How to extract weak values from a mesoscopic electronic system

Abstract

Weak value (WV) protocols may lead to extreme expectation values that are larger than the corresponding orthodox expectation values. Recent works have proposed to implement this concept in nano-scale electronic systems. Here we address the issue of how one calibrates the setup in question, maximizes the measurement’s sensitivity, and extracts distinctly large WVs. Our concrete setup consists of two Mach–Zehnder interferometers: the “system” and the “detector”. Such setups have already been implemented in experiment.

Appendices

Appendix 1: Explicit expressions for electronic trajectories

1.1 A.1: Solution to the single-particle Hamiltonian

Here we solve the single-particle problem for a single MZI (suppose MZI$_\text {sys}$ from Fig. 2) and expand the solution in partial electronic trajectories (cf. Fig. 3). We begin with the single-particle Schrödinger equation

$$\begin{aligned} \left( i\hbar \frac{\partial }{\partial t}-\hat{H}\right) {\Psi }=0, \end{aligned}$$

(13)

where ${\Psi }=\begin{pmatrix}{\psi }_1(x,t)\\ {\psi }_2(x,t)\\ \end{pmatrix}$, and

$$\begin{aligned} \hat{H}{\Psi }=\begin{pmatrix}iv\partial _x{\psi }_1(x,t)+\sum _{{\alpha }=A,B}\left[ \frac{{\Gamma }_{\alpha }^*}{2}{\delta }(x-x_{1{\alpha }}^+){\psi }_2(x_{1{\alpha }}^-,t)+\frac{{\Gamma }_{\alpha }^*}{2}{\delta }(x-x_{1{\alpha }}^-){\psi }_2(x_{1{\alpha }}^+,t)\right] \\ iv\partial _x{\psi }_2(x,t)+\sum _{{\alpha }=A,B}\left[ \frac{{\Gamma }_{\alpha }}{2}{\delta }(x-x_{2{\alpha }}^+){\psi }_1(x_{2{\alpha }}^-,t)+\frac{{\Gamma }_{\alpha }}{2}{\delta }(x-x_{2{\alpha }}^-){\psi }_1(x_{2{\alpha }}^+,t)\right] \\ \end{pmatrix}. \end{aligned}$$

(14)

Here ${\psi }_1(x,t)$ and ${\psi }_2(x,t)$ denote the wavefunctions in the corresponding arms 1 and 2, ${\Gamma }_{\alpha }$ represents the tunneling term associated with the ${\alpha }$th QPC, connecting points $x_{1{\alpha }}$ and $x_{2{\alpha }}$, ${\alpha }=A,B$ (cf. Fig. 6); $x^{\pm }= \lim _{{\varepsilon }\rightarrow 0}x\pm {\varepsilon }$. ${\Gamma }_{\alpha }$ may be related to the scattering amplitudes through Eqs. (16a), (16b).

This problem is diagonal in the scattering basis

$$\begin{aligned} {\Psi }_{k,l}(x,t)=\frac{1}{\sqrt{L}}e^{ik(x-vt)}\left\{ \begin{array}{lc} \mathbf {{\nu }}_l,&{}\quad x\in \text {I}\\ \breve{S}_A\mathbf {{\nu }}_l,&{}\quad x\in \text {II}\\ \breve{S}_B\mathbf {{\nu }}_l,&{}\quad x\in \text {III} \\ \end{array}\right. \end{aligned}$$

(15)

Here the Latin numerals denote the various sectors of the MZI: I–left to QPC A, II–between QPC A and B, III–right of QPC B, $\breve{S}_{\alpha }=\begin{pmatrix}r_{\alpha }&{}-t_{\alpha }^*\\ t_{\alpha }&{}r_{\alpha }\\ \end{pmatrix}$ is the scattering matrix at QPC ${\alpha }$ and the scattering amplitudes are

$$\begin{aligned} r_{\alpha }= & {} \frac{(2v^2)-\left| {\Gamma }_{\alpha }\right| ^2}{(2v^2)+\left| {\Gamma }_{\alpha }\right| ^2}\end{aligned}$$

