aa r X i v : . [ qu a n t - ph ] O c t Schr¨odinger Cat States in Circuit QED
Lectures presented at the Les Houches SummerSchool, Session CVII–Current Trends in AtomicPhysics, July 2016.
S. M. Girvin
Yale Quantum InstitutePO Box 208334New Haven, CT 06520-8263 USA reface The last 15 years have seen spectacular experimental progress in our ability to create,control and measure the quantum states of superconducting ‘artificial atoms’ (qubits)and microwave photons stored in resonators. In addition to being a novel testbed forstudying strong-coupling quantum electrodynamics in a radically new regime, ‘circuitQED,’ defines a fundamental architecture for the creation of a quantum computerbased on integrated circuits with semiconductors replaced by superconductors. Theartificial atoms are based on the Josephson tunnel junction and their relatively largesize ( ∼ mm) means that the couple extremely strongly to individual microwave pho-tons. This strong coupling yields very powerful state-manipulation and measurementcapabilities, including the ability to create extremely large ( >
100 photon) ‘cat’ statesand easily measure novel quantities such as the photon number parity. These new ca-pabilities have enabled a highly successful scheme for quantum error correction basedon encoding quantum information in Schr¨odinger cat states of photons. cknowledgements
The ideas described here represent the collaborative efforts of many students, postdocsand faculty colleagues who have been members of the Yale quantum information teamover the past 15 years. The author is especially grateful for the opportunities he has hadto collaborate with long-time friends and colleagues Michel Devoret, Leonid Glazman,Liang Jiang and Rob Schoelkopf as well as frequent visitor, Mazyar Mirrahimi. Most ofthe ideas presented in these notes originated primarily with them and not the author.This work was supported by the National Science Foundation through grant DMR-1609326, by the Army Research Office and the Laboratory of Physical Sciences throughgrant ARO W911NF1410011, and by the Yale Center for Research Computing andthe Yale Quantum Institute. ontents
Appendix A The Wigner Function and Displaced Parity Mea-surements References Schr¨odinger Cat States in CircuitQED
Schr¨odinger Cat States in Circuit QED
Circuit quantum electrodynamics (‘circuit QED’) (Blais et al. , 2004; Wallraff et al. , 2004;Blais et al. , 2007; Schoelkopf and Girvin, 2008; Girvin, 2014; Devoret and Schoelkopf, 2013;Nigg et al. , 2012; Solgun and DiVincenzo, 2015; Vool and Devoret, 2016) describes thequantum mechanics and quantum field theory of superconducting electrical circuitsoperating in the microwave regime near absolute zero temperature. It is the analogof cavity QED in quantum optics with the role of the atoms being played by su-perconducting qubits. A detailed pedagogical introduction to the subject with manyreferences is available in the author’s lecture notes from the 2011 Les Houches Schoolon Quantum Machines (Girvin, 2014). The present notes will therefore provide onlya brief introductory review of the subject and will focus primarily on novel quantumstates that can be produced using the strong coupling between the artificial atom andone or more cavities.It is a basic fact of QED that each normal mode of the electromagnetic field is anindependent harmonic oscillator with the quanta of each oscillator being photons inthe corresponding mode. For an optical or microwave cavity, such modes are standingwaves trapped inside the cavity. Cavities typically contain many-modes, but for sim-plicity we will focus here on the case of a single mode, coupled to an artificial atomapproximated as having only two-levels (and hence described as a pseudo spin-1/2).This leads to the Jaynes-Cummings model (Blais et al. , 2004; Girvin, 2014) H = ˜ ω c a † a + ˜ ω q σ z + g [ aσ + + a † σ − ] (1.1)describing the exchange of energy between the cavity and the atom via photon ab-sorption and emission within the RWA (rotating wave approximation). Here ˜ ω c is the(bare) cavity frequency, ˜ ω q is the (bare) qubit transition frequency and g is the vacuumRabi coupling which, because of the large qubit size and small cavity volume can beenormously large ( ∼ ∼ et al. , 2007) for the Jaynes-Cummings model for dif-ferent realizations of cavity and circuit QED is illustrated in Fig. (1.1). The verticalaxis represents the strength g of the atom-photon coupling in the cavity and the hori-zontal axis represents the detuning ∆ = ˜ ω q − ˜ ω c between the atomic transition and thecavity frequency. Both are expressed in units of Γ = max [ γ, κ, /T ], where γ and κ arethe qubit and cavity decay rates respectively, and for the case of (‘real’) atoms, T is thetransit time for the atoms passing through the cavity In the region g ≥ ∆ the qubitand cavity are close to resonant and first-order degenerate perturbation theory applies.The lowest two excited eigenstates are the upper and lower ‘polaritons’ which are co-herent superpositions of atom+photon excitations (Blais et al. , 2004; Girvin, 2014).They are essentially the bonding and anti-bonding combinations of ‘atom ± photon.’An excited state atom introduced into the cavity will ‘Rabi flop’ coherently betweenbeing an atomic excitation and a photon at the ‘vacuum Rabi rate’ 2 g (which is the Another benefit of circuit QED is that our artificial atoms are ‘glued down’ and stay in the cavityindefinitely. ntroduction to Circuit QED energy splitting between the upper and lower polariton). Within this degenerate re-gion there is a ‘strong-coupling’ regime g ≫ Γ, in which the cavity and qubit undergomultiple vacuum Rabi oscillations prior to decay.
Fig. 1.1
A phase diagram for cavity QED. The parameter space is described by the atom--photon coupling strength, g , and the detuning ∆ between the atom and cavity frequencies,normalized to the rates of decay represented by Γ = max [ γ, κ, /T ]. Here γ and κ are thequbit and cavity decay rates respectively, and T is the transit time for atoms passing throughthe cavity. Different cavity QED systems, including Rydberg atoms, alkali atoms, quantumdots, and circuit QED, are represented by dashed horizontal lines. Since the time this graphwas first constructed Γ has decreased dramatically putting circuit QED systems much deeperinto the strong dispersive regime. Adapted from (Schuster et al., 2007). We will focus here on the case where the detuning ∆ between the qubit and thecavity is large (∆ ≫ g ). Because of the mismatch in frequencies, the qubit can only vir-tually exchange energy with the cavity and the vacuum Rabi coupling can be treatedin second-order perturbation theory. Applying a unitary transformation which elim-inates the first-order effects of g and keeping only terms up to second-order yields(Blais et al. , 2004; Girvin, 2014) H = ω c a † a + ω q | e ih e | − χa † a | e ih e | , (1.2)where we have shifted the overall energy by an irrelevant constant, ω c and ω q arerenormalized cavity and qubit frequencies and | e ih e | is the projector onto the excitedstate of the qubit. The quantity χ ≈ g ∆ ∼ π ∗ (2 − In the dispersive regime, the qubit acts like adielectric whose dielectric constant depends on the state of the qubit. This causes thecavity frequency to depend on the state of the qubit. The simple expression given here for the dispersive shift is not accurate for the so-called transmonqubit because it is a weakly anharmonic oscillator and not necessarily well-approximated as a two-levelsystem. For a quantitatively more accurate approach see (Nigg et al. , 2012; Girvin, 2014).
