Signature of Generalized Gibbs Ensemble deviation from Equilibrium: Negative absorption induced by a local quench

When a parameter quench is performed in an isolated quantum system with a complete set of constants of motion, its out of equilibrium dynamics is considered to be well captured by the Generalized Gibbs Ensemble (GGE), characterized by a set of coefficients $\lambda_\alpha$'s related to the constants of motion. We determine the simplest GGE deviation from the equilibrium distribution that leads to detectable effects: By quenching a suitable local attractive potential in a one-dimensional electron system, the resulting GGE differs from equilibrium by only one single $\lambda_\alpha$, corresponding to the emergence of an only partially occupied bound state lying below a fully occupied continuum of states. The effect is shown to induce optical gain, i.e., a negative peak in the absorption spectrum, indicating the stimulated emission of radiation, enabling one to identify GGE signatures in fermionic systems through optical measurements. We discuss the implementation in realistic setups.


I. INTRODUCTION
The concept of quantum quench, i.e. the sudden change in the Hamiltonian parameters of an isolated quantum system [1][2][3][4] , has an extraordinary impact in both technological applications and fundamental physics. Not only it represents a basic operational tool for quantum computing, it also enables one to tailor material properties 5 and quantum phases 6 . Furthermore, because a quench drives the system out of equilibrium, challenging questions have intrigued many scientists in the last years: Can the system "thermalize" in some sense at long times and, if so, what are the properties of the steady state? The answers to these non trivial problems mainly depend on two aspects. First, the type of quench: While early studies considered quenches of spatially homogeneous parameters [7][8][9][10][11][12][13][14][15][16] , recent works have focussed on inhomogeneous quenches such as extensive disorder potentials 17,18 , e.g. in view of many-body localization 19,20 , and spatially localized perturbations [21][22][23][24][25][26][27][28][29][30][31] , which can for instance generate persistent oscillations in physical observables 17,18,29 . The second important ingredient in the problem is the type of system. In particular, in the case of integrable quantum systems 32 , the post-quench dynamics necessarily takes place in agreement with the existence of a complete set {Î α } of local constants of motions commuting with the post-quench Hamiltonian 33 . This typically implies that, in the long time limit, such systems exhibit an out of equilibrium steady state described by the Generalized Gibbs Ensemble (GGE) density matrix 7,34-39 where the Lagrange multipliers {λ α } are determined by the pre-quench state and uniquely characterize the GGE. On the theoretical side, there is a growing consen-sus that the GGE hypothesis works both for homogeneous 7,34-39 and inhomogeneous 17,18,[40][41][42][43][44] quenches. However, only a few experimental GGE signatures have been observed so far, mostly limited to trapped onedimensional Bose gases 45 , while proposals for detection in Fermi systems are needed.
A particularly illuminating case where sound results concerning GGE are known is when the post-quench HamiltonianĤ is a one-body operator. In such a case, the latter can always be brought into a diagonal form H = α ε αγ † αγα through a change of basis to suitable creation/annihilation operatorsγ † α ,γ α of single particle states α, and the complete set of constants of motion {Î α } are identified as the number operatorsn α ≡γ † αγα . For these systems, it has been proven that the long-time dynamical average of any one-body operator does equal the GGE statistical average over Eq.(1) 17,18 . Importantly, in this case one can also quantify the deviation of GGE from thermal equilibrium. This can be done through the single-particle reduced density matrix stemming from ρ GGE , which is explicitly given byρ D = α |α α|f α and is thus called the "diagonal ensemble" in the αbasis. Here f α ≡ n α • = Tr[n αρ• ] represent the occupancies of the post-quench constants of motion over the pre-quench stateρ • . They are in one-to-one correspondence with the {λ α }, which are fixed through the relation n α GGE = n α • . In particular, for fermionic systems, this implies f α = (1 + exp[λ α ]) −1 .
Thus, while the equilibrium state at temperature T and chemical potential µ corresponds to the Fermi dis- , or equivalently to λ eq α = (ε α − µ)/k B T , the out of equilibrium state is characterized by the actual set {f α }, or equivalently by the set {λ α }, and is thus quantified in terms of "how many" occupancies f α deviate from f eq α and by "how much". In this Article we focus on Fermi systems and address the following question: what is the simplest deviation from equilibrium that can produce ob-arXiv:2009.01032v1 [cond-mat.mes-hall] 2 Sep 2020 servable effects? We shall show that quenching a spatially localized potential can lead, under suitable circumstances, to a striking out of equilibrium distribution, where only one GGE parameter λ α deviates from equilibrium, corresponding to an only partially occupied bound state lying below a continuum of fully occupied extended states. Furthermore, we show that such condition yields a negative absorption spectrum, also known in optoelectronics as the optical gain, thereby paving the way to observe signatures of GGE through optical measurements.

