Dissipative dynamics of circuit-QED in the mesoscopic regime
We investigate the behavior of a circuit QED device when the resonator is initially populated with a mesoscopic coherent field. The strong coupling between the cavity and the qubit produces an entangled state involving mesoscopic quasi-pointer states with respect to cavity dissipation. The overlap of the associated field components results in collapse and revivals for the Rabi oscillation. Although qubit relaxation and dephasing do not preserve these states, a simple analytical description of the dissipative dynamics of the circuit QED device including cavity relaxation as well as qubit dissipation is obtained from the Monte-Carlo approach. Explicit predictions for the spontaneous and induced Rabi oscillation signals are derived and sucessfully compared with exact calculations. We show that these interesting effects could be observed with a 10 photon field in forthcoming circuit QED experiments.
Cavity quantum electrodynamics is an ideal playground for the realization of tests of quantum processes, simple quantum information processing and exploration of the quantum/classical boundary. Mesoscopic superpositions made of coherent field components with different classical attribues (phase and amplitude) and their decay as a result of decoherence have been observed in experiments involing circular Rydberg atoms Brune et al. (1996a). Recently, an experimental scheme for cavity QED experiments using Josepshon qubits embedded in a superconducting microstrip planar resonator has been proposed Blais et al. (2004) and realized Wallraff et al. (2004) by the Schoelkopf group at Yale. Relying on conventional lithography and nanofrabrication techniques, these circuit-QED devices open the way to scalable quantum circuits coupling qubits to high- cavities acting as quantum buses. Recent experimental progresses on these devices have been dramatic, leading to the experimental demonstration of quantum non-demolition measurements of the qubit state in the dispersive regime Wallraff et al. (2005) and of the resolution of photon number states of the cavity Schuster et al. (2007).
These recent developements provide a strong motivation toward studying the dynamics of circuit QED devices at the quantum/classical boundary. In particular, it is important to analyze their ability to produce mesoscopic Schrődinger cat states involving coherent components with a mesoscopic number of photons. Such states can be produced in the dispersive regime where the off resonant qubit acts as a state dependant transparent dielectrics. In this regime, the interaction between a coherent field and a qubit, initially prepared in a superposition of states naturally leads to a quantum superposition of phase shifts.
A much faster way to produce entangled qubit + cavity states is to use a resonant qubit + cavity interaction. In the mesoscopic regime, photon graininess rapidely casts the initially coherent cavity field into a superposition of two components with different phases Gea-Banaloche (1991). This phase splitting is a mesoscopic effect that disappears in the classical limit of a very large field, which is then left unaffected by the atoms. Thus, in the mesoscopic limit, the cavity field acts as a which-path detector for the atomic states. The overlap of field components of the qubit + cavity state is responsible for the collapse and revivals of Rabi oscillations which can be either spontaneous or induced by an echo sequence. This resonant phase splitting effect has been evidenced in Rydberg atom experiments for fields containing up to a few tens of photons Auffeves et al. (2003). Its coherence has been checked using an echo technique borrowed from NMR Meunier et al. (2005), following a proposal by Morigi et al Morigi et al. (2002).
In the presence of dissipation, the entangled qubit + cavity state is subject to decoherence which leads to a damping of the spontaneous and induced Rabi oscillation revivals. In a previous work Meunier et al. (2006), we have developped a simple analytical model that describes the behaviour of identical non dissipative qubits resonantly and symmetrically coupled to a high- cavity. Considering cavity relaxation as the source for decoherence, this model is perfectly well suited for describing Rydberg atom experiments. But this is not sufficient for circuit QED since it does not take into account qubit relaxation and dephasing.
The purpose of the present paper is to present a simple analytical model that takes into account qubit relaxation and dephasing for a single qubit in a cavity thus generalizing results previously obtained by Gea-Banacloche Gea-Banacloche (1993) and ourselves Meunier et al. (2006). Using the physical insight provided by the stochastic wave function approach Dalibard et al. (1992), an analytic description for the decoherence of the mesoscopic qubit + cavity state is derived. Our derivation also sheds light on the range of validity of our previous analysis: whenever the internal interactions within a mesoscopic device tend to produce pointer states with respect to the dominant coupling to the environment, its evolution can described in term of these pointer states and decoherence functionals reflecting the cumulative imprints they left in the environment. In circuit-QED devices, this simple image is broken by qubit dissipative processes but a careful analysis of the dynamics enables us to obtain simple analytical results for the spontaneous and induced Rabi oscillation signals. In principle our analysis can be also used to discuss field tomography and can also be extended to the case of several qubits.
This paper is organized as follows. In Sec. II, the basic model for cavity-QED is presented, its dynamics in both resonant and dispersive regimes are briefly recalled. In Sec. III, circuit-QED devices are presented and dissipation sources are discussed. In Sec. IV, the dissipative dynamics of a one qubit circuit-QED device is studied within the stochastic wave function framework. For completeness, analytical results are derived both for the resonant and dispersive regimes, recovering in the latter case results from the Yale group Gambetta et al. (2006). Section V presents numerical results obtained from quantum Monte-Carlo simulations in the resonant regime. These are used to discuss the validity of our analytical model and to derive experimentally accessible windows for the observation of mesoscopic qubit + cavity states in forthcoming circuit-QED devices. In the conclusion, we comment on the inclusion of other dissipative effects which may be relevant in the study of other cavity QED superconducting circuits Buisson and Hekking (2001).
