Converting quasiclassical states into arbitrary Fock state superpositions in a superconducting cavity

TL;DR

This paper introduces a new method for generating arbitrary superpositions of Fock states in a superconducting cavity, leveraging a dispersively coupled qubit to modulate state amplitudes without frequency tuning, enhancing robustness and fidelity.

Contribution

The authors develop and demonstrate a novel technique that avoids fine-frequency tuning and photon-by-photon pumping, enabling high-fidelity superpositions in superconducting cavities.

Findings

Successfully generated high-fidelity phase eigenstates.

Produced various superpositions including squeezed states.

Method shows robustness to noise and experimental errors.

Abstract

We propose and experimentally demonstrate a new method to generate arbitrary Fock-state superpositions in a superconducting quantum circuit, where a qubit is dispersively coupled to a microwave cavity mode without the need of fine-frequency tuning. Here the qubit is used to conditionally modulate the probability amplitudes of the Fock state components of a coherent state to those of the desired superposition state, instead of pumping photons one by one into the cavity as in previous schemes. Our method does not require the adjustment of the qubit frequency during the cavity state preparation, and is more robust to noise and accumulation of experimental errors compared to previous ones. Using the method, we experimentally generate high-fidelity phase eigenstates under various Hilbert-space dimensions and squeezed states, which are useful for quantum walk and high-precision measurements.

Tables

Table 1. Table 1: Amplitudes α 𝛼 \alpha of the initial coherent states, Wigner function reduction factors R 𝑅 R , success probabilities P 𝑃 P , and fidelities F 𝐹 F of the produced truncated phase states with θ N , k = 0 subscript 𝜃 N k 0 \theta_{\mathrm{N,k}}=0 (Fig.  2 ). The uncertainties are determined by bootstrapping on the measured Wigner functions  Vlastakis et al. ( 2015 ) . Bootstrapping allows random sampling with replacement at each pixel point of the measured Wigner function to obtain a pool of new Wigner functions that are used to extract R 𝑅 R and F 𝐹 F for a statistical analysis.

N, $k$	5, 0	6, 0	7, 0
$\left|\alpha \right|$	1.63	1.74	1.85
$R$	$0.84\pm 0.01$	$0.82\pm 0.01$	$0.80\pm 0.01$
$P$	$0.37$	$0.31$	$0.23$
$F$	$0.97\pm 0.01$	$0.96\pm 0.01$	$0.96\pm 0.01$

Table 2. Table 2: R 𝑅 R , P 𝑃 P , F 𝐹 F , and ⟨ ( Δ ​ X ​ ( θ / 2 ) ) 2 ⟩ delimited-⟨⟩ superscript Δ 𝑋 𝜃 2 2 \langle(\Delta X(\theta/2))^{2}\rangle for the produced squeezed states with r = 0.8 𝑟 0.8 r=0.8 (Fig.  3 ). All have an initial coherent state amplitude | α | = 1.45 𝛼 1.45 |\alpha|=1.45 .

$\theta$	0	$\pi$/2	$\pi$
$R$	$0.82\pm 0.01$	$0.84\pm 0.01$	$0.82\pm 0.01$
$P$	$0.15$	$0.14$	$0.14$
$F$	$0.96\pm 0.01$	$0.94\pm 0.01$	$0.96\pm 0.01$
$⟨{\left(\Delta X\left(\theta /2\right)\right)}^2⟩$	$0.067\pm 0.001$	$0.070\pm 0.001$	$0.070\pm 0.001$

Equations

$$H_I=-{\chi }_{qc}{a}^{†}a|e⟩⟨e|+\left[\sum _{n=0}^N{\Omega }_n{e}^{i({\delta }_{n}t+{\varphi }_n)}|e⟩⟨g|+h.c.\right],$$

$$|\alpha ⟩=\sum _{n=0}^{\infty }{c}_n|n⟩,$$

$$|{\psi }_d⟩=\sum _{n=0}^N{d}_n|n⟩,$$

$$|{\psi }_{qc}(\tau )⟩=\sum _{n=0}^{\infty }{c}_n\left[\cos {\beta }_n|g,n⟩-i{e}^{i({\varphi }_n+n{\chi }_{qc}\tau )}\sin {\beta }_n|e,n⟩\right],$$

$$|{\psi }_c⟩=\mathcal{N}\sum _{n=0}^N{c}_n{e}^{i({\varphi }_n+n{\chi }_{qc}\tau )}\sin {\beta }_n|n⟩,$$

$$|{\theta }_{\mathrm{N},\mathrm{k}}⟩=\frac{1}{\sqrt{N+1}}\sum _{n=0}^N{e}^{in{\theta }_{\mathrm{N},\mathrm{k}}}|n⟩$$

$$|\xi ⟩=\frac{1}{\sqrt{\cosh r}}\sum _{n=0}^{\infty }\frac{(-e^{i\theta }\tanh r{)}^n\sqrt{(2n)!}}{n!2^n}|2n⟩,$$