II. MODEL AND POST-QUENCH OCCUPANCIES
In order to illustrate the effect, we consider as a prequench system a homogeneous one-dimensional gas of free spinless electrons, described by the Hamiltonian H pre = dxΨ † (x)p 2Ψ (x)/2m, withΨ denoting the electron field operator andp = −i ∂ x the momentum operator. The system is at equilibrium with a reservoir, at a temperature T and a chemical potential µ. This entails that the Fourier mode operatorsĉ(k) diagonalizing the Hamiltonian,Ĥ pre = dk ε(k)ĉ † (k)ĉ(k), are characterized by where ε(k) = 2 k 2 /2m is the pre-quench spectrum. Then, the system is disconnected from the reservoir and, at the time t = 0, a localized attractive potential V (x) < 0 is switched on near the origin x = 0, so that the post-quench Hamiltonian isĤ post =Ĥ pre + dxΨ † (x)V (x)Ψ(x). Notably, whileĤ pre has a purely continuous spectrum,Ĥ post also displays a discrete set of bound states, spatially localized around the origin, and with energies ε n < 0 (n = 0, 1, 2 . . .) lying below the continuum branch ε > 0.
Two arguments make the post-quench dynamics of this isolated system intriguing. On the one hand, as the quench potential is local, the system only experiences a negligibly small energy change in the thermodynamic limit. On the other hand, in such a limit, the Anderson orthogonality catastrophe 46 ensures that the many-body ground state of the post-quench Hamiltonian is orthogonal to the pre-quench one, suggesting quite a different behavior. In order to characterize the out of equilibrium dynamics, we first bring the post-quench Hamiltonian, quadratic in the fermionic fieldsΨ andΨ † , to its diagonal formĤ post = α ε αγ † αγα through a unitary transformation. Here the symbol is a compact notation indicating a summation over the discrete spectrum branch and an integral over the continuous spectrum branch. This shows, as observed above, that the out of equilibrium dynamics of the system is governed by a GGE, which is characterized by the set of post-quench occupancies f α of the constants of motion.
However, because the post-quench spectrum contains both a discrete and a continuum branch, care must be taken in identifying the occupancies f α , which in this case are determined from the diagonal ensemble density matrix through the relation (ρ D ) α α ≡ γ † αγα GGE = d αα f α , where d αα ≡ δ αα for α, α ∈ discrete spectrum, while d αα ≡ δ(α − α ) for α, α ∈ continuum spectrum, and d αα = 0 otherwise 47 . In turn, theρ D entries can be computed by exploiting the transformation γ α = dk U (α, k)ĉ(k) linking the post-to the pre-quench operators, where U (α, k) = dx ψ * α (x)ϕ k (x) is the overlap integral between the post-quench eigenfunctions ψ α and the pre-quench eigenfunctions ϕ k . By recalling the expectation values (2) of the pre-quench operators, it is straightforward to show that whence the post-quench occupancies f α are obtained through the above prescription.

