Deterministically Encoding Quantum Information Using 100...

/ http://www.sciencemag.org/content/early/recent / 26 September 2013 / / 10.1126/science.1243289

Cavity quantum electrodynamics is a test-bed system for quantum optics allowing the observation of strong interactions between photons and (artificial) atoms (1–3). Techniques utilizing these systems allow the production of non-classical states of light, which have important uses for quantum communication, quantum computation, and investigations of fundamental quantum theory. For superconducting quantum circuits, cavity resonators have proven a valuable resource serving several roles: a quantum bus to generate entanglement between qubits (4); a quantum non-demolition probe allowing efficient quantum measurements (5, 6); a generator of single microwave photons (7, 8); and a quantum memory to store and shuttle information (9, 10). With its large Hilbert space, a cavi-ty resonator also has the potential to store multiple quantum bits or re-dundantly encode information as necessary for quantum error correction. With the proper controls, a single cavity could be made equivalent to a multi-qubit register allowing for simplifications of hardware design (11, 12). While there have been some investigations of complex, multi-photon superpositions in superconducting cavity resonators, most tech-niques developed so far require fast qubit frequency tunability and are based on controlling individual photons one-by-one (13, 14). These implementations become increasingly burdensome for complex states or large photon numbers, making it desirable to develop a more natural method for controlling the large cavity Hilbert space.

We demonstrate a set of multi-photon operations using a fixed-frequency superconducting transmon qubit coupled to a waveguide cavi-ty resonator. We realize a highly ideal strong- dispersive coupling, where the strength of the off-resonant qubit-cavity interactions are sev-eral orders of magnitude greater than the cavity decay rate and higher

order non-linearities. This creates a set of qubit-cavity entangling operations allowing for control over the large cavi-ty phase space. We implement two of these operations: the qubit-state condi-tional cavity phase shift (15) and the photon-number conditional qubit rota-tion (14, 16). We combine these with unconditional qubit and cavity opera-tions to perform direct measurements of the cavity Wigner function (17) and to efficiently generate large superposition states. Using these tools, we realize a recently proposed protocol (18) to de-terministically encode quantum infor-mation in a cat state by creating an arbitrary superposition of quasi-orthogonal coherent states conditioned on an initial qubit state. We concatenate these entangling operations to encode quantum information into multiple phases of the cavity state thereby creat-ing multi-component cat states and producing example states proposed for high-precision measurements surpas-sing the quantum-noise limit (19, 20). Unlike previous demonstrations of cat states (16, 21, 22), the procedures real-ized here allow for the deterministic generation of superimposed coherent states with arbitrary phase and ampli-tude. The set of operations demonstrat-ed provides an efficient method to manipulate coherent states and could enable a variety of powerful methods for employing cavity states in quantum

information processing. We realize qubit-photon quantum logic utilizing a strong off-

resonant coupling of a qubit and cavity which can be described by the dispersive Hamiltonian (omitting higher order non-linear terms)

† †q s qsH e e a a a a e e= ω + ω − χ (1)

where |e⟩ is the excited state of the qubit, a†(a) are the raising (lowering) ladder operators of the cavity resonator, ωq,s are qubit and cavity transi-tion frequencies, and χqs is the dispersive interaction between these modes. This interaction produces a state-dependent shift in either the qubit or cavity transition frequency. We exploit this conditional frequen-cy shift to produce qubit-photon entanglement with two operations: con-ditional cavity phase shifts and conditional qubit rotations. The conditional cavity phase shift can be described as

(2)

where |g⟩ is the ground state of the qubit and Φ is the conditional phase shift induced on the cavity state. This conditional phase appears from the free evolution of the dispersive Hamiltonian for a time τ where Φ = χqsτ. For example, acting a conditional cavity phase shift on a coherent state while the qubit is in a superposition state produces an entangled qubit/cavity state, ( ){ } , ,iC g e g e eΦ

Φ α ⊗ + = α + α (disregarding

normalization) (15). Noting that a coherent state 2

20 !

n

ne nn

α− ∞

=α

α = ∑

is a superposition of Fock states |n⟩ represented by a complex value α,

Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States Brian Vlastakis,1∗ Gerhard Kirchmair,1† Zaki Leghtas,1,2 Simon E. Nigg,1‡ Luigi Frunzio,1 S. M. Girvin,1 Mazyar Mirrahimi,1,2 M. H. Devoret,1 R. J. Schoelkopf1 1Departments of Physics and Applied Physics, Yale University, New Haven, CT 06511, USA. 2INRIA Paris-Rocquencourt, 78153 Le Chesnay Cedex, France.

∗Corresponding author. E-mail: [email protected]

†Present address:Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria and Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria.

