# Hamiltonian Amplification: Another Application of Parametric Amplification

This is a Perspective on "Amplification of quadratic Hamiltonians" by Christian Arenz, Denys I. Bondar, Daniel Burgarth, Cecilia Cormick, and Herschel Rabitz, published in Quantum 4, 271 (2020).

By Wenchao Ge (Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA).

#### Motivation

Enhancing coherent interactions against decoherence in quantum systems is the holy grail for implementing various quantum technologies, such as quantum computation, quantum simulation, and quantum sensing. In addition to improve the performance of these applications, enhancing coherent interactions would also allow the exploration of new parameter regimes.

A generic paradigm for coupling different quantum objects is achieved via boson-mediated interactions, where photons or phonons couple to qubits or another bosonic mode. A typical boson-mediated interaction is described by ($\hbar=1$)

H_0=\frac{\omega^{(x)}}{2} x(s^{\dagger}+s)+i\frac{\omega^{(p)}}{2} p(s^{\dagger}-s),

where $\omega^{(x)},\ \omega^{(p)}$ are the coupling strengthes, $x \ (p)$ is the position (momentum) operator of a bosonic mode, and $s$ is a platform-dependent operator of the other quantum object, e.g. $s=\sigma_{+}$ for a qubit system. Stronger coupling strength maybe possible with advanced technologies in these systems or new conceptual platforms. However, with current technology, can we enhance the interaction strengths $\omega^{(x,p)}$ coherently?

#### Detuned Parametric Amplification

Recently, there has emerged a useful tool to achieve this goal by parametrically amplifying the bosonic mode in various coupled systems ranging from optomechanics [1], to phonon-mediated superconductivity [2],
to cavity or circuit QED [3,4,5,6], to trapped ions [7] and hybrid systems [8]. Parametric amplification is the process of modulating the potential of the bosonic mode at close to twice its harmonic frequency. With detuning $\delta$ and the parametric driving strength $g$, the process can be described by [9]

H_c=\delta a^{\dagger}a+\frac{g}{2}\left(a^2e^{-i\theta}+a^{\dagger2}e^{i\theta}\right),

where $a=(x+ip)/\sqrt{2}$ is the annihilation operator of the bosonic mode and $\theta$ is the phase of the parametric amplification. The mechanism of detuned parametric amplification is that the parametric driving term can amplify a quadrature of the mode periodically due to the rotation from the detuning term [9]. Effectively, the amplification of a particular quadrature ($x$ or $p$) is achieved for a corresponding $\theta$ such that the interaction strength can be amplified accordingly.

The physics can be understood using a Bogoliubov transformation [10] that $b=a\cosh r -a^{\dagger}e^{i\theta}\sinh r$. By choosing $\theta=0$, the detuned parametric amplification becomes harmonic oscillation of the Bogoliubov mode as $H_c=\delta^{\prime}b^{\dagger}b$ with $\delta^{\prime}=\sqrt{\delta^2-g^2}$, where $|\delta|>|g|$. The boson-mediated interaction is transformed to

H_0=\frac{\omega^{(x)}e^r}{2} x_b(s^{\dagger}+s)+i\frac{\omega^{(p)}e^{-r}}{2} p_b(s^{\dagger}-s),

where $x_b \ (p_b)$ is the position (momentum) operator of the Bogoliubov mode and the enhancement factor $e^r=\left(\frac{\delta+g}{\delta-g}\right)^{1/4}$ with the squeezing parameter $r$. Ideally, the factor $\omega^{(p)}$ can be zero for some platforms [7]; otherwise, the term proportional to $\omega^{(p)}$ can be viewed as a source of error whose contribution is negligible in the limit of $e^r\gg1$ [1,5]. Therefore, the interaction is coherently enhanced by the factor $e^r$. After eliminating the bosonic bus, the effective interaction enhancement can be calculated by taking into account of the effective detuning $\delta^{\prime}$ of the Bogoliubov mode [7,1,6].

Very recently, the first experimental realization has been demonstrated in trapped ions [11] showing a $3.3-$fold enhancement in the coherent interaction strength and demonstrated the potential of this approach to improve entangling-gate fidelities against errors from the qubit.

With the full knowledge of the Hamiltonian $H_0$, the detuned parametric amplification can be well controlled to enhance the coherent interaction. Then a conceptual question arises that whether it is possible to enhance the interaction in Hamiltonian $H_0$ if we have no knowledge about its specific form, e.g., we don’t know whether $\omega^{(x)}=0$ or $\omega^{(p)}=0$?

