with Quasiprobabilities

**Justin Dressel**

Institute for Quantum Studies, Chapman University

QuiDiQua Conference, 2023/11/10

How do we **measure **what the Electric Field is at some point \(x\) in space?

We put a "**test charge**" at \(x\) and look at its *response*.

**Electric Field: **\(\displaystyle \vec{E} \equiv \frac{\vec{F}}{q}\Bigg|_{q\to 0} \)

We take the **weak interaction limit** where *the test charge is small*, so the field we are measuring is **not disturbed** by the test charge itself.

**Yakir Aharonov: **

"Let's do *exactly the same thing* with quantum theory."

**Probe charge: ** \(|\phi\rangle\) s.t. \(\langle \hat{x}\rangle = 0\)

**System:** \(|\psi\rangle\)

**Interaction:** \(\hat{H} = q\hat{A}\otimes\hat{p}\)

**Response:**

\(e^{-iqt\hat{A}\hat{p}/\hbar}|\psi\rangle|\phi\rangle = \sum_a\int dx |a\rangle|x\rangle\, \langle a|\psi\rangle\,\langle x-qat|\phi\rangle \)

Mean velocity per charge yields:

\(\displaystyle \frac{\partial_t\langle \hat{x}\rangle}{q}\Bigg|_{q\to 0} = \sum_a |\langle a|\psi\rangle|^2 a = \langle \hat{A}\rangle \)

*Expectation value* of \(\hat{A}\) measured *without probe disturbing the system (much)*.

**Yakir Aharonov: **

"Let's do *exactly the same thing* with quantum theory, but also *add a final postselection measurement*."

**System Postselection:** \(\langle f |\)

**Small test charge response:**

\(\begin{aligned}\langle f |e^{-iqt\hat{A}\hat{p}/\hbar}|\psi\rangle|\phi\rangle &\approx \langle f |\psi\rangle(1 -iqt A_w\hat{p}/\hbar)|\phi\rangle \\ &\approx \langle f|\psi\rangle \int dx |x\rangle\, \,\langle x-q A_w t|\phi\rangle \end{aligned}\)

Mean velocity per charge yields:

\(\displaystyle \frac{\partial_t\langle \hat{x}\rangle}{q}\Bigg|_{q\to 0} = \text{Re}A_w \equiv \text{Re} \frac{\langle f | \hat{A}|\psi\rangle}{\langle f | \psi \rangle} \)

The expectation value of \(\hat{A}\) *conditioned on a final postselection* \(\langle f|\) is a **weak value**.

*Aharonov et al. PRL 1988*

A "conditioned expectation value" measured in the *weak interaction* ("test charge") limit.

\displaystyle A_w \equiv \frac{\langle f | \hat{A} | \psi \rangle}{\langle f | \psi \rangle} = \frac{\langle \psi | f\rangle\!\langle f | \hat{A} | \psi \rangle}{|\langle f | \psi \rangle|^2}

An expectation value is partitioned into a convex mixture of weak values,

each specific to a particular "postselection".

\displaystyle \langle A \rangle \equiv \langle \psi | \hat{A} | \psi \rangle = \sum_f \langle \psi | f \rangle\!\langle f |\hat{A} | \psi \rangle = \sum_f |\langle f | \psi \rangle|^2 \frac{\langle f | \hat{A} | \psi \rangle}{\langle f | \psi \rangle} = \sum_f P_{f|\psi}\, A_{w,f}

Weak values are *not necessarily constrained* by the spectrum of the operator \(\hat{A}\).

The "conditioned probabilities" that average the eigenvalues of \(\hat{A}\) are *quasiprobabilities*.

\displaystyle A_w \equiv \frac{\langle \psi | f\rangle\!\langle f | \hat{A} | \psi \rangle}{|\langle f | \psi \rangle|^2} = \sum_a a \frac{\langle \psi | f\rangle\!\langle f | a\rangle\!\langle a | \psi \rangle}{|\langle f | \psi \rangle|^2} = \sum_a a \frac{Q_{a,f|\psi}}{P_{f|\psi}}

The Kirkwood-Dirac quasiprobabilities are *conditioned* by the postselection likelihoods.

