Quantum nondemolition measurements in the relativistic Dirac oscillator

In the weakly relativistic limit $\epsilon\ll 1$ we have $E_{n}^{+}\approx mc^{2}+n\hbar\omega$, $A_{n}\approx 1$ and $B_{n}\approx 0$, so that the superpositions of opposite spin states disappear, the eigenstates reducing to $\left|n,\uparrow\right\rangle$ and $i\left|n,\downarrow\right\rangle$, respectively. This can be seen more explicitly when eliminating $\psi_{2}$ or $\psi_{1}$ from these two equations by multiplying them by $(E+mc^{2})$ and $(E-mc^{2})$, respectively, resulting in the Klein-Gordon-like equations

Consider first the component $\psi_{1}$ of the Dirac spinor. With $E=mc^{2}+{\cal E}$, the weakly relativistic regime corresponds to ${\cal E}\ll mc^{2}$, so that

so that

with corresponding effective positive energy Hamiltonian

Similarly, for the component $\psi_{2}$ we set $E=-mc^{2}+{\cal E}$. The weakly relativistic limit $|{\cal E}|\ll mc^{2}$ yields then

so that

with corresponding negative energy Hamiltonian

Combining Eqs. (22) and (25) gives finally the effective Hamiltonian characterizing the weakly relativistic limit of the (1+1)-dimensional Dirac oscillator as

In that limit the two components of the spinor $\psi$ become therefore approximately decoupled and $\hat{\sigma}_{z}$ is a constant of motion, hence a QND observable. In addition, it also follows that the system possesses an additional pair of QND observables $\hat{X}_{\ell,\text{nr}}$, $\ell=(1,2)$, of precisely the same form as the quadrature operators $\hat{Y}_{\ell}$ of Eqs. (1) but with the important difference that the frequency $\omega$ becomes now a spin-dependent frequency operator

so that

Since $[\hat{a},\hat{\omega}_{\text{nr}}]=0$ these operators obey the same commutation relations as the harmonic oscillator quadratures $\hat{Y}_{1,2}$.

In the other extreme limit, $\epsilon\gg 1$, the spin-orbit coupling term $\eta\hat{\sigma}_{-}\hat{a}+\eta^{\ast}\hat{\sigma}_{+}\hat{a}^{\dagger}$ dominates the Hamiltonian so that $E_{n}^{+}\approx mc^{2}\sqrt{2n\epsilon}$ and $A_{n}\approx B_{n}\approx 1/\sqrt{2}$. Moreover, the energy difference between neighboring motional states $\left|n\right\rangle$ and $\left|n-1\right\rangle$ scales as $1/\sqrt{n}$ and approaches zero for large excitations $n\gg 1$. The eigenstates of $\hat{H}_{\rm D}$ can then be approximated by the factorized motional and spin states $\left|E_{n}^{\pm}\right\rangle\approx|n\rangle(\left|\uparrow\right\rangle\mp i\left|\downarrow\right\rangle)/\sqrt{2}$, corresponding to the effective Hamiltonian

In contrast to $\hat{H}_{\rm nr}$ the eigenenergy spectrum of $\hat{H}_{\rm r}$ are not equidistant, and $\hat{\sigma}_{y}$ rather than $\hat{\sigma}_{z}$ is now a constant of motion, hence a QND observable. Like in the weakly relativistic case $\hat{H}_{\rm r}$ also possesses a pair of QND observables $\hat{X}_{\ell,\rm r}$ that are formally identical with the quadrature operators (28), but now with the replacement of $\omega$ by a frequency operator $\hat{\omega}_{\rm r}$ with both a motional and spin dependence,

With the equality $\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a}$ one can readily show that the dynamical evolution and uncertainty relation of $\hat{X}_{\ell,\rm r}$ are the same as those of the quadrature operators $\hat{Y}_{\ell}$. (Note that since $[\hat{a},\hat{\omega}_{\text{r}}]\neq 0$ the operator ordering in $\hat{X}_{\ell,\rm r}$ cannot be arbitrarily changed.)

It is easily verified that the formal similarity between the QND observables $\hat{X}_{\ell}$ of the Dirac oscillator in the limits $\epsilon\ll 1$ and $\epsilon\gg 1$ and the quadratures $\hat{Y}_{\ell}$ of the simple harmonic oscillator follows from the fact that in both limits $\hat{n}$ is a conserved quantity (approximately so for the case $\epsilon\gg 1$), in which case QND observables in that general form, with an appropriately chosen frequency operator $\hat{\omega}$, can always be found. However, in the intermediate regime of the Dirac oscillator $\hat{n}$ is no longer a conserved quantity, a consequence of the spin-orbit induced exchange of energy between the motional and spin degrees of freedom.

