How long-range coherence is encoded in the Weyl quasicharacteristic function

In this post, I derive an identity showing the sense in which information about coherence over long distances in phase space for a quantum state \rho is encoded in its quasicharacteristic function \mathcal{F}_{\rho}, the (symplectic) Fourier transform of its Wigner function. In particular I show

(1)   \begin{align*} \int \mathrm{d}^2 \gamma |\langle \alpha|\rho|\beta\rangle|^2 = (|\mathcal{F}_\rho|^2 \ast |G|^2)(\beta-\alpha) \end{align*}

where |\alpha\rangle and |\beta\rangle are coherent states, \gamma:=(\alpha+\beta)/2 is the mean phase space position of the two states, “\ast” denotes the convolution, and G(\alpha) = e^{-|\alpha|^2} is the (Gaussian) quasicharacteristic function of the ground state of the Harmonic oscillator.

Definitions

The quasicharacteristic function for a quantum state \rho of a single degree of freedom is defined as

    \[\mathcal{F}_\rho(\xi) := \mathrm{Tr}[\rho D_\xi] = \langle\rho,D_\xi\rangle_{\mathrm{HS}},\]

where D_\xi = e^{i \xi \wedge R} is the Weyl phase-space displacement operator, \xi = (\xi_{\mathrm{x}},\xi_{\mathrm{p}}) \in \mathbf{R}^2 are coordinates on “reciprocal” (i.e., Fourier transformed) phase space, R:=(X,P) is the phase-space location operator, X and P are the position and momentum operators, “\langle \cdot,\cdot\rangle_{\mathrm{HS}}” denotes the Hilbert-Schmidt inner product on operators, \langle A,B\rangle_{\mathrm{HS}}:=\mathrm{Tr}[A^\dagger B], and “\wedge” denotes the symplectic form, \alpha\wedge\beta := \alpha_{\mathrm{x}}\beta_{\mathrm{p}} - \alpha_{\mathrm{p}}\beta_{\mathrm{x}}. (Throughout this post I use the notation established in Sec. 2 of my recent paper with Felipe Hernández.) It has variously been called the quantum characteristic function, the chord function, the Wigner characteristic function, the Weyl function, and the moment-generating function. It is the quantum analog of the classical characteristic function.

Importantly, the quasicharacteristic function obeys |\mathcal{F}_{\rho}(\xi)|\le 1 and \mathcal{F}_{\rho}(0)=1, just like the classical characteristic function, and provides a definition of the Wigner function where the linear symplectic symmetry of phase space is manifest:

(2)   \begin{align*} \mathcal{W}_{\rho}(\alpha) &:= (2\pi)^{-1}\int\! \mathrm{d}^2\xi \, e^{-i\alpha\wedge\xi}\mathcal{F}_{\rho}(\xi)\\ &= (2\pi)^{-1}\int\! \mathrm{d}\Delta x \, e^{ip\Delta x} \rho(x-\Delta x/2,x+\Delta x/2) \end{align*}

where \alpha = (\alpha_{\mathrm{x}},\alpha_{\mathrm{p}}) = (x,p) \in \mathbf{R}^2 is the phase-space coordinate and \rho(x,x')=\langle x | \rho|x'\rangle is the position-space representation of the quantum state. This first line says that \mathcal{W}_{\rho} and \mathcal{F}_{\rho} are related by the symplectic Fourier transform. (This just means the inner product “\cdot” in the regular Fourier transform is replaced with the symplectic form, and has the simple effect of exchanging the reciprocal variables, (\xi_{\mathrm{x}},\xi_{\mathrm{p}})\to (-\xi_{\mathrm{p}},\xi_{\mathrm{x}}), simplifying many expressions.) The second line is often taken as the definition of the Wigner function, but it suffers from explicitly breaking symmetry in phase space, unnecessarily privileging position over momentum. The above relations make it clear that \mathcal{F}_{\rho} is yet another 1-to-1 representation of a quantum state.

Preliminaries

First, we will need these checkable properties of the displacement operator,

(3)   \begin{align*} D^\dagger_\xi = D_{-\xi}, \qquad\qquad D_\xi D_\eta = e^{-i\xi\wedge\eta/2} D_{\xi+\eta}, \qquad\qquad \mathrm{Tr}[D_\xi^\dagger D_\eta] = \delta(\eta-\xi) \end{align*}

from which we can invert the definition of the quasicharacteristic function:

(4)   \begin{align*} \rho = \int \!\mathrm{d}^2\xi \, \mathcal{F}_\rho(\xi) D^\dagger_\xi. \end{align*}