Ii Cavity QED
ii.1 The model
The effective Hamiltonian describing the cavity QED systems is the Jaynes-Cummings Hamiltonian Jaynes and Cummings (1963) involving the coupling of an harmonic mode to a two level system (the qubit):
In the strong coupling regime, the coupling energy is assumed to be much larger than all energy scales characterizing dissipative processes in the system, i.e. the cavity relaxation rate and the atomic relaxation and dephasing rates. Neglecting damping, the Jaynes-Cummings Hamiltonian can be exactly diagonalized since the Hilbert space decomposes into two dimensional multiplets generated by the states and . As a function of the qubit / cavity detuning , the cavity + qubit system has two different regimes.
ii.2 The dispersive regime
The dispersive regime is reached when the cavity and the qubit are out of resonance. For an initally empty cavity, it is obtained for . In this regime, the eigenstate are very close to the ones of the uncoupled cavity + qubit states. Performing a second order expansion in leads to
where represent the ac-Stark shift per photon. In the presence of a mesoscopic coherent state with average photon number , the cavity + qubit coupling is enhanced by the coherent field. Off diagonal terms in the Jaynes Cummings multiplet are small compared to diagonal ones when .
In the dispersive regime, qubit flips can only occur because of qubit relaxation or external driving of the system. With respect to the cavity mode, the qubit behaves as a transparent medium with a state-dependent refraction index. In Rydberg atom experiments, this regime has been used to produce Schrődinger cat states Brune et al. (1996a), to perform QND measurement of the photon number Gleyzes et al. (2007) and a measurement of the Wigner function of the field Bertet et al. (2002). In circuit QED devices, the cavity has been used to performed a QND measurement of the qubit state Wallraff et al. (2005). More recently, the strong coupling dispersive regime where a single photon drastically alters the qubit absorbtion spectrum has been has been studied theoretically Gambetta et al. (2006) and demonstrated experimentally Schuster et al. (2007).
ii.3 The resonant regime
The resonant regime is obtained when . In this case, the Jaynes-Cummings eigenstates are symmetric and antisymmetric combinations of the form separared by an energy (vacuum Rabi splitting). This leads to vacuum Rabi oscillations between states and which have been observed Brune et al. (1996) and used to transfer the qubit state in the cavity Maitre et al. (1997). In the mesoscopic regime , the resonant interaction leads to an entangled atom-field state with two quasi-coherent field components with different classical phases Gea-Banaloche (1991). The approximate solution by Gea-Banacloche precisely provides explicit expression for these states and provides the basic framework for discussing the complete dissipative dynamics of circuit QED devices.
It has proven very convenient to describe the dynamics using an effective Hamiltonian approach that involves effective spin operators acting within the Jaynes-Cummings multiplet Klimov and Chumakov (1995); Meunier et al. (2006). Within the framework of this mesoscopic approximation ( and ), the evolution of the cQED system can be computed exactly. Under the Jaynes-Cummings Hamiltonian, in the mesoscopic regime, the states
where remain of the same form with a time dependent angle . In the mesoscopic approximation, can be approximated by a factorized state of the form Meunier et al. (2006):
where the atomic polarization is given by:
and the field component is
The field state is proportional to a quasi coherent state whose parameter is roughly equal to . At fixed , the phase, in Fresnel plane, associated with the generalized Gea-Banacloche state is . The atomic polarization evolves in the equatorial plane of the Bloch spere, its phase being perfectly correlated to the electromagnetic phase.
The states act as a path detector for the atomic polarizations and this explains the pattern of collapses and revivals of Rabi oscillations. When , the field components still overlap and therefore, Rabi oscillations arising from interference between the and are observed. Once , the field components do not overlap and Rabi oscillations disappear. Only when is a multiple of , the two states overlap again. The information about the qubit state stored by into the cavity is erased, thus leading to a revival of Rabi oscillations. Henceforth, collapses and revivals of Rabi oscillations in the mesoscopic regime are a direct illustration of the complementarity principle.
When , the qubit polarizations will coincide. Then, the atom + cavity state disentangles and the resulting cavity state is a mesoscopic Schrődinger cat states involving two quasi-coherent components with opposite phases. This occurs much faster than in the dispersive regime where such a Schrődinger cat state would be generated in .
Iii Circuit QED devices
Recent experiments performed at Yale involve the coupling of a superconducting Josephson qubit and a superconducting microstrip resonator with high quality factor Wallraff et al. (2004). We will first present these devices and recall how they can be described by the Jaynes-Cummings Hamiltonian. Dissipation sources will then be discussed.
iii.1 The cavity: a superconducting microstrip resonator
The resonator is 1D a transmission line made by photolithography whose lowest mode lies within the 1-10 GHz frequency range and whose quality factors can be as high as Frunzio et al. (2005). Besides this, the electromagnetic field associated with their eigenmodes is confined within a relatively small volume, thus leading to high electric fields between the center and ground planes (typically V/m). Such a resonator is usually characterized by a wave velocity , a real impedance ans its length .