‡Present address: Department of Physics, University of Basel, CH-4056 Basel, Switzerland.

In contrast to a single quantum bit, an oscillator can store multiple excitations and coherences provided one has the ability to generate and manipulate complex multi-photon states. We demonstrate multi-photon control using a superconducting transmon qubit coupled to a waveguide cavity resonator with a highly ideal off-resonant coupling. This dispersive interaction is much greater than decoherence rates and higher-order non-linearities to allow simultaneous manipulation of hundreds of photons. With a toolset of conditional qubit-photon logic, we map an arbitrary qubit state to a superposition of coherent states, known as a ‘cat state’. We create cat states as large as 111 photons and extend this protocol to create superpositions of up to four coherent states. This control creates a powerful interface between discrete and continuous variable quantum computation and could enable applications in metrology and quantum information processing.

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this operation enables us to encode the qubit state information into the cavity phase and entangle the qubit with many photons simultaneously. For a special case, CΦ=π, the cavity state attains a conditional π shift per photon. This, in turn, causes the qubit state to acquire a phase shift con-ditional on there being exactly an odd number of photons in the cavity, resulting in a mapping of the cavity photon number parity to the qubit state. By using Ramsey interferometry, we utilize this gate to measure the cavity photon parity and ultimately the cavity Wigner function (17, 23, 24). The Wigner function is a representation of a quantum state in a continuous variable basis and can be expressed as

( ) †2 TrW D D Pα α α = ρ πwhere

† *a aD eα −αα = is the cavity displacement

operator, †i a aP e π= is the photon number parity operator, and ρ is the

cavity state density matrix. A direct Wigner measurement is produced by measuring the mean photon parity ⟨P⟩ at many points in the cavity phase space which we use to completely determine the quantum state of the cavity.

The second operation, the conditional qubit rotation, is a rotation on the qubit state conditional on the photon number of the cavity state. Because the qubit transition frequency is strongly photon number de-pendent, we can drive a particular transition selective on a cavity Fock state (14, 16). A rotation on the qubit state conditioned on the mth photon Fock state can ideally be described as

(3)

where ˆ,nR θ is a qubit rotation about a vector n̂ with rotation angle θ. In practice, this operation will result in residual photon-dependent phase shifts due to the AC Stark effect, which we can correct for the purposes of this experiment (24). In order to realize these two entangling operations, we must achieve dispersive shifts much greater than the qubit and cavity decoherence rates, max,»qs snχ γ κ where γ is the qubit decay rate, κs is the storage cavity decay rate, and nmax is the maximum occupied pho-ton number. This dispersive approximation (Eq. 1) is valid in a low pho-ton number regime where the dispersive interaction †

qsa a e eχ is much greater than higher order non-linear terms such as the cavity self-Kerr † 2 2

sK a a and the non-linearity of the dispersive shift † 2 2

qsa a e e′χ (25). Combined with unconditional qubit/cavity manipu-lations,these two entangling operations give us a powerful toolset for controlling the joint qubit/cavity system (12, 18, 24).

Our experiment consists of two waveguide cavity resonators (16, 26) coupled to a transmon qubit (Fig. 1A). Cavity 1 is used for photon state manipulation, preparation, and storage with transition frequency

8.18 GHz2

sω =π

and decay rate 17.2 kHz2 2 22.1 s

sκ = =π π× µ

(limited by

internal losses). Cavity 2 is used for qubit state detection with transition

frequency 9.362

rω =π

GHz and decay rate 1330 kHz2 2 480ns

rκ = =π π×

(limited by output coupling for increased readout fidelity). Both cavities are coupled to a ‘vertical’ transmon qubit (16, 24) with transition fre-

quency 7.46 GHz2

qω =π

and decay rate 136 kHz2 2 4.4 sγ= =

π π× µ (lim-

ited by internal losses). This system creates a dispersive interaction between the qubit and storage cavity mode resulting in a state-dependent

frequency shift 2.4 MHz2

qsχ=

π. We independently measure (24) higher

order non-linear terms Ks and qs′χ allowing us to put a limit on the max-

imum accessible photon number for this experiment:

max min / 560, / 650, / 330« qs qs qs s qs sn n K ′= χ χ = χ = χ κ = .

By combining a conditional cavity phase shift with a conditional qubit rotation, we sequentially entangle then disentangle the qubit and cavity to map a qubit state to a superposition of quasi-orthogonal coher-ent states (18). Following the sequence outlined in Fig. 1B, we start with an unentangled qubit/cavity state ( )0 g eψ = β ⊗ + (disregarding

normalization) where |β⟩ is a coherent state. Performing a conditional cavity π phase shift on the initialized state creates an entangled qubit-cavity state 1 0 , ,C g eπψ = ψ = β + −β . This state, where the qubit state is entangled with the phases of the superimposed coherent states, is often referred to as a ‘Schrödinger cat’ which has been studied in other quantum systems (15, 27). We can unconditionally displace this state to obtain 2 1 2 , 0,D g eβψ = ψ = β + .