(16a)

$$\begin{aligned} t_{\alpha }= & {} \frac{4iv{\Gamma }_{\alpha }}{(2v^2)+\left| {\Gamma }_{\alpha }\right| ^2}, \end{aligned}$$

(16b)

for the symmetric MZI case ($x_{1B}-x_{1A}=x_{2B}-x_{2A}$). The index $l=1,2$ denotes two orthogonal solutions $\mathbf {{\nu }}_1=\begin{pmatrix}1\\ 0\\ \end{pmatrix}$ and $\mathbf {{\nu }}_2=\begin{pmatrix}0\\ 1\\ \end{pmatrix}$ that correspond to the scattering state incident from S1 or S2.

1.2 A.2: Explicit expressions for the coefficients $c_i$

The probability of the particle incident from S1 to be detected in D1 can be presented using the path integral formalism as

$$\begin{aligned} P_{S1\rightarrow D1}=\left| \int _\mathcal {C}\mathcal {D}{\Psi }e^{iS\left\{ {\Psi }\right\} /\hbar }\right| ^2, \end{aligned}$$

(17)

where $\mathcal {C}$ represents all the trajectories from S1 to D1, and $c_i{\triangleq }e^{iS\left\{ {\Psi }_i\right\} /\hbar }$ is the weight of the corresponding trajectory. The same argument may be repeated for probabilities $P_{S1\rightarrow D2}$, $P_{S2\rightarrow D1}$, and $P_{S2\rightarrow D2}$ to include all the trajectories depicted in Fig. 3. The explicit expressions for the trajectory weights, $c_i$, may be found from the exact solution, Eq. (15). Here we summarize the results. Up to unimportant orbital phases

$$\begin{aligned} c_1^{\mathrm{sys}}= & {} -t_A^*r_B\end{aligned}$$

(18a)

$$\begin{aligned} c_2^{\mathrm{sys}}= & {} r_A r_B\end{aligned}$$

(18b)

$$\begin{aligned} c_3^{\mathrm{sys}}= & {} -r_A t_B^*\end{aligned}$$

(18c)

$$\begin{aligned} c_4^{\mathrm{sys}}= & {} -t_A^*t_B\end{aligned}$$

(18d)

$$\begin{aligned} c_1^{\mathrm{det}}= & {} -t_C t_D^*\end{aligned}$$

(18e)

$$\begin{aligned} c_2^{\mathrm{det}}= & {} r_C t_D\end{aligned}$$

(18f)

$$\begin{aligned} c_3^{\mathrm{det}}= & {} r_C r_D\end{aligned}$$

(18g)

$$\begin{aligned} c_4^{\mathrm{det}}= & {} t_C r_D. \end{aligned}$$

(18h)

Fig. 6

(Color online) Schematics of an MZI. The points $x_{1{\alpha }}$ and $x_{2{\alpha }}$, ${\alpha }=A,B$ at QPCs A and B are connected by the tunneling terms ${\Gamma }_{\alpha }$. The Latin numerals denote the various sectors of the MZI: I left to QPC A, II between QPC A and B, III right of QPC B

Appendix 2: Derivation of expectation values and correlators

Here we derive explicit expressions for expectation values of operators that appear throughout the manuscript. For simplicity we assume a symmetric MZI ($L_1=L_2$, $L_3=L_4$) operating in the low frequency, zero temperature regime, where the quantum state does not vary much over the time of the experiment, hence is almost steady. Thus, in this regime, all quantities are essentially time independent. We evaluate the expectation values by computing a trace of the operator with respect to the initial density matrix, ${\rho }_i=\left| {s1,s4\rangle \langle s1,s4} \right|$ describing two particles, which are taken from out-of equilibrium distribution,and which are incident from the biased sources S1 and S4 (cf. Fig. 2). At the end of this appendix (cf. Section B.4) we add a contribution from the equilibrium, background sea of electrons below the Fermi level. Throughout this section, the charge operators associated with each MZI are normalized to have no physical units. The latter may be recovered at the end, multiplying the normalized charge by $\left\langle Q_i\right\rangle _0$ ($i=2,3$) [those are defined following Eqs. (6) and (7)].