Schr¨odinger Cat States in Circuit QED
As illustrated in Fig. 1.2, when the qubit is in the excited state, the cavity frequencyshifts by − χ . As we will soon discuss, this dispersive shift has a number of usefulapplications, but the simplest is that it can be used to read out the state of the qubit(Blais et al. , 2004). We can find the qubit state by measuring the cavity frequency bysimply reflecting microwaves from it and measuring the resulting phase shift of thesignal. Because the cavity frequency is very different from the qubit frequency, thephotons in the readout do not excite or deexcite the qubit and the operation is non-destructive. Even though χ is a second-order effect in g it still can be several thousandtimes larger than the respective cavity and qubit decay rates, κ and γ . This is theso-called ‘strong-dispersive’ regime (Schuster et al. , 2007; Girvin, 2014) of the phasediagram. The ability to easily enter this regime gives circuit QED great advantagesover ordinary quantum optics and will prove highly advantageous for quantum statemanipulation, control and error correction.The very same dispersive coupling term can also be viewed as producing a quantized‘light shift’ of the qubit transition frequency by an amount − χ for each photon that isadded to the cavity. Fig. 1.3 shows quantum jump spectroscopy data illustrating thisquantized light shift. In ordinary spectroscopy of a medium (a gas, say) one measuresthe spectrum by the absorption of the spectroscopic light at different frequencies. Herewe instead use quantum jump spectroscopy in which we use the dispersive readout ofthe state of the qubit to determine when the qubit jumps to the excited state as aresult of excitation by the spectroscopy tone.The dispersive term commutes with both cavity photon number and qubit exci-tation number. It is thus ‘doubly QND,’ and can be utilized to make quantum non-demolition measurements of the cavity (using the qubit) and the qubit (using thecavity). In the ‘strong-dispersive’ region χ ≫ Γ, quantum non-demolition measure-ments of photon number are possible. We will see later that in this regime it is evenpossible to make a QND measurement of the photon number parity without learningthe value of the photon number!In the ‘weak-dispersive regime’ ( χ ≪ Γ) of the phase diagram, the light shift perphoton is too small to dispersively resolve individual photons, but a QND measurementof the qubit can still be realized by using many photons. The qubit-dependent cavityfrequency shift is less than the linewidth of the cavity but with enough photons over along enough period, this small frequency shift can be detected (using the small phaseshift of the reflected signal, assuming that the qubit lifetime is longer than the requiredmesaurement time) (Clerk et al. , 2010).
In the strong-dispersive regime, if we measure the qubit transition frequency the stateof the cavity collapses to a Fock state with definite photon number determined by themeasured light-shift of the qubit transition frequency. Unlike a photomultiplier, thismeasurement is not only photon-number resolving, it is QND. The photon is not ab-sorbed and we can repeat the measurement many times to overcome any imperfectionsin the measurement (improving the quantum efficiency and lowering the dark count).This novel feature has the potential to dramatically accelerate searches for axion dark easurement of Photon Number Parity (cid:90) ge c (cid:90) c (cid:90) (cid:70)(cid:16) (cid:78) ‘strong-dispersive’ limit ~ 10 (cid:70) (cid:78) Fig. 1.2
Illustration of cavity response (susceptibility) depending on the state of the qubit.In the qubit ground state, the cavity resonance frequency is ω c . When the qubit is in theexcited state, the cavity frequency shifts by − χ , which can be thousands of times larger thanthe cavity linewidth κ . This is the ‘strong-dispersive’ regime. When the dispersive shift is lessthan a linewidth of the cavity (or the qubit) we are in the ‘weak-dispersive’ regime. (power broadened 100X) Q ub i t s p e c t r a l d e n s i t y (cid:70) Fig. 1.3
Quantum jump spectroscopy of a transmon qubit dispersively coupled to a cavityillustrating the quantized light shift of the qubit transition frequency depending on how manyphotons are in the cavity. The weakly damped cavity is driven to produce a coherent statewhose time-averaged photon number distribution is Poisson. A weak spectroscopy tone is usedto excite the qubit at different frequencies. The spectral density of the qubit is determinedby the probability that the spectroscopy tone excites the qubit. In this case a relativelystrong spectroscopy tone was used, resulting in a power-broadening of the spectral line byapproximately 100x. The spectral peaks are actually about 10 times narrower than theirseparation. The symbol | n i denotes the number of photons in the cavity associated with eachspectral peak of the qubit. The quantization of the light shift clearly shows that microwavesare particles! Courtesy R. Schoelkopf group. matter particles which convert into microwave photons in the presence of a strongmagnetic field (Zheng et al. , 2016).Another extremely powerful feature of the strong-dispersive regime is that it givesus the ability to measure the parity of the photon number without learning the photonnumber itself. Why is this interesting? It turns out that in quantum systems, what we Schr¨odinger Cat States in Circuit QED do not measure is just as important as what we do measure! One way to measure theparity would be to measure the eigenvalue of the photon number operator ˆ n . If theresult is m , then we assign parity Π = ( − m . This process works, but it does a greatdeal of ‘damage’ to the state by collapsing it to the Fock state | m i . This particularmeasurement has strong ‘back action.’ Let us formalize this by considering an arbitrarycavity state | ψ i . The state of the system conditioned on the measurement result | ψ i m = | m ih m | ψ i [ h ψ | m ih m | ψ i ] = | m i (1.3)is completely independent of the starting state. (Only the probability of obtaining themeasurement outcome m depends on | ψ i .)If we somehow had a way to directly measure the eigenvalue of the photon numberparity operator ˆΠ = e iπa † a = ( − ˆ n , (1.4)the back action would be much weaker. This is because there are only two measurementoutcomes and the projection onto the even or odd subspaces only cuts out half of theoverall Hilbert space, rather than all but one Fock state. Conditioned on the paritymeasurement being ± | ψ ± i = ˆΠ ± | ψ i [ h ψ | ˆΠ ± | ψ i ] , (1.5)where ˆΠ + = X j ∈ even | j ih j | , (1.6)ˆΠ − = X j ∈ odd | j ih j | . (1.7)Because we don’t learn the value of the photon number, only its parity, the back actionis minimized to that associated with the information we wanted to learn. In the firstmethod we learned too much information (namely the value of the photon numbernot just its parity). The importance of what you do not measure will come to the forewhen we discuss how Schr¨odinger cat states can be used for quantum error correction.The parity operator in eqn 1.4 is a non-trivial function of the number operator. Atfirst sight it appears to be quite difficult to measure. It turns out however to be verystraightforward to create a situation in which we entangle the state of the qubit withthe parity of the cavity. Measuring the qubit then tells us the parity and nothing more.To see how this works, let us move to the interaction picture (i.e. go to an appropriate More formally, the projector onto Fock state | m i has unit trace, Tr | m ih m | = 1, while the parity-subspace projectors defined in eqns 1.6-1.7 have infinite trace, Tr ˆΠ ± = Tr ˆ I = ∞ . Once could (inprinciple) map the entire Hilbert space into the even or the odd subspaces! easurement of Photon Number Parity rotating frame) where the time evolution is governed solely by the dispersive coupling.Moving to a rotating frame via the time-dependent unitary U ( t ) = e iH t , (1.8)where H = ω c a † a + ω q | e ih e | , (1.9)the new Hamiltonian in the rotating frame becomes V = U HU † − U i ddt U † = − χa † a | e ih e | . (1.10)Now