III. THE CASE OF A QUANTUM WELL
For definiteness, we shall evaluate the post-quench occupancies for the case of a rectangular quantum well (QW) potential V (x) = −V 0 θ(a/2 − |x|), characterized by a potential depth V 0 and a width a around the origin. Here θ denotes the Heaviside function. In this case, space parity is conserved across the quench, the postquench eigenfunctions ψ α are well known, just like the pre-quench free-particle eigenfunctions ϕ k , and the occupancies Eq.(3) can be evaluated for all the post-quench states.
As far as the continuous spectrum is concerned, it is worth recalling that the presence of the QW does modify the continuum states with respect to the free-particle waves, especially at small energies (0 < ε < V 0 ). Nevertheless, a lengthy but straightforward calculation (see Appendix for details), shows that in the thermodynamic limit, the post-quench occupancy of the continuum is f α = f eq (ε α ), i.e. it coincides with the equilibrium Fermi function of the pre-quench state, with the same temperature and chemical potential, regardless of the values a and V 0 of the QW parameters. This is the hallmark of the locality of the quench. In particular, at zero temperature all continuum states are fully occupied up to the chemical potential µ.
The situation is different for the bound states. As is well known, the number of bound states in a rectangular QW depends on the ratio between the well potential depth V 0 and the kinetic energy E a = π 2 2 /2ma 2 associated to the confinement in the well width a. The smallest deviation from equilibrium is when one single discrete level, lying below the continuous spectrum of occupied states, is not fully occupied. Focussing then on the regime V 0 < E a , where the QW hosts only one bound state, one can exploit the well known expression for the bound state of a rectangular QW and evaluate its occupancy f bs = (ρ D ) bs,bs numerically from Eq.(3).
The result is shown in Fig.1(a), where f bs is plotted as a function of the ratio V 0 /E a , at zero temperature, for four values of chemical potential µ. While for an extremely shallow and thin well (V 0 /E a 1) one has f bs 1, i.e., the value one would obtain if the post-quench system were at equilibrium, for V 0 /E a 1 the occupancy decreases. Notably, such a reduction is the more pronounced the lower µ is, which can be understood from the following arguments. Since the pre-quench eigenfunctions ϕ k are essentially plane waves, the U (bs, k) coefficient is the Fourier transform of the bound state wavefunction ψ bs and becomes negligible for k 1/ , where a is the lengthscale over which ψ bs is localized. The chemical potential µ of the pre-quench state appearing in the Fermi function, cuts the integral in Eq.(3) at the Fermi wavevector dk |U (bs, k)| 2 = 1 (unitarity of the U transformation), for small chemical potential, such that k F 1/ , the integral is cut before yielding the occupancy 1.
The resulting occupancy of the post-quench spectrum is sketched in Fig.1(b) at zero temperature: While the µ < l a t e x i t s h a 1 _ b a s e 6 4 = " n c j M 6 f q o o 6 9 s t s t x X + e Z g X A L u P M = " < l a t e x i t s h a 1 _ b a s e 6 4 = " t 5 r h g m + r K M W f i w 1 f V / n + D a Q L n G s = " > A A A B 8 n i c b Z D L S g M x F I Y z X m u 9 V V 0 K E i y C q z J T F 7 o Q L L h x 2 Y K 9 w L S U T H q m D c 0 k Q 5 I p l K F L H 8 G N C 0 X c + g B u f Q V 3 P o M + h O l l o a 0 / B D 7 + / x x y z g l i z r R x 3 U 9 n a X l l d W 0 9 s 5 H d 3 N r e 2 c 3 t 7 d e 0 T < l a t e x i t s h a 1 _ b a s e 6 4 = " f e 4 p p g 2 x g S q l q g

b 4 T D Q E F h w 8 U 3 5 + 4 J c = " > A A A B 8 3 i c b V D L S g N B E O y N r x g f i X r 0 M h g E v Y T d K O h B M C C C x w g m E b J L m J 3 M J k N m Z p e Z W S E s + Q 0 v H h T
x 6 h f 4 F 9 7 8 B P / C y e O g i Q U N R V U 3 3 V 1 h w p k 2 r v v l 5 J a W V 1 b X 8 u u F j c 2 t 7 W J p Z 7 e p 4 1 Q R 2 i A x j 9 V 9 i D X l T N K G Y Y b T + 0 R R L E J O W + H g a u y 3 H q j S L J Z 3 Z p j Q Q O C e Z B E j 2 F j J j z q Z r w Q K 9 e j C 6 5 T K b s W d A C 0 S b 0 b K t e L 3 5 X H h 4 7 r e K X 3 6  continuum states ε > 0 are characterized by the very same Fermi function as the equilibrium pre-quench state and are thus fully occupied up to the chemical potential µ for any QW parameter, the bound state ε bs < 0 is only partially occupied, despite being energetically more favorable than the continuum. This striking out of equilibrium effect thus realizes the simplest GGE deviation from equilibrium: only the bound state λ bs = ln[(1 − f bs )/f bs ] deviates from the equilibrium value. In particular, this is quite different from the case of a homogeneous quench, where typically an extensive number of post-quench occupancies deviate from equilibrium 38 . Note that, because of particle conservation, the partial occupation of the quench-induced bound state corresponds to a depletion by (at most) one electron of the continuous spectrum, an effect that is not sizable directly from the continuum itself, since in the thermodynamic limit the number of continuum states is infinite. In contrast, the emergence of an only partially occupied bound state, energetically separated from the fully occupied continuum above, has a remarkable consequence: It realizes the condition of population-inversion, well known in optoelectronics. While at equilibrium a radiation impinging onto an electron system yields the absorption of an energy quantum causing a transition from energetically lower and more populated levels to upper and less populated levels, the out of equilibrium population obtained here leads to a release of energy, causing a stimulated emission or, a "negative" absorption. This opens up the possibility to observe this GGE signature through optical measurement, as we shall describe in the next Section.

