Sub-Planck structures: Analogies between the Heisenberg-Weyl and SU(2) groups

Coherent states were introduced by Schrödinger as non-spreading wave packets of the quantum harmonic oscillator Schrödinger (1935). They are defined Glauber (1963); Gerry and Knight (2005); Navarrete-Benlloch (2015) as the eigenstates of the annihilation operator $\hat{a}\ket{\alpha}=\alpha\ket{\alpha}$, with $\alpha\in\mathbb{C}$, and are obtained from the vacuum state by acting on the displacement operator $\hat{D}(\alpha)$ as

where

In this work we are interested in states built as superpositions of coherent states

The superposition of two coherent states with the same amplitude and opposite phase leads to the so-called cat state Dodonov et al. (1974); Gerry and Knight (2005), while the superposition of four coherent states along the same phase-space line is known as the quantum tetrachotomous state Shukla et al. (2019). On the other hand, superposing four coherent states with the same amplitude and maximally-spread phases we obtain the so-called compass states Zurek (2001). We provide specific examples shortly.

The Wigner function provides a useful way to visualize all these states in phase space Hillery et al. (1984); Schleich (2001); Leonhardt (2010); Navarrete-Benlloch (2015). For a generic quantum state $\hat{\rho}$, it can be written as the expectation value of the displaced parity operator Royer (1977)

where

is the coordinate vector in phase space and $\hat{\Pi}:=(-1)^{\hat{a}^{\dagger}\hat{a}}$ is the parity operator. Using the well-known properties

it is easy to find for the arbitrary coherent-state superposition (4) that

where the Wigner function of the operator $\ket{\alpha_{n}}\bra{\alpha_{m}}$ is the complex Gaussian

where

represents the locations of the coherent states in phase space and $\Omega:=\big{(}\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\big{)}$ is known as the symplectic form.

The product of quadrature uncertainties has a lower limit $\Delta x\Delta p=1$, which is sometimes denoted by the Planck action in phase space (note that it is equal to $\nicefrac{{\hbar}}{{2}}$ for position and momentum with proper units). This led to the belief that phase-space structures with areas below this Planck scale either do not exist or pose no observational consequences for physical quantum states. This was challenged by Zurek Zurek (2001), who showed that compass states not only have sub-Planck structures, but play a crucial role in enhancing their sensitivity to phase-space displacements, which is related to their metrological power as we discuss later.

In order to discuss in more detail the origin of sub-Planck structures, we consider next some specific classes of coherent-state superpositions. Let us start with cat states Gerry and Knight (2005), in particular the horizontal-cat defined along the position phase-space axis (in the following we omit normalizations of states and Wigner functions to simplify the notation)

with $x_{0}\in\mathbb{R}$. Particularizing the general Wigner function (8) to this state we obtain

for

which we plot in Fig. 1(a). The figure presents two Gaussian lobes centered at positions $(\pm x_{0},0)$, which correspond to the Wigner functions of the underlying coherent states $\ket{\pm\nicefrac{{x_{0}}}{{2}}}$. In addition, around the origin of phase space, it shows oscillations along the momentum direction with period $\nicefrac{{2\pi}}{{x_{0}}}$, generated from the quantum interference between the coherent states.

We can also build vertical cat states as a superposition of two coherent states in the momentum axis, and hence just as a $\pi/2$ rotation of the horizontal cat states, that is,

The corresponding Wigner function has the same form as Eq. (12), but swapping $x$ and $p$ as

Let us consider now the superposition of horizontal and vertical cat states, leading to Zurek’s compass state Zurek (2001)

The corresponding Wigner function is represented in Fig. 1(b), which using Eq. (8) can be written as

where

corresponds to the Wigner functions of the four coherent states underlaying the compass state (Gaussian lobes at the north, south, east, and west positions),

corresponds to their interference pattern at the center of phase space, and

for

contains the interference terms generated far from the phase-space origin (cat-like interference patterns located at the northeast, northwest, southeast, and southwest positions).