consider time evolution under this Hamiltonian for a period t = π/χ . Theunitary evolution operator is U π = e + iπa † a | e ih e | . (1.11)Using the fact that | e ih e | is a projector, we can reformulate this in terms of the photonnumber parity operator as U π = | g ih g | + ˆΠ | e ih e | = ˆΠ + ˆ I + ˆΠ − σ z , (1.12)where | g ih g | is the projector onto the qubit ground state. Now sandwich this withHadamard operators. These operators interchange the roles of the x and z componentsof the spin: H | g i = | + x i and H | e i = | − x i ). This yields HU π H = ˆΠ + ˆ I + ˆΠ − σ x . (1.13)Thus if the photon number is even, nothing happens to the qubit, whereas if the photonnumber is odd the qubit is flipped. Hence we have successfully entangled the paritywith the qubit state. Starting with the qubit in the ground state, applying HU π H andthen measuring the state of the qubit tells us the parity but not the photon number.Essentially what we have done is the following. We use the Hadamard gate to putthe spin in the + x direction on the Bloch sphere. We then allow the qubit to precessaround the z axis at a rate that depends on the quantized light shift. If the parityis even, the precession is through an even integer multiple of π radians and returnsthe qubit to the starting point (and erasing the information on the value of the eveninteger). If the parity is odd, the qubit precesses through an odd integer multiple of π radians and ends up pointing in the − x direction on the Bloch sphere. The secondHadamard gate converts σ x to σ z which we then measure to determine the parity ofthe cavity photon number.Provided that we are in the strong-dispersive limit (more precisely, provided that χ ≫ ¯ nκ where ¯ n = h ψ | ˆ n | ψ i ) then there is very little chance that an error will resultfrom the cavity losing a photon during the course of the parity measurement. Onecould also imagine other errors which would make the measurement non-QND. Themanipulations of the qubit could for example accidentally add a photon to the cavity.This is unlikely however because the qubit and cavity are strongly detuned from eachother. Ref. (Ofek et al. , 2016) were able to make parity measurements of a high-Qstorage cavity that were 99.8% QND and hence could be repeated hundreds of timeswithout extraneous damage to the state. (The fidelity with which the parity could bedetermined in a single measurement was 98.5% due to uncertainties in the readout, butthese uncertainties were not associated with any adverse back action on the cavity.) Schr¨odinger Cat States in Circuit QED
Quantum computation relies on the ability to create and control complex quantumstates. The dimension of the Hilbert space grows exponentially with the number ofqubits and the task of verifying the accuracy of a particular state (state tomography)or verifying a particular transformation of that state (process tomography) becomesexponentially difficult. To fully determine the state, one has to be able to measure notjust the states of individual qubits, but also measure non-local multi-qubit correlatorsof arbitrary weight h σ zi σ zj σ xk σ yl ....σ zn i , a task which is quite difficultFor bosonic states of cavities, the quantum jump spectroscopy illustrated in Fig. 1.3gives an excellent way to measure the photon number distribution in the state ofthe cavity. This however is far short of the information required to fully specify thequantum state. We need not just the probabilities of different photon numbers butthe probability amplitudes . In general, the state need not be pure (and the processeswe are studying need not be unitary) and so we need to be able to measure the fulldensity matrix which has both diagonal and off-diagonal elements.It turns out that the ability to make high-fidelity measurements of the photon num-ber parity gives an extremely simple and powerful way to measure the Wigner func-tion. As shown in App. A, the Wigner function provides precisely the same informationstored in the density matrix but displays it in a very convenient and intuitive format.The Wigner function W ( X, P ) is a quasi-probablilty distribution in phase space andcan be obtained by the following simple and direct recipe (Lutterbach and Davidovich, 1997):(1) displace the oscillator in phase space so that the point (
X, P ) moves to the origin;(2) then measure the value of the photon number parity. By repeating many times,one obtains the expectation value of this ‘displaced parity’ W ( X, P ) = 1 π ~ Tr n D ( − X, − P ) ρ D † ( − X, − P ) ˆΠ o , (1.14)where D is the displacement operator. As shown in App. A, this is precisely thedesired Wigner function which fully characterizes the (possibly mixed) state of thesystem. Related methods in which one measures not the parity but the full pho-ton number distribution of the displaced state (using quantum jump spectroscopy)can in principle yield even more robust results in the presence of measurement noise(Shen, Heeres, Reinhold, Jiang, Liu, Schoelkopf and Jiang, 2016).The Wigner function is a quasi-probability distribution. Unlike wave functions it isguaranteed to be real, but unlike classical probabilities, it can be negative. In quantumoptics states with negative-valued Wigner functions are defined to be ‘non-classical.’It should be noted however that the marginal distributions of momentum and positionare always ordinary positive-valued probability distributions (much like the square ofthe wave function) ρ ( P ) = Z + ∞−∞ dX W ( X, P ) (1.15) ρ ( X ) = Z + ∞−∞ dP W ( X, P ) . (1.16) pplication of Parity Measurements to State Tomography Just as the density matrix contains all the information ever needed to compute theexpectation value of any observable via h ˆ Oi = Tr [ ˆ O ρ ], the Wigner function of ρ canbe used to obtain the same quantity (Cahill and Glauber, 1969). Exercise 1.1
From the definition of the Wigner function in App. A prove that
Z Z dXdP W ( X, P ) = 1 . Since we are dealing with photons in a resonator, it is convenient to replace theposition and momentum coordinates of phase space by a single dimensionless complexnumber β that expresses both position and momentum in units of ‘square root ofphotons.’ The displacement operator is then given by D ( β ) = e βa † − β ∗ a . (1.17)We can check this expression by noting that D ( − β ) a D (+ β ) = a + β (1.18)and that the displacement of the vacuum yields the corresponding coherent state D ( β ) | i = e βa † e − β ∗ a e − [ βa † , − β ∗ a ] | i = e − | β | e βa † | i = | β i (1.19)with mean photon number ¯ n = | β | . In these units, the (now-dimensionless) Wignerfunction is given by W ( β ) = 2 π Tr n ρ D † ( − β ) ˆΠ D ( − β ) o , (1.20)where we have used the fact that the Jacobian for the transformation from X, P to β = β R + iβ I obeys dX dP = 2 ~ dβ R dβ I . (1.21) Exercise 1.2
Verify eqn 1.17. Hint: differentiate both sides of the equation with respect to(the magnitude of) β . Exercise 1.3 a) Verify eqn 1.21. b) Verify the normalization RR dβ R dβ I W ( β ) = 1 directlyfrom eqn 1.20. Exercise 1.4
Using eqn 1.17, show that the Wigner function of the coherent state | α i is aGaussian centered at the point α and given by: W ( β ) = π e − | α − β | . Schr¨odinger Cat States in Circuit QED
As described in App. A, the experimental procedure to measure the Wigner func-tion is very simple. One applies a microwave tone resonant with the cavity and havingthe appropriate phase, amplitude and duration to displace the cavity in phase spaceby the desired amount. One then uses the standard procedure described above to de-termine whether the parity is +1 or −
1. This procedure is repeated and the resultsaveraged to obtain the mean value of the parity operator. This ‘continuous variable’quantum tomography procedure is vastly simpler and more accurate than having tomeasure different combinations of multi-qubit correlators as required to do tomogra-phy in the discrete variable case. Furthermore the microwave cavity is simple (literallyan ‘empty box’), and the tomography can be performed with only a single ancillatransmon and cavity. This hardware efficiency is extremely powerful.