The effects of this spin-orbit coupling can be formally eliminated in the Foldy-Wouthuysen (FW) representation Moreno and Zentella (1989), as we now show. For the one-dimensional Dirac oscillator, the FW and the original Dirac representation are related by the unitary transformations Verlan and Razumova (2001),

where the operators $\hat{A}_{n}$ and $\hat{B}_{n}$ are obtained by replacing $n$ by $\hat{n}$ in the expressions for $A_{n}$ and $B_{n}$ of Eqs. (17). The transformed Hamiltonian is then the sum of $\hat{n}$-conserving Hamiltonians $\hat{H}_{\text{F}}(\hat{n})$ of the form

with eigenstates $\left|n,\uparrow\right\rangle$ and $\left|n,\downarrow\right\rangle$ and associated positive and negative eigenenergies $E_{n}^{+}$ and $E_{n}^{-}$, respectively. Importantly, the FW eigenstates do not involve superpositions of the two spin states, that is, the positive and negative energy eigenstates are uniquely associated with spin up and spin down, respectively, as was the case in the weakly relativistic limit $\epsilon\ll 1$. Following a similar approach we can therefore again construct a pair of QND observables

where the frequency operator is now

and accounts for the unequally spaced energy spectrum. As in the previous cases these quadratures obey the commutation relation $[\hat{X}_{1,\text{F}},\hat{X}_{2,\text{F}}]=i\hbar/m\omega$.

From Eq. (34), the eigenvalues $\omega_{n}^{\pm}$ of $\hat{\omega}_{\epsilon}$, given by $\hat{\omega}_{\epsilon}|n,\uparrow\rangle=\omega_{n}^{+}|n,\uparrow\rangle$ ($n\geq 1$) and $\hat{\omega}_{\epsilon}|n,\downarrow\rangle=\omega_{n}^{-}|n,\downarrow\rangle$ ($n\geq 0$), are simply the differences in frequency of neighboring eigenstates of the FW-transformed Dirac oscillator. In the weakly relativistic limit $\epsilon\ll 1$ they approach the constant value $\omega$ characteristic of a harmonic oscillator, while for a system deep in the relativistic regime we have $\left|\omega_{n}^{\pm}\right|/\omega\to 0$ for large $n$, a signature of the increasing anharmonicity of the system.

If one were to ignore the important point that $\hat{X}_{1,\rm F}$ needs to be transformed back to the Dirac representation to assess possible practical measurement schemes, one might argue that it could formally be monitored e.g. by continuous QND measurements described by the total Hamiltonian

where $\hat{b}$ represents the annihilation operator of an optical mode of frequency $\omega_{b}$, a form reminiscent of the optomechanical Hamiltonian of quantum optics. The associated Heisenberg equations of motion $d\hat{X}_{1,\rm F}/dt=0$, $d\hat{X}_{2,\rm F}/dt=-g\hat{b}^{\dagger}\hat{b}/m\omega$ and $d\hat{b}/dt=-i(\omega_{b}+g\hat{X}_{1,\rm F}/\hbar)\hat{b}$ show that the QND variable $\hat{X}_{1,\rm F}$ is imprinted on the phase of the optical mode $\hat{b}$, while $\hat{X}_{2,\rm F}$ is subject to the the measurement backaction from the radiation pressure force $g\hat{b}^{\dagger}\hat{b}$. For example, for a Dirac oscillator initially in the state $|\phi\rangle_{\rm F}=|\alpha\rangle\otimes(c_{1}\left|\uparrow\right\rangle+c_{2}\left|\downarrow\right\rangle)$, with $|\alpha\rangle$ a coherent state and $c_{1}$ and $c_{2}$ arbitrary probability amplitudes, we have that $\Delta\hat{X}_{1,\rm F}^{2}(0)=\Delta\hat{X}_{2,\rm F}^{2}(0)=\hbar/2m\omega$. In that case, as shown in Fig. 1(a), the mean value and the uncertainty in the measured observable $\hat{X}_{1,\rm F}$ remain constant in time as expected of QND measurements, while the mean value and variance of $\hat{X}_{2,\rm F}$ become$\langle\hat{X}_{2,\rm F}(t)\rangle=\langle\hat{X}_{2,\rm F}(0)\rangle-(g/m\omega)\langle\hat{n}_{b}\rangle t$ and $\Delta\hat{X}_{2,\rm F}^{2}(t)=\hbar/2m\omega+(g/m\omega)^{2}\Delta\hat{n}_{b}^{2}t^{2}$.