Full text

The creation and manipulation of complex superposition states of a harmonic oscillator (e.g., a light field) is an important subject in quantum optics Bužek and Knight (1995), and has various applications in quantum technologies. In contrast to nonlinear quantum systems (e.g., atoms), a harmonic oscillator possesses an infinite-dimensional Hilbert space where the energy is linear in the quantum number. Due to this feature, a harmonic oscillator has extremely rich nonclassical effects and serves as an ideal candidate for investigation of fundamental quantum theory. In addition to fundamental interest, it plays important roles in many quantum computation protocols. On one hand, it acts as a data bus to link different qubits, as in the ones with trapped ions Cirac and Zoller (1995), cavity quantum electrodynamics (QED) Raimond et al. (2001), and circuit QED Sillanpää et al. (2007); Majer et al. (2007); DiCarlo et al. (2009). It can also be used as a quantum memory to store quantum bits (qubits), offering the possibility to construct a quantum processor with fewer resources Santos (2005) and to realize quantum error correction Leghtas et al. (2013); Ofek et al. (2016); Michael et al. (2016). On the other hand, harmonic oscillators are the basic building block for implementation of quantum computation based on continuous variables Lloyd and Braunstein (1999); Gottesman et al. (2001). In addition, the nonclassical effects of harmonic oscillators have applications in other quantum technologies, e.g., when a quantum state exhibits squeezing, i.e., the reduction of the fluctuation in one certain quadrature below the vacuum level, it can be used in high-precision measurement Caves et al. (1980). Many of these applications rely on the ability to generate an arbitrary superposition state of a harmonic oscillator, however, its linear energy spectrum prevents one from doing so by using classical drives only; the application of a classical drive produces a displacement in phase space without enhancing the nonclassical behaviors.

Previous methods employed a step-by-step algorithm Vogel et al. (1993); Law and Eberly (1996), where in each step the quantum state of a two-level system is manipulated in a controllable manner and then transferred to the harmonic oscillator, increasing the quantum number of the largest Fock state component in the superposition by $1$. After $N$ steps, the oscillator evolves from the ground state to any superposition of the first $N+1$ number states. Such a technique was first demonstrated on the harmonic motion of a trapped ion trap Ben-Kish et al. (2003), and then implemented in a superconducting resonator Hofheinz et al. (2008, 2009) using a qubit with fast frequency tunability to coherently pump photons from external microwave drive pulses into the resonator. More recently, it was shown that $2N$ selective number-dependent arbitrary phase gates, together with $2N+1$ displacement operations, can be used to construct such superposition states Krastanov et al. (2015). Following this proposal, a one-photon Fock state was created in a circuit QED system Heeres et al. (2015). With a similar setup, the Gradient Ascent Pulse Engineering method Khaneja et al. (2005); de Fouquieres et al. (2011) was recently used for creating the six-photon state and four-component cat states Heeres et al. (2016). In the context of cavity QED, Fock states and cat states of a cavity were conditionally generated with Rydberg atoms dispersively coupled to the cavity mode Deléglise et al. (2008).

Here we propose and experimentally demonstrate a distinct method to synthesize any superposition of Fock states for the field stored in a cavity that is dispersively coupled to a qubit. Our method relies on modulating the probability amplitudes associated with the superposed Fock states in an initial coherent state, a superposition of photon number states with a Poissonian distribution, to those of the desired target state. Coherent states are quasiclassical states characterized by a complex amplitude Deléglise et al. (2008), and can be generated by using a classical drive. The photon-number-dependent shift in the qubit frequency, arising from the dispersive qubit-cavity coupling, enables one to use carefully tailored microwave drive signals to individually control the qubit transition amplitude associated with each Fock state. The subsequent measurement of the qubit transition conditionally projects the cavity to a superposition of Fock states, with the probability amplitude of each state component being proportional to the associated qubit transition amplitude, controllable by the amplitude and phase of the corresponding drive signal. Unlike the previous methods, it is unnecessary to individually pump photons into the cavity one by one, and the modulating drive signals can be simultaneously applied and incorporated into a single pulse. Furthermore, during the preparation process the qubit-cavity detuning needs not to be adjusted, which is important for quantum state engineering in circuit QED systems with three-dimensional cavities Heeres et al. (2015); Paik et al. (2011); Vlastakis et al. (2013); Sun et al. (2014), where it is difficult to adjust the qubit frequency. We implement the experiment in a superconducting circuit, where a three-dimensional cavity is coupled to a transmon qubit. We analyze the produced states by measuring the Wigner functions, showing good agreement with the desired ones.

