Quantum dots as a probe of fundamental physics: deviation from exponential decay law

Abstract

We explore a possibility of measuring deviation from the exponential decay law in pure quantum systems. The power law behavior at late times of decay time profile is predicted in quantum mechanics and has been experimentally attempted to detect, but with failures except a claim in an open system. It is found that electron tunneling from resonance state confined in man-made atoms, quantum dots, has a good chance of detecting the deviation and testing theoretical predictions. How initial unstable state is prepared influences greatly the time profile of decay law, and this can be used to set the onset time of the power law at earlier times. Comparison with similar process of nuclear alpha decay to discover the deviation is discussed, to explain why there exists a difficulty in this case.

Graphic Abstract

Appendices

Appendix A: Exact solutions of the Schrodinger equation for parabolic potential

The stationary Schrodinger equation of energy E in a parabolic potential

$$\begin{aligned}&- \frac{1}{2m} \frac{d^2 \psi }{d x^2} + \left( V(x) - E \right) \psi = 0 \,, \nonumber \\&V_2(x) = V_1 - \frac{ \left| V_0'' \right| }{2} x^2 \,, \end{aligned}$$

(A1)

with $V_1 > 0$, may be transformed to Weber’s differential equation,

$$\begin{aligned}&\frac{d^2 w}{d z^2} + (\lambda + \frac{1}{2} - \frac{z^2}{4} ) w = 0 \,, \end{aligned}$$

(A2)

by a change of coordinate variable and their parameters,

$$\begin{aligned}&z = \alpha x \,, \alpha = e^{ i \pi /4} (m \left| V_0'' \right| )^{1/4} \,, \nonumber \\&\lambda = - \frac{1}{2} - 2i \left( \frac{ m}{ \left| V_0'' \right| } \right) ^{1/2} (V_1 - E) \,. \end{aligned}$$

(A3)

Classical turning points for electron moving with energy $E < V_1$ are at

$$\begin{aligned}&\pm a \,, a = \sqrt{\frac{ 2(V_1 - E)}{ \left| V_0'' \right| }} \,. \end{aligned}$$

(A4)

Two independent solutions of Weber’s differential equations are Weber’s functions denoted by $D_{\lambda }(z)$ and $D_{-\lambda - 1}(iz)$ [14]. These functions can be written in terms of linear combinations of confluent hypergeometric functions $F(\alpha , \gamma ; z^2/2)$ of parameters, $(\alpha , \gamma ) = (-\lambda /2, 1/2)\,, (1/2 - \lambda /2, 3/2)$.

Using general formulas of asymptotic expansion for confluent hypergeometric functions, one confirms that a specific linear combination of two fundamental solutions gives the boundary condition of no incoming wave at the far right. This solution at $x>a$ is given by, disregarding normalization,

$$\begin{aligned} \psi (x; E)= & {} \frac{\sqrt{2\pi }e^{i\pi /4} }{\Gamma ( \frac{1}{2} - i a_u^2) } e^{- \frac{\pi }{2} a_u^2}\, D_{ - \frac{1}{2} - i a_u^2} ( e^{i \pi /4}\, u ) \nonumber \\&- D_{ - \frac{1}{2} + i a_u^2} ( e^{ i 3 \pi /4}\, u) \,, \end{aligned}$$

(A5)

and its leading term at $x \gg a$ is

$$\begin{aligned}&\psi _R (x; E) = \frac{- i }{\left( ( m \left| V_0'' \right| )^{1/4} x \right) ^{1/2} } \frac{e^{-\pi a_u^2} }{(u^2 - a_u^2)^{1/4} } \nonumber \\& \cdot \exp [ \frac{i}{2} \left( u \sqrt{u^2- a_u^2} + a_u^2 \ln u^2 \right) ] \,, \end{aligned}$$

(A6)

$$\begin{aligned}&a_u = (mV_0'')^{1/4} a = (\frac{m}{ \left| V_0'' \right| })^{1/4} \sqrt{2 (V_1 - E)} \,, \nonumber \\&u = (m \left| V_0'' \right| )^{1/4} x \,. \end{aligned}$$

(A7)

