Cross-Kerr nonlinearity between continuous-mode coherent states and single photons
Weak cross-Kerr nonlinearities between single photons and coherent states are the basis for many applications in quantum information processing. These nonlinearities have so far mainly been discussed in terms of highly idealized single-mode models. We develop a general theory of the interaction between continuous-mode photonic pulses and apply it to the case of a single photon interacting with a coherent state. We quantitatively study the validity of the usual single-mode approximation using the concepts of fidelity and conditional phase. We show that high fidelities, non-zero conditional phases and high photon numbers are compatible, under conditions where the pulses fully pass through each other and where unwanted transverse-mode effects are suppressed.
Photons are among the main candidate systems for the implementation of quantum information processing. One potential avenue in this context is to implement quantum gates between individual photonic qubits. Although there has been considerable experimental progress in realizing photonic pulse cross-phase modulation (XPM) in recent years zhu ; chen ; mat ; Lo , this is still an extremely challenging goal, because the interaction between single photons in nonlinear media is generally weak.
An alternative attractive approach involves the interaction of individual photons with intense coherent states. This approach can be used for the implementation of nondemolition measurements of the photon number imoto ; munroND . It also forms the basis for quantum gate proposals between individual photons that use the coherent state as an auxiliary system n-m-04 ; b-s-05 ; m-05 . For all of these applications, given an idealized single-mode coherent state and a single photon state , one aims to perform the transformation , where can be small. It was first proposed in n-m-04 ; b-s-05 that a deterministic parity gate for single-photon qubits could be realized by such weak nonlinearity. This idea has been widely employed in the researches on physical realization of quantum communication and quantum computing, see, e.g., j-05 ; k-p-07 ; h-b ; t-k ; s-d ; l-l ; l-h ; n-k-k for recent studies.
Most of the previous studies adopt a single mode assumption for XPM. Under this assumption the interaction term for the photonic states is given as , where and represent the mode of single photon and coherent state, respectively, and is the interaction strength ( is used throughout this paper). This actually models an ideal XPM mentioned above as the coupling of two harmonic oscillators, one of which is in the state and the other is in the state , through the interaction described by the nonlinear term .
This type of single-mode description is highly idealized. In reality, even if the light pulses start out as single-mode, interactions will generically create continuous-mode entanglement. This has been analyzed for photon-photon gates, studying both the longitudinal eit-review ; lukin-imamoglu ; petrosyan and, more recently, the transverse trans degrees of freedom. However, to our knowledge, no such study has been made for the equally important case of single photons interacting with coherent states.
Here we develop an effective quantum field theory description of the XPM between continuous-mode photonic pulses. The photonic degrees of freedom are described by quantum fields over the whole space that interact with each other through a general potential . The interaction is thus treated as instantaneous, which corresponds to an adiabatic elimination of the degrees of freedom of the nonlinear medium. This approximation is well established for nonlinearities based on electromagnetically induced transparency (EIT) eit-review ; lukin-imamoglu ; petrosyan ; xpm . It has also been used to describe interactions between Rydberg polaritons headon-1 , and photonic nonlinearities due to collisions in Bose-Einstein condensates (BECs) rispe . We will furthermore focus on very short-range interactions, which are typically modeled by the contact potential . For a discussion of non-instantaneous effects see Ref. sh-07 .
This paper is organized as follows. In Sec. II we discuss how the quantum states of continuous-mode pulses can be expressed in terms of quantum field operators. In Sec. III we describe their interactions with the help of the above-mentioned interaction potential approach. In Sec. IV we introduce an interaction picture for the interacting quantum field model. In Sec. V we apply the developed formalism to several relevant cases. In particular in Sec. V.A we study the case of two interacting single photons, introducing the concepts of fidelity and conditional phase. In Sec. V.B we treat the case of a coherent state interacting with a single photon, adapting the aforementioned concepts. In Sec. VI we study the performance of cross-Kerr nonlinearity based on pulses interacting through contact potential. In particular, we discuss the the different types of XPM with pulse of unequal and equal group velocities. Finally in section VII we give our conclusions.