1.1 B.1: Derivation of Eqs. (6) and (7)

Writing explicitly the expectation values appearing in Eq. (1), we come up with the expression

$$\begin{aligned} \left\langle Q_2\right\rangle _{\mathrm{WV}}=\frac{\left<{S1,S4}\left| {Q_2 I_{D1}}\right| {S1,S4}\right>}{\left<{S1,S4}\left| {I_{D1}}\right| {S1,S4}\right>}, \end{aligned}$$

(19)

where $I_{D1}=\left| {D1\rangle \langle D1} \right|$. To leading order in the coupling, ${\lambda }$ [cf. Eq. (5)], the two MZIs may be considered as decoupled, and the trace over MZI$_\text {det}$states ($\left| {S4\rangle \langle S4} \right|$) is trivial. Hence the expression reads

$$\begin{aligned} \left\langle Q_2\right\rangle _{\mathrm{WV}}=\frac{\left<{S1}\left| {Q_2}\right| {D1}\right>}{\left<\left. {S1}\right| {D1}\right>}. \end{aligned}$$

(20)

The operator $Q_2$ is proportional to a projection operator that selects only partial wavepackets $|{\psi }_{\mathrm{sys}}^1\rangle$ and $|{\psi }_{\mathrm{sys}}^4\rangle$ (cf. Fig. 3). Only $|{\psi }_{\mathrm{sys}}^1\rangle$ has support at D1 and contributes to the numerator. The denominator includes two contributions, $|{\psi }_{\mathrm{sys}}^2\rangle$ and $|{\psi }_{\mathrm{sys}}^4\rangle$. It follows that

$$\begin{aligned} \left\langle Q_2\right\rangle _{\mathrm{WV}}=\frac{c^{\mathrm{sys}}_4}{c^{\mathrm{sys}}_4+c^{\mathrm{sys}}_2}. \end{aligned}$$

(21)

The derivation of an expression for $\left\langle Q_3\right\rangle _{\mathrm{WV}}{\triangleq }\frac{\left\langle Q_3\cdot I_{D4}\right\rangle }{\left\langle I_{D4}\right\rangle }$ [Eq. (7)] is similar.

1.2 B.2: Expression for the system–detector current correlator

Here we derive the expression for $\left\langle I_{D1}I_{D4}\right\rangle$ to leading order in the coupling ${\lambda }$. The expression for the non-equilibrium currents correlator reads

$$\begin{aligned} \left\langle I_{D1}I_{D4}\right\rangle =\left<{S1,S4}\left| {I_{D1}I_{D4}}\right| {S1,S4}\right>. \end{aligned}$$

(22)

We plug in the explicit expressions for the currents, ${I_{D1}} = |D1\rangle \,\langle D1|\,$ and ${I_{D4}} = |D4\rangle \,\langle D4|\,$ to obtain,

$$\begin{aligned} \left\langle I_{D1}I_{D4}\right\rangle =\left| \left<\left. {S1,S4}\right| {D1,D4}\right>\right| ^2. \end{aligned}$$

(23)