The finite length superconducting resonator has many stationary modes up to a certain high frequency cutoff but in the present situation, only the coupling to the lowest energy mode The voltage difference between the inner and outer electrodes at point is then expressed in terms of the mode creation and destruction operators:
where denotes the maximum voltage felt by the qubit.
iii.2 The artificial atom: a Josephson qubit
Within the resonator lies a superconducting Josephson charge qubit (initially a Cooper pair box) coupled to the cavity electric field. It consists in a small superconducting island connected to a superconducting reservoir through a very thin insulating barrier. When the temperature and the the Coulomb charging energy of the island are smaller than the superconducting gap, the qubit degrees of freedom are encoded by the charge state of the superconducting island. This device can be characterized by the Josephson amplitude and the Coulomb energy . Control parameters are the voltage potential imposed to the superconducting island and the Josephson amplitude which can be tuned using a SQUID instead of a single Josephson junction Makhlin et al. (2001): where denotes the flux quantum and the magnetic flux through the SQUID.
In the charge regime and for gate charge restricted to a unit charge interval , the system effectively behaves like a two level system (TLS) involving two adjacent charge states of the island. In this charge basis, its effective Hamiltonian is given by:
where and .
iii.3 The qubit/cavity coupling
The electromagnetic mode of the 1D cavity naturally couples to the qubit charge through capacitance between the resonator inner and outer electrodes. The coupling energy is limited by the dipolar energy of the qubit within the resonator: where is a capacitance ratio and is the quantum of resistance. Classical electrodynamics shows that is of the order of where is the fine structure constant, is the relative permittivity and a geometric factor, usually logarithmic in the microstrip aspect ratios. Within the 1 to 10 GHz frequency range, values of range from 11 to 105 MHz (see table 1) the latter value being obtained using the ”transmon” Koch et al. (2007). Assuming that the two level description of the qubit is valid, the coupling Hamiltonian between the qubit and the cavity is given by:
The coupling being small compared to the resonator and qubit eigenfrequencies, the rotating wave approximation is valid and close to the charge degeneracy point , eq. (9) reduces to the Jaynes-Cummings Hamiltonian (1).
iii.4 Measurement protocols
In these experiments, the Josephson qubit cannot be measured directly. This circuit-QED system is probed through the resonator which is then connected to external ports at its ends. A network analyzer is used to analyzed the phase and amplitude of a classical electromagnetic wave transmitted through the device. This method has been used to perform a non-destrictive measurement of the qubit state using a dispersive measurement technique Wallraff et al. (2005). Thus, the relaxation time scale of the resonator sets a lower bound on the measurement time. Probing the state of the qubit thus requires that where . Although cavities with quality factors of the order of have been manufactured Frunzio et al. (2005), devices used in experiments have a lower in order to satisfy the fast measurement constraint.
In order to avoid this limitation, one has to rely on an alternative method for measuring the qubit state. Recently, a new detection scheme based on the dynamical bifurcation of Josephson junction has been realized and provides a high contrast, low backaction, dissipationless and fast measurement of the qubit state Siddiqi et al. (2005). Thus, rapid improvement of Josepshon circuit technology suggests that alternative methods of detection might become available in the near future.
In the present paper, we focus on the evolution of the qubit + cavity system prior to measurement. In particular, our main objective is to describe the dynamics of the circuit QED system at resonance in the mesoscopic regime and shed light on the main physical effects independantly from the measurement protocol. Therefore, for the sake of simplicity, we will not attempt modeling the subsequent measurement process.
iii.5 Dissipation mechanisms
Qubit relaxation and dephasing
Qubit dissipation and decoherence arise from their coupling to environmental degrees of freedom either extrinsinc (measurement circuit) or intrinsic (structural defects inside the material). Relaxation involves energy exchange between the qubit and its environment and, as such, is generically sensitive to the low frequency part of the environmental spectrum. Relaxation also leads to decoherence defined as the decay of off diagonal matrix element in the qubit’s eigenbasis.
But dephasing can also occur without energy exchange (pure dephasing). Early circuit-QED experiments used a Cooper pair box. The coherence properties of these devices are limited by pure dephasing induced by low frequency noise. It arises from fluctuating charges in the insulating amorphous used to make the Josephson junctions Astafiev et al. (2004). It is known that the effect of this low frequency noise cannot be described within a Markovian framework Makhlin and Shnirman (2003). A possible escape to this problem is to operate the qubit at special point where sensitivity to voltage fluctuations is at second order instead of first order Vion et al. (2002). At this working point, treating the effect of low frequency noise requires going beyond the Markovian approximation Makhlin and Shnirman (2004); Ithier et al. (2005). Fortunately recent circuit QED experiments use another type of qubit, called the ”transmon” which has very low sensitivity to voltage fluctuations (sweet spot everywhere), can be operated as a two level system and the low frequency noise Koch et al. (2007). Therefore, having in mind future experiments performed with transmons, qubit relaxation and dephasing will be treated within the Markovian approximation in the present paper.
Under this hypothesis, the evolution of the qubit + cavity system reduced density operator is described by a master equation of the form:
where denotes the Jaynes-Cummings Hamiltonian (1) and are the Lindbladian superoperators associated with each markovian dissipation channel.
The relaxation Lindbladian operator is given by:
where represents the relaxation rate of the isolated qubit. Markovian pure dephasing corresponds to the difusion of the relative phase between states . The pure dephasing Lindbladian operator is then given by:
where is the pure dephasing rate of the qubit defined as the decaying rate of the off diagonal matrix element for a standalone qubit.