Remarkably, the central chessboard-like pattern $W_{\text{cent}}(\bm{r})$ contains tiles of alternating sign (denoted by different colors in the figure) with areas proportional to $x_{0}^{-2}$, hence below the Planck scale for $x_{0}\gg 1$. These are the sub-Planck structures first identified in Zurek (2001), whose size is limited in all phase space directions. In contrast, the structures appearing in the interference pattern of the cat state of Fig. 1(a) are limited only in the vertical direction, but not in the horizontal direction, where they show a Gaussian profile.

It is interesting for our purposes to consider one more type of states, built as incoherent mixtures of the cat states introduced above

The corresponding Wigner function, shown in Fig. 1(c), reads

which is almost the same as that of the compass state, just missing the $W_{\text{int}}(\bm{r})$ interference terms. Remarkably, this means that the same chessboard pattern with sub-Planck structures appears for the cat mixture Kumari et al. (2015). Now, since the interference structures of cat states are not considered to be sub-Planck (since they are not limited in all phase-space directions), and an incoherent mixing of two states cannot enhance their individual quantum properties, this raises legitimate doubts about up to what point the presence of sub-Planck structures in the Wigner function is enough to claim that any interesting quantum properties appear.

In order to show the differences between the three types of states introduced in the preceding section, it is interesting to consider their sensitivity to phase-space displacements. In particular, given a state $\hat{\rho}$ and its displaced version $\hat{D}(\delta\alpha)\hat{\rho}\hat{D}^{\dagger}(\delta\alpha)$, where $\delta\alpha\in\mathbb{C}$ is an arbitrary displacement, we consider their overlap

where the last equality applies when the state is pure, $\hat{\rho}=\ket{\psi}\bra{\psi}$. This quantity provides a measure Audenaert (2014) for the distinguishability of the state and its displaced version. The smaller the displacement $\delta\alpha$ needs to be in order to bring the overlap to zero, the more sensitive the state is said to be against displacements. This has immediate implications for quantum metrology: Imagine that a signal that we want to measure is linearly coupled to our harmonic oscillator, which then experiences a displacement proportional to the signal’s strength; the sensitivity of the oscillator to displacements is then translated into a signal resolution, such that quantum states with higher sensitivity are able to resolve weaker signals Toscano et al. (2006).

Let us consider the example of coherent states. Using (7), we easily find the overlap as

We see that displacements above the Planck scale $|\delta\alpha|>1$ are required in order to make this overlap vanish. This determines the resolution of coherent states as metrological tools. It is interesting to note that, in the case of coherent states, the sensitivity to displacements is independent of the number of quanta present in the state, $N=\braket{\hat{a}^{\dagger}\hat{a}}=|\alpha|^{2}$. Hence, the sensitivity cannot be improved by increasing $N$ and is solely limited by the shot noise introduced by vacuum fluctuations.

Consider next cat states, in particular the horizontal cat state defined in Eq. (11). Decomposing the displacement in real and imaginary parts as

the overlap (24) is easily found to be

Here and in the following we make heavy use of the properties (7). The overlap function $F_{\ket{\psi_{\text{H}}}}\left(\delta\alpha\right)$ vanishes identically when

Note that the number of excitations in the cat state is $N\approx\nicefrac{{x_{0}^{2}}}{{4}}$ for $x_{0}\gg 1$. Hence, in contrast to coherent states, the sensitivity of cat states to displacements $\delta\alpha=\text{i}\delta_{p}$ in the momentum direction scales as $\nicefrac{{1}}{{\sqrt{N}}}$, which is well beyond the Planck scale for “macroscopic” cat states with $x_{0}\gg 1$ (this scaling is known as the Heisenberg limit of the sensitivity). On the other hand, for displacements $\delta\alpha=\delta_{x}$ in the position direction, cat states pose no advantage with respect to coherent states. This is illustrated in Fig. 2(a), where we plot the overlap as a function of the displacement.