The Schr¨odinger Cat paradox has a long and storied history in quantum mechanics. Incircuit QED the cat being dead or alive is represented by a qubit being in the groundstate | g i or the excited state | e i . The role of the poison molecules in the air is playedby a coherent state | α i of photons in the cavity. The quantum state we want to createis | Ψ i = 1 √ | e i| i ± | g i| α i . (1.22)This is a coherent superposition of ‘cat alive, no poison in the air’ and ‘cat dead, poisonin the air.’ Notice that this is not a superpostion of ‘cat dead’ and ‘cat alive.’ (At thispoint the choice ± entangled state between the cat and the poison. Forlarge α the two states of the cavity are macroscopically distinct and orthogonal. Notethat it is important that the cavity be long-lived because the (initial) rate of photonsleaking out from the coherent state is κ ¯ n = κ | α | , where κ is the cavity dampingrate. If a photon ever leaks out of the cavity into the environment, the environmentimmediately collapses the state to a product state in which the cat is dead. On theother hand, if no photons are detected after a long time, we grow more and morecertain that they were never there in the first place and the second component of thestate gradually damps out collapsing the system to | e i| i . This quantum back actionfrom not observing a photon entering the environment is quite a novel and subtleeffect. See (Haroche and Raimond, 2006; Michael et al. , 2016) for further discussion. How do we create this Schr¨odinger cat state? It is surprisingly easy in the strong-dispersive coupling regime. The recipe is as follows. First apply a π/ | + x i = √ [ | g i + | e i ]. Then apply a drive tone to the cavityat frequency ω c . As we can see from Fig. 1.2, this drive will be on resonance with thecavity (and hence able to displace the cavity state from | i to | α i ) if and only if thequbit is the ground state. If the qubit is in the excited state, the cavity remains in thevacuum state (to a very good approximation since the drive is thousands of linewidthsoff resonance in this case). Hence we immediately obtain the cat of Eqn 1.22. It is See also the short story ‘Silver Blaze,’ by Sir Arthur Conan Doyle in which Sherlock Holmessolves a crime by noting the curious fact that a dog did not bark in the night. reating Cats straightforward to produce very ‘large’ cats with ¯ n = | α | corresponding to hundredsof photons but our ability to do high-fidelity state tomography begins to degrade aboveabout 100 photons (Vlastakis et al. , 2013 a ).Another interesting state, confusingly referred to in the literature not as a ‘Schr¨odingercat’ but as a ‘cat state’ of photons is given by | Ψ ± i = | g i √ | + α i ± | − α i ] . (1.23)This is a product state in which the qubit is not entangled with the cavity but thecavity is in a quantum superposition of two different (and for large α , macroscopicallydistinct) coherent states. (The normalization is approximate and only becomes exact inthe asymptotic limit of large | α | . We will ignore this detail throughout our discussion.)Because the qubit state factors out we will drop it from further discussion. The ± signin the superposition labels the photon number parity of the state. That is, these areeigenstates of the parity operator: ˆΠ | Ψ ± i = ±| Ψ ± i . (1.24)The parity of the state can be directly verified using the defining property of coherentstates in eqn 1.19 but is perhaps best visualzed in terms of the first-quantization wavefunction for the cat state. Since the ground state wave function is a Gaussian, thecat state consists of a sum or difference of two displaced Gaussians as illustrated inFig. 1.4. We see that the spatial parity symmetry under coordinate reflection and thephoton number parity are entirely equivalent. Mathematically this is because photonnumber parity reverses the position and momentum of the oscillator (i.e. inverts phasespace), a fact which is readily verified using, e.g. ˆΠˆ x ˆΠ ∼ ˆΠ( a + a † ) ˆΠ = − ( a + a † ). x (cid:68)(cid:14) (cid:68)(cid:16) ( ) x (cid:14) (cid:60) x (cid:68)(cid:14) (cid:68)(cid:16) ( ) x (cid:16) (cid:60) Fig. 1.4
First quantization wave functions for even and odd parity cat states.
The Wigner function of cat states of photons is very interesting. As one can see infig. 1.5, there are peaks of quasi-probability in phase space at ± α as expected, but inaddition the cat has ‘whiskers.’ These periodic oscillations are a kind of interferencepattern (much like a two-slit interference pattern between the two lobes) that arepresent for cat states but not for incoherence mixtures of | + α i and | − α i . Schr¨odinger Cat States in Circuit QED
Fig. 1.5
Wigner functions for even- (left panel) and odd-parity (right panel) cat states of size α = 2 .
5. The foreground and background peaks are associated with the coherent states | ± α i .Note that the parity oscillations near the origin have opposite sign and represent interferencefringes associated with the coherence of the superposition of the two distinct states. Exercise 1.5
Using eqn 1.20 to write down an (approximate) analytic expression for theWigner function W ( β ) of a cat state. Assume an even (odd) cat of the form in eqn 1.23with α real and positive, and sufficiently large that the normalization of the states is well-approximated by the √ W ( β ) = 2 π e − | β | n ± cos[4 α (Im β )] + cosh[4 α (Re β )] e − α o . (1.25)Using this result show that there are no interference fringes in the Wigner function of anincoherent mixture of an even and an odd cat. There is a simple recipe for deterministically creating cat states of photons beginswith the Schr¨odinger cat of eqn 1.22 except that the coherent state amplitude shouldbe 2 α instead of α . The trick is to figure out how to disentangle the qubit from thecavity. From eqn 1.22 we see that we need to be able to flip the qubit if an only thecavity is in the vacuum state. When the cavity is in the state | α i it has (for large | α | )negligible amplitude to have zero photons. We can carry out this special disentanglingoperation again using the strong-dispersive coupling. We rely on the quantized lightshift of the qubit transition frequency by applying a π pulse to the qubit at frequency ω q which is the transition frequency of the qubit when there are zero photons in thecavity. This ‘number selective π pulse yields the product state | Ψ ′ i = | g i √ | i ± | α i ] . (1.26)The final step is simply to apply a drive to displace the cavity by a distance − α in phase space to produce the cat state shown in eqn 1.23. The creation of this cat reating Cats can be verified via quantum state tomography via measuring the Wigner function asdescribed above. One can also use quantum jump spectroscopy to find the photonnumber distribution and see that even (odd) cats contain only even (odd) numbersof photons. For the case of circuit QED, both of these checks have been carried outin Ref. (Vlastakis et al. , 2013 b ) with negative Wigner function fringes measured instates with size up to d = 111 photons, where d = 2 α is the distance between the twocoherent states of the cat.Large cat states can be readily produced for microwave photons in circuit QED(Vlastakis et al. , 2013 a ) and cavity QED with Rydberg atoms (Haroche, 2013) andproduced in the phonon modes of trapped ions using spin-dependent optical forces act-ing on the ion motional degree of freedom (Monroe et al. , 1996; McDonnell et al. , 2007;Poschinger et al. , 2010; Wineland, 2013; Kienzler et al. , 2016).There is another interesting recipe for producing cat states. While simple, thisrecipe produces cats with non-deterministic parity. Start with a coherent state | α i and write it as a coherent superposition of an even and an odd cat (again assuming | α | is large for simplicity) | α i = 1 √ | Ψ + i + | Ψ − i ] , (1.27)where | Ψ ± i is given by eqn 1.23 (except we continue to drop the qubit state since itfactors out). Now simply follow the parity measurement protocol described above. Thiscollapses the state onto definite (but random) parity (Brune et al. , 1992; Sun et al. , 2014)and hence the measurement back action creates a cat state! Fig. 1.6 shows the Wignerfunction for this process conditioned on the state of the qubit used to perform theparity measurement (Sun et al. , 2014). a) b) c) Fig. 1.6
Non-deterministic production of a cat state via the back action of a parity measure-ment of a coherent state with amplitude α = 2 (¯ n = 4). a) Wigner function of the incoherentmixture state resulting from tracing over the outcome of the parity measurement; b) Wignerfunction when the outcome of the parity measurement is odd; c) Wigner function when theoutcome of the parity measurement is even. The phase of the fringe oscillations is oppositein the even and odd cats. Adapted from Sun et al. (2014). Exercise 1.6
For large | α | the non-deterministic procedure above produces even and oddcats randomly but with equal probability. Show that for small | α | there is a bias towards Schr¨odinger Cat States in Circuit QED producing even cats more frequently than odd cats. Compute the probability. (Hint: in thelimit | α | →
0, we obtain the vacuum state which is definitely even parity.)