In the case of the circuit-QED devices, relaxation mainly comes from capacitives losses at the ends of the resonator. The temperature dependance can be explained by considering that the inverse quality factor is a sum of two contributions. The first contribution represents thermal breaking of Cooper-pairs in the superconductor and it scales as . The other contribution has a much weaker temperature dependance and is probably associated with intrinsic dissipation mechanisms such as dielectric losses or magnetic vortices Frunzio et al. (2005). This term is responsible for the low temperature value of quality factor. Note that the current limitation in the quality factor of microwave cavities used in Rydberg atom experiments comes from diffraction losses, surface rugosity and residual resistance of the superconducting mirrors coming from defects, impurities and magnetic vortices Kuhr et al. (2006).
The corresponding Lindbladian is then:
where denotes the total relaxation rate of the oscillator.
Iv Dissipative dynamics
iv.1 The stochastic wave function method
In principle, eq. (10) can be solved numerically in order to obtain the quantum dynamics. However, an analytical ansatz for the reduced density matrix can be found within the mesoscopic approximation using the the quantum jump approach Dalibard et al. (1992) to the dissipative dynamics of the atoms + cavity system. As we shall see extensively in this paper, it provides a deep and useful insight into the full dissipative dynamics of the cQED system and will enable use to find simple analytical results for the Rabi oscillation signals.
The quantum jump method provides a solution to the master equation (10) by assuming that the environment of the system is continuously monitored so that any emission or absorption of quanta by the system can be assigned a precise, although stochastic date. Each time such an event occurs, the system undergoes a quantum jump:
The probability rates at a given time for the various quantum jumps are directly obtained as averages where the denote the quantum jump operator of type . Here, the quantuum jump operators associated to pure dephasing, qubit relaxation and cavity relaxation are respectively equal to: corresponding to a rotation around the axis, and .
Between these jumps, the evolution is described by an effective Hamiltonian that describes both its intrinsic dynamics and the acquisition of information arising from the fact that no quanta has been detected:
The reduced density matrix is then recovered by averaging over the set of stochastic trajectories associated with a large set of quantum jumps sequences. The weight of a given trajectory can be directly related to the dates and types of the various quantum jumps. The corresponding Linbladian is then given by:
This method proves to be very convenient numerically since the number of variables involved is of the order of the dimension of the system’s Hilbert space whereas it scales as in the master equation approach.
iv.2 Pointer states dynamics
A general stochastic dynamics tends to produce arbitrary mixtures of states. However, it takes a very simple form for states that are preserved (up to a phase) between and during quantum jumps. These so called pointer states entangle minimally with the system’s environment. In some cases, they provide a basis of the system’s Hilbert space, thus leading to a simple description of the stochastic dynamics which we call a pointer state dynamics.
Since coherent states are pointer states with respect to cavity losses, such a description is directly relevant for the describing the effect of cavity relaxation on cavity QED systems in both dispersive and resonant regime.
We assume that each eigenvalue of the quantum jump operator is non degenerate and choose a fixed section () of the corresponding vector bundle over ’s spectrum. Pointer states remain pointer states in the evolution between quantum jumps as soon as and (sufficient condition). In this case, evolution beween quantum jumps of produces a single quantum trajectory where:
with initial condition . The phase contains an Hamiltonian and a Berry phase contribution:
Then, starting from a single pointer state , the stochastic dynamics taking into account all possible sequences of quantum jumps occuring at times produces a single trajectory but encodes the sequence of quantum jumps in a dependent phase. The resulting state at time is of the form and therefore, any initial state which is linear combination of pointer states will experience decoherence because of the averaging of these random phases.
Summing over all quantum jumps sequences leads to the system’s reduced density operator obtained from an initial state :
where the decoherence functional is given by:
As expected, this expression coincides with the accumulated decoherence of a pair of coherent states driven along trajectories .
Application to cQED systems
In the dispersive regime of cQED, the effective dissipative dynamics is described by the effective Hamiltonian (2). In the strong coupling regime , the effective quantum jump operator associated with cavity losses can still be approximated by . Therefore, in this scheme, the pointer states with respect to cavity losses are states of the form . Since any state can be expanded as a linear combination of these pointer states, the dispersive regime of cQED realizes the above describes pointer state dynamics. Their respective coherent state parameters evolve according to:
where is the ac-Stark shifted cavity frequency and is a driving of the cavity. Starting from an initial state of the form leads to a qubit + cavity reduced density operator of the form (20) involving two pointer states and where denote solutions of (22) with initial condition . The decoherence functional induced by cavity losses and relative to these states is obtained by substituting into (21). As we shall see in more details in section IV.4.2, this is exactly the dynamics solved in Ref. Gambetta et al. (2006) using a different method.
In the case of a qubit resonantly coupled to the cavity, generalized Gea-Banacloche states (3) are approximate pointer states with respect to cavity losses. Let us first consider an initial state of the form where is the qubit initial state and where the parameter of the initial coherent field is taken real. Cavity losses will then produce a qubit + cavity reduced density operator involving two generalized Gea-Banacloche states . The decoherence coefficient in front of the is obtained by substituting in (21). Introducing one or several -pulses (single echo or bang-bang control) only changes the trajectories of the Gea-Banacloche parameters in the Fresnel plane but eq. (21) remains valid. To deal with more general initial condition such as , one just has to decompose the initial state as a sum of rotated generalized Gea-Banacloche states where the direction is replaced by the directions given by the phases of the coherent states present in the initial condition.
