Quantum states of dark solitons in the 1D Bose gas
We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross-Pitaevskii equation) for the one-dimensional (1D) Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.
The experimental technique of trapped one-dimensional atomic gases [1, 2] has now become a fundamental tool for revealing nontrivial effects in quantum many-body systems [3, 4]. For the interacting Bose gas in one dimension (1D), the first set of exact results goes back to the pioneering work of Girardeau  on the impenetrable Bose gas where the strong interacting limit is considered. The 1D Bose gas interacting with the delta-function potentials, i.e. the Lieb-Liniger (LL) model, gives a solvable model for interacting bosons in 1D , where it is integrable even when the interaction parameter is generic. For the impenetrable Bose gas which corresponds to the Tonks-Girardeau (TG) limit, i.e. the strong interacting limit of the LL model, the one-body reduced density matrix is derived and successfully expressed in terms of the determinant of a Fredholm operator . The exact result is followed by several important developments in mathematical physics [8, 9]. Furthermore, dynamical correlation functions of the LL model are now systematically derived .
Soliton-like localized excitations in a periodic 1D Bose gas have attracted much interest recently and have been studied theoretically [11, 12, 13]. Here we remark that dark solitons are created experimentally in cold atoms , for instance, by the phase-imprinting method [15, 16] (See also ). Localized quantum states are important for investigating dynamical responses of interacting quantum systems. Quantum dark solitons in confining potentials are studied by semiclassical quantization , and those in thermal equilibrium of a quasi-1D Bose gas by generating classical field ensembles . However, it is not clear even at zero temperature how we can construct quantum states associated with dark solitons in the many-body system of the LL model.
Let us consider the Gross-Pitaevskii (GP) equation, which describes Bose-Einstein condensation (BEC) in the mean-field approximation . We also call it the nonlinear Schrödinger equation. The GP equation has dark soliton solutions for the repulsive interactions, while it has bright soliton solutions for the attractive interactions . It was conjectured that dark solitons are closely related to Lieb’s type-II excitations, i.e. one-hole excitations, by carefully studying the dispersion relations . The dispersion relations of the LL model are briefly compared with those of the classical nonlinear Schrödinger equation in the weak coupling limit . However, it has not been shown how one can construct such a quantum state that leads to a dark soliton in the classical limit or what kind of physical quantity can show a property of a dark soliton for some quantum state. Here we remark that each of the type-II eigenstates has a flat density profile since the Bethe ansatz eigenstates are translationally invariant. Moreover, we consider dark solitons under the periodic boundary conditions, which are expressed in terms of elliptic functions [11, 12, 13].
In this paper we demonstrate that a quantum state derived from the Bethe ansatz eigenvectors of the LL model by taking the Fourier transform of the type-II excitations over a branch  has many aspects closely related to classical dark solitons of the GP equation under the periodic boundary conditions. We call the state the quantum dark soliton state and a classical dark soliton under the periodic boundary conditions simply a classical dark soliton. Through the Bethe ansatz we show several corresponding aspects in the weak coupling regime. Firstly, the density profile of the quantum dark soliton state at an initial time is consistent with that of a classical dark soliton. Here we call the graph of the expectation value of the local density operator for a given state versus the position coordinate the density profile of the state, and for a quantum dark soliton state we simply call it the density profile of quantum dark soliton; we call the graphs of the square amplitude and phase in the complex scalar field of a classical dark soliton versus the position coordinate the density profile and phase profile of classical dark soliton, respectively. Secondly, in time evolution the density profile of quantum dark soliton coincides with that of the corresponding classical dark soliton over the whole graph and they move together with the same velocity for a long period of time. Thirdly, for the matrix element of the bosonic field operator between two quantum dark soliton states where one state has particles and another particles, the profiles of the square amplitude and phase at the initial time graphically agree with those of classical dark soliton, respectively. In time evolution the profiles of square amplitude and phase overlap with those of classical dark soliton, respectively, over the whole region and the corresponding profiles move together in time for a long period of time. Here we remark that a classical dark soliton parametrized by elliptic functions becomes a standard dark soliton with open boundaries by simultaneously sending the system size to infinity and the elliptic modulus to unity. Furthermore, in order to illustrate the method for constructing quantum dark solitons, in the 1D free fermions we show from the anti-commutation relations that a notch appears in the density profile of some superposition of one-hole excitations. Interestingly, the density profile of the fermionic state coincides with that of quantum dark soliton for the 1D Bose gas in the large coupling case, i.e. near the TG limit, not only at an initial time but also during the quantum dynamics for some period of time.