Consider now the compass state of Eq. (16), for which we easily obtain

with $\delta_{\pm}=\delta_{x}\pm\delta_{p}$. This overlap is shown in Fig. 2(b). The vanishing condition is now either of the following

As appreciated in Fig. 2(b), this is satisfied for displacements with $\left|\delta\alpha\right|\sim x_{0}^{-1}$ and arbitrary phase. Hence, as compared to coherent states, a compass state with $N$ excitations (approximately equal to $\nicefrac{{x_{0}^{2}}}{{4}}$ for $x_{0}\gg 1$) has a $\sqrt{N}$-enhanced sensitivity against displacements, but now in any phase-space direction, making it much more valuable for quantum metrology than cat states Toscano et al. (2006).

There is a natural impulse to associate this property of compass states with their sign-alternating sub-Planck structures. However, only by themselves, these structures are not enough to generate such enhancement of the sensitivity to arbitrary displacements. In order to show this, consider now the cat-state mixture (22), which as mentioned above also presents the same sub-Planck structures in the center of phase space. The overlap (24) reads in this case

In order for this expression to vanish, the following two conditions need to be satisfied simultaneously:

Hence, the cat-state mixture also has $\sqrt{N}$-enhanced sensitivity against displacements, but only for those performed along the directions diagonal with respect to the axis system formed by the cat states ($\pm 45^{\circ}$ in our case). This is illustrated in Fig. 2(c). Hence, in spite of also having sub-Planck structures in the Wigner function, cat-state mixture do not have the potential for metrology of compass states, for which the extra quantum coherence of the cat-state superposition plays a crucial role.

In summary, as compared to coherent states, for a given number of quanta $N$, compass states show a $\sqrt{N}$-enhancement of the sensitivity against displacements in arbitrary directions in phase space. In contrast, cat states and cat-state mixtures show this sensitivity enhancement only in specific phase-space directions. Interestingly, this shows that even though the same sub-Planck scales are present in the Wigner function of cat-state mixtures and superpositions, the latter have way more potential for quantum metrology.

Let us introduce the SU(2) coherent states Radcliffe (1971); Perelomov (1972); Gilmore (1972); Arecchi et al. (1972); Perelomov (1986). Our starting point is the SU(2) displacement operator, which admits either of the forms

parametrized through $\gamma=\text{e}^{i\phi}\tan(\theta/2)$ for convenience, with $\phi\in[0,2\pi[$ and $\theta\in[0,\pi]$, in which case $\alpha=\text{e}^{\text{i}\phi}\theta/2$, $\eta=\ln(1+|\gamma|^{2})$, and $\bm{n}=(\sin\phi,-\cos\phi,0)$. The raising and lowering operators are

The SU(2) displacements correspond to rotations of angle $-\theta$ around the $\bm{n}$ axis. Throughout the article, we make use of their composition rule Perelomov (1986)

where

As Eq. (37) does not depend on $j$, it is easily proven by particularizing it to the $j=\nicefrac{{1}}{{2}}$ representation, where all operators correspond to $2\times 2$ matrices (Pauli matrices for the components of $\hat{\bm{J}}$).

Applying the displacement operator to the reference state $\ket{j,j}$, we obtain a SU(2) coherent state

where

is easily found by using the intermediate form of the displacement (35), taking into account that

where $N_{\mu}=\sqrt{(j+\mu)(j-\mu+1)}$. The coherent states can be associated with the points

on a unit sphere, such that $\ket{j,j}$ corresponds to the north pole. The overlap between two coherent states is

as easily proven from Eqs. (37) and (39).