Q_{a,f|\psi} \equiv \langle \psi | f\rangle \!\langle f | a \rangle\!\langle a | \psi \rangle = |Q_{a,f|\psi}|\,e^{iS}

The phase is a **geometric Berry/Pancharatnam phase** determined by the

*oriented area* enclosed by *geodesics *connecting three quantum states.

Classically compatible states enclose zero area.

Non-zero phases encode *dynamical *incompatibility and a *temporal directionality* for closing the loop.

2\(S\)

\(|\psi\rangle\)

\(|a\rangle\)

\(|f\rangle\)

*Hofmann, NJP 2011*

Weak Values

Due to their close connection to the geometry of quantum state space, weak values (and thus the KD QP) can appear **even without making any explicit weak measurements**.

Let's explore several examples.

\hat{H}|E_k\rangle = E_k |E_k \rangle

(\hat{H}+\hat{\Delta})|E'_j\rangle = E'_j |E'_j \rangle

\langle E'_j |(\hat{H}+\hat{\Delta})|E_k\rangle = E'_j \langle E'_j | E_k \rangle = E_k \langle E'_j | E_k \rangle + \langle E'_j |\hat{\Delta}| E_k \rangle

\displaystyle E'_j - E_k = \frac{\langle E'_j | \hat{\Delta} | E_k \rangle}{\langle E'_j | E_k \rangle}

*Measurable* energy shifts caused by a perturbation are always (purely real) weak values.

"Strange weak values" outside the spectrum of the perturbation are *very common*.

*JD, PRA 91 032116 (2015)*

**Trivial Example of Dynamical Weak Values**:

When a *perturbation *is added to a *Hamiltonian*, the energy spectra will shift by *weak values* of the perturbation.

\displaystyle \hat{H} = \frac{\omega_q}{2}\hat{\sigma}_z + \omega_r \hat{a}^\dagger \hat{a} + \chi \hat{\sigma}_z \hat{a}^\dagger \hat{a}

At **steady state**, the balance of pump and decay from the resonator leaves the qubit entangled with distinct coherent states in the resonator:

**Dispersive Coupling Hamiltonian:**

|\Psi\rangle = c_0 |0\rangle |\psi_0\rangle + c_1 |1\rangle |\psi_1\rangle

This **reduced state** qubit coherence evolves as:

\rho_{01}(t) = c_1^* c_0 \langle \psi_1 | \psi_0\rangle

The **coherence** of the **reduced qubit state** is thus:

\partial_t \rho_{01}(t) = i[\omega_q + 2\chi \,n_w]\rho_{01}(t)

\displaystyle n_w \approx \frac{4\varepsilon^2}{\kappa^2}\left[1 + i\frac{4\chi}{\kappa}\right] \equiv \bar{n} + i\frac{4\chi \bar{n}}{\kappa}

Photon number **weak value**!

- Real part :
**AC Stark shift**

- Imaginary part :
**ensemble dephasing**

\Delta\omega_q = 2\chi\,\text{Re}\,n_w = 2\chi\bar{n}

\displaystyle \Gamma = 2\chi \,\text{Im}\,n_w = \frac{8\chi^2\bar{n}}{\kappa}

*JD, PRA 91 032116 (2015)*

For pump \(\varepsilon\) and decay rate \(\kappa\), the steady states yield:

\displaystyle i\hbar \partial_t |\psi(t)\rangle = \left[ \frac{\hat{p}^2}{2m} + V(\hat{x}) \right] |\psi(t)\rangle

Schrodinger Equation:

**Hamilton's Principle Function** :

S(t, x) \equiv -i\hbar \ln\langle x | \psi(t) \rangle

\displaystyle p_w(t,x) \equiv \partial_x S(t,x) = \frac{-i\hbar\partial_x\psi(t,x)}{\psi(t,x)} = \frac{\langle x | \hat{p} | \psi(t) \rangle}{\langle x | \psi(t) \rangle}

Momentum *defined in the usual way* is the **weak value** of the momentum operator:

Schrodinger's Equation is equivalent to a **quantum Hamilton-Jacobi Equation**:

\partial_t S(t, x) + H_w[t,x, p_w(t,x)] = 0

\displaystyle \text{Re} H_w[t,x, p(t,x)] = \frac{(\text{Re}\,p_w(t,x))^2}{2m} + V(x) + Q(x)

