Transport of a quantum particle in a time-dependent white-noise potential Free

A quantum particle in a random potential can move diffusively, ballistically, or superballistically depending on the circumstances. In this note, we derive some results about the mean square displacement of a quantum particle subject to a time-dependent white-noise Gaussian potential V_ω(x, t) that is correlated in space and uncorrelated in time. We prove that the mean square displacement is superballistic for models on $Rd$ and diffusive for models on the lattice $Zd$⁠.

We consider the Schrödinger equation with a time-dependent potential. Due to the singular nature of the potential, the Schrödinger equation is a stochastic differential equation, as we describe in  Appendix A. Here, however, we proceed formally and write the single-particle Schrödinger equation as

where x is in $Rd$ (resp. $Zd$⁠) and the operator −Δ is the Laplacian on $Rd$ (resp. discrete Laplacian on $Zd$⁠). The potential V_ω(x, t) is a mean zero Gaussian stochastic process with covariance

where the strength of the disorder is V₀ > 0, and the spatial correlation function $g∈C2(Rd;R)$ is a real, even function with sufficiently rapid decay at infinity. We assume the physically reasonable convexity condition that |(∇g) (0)| = 0 and that the Hessian matrix Hess(g) (0) is negative definite. The angular brackets in (2) denote averaging with respect to the Gaussian probability measure.

Generalizing from the single particle wave function ψ, the evolution of a density matrix $ρ∈I$⁠, where $I$ is the ideal of trace class operators, is formally governed by the quantum stochastic Liouville equation

and an initial condition $ρ(t=0)=ρ0∈I$⁠, a non-negative trace class operator. We assume that the density matrix $ρ∈I$ has a kernel ρ(x′, x, t). It follows from (3) that the kernel ⟨ρ(x′, x, t)⟩ formally satisfies the quantum stochastic Liouville equation

with initial condition ρ(x′, x, 0) corresponding to $ρ(t=0)=ρ0∈I$⁠. The generator L of the time evolution of the kernel ρ(x′, x, t) appearing in (4), called the Liouvillian, is given by

For example, if ρ is a pure state density matrix ρ = P_ψ, where P_ψ projects onto the normalized state ψ, the kernel $ρ(x′,x,t)=ψ¯(x′,t)ψ(x,t)$⁠.

In general, we will consider kernels ρ(x′, x, t) of density matrices ρ solving the stochastic Liouville equation (4) with initial kernels ρ₀(x′, x), corresponding to a non-negative trace-class operators ρ₀. We assume that ρ₀ has a well-behaved kernel so that $⟨‖x‖2⟩ρ0<∞$⁠. The solution ρ_t has a well-behaved kernel ρ(x′, x, t) so that $Trρt=∫Rdρ(x,x,t)dx<∞$ for all $t∈R$⁠. We set the disorder λ = 1. We denote by ⟨A⟩_ρ(t) ≔ ⟨Tr{Aρ(t)}⟩ the average of the expectation of an observable A in the state ρ(t).

Equations (1) and (4) are stochastic partial differential equations. The product of the white-noise potential and the solution is interpreted in the Stratonovich sense. The details of the derivation of the partial differential equation for the disorder-averaged kernel ⟨ρ(x′, x, t)⟩ of the density matrix (12) are presented in  Appendixes A and  B. Our main results concern the evolution of this disorder-averaged kernel ⟨ρ(x′, x, t)⟩ that satisfies the deterministic partial differential equation (12).

Fischer, Leschke, and Müller^7,8 proved a result similar to part (1) of Theorem 1. They work in the Weyl-Wigner-Moyal formulation of quantum mechanics on phase space $Rd×Rd$⁠. Their Hamiltonians have the form H(p, q) + N_t(p, q), where the white-noise potential N_t(p, q) satisfies a covariance relation similar to (7) where the covariance function g depends on p as well as q. For the case H(p, q) = p²/(2m), they prove that $q(t)∼t32$⁠; see Eq. (2.60e) of Ref. 7. We remark that in a related paper, Fischer, Leschke, and Müller⁸ state if one adds an external magnetic field to the model on $R2$⁠, then the superballistic motion is suppressed and one recovers diffusive motion. This is proved by Müller in his thesis (Beispiel 2.36 of Ref. 17).