At this point, we can apply a qubit π rotation conditional on the cavi-ty vacuum state |0⟩ which produces the unentangled cat state

( )0ˆ3 2, 2 0yR gπψ ≈ ψ = β + ⊗ . Due to the non-orthogonality of

coherent states, this operation will leave some remaining entanglement which rapidly decreases with cat state size and can be neglected com-pared to other experimental imperfections. An additional displacement results in the final state ( )4 3D g−βψ = ψ = β + −β ⊗ . As an ex-

ample, we create the target cavity state ( )targψ = β + −β where

12

≈ with | | 7β = , resulting in a fidelity F = ⟨ψtarg|ρ|ψtarg⟩ = 0.81,

which we confirm by direct Wigner tomography (Fig. 2A). This proce-dure can be generalized to any arbitrary qubit state and cavity phase which maps as

( ) ( ){ } ( ) ( ){ }2 2 2 20 cos sin cos sini i ig e e e e gφ φ Φθ θ θ θ⊗ + → β + β ⊗ (4)

where θ and φ are parameters of the initial qubit state and when the su-perimposed coherent states are sufficiently orthogonal 2| 1«| | iΦ⟨β β ⟩ . Figure 2B shows the creation of cat states conditioned by qubit states prepared at the six cardinal points of the Bloch sphere. Two special forms of cat states result in complete destructive interference of either the odd or even Fock state amplitudes. Known as the even (odd) cat states |β⟩ ± |−β⟩, these states produce superpositions of only even (odd) photon numbers. This interference can be showcased in the dispersive regime by performing qubit spectroscopy after the creation of one of these cavity states. Due to the strong-dispersive interaction, each spectral peak reveals a photon number probability of the prepared cavity state (28). For a coherent state |β⟩, the qubit spectrum will represent a Pois-

sonian photon number distribution 2| | 22 | |( )

!

n

neP n

n

− β ββ = β = . An

even and odd cat state of equivalent amplitude follows this same enve-lope but with destructive interference for the odd and even photon num-

ber states respectively, ( ) ( )2| | 2| |1

!

ni n

neP e

n

− βπ β

β ± −β ≈ ± . We perform

spectroscopy on the qubit with three prepared states: |β, g⟩, {|β⟩ + |−β⟩} ⊗ |g⟩, and {|β⟩ − |−β⟩} ⊗ |g⟩ for |β| = 2.3, illustrating the discreteness of the electromagnetic signals in the cavity and revealing the non-classical nature of the generated cat states (Fig. 3A).

Unlike building photon superpositions one-by-one (13, 14), this mapping protocol can scale to cavity states with larger quantum super-positions by merely increasing the displacement amplitude. The size of a quantum superposition in a cat state S = |β1 − β2|2 is determined by its square distance in phase space between the two superimposed coherent states |β1⟩, |β2⟩ (21). To characterize S without performing full state to-


mography, we measure cuts of the Wigner function along the axis per-pendicular to its quantum interference, W (Re(α) = 0, Im(α)). The inter-ference fringes in these cuts appear as

( ) ( )22|Im( )|0, Im( ) cos 2 Im( )W Ae S− αα = α + δ , where A and δ are the

fringe amplitude and phase (29). Using this method, we create and con-firm cat states with sizes from 18 to 111 photons (Fig. 3B). The in-creased oscillation rate of these fringes with S shows the increased sensitivity to small displacements in cavity field due to larger quantum superpositions (see (24) for proof-of-principle Heisenberg-limited phase estimation). Reduced fringe visibility with larger sizes is due to the in-creased sensitivity to cavity decay. Other factors also contribute, namely infidelity in Wigner tomography and qubit decoherence during prepara-tion.

Since the methods outlined here are deterministic, entangling opera-tions can be combined to create complex non-classical cavity states. Using a conditional cavity phase shift for various phases Φ, we can en-code quantum information to a particular phase of a coherent state. Us-ing this operation, we can create superpositions of multiple coherent states, multi-component cat states. We use gates C2π/3 and Cπ/2 to create three- and four-component cat states (Fig. 4A,B,C) with fidelity FA = 0·60, FB = 0·58, and FC = 0·52 (24). The skewing of these states is caused by the inherited cavity self-Kerr. Additional factors contributing to infidelity include photon decay during preparation and measurement as well as tomography pulse errors. Note the state in Fig. 4C, also known as the ‘compass state’, contains overlapping interference fringes reveal-ing increased sensitivity to cavity displacements in both quadratures simultaneously (19).