Ii Continuous-mode photonic states
We first clarify the quantum states for the realistic pulses. Any classic electric field (magnetic field ) can be expressed in terms of the plane wave expansion
where with represent the polarization vectors, and stands for the complex conjugate. If the spatial volume tends to infinity, the discrete sum with respect to will be replaced by the integrals over the continuous spectrum. The quantization of the field is straightforwardly performed by replacing the amplitudes with the annihilation operators s-z :
where stands for the Hermitian conjugate. A current distribution acted by a field with the electromagnetic potential , where and , will radiate an electromagnetic field in coherent state with multiple modes s-z :
A pulse generated in the above process carries a continuous spectrum of the modes .
We here adopt a systematic way to express the states of photonic pulses in terms of the polarization components of a slowly varying field . For any field sharply peaked around a certain frequency, the slowly varying frequency in the square-root factor of (2) can be regarded as constant s-z . Then the dynamical behavior of the slowly varying field will be simply reduced to that of the slowly varying envelope corresponding to its polarization components g-c-08 . The detailed relation between the field operators and can be found in g-c-08 . For simplicity we will not distinguish between different polarization components. The field operator satisfies the simple equal-time commutation relation g-c-08
Using the field operator , a multi-mode coherent state generated in the process of Eq. (3) can be written as
The amplitudes of the modes are the Fourier transforms of :
where we have considered a continuous pulse spectrum. Similarly a continuous-mode single photon state can be expressed as
The function and depict the pulse shapes.
Iii dynamics of interacting pulses
To completely understand the XPM between photonic pulses, it is necessary to find the output state of the pulses from the initial product state, e.g., . The formalism provided above allows one to obtain the output state from the dynamical evolution of the corresponding quantum fields , where stands for the pulse in a coherent state and a single photon state, respectively.
Let’s consider a general interaction between the two light fields in a nonlinear medium. For the slowly varying and paraxially approximated light fields, the system evolves with the Hamiltonian , where
is the kinetic term g-c-08 , and the interaction term
describes the general two-body field interaction fetter . Here the pulses are assumed to propagate along the axis with the group velocities , and is the central wave number. For example, can be effectively given as for XPM between pulses in media under EIT conditions, where the nonlinear interacting rate is determined by the atomic structures and pulse properties in various systems, see, e.g., h-h-99 ; lukin-imamoglu ; petrosyan ; p-k-02 ; m-z-03 ; o-03 ; r-04 ; w-s-06 .
The evolution equation of the slowly varying field operators ,
The formal solutions to Eq. (13) are the unitary transformation
of the fields, where . In what follows, we will focus on the unitary evolution of the interacting pulses by neglecting the decoherence effects such as pulse absorption, etc. For XPM between pulses in EIT media, the pulse loss can be neglected given an EIT transparency window much larger than pulse bandwidth. The assumption of a real-number interaction potential is also adopted in the discussions below (i.e. we assume that there is no interaction-induced loss).
Iv interaction picture
We will use the interaction picture to study the evolution of photonic states. By using this picture one can eliminate the effects of pulse propagation and pulse diffraction, which are due to the kinetic term , from the relevant calculations, cf. below. The interaction picture is especially helpful to simplify the description of interacting pulses with unequal group velocities .
In the interaction picture an operator is transformed to , where . For example, the field operators in the interaction picture will be
The interaction Hamiltonian will be correspondingly
The transverse Laplacians in of (16) modify the interaction potential in the situation of as follows (the expression for the general situation of is similar though more complicated):
where is the relative coordinate. This effect is discussed in a different way in trans . If the propagation length of the pulses is much shorter than their Rayleigh length, the correction terms to will be insignificant. Moreover, given a weak interaction where the average of in Eq. (14) is a small term, the correction terms from the interplay between the transverse Laplacian and pulse interaction will be even less and can be well neglected in our discussions.