Concerning lattice models, Ovchinnikov and Érikhman¹⁹ proved a result similar to part (2) of Theorem 1 for one-dimensional linear random chains using a Laplace transform method similar to that used later by Jayannavar and Kumar.¹³ We discuss their work further at the end of Sec. III. As mentioned in the abstract, our results for the lattice are related to the work of Kang and Schenker¹⁴ who proved diffusive motion for a quantum particle on the lattice under the influence of a time-dependent Markovian potential. Their work utilizes the framework developed by Pillet.²⁰ 

Our methods are very different from those of Refs. 7 and 8, and generalize many results of Ref. 19 to all dimensions. Furthermore, our methods are applicable to both the continuum and lattice models.

There has been some discussion in the physics literature concerning models for which the delta correlation in time in (2) is replaced by a more general function h(t − t′), the so-called colored noise (see  Appendixes A and  B). Golubović, Feng, and Zeng¹¹ studied Gaussian random variables with a covariance

where ℓ and τ are effective spatial and temporal correlation lengths. Beginning with the Schrödinger equation on $Rd$⁠, they derive an effective Fokker-Planck equation for the velocity distribution. Their main result is that temporal correlations cause superballistic behavior but with different exponents than in Theorem 1. Golubović, Feng, and Zeng¹¹ showed that the mean square displacement is superballistic: for d = 1, it is $t125$⁠, while for d > 1, it is $t94$⁠. Rosenbluth,²¹ disagreeing with the derivation of the Fokker-Planck equation in Ref. 11, derived another equation and used it to show that the mean square displacement for d = 1 is $t125$⁠, while for d > 1, it is t², ballistic motion.

The subject of stochastic acceleration for classical systems has been frequently discussed in the literature. For example, in the work of Aguer et al.,¹ the authors present theoretical and numerical results that indicate, among other results, that the mean square displacement in a space and time homogeneous random field with rapid decay in space but not necessarily in time, for which the force field is not a gradient field, is superballistic with $⟨‖q(t)‖2⟩∼t83$ for dimensions d ⩾ 1. When the force field is a gradient field, the motion is superballistic only in dimension one, and ballistic for d ⩾ 2. The models include the inelastic nondissipative soft Lorentz gas. In Soret and De Bièvre,²² the authors treat a simplified random, inelastic, Lorentz gas model and prove, roughly speaking, that the average velocity $⟨‖q′(t)‖⟩∼t15$ for dimensions d ⩾ 5, provided the initial velocity is sufficiently large. The temporal exponent for this superballistic motion differs from the one in Theorem 1. This may be attributed to the fact that the model describes a classical particle moving at high velocity under the influence of a time-dependent potential with a finite temporal correlation length, similar to the results for colored noise described above.

In Sec. II, we prove superballistic motion for the model on $Rd$ extending the Laplace transform method introduced by Jayannavar and Kumar¹³ for one-dimensional models. After reviewing this approach, we extend it to multidimensional models using the method of characteristics. We then discuss in Sec. III the slowing effect of the lattice $Zd$⁠. The proof of diffusive motion on the lattice requires modification of the Laplace transform technique. We conclude with two appendices. In  Appendix A, we discuss the Stratonovich interpretation of the stochastic differential equation, and in  Appendix B, we present a calculation on Gaussian correlations.

Jayannavar and Kumar¹³ studied the evolution of a pure state density matrix in one dimension. They chose a Gaussian initial state $ψ(x,0)=Cσe−x2/4σ2$ so that ρ₀(x′, x) = ψ(x′, 0)ψ(x, 0). The solution ρ(x′, x, t) satisfies the stochastic Liouville equation (4) and (5). Averaging over the Gaussian disorder, Jayannavar and Kumar¹³ solved the resulting equation for the averaged density kernel ⟨ρ(x′, x, t)⟩. We first review the construction of the solution in one-dimension. After a review of the method of characteristics in Sec. II B 1, we construct an explicit solution for the averaged kernel of the density matrix in any dimension for any initial density matrix, not necessarily a pure state. We prove that the quantum motion behaves like $⟨‖x(t)‖⟩∼t32$ in any dimension d on $Rd$⁠. This exponent $32$ is independent of the dimension and appears to be independent of the type of disorder.

• H1:

  The potential V_ω(x, t) is a mean zero Gaussian stochastic process ⟨V(x, t)⟩ = 0 with covariance

• H2:

  The strength of the potential is V₀ > 0.

• H3:

  The spatial correlation function $g∈C2(Rd,R)$ is assumed to be a real, even function with sufficiently rapid decay at infinity. It satisfies the physically reasonable convexity condition that |(∇g)(0)| = 0 and that Hess(g)(0) is negative definite.