The formula $e^{-\mathcal{P}_b}\,, \mathcal{P}_b = \pi ( m \left| V_0'' \right| )^{1/2} a^2$ may be regarded as an effective barrier penetration factor for the parabolic potential. On the other hand, the leading term at $x \ll -a$ is given by

$$\begin{aligned}&\frac{2 i e^{i\pi /4} }{\left( ( m \left| V_0'' \right| )^{1/4} \left| x\right| \right) ^{1/2} }\nonumber \\&\quad \cdot \cos \left( \frac{1}{2} ( u \sqrt{u^2- a_u^2} + a_u^2 \ln u^2 ) + \frac{\pi }{4} \right) \,. \end{aligned}$$

(A8)

Using the asymptotic formula of exact solution, one may estimate the probability at turning points, $x = \pm \sqrt{ 2(V_1-E)/ \left| V_0'' \right| } \equiv \pm a$, to give

$$\begin{aligned}&\frac{1}{ (m \left| V_0'' \right| )^{1/4} a} \exp [- 2\pi (m \left| V_0'' \right| )^{1/2} a^2 ]\nonumber \\&\quad = \sqrt{\frac{\pi }{ \mathcal{P}_b}} \exp [ - 2 \mathcal{P}_b] \,. \end{aligned}$$

(A9)

In Fig. 9, we illustrate exact solution given by Weber’s function, which shows that the left side sinusoidal wave goes out of a parabolic potential to a right-moving component alone. The penetration factor $\exp [- \pi (m \left| V_0'' \right| )^{1/2} a^2/2 ] = 0.21$ is only modestly small in this example.

Fig. 9

Solution given by Weber’s function for $a_u = (m \left| V_0'' \right| )^{1/4} a = 2$: absolute value in sold black, imaginary part in dashed red, and real part in dashed–dotted blue

Although exact solutions are valuable, its mathematical complexity is often demanding for its full understanding. The semi-classical or WKB approximation is intuitively appealing, giving a clear relation to classical behaviors. The wave functions in classically allowed region of $\left| x \right| > a$ are described by linear combinations of running waves; at $x>a$

$$\begin{aligned}&\frac{1}{\sqrt{ 2m \left( E - V_2(x) \right) }}\nonumber \\&\quad \cdot \exp [\pm i \int _a^x dy\, \sqrt{ 2m \left( E - V_2(x) \right) } ] \,. \end{aligned}$$

(A10)

We impose the boundary condition of no incoming wave from the far right, which chooses one of these waves. Connection passing turning points to potential region under the parabolic barrier and then to the left of $x < -a$ can be carried out, following the method of complex coordinate [11]. This connection gives wave function in the far left region of the form similar to Eq. (A8), except the phase term $\propto \ln u^2 \propto \ln x^2$. In Fig. 10, we compare thus modified semi-classical result relative to exact result.

Fig. 10

Ratio of exact solution to the semi-classical result in the case of $a_u = (mV_0'')^{1/4} a = 2$: absolute value in solid black, imaginary part in dashed red, real part in dashed–dotted blue, and the semi-classical result without $\propto \ln x^2$ phase factor in dotted black

Appendix B: Problem of partially linear potential

We consider a partially linear potential as depicted in Fig. 11. It is a $x\leftrightarrow -x$ symmetric potential of the form,

$$\begin{aligned}&V_1(x) = V_1^+(x) + V_1^+(-x) \,, \nonumber \\&V_1^+(x) = - V_0 \theta (a - x ) \theta (x) \nonumber \\& + \left( V_1 - \left| V_a' \right| (x - a) \right) \theta ( x-a ) \theta (b- x) \theta (x) \,, \qquad \end{aligned}$$

(B11)

with $V_0\,, V_1$ taken positive. This potential has an aspect of cusp structure (at $x= \pm a$) common to Gamow potential often used in discussion of nuclear $\alpha$ decay. The cusp in Gamow model occurs at nuclear radius.

Fig. 11

Example of potential given by eq.(B11) for the case of $a = 5$nm, $V_0 = 20\,, V_1= 20$ eV’s, and $V_a' = 20\mathrm{eV}/ 2\mathrm{nm}$

Consider the right half $x>0$ of this potential. Resonance solution is characterized by no incoming wave boundary condition from $x>b$, proportional to $e^{i k x}\,, k > 0$. Such a solution is given by a linear combination of two Airy functions, $A_i(z)\,,B_i(z)$: at $a < x \le b$:

$$\begin{aligned} C_i(x)= & {} N \left( A_i (\zeta (x)\,) + i B_i (\zeta (x)\, ) \right) \,, \nonumber \\ \zeta (x)= & {} \left( \frac{2}{3} \frac{\sqrt{2m} }{ \left| V'_a \right| } \right) ^{2/3}\, \left( E - V_1 + \left| V_a' \right| (x - a) \right) \,, \nonumber \\ \end{aligned}$$