The system under investigation consists of a qubit dispersively coupled to a cavity and driven by a classical microwave containing $N+1$ frequency components. In the interaction picture, the Hamiltonian for the total system is

[TABLE]

where $\left|e\right\rangle$ and $\left|g\right\rangle$ denote the excited and ground states of the qubit, $a^{\dagger}$ and $a$ are the creation and annihilation operators of the photonic field in the cavity, $\chi_{qc}$ is the qubit’s frequency shift induced by per photon in the cavity due to the dispersive qubit-cavity coupling, $\delta_{n}$ is the detuning between the qubit and the $n$th driving component with a Rabi frequency $\Omega_{n}$ and a phase $\phi_{n}$, and $h.c.$ denotes the Hermitian conjugate. The qubit is initially in the ground state $\left|g\right\rangle$, and the cavity in a coherent state

[TABLE]

where $c_{n}=\exp(-\left|\alpha\right|^{2}/2)\alpha^{n}/\sqrt{n!}$ is the probability amplitude for having $n$ photons. The photon-number-dependent shift of the qubit transition frequency enables individually modulating the probability amplitudes of the superposed Fock states in $\left|\alpha\right\rangle$ with a carefully tailored microwave drive addressing the qubit. Our aim is to convert such a quasiclassical state to a target quantum state

[TABLE]

with the desired complex amplitudes $d_{n}$ for the corresponding Fock state components.

Under the condition $\delta_{n}=n\chi_{qc}$ and $\Omega_{n}\ll\chi_{qc}$, the $n$th component resonantly drives the qubit transition when the cavity is in the Fock state $\left|n\right\rangle$, but does not affect the qubit state if the cavity is in any other Fock state due to large detunings. The dynamics governed by $H_{I}$ enables individually controlling the transition in each subspace {$\left|g,n\right\rangle$, $\left|e,n\right\rangle$} (the notation denotes $\left|qubit,cavity\right\rangle$); there is no interaction between different subspaces. After an interaction time $\tau$ the qubit-cavity system evolves to

[TABLE]

where $\beta_{n}=\int_{0}^{\tau}\Omega_{n}dt$ is the pulse area associated with the $n$th frequency component. We take $\beta_{n}=0$ for $n>N$, where $N$ is the maximum photon number in the desired target state. The detection of the qubit in the state $\left|e\right\rangle$ projects the cavity to the superposition state

[TABLE]

where $\mathcal{N=}\left[\sum_{n=0}^{N}\left|c_{n}\right|^{2}\sin^{2}\beta_{n}\right]^{-1/2}$ is a normalization factor. The final cavity state $\left|\psi_{c}\right\rangle$ is equivalent to the desired state $\left|\psi_{d}\right\rangle$ when the parameters $\beta_{n}$ and $\phi_{n}$ are appropriately chosen so that $d_{n}=\mathcal{N}c_{n}\sin\beta_{n}e^{i(n\chi_{qc}\tau+\phi_{n})}$. To maximize the success probability $P=1/\mathcal{N}^{2}$, we numerically optimize the parameters $\left|\alpha\right|$ and $\beta_{n}$, with the optimal values depending upon the desired state.