IV. ABSORPTION SPECTRUM
For an electron system coupled to an electromagnetic radiation of frequency ω, the non-linear absorption spectrum A(ω) is given, within the conventional perturbation-theory based on a Fermi's golden rule treatment of the light-matter interaction 48 , by where n denotes the real part of the refraction index, c the speed of light, 0 the vacuum dielectric constant, m e the bare electron mass, and V the volume. Equation (4) describes all transitions from initial states α to final states α compatible with the transition energy ω, and its non-linear nature is determined by the factor f α − f α . While at equilibrium the final state α is necessarily less populated than α (f α > f α ), causing an actual absorption, A(ω) > 0, in the population-inversion regime induced by the quench, one has f α > f α for α = bs and α in the occupied continuous spectrum, opening up the possibility of a negative absorption coefficient, A(ω) < 0, i.e., to the emission of an electromagnetic radiation stimulated by the quench. This is known in optoelectronics as the optical gain effect 48 . However, unlike the more conventional inter-band transitions, the effect described here can be considered as "intraband", as it originates from a quench on one single pre-quench band.

A. Implementation
As can be deduced from Fig.1(a), the optimal regime to obtain a population-inversion is in principle µ V 0 E a . However, a too small chemical potential reduces screening effects and makes electron-electron interaction effects relevant. A still quite acceptable regime is µ V 0 E a , which can be achieved e.g. with a InSb nanowire (NW), characterized by a small effective mass m = 0.015m e , and a realistic QW realized by a finger gate deposited on a NW portion with size a = 150 nm and biased by a gate voltage V 0 < 0. This yields E a 1.12 meV and, by taking a realistic value µ = 0.2 meV, one still has an energy window for the QW depth V 0 . Since Eq.(4) cannot be computed analytically, we have performed a numerically exact evaluation on a finite system, whose total length L = 16 µm is two orders of magnitude bigger than the QW width a, at a realistic temperature of T = 250 mK. Furthermore, the unavoidable presence of inelastic processes broadening the otherwise sharp energy levels has been taken into account by replacing the ideal Dirac δ-function appearing in Eq.(4) with a broadened function of Gaussian shape The ratio R between the out of equilibrium absorption spectrum A(ω) induced by the quench and the equilibrium absorption spectrum A eq (ω) of the postquench system, for a InSb NW with a QW width a = 150 nm (Ea 1.12 meV) and depth V0 = 0.5 meV (black curve) and V0 = 1.0 meV (red curve). The pre-quench temperature and chemical potential are T = 250mK and µ = 0.2 meV, respectively. While at low frequencies the quench does not induce any deviation from equilibrium (R → 1), a significant negative peak appears at ω * = |ε bs | corresponding to the energy separation between the continuum and the bound state.
where the value of broadening energy has been taken as ε b = 20 µeV. This roughly corresponds to k B T , i.e. the typical broadening related to electron-acoustic phonon energy exchange. The result is illustrated in Fig.2, where we have plotted the ratio R( ω) ≡ A(ω)/A eq (ω) between the out of equilibrium absorption spectrum induced by the quench and the equilibrium case corresponding to the situation where the post-quench system is at equilibrium, for two different values of QW depth V 0 .
At low frequencies one has R( ω) 1, indicating that the spectrum of the quench-induced absorption coefficient is just like the equilibrium one. In this regime the intraband absorption processes are caused by continuum→continuum transitions from energetically lower and almost fully occupied states 0 < ε < µ to energetically higher and almost empty states ε > µ. It is worth pointing out that such transitions occur because of the presence of the QW, which makes the dipole matrix entries α |p|α non vanishing for α = α .
The most interesting effect, however, arises as the frequency approaches the value ω * ≡ |ε bs |/ , where transitions can occur from the fully occupied lowest continuum states to the only partially occupied bound state lying underneath. This is how the population-inversion regime causes a negative absorption, i.e. the stimulated emission of an electromagnetic radiation. The hallmark of this optical gain effect is the negative peak located around ω * . Note that, just like the value of such resonance frequency, also the depth R * of the negative peak is controlled by the value of the potential depth V 0 , and its magnitude can be significantly higher than 1, so that the negative absorption is significantly stronger than the equilibrium positive absorption contribution. For higher frequencies, the ratio R(ω) becomes positive again. This corresponds to an actual absorption, arising from transitions to the energetically higher and almost empty continuum states from both the bound state and the energetically lower and occupied continuum states.