The time evolution of the expectation value of the local density operator in the 1D Bose gas should be important also from the renewed interest on the fundamentals of quantum statistical mechanics [25, 26, 27, 28, 29]. The density profile of a quantum dark soliton state has initially a localized notch but collapses slowly to a flat profile in time evolution . The relaxation behavior is consistent with the viewpoints of equilibration of an isolated quantum system [30, 31] and thermalization due to the typicality of quantum states .
We now argue that the density profile of quantum dark soliton has a finite lifetime but it is much longer than the quantum speed limit time of the quantum state in the weak coupling case. Here we remark that the lifetime of a generic state is given by the quantum speed limit time for it [33, 34, 35]. By observing the exact time evolution of the density profile of quantum dark soliton we estimate its lifetime. We shall show in section 5 that the observed life time is inversely proportional to the coupling constant. We thus suggest that the localized density profile of a quantum dark soliton is much more stable than the density profile of a generic quantum state in the weak coupling case. Here we remark that Girardeau and Wright discussed a permanent quantum soliton for impenetrable bosons in 1D , which corresponds to the infinite coupling case of the LL model.
We also argue that the behavior of the wavefunctions is nontrivial in the weak coupling limit for the LL model. The wavefunctions do not simply become close to such wavefunctions of the 1D free bosons that could be consistent with the mean-field picture, when the coupling constant approaches zero but it takes a nonzero value. The exact Bethe ansatz wavefunctions consist of a large number of terms such as . Furthermore, we can show that many-body correlations increase if we increase the particle number while we keep the coupling constant small but fixed and the particle density constant. For instance, the zero mode fraction in the ground state of the 1D Bose gas, which we regard as the condensate fraction of BEC, becomes small and decreases to zero if the number of particles increases while the coupling constant and the particle density are fixed . We suggest that it is also the case for quantum dark solitons i.e., that some properties of quantum dark solitons corresponding to classical dark solitons do not necessarily hold if we increase the particle number while we keep the coupling constant small but fixed and the density of particles fixed. In fact, it is not a priori clear how valid the mean-field approximation is for quantum dark solitons even in the weak coupling case. Here we remark that the breakdown of mean-field theory is addressed for impenetrable bosons , and a long-wavelength theory beyond the mean-field approximation is discussed for low-dimensional bose liquids .
Let us consider the Hamiltonian of the LL model for bosons with coupling constant
Here we impose periodic boundary conditions with length . We employ a system of units with , where is the particle mass. We introduce the canonical Bose field operator with the commutation relations 
The second-quantized Hamiltonian of the LL model is written in terms of the field operator as
where is the chemical potential. The Heisenberg equation of motion of this system has the form
In the classical limit where the quantum field operator is replaced by a complex -number field , equation (5) becomes the following partial differential equation
We call it the nonlinear Schrödinger equation. The equation (6) can be solved by the inverse scattering method and has soliton solutions  (see also ). We recall that it has dark soliton solutions for the repulsive interactions , while it has bright soliton solutions for the attractive interactions . For the attractive case, bright solitons are analytically derived from the quantum wave packets constructed through the Fourier transform of the bound states . The construction was extended for optical fibers . Time evolution of quantum states of bright solitons was investigated theoretically  and experimentally . The effect of quantum noises on the quantum states of bright solitons was also studied [45, 46, 47].