(B12)

with N a normalization constant. This combination of Airy functions has asymptotic behavior given by

$$\begin{aligned}&C_i(x) \propto x^{-1/4} \exp [i \sqrt{2m \left| V_a' \right| }\, x^{3/2} ] \,. \end{aligned}$$

(B13)

As shown in [11], this wave function has a constant flux despite of a fast phase variation $\propto x^{3/2}$.

At a far right of $x > b$, the function $C_i(x)$ is connected to outgoing plane-wave D(x) given by

$$\begin{aligned}&D(x) = C_i(b) \exp [i \sqrt{2m E} (x-b)] \,. \end{aligned}$$

(B14)

Within the well at $x < a$, the wave function is given by sinusoidal function, either a cosine or sine function depending on parity of states, of variable $\sqrt{2m (E+V_0)}\, x$. Connection of wave functions at turning points $x= a$ requires matching of logarithmic derivatives:

$$\begin{aligned}&\sqrt{2m (E+V_0)} \cdot \left( - \tan (\sqrt{2m (E+V_0)}\,a ) \,, \;\right. \nonumber \\&\quad \left. \mathrm{or}\; \cot (\sqrt{2m (E+V_0)}\,a ) \right) \nonumber \\& = \left( \frac{2}{3} \frac{\sqrt{2m} }{ \left| V'_a \right| } \right) ^{2/3}\, \left| V'_a \right| \left( A_i' (\zeta (a)\,) + i B_i' (\zeta (a)\,) \right) \,, \nonumber \\ \end{aligned}$$

(B15)

two respective equations corresponding to parities of states. This equation determines the resonance energy $E= E_*$.

The asymptotic form of Airy functions is adequately described by semi-classical approximation [11]. The space–time evolution at late times may thus be investigated by working out the semi-classical wave function:

$$\begin{aligned}&\psi (x, t) = \int dE f(E) \frac{ c_+(E)}{ \sqrt{ k(x)}} \exp [ i \Psi (E, x) ] \,, x > b \,, \nonumber \\ \end{aligned}$$

(B16)

$$\begin{aligned}&\Psi (E, x) = W(x; E) - E t \,, W(x; E) = \int _b^x dy\, k(y) \,, \nonumber \\&k(y) = \sqrt{ 2 m \left( E - V_1(y) \right) } \,. \end{aligned}$$

(B17)

Estimate of the energy integral in this equation may be given by the saddle point method. Saddles are found by vanishing condition of the phase derivative,

$$\begin{aligned}&\left( \frac{\partial \Psi (E, x)}{ \partial E} \right) _{E=E_s}= \pm \frac{ \sqrt{2m}}{ \left| V'_a \right| } (E_s - V_1)^{1/2}\nonumber \\&\quad + \sqrt{\frac{m}{2E_s}} x - t =0 \,. \end{aligned}$$

(B18)

Time t is taken large at late times, and the saddle point energy $E_s$ is found of order, $E_s = O( t^2 \left|V'_a\right|^2/m)$. The gaussian width at the saddle necessary for the energy integral is given by

$$\begin{aligned}&\frac{\partial ^2 \Psi }{ \partial E_s^2} = \pm \frac{1}{2} \frac{ \sqrt{2m}}{ \left| V'_a \right| } (E_s - V_1)^{-1/2} - \frac{1}{4} \frac{ \sqrt{2m} }{ E_s^{1/2}}\,x \,. \nonumber \\ \end{aligned}$$

(B19)

This gaussian width $\left| \frac{\partial ^2 \Psi }{ \partial E_s^2} \right|^{-1/2}$ is in proportion to $E_s^{1/4} \propto t^{1/2}$ for large $E_s$, which implies, combined with the pre-factor $1/\sqrt{k}=1/(2m E_s)^{-1/4}$, a constant decay probability $\propto t^0$, implying that the decay process terminates at latest times after the exponential period. We regard this result inappropriate for description of decay processes. The result was confirmed by using exact Airy solutions as well.