Hence, for $j\gg 1$ the overlap between the reference state $\ket{0}=\ket{j,j}$ and any coherent state $\ket{\gamma}$ is approximately Gaussian as a function of $\gamma$,

similarly to the HW coherent states. Note that with this parametrization of SU(2) coherent states in terms of $\gamma$, the width of the Gaussian scales as $\nicefrac{{1}}{{\sqrt{j}}}$. In general, the overlap between two coherent states pinned at different points of the unit sphere decreases with $j$ as

where

for

Indeed, we see later that the effective support of the SU(2) coherent-state’s Wigner function (which can be defined as a quasi-probability distribution on the sphere) decreases following the corresponding scaling in $j$.

In the next section we consider different coherent-state superpositions of the type

which generalize the ones studied for the $\mathfrak{hw}(1)$ algebra. Just as in that case, the Wigner function is a very useful tool for the characterization of these states Várilly and Gracia-Bondía (1989); Heiss and Weigert (2000); Klimov et al. (2017); Koczor et al. (2020). In the SU(2) case, for any state $\hat{\rho}$, it can be evaluated via the Stratonovich-Weyl correspondence as Davis et al. ,

with

where

with $\braket{j,\mu;l,0}{j,\mu}$ the Clebsch-Gordan coefficients. As for the Heisenberg-Weyl group, this Wigner function is real and bounded (but not necessarily positive, not even for coherent states in this case, although their negativity decreases with $j$ Davis et al. ). Hence, we can use it to visualize the state, by plotting it either over the surface of the unit sphere or, as we do, in the stereographic plane $(x,p)$, where

Note that the origin of this plane corresponds to the north pole of the sphere, while the equator is represented as the circle of unit radius, and the south pole as a circle of infinite radius.

The Wigner function of the generic superposition (48) can be written as

where, using Eqs. (37) and (39), the Wigner function of the operator $\ket{\gamma_{n}}\bra{\gamma_{m}}$ is easily found to be

with

This form is suited for numerical computation, but, for analytical purposes, sometimes alternative derivations are more convenient, as we show in the Appendix for the states that we consider in the next section.

Let us now consider specific coherent-state superpositions, and show how the concept of sub-Planck structures presented in §II for the $\mathfrak{hw}(1)$ algebra, also extends to the $\mathfrak{su}(2)$ case. To this aim, we focus on coherent states distributed along the equator, that is, $\left|\gamma_{n}\right|=1$ in Eq. (48).

Superposing two antipodal SU(2) coherent states we obtain a SU(2) cat state Sanders (1989b); Huang et al. (2015, 2018); Maleki and Zheltikov (2020); Davis et al. . Consider in particular the coherent states along the horizontal axis of the stereographic plane (as in §II, we omit normalizations in states and Wigner functions), defining the horizontal cat state

We derive the Wigner function of this state in the Appendix, obtaining

where the last term provides the interference between the underlying coherent states

with

while the first terms correspond to the Wigner functions of the coherent states

where $P_{l}(x)$ is the $l\mathrm{th}$ Legendre polynomial Várilly and Gracia-Bondía (1989).

In Fig. 3(a) we plot this Wigner function in the stereographic plane $(x,p)$, as explained in the preceding section. We can clearly see two lobes centered at positions $(\pm 1,0)$, which correspond to the coherent states. Increasing $j$ has the effect of reducing the extension (effective support) of these lobes. In addition, the interference between these coherent states generates an oscillating pattern in the vertical direction, similarly to what happens for the horizontal HW cat states. The zeros of this interference pattern $I_{\text{H}}(\gamma)$, occur when

Along the $p$ axis ($x=0$), this means that the first zeros are located at $p=\pm\tan(\pi/8j)\approx\pm\pi/8j$ for $j\gg 1$. On the other hand, in the horizontal direction for any fixed $p$, the interference pattern simply decays as a Gaussian $I_{\text{H}}(x)/I_{\text{H}}(0)\approx\text{e}^{-4jx^{2}}$ for $j\gg 1$, hence with a width proportional to $\nicefrac{{1}}{{\sqrt{j}}}$. This is precisely the same scaling that we found for the width of the overlap between coherent states. Hence, similarly to the HW case, we see that the support of the structures appearing in the interference pattern of horizontal SU(2) cat states is limited only in the $p$ direction.