We note that due to charge conservation and the assumption of a steady state, the sum of the currents on the two arms (between the QPCs) of any MZI, is equal to the total current that flows into the MZI. For example, for MZI$_\text {sys}$,

$$\begin{aligned} I_{S1}=I_1+I_2, \end{aligned}$$

(24)

where $I_{S1}$ denotes an operator that measures current flow through S1, and $I_{1(2)}$ measure currents at arbitrary points on arm 1 (2) (cf. Fig 2). Next, we integrate Eq. (24) over the time-of flight of electron in the MZI. Integration of $I_{S1}$ over this time yields the total charge that flows into the MZI during the time-of-flight. The latter is noiseless according to our assumptions and therefore is proportional to the identity (exactly one electron has been injected from S1). The integration of $I_{1(2)}$ yields the fraction of charge that when to arm 1(2), $Q_{1(2)}$. It follows from the above that

$$\begin{aligned} Q_1+Q_2=\mathbbm {1}, \end{aligned}$$

(25)

where we have used ${\tau }_\mathrm{{fl}}I_{S1}=\mathbbm {1}$ in dimensionless units. A similar identity may be obtained for MZI$_\text {det}$,

$$\begin{aligned} Q_3+Q_4=\mathbbm {1}. \end{aligned}$$

(26)

Next, we insert those unit operators into Eq. (23), which yields the expression

$$\begin{aligned} \begin{array}{l} \langle {I_{D1}}{I_{D4}}\rangle = {\Big | {\langle S1,S4|{Q_1}{Q_3} +{Q_1}{Q_4}+ {Q_2}{Q_3} + {Q_2}{Q_4}|D1,D4\rangle } \Big |^2} \end{array} \end{aligned}$$

(27)

Each element in the sum is diagonal in the basis of trajectories, and can be evaluated employing the wavefunction (5). It follows that

$$\begin{aligned} \begin{array}{ll} \left\langle I_{D1}I_{D4}\right\rangle &{}=\Big |\left\langle Q_1\right\rangle _{S1;D1} \left\langle Q_3\right\rangle _{S4;D4}+ \left\langle Q_1\right\rangle _{S1;D1}\left\langle Q_4\right\rangle _{S4;D4}\\ &{}\quad +\,e^{i{\lambda }}\left\langle Q_2\right\rangle _{S1;D1}\left\langle Q_3\right\rangle _{S4;D4}+\left\langle Q_2\right\rangle _{S1;D1}\left\langle Q_4\right\rangle _{S4;D4}\Big |^2, \end{array} \end{aligned}$$

(28)

where we have introduced the notation $\left\langle Q\right\rangle _{S;D}=\left<{S}\left| {Q}\right| {D}\right>$. Using the identities (25) and (26), and expanding to leading order in ${\lambda }$, we arrive at the expression

$$\begin{aligned} \begin{array}{ll} \left\langle I_{D1}I_{D4}\right\rangle =\Big |&\left\langle \mathbbm {1}\right\rangle _{S1;D1} \left\langle \mathbbm {1}\right\rangle _{S4;D4}+i{\lambda }\left\langle Q_2\right\rangle _{S1;D1}\left\langle Q_3\right\rangle _{S4;D4}\Big |^2. \end{array} \end{aligned}$$

(29)

Simplifying it, employing the relations $\left\langle I_{D1}\right\rangle _0{\equiv }\left| \left\langle \mathbbm {1}\right\rangle _{S1;D1}\right| ^2$, $\left\langle I_{D4}\right\rangle _0{\equiv }\left| \left\langle \mathbbm {1}\right\rangle _{S4;D4}\right| ^2$ [see for example Eqs. (23) and (20)], we obtain

$$\begin{aligned} \begin{array}{ll} \left\langle I_{D1}I_{D4}\right\rangle =&\left\langle I_{D1}\right\rangle _0\left\langle I_{D4}\right\rangle _0\left( 1+2{\lambda }\mathfrak {Im}\left\{ \left\langle Q_2\right\rangle _{\mathrm{WV}}\left\langle Q_3\right\rangle _{\mathrm{WV}}\right\} \right) . \end{array} \end{aligned}$$

(30)

1.3 B.3: Expressions for the current expectation values

Here we compute the expressions for the current expectation values. To do so, we employ the charge conservation in a steady-state regime. We repeat the discussion following Eq. (25) that led to the identities

$$\begin{aligned} I_{D1}+I_{D2}= & {} \mathbbm {1}\end{aligned}$$

(31a)

$$\begin{aligned} I_{D3}+I_{D4}= & {} \mathbbm {1}. \end{aligned}$$

(31b)