By sufficiently rapid decay in [H3], we need that, at a minimum, g and its first and second partial derivatives belong to $L1(Rd)$⁠. Since a correlation function must be the Fourier transform of a positive measure, the reality of g forces it to be an even function. A common example of a correlation function g satisfying [H3] is a Gaussian function $g(x)=e−∑i,j=1dAijxixj$ for a positive definite matrix A.

For a random variable K, we denote by ⟨K⟩ the expectation with respect to the Gaussian process.

Since the Schrödinger equation (1) and the Liouville equations (4) and (5) with a white-noise potential (7) are stochastic partial differential equations, the singular term V_ω(x, t)ψ(x, t) requires some interpretation. The Gaussian nature of the process, however, allows us to easily derive a partial differential equation for the averaged density ρ.

The basic approach of Jayannavar and Kumar¹³ is as follows. If ψ(x, t) satisfies the Schrödinger equation

then the pure state density matrix ρ(x′, x, t), given by $ρ(x′,x,t)=ψ¯(x′,t)ψ(x,t)$⁠, satisfies the equation

the Liouvillian equations (4) and (5). In general, we now assume that ρ(x′, x, t) is the kernel of a density matrix that solves the Liouville equation (3) so that the kernel satisfies (9) with an initial condition ρ₀(x′, x). We normalize the initial conditions so that $Trρ0=∫Rdρ0(x,x)dx=1$⁠. We will always assume that the diagonal of the kernel of the initial density matrix ρ₀ satisfies

in the continuum case, or, for the $Zd$ case,

Taking the expectation of (9), we obtain

We next use Novikov’s Theorem¹⁸ to compute the average ⟨V_ω(z, t)ρ(x′, x, t)⟩, where z denotes x or x′, with respect to the Gaussian random variables under the covariance assumption (7). From  Appendix B, we obtain

Using this result, we find that (10) may be written as

We introduce new variables X ≔ x + x′ and Y ≔ x − x′ into Eq. (9). We write R(X, Y, t) for ⟨ρ(x′, x, t)⟩. The Laplace transform of R(X, Y, t) with respect to t is

Using the initial condition R(X, Y, 0) ≔ ⟨ρ(x′, x, 0)⟩ and defining

we obtain

We define the Fourier transform of the density matrix with respect to X

Taking the Fourier transform of Eq. (15) with respect to X, we obtain

We will solve this equation for $R̃^(k,Y,s)$ in one dimension in Sec. II A and in any dimension in Sec. II B. Setting Y = 0, this yields the Laplace transform of the function $R̃(k,0,t)$⁠. Combining (16) and the fact that R(X, 0, t) = ⟨ ρ(x, x, t), we find that the second moments of the position vector x are given by

The mean square displacement is given by

In Sec. III, we will prove that quantum transport with the white-noise potential on $Zd$ is diffusive. The diffusion coefficient matrix [d_ij] is defined by

and the diffusion constant D is the trace of this matrix

In the one-dimensional case, Eq. (15) for $R̃(X,Y,s)$ agrees with Eq. (10) of Ref. 13. We now take the Fourier transform with respect to X. This results in the ordinary differential equation

We integrate this equation with the boundary condition $R̃^(k,Y=b,s)=0$ and obtain

Taking b = −∞ imposes the physically reasonable boundary condition $limY→−∞R̃^(k,Y,s)=0$ corresponding to the decay of the kernel of the density matrix. We now take Y = 0. Using the definition of h in (14) and changing variables with $z̃≔−(m/(2ℏk))z$⁠, we obtain

The integral in (24) is in the form of a Laplace transform, so

Note that one does not have to specify the spatial correlation function g or the initial density matrix ρ₀.

As follows from (19), the second moment of the position operator is calculated from two derivatives with respect to k of $R^$

We define the phase function in (25) to be

and note that Φ(0, t) = Φ′(0, t) = 0 due to [H3]. In terms of the phase function, we have

The computation of the second derivative yields

The crucial part of the calculation that gives the leading behavior in t is the second derivative of the phase function in (27)

where integration over $w$ ∈ [0, t] gives the t³ term. This shows that

We note that the evenness of g is important here: If g′(0) ≠ 0 then the term

is g′(0)²t⁴ and dominates the behavior of the second moment.

If the random potential vanishes, the correlation function g = 0 and the phase Φ = 0. We see that the last term on the right of (28) behaves like t² so the motion is ballistic.

