Spectroscopic probes of isolated nonequilibrium quantum matter: Quantum quenches, Floquet states, and distribution functions

We investigate radio-frequency (rf) spectroscopy, metal-to-superconductor tunneling, and angle-resolved photoemission spectroscopy (ARPES) as probes of isolated out-of-equilibrium quantum systems, and examine the crucial role played by the nonequilibrium distribution function. As an example, we focus on the induced topological time-periodic (Floquet) phase in a two-dimensional $p+ip$ superfluid, following an instantaneous quench of the coupling strength. The post-quench Cooper pairs occupy a linear combination of “ground” and “excited” Floquet states, with coefficients determined by the distribution function. While the Floquet band structure exhibits a single avoided crossing relative to the equilibrium case, the distribution function shows a population inversion of the Floquet bands at low energies. For a realization in ultracold atoms, these two features compensate, producing a bulk average rf signal that is well captured by a quasiequilibrium approximation. In particular, the rf spectrum shows a robust gap. The single crossing occurs because the quench-induced Floquet phase belongs to a particular class of soliton dynamics for the BCS equation. The population inversion is a consequence of this, and ensures the conservation of the pseudospin winding number. As a comparison, we compute the rf signal when only the lower Floquet band is occupied; in this case, the gap disappears for strong quenches. The tunneling signal in a solid-state realization is ignorant of the distribution function, and can show wildly different behaviors. We also examine rf, tunneling, and ARPES for weak quenches, such that the resulting topological steady state is characterized by a constant nonequilibrium order parameter. In a system with a boundary, tunneling reveals the Majorana edge states. However, the local rf signal due to the edge states is suppressed by a factor of the inverse system size, and is spatially deconfined throughout the bulk of the sample.

Figure 1

Quench phase diagram [31] showing three distinct dynamical phases characterizing the out-of-equilibrium states induced by sudden quench of the interaction strength in a 2D $p+ip$ fermion superfluid [8, 9]. The order-parameter amplitude ${\mathrm{\Delta }}_{\infty }\left(t\right)$ either vanishes, converges to a nonzero constant, or oscillates persistently for quenches in phases I, II, or III, respectively. The vertical and horizontal axes show ${\mathrm{\Delta }}_0^{(i)}$ and ${\mathrm{\Delta }}_0^{(f)}$, the order-parameter amplitudes associated to the ground states of the pre- and post-quench Hamiltonians. ${\mathrm{\Delta }}_{\mathrm{QCP}}$ indicates the critical point where the ground-state chemical potential vanishes. It separates the topologically nontrivial BCS and trivial BEC phases in equilibrium. The purple dashed curve is its nonequilibrium extension. Each point to the left (right) of this line in II represents a topologically nontrivial (trivial) nonequilibrium state with positive (negative) ${\mu }_{\infty }$. Phase III is a quench-induced Floquet topological superfluid state, which is our main focus here. The labeled points are particular quenches discussed in the text. The amplitudes ${\mathrm{\Delta }}_0^{(i,f)}$ and ${\mathrm{\Delta }}_{\mathrm{QCP}}$ are measured in units of the inverse length $\sqrt{n}$, where $n$ is the fixed particle density [9].

Figure 2

Bulk rf spectra (red solid curves) for nonequilibrium states induced by (a) BCS to BCS quench “a,” (b) BEC to BCS quench “b,” as indicated in the quench phase diagram Fig. 1. The green dashed curves show the relative weight imparted by the distribution function equal to $\frac{1-\gamma \left(k_{\omega }\right)}{2},\frac{1+\gamma \left(k_{\omega }\right)}{2}$ in the positive- and negative-frequency domains, respectively. The distribution function ${\gamma }_k$ is illustrated in the inset. It winds from $+1$ at $k=0$ to $-1$ as $k\to \infty$ for the BEC to BCS quench b, but goes from $-1$ to $-1$ for BCS to BCS quench a. For both quenches, the gap is located between $-{\mathrm{\Delta }}_{\infty }^2$ and 0 in the spectrum, indicating the (topologically nontrivial) weak-pairing BCS nature of the post-quench state. A discontinuous jump appears at $\omega =0$ in (a) which is associated to a quench from a BCS initial state. The coordinates for quenches a and b are $\{{\mathrm{\Delta }}_0^{(i)},{\mathrm{\Delta }}_0^{(f)}\}=\{0.8,0.5\}{\mathrm{\Delta }}_{\mathrm{QCP}}$ and $\{1.1,0.8\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, respectively.

