density for Fermi surface nesting effect. We compared the crossover region of the universal critical pumping strength curve and localization effect shown by the typical width of wave function $\phi_{n_{x}}(x)$ at the Fermi level. We find these two regions coincide with each other. Here we define $L_{1.33}$ as the length of $1.33$ scaling in $V_{P}^{\rm cr}(N_{\rm at})$. Therefore we conclude that the Pauli blocking scaling $0.35$ is not affected by localization and a new scaling $1.33$ emerges due to localization effect. Thus we find the evidence of Fermi surface nesting effect can be magnified by localization effect induced by the trap.

To go further, we will now present a theoretical effective theory to understand why the Pauli blocking effect is immune to the localization effect in large $N_{\rm at}$ limit and whether the new scaling $1.33$ can be interpreted as the Fermi surface nesting effect. Before we give our analysis, we will first give our conclusion. The predictions of our effective theory are 1) the large $N_{\rm at}$ limit of $V_{P}^{\rm cr}$ is $V_{P}^{\rm cr}\propto N^{-1/3}$ and it is not changed by localization effect; 2) there is a kink at some middle range $N_{\rm at}$, and the critical pumping strength scaling of $N_{\rm at}$ becomes divergent before the kink in $\omega_{z}\rightarrow 0$ limit. The second phenomenon is a signature of Fermi surface nesting effect.

Now we will present the assumptions of our effective theory. First of all, $\phi_{n_{z}}(z)$ is approximated by its large $n_{z}$ asymptotic expression $\phi_{n_{z}}(z)\sim\cos(\sqrt{(2n_{z}+1/2)\omega_{z}}z)$ for even $n_{z}$ and $\sin(\sqrt{(2n_{z}+1/2)\omega_{z}}z$ for odd $n_{z}$. Although the original approximation is only correct at large $n_{z}$ limit, here we take an approximation for every $n_{z}$. Further we drop the constant $1/2$ and employ $k_{z}^{2}=2n_{z}\omega_{z}$. Thus $\sum_{n_{z}}=\int\frac{|k_{z}|}{m\omega_{z}}dk_{z}$ and $\phi_{n_{z}}(z)=\cos(k_{z}z)$ or $\sin(k_{z}z)$. Second, in $x$ direction, if $\phi_{n_{x}}(x)$ is localized at $x_{0}$, then $\phi_{n_{x}}(x)=\delta(x-x_{0})$. If the wave function is not localized, we apply a local density approximation and $\phi_{n_{x}}(x)$ is characterized by both position $x_{0}$ and wave vector $k_{x}$. The eigen-energy is $\varepsilon_{x_{0},k_{x}}=2t\cos k_{x}+\frac{1}{2}\omega_{x}x_{0}^{2}$. Here we introduce $x_{\rm Mob}$ as the mobility edge in $x$ direction. For $x<x_{\rm Mob}$, the wave function $\phi_{n_{x}}(x)$ is extended and for $x>x_{\rm Mob}$, the wave function is localized. $\sum_{n_{x}}$ is approximated as a phase space integral $\int dk_{x}dx_{0}$. With all these approximations, Eq. 9 and Eq. 10 can be calculated analytically and we find $V_{P}^{\rm cr}\propto N^{-1/3}_{\rm at}$ at large $N_{\rm at}$ limit irrelevant to the position of $x_{\rm Mob}$. The detailed calculation is given in supplementary materialsupp and the theoretical prediction is shown in Fig. 4(a) in solid black curve. A kink at Fermi surface nesting atom number can be observed.

Since the approximation in $z$ direction deviate from reality, we find the approximation is better when $E_{R}/\omega_{z}\gg 1$. Therefore we expect the critical pumping strength scaling of $N_{\rm at}$ in the middle range $N_{\rm at}$ will increase when $\omega_{z}/E_{R}$ is decreasing. Thus this prediction can be numerically checked by numerics. Interestingly, we find except for scaling around the Fermi surface nesting atom number, other scalings of critical pumping strength is invariant for different $\omega_{z}$. The scaling of critical pumping strength $|\log V_{P}^{\rm cr}/\log N_{\rm at}|$ is increasing when $\omega_{z}$ is decreasing. We are restricted by the system size and particle number in a numerical calculation, therefore we can only prove that the scaling is increasing when $\omega_{z}$ becomes smaller. A finite size scaling is carried out, and a divergent scaling is obtained in $\omega_{z}\rightarrow 0$ limit. All of the present data analysis can be equally done for experimental data. These signatures in critical pumping strength can then be identified as Fermi surface nesting effect.

Conclusion To summarize, we find the Fermi surface nesting effect in fermionic superradiant transition in a cavity can be verified with the help of trap induced localization. We find when the harmonic trap depth is effectively changed, there are two typical curves for superradiant transition critical pumping strength as a function of particle number. For shallow trap without localization effect, there are no signs of Fermi surface nesting, and for tight trap with localization effect, there is Fermi surface nesting signal which manifests as $\varkappa>1$ for $\varkappa=\log(V_{P}^{\rm cr})/\log N_{\rm at}$. We also verified the tendency for $\varkappa\rightarrow\infty$ in zero trap frequency limit. We find the interplay between trap or localization and superradiance is quite interesting, and statistical effect can be magnified and thus benefits the generation of superradiance.

Acknowledgments.— Y. C is supported by NSFC under Grant No. 12174358 and No. 11734010, and Beijing Natural Science Foundation (Z180013).

Supplementary Material