So far, only the effect of cavity losses has been discussed. How do other sources of dissipation affect this picture in the dispersive and the resonant regimes ?
To answer this question, let us first remark that, in the dispersive regime, pointer states with respect to cavity losses are also pointer states with respect to pure dephasing of the qubit. Qubit relaxation sends a pointer state of the form on a pointer state . Therefore, in the dispersive regime, a pointer state is sent on another pointer state. But the situation is more involved in the resonant or quasi-resonant regimes. However, we shall see that quasi pointer states are sent on linear combinations of quasi pointer states. Thus, the complete stochastic dynamics can still be described in terms of generalized Gea-Banacloche states (3) and for this reason, simple analytical results can still be obtained.
The simpler case of pure dephasing will be considered first and we will then turn to the more complicated case of qubit relaxation. Explicit analytical results for the spontaneous and induced Rabi oscillation signals are given in section IV.5. For the sake of simplicity, the stochastic dynamics will be discussed using the factorized form (4) of generalized Gea-Banacloche states, assuming that their coherence is not broken by the dissipative processes under consideration. This assertion will be justified by a more precise discussion of the dynamics postponed in appendix B.
iv.3 Pure dephasing
First of all, let us note that the corresponding quantum jump operator has no effect between quantum jumps. Moreover, the rate of pure dephasing jumps is constant in time, independent of the state and equal to . Next, each quantum jump acts as a rotation around the axis, exactly like an echo pulse. As such, it sends the atomic polarization defined in eq. (5) on not affecting the field component (6). After jumps occuring at times , an initial state has turned into where and is defined as
where and .
A typical trajectory of the associated phase in Fresnel plane is depicted on figure 1, with slope changing of sign at each quantum jump. The relative phase between the two quasi-coherent field components follows a random trajectory directly related to the integral of a telegraphic noise:
where starts with and jumps each time there is a quantum jump. The probability distribution of waiting times between quantum jumps is exponential . The characteristic function of the probability distribution for is nothing but the decoherence coefficient of a qubit in a longitudinal telegraphic noise Paladino et al. (2002). Its behavior depends on the dimensionless parameter which is equal to where is the average number of quantum jump events occuring before the date of the first expected Rabi oscillation revival.
For , only a single event is necessary to spread the phase over . In this regime, the evolution of the cQED system up to the first revival is dominated by quantum trajectories having no pure dephasing quantum jumps. Their total weight is . Trajectories with one quantum jump or more have a total weight and can therefore be treated perturbatively. For fixed and , this ”weak dephasing limit” is realized for . In practice, this is the regime of interest for present and forthcoming experiments since is above 40.
For , a large number of events are necessary to spread significantly the random relative phase . In this case, rapid dephasing of the qubit prevents the observation of an atom + cavity entangled state. As shown in appendix B, the incoherent qubit ends up breaking the coherence of the coherent field, selecting Fock states as pointer states. This is the ”strong dephasing limit” which, as already stressed, can only be reached for very large photon numbers in the strong coupling regime of cQED.
Nevertheless, in the strong dephasing limit, even if the spreading of the field phase takes place over a time scale the Rabi oscillation signal will be strongly damped in a much shorter time. Indeed for , although the Fresnel angle of Gea-Banacloche states are weakly dispersed around zero and thus have a rather strong overlap for a typical given quantum trajectory, the classical Rabi oscillation phase averages to zero over a time scale of the order of . This means that the qubit dephasing time is anyway the upper limit to the Rabi oscillation visibility in the mesoscopic regime. Nevertheless, for qubits in high quality resonators but rather poor dephasing properties , the crossover from weak to strong dephasing regimes might be observed by performing a tomography of the field state.
Finally, the result of this analysis is that, in the strong coupling regime of cQED, the main contribution to the Rabi oscillation signal comes from quantum histories without any pure dephasing quantum jump. As far as we are interested by this signal, the effect of qubit dephasing can be accounted for through a supplementary decoherence factor on top of the decoherence factor associated with cavity losses:
In the dispersive regime, the analysis is much simpler since pointer states with respect to cavity losses are also pointer states with respect to pure dephasing of the qubit. Therefore, starting with an initial state of the form , the only effect of pure dephasing is to multiply the decoherence functional associated with cavity losses by:
iv.4 Qubit relaxation
For a standalone qubit, the non hermitian part of the Hamiltonian tends to bring the state of the qubit towards the state, thus reflecting the acquisition of information associated with the absence of relaxation quantum jump. But at resonance, in the presence of a mesoscopic state, the pumping of the qubit by the cavity photons alters the dynamics between quantum jumps. As discussed in appendix A, in the regime, the dynamics is dominated by the strong atoms + cavity coupling. Contrarily to the case of a the dispersive regime where no photon can excite the qubit after relaxation, several relaxation jump can occur. The statistics of relaxation quantum jumps is indeed described by a renewal process with effective waiting time distribution . The effective rate follows from the localization of generalized Gea-Banacloche states in the equatorial plane of the Bloch sphere.