Figure 3

Bulk rf spectra for nonequilibrium states induced by (a) BEC to BEC quench “c,” (b) BCS to BEC quench “d,” as indicated in the quench phase diagram Fig. 1. As in Fig. 2, the red solid and green dashed curves respectively correspond to the actual spectrum and relative weight, while the inset shows the distribution function ${\gamma }_k$. The distribution function contains an even and odd number of zeros for quenches c and d, respectively. The post-quench states corresponding to both (a) and (b) are topologically trivial, i.e., there would be no Majorana edge modes in a system with a boundary, and the nonequilibrium phase is of “BEC type.” This is indicated by the gaps which appear between 0 to $|2{\mu }_{\infty }|$. In (b), the rf spectrum exhibits a discontinuous jump at $\omega =0$, indicative of the BCS initial condition. The coordinates for quenches c and d are $\{{\mathrm{\Delta }}_0^{(i)},{\mathrm{\Delta }}_0^{(f)}\}=\{1.02,1.2\}{\mathrm{\Delta }}_{\mathrm{QCP}}$ and $\left\{0.7,1.1\right\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, respectively.

Figure 4

The bulk tunneling spectra of the nonequilibrium state following quenches (a) “a,” (b) “f,” (c) “d,” as indicated in the quench phase diagram Fig. 1. The corresponding coordinates are $\{{\mathrm{\Delta }}_0^{(i)},{\mathrm{\Delta }}_0^{(f)}\}=\{0.8,0.5\}{\mathrm{\Delta }}_{\mathrm{QCP}},\{0.8,0.9\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, and $\{0.7,1.1\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, respectively. In (a), the long-time asymptotic nonequilibrium state has ${\mu }_{\infty }>{\mathrm{\Delta }}_{\infty }^2$; two coherence peaks appear at $\tilde{V}=\pm V_{\min }$. ${\mathrm{\Delta }}_{\infty }^2>{\mu }_{\infty }>0$ and ${\mu }_{\infty }<0$ for quenches f and d, respectively. In (b), (c), the tunneling signal jumps from 0 at one edge of the gap $\tilde{V}={\mu }_{\infty }$ and grows continuously at the other edge $\tilde{V}=-{\mu }_{\infty }$. The discontinuous jump occurs at the right (left) side of the gap in (b) [(c)].

Figure 5

The ARPES spectra of the nonequilibrium state induced by quenches (a) “a,” (b) “b,” (c) “c,” (d) “d.”

Figure 6

Oscillating order parameter ${\mathrm{\Delta }}_{\infty }\left(t\right)$ induced by two phase III quenches, labeled “A” and “B” in the quench phase diagram Fig. 1. The left panels show the time dependence of the modulus and phase, while the right depicts the ${\mathrm{\Delta }}_{\infty }\left(t\right)$ orbits in the complex $\mathrm{\Delta }$ plane. The magnitude of $|{\mathrm{\Delta }}_{\infty }\left(t\right)|$ oscillates between the two turning points ${\mathrm{\Delta }}_2\pm {\mathrm{\Delta }}_1$ [cf. Eq. (4.1)]. ${\mathrm{\Delta }}_2$ is indicated by the center point of each orbit on the right. The coordinates for quenches A and B are $\{{\mathrm{\Delta }}_0^{(i)},{\mathrm{\Delta }}_0^{(f)}\}=\{0.119,0.514\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, and $\{0.00651,0.825\}{\mathrm{\Delta }}_{\mathrm{QCP}}$, respectively.