#### Hamiltonian Amplification

In this work of Arenz et al. [12], a generic model of Hamiltonian amplification (HA) is proposed, answering positively to this conceptual question. The main idea of their procotocl is to apply very fast alternating squeezing $S$ and anti-squeezing $S^{\dagger}$ operations during the evolution of the original Hamiltonian $H_0$. The main result is that a class of quadratic Hamiltonian of continuous-variable systems can be literally amplified, i.e.,

H_0\rightarrow \lambda H_0,

where $\lambda>1$. The amplification rate $\lambda$ is controlled by the strength of the squeezing operation.
Besides the conceptual difference, there are two operational differences compared to the detuned parametric amplification. First, in HA, the squeezing operation is resonant with the bosonic mode, i.e., $S(r)=e^{-iH_c\tau}$ with $\delta=0$ in $H_c$. Here the parametric driving time is $\tau$, the squeezing parameter $r=g\tau$, and the phase $\theta=-\pi/2$. Second, $S(r)$ and $S^{\dagger}(r)$ are applied alternatively for very short intervals instead of having parametric amplification on continuously in detuned parametric amplification protocols. Such stroboscopic approach of squeezing has also been considered in enhancing coherent interactions in trapped-ion quantum information processing [13].

The main mechanism for this protocol to work is to realize phase independent quadrature amplification. This can be achieved using the following pattern:

\left[S(r)e^{-iH_0\Delta t}S^{\dagger}(r)S^{\dagger}(r)e^{-iH_0\Delta t}S(r)\right]^n,

where $\Delta t$ is a very short time interval and $n$ is the number of operation sequences during time $t$ such that $t=2n*\Delta t$.
In the first half of a sequence, $S(r)x S^{\dagger}(r)=xe^r$ and $S(r)p S^{\dagger}(r)=pe^{-r}$, while in the second half, we have the opposite effect, i.e., $S^{\dagger}(r)x S(r)=xe^{-r}$ and $S^{\dagger}(r)p S(r)=pe^{r}$. Although either half of the sequence is phase-sensitive amplification, the whole sequence is phase insensitive. In the limit of $\Delta t\rightarrow 0$, $n\rightarrow \infty$ and $t$ is fixed, both the operators $x$ and $p$ are amplified by the same rate $\cosh(r)$. For quadratic operators, $x^2$ and $p^2$, they can be amplified by a factor of $\lambda=\cosh(2r)$. More interestingly, the idea can be generalized to amplify an arbitrary multi-mode quadratic Hamiltonian of the form $H_0=\sum_{i,j}\omega_{i,j}^{(x)}x_ix_j+\omega_{i,j}^{(p)}p_ip_j$.

Arenz et al. have further supported their idea with some mathematical theorems and practical considerations. First, they proved that HA can only work for systems with infinite dimensions, such as bosonic modes, according to unitarily invariant norm for finite-dimensional systems [14]. Second, they considered finite time $\Delta t$ implementation instead of instantaneous and infinite fast alternation as previously described. They showed that HA can be achieved with negligible higher-order contributions from non-commuting time-dependent hamiltonian in the interaction picture of the parametric amplification during the finite time. They also quantified the error between the target Hamiltonian and the actual Hamiltonian under this practical consideration.

In this work, two applications of HA have been investigated by Arenz et al. First, they proved the possibility of implementing both system HA and system-environment coupling suppression. They pointed out that this can be done by combining parametric amplification with dynamical decoupling, a procedure of using instantaneous control pulses to average out unwanted system-environment coupling. This is based on a previous idea on dynamical decoupling for continuous variable systems with quadratic Hamiltonians [15]. Second, they proposed a possible implementation of Hamiltonian amplification for an oscillator-qubit system to speedup two-qubit gates by several times. To achieve this application, one requirement is that the time interval $\Delta t\approx 1$ ps, a couple of orders smaller than the typical oscillator period.

The essential ingredient to demonstrate this idea is very fast alternation between squeezing and anti-squeezing operations. The technology of alternation between squeezing and anti-squeezing has been realized in trapped ions for amplifying mechanical oscillator motion by Burd et al from NIST Ion Storage Group [16]. Promisingly, the first proof-of-principle demonstration of this protocol has also been realized in the same group to obtain a $1.5$-fold amplification.

#### Summary and Outlook

The merit of this work is to form a conceptual new idea of Hamiltonian amplification without full knowledge of the system. This protocol could lead to another avenue that parametric amplification permits. Considering recent theoretical and experimental progress with parametric amplification, it is encouraging that this nonlinear process could bring in new protocols and new possibilities in engineering coherent interactions in various quantum platforms.

### ► References

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