Our experiment is implemented with a superconducting circuit, where one transmon qubit is dispersively coupled to two three-dimensional cavities Wallraff et al. (2004); Paik et al. (2011); Vlastakis et al. (2013); Sun et al. (2014); Liu et al. (2016). The qubit, with a transition frequency $\omega_{q}/2\pi=5.345$ GHz, has an energy relaxation time $T_{1}=13.0$ $\mu$s and a pure dephasing time $T_{\phi}=13.8$ $\mu$s. Cavity 1, with a frequency $\omega_{s}/2\pi=8.230$ GHz and a photon lifetime $\tau_{s}=80$ $\mu$s, is used for storage of the photonic state. Cavity 2, with a frequency $\omega_{r}/2\pi=7.290$ GHz and a photon lifetime $\tau_{r}=44$ ns, is connected to a Josephson parametric amplifier Hatridge et al. (2011); Roy et al. (2015); Kamal et al. (2009); Murch et al. (2013), allowing for a high-fidelity (nearly-unity) and quantum non-demolition single-shot readout of the qubit state. The average readout photon number in the readout cavity is about five and the readout time is $240$ ns. For simplicity, the term “cavity” refers to the cavity 1. The dispersive frequency shift is $\chi_{qc}/2\pi=-1.44$ MHz. The experimental apparatus and readout properties can be found in the Supplementary Material and Ref. Liu et al. (2016).

The experimental sequence is outlined in Fig. 1. The initialization of the system to $\left|g,0\right\rangle$ is realized by the post-selection on the qubit’s ground state and the subsequent cavity parity measurement. The initial coherent state of the cavity is achieved by application of a microwave pulse, which produces a displacement operation $D(\alpha)$, transforming the cavity from the ground state $\left|0\right\rangle$ to $\left|\alpha\right\rangle$. This is followed by driving the qubit, initially in the ground state $\left|g\right\rangle$, with a pulse with $N+1$ frequency components resonant with the corresponding qubit-cavity transition frequencies, simultaneously implementing $N+1$ conditional qubit rotations: $\prod_{n=0}^{N}R_{e_{n}|n}(\beta_{n})$, where $R_{e_{n}|n}(\beta_{n})$ represents the qubit rotation on the Bloch sphere by an angle $\beta_{n}$ around the $e_{n}$ direction conditional on the cavity containing $n$ photons, with $e_{n}$ being on the equatorial plane with the azimuth angle $\phi_{n}+n\chi_{qc}\tau$. In order to have enough selectivity, each frequency component in the qubit pulse simply has a broad Gaussian envelop truncated to 4$\sigma=1.44~{}\mu$s ($\sigma_{f}=0.44$ MHz). Post-selecting on the excited qubit state, we obtain the desired cavity state. After preparation of the desired cavity state, the qubit is disentangled with the cavity and can be used to measure the Wigner function of the cavity for examination of the produced state. Following a technique devised by Lutterbach and Davidovich Lutterbach and Davidovich (1997) and demonstrated in cavity QED Bertet et al. (2002) and circuit QED Vlastakis et al. (2013); Sun et al. (2014); Liu et al. (2016), the Wigner quasiprobability distribution $W(\beta)$ is measured by a combination of the cavity’s displacement operation $D(-\beta)$ and qubit Ramsey-type measurement, where a conditional cavity $\pi$ phase shift $C(\pi)$ is sandwiched in between two unconditional qubit rotations $R_{y}(\pi/2)$.

As an example, we prepared the truncated phase states Pegg and Barnett (1997)

[TABLE]

with $N=5$ to $7$, where $\theta_{\mathrm{N,k}}=2k\pi/(N+1)$. Apart from the fundamental interest, such states are useful for implementation of quantum walk Sanders et al. (2003). The Wigner functions of the ideal states and the produced states with $\theta_{\mathrm{N,k}}=0$ are shown in the first and second rows of Fig. 2, respectively. We note that, mainly due to the decoherence of the system (see Supplementary Material), the contrast of the measured Wigner function is reduced by a factor $R$: $W_{m}(\beta)=RW_{r}(\beta)$, where $W_{m}(\beta)$ and $W_{r}(\beta)$ are the measured and real Wigner functions of the produced state Vlastakis et al. (2015). The reduction factor is $R=\int W_{m}(\beta)d^{2}\beta$. To make the results physically meaningful, we normalize the measured Wigner functions, taking $W_{m}(\beta)/R$ as the real Wigner functions of the produced states. The shapes of the Wigner functions of the produced states agree very well with those of the ideal ones, implying that the quasiclassical states have been converted into the desired quantum states with high accuracy. The amplitudes $\alpha$ of the initial coherent states, Wigner function reduction factors $R$, success probabilities $P$, and fidelities $F$ between the produced states and desired ones are detailed in Table 1. Here the fidelity is defined as $F=\left\langle\theta_{\mathrm{{N,k}}}\right|\rho_{p}\left|\theta_{\mathrm{{N,k}}}\right\rangle$, where $\rho_{p}$ is the density operator of the produced state reconstructed by least-square regression using a maximum likelihood estimation Smolin et al. (2012); Vlastakis et al. (2015). The infidelity is mainly due to the decoherence of the system and imperfection of the conditional qubit rotations. However, since the desired cavity state is obtained by post-selection on the qubit’s excited state $\left|e\right\rangle$, its fidelity is actually insensitive to the decoherence of the qubit (see Supplementary Material) as demonstrated by the measured results in Table 1.