The latter may be employed to write

$$\begin{aligned} \left\langle I_{D1}\right\rangle= & {} \left\langle I_{D1}I_{D3}\right\rangle +\left\langle I_{D1}I_{D4}\right\rangle \end{aligned}$$

(32a)

$$\begin{aligned} \left\langle I_{D4}\right\rangle= & {} \left\langle I_{D2}I_{D4}\right\rangle +\left\langle I_{D1}I_{D4}\right\rangle . \end{aligned}$$

(32b)

Since $\left\langle I_{D1}I_{D4}\right\rangle$ was found in Eq. (30), one may follow the derivation in Appendix B.2 to obtain an expression for $\langle {I_{D1}I_{D3}\rangle }$ and $\langle {I_{D2}I_{D4}\rangle }$. Plugging those results into Eqs. (32a), (32b) one ends up with the equalities

$$\begin{aligned}&\left\langle I_{D1}\right\rangle =\left\langle I_{D1}\right\rangle _0\left( 1+2{\lambda }\mathfrak {Im}\left\{ \left\langle Q_2\right\rangle _{\mathrm{WV}}\left\langle Q_3\right\rangle \right\} \right) \end{aligned}$$

(33a)

$$\begin{aligned}&\left\langle I_{D4}\right\rangle =\left\langle I_{D4}\right\rangle _0\left( 1+2{\lambda }\mathfrak {Im}\left\{ \left\langle Q_3\right\rangle _{\mathrm{WV}}\left\langle Q_2\right\rangle \right\} \right) . \end{aligned}$$

(33b)

1.4 B.4: The contribution of the background charge

Above we have derived expressions for the average currents and current–current correlator in the presence of single particle taken from out-of-equilibrium distribution. In real life, there is background charge. The latter may interact with the incoming electrons, producing a shift in the measured signal. We consider the background charge as a noiseless constant that shifts the charge operator on each arm ($Q_2\rightarrow Q_2+\left\langle Q^{\mathrm{bg}}_2\right\rangle$, $Q_3\rightarrow Q_3+\left\langle Q^{\mathrm{bg}}_3\right\rangle$). We may now rewrite now Eqs. (30) and (33a), (33b) with the contribution of the background charge:

$$\begin{aligned} \left\langle I_{D1}\right\rangle= & {} \left\langle I_{D1}\right\rangle _0\left( 1+2{\lambda }\mathfrak {Im}\left\{ \left\langle Q_2\right\rangle _\mathrm{{WV}}\left[ \left\langle Q_3\right\rangle +\left\langle Q_3^{\mathrm{bg}}\right\rangle \right] \right\} \right) \end{aligned}$$

(34a)

$$\begin{aligned} \left\langle I_{D4}\right\rangle= & {} \left\langle I_{D4}\right\rangle _0\left( 1+2{\lambda }\mathfrak {Im}\left\{ \left\langle Q_3\right\rangle _\mathrm{{WV}}\left[ \left\langle Q_2\right\rangle +\left\langle Q_2^{\mathrm{bg}}\right\rangle \right] \right\} \right) \end{aligned}$$

(34b)

$$\begin{aligned} \left\langle I_{D4}I_{D1}\right\rangle =\left\langle I_{D4}\right\rangle _0\left\langle I_{D1}\right\rangle _0\left[ 1+2{\lambda }\mathfrak {Im}\left\{ \left[ \left\langle Q_2\right\rangle _\mathrm{{WV}}+\left\langle Q_2^{\mathrm{bg}}\right\rangle \right] \left[ \left\langle Q_3\right\rangle _\mathrm{{WV}}+\left\langle Q_3^{\mathrm{bg}}\right\rangle \right] \right\} \right] \end{aligned}$$

(35)

[cf. Eq. (8)].