For the case of physical objects like trapped ions, the cat state splits the ion wavepacket into two positions which can be separated by an amount much larger thantheir zero-point position uncertainty. If environmental degrees of freedom are ableto gain information about the position of the ions then of course the superpositioncollapses to a single coherent state. For example, a stray gas atom might bounce offone of the trapped ions or the electric field from the moving ions may induce dampingvia energy transfer into nearby metallic structures. For the electromagnetic oscillatorthe ‘position’ of the ‘object’ that is oscillating is the electric field strength (say) atsome selected point inside the cavity. (That point can be selected arbitrarily as longas it is not a nodal point of the mode.) Because the frequency of the cavity is setby geometry it is very stable and there is very little dephasing of the electromagneticoscillations. The primary interaction with the environment is through weak dampingof the resonator via the port that brings in the drive tones. When a photon leaks outof the cavity the parity of the cat must of course change. Equivalently this follows fromthe property of coherent states that they are eigenstates of the destruction operator: a | + α i = (+ α ) | + α i (1.28) a | − α i = ( − α ) | − α i (1.29)Interestingly this means that photon loss does not dephase a coherent state (indeed itdoes nothing to a coherent state!) but photon loss is a dephasing error on cat states.Since the rate of photon loss κ | α | grows with the ‘size’ of the cat, macroscopic catsdephase quickly and become classical mixtures of two coherent states.If photon loss does nothing to a coherent state how does its energy decay? Thiscomes from the back action associated with the intervals in which photons are notobserved leaking out that was mentioned above. This leads to a deterministic decayof the amplitude | α ( t ) i = | e − κ t α (0) i . (1.30)One remakrable feature of this is that even the cavity is emitting photons into theenvironment, it remains in a pure state and does not become entangled with theenvironment. This is a unique feature of coherent states in simple harmonic oscillators(Haroche and Raimond, 2006).Let us now move beyond simple coherent states to more general (and possiblymixed) states of the damped oscillator. The time evolution of the density matrix can besolved exactly by considering all the possible quantum trajectories specified by the ran-dom instants in time that photons leak out of the cavity (Haroche and Raimond, 2006;Michael et al. , 2016). The so-called Kraus operators that describe the CPTP (com-pletely positive trace-preserving) map can be organized according to the total number ecoherence of Cat States of Photons of photons lost in time t . In the absence of any Kerr non-linearities, the actual timethat the photons are lost has no effect on the time evolution: ρ ( t ) = ∞ X ℓ =0 ˆ E ℓ ρ (0) ˆ E † ℓ , (1.31)where the ℓ th Kraus operator describing the loss of ℓ photons isˆ E ℓ ( t ) = r (1 − e − κt ) ℓ ℓ ! e − κt ˆ n a ℓ . (1.32) Exercise 1.7
Prove that the CPTP in eqn 1.31 is in fact trace-preserving by showing that ∞ X ℓ =0 ˆ E † ℓ ˆ E ℓ = ˆ I, where ˆ I is the identity operator. Hint: Prove the identity is true for an arbitrary Fock state | n i and use the fact that the Fock states are complete. Exercise 1.8
Compute the expectation value of the parity over time as a damped oscillatordecays starting from: a) an initial even cat state, and b) an initial odd cat state. The expec-tation value of the parity will begin at ±
1, then (for large cats) decay close to zero and thenend up at +1 because the vacuum is even parity.
Exercise 1.9
Compute the Wigner function over time for a damped oscillator decaying from:a) an initial even cat state, and b) an initial odd cat state.
Cat states are well-known to be delicate and their phase coherence is notoriously sub-ject to rapid decay due to photon loss. It therefore seems like a very bad idea to try touse them to store quantum information. In fact the opposite is true. Refs. (Leghtas et al. , 2013;Mirrahimi et al. , 2014) propose and Ref. (Ofek et al. , 2016) demonstrates a cleverscheme for encoding quantum information in two (nearly) orthogonal code words con-sisting of even-parity cat states | W i = 1 √ | + α i + | − α i ] (1.33) | W i = 1 √ | + iα i + | − iα i ] , (1.34)where (without loss of generality) α is real and positive. (We assume α is large enoughthat the states are nearly orthonormal.) These states are indeed highly sensitive tophoton loss and quickly dephase. However as noted above, the source of the incoherence Schr¨odinger Cat States in Circuit QED is the loss of parity information when we do not know how many photons have leakedout of the resonator. Fortunately, this problem can be overcome because, as notedabove, we have the abiity to make rapid, high fidelity and highly QND measurementsof the photon number parity that can be repeated hundreds of times without backaction damage to the state.Let us now examine in detail how by (semi-) continuously monitoring the parityjumps in the system, we can recover the stored quantum information. Under photonloss the two code words obey the following relations a | W i → √ | + α i − |− α i ] (1.35) a | W i → + | W i (1.36) a | W i → i √ | + iα i − |− iα i ] (1.37) a | W i → −| W i . (1.38)After the loss of two photons the parity has returned to being even but we have a phaseflip error. Thus it takes the loss of four photons for an arbitrary superposition of thetwo code words to return to the original state. We do not need to correct immediatelyfor each photon loss. We need only keep track of the total number lost modulo 4and then conditioned on that, apply one of 4 unitaries to restore the state. This isextremely powerful and extremely simple and has allowed circuit QED to be the firsttechnology to reach the break-even point for quantum error correction in which thelifetime of the quantum information exceeds that of the best single component of thesystem (Ofek et al. , 2016). Assuming perfect parity tracking (and no dephasing of thecavity) the only source of error from parity jumps would be the possibility that theparity jumps more than once in the short interval ∆ t between measurements. Twojumps would leave the parity untouched and the error monitoring would miss thisfact. If photon loss occurs at rate Γ ∼ κ ¯ n , then parity monitoring at intervals ∆ t would reduce the error rate from first order Γ to second order O (Γ ∆ t ).As noted above, in addition to the photon number jumps which occur at randomtimes, the amplitude of the coherent state decays at rate κ/
2. Because this is fullydeterministic, the logical qubit decoding circuit can simply take this into account. Thecode fails at long times because the amplitude αe − κ t becomes sufficiently small thatthe two code words are no longer orthonormal. Novel ideas for ‘cat pumping’ to keepthe coherent states energized indefinitely have already been demonstrated experimen-tally for two-legged cats (Leghtas et al. , 2015; Touzard et al. , 2017). Bosonic codes forquantum error correction are now an active area of research (Gottesman et al. , 2001;Terhal and Weigand, 2016; Chuang et al. , 1997; Michael et al. , 2016; Li et al. , 2017;Albert et al. , 2017) and will be discussed in detail at a future Les Houches School. Circuit QED offers access to strong-coupling cavity QED with artificial atoms basedon Josephson junctions coupled via antenna elements to microwave photons. In thedispersive regime where the atom and cavity are strongly detuned from each other, the onclusions and Outlook dispersive coupling between atom and cavity can be several orders of magnitude largerthan dissipation rates. This ‘strong-dispersive’ regime allows robust universal quantumcontrol of the coupled system and permits creation of novel entangled Schr¨odingercats as well as cat states of photons. In this regime it is easy to make high-fidelityand highly QND measurements of photon number parity. This allows production ofcat states by measurement back action on coherent states and also allows repeatedmeasurements of the primary error syndrome for cavity decay. This in turn permitshighly hardware-efficient quantum error correction protocols and very simple quantumstate tomography through measurement of the Wigner function of the cavity.In addition to yielding a wonderful new regime to explore cavity QED, these capa-bilities will be key to a novel quantum computer architecture in which the logical qubitsare stored in microwave resonators using bosonic codes and controlled by Josephson-junction based non-linear elements. Rapid progress towards this goal is being made asevidenced by recent demonstration of a universal gate set for a logical qubit encodedin a cavity (Heeres et al. , 2017), entangling photon states (Wang et al. , 2011) and catstates (Wang et al. , 2016) between separate cavities, implementation of a CNOT gatebetween bosonic codes words stored in separate cavities (Rosenblum et al. , 2017) anddevelopment of a ‘catapult’ to launch cat states stored in cavity into flying modes(Pfaff et al. , 2017) for use in error correction in quantum communication (Li et al. , 2017)and remote entanglement. ppendix A The Wigner Function and DisplacedParity Measurements
The density matrix for a quantum system is defined by ρ ≡ X j | ψ j i p j h ψ j | (A.1)where where p j is the statistical probability that the system is found in state | ψ j i .As we will see further below, this is a useful quantity because it provides all theinformation needed to calculate the expectation value of any quantum observable O . In thermal equilibrium, | ψ j i is the j th energy eigenstate with eigenvalue ǫ j and p j = Z e − βǫ j is the corresponding Boltzmann weight. In this case the states are allnaturally orthogonal, h ψ j | ψ k i = δ jk . It is important to note however that in generalthe only constraint on the probabilities is that they are non-negative and sum to unity.Furthermore there is no requirement that the states be orthogonal (or complete), onlythat they be normalized.The expectation value of an observable O is given by hhOii = Tr O ρ (A.2)where the double brackets indicate both quantum and statistical ensemble averages.To prove this result, let us evaluate the trace in the complete orthonormal set of eigen-states of O obeying O| m i = O m | m i . In this basis the observable has the representation O = X m | m i O m h m | (A.3)and thus we can write Tr O ρ = X m h m |O ρ | m i = X m O m X j h m | ψ j i p j h ψ j | m i = X j p j X m h ψ j | m i O m h m | ψ j i = X j p j h ψ j |O| ψ j i ≡ hhOii . (A.4) he Wigner Function and Displaced Parity Measurements The considerations are quite general. We now specialize to the case of a single-particle system in one spatial dimension. A useful example is the harmonic oscillatormodel which might represent a mechanical oscillator or the electromagnetic oscillationsof a particular mode of a microwave or optical cavity. Our first task is to understandthe relationship between the quantum density matrix and the classical phase spacedistribution.In classical statistical mechanics we are used to thinking about the probabilitydensity P ( x, p ) of finding a particle at a certain point in phase space. For example, inthermal equilibrium the phase space distribution is simply P ( x, p ) dxdp π ~ = 1 Z e − βH ( x,p ) dxdp π ~ , (A.5)where the partition function is given by Z = Z Z dxdp π ~ e − βH ( x,p ) , (A.6)and where for convenience (and planning ahead for the quantum case) we have madethe phase space measure dimensionless by inserting the factor of Planck’s constant.The marginal distributions for position and momentum are found by ρ ( x ) = 12 π ~ Z + ∞−∞ dp P ( x, p ) (A.7) ρ ( p ) = 12 π ~ Z + ∞−∞ dx P ( x, p ) . (A.8)Things are more complex in quantum mechanics because the observables ˆ x and ˆ p do not commute and hence cannot be simultaneously measured because of Heisenberguncertainty. To try to make contact with the classical phase space distribution it isuseful to study the quantum density matrix in the position representation given by ρ ( x, x ′ ) = h x | ρ | x ′ i = X j p j ψ j ( x ) ψ ∗ j ( x ′ ) , (A.9)where the wave functions are given by ψ j ( x ) = h x | ψ j i . It is clear from the Born rulethat the marginal distribution for position can be found from the diagonal element ofthe density matrix ρ ( x ) = ρ ( x, x ) = X j p j | ψ j ( x ) | . (A.10)Likewise the marginal distribution for momentum is given by the diagonal element ofthe density matrix in the momentum representation ρ ( p ) = h p | ρ | p i . (A.11)We can relate this to the position representation by inserting resolutions of the identityin terms of complete sets of position eigenstates Note that we are using unnormalized momentum eigenstates h x | p i = e ipx/ ~ . Correspondingly weare not using a factor of the system size L in the integration measure (‘density of states in k space’)for momentum. The Wigner Function and Displaced Parity Measurements ˜ ρ ( p, p ′ ) = 12 π ~ Z + ∞−∞ dxdx ′ h p | x ih x | ρ | x ′ ih x ′ | p ′ i = 12 π ~ Z + ∞−∞ dxdx ′ e − ipx/ ~ ρ ( x, x ′ ) e + ip ′ x ′ / ~ . (A.12)Thus the momentum representation of the density matrix is given by the Fouriertransform of the position representation. We see also that the marginal distributionfor the momentum involves the off-diagonal elements of the real-space density matrixin an essential way ρ ( p ) = 12 π ~ Z + ∞−∞ dxdx ′ e − ip ( x − x ′ ) / ~ ρ ( x, x ′ ) . (A.13)For later purposes it will be convenient to define ‘center of mass’ and ‘relative’ coor-dinates y = x + x ′ and ξ = x − x ′ and reexpress this integral as ρ ( p ) = 12 π ~ Z + ∞−∞ dy Z + ∞−∞ dξ e − ipξ/ ~ ρ ( y + ξ/ , y − ξ/ . (A.14)Very early in the history of quantum mechanics, Wigner noticed from the aboveexpression that one could write down a quantity which is a natural extension of thephase space density. The so-called Wigner ‘quasi-probability distribution’ is definedby W ( x, p ) ≡ π ~ Z + ∞−∞ dξ e − ipξ/ ~ ρ ( x + ξ/ , x − ξ/ . (A.15)Using this, eqn (A.14) becomes (changing the dummy variable y back to x for nota-tional clarity) ρ ( p ) = Z + ∞−∞ dx W ( x, p ) . (A.16)Similarly, by using eqn (A.15), we can write eqn (A.10) as ρ ( x ) = Z + ∞−∞ dp W ( x, p ) . (A.17)These equations are analogous to Eqs. (A.7,A.8) and show that the Wigner distributionis analogous to the classical phase space density P ( x, p ). However the fact that positionand momentum do not commute turns out to mean that the Wigner distribution neednot be positive. In fact, in quantum optics one often takes the defining characteristicof non-classical states of light to be that they have Wigner distributions which arenegative in some regions of phase space.The Wigner function is extremely useful in quantum optics because, like the den-sity matrix, it contains complete information about the quantum state of an elec-tromagnetic oscillator mode, but (at least in circuit QED) is much easier to mea-sure. Through a remarkable mathematical identity (Lutterbach and Davidovich, 1997;Haroche and Raimond, 2006) we can relate the Wigner function to the expectation he Wigner Function and Displaced Parity Measurements value of the photon number parity, something that can be measured (Bertet et al. , 2002)and is especially easy to measure (Vlastakis et al. , 2013 b ) in the strong-coupling regimeof circuit QED (a regime not easy to reach in ordinary quantum optics).We are used to thinking of the photon number parity operator in its second quan-tized form ˆΠ = e iπa † a (A.18)in which its effect on photon Fock states is clearˆΠ | n i = ( − n | n i , (A.19)and indeed it is in this form that it is easiest to understand how to realize the operationexperimentally using time evolution under the cQED qubit-cavity coupling χσ z a † a in the strong-dispersive limit of large χ relative to dissipation. However because theWigner function has been defined in a first quantization representation in terms of wavefunctions, it is better here to think about the parity operator in its first-quantized form.Recalling that the wave functions of the simple harmonic oscillator energy eigenstatesalternate in spatial reflection parity as one moves up the ladder, it is clear that photonnumber parity and spatial reflection parity are one and the same. That is, if | x i is aposition eigensate ˆΠ | x i = | − x i , (A.20)or equivalently in terms of the wave functionˆΠ ψ ( x ) = h x | ˆΠ | ψ i = ψ ( − x ) . (A.21)To further cement the connection, we note that since the position operator is lin-ear in the ladder operators, it is straightforward to verify from the second-quantizedrepresentations that ˆΠˆ x = − ˆ x ˆΠ . (A.22)We now want to show that we can measure the Wigner function W ( X, P ) by thefollowing simple and direct recipe (Lutterbach and Davidovich, 1997): (1) displace theoscillator in phase space so that the point (
X, P ) moves to the origin; (2) then measurethe expectation value of the photon number parity W ( X, P ) = 1 π ~ Tr n D ( − X, − P ) ρ D † ( − X, − P ) ˆΠ o , (A.23)where D is the displacement operator. Related methods in which one measures not theparity but the full photon number distribution of the displaced state can in principleyield even more robust results in the presence of measurement noise (Shen et al. , 2016).Typically in experiment one would make a single ‘straight-line’ displacement. Tak-ing advantage of the fact that ˆ p is the generator of displacements in position and ˆ x isthe generator of displacements in momentum, the ‘straight-line’ displacement operatoris given by D ( − X, − P ) = e − i ~ ( P ˆ x − X ˆ p ) . (A.24)In experiment, this displacement operation is readily carried out by simply applying apulse at the cavity resonance frequency with appropriately chosen amplitude, duration The Wigner Function and Displaced Parity Measurements and phase. In the frame rotating at the cavity frequency the drive corresponds to thefollowing term in the Hamiltonian V ( t ) = i [ ǫ ( t ) a † − ǫ ∗ ( t ) a ] (A.25)where ǫ ( t ) is a complex function of time describing the two quadratures of the drivepulse. The Heisenberg equation of motion ddt a = i [ V ( t ) , a ] = ǫ ( t ) , (A.26)has solution a ( t ) = a (0) + Z t −∞ dτ ǫ ( τ ) , (A.27)showing that the cavity is simply displaced in phase space by the drive. For the‘straight-line’ displacement discussed above, ǫ ( t ) has fixed phase and only the magni-tude varies with time. Exercise A.1
Find an expression for ǫ ( t ) such that time evolution under the drive ineqn (A.25) will reproduce eqn (A.24). Ignore cavity damping (an assumption which is validif the pulse duration is short enough). For theoretical convenience in the present calculation, we will carry out the dis-placement in two steps by using the Feynman disentangling theorem e ˆ A + ˆ B = e ˆ A e ˆ B e [ ˆ B, ˆ A ] (A.28)(which is valid if [ ˆ B, ˆ A ] itself commutes with both ˆ A and ˆ B ) to write D ( − X, − P ) = e iθ D (0 , − P ) D ( − X,
0) = e iθ e − i ~ P ˆ x e + i ~ X ˆ p , (A.29)where θ ≡ i ~ XP . This form of the expression represents a move of the phase spacepoint ( X, P ) to the origin by first displacing the system in position by − X and thenin momentum by − P . This yields the same final state as the straightline displacementexcept for an overall phase θ which arises from the fact that displacements in phasespace do not commute. For present purposes this overall phase drops out and we willignore it henceforth.Under this pair of transformations the wave function becomes ψ ( x ) → ψ ( x + X ) → e − iP x/ ~ ψ ( x + X ) . (A.30)More formally, we have two results which will be useful in evaluating eqn (A.23) h ξ |D (0 , − P ) D ( − X, | ψ i = e − iP ξ/ ~ ψ ( ξ + X ) (A.31) h ψ |D † ( − X, D † (0 , − P ) | ξ i = e + iP ξ/ ~ ψ ∗ ( ξ + X ) . (A.32)Taking the trace in eqn (A.23) in the position basis yields he Wigner Function and Displaced Parity Measurements W ( X, P ) = 1 π ~ X j p j Z + ∞−∞ dξ h ξ |D ( − X, − P ) | ψ j ih ψ j |D † ( − X, − P ) ˆΠ | ξ i = 1 π ~ X j p j Z + ∞−∞ dξ h ξ |D ( − X, − P ) | ψ j ih ψ j |D † ( − X, − P ) | − ξ i = 1 π ~ X j p j Z + ∞−∞ dξ e − iP ξ/ ~ ψ j ( ξ + X ) ψ ∗ j ( − ξ + X )= 12 π ~ X j p j Z + ∞−∞ dξ e − iP ξ/ ~ ψ j ( X + ξ/ ψ ∗ j ( X − ξ/ , (A.33)which proves that the displaced parity is indeed precisely the Wigner function. eferences Albert, Victor V., Noh, Kyungjoo, Duivenvoorden, Kasper, Brierley, R. T., Rein-hold, Philip, Vuillot, Christophe, Li, Linshu, Shen, Chao, Girvin, S. M., Terhal,Barbara M., and Jiang, Liang (2017). Performance and structure of bosonic codes. arXiv: , 1708.05010.Bertet, P., Auffeves, A., Maioli, P., Osnaghi, S., Meunier, T., Brune, M., Raimond,J. M., and Haroche, S. (2002). Direct measurement of the wigner function of aone-photon fock state in a cavity.
Phys. Rev. Lett. , , 200402.Blais, Alexandre, Gambetta, Jay, Wallraff, A., Schuster, D. I., Girvin, S. M., Devoret,M. H., and Schoelkopf, R. J. (2007). Quantum-information processing with circuitquantum electrodynamics. Phys. Rev. A , , 032329.Blais, Alexandre, Huang, Ren-Shou, Wallraff, Andreas, Girvin, S. M., and Schoelkopf,R. J. (2004). Cavity quantum electrodynamics for superconducting electrical cir-cuits: an architecture for quantum computation. Phys. Rev. A , , 062320.Brune, M., Haroche, S., Raimond, J. M., Davidovich, L., and Zagury, N. (1992, Apr).Manipulation of photons in a cavity by dispersive atom-field coupling: Quantum-nondemolition measurements and generation of “schr¨odinger cat” states. Phys. Rev.A , , 5193–5214.Cahill, K. E. and Glauber, R. J. (1969, Jan). Density operators and quasiprobabilitydistributions. Phys. Rev. , , 1882–1902.Chuang, Isaac L., Leung, Debbie W., and Yamamoto, Yoshihisa (1997). Bosonicquantum codes for amplitude damping. Phys. Rev. A , (2), 1114.Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F., and Schoelkopf, R. J.(2010). Introduction to quantum noise, measurement and amplification. Rev. Mod.Phys. , , 1155–1208. (Longer version with pedagogical appendices available at:arXiv.org:0810.4729).Devoret, M. H. and Schoelkopf, R. J. (2013). Superconducting circuits for quantuminformation: An outlook. Science , (6124), 1169–1174.Girvin, Steven M. (2014). Proceedings of the 2011 Les Houches Summer School onQuantum Machines , Chapter Circuit QED: Superconducting Qubits Coupled toMicrowave Photons. Oxford University Press.Gottesman, Daniel, Yu. Kitaev, Alexei, and Preskill, John (2001, June). Encoding aqubit in an oscillator.
Phys. Rev. A , (1), 012310.Haroche, Serge (2013, Jul). Nobel lecture: Controlling photons in a box and exploringthe quantum to classical boundary. Rev. Mod. Phys. , , 1083–1102.Haroche, Serge and Raimond, Jean-Michel (2006). Exploring the Quantum: Atoms,Cavities and Photons . Oxford University Press.Heeres, Reinier W., Reinhold, Philip, Ofek, Nissim, Frunzio, Luigi, Jiang, Liang,Devoret, Michel H., and Schoelkopf, Robert J. (2017). Implementing a universal eferences gate set on a logical qubit encoded in an oscillator. Nat Commun. , , 94.Kienzler, D., Fl¨uhmann, C., Negnevitsky, V., Lo, H.-Y., Marinelli, M., Nadlinger,D., and Home, J. P. (2016, Apr). Observation of quantum interference betweenseparated mechanical oscillator wave packets. Phys. Rev. Lett. , , 140402.Leghtas, Zaki, Kirchmair, Gerhard, Vlastakis, Brian, Schoelkopf, Robert J., De-voret, Michel H., and Mirrahimi, Mazyar (2013, September). Hardware-EfficientAutonomous Quantum Memory Protection. Phys. Rev. Lett. , (12), 120501.Leghtas, Z., Touzard, S., Pop, I. M., Kou, A., Vlastakis, B., Petrenko, A., Sliwa,K. M., Narla, A., Shankar, S., Hatridge, M. J., Reagor, M., Frunzio, L., Schoelkopf,R. J., Mirrahimi, M., and Devoret, M. H. (2015). Confining the state of light to aquantum manifold by engineered two-photon loss. Science , (6224), 853–857.Li, Linshu, Zou, Chang-Ling, Albert, Victor V., Muralidharan, Sreraman, Girvin,S. M., and Jiang, Liang (2017, Jul). Cat codes with optimal decoherence suppressionfor a lossy bosonic channel. Phys. Rev. Lett. , , 030502.Lutterbach, L. G. and Davidovich, L. (1997). Method for direct measurement of thewigner function in cavity qed and ion traps. Phys. Rev. Lett. , , 2547–2550.McDonnell, M. J., Home, J. P., Lucas, D. M., Imreh, G., Keitch, B. C., Szwer, D. J.,Thomas, N. R., Webster, S. C., Stacey, D. N., and Steane, A. M. (2007, Feb). Long-lived mesoscopic entanglement outside the lamb-dicke regime. Phys. Rev. Lett. , ,063603.Michael, Marios H., Silveri, Matti, Brierley, R. T., Albert, Victor V., Salmilehto, Juha,Jiang, Liang, and Girvin, S. M. (2016, Jul). New class of quantum error-correctingcodes for a bosonic mode. Phys. Rev. X , , 031006.Mirrahimi, Mazyar, Leghtas, Zaki, Albert, Victor V., Touzard, Steven, Schoelkopf,Robert J., Jiang, Liang, and Devoret, Michel H. (2014, April). Dynamically pro-tected cat-qubits: a new paradigm for universal quantum computation. New J.Phys. , (4), 045014.Monroe, C., Meekhof, D. M., King, B. E., and Wineland, D. J. (1996). A “schr¨odingercat” superposition state of an atom. Science , (5265), 1131–1136.Nigg, Simon E., Paik, Hanhee, Vlastakis, Brian, Kirchmair, Gerhard, Shankar, S.,Frunzio, Luigi, Devoret, M. H., Schoelkopf, R. J., and Girvin, S. M. (2012, Jun).Black-box superconducting circuit quantization. Phys. Rev. Lett. , , 240502.Ofek, Nissim, Petrenko, Andrei, Heeres, Reinier, Reinhold, Philip, Leghtas, Zaki,Vlastakis, Brian, Liu, Yehan, Frunzio, Luigi, Girvin, S. M., Jiang, Liang, Mirrahimi,Mazyar, Devoret, M. H., and Schoelkopf, R. J. (2016, February). Extending the life-time of a quantum bit with error correction in superconducting circuits. Nature , ,441.Pfaff, Wolfgang, Axline, Christopher J., Burkhart, Luke D., Vool, Uri, Reinhold,Philip, Frunzio, Luigi, Jiang, Liang, Devoret, Michel H., and Schoelkopf, Robert J.(2017). Controlled release of multiphoton quantum states from a microwave cavitymemory. Nature Physics , , 882887.Poschinger, U., Walther, A., Singer, K., and Schmidt-Kaler, F. (2010, Dec). Ob-serving the phase space trajectory of an entangled matter wave packet. Phys. Rev.Lett. , , 263602.Rosenblum, S., Gao, Y.Y., Reinhold, P., Wang, C., Axline, C.J., Frunzio, L., Girvin, References
S.M., Jiang, Liang, Mirrahimi, M., Devoret, M.H., and Schoelkopf, R.J. (2017). Acnot gate between multiphoton qubits encoded in two cavities. arXiv:1709.05425 .Schoelkopf, RJ and Girvin, SM (2008). Wiring up quantum systems.
Nature , ,664.Schuster, D.I., Houck, A.A., Schreier, J.A., Wallraff, A., Gambetta, J., Blais, A.,Frunzio, L., Johnson, B., Devoret, M.H., Girvin, S.M., and Schoelkopf, R.J. (2007).Resolving photon number states in a superconducting circuit. Nature , , 515–518.Shen, Chao, Heeres, Reinier W., Reinhold, Philip, Jiang, Luyao, Liu, Yi-Kai,Schoelkopf, Robert J., and Jiang, Liang (2016, Nov). Optimized tomography ofcontinuous variable systems using excitation counting. Phys. Rev. A , , 052327.Solgun, Firat and DiVincenzo, David P. (2015). Multiport impedance quantization. Annals of Physics , , 605–669.Sun, L., Petrenko, A., Leghtas, Z., Vlastakis, B., Kirchmair, G., Sliwa, K. M., Narla,A., Hatridge, M., Shankar, S., Blumoff, J., Frunzio, L., Mirrahimi, M., Devoret,M. H., and Schoelkopf, R. J. (2014, July). Tracking photon jumps with repeatedquantum non-demolition parity measurements. Nature , (7510), 444–448.Terhal, B. M. and Weigand, D. (2016, January). Encoding a qubit into a cavity modein circuit QED using phase estimation. Phys. Rev. A , (1), 012315.Touzard, S., Grimm, A., Leghtas, Z., Mundhada, S.O., Reinhold, P., Heeres, R.,Axline, C., Reagor, M., Chou, K., Blumoff, J., Sliwa, K.M., Shankar, S., Frunzio, L.,Schoelkopf, R..J., Mirrahimi, M., and Devoret, M.H. (2017). Coherent oscillationsin a quantum manifold stabilized by dissipation. arXiv:1705.02401 .Vlastakis, Brian, Kirchmair, Gerhard, Leghtas, Zaki, Nigg, Simon E., Frunzio, Luigi,Girvin, S. M., Mirrahimi, Mazyar, Devoret, M. H., and Schoelkopf, R. J. (2013 a ,November). Deterministically Encoding Quantum Information Using 100-PhotonSchr¨odinger Cat States. Science , (6158), 607–610.Vlastakis, Brian, Kirchmair, Gerhard, Leghtas, Zaki, Simon E. Nigg, and Luigi Frun-zio, Girvin, S. M., Mirrahimi, Mazyar, Devoret, M. H., and Schoelkopf, R. J.(2013 b ). Deterministically encoding quantum information in 100-photon schr¨odingercat states. Science , , 607.Vool, Uri and Devoret, M.H. (2016). Introduction to quantum electromagnetic cir-cuits. arXiv:1610.03438 .Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Huang, R.-S., Majer, J., Kumar,S., Girvin, S. M., and Schoelkopf, R. J. (2004). Circuit quantum electrodynamics:Coherent coupling of a single photon to a cooper pair box. Nature , , 162–167.Wang, Chen, Gao, Yvonne Y., Reinhold, Philip, Heeres, R. W., Ofek, Nissim, Chou,Kevin, Axline, Christopher, Reagor, Matthew, Blumoff, Jacob, Sliwa, K. M., Frun-zio, L., Girvin, S. M., Jiang, Liang, Mirrahimi, M., Devoret, M. H., and Schoelkopf,R. J. (2016, May). A Schr¨odinger cat living in two boxes. Science , (6289),1087–1091.Wang, H., Mariantoni, Matteo, Bialczak, Radoslaw C., Lenander, M., Lucero, Erik,Neeley, M., O’Connell, A., Sank, D., Weides, M., Wenner, J., Yamamoto, T., Yin, Y.,Zhao, J., Martinis, John M., and Cleland, A. N. (2011). Deterministic entanglementof photons in two superconducting microwave resonators. Phys. Rev. Lett. , ,060401. eferences Wineland, David J. (2013, Jul). Nobel lecture: Superposition, entanglement, andraising schr¨odinger’s cat.
Rev. Mod. Phys. , , 1103–1114.Zheng, Huaixiu, Silveri, Matti, Brierley, R. T., Girvin, S. M., and Lehnert, K. W.(2016). Accelerating dark-matter axion searches with quantum measurement tech-nology. arXiv:1607.02529arXiv:1607.02529