The possibility of a constant decay probability after the exponential period has been noted in a solvable model [15]. In this particular model, the non-decay at latest times is attributed to existence of bound state which appears in a strong coupling regime beyond perturbation theory. Our case of partially linear potential model does not correspond to this case of emergent bound state. It is thus not clear under what conditions the behavior of constant decay probability at late times emerges.

Appendix C: Late time behavior of nuclear $\alpha$ decay

We clarify late-time profile of $\alpha$ decay by calculating probability amplitude based on Woods–Saxon attractive $\alpha -$nucleus potential more realistic than the Gamow potential often used in textbooks of nuclear physics. A modified potential is also necessary from result of Appendix B that led to termination of decay process.

1.1 C.1 $\alpha$-nucleus interaction based on Woods–Saxon potential

Consider interaction of $\alpha$ particle ($^4$He nucleus) with residual nucleus, consisting of attractive force inside nucleus and electrostatic force outside nucleus. We assume that proton or charge distribution inside the residual nucleus has the same form as an attractive Woods–Saxon potential $V_{WS} (r)$,

$$\begin{aligned}&V_{WS} (r) = - V_0 v(r) \,, \nonumber \\&v(r) = \frac{1}{ 1 + \exp [(r-R)/a] } = \frac{1}{2} ( 1 - \tanh \frac{r-R }{ 2a} ) \nonumber \\ \end{aligned}$$

(C20)

$$\begin{aligned}&V_0 = ( 51 - 33 \frac{N-Z }{ A} )\, \mathrm{MeV} \,, \nonumber \\&a = 0.67\, \mathrm{fm} \,, R = 1.27 A^{1/3} \, \mathrm{fm} \,. \end{aligned}$$

(C21)

This potential incorporates a thin nuclear effect instead of cusp at nuclear radius R in the Gamow potential A, Z, N are atomic number, atomic nuclear charge, and neutron number of nucleus.

Electro static potential $V_C$ between $\alpha$ and residual nucleus can be calculated from the Poisson equation,

$$\begin{aligned} \vec {\nabla }^2 V_C= & {} 2 (Z-2) e^2 N v(r) \,, \nonumber \\ N^{-1}= & {} \int d^3 r v(r) = \frac{R^3}{c} \,, \end{aligned}$$

(C22)

$$\begin{aligned} \frac{1}{c}= & {} 2\pi \int _0^{\infty } dx \, x^2 \left( 1- \tanh \frac{R}{2a} (x-1) \right) \nonumber \\&\quad = \frac{1}{0.2216} \; \mathrm{for}\; A = 212 \,, \end{aligned}$$

(C23)

$$\begin{aligned} V_C (r)= & {} \frac{ 2 (Z-2) e^2 }{4\pi } N \int d^3 \rho \frac{v(r) }{ \left| \vec {r} - \vec {\rho } \right|} \nonumber \\= & {} \frac{(Z-2) e^2 c}{R^3} \left( \frac{1}{r} \int _0^r dy\, y^2 + \int _r^{\infty } dy\, y\right) \nonumber \\&\quad \cdot \frac{1}{ 1 + e^{(y-R)/a}} \,. \end{aligned}$$

(C24)

The constant N is determined by the total charge of $\alpha -$residual nucleus. The remaining radial integral is expressed in terms of polylogarithmic functions of n-th order, $Li_n (z)$:

$$\begin{aligned}&V_C (r) = - \frac{ (Z-2) e^2\,c }{ Li_3 (- e^{R/a} )} \frac{r^2}{a^3} J(r) \,, \end{aligned}$$

(C25)

$$\begin{aligned}&(\frac{r}{a})^2 J(r) = \frac{\pi ^2}{6} + \frac{1}{2} (\frac{R}{a})^2 - \frac{1}{6} (\frac{r}{a})^2 - Li_2 ( - e^{(r-R)/a} ) \nonumber \\& + 2 \frac{r}{a} Li_3 ( - e^{(r-R)/a} ) - 2 \frac{r}{a}Li_3 ( - e^{-R/a} ) \,. \end{aligned}$$

(C26)

The total potential acting on $\alpha$ is given by

$$\begin{aligned}&V_{\alpha } (r) = V_{WS} (r) + V_C (r) \,. \end{aligned}$$

(C27)

Example of $^{212}_{84}$Po is illustrated in Fig. 12.