The contents of the paper consists of the following. In section 2 we briefly introduce the notation for the Bethe ansatz, characterize the type II excitation branch, and then construct quantum dark soliton states. We argue that the structure of quantum dark soliton states is analogous to the Dirichlet kernel, and show for free fermions that the density profile of a superposition of one-hole excited states has a notch. It illustrates the construction of quantum dark soliton states. In section 3 we derive one-soliton solutions of the nonlinear Schrödinger equation (6) under the periodic boundary conditions. We express the classical dark solitons in terms of elliptic functions similarly as in Ref. [11, 13]. By sending the elliptic modulus to 1, the classical dark solitons under the periodic boundary conditions become the standard classical dark solitons of the nonlinear Schrödinger equation with open boundaries. Furthermore, we calculate the logarithmic corrections to them, in particular, to the chemical potential and the soliton velocity, by which we can confirm the numerical estimates of the soliton parameters shown in section 4 at least approximately. In section 4 we show several corresponding aspects between the quantum dark soliton states and the corresponding classical dark solitons, which hold in the weak coupling case and should be rigorously valid when the coupling constant approaches zero with the density and the particle number fixed. Firstly, we show that the density profile of quantum dark soliton at an initial time is consistent with that of classical dark soliton. Secondly, we show that in exact time evolution the density profile of quantum dark soliton as a whole coincides with that of classical dark soliton for a long period of time. Thirdly, we show that both the square amplitude and phase profiles for the matrix element of the quantum field operator in the LL model between the quantum dark soliton states with and particles overlap with those of classical dark soliton, respectively, all over the profiles. The square amplitude and phase profiles move in time evolution with exactly the same speed as those of classical dark soliton, respectively. Furthermore, we remark that the density profile of a sum of one-hole excitations in the 1D free fermions overlap with that of quantum dark soliton in the 1D Bose gas with the large coupling constant, i.e. in a regime close to the TG limit. For the matrix element of the fermionic field operator between two such fermionic states the square amplitude and phase profiles have several features in common with those of quantum dark soliton in the 1D Bose gas for the large coupling case, respectively, although they do not overlap each other. We thus obtain useful tools to describe approximately the profiles of quantum dark soliton in the large coupling case. However, it is still quite nontrivial that the profiles of quantum dark soliton in the weak coupling case are consistent with those of classical dark soliton. In section 5 we argue that the lifetime of a notch in the density profile of quantum dark soliton is much longer than that of a generic state when the coupling constant is very small. We compare the lifetime of a notch in the density profile of quantum dark soliton with the quantum speed limit time, and show that the ratio of the former to the latter increases as the coupling constant decreases. It becomes infinitely large as the coupling constant approaches zero. Thus, although the localized density profile collapses in time evolution, we call the quantum states quantum dark soliton states.
2 Excited states of the 1D Bose gas with delta-function potentials
2.1 Bethe ansatz eigenwavefunctions
In the framework of the Bethe ansatz method, the Bethe ansatz wavefunction which is expressed in terms of quasi-momenta
gives an eigenstate of the LL model (1), if the quasi-momenta satisfy the Bethe ansatz equations
In the logarithmic form, we have
Here ’s are integers for odd and half-odd integers for even . We call them the Bethe quantum numbers. The total momentum and energy eigenvalue are written in terms of the quasi-momenta as
The configuration of the Bethe quantum numbers for the ground state is given by the regular array around the center:
It is depicted in the top panel of Figure 1 (a) in the case of .
We can construct low-lying excited states systematically by putting holes and placing particles in the regular array of the Bethe quantum numbers for the case of the ground state. The arrays of the Bethe quantum numbers for one-hole excitations are shown in the second panel of Figure 1 (a). In each sequence there exists an unoccupied number in the middle of the sequence of the five red circles, which we call the one-hole. We remark that the one-hole excitations are also called the type II excitations . Similarly as in the second panel, the arrays of the Bethe quantum numbers for one-particle excitations are shown in the third panel of Figure 1 (a). The right most red particle corresponds to the one-particle in each array. The one-particle excitations are also called the type I excitations .
The dispersion relation of the one-particle excitations and that of the one-hole excitations are plotted with filled circles and filled squares, respectively, in Figure 1 (b). The former branch corresponds to the type I excitations and the latter branch the type II excitations.
2.2 Construction of quantum dark soliton states through type II excitations
We now explain how we construct a series of quantum states  through the type II excitations. Furthermore, we call them quantum dark soliton states.
In the type II branch, for each integer in the set , we consider momentum . We denote the normalized Bethe eigenstate of particles with total momentum by , which we call a one-hole excited state. The Bethe quantum numbers ’s of the one-hole excitation are given by
For any value of satisfying we define the coordinate state by the discrete Fourier transformation:
The density profile of this state shows a density notch as will be shown in section 4.
2.3 Quantum state analogue of a truncated series of the delta function
The formal structure of quantum dark soliton states (13) is analogous to a truncated series of the delta function. For periodic functions with period the delta function is expressed in the form of an infnite series
We now truncate it by keeping only such terms with integers satisfying
It is called the Dirichlet kernel in the Fourier series analysis. It has the peak around at with peak height and oscillating behavior in the region for with period . Here we assume that is independent of . For , the oscillations in the graph of the Dirichlet kernel become of very short wavelength, and we may call them ripples.
The integral of the squared amplitude of the Dirichlet kernel over the period is given by
We therefore normalize the function by dividing it by the square root
In terms of the normalized momentum eigenfunctions where the normalized function is expressed as
It is analogous to the structure of the states of quantum dark solitons (13), where the number in (18) corresponds to the number of states in (13), and the single-particle momentum eigenstates to the one-hole excitations consisting of particles.
By considering the phase factors in (13) the graphs are shifted by in the -direction. We have for the truncated series of the delta function modified with the phase factors.