Next, take \chi(x) to be an arbitrary normalized pure wavefunction (i.e., \chi \in L^2(\mathbf{R})) that will serve as a “reference wavepacket”. This is typically taken to be a wavepacket with minimal amounts of momentum well localized around the origin in configuration space, that is, a state whose Wigner function \mathcal{W}_{\rho}(\alpha) is mostly concentrated around the origin in phase space. Then we define \chi_\beta (x) :=(D_{\beta} \chi)(x) = e^{i (x-\beta_{\mathrm{x}}/2)\beta_{\mathrm{p}}}\chi(x-\beta_{\mathrm{x}}) to be the reference wavepacket displaced in phase space by the vector \beta. We call the set \{\chi_\beta\} the “wavepacket basis”; it forms an overcomplete basis (formally, a frame) of the Hilbert space, in particular providing a resolution of the identity I = (2\pi)^{-1}\int\!\mathrm{d}^2\beta\,|\chi_\beta\rangle\!\!\langle\chi_\beta|. For concreteness, you can if you like take \chi to be the ground state of the Harmonic oscillator, i.e., a Gaussian with zero expectation of position and momentum: \chi(x) = (\pi\sigma^2)^{-1/4}\exp[-x^2/(2\sigma^2)] for some characteristic spatial scale \sigma; this makes \{\chi_\beta\} the set of coherent states.

Now we consider the matrix elements \mathcal{M}^{\chi}_{\rho}(\alpha,\beta):=\langle \chi_\alpha|\rho|\chi_\beta\rangle in the wavepacket basis. Unlike for an orthonormal basis, there is no sharp distinction between off-diagonal and on-diagonal matrix elements. Rather, \mathcal{M}^{\chi}_{\rho}(\alpha,\beta) can be considered roughly off-diagonal whenever \alpha and \beta are sufficiently far apartSomewhat more precisely, we can require \mathcal{W}_\chi(\gamma)\approx 0 for \gamma \cdot M \cdot \gamma > (\alpha-\beta) \cdot M \cdot (\alpha-\beta) for some positive, symmetric, real-valued, and invertible 2×2 matrix M. In the Gaussian case, M^{-1} is basically the covariance matrix of the wavepacket and, per the uncertainty principle, will necessarily obey |M^{-1}| \gtrsim \hbar. It acts to define an inner product on phase space, which arises from the chosen reference state \chi, not the bare mechanical structure of phase space (which only knows about the symplectic form). In the special case of the coherent state, M=I (up to factors of 2 that depend on choice of normalization), and the above requirement for off-diagonality is just that \mathcal{W}_\chi(\gamma)\approx 0 for |\gamma| > |\alpha-\beta|.a   that \langle \chi_\alpha|\rho|\chi_\beta\rangle \approx 0. Large off-diagonal terms are indicative of long-range coherence in phase space, where “large” is relative how closely it saturates the Cauchy-Schwartz inequality

(5)   \begin{align*} |\mathcal{M}^{\chi}_{\rho}(\alpha,\beta)|^2 \le \mathcal{M}^{\chi}_{\rho}(\alpha,\alpha)\mathcal{M}^{\chi}_{\rho}(\beta,\beta). \end{align*}

For instance, if \rho_{\mathrm{sup}} \approx (|\chi_\alpha\rangle+|\chi_\beta\rangle)(\langle\chi_\alpha|+\langle\chi_\beta|)/2 is the coherence superpositionWe use an approximate symbol “\approx” here because the state would need its normalization slightly fixed in light of the fact that \langle \chi_\alpha|\chi_\beta\rangle gets exponentially small but does not completely vanish.b   of two widely separated wavepackets |\chi_\alpha\rangle and |\chi_\beta\rangle, and if \rho_{\mathrm{mix}} = (|\chi_\alpha\rangle\!\!\langle\chi_\alpha|+|\chi_\beta\rangle\!\!\langle\chi_\beta|)/2 is the corresponding incoherent mixture, then |\mathcal{M}^{\chi}_{\rho_{\mathrm{sup}}}(\alpha,\beta)|^2 \approx \mathcal{M}^{\chi}_{\rho_{\mathrm{sup}}}(\alpha,\alpha)\mathcal{M}^{\chi}_{\rho_{\mathrm{sup}}}(\beta,\beta) but |\mathcal{M}^{\chi}_{\rho_{\mathrm{mix}}}(\alpha,\beta)|^2 \approx 0, with all on-diagonal elements the same: \mathcal{M}^{\chi}_{\rho_{\mathrm{mix}}}(\alpha,\alpha) = \mathcal{M}^{\chi}_{\rho_{\mathrm{mix}}}(\beta,\beta) = 1/2 \approx \mathcal{M}^{\chi}_{\rho_{\mathrm{sup}}}(\alpha,\alpha) = \mathcal{M}^{\chi}_{\rho_{\mathrm{sup}}}(\beta,\beta).