We can also define cat states along the vertical axis of the stereographic plane as

whose Wigner function is like the one of the horizontal cat state, but rotated by $\pi/2$ in the stereographic plane, that is,

Note that, similarly to Eq. (57) for $\ket{\psi_{\text{H}}}$, we write the Wigner function of $\ket{\psi_{\text{V}}}$ as the sum of the Wigner function of its underlaying coherent states $W_{\ket{\pm\text{i}}}(\gamma)=W_{\ket{\pm 1}}(p+\text{i}x)$, plus their interference $I_{\text{V}}(\gamma)=I_{\text{H}}(p+\text{i}x)$.

Given these cat states, we then define the SU(2) compass states as we already did for the HW case: the balanced superposition defined by

The corresponding Wigner function, which we show in Fig. 3(b), is equal to the sum of the individual Wigner functions of each cat state, plus the terms coming from the interference between these (the northwest, northeast, southeast, and southwest structures shown in the figure). We discuss the analytic form of this Wigner function in the Appendix. For our purposes here, it is interesting to note that for $j\gg 1$ the interference pattern close to the origin of the stereographic plane (say $|\gamma|<1$) is the sum of the interference patterns of the cat states, that is,

leading to the chessboardlike pattern shown in Fig. 3(b). This pattern is reminiscent of the one found for the $\mathfrak{hw}(1)$ algebra, and shows that structures limited in all directions of the stereographic plane appear in this SU(2) compass state. This alternating-sign tiles have an extension proportional to $\nicefrac{{1}}{{j}}$ in any direction, which is a factor $\nicefrac{{1}}{{\sqrt{j}}}$ smaller than the extension found for coherent states. This generalizes the concept of sub-Planck structures to SU(2): structures with a support of area $\nicefrac{{1}}{{j}}$ smaller than the support of coherent states.

From the previous discussion, it is clear that a mixture of cat states also present the same chessboard pattern. Indeed, considering the mixture,

the corresponding Wigner function of above cat-state mixture can be simply written as,

which is shown in Fig. 3(c), where the chessboard can be appreciated. In fact, just like in the case of the $\mathfrak{hw}(1)$ algebra, only the inter-cat-state interference regions are missing with respect to the Wigner function of the compass state.

In summary, we have shown that the same phase-space features found for HW cat and compass states, as well as cat-state mixtures, are found for their SU(2) counterparts in the stereographic plane, provided we restrict the underlying coherent states to the equator. Interestingly, the role of $x_{0}$ in the HW case (distance of the coherent states from the origin) is played by $\sqrt{j}$ in the SU(2) case. Let us remark that, around the equator, the SU(2) group is contracted in the $j\rightarrow\infty$ limit to the two-dimensional Euclidean group E(2) group Rowe et al. (2001), which makes the results that we are exposing in this section nontrivial. In contrast, around the poles SU(2) is contracted to the HW group (Holstein-Primakoff approximation) Arecchi et al. (1972); Rowe et al. (2001), in which case all the results of the preceding section are trivially generalized, since $\lim_{j\rightarrow\infty}\nicefrac{{\hat{J}_{-}}}{{\sqrt{j}}}$ simply becomes an annihilation operator.

Let us now show that the SU(2) coherent-state superpositions discussed above also behave similarly to their HW counterparts regarding their sensitivity against displacements. For this, again we compute the overlap between the states and their $\delta\gamma$-displaced versions, as given by Eq. (24). This overlap is easily evaluated for coherent-state superpositions by making use of the property

which is proven by using the composition rule (37) and the coherent-state overlap (43).

In the case of a single coherent state, we have already established in §II that, for $j\gg 1$, this overlap has an approximately Gaussian form $\text{e}^{-j|\delta\gamma|^{2}}$. Hence, in the SU(2) case, the sensitivity of coherent states to displacements scales as $\nicefrac{{1}}{{\sqrt{j}}}$. We then have to compare the sensitivity of coherent-state superpositions against this scaling.