However, when expressed in the Dirac representation the observables $\hat{X}_{\ell}=\hat{U}^{\dagger}\hat{X}_{\ell,{\rm F}}\hat{U}$ become in general complex combinations of $\hat{x}$, $\hat{p}$, and $\bm{\sigma}$, so that realizing the system-probe interaction (35) is anything but obvious in general. That this is the case is also apparent from Fig. 1(b), which shows the evolution of $\langle\hat{x}\rangle$ and $\langle\hat{p}\rangle$ under the measurement scheme and the same parameters as Fig 1(a). In addition to slow oscillations at a frequency of order $\omega$, the evolution of these observables is characterized by high-frequency Zitterbewegung oscillations induced by the spin-orbit coupling induced interferences between the positive and negative energy states Zhang et al. (2018), as well as by significant measurement backaction. This situation is considerably more complex than for the simple harmonic oscillator, where the QND nature quadratures (1) follows readily and intuitively from the simple oscillatory evolution of $\hat{x}$ and $\hat{p}$.

For $\epsilon\ll 1$ we have

so that in addition to the measurement of $\hat{x}$ the composite observable $\hat{p}\hat{\sigma}_{z}$ requires only spin-sensitive momentum measurements. This is because in this limit the spin-orbit coupling is so weak as not to couple the components $\psi_{1}$ and $\psi_{2}$ of the spinor, as we have seen. $\hat{\sigma}_{z}$ is then a constant of motion and commutes with $\hat{p}$. Consequently, if the initial state of the Dirac oscillator is known not to exhibit spin-motion correlations, these measurements can be performed by measuring the spin and the momentum independently.

The situation is somewhat more complex, but still manageable in principle in the extreme relativistic limit $\epsilon\gg 1$ where

As we have seen, that regime is characterized by a relativistically induced anharmonicity of the oscillator which results in an excitation frequency that is no longer a constant, but rather a quantum observable $\hat{\omega}_{\rm r}$ that does not commute with $\hat{x}$ and $\hat{p}$, see Eq. (30). In that case the measurement of the composite observables that comprise $\hat{X}_{1,\rm r}$ requires a coupling between the probe and $\hat{x}$ and $\hat{p}$ that is sensitive to the excitation frequency. However, as we shall see the time evolution of $\hat{\omega}_{\rm r}$ is far slower than the other observables, so that this coupling can be approximated by introducing an additional probe of $\hat{\omega}_{\rm r}$ and then combining with a probe of the motional observables.

To see more concretely how this scheme can be implemented we consider a probe with internal degree of freedom described by the Pauli spin operators $\hat{s}_{x,y,z}$, and external degree of freedom by the bosonic operators $\hat{b}$ and $\hat{b}^{\dagger}$. The measurement scheme is described by the total Hamiltonian

where $\omega_{b}$ and $\omega_{s}$ are the external and the internal characteristic frequencies of the probe. in the extreme relativistic limit considered here $\hat{H}_{\rm D}\approx\hat{H}_{\rm r}$. Finally $\hat{V}_{\rm{m}}$ describes the measurement interaction. Its explicit form is

where the first term accounts for the rotation of the probe internal spin due to $\hat{\omega}_{r}$, and the second describes the Dirac oscillator driven modification of the spin-orbit coupling of the probe. The Heisenberg equations of motion ot the oscillator-probe system are

where the coupling between Eqs. (43) and (44) is due to the fact $[\hat{a},\hat{\omega}_{\text{r}}]\neq 0$, which leads to the difficulty of measuring $\hat{x}$, $\hat{p}$, and $\hat{\omega}_{\rm r}$ independently, in contrast to the weakly relativistic limit.

Assuming that $\langle\hat{a}\rangle$ scales as $\sqrt{n}$, Eq. (43) shows that $\langle\dot{\hat{\omega}}_{\rm r}\rangle$ scales then as $1/n$, so that $\langle\dot{\hat{\omega}}_{\rm r}\rangle/\langle\hat{\omega}_{\rm r}\rangle$ scales as $g/\sqrt{n\hbar\omega m}$, where we have used Eq. (30). For weak enough measurement strength $g$, large enough oscillator excitation $n$, and assuming further that $\langle\hat{\omega}_{r}\rangle\gg\omega_{s}$, so that it dominates the evolution of the probe field, see Eqs. (40), (41), and (42), the Heisenberg equations for $\hat{s}_{x,y,z}$ can be approximated as

so that

Substituting these expressions into $\hat{V}_{\rm m}$, and for the probe initially in the internal state $\left|\uparrow\right\rangle$ so that $\langle\hat{s}_{x,y}(0)\rangle=0$ and $\langle\hat{s}_{z}(0)\rangle=1$, the effective form of the measurement interaction becomes

so that this model detection scheme performs indeed approximate QND measurements on that observable.