Result

We can then compute

(6)   \begin{align*} \mathcal{M}^{\chi}_{\rho}(\alpha,\beta) &= \langle \chi_\alpha|\rho|\chi_\beta\rangle \\ &= \int \mathrm{d}^2\xi\, \mathcal{F}_\rho(\xi) \langle \chi_\alpha|D^\dagger_\xi |\chi_\beta\rangle\\ &= \int \mathrm{d}^2\xi\, \mathcal{F}_\rho(\xi) \langle \chi|D^\dagger_\alpha D^\dagger_\xi D_\beta |\chi\rangle \\ &= \int \mathrm{d}^2\xi\, \mathcal{F}_\rho(\xi) e^{i(\xi\wedge\alpha+\xi\wedge\beta+\alpha\wedge\beta)/2}\langle \chi|D_{\beta-\alpha-\xi} |\chi\rangle \end{align*}

and using the shorthands \bar{\gamma}:=(\alpha+\beta)/2 and \Delta \gamma:=\beta-\alpha for the phase-space mean and separation between the two wavepackets \chi_\alpha and \chi_\beta, we have

(7)   \begin{align*} \mathcal{M}^{\chi}_{\rho}(\bar\gamma-\Delta\gamma/2,\bar\gamma+\Delta\gamma/2) &= \int \mathrm{d}^2\xi\, \mathcal{F}_\rho(\xi) e^{-i\bar\gamma\wedge(\xi-\Delta\gamma/2)} \mathcal{F}_\chi(\Delta\gamma-\xi) \end{align*}

where \mathcal{F}_\chi(\eta) = \mathrm{Tr}[|\chi\rangle\!\!\langle\chi| D_\eta ] = \langle| \chi D_\eta | \chi\rangle is the quasicharacteristic function of the reference wavepacket \chi. Several things could be said about this expression, especially if we introduced the twisted convolution, but let’s just observe that if \mathcal{F}_\chi (\eta)\approx 0 for \eta outside some region in reciprocal phase space then \mathcal{M}^{\chi}_{\rho}(\bar\gamma-\Delta\gamma/2,\bar\gamma+\Delta\gamma/2) only “knows” about \mathcal{F}_\rho(\xi) when \xi is in that region translated so it’s centered around \Delta\gamma. Furthermore, it only knows about the part proportional to the local Fourier component e^{i\bar\gamma\wedge\xi}. In particular, if our reference wavepacket is Gaussian, \chi(x)=(\pi\sigma^2)^{-1/4}\exp[-x^2/(2\sigma^2)], then \mathcal{F}_\chi(\xi) = \exp[-\xi_{\mathrm{x}}^2/2\sigma^2-\xi_{\mathrm{p}}^2\sigma^2/2], so that \mathcal{M}^{\chi}_{\rho}(\alpha,\beta) is essentially determined by the values \mathcal{F}_\rho takes in a \sigma \times \sigma^{-1}-sized region centered around \xi=\Delta\gamma = \beta-\alpha.

From this one can also quickly check that if we take the squared norm of this off-diagonal matrix element and integrate over the entire phase space \overline\gamma with a fixed value of the separation \Delta\gamma between the two points \alpha and \beta, we get

(8)   \begin{align*} \int \mathrm{d}^2 \bar\gamma |\mathcal{M}^{\chi}_{\rho}(\bar\gamma-\Delta\gamma/2,\bar\gamma+\Delta\gamma/2)|^2 &= \int \mathrm{d}^2\xi\, |\mathcal{F}_\rho(\xi)|^2 |\mathcal{F}_\chi(\Delta\gamma-\xi)|^2\\ &= (|\mathcal{F}_\rho|^2 \ast |\mathcal{F}_\chi|^2)(\Delta\gamma) \end{align*}

where “\ast” denote the convolution. So we find that the “total amount of coherence” over the phase-space distance \Delta\gamma (i.e., the summed amount of coherence between all pairs of wavepackets separated by \Delta\gamma) is encoded in the value of \mathcal{F}_\rho in a small \hbar^{-1}-sized region around \Delta\gamma. In the aforementioned Gaussian case, we have |\mathcal{F}_\chi(\xi)|^2 = \exp[-\xi_{\mathrm{x}}^2/\sigma^2-\xi_{\mathrm{p}}^2\sigma^2].

Footnotes

(↵ returns to text)

  1. Somewhat more precisely, we can require \mathcal{W}_\chi(\gamma)\approx 0 for \gamma \cdot M \cdot \gamma > (\alpha-\beta) \cdot M \cdot (\alpha-\beta) for some positive, symmetric, real-valued, and invertible 2×2 matrix M. In the Gaussian case, M^{-1} is basically the covariance matrix of the wavepacket and, per the uncertainty principle, will necessarily obey |M^{-1}| \gtrsim \hbar. It acts to define an inner product on phase space, which arises from the chosen reference state \chi, not the bare mechanical structure of phase space (which only knows about the symplectic form). In the special case of the coherent state, M=I (up to factors of 2 that depend on choice of normalization), and the above requirement for off-diagonality is just that \mathcal{W}_\chi(\gamma)\approx 0 for |\gamma| > |\alpha-\beta|.
  2. We use an approximate symbol “\approx” here because the state would need its normalization slightly fixed in light of the fact that \langle \chi_\alpha|\chi_\beta\rangle gets exponentially small but does not completely vanish.
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