Let us start with cat states. Defining $\delta\gamma:=\delta_{x}+\text{i}\delta_{p}$, with $(\delta_{x},\delta_{p})\in\mathbb{R}^{2}$, using Eq. (68), for the horizontal cat (56) we easily obtain

We show this overlap in Fig. 4(a). As long as $j\gg 1$ and $\delta\gamma<1$, Eq. (69) shows that $\delta_{x}$ plays no relevant role in the overlap, which vanishes for displacements

For large $j$, minimum displacement that turns the cat state into an orthogonal state scales as $\left|\delta\gamma\right|\sim\nicefrac{{1}}{{j}}$ and must occur in the vertical direction of the stereographic plane. Therefore, in this direction, as compared to coherent states, cat states show a $\sqrt{j}$-enhanced sensitivity to displacements. In contrast, for horizontal displacements they show no enhancement of the sensitivity compared to coherent states.

Let us now consider the compass state, for which the overlap (68) leads to

We show this overlap in Fig. 4(b). Similarly to the HW compass state, now the $\sqrt{j}$-enhanced sensitivity to displacements is independent of the displacement direction.

Finally, we calculate the overlap for the SU(2) cat-state mixture (66). The key point here is to note that, for $j\gg 1$ and $\left|\delta\gamma\right|<1$, the contribution of the cross terms between the cat states to the overlap, e.g., $\bra{\psi_{\text{H}}}\hat{D}(\delta\gamma)\ket{\psi_{\text{V}}}$, is negligible. Then, for small displacements the overlap is the sum of overlaps of the individual cat states, that is,

where $F_{\ket{\psi_{\text{V}}}}(\delta\gamma)=F_{\ket{\psi_{\text{H}}}}(\delta_{p}+\text{i}\delta_{x})$. The corresponding overlap is shown in Fig. 4(c), and shows the same properties as its HW counterpart: the $\sqrt{j}$-enhanced sensitivity is only present for displacements in the $\delta_{p}=\pm\delta_{x}$ directions.

These results show that, as promised, all the features we found for HW coherent-state superpositions are exported to SU(2) coherent-state superpositions, as long as the states are restricted to the equator, where the SU(2) group contracts to the Euclidean group in two dimensions E(2) Rowe et al. (2001), as already mentioned above. In particular, we have shown that SU(2) compass states have a $\sqrt{j}$-enhanced sensitivity against displacements as compared to coherent states. This occurs for displacements in arbitrary directions in the stereographic plane, while cat states and mixtures show such an enhanced sensitivity only in special directions.

Note that one of the main motivations for the study of SU(2) compass states is their potential application to quantum metrology, in particular as sensors of rotations. When the rotation axis is known, cat states along that axis are known to provide the desired quantum advantage over coherent states Giovannetti et al. (2004); Dowling (2008); Giovannetti et al. (2011); Pezzè et al. (2018). In contrast, when the axis of rotation is completely unknown, it has been found that the optimal quantum rotosensor states are complicated in general Björk et al. (2015, 2015); Chryssomalakos and Hernández-Coronado (2017); Goldberg and James (2018); Martin et al. (2020), but well approximated for small angles by anticoherent states, whose so-called Majorana constellation is uniformly spread over the unit sphere Giraud et al. (2010). This has been checked so far up to $j=5$, but it is believed to be true for arbitrary $j$ based on physical arguments Chryssomalakos and Hernández-Coronado (2017); Martin et al. (2020). Our compass states can be very useful for intermediate situations in which the rotation axis is bound to a given plane, but is otherwise unknown. This is because the SU(2) displacements defined in Eq. (35) effect rotations around any axis defined on the equator of the unit sphere so that by orienting the sphere appropriately, compass states will provide $\sqrt{j}$-enhanced sensitivity to the rotations we want to measure as compared to coherent states.