We have demonstrated the efficient generation and detection of co-herent state superpositions using off-resonant interactions inherent in the cavity QED architecture. The tools and techniques described here re-quire only a fixed-frequency, strong-dispersive interaction and realize an interface between discrete and continuous variable quantum computation (30). This can lead to simplified methods for individual storage and re-trieval of multi-qubit states in a cavity resonator and creates ways to perform multi-qubit stabilizer measurements (31) or to redundantly en-code information for quantum error correction (12) using minimal hard-ware. Additional applications include Heisenberg-limited measurement (19, 20, 24) and quantum information storage in thermally excited reso-nator states (32).

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discussions. This research was supported by the National Science Foundation (NSF) (PHY-0969725), the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) through the Army Research Office (W911NF-09-1-0369), and the U.S. Army Research Office (W911NF-09-1-0514). Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering (YINQE) and the NSF (MRSECDMR 1119826). SMG and ZL acknowledge support from the NSF (DMR-1004406). MM and ZL acknowledge support from French Agence Nationale de la Recherche under the project EPOQ2 (ANR-09-JCJC-0070).

Supplementary Materials www.sciencemag.org/cgi/content/full/science.1243289/DC1 Materials and Methods Supplementary Text Figs. S1 to S10 Tables SI and SII References (33–37)

16 July 2013; accepted 18 September 2013 Published online 26 September 2013 10.1126/science.1243289


Fig. 2. Wigner tomography of coherent state superpositions. (A) Wigner tomography of the cavity state ( )ψ = β + −β with | | 7β = and 1

2≈ using an 81 × 81 grid of tomography displacements showcases the

interference fringes characteristic of a quantum superposition. Cuts along the real and imaginary axes reveal the relative population and quantum interference of the superimposed coherent states. The visibility of these interference fringes is reduced due to cavity decay during preparation and measurement (a perfect superposition would achieve unity mean photon parity). (B) Using qubit states initially prepared in the six cardinal points of the Bloch sphere, we map populations and phases in a resulting cat state.

Fig. 1. Experimental device and protocol. (A) A cross-section shows the device, machined from two halves of aluminum alloy, which contains two cavity resonators and holds a sapphire chip with a lithographically patterned transmon qubit. (B) The protocol for mapping and measuring a qubit state into a superposition of coherent states is performed in three steps: qubit state preparation (red) using a single qubit rotation ˆ ,nR θ ; qubit to cavity state mapping (blue) using conditional operations Cπ and 0

ˆ,xR π with cavity displacements Dβ; and cavity state Wigner tomography (green) using Ramsey interferometry with unconditional qubit rotations

ˆ ,2

xR π .


Fig. 4. Multi-component cat states. Using conditional cavity phase shifts and Cπ/2, we create superpositions of three and four coherent states. Shown here is Wigner tomography of cavity states: (A) 1 22 /3 4 /3i ii ie e e eλ λπ πβ + β + β where | | 7β = , 1 0.6λ = π , and λ2 = −0.3π; (B)

31 2 /3 2 /30 ii i i ie i e e e eµµ µ π π+ − β + β + β where | | 7β = , μ1 = 0.5π, μ2 = −0.4π, and μ3 = −0.2π; and (C)

1 2i ie i e iν νβ + β + −β + − β where | | 7β = , ν1 = π, and ν2 = −0.2π. (D) A closer inspection of the quantum interference in (C) reveals increased sensitivity to cavity displacements in both quadratures simultaneously.

Fig. 3. Qubit spectroscopy and scaling to large photon superpositions (A) Photon number splitting is observed when performing spectroscopy of a qubit dispersively coupled to the storage cavity with three different prepared states: a coherent state |β⟩, even cat state |β⟩ + |−β⟩, and odd cat state |β⟩ − |−β⟩ with amplitude |β| = 2.3. Each spectral peak corresponds to the probability for a photon number state following a Poissonian distribution. Dashed bar plots show the expected photon probabilities for each of these states. Notice that even and odd cat states show destructive interference for the odd and even photon numbers. These spectra are acquired by deconvolving the measured signal with the Fourier spectrum of the finite-width spectroscopy pulse. (B) Cuts along the imaginary axis of the measured Wigner function for each prepared cat state reveal quantum superpositions with up to 111 photons in size. Cat state size S is determined by these measured interference fringes following the relation

( )22|Im( )| cos 2 Im( )Ae S− α α + δ where S, A and δ are fit parameters.

Deterministically Encoding Quantum Information Using 100...

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