For the state vectors there is the relation
between the state vectors in the Schrödinger picture and in the interaction picture. The states in the interaction picture evolve according to the equation
An evolved state in the interaction picture is therefore
V cross-phase modulation between continuous-mode pulses
v.1 Single photon pair
We will study XPM between photonic pulses with the interaction picture introduced in the last section. For a clearer illustration of the technical steps, we first look at the XPM between two individual photons with the input state as
where are the pulse profiles. The state assumes the same form in the interaction picture because at . According to Eq. (21), the output state due to the photon-photon interaction will be
where we have considered the invariance of the vacuum state . The transformation of the field operators in the above equation is
where is defined in Eq. (18). Here we have used the commutator of the field operator, , in computing the commutators such as
Therefore, we obtain the output state
considering the relation . The operator of the under-brace in the above equation can be reduced to the form of a field operator multiplied by a c-number phase:
The commutator, , is used in deriving the above result. Under unitary evolution, the output state of an interacting photon pair therefore reads
which is inseparable with respect to and due to the induced phase . Such inseparability shows the entanglement between the single photon pulses, impairing the ideal performance of a quantum phase gate, . The conditional phase and fidelity of a quantum phase gate based on such XPM between two single photons can be determined by the following overlap trans :
Obviously a more entangled output state means a lower fidelity of the gate operation.
The interaction picture adopted in the above discussions makes it unnecessary to consider the pulse propagation and pulse diffraction effects described by Eq. (10), which in any case have no impact on the fidelity and conditional phase defined in Eq. (LABEL:p-f), cf. Ref. trans . The pulse profiles and in computing the overlap with Eq. (LABEL:p-f) are just those of the initial state at . This helps to simplify the treatment of XPM especially when two pulses have unequal velocities .
v.2 Coherent state and single photon state
We now turn to the main topic of this paper, namely the interaction of a coherent state and a single photon. In the interaction picture, the input of a coherent state and a single photon state is given as
The above input state will evolve under the interaction as
In the -th term of the above expansion there are phase factor operators , see the second step of the equation. Similar to the procedure in Eq. (27), the commutations of these phase factor operators with the field operator of the single photon will give rise to c-number phase functions , where is defined in Eq. (27). The last equality in Eq. (31) is one of the main results of this paper. It gives a compact expression for the output state, which forms the basis for all of our results below.
One can see that in general the output state is different from the ideal output state described in the introduction. The output state in Eq. (31) can be expanded as
The Fourier transform of the coefficient functions, such as that of the function in the second term of bi-photon component, will be generally inseparable with respect to the wave-vector modes of the coherent state and the modes of the single photon. This is due to the phases (inseparable with respect to and ) arising from pulse interaction. However, the XPM for realizing an ideal quantum phase gate is that , i.e. all components should pick up the same constant phase . Each expanded term in the ideal output state should be still separable with respect to and as in the input state. The entanglement between the modes and in a realistic output, as well as the inhomogeneity in pulse interaction, will generally impair such ideal performance for a quantum phase gate.
Another interesting observation is the existence of the self-phase modulation (SPM) effect in a more general situation with the factor in Eq. (14) being replaced by
Here the potentials and give rise to SPM and XPM, respectively. The contribution of the SPM effect to the output state in Eq. (31) is manifested by the commutations of , where
with the field operator of the coherent state, giving the extra phases due to the self interaction of the pulse in coherent state. This is totally different from the interaction between single photons, where only XPM effect exists trans . There is, however, no effect of SPM in multi-photon pulse interaction inside EIT media eit-review , and we will not consider SPM below.
Now we look in more detail at the difference of the actual XPM from an ideal XPM which realizes the transformation . In the spatial coordinate an ideal output state is given as
One can measure the closeness between an actual given by Eq. (31) and an ideal output state via their overlap