2.4 Density notch in the sum over one-hole excitations for free fermions
Through the anti-commutation relation of the field operators we now show that a notch appears in the density profile of a linear combination of one-hole excited states in the 1D free fermions.
Under the periodic conditions of length the momenta are given by with some integers in the 1D free fermions. We denote by and for some integers the creation and annihilation operators of the one-dimensional free fermions with momenta , respectively. We assume that an infinite series of integers for cover all integers. The field operator and its conjugate are given by
Let us take a set of arbitrary integers and consider the state of fermions with momenta ’s
For simplicity, we may assume that is the ground state. By applying the field operator to it, we have
Here we have assumed the ordering of fermionic operators as . We regard states as ‘one-hole excitations’ of . We define the state by the sum over all one-hole excitations derived from
Let us define the local density for the state of the 1D free fermions by
We now show that the local density vanishes at the origin: for the state. We first recall that the square of the field operator at the origin vanishes, , due to the anti-commutation relations of the field operators. Thus, we have
We can show that the integral of the local density over the entire interval is given by a large positive value such as
It follows that the local density cannot be always equal to zero, although it vanishes at the origin. Therefore, for the state which is given by the sum over all the one-hole excited states of , the local density has a notch at least at the origin.
In section 4 we shall construct the free-fermionic analog of quantum dark soliton states for the 1D free fermions and show that the density profile coincides with those of the LL model if the coupling constant is large enough such as .
3 Classical dark solitons in the ring
3.1 Traveling-wave solutions expressed in terms of elliptic functions
Let us assume the traveling wave solution with velocity : to the nonlinear Schrödinger equation (6). Hereafter in section 3 we denote simply by . We denote the time derivative by and the second spatial derivative by , where denotes with . We therefore have
We denote the solution of eq. (26) by as a function of , for simplicity. We express the complex scalar field in terms of the amplitude and the phase , both of which are real, as
It follows from equation (29) that we have . By integrating it once with respect to , we have
Integrating this equation with respect to , we have
where is another constant of integration. Thus, we have
where the potential function is given by
Let us assume that the cubic equation: has three distinct real roots, , , and . We put them in increasing order: . The coefficients of the cubic equation are expressed in terms of the roots as follows.
Let us define the modulus of Jacobi’s elliptic functions, , by
It is clear that . Through the transformation of variables from to
we express the integral (36) in terms of the elliptic integral of the first kind:
We therefore have the solution to eq. (33) in terms of Jacobi’s elliptic function
3.2 Classical dark solitons expressed in terms of elliptic integrals
We now express , and in terms of the complete elliptic integrals. Here we consider the periodic boundary conditions (PBC) for the square amplitude and the phase of the classical complex scalar field , and the normalization condition of the square amplitude . Then, the elliptic modulus is related to the depth of the classical dark soliton, i.e. the minimum value of the square amplitude . We express the soliton velocity in terms of the elliptic integral of the second kind, and determine the chemical potential .
PBCs for the square amplitude and the phase of classical dark soliton
Suppose that the square amplitude of a classical dark soliton increases from to and then returns back to with period . We therefore assign the conditions at the initial and the middle points as
It follows from (39) that in terms of the complete elliptic integral of the first kind, , we have
We have . Here and hereafter we often denote by , suppressing the modulus .
We assume the periodic boundary condition for the phase: . Suppose that the soliton velocity is positive: . It follows that we have and the velocity is expressed in terms of the complete elliptic integral of the third kind as
Normalization condition for the square amplitude
We impose the normalization condition to the square amplitude
Thus, by the formula (A.11) in terms of the complete elliptic integral of the second kind we have
Parametrization of , and in terms of complete elliptic integrals
Since the minimum value of the amplitude must be non-negative, i.e. , we assign the following condition
Thus, if the coupling constant , the system size , the density and the elliptic modulus satisfy condition (51), the classical dark soliton exists as a solution of the nonlinear Schrödinger equation (6) (i.e. the GP equation). The soliton velocity and the chemical potential are given by
Numerical determination of the parameters of classical dark solitons
Let us explain how we evaluate the parameters of a dark soliton such as and in order to compare the profiles of the dark soliton with those of a given quantum dark soliton. Here we remark that when a quantum state is given, parameters , and the particle number have been specified. We determine the elliptic modulus from the depth of the dark soliton, i.e. : We solve equation (48) as an equation for modulus . We assign the value on the parameter of density for the density profile of a quantum state. However, we determine the density numerically for the squared amplitude profile of the matrix element of the bosonic field operator between different quantum states, since it may take a value smaller than .