Figure 7

Quasienergy spectrum $E^{\left(F\right)}\left(\varepsilon \right)$ in the extended zone scheme for quenches (a) “A” and (b) “B”, as indicated in the quench phase diagram Fig. 1. Here, $\varepsilon =k^2$, where $k$ is the momentum. For both quenches, the quasienergy $E^{\left(F\right)}\left(\varepsilon \right)$ (red solid curve) is quite close to the static phase II approximation for the quasiparticle excitation energy $E_2\left(\varepsilon \right)$ (green dotted curve) for large $\varepsilon$, and to $-E_2\left(\varepsilon \right)+\mathrm{\Omega }$ (blue dashed curve) for small $\varepsilon$. Here, $\mathrm{\Omega }$ denotes the oscillation frequency of ${\mathrm{\Delta }}_{\infty }\left(t\right)$, determined by the quench. The crossover of $E^{\left(F\right)}\left(\varepsilon \right)$ between “ground-state” and “excited-state” branches occurs around ${\varepsilon }_{\pm }$, defined in the text.

Figure 8

Oscillation of the first component of the phase III Floquet solution $|u^{\left(F\right)}(\varepsilon ,t)|$, associated to quenches A (a) and B (b). The top green (bottom orange) solid curve shows the maximum (minimum) of $|u^{\left(F\right)}(\varepsilon ,t)|$ within one period. The yellow (shaded) area enclosed by these two curves marks the region swept out by the periodic modulation. We compare these to the static phase II approximation for the coherence factors $|u_2\left(\varepsilon \right)|$ (pink dotted curve) and $|v_2\left(\varepsilon \right)|$ (cyan dashed curve). The magnitude of the Floquet component $|u^{\left(F\right)}(\varepsilon ,t)|$ oscillates within a narrow region around the phase II approximation for most single-particle energies $\varepsilon$, and switches branches near ${\varepsilon }_{\pm }$, where the oscillation amplitude is maximized.

Figure 9

Distribution function $\gamma \left(\varepsilon \right)$ for quenches (a) A and (b) B. Due to the conservation of the pseudospin winding number [9], both wind from $+1$ at $\varepsilon =0$ to $-1$ at $\varepsilon \to \infty$.

Figure 10

Time-averaged bulk rf spectra (red solid curves) from a topological Floquet system induced by the phase III quenches (a) “A” and (b) “B,” as indicated in the quench phase diagram Fig. 1. For comparison, the corresponding phase II (static quasiequilibrium) approximations [Eq. (4.5)] are plotted with dashed blue curves. Unsurprisingly, the two curves agree well for quench A (a), which is close to the phase II-III border. More interesting is the close agreement for quench B (b), which resides deep in phase III and is characterized by the large anharmonic oscillations in ${\mathrm{\Delta }}_{\infty }\left(t\right)$ shown in Fig. 6. The good agreement can be largely attributed to the nonequilibrium distribution functions. These show a population inversion in the Floquet bands at small $\varepsilon$, as shown in Fig. 9. The small deviation in (b) consists of a sequence of evenly spaced peaks (Floquet copies [18]), with the spacing equal to the oscillation frequency $\mathrm{\Omega }$.

Figure 11

Time-averaged bulk rf spectra of the lower Floquet band associated to quenches (a) A and (b) B. These spectra are calculated using the same Floquet states and quasienergies utilized in Fig. 10 but assuming $\gamma \left(\varepsilon \right)=-1$, that is, only the lower Floquet band is occupied. Although these only differ in the distribution function, the spectra are completely unlike the corresponding ones in Fig. 10. In particular, the case (b) corresponding to the Floquet band structure induced by quench B gives a rf spectrum from the lower Floquet band that exhibits no gap, in contrast to quench-induced spectrum in Fig. 10, which takes into account the physical distribution function.

Figure 12

Real part of the bulk first-order rf harmonics $Re{I}_{\text{rf}}\left(1\right)$ for quenches (a) A, (b) B.

Figure 13

Time-averaged bulk tunneling conductance for a topological Floquet system following quenches (a) A, (b) B. The red solid curve and blue dashed curve in this figure respectively show the phase III conductance $\overline{G}\left(\tilde{V}\right)$ and its phase II approximation. These two curves agree well in (a) but not in (b). The phase II approximation fails for the strong quench B as the distribution function does not influence the tunneling signal (cf. the rf spectra in Figs. 10 and 11).

Figure 14

Real part of the bulk first-order tunneling harmonics $ReG\left(1\right)$ for quenches (a) A, (b) B.

Figure 15

The ARPES spectra for the topological Floquet system induced by phase III quenches (a) A, (b) B. Their phase II approximations are depicted in (c) and (d), respectively.