In Fig. 2, we also display the real and imaginary parts of the density matrices in the Hilbert space obtained from the normalized measured Wigner functions (the third and fourth rows, respectively). As expected, the first $N+1$ photon number states are almost equally populated, while the others remain unpopulated. The quantum coherence of the superposed Fock states is manifested in the off-diagonal elements $\rho_{m,n}$ with $m\neq n$. The non-vanishing imaginary parts of the off-diagonal elements are mainly due to the imperfect setting of the relative phases of different driving microwave components, and they can be minimized further by more careful optimization. The value of the matrix element $\rho_{m,n}$ ($0\leq m,n\leq N$) slightly decreases as $m$ and $n$ increases since the decay rate is proportional to $(m+n)/2$. The results further show that the produced states are in good agreement with the ideal ones, demonstrating high-fidelity generation of the desired states.

As another example to demonstrate the ability to generate any superposition state, we produce the squeezed state

[TABLE]

with the squeeze parameter $\xi=re^{i\theta}$. For such a state the fluctuation of the quadrature $X(\theta/2)=(a^{\dagger}e^{i\theta/2}+ae^{-i\theta/2})/2$ is reduced by a factor $e^{-2r}$ compared to the vacuum level, so that it can be used for high-precision measurement Caves et al. (1980). The squeezed state can be well approximated by truncating the expansion into a superposition of the first $N+1$ Fock states, with $N$ depending on $r$. We approximately produce the squeezed states for $\xi=0.8$, $0.8i$, and $-0.8$ with a cutoff of $N=8$. The good agreement in the calculated and measured Wigner functions (Fig. 3) indicates the full controllability of both the amplitude and phase associated with each Fock state component. Table 2 shows the corresponding $R$, $P$, $F$, and $\langle(\Delta X(\theta/2))^{2}\rangle$ [obtained from the Gaussian fit of the probability distribution of $X(\theta/2)$ calculated by integrating the measured Wigner function over $X(\theta/2+\pi/2)$] in the produced states. The measured $\langle(\Delta X(\theta/2))^{2}\rangle\ll 1/4$, showing that the initial quasiclassical states have been converted into quantum states with strong squeezing.

We note that, in the effective Hamiltonian of Eq. 1, we have ignored the cross-Kerr terms between the cavity and the readout cavity, $-\chi_{cr}a^{\dagger}aa_{r}^{\dagger}a_{r}$, where $a_{r}^{\dagger}$ and $a_{r}$ are the creation and annihilation operators of the readout cavity. This effect can either be corrected by properly setting the initial phase of $\left|\alpha\right\rangle$ or the qubit driving phases. We have also ignored higher order terms, among which the dominant terms are the cavity’s self-Kerr effect $-Ka^{\dagger 2}a^{2}/2$ and the correction to the dispersive qubit-cavity coupling $\chi_{qc}^{{}^{\prime}}a^{\dagger 2}a^{2}\left|e\right\rangle\left\langle e\right|/2$. Each of these terms causes additional photon-number-dependent phase shifts; however, the resulting effect is small, as evidenced by the high-fidelity of our experimental result.

In summary, we have proposed and experimentally demonstrated a new single-step scheme to produce arbitrary superposition of Fock states for a cavity. We show that a coherent state can be conditionally converted to any desired superposition state by tailoring the probability amplitudes associated with the superposed Fock state components. This is achieved based on the experimentally convenient dispersive coupling between the cavity and the qubit. The success probability of the scheme depends on the desired state. The achieved high fidelity of the experimental states prepared with this method opens up interesting applications of such non-classical states for implementation of quantum information and precision measurements. Of particular interest are the demonstrated truncated phase states that are directly applicable in quantum walk, which allows exponential speedup over classical computation for certain problems Childs et al. (2003) and more importantly, can be used as a primitive for universal quantum computation Childs (2009).