The fact that in the weakly and extreme relativistic limits $\epsilon\ll 1$ and $\epsilon\gg 1$ the QND variables $\hat{X}_{1,\rm nr}$ and $\hat{X}_{1,\rm r}$ are amenable to direct detection suggests exploiting continuous measurement departures from their QND character as probes of relativistic effects, that is, of spin-orbit coupling.

This is illustrated for the example of $\hat{X}_{1,\rm nr}$ in Fig. 2, which shows both its evolution and the evolution of its conjugate variable $\hat{X}_{2,\rm nr}$ for three values of $\epsilon$. In the weakly relativistic regime $\epsilon=10^{-3}$ of Figs. 2(a) and 2(d), $\hat{X}_{1,\rm nr}$ behaves as expected as a QND observable, but it starts departing from this behavior as $\epsilon$ increases. This is apparent also from the temporal increase in its variance, as well as by the onset of Zitterbewegung superposed to damped oscillations at frequency $2\omega$ for large enough $\epsilon$. Fig. 2(g) shows the dependence on $\epsilon$ of $X_{1,{\rm nr}}$ and $X_{2,{\rm nr}}$ at a fixed time $t=4\pi/\omega$, for the same parameters, illustrating the sensitivity of both $\langle\hat{X}_{1,\rm nr}\rangle$ and its variance on $\epsilon$.

More quantitatively, on the weakly relativistic side, and keeping terms up to the first order of $\epsilon$ in $\hat{H}_{\rm D}$, the Heisenberg equations for $\hat{X}_{\ell,\rm{nr}}$ become

which illustrates the $\epsilon$-dependence of measurement induced backaction on $\hat{X}_{1,\rm{nr}}$. (The results of Fig. 2 were obtained numerically, keeping $\epsilon$ to all orders.)

Similarly, slightly departing from the extreme relativistic limit the Heisenberg equations for $\hat{X}_{\ell,\rm{r}}$ are

Turning finally to order-of-magnitude estimates of relevance for a potential laboratory implementation of this system, we note first that for a sample of electrons with mass $m_{e}\sim 10^{-31}$kg in the relativistic limit with $\epsilon\sim 10^{3}$ and in highly excited states $n\sim 10^{4}$, the frequency of the Dirac oscillator is $\omega\sim 10^{23}$Hz, and the uncertainty $\Delta X_{1}\sim 10^{-13}$m, This corresponds to electron energies of the order of $E_{e}\sim\langle\hat{H}_{\rm r}\rangle\sim 10$MeV, rendering an experimental implementation challenging. On the other hand, these numbers are significantly improved in tabletop simulations of relativistic quantum systems. For example, for a Dirac oscillator simulated by a spin-orbit coupled ultracold atomic sample Zhang et al. (2018) with effective mass $m\sim 10^{-27}$kg and effective velocity of light $c\sim 10^{-2}$m/s, and for a same relativistic limit condition, the oscillating frequency $\omega\sim 10^{6}$Hz and $\Delta X_{1}\sim 10^{-7}$m, with characteristic energy levels decreased to a manageable $10^{-9}$eV.

In the Dirac representation the Hamiltonian of a (1+1)-dimensional Dirac oscillator can be written in the form of anti-Jaynes-Cummings model,

where the matrix forms of the Pauli operators are

Then its eigenstates are in the form of dressed states as shown in Eq. (15),

In the Foldy-Wouthuysen (FW) representation, the complications associated with the spin-orbit coupling are formally eliminated. The form of the Hamiltonian becomes into a diagonal $2\times 2$ matrix

where the unitary transformation operators $\hat{U}$ is given in Eq. (31) and we have used the equality

The eigenstates become $|n,\uparrow\rangle$ and $|n,\downarrow\rangle$ corresponding to the positive eigenenergies $E^{+}_{n}$ and the negative eigenenergies $E^{-}_{n}$, respectively. The eigenstates in the FW and the Dirac representations are related by the unitary transformation

Then the QND observables for a (1+1)-dimensional Dirac oscillator can be concisely expressed in the FW representation,