B. Finite switching time
So far, we have considered a sudden quench. In realistic implementations, however, the quench is applied over a finite switching time τ sw . To take this issue into account, we have considered a time-dependent Hamil- is a switching function ranging from 0 to 1, up to 2%, within a time scale τ sw . By solving numerically the Liouville-von Neumann Equation i ∂ tρ = [H,ρ] for the single-particle density matrixρ, the related diagonal density matrixρ D is extracted, and the 'post-quench' absorption spectrum, i.e. the value of Eq.(4) evaluated at time t τ sw , is computed. By increasing τ sw the shape of the negative peak is roughly unaltered, whereas its depth R * is reduced. Taking e.g. V 0 = 1 meV, the value R * −20 obtained for an ideally instantaneous quench (red curve in Fig.2) reduces to R * −9 and R * −3 for realistic τ sw values of 5 and 10 ps, respectively. Yet, the value |R * | > 1 indicates that the out of equilibrium contribution of the negative peak is still larger than the positive equilibrium one. For a very slow switching, the occupancies adiabatically follow the localized potential and the distribution cannot be distinguished from the equilibrium one.

V. CONCLUSIONS
We have shown that, by quenching a suitable local attractive potential in an isolated one-dimensional free electron gas, the out of equilibrium dynamics is determined by a GGE describing the elementary deviation from equilibrium, where only one Lagrange multiplier λ bs deviates from its equilibrium value. The occupancy of the continuum states is unaltered by the quench and is still described by an equilibrium Fermi function, so that all such states are occupied up to the chemical potential at zero temperature. In striking contrast, the bound state generated by the quench is only partially occupied, despite being energetically more favorable than the continuum (see Fig.1). Then, such population-inversion regime has been shown to cause a negative peak in the absorption spectrum, realizing an optical gain (see Fig.2). The implementation with QWs in realistic InSb NWs has been discussed and shown to be robust to finite switching time of the local potential. These results could pave the way to observe signature of GGE in fermionic systems via optical measurements.
Appendix: Post-quench occupancies of the continuum spectrum

Basis change coefficients (continuum spectrum)
In computing the coefficients of the pre-post basis change appearing in Eq.(3), we observe that where in Eq.(A.3) we have neglected the second contribution C in η , which is negligible with respect to the first contribution C out η , because the space region outside the quantum well is infinitely long in the thermodynamic limit and because we are focusing on the continuum spectrum wavefunctions. Thus the U ηη (q, k) coefficients can be straightforwardly evaluated by inserting the definitions of ϕ andφ into Eq.(A.3), and by exploiting the identity are the regularized versions of the δ-function and the Principal Value (P.V.), respectively, while k min is an infrared cut-off controlling the integral divergences and mimicking the inverse total length of the system (k min ∼ 2/L → 0 in the thermodynamic limit L → ∞). Within a few algebraic steps one obtains U ηη (q, k) = δ ηη cos θ η,q δ(q − k) − η π sin (q + k) a 2 + θ η,q P.V.
In particular δ(0) = δ(q = 0) = (πk min ) −1 ∼ L/2π → ∞, as expected, since the total number of electrons in the continuum should scale extensively with the system size. By singling out a δ(0) pre-factor, one can see that, apart from the cos 2 θ η,q contribution in the first line of Eq.(A.8), the only term yielding a finite contribution is the squared Principal Value appearing on the third line, due to the relation → π 2 δ(q − k) . (A.10) In conclusion, one obtains that the occupancy of the postquench continuum states equals the equilibrium one.