When the first relaxation quantum jump occurs at time , the qubit gets projected on the state leaving the cavity in a superposition of quasi-coherent states . The subsequent Rabi oscillation signal then shows rapid oscillations of frequency which correspond to the superposition of the Rabi oscillation signals associated with the evolution of . Therefore, the first relaxation jump leaves us with four quasi-coherent components in the Fresnel plane as shown in fig. 2:
The Rabi oscillations starting right after the first jump collapse in time due to the splitting of the pairs of counter-rotating quasi components in the Fresnel plane. Revivals then occur when two of them recombine.
Assuming that , the first recombination occurs at time and involves components (b) and (c) on fig. 2). Components (a) and (b) will recombine at time as well as components (c) and (d). But in these three cases, the rapid oscillations are simply washed out by the averaging over as in the pure dephasing case. Only the recombination of components (a) and (d) will not be averaged to zero since it takes place at time , independent of .
A careful discussion of of all contributions is presented in appendix B. It shows that keeping only the first term in the r.h.s. of (27) corresponds to retaining only trajectories for rapid oscillations survive the averaging over dates of relaxation jumps. Therefore, we are brought back to the case of a single pair of trajectories within the Fresnel plane. But with each relaxation jump comes not only a pure phase as in eq. (38) of Meunier et al. (2006) but a coefficient . Following Meunier et al. (2006), these coefficients can be resummed thus leading to the decoherence coefficient associated with relaxation in front of the coherence :
where denotes the angular separation in the Fresnel plane of the quasi-component components of .
As a final comment of this discussion, let us point out that this decoherence coefficient alone is not sufficient to keep track of the complete state of the atoms + cavity system. More precise results could be obtained by perfoming a pertubative expansion in keeping track of all contributions arising from (27). Cavity losses and dephasing jumps can then be taken into account. The resulting formula are appropriate in the weak relaxation regime but rapidely require numerical evaluation.
Being off resonance, the cavity cannot send back the qubit to the state after a relaxation quantum jump. As stressed before, since the states are pointer states with respect to cavity relaxation, the relaxation dynamics is most conveniently studied in term of these states.
The state are obviously left invariant by the relaxation dynamics since . The state is sent on by the relaxation jump. The probability for such a jump to occur between and is , independent of . With this in mind, an exact solution for the dynamics in the dispersive regime, taking into account all dissipative processes (relaxation of the cavity, qubit relaxation and dephasing) can now be given for an initial state of the form .
Following Ref. Gambetta et al. (2006), the qubit + cavity reduced density operator can be decomposed with respect to the qubit state . Note that and . First of all, diagonal terms are not affected by pure dephasing. Next, the operator only contains contributions coming from trajectories without any relaxation quantum jump whereas is fed from trajectories that have had no quantum jump and trajectories having a single jump at time . This leads to:
Because coherence operators and involves a state, they only receive contributions from trajectories without any quantum jump. Taking into account pure dephasing leads to:
These formula are nothing but generalizations of eqs. (5.14) to (5.18) of Gambetta et al. (2006) that take into account relaxation of the qubits and for a general driving of the cavity.
Our derivation, not relying on the -function formalism, provides a simple view of the underlying quantum dynamics showing the importance of pointer states with respect to to cavity losses and qubit pure dephasing. Following the same line of reasoning, the reduced density operator for several qubits coupled to a driven cavity all in the dispersive regime can be computed although it leads to more complicated expressions.
iv.5 Rabi oscillation signals
General form of the signal
Let us now use these results by computing the Rabi oscillation signal and its enveloppe assuming that the average photon number in the cavity remains equal to . The result for the Rabi oscillation signal defined as the probability of finding the qubit in the state is given by:
where measures the overlap between the two Gea-Banacloche quasi coherent component of the field. Decoherence is contained in . Given these coefficients, the upper and lower enveloppes of the Rabi oscillation signals are given, in the case of a single qubit initially in the excited state, by Meunier et al. (2006):
Explicit expressions for the overlap and decoherence coefficients will now be given for the case of free evolution of the cQED system (spontaneous revivals) and for the case of an echo experiment (induced revivals).
In the case of a free evolution (spontaneous revivals), the overlap factor is given by:
Taking into account all decoherence sources, the decoherence factor is given by:
Decoherence is mainly dominated by an exponential decay with rate . The appearance of reflects the photon emission rate enhancement by stimulated emission. The reduction of the relaxation rate comes from the averaging over classical Rabi oscillations of the qubit relaxation rate. To explain the reduction of the dephasing let us note that, for a standalone qubit, contributions from all quantum history pile up to give . On the contrary, for a qubit coupled to a mesoscopic field in a cavity, only quantum histories without any quantum jump before the measurement time contribute to the Rabi oscillation signal.
In the echo experiment with echo pulse performed at time , the Rabi oscillation signal at time is determined by eqs. (34), (35) and (36). For , the overlap factor is given by where the r.h.s. involves the free evolution overlap factor since the echo pulse reverses the dynamics of the qubit + cavity state Morigi et al. (2002).
The decoherence factor for is now given in terms of:
where . Its phase is the sum of a contribution due to cavity relaxation (eq. (51) of Meunier et al. (2006)) and a contribution due to qubit relaxation:
Note that the dephasing exponential factor is the same than in the free evolution since the echo pulse is not able to reverse the effect of a Markovian qubit dephasing. Besides the overlap coefficient , the echo pulse mainly affects the contribution arising from the accumulation of slow phases either associated to cavity relaxation or to qubit relaxation.
V Discussion of the results
v.1 Methods and parameters
The Yale group reports of values of between
and MHz Blais et al. (2006).