For the density profile of a quantum state , the integral of the expectation value of the local density operator over the whole space is equal to . However, for the matrix element of the quantum field operator between two states and , the integral of the squared amplitude over the whole space may be smaller than the value of . In this case we determine the parameter numerically by taking the integral of the squared amplitude over the whole space. Here we remark that the symbols will be defined in section 4.1.1.
In summary, after we fix the value of density , we solve equation (48) as a function of modulus and estimate its numerical value very precisely. In section 4 we shall make the profiles of classical dark soliton for given density profiles of quantum dark soliton by the method explained in the above.
We define the function by . It appears in the third term of the expression (53) for the chemical potential . We can show that it is a monotonically decreasing function of modulus , which yields the inequality . This gives an upper bound for the absolute value of soliton velocity : , where
Here we remark that the critical velocity is defined for the system of a finite size. We shall denote by the critical velocity for an infinite system.
3.3 Asymptotic behavior of a classical dark soliton
Square amplitude and the phase expressed in terms of elliptic functions
We now express a classical dark soliton explicitly in terms of the complete elliptic integrals. The square amplitude (46) is given by
We shall show that the second term in the right-hand side of equation (55) leads to the logarithmic correction associated with the conservation of the number of particles. Let us introduce a parameter by
We denote the argument in (40) by . Then, the square amplitude is expressed as follows:
We now show that the phase is expressed in terms of Jacobi’s Theta function. Let us define a pure imaginary number with by
Parameters in the asymptotic expansion with respect to the system size
Let us consider the limit of sending the system size to infinity () and the modulus to 1 () simultaneously so that the ratio is kept constant. We denote the ratio by : . Here we remark that we keep the density constant. We define by the simultaneous limit of sending to infinity and to 1 with .
Let us introduce the complementary modulus by . We define and by and , respectively. We remark that and give quarter periods of Jacobi’s elliptic functions. We introduce a parameter by , which is small when modulus is close to 1. Making use of formulae (A.5), (A.13), (A.14) and Legendre’s relation (A.12) we expand ’s and ’s with respect to as follows:
We have , since we have fixed the ratio .
Asymptotic expansion of the velocity and the chemical potential when approaches 1
Here we recall that parameter has been defined by (58).
Let us now evaluate . We first express the left-hand side of (58) in terms of as
When the modulus is close to 1, by making use of Jacobi’s imaginary transformation of sn function: and then by expressing it in terms of Jacobi’s Eta and Theta functions, we show
Through (A.30) we express the value of Zeta function in terms of as follows.
and through (53) we evaluate the chemical potential up to the order of logarithmic terms
It follows that chemical potential approaches when we send system size to infinity. We denote the limiting value by . Here we remark that the logarithmic corrections correspond to the finite size corrections since we have . Moreover, the chemical potential is larger than when the system size is finite since we have for . We can confirm such behavior by checking the numerical estimates for the parameters of classical dark solitons listed in section 4.
Asymptotic expansion of the square amplitude of the classical complex scalar field
Through Jacobi’s imaginary transformation for dn function we have
We therefore have
It is easy to show that the asymptotic expansion of the square amplitude up to the order of , i.e. the order of , satisfies the normalization condition (45).
At , the square amplitude is expressed in terms of by
Thus, the critical velocity with the infinite system size () is given by . Here we recall . It follows that we have
Here is expressed as . We have thus derived the square amplitude of one-soliton solution due to Tsuzuki .
Asymptotic expansion of the phase of the classical complex scalar field
Making use of (64) we derive the expansion of up to the order of the inverse of as follows:
Therefore, in the limit of sending to 1, we obtain the phase of the dark soliton solution by Tsuzuki :
Thus, we have shown that the classical dark soliton under the PBCs approaches the dark soliton solution by Tsuzuki  through the simultaneous limit of sending the system size to infinity and the modulus to 1.
4 Aspects of quantum states corresponding to classical solitons
4.1 Density profile of quantum and classical dark solitons
Formula of form factors of the local density operator
We define the local density operator by , in the second-quantized system of the 1D Bose gas interacting through the delta-function potentials (4). Here we denote by the field operator at the initial time : . We now consider the graph of the expectation value of the local density operator for a quantum state (i.e. ) versus the position coordinate . Here we recall that we call it the density profile of the state .
We evaluate the expectation value of the local density operator for the state by the form factor expansion. We express the expectation value as the sum over the form factors between the Bethe eigenstates
Here and are the normalized Bethe eigenstates of particles in the type II branch (i.e. they are one-hole excitations) and have total momentum and , respectively. Here the form factors are effectively calculated  by the determinant formula for the norms of Bethe eigenstates [49