Imaginary part is a *continuity equation* for probability. Real part is *classical *HJ Equation:

\displaystyle Q(x) = \frac{\langle x| (\hat{p} - \text{Re}\,p_w(t,x))^2 | \psi(t) \rangle}{\langle x | \psi(t) \rangle} = -\frac{\hbar^2}{2m}\frac{\partial_x^2 |\psi(x,t)|}{|\psi(x,t)|}

**"Quantum Potential"**: *Weak Variance* of momentum away from mean weak value

**Nontrivial example: Hamilton-Jacobi Theory**

\displaystyle H_w[t,x, p_w(t,x)] = \frac{\langle x | \hat{p}^2/2m + V(\hat{x}) | \psi \rangle}{\langle x | \psi \rangle}

*Real part: Bohmian momentum
Imaginary part: Nelson Osmotic momentum*

*JD, PRA 91 032116 (2015)*

**Nontrivial example: Classical field streamlines**

Weak values also appear as **physical properties** of a classical field, even when there is not an obvious "weak measurement" at the level of individual field quanta.

Momentum weak value proportional to the local (orbital part of the) **Poynting vector** \(S_O\) of an optical field, scaled by the frequency \(\omega\) and energy density \(W\).

This can be *measured *and used to reconstruct "averaged trajectories" for the mean momentum streamlines.

The average local momentum corresponds to the local *optical pressure *felt by* small probe particles,* and thus the momentum part of the *stress-energy tensor.*

*Kocsis et al., Science (2011)
Bliokh et al. NJP (2013)*

\displaystyle \text{Re}\,\mathbf{p}(\mathbf{r}) = \text{Re} \frac{\langle x | \hat{p} | \psi \rangle}{\langle x | \psi \rangle} = \frac{\omega}{c^2} \frac{S_o(\textbf{r})}{W(\textbf{r})}

**Nontrivial Example: Bessel beams**

Both *real *and *imaginary *parts of this local momentum average also describe physical properties of classical "vortex beams" like Bessel beams.

\displaystyle \mathbf{p}_w(\mathbf{r}) = \frac{\langle x | \hat{p} | \psi \rangle}{\langle x | \psi \rangle}

The **real part** of the momentum weak value appears as the circulating local *orbital momentum* that can be transferred to probe particles by pushing them around in circular orbits.

The **imaginary part** is directed *radially* and *confines *the optical intensity into concentric rings, similarly trapping probe particles to orbit only within the optical rings.

*Bliokh et al., NJP (2013)*

**Boundary conditions:**

- Past time \(t_I\): \(|I\rangle\)
- Future time \(t_F\): \(\langle F |\)

**"Test charge" current:**

- Intermediate time \(t\): \(J(x)\)
- Take limit as \(J \to 0\)

But wait, there's more!

**Schwinger **taught us to do *exactly the same thing in quantum field theory *to probe *mean (classical) *fields and their *correlations*

*Schwinger, PR (1951)*

**Generating Functional:**

\(W[J] = -i\hbar \ln\langle F |\hat{U}[J] | I \rangle \)

**Effective Action:**

\(\Gamma[\varphi] = W[J] - \int d^4x J(x)\varphi(x)\)

**"Classical" Field:**

\(\displaystyle \varphi(x) \equiv \frac{\delta W[J]}{\delta J(x)}\Bigg|_{J\to 0} \!\! = \frac{\langle F(t)|\hat{\varphi}(x)|I(t)\rangle}{\langle F(t)|I(t)\rangle}\)

**"Classical" Equation of Motion:**

\(\displaystyle \frac{\delta \Gamma[\varphi]}{\delta \varphi(x)} = -J(x) \to 0\)

All *classical *fields according to QFT are **weak values **of the field operators.

*JD, PRL (2014)*

*Weak values* are prevalent in the quantum formalism,

even without performing weak measurements,

which means that *Kirkwood-Dirac quasiprobabilities* are also!

- Conditioned average of weakly measured observable
- Spectral shifts due to perturbations
- Ensemble-averaged dynamical parameters
- Classical mean field properties
- Classical limit for observable values
- ...and many more examples

**Thank you!**