A similar approach may be taken in order to compute the mean square displacement on $Rd$ in any dimension. The additional component required for this is the solution of a nonhomogeneous transport equation.

In this section, we review the method of characteristics for a semilinear transport equation

In our case, the function c(x, u) on the right of (31) has the form c(x, u) = −h(x)u(x) + m(x). The characteristic equations for the pair (x(s), z(s)) are as follows:

and

The first equation integrates to give x(s) = ks + k₀, with $k,k0∈Rd$⁠, and $s∈R$⁠. With this solution x(s), the first-order ordinary differential equation (33) for z(s) becomes,

and the solution z(s) has the form

with the boundary condition z(b) = 0 at b that is determined by the problem. We recall that z(s) = u(x(s)) solves the original Eq. (31).

We apply the method of characteristics to the semilinear transport equation (17). We let (Y (⁠$v$⁠), z(⁠$v$⁠)) be the solutions of the corresponding characteristic equations (32) and (33) so that $z(v)=R̃^(k,Y(v),s)$⁠. The function Y (⁠$v$⁠) = k$v$ + k₀, for $w∈R$ and $k,k0∈Rd$⁠, solves the characteristic equation (32). The function z(⁠$v$⁠) solves the second Eq. (34)

where, as before in (14), we define

As in (35), the solution to (36) is

with the boundary condition $R̃^(k,Y(−∞),s)=0$ as described below (23).

Following the reductions used in the one-dimensional case, we re-express the integral in (37) as a Laplace transform. We finally obtain

We now compute the second moment of the position operator following (19). The integration constant k₀, appearing in the solution of the characteristic equation (32), is set equal to zero. As in (27), we define the phase function Φ(k, t) as

In analogy with (25), the crucial computation of the derivative with respect to k relies on the fact that Y(⁠$v$⁠) = $v$k, $v∈R$⁠. Consequently, we have a term identical to (29) which gives the superballistic behavior (28). In analogy with the calculation (28), we find

In light of (19), this yields

The motion of a quantum particle restricted to a square lattice $Zd$ differs from the results in Sec. II primarily due to the fact that the lattice Laplacian is a bounded operator. The Gaussian white-noise potential results in diffusive motion on the lattice. This is in keeping with the result of Kang and Schenker.¹⁴ They studied a similar problem on $Zd$ for which the potential V_ω(x, t) is a Markovian potential. They proved that the motion is diffusive. We now turn to the proof of the second part of Theorem 1.

To prove part (2) of Theorem 1, we will pursue the same basic strategy as in Sec. II, except an exact solution for the Laplace transform as in (24) is no longer possible. Instead, we will prove that

for C₀ > 0 depending on V₀ and g. Condition (42) is equivalent to $⟨‖x‖2⟩ρt∼t$ indicating diffusive motion. We will use below the fact that the expressions in (42) are real.

Let {ê_j} be the standard orthonormal basis of $Zd$⁠. The discrete Laplacian acting on a function f at site $x∈Zd$ sums f over all 2d nearest neighbors to x. We write ${f^j}j=12d$ for the 2d nearest neighbor directions at the origin so that $f^j=êj$⁠, for j = 1, …, d and $f^d+j=−êj$⁠, for j = 1, …, d. With this notation, the discrete Laplacian Δ on $ℓ2(Zd)$ is the finite-difference operator

The Laplacian is normalized so its spectrum is [−2d, 2d]. The Laplacian may be factored via directional derivatives $∇i±$ defined by

These two finite-difference operators commute. The adjoint of $∇i±$ is $∇i∓$⁠. In terms of these, the discrete Laplacian (43) may be written as

It is convenient to introduce new variables X = x + x′ and Y = x − x′. In terms of these variables, we obtain

As mentioned above, we write R(X, Y, t) for the averaged density ⟨ρ(x′, x, t)⟩. We then obtain from the fundamental equation (12) the equation for R(X, Y, t)

We write $R̃(X,Y,s)$ for the Laplace transform of R(X, Y, t) with respect to t. Taking the Laplace transform of (47) we obtain

where R(X, Y, 0) = ⟨ρ₀(x, x′)⟩ and $h(Y,s)≔s+V022g(0)−g(Y)$⁠.

We next take the Fourier transform with respect to X,

The Fourier transform of the differential operator term on the left in (48) may be written as

By means of (50), the Fourier transform of (48) with respect to X is