Low relaxation and dephasing rates have recently been
measured Wallraff et al. (2005) at the magic point : MHz,
for a low value of the coupling
|Type of device|
|Rydberg atoms (1)|
|Rydberg atoms (2)|
|Circuit QED (1)|
|Circuit QED (2)|
|Circuit QED (3)|
Analytical results presented in section IV.5 have been compared to quantum Monte-Carlo simulations of the qubit + cavity evolution. For these simulations, the Adams-Blashford scheme of order four has been used to compute the evolution under the non-hermitian Hamiltonian between quantum jumps. All simulation results presented here have been obtained using an average over 10000 trajectories.
v.2 Free evolution
As a first step, it is interesting to see how our analytical model provides accurate predictions in the presence of qubit relaxation and pure dephasing on top of cavity relaxation. Figure 3 presents a comparison of the analytical enveloppes for the Rabi oscillation signal in the presence of (a) cavity relaxation alone, (b) cavity and qubit relaxation but no pure dephasing, (c) cavity relaxation and pure dephasing and finally (d) including all dissipative mechanisms. The dissipative parameters used for this comparison correspond to the Circuit QED (2) row of table 1: , and . With these parameters, the analytical model correctly describes the enveloppe of the Rabi oscillation signal.
Figure 4 compares the Rabi oscillation signals for the experimental parameters corresponding to Circuit QED lines in table 1. In cases (a) to (c), our anaytical model predicts the upper and lower enveloppes for the Rabi oscillation signal with good precision. But in case (d), only the initial collapse is correctly accounted for by the analytical enveloppes. Our Monte-Carlo simulation shows a relaxation of towards at longer times which is not described by our analytical model. This result is not surprising since for , at , the average photon number in the cavity should have decayed from 10 to 2 because of cavity relaxation alone. Therefore, at , the qubit + cavity system is already out of the mesoscopic regime and our model is not expected to be valid in this regime.
Finally, these numerical and analytical results show that observing spontaneous Rabi oscillation revivals in circuit QED is not possible with the Circuit QED (1) parameters of table 1. Only the recent improvements on the value of (Circuit QED (2) parameters) open the possibility for observing this phenomenon.
The influence of photon number can be visualized through the effective decoherence coefficient as a function of and . It contains the effect of the environment on the contrast of Rabi oscillation revivals. Contour plots , , , and are shown on figure 5 for the Circuit QED parameter sets of table 1. The fourth graph shows these coutours for the case of a Rydberg atom experiment assuming a cavity damping time of 14 ms.
At high photon number, the various coutour curves tend to become vertical. This is easily understood by noticing that, at high photon number, the cavity decoherence coefficient becomes independant of and dominated the decoherence process. In this regime, and the decoherence time scales typically as . Figure 5 shows that a coherent mesoscopic dynamics could be observed in really good conditions in cases (c) and (d): at corresponding to the generation of a mesoscopic field Schrődinger cat state involving up to 15 photons.
v.3 Echo experiment
First of all, we have considered a circuit QED echo experiment obtained with 10 photons initially and dissipative parameters , and (Circuit QED (2) of table 1). Figure 6 shows the echo signals for as well as the analytical enveloppes obtained for when the two quasi-coherent components of the field recombine.
Because the echo experiment reverses exactly the dyamics of the qubit + cavity system, the reduction of the induced Rabi oscillation revivals contrast is exactly given by the environmental decoherence: . Figure 7 shows coutour plots for as a function ot and using the sets of parameters given in table 1 for circuit QED devices. As in the previous paragraph, the corresponding plot for Rydberg atom experiments with cavity damping time of 14 ms are given.
v.4 Discussion of the results
Within the context of circuit QED experiments, our results suggest that tests of quantum coherence in the resonant mesoscopic regime could be performed using the latest generation of circuit QED devices. Echo experiments could be performed with up to 10 photons and echo times up to five vacuum Rabi periods. A more precise insight into the field dynamics could be obtained by performing a field tomography giving access to either the or the Wigner function of the field. The function probes the phase distribution of the field and its experimental determination would test the splitting of the field into two quasi-coherent components with opposite phases. Observing the induced revival then provides an experimental proof of the generation of a mesoscopic superposition superposition in the cavity Meunier et al. (2005).
On the other hand, Rydberg atom devices Raimond et al. (2001) are characterized by very long atomic relaxation time (33 ms) and the absence of atomic dephasing. The main limitation comes from cavity relaxation and time of flight. Up to 2005, the LKB team was able to work with a ms cavity relaxation time corresponding to Meunier et al. (2005) (see row Rydberg atoms (1) in table 1). Cavity damping times up to 14 ms have been considered in our recent publication Meunier et al. (2006) an are recalled in row Rydberg atoms (2) of table 1. Recently, the LKB team reported on the realization of a ultra-high finesse cavity reaching Hz Kuhr et al. (2006) corresponding to ! With such performances, the only limitation is the atomic time of flight. This is an important limitation for the study of spontaneous revivals. In practice, only partial spontaneous revivals arising from small atomic ensembles could be observed Meunier et al. (2006). It is worth mentioning that circuit QED experiments do not suffer from this limitation although their dissipative properties are still behind the Rydberg atom ones.
