Spectral densities of superconducting qubits with environmental resonances
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Spectral densities of superconducting qubits with environmental resonances.
Kaushik Mitra, C. J. Lobb, and C. A. R. S´a de Melo
Joint Quantum Institute and Department of PhysicsUniversity of Maryland College Park MD 20742 (Dated: November 1, 2018)
PACS numbers: 74.50.+r, 85.25.Dq, 03.67.Lx
In this paper we derive the environmental spectral den-sity for flux [1], phase [2] and charge [3] qubits when eachof them is coupled to an environment with a resonance.From the spectral density we obtain the characteristicspontaneous emission (relaxation) lifetimes T for eachof these qubits, and show that there is a substantial en-hancement of T beyond the resonant frequency of the en-vironment. The circuits considered are shown in Fig. 1, 2and 3.The flux qubit shown in Fig. 1 is measured by a dc-SQUID. Hence to study decoherence and relaxation timescales, one has to consider the noise that is transferredfrom the qubit to the dc-SQUID. φ C s φ I b I c , C J I c0 , C Rdc SQUID
FIG. 1: Flux qubit measured by a dc-SQUID (gray line).The qubit is in the inner SQUID loop with critical current I c and capacitance C J for both junctions. The inner SQUID isshunted by a capacitance C s , and environmental resistance R and is biased by a ramping current I b . The dc-SQUID loophas junction capatitance C and critical current I c . The classical equation of motion for the dc-SQUID is C ¨ φ + 2 π Φ I c sin φ − π Φ I + Z t dt ′ Y ( t − t ′ ) ˙ φ ( t ′ ) = 0 (1)where φ is the gauge invariant phase across the Josephsonjunction of the outer dc-SQUID loop, I c is the criticalcurrent of its junction, Φ = h/ e is the flux quantum.and the total induced current in the outer dc-SQUID is I = 4 L dc h δφ σ z i + 4 L dc h φ m i + (cid:18) π Φ (cid:19) J h φ p i . (2)Here φ p and φ m are the sum and difference of the gaugeinvariant phases across the junctions of the inner SQUID, L dc is the self-inductance of the inner SQUID, and J isthe bilinear coupling between φ m and φ p at the potentialenergy minimum. The term δφ = πM q I cir / Φ , where I cir is the circulating current of the localized states of the qubit (described in terms of Pauli matrix σ z ), and M q isthe mutual inductance between the qubit and the outerdc-SQUID. The last term in Eq. (1) is the dissipationterm due to effective admittance Y ( ω ) felt by the outerdc-SQUID.For the charge qubit shown in Fig. 2 the classical equa-tion of motion for the charge Q is V g ( ω ) = (cid:18) − ω L J ( ω )1 − ω L J ( ω ) C J + 1 C g + iωZ ( ω ) (cid:19) Q ( ω ) (3)Here, V g is the gate voltage, C g is the gate capacitance, L J and C J are the Josephson inductance and capacitancerespectively, Z ( ω ) is the effective impedance seen by thecharge qubit due to a transmission line resonator (cavity),Ω is the resonant frequency of the resonator, and Q is thecharge across C g . Z ( ω ) C g E J , C J CavityV g FIG. 2: Circuit diagram of the Cooper-pair box. The super-conducting island is connected to a large reservoir througha Josephson junction with Josephson energy E J and capaci-tance C g . The voltage bias V g is provided through a resonator(cavity) having environmental impedance Z ( ω ), which is con-nected to C g as shown. For the phase qubit shown in Fig. 3 the classical equa-tion of motion is C ¨ γ + 2 π Φ I c sin γ − π Φ I + Z t dt ′ Y ( t − t ′ ) ˙ γ ( t ′ ) = 0 (4)where I c is the critical current of Josephson junction J in Fig.1, I is the bias current, and Φ = h/ e is the fluxquantum. The last term is the dissipation term due to Y ( ω ) which is the effective admittance as seen by thedc-SQUID.The equations of motion described in Eqs. (1), (3), and(4), can be all approximatelly described by the effectivespin-boson Hamiltonian e H = ¯ hω σ z + X k ¯ hω k b † k b k + H SB , (5) I L C Φ a O C L R L R qubit J Y int ( ω ) Y iso ( ω ), isolation network FIG. 3: Schematic drawing of the phase qubit with an RLCisolation circuit. written in terms of Pauli matrices σ i (with i = x, y, z )and boson operators b k and b † k . The first term in Eq. (5)represents a two-level approximation for the qubit (sys-tem) described by states | i and | i with energy differ-ence ¯ hω . The second term corresponds to the isolationnetwork (bath) represented by a bath of bosons, where b k and b † k are the annihilation and creation operator ofthe k -th bath mode with frequency ω k . The third termis the system-bath (SB) Hamiltonian which correspondsto the coupling between the environment and the qubit.At the charge degeneracy point for the charge qubit(gate charge N g = 0), at the flux degeneracy point (ex-ternal flux Φ ext = π Φ ) for the flux qubit, and the suit-able flux bias condition for the phase qubit (external fluxΦ a = L φ ) H SB reduces to, H SB = 12 σ x ¯ h h | v | i X k λ k (cid:16) b † k + b k (cid:17) (6)where v = φ for the flux qubit, Q for the charge qubit,and γ for the phase qubit. The spectral density of thebath modes J ( ω ) = ¯ h P k λ k δ ( ω − ω k ) has dimensions ofenergy and can be written as J ( ω ) = ω Re Y ( ω )(Φ / π ) for flux and phase qubits and J ( ω ) = 2¯ hωe / ¯ h Re Z ( ω )for charge qubits.For the flux qubit circuit shown in Fig. 1, the shuntcapacitance C s is used to control the environment, whilethe Ohmic resistance of the circuit is modelled by R . Inthis case, the environmental spectral density is J ( ω ) = α ω (1 − ω / Ω ) + 4 ω Γ / Ω . (7)when ω m ≫ max( ω p , ω ), and when the dc-SQUID is faraway from the switching point to be modelled by an idealinductance L J . Here, Ω = 1 / √ L J C s = p πI c / ( C s Φ )is the plasma frequency of the inner SQUID and playsthe role of the resonant frequency, where I c is the crit-ical current for each of two Josephson junctions. Also,Γ = 1 / ( C s R ) plays the role of the resonance width, and α = 2( eI p I b M q ) / ( C s ¯ h R Ω ) reflects the low frequencybehavior. The coupling between the flux qubit and theouter dc-SQUID occurs emerges from the interaction ofthe persistent current I p of the qubit and the bias current I b of the dc-SQUID via their mutual inductance M q . The spectral density for the charge qubit shown inFig. 2 is obtained by solving for the normal modes ofthe resonator and transmission lines, including an inputimpedance R at each end of the resonator. It is given by J ( ω ) = e Ω ℓc Γ ( ω − Ω ) + (Γ / (8)were Ω is the resonator frequency, ℓ is resonator length, c is the capacitance per unit length of the transmissionline, C g is the gate capacitance, and C J is the junctioncapacitance. The quantity Γ = Ω /Q where Q is thequality factor of the cavity.For negligeable Y int ( ω ), the spectral density of thephase qubit shown in Fig. (3) is J ( ω ) = (cid:18) Φ π (cid:19) α ω (1 − ω / Ω ) + 4 ω Γ / Ω , (9)where α = L / (( L + L ) R ) ≈ ( L/L ) /R is theleading order term in the low frequency ohmic regime,Ω = p ( L + L ) / ( LL C ) ≈ / √ LC is essentially theresonance frequency, and Γ = 1 / (2 CR ) plays the role ofresonance width. Here, we used L ≫ L correspondingto the relevant experimental regime.Once the spectral functions of the environments areknown, the relaxation rates are derived following stan-dard methods [4]. At finite temperatures, the relaxationtime ¯ hT ,i = β i J i ( ω ) coth ¯ hω k B T (10)is directly related to the environmental spectral den-sity. Here, the index i = 1 , , β i takes values β = 1 for the flux qubit of Fig. 1, β = C g / ( C g + C J ) for the charge qubit of Fig. 2 and β = 1 / h (Φ / π ) C ω i for the phase qubit of Fig. 3.An inspection of Eq. (10) and the corresponding spectralfunctions J i ( ω ) shows that suitably detuning the qubit tohigher frequencies beyond the environmental resonancecan enhance the characteristic spontaneous emission life-time (relaxation time) T ,i of the qubit by a couple oforders of magnitude as discussed in Ref. [5]. [1] J. E. Mooij et al , Science , 1036 (1999)[2] J. Martinis et al. , Phys. Rev. Lett.
9, 117901 (2002)[3] A. Wallraff et al , Nature
162 (2004)[4] U. Weiss,