Fig. 12

$\alpha$ decay potential of $^{212}_{84}$Po and its components: modified Coulomb in sold black, Woods–Saxon attraction in dashed red, and their sum in dashed–dotted blue

1.2 C.2 Exponential decay law

The time profile of $\alpha -$decay viewed far away from nucleus is derived using transition amplitude $\mathcal{T}(E)$, a function of $\alpha$ energy E,

$$\begin{aligned}&\psi _{\alpha } (r, t) = \int _{-\infty }^{\infty } dE f(E) \mathcal{T} (E) e^{i k(E) ( r -R ) - i Et } \,, \nonumber \\ \end{aligned}$$

(C28)

and is dominated by barrier penetration of $\alpha$ particle. Mechanism of resonant $\alpha$ particle formation is not well understood, but the time evolution is essentially dictated by the boundary condition of no incoming wave from the outside of Coulomb repulsion. This time interval is described in a similar way to resonance formation of electronic state in man-made atoms, and one can assume the formula of Eq. (14) adapted to this potential case,

$$\begin{aligned}&\psi _{\alpha } ( x,t; E_*) \sim \frac{ f(E_*) C_+( E_*) \Delta E }{\sqrt{2\pi } \sqrt{k_*} \theta } e^{ i k_* ( x-b) - i E_* t} \nonumber \\& \cdot \exp [- \frac{\Gamma }{2} \left( t - \frac{x- b}{v_*} \right) ] \,, \end{aligned}$$

(C29)

$$\begin{aligned}&\left| \psi _{\alpha } ( x,t; E_*) \right| ^2 \simeq \frac{ \left| f(E_*) C_+( E_*) \right|^2 \Delta E^2}{k_*\, \theta ^2} \nonumber \\& \cdot \exp [ -\Gamma \left( t - \frac{x- b}{v_*} \right) ] \,, \end{aligned}$$

(C30)

$$\begin{aligned}&\theta ^{-2} = 4M R^2 \Gamma \nonumber \\& = \exp [- 2 \int _{-R}^{R} d\rho \sqrt{ 2M \left( V_{\alpha } - E \right) }] \,, \end{aligned}$$

(C31)

with $C_+( E)$ being the amplitude hitting at the barrier. Details of the potential are irrelevant to the exponential decay law, and the most important factor of barrier penetration factor $1/\theta$ is given in terms of imaginary momentum $\kappa (\rho ) = \sqrt{ 2M \left( V_{\alpha }(\rho ) - E \right) }$ integral.

1.3 C.3 Late time behavior in the semi-classical approximation and estimate of transition time

Global features of $\alpha$ decay potential are similar to potential form given in Section 4, except energy (MeV vs eV) and size (fm vs nm) scales. The decay probability formula given for electron resonance decay, Eq. (32), may be adopted in $\alpha$ decay by appropriate changes of physical quantities from atoms to nuclei.

Estimate of transition time T using $\xi = \Gamma T$ is thus given by

$$\begin{aligned}&\xi - 2 \ln \xi = \ln \frac{\pi r }{2 v(V_m) M R^2 } + 2 \ln \frac{f(Q)}{ f(V_m)} \,,\qquad \end{aligned}$$

(C32)

$$\begin{aligned}&\frac{f(Q)}{ f(V_m)} \simeq 1 + \frac{ 4 (V_m - Q)^2}{ (\Delta E)^2} \,. \end{aligned}$$

(C33)

We imagine observations of $\alpha$ particle at a fixed distance r from nuclear targets, taken 1 m in the calculation. The largest target number at transition time is estimated as $e^{-42.98} = 2.16 \times 10^{-19}$ in this example.

A few important features of Eq. (C32) are

• (1) independence of both Q and $\Gamma$, which suggests that lifetime of order 1 sec to 1 hour may be of experimental interest.

• (2) choice of measurement distance r: distance closer to target region is favored. Additional dependence away from 10 m is $\ln (r/ 10 \mathrm{m})$. At $r= 10$cm, $\Gamma T =40.56 \,, e^{-\Gamma T} \sim 2.4 \times 10^{-18}$.

• (3) atomic mass number dependence $- (2/3)\ln A$, which favors large mass number

Experimental efforts of searching for deviation from the exponential decay have been most extensively focused on nuclear beta and alpha decays. Record search time is $45 T_{1/2}$ in nuclear beta decay [5]. and $40 T_{1/2}$ in nuclear alpha decay [4] without any positive result for the deviation. There are reasons why the search has met difficulties in these cases.