Probing two Higgs oscillations in a one-dimensional Fermi superfluid with Raman-type spin-orbit coupling

Collective excitation is important dynamics of many-body quantum system, and becomes an interesting research topic in all realm of physics. As a kind of gapped collective excitation, the Higgs mode is a quantum phenomenon investigated in superconductors Littlewood and Varma (1981, 1982); Sooryakumar and Klein (1980, 1981), magnetic materials Matsumoto et al. (2004); Rüegg et al. (2008), and ultracold atoms in continuous or lattice system Scott et al. (2012); Altman and Auerbach (2002); Pollet and Prokof’ev (2012); Bissbort et al. (2011); Endres et al. (2012). A review paper about Higgs mode in condensed matter physics can be found in Ref. Pekker and Varma (2015). Physically the Higgs mode is described by the amplitude fluctuation of the order parameter, which is different from the gapless Goldstone excitation related to the phase fluctuation of order parameter.

While the appearance of Goldstone mode is easy to observe when continuous symmetries are broken, stable Higgs modes require additional symmetry to stop them from rapidly decaying into other low-energy excitations. In high-energy physics, the stability of Higgs mode is ensured by Lorentz invariance, whose role is replaced by particle-hole symmetry in condensed matter physics. The famous Bardeen-Cooper-Schrieffer (BCS) Hamiltonian describing a weakly interacting superconductor is a typical example of hosting a stable Higgs mode with particle-hole mechanism Littlewood and Varma (1981, 1982), and related evidence has also been found in conventional BCS superconductors Sooryakumar and Klein (1980); Matsunaga et al. (2013); Sherman et al. (2015). The same BCS theory, which is usually called Bogoliubov-de Gennes (BdG) mean field theory, is also widely used to study the ultracold Fermi gases. The Higgs mode has also been theoretically investigated in the BCS-Bose Einstein Condensate (BEC) crossover of Fermi superfluid Yuzbashyan and Dzero (2006); Scott et al. (2012); Hannibal et al. (2015) with a time-dependent Bogoliubov-de Gennes (BdG) simulation. The order parameter has a close connection with the condensate fraction Altman and Vishwanath (2005); Perali et al. (2005); Matyjaśkiewicz et al. (2008). Following this relation, experimentally the Higgs mode has been observed in a strongly interacting fermionic superfluid with radiofrequency field technique Behrle et al. (2018).

To excite the Higgs mode in ultracold Fermi gases, generally one can resort to the modulation of all parameters which can decide the order parameter, e.g., the interaction parameter $1/\left(k_{F}a\right)$ in the BCS-BEC crossover Scott et al. (2012); Liu et al. (2016); Han et al. (2016); Kurkjian et al. (2019). In 1974, Volkov and Kogan studied the response of superconductors in the presence of a small perturbation with the Green’s functions, and found that the order parameter $\left|\Delta\right|$ oscillates with the period $\pi\hbar/\Delta_{\rm{gap}}$, where the $\Delta_{\rm{gap}}$ is the energy gap in the spectrum of fermionic excitations Volkov and Kogan (1974). This also indicates that the Higgs mode is greatly influenced by the single-particle excitation. Usually the Higgs mode is mixed and coupled with the continuum spectrum of the single-particle excitation in many Fermi superfluids and thus we will call it a Higgs oscillation instead in the remaining text. Since the development of artificial gauge field in Fermi superfluid Wang et al. (2012); Cheuk et al. (2012), more control knobs, like effective Zeeman field and spin-orbit coupling strength, can be brought in to perturb the amplitude of order parameter. Higgs oscillation is expected to display richer and much interesting dynamical behavior in spin-orbit coupled (SOC) degenerate Fermi gases. Previously topological phase transition of quench dynamics and dynamical phases had been studied in SOC Fermi superfluid Wang et al. (2015); Dong et al. (2015). But to date there are quite few discussions to introduce the Higgs oscillation and its physical properties in SOC Fermi superfluid. In this paper, we will introduce two kinds of Higgs oscillations with different periods in SOC Fermi superfluid.

In this work, motivated by previous theoretical studies and recent experiments, we explore the fascinating Higgs oscillation in a one-dimensional (1D) SOC Fermi superfluid and aim to characterize two distinct Higgs oscillations by studying the related dynamic behaviour. With the help of time-dependent BdG equation, we first investigate the properties of the order parameter as well as the excitation spectrum on the tunable effective Zeeman field in different phase regimes. By introducing three time-dependent ways to tune the effective Zeeman field, we then obtain the oscillating behaviours of the amplitude of the order parameter in both the BCS and topological phases. Finally, by means of a Fourier analysis, we numerically calculate the oscillating frequency or period to straightforwardly characterize the Higgs oscillation, and compare it with the previous theoretical prediction of Volkov and Kogan in both two phases.

The paper is organized as follows. In the next section, we will briefly introduce the model and Hamiltonian of a SOC Fermi superfluid with the mean-field theory. In Sec. III, we probe and investigate two Higgs oscillations in both the BCS and topological states, by tuning the effective Zeeman field in three different ways and investigating the oscillating behaviors of the amplitude of the order parameter. Finally, we summarize and draw conclusions in Sec. IV.

We consider a 1D superfluid Fermi gas with Raman-type spin-orbit coupling effect. The system can be described by a single-channel model Hamiltonian $H=\int dx\left[\mathcal{H}_{0}+\mathcal{H}_{int}\right]$, where

is the SOC single-particle part in a uniform system and

is the interaction Hamiltonian with $g_{{}_{\mathrm{1D}}}=-\gamma\hbar^{2}n_{0}/m$ describing an attractive $s$-wave contact interaction between two spin states $\left(\sigma=\uparrow,\downarrow\right)$. $n_{0}$ is the bulk density, and $\gamma$ denotes a dimensionless interaction parameter. Here $\psi_{\sigma}$ or $\psi_{\sigma}^{\dagger}$ is the field operator that annihilates or creates a spin $\sigma$ atom with mass $m$. $\mathcal{H}_{s}=-\hbar^{2}\partial_{x}^{2}/\left(2m\right)-\mu$ describes the motion of free atoms with chemical potential $\mu$. The term $\lambda\hat{k}_{x}\sigma_{y}-h\sigma_{z}$, with the momentum operator $\hat{k}_{x}=-i\partial/\partial_{x}$ and Pauli matrices $\sigma_{y}$ and $\sigma_{z}$, is induced by the Raman process, describing a synthetic spin-orbit coupling with a strength $\lambda\equiv\hbar^{2}k_{R}/m$ and an effective Zeeman field $h=\Omega_{R}/2$. Here $k_{R}$ and $\Omega_{R}$ are the recoil momentum and the Rabi frequency of Raman laser beams, respectively. In the following, we will set $\hbar=1$ for simplicity.

We use the standard mean-field theory to solve the single-channel model Hamiltonian. Within the mean-field approximation, we define an order parameter $\Delta\left(x\right)\equiv-g_{{}_{\mathrm{1D}}}\left\langle\psi_{\downarrow}\left(x\right)\psi_{\uparrow}\left(x\right)\right\rangle$, and the interaction Hamiltonian is decoupled as

After the standard Bogoliubov transformation $\psi_{\sigma}=\sum_{\eta}\left[u_{\sigma\eta}e^{-iE_{\eta}t}c_{\eta}+v_{\sigma\eta}^{*}e^{iE_{\eta}t}c_{\eta}^{\dagger}\right]$ to all field operators of mean-field Hamiltonian, we obtain the BdG equations

in a Nambu spinor representation with BdG Hamiltonian

the quasi-particle wave function $\phi_{\eta}=[u_{\uparrow\eta},u_{\downarrow\eta},\upsilon_{\uparrow\eta},\upsilon_{\downarrow\eta}]^{T}$ and the corresponding quasi-particle eigenenergy $E_{\eta}$. The BdG equations above should be solved self-consistently with the order parameter equation

and the density equation

where $f(E)=1/[e^{E/k_{B}T}+1]$ is the Fermi-Dirac distribution function at a temperature $T$. It is important to note that the use of Nambu spinor representation doubles the size of Hilbert space of the system. As a result, there is a particle-hole symmetry in the Bogoliubov solutions: for any “particle” solution with the wave function $\phi_{\eta}^{\left(p\right)}=\left[u_{\uparrow\eta},u_{\downarrow\eta},\upsilon_{\uparrow\eta},\upsilon_{\downarrow\eta}\right]^{T}$ and energy $E_{\eta}^{\left(p\right)}\geq 0$, we can always find the other “hole” solution with a wave function $\phi_{\eta}^{\left(h\right)}=\left[\upsilon_{\uparrow\eta}^{*},\upsilon_{\downarrow\eta}^{*},u_{\uparrow\eta}^{*},u_{\downarrow\eta}^{*}\right]{}^{T}$ and energy $E_{\eta}^{\left(h\right)}=-E_{\eta}^{\left(p\right)}\leq 0$. Generally these two states describe the same physical state. To remove this redundancy, we have added an extra factor of $1/2$ in the expressions for the order parameter (6) and total density (7). In the following discussions, we focus at zero temperature with a typical interaction strength $\gamma=\pi$, the SOC strength $\lambda=1.5E_{F}/k_{F}$. As shown in Fig. 1, the system undergoes a phase transition from a BCS superfluid to a topological superfluid when continuously increasing the effective Zeeman field $h$ over a critical value $h_{c}\simeq 0.92E_{F}$ Kong et al. (2021), where the chemical potential $\mu$ and the bulk order parameter $\Delta_{\mathrm{bulk}}$ present a jump change. We need to emphasize that the value of $h_{c}$ will be slightly influenced by some parameters in the numerical calculation such as the size of box and the energy cutoff.

In a uniform and infinite system, the continuous momentum $k$ is a good quantum number. Thus, it is possible to get an analytic expression of four quasi-particle eigenenergy $E_{\pm}^{\left(p\right)}\left(k\right)=-E_{\pm}^{\left(h\right)}\left(k\right)\equiv E_{\pm}\left(k\right)$ in Eq. (4) with Wei and Mueller (2012)

where $\xi_{k}=k^{2}/\left(2m\right)-\mu$ and $E_{k}=\sqrt{\xi_{k}^{2}+\Delta^{2}}$. The excitation of the Higgs oscillation is closely related to the minimum of quasi-particle energy Volkov and Kogan (1974). In Fig. 2, we present the positions and values of the minimum in two positive quasi-particle energy branches $E_{\pm}\left(k\right)$ as a function of the Zeeman field $h$. Typically there are three regimes separated by $h_{c}\approx 0.92E_{F}$ and $h_{sp}\approx 1.1E_{F}$, where the locations of minimum in two energy branches marked by blue arrows in three upper panels are different. The locations of minimum are both at $k=0$ when $h<h_{c}$, then the minimum of $E_{+}(k)$ is shifted to a nonzero momentum while the one in $E_{-}(k)$ is still at $k=0$ when $h_{c}<h<h_{sp}$, and finally both of them are shifted to a nonzero momentum when $h>h_{sp}$.

In this work, a 1D SOC Fermi superfluid is taken into account where the quasi-1D geometry is usually realized by applying a strong confinement along both $y$ and $z$ axes in a three-dimensional (3D) system Cazalilla et al. (2011); Guan et al. (2013). In general, the order parameter is determined by the realistic parameters of the system, such as the interaction strength $\gamma$ , the SOC strength $\lambda$ and the effective Zeeman field $h$. The interaction strength can be well controlled by both confinement and Feshbach resonances as discussed in references Olshanii (1998); Haller et al. (2009, 2010); Peng et al. (2010, 2011, 2014). However, it is tough to change the interaction strength rapidly or in a very short time scale. In addition, the SOC strength $\lambda$ can not be tuned over a large range in ultracold atoms experiments. Thus, we choose the effective Zeeman field $h$ as the ramping parameter in this work. First, the Zeeman field determines directly the topological structure of the ground state as shown in Fig. 1 and we can discuss for different cases. Besides, the effective Zeeman field can be feasibly tuned in a long or short time scale by the laser intensity or the detuning in the experiments of the SOC Fermi gases. These experiment features have been introduced in Refs. Lin et al. (2011); Wang et al. (2012); Cheuk et al. (2012); Zhang et al. (2012); Williams et al. (2013); Qu et al. (2013). In order to excite and investigate the Higgs oscillation, we begin with an initial Zeeman field $h_{i}$ to calculate self-consistently its ground state, and then vary $h$ over time in a slow linear ramp way or by abruptly changing to reach a final Zeeman field $h_{f}$. Therefore, the dynamics of the order parameter can be then studied by solving the time-dependent BdG equation

In summary, we theoretically probe and study two Higgs oscillations in a one-dimensional Raman-type spin-orbit-coupled Fermi superfluid, by solving the time-dependent BdG equations with three different ways to tune the effective Zeeman field. In contrast to the single Higgs oscillation in the conventional Fermi superfluid, we find two distinct Higgs oscillations in both the BCS and topological states when investigating numerically the oscillation of the order parameter. The Higgs oscillation can be well explained from the excitation in two quasi-particle energy spectrum, whose oscillation periods exhibit a great agreement with previous Volkov and Kogan’s theoretical prediction over a large range of the Zeeman field except for crossing the phase transition point. Further research could be undertaken to thoroughly explore the oscillation behaviours related to other physical observables, such as density and spin polarization.