Motivated by rapid experimental progresses of circuit QED devices, we have studied the dissipative dynamics of one qubit resonantly coupled to a mesoscopic field in a cavity taking into account cavity relaxation as well as qubit relaxation and pure dephasing. The resonant coupling between the qubit and the cavity produces an entangled state involving quasi-pointer states with respect to cavity relaxation. This very precise fact ultimately justifies in full generality the simple analytical model previously used to describe the dynamics of Rydberg atom ensembles coupled to a high-Q cavity Meunier et al. (2006). Although quasi-pointer states naturally produced by the resonant interaction are not preserved by qubit relaxation and dephasing, we have obtained a simple description of the dynamics of the cavity + qubit system in the presence of qubit dissipative processes. Simple analytical expressions for the spontaneous and induced Rabi oscillation signals have been obtained and shown and to be in good agreement with quantum Monte-Carlo simulations.
Our study shows that these effects and the generation of mesoscopic Schrödinger cat states could be within experimental reach in forthcoming circuit-QED experiments. However, our prediction may still be over optimistic since we did not attempt at modeling precisely the qubit readout. Thus, as done in Ref. Wallraff et al. (2005) for the Ramsey fringes measurement, the measurement beam can only be switched on after the qubit + cavity resonant interaction period, once the qubit has been sent off resonance for dispersive readout. Obtaining more precise quantitative results suitable for describing an actual experiment will probably require a more complete modeling of the experimental run, from preparation to measurement. The same remark is also relevant for obtaining precise predictions for the result of a cavity state tomography following for example a proposed protocol recently proposed in Ref. de Melo et al. (2006). Ultimately, we may probably have to rely on quantum Monte-Carlo simulations.
Other superconducting circuit for on-chip cavity QED have been proposed. In these proposals, the microstrip resonator is replaced by another oscillator such as an LC circuit or a biased large Josepshon junction in the appropriate parameter regime. Buisson and Hekking Buisson and Hekking (2001) have proposed a circuit that couples a current biased dc-SQUID to a Cooper pair box. In principle, values of of the order of 100 to 200 MHz are within reach with such a device. However, modeling its dissipative dynamics requires taking into account the effect of low frequency noises which act on the qubit through low frequency dephasing and on the resonator through slow fluctuations of its resonant frequency. These effects are easy to account for using our formalism and a quasi-static approximation. But the resonator is also coupled to high frequency current noise through the SQUID Claudon et al. (2006). Work is in progress to describe the effect of this coupling on the Buisson-Hekking cavity-QED circuit.
Acknowledgements.This work was supported in part by the Quantum Condensed Matter Visitors Program of Boston University. M. Clusel and J.-M. Raimond are thanked for their careful reading of the manuscript.
Appendix A Non unitary evolution between quantum jumps
The non hermitian Hamiltonian describing the evolution between two quantum jumps is:
Obviously, the pure dephasing part leads to an trivial overall factor. In the mesoscopic regime, the resulting Hamiltonian can then be rewritten in terms of the generators suitable for describing the dynamics of generalized Gea-Banacloche states (3) Meunier et al. (2006):
This Hamiltonian is of the same form as the one discussed in appendix D of Meunier et al. (2006). The resulting dynamics can then be inferred straightforwardly. In Meunier et al. (2006), since , the backaction of cavity relaxation on the qubit has been neglected and since , the term proportional to shall also be dropped out. Finally, starting from a state of the form
and assuming that , so that the average photon number in the cavity remains constant between and , we get:
The exponentially decaying prefactors are due to the non-hermiticity of the Hamiltonian. They reflect the decaying weight in time of quantum trajectories without any quantum jump between and . The apparently surprising appearance of instead of for relaxation quantum jumps corresponds to time averaging to of the qubit populations induced by the strong coupling with the cavity. Physically, we assume that the number of photons is so high and relaxation so weak that the cavity mode keeps driving the qubit with the same intensity. Note that the resulting effective dynamics correspond to an effective Poissonian statistics of relaxation quantum jumps. But when the qubit state is close to , the probability rate for such a jump is much smaller than when it is close to . Indeed deviations from the effective Poissonian statistics average out since dissipative processes take place over a much longer time scale than rapid oscillations induced by the resonant qubit/cavity coupling.
Appendix B Effects of dissipation on the resonant cQED system
In this section, the evolution of generalized Gea-Banacloche states (3) under the stochastic dynamics involving qubit relaxation and dephasing is discussed. We first derive the effective dynamics resulting from relaxation and pure dephasing quantum jumps. The case of pure dephasing can then be treated exactly using renewal theory. A formal solution including relaxation in the low dissipation, large photon number limit is provided and used to extract the dominant contributions to the Rabi oscillation signal. This appendix provides a firm basis for the disscusions of sections IV.3 and IV.4.
b.1 Stochastic dynamics
Evolution of the coherences between generalized Gea-Banacloche states
In order to compute the evolution of the qubit + cavity state, it is convenient to look at the evolution of a coherence between a given pair of generalized Gea-Banacloche states with respective indices . Such a coherence is initially of the form
As shown in appendix A, in the weak dissipation limit and in the mesoscopic regime, the evolution between quantum jumps is exactly the same as in the dissipationless case apart from the normalization prefactor that reflects the weight of quantum trajectories not presenting any quantum jump in the considered time interval. Moreoever, in the following, we shall restrict ourselved to short times () so that the average photon number in the cavity remains